IEEE Transactions on Antennas and Propagation [volume 58 number 2]

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FEBRUARY 2010

VOLUME 58

NUMBER 2

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Bandwidth Limitations on Linearly Polarized Microstrip Antennas . . . . . . A. Ghorbani, M. Ansarizadeh, and R. A. Abd-alhameed Investigation Into the Effects of the Patch-Type FSS Superstrate on the High-Gain Cavity Resonance Antenna Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Foroozesh and L. Shafai Study of a Uniplanar Monopole Antenna for Passive Chipless UWB-RFID Localization System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Hu, Y. Zhou, C. L. Law, and W. Dou Compact, Dual-Polarized UWB-Antenna, Embedded in a Dielectric . . . . . . . . . . . . . . . . . . . . . . G. Adamiuk, T. Zwick, and W. Wiesbeck Novel Compact Model for the Radiation Pattern of UWB Antennas Using Vector Spherical and Slepian Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Dullaert and H. Rogier Time Domain Analysis of the Near-Field Radiation of Shaped Electrically Large Apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Quan A Novel Beam Squint Compensation Technique for Circularly Polarized Conic-Section Reflector Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Xu and Y. Rahmat-Samii Broadband, Efficient, Electrically Small Metamaterial-Inspired Antennas Facilitated by Active Near-Field Resonant Parasitic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Jin and R. W. Ziolkowski Coaxial-to-Waveguide Matching With "-Near-Zero Ultranarrow Channels and Bends . . . . . . . . . . . . . . . . . . . . . . . A. Alù and N. Engheta Periodic Leaky-Wave Antenna for Millimeter Wave Applications Based on Substrate Integrated Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Xu, K. Wu, and X. Zhang Multiband Handset Antenna With a Parallel Excitation of PIFA and Slot Radiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Anguera, I. Sanz, J. Mumbrú, and C. Puente Arrays On the Design of a Compact Neural Network-Based DOA Estimation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. J. G. Fonseca, M. Coudyser, J.-J. Laurin, and J.-J. Brault Design of a Beam Switching/Steering Butler Matrix for Phased Array System . . . . . . . . . . . . C.-C. Chang, R.-H. Lee, and T.-Y. Shih Novel Composite Phase-Shifting Transmission-Line and Its Application in the Design of Antenna Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Q. Lin, D. Bao, H. F. Ma, and T. J. Cui Hermite-Rodriguez UWB Circular Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Marrocco and G. Galletta Analysis and Design of a Novel Dual-Band Array Antenna With a Low Profile for 2400/5800-MHz WLAN Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. He and J. Xie Dual Polarization Interleaved Spiral Antenna Phased Array With an Octave Bandwidth . . . . . . . . . . . R. Guinvarc’h and R. L. Haupt A Novel Geometrical Technique for Determining Optimal Array Antenna Lattice Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. R. Zinka, I.-B. Jeong, J. H. Chun, and J.-P. Kim

250 258 271 279 287 300 307 318 328 340 348

357 367 375 381 391 397 404

(Contents Continued on p. 249)

(Contents Continued from Front Cover) Electromagnetics A Numerical Methodology for Efficient Evaluation of 2D Sommerfeld Integrals in the Dielectric Half-Space Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Hochman and Y. Leviatan Surface Current Source Reconstruction for Given Radiated Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Mohajer, S. Safavi-Naeini, and S. K. Chaudhuri Imaging Microwave Imaging in Layered Media: 3-D Image Reconstruction From Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Yu, M. Yuan, Y. Zhang, J. Stang, R. T. George, G. A. Ybarra, W. T. Joines, and Q. H. Liu Viable Three-Dimensional Medical Microwave Tomography: Theory and Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q. Fang, P. M. Meaney, and K. D. Paulsen Far Field Subwavelength Source Resolution Using Phase Conjugating Lens Assisted With Evanescent-to-Propagating Spectrum Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. Malyuskin and V. Fusco Numerical Methods An E-J Collocated 3-D FDTD Model of Electromagnetic Wave Propagation in Magnetized Cold Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Yu and J. J. Simpson Calculation of the Impedance Matrix Inner Integral to Prescribed Precision . . . . . . . J. S. Asvestas, S. P. Yankovich, and O. E. Allen Method of Moments Solution of Electromagnetic Scattering Problems Involving Arbitrarily-Shaped Conducting/Dielectric Bodies Using Triangular Patches and Pulse Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. I. Mackenzie, S. M. Rao, and M. E. Baginski GPU-Based Shooting and Bouncing Ray Method for Fast RCS Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Tao, H. Lin, and H. Bao Propagation A Body Area Propagation Model Derived From Fundamental Principles: Analytical Analysis and Comparison With Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Fort, F. Keshmiri, G. R. Crusats, C. Craeye, and C. Oestges Exploration of Whole Human Body and UWB Radiation Interaction by Efficient and Accurate Two-Debye-Pole Tissue Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Fujii, R. Fujii, R. Yotsuki, T. Wuren, T. Takai, and I. Sakagami Efficient Numerical Modal Solutions for RF Propagation in Lossy Circular Waveguides . . . . . . . . . . . . . . R. W. Moses and D. M. Cai

413 432

440 449 459

469 479 488 494

503 515 525

Scattering Acceleration of Ray-Based Radar Cross Section Predictions Using Monostatic-Bistatic Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Buddendick and T. F. Eibert Scattering by Polygonal Cross-Section Dielectric Cylinders at Oblique Incidence . . . . M. Lucido, G. Panariello, and F. Schettino Backscattering From a Two Dimensional Rectangular Crack Using FIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Bozorgi, A. Tavakoli, G. Monegato, S. H. H. Sadeghi, and R. Moini

552

Wireless Orbital Angular Momentum in Radio—A System Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. M. Mohammadi, L. K. S. Daldorff, J. E. S. Bergman, R. L. Karlsson, B. Thidé, K. Forozesh, T. D. Carozzi, and B. Isham On-Body Diversity Channel Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Khan, Y. I. Nechayev, and P. S. Hall

565 573

531 540

COMMUNICATIONS

Switchable Frequency Selective Surface for Reconfigurable Electromagnetic Architecture of Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. I. Kiani, K. L. Ford, L. G. Olsson, K. P. Esselle, and C. J. Panagamuwa Performance Improvement for a Varactor-Loaded Reflectarray Element . . . . . . . . . . . . . . . L. Boccia, G. Amendola, and G. D. Massa Collocated Microstrip Antennas for MIMO Systems With a Low Mutual Coupling Using Mode Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Sarrazin, Y. Mahé, S. Avrillon, and S. Toutain A Multiband Quasi-Yagi Type Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.-J. Wu, C.-H. Kang, K.-H. Chen, and J.-H. Tarng A Novel Wideband and Compact Microstrip Grid Array Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Chen, G. Wang, and K. Huang Modular Broadband Phased-Arrays Based on a Nonuniform Distribution of Elements Along the Peano-Gosper Space-Filling Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. G. Spence, D. H. Werner, and J. N. Carvajal Reducing the Number of Elements in the Synthesis of Shaped-Beam Patterns by the Forward-Backward Matrix Pencil Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Liu, Q. H. Liu, and Z. Nie Stability Optimization of the Coupled Oscillator Array Steady State Solution . . . . . . . . . . . . . . . . . . . . . . . A. Georgiadis and K. Slavakis Wireless Transmission in Tunnels With Non-Circular Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. F. Mahmoud Numerical Evaluation of the Scattering of Brillouin Precursors From Targets Inside Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Safian Extraction of the Wideband Dielectric Properties of a Material Layer Using Measured Natural Frequencies . . . . . . . . E. J. Rothwell

581 585 589 593 596 600 604 608 613 616 620

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

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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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Digital Object Identifier 10.1109/TAP.2010.2041863

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Bandwidth Limitations on Linearly Polarized Microstrip Antennas A. Ghorbani, M. Ansarizadeh, and R. A. Abd-alhameed

Abstract—The Bode-Fano integral can be used as an objective tool for assessing the bandwidth of antennas, and especially schemes for bandwidth improvement. Results for U-slot and E-slot dual resonant patch antennas suggest that the Fano integral is invariantly related to the overall volume. The Bode-Fano and Youla theories of broadband matching have been applied to the narrowband and wideband lumped equivalent circuit of microstrip antennas to calculate the maximum achievable return loss-bandwidth product of linearly polarized microstrip antennas. Curves are presented showing the relation between the antenna bandwidth, maximum achievable return loss, and parameters of the equivalent circuit. It has been shown that creating parallel slots on the patch despite all potential advantages, may reduce the potential bandwidth of patch antennas. Index Terms—Bandwidth, Fano theory, microstrip antennas, Youla theory. Fig. 1. (a), (b) double and multi-resonant equivalent circuit of microstrip patch antennas.

I. INTRODUCTION

T

HE Q-factor is often used for evaluating the impedance bandwidth of microstrip antennas. This approach is based on computing the radiation Q-factor by calculating the average stored electromagnetic energy and radiated power or using the antenna input impedance [1], [2]. The Q-factor is then assumed to be the inverse of the normalized bandwidth of the antenna. However, this relationship is valid for narrow bandwidths only because determination of antenna bandwidth using Q-factor does not take into account double or multiple resonant behaviors as in the case of broadband antennas. In [3] the Bode-Fano theory [4]–[12] has been applied to the double resonant equivalent circuit of microstrip antennas, shown in Fig. 1. With refer to this figure, [3] showed that the potential bandwidth of patch antennas is related which can directly be calculated from to measurement data. Moreover, the effect of the feed probe on the maximum achievable return loss-bandinductance width product has been taken into account; showing its limiting effect on the potential bandwidth if it exceeds a specific value Manuscript received October 08, 2008; revised August 04, 2009. First published December 04, 2009; current version published February 03, 2010. A. Ghorbani is with the Electrical Engineering Department, Amirkabir University of Technology, Tehran 15914, Iran (e-mail: [email protected]). M. Ansarizadeh was with the Electrical Engineering Department, Amirkabir University of Technology, Tehran, 15914 Iran. He is now with the Electrical and Computer Engineering Department, Concordia University, Montreal QC H3G 1M8, Canada. R. A. Abd-alhameed is with the Electrical Engineering Department, Bradford University, West Yorkshire BD7 1DP, U.K. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037768

[3]. However, the effect of has been ignored so far to avoid further complications in the equations given by the Bode-Fano may reduce the potential bandwidth of mitheory. Since we extend the crostrip antennas regardless of the value of method that was described in [3] to include the effect of in computing the potential bandwidth of microstrip antennas. One of the drawbacks of the Bode-Fano theory is the requirement for the equivalent circuit of loads in the Darlington canonical form which is hard to obtain, especially for multiple resonant antennas [4], [5]. In this paper the Youla’s broadband matching theory has been applied to the multi-resonant equivalent circuit of microstrip antennas which is shown in Fig. 2 to circumvent such limitations [5], [6].This communication is cited as follows, in Section II, derivation of the double and multiple resonant lumped equivalent circuits of microstrip antennas is presented. In Section III, the Bode-Fano theory is applied to double resonant equivalent circuits of a rectangular and E-shaped patch antenna and the maximum achievable return loss of the TM01 radiating modes of these antennas is calculated versus bandwidth. Also in this section, Bode-Fano integral was carried out for U-shaped antenna. Next, in Section IV, the Youla theory is presented and applied to multi-resonant equivalent circuits of the above mentioned microstrip antennas and results are presented and discussed. Finally, conclusions are made in Section V. II. ANTENNA EQUIVALENT CIRCUIT Application of the Bode-Fano or Youla theories to an antenna involves derivation of the lumped equivalent circuit of the antenna under consideration [4]–[12], such an equivalent circuit must model the antenna input impedance over the frequency

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where

(2) (3) in modeling the input impedance of an The RMS error antenna can be defined by

Fig. 2. Rectangular and E-shaped patch antennas (L ; W ; P) = (4; 35; 9).

(4)

Fig. 3. Measured reflection coefficient of the antennas shown in Fig. 2.

range of desired radiating modes and be consistent with the behavior of the antenna at frequencies of zero and infinity. The E-shaped patch antenna reported in [13] and its slotless rectangular patch has been chosen in our investigation so that variations in potential bandwidth due to parallel slots on the patch can be analyzed. The measured reflection coefficients of these antennas are shown in Fig. 3. In the following, we present the double and multi-resonant equivalent circuits of the chosen patch antennas. A. Double Resonant Circuit The double resonant equivalent circuit of microstrip antennas shown in Fig. 1(a) is capable of modeling the input impedance of patch antennas over up to 30% bandwidth at center frequency of the dominant radiating mode [6]. R and model the radiation resistance and the feed probe inductance, respectively. In case resonates with to enhance the of the E-shaped antenna impedance bandwidth. With refer to Fig. 1(a) the input impedance Zeq, can be expressed as

(1)

is the number of measured samples; where and are the reference and measured impedances, respectively. Various methods have been proposed to determine the best values of equivalent circuit parameters. Generally, measured or simulated data of the input reflection coefficient of patch antennas are required to derive their equivalent circuit however, in case of rectangular patch antennas with thin substrates the equivalent circuit parameters have been related to the physical dimensions of the antenna [14]–[17]. In this paper, equivalent circuit parameters are found through nonlinear curve fitting such that the RMS error defined by (4) is minimized. Using measurement data and then following the method described in [17], equivalent circuits of the E-shaped and rectangular patch antennas are calculated as given in Table I. Fig. 4 shows reasonable agreement between equivalent circuit and measured impedances over the TM01 frequency band. B. Multiple Resonant Equivalent Circuits The general topology of the multi-resonant equivalent circuit of patch antennas has been shown in Fig. 1(b). With refer to this figure, represents power loss in the antenna structure and is represents the found to be very small and thus is neglected. represents TM00 mode of two parallel conducting planes and higher order modes [18]. The input impedance of the equivalent circuit can be written as [17]

(5)

where is the number of radiating modes in the frequency and are the resonant frequency, band of interest. -factor and radiation resistance of the th-radiating mode, respectively. We have calculated the equivalent circuits of the antennas shown in Fig. 1 and results are presented in Table II [17]. Good agreement between measured and equivalent circuit

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has second order power transmission zeros at the frequencies of zero and infinity. According to [4] the optimum value of the matching exponent in the passband (K) must satisfy the following system of equations: (6) (7) (8) (9) where (10) is the magnitude of the reflection coefficient and is and unity for all frequencies except for the ones in the passband is constant. are the right half plane (RHP) in which and should be selected optimally to achieve the zeros of maximum value of return loss bandwidth product. The matching exponent K is related to return loss by Fig. 4. (a), (b) The measured and equivalent circuit resistance and reactance for the E-shaped microstrip antenna given by (1) and its slotless form.

TABLE I EQUIVALENT CIRCUIT PARAMETERS

(11) Moreover,

and

are determined by [4]

(12)

(13)

values of input impedances was achieved in the frequency range of 0 to 6 GHz. III. BODE-FANO THEORY A. Application of the Bode-Fano Theory Fano extended the work of Bode and calculated maximum achievable return loss bandwidth product of arbitrary passive lumped loads when the load is matched by passive lossless network to a generator with pure resistive, frequency independent internal impedance [4]. To apply the Bode-Fano theory to the equivalent circuit shown in Fig. 1(a) the reflection coefficient must be defined as shown in Fig. 5 [4]. This equivalent circuit

Applying (12) and (13) to Fig. 5 and performing symbolic math computations in MATLAB we obtain (14) (15) (16) (17) and are the first order time constants of the antenna and are independent from external matching networks. However, the right-hand side of (16) and (17) may be reduced by the matching

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TABLE II WIDEBAND EQUIVALENT CIRCUIT PARAMETERS OF ANTENNAS DEPICTED IN FIG. 1

bandwidth of the antenna. Moreover, (23) reveals that the geometrical mean of the lower and higher edges of the optimum passband is equal to the resonant frequency of the parallel resonant section. It can be shown that (21) leads to

(24) Fig. 5. Definition of the input reflection coefficient.

If network and thus, in solving (6)–(9) they are considered as degrees of freedom which can be changed in only one direction. In order to solve (6)–(9) we must first consider the case in which the following non-equalities are satisfied

(18) (19)

and

(25) then, it is necessary that the feed probe If inductance be resonated out by a series capacitor in order to achieve the optimum return loss bandwidth product. When the desired bandwidth exceeds the right hand side of (20) or either (18) or (19) does not hold then solving (6)–(9) requires incorporation of optimum . Having this in mind (6)–(9) are rewritten as

(20) (26) If (18)–(20) hold then the antenna equivalent circuit should contain a degenerate element (adding degenerate elements to the equivalent circuit does not add any transmission zeros in the antenna structure) such that (21) is met

(27) (28)

(21)

(29)

Assuming (18)–(20) are satisfied and (21) is maintained then (6)–(9) are solved in a straightforward manner and result in

As pointed out by Fano [4] minimum number of maximizes the return loss bandwidth product. Four degrees of or K freedom are necessary to solve (26)–(29). Either can be selected as one degree of freedom and in the simplest case, three right half plane zeros are required to solve (26)–(29). which satisfy (30) However, a pair of complex conjugate solves (26)–(29) optimally

(22) (23) Equation (22) shows the maximum return loss bandwidth product of the parallel resonant section is only related to . or the matching can not increase the potential Therefore,

(30)

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Putting (30) in (26)–(29) and doing straightforward manipulations one arrives at

(31) (32) where

(33) Since the magnitude of have

is already specified by (30) we

Fig. 6. Maximum achievable return loss versus bandwidth for the E-shaped antenna and its slotless form depicted in Fig. 1.

TABLE III BODE-FANO COEFFICIENTS OF THE EQUIVALENT CIRCUITS OF ANTENNAS SHOWN IN FIG. 1

(34) Substituting (34) in (31) and (32) and eliminating rives at

one ar-

(35) where

is the normalized bandwidth defined as (36)

By solving (35) and using (11) the maximum achievable return loss of patch antennas versus bandwidth is obtained. In the following, a numerical result of application of the Bode-Fano theory to antennas shown in Fig. 2 is presented. B. Numerical Results 1) E-Shape Antenna: The Fano coefficients of the double resonant equivalent circuit of antennas shown in Fig. 1 are calculated using (14)–(17) and are given in Table III. Results of solving (35) for the antennas under consideration are shown in Fig. 6. This figure shows that parallel slots on the patch may reduce the maximum achievable return loss bandwidth product. In which fact, the potential bandwidth of antenna is related to has been reduced when parallel slots are created on the patch. However, parallel slots act as equalizers that reduce the differand so that the impedance ence between is increased. Although advanbandwidth for which tages of incorporation of matching networks in the structure of the patch antenna itself overweight reduction in potential bandwidth the Bode-Fano theory determines the upper limit on the achievable bandwidth by incorporation of parallel slots on the patch. 2) U-Shape Antenna: In order to show the Bode-Fano inte. We try to gral is invariantly related to the overall volume or calculate the first order Bode-Fano integral for un-slotted and slotted antennas as well. From many pro-

Fig. 7. Computed return loss for slotted and non-slotted antennas.

posed broadbanded designs [19]–[21], we chose to investigate the U-slot design of [19], [20] since a singly resonant antenna of the same overall volume, and almost the same operating frequency, can easily be constructed. It suffices to remove the slots from the patch while leaving the geometry otherwise unchanged, apart from a change in feed probe position. The has been found patch geometry is as in [19], [20] and raw for both slotted and un-slotted forms from Ansoft Designer simulation software. Predictions of several packages and measured results of [19], [20] agreed closely. Fig. 7 shows computed return loss for the two designs, with additional impedance poles evidently present in the slotted form. Higher resonances, inevitably present, do not contribute usefully to the performance of a conventional band-limited communication system. Clearly judgment must be applied in truncating the Bode-Fano integral to remove the higher order modes.

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TABLE IV PROPERTIES OF SLOTTED AND NON-SLOTTED ANTENNA

Fig. 8. Broadband matching of ZL(s) using the Youla theory.

TABLE V TRANSMISSION ZEROS OF THE RECTANGULAR AND E-SHAPED ANTENNAS DEPICTED IN FIG. 2

On the basis of Fig. 7, we have computed integrals by straightforward numerical integration, but setting the upper integration and corresponding integrals were also limit at 7 GHz. The recalculated for the same antenna connected to a generator of varying impedance . A confusing factor is the feed probe inductance which in the un-slotted form prevents a critical match for any source impedance and gives a false impression of the intrinsic antenna quality. We considered it valid to add (in circuit but not field simulation) a single series lumped capacitor to exactly cancel the probe inductance; this can in any case be done internally by a capacitive gap between the probe and the patch. Table IV shows very similar values of the Bode-Fano integral for the slotted and un-slotted forms of the antenna, and even when the generator impedance is changed. For both forms, a left hand plane zero of appears above the critical value of generator impedance. This condition is recognized by the antenna’s becoming under coupled so that its Smith chart locus does not enclose the origin. At this point reduction of the integral is expected, and indeed observed. Thus results are similar for the E-shaped slot antenna as mentioned. Reducing the generator impedance below critical should give a constant integral, but in this case produces very broad responses whose integral is reduced by the truncation used. For some structures we have observed a constant truncated integral for a very wide range of sub-critical generator impedances. Empirically this is also true for the present structure if a much higher truncation limit is used, although higher resonances now contribute to the integral. A further interesting result is that adding a series inductor to the un-slotted patch (with tuning capacitor) raises the network order by one, as does the incorporation of the U-slot. It is observed in Fig. 7 that the same percentage bandwidth (33%) can be produced with this arrangement as with the U-slot. As also shown, a matching network of order one higher can raise this figure to 38%. Therefore we have shown that the Bode-Fano integral can be usefully calculated numerically for a small antenna, and can be used as an objective measure of antenna quality. Our results suggest that the intrinsic quality of a small antenna is invariantly

related to its volume when broad banding is attempted. In the U-slot antenna considered, the structural elaboration can give a very convenient practical result but in a sense is not increasing the inherent quality, as it does not change the integral and the same bandwidth could be achieved by external matching of the same network order. In a U-slot antenna, for at least one resonant mode the radiation contributions from oppositely directed currents on adjacent slot edges are tending to cancel. Possibly a similar property occurs in all broad banded patch designs. Therefore the Bode-Fano integral is suggested as a useful objective tool for this purpose. IV. YOULA THEORY A. Application of the Youla Theory Youla has proposed a new theory of broadband matching that unlike the Bode-Fano theory, does require the Darlington canonical form of the antenna equivalent circuit [5], [22]. Youla theory can be represented in differential or integral from which is used in designing matching networks or calculation of the maximum return loss bandwidth product, respectively. This theory also fixes some problems that are associate with the Bode-Fano theory [5], [22]. The schematic of broadband matching an antenna to a pure resistive generator using the Youla theory has been depicted in Fig. 8 where the generators internal impedance is considered to be pure resistive and independent from frequency, The driving point input impedance of the load should be represented by a positive-real rational function of the complex frequency variable. The first step in the application of the Youla theory is finding represent the RHP transmission zeros of the antenna. Let the input resistance of the antenna, that is

(37)

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where is the input impedance of the antenna. The RHP and their order are the same as the transmission zeros of RHP zeros of [4]

(38) Youla has categorized RHP transmission zeros of the load into four mutually exclusive classes. Each class is recognized by its location in the complex s-plane and the behavior of the input impedance at that particular point. For each class of transmission zeros there is a set of equations and non-equalies that are satisfied regardless of the matching network [5], [22]. Multiresonant equivalent circuits of the antennas shown in Fig. 2 have been calculated in Section II. The RHP transmission zeros, their multiplicity and their associated category are calculated using (38) and are given in Table V. For transmission zeros of class 1 with orders 1 Youla theory states that [5]

Fig. 9. Maximum return loss of the antenna versus bandwidth.

(47) (39) (40)

(48)

(40) (49) (41)

(50)

(42)

(51)

and are the poles of in the right half . s-plane, are zeros of For microstrip antennas, and the input reflection coefficient of (s) in the RHP. Transmission zeros at the frequency of zero and infinity are of class 4 and order k. According to [5] the following equations are satisfied:

(52)

(43) and (44) where

(45) (46)

and in the passband of and 0 else where. The rectangular shape of maximizes the return loss-bandwidth product. The equality sign in (44) is valid if and only if the matching network is non-degenerative [5]. Again, incorporation of minimum number of s maximizes the return loss bandwidth product.

B. Numerical Results The set of equations and non-equalities given by the Youla theory has been obtained (37)–(50). The maximum return loss of the E-shaped microstrip antenna and its slotless form (Fig. 2) has been calculated versus bandwidth using MATLAB software as shown in Fig. 9. It can be seen that potential bandwidth of the multi-resonant E-shaped microstrip antenna has been reduced by creating parallel slots on antenna patch. This result is in agreement with that of the Fano theory. Moreover, utilization of a multiresonant matching network is a novel method which, to our knowledge, has not been practiced yet.

GHORBANI et al.: BANDWIDTH LIMITATIONS ON LINEARLY POLARIZED MICROSTRIP ANTENNAS

V. CONCLUSION It was shown that the Bode-Fano integral can be usefully calculated numerically for a small antenna, and can be used as an objective measure of antenna quality. Our first results suggest that the intrinsic quality of a small antenna is invariantly related to its volume when broad banding is attempted. Also the maximum achievable return loss of linearly polarized microstrip antennas has been calculated versus bandwidth using an equivalent circuit and the Bode-Fano or Youla theory. The topology of the equivalent circuit is the result of application the mode expansion technique to microstrip antennas. The lumped elements of the equivalent circuit have been computed using the curve fitting techniques. The effect of creating parallel slots on the maximum achievable bandwidth of rectangular microstrip antennas has been investigated and it has been shown that parallel slots can reduce the potential bandwidth of microstrip antennas. Although advantages of creating parallel slots outweighs this problem we have demonstrated an upper limit of bandwidth on the achievable return loss bandwidth product. Moreover, an upper limit for the feed inductance was obtained that above which the potential bandwidth starts to reduce. Utilizing the Youla theory of broadband matching, the most precise estimate of the maximum theoretical bandwidth of an antenna can be attained. Also, this theory can be applied to other wideband structures in the future. REFERENCES [1] H. A. Wheeler, “The wide-band matching area for small antennas,” IEEE Trans. Antennas Propag., vol. AP-31, no. 2, pp. 364–367, Mar. 1983. [2] H. A. Wheeler, “Small antennas,” IEEE Trans. Antennas Propag., vol. AP-23, pp. 462–469, Jul. 1975. [3] A. Ghorbani and R. A. Abd-Alhameed, “An approach for calculating the limiting bandwidth—Reflection coefficient product for microstrip patch antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1328–1331, Apr. 2006. [4] R. M. Fano, Theoretical limitations on the broadband matching of arbitrary impedances [Online]. Available: http://dspace.mit.edu/handle/ 1721.1/12909 [5] D. C. Youla, “A new theory of broadband matching,” IEEE Trans. Circuit Theory, vol. CT-11, pp. 30–50, Mar. 1964. [6] H. J. Carlin and P. J. Crepeau, “Theoretical limitations on the broadband matching of arbitrary impedances,” IRE Trans. Circuit Theory, p. 165, 1961. [7] M. Ansarizadeh and A. Ghorbani, “Maximum theoretical bandwidth of microstrip patch antennas,” in Proc. iWAT2008, Chiba, Japan, Mar. 2008, pp. 310–313. [8] A. Ghorbani and M. A. Ansarizadeh, “The Fano integrals as an objective measure of antenna bandwidth reflection coefficient product limit,” presented at the Int. RF and Microwave Conf., Putrajaya, Malaysia, Sep. 2006. [9] A. Hujanen, J. Holmberg, and J. C.-E. Sten, “Bandwidth limitations of impedance matched ideal dipoles,” IEEE Trans. Antennas Propag., vol. 53, no. 10, Oct. 2005. [10] Z. Lizhong and Q. Yihong, “A novel approach to evaluating the gainbandwidth potential of antennas,” in Proc. Antennas Propag. Society Int. Symp. AP-S. Digest, Jul. 1996, vol. 3, pp. 2058–2061. [11] A. R. Lopez, “More on narrowband impedance-matching limitations,” IEEE Antennas Propag. Mag., vol. 46, no. 6, p. 102, Dec. 2004. [12] R. C. Hansen, “Fano limits on matching bandwidth,” IEEE Antennas Propag. Mag., vol. 47, no. 3, Jun. 2005. [13] F. Yang, X. Z. Zhang, and Y. Rahmat Samii, “Wideband E-shaped patch antennas for wireless communications,” IEEE Trans. Antenna Propag., vol. 49, no. 7, Jul. 2001. [14] F. Abboud et al., “Simple model for the input impedance of coax-fed rectangular microstrip patch antenna for CAD,” Proc. Inst. Elect. Eng. Microw., Antennas Propag., vol. 135, no. 5, pp. 323–326, Oct. 1988.

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[15] D. Kajfez, “De-embedding of lossy foster networks,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 1328–1331, Oct. 2005. [16] Y. Kim and H. Ling, “Equivalent circuit modeling of broadband antennas using a rational function approximation,” Microw. Opt. Technol. Lett., vol. 48, no. 5, pp. 950–953, May 2006. [17] M. Ansarizadeh and A. Ghorbani, “An approach for equivalent circuit modeling of rectangular microstrip antennas,” Progr. Electromagn. Res. B, vol. 8, pp. 77–86, 2008. [18] W. Richards, “An improved theory for microstrip patches,” IEEE Trans. Antennas Propag., vol. AP-29, pp. 38–46, Jan. 1981. [19] A. K. Bhattacharyya and R. Garg, “Generalized transmission line model for microstrip patches,” Proc. Inst. Elect. Eng., vol. 132, pt. H, pp. 93–98, 1985. [20] K. F. Lee et al., “Experimental and simulation studies of the coaxially fed U-slot rectangular patch antenna,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 144, no. 5, Oct. 1997. [21] R. Bhalla and L. Shafai, “Resonance behavior of single U-slot and dual U-slot antennas,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2001, vol. 2, pp. 700–703. [22] D. M. Kokotoff and J. T. Aberle, “Rigorous analysis of probe-fed printed annular ring antennas,” IEEE Trans. Antennas Propag., vol. AP-47, no. 2, pp. 384–388, Feb. 1999. [23] D. C. Youla, “A new theory of cascade synthesis,” IRE Trans. Circuit Theory, vol. 8, no. 3, pp. 244–260, Sep. 1961.

A. Ghorbani received the Postgraduate Diploma, M.Phil., and Ph.D. degrees in electrical and communication engineering and a postdoctoral degree from the University of Bradford, Bradford, U.K., in 1984, 1985, 1987, and 2004, respectively. Since 1987, he has been teaching various courses in the Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran. In 2004, he was on sabbatical leave at Bradford University. He is the author or coauthor of more than 100 papers in various conferences as well as journals. Dr. Ghorbani was the recipient of a John Robertshaw Travel Award, a URSI Young Scientists Award from the General Assembly of URSI, Prague, Czech Republic, in 1990, and the Seventh and Tenth Kharazmi International Festival Prizes, in 1993 and 1995, respectively, for design and implementation of antiecho chamber and microwave subsystems.

M. Ansarizadeh received the B.Sc. degree from Sistan Baluchestan University, Sistan Baluchestan, Iran and M.Sc. degree from AmirKabir University, Tehran, Iran, in 2004 and 2008, respectively. Currently, he is with the Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada.

Raed A. Abd-Alhameed received the B.Sc. and M.Sc. degrees from Basrah University, Basra, Iraq, in 1982 and 1985, respectively, and the Ph.D. degree from the University of Bradford, Bradford, U.K., in 1997, all in electrical engineering. From 1997 to 1999, he was a Postdoctoral Research Fellow at the University of Bradford, specializing in computational modeling of electromagnetic field problems, microwave nonlinear circuit simulation, signal processing of preadaption filters for adaptive antenna arrays and simulation of active inductance. From 2000 to 2003, he was a Lecturer, in August 2003, he was appointed Senior Lecturer in applied electromagnetics, in September 2005, he was appointed Reader in radio frequency engineering in the School of Engineering, Design and Technology, and in November 2007, he was appointed Professor of electromagnetics and radio frequnecy engineering, in the same school.

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Investigation Into the Effects of the Patch-Type FSS Superstrate on the High-Gain Cavity Resonance Antenna Design Alireza Foroozesh, Member, IEEE, and Lotfollah Shafai, Life Fellow, IEEE Abstract—Results of modeling, design, simulation and fabrication are presented for a high-gain cavity resonance antenna (CRA), employing highly-reflective patch-type superstrates. In order to determine the resonant conditions, the antenna is first analyzed using the transverse equivalent network (TEN) model, as well as the well known simple ray-tracing method. Prior to that, a highly-reflective patch-type frequency selective surface (FSS) is designed in order to be employed as the superstrate layer of the CRA. Next, a 2.5-D full-wave analysis software package, based on the method of moments (ANSOFT Designer v4.0), is utilized to analyze the antenna structure. Using this full-wave analyzer, the input impedance properties of an actual antenna are investigated as well. Then, a 3-D full-wave analyzer, based on the finite element method (ANSOFT HFSS), is used to extract the directivity and radiation patterns of the CRA, taking into account the finiteness of the substrate, superstrate and ground plane. Some previously unaddressed issues, such as the effects of the FSS superstrate on the input impedance characteristics of the probe-fed microstrip patch antenna, acting as the excitation source of the CRA are also studied. The effects of the highly-reflective FSS superstrate size on the CRA directivity, and explicitly its aperture efficiency, are investigated as well. A comparative study is also performed between CRAs with patch-type FSS and high permittivity dielectric superstrates. Measurement results are provided to support the modelings and simulations. Index Terms—Cavity resonant antenna, frequency selective surface (FSS), transmission line.

I. INTRODUCTION

A

HIGHLY-REFLECTIVE surface can be used as the superstrate layer of an antenna to enormously increase its directivity [1]. The principle which results in the significant high directivity has been described in [1], based on the multiple reflections occurring between the reflective superstrate surface above and the ground plane beneath the source antenna, using a simple ray-tracing method. The phenomenon is very similar to the Fabry-Perot resonator, except that one of the reflecting surfaces is allowed to slightly leak out the wave. Because of this similarity, this class of antennas is also called cavity resonant antennas (CRAs). Later in [2], it was shown that this class of antennas can also be modeled using the transverse equivalent network (TEN) model. The TEN model had earlier been

Manuscript received December 09, 2008; revised July 29, 2009. First published December 04, 2009; current version published February 03, 2010. The authors are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, 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.2009.2037702

introduced and utilized in [3], to analyze the multilayer substrate-superstrate structures when the source is a horizontal electric Hertzian dipole. It was shown in [4] that this method produces identical results as the other more complicated methods based on multilayer Green’s functions. The TEN model, then, was extended to arbitrarily oriented multilayer dielectric structures, as well as arbitrary feeding-source antennas in [5], [6]. Basically in [5], [6], the feeding-source antenna was replaced by a set of Hertzian dipoles that can replicate the far-field properties of the original antenna. The extracted far-field properties produced by this method have shown good agreement with those obtained by the full-wave simulations [5], [6]. The advantage of the TEN model is its extremely shorter computational time than those of the full analyzers, while giving the peak directivity and resonance conditions with adequate accuracy. In addition to these, one is able to explain relevant radiation mechanisms, in terms of the excitation of cylindrical leaky waves supported by these structures, using the TEN model [7]–[9]. Its shortcoming, however, is the inability of extracting near-field properties of the antenna, such as the input impedance and aperture-field distributions above the superstrate layer. Therefore, many researchers have employed full-wave analyzers to simulate CRAs, in order to investigate the properties of these antennas more thoroughly [10]–[15]. Many of these studies have also been provided with measurement data, not only to provide a comparison between simulation and fabrication results, but also to propose an application for these antennas [10]–[12]. Furthermore, antenna arrays have been used as the CRA feeds in [12] and [13], in order for the gains to reach as high as 30 dBi and higher. In this paper, a highly-reflective frequency selective surface (FSS) consisting of the conducting periodic patches, etched on a dielectric slab, is designed to operate in the desired frequency band. A cavity resonant antenna is considered, in which a horizontal electric Hertzian dipole is embedded in a multilayer structure, consisting of a grounded dielectric slab and the aforementioned FSS superstrate. The far-field characteristics of the antenna, such as the directivity and radiation patterns in cardinal planes, are predicted using the TEN model. As well, the ray-tracing approximation for the peak directivity is presented. These models give appropriate initial design values for the air-gap length at the desired frequency. They can also predict the boresight directivity to a good extent. Next, the antenna is simulated using a MoM-based software package (ANSOFT Designer v4.0). In this case, a finite number of the FSS unit cells can be utilized in the superstrate surface. This is also a practical constraint which is relaxed in the TEN modeling. In addition to the investigation of the antenna di-

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rectivity versus various parameters, such as the air-gap height and superstrate size, an extensive study of the antenna input impedance characteristics is performed, which have not been explored before in the literature. It is illustrated that the superstrate size and CRA air-gap height have considerable impact on the input impedance behavior, such as shifting the resonant freat the input port. A quency and changing the level of the comparative analysis on the effects of the microstrip patch an, when the FSS superstrate is present tenna feed point on the or absent is also provided. In order to investigate the effects due to finite sizes of the superstrate, substrate and ground plane, a FEM-CAD 3-D fullwave analyzer (ANSOFT HFSS 11.0) is employed. Excluding the return loss effects, it is shown that the aperture efficiency of the antenna remains almost the same, over a wide range of the superstrate size. This observation is elucidated in details. After that, a comparative study between CRAs with high permittivity dielectric superstrates and the designed patch-type FSS superstrate is performed. As known, CRAs can also be realized using superstrates with high contrast [3]–[5], [15], [17], [20]. Therefore, it is worthwhile illustrating the advantages of the FSS-type superstrate over the dielectric ones. Then, measurement results are presented. Peak gains as high as 20.07 dBi is achieved at the frequency of 8.44 GHz. Measurement results are compared against theoretical ones, to thoroughly evaluate the performance of the designed antennas. Effects of the CRA air-gap heights on directivity, radiation patterns and input impedance characteristics are addressed through a comprehensive experimental parametric studies. Finally, conclusions are drawn. II. ANTENNA MODELLING A CRA is depicted in Fig. 1(a) whose excitation source is a horizontal electric Hertzian dipole. The TEN model of this antenna is shown in Fig. 1(b), and ray-tracing description of the phenomenon, due to the multiple reflections inside the cavity and wave leakage to outside the cavity from the FSS superstrate is portrayed in Fig. 1(c). The FSS design procedure and the antenna modeling are discussed as follows. A. FSS Design and Modelling The key part of the high-gain CRA is its highly-reflective superstrate [1], [10]. In this study, the capacitive screen FSS, consisting of conducting periodic patches supported by a dielectric slab, is selected to serve our purpose. The choice of a capacitive (inductive) screen produced using periodic patches (apertures), is very suitable for the wideband highly-reflective superstrate, since it introduces a fairly constant high reflectivity over a wide frequency band. In the same frequency band, the linear behavior of the reflection coefficient phase also aids in producing a wideband high gain antenna, since the resonance length is proportional to the reflection phase of the FSS superstrate [10]. This has been explained in [10]. Some other FSS superstrates, such as concentric double square rings, have been used in previous works that have exhibited narrowband characteristics [16]. The dielectric support is Arlon Diclad 522 material, having the relative permittivity and loss tangent of 2.5 and 0.002, respectively. The FSS characteristics vs. frequency are shown in

Fig. 1. (a) Cavity resonant antenna (CRA) fed by a horizontal electric Hertzian dipole. (b) TEN model of the CRA. FSS superstrate is modeled by a two port network functioning at two different modes (TE and TM). (c) Illustration of the phenomenology due to the multiple reflections and leaky waves. (d) Dimensions of the utilized FSS as the superstrate layer in the design.

Fig. 2, for normal incidence upon the screen. As shown, it is highly-reflective (the reflection coefficient ) over a wide frequency range. In order for the FSS to be utilized in the TEN method, it is required to model it as a multi-port network. The unit cell of such a structure is depicted in Fig. 1(d). The cross-polarized transmission and reflection coefficients (TE-TM and TM-TE) are not plotted because they are negligible , and therefore, the two-port network suffices to model the FSS (Fig. 1(b)). Otherwise, the multi-port network has to be considered to account for the FSS. The angular properties of the FSS, at the design frequency (8.28 GHz), are drawn in Fig. 3. It is worthwhile noting that transmission coefficients calculated at and ) are identical both in magboth port 1 and port 2 ( nitude and phase. This fact restates that this FSS is a reciprocal structure. However, for the reflection coefficients, only the magnitudes are identical at port 1 and 2. To be more clear on port numbering, one should note that and T, shown in Fig. 2., represents and in Fig. 1(b). These reflection and transmission coefficients serve as the scattering matrix of the two-port network depicted in Fig. 1(b). The above-mentioned simulation results were obtained using the MoM-CAD (ANSOFT Designer)

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and are characteristics impedance of the substrate where dielectric and air, respectively, is the dielectric phase constant and is the dielectric substrate thickness. Considering the above-mentioned materials, is found to be 146.94 . Thereof 17.03 mm is obtained fore, the resonant air-gap length at the resonant frequency of 8.28 GHz. An expression for estidirectivity, relative to that of the mating the boresight primary antenna (feed antenna) has been derived in [1], [10]

(3)

Fig. 2. Magnitude and phase of the reflection (0) and transmission (T) coefficients of the FSS whose unit cells are depicted in Fig. 1(d).

software package than can deal with infinite periodic structures.

At 8.28 GHz, FSS reflection coefficient is 0.95. Thus, the relative directivity is estimated to be about 15.9 dBi. Therefore, when a primary source antenna has 6 to 7 dBi gain (such as microstrip antenna), placing this FSS superstrate above the ground plane, at the appropriate distance, can increase the CRA gain to as high as 21.9 to 22.9 dBi. Interestingly, this estimated gain is in good agreement with what was predicted by the TEN model. III. FULL-WAVE ANALYSIS OF THE CRA

B. Antenna Modelling Using TEN The cavity resonance antenna and its TEN modeling are shown in Fig. 1. The theory is well explained in the literature [2]–[6] and is not brought here, for the sake of brevity. In this study, both substrate and superstrate (supporting the FSS conducting patches) dielectric materials are Arlon Diclad-522, with relative permittivity and loss tangent of 2.5 and 0.0022, respectively. Their thicknesses are 1.6 mm. The ground plane is a perfect electric conductor, while the FSS metallic patches are copper. The peak directivity at boresight, vs. the air-gap height , and the far-field radiation patterns for various air-gap heights in the cardinal planes are drawn in Fig. 5(a) and (b), respectively. As can be observed, the maximum boresight directivity occurs at the air-gap length of 17.3 mm. The peak directivity is as high as 21.7 dBi. C. Resonance Estimation Using Ray-Tracing

In this section, the cavity resonance antenna proposed in Section II is fully analyzed using the MoM-CAD software package (ANSOFT Designer v.4.0). In the TEN model, one deals with ideal conditions such as the infiniteness of the FSS superstrate, modeled by a two-port network and a horizontal electric Hertzian dipole as the excitation source. In order for the antenna to be practical, the electric Hertzian dipole is replaced by a probe-fed microstrip patch antenna, and the number of FSS unit cells is truncated. The 3-D and top-views of this antenna are depicted in Fig. 5. Various parameters of the antenna, such as the air-gap length, number of the FSS unit cells employed within the superstrate, size of the source microstrip patch antenna and the location of the feeding probe, are considered in this study. All conductors are considered as copper. The study on effects of various parameters is conducted as follows. A. Air-Gap Height

The resonance condition for the antenna structure at the borein this study) and the operating frequency sight angle ( can be obtained as [14]

(1) is an integer number, is the distance between the where and are the reflection phases ground plane and the PRS, is obtainof the FSS and the ground plane, respectively. . The reflection coefficient able from Fig. 3(b) which is phase of a grounded dielectric slab, , is obtained as

(2)

The accepted gain and reflection coefficient magnitude of the , versus frequency, are plotted antenna at the input port for different airgap lengths in Fig. 6(a) and (b), respectively. The accepted gain is the actual gain, excluding the mismatch loss. In fact, it signifies the potential directivity of the antenna, when ohmic and dielectric losses are included. As the air-gap length decreases, the accepted gain and resonant frequency increases. The former is because of the fact that the reflection coefficient magnitude of the FSS superstrate increases with frequency (Fig. 2), since the higher reflective is the superstrate, the more directive becomes the antenna. The latter is attributed to the well-known effect that the resonant frequency is inversely proportional to the effective air-gap length of the cavity resonance antenna. One should note the importance of impedance matching of the source antenna. As can be seen, the microstrip patch antenna is fairly well-matched for all air-gap heights. On the other hand, this study shows that source antenna input impedance also varies with the air-gap height. Therefore, in all

FOROOZESH AND SHAFAI: INVESTIGATION INTO THE EFFECTS OF THE PATCH-TYPE FSS SUPERSTRATE

Fig. 3. Reflection and transmission coefficients of the FSS versus incident angle () at 8.28 GHz. The FSS is depicted in Fig. 1 and these results are utilized in the TEN model shown in Fig. 1(b).

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Fig. 4. (a) Directivity vs. air-gap length (l). (b) Far-field radiation patterns for different air-gap lengths. Frequency is fixed at 8.28 GHz.

designs one must insure impedance matching, in order for the CRA to work efficiently. B. Microstrip Patch Source Effect As mentioned before, the input impedance matching of the source antenna is an important factor for the CRA to radiate efficiently. This can be done by adjusting the source antenna patch size, and the location of its feeding probe. It is worthwhile mentioning that the source patch antenna, without the superstrate, usually resonates at a higher frequency. This may be due to the fact that the effective permittivity of the medium without the superstrate is less because of the absence of both FSS capacitive screen and supporting dielectric superstrate. Moreover, the location of the feeding probe is closer to the patch center, for obtaining the best input impedance matching. This can also be attributed to the same phenomenon, along with invoking the transmission line modeling approach of the microstrip antenna. As the effective permittivity of the medium increases the intrinsic impedance characteristic of the microstrip line decreases,

resulting in a lower antenna input impedance. Therefore, the impedance seen by the input port will decrease. Moving the feeding probe location away from the patch center compensates for this effect. Through a parametric study, this is rigorously illustrated as follows. As can be seen in Fig. 7(a), the best input impedance matching for the antenna without superstrate occurs mm, where is the distance from at 8.6 GHz and the feeding probe to the rectangular patch center. However, for the antenna with the FSS superstrate this occurs at 8.35 GHz mm. The air-gap length is set to be 17.5 mm when for this case. So, this frequency corresponds to the accepted gain of about 21.7 dBi, shown in Fig. 6(a). Of course, there is another minimum at 8.8 GHz, but the latter does not produce the maximum gain. Interestingly, it is observed that the FSS superstrate has considerably improved the input impedance bandwidth as well. In this case, the input impedance bandwidth is about 9.8%, which is more than twice that of the simple patch

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Fig. 5. Cavity resonance antenna with truncated FSS superstrate (15 by 15 unit cells) and microstrip patch antenna, (a) 3-D view (b) top-view. L = 10 mm and W = 8:5 mm.

(4.65%) in Fig. 7(a). One should note that the input impedance bandwidth, here, is defined as the frequency range, where of the antenna is below dB. The accepted gain of a typical microstrip antenna is also drawn in Fig. 7(b). This value is about 6.2 dBi. Utilizing a highly-reflective superstrate improves the gain significantly, up to 22.0 dBi, as illustrated in Fig. 6(a). However, the gain bandwidths of the CRAs are around 4.65% whereas that of the microstrip antennas is about 13.2%. The gain bandwidth, here, is defined as the frequency range where the gain drops 3 dB from its maximum. The study performed above shows clearly that the input impedance and gain bandwidths do not necessarily coincide and match. C. Effects of the FSS Size One of the most important parameters that considerably influences the CRA directivity is its superstrate size [10] and [15]. This has been studied in [10], for a highly-reflective superstrate consisting of strip-dipoles. The test for efficiency has been examined in [10], which is based on the realized antenna gain, including both mismatch loss and the aperture efficiency. In the previous section, it was observed that the input impedance characteristics of an antenna, with and without superstrate, are noticeably different. Therefore, it is fair to compare the aperture efficiency of the antennas with different superstrate sizes separately, excluding the characteristics of the input impedance. In order to include mismatch efficiency in the studies dealing with different superstrate sizes, one should match the antenna input impedance in each and every case to show the realizable gain in that case. Otherwise, the aperture efficiency is a more reliable figure of merit of the functionality of the antenna. Moreover,

Fig. 6. (a) Accepted gain at boresight and (b) quency for various air-gap lengths.

j

S

j

of the CRA versus fre-

by changing the size of the superstrate layer, the resonant frequency may shift, as well. It is thus better to consider the peak resonant gain in the frequency range under study. In this work, initially both accepted gain (the gain excluding mismatch loss) and input impedance of the antennas are studied using a commercial MoM-CAD full-wave analyzer (ANSOFT Designer). It is observed that due to the assumption of infinite dielectric layer, this method may not be a suitable tool for investigating accurately the effects of the FSS superstrate size on the antenna directivity. Therefore, a FEM-CAD full-wave analyzer (ANSOFT HFSS) is next employed to extract radiation characteristics of the antenna versus the superstrate size. versus freIn Fig. 8(a), directivities at boresight quency are plotted for different numbers of FSS unit cell. As the unit cell number increases, the peak directivity increases which is expected. However, as the superstrate becomes very large (extended from a certain value) the differential increase in the peak

FOROOZESH AND SHAFAI: INVESTIGATION INTO THE EFFECTS OF THE PATCH-TYPE FSS SUPERSTRATE

Fig. 7. (a) jS j of a CRA consisting of a source microstrip patch antenna and the uncovered microstrip patch antenna versus frequency for various feeding probe positions. f is the distance from the feeding probe to the rectangular patch center. For the case of the CRA, FSS superstrate is placed above the source microstrip patch at the distance of l = 17:5 mm. (b) Accepted and realized gain and jS j of a microstrip patch antenna (without any cover) for f = 1:6 mm.

directivity is not as significant. It appears that the peak directivity is asymptotically saturated, after a certain FSS size. In Fig. 8(b), reflection coefficients at the input port of the antennas versus frequency are drawn for the same cases of FSS size. It shows that CRAs with larger FSS areas tend to have lower resonant frequencies. Interestingly, the input impedance bandwidth is wider for the larger size FSS superstrate, but narrower for the directivity bandwidth (Fig. 8). In other words, the CRAs with smaller FSS superstrates demonstrate lower directivities but wider directivity bandwidths. These results are obtained by employing a commercial MoM-CAD software package (ANSOFT Designer). versus frequency are Directivities at boresight plotted in Fig. 9 for different numbers of FSS unit cells, using a FEM-CAD software package (ANSOFT HFSS v11.11). Although the trends of the graphs are similar to those obtained

Fig. 8. Simulated (a) directivity and (b) jS superstrate sizes using ANSOFT Designer.

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j

of the CRA for different FSS

by the MoM-CAD full-wave analyzer shown in Fig. 9(a), their level of directivities are significantly different for the smaller FSS superstrate sizes. Comparing the graphs shown in Figs. 8(a) and 9, one can find out that when the size of the FSS superstrate is 15 15 cm the peak directivities, obtained from both MoM and FEM methods, are almost the same and slightly higher than 21.0 dBi. However, when the FSS superstrate size is small, this difference is considerable. For example, when the FSS superstrate size is 5 5 cm , the peak directivity obtained by the MoM-CAD software package is about 15 dBi at 8.55 GHz, whereas that obtained by the FEM-CAD software package is around 12 dBi at 8.8 GHz. Due to the assumption of an infinite dielectric layer in the MoM analysis, the aperture efficiency may be calculated inaccurately using this method. Aperture efficiencies obtained by both ANSOFT Designer and HFSS are plotted versus FSS superstrate side dimension in Fig. 10. As can be seen, using the MoM-CAD analyzer results in an over-estimation of 140% aperture efficiency for the smallest FSS superstrate size. On

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Fig. 10. Aperture efficiency of the CRA versus FSS superstrate side dimension. Fig. 9. Simulated directivity of the CRA at boresight for different FSS superstrate sizes using ANSOFT HFSS.

the other hand, as FEM-CAD simulations show, the aperture efficiency drops when the FSS superstrate size is significantly reduced. In fact the lowest aperture efficiency belongs to the case with smallest FSS superstrate size (case of 5 5 cm ). An interesting observation which is worthwhile pointing out is the attainable aperture efficiency, as high as around 65% over a wide range of FSS superstrate sizes. This observation differs from that reported in [10] by the fact that in this work only the directivity is considered to calculate the aperture efficiency, and not the realized gain. Whereas in [10], the efficiency of the antennas are reported and hence both mismatch loss and aperture efficiency are included in the calculations. Therefore, it may be possible to have highly-directive gain antennas for several FSS superstrate sizes, if their input impedance is well-matched. In fact, since it was illustrated earlier that changing the FSS superstrate size may alter the input impedance characteristics, such , the CRA has to be matched as resonant frequency and for each and every case to perform its best. Thus, in this work mismatch losses are excluded from the aperture efficiency of the antenna. D. Radiation Properties Far-field radiation patterns of the antennas are plotted in Figs. 11(a) and (b) using the MoM-CAD and FEM-CAD full-wave analyzers, respectively. Since in the MoM-CAD simulations, the ground plane is assumed to be infinite, no radiation in the lower half space occurs. As well, at the grazing dBi). This angle the radiation patterns approach to zero ( trend is similar to the one exhibited by the TEN method in Fig. 4(b). This is mainly due to the calculation of spectral Green’s functions for the infinite dielectric layers, involved in the planar antenna structure. However, a peak accepted gain as high as 21.8 dBi is obtained, which is fairly close to that obtained by the TEN method. On the other hand, FEM analysis reveals that the CRA introduced a backlobe level as high as about 8 dBi, as shown in Fig. 11(b). A peak accepted gain of 21.4 dBi is obtained using this full-wave analyzer.

It is noteworthy mentioning that the cross-polar levels in the radiation patterns are higher in the cases that FEM-CAD is utilized. The electric fields of the near-zone, which implies the aperture field distribution on the superstrate, is depicted in Fig. 12. The aperture is assumed to be 2 mm above the dielectric support of the FSS superstrate. As illustrated in the color bars of Fig. 12(a) and (b), the amplitude of maximum x-directed electric field is higher than that of the y-directed one by an order of magnitude. This indicates that the cross-polarization level of the CRA in the far-field is not high, which has already been verified in simulation results shown in Fig. 11. Moreover, the high concentration of the field is in the middle of the CRA upper surface, illustrating the high electric field intensity there. The aforementioned results are for the antenna with the superstrate layer consisting of 15 15 FSS unit cells, supported by a dielectric with constitutive parameters and dimensions mentioned earlier in Section II.B of this paper. IV. CRA WITH HIGH PERMITTIVITY DIELECTRICS One of the well-known and widely utilized highly reflective superstrate is a slab of a material with high contrast in either permittivity or permeability [3]–[5]. The purpose of this section is to give a brief comparative study of the input impedance and gain bandwidth of this kind of antennas with respect to those of the CRAs with FSS superstrates, studied in the previous section. An antenna with a high permittivity dielectric superstrate is shown in Fig. 13. The resonance conditions for this kind of CRAs to work optimally and produce the highest gain have been elucidated in [3], [4], [17]. When the air-gap length and dielectric thickness are approximately half and quarter electric wavelength of their corresponding media, the resonance conditions are satisfied. Parametric studies using the TEN has verified these resonance conditions [18], [19]. However, in many practical designs, the microstrip patch antenna is placed on a dielectric coated ground plane, in the same manner as the cases studied in the previous section. Therefore, the resonance condition may be altered due to the additional substrate dielectric,

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Fig. 12. Aperture electric field distribution, sketched on a surface which is and (b) . placed 2 mm above the FSS superstrate dielectric support. (a)

E

E

= 17 5

Fig. 11. Radiation gain patterns of the CRA with l : mm. (a) MoM-CAD (ASOFT Designer) and (b) FEM-CAD (ANSOFT HFSS) simulation results.

which decreases the air-gap length [20]. In this case, the effective length of the substrate plus the air-gap is approximately half a wavelength [21]. The thickness of the superstrate dielectric, however, remains almost unchanged and about quarter wavelength of its corresponding medium [20]. Thorough and detailed study of the gain and input resistance bandwidths has been performed in [20]. Effects of the finite size superstrate, on the gain enhancement of the antennas, have been reported in [15]. Some unaddressed issues such as the effects of the probe position and a comparison between the patch-type FSS superstrate and dielectric slab are discussed here. A. Air-Gap Height of these The effects of the air-gap length on the gain and antennas are shown in Fig. 14. The permittivity of the dielectric has been assumed to be 10. Form Fig. 14(a), it superstrate is evident that the resonant length for the antenna to produce the

Fig. 13. A CRA with high permittivity dielectric superstrate.

maximum gain, at about 8.3 GHz, is mm. It is clear that this air-gap resonant length is different from that obtained when FSS superstrate was used in the previous section, which was about 17.5 mm. However, as can be observed in Fig. 14(b), the antenna is not well matched at this frequency. The matching can be achieved by varying the probe position which is discussed in the next part. One should note that all the results are obtained mm) and under condition of a fixed probe position ( fixed superstrate dielectric thickness of mm (quarter wavelength). B. Microstrip Patch Source Effects Effects of the probe position on the characteristics of the CRAs are shown in Figs. 16, 17, and 18 when the superstrate

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Fig. 14. (a) Gain and (b) jS j of the CRA, shown in Fig. 13, versus frequency for different air-gap lengths. Relative permittivities of the substrate and superstrate are 2.5 and 10, respectively.

Fig. 15. (a) Gain and (b) jS j of the CRA, shown in Fig. 13, versus frequency for different probe feed position. Relative permittivities of the substrate and superstrate are 2.5 and 10, respectively.

dielectric relative permittivities are 10, 20 and 30, respectively. Maximum gains of 15.45, 18.5 and 20.12 dBi are obtained for are 10, 20 and 30, respectively. One should the cases that note that in each case, the thickness of the dielectric superstrate is chosen as a quarter wavelength, i.e., 2.86, 2.02 and 1.65 mm, respectively for the above-mentioned cases. One should note that, it is possible to achieve slightly higher gains using the aforementioned dielectric superstrates, provided that a thinner dielectric substrate with lower permittivity is used. The maximum gain is not only proportional to the permittivity of the superstrate [3], [20], but it also is inversely proportional to the effective permittivity of the substrate using asymptotic formulas introduced in [3]. For example, using a 0.52 mm substrate, with a relative permittivity of 2.17 and superstrate with a relative permittivity of 10.2, a maximum gain of 16.65 dBi has been achieved in [20]. In [19], the peak gain of 16.88 dBi

has been achieved for a CRA with a suspended horizontal electric Hertzian dipole in the air-gap (no other substrate), when the superstrate relative permittivity is 9.8. FEM-CAD simulations (Ansoft HFSS), when a half-wavelength dipole placed horizontally in the air-gap is employed as a source, and a high permittivity dielectric with relative permittivity of 9.8 is used as a superstrate of the CRA, exhibit a gain as high as 17.79 dBi [21]. On the other hand, a maximum gain of only 15.20 dBi has been achieved in [22] for a CRA whose substrate and superstrate relative permittivities are 3.2 and 9.8, respectively. The latter example shows the adverse effect of using a relatively thicker and higher permittivity dielectrics as the CRA substrate. Nevertheless, since the intention here is on a fair comparative study between CRAs with FSS and high permittivity dielectric superstrates, the condition on substrate has been kept the same as those presented in the previous section. All reported

FOROOZESH AND SHAFAI: INVESTIGATION INTO THE EFFECTS OF THE PATCH-TYPE FSS SUPERSTRATE

Fig. 16. (a) Gain and (b) jS j of the CRA, shown in Fig. 13, versus frequency for different probe feed position. Relative permittivities of the substrate and superstrate are 2.5 and 20, respectively.

results in Figs. 14 to 17 are for the cases where the substrate dielectric is Arlon Diclad 522, with the properties introduced in Section II.B of this paper. Dimensions of the microstrip source patch are the same as the previous cases and shown in Fig. 5. Applying the above-mentioned considerations, the comparative studies for CRA with high permittivity dielectric is provided next. C. Comparision Between CRAs Having FSS and High Permittivity Superstrates Based on the curves shown in Figs. 15, 16 and 17, the best and high gain, performances of the CRAs in terms of low at the resonant frequency of 8.30 GHz, are obtained for the cases that the feeding probe is located at 1.5 mm away from the patch center. Although, increasing the superstrate permittivity enhances the gain significantly, it reduces both impedance

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Fig. 17. (a) Gain and (b) jS j of the CRA, shown in Fig. 13, versus frequency for different probe feed position. Relative permittivities of the substrate and superstrate are 2.5 and 30, respectively.

and gain bandwidths dramatically. The input impedance and gain bandwidths of the CRA for the various cases studied in Figs. 15, 16 and 17 are summarized in Table I. One can see that the CRA with the patch-type FSS, designed in Section II, offers better performance both in terms of gain and impedance bandwidth, while giving a gain as high as 21.7 dBi which is even higher than the case of the CRA with of 30. Furthermore, the thickness of the superstrate layer is basically a limitation for designing superstrate-substrate CRAs. This point is also important from manufacturing point of view. High permittivity dielectrics are usually ceramic-type materials. When high-gain is desired the permittivity of the superstrate becomes extremely high and therefore the dielectric slab needs to be very thin. This makes the fabrication of the actual design very costly. Another interesting point is that in the input impedance characteristics of the CRA, with high-permittivity dielectrics, only one resonance can be detected, while when the FSS-type superstrate is used double

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TABLE I GAIN AND INPUT IMPEDANCE BANDWIDTHS OF CRA HAVING HIGH-PERMITTIVITY DIELECTRIC SUPERSTRATE FOR VARIOUS "

Fig. 18. Actual antenna under test placed in the Compact Antenna Test Range measurement system of the Antenna Laboratory at the University of Manitoba.

resonance occurs resulting in a wider bandwidth. Therefore, the double resonance can be due to the existence of FSS metallization on the superstrate support. The above discussion indicates that the flexibility and advantages offered by the FSS-type superstrate can alleviate the shortcomings and relax the limitations of the high-permittivity dielectric superstrate, thereby make it a superior choice in many applications. V. MEASUREMENTS AND EXPERIMENTAL RESULTS Fabricated antenna under the test is shown in Fig. 18. The and normalized input impedance with respect to the 50 are plotted in Figs. 19(a) and (b), respectively, in the rectangular coordinate and on the Smith Chart, for various air-gap lengths . As can be observed, better input impedance matching is attainable for larger air-gap lengths. Also, the resonant frequency shifts to lower values, as the air-gap length increases. These two phenomena are consistent with the simulation results, shown in Fig. 6(b). The resonant frequency in simulations, however, is 8.35 GHz whereas in the experimental result is 8.45 GHz for the air-gap length of 17.5 mm. A closer look at Fig. 6(b) indicates in simulations a resonant frequency of 8.45 GHz, corresponds to the air-gap length of 17.3 mm. Although, a firm foam was placed on top of the FSS supersupport layer strate, and a thick aluminum ground plane was utilized to ensure the superstrate and substrate layers are as flat as possible, respectively, the perfect flatness may not have been achieved. Therefore, the air-gap length in the middle of the CRA may be less than 17.5 mm, due to sagging and warping of the superstrate and substrate dielectric layers. Moreover, all length adjustments were done manually and machine precision has not been involved. Thus, the input impedance in the experimental results is slightly different from those of the simulation ones. The antenna input impedances, sketched on the Smith

Fig. 19. Measurement results of the (a) jS j and (b) Smith Chart demonstration of the CRA input impedance normalized to 50 , for different air-gap lengths.

Chart in Fig. 19(b), demonstrate interesting characteristics of the CRAs. As the air-gap length increases, the large circle-shape impedance curve shrinks to smaller ones, in the vicinity of the Smith Chart center circle (reference impedance value) and exhibit two loops. This implies that the CRA can potentially have a wider input impedance bandwidth than the conventional microstrip patch antennas. This can be achieved by placing the FSS superstrate layer in an appropriate distance above the ground

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Fig. 20. Measurement results of the realized gain at boresight versus frequency for different air-gap heights. Fig. 21. Measured radiation patterns for CRAs with different air-gap heights. : mm and frequency of 8.33 GHz, (b) l : mm and frequency (a) l of 8.44 GHz, (c) l : mm and frequency of 8.69 GHz, and (d) : mm and frequency of 8.89 GHz.

= 18 0

plane. Moreover, by moving the feeding probe of the source microstrip antenna, the input impedance can be tuned. In this case, since the input impedances are smaller than the reference value (50 ), the feeding probe should be moved away from the centre of the rectangular microstrip patch antenna. Far-field properties of the aforementioned CRAs were also measured using the Compact Antenna Test Range (CATR facilities at the Antenna Laboratory of the University of Manitoba as shown in Fig. 18. A larger aluminum ground plane, with dimensions of 30 30 cm , was used in the measurements. Maximum realized gains versus frequency for various air-gap lengths are plotted in Fig. 20. The highest realized gain is 20.07 dBi which mm and frequency of 8.44 GHz. belongs to the case of Corresponding radiation patterns to each peak frequency are plotted in Fig. 21. Back radiations were measured for the angle variations in the range. Side-lobe levels as high as dB below the peak gain aroused in the copolar H-plane radiation patterns. Another important observation is the level of cross polarizations. As can be seen, the FEM-CAD software package delivers a better estimation of the cross-polarization levels than those given by the MoM-CAD software package. However, about 1.4 dB loss in the gain level is observed in the measurement results compared to simulations. These discrepancies can be due to imperfectness of the CRAs implementations as discussed above.

= 17 5

= 17 0

= 16 5

utilized to perform the full-wave analysis on the designed CRA antenna. The effects of the air-gap heights, position of the feeding probe and the FSS superstrate size on the microstrip patch source and CRA were studied in details, some of which had not previously been investigated or addressed in the literature. The antenna was fabricated and successfully tested demonstrating a gain, as high as 20.07 dBi. It was illustrated that not only this antenna is a high-gain one but it also can potentially offer wider input impedance bandwidths, in comparison to conventional microstrip patch antennas. Through adequate simulations, it was shown that the CRA with FSS-type superstrates are advantageous to the CRAs with high permittivity superstrates, with respect to achieving higher peak gains and wider input impedance and gain bandwidths, using relatively simple FSS surfaces that are low cost and easy to fabricate. A double resonance behavior was observed in the input impedance characteristics of the CRA, with FSS-type superstrates, both in simulation and measurement results, which potentially can be used in making wider band antennas in the input impedance sense. This behavior was not observed in CRAs with high permittivity dielectric superstrates. REFERENCES

VI. CONCLUSION The cavity resonant antenna (CRA) with highly reflective patch-type FSS superstrates was modeled, simulated, designed and fabricated. First, an FSS was designed and fabricated demonstrating an excellent reflectivity in the desired frequency band. Then, in order to find the resonant air-gap length the simple well-known ray-tracing formula and the TEN model of the CRA were employed. The TEN showed a better estimation on both air-gap length and directivity of the CRA. Using the initial values obtained from the TEN model, the MoM-CAD (ANSOFT Designer) and FEM-CAD (ANSOFT HFSS) were

[1] G. Von Trentini, “Partially reflecting sheet arrays,” IRE Trans. Antennas Propag., vol. AP-4, pp. 666–671, Oct. 1956. [2] T. Zhao, D. R. Jackson, J. T. Williams, and A. A. Oliner, “General formulas for 2-D leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. AP-53, pp. 3525–3533, 2005. [3] D. R. Jackson and N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas,” IEEE Trans. Antennas Propag., vol. AP-33, pp. 976–987, 1985. [4] N. G. Alexopoulos and D. R. Jackson, “Fundamental superstrate (cover) effects on printed circuit antennas,” IEEE Trans. Antennas Propag., vol. AP-32, pp. 807–816, 1984. [5] X. H. Wu, A. A. Kishk, and A. W. Glisson, “A transmission line method to compute the far-field radiation of arbitrarily directed Hertzian dipoles in multilayer dielectric structure,” IEEE Trans. Antennas Propag., vol. AP-54, pp. 2731–2741, 2006.

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[6] X. H. Wu, A. A. Kishk, and A. W. Glisson, “A transmission line method to compute the far-field radiation of arbitrarily directed Hertzian dipoles in multilayer dielectric structure embedded with PEC interfaces,” IEEE Trans. Antennas Propag., vol. AP-55, pp. 3191–3198, 2007. [7] A. Ip and D. R. Jackson, “Radiation from cylindrical leaky waves,” IEEE Trans. Antennas Propag., vol. AP-38, pp. 482–488, Apr. 1990. [8] T. Zhao, D. R. Jackson, J. T. Williams, H.-Y. D. Yang, and A. A. Oliner, “2-D Periodic leaky-wave antennas—Part I: Metal patch design,” IEEE Trans. Antennas Propag., vol. AP-53, pp. 3505–3514, 2005. [9] 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. AP-54, pp. 1442–1452, 2006. [10] A. P. Feresidis and J. C. Vardaxoglou, “High gain planar antenna using optimised partially reflective surfaces,” Proc. Inst. Elect. Eng. Microw. Antennas. Propag., vol. 148, no. 6, pp. 345–350, 2001. [11] E. Rodes, M. Diblanc, E. Arnaud, T. Monédière, and B. Jecko, “Dualband EBG resonator antenna using a single-layer FSS,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 368–371, 2007. [12] L. Leger, T. Monediere, and B. Jecko, “Enhancement of gain and radiation bandwidth for a planar 1-D EBG antenna,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 9, pp. 573–575, 2005. [13] R. Gardelli, M. Albani, and F. Capolino, “Array thinning by using antennas in a Fabry-Perot cavity for gain enhancement,” IEEE Trans. Antennas Propag., vol. AP-54, no. 7, pp. 1979–1990, 2006. [14] A. Foroozesh and L. Shafai, “2-D truncated periodic leaky-wave antennas with reactive impedance surface ground,” in Proc. IEEE AP-S Int. Symp., Albuquerque, NM, Jul. 9–14, 2006, pp. 15–18. [15] F. Kaymaram and L. Shafai, “Enhancement of microstrip antenna directivity using double-superstrate configurations,” Canadian J. Elect. Comput. Eng., vol. 32, no. 2, pp. 77–82, Spring 2007. [16] A. Foroozesh, L. Shafai, and M. Ng Mou Kehn, “Application of polarization and angular dependent artificial ground planes in compact planar high-gain antenna design,” Radio Sci., vol. 43, 2008, RS6S03, doi:10.1029/2007RS003795. [17] Y. Sogio, T. Makimoto, S. Nishimura, and H. Nakanishi, “Analysis for gain enhancement of multiple-reflection line antenna with dielectric plates,” Trans. IECE, pp. 80–112, Jan. 1981. [18] A. Foroozesh, M. Ng Mou Kehn, and L. Shafai, “Application of artificial ground planes in dual-band orthogonally-polarized low-profile high-gain planar antenna design,” Progr. Electromagn. Res., vol. PIER-84, pp. 407–436, 2008. [19] A. Foroozesh and L. Shafai, “Effects of artificial magnetic conductors in the design of low-profile high-gain planar antennas with high-permittivity dielectric superstrate,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 10–13, 2009. [20] X.-H. Shen, G. A. E. Vandenbosch, and A. R. Van de Capelle, “Study of gain enhancement method for microstrip antennas using moment method,” IEEE Trans. Antennas Propag., vol. AP-43, pp. 227–231, 1995. [21] A. Foroozesh and L. Shafai, “Size reduction of a microstrip antenna with dielectric superstrate using meta-materials: Artificial magnetic conductors versus magneto-dielectrics,” in Proc. IEEE AP-S Int. Symp., Albuquerque, NM, July 9–14, 2006, pp. 11–14. [22] A. Foroozesh and L. Shafai, “Effects of the excitation source position on the radiation characteristics of the antennas with a cover layer: A few case studies,” in Proc. IEEE AP-S Int. Symp., Albuquerque, NM, Jul. 9–14, 2006, pp. 1507–1510.

Alireza Foroozesh (M’08) received the B.Sc. degree from Tehran Polytechnic, Tehran, Iran, in 1996, the M.Sc. from the Iran University of Science and Technology, Tehran, in 1999, and the Ph.D. degree from the University of Manitoba, Winnipeg, Manitoba, Canada, in 2007, all in electrical engineering. From May 2000 to July 2002, he was a Researcher with the Antenna Laboratory, Iran Telecommunication Research Center (ITRC), where he was involved in projects related to antenna design and measurement. He is currently a Postdoctoral Fellow with the Department of Electrical and Computer Engineering, University of Manitoba. His main research interest is the analysis and modeling of periodic structures and their applications to antennas and microwave systems. Dr. Foroozesh was the recipient of the Best Student Paper Award at the International Symposium on Antennas and Propagation (ISAP 2007) in Niigata, Japan. He received a Young Scientist Travel Grant at ISAP 2007, Niigata, Japan, and a Young Scientist Award at the Electromagnetic Theory Symposium (EMTS 2007), Ottawa, Canada.

Lotfollah Shafai (LF’07) completed the B.Sc. degree at the University of Tehran, Tehran, Iran, in 1963 and the M.Sc. and Ph.D. degrees from the University of Toronto, Toronto, ON, Canada, in 1966 and 1969, all in electrical engineering. In November 1969, he joined the Department of Electrical and Computer Engineering, University of Manitoba as a Sessional Lecturer, becoming an Assistant Professor in 1970, Associate Professor in 1973, and Professor in 1979. Since 1975, he has made special efforts 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 on the Review Committees. He is a member of URSI Commission B and was its Chairman during 1985–88. 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 3rd 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 IEE (London) 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 is an elected Fellow of IEEE since 1987 and was elected Life Fellow in 2007. He is a Fellow of The Royal Society of Canada in 1998. He was a recipient of the IEEE Third Millenium Medal in 2000 and in 2002 was elected a Fellow of The Canadian Academy of Engineering and Distinguished Professor at The University of Manitoba. He holds a Canada Research Chair in Applied Electromagnetics and is International Chair of URSI Commission B.

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Study of a Uniplanar Monopole Antenna for Passive Chipless UWB-RFID Localization System Sanming Hu, Student Member, IEEE, Yuan Zhou, Student Member, IEEE, Choi Look Law, Senior Member, IEEE, and Wenbin Dou, Member, IEEE

Abstract—The principle of a passive chipless ultrawideband-enabled radio-frequency identification (UWB-RFID) and localization system is firstly introduced. A uniplanar monopole antenna with a 23 0 508 mm is then designed. Six passive size of only 23 chipless tags based on the antenna structure are subsequently developed for the system application. The backscattering characteristics of the tags are theoretically and experimentally studied in both the frequency and time domain. Following that, a simple receiver structure is proposed to simultaneously identify these tags and find their ranges. Results show that the proposed tags are excellent candidates for passive chipless UWB-RFID localization system applications. Index Terms—Backscattering, passive chipless tag, radar crosssection (RCS), radio-frequency identification (RFID), ultrawideband (UWB) antennas.

I. INTRODUCTION UE to its numerous expected distinct advantages such as high resolution, multipath immunity, and simultaneous ranging and communication, ultrawideband (UWB) technology has been widely investigated since the early 1960s, especially after 2002 [1]–[6]. Another hot topic in wireless technology is radio-frequency identification (RFID) which is finding more and more applications in various fields [7]–[13]. Nevertheless, the operating frequencies of most RFID systems are narrowband. More recently, increasing interest is being paid to apply UWB technology for RFID, tagging, and localization applications to obtain very high positional accuracy unachievable by conventional narrowband RFID or Wi-Fi technologies. Several UWB enabled RFID (UWB-RFID) and localization systems have been studied and developed by researchers from both academia and industry [5], [14]–[20]. However, the tags adopted in these systems have complex structures.

D

Manuscript received November 05, 2008; revised July 03, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported by the ASTAR SERC under Grant 052-121-0086. S. Hu is with the State Key Laboratory of Millimeter Waves, Southeast University, Nanjing 210096, China and also with the Positioning and Wireless Technology Centre, Nanyang Technological University, Singapore 637553, Singapore (e-mail: [email protected]; [email protected]). Y. Zhou and C. L. Law are with the Positioning and Wireless Technology Centre, Nanyang Technological University, Singapore 637553, Singapore (e-mail: {zhou0119, ecllaw}@ntu.edu.sg). W. Dou is with the State Key Laboratory of Millimeter Waves, Southeast University, Nanjing 210096, 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.2009.2037760

It is well known that, based on the power supply of the tag, RFID systems can be generally divided into three types: active, semipassive, and passive systems [8]–[11]. The aim of our project was to develop a passive chipless UWB-RFID and localization system which employs tags containing neither battery nor chip. Compared with the traditional active and/or passive tag which employs chip to generate identification (ID) code, this passive chipless UWB-RFID tag introduces meandrous timedelay line integrated within the antenna to produce a sort of ID data using pulse-position and pulse-polarity. It not only has advantages such as low cost, long life-time, and compact size, but also is suitable for some applications in harsh environments where the battery and chip are undesirable. In the reminder of this paper, the passive chipless UWB-RFID localization system is firstly introduced in Section II. Following that, in Section III, a CPW-fed monopole antenna and 6 passive chipless tags are proposed. Their backscattering characteristics including frequency-domain radar cross-section (RCS) and time-domain scattering waveforms are then studied numerically [21] and experimentally [22]. Subsequently, a simple receiver structure is developed for tag acquisition. Discussions on the general scattering characteristics of UWB antennas are carried out in Section IV. II. PASSIVE CHIPLESS UWB-RFID LOCALIZATION SYSTEM Similar to the conventional passive RFID systems [9]–[11], the passive chipless UWB-RFID localization system is based on electromagnetic backscattering to transfer data from the tag to the reader. In this kind of system demonstrated in Fig. 1, the position of the locators (transceivers) are known, and the objective is to identify and locate each tag. The identification and localization starts with transmission of a UWB pulse. When the pulse arrives at the passive chipless tag, it is backscattered. Each tag is designed to give a unique backscattered waveform as signature/ID. The reflected pulses are captured by the locator, where the tag ID and time-of-arrival (TOA) are computed. With such information from multiple receivers, the tag can be identified and located. The fundamental difference between the proposed UWB-RFID localization system and the abovementioned systems [5], [14]–[20] is the configuration of the tags. Different kinds of tags bring about different system benefits: the active tags make the location system work efficiently over a relative longer range; whereas the proposed passive chipless UWB-RFID tags feature compact size, long life-time (because no battery is needed), and low cost (it can be reduced further by employing a cheaper substrate), what is more, this kind of tag may be “printable” on a thin flexible substrate such as

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Fig. 2. Geometry of the CPW transmission line with smoothed bends.

Fig. 1. An overview of a passive chipless UWB-RFID localization system.

(2) polyester [9] due to its simple configuration and uniplanar structure. The underlying idea of the proposed passive chipless UWB-RFID tag is similar to the design concept of surface acoustic wave (SAW) tags [23], wherein the interdigital transducer (IDT) is replaced by a transmission delay line, i.e., feed line of the antenna. SAW tags have started appearing in the market [24] due to some distinct advantages when compared with their chip-based counterparts. Nevertheless, the SAW tag is high-loss and difficult to fabricate. Moreover, it operates on narrowband and hence has poor temporal resolution. III. DESIGN AND PERFORMANCES OF PASSIVE CHIPLESS UWB-RFID TAGS In this section, a coplanar waveguide (CPW)-fed monopole antenna is studied and designed for passive chipless tag applications, and the scattering characteristics of the proposed tags are studied in both the time and frequency domain. The CPW structure rather than microstrip is employed as the transmission line to feature the passive chipless UWB-RFID tag several advantages: (1) easy surface mounting of objects under tagging; (2) easy control of desirable characteristic impedance; (3) easy adjustment of the feed-line length; and (4) easy fabrication of matched-load and short-circuit structures for the delay line, i.e., no via hole is needed. A. Design and Characteristics of CPW Transmission Line The proposed meandrous CPW line with smoothed bends created using 0.5 miters [25] is illustrated in Fig. 2. To enhance the , transmission efficiency, Rogers RO4003C ( ) is chosen as the substrate because of its low loss tangent and moderate cost. The substrate is with thickness of 0.508 mm and size of 23 8 mm. The length of the meandrous strip whereas the width of strip and slot are 0.6 and is 0.2 mm, respectively. Theoretically, the per unit length effective and characteristic impedance can dielectric constant be calculated by analytical expressions [26]

(1)

is the ratio of complete elliptic integral of where and the first kind and its complement, the two variables are defined in terms of the dimensions of the CPW line [26]. , , For the design illustrated in Fig. 2, , and . In a typical narrowband RFID system, proper impedance match between the antenna and the chip is of paramount importance. Due to cost and fabrication issues, the antenna is usually directly connected to the chip [9]. However, an RFID chip is a nonlinear load which can drastically affect the performance of the tag [11]. Fortunately, in this paper, the CPW-fed monopole shown in the following subsection is designed to work as a passive chipless tag and there is no need to connect it to a chip. Therefore, from the matching point of view, the characteristic of CPW transmission delay line should only impedance be matched to the input impedance of the antenna to prevent re-reflection between the CPW line and radiation element. Furthermore, it is not necessary to be fixed at 50 . This freedom makes the meandrous feed-line structure more compact. The of 60 (which is transmission line illustrated in Fig. 2 has the input impedance of the designed radiator). CPW transmission lines and bends are conventionally studied in the frequency-domain. Nevertheless, impulse radio UWB (IR-UWB) technology is based on the time-domain electromagnetics. The reminder of this subsection presents the temporal transmission and reflection characteristics of the meandrous CPW line shown in Fig. 2. Fig. 3 illustrates the reflection characteristics of the CPW line with three kinds of terminations: open-circuit, short-circuit, and matched-load. The reflected pulses have a phase difference of 180 between the cases of open-circuit and short-circuit terminations. For a matched-load CPW line, there should be no reflection pulse (the small one shown in Fig. 3 is due to the imperfect matching). beAs shown in Fig. 3, the simulated time interval tween the incident wave and the reflected one can be easily calculated by

(3)

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TABLE I SUMMARY OF TIME INTERVAL

Fig. 4. Geometry and prototype of a monopole antenna with a meandrous CPW line feed.

Fig. 3. Time-domain reflection waveforms of the meandrous CPW transmission line with 3 different kinds of loads.

where and are defined to be the time points where the first peaks of the incident and reflected Gaussian pulses arrive, is respectively. Theoretical time interval (4) where is the length of the meandrous strip, is the speed is the per unit length effective of light in free space, and dielectric constant which can be calculated by (1). As tabulated in Table I, the theoretical and simulated results which are 0.474 and 0.472 ns respectively, are in good agreement. B. Antenna Geometry, Gain, and Impedance Bandwidth Following the investigation of the abovementioned CPW line, a CPW-fed antenna is developed for the passive chipless UWBRFID tag applications and illustrated in Fig. 4. A trimmed elliptical-ring patch is employed as the radiation element. Therefore the size of the CPW-fed monopole becomes small, and its structural mode scattering decreases whereas the antenna mode almost keep the same. Conventional rectangular ground is curved at the feed port with optimized radius of 1.2 mm to enhance the impedance bandwidth. curves. When Fig. 5 presents the antenna gain and the antenna impedance is normalized to 50 and 60 , the

Fig. 5. Simulated jS 11j and boresight antenna gain of the CPW-fed monopole antenna.

10-dB impedance bandwidths (3.55–8.39 and 3.55–16.11 GHz, respectively) are different whereas the antenna gain almost is dependent keeps the same. This is due to the fact that, on the normalized impedance, whereas the antenna gain in simulation is based on the assumed perfect matching at the feeding port. The antenna gain varies from -0.1 to 5.2 dBi over the frequency range of 3.1 to 10.6 GHz. C. Passive Chipless UWB-RFID Tags Based on the designed CPW-fed monopole antenna, 6 passive chipless UWB-RFID tags are proposed, fabricated, and shown in Fig. 6. These tags are with same size

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Fig. 6. Prototype of the passive chipless UWB-RFID Tags.

and radiation elements, but with different terminations and CPW transmission lines. To name them more conveniently, they are divided into 2 categories according to the length of and category B the CPW line: category A . Under each category, there are 3 different tags whose the terminations are open-circuit, short-circuit, and 56 resistor that is prevalent in the market. As discussed in Section I, the scattering properties of these tags are very important for our system application. In the following subsections, the backscattering characteristics are numerically studied [21] and experimentally verified [22] in both the frequency and time domain. It should be noted that, all simulated and measured results on scattering are for vertical polarization, the matched-load is set to 56 , and the distance between which satisfies the far-field the locator and tag is condition. D. Frequency-Domain Characteristics of Backscattering From Passive Chipless UWB-RFID Tags RCS is the most important index to describe the scattering characteristics of radar targets, especially in the frequency-domain [27]. This subsection presented the backscattering crosssection values of these 6 passive chipless tags. As illustrated in Figs. 7 and 8, the measured results agree well with the simulated ones, except for the matched-load cases around 4 GHz. The peaks for the cases of open and short-circuit occur when the structural mode and reflected antenna mode signals add in phase. In the matched-load case, there is no reflected antenna mode signal and hence the RCS do not fluctuate as much with frequency. These features are further analyzed in the following paragraph. The passive chipless tags are actually terminated antennas, of the tags can be divided the total scattering cross section into the following two categories: RCS of structural mode and antenna mode . Their relationship is given by [27]

(5)

Fig. 7. RCS of the tags (category A): (a) simulated and (b) measured.

where is the phase difference between these two modes. According to (5), there is

(6) then, we can get

(7) of these As shown in Figs. 7 and 8, the total RCS variation tags is around 25 dB. Therefore, we can predict that the antenna mode and structure mode have comparable peak-to-peak amplitude. This prediction is validated by the following time-domain waveforms of these passive chipless tags.

HU et al.: STUDY OF A UNIPLANAR MONOPOLE ANTENNA FOR PASSIVE CHIPLESS UWB-RFID LOCALIZATION SYSTEM

Fig. 8. RCS of the tags (category B): (a) simulated and (b) measured.

E. Time-Domain Characteristics of Backscattering From Passive Chipless UWB-RFID Tags With a normalized fifth-order Gaussian pulse (the incident wave shown in Fig. 3) as the incidence along the boresight direction of the tag, the time-domain backscattered waveforms are obtained and illustrated in Figs. 9 and 10. The simulated and measured pulses are almost identical. There are two obviously separated clusters of pulses: earlytime and late-time pulses. Theoretically, the point at which the first cluster begins is the time it takes for electromagnetical wave to backscatter from the tag to RFID locator, i.e. , where is the speed of light in free space. As shown in Figs. 9 and 10, the simulated and measured values are 5.335 and 5.328 ns, respectively. They are in good agreement. Therefore, it is theoretically possible and reasonable to use this first cluster (early-time pulses) for tag ranging with acceptable accuracy. The interval between the two separated clusters is the time it takes for quasi-TEM wave traveling along the meandrous CPWline and reflecting back to the radiation element, it can be calis within culated by (4). As tabulated in Table I, 0.005 ns, i.e., the measured values agree very well with the

275

Fig. 9. Boresight backscattering waveforms of the tags (category A) with 3 different kinds of load: (a) simulated and (b) measured.

simulated results. However, there is around 0.16 ns difference between the measured results compared with the theoretical ones which are calculated by (4). This is due to the between the antenna mode and struccharacteristic phase tural mode RCS which (4) does not account for. As formulated exists even if [27]. Actually, for in (5), this phase narrowband antennas, it is difficult to get this phase information [27]. However, the tabulated results show that we can efficiently obtain this delay of UWB antennas by comparing the theoretical value with the simulated and/or measured ones. As illustrated in Figs. 9 and 10, the early-time pulses of the open-circuited, short-circuited, and match-loaded tags keep the same pulse shape and amplitude, whereas the late-time ones change a lot. Similar to the reflected waveforms shown in Fig. 3, the late-time pulses have a phase difference of 180 between the cases of open-circuit and short-circuit terminations. In the case of matched-load, there are small late-time pulses solely due to the imperfect matching. Actually, the early-time and latetime temporal pulses are corresponding to the frequency-domain structural and antenna mode RCS, respectively. Therefore, the abovementioned phenomena can be easily interpreted by the

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Fig. 11. A receiver structure for tag identification and TOA estimation.

F. Tag Acquisition

Fig. 10. Boresight backscattering waveforms of the tags (category B) with 3 different kinds of load: (a) simulated and (b) measured.

fact that the structural mode RCS of an antenna is dependant on the structural characteristics such as antenna type, geometry, and material constitutions, whereas the antenna mode is linearly proportional to the antenna reflection coRCS efficient as [27]

(8) where is the termination and is the antenna impedance. Based on both simulated and measured results, the ratio between the peak-to-peak amplitude of the two pulses is around 3.0 dB which strongly verifies the prediction based on (7). Both the waveforms presented in Figs. 9 and 10 and data tabulated in Table I show that, it is feasible to identify the passive chipless UWB-RFID tags employing different pulse-polarity and pulse-position simultaneously, i.e., different feed termination is adopted to modulate the polarity of the late-time pulse, and the transmission line length is employed to control the time interval between the early-time and late-time pulses.

As discussed in the previous subsection, identification of the passive chipless tags is based on the position and polarity of the late-time pulse. The dual pulse reflection characteristic enables the antenna to act as a transmitted reference (TR) system [28]. Hence, a low-cost receiver structure that can be implemented in analog circuit easily is proposed and shown in Fig. 11. We use the received early-time pulse as the template and coris direlate it with the late-time pulse. The received signal vided into parallel branches. In each branch, the signal is mulat each tiplied with a delayed version of itself. The delay branch corresponds to the unique interval between the earlytime and late-time pulses for Tag . The total number of tags is assumed to be . Following each mixer is a low pass filter (LPF) that detects the mixer output envelope. It is straightforward that the tag ID corresponds to the branch where the LPF output peak has the largest amplitude with correct polarity, and tag signal TOA can be inferred from the position of the LPF output peak. Once TOA is measured at each locator, the tag position can be solved by some well-known positioning algorithms [29]–[31]. Extending the analysis framework for TR receiver, we can analyze the TOA detection performance and adapt the results in [28] to our receiver proposed in this paper. As an example, we use UWB pulses centered at 7.5 GHz with effective duration of 0.3 ns at a pulse repetition frequency (PRF) of 10 MHz. The emission power is 32 dBm and obeys the FCC peak power limit [34]. The tag is located 1.5 meters away from the locator (transceiver). The channel can be considered free-space as the tag is in the proximity of the locator. Assuming the antenna gain values of both the tag and the locator are 3 dBi, the received pulse encan be calculated to be about 12 dB, ergy to noise ratio which corresponds to 1 percent TOA detection error probability [28]. This result also agrees with our measurement. We may increase the reading range further by averaging the received signal . to improve the Based on the required detection error probability, the amount of different tag IDs supported can also be determined. As shown in Fig. 12, where the tag IDs are assumed to be uniformly asof 30 dB, 20 tags can be signed (i.e. uniform delay). For supported with a probability of detection error less than 1 percent.

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V. CONCLUSION A passive chipless UWB-RFID localization system is introduced in this paper. A CPW-fed monopole antenna is then proposed for the passive chipless UWB-RFID tag applications. The backscattering characteristics of 6 passive chipless tags are numerically and experimentally studied in both the frequency and time domain. The IDs and TOA of the open-circuited and shortcircuited tags are efficiently detected by a simple differential delay-and-integrate receiver. Finally, the scattering characteristics are investigated and some interesting phenomenon are briefly explained and discussed.

REFERENCES Fig. 12. Probability of detection error for different E (M ).

=N

and number of tags

IV. DISCUSSION ON THE GENERAL SCATTERING CHARACTERISTICS OF UWB ANTENNAS It is well-known that narrowband antennas are generally characterized by the frequency-domain parameters such as antenna gain and radiation patterns, and their properties are considered reciprocal when the same antenna is used for transmitting and receiving. However, in UWB systems, the transient response of the same antenna when transmitting and receiving are not identical [35], [36]. In this section, the general scattering characteristics of UWB antennas are briefly investigated and discussed by comparing the above illustrated results of CPW-fed monopole antennas/tags with the conventional narrowband antennas. Firstly, there are at least 3 kinds of definitions of structural and antenna mode RCS although the concept of dividing antenna RCS into the two components is easily grasped [27]. From the narrowband antenna point of view, it is difficult to decide which definition is the best. Fortunately, based on the time-domain scattering waveforms of the proposed tags, which are shown in Figs. 9 and 10, we can easily conclude that, the most reasonable and straightforward one is given by R. B. Green [37]: the structural mode is the early-time pulse which is scattered from a matched-loaded antenna, whereas the antenna mode is the late-time pulse which is due to the termination mismatching and re-radiation. Besides, the RCS of a narrowband antenna usually obey this law: short-circuited case corresponds to the least RCS value, whereas the largest RCS can be obtained when terminated with open-circuit [38]. However, the results illustrated in Figs. 7 and 8 tell us the law might only be applicable for narrowband antenna but might be unsuitable for UWB antennas (at least for the UWB antenna proposed in this paper). Finally, for a narrowband antenna, it is difficult to obtain the characteristic phase between the structural and antenna mode RCS, however, as discussed in the Subsection E of Section III, it is relatively easy to get the characteristic time delay between the two modes of UWB antennas.

[1] C. L. Bennett and G. F. Ross, “Time-domain electromagnetics and its applications,” Proc. IEEE, vol. 66, no. 3, pp. 299–318, Mar. 1978. [2] T. W. Barrett, “History of ultrawideband (UWB) radar and communications: Pioneers and innovators,” presented at the Progress. Electromagn. Symp.(PIERS’00), Cambridge, MA, Jul. 2000. [3] Federal Communications Commission, Revision of Part 15 of the Commission’s Rules Regarding Ultra-Wideband Transmission Systems Report and order, adopted February 14, 2002, released Jul. 15, 2002. [4] D. Porcino and W. Hirt, “Ultra-wideband radio technology: Potential and challenges ahead,” IEEE Commun. Mag., vol. 41, no. 7, pp. 66–74, 2003. [5] R. J. Fontana, “Recent system applications of short-pulse ultrawideband (UWB) technology,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2087–2104, 2004. [6] R. A. Scholtz, D. M. Pozar, and W. Namgoong, “Ultra-wideband radio,” EURASIP J. Appl. Signal Process., vol. 3, pp. 252–272, 2005. [7] K. Finkelzeller, The RFID Handbook, 2nd ed. London, U.K.: Wiley, 2003. [8] R. Want, “An introduction to RFID technology,” IEEE Pervasive Computing, vol. 5, no. 1, pp. 25–33, 2006. [9] K. V. S. Rao, P. V. Nikitin, and S. F. Lam, “Antenna design for UHF RFID tags: A review and a practical application,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 3870–3876, 2005. [10] G. D. Vita and G. Iannaccone, “Design criteria for the RF section of UHF and microwave passive RFID transponders,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2978–2990, 2005. [11] P. V. Nikitin and K. V. S. Raoc, “Theory and measurement of backscattering from RFID tags,” IEEE Antennas Propag. Mag., vol. 48, no. 6, pp. 212–218, 2006. [12] E. W. T. Ngaia, K. K. L. Moonb, F. J. Rigginsc, and C. Y. Yi, “RFID research: An academic literature review (1995–2005) and future research directions,” Int. J. Prod. Econom., vol. 112, no. 2, pp. 510–520, 2008. [13] S. Preradovic, N. C. Karmakar, and I. Balbin, “RFID transponders,” IEEE Microw. Mag., vol. 9, no. 5, pp. 90–103, 2008. [14] J. Reunamaki, “Ultra Wideband Radio Frequency Identification Techniques,” U.S. Patent 7 154 396, Dec. 26, 2006. [15] Z. Zou, M. Baghaei-Nejad, H. Tenhunen, and L. R. Zheng, “An efficient passive RFID system for ubiquitous identification and sensing using impulse UWB radio,” Elektrotechnik und Informationstechnik, vol. 124, no. 11, pp. 397–403, 2007. [16] M. R. Mahfouz, C. Zhang, B. C. Merkl, M. J. Kuhn, and A. E. Fathy, “Investigation of high-accuracy indoor 3-D positioning using UWB technology,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 6, pp. 1316–1330, 2008. [17] Ubisense [Online]. Available: http://www.ubisense.net/ [18] Time Domain Corporation [Online]. Available: http://www.timedomain.com/ [19] SandLinks [Online]. Available: http://www.sandlinks.com/ [20] Aether Wire and Location, Inc. [Online]. Available: http://www.aetherwire.com/ [21] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [22] S. Hu, C. L. Law, and W. Dou, “Measurements of UWB antennas backscattering characteristics for RFID systems,” in Proc. IEEE Int. Conf. Ultra-Wideband, Singapore, Sep. 2007, pp. 94–99.

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[23] L. Reindl, G. Scholl, T. Ostertag, H. Scherr, U. Wolff, and F. Schmidt, “Theory and application of passive SAW radio transponders as sensors,” IEEE Trans. Ultraso., Ferroelect. Freq. Contr., vol. 45, no. 5, pp. 1281–1292, 1998. [24] RF SAW, Inc. [Online]. Available: http://www.rfsaw.com/ [25] J. Lee, J. Kim, S. Yu, and J. Kim, “Picosecond time-domain characterization of CPW bends using a photoconductive near-field mapping probe,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 11, pp. 453–455, Nov. 2007. [26] L. Zhu, “Coplanar waveguide (CPW) transmission lines,” in Encyclopedia of RF and Microwave Engineering, K. Chang, Ed. Hoboken, NJ: Wiley, 2005, pp. 821–833. [27] E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section, 2nd ed. Raleigh: SciTech Publishing, 2004. [28] I. Guvenc, Z. Sahinoglu, and P. V. Orlik, “TOA estimation for IR-UWB systems with different transceiver types,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1876–1886, 2006. [29] J. Smith and J. Abel, “Closed-form least-squares source location estimation from range-difference measurements,” IEEE Trans. Acoust., Speech, Signal Process., vol. 35, no. 12, pp. 1661–1669, 1987. [30] Y. T. Chan and K. C. Ho, “A simple and efficient estimator for hyperbolic location,” IEEE Trans. Signal Process., vol. 42, no. 8, pp. 1905–1915, 1994. [31] Y. Huang, J. Benesty, G. W. Elko, and R. M. Mersereati, “Real-time passive source localization: A practical linear-correction least-squares approach,” IEEE Trans. Speech, Audio Process., vol. 9, no. 11, pp. 943–956, 2001. [32] M. Ho, V. S. Somayazulu, J. Foerster, and S. Roy, “A differential detector for an ultra-wideband communications system,” in Proc. IEEE Vehicular Technology Conf., May 2002, pp. 1896–1900. [33] S. Niranjayan and N. C. Beaulieu, “On the integrated cross-noise component in correlation receivers,” in Proc. IEEE Global Telecommunications Conf., San Francisco, CA, Dec. 2006, pp. 1–6. [34] D. D. Wentzloff and A. P. Chandrakasan, “Gaussian pulse generators for subbanded ultra-wideband transmitters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1647–1655, Apr. 2006. [35] M. Kanda, “Time-domain sensors and radiators,” in Time-Domain Measurements in Electromagnetics, E. K. Miller, Ed. New York: Springer, 1986, pp. 122–174. [36] X. Qing, Z. N. Chen, and M. Y. W. Chia, “Characterization of ultrawideband antennas using transfer functions,” Radio Sci., vol. 41, no. 1, p. RS1002, Jan. 2006. [37] R. B. Green, “The General Theory of Antenna Scattering,” Ph.D. dissertation, Ohio State Univ., Columbus, 1963. [38] D. M. Pozar, “Radiation and scattering from a microstrip patch on a uniaxial substrate,” IEEE Trans. Antennas Propag., vol. AP-35, no. 6, pp. 613–621, Jun. 1987.

Sanming Hu (S’07) was born in Tongcheng, Hubei, China. He received the B.Eng. degree in communication engineering from Nanjing University of Posts and Telecommunications (NUPT), Nanjing, China, in July 2004 and the Ph.D degree in microwave engineering from the State Key Laboratory of Millimeter Waves, Southeast University (SEU), Nanjing, China, in March 2009. From March 2006 to July 2009, he was a visiting Ph.D. student and then Research Engineer at the Positioning and Wireless Technology Centre (PWTC), Nanyang Technological University (NTU), Singapore. Currently, he is working at Institute of Microelectronics (IME), Agency for Science, Technology, and Research (A ? STAR), Singapore, as a senior research engineer. His main research interests include antennas and circuits for wireless applications such as ultrawideband (UWB), THz, radio-frequency identification (RFID), and phased

arrays. He is also interested in computational electromagnetics (CEM). He has been served as a reviewer for several international journals and conferences.

Yuan Zhou (S’05) was born in China in 1983. He received the B.Eng. degree in electrical and electronic engineering from Nanyang Technological University, Singapore, in 2007, where he is currently working toward the Ph.D. degree. His research interests include ultrawideband radio communication and signal processing with applications to ranging, positioning, and tracking.

Choi Look Law (SM’04) received the B.Eng. and Ph.D. degrees from King’s College, London, U.K., in 1983 and 1987, respectively. From 1986 to 1988, he was a Senior Research Engineer at ERA Technology in UK. He joined the School of EEE, Nanyang Technological University, Singapore, in 1988 where he is currently an Associate Professor in the Communication Engineering Division and Founding Director of the Positioning and Wireless Technology Centre. His research interests are in ultrawideband microwave circuit characterization, design and modeling, wideband channel characterization and effects on high speed wireless communication, radio frequency identification, wireless networking and positioning systems. He co-founded RFNET in 2001, a company specializing in wireless LAN and RFID products and services. He has given numerous continuing educational courses to industries and acted as consultant to a number of companies and government agencies. He has published over 100 international conference and journal papers.

Wenbin Dou (M’94) graduated from the University of Science and Technology of China, Hefei, in 1978 and received the M.S. and Ph.D. degrees from University of Electronic Science and technology of China, Chengdu, in 1983 and 1987, both in electronics and communications. From 1987 to 1989, he worked at Southeast University as a Postdoctoral Fellow. Since 1989, he has been with the Department of Radio Engineering, Southeast University, Nanjing, China, where, in 1994, he was promoted to Professor. He is also a Vice Director of State Key Laboratory of Millimeter Waves. His research interests include ferrite devices, millimeter wave quasi-optics, millimeter wave focal imaging, antennas and scattering, millimeter wave binary optics, and so on. He has completed many projects on millimeter waves from the State Ministries and Foundation and is now in charge of some key projects. He has published over 100 technical papers in journals. Two books on ferrite devices and millimeter wave quasi-optical techniques have been published in 1996 and 2000, respectively, and the book on millimeter wave quasi-optics was republished as a second edition in 2006. Prof. Dou received many awards from State Ministry, Foundation and Southeast University. He is member of State Ministry Expert Committee. He is an editor of PIER (USA) and an invited reviewer for journals such as Applied Optics, Journal of Optical Society of America (A), Optical Express, etc., by the Optical Society of America and other magazines. He is senior member of CIE and a member of Microwave Institute of CIE. He is Co-Chairman of the Program Committee of IRMMW-THz 2006 and a member of the International Advisory Committee of IRMMW-THz 2009.

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Compact, Dual-Polarized UWB-Antenna, Embedded in a Dielectric Grzegorz Adamiuk, Student Member, IEEE, Thomas Zwick, Senior Member, IEEE, and Werner Wiesbeck, Fellow, IEEE

Abstract—A compact dual-polarized antenna is described for ultrawideband (UWB) applications. The main features of the antenna, besides an ultrawidebandwidth are low cross-polarization and small dimensions. The feed of the antenna is based on the tapered slot antennas, enclosed in a dielectric. The new antenna concept, including the feed of the antenna for dual-polarization and its integration, is described in detail. Prototypes are shown and their performances are demonstrated, based on simulation and measurement results in frequency and time domain. The antenna prototype input match is better than 10 dB, a maximal gain of 10.5 dBi and a mean polarization decoupling of approximately 17 dB in the main beam direction in the 3.1–10.6 GHz UWB band are achieved. In the time domain a peak value of 0.4 m/ns, a full width at half maximum of 100 ps and a ringing of 145 ps are measured. With a maximum antenna diameter of 35 mm and 53 mm length this new antenna is also suited for UWB antenna arrays. Index Terms—Antenna arrays, antennas, dielectric loaded antenna, dual-polarized antennas, polarization diversity, ultrawideband (UWB).

I. INTRODUCTION LTRAWIDEBAND (UWB) has become a very interesting area of research. A license free usage of a wide frequency spectrum opens many possible applications investigated by the researchers and the industry. Remarkable is the bandwidth of the system, which, originally defined by FCC [1], is the widest of all specified wireless systems in the microwave spectrum. Systems with signals covering the frequency range from 3.1 to 10.6 GHz may result in high data rates for communication applications [2], very high resolution in radars [3] or high accuracy in localization systems [4]. On the other hand the low limit for the power spectral density of allows only for short-range applications, e.g., indoors. An important component of such a UWB system is the antenna. A variety of UWB antennas have been published [7]–[10]. It has been shown that traveling-wave antennas exhibit a very good time domain performance, which is important for UWB impulse systems. Additionally, in most cases they exhibit a reasonable gain. Examples of such antennas are horn

U

Manuscript received February 23, 2009; revised July 14, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant Wi 1044-23. The authors are with the Institut fuer Hochfrequenztechnik und Elektronik, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany (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.2009.2037712

antennas, tapered slot antennas or dielectric rod antennas. For several of the envisioned UWB systems (e.g., localization or radar [5], [6]) antenna arrays are required to enable imaging, digital beam forming or just to achieve higher antenna directivity. The major requirements for the antenna elements of an UWB antenna array, beside the wide bandwidth, are a small size and a directive radiation in order to suppress the grating lobes. The spacing between two antenna elements of an array for the FCC UWB band should not exceed approx. 40 mm in order to avoid grating lobes at higher frequencies [11]. This limits the dimensions in transversal direction of a single antenna element to less than 40 mm. Additionally to the above characteristics a good time domain behavior is required (described in Section IV-B). In radar/sensor and imaging applications a significant improvement of system performance can be achieved by the application of polarization diversity. For that purpose dual-polarized antennas with a common radiation phase center are mandatory. The common phase center for both polarizations is important, since offset phase centers lead to systematic errors in radar or imaging, which are often difficult to compensate. In this paper a new concept for a dual-polarized antenna suitable for UWB antenna arrays is presented. In Section II the principle of the antenna is briefly explained, followed by a detailed description of the feed network. Shaping the embedding dielectric, which is explained in Section III, optimizes the radiation characteristics. Finally in Sections IV and V the measurement results in the frequency and the time domain are presented and important conclusions are drawn. II. DIELECTRIC ROD ANTENNA WITH INTEGRATED FEED In [15] a dual polarized traveling-wave antenna for the UWB band is presented. This dual-polarized, tapered slot antenna shows a promising performance: narrow beam width, good decoupling between the ports and polarization purity in the far field of approx. 20 dB. However the width of the antenna is 62 mm, which leads to strong grating lobes in antenna arrays because of the required spacing close to two wavelengths in the upper FCC-UWB frequency range. The proper way to further miniaturize the antenna is the reduction of the wavelength by integrating it in dielectric material. Such solution is presented in [16]. The achieved width is 35 mm, which is sufficient for array applications. However the complexity of the structure (separate feed, launcher and antenna) causes problems in manufacturing of the device, which result in a lower performance and sensitivity to mechanical vibrations. The other dual polarized antennas, which can be found in literature, are dielectric rod antennas. They use metallic strips and ohmic sheets to launch

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Fig. 2. Aperture coupled, tapered slot antenna elements (left: top view; right: bottom view).

TABLE I DIMENSIONS OF THE FEEDING ELEMENT IN FIG. 2

Fig. 1. Concept of the dual-polarized UWB-antenna embedded in a dielectric, with crossed, tapered slot antennas as feed elements (top: left side, center: right side, bottom: 3D view). All units are in millimeters.

the electromagnetic wave into the dielectric rod [12]. This realization needs long launcher sections and additional feeding networks, which split the signal into differential signals. Another possibility for feeding these dielectric rod antennas is a tapered slot antenna as shown in [13]. The dielectric rod used in this antenna aims at gain enhancement for a single polarized solution, but with transversal dimensions, which do not satisfy the requirements for UWB arrays. A relatively compact solution of a dielectric rod antenna is presented in [14], but this feeding technique is not well suited for dual polarization. Therefore a new solution for a compact dual-polarized UWB array antenna is needed. To overcome the above-mentioned problems, the new concept for an antenna embedded in a dielectric is proposed with the feed, the launcher and the dielectric integrated into one compact device. This yields easy manufacturing of the antenna and improves the properties like matching and spurious radiation. The new antenna concept is shown in Fig. 1. The antenna was designed, simulated and optimized with CST Microwave Studio [17]. The antenna has an overall length of about 10 cm of which 7 cm are taken by the dielectric cone. The cone enables the smooth transmission of the waves to free space and also forms the radiation pattern. The remaining length of the dielectric houses the feed. The diameter is 35 mm, well suited for the usage in UWB antenna arrays. As a dielectric the Polytetrafluois roethylene (Teflon) with a dielectric permittivity of used. The material is chosen because of its very low losses and the good mechanical properties. The feed of the antenna is a scaled aperture coupled, tapered slot antenna, which is also often called a Vivaldi antenna [20]. The top and bottom views of the element for a single polarization are shown in Fig. 2 and the dimensions are listed in Table I.

It consists of a feeding micro-strip line, a balun and a tapered slot, which couples the wave into the surrounding material. The height of the element is 35 mm, which is equal to the diameter of the dielectric rod. As substrate Duroid 5880 with a dielecand a thickness of 0.79 mm was tric permittivity of used. Integration of this feed as launcher in the Teflon enables the coverage of the full frequency band from 3.1 to 10.6 GHz with a transversal dimension of 35 mm. The shape of the dielectric cone yields a good matching and a focused radiation with a reasonable gain. For the dual-polarized antenna, two launching elements are crossed as shown in Fig. 1. The procedure is similar to that described in [15]. To avoid the intersection of the feeding microstrip lines a small longitudinal shift between both elements is implemented. This shift is optimized to 3 mm and can be seen in Fig. 1. After cutting two crossed slots into the dielectric, the final launcher is shifted into the rod, as shown in Fig. 1 as well. III. OPTIMIZATION OF THE RADIATION CHARACTERISTICS The far field properties of the antenna are measured in an anechoic chamber in the frequency domain with a VNA. The measurement shows that both polarizations exhibit nearly the same radiation properties. The results of the gain measurements in the -plane (Co-pol) for the port 1 are shown in Fig. 8, for the geometry it is referred to Fig. 5. The radiation is focused . In the frequency range below 5 GHz the anat tenna becomes small compared to the wavelength and shows a reduced gain. In the frequency range between 6 and 11 GHz the gain is in the main beam direction relatively constant. Above 6 GHz side lobes build up close to the main beam and above 7 GHz, up to 11 GHz, a week, angular wide spread backwards radiation is visible. Narrowband numerical simulations show that the over the very wide bandwidth changing field distribution

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Fig. 3. Electric near field distributions in the H-plane at 10 GHz. Top: antenna with the shortened cone, see Fig. 4; bottom: antenna with the original cone, see Fig. 1.

Fig. 6. Simulated and measured S-parameters S , S with the full cone, see Fig. 1.

and S

of the antenna

Fig. 4. Prototype of the antenna with the shortened cone for improved radiation characteristics.

Fig. 7. Measured S-Parameters of the dual polarized antenna with shortened cone (Fig. 4).

Fig. 5. Coordinates for the antenna measurements.

causes this behavior A similar effect of side lobe generation was observed in [14]. The electric field distribution in the antenna near field can be influenced e.g., by an optimization of the embedding dielectric. Simulations show the influence of the shape of the dielectric cone. Different shapes of the dielectric cone were investigated. Relatively good improvement of the radiation characteristic is achieved by cutting off the cone to a length of 2 cm. The electric field distributions in this antenna at 10 GHz are shown in Fig. 3 for the original cone and the reduced cone. As a result of the cut off cone the electric field distribution changes, the side lobe

radiation reduces and the main beam becomes wider. The prototype of the shortened antenna is shown in Fig. 4. It becomes even more compact with an overall length of 53 mm.

IV. MEASUREMENT RESULTS OF THE PROTOTYPE ANTENNAS For the verification of the simulations two prototypes were built, one for the full cone, one for the cut cone. Fig. 5 explains the geometry and definitions for the measurements for the two orthogonal antennas. The definitions for the full cone antenna are identical. The measurements are performed in the frequency domain with Fourier transform into the time domain for the investigation

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of the time domain properties. The scattering parameters are defined as follows:

(1) represent the input matching; the parameters The parameters are the coupling between the ports. The characteristics for the antenna diagrams are defined for antenna 1 in (2) and are correspondingly defined for antenna 2

G

f; 

;

Fig. 8. Measured gain ( =0 ) of the antenna with the full cone, see Fig. 1, in the H-plane (co-pol) versus frequency (for coordinates see Fig. 5).

(2) for example is the characteristic of antenna 1 in co-polarization, -plane, along the angle with . The measurements have verified the symmetry; therefore the results for only one antenna are presented where appropriate

(3) A. Frequency Domain Characteristics

G

f; 

Fig. 9. Measured (a) and simulated (b) gain ( = 0 of the antenna 1 from Fig. 4 in the H-plane (co-pol) versus frequency.

;

)

In Fig. 6 the simulated and measured -parameters for the antenna with the original cone (cf. Fig. 1) are shown. The antenna , is better than for both polarizations matching over almost the whole FCC frequency range. The agreement between simulated and measured parameters is relatively good. A significant difference between the simulated and the meabetween the ports 1 and 2 can be observed. sured coupling The reason is that all materials were assumed lossless in the simulation. The low coupling between the two ports of over the whole UWB band is important for polarimetric UWB radar systems. The matching and coupling measurements for the shortened cone in Fig. 7 show that the input matching, as well as the decoupling between the ports, remain nearly unaffected over the whole UWB frequency band. The small ripple is due to reflections at the cone front, which do not exist for the full cone. The measured and simulated gains of the antenna 1 with the short cone (cf. Fig. 4) are plotted in Fig. 9. A comparison of the measurement results with the results for the long cone in Fig. 8 shows that the gain is well retained, although the beam becomes wider at higher frequencies. The side-lobes are reduced and spread over a wider angular range. At the lower frequency end weak resonances show up. These effects are rather well predicted by the simulation. A very similar radiation characteristic is measured in the -plane as shown in Fig. 10 with low side lobes in the whole frequency band. For radar applications the polarization purity of the dual polarized antennas in the far field is vital. The simulated and

ADAMIUK et al.: COMPACT, DUAL-POLARIZED UWB-ANTENNA, EMBEDDED IN A DIELECTRIC

G

f; ;

Fig. 10. Measured (a) and simulated (b) gain ( =0 ) of the antenna 1 from Fig. 4 in the E-plane (co-pol) versus frequency.

Fig.

G

283

11. Measured

f; ;

(

(a)

and

simulated

(b)

cross-polarization

gain

= 0 ) of the antenna 1 from Fig. 4 in the E-plane

versus frequency.

measured results for cross polarization in the -plane are shown in Fig. 11. It can be seen that the radiated cross-polarized components are much weaker than the co-polarized ones. For a better evaluation of the decoupling between both polarizations in the far field the co- and cross-polarized gains of the antenna in the main beam direction , are plotted versus frequency in Fig. 12. For most frequencies the measurements show a decoupling better than 15 dB, which means that the power ratio of radiated cross-polarized components to the co-polarized is less than approx. 3 %. The maximal gain increases with frequency and exceeds 10 dBi at upper band limit. The measurements verify the simulations. B. Time Domain Characteristics For an UWB antenna the time domain characteristics are important for a pulsed mode operation. The mean gain

G

f; 

;

Fig. 12. Measured and simulated gain ( = 0 = 0 ), ( =0 = 0 ) of the antenna 1 in Fig. 4 in Co- and cross-polarization in the main beam direction versus frequency.

G

f; 

;

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main and transformed into the time domain. The exact description of the procedure is given in [14], [19]. For pulse mode operation a very short impulse response with a low dispersion is required. The impulse refor the desired polarization is characterized sponse by the following parameters. • The peak value (5)

G

f; ;

Fig. 13. Measured and simulated mean gain ( = 0 ), ( = 0 ) of the antenna 1 in Fig. 4 in the E-plane.

G

f; ;

describes the amount of power radiated in a specified direction. Typical values for compact UWB antennas are in to 0.5 m/ns [10], [14]. the range of is • The full width at half maximum (FWHM value) mathematically defined by the following formula (6) This parameter describes the spread of the antenna impulse response in the time domain. The higher this value is, the more distorted is the radiated signal. Hence for UWB devices a short FWHM value is desirable. This corresponds to a linear phase response of the antenna. • The ringing is defined as duration of the decay of the from the peak value to transient response , where is chosen between 0 and 1 the certain value (7)

t;  = 0 ;

Fig. 14. Impulse response jh ( from Fig. 4 for co-polarization in the H-plane.

)j of the antenna 1

describes the amount of energy radiated in a specified direction [18], [19]. The angular dependent arithmetical mean gain is defined as

(4) with being the angle and frequency dependent gain of the antenna, and being the low and high cutoff frequencies of the considered frequency range. Based on (4) the measured and simulated mean gain , are shown in Fig. 13 in the -plane for the co- and cross-polarization of the antenna. The measured maximum of the mean gain is ca. 7 dBi, the cross-polarization (X) in the main beam degree) is suppressed by ca. 17 dB w.r.t. the direction ( co-polarization (Co). The second information required for time domain operation of the antenna, which is the is the impulse response in freFourier transform of the transfer function may be perquency domain. The measurement of formed in time domain, usually with less precision than in frequency domain. The results here are measured in frequency do-

Ringing is an undesired effect that results from an oscillation of the antenna’s impulse response after the peak value and may result in an inter symbol interference in communication systems or ambiguities in radars. Since there exist no standardized values for the time domain antenna parameters, no specified value for the parameter is available. Practical values for are . Here is set to , which corresponds to the decay of the temporal power density to approx. 5 % of the peak temporal power density in the specified direction. The absolute value of the impulse response of the antenna 1 from Fig. 4 in the -plane for co-polarization is shown in Fig. 14. For the exact representation the real part of the impulse in the directions 0 , 45 and response 90 from the same measurement are plotted in Fig. 15. The transient response in the main beam direction is short and strong. Also the ringing is very short. The values of the time domain parameters for the main beam direction are given in Table II. All the quantities indicate a very good performance of the antenna in the time domain. Like in frequency domain, the time domain characteristics are almost identical for both polarizations. V. CONCLUSION In this publication a new, dual polarized UWB antenna is presented. The antenna is based on the tapered slot antennas as feeds embedded in a dielectric with a diameter of 35 mm and

ADAMIUK et al.: COMPACT, DUAL-POLARIZED UWB-ANTENNA, EMBEDDED IN A DIELECTRIC

Fig. 15. Impulse response h (t;  = 0 ; ) of the antenna 1 from Fig. 4 for co-polarization in the H-plane for three different angles.

TABLE II TIME DOMAIN PARAMETERS OF THE ANTENNA FROM FIG. 4 IN THE MAIN BEAM DIRECTION

a length of 53 mm. The measurements in the frequency domain show a directive radiation pattern with an antenna gain up to 10 dBi. The decoupling between the 2 orthogonal ports is better than 20 dB and the mean decoupling of the co- and cross-polarization is 17 dB. In the time domain the measured antenna , a high peak impulse response has a short magnitude and short ringing . This makes the antenna suitable for polarimetric, impulse based UWB systems in communications and radar systems. Because of the small size it can be applied in UWB antenna arrays. REFERENCES [1] “Revision of Part 15 of the Commission’s Rules Regarding Ultra Wideband Transmission” Systems Federal Communications Commission (FCC), 2002, First Rep. and Order, ET Docket 98-153, FCC 02-48. [2] J. H. Reed, An Introduction to Ultra Wideband Communication Systems. Englewood Cliffs, NJ: Prentice Hall, 2005. [3] S. Gezici, Z. Tian, G. B. Giannakis, H. Kobayashi, A. F. Molisch, H. V. Poor, and Z. Sahinoglu, “Localization via ultra-wideband radios: A look at positioning aspects for future sensor networks,” IEEE Signal Processing Mag., vol. 22, no. 4, pp. 70–84, Jul. 2005. [4] R. Zetik, J. Sachs, and R. Thomä, “UWB short range radar sensing,” IEEE Instrum. Meas. Mag., vol. 10, pp. 39–45, Apr. 2007. [5] X. Zhuge, T. G. Savelyev, A. G. Yarovoy, L. P. Ligthart, J. Matuzas, and B. Levitas, “Human body imaging by microwave UWB radar,” in Proc. European Radar Conf. EuRAD, Amsterdam, Oct. 30–31, 2008, pp. 148–151. [6] C. Senger and T. Kaiser, “Beamloc – An approach for NLOS localization in UWB indoor environments,” in Proc. Institution of Engineering and Technology Seminar on Ultra Wideband Systems, Technologies and Applications, London, U.K., Apr. 20–20, 2006, pp. 176–180. [7] Ultra-Wideband Antennas and Propagation for Communications, Radar and Imaging B. Allen, M. Dohler, E. E. Okon, W. Q. Malik, A. K. Brown, and D. J. Edwards, Eds. London, U.K., Wiley, 2006. [8] D. Ghosh, A. De, M. C. Taylor, T. K. Sarkar, M. C. Wicks, and E. L. Mokole, “Transmission and reception by ultra-wideband (UWB) antennas,” IEEE Antennas Propag. Mag., vol. 48, no. 5, pp. 67–99, Oct. 2006. [9] H. Schantz, The Art and Science of UWB Antennas. : Artech House, 2005. [10] W. Wiesbeck and G. Adamiuk, “Antennas for UWB-systems,” in Proc. 2nd Int. ITG Conf. on Antennas INICA’07, Mar. 28–30, 2007, pp. 67–71.

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[11] W. Sörgel, C. Sturm, and W. Wiesbeck, “Impulse responses of linear UWB antenna arrays and the application to beamsteering,” in IEEE Int. Conf. on Ultra-Wideband, ICU, Sep. 5–8, 2005, pp. 275–280. [12] J.-Y. Chung and C.-C. Chen, “Ultra-wide bandwidth two-layer dielectric rod antenna,” in Proc. IEEE Antennas and Propag. Int. Symp., Jun. 9–15, 2007, pp. 4889–4892. [13] A. Elsherbini, C. Zhang, L. Song, M. Kuhn, A. Kamel, A. E. Fathy, and H. Elhennawy, “UWB antipodal vivaldi antennas with protruded dielectric rods for higher gain, symmetric patterns and minimal phase center variations,” in Proc. IEEE Int. Antennas and Propag., Jun. 9–15, 2007, pp. 1973–1976. [14] M. D. Blech and T. F. Eibert, “A dipole excited ultrawideband dielectric rod antenna with reflector,” IEEE Trans. Antennas Propag., vol. 55, pp. 1948–1954, Jul. 2007. [15] G. Adamiuk, T. Zwick, and W. Wiesbeck, “Dual-orthogonal polarized Vivaldi antenna for ultra wideband applications,” in Proc. 17th Int. Conf. on Microwaves, Radar and Wireless Communications, MIKON, Wroclaw, Poland, May 19–21, 2008. [16] G. Adamiuk, C. Sturm, T. Zwick, and W. Wiesbeck, “Dual polarized traveling wave antenna for ultra wideband radar application,” presented at the Proc. Int. Radar Symp. IRS, Wroclaw, Poland, May 2008. [17] CST Microwave Studio, [Online]. Available: www.cst.com [18] E. G. Farr and C. E. Baum, “Extending the Definitions of Antenna Gain and Radiation Pattern into the Time Domain” Sensor and Simulation Notes Air Force Research Laboratory, Directed Energy Directorate, Kirtland, New Mexico, 1992, Note 350. [19] W. Sörgel, “Charakterisierung von Antennen für die Ultra-WidebandTechnik,” , Forschungsberichte aus dem Institut für Höchstfrequenztechnik und Elektronik der Universität Karlsruhe, , 2007. [20] W. Sörgel and W. Wiesbeck, “Influence of the antennas on the ultrawideband transmission,” EURASIP J. Appl. Signal Processing, no. 1, pp. 296–305, Jan. 2005.

Grzegorz Adamiuk (S’07) received the Dipl.-Ing. degree in electrical engineering (M.Sc. E. E.) from the Technical University of Gdansk, Poland and from the Universität Karlsruhe (TH), Germany, both in 2006. He is currently working toward the Dr.-Ing. (Ph.D.E.E.) degree at the Institut für Hochfrequenztechnik und Elektronik, Karlsruhe Institute of Technology (KIT). His main research topic is UWB technology with the focus on UWB antennas, antenna arrays and UWB radar. Mr. Adamiuk is a member of the German Association for Electrical, Electronic and Information Technologies (VDE). Since 2008, he has chaired a working group on RF Aspects in UWB Communication and Localization within the framework of COST 2100 (European Cooperation in the field of Scientific and Technical Research) - “Pervasive Mobile & Ambient Wireless Communication.” In 2003 he was granted a scholarship from the Universität Karlsruhe (TH), Germany. He received the award for the Best Young Scientist Presentation at the 17th IEEE International Conference on Microwaves, Radar and Communications MIKON 2008 and the Best Paper Award at the 4th IEEE UWB Forum on Sensing and Communication.

Thomas Zwick (SM’06) received the Dipl.-Ing. (M.S.E.E.) and Dr.-Ing. (Ph.D.E.E.) degrees from the Universität Karlsruhe (TH), Karlsruhe, Germany, in 1994 and 1999, respectively. From 1994 to 2001, he was a Research Assistant at the Institut für Höchstfrequenztechnik und Elektronik (IHE), Universität Karlsruhe (TH). In February 2001, he became a Research Staff Member at the IBM T. J. Watson Research Center, Yorktown Heights, NY. From October 2004 to September 2007, he was with Siemens AG, Lindau, Germany. During this period he managed the RF development team for automotive radars. In October, he was appointed Full Professor at the Universität Karlsruhe (TH), where he is Director of the Institut für Hochfrequenztechnik und Elektronik (IHE). His research topics include wave propagation, stochastic channel modeling, channel measurement techniques, microwave material measurements, microwave techniques, millimeter wave antenna design, ultrawideband systems, wireless communication and radar system design.

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Prof. Zwick participated as an expert in the European COST231 Evolution of Land Mobile Radio (Including Personal) Communications and COST259 Wireless Flexible Personalized Communications. For the Carl Cranz Series for Scientific Education he served as a Lecturer for Wave Propagation. He received the Best Paper Award at the International Symposium on Spread Spectrum Techn. and Appl. ISSSTA 1998. In 2005, he received the Lewis Award for the outstanding paper at the IEEE International Solid State Circuits Conference. Since 2008, he is President of the Institute for Microwaves and Antennas (IMA).

Werner Wiesbeck (F’94) received the Dipl.-Ing. (M.S.E.E.) and Dr.-Ing. (Ph.D.E.E.) degrees from the Technical University Munich, Germany, in 1969 and 1972, respectively. From 1972 to 1983, he was with AEG-Telefunken in various positions, including that of Head of R&D of the Microwave Division in Flensburg and Marketing Director Receiver and Direction Finder Division, Ulm. During this period, he had product responsibility for millimeter-wave radars, receivers, direction finders, and electronic warfare systems. From 1983 to 2007, he was Director of the Institut für Höchstfrequenztechnik und Elektronik, University of Karlsruhe (TH), where he had been Dean of the Faculty of Electrical Engineering and is now Distinguished Scientist with the

Karlsruhe Institute of Technology. Research topics include electromagnetics, antennas, wave propagation, communications, radar, and remote sensing. In 1989 and 1994, respectively, he spent a six-month sabbatical with the Jet Propulsion Laboratory, Pasadena, CA. He is a member of an Advisory Committee of the EU-Joint Research Centre (Ispra/Italy). He is an advisor to the German Research Council (DFG), to the Federal German Ministry for Research (BMBF), and to industry in Germany. Dr. Wiesbeck is an Honorary Life Member of IEEE GRS-S, a member of the Heidelberger Academy of Sciences, and a member of acatech (German Academy of Engineering and Technology). He is a member of the IEEE GRS-S AdCom (1992–2000), Chairman of the GRS-S Awards Committee (1994–1998, 2002-), Executive Vice President IEEE GRS-S (1998–1999), President of IEEE GRS-S (2000–2001), Associate Editor IEEE-AP Transactions (1996–1999), and past Treasurer of the IEEE German Section (1987–1996, 2003–2007). He has been General Chairman of the 1988 Heinrich Hertz Centennial Symposium, the 1993 Conference on Microwaves and Optics (MIOP ’93), the Technical Chairman of International mm-Wave and Infrared Conference 2004, Chairman of the German Microwave Conference GeMIC 2006, and a member of the scientific committees and Technical Program Committees of many conferences. He has received many awards, most recently the IEEE Millennium Award, the IEEE GRS Distinguished Achievement Award, the Honorary Doctorate (Dr. h.c.) from the University Budapest/Hungary, the Honorary Doctorate (Dr.-Ing. E.h.) from the University Duisburg/ Germany, and the IEEE Electromagnetics Award 2008.

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Novel Compact Model for the Radiation Pattern of UWB Antennas Using Vector Spherical and Slepian Decomposition Wouter Dullaert and Hendrik Rogier, Senior Member, IEEE

Abstract—A new compact model is described for the 3D radiation pattern of an ultrawideband antenna, using a vector spherical and Slepian decomposition. Vector spherical modes are known to provide a good basis for the angular dependency of the radiation pattern. This paper is the first to extend such a model to also incorporate the frequency dependency of a radiation pattern. This is achieved by using a Slepian mode expansion. It is shown that this model requires considerably less coefficients than traditional sampling to accurately describe a frequency-dependent 3D radiation pattern. Also, generating the Slepian modes is computationally more efficient than comparable techniques, such as the singularity expansion method (SEM). The coefficients can then directly be used to efficiently calculate performance measures such ) without reconas the antenna Fidelity Factor for all angles ( structing the radiation pattern, or to reduce the noise contribution. Index Terms—Antenna radiation patterns, discrete prolate spheroidal sequences (DPSS), fidelity factor, modeling, spherical modes, ultrawideband (UWB) antennas, ultrawideband (UWB) radiation.

I. INTRODUCTION LTRAWIDEBAND (UWB) communication is a very hot topic at the moment in both industrial and academic research. The protocol already exists for quite some time in military radar applications, but since the frequency band allocation by the FCC in 2002, [1], interest for consumer applications gained momentum. In [1] a UWB system is defined as a communication system that has an absolute bandwidth larger than 500 MHz or a relative bandwidth larger than 0.2 times the center frequency. The FCC allows unlicensed use of these systems between 3.1 GHz and 10.6 GHz with a spectral mask of . In this paper a special form of UWB communication will be used: pulse modulated UWB. It is assumed that the pulse uses the entire FCC band. The large bandwidth, in combination with the fact that pulse modulated UWB operates in time domain, is a big challenge for the predominantly small-band and frequency-domain oriented antenna design. [2] shows that pulse distortion deserves

U

Manuscript received December 05, 2008; revised June 19, 2009. First published December 04, 2009; current version published February 03, 2010. W. Dullaert is with the Department of Information Technology, Ghent University, Ghent B-9000, Belgium (e-mail: [email protected]). H. Rogier is with the Vakgroep Informatietechnologie (INTEC), Ghent University, Ghent B-9000, Belgium (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037700

special care when designing a UWB antenna. Unfortunately pulse distortion does not translate to traditional antenna performance measures such as return loss or gain in a straightforward manner. Therefore various performance measures for UWB antennas have been presented in literature: [3] proposed the fidelity factor, which correlates the applied voltage pulse with the pulse shape of the transmitted electric field. In this paper, for the first of azimuth and time, this factor is evaluated for each pair elevation angles. [4] defined the pulse width stretch ratio, which measures how much a pulse gets smeared out in time by the antenna. [5] extended the fidelity factor by replacing the applied voltage pulse in the correlation by a template function that is tailored to the situation. Finally [6] combines previous research on the pattern stability factor and pattern stability bandwidth. All these performance measures require full knowledge of the frequency response of the antenna, which is closely related to the frequency-dependent radiation pattern. For an antenna covering the entire FCC band this is an impractically large amount of data. An efficient, compact representation of the radiation pattern will decrease the calculation time and the memory requirements when determining antenna performance measures. In this paper such an efficient model is presented. As in [7], the angular dependency of the radiation pattern is modeled using a decomposition into vector spherical modes. The frequency dependency is expanded into a series of Slepian modes, which are known to be a good set of basis functions for bandlimited signals, [8]. The performance of this Slepian expansion is compared to a vector fitting technique: the singularity expansion method (SEM). Another model based parameter estimation technique is discussed in [9]. The decomposition of the radiation pattern into vector spherical modes is discussed in Section II. Section III provides some theoretical background for the Slepian expansion. In Section IV this knowledge is applied to decompose the frequency dependency of the radiation pattern into Slepian modes. The model is then used to evaluate the fidelity factor in Section V. Section VI briefly introduces the SEM technique, used for validation. In Section VII the model is validated by applying it to an in-house developed antenna and comparing the results to those obtained with the SEM method.

II. VECTOR SPHERICAL DECOMPOSITION Because of the very large bandwidths considered, the radiation pattern of an ultrawideband antenna must be described as

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a function of three variables: the azimuth angle , the elevation angle and the frequency . Adding the frequency variable to the standard angular and variables easily increases the amount of required sample points by a factor of 50. is considIn this paper a full 3D radiation pattern is deered. In a first step the 3D radiation pattern composed into a series of vector spherical modes. Vector spherical modes are a combination of phase modes, described in [10], and spherical modes, described in [11]. Other ways to efficiently describe the angular dependency of a radiation pattern, such as [12], have been presented, but will not be covered further. [13] and [7] show that the decomposition into vector spherical modes results in a very compact description of the angular dependencies of the radiation pattern. In [14] this decomposition is applied to a UWB radiation pattern. The vector spherical decomposition of the 3D radiation pattern is given by:

(1)

for either of them, which makes them rather difficult to generate. However, as is explained later on, they can be generated efficiently by calculating the eigenvectors of a carefully constructed matrix. They form the mathematical basis of the Slepian decomposition discussed in Section IV. For more information, the interested reader is referred to [15]. Driven by the mathematical, but unnatural, certainty that a non-trivial function cannot have limited support in both time and is defined as frequency domain, the zeroth-order PSWF the solution to the following maximization problem, for ban: dlimited

(5) where denotes the considered time-interval of the PSWF. The is the solution to (5) for all W-bandlimfirst-order PSWF ited orthogonal to , where orthogonality is defined using the standard L2-norm. The higher-order PSWFs are defined in a similar fashion, [16]–[19] Some mathematical manipulation on (5) shows that the PSWFs are also the eigenfunctions of the sinc-kernel

Where and are the harmonic vector spherical basis functions defined by:

(2)

(6) , the time-bandwidth product, the where and is the order of the PSWF, bandwidth of resulting in a second way to define the PSWFs. The PSWFs are also the solution to the following secondorder differential eigenvalue problem:

(7)

(3) with

representing the associated Legendre polynomials, and given by:

(4) [7] showed that the amount of relevant phase modes, and thus by extension vector spherical modes, is limited by the dimensions of the antenna. This allows us to truncate the series by limiting the first summation in (1) to M instead of . [13] suggests that this truncated series is a good approximation for the , with the largest dimension of radiation pattern if the antenna and the wave number of the highest considered frequency.

is a real and positive eigenvalue: . where This is the third and final way to define the PSWFs. Whereas the first method is the most intuitive one, this last method gives much more information about the functions. PSWFs form a complete and orthonormal basis for all W-ban, they are orthonormal on the real timedlimited functions . The finite-Fourier transform axis and orthogonal over of a PSWF satisfies

(8) whereas their Fourier transform satisfies

III. PSWF AND DPSS THEORY In this section, some properties are described of the prolate spheroidal wave functions (PSWF) and the discrete prolate spheroidal sequences (DPSS), the discrete counterpart of the PSWF. There is no closed-form analytic expression known

(9) Because of the numerous properties associated with these functions, they have been used in a multitude of different fields. [20]

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proposed to use them as the basis for a series expansion which compares favorably to other expansions in a lot of cases. The functions have also proven useful in channel estimation [21], wavelets [22], filter design [23], etc. In [24] Slepian presented the discrete equivalent of the PSWF: the discrete prolate spheroidal sequence (DPSS) and the order of the DPSS. He proved that they can be defined as the solution to a discrete energy maximization problem. As in (6), they are also the sinc-matrix with elements eigenvectors of a

(10) where is both the number of DPSSs in a set and the amount is a bandwidth with the constraint of samples per DPSS. , resulting from the Nyquist theorem dictating that only frequencies up to half the sampling frequency can be accurately represented. In theory it is possible to calculate the DPSSs from (10), but unfortunately the problem is ill-conditioned. Although it can be proven that the eigenvalues of (10) are all distinct, they are all concentrated around either 0 or 1, and the difference between two eigenvalues can be smaller then the numerical precision used in the calculations. Fortunately there is, similar to (7), a third way to define them by using a difference equation, [24], [25]

(11) This difference equation can be written as an eigenvalue , defined as follows: problem for the matrix

(12) otherwise defined as before. Because this matrix commutes with and from (10), it has the same eigenvecwith the matrix are tors, but different eigenvalues: the eigenvalues of not clustered, eliminating the possibility that the eigenvector problem becomes ill-conditioned. This method, which has been used in this article, provides a robust and fast numerical way to generate the DPSSs. As for PSWFs, the DPSSs are orthonormal, , and they form a complete basis for vectors of length . Remark that no equivalent for properties (8) and (9) of the PSWFs could be found. It would be very interesting if such properties would exist, and this remains an interesting topic for further research. In Fig. 1 the magnitude of the first 3 orders of the DPSS is shown for a number of samples .

Fig. 1. Absolute value of the DPSS’s with length L

= 73.

IV. SLEPIAN DECOMPOSITION Because the PSWFs and DPSSs are the solutions to an energy maximization problem for bandlimited functions, see (5) and (10), they are extremely well suited to model band limited signals. The Slepian decomposition used here assumes that discrete frequency the radiation pattern is known at samples. The decomposition starts by expanding the radiation pattern following (1). The frequency-depenand dent coefficients are expanded into a seis the DPSS of order , calculated ries of DPSSs, where as the eigenvector of matrix defined by (12). is a free paramFor our purpose the bandwidth parameter eter which can be chosen to best suit the application. An optimal is determined by using a particle swarm optimizavalue for tion algorithm, [26], which maximizes the energy contained in the first 20 orders of the Slepian expansion. The algorithm perwas formed 3000 iterations, after which a value of obtained. A value within 0.1% of the final value was obtained after 200 iterations, so it is safe to assume that the optimal value for this problem was found. If the radiation pattern is known as a continuous function of frequency, PSWFs should be taken as basis functions for the Slepian expansion. Since the DPSSs are the natural vectors for a bandlimited series, [21], the energy of the coefficients decreases with increasing order of the Slepian modes. We can therefore truncate coefficients without great loss the series to a maximum of of precision, by replacing with in both formulas. A rough can be found using the folvalue for the maximum order lowing rule of thumb:

(13) where is the upper limit of the frequency band of interest and the lower limit. The final expression for the radiation pattern

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then becomes (14), shown at the bottom of this page. V. FIDELITY FACTOR The model presented in the previous sections would have little advantages other than data compression, if the radiation pattern needs to be explicitly reconstructed to perform calculations. However UWB performance measures based on the radiation pattern can directly be calculated using the model coefficients. In this section this is shown for the fidelity factor, defined in [3]. is one of the most often used pulseThe fidelity factor modulated-UWB performance measures, defined in [3] as follows:

(15) where normalized

is version

of

the

input

pulse

the ,

is the normalised version of the output pulse and the impulse response of the antenna. The Fourier , i.e., the frequency response , transform of is given by the radiation pattern of the antenna. Knowledge of the radiation pattern over all angles as a function of frequency and the input-pulse as a function of time is necessary to compute the fidelity factor. and , (15) beMaking use of the definitions of comes (16), shown at the bottom of the page, where the cordenotes relation is calculated in the frequency domain, the inverse Fourier transform of its arguments, the superscript *

stands for the complex conjugate and the denominator has been calculated using Parceval’s theorem. Substituting (14) in (16) results after some simplifications in (17), shown at the bottom of the page, where (18) only depends on the input pulse and

(19) only depends on the antenna radiation pattern. The effects of the antenna and the input pulse are clearly separated. This property is a great advantage during optimizations as only or needs to be calculated in each run, respectively depending on whether the antenna or input pulse are being optimized. Equation (17) expresses the fidelity factor as a function and without explicitly of the model coefficients reconstructing the entire radiation pattern, which makes the fieasier to calculate because a delity factor as a function of much smaller number of samples needs to be evaluated. Despite the fact that the definition of the fidelity factor (15) , most papers, such as [27] and even the origdepends on inal paper by Lamensdorf and Susman [3], present one fidelity factor for the entire antenna, along a single angle of departure. It is measured by correlating the input pulse with the output pulse of the entire system, where all other elements of the system except for the Antenna Under Test are assumed distortionless. This approach only yields useful results for highly directive antennas, which have one well-defined direction of radiation, along which

(14)

(16)

(17)

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the fidelity factor is calculated. However, most UWB antennas are meant for mobile use and have rather omnidirectional radiation patterns, as the locations of the transmitting and receiving antenna are not fixed. More recent papers, such as [28], recognize this flaw and evaluate the fidelity factor for a few different directions. Here the fidelity factor is for the first time evaluated . for all angles VI. SINGULARITY EXPANSION METHOD To validate the Slepian mode expansion we will also consider the frequently used singularity expansion method, [29], which can be computed by Prony’s method [30]–[32] or the matrix pencil method (MPM) [33]. In antenna modeling the MPM is often used because of its robustness and computational efficiency: [34] uses the MPM to model the impulse response of an antenna in one direction. [35] is similar to [34] but models the effective length instead of the antenna transfer function. A modified version of the MPM is proposed in [36] to model the radiation pattern for multiple directions. [37] applies this algorithm together with a spherical wave decomposition to model an antenna radiation pattern in multiple directions. The SEM expansion for one polarization and multiple directions looks as follows:

Fig. 2. Planar monopole topology.

will have a lot of coefficients with a very low energy level and thus a very significant noise contribution. These coefficients will make it much more difficult for the SEM to find the correct antenna poles, greatly reducing the accuracy of the model. VII. VALIDATION A. Flexible UWB Antenna

(20) where is theoretically , but can be truncated to a finite number with minimal loss in accuracy because of the bandlimited character of antennas. The poles are the natural frequencies of the object and are theoretically independent of the direction of observation. Unfortunately, because of noise and inaccuracies, in both simulation and measurements, the resulting poles will differ for different look directions. This is why a separate algorithm to calculate the poles for multiple directions, such as the one presented in [36], is needed. It should be noted that the extraction of the poles revolves around an SVD expansion of a matrix that grows linearly with the number of directions used in the expansion. For this reason the poles are calculated using a 2D cut of the radiation pattern, after which these poles are used to calculate the residues for the entire 3D radiation pattern. and residues are known, the Once the poles angular dependencies of the residues can be decomposed into spherical waves, using the same approach outlined in Section II, where the residues of both components of the radiation pattern are recombined as a vector

(21) Because the SEM is a form of rational modeling, it is very sensitive to discontinuities, such as noise, in the dataset. It is therefore important to start with the SEM decomposition before decomposing the angular dependencies of the residues into spherical wave functions. The spherical wave decomposition

The model proposed in this paper is now validated based on an in house developed UWB antenna for use in the 3.1 GHz to 10.6 GHz FCC band. The antenna uses the popular planar monopole topology, amongst others also used in [38], with an inverted tear top-element. This topology has numerous advantages. Its planar nature and small size allow the antenna to be easily integrated into other devices, it offers a very large bandwidth and has a small group delay. The antenna was fabricated on a flexible polyimide substrate, polyimide layer and a 18 copper layer. consisting of a 25 at 10 kHz. The polyimide has a dielectric constant of Because the substrate is extremely thin, the exact in the frequency band will be of little importance. The flexibility and extremely thin nature of the antenna allows it to be very easily integrated in, for example, modern cell phones with strange form factors. A schematic representation of the antenna is shown in Fig. 2. The produced antenna is shown in Fig. 3. The antenna was designed and optimised for a minimal reflection coefficient in the frequency band of interest using ADS Momentum. The corresponding dimensions are shown in Table I. The results were later verified by simulating the design in CST Microwave studio. Fig. 4 shows both the simulated and measured reflection coefficient. It can be seen that the measured and simulated curves show a good agreement: they have roughly the same shape, but the curve of the measured antenna has shifted to higher frequencies. The simulated antenna has a reflection coefficient dB in the entire FCC band. The differences between the simulated and measured antenna are due to imperfections in the manual soldering of the SMA connector. The antenna is very sensitive to small misalignments between antenna and connector.

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Fig. 5.

F

simulated for 

= 90

.

Fig. 6.

F

simulated for 

= 90

.

Fig. 3. Produced planar monopole antenna.

Fig. 4. Simulated and measured reflection coefficient.

TABLE I ANTENNA DIMENSIONS

the simulated pattern, but the noise added by the measurement makes a direct comparison impossible, especially at the higher frequencies. B. Noise Reduction

Figs. 5 and 6 show two cuts of the simulated radiation pattern as a function of frequency, for the and polarization respectively. Fig. 7 shows the measured radiation pattern for a fixed , with 121 and 51 frequency samples. elevation angle The measured radiation pattern has roughly the same shape as

As a first application of the new expansion proposed in this paper, we show that noise can be removed just by neglecting the higher order (spherical) phase modes and/or Slepian modes. Fig. 7 presents a 2D cut of the radiation pattern. Here the angular dependency of this radiation pattern is modeled using phase modes in azimuth only instead of a complete vector spherical expansion. The model coefficients are shown in Fig. 8 for the measured data and in Fig. 9 for the simulated data. Despite the considerable noise contribution at higher frequencies, the energy of the coefficients still decreases exponentially with increasing

DULLAERT AND ROGIER: NOVEL COMPACT MODEL FOR THE RADIATION PATTERN OF UWB ANTENNAS

Fig. 7.

F

measured for 

= 90

293

.

Fig. 9. Coefficients of the simulated 2D radiation pattern.

Fig. 10. Reconstructed measured radiation pattern. Fig. 8. Coefficients of the measured 2D radiation pattern.

C. Data Compression order. The noise in the measurement is visible by the increased noise floor in the coefficient data. Fig. 10 shows the reconstructed radiation pattern. It has good resemblance with the original measured data, but a greatly reduced noise contribution. Truncating the Slepian series and phase modes kept the frequency dependency of the radiation pattern but filtered out the measurement noise. The model is, unlike rational modeling methods, so resistant to noise that it can effectively be used as a filter.

Next the model constructed in Section IV is applied to the simulated full 3D radiation pattern of the antenna from Section VII-A using Matlab R2007b. The radiation pattern and is sampled every degree for both the azimuth angle elevation angle for both polarizations. The radiation pattern was sampled every 125 MHz. This results in a radiation pattern described by 2 73 360 180 samples. 1) Slepian Expansion: The parameters of the model were and . This choice results in a chosen to be

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Fig. 11. Fixed Slepian mode cut: A

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for k

= 1. Fig. 13. Fixed vector spherical mode cut: A

Fig. 12. Fixed phase mode cut: A

for m

for n

= 6.

= 3.

model of 2 coefficient cubes with each 30 8 17 samples, a reduction with a factor of more than 1159. The amount of samples can be reduced even further by taking into account that . no vector spherical modes exist for Since showing two 3D cubes of data is not practical, 3 cuts cube for a of one cube will be shown: Fig. 11 shows the , Fig. 12 shows the cube fixed Slepian Mode order, , and Fig. 13 shows the for a fixed phase mode, cube for a fixed value, . It can be visually verified that the energy of the coefficients has become sufficiently low at the truncation boundaries. The relative error between the original sampled radiation pattern and the radiation pattern reconstructed from the model coefficients for a given polarization is evaluated using the following error function:

(22)

and denote the original and where reconstructed radiation pattern, respectively. Fig. 14 shows the relative error for both polarizations. The two relative errors remain smaller than 2%. Both errors are almost equal because the information for both polarizations is present in equal amounts

Fig. 14. Relative error for both polarizations.

in the and coefficients: leaving out the higher order coefficients affects both polarizations at the same time in the same way. 2) SEM Expansion: For validation the SEM model from Section VI is used to model the same radiation pattern. Because the SEM expansion performs a SVD expansion there is a limit to the size of the matrix used in the model. For this reason the radiation pattern is sampled every 3 degrees instead of every single degree. Combined with a frequency sampling every 125 MHz, this results in a radiation pattern of 2 73 120 60 samples. The SEM expansion has another major drawback: it can only model reasonably smooth data. This is in general not a problem for simulated antenna radiation patterns, except when the radiation pattern contains a very steep null. The -polarization of the radiation pattern contains a zero that is not smooth over frequency, due to numerical noise in the simulations. The very small values are very noisy as a function of frequency, but are continuous as a function of the angle . Because of this reason

DULLAERT AND ROGIER: NOVEL COMPACT MODEL FOR THE RADIATION PATTERN OF UWB ANTENNAS

Fig. 15. Fixed pole cut: Asem

for p

295

= 3.

the SEM expansion cannot model the -polarization of the radiation pattern and in what follows only the -polarization is considered. The parameters of the vector spherical expansion in the SEM procedure are chosen in the same way as in the Slepian expan. At this point the energy contained in sion procedure: the vector spherical modes has decreased sufficiently to neglect the higher order modes. The number of poles to be included in the SEM expansion was determined by trial and error: the number of poles was increased until the radiation pattern could be reasonable well reconstructed. This results in a number of . It should also be noted that the reconstruction poles error decreases very steeply when the number of poles increases: the reconstructed radiation pattern looks nothing like the original radiation pattern when only 34 poles are used. This behavior renders the SEM expansion useless for filtering purposes. The model now contains 2 (35 31 15 35) or 32620 coefficients. This is a reduction by a factor 32. Fig. 15 displays a cut of the coefficient cube for a fixed . We can clearly see that the energy contained in pole: the residues can still be accurately modeled with a spherical wave expansion: the energy of the coefficients exponentially and increase. Fig. 16 displays decreases when the orders . The energy vector the coefficient cube for a fixed order spherical modes still decreases exponentially with increasing order . The energy in the coefficients does not decrease when the order of the poles increases, but this is not a problem, as the energy of the residues is plotted and the actual frequency dependency is modeled in the poles. Fig. 17 shows the poles for the -polarization by means of a scatter plot in the complex plane. To compare the relative error between the reconstructed radiation pattern and the original radiation pattern, the same error function (22) is used. Fig. 18 shows the relative error on . The error varies evenly over frequency with a maximum of The calculation times are compared on a quad core linux system (two dual core AMD Opteron™ processors model 270

Fig. 16. Fixed phasemode cut: Asem

for m

= 3.

Fig. 17. Poles of the  polarization.

with 2 GHz per core and 8 GB of RAM) using Matlab, by executing each algorithm 100 times on a 2 73 120 60 radiation pattern and taking the mean calculation time. It takes on average 0.1846 seconds to calculate the Slepian model coefficients for a radiation pattern of this size and 29.7265 seconds to expand the radiation pattern in SEM coefficients. D. Fidelity Factor Evaluation As a third application, the fidelity factor, defined in (15), was calculated using a fifth-order derivative of a Gaussian monocycle, which is known to fit into the FCC spectral mask, [39]

(23)

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Fig. 18. Relative error for E using SEM expansion.

Fig. 20. UWB pulse spectrum and FCC spectral mask.

Fig. 19. FCC compliant UWB pulse.

Fig. 21. Fidelity Factor 8(; ) for the  polarization.

where is a normalisation constant to fit the pulse to the FCC . The FCC compliant pulse is spectral mask, and shown in Fig. 19. Fig. 20 shows the spectrum of the pulse and the FCC spectral mask for indoor UWB communications. The fidelity factor will not be evaluated at the locations of the nulls in the radiation pattern. The fidelity factor suffers from the fact that it is only defined for a single polarization and that a null in the radiation pattern deteriorates the fidelity factor significantly, even though these directions do not contribute to the these directions are and , radiation. For these directions are and . The fidelity for factors for all other angles and and both polarizations, calculated using direct application of (15) are shown in Figs. 21 and 22. is larger than 0.8 for most of the The fidelity factor directions and is 0.94 in the main radiation direction for both

polarizations. These are good results: the maximum value of compares well to the results presented in [28], and espeonly varies by a small amount cially the -component of . It can also be seen that the antenna as a function of performs a lot better for -polarization than for -polarization: is very low for a few angles. If only the system fidelity factor, or the fidelity factor at 5 discrete directions, had been evaluated, these directions with high distortion would not have been known. The fidelity factor is now evaluated using the Slepian model, using (17). Fig. 23 shows the relative error between the original -polarised fidelity factor and the one calculated using the Slepian model. Fig. 24 shows the relative error for the -polarization fidelity factor. The relative error is well below 1% for both polarizations. The peaks in both figures are located near the nulls of the radiation pattern: the radiated energy in these

DULLAERT AND ROGIER: NOVEL COMPACT MODEL FOR THE RADIATION PATTERN OF UWB ANTENNAS

Fig. 22. Fidelity Factor 8(; ) for the  polarization.

Fig. 23. Relative Error for the  -polarization fidelity factor.

directions is smaller which means that the error made by the model becomes relatively larger. calculated from the Fig. 25 shows the relative error on SEM decomposition. We notice that the error is of the same magnitude as the error made on the radiation pattern itself: 4%. We now compare the computation speed of the direct algorithm, based on the definition of the fidelity factor (15), which evaluates the correlations and convolutions in the frequency domain using the convolution theorem of the DFT, with the method based on the Slepian expansion, using (17). The direct method requires a CPU-time of 52.67s. The method using model coefficients takes 10.37s to calculate a fidelity factor for all direc. This time includes the decomposition of the comtions plete frequency-dependent 3D radiation pattern into coefficients . Using the new compact model, the fidelity factor can be calculated roughly five times faster, which is a considerable increase in performance.

297

Fig. 24. Relative error for the -polarization fidelity factor.

Fig. 25. Relative error on the fidelity factor for the  -polarization.

E. Comparison The comparison between the two models yields the following observations. 1) The Slepian expansion can handle much larger datasets than the SEM expansion 2) The SEM expansion is very sensitive to noise: it is unable to model the -polarization which contains a deep null. The Slepian expansion handles this without problems. 3) The Slepian expansion modeled a 2 73 360 180 radiation pattern by means of 2 30 17 8 coefficients with a maximum relative error smaller than 2%. The SEM expansion modeled a 2 73 120 60 using 2 35 31 15 35 coefficients with a maximum relative error smaller than 4.5%. This means that the Slepian expansion modeled a more detailed radiation pattern with less coefficients and greater accuracy.

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4) The Slepian expansion modeled the radiation pattern on average in 0.1846 seconds. The SEM expansion in 29.7265 seconds. For a given dataset the Slepian expansion is roughly a factor 161 faster. VIII. CONCLUSION In this paper a novel model to describe the radiation pattern of wideband antennas, using vector spherical wave functions and discrete prolate spheroidal sequences, was proposed. This model compresses a radiation pattern with a factor of more than 1159, while having a relative error smaller than 2%. The model for has also been used to evaluate the fidelity factor . Using the model, the fidelity the first time for all angles is evaluated a factor 5 faster, with a relative error factor smaller than 1%. Modeling a measured 2D cut of the radiation pattern shows that the model is also very noise resistant and can be used to filter out measurement noise. The model was also compared to a model based on the often used SEM expansion. The Slepian model suffers from none of the known drawbacks of the SEM expansion, is faster to evaluate and models the radiation pattern more accurately with less coefficients. REFERENCES [1] New Public Safety Applications and Broadband Internet Access Among Uses Envisioned by FCC Authorization of Ultra-Wideband Technol.-FCC News Release 2002 [Online]. Available: http://www.fcc.gov/Bureaus/Engineering_Technology/News_Releases/2002/nret0203.html [2] W. Lauber and S. Palaninathan, “Ultra-wideband antenna characteristics and pulse distortion measurements,” in Proc. IEEE Int. Conf on Ultra-Wideband, 2006, pp. 617–622. [3] D. Lamensdorf and L. Susman, “Baseband-pulse-antenna techniques,” IEEE Antennas Propag. Mag., vol. 36, no. 1, pp. 20–30, Feb. 1994. [4] D.-H. Kwon, “Effect of antenna gain and group delay variations on pulse-preserving capabilities of ultrawideband antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2208–2215, Aug. 2006. [5] J. McLean, H. Foltz, and R. Sutton, “Pattern descriptors for UWB antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 553–559, Jan. 2005. [6] T. Dissanayake and K. P. Esselle, “Correlation-based pattern stability analysis and a figure of merit for UWB antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 3184–3191, Nov. 2006. [7] H. Rogier and E. Bonek, “Analytical spherical-mode-based compensation of mutual coupling in uniform circular arrays for direction-of-arrival estimation,” Archiv fur Elektronik undÜbertragungstechnik (AEÜ)-Int. J. Electron. Commun., vol. 60, no. 3, pp. 179–189, Mar. 2006. [8] H. Xiao, V. Rokhlin, and N. Yarvin, “Prolate spheroidal wavefunctions, quadrature and interpolation,” Inverse Problems, vol. 17, pp. 805–838, Aug. 2001. [9] D. H. Werner and R. J. Allard, “The simultaneous interpolation of antenna radiation patterns in both the spatial and frequency domains using model-based parameter estimation,” IEEE Trans. Antennas Propag., vol. 48, no. 3, pp. 383–392, Mar. 2000. [10] D. E. N. Davies, The Handbook of Antenna Design, ser. IEE Electromagnetic Wave Series, A. Rudge, K. Milne, A. Olver, and P. Knight, Eds. London: Peregrinus, 1983, ch. 12. [11] R. F. Harrington, Time-Harmonic Electromagnetic Fields, ser. Electrical and Electronic Engineering Series. New York: McGraw-Hill, 1961. [12] O. Bucci, G. D’Elia, G. Franceschetti, and R. Pierri, “Efficient computation of the far field of parabolic reflectors by pseudo-sampling algorithm,” IEEE Trans. Antennas Propag., vol. 31, pp. 931–937, Nov. 1983. [13] H. Rogier, “Spatial correlation in uniform circular arrays based on a spherical-waves model for mutual coupling,” Archiv fur Elektronik undÜbertragungstechnik (AEÜ)-Int. J. Electron. Commun., vol. 60, no. 7, pp. 521–532, Mar. 2006.

[14] C. Roblin and A. Sibille, “Ultra compressed parametric modeling of UWB antenna measurements using symmetries,” presented at the XXIX URSI General Assembly, Chicago, IL, Aug. 2008. [15] D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Review, vol. 25, no. 3, pp. 379–393, 1983. [16] D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I,” Bell Syst. Technol. J., vol. 40, no. 1, pp. 43–64, Jan. 1961. [17] H. Landau and H. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II,” Bell Syst. Technol. J., vol. 40, no. 1, pp. 65–84, Jan. 1961. [18] H. Landau and H. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III,” Bell Syst. Technol. J., vol. 41, no. 4, pp. 1295–1336, Jul. 1962. [19] D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV,” Bell Syst. Technol. J., vol. 43, no. 6, pp. 3009–3058, Nov. 1964. [20] I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties,” Appl. Comput. Harmon. Anal., vol. 16, pp. 208–230, 2004. [21] T. Zemen and C. Mecklenbrauker, “Time-variant channel estimation using discrete prolate spheroidal sequences,” IEEE Trans. Signal Processing, vol. 53, pp. 3597–3607, Sep. 2005. [22] G. G. Walter and X. Shen, “Wavelets based on prolate spheroidal wave functions,” J. Fourier Anal. Appl. J. Fourier Analy. and Applicat., vol. 10, no. 1, pp. 1–26, 2004. [23] L. Varshney, “On the use of discrete prolate spheroidal windows for frequency selective filter design,” Applicat. Signal Processing, Feb. 2004. [24] D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty. V—The discrete case,” AT&T Technical J., vol. 57, pp. 1371–1430, Jun. 1978. [25] F. A. Grünbaum, “Eigenvectors of a Toeplitz matrix: Discrete version of the prolate spheroidal wave functions,” SIAM J. Algebraic Discrete Methods, vol. 2, no. 2, pp. 136–141, 1981. [26] P. Demarcke, H. Rogier, R. Goossens, and P. D. Jaeger, “Beamforming in the presence of mutual coupling based on constrained particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 57, pp. 1655–1666, Jun. 2009. [27] A. Abbosh and M. Bialkowski, “A UWB directional antenna for microwave imaging applications,” in Proc. IEEE Antennas and Propag. Int. Symp., Jun. 9–15, 2007, pp. 5709–5712. [28] R. Aylo, K. Kabalan, A. El-Hajj, M. Al-Husseini, and J. Costantine, “An investigation of the wideband properties of a resistively-loaded v-shaped conical antenna,” in Proc. IEEE Antennas and Propag. Society Int. Symp. AP-S, 2008, pp. 1–4. [29] C. E. Baum, “On the singularity expansion method for the solution of electromagnetic interaction problems,” EMP Interaction Note 8, vol. 8, pp. 112–112, Dec. 1971. [30] B. G. R. d. Prony, “Essay Experimental et Analytique: Sur les lois de la dilatabilité de fluides élastiques et sur celles de la force expansive de la vapeur de l’alcool à differentes temperatures,” J. de l’cole Polytechnique, vol. 1, no. 22, pp. 24–76. [31] F. Hildebrand,Introduction to Numerical Analysis, 1974 [Online]. Available: http://sfxit.ugent.be/sfx_local?sid=google;auinit=FB;aulast =Hildebrand;title=Introduction%20to%20numerical%20analysis; genre =book;isbn =0486653633;date=1974 [32] M. L. Van Blaricum and R. Mittra, “Problems and solutions associated with Prony’s method for processing transient data,” IEEE Trans. Electromagn. Compat., pt. I, pp. 174–182, Feb. 1978. [33] T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag., vol. 37, no. 1, pp. 48–55, Feb. 1995. [34] Y. Duroc, T. P. Vuong, and S. Tedjini, “Realistic modeling of antennas for ultra wide band systems,” in Proc. IEEE Radio and Wireless Symp., Oct. 17–19, 2006, pp. 347–350. [35] S. Licul and W. A. Davis, “Unified frequency and time-domain antenna modeling and characterization,” IEEE Trans. Antennas Propag., vol. 53, pp. 2882–2888, Sep. 2005. [36] T. K. Sarkar, S. Park, J. Koh, and S. M. Rao, “Application of the matrix pencil method for estimating the SEM (singularity expansion method) poles of source-free transient responses from multiple look directions,” IEEE Trans. Antennas Propag., vol. 48, pp. 612–618, Apr. 2000. [37] C. Roblin, “Ultra compressed parametric modelling of UWB antenna measurements,” in Proc. 1st Eur. Conf. on Antennas and Propag. EuCAP, Nov. 6–10, 2006, pp. 1–8.

DULLAERT AND ROGIER: NOVEL COMPACT MODEL FOR THE RADIATION PATTERN OF UWB ANTENNAS

[38] G. Brzezina, L. Roy, and L. MacEachern, “Planar antennas in LTCC technology with transceiver integration capability for ultra-wideband applications,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 2830–2839, Jun. 2006. [39] H. Sheng, P. Orlik, A. Haimovich, L. J. Cimini Jr., and J. Zhang, “On the spectral and power requirements for ultra-wideband transmission,” in Proc. IEEE Int. Conf. on Commun. ICC ’03, May 11–15, 2003, vol. 1, pp. 738–742.

Wouter Dullaert was born in 1985. He received the M.Sc. degree in electrical engineering from Ghent University, Ghent, Belgium, in 2007, where he is currently working toward the Ph.D. degree. His current research interests comprise ultrawideband antennas and antenna modeling.

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Hendrik Rogier (SM’06) was born in 1971. He received the Electrical Engineering and the Ph.D. degrees from Ghent University, Gent, Belgium, in 1994 and in 1999, respectively. He is currently a Postdoctoral Research Fellow of the Fund for Scientific Research—Flanders (FWO-V), Department of Information Technology, Ghent University where he is also Associate Professor with the Department of Information Technology. From October 2003 to April 2004, he was a Visiting Scientist at the Mobile Communications Group of Vienna University of Technology. He authored and coauthored about 50 papers in international journals and about 70 contributions in conference proceedings. His current research interests are the analysis of electromagnetic waveguides, electromagnetic simulation techniques applied to electromagnetic compatibility (EMC) and signal integrity (SI) problems, as well as to indoor propagation and antenna design, and in smart antenna systems for wireless networks. Dr. Rogier was twice awarded the URSI Young Scientist Award, at the 2001 URSI Symposium on Electromagnetic Theory and at the 2002 URSI General Assembly. He is serving as a member of the Editorial Boarding of IET Science, Measurement & Technology and acts as the URSI Commission B representative for Belgium.

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Time Domain Analysis of the Near-Field Radiation of Shaped Electrically Large Apertures Shaohui Quan

Abstract—The near-field radiation of several electrically large apertures has been computed based on the aperture field convolution method, including a square aperture, a circular aperture, and a serrated edge aperture. The frequency range is 8.0–12.0 GHz. The aperture diameters are limited to 5.00 m and the corresponding electrical size is 133–200 wavelengths. The time domain spectrum has been obtained by chirp z-transform with a Hamming window and the mechanism for the generation of the aperture near-field radiation in time domain is analyzed. The calculation results demonstrate that the near-field radiation of the aperture can be approximately seen as the synthesis of a plane wave from the aperture, cylindrical waves from the edges, and spherical waves from the corners. By this method, the direction, position, and magnitude of incoming waves in the near-field region can be estimated, and the aperture design can be modified to meet the requirements of the near-field radiation. The near field of the three apertures mentioned above has been compared, and it is shown that the serrated edge aperture has more uniform direct wave and lower diffraction waves, making it a candidate for compact range (CR) aperture design. Index Terms—Aperture antennas, apertures, compact range (CR), near fields, spectral analysis, time domain analysis.

I. INTRODUCTION N compact range (CR) project, it is essential to design the shape, edge, and distribution of the incident field of the reflectors or lens in order to meet the requirement of the farfield condition for antenna and scattering measurements. The finite size of the CR aperture might result in significant errors in some low sidelobe antennas and scattering pattern measurements [1]–[3]. The ultimate aim of the CR design is to eliminate the edge diffraction waves and make the quiet zone field close to a uniform plane wave (UPW) as much as possible. The aperture can be obtained by projecting the reflector or lens along the main axis. According to the equivalence principle, the field produced by the reflector or lens current could be obtained by computing the field of the aperture equivalent source. In this way, the design of the reflector or lens can be converted to the design of the aperture, including the aperture shape, the aperture field distribution, and so forth.

I

Manuscript received December 16, 2008; revised October 20, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported by the National Natural Science Foundation of China (NSFC) Project 60771011 and 60401014. The author is with the Electromagnetic Engineering Laboratory, School of Electronics and Information Engineering, Beihang University (BUAA), Beijing 100191, 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.2009.2037705

In practical engineering, in order to reduce the diffraction effect of the edge, the serrated and rolled edges have been investigated for many years [4]–[8]. The serrated edge is more popular for its easier manufacturing and lower cost [9]. A complete CR quiet zone prediction or evaluation can be made by comparing the amplitude and phase specifications of the quiet zone, which can be obtained by numerical computation in the CR designing process or planar near-field measurements after it has been mounted [10], [11]. For a CR, the aperture is generally electrically large and its radiation or scattering field can be seen as the synthesis of a direct wave from the aperture and diffraction waves from the edges and corners. Once the wide band frequency domain field has been obtained, the time domain field, which is also called the time domain spectrum, can be computed by a Fourier transform. The design characteristics of the aperture near field can be presented in the time domain spectrum clearly and completely [12]. Because the aperture direct wave and diffraction waves have different distances in the time domain, it is possible to separate them in the time domain spectrum, and to design and diagnose the aperture by means of the time domain spectrum [13]–[15]. For a CR, the quiet zone should be located in the central region of the aperture near field. The amplitude and phase should be kept as uniform as possible inside the aperture. The amplitude of the incoming wave should be tapered rapidly outside the aperture to avoid the direct radiation to the feed and the chamber. All the characteristics mentioned above can be presented by the analysis of the aperture near field in the time domain. To study the diffraction characteristics of different apertures and obtain the magnitude of edges and corners diffraction waves quantitatively, the near-field radiation of several electrically large shaped apertures has been computed. The frequency range is 8.0–12.0 GHz. The frequency domain sampling points are selected to be 134 for Fig. 3–8 and 151 for Figs. 9 and 10. The aperture diameters are limited to 5.00 m, and the corresponding electrical size is 133–200 wavelengths. Many methods have been adopted to predict the quiet zone field for CR design, such as physical optics (PO) [10], [16], geometrical optics (GO), geometrical theory of diffraction (GTD) [7], [9], [17], and plane wave spectrum method (PWS) [18]. For electrically large apertures, the near-field computation for wide frequency band is generally complicated and time-consuming. The aperture field convolution method adopted in this article solves this problem effectively because it is based on FFT algorithm. Once an aperture field convolution computation has been performed, the near field in a double-aperture-sized region is obtained.

0018-926X/$26.00 © 2009 IEEE

QUAN: TIME DOMAIN ANALYSIS OF THE NEAR-FIELD RADIATION OF SHAPED ELECTRICALLY LARGE APERTURES

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Fig. 1. Several shaped apertures: square, circular, and serrated.

Fig. 5. Time domain spectrum of P .

Fig. 2. Aperture and its near-field region observation points.

Fig. 6. Direct wave (at 5.00 m) amplitude inside the aperture.

Fig. 3. Time domain spectrum of P .

Fig. 7. Direct wave (at 5.00 m) amplitude outside the aperture.

Fig. 4. Time domain spectrum of P .

of the arbitrary selection of the observation time distance region and the suppressed sidelobes. II. PRINCIPLES

The chirp z-transform with a Hamming window is used to obtain the near-field time domain spectrum, for the advantage

Several shaped apertures, including a square aperture, a circular aperture, and a serrated edge aperture, are shown in Fig. 1.

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TABLE I THE BASE LENGTH OF SERRATION TRIANGLES (UNIT: METER)

Fig. 8. Left and right edge diffraction wave (at 5.59 m) amplitude.

along the -axis and -axis, respectively. The height of each triangle is 0.75 m and the length of each base connected with the solid part of the aperture is decreased from the two corners to the center, as shown in Table I. Considering the symmetry, the lengths of the first six serration triangles are listed out. The labeling number is from corner to center. and , as The selected observation points are and are located on the -axis. The disshown in Fig. 2. and are 5.00 m and 10.00 m, respectively. tances and are located directly under . The distances and are 1.25 m and 3.75 m, respectively. A. Convolution Method for Near-Field Computation [19] Assume that the aperture tangential electric field is , where is the coordinate of the equivalent source point. According to the equivalence principle, the equivalent magnetic current is

(1) The electrical vector potential is Fig. 9. The aperture direct wave amplitude versus z -coordinate.

(2) The electric field at an arbitrary observation point

is

(3) is the distance between the source and field where point. It can be obtained by decomposing (3) as

Fig. 10. The aperture edge diffraction wave amplitude versus z -coordinate.

(4a) (4b)

Both the width and the height are 5.00 m, that is, m, m. A coordinate system must be established as shown in Figs. 1 is located in the coordinate and 2. The aperture plane. For the serrated aperture, there are 12 isosceles triangle serrations at each edge, which are symmetrically distributed

(4c)

QUAN: TIME DOMAIN ANALYSIS OF THE NEAR-FIELD RADIATION OF SHAPED ELECTRICALLY LARGE APERTURES

For a certain observation plane at a fixed coordinate, this equation can be transformed into a two-dimensional convolution, that is (5) is the input excitation function, which is related where is the output response functo the aperture field tion, which is the field in the observation plane orthogonal to is the spatial network response function, the -axis. which is already known and can be expressed as

2)

(6) is known, with arbitrary -coordinate can If be computed with (5). A high-resolution near-field time domain spectrum can be obtained by the transformation of the wide band frequency domain . near field B. Time Domain Spectrum of Aperture Near Field , the obserAssume that the frequency domain field is vation point position vector is , the aperture equivalent source position vector is , and is the velocity of light in free space. Then the arrival time from the source point to the observation point is:

3)

(7) 4) Assume that the time domain field can be expressed as . It can be obtained by the inverse Fourier transform of through

(8) Once the wide band frequency domain field has been computed, the time domain field can be obtained. To improve the local resolution in the time domain, the chirp z-transform is adopted. Furthermore, considering that the magnitude of the diffraction wave might be lower, a Hamming window function has been used to suppress the sidelobes. C. Plane Wave, Cylindrical Wave, and Spherical Wave Assumptions The position and amplitude of the direct and diffraction waves in the time domain can be estimated according to the distance between the aperture and the observation points. In this process, the wave from the aperture can be approximately seen as a plane wave, the wave from the edge can be approximately seen as a cylindrical wave, and the wave from the corner can be approximately seen as a spherical wave. The above assumptions can be corroborated by the following facts, about which some points must also be noted: 1) Based on the electromagnetic equivalence principle, for a finite-sized aperture, the aperture field is equivalent to sur-

5)

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face current source. Because the aperture field is truncated at the edge, it can be seen that the equivalent edge line currents are formed on the basis of the diffraction theory [20]–[23]. Furthermore, the equivalent point current elements can be seen as forming at the corners of the edges. The plane wave, the cylindrical wave, and the spherical wave in the near-field central region emanate from the surface current source, the edge line current source, and the point current element source on the aperture, respectively. According to the time domain spectrum results, for the square aperture, the time domain spectrum of the observation points on the -axis can be clearly divided into three waves (see the following Fig. 4). The first wave corresponds to the distance from the observation point to the aperture plane, which is the plane wave propagation distance from the aperture plane. The second wave corresponds to the distance from the observation point to the four edges, which is the cylindrical wave propagation distance from the edge. The third wave corresponds to the straight-line distance from the observation point to the four corners, which is the spherical wave propagation distance from the corner. Considering that the near-field computation and frequency-time transformation are independent, these results further confirm the assumptions stated in the above paragraph. Because a finite-sized aperture corresponds to finite-sized equivalent surface or line currents, the plane and cylindrical wave assumptions are more suitable for the nearfield central region, especially for the region inside the aperture. Generally, the distribution of the aperture equivalent currents is related to the aperture field distribution, the aperture size, the aperture shape, and the observation point. For the CR aperture design, if the direct wave and an in-dB interterference wave exist, an approximately ference wave relative to the direct wave can lead to 1.0 dB amplitude peak-to-peak ripple of the synthetic wave. Thus, dB can be seen as a standard for estimating the value the edge design. When the interference wave amplitude is dB, then the amplitude peak-to-peak ripple less than of the synthetic wave is less than 1.0 dB.

III. TIME DOMAIN CHARACTERISTICS OF SHAPED APERTURES As mentioned above, the total field of the aperture near field can be approximately seen as the synthesis of a plane wave from the aperture, cylindrical waves from the edges, and spherical waves from the corners. Based on this thought, the equivalent source distance of the square aperture has been computed, as shown in Table II. These distances can also be suitable for the circular and serrated apertures to some extent. A. Time Domain Spectrum Inside the Aperture Fig. 3 is the time domain spectrum of . It is clear that the direct wave is located at the position 5.00 m from the aperture. m, which There is an interference wave at distance represents the diffraction wave from the four edges. By comparing three apertures, it can be seen that the circular edge has the maximum interference wave magnitude, which is

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TABLE II THE EQUIVALENT DISTANCE OF THE SQUARE APERTURE (UNIT: METER)

dB at m and dB at two, which are m, respectively. The influence of the upper edge (supposed to be at m) for has disappeared. The synthesis field of mainly comes from the lower edge and the left and right edges. C. Direct Wave Versus Transverse Distance

about dB. The square edge is the second with dB dB. For the square and the serrated edge is the last with -dB residual diffraction wave at aperture, there is a m, which can be seen as diffraction wave from four corners. -dB interference For the serrated aperture, there are a m and a -dB interference wave at wave at m, which can be seen as the diffraction wave from the serration triangle base and the hypotenuse, respectively. By comparing time domain spectrum of three apertures, it can be seen that the serrated edge can reduce the edge diffraction wave by dividing it into several weaker waves, therefore avoiding larger destruction of the aperture near-field uniformity. Fig. 4 is the time domain spectrum of . It can be seen that m decreases for the interference wave amplitude at all the three apertures because only the left and right edge take dB for the square aperture, effect. It decreases to dB for the circular aperture, and dB for the serrated edge aperture. is offset from the -axis, so the distance to the upper and lower edges is not the same and the edge diffraction wave is -dB interference split. For the serrated edge, there is a m and a -dB interference wave at wave at m, which represents the diffraction wave of the upper and lower edges respectively.

B. Time Domain Spectrum Outside the Aperture Fig. 5 is the time domain spectrum of . For the square and -dB interference wave at circular aperture, there is a m and a -dB interference wave at m, which represent the diffraction wave from the lower edge and the left and right edge, respectively. For the serrated edge aperture, the lower edge diffraction m and the magniwave moves backward slightly to tude is dB. The left and right edge diffraction wave is m and the magnitude is dB. located at -dB interference For the square aperture, there is a m. For the circular aperture, there is a wave at -dB interference wave at m. Both represent the residual diffraction effect of the aperture, whose equivalent source position is down slightly from the aperture center. For the serrated aperture, the residual diffraction wave is split into

According to the time domain spectrum method, the direct wave amplitude versus transverse distance relation curve can be obtained, as shown in Figs. 6 and 7. For Fig. 6, it is the direct wave amplitude versus transverse distance relation curve inside the aperture. The transverse distance is from the center point and the direction is along the negative -axis. It can be seen that all three apertures keep uniform m. There in the region from the center to the distance of are large oscillations for the square and circular apertures in the m to m due to the lower edge region from diffraction effect. The field tapers rapidly at the edge area for the serrated aperture due to the effect of the serrations. Fig. 7 is the curve of the direct wave amplitude against the -coordinate outside the aperture. It can be seen that the direct wave tapers rapidly for all three apertures. D. Edge Diffraction Wave Versus Transverse Distance The edge diffraction wave amplitude versus transverse distance relation can also be obtained by the time domain spectrum. Comparing Figs. 3, 4, and 5, it can be seen that there is m, which represents the a fixed interference wave at diffraction wave from the left and right edges (four edgexs for ). Fig. 8 is the diffraction wave amplitude curve along the negative -axis. The time domain distance of the diffraction wave m. The transverse distance scope is 5.00 m. For the is square aperture, it can be seen that it stays approximately uniform inside the aperture and that the average amplitude is dB. It tapers rapidly to an average dB outside the aperture. For the circular aperture, it can be seen that the diffraction wave amplitude is extremely strong in the central area and dedB rapidly. It stays approximately creases to an average of invariant in other areas. For the serrated aperture, the amplitude dB in the whole area. stays at an average of about E. Direct Wave Versus Z-Axis Distance When the observation point moves along the -axis, the position of the direct wave in the time domain will change consequently. Fig. 9 is the aperture direct wave amplitude along the -axis which is extracted from the time domain spectrum. The scope of the -axis distance is 5.00 to 10.00 m. It can be seen that the direct wave amplitude stays approximately uniform for dB, therefore all three apertures. The ripples are all within it can be deduced that this scope is suitable for CR quiet zone selection. F. Edge Diffraction Wave Versus Z-Axis Distance When the observation point moves along the -axis, the position of the edge diffraction wave in the time domain will also

QUAN: TIME DOMAIN ANALYSIS OF THE NEAR-FIELD RADIATION OF SHAPED ELECTRICALLY LARGE APERTURES

change consequently. Considering the cylindrical wave characteristic of the edge diffraction wave, the time domain position of the edge diffraction wave can be computed by

(9) Fig. 10 is the extracted edge diffraction wave amplitude curve along the -axis. It can be seen that in the -axis distance scope of 5.00 to 10.00 m, the circular aperture has the maximum edge and 0.0 dB. The diffraction wave, with amplitude between square aperture is second, with the edge diffraction wave lower dB. The serrated edge aperture is the minimum, with than dB. the edge diffraction wave lower than Comparing the three apertures, the serrated edge aperture has the minimum edge diffraction wave amplitude, both in the transverse direction and -axis direction, thus it is the best candidate for CR design in the three apertures. Further optimization must be done for practical CR aperture design [24]. IV. CONCLUSION The wide band radiation near field of three electrically large apertures has been computed based on the aperture field convolution method. The time domain spectrum has been obtained by the chirp z-transform with a Hamming window. The following conclusions can be obtained: 1) The near-field radiation of electrically large apertures can be approximately seen as the synthesis of a plane wave from the aperture, cylindrical waves from the edges, and spherical waves from the corners. The distance and amplitude of direct and diffraction waves can be obtained by the time domain spectrum. 2) Compared with the square and the circular apertures, the serrated aperture can divide the stronger edge diffraction wave into several weaker interference waves, therefore reducing the disturbance on the direct wave caused by the diffraction waves. 3) Compared with the square and circular apertures, the direct wave amplitude of the serrated aperture is more uniform in the central area and tapers more rapidly in the edge area and outside the aperture. 4) For the fixed left and right edge diffraction waves, the circular aperture has the strongest amplitude in the central area, while in the edge area, the amplitude is reduced. The square aperture has stronger rippled amplitude both in the central and edge areas, while in other areas, the amplitude stays approximately uniform. The serrated aperture is low in amplitude in the whole area, which indicates that the edge diffraction has been weakened effectively. 5) For the -axis distance of 5.00 to 10.00 m, the aperture direct wave amplitude can be kept approximately uniform, therefore this scope is suitable for the CR quiet zone selection. In the same -axis distance, the serrated edge aperture has the minimum edge diffraction wave amplitude. Because the time domain analysis of aperture near field can describe the aperture design characteristics completely and indicate changes of all near-field indexes effectively, it can be ap-

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plied in the CR design and testing. It is also useful in reflector antenna and aperture antenna design. On the other hand, it must be noted that the three apertures proposed in the article are mainly used to present the application of time domain analysis method. For practical CR design, further aperture optimization must be performed. Also, the discretization and truncation in numerical computation might result in quantization errors. A trade-off has to be made between computation precision and computation time. REFERENCES [1] T. H. Lee and W. D. Burnside, “Stray signal requirements for compact range reflectors based on RCS measurement errors,” IEEE Trans. Antennas Propag., vol. 39, no. 8, pp. 1193–1202, 1991. [2] M. Philippakis and G. Parini, “Compact antenna range performance evaluation using simulated pattern measurements,” IEE Proc., Microw. Antennas Propag., vol. 143, no. 3, pp. 200–206, 1996. [3] T. H. Lee and W. D. Burnside, “Compact range reflector edge treatment impact on antenna and scattering measurements,” IEEE Trans. Antennas Propag., vol. 45, no. 1, pp. 57–65, 1997. [4] I. J. Gupta, K. P. Ericksen, and W. D. Burnside, “A method to design blended rolled edges for compact range reflectors,” IEEE Trans. Antennas Propag., vol. 38, no. 6, pp. 853–861, 1990. [5] R. Johnson and D. Hess, “Performance of a compact antenna range,” in Antennas and Propagation Society Int. Symp. Digest, June 1975, vol. 13, pp. 349–352. [6] D. Hess, F. Willwerth, and R. Johnson, “Compact range improvements and performance at 30 GHz,” in AAntennas and Propagation Society Int. Symp. Digest, June 1977, vol. 15, pp. 264–267. [7] J. Hartmann and D. Fasold, “Improvement of compact ranges by design of optimized serrations,” presented at the Millenium Conf. on Antennas and Propagation, Davos, Switzerland, Apr. 9–14, 2000. [8] W. D. Burnside, M. C. Gilrethe, B. M. Kent, and G. L. Clerici, “Curved edge modification of compact range reflectors,” IEEE Trans. Antennas Propag., vol. 35, no. 2, pp. 176–182, 1987. [9] T.-H. Lee and W. D. Burnside, “Performance trade-off between serrated edge and blended rolled edge compact range reflectors,” IEEE Trans. Antennas Propag., vol. 44, no. 1, pp. 87–96, 1996. [10] G. Parini and M. Philippakis, “Use of quiet zone prediction in the design of compact antenna test ranges,” IEE Proc., Microw. Antennas Propag., vol. 143, no. 3, pp. 193–199, 1996. [11] A. Repjar and D. Kremer, “Accurate evaluation of a millimeter wave compact range using planar near-field scanning,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 419–425, 1982. [12] S.-H. Quan, “Diagnostic spectrum of aperture near field[J].,” Acta Aeronoutica et Astronautica Sinica, vol. 29, no. 1, pp. 136–140, 2008, (in Chinese). [13] Mart-Canales et al., “Performance analysis of a compact range in the time domain,” IEEE Trans. Antennas Propag., vol. 50, no. 4, pp. 511–516, 2002. [14] S.-H. Quan, G.-Y. He, and Y.-B. Xu, “Optimization design of CR aperture based on time domain spectrum[J].,” Chinese J. Radio Sci., vol. 21, no. 4, pp. 601–605, 2006, (in Chinese). [15] S.-H. Quan, G.-Y. He, and Y.-B. Xu, “CATR extraneous waves analysis using time domain approach[J].,” Chinese J. Radio Sci., vol. 18, no. 4, pp. 389–392, 2003, (in Chinese). [16] P. A. Beeckman;, “Prediction of the Fresnel region field of a compact antenna test range with serrated edges,” IEE Proc., Microw. Antennas Propag., vol. 133, no. 2, pp. 108–114, 1986. [17] M. S. A. Sanad and L. Shafai, “Dual parabolic cylindrical reflectors employed as a compact range,” IEEE Trans. Antennas Propag., vol. 38, no. 6, pp. 814–822, 1990. [18] J. P. McKay and Y. Rahmat-Samii, “Compact range reflector analysis using the plane wave spectrum approach with an adjustable sampling rate,” IEEE Trans. Antennas Propag., vol. 39, no. 6, pp. 746–753, 1991. [19] G. He, H. Fang, X. Jiang, and Z. Dai, “Study of diffraction field of aperture antenna[J].,” Acta Aeronoutica et Astronautica Sinica, vol. 17, no. 4, pp. 404–409, 1996, (in Chinese). [20] A. Michaeli, “Equivalent edge currents for arbitrary aspects of observation,” IEEE Trans. Antennas Propag., vol. 32, no. 3, pp. 252–258, 1984. [21] O. Breinbjerg, “Higher order equivalent edge currents for fringe wave radar scattering by perfectly conducting polygonal plates,” IEEE Trans. Antennas Propag., vol. 40, no. 12, pp. 1543–1554, 1992.

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[22] G. James and V. Kerdemelidis, “Reflector antenna radiation pattern analysis by equivalent edge currents,” IEEE Trans. Antennas Propag., vol. 21, no. 1, pp. 19–24, 1973. [23] A. Altintas and P. Russer, “Time-domain equivalent edge currents for transient scattering,” IEEE Trans. Antennas Propag., vol. 49, no. 4, pp. 602–606, 2001. [24] S.-H. Quan, G.-Y. He, Y.-B. Xu, and Y.-L. Dong, “A high performance single reflector compact range[J].,” J. Beijing Univ. Aeronautics and Astronautics, vol. 29, no. 9, pp. 767–769, 2003, (in Chinese).

Shaohui Quan received the Ph.D. degree in electromagnetic field theory and microwave technology from Beihang University (BUAA), Beijing, China, in 2003. Currently, he works as an Associate Professor in the Electromagnetic Engineering Laboratory, School of Electronics and Information Engineering, BUAA. His research interests are in the areas of microwave and millimeter wave measurement theory and technology, antenna theory and engineering, scattering theory and engineering.

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A Novel Beam Squint Compensation Technique for Circularly Polarized Conic-Section Reflector Antennas Shenheng Xu, Student Member, IEEE, and Yahya Rahmat-Samii, Fellow, IEEE

Abstract—Beam squint generally exists in offset reflector antennas with circularly polarized feeds. It is manifested by a small beam shift of the radiation pattern in the plane perpendicular to the principal offset plane, which can significantly affect the beam pointing accuracy. In this paper a practical and widely applicable compensation technique for the beam squint is proposed. Simulation results show that a small lateral feed displacement in the perpendicular plane can effectively minimize or eliminate the linear phase shift caused by the depolarization effect, thus compensating for the beam squint effect. This is in practice very useful for offset reflector antennas where the previously suggested method based on feed tilting may not be proper. A simple formula is derived to quickly estimate the optimal feed displacement for both right- and left-hand circularly polarized feeds. Three representative examples: a single offset parabolic reflector, a suboptimal offset Cassegrain reflector, and an axially symmetric Cassegrain reflector with an off-focus feed, are presented to validate the proposed method. Satisfactory results are achieved for all three examples. Index Terms—Compensation, offset reflector antennas, reflector antennas, reflector antenna feeds.

I. INTRODUCTION

C

IRCULAR polarizations are extensively used in modern reflector antenna applications. The beam squint phenomenon, however, generally exists in offset parabolic, Cassegrain, and Gregorian reflector antennas illuminated by circularly polarized feeds [1]–[6]. It is manifested by a small beam shift of the radiation pattern in the plane perpendicular to the principal offset plane. This phenomenon can also be observed when the circularly polarized feed is offset from the focus of a symmetric parabolic reflector. The squint angle can significantly affect the beam pointing accuracy, and must be taken into account for advanced reflector antenna applications, such as satellite communications, deep-space telemetry, radio astronomy, and compact range measurements. Beam squint is caused by the depolarization effect on the polarized components of the incident field, which results in a linear phase shift across the reflector aperture [1]–[4]. A right-hand

Manuscript received February 20, 2009; revised July 14, 2009. First published December 04, 2009; current version published February 03, 2010. The authors are with the Department of Electrical Engineering, University of California, Los Angeles, Los Angeles, CA 90095 USA (e-mail: [email protected]. edu; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037711

circularly polarized (RCP) beam squints toward the left in the perpendicular plane if one is looking in the direction of wave propagation, while a left-hand circularly polarized (LCP) beam squints toward the right (Fig. 1). Adatia and Rudge [3] derived a simple formula that accurately predicts the squint angle for a single offset parabolic reflector with an on-focus circularly polarized feed

(1) where is the tilt angle of the feed, is the focal length of the paraboloid (Fig. 1), and is the wavenumber in free space. The minus sign is for RCP beams, referring to a negative in the plane in —the corresponding spherical coordinates of the global coordinate system . Conversely, the plus plane for LCP beams. A sign means a positive in the more generalized formula was obtained by Duan and RahmatSamii [4] which is applicable to symmetric or offset parabolic reflectors with circularly polarized feeds positioned on focus or displaced off focus

(2) is the tilt angle of the radiated beam due to the feed where plane, and the angle is interdisplacement in the preted as the angle between the incident beam and the radiated beam (Fig. 2). Although the beam squint can be used to effectively separate two RCP and LCP beams, it is undesirable in most applications, and some possible approaches to achieve a squintfree design have been proposed in literature [4], [6]. Duan and Rahmat-Samii [4] proposed a method to correct the beam squint for single parabolic reflectors by properly tilting the feed so that (3) This condition becomes a natural choice by observing that the predicted by (2) is always zero given (3) is satsquint angle isfied. The proposed method works well for symmetric parabolic reflectors with off-focus feeds, especially when the feed displacement is small. However, it is impractical when applied to offset parabolic reflectors. That is because is usually much larger than in offset parabolic reflectors, and the feed is tilted away from the reflector if (3) is satisfied. The illumination over

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TABLE I BEAM SQUINT COMPENSATION FOR REFLECTOR ANTENNAS WITH CIRCULARLY POLARIZED FEEDS

Based on the concept of equivalent paraboloid [7], it is found that the squint angle can be eliminated when the feed is tilted direction (the direction of the equivalent paraboloid in axis, which, as indicated in (5), depends on the subreflector eccentricity and the tilt angle of the subreflector axis )

(4) (5)

Fig. 1. Beam squint in a single offset parabolic reflector antenna with an on-focus circularly polarized feed. The tilt angle of the feed is  . The observer is looking in the direction of wave propagation.

Fig. 2. Beam squint in a single offset parabolic reflector antenna with an offfocus circularly polarized feed. The tilt angle of the feed is  . The observer is looking in the direction of wave propagation.

the reflector surface becomes so bad that the spillover dominates the antenna performance. A squint-free condition for Cassegrain and Gregorian reflector antennas with on-focus feeds was also provided in [4].

where is the focal length of the equivalent paraboloid. This condition is the same as the Mizugutch condition that is used to minimize the cross-polarized field of dual-reflector antennas [8], [9]. It is expected because the depolarization effect is eliminated when the feed axis is aligned with the equivalent paraboloid axis. A feed is said to be optimally oriented if this condition is satisfied for a dual-reflector design. The main drawback is that great limitations are imposed in order to satisfy this condition. Eilhardt et al. [6] proposed another approach to compensate for the beam squint in axially symmetric dual reflector antennas with off-focus feeds. A comprehensive parametric study shows that by properly choosing geometrical parameters such as the feed tilt angle, the subreflector half-subtended angle, or the focal ratio , a squint-free dual-reflector configuration can be achieved. However, this method greatly limits the freedom in choosing geometrical parameters, and a design often may be suboptimal for other reasons if one parameter is fixed. Furthermore, no general guidelines are provided for other reflector configurations. A summary of the beam squint in reflector antennas with circularly polarized feeds is presented in Table I. Two approaches—feed tilting [4] and fixed geometrical parameters [6]—may be used to compensate for the beam squint in some specific scenarios, but with some obvious drawbacks. Neither of them is applicable to all kinds of scenarios. In particular, previously no good solution had been identified for the beam squint in single offset parabolic reflectors, even when the circularly polarized feed is on focus. In this paper, we propose a practical and widely applicable compensation technique for the beam squint effect by optimally displacing circularly polarized feeds in the perpendicular plane. In Section II a simple formula is derived to quickly estimate the optimal feed displacement, which effectively corrects the linear phase shift across the reflector aperture caused by the depolarization effect. Three representative examples: a single offset parabolic reflector, a suboptimal offset Cassegrain reflector, and an axially symmetric Cassegrain reflector with an off-focus feed,

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Fig. 4. Geometry of a single offset parabolic reflector antenna. The tilt angle of the RCP feed is 43.18 .

Fig. 3. A feed displacement in the perpendicular direction can effectively correct the linear phase shift across the reflector aperture.

are presented in Section III to validate the proposed method. Satisfactory results are achieved for all three examples, and potential applications are discussed. Finally some concluding remarks are presented in Section IV. II. THE SIMPLE FORMULA First consider the simplest case of beam squint, a single offset parabolic reflector antenna. The reflector geometry and the associated coordinate systems are shown in Fig. 1. The reflector is a standard conic-section-generated offset parabolic reflector with a circular projected aperture [10]. The diameter of the projected aperture is , the focal length is , and the offset height is . A circularly polarized feed is situated at the focal point , with a tilt angle of . The reflector surface can be mathematically described as

plane because of the depolarization effect of offset reflector configurations on the polarized components of the incident field from the feed. Without losing generality, the phase at point is assumed to be leading by compared with the phase at point , and , the diameter of the projected aperture. The linear phase shift , which causes the radiated beam to squint in direction, can be determined by (7) can be either predicted using (1) or obtained through where measurements or numerical simulations. In order to compensate for the beam squint, one can displace the feed along the axis from point to point . By optimally choosing the feed location, the feed displacement creates a new with opposite phase variations, and linear phase shift along effectively minimizes or even eliminates the linear phase shift described in (7). The optimal feed displacement is

(8) where is the half-subtended angle of the feed in the perpendicular plane. It can be determined by

(6) are the global coordinates shown in Fig. 1, and where the coordinates of the focal point are . Fig. 3 presents a view from the top of the perpendicular plane axis only aligns with the axis when in Fig. 1. Note that the equals zero. For a circularly polarized feed, a linear phase shift [1], [3], [4] exists across the reflector aperture in the

(9) are where the coordinates of point . Note that an assumption is made that the reflector is located far away from the feed and is much smaller

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Fig. 5. (a) Simulated far-field pattern of the single offset parabolic reflector antenna operating at 8.45 GHz. The pattern is cut through its boresight. (b) A : . close-up view. The simulated squint angle 

= 0 0604

compared with and so that line is approximately parallel to line . It is a good approximation considering the squint angle usually is very small and consequently is small, too. Combining (7) and (8), the formula to estimate the optimal feed displacement can be expressed as

(10) Although it is derived based on a single offset parabolic reflector antenna with an on-focus feed, the formula (10) can also be applied to other configurations. For example, when a single offset parabolic reflector antenna with an off-focus feed is considered, (2) instead of (1) should be used to predict the beam squint. Once the squint angle is predicted or obtained through

50 2 50

Fig. 6. Plots of the x-component of the simulated near-field in a  aperture plane: (a) normalized amplitude in dB scale, (b) phase in degrees, and . The dash line represents the projected (c) a line cut of phase through x aperture boundary.

=0

simulation, (10) is still applicable since no further conditions are imposed in the derivation of the formula.

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Fig. 8. (a) Simulated far-field pattern of the offset reflector antenna compensated by the feed displacement of 0:0658 . The pattern is cut through its boresight. (b) A close-up view. No squint angle is observed.

0

diameter , the focal length , and the offset height of the can be calculated (see [7, Eqs. (20), equivalent paraboloid (31), and (32)])

(11) (12)

2

Fig. 7. Plots of the y -component of the simulated near-field in a 50 50  aperture plane: (a) normalized amplitude in dB scale, (b) phase in degrees, and (c) a line cut of phase through x = 0. The dash line represents the projected aperture boundary.

With the help of the equivalent paraboloids, the application of the formula (10) can even be extended to Cassegrain and Gregorian reflector antennas with on-focus or off-focus feeds. The

(13)

and the same approach is then applied to obtain the optimal feed displacement using the equivalent paraboloid. The minus sign in (10), however, needs to be modified due to the fact that the polarized signal from the feed is reflected twice; an RCP feed results in an RCP beam squinted toward the left, and an

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Fig. 10. Geometry of an offset Cassegrain reflector antenna.

III. SIMULATION RESULTS OF THREE REPRESENTATIVE EXAMPLES Three representative examples: a single offset parabolic reflector, a suboptimal offset Cassegrain reflector, and an axially symmetric Cassegrain reflector with an off-focus feed, are chosen to validate the proposed compensation technique for the beam squint of reflector antennas. A. Single Offset Parabolic Reflector Antenna

2

Fig. 9. Line cuts of the simulated phase distribution in the 50 50  aperture plane after the feed displacement compensation: (a) x-component and (b) y -component.

LCP feed results in an LCP beam squinted toward the right. Therefore, the formula becomes (14) Furthermore, this approach is believed to be applicable to any multi-reflector antenna with a parabolic main reflector when there exists an equivalent paraboloid, and the optimal feed displacement is (15) where is the number of reflectors in the antenna system. Note that the sign of is determined by both and . For a fixed , the optimal feed position is located at opposite sides of the offset plane, depending on the polarization of the radiated beam.

A single offset parabolic reflector antenna operating at 8.45 GHz [11] is first chosen to examine the effectiveness of this technique. As shown in Fig. 4, the diameter of the projected aperture is , the focal length is , and the offset height is . An RCP feed is positioned on focus, tilted toward the center of the projected aperture at a tilt angle of 43.18 . The feed type feed with dB feed taper, which is modeled by a provides optimal performance with balanced illumination and spillover. Physical optics (PO) analysis [12] is employed to obtain the far-field radiation pattern of the reflector because it has been confirmed [4], [5] that PO is capable of adequately capturing the beam squint feature through numerical simulations. Fig. 5(a) shows the simulated far-field pattern of the offset reflector antenna, and a close-up view is presented in Fig. 5(b). A cut, squint angle of 0.0604 is clearly observed in the where the beam maximum is 42.535 dB. The simulated directivity at boresight (the boresight direction is defined as which may not coincide with the beam peak) is 42.514 dB, and a directivity loss of 0.021 dB is resulted due to the beam squint aperture plane (Fig. 4) is also effect. The near-field in a simulated, and the plots of the - and -components of the aperture field are presented in Figs. 6 and 7, respectively. An LCP feature in the near-field can be observed: equal amplitudes of - and -components with dB edge taper (including path loss effects) and the 90 phase lagging of -component. The

XU AND RAHMAT-SAMII: A NOVEL BEAM SQUINT COMPENSATION TECHNIQUE FOR CIRCULARLY POLARIZED

Fig. 11. (a) Simulated far-field pattern of the suboptimal offset Cassegrain reflector antenna. The pattern is cut through its boresight. (b) A close-up view. The simulated squint angle  : .

= 00 0053

line cuts of the phase distribution through [Figs. 6(c) and 7(c)] clearly show the linear phase shift predicted in [1], [3], [4]. Using linear regression, (defined as the phase leading at compared with the phase at ) is cal. A slightly different phase shift may be culated to be obtained depending on how the linear regression is performed. Considering that the phase shift [Figs. 6(c) and 7(c)] shows more volatility near the reflector edge where the field is considerably tapered, a linear regression based on weighted phases may be appropriate. Invoking (1) the squint angle due to the feed tilt angle of 43.18 is determined to be 0.0605 , well in agreement with the simulated squint angle of 0.0604 . is calculated to be using (7), matching well with simulation, too. is obtained The optimal feed displacement of using (10), where , the half-subtended angle in the perpendicular plane, is 22.28 . The simulated far-field pattern (Fig. 8) shows the squint angle is successfully compensated, and the directivity at boresight is 42.535 dB. The line cuts of the aperture plane simulated phase distribution in the

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Fig. 12. (a) Simulated far-field pattern of the suboptimal offset Cassegrain re: . The patflector antenna compensated by the feed displacement of tern is cut through its boresight. (b) A close-up view. No squint angle is observed.

00 0287

is calculated to be 0.01 using are shown in Fig. 9, and linear regression. Compared with the simulation results without compensation, the feed displacement compensation technique effectively eliminates the squint angle, remarkably reduces the linear phase shift, and fully recovers the directivity loss due to the beam squint. Note that the laterally displaced feed does not necessarily need to be realigned toward the center of the reflector, because the feed displacement is so small that the illumination on the reflector surface is almost not affected. Bandwidth performance is an important parameter for practical applications. The criteria for this beam squint compensation technique is defined as

%

(16)

where and are the squint angle before and after compensation, respectively. Simulation results show that

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Fig. 13. Geometry of an axially symmetric Cassegrain reflector antenna. Only half of the Cassegrain reflector is plotted. The feed is displaced by 10  in the  = 0 plane.

for this specific example, (16) is satisfied from 7.5 GHz to 9.1 GHz, 18.9% at 8.45 GHz. It is also worthwhile to point out that the squint angle and therefore the optimal feed displacement depend on the angle between the incident and radiated beams . For an array of feeds (independently operating at the same frequency), a slightly different feed displacement is required to compensate for the squint angle of each feed. In practice, however, it may not be achievable. Simulation results show that the optimal feed displacement calculated based on the center array element provides a good trade-off which will considerably reduce the squint angle of an individual feed. B. Suboptimal Offset Cassegrain Reflector Antenna The geometry of the second example, an offset Cassegrain reflector antenna [4], is presented in Fig. 10. The diameter of the , the focal length is , main parabolic reflector is and the offset height is . The hyperboloid eccentricity , and the tilt angle of the subreflector axis is zero. A type RCP feed with dB feed taper is located at the focal point , tilted toward the center of subreflector at a tilt angle of 20.20 . Using (5), (11), (12), and (13), the parameters of the equivalent paraboloid are calculated. , and . Because the feed axis is not aligned with the equivalent paraboloid axis, the beam squint phenomenon exists for the on-focus RCP feed. PO analysis is performed on both reflectors in order to better capture the diffraction effects. Simulated far-field pattern (Fig. 11) shows , exactly the same as predicted that the squint angle is by (4). Note that an RCP feed produces an RCP beam in a dual-reflector antenna. That is why the minus sign is chosen using in (4). The optimal feed displacement is (14), where for the equivalent paraboloid. The

Fig. 14. (a) Simulated far-field pattern of the axially symmetric Cassegrain reflector antenna with a laterally displaced feed. The pattern is cut through the beam maximum in the  = 0 cut ( = 0:7918 ). (b) A close-up view. The simulated squint angle  = 1:9 arcseconds.

0

0

compensated far-field pattern is shown in Fig. 12. The beam squint effect is successfully eliminated. C. Axially Symmetric Cassegrain Reflector With an Off-Focus Feed The last example examines the validity of the proposed technique for reflectors with off-focus feeds. The geometry of an axially symmetric Cassegrain reflector antenna [6] is shown in Fig. 13, where the diameter of the main reflector is (100 m at the operating frequency of 1.5 GHz), and the focal length . The subreflector eccentricity , the diameter is , and the inter-foci distance is .A type is dB is used to illuminate the RCP feed with feed taper of subreflector. Because of the symmetry of the reflector configuration, the radiation pattern is free of beam squint if the RCP in feed is situated on focus. When the feed is displaced by plane to achieve a scanned beam, the arcseconds is a squint angle of observed in the cut (the far-field pattern is cut through

XU AND RAHMAT-SAMII: A NOVEL BEAM SQUINT COMPENSATION TECHNIQUE FOR CIRCULARLY POLARIZED

Fig. 15. (a) Compensated far-field pattern of the axially symmetric Cassegrain reflector antenna with an off-focus feed. The feed tilt angle is 2 . The pattern is cut through the beam maximum in the  = 0 cut ( = 0:7918 ). (b) A close-up view. No squint angle is observed.

0

the beam maximum in the cut), as shown in Fig. 14, and the simulated directivity is 62.331 dB. Note that the feed is tilted toward the focal point in order to optimize the directivity performance of the overall antenna system. As indicated in Fig. 13, . Note that in this example, the feed tilt angle does not refer to the direction of the equivalent paraboloid axis. Eilhardt et al. [6] suggested that the beam squint in this specific reflector configuration can be compensated when the feed . The simulated far-field pattern after feed tilt angle equals tilting compensation is presented in Fig. 15. It can be observed that this technique effectively eliminates the squint angle. The simulated directivity, however, is decreased to 61.934 dB, resulting in a 0.397 dB directivity loss. It is because that the large feed tilt angle causes more spillover around the lower part of the subreflector. The unbalanced illumination also distorts the cut [see Fig. 15(a)]. radiation pattern, especially in the Again the feed displacement compensation technique discussed in this paper is applied to compensate for

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Fig. 16. (a) Compensated far-field pattern of the axially symmetric Cassegrain reflector antenna with an off-focus feed. The optimal feed displacement is 0:00676 . The pattern is cut through the beam maximum in the  = 0 cut ( = 0:7918 ). (b) A close-up view. No squint angle is observed.

0

0

the beam squint. The equivalent paraboloid is a symmetric parabolic reflector with the same aperture size, and . The off-focus feed is still aligned in direction. Using (14) the optimal feed displacement is calculated to be , and the compensated far-field pattern is plotted in Fig. 16, where the simulated directivity is 62.331 dB. Therefore the proposed approach not only corrects the squint angle, but also fully retrieves the maximum directivity of the overall antenna system. Compared with Fig. 15(a), the envelope of the radiation pattern is improved as well, especially cut. in the first sidelobe region in the D. Potential Applications Simulation results of all three representative examples illustrate the effectiveness and universal applicability of the proposed compensation technique for the beam squint. Most importantly, this novel approach works quite well for the beam squint compensation for single offset reflector antennas, which to our best knowledge has not been solved so far. It has been

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shown that the feed displacement can significantly reduce the linear phase shift in the aperture plane. One potential application is to improve the measurement accuracy of circularly polarized operating compact range antennas [13], [14]. A more uniform phase distribution can now be achieved to better illuminate the antenna under test in a much larger region. Moreover, no constraints are imposed on selecting geometrical parameters when applying this approach. Reflector and feed configurations can be optimized for any design objectives without worrying about the beam squint phenomenon, which now can be compensated by a small lateral feed displacement in the last step without changing existing parameters. In particular it is suitable for improving the pointing accuracy of an existing reflector antenna for deep-space telemetry or radio astronomy. A lateral feed displacement, which can be optimally determined within a few steps of calculation, is the only modification necessary to accomplish the compensation. With the help of the equivalent paraboloids, the same compensation technique can be applied to a wide variety of reflector antennas: symmetric or offset; single paraboloid, Cassegrain, Gregorian, or even multi-reflectors; on-focus or off-focus feeds; RCP or LCP feeds. The obvious drawback of the proposed technique, however, is that the optimal feed displacement is at opposite sides of the offset plane for RCP and LCP feeds. The feed must be readjusted when the feed polarization is changed. The optimal feed displacement is usually very small, ranging from a couple of millimeters to even sub-millimeter region, depending on the operating frequency and the reflector configuration. An accurate adjustment is therefore required. IV. CONCLUSION The beam squint phenomenon is carefully revisited and a practical and widely applicable compensation technique for the beam squint is proposed in this paper. Beam squint generally exists in offset reflector antennas with circularly polarized feeds, manifested by a small beam shift of the radiation pattern in the plane perpendicular to the principal offset plane. It is understood that the linear phase shift caused by the depolarization effect on polarized components is the primary factor responsible. Accordingly a simple formula is derived to quickly estimate the optimal feed displacement for both RCP and LCP cases, which can effectively minimize or even eliminate the phase variation. Simulation results of three representative examples: a single offset parabolic reflector, a suboptimal offset Cassegrain reflector, and an axially symmetric Cassegrain reflector with an off-focus feed, illustrate the validity and universal applicability of the proposed method. In particular, this novel technique is capable of compensating for the beam squint of offset reflector antennas where the previously suggested method based on feed tilting may not be desirable due to the excessive illumination spillover at large tilt angles. REFERENCES [1] T. S. Chu and R. H. Turrin, “Depolarization properties of offset reflector antennas,” IEEE Trans. Antennas Propag., vol. 21, no. 3, pp. 339–345, May 1973.

[2] A. W. Rudge, “Multi-beam antennas: offset reflectors with offset feeds,” IEEE Trans. Antennas Propag., vol. 23, no. 3, pp. 317–322, May 1975. [3] N. A. Adatia and A. W. Rudge, “Beam squint in circularly polarized offset reflector antennas,” Electron. Lett., vol. 11, no. 21, pp. 513–515, Oct. 1975. [4] D. W. Duan and Y. Rahmat-Samii, “Beam squint determination in conic-section reflector antennas with circularly polarized feeds,” IEEE Trans. Antennas Propag., vol. 39, no. 5, pp. 612–619, May 1991. [5] D. Fiebig, R. Wohlleben, A. Prata, and W. V. T. Rusch, “Beam squint in axially symmetric reflector antennas with laterally displaced feeds,” IEEE Trans. Antennas Propag., vol. 39, no. 6, pp. 774–779, June 1991. [6] K. Eilhardt, R. Wohlleben, and D. Fiebig, “Compensation of the beam squint in axially symmetric, large dual reflector antennas with largeranging laterally displaced feeds,” IEEE Trans. Antennas Propag., vol. 42, no. 10, pp. 1430–1435, Oct. 1994. [7] W. V. T. Rusch, A. Prata, Jr., Y. Rahmat-Samii, and R. A. Shore, “Derivation and application of equivalent paraboloid for classical offset Cassegrain and Gregorian antennas,” IEEE Trans. Antennas Propag., vol. 38, no. 8, pp. 1141–1149, Aug. 1990. [8] Y. Mizugutch, M. Akagawa, and H. Yokoi, “Offset dual reflector antennas,” in IEEE. Antennas Propag. Soc. Symp. Dig., Oct. 1976, pp. 2–5. [9] T. S. Bird, “Investigation of crosspolarization in offset Cassegrain antennas,” Electron. Lett., vol. 17, no. 17, pp. 585–586, Aug. 1981. [10] V. Jamnejad-Dailami and Y. Rahmat-Samii, “Some important geometrical features of conic-section-generated offset reflector antennas,” IEEE Trans. Antennas Propag., vol. 28, no. 6, pp. 952–957, Nov. 1980. [11] Y. Rahmat-Samii, “Array feeds for reflector surface distortion compensation: Concepts and implementation,” IEEE Antennas Propag. Mag., vol. 32, no. 4, pp. 20–26, Aug. 1990. [12] D. W. Duan and Y. Rahmat-Samii, “A generalized diffraction synthesis technique for high performance reflector antennas,” IEEE Trans. Antennas Propag., vol. 43, no. 1, pp. 27–40, Jan. 1995. [13] R. Johnson, H. Ecker, and R. Moore, “Compact range techniques and measurements,” IEEE Trans. Antennas Propag., vol. 17, no. 5, pp. 568–576, Sept. 1969. [14] J. P. McKay and Y. Rahmat-Samii, “An array feed approach to compact range reflector design,” IEEE Trans. Antennas Propag., vol. 41, no. 4, pp. 448–457, Apr. 1993. Shenheng Xu received the B.S. and M.S. degrees (with distinction) in electrical engineering from Southeast University, Nanjing, China, in 2001 and 2004, respectively, and the Ph.D. degree in electrical engineering from the University of California, Los Angeles (UCLA), in 2009. From 2001 to 2004, he was a Research Assistant in the State Key Laboratory of Millimeter Waves at Southeast University. In 2004, he joined the Antenna Research, Analysis, and Measurement Laboratory at UCLA, where he worked as a Graduate Student Researcher under the direction of Prof. Yahya Rahmat-Samii. He is currently a Postdoctoral Scholar at UCLA. His research interests include novel designs of modern reflector antennas for advanced applications, various compensation techniques for reflector antennas, and evolutionary algorithms for electromagnetic applications.

Yahya Rahmat-Samii (S’73–M’75–SM’79–F’85) received the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois, Urbana-Champaign. He is a Distinguished Professor, holder of the Northrop Grumman Chair in Electromagnetics, and past Chairman of the Electrical Engineering Department, University of California, Los Angeles (UCLA). He was a Senior Research Scientist with the National Aeronautics and Space Administration (NASA) Jet Propulsion Laboratory (JPL), California Institute of Technology prior to joining UCLA in 1989. In summer 1986, he was a Guest Professor with the Technical University of Denmark (TUD). He has also been a consultant to numerous aerospace and wireless companies. He has been Editor and Guest editor of numerous technical journals and books. He has authored and coauthored over 750 technical journal and conference

XU AND RAHMAT-SAMII: A NOVEL BEAM SQUINT COMPENSATION TECHNIQUE FOR CIRCULARLY POLARIZED

papers and has written 25 book chapters. He is a coauthor of Electromagnetic Band Gap Structures in Antenna Engineering (New York: Cambridge, 2009), Implanted Antennas in Medical Wireless Communications (Morgan & Claypool Publishers, 2006), Electromagnetic Optimization by Genetic Algorithms (New York: Wiley, 1999), and Impedance Boundary Conditions in Electromagnetics (New York: Taylor & Francis, 1995). He has received several patents. He has had pioneering research contributions in diverse areas of electromagnetics, antennas, measurement and diagnostics techniques, numerical and asymptotic methods, satellite and personal communications, human/antenna interactions, frequency selective surfaces, electromagnetic band-gap structures, applications of the genetic algorithms and particle swarm optimization, etc., (visit http://www. antlab.ee.ucla.edu/). Dr. Rahmat-Samii is a Fellow of the Institute of Advances in Engineering (IAE) and a member of Commissions A, B, J and K of USNC/URSI, the Antenna Measurement Techniques Association (AMTA), Sigma Xi, Eta Kappa Nu and the Electromagnetics Academy. He was Vice-President and President of the IEEE Antennas and Propagation Society in 1994 and 1995, respectively. He was appointed an IEEE AP-S Distinguished Lecturer and presented lectures internationally. He was a member of the Strategic Planning and Review Committee (SPARC) of the IEEE. He was the IEEE AP-S Los Angeles Chapter Chairman (1987–1989), and chapter won the best chapter awards for two consecutive years. He is listed in Who’s Who in America, Who’s Who in Frontiers of Science and Technology and Who’s Who in Engineering. He has been the plenary and millennium session speaker at numerous national and international symposia. He has been the organizer and presenter of many successful short courses worldwide. He was a Directors and Vice President of AMTA for three years. He has been Chairman and Co-chairman of several national and international

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symposia. He was a member of the University of California at Los Angeles (UCLA) Graduate council for three years. For his contributions, he has received numerous NASA and JPL Certificates of Recognition. In 1984, he received the Henry Booker Award from URSI, which is given triennially to the most outstanding young radio scientist in North America. Since 1987, he has been designated every three years as one of the Academy of Science’s Research Council Representatives to the URSI General Assemblies held in various parts of the world. He was also invited speaker to address the URSI 75th anniversary in Belgium. In 1992 and 1995, he received the Best Application Paper Prize Award (Wheeler Award) for papers published in 1991 and 1993 IEEE Transactions on Antennas and Propagation. In 1999, he received the University of Illinois ECE Distinguished Alumni Award. In 2000, he received the IEEE Third Millennium Medal and the AMTA Distinguished Achievement Award and in 2001, an Honorary Doctorate in physics from the University of Santiago de Compostela, Spain. In 2001, he became a Foreign Member of the Royal Flemish Academy of Belgium for Science and the Arts. In 2002, he received the Technical Excellence Award from JPL. He received the 2005 URSI Booker Gold Medal presented at the URSI General Assembly. He is the recipient of the 2007 Chen-To Tai Distinguished Educator Award of the IEEE Antennas and Propagation Society. In 2008, he was elected to the membership of the National Academy of Engineering (NAE). In 2009, he was selected to receive the IEEE Antennas and Propagation Society highest award, Distinguished Achievement Award, for his outstanding career contributions. He is the designer of the IEEE Antennas and Propagation Society (IEEE AP-S) logo displayed on all IEEE-AP-S publications.

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Broadband, Efficient, Electrically Small Metamaterial-Inspired Antennas Facilitated by Active Near-Field Resonant Parasitic Elements Peng Jin, Student Member, IEEE, and Richard W. Ziolkowski, Fellow, IEEE

Abstract—The possibility of using an active internal matching element in several types of metamaterial-inspired, electrically small antennas (ESAs) to overcome their inherent narrow bandwidths is demonstrated. Beginning with the Z antenna, which is frequency tunable through its internal lumped element inductor, a circuit model is developed to determine an internal matching network, i.e., a frequency dependent inductor, which leads to the desired enhanced bandwidth performance. An analytical relation between the resonant frequency and the inductor value is determined via curve fitting of the associated HFSS simulation results. With this inductance-frequency relation defining the inductor values, a broad bandwidth, electrically small Z antenna is established. This internal matching network paradigm is then confirmed by applying it to the electrically small stub and canopy antennas. that An electrically small canopy antenna with has over a 10% bandwidth is finally demonstrated. The potential implementation of the required frequency dependent inductor is also explored with a well-defined active negative impedance converter circuit that reproduces the requisite inductance-frequency relations.

= 0 0467

Index Terms—Active antennas, bandwidth, electrically small antennas, metamaterials, parasitic antennas, factor.

I. INTRODUCTION LECTRICALLY small antennas (ESAs) have been studied extensively in the past and have many potential applications in all wireless communication and sensor systems because of their compact dimensions. It is well known that the performance characteristics of an ESA are limited by its physical dimensions [1]–[3]. For instance, the bandwidth performance of an ESA can be estimated by its value in relation is its to the Chu-based lower bound. In particular, if value is given half power VSWR fractional bandwidth, its by . If its radiation efficiency is , then , the Chu-based lower bound is where is the free space wavenumber and is the minimum

E

Manuscript received January 22, 2009; revised May 27, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported in part by DARPA Contract HR0011-05-C-0068. P. Jin is with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721 USA (e-mail: [email protected]). R. W. Ziolkowski is with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721 USA and also with the College of Optical Sciences, University of Tuscon, Tucson, AZ 85701 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.2009.2037708

radius of a sphere that completely encloses the antenna. Then the natural figure of merit associated with the bandwidth is . An antenna is generally the ratio, i.e., . However, if , its comclassified as an ESA if pact electrical dimension comes at the cost of a very narrow . bandwidth, which is limited approximately by , the bandwidth can at most be For example, when . Moreover, a resonant ESA usually has an associated low radiation resistance and usually requires an external matching network to achieve a high accepted power level. Such a matching network will add additional size to the ESA, and usually, it will further limit the overall system bandwidth. To surpass the Chu limit, non-Foster (NF) matching networks have been proposed, e.g., see [4]. A NF matching network realizes negative inductance and capacitance values with active elements; these values are designed to bring the antenna into resonance (reactance matching) and to optimize the power delivered to its terminals from the source (resistance matching). As depicted, for instance, in [4], the NF matching network is implemented between the source and the antenna. We will refer to it as an external matching network. The internal matching network, which we introduce below, is internal to and part of the actual radiating element. Metamaterial-inspired, efficient ESAs have been introduced in [5]–[7]. These ESAs are constructed as a driven element and a resonant parasitic element in the very near field of the driven element. These ESAs are nearly completely matched to a real source and have a very high overall efficiency. These properties are achieved through the parasitic element, which replaces the need for an external matching network and which works with the driven element to enhance the radiation process. Based on these works, the Z antenna, which uses an internal lumped element, was then introduced in [8] and [9]. In these works, the Z antenna was tuned to resonate at different frequencies by changing the value of the lumped element, but without changing the overall dimensions of the antenna system. From these results, we realized that if one could develop self-tuned lumped elements fulfilling the resonance requirements at all frequencies in a certain , the Z antenna frequency band of interest, i.e., for . would have an instantaneous bandwidth of In this paper, we develop such a self-tuning lumped element, its frequency dependent behavior, and ways to implement the resulting frequency dependent internal matching element, to achieve an active, broad band ESA. Our work is assembled as follows. In Section II, the results for ANSOFT HFSS-Designer co-simulations of the Z antenna are detailed and a circuit

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JIN AND ZIOLKOWSKI: BROADBAND, EFFICIENT, ELECTRICALLY SMALL METAMATERIAL-INSPIRED ANTENNAS

Fig. 2. The HFSS-predicted magnitude of S

Fig. 1. The Z antenna configuration.

model equivalent is developed. The impedance of the lumped element required to achieve a broad bandwidth is then revealed numerically. The relation between the lumped element and the resonant frequency of the antenna is obtained in Section III. It is used to define a circuit model that could be used to implement the desired self-tuning lumped element, i.e., the internal matching network (IMN). In Section IV, this self-tuning lumped element design is applied to several ESAs, including the stub and canopy antennas introduced, respectively, in [10] and [11]. Approximately a 10% fractional bandwidth is achieved for each case. Our conclusions are given in Section V. II. ANSOFT HFSS AND DESIGNER SIMULATIONS OF THE Z ANTENNA Fig. 1 shows the Z antenna loaded with a lumped element, 1000 nH inductor. Its HFSS-predicted (version 11.1.3) values for a 50 source are shown in Fig. 2. All of the materials are treated as lossless to simplify the bandwidth considerations. The minimum enclosing sphere for this Z antenna has a so that , where , radius being the speed of light in vacuum and being its resonant frequency. The overall efficiency, as expected, . The 3 dB fractional bandwas approximately width was , and the 10 dB fractional . Thus, one finds bandwidth was . This value is rather far from the Chu-based lower bound because the Z antenna physically occupies only a small portion of its minimum enclosing sphere. While these HFSS simulations show that this very electrically small Z antenna is well matched to the 50 source and has a high overall efficiency, its potential for applications is limited by its narrow fractional bandwidth. In fact, even if this very small Z antenna could achieve the Chu limit with a similar overall efficiency, its 3 dB bandwidth would remain less than 0.02%. Thus, a means to increase its bandwidth was sought.

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values of the Z antenna.

We note that the Z antenna components were treated in this design as lossless. As noted above, this choice was made to simplify the and bandwidth calculations. Naturally, a lossy Z antenna design would have exhibited a lower overall efficiency and a broader bandwidth. However, its value would have been approximately the same when its lower radiation efficiency was properly taken into account. Losses are inherent with any real electrically small design and can significantly impact its performance. In this context, all of the designs reported in the remaining sections of this paper are simulated using lossy copper, . Consequently, conductive its conductivity being 5.8 loss effects are incorporated into those designs. On the other hand, the lumped element inductors will remain modeled as ideal lossless components. Both the losses in the lumped element inductors and the copper losses were included in the Z antenna designs reported recently in [12]. These antennas were fabricated and measured. The measured results were in very good agreement with their predicted, reasonably high values. Because these designs differ from their ideal lossless inductor cases only by the monopole height to achieve nearly complete matching and because we found that the overall efficiency was closely coupled to the availability of specific component values and to how much one was willing to pay for high quality components, we remain focussed here on the bandwidth issues. In any practical realization of the antennas discussed below, the final value of the overall efficiency will thus be closely tied to the quality of their lumped element components selected from a vendor for implementation. Consequently, the efficiency values reported below are upper bounds, while the corresponding bandwidth values will be lower bounds. Consistent results among HFSS simulations, HFSS-Designer co-simulations, and measurements for the two-dimensional magnetic EZ antenna with a lumped capacitor were described in [13]. Consequently, we decided to employ the HFSS-Designer co-simulation approach, which relies on a circuit model of the antenna system shown in Fig. 3, to study the bandwidth behavior of the Z antenna. The antenna block in Designer is treated as an N-port sub-circuit which is imported as an matrix from the HFSS simulation in which the lumped LRC element is replaced with a lumped port. One port is treated as the source wave port. The lumped RLC element is reintroduced into the Designer model as a circuit element with the corresponding combination of , , and . The frequency can then be swept

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Fig. 6. Antenna circuit model with the NET load. Fig. 3. Z antenna circuit model.

Fig. 4. ANSOFT Designer circuit model for the Z antenna. The IMN considered for this case is simply the idealized 1000 nH inductor.

Fig. 7. ANSOFT Designer predicted magnitude of S having the designed frequency dependent IMN.

vales for the Z antenna

equal or nearly equal to the source impedance . Based on this consideration, the internal matching network was designed such that

(1) Fig. 5. ANSOFT Designer predicted magnitude of S loaded with the passive 1000 nH inductor.

values for the Z antenna

in the Designer simulation to find the resonance frequency. Unfortunately, we have found that for our very narrow bandwidth antenna systems, the resonant frequencies predicted by the co-simulation approach and by HFSS for the same lumped RLC element values are consistently offset by 1%–2%. For instance, for the antenna shown in Fig. 1, the co-simulation with ANSOFT Designer 3.5.2 was performed with the model shown in Fig. 4 where the IMN is the idealized 1000nH inductor; and, as shown in Fig. 5, predicted the resonance frequency to be 197.16 MHz. As noted, the resonance frequency, predicted by , is 1.9208 MHz lower than that HFSS: value, i.e., 0.98% lower. Nonetheless, because this offset is consistent, the circuit model of the Z antenna shown in Fig. 3 can still be used to find what kind of internal matching network is required to achieve a broad bandwidth. , , , and terms are the eleIn Fig. 3, the matrix which represents the antenna block ments of the and the is the lumped element inductor. The proposed internal matching network would replace the block between the dashed lines in Fig. 3. The resulting circuit model is shown in Fig. 6, , , , and are the matrix where parameters of the internal matching network. Then, to obtain values, the antenna input impedance must be made low

i.e., the antenna structure combined with the internal matching network was designed to have the source resistance value. The matrix of the internal matching network can then be determined analytically as

(2) matrix of the internal matching netTo calculate the work, the or parameters were obtained for the frequencies in the interval of interest from the HFSS simulations which included the lumped port element. These values were then conparameters. The values verted to the requisite antenna for the IMN block shown in Fig. 4 were then calculated from (2) in MATLAB. These results were then reconverted into their S-parameter form and incorporated into the Designer model as values for the the -port element. The Designer predicted IMN-based Z antenna system are shown in Fig. 7. The values are below for over a 20% fractional bandwidth. These results clearly demonstrate that an appropriately designed IMN can lead to a matched electrically small antenna over a very broad frequency range. We note that a real IMN circuit could be developed from these matrix results. However, it could be very challenging because there are four independent variables. Note that in Fig. 6, the matching network is connected to the antenna on one port

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Fig. 8. Antenna circuit model with equivalent load.

and is shorted on the other port. According to the definition

matrix

(3) and are the voltage and current at the left where and port of the matching network in Fig. 6 and are the voltage and current at its right port. Since , this model yields the relation

(4) Utilizing this relation, the circuit model given in Fig. 6 with the shorted matching network can be simplified to the circuit model in Fig. 8. Both HFSS and Designer co-simulations were prevalues formed for the simplified circuit model; the same values in shown in Fig. 7 were obtained. The calculated the whole sweep range are found to be pure imaginary values, whose imaginary part is always positive and changing with frequency. Consequently, one finds that the IMN can be implemented by a frequency dependent inductor, i.e., to have the Z antenna resonate at different frequencies and, hence, to expand its very limited bandwidth, an inductor with frequency dependent values is needed.

Fig. 9. Resonant frequency of the Z antenna as its lumped element inductor value is varied.

the inductance of the lumped element, , and of all of the radidoes not change ating elements, . For a fixed geometry, its value. Moreover, it is found that the lumped element induc. Consetance is much larger than , which means quently, the resonance frequency of the Z antenna can be controlled simply by changing the value of the lumped element inductor. These properties are also true for the electrically small stub and canopy antennas to be discussed below. relation (6) was readily demonSatisfaction of the strated with the set of discrete HFSS simulation results shown in Fig. 9. The frequency was swept from 60 MHz to 1.0GHz. In this sweep the inductor values were varied and only small adjustments to the height of the monopole antenna were made to bring the Z antenna radiation resistance back into match with the source. Comparing these discrete results with those given by the analytical expression (6), one finds very good agreement.

IV. BANDWIDTH ENHANCEMENT FOR METAMATERIAL-INSPIRED ESAs

III. INDUCTOR VERSUS RESONANT FREQUENCY The Z antenna results were used to establish a relation between its inductor value and its resonant frequency . For such an antenna structure, one finds that its resonant frequency

(5) and are, respectively, its effective inductance where and capacitance. According to this relation, if remains the same, then the effective inductance must satisfy the relation

(6) where is a constant. Thus, the Z antenna will if its effective inductance satisfies (6). It be resonant at should be noted that this effective inductance is composed of

Having established that the Z antenna can be predictably tuned by varying its lumped element value and the monopole height, we investigated its achievable bandwidth by only varying the inductor value. Moreover, because the stub and canopy antennas are also realized as lumped element controlled near-field resonant parasitics, they were also included in our studies. We have found that when all of these resonant near field parasitic antennas are designed with passive inductors, their bandwidths are restricted by the Chu-lower bound. However, when active inductors are included, significant enhancements of their bandwidths can be realized. This observation is consistent with several previous investigations into active metamaterials. For instance, active artificial molecules were considered in [14] for several scattering applications. Active unit cells were considered in [15] to achieve wide bandwidth negative permittivity and permeability metamaterials. An active metamaterial was designed in [16] to recover the large bandwidths associated with the idealized, dispersion-free metamaterial-based

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Fig. 11. Results obtained by curve fitting of the inductor values. Fig. 10. Z Antenna with ka

= 0:266.

antennas. Active metamaterials at visible wavelengths were considered in [17]. A. Z Antenna The Z antenna was first studied to show its behavior as the value of the lumped element inductor was varied. Because the Z antenna in Fig. 1 has a very small value and a high ratio, its bandwidth was found to be very limited. As will be discussed below, this made it difficult to enhance its value even with value Z antenna an active element. Consequently, a larger , was designed. In particular, this Z antenna had , , and for an 100 nH inductor. Twenty four different inductor values were considered in the neighborhood of this original value. The HFSS simulations predicted the resonance frequencies shown in Fig. 11. The variation of these resonant frequency values as a function of the inductance was curve fit with a minimum mean square error (MMSE) approach. It was found that the frequency dependent inductor value can be expressed by the relation:

(7) and . The units of the where inductance, , and the frequency, , are, respectively, nH and MHz, For the corresponding metric units, respectively, H and Hz, this relation becomes

Fig. 12. Negative lumped element circuit model.

to show that an equivalent circuit can be synthesized with active elements to provide the same impedance values. The frequency , corresponding to the inductance dependent impedance, can be written in the form

where we have introduced the equivalent capacitor and inductor and , respectively. The series circuit shown terms, in Fig. 12, which consists of this equivalent capacitor and inductor, produces the desired frequency dependent impedance. According to (9), the component values in Fig. 12 that reproduce the curve fit in Fig. 11 are

(8) The curve fitting results shown in (7) and (8) are consistent with the relation (6) when , i.e., note that is . negative and recall that The frequency dependent inductor values predicted by (7) or (8) cannot be generated by a simple circuit element. In particular, we note that these values have a non-Foster reactance behavior, i.e., the inductance is decreasing quickly enough with . It is straightforward increasing frequency so that

The predicted negative values for this negative capacitor and inductor circuit can be realized with a negative impedance converter circuit [18]. In particular, the negative impedance converter (NIC) element shown in Fig. 13 produces the following relation between the input impedance and the desired load:

(9)

JIN AND ZIOLKOWSKI: BROADBAND, EFFICIENT, ELECTRICALLY SMALL METAMATERIAL-INSPIRED ANTENNAS

Fig. 13. Circuit with negative impedance converter that is equivalent to the negative element circuit.

Fig. 14. Floating negative impedance converter circuit.

and is a positive constant. A typical NIC circuit [18] that produces the desired values is shown in Fig. 14, i.e., its input impedance is defined as

(10) In our set of twenty four fine resolution HFSS simulations about the original resonance frequency, we considered only variations of the lumped inductor values with no changes in the monopole height or any other design parameter for this specific set of nearby resonant frequencies. The criterion for deciding how far away from the original resonance frequency, all of these resonance frequencies could be, was that the antenna system remain nearly completely matched to the source, i.e., that for all of values were smaller than . these frequencies, the , the built in inductance of the Z anWe note that like tenna is also constant since the antenna structure was maintained without any changes. The derivative with respect to the inductance value of (6) shows that the rate of change of the resonant frequency with respect to the inductor value is given by the expression

(11) which can also be re-written approximately (i.e., recall that since ) as

(12) According to (12), it can then be concluded that to obtain a 10% bandwidth, the change of the inductor value must be approximately 20%, which is confirmed by the results given in Fig. 11. The - relation provides a guideline to determine the values in the design and implementation of the Z antenna to allow for adequate variation in the parameters so that an actual implementation of the active circuit design might be realized.

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Because of the nature of the curve fitting, there are always errors between the values specified by the resulting curve and the exact (as specified by the HFSS simulations) values. A meaningful curve fit should provide inductor values close enough to the exact values that one could fulfill some additional practical at every frequency in the range criteria, e.g., of interest. at frequency . Then asAssume the inductor value is , sume that the curve fitting yields an inductor value . Assume which corresponds to the resonant frequency . If that the resulting Z antenna has a 10 dB bandwidth at frequency , then one will have . must be broad enough to acConsequently, the bandwidth commodate the maximum error in the curve fitting. It thus revalue should be large enough to insure this quires that the requisite bandwidth. In an actual implementation the actual errors in the inductor value may be much larger simply because of the manufacturer’s tolerance values (manufacturing imperfections) of the components in Fig. 14. The Z antenna has a 10 dB bandwidth of 0.1% . Neglecting the curve fitting error, the circuit for in Fig. 14 must then be constructed with inductor values having a 0.1% accuracy. Such components would most likely be extremely expensive since generally a 1% tolerance is considered to be very good. Note that the bigger the error in the inductor value, the broader the bandwidth has to be to guarantee the overall performance of the antenna system; and, therefore, the values and/or lower values when this need for larger occurs. B. Stub Antenna In the bandwidth enhancement process for the active IMNbased Z antenna, it was emphasized that it is necessary to minimize the error between the curve fit values and the actual inductor values, which yielded the resonant, matched conditions, values resulting from the curve-fit inductor so that the values fall below the 10 dB bandwidth criterion. For the Z antenna, this criterion was met by increasing the Z antenna size to a . This led to a broader 10 dB bandwidth larger value: and, hence, a larger fractional bandwidth limit. The curve fitting errors associated with defining the active inductor can then be accommodated by the design. In the Z antenna, the meander line, i.e., the “Z” portion of the parasitic element, was designed originally in [9] to provide additional inductance to the system, as well as to enhance the radiation mechanism. However, in our active inductor design studies, it was found that the meander line inductance is actually negligible when compared to the lumped element inductance. On the other hand, the complexity of the meander line itself caused some difficulties in the convergence of the HFSS simulations, and thus produced numerical sensitivities in their predicted values. In addition, it was recognized that a structure, which has a more complex design, will generally lead to non-trivial fabrication sensitivities. Consequently, we felt that the curve fitting errors associated with parasitic elements whose designs were simpler, would be lower and would thus more readily lead to a successful active element design. We thus decided to investigate whether the stub antenna introduced in [10], which has a simpler near-field parasitic element

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Fig. 16. Inductor-frequency (L-F) sweep for the stub antenna cases.

Fig. 15. One-leg stub antenna.

and can be designed to have a lower -ratio value, would lead to improved curve fits and, hence, to lower curve fitting error values. The stub antenna is a metamaterial-inspired ESA which was first introduced in [10] and whose ratio behavior was further studied in [11]. Although a stub antenna in [11] with was introduced, its leads to a bandwidth which is too narrow for our purposes. We thus designed the stub antenna shown in Fig. 15. It has a coax-fed monopole whose radius and height are, respectively, 0.5 mm and 9.2 mm, and a parasitic whose radius and height are, respectively, 1.205 mm and 17.35 mm, and whose center is located 10 mm from the center of the monopole. The length of the inductor and the conductor of the parasitic are, respectively, 3.35 mm and 14 mm. This stub , the radius being measured from antenna has the center of the parasitic; it resonates at 299.6839 Mhz; and when the lumped element inductor it has . A discrete set of 20 additional HFSS simulations, symmetrically located about the center resonant frequency, based on 6 nH increments of the inductor value at the center frequency ) were then run. The re(i.e, approximetely 1% of sulting HFSS-predicted inductor-frequency sweep and the corresponding curve fit results for this antenna are labeled as Ant1 in Fig. 16. The associated curve fitting error percentage and the corresponding fractional bandwidth limiting values are given in Fig. 17. Although this one-stub antenna has a ratio similar to the Z antenna, one observes from Fig. 17 that it has, as expected, a lower error level and, hence, a further separation from the limiting fractional bandwidth values. One finds that a 10% stub fractional bandwidth can be achieved for this antenna. To lower the ratio, the radius of the parasitic element shown in Fig. 15 was increased from 1.205 mm to 3 mm and the

Fig. 17. Comparison of the curve fitting errors and the F BW the stub antenna cases.

values for

inductor value was decreased to . This thicker parand resonates asitic element one-stub antenna has at 300.3901 MHz. This means . A similar set of HFSS simulations were run based on 3 nH increments of the inductor value. The results for this antenna are labeled as Ant2 in Figs. 16 and 17. One observes that with a lower -ratio, the curve fitting errors have been decreased while the fractional bandwidth error limits have been increased. These results imply that it will be easier to design and achieve an active inductor element version of the lower -ratio passive antenna. To emphasize this point further, the four parasitic element stub one-stub antenna was obantenna version of the tained. It is shown in Fig. 18. From [11] it was known that this four-stub antenna has a lower ratio than the one-stub case. In particular, setting each inductor to , adjusting the copper portion of the parasitic so that its overall height was 13.30 mm, and decreasing the monopole height to 7.3 mm, the giving resonant frequency was , the now being taken with respect to the center of the for the four-stub antenna. Anmonopole. Thus, other similar set of HFSS simulations were run based on 1% ininductor value. The results for this crements of the antenna are labeled as Ant3 in Figs. 16 and 17. The fractional bandwidth criterion values are increased further while the curve fitting errors are decreased further. It is now clear that for the same electrical size and a similar parasitic structure, a lower value results in a larger tolerance between the curve fitting ervalues. This means that there is rors and the limiting a smaller accuracy requirement for the active internal matching

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Fig. 19. One-leg canopy antenna.

Fig. 18. Four-leg stub antenna.

network implementation. We note that in all three stub antenna cases with , one finds that their active versions can have more than a 10% fractional bandwidth. C. Canopy Antenna -valued antenna that achieves To achieve yet a smaller more than a 10% fractional bandwidth, the one-, two-, and four-leg canopy antennas introduced in [11] were considered. ratio: All of these antennas have the same, even lower . The curve fitting procedure was performed explicitly for the one-leg canopy antenna shown in Fig. 19 and for the four-leg canopy antenna shown in Fig. 20. For both cases, each leg was treated as an ideal inductor; the canopy was treated as copper whose thickness coincided with the diameter , of the inductor. For an outer radius an inductor , a 0.2 mm shell thickness, a 4.4 mm inductor height, a 0.5 mm monopole radius, and a 1.98 mm monopole height, the one-leg canopy antenna has a res, and, hence, has onance frequency, . With a passive inductor, its fractional bandwidth is 0.0133%. We take this resonance frequency as the center frequency of the active inductor sweep. The results for a discrete set of 20 more HFSS simulations symmetrically located about the center resonant frequency, based on 4 nH increments of the inductor value at the center frequency (i.e., approximately 1% ) are shown in Fig. 21. The curve fit developed of from these HFSS simulation results is also shown in Fig. 21. The derived constants for the curve fit (7) are and . The errors in the resonance frequencies as calculated with the curve fit are shown in Fig. 22 along with the corresponding HFSS-predicted limiting values. One observes that the curve fitting resonance frequency

Fig. 20. Four-leg canopy antenna.

errors are even more separated from their limiting values than they were for the stub antenna cases, thus ensuring canopy antenna would be that the active one-leg resonantly well-matched to the source over more than a 10% fractional bandwidth. The same procedure was also performed for the four-leg canopy antenna. It should be noted that although the canopy , its actual antenna with four inductors also has bandwidth is slightly narrower than that of the one-leg canopy antenna because its radiation efficiency is a little higher. However, because of the symmetry of the four-leg canopy antenna, the HFSS simulations could be run with two perfect H symmetry planes, which reduced these problems to a quarter of their original sizes and, hence, allowed us to enhance the discretization used to further reduce their numerical errors. Except for two changes, all of the constituent elements remained the same.

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Fig. 21. Inductor-frequency (L-F) sweep for the one- and four-leg canopy antennas.

Fig. 22. Comparison of the curve fitting errors and the F BW the one- and four-leg canopy antennas.

values for

With the inductor value now and the height of the monopole now 1.88 mm, the HFSS-predicted value for the ; the value center frequency was was thus . The active inductor sweep was again taken to be a discrete set of 20 more HFSS simulations symmetrically located about the center resonant frequency with 16nH (1%) increments of the inductor value, which now of course, . The results for this active inductor sweep was are also shown in Fig. 21; the corresponding curve fit results are also shown. The derived constants for the curve fit (7) are and . The curve fitting errors in the resonance frequencies are shown in Fig. 22 along with limiting values. the corresponding HFSS-predicted As anticipated, because of the smaller modeling errors, one does find a further separation between the curve fitting errors values in Fig. 22. and the corresponding limiting These results demonstrate that the active four-leg canopy antenna would also be resonantly well-matched to the source over more than a 10% fractional bandwidth. V. CONCLUSION In this paper, the use of an active internal matching network for several near-field resonant parasitic element antennas was considered. Simulations of the electrically small Z antenna led to the development of its circuit model representation. An internal matching network version of this model was then proposed and validated. This result revealed that the requisite IMN

was simply a frequency dependent inductor and that with such an element, the electrically small Z antenna could be resonantly matched to the source over a specified frequency range. A relation between the required inductor value and the resonance frequency at which matching was maintained was developed. It was found that it had a non-Foster behavior which could be realized with a negative impedance converter. A curve-fitting procedure based on a set of HFSS simulations in which the inductor value was varied and the frequency at which resonant matching was obtained then led to the definition of the inductor values that needed to be obtained with the active inductor. Comparisons of the errors between the resonance frequencies predicted by the HFSS simulations based on these curve fitting results and the original discrete set of values and the corresponding 10 dB fractional bandwidths obtained from the latter defined whether the IMN-based (active inductor) near-field resonant parasitic antenna would work or not. This active internal matching network procedure was then applied to three specific ESAs: the Z, stub and canopy antennas. It was demonstrated that more than a 10% fractional bandwidth Z antenna, a could be realized with a stub antenna, and a canopy antenna. By considering these passive antenna systems with ever decreasing -ratio values and with a corresponding ever increasing accuracy of their HFSS simulations, it was shown that the separation between the curve fitting errors and the 10 dB fractional bandwidth limits could be increased. Consequently, the even lower valued active versions of these systems surpass the desired bandwidth goals. We further note that the active elements may also help with any reduced efficiency issues that would arise when real components are introduced into the IMN circuits, i.e., they could be used to compensate for the losses inherent in real components. This active loss compensation has become an important aspect of metamaterials research, particularly at optical frequencies where gain media are more readily available. While theoretical implementations of the requisite NIC circuit realizations of the active inductors considered here were presented, we are now concentrating on fabricating actual prototypes to validate these designs. We hope to present these results in the future. REFERENCES [1] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., pp. 1163–1175, Dec. 1948. [2] H. A. Wheeler, “Fundamental limitations of small antennas,” Proc. IRE, pp. 1479–1484, Dec. 1947. [3] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., pp. 1298–1324, Apr. 2005. [4] J. T. Aberle and R. Lopesinger-Romak, Antenna With Non-Foster Matching Networks. San Rafael, CA: Morgan & Claypool Publishers, 2007. [5] A. Erentok and R. W. Ziolkowski, “An efficient metamaterial-inspired electrically-small antenna,” Microw. Opt. Tech. Lett., pp. 1287–1290, Jun. 2007. [6] A. Erentok and R. W. Ziolkowski, “Two-dimensional efficient metamaterial-inspired electrically-small antenna,” Microw. Opt. Tech. Lett., pp. 1669–1673, Jul. 2007. [7] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficient electrically-small antennas,” IEEE Trans. Antennas Propag., pp. 691–707, Mar. 2008. [8] R. W. Ziolkowski and P. Jin, “Introduction of internal matching circuit to increase the bandwidth of a metamaterial-inspired efficient electrically-small antenna,” presented at the IEEE Int. Symp. Antennas Propag., San Diego, CA, Jul. 2008.

JIN AND ZIOLKOWSKI: BROADBAND, EFFICIENT, ELECTRICALLY SMALL METAMATERIAL-INSPIRED ANTENNAS

[9] R. W. Ziolkowski, “An efficient, electrically small antenna designed for VHF and UHF applications,” IEEE Antennas Wireless Propag. Lett., pp. 217–220, 2008. [10] A. Erentok and R. W. Ziolkowski, “Efficient electrically small antenna facilitated by a near-field resonant parasitic,” IEEE Antennas Wireless Propag. Lett., pp. 580–583, 2008. [11] P. Jin and R. W. Ziolkowski, “Low- , electrically small, efficient nearfield resonant parasitic antennas,” IEEE Trans. Antennas Propag., vol. 57, pp. 2548–2563, 2009. [12] R. W. Ziolkowski, P. Jin, J. A. Nielsen, M. H. Tanielian, and C. L. Holloway, “Design and experimental verification of Z antennas at UHF frequencies,” Antennas Wireless Propag. Lett., vol. 8, pp. 1329–1333, 2009. [13] A. Erentok, “Metamaterial-Based Electrically Small Antennas,” Ph.D. dissertation, Univ. of Arizona, Tucson, 2007. [14] F. Auzanneau and R. W. Ziolkowski, “Artificial composite materials consisting of linearly and nonlinearly loaded electrically small antennas: Operational amplifier based circuits with applications to smart skins,” IEEE Trans. Antennas Propag., pp. 1330–1339, Aug. 1999. [15] S. A. Tretyakov, “Metamaterials with wideband negative permittivity and permeability,” Microw. Opt. Tech. Lett., pp. 163–165, Nov. 2001. [16] R. W. Ziolkowski and A. Erentok, “At and beyond the Chu limit: Passive and active broad bandwidth metamaterial-based efficient electrically small antennas,” IET Microw. Antennas Propag., pp. 116–128, Feb. 2007. [17] J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Exp., pp. 6692–6716, Apr. 2008. [18] A. Larky, “Negative-impedance converters,” IRE Trans. Circuit Theory, pp. 124–131, Sep. 1957.

Q

Peng Jin (S’05) received the B.Sc. degree from the University of Science and Technology of China, HeiFei, in 1999 and the M.Sc. degree from the North Dakota State University, Fargo, in 2004. Currently, he is working toward the Ph.D. degree at the University of Arizona, Tucson. His research interests include electrically small antennas and metamaterial applications to antenna designs.

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Richard W. Ziolkowski received the Sc.B. degree in physics (magna cum laude with honor,) from Brown University, Providence, RI, in 1974 and the M.S. and Ph.D. degrees in physics from the University of Illinois at Urbana-Champaign, in 1975 and 1980, respectively. He was a member of the Engineering Research Division, Lawrence Livermore National Laboratory, from 1981 to 1990, and served as the leader of the Computational Electronics and Electromagnetics Thrust Area for the Engineering Directorate. He joined the Department of Electrical and Computer Engineering, University of Arizona, Tuscon, as an Associate Professor in 1990, and was promoted to Full Professor in 1996. He was selected by the Faculty to serve as the Kenneth Von Behren Chaired Professor for 2003-2005. He currently is serving as the Litton Industries John M. Leonis Distinguished Professor. He holds a joint appointment with the College of Optical Sciences. His research interests include the application of new mathematical and numerical methods to linear and nonlinear problems dealing with the interaction of acoustic and electromagnetic waves with complex media, metamaterials, and realistic structures. Prof. Ziolkowski is a member of Tau Beta Pi, Sigma Xi, Phi Kappa Phi, the Institute of Electrical and Electronics Engineers (IEEE), the American Physical Society, the Optical Society of America, and Commissions B (Fields and Waves) and D (Electronics and Photonics) of URSI (International Union of Radio Science). He is an IEEE Fellow. He was awarded the 1993 Tau Beta Pi Professor of the Year Award and the 1993 and 1998 IEEE and Eta Kappa Nu Outstanding Teaching Awards. He was the IEEE Antennas and Propagation Society (AP-S) Vice President in 2004 and President in 2005. He has served as a member of the IEEE AP-S Administrative Committee (ADCOM), as the Vice Chairman of the 1989 IEEE AP-S International Symposium and URSI Meeting, and as the Technical Program Chairperson for the 1998 IEEE Conference on Electromagnetic Field Computation. He is currently the Chair of the IEEE Electromagnetics Award Committee. He was an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and a Co-Guest Editor for the October 2003 Special Issue on Metamaterials. He was a Steering Committee Member for the 2004 ESA Antenna Technology Workshop. He was a Co-Chair of the International Advisory Committee for the inaugural IEEE International Workshop on Antenna Technology: Small Antennas and Novel Metamaterials, IWAT2005. He has been a member of the International Advisory Committees for IWAT 2006-2010, MAPE2005, Meta’08 and Meta’10. He was an Overseas Corresponding Member of the ISAP2007 Organizing Committee. He was the Technical Program Committee Chair for the 2008 Metamaterials Congress. He is currently a member of the Steering Committee for the 2009 and 2010 Metamaterials Congresses. For the US URSI Society he served as Secretary and as Chairperson of the Technical Activities Committee for Commission B and as Secretary for Commission D. He was a Member-at-Large of the U.S. National Committee (USNC) of URSI and is now serving as a member of the International Commission B Technical Activities Board. He is a Fellow of the Optical Society of America. He was a Co-Guest Editor of the 1998 Feature Issue of Journal of the Optical Society of America, on Mathematics and Modeling in Modern Optics. He was a Co-Organizer of the Photonics Nanostructures Special Symposia at the 1998-2000 OSA Integrated Photonics Research (IPR) Topical Meetings. He was the Chair of the 2001 IPR sub-committee IV, Nanostructure Photonics. He has served as a Co-Chair of the 2008 and 2010 SPIE Europe Conferences on Metamaterials.

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Coaxial-to-Waveguide Matching With "-Near-Zero Ultranarrow Channels and Bends Andrea Alù, Member, IEEE, and Nader Engheta, Fellow, IEEE

Abstract—We propose the use of metamaterial-inspired ultranarrow channels at cutoff to realize an interesting matching between a coaxial antenna and a waveguide. The anomalous properties of a channel at cutoff, analogous to those of zero permittivity materials, allow a simple matching design, valid for arbitrary waveguides, with large degrees of freedom in terms of geometry, length and possible bending of the connecting channel. Moreover, the static-like properties of the channel allow such matching, independent of the relative position of the antenna and possible bending and abruptions along the channel. Index Terms—Coaxial, metamaterials, matching, waveguides.

I. INTRODUCTION ETAMATERIALS have been a research topic of particular interest in the past several years [1]–[4], mainly due to their unconventional electromagnetic features that result in exciting novel potential applications in a wide range of fields. Among various classes of metamaterials, those with simultaneously negative permittivity and permeability have attracted the most attention [5]. However, since the metamaterial parameters may in principle be designed to achieve unusual values, other sets of metamaterials, such as -near-zero (ENZ), have also been thoroughly considered, offering exciting possibilities and potentials. Indeed, since the pioneering work of Rotman [6] on plasma-like microwave artificial dielectrics, several antenna applications of ENZ metamaterials have been proposed, mainly with the purpose of directivity enhancement [7]–[10]. In our earlier works, our group has been particularly interested in the anomalous features of these low-index metamaterials, highlighting their exciting properties and potentials in several different antenna, waveguide and radiation applications. In [11] we have shown how a uniform low-index slab may drastically enhance the transmission through a sub-wavelength aperture, whereas in [12], [13] we have applied the anomalous

M

Manuscript received April 28, 2009; revised June 26, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported in part by the U.S. Office of Naval Research (ONR) under Grant N 00014 -07-1-0622. A. Alù was with the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA. He is now with the Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78712 USA (e-mail: [email protected]). N. Engheta is with the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104 USA (e-mail: engheta@ee. upenn.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037714

properties of ENZ materials to design sub-wavelength planar and cylindrical antennas with directive properties. Moreover, in [14] we have speculated that the fascinating properties of ENZ metamaterials may give rise to arbitrarily pattering and tailoring the phase of radiation in various radiation problems and in [15]–[19] we have shown how such ENZ materials may be effectively used as electromagnetic cloaks. These various applications are based on the anomalous response of the electromagnetic wave in media with low refractive index, which implies high phase velocity and thus “static-like” features, despite a time-varying excitation. As another intriguing application based on these ENZ properties, it has been recently shown how an ultranarrow waveguide channel connecting two larger waveguide sections may become totally transparent to the impinging wave front when filled with ENZ materials [20], [21]. In such a setup, the wave may be squeezed and tunneled through an ultrananrrow ENZ transition region with negligible reflections, despite sharp abruptions and bends along the channel, as long as its total longitudinal cross sectional area, defined as the total area enclosed between the channel length and its height, remains electrically small. This counterintuitive phenomenon, coined as supercoupling, does not depend on the shape, geometry and length of the transition channel (as long as its longitudinal cross section remains small), leading to several exciting possibilities and potential applications in microwave and optics [22]–[29]. Realizing an ENZ metamaterial in an ultranarrow region may be technologically challenging, although Smith’s group at Duke University has indeed realized a metamaterial setup that proves experimentally the supercoupling phenomenon, based on resonant inclusions [22]. Our experimental approach has arguably been simpler: since the natural dispersion of waveguides may well mimic the anomalous electromagnetic properties of metamaterials and it may be used to effectively realize their properties without relying on resonant inclusions [30]–[32], we have experimentally verified the supercoupling concepts in a very simple setup: a hollow ultranarrow rectangular channel operating at cutoff, which has been shown to behave in many senses equivalently to a zero-permittivity channel [23], [26]. Indeed, in this 3-D experimental setup, without a need for any metamaterial inclusion, we have verified total transmission through an ultranarrow channel with static-like properties, making the tunneling independent of the shape, length, geometry and possible bending of the transition channel, exactly as if it were filled by an ENZ material. In our works, we have shown how the wave energy is effectively “squeezed” through the ultranarrow channels and bends with essentially no reflection, strongly increasing the value of electric field and Poynting vector inside the transition region

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ALÙ AND ENGHETA: COAXIAL-TO-WAVEGUIDE MATCHING WITH ENZ ULTRANARROW CHANNELS AND BENDS

with almost no phase variation. In other words, with a modest amount of input energy, one may achieve highly enhanced electric field in the ENZ region, almost independent of the channel shape. Such enhancement may provide several interesting opportunities for sensing [29] and nonlinear harmonic generation [28]. Applying the principle of reciprocity, we may heuristically consider exploiting this strong field enhancement in coupling an antenna to the ultranarrow ENZ transition channel, which may feed the larger waveguide sections connected to it. We may expect that by exciting this waveguiding structure at its ENZlike transition channel, one may effectively couple an antenna to this setup, due to the large field enhancement experienced inside the channel in the reciprocal problem. This may provide an interesting method for matching and coupling an antenna to a waveguide system at the frequency of interest. Due to the staticlike properties of ENZ-based structure, this anomalous coupling and matching is expected to be interestingly independent of the shape, geometry, length and possible abruptions of the transition region, as well as of the actual position of the antenna inside the channel, properties that are usually hard to achieve in a matching device. In the following, using the same analogy as in [23], we verify these possibilities in a hollow rectangular channel at cutoff, connecting two larger waveguide sections. We will formulate the theory supporting this anomalous matching phenomenon and we support these findings with full-wave numerical simulations. It is worth mentioning that techniques for matching a coaxial cable to rectangular or parallel-plate waveguides are well known [33]–[36], but they require specific matching networks, careful design of the connection, presence of parasitic loops and specific positioning of the probe. Inspired by the anomalous and exciting electromagnetic properties of these ENZ channels, our findings here may indeed provide a novel, simpler and more elegant way to connect and couple coaxial cables and antennas to waveguides of arbitrary geometry, by using ultranarrow hollow channels at their cutoff. This may provide larger and new degrees of freedom in the design.

II. GEOMETRY AND TUNNELING PROPERTIES Consider the geometry of Fig. 1, i.e., two rectangular waveguide sections of width , height , filled by a material with permittivity , and connected by a narrower channel of width , height much smaller than , permittivity and length . For the moment, we disregard the coaxial cable depicted in the figure. This geometry is consistent with the setup analyzed in our recent theoretical and experimental works devoted to the supercoupling phenomenon [23]–[29], for which anomalous tunneling through the ultranarrow channel is expected near its cutoff frequency. At this frequency, the propagation properties of the channel closely resemble those of a metamaterial with near-zero permittivity (ENZ), allowing an anomalous matching between the outer waveguide sections and the channel itself, quite different from any other Fabry-Perot tunneling resonance [24], as we discuss in the following. The distinct fast-wave and quasi-static properties of the channel at cutoff, related to its

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Fig. 1. Geometry of the problem: top view, side view and perspective view of an ultranarrow rectangular channel at cutoff connecting two much thicker waveguide sections. The channel may be fed by a coaxial cable.

near-zero effective permittivity, produce tunneling that is independent of the channel length, geometry and possible presence of bending and abruptions. The tunneling frequency is uniquely determined by the cutoff condition

(1) is the wavelength in a material with permittivity . where mode supported by such a Since the propagation of the rectangular channel may be resembled to propagation of a plane wave inside a medium with effective permittivity [6], [30]–[32]

(2) it is evident that the tunneling frequency arises near the frequency for which the effective permittivity of the channel is close to zero. This generalizes the supercoupling phenomenon, originally formulated for parallel-plate channels filled with zero permittivity metamaterials [20]–[22] to rectangular waveguides filled with conventional materials (or simply hollow), as reported theoretically and experimentally in [23]–[29]. Consider, for instance, the case in which the left waveguide mode (nesection is somehow excited with its dominant glect for now the presence of the coaxial cable in Fig. 1). Fig. 2 reports the transmission properties of the connecting channel, as evaluated from time-domain full-wave simulations [37], for two sample designs. In both cases the ultranarrow channel has , and , and . In the outer waveguide sections have height the first case (solid curves) the outer waveguide sections have as the channel and they are filled the same width , to ensure propagation at the frequency with Teflon for which the channel is at cutoff and manifests its ENZ properties. In the second scenario (dashed curves), the outer waveguides are hollow , but the width has been with the same goal of ensuring modal increased to

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properties, due to the static-like behavior of the field inside the channel [24]. In preparation for the analysis of the following sections, it is instructive to determine the nature of this tunneling phenomenon, which is very distinct from that of a conventional FabryPerot tunneling. As we have discussed in [24], the supercoupling phenomenon is indeed caused by an interesting matching phenomenon between the outer waveguide sections and the inner impedance of the outer waveguide sections channel. The is given by

(3) where is written similar to (2), but using the outer waveguide parameters. Entering the channel, the mode sees a usually mismatched load, with impedance

(4)

Fig. 2. Transmission (amplitude and phase) through the ultranarrow channel ,w ," " , in the geometry of Fig. 1 with a a : . The figure compares the transmission for the case of a Teflon " , solid line) and the case of a hollow waveguide waveguide (w w , " (w ," " , dashed line).

= 44 5 mm = = 15 cm =

= 1 mm

= 10 cm

=

=2

propagation around the ENZ frequency. In both cases, the outer waveguide sections support propagation above 1 GHz. It is evident how in both scenarios anomalous total transmission is achieved through the ultranarrow channel near the ENZ frequency . As extensively discussed in [24], the E-plane step adds in both cases a small capacitive load that slightly detunes the tunneling frequency to a lower frequency. The H-plane discontinuity, necessary in the second scenario (dashed lines), contributes with a small inductive load that tunes back the tunneling to the original ENZ frequency.1 Indeed, in the Teflon scenario and in the total transmission is achieved at hollow scenario the tunneling arises at , both very close to the ideal cutoff frequency of the channel. In both cases, the phase delay across the channel is identically zero at the tunneling frequency, ensuring that the ENZ-like properties of the channel force the wave to be reproduced uniformly across the channel with no phase propagation. This is consistent with our previous theoretical and experimental findings [23]–[29]. It is worth underlining here that the main parameter that tunes , in agreement the tunneling properties of the channel is and possibly the width ratio with (1). The height ratio (hollow case), mainly determine the tunneling bandwidth, whereas presence of steps, bending or abruptions in the channel are expected to only weakly affect the transmission 1It is evident that the small detuning due to the E-plane capacitive step may be fully taken into account in the design, and the tunneling at the frequency of interest may be achieved by fine tuning the channel width w .

. which causes strong reflections, due to the high ratio However, near the ENZ frequency , the low value of may affect and compensate this strong mismatch, producing a non-resonant total tunneling phenomenon based on a unique matching between waveguides with strongly different geometries.2 It is interesting to see that an ENZ material slab would be strongly mismatched to free-space, due to its low permittivity, but in this scenario the low permittivity allows to exactly compensate the strong geometrical mismatch at the entrance and exit faces of the connecting channel. This interpretation is fully consistent, in the limit, with the electromagnetic analysis provided in [20] in the case of identically zero permittivity. As an example, Fig. 3 reports the ideal impedance dispersion for the outer waveguide sections and for the connecting channel, as evaluated from (3), (4) for the two scenarios of Fig. 2. It is evident that the channel is highly mismatched at all frequencies, apart from one frequency region very close to its cutoff frequency, where the impedance suddenly grows with a vertical asymptote at frequency . Evidently, for both sections, and for a wide range of geometries, a matching condition arises very close to the frequency , which is responsible for the anomalous tunneling highlighted in Fig. 2. At higher frequencies, Fabry-Perot tunneling is possible due to the multiple reflections of the propagating modes at the entrance and exit faces of the channel, as it can be seen from the growing transmission curves in the right side of Fig. 2. Clearly, the frequencies associated with Fabry-Perot tunneling strongly depend on the presence of the exit channel, on the length and geometry of the connecting channel and on the possible presence of bends and abruptions, differently from 2In order to underline the drastic difference in nature between this tunneling phenomenon and a regular Fabry-Perot resonant tunneling, in [24] we have shown that the ENZ tunneling may be achieved even in a channel without exit face, proving that the ENZ properties of the channel are indeed able to completely match the drastic abruption at the entrance face. Clearly Fabry-Perot tunneling would not be achievable in such scenario.

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Fig. 3. Impedances of the outer waveguide sections for the geometries of Fig. 2 (thinner lines) compared to the impedance of the inner channel (thicker dotted line).

the ENZ scenario. Moreover, for the Fabry-Perot case the field has specific standing-wave distributions along the channel, with sharp maxima and dips. The nature of these resonances is clearly very distinct from the ENZ tunneling phenomenon, as detailed in [24]. For the purposes of the present problem, these are not of main interest. Another aspect of interest of the ENZ tunneling, crucial for the following analysis, consists in the uniform distribution and amplification of electric field along the central axis inside the channel (with conventional sinusoidal distribution in the transverse plane). Since the voltage across the abruptions at the entrance and exit faces is continuous, the electric field inside the channel at frequency is necessarily enhanced with respect to . the impinging field by a factor proportional to Fig. 4 reports the magnitude and phase of the transverse electric field, normalized to the amplitude of the electric field in the impinging mode, in the two scenarios of Fig. 2 at the ENZ tunneling frequencies. Large enhancement of the electric field in the channel (shadowed region) in both cases is noticed. In , conservation of enthe Teflon geometry, for which ergy and voltage continuity imply that the field enhancement . at the entrance and exit of the channel equals the ratio In the hollow geometry, the additional squeezing in the H plane implies an even stronger enhancement of the electric field entering the channel. In this last case, since the tunneling frequency effectively coincides with the ENZ frequency , the electric field all over the central axis of the channel has uniform amplitude, whereas in the Teflon case the slight detuning associated with the E-plane abruption produces a slight variation along the channel, associated with the fact that it is slightly below cutoff at the tunneling frequency. The phase plot confirms absence of standing waves and infinite phase velocity inside the channel, with constant phase distribution. This ensures that the electric field is uniformly enhanced inside the ENZ channel at the tunneling frequency. At this point, we are ready to apply the reciprocity principle: if feeding the outer waveguide sections may produce a uniform strongly enhanced transverse electric field inside the ultranarrow connecting channel, we may argue that an antenna, or a coaxial cable feeding such a channel, should produce an

Fig. 4. Magnitude and phase of the normalized transverse electric field along the central line of the waveguide of Fig. 1 (evaluated on the bottom plate) for the two geometries of Fig. 2, at the corresponding ENZ tunneling frequencies. The shadowed region indicates the channel.

equally good radiation towards the outer waveguide sections at frequency . For the previous considerations, this matching phenomenon is heuristically expected to be surprisingly independent of the position of the antenna along the central axis of the channel, and of the geometry, length and possible bending of the matching channel, despite the overall simplicity of the setup. We verify these assertions in the following. III. COAXIAL MATCHING IN THE ENZ CHANNEL Consider again the geometry of Fig. 1, this time including the coaxial feeding line. In this section, we analyze the possibility of matching and tuning such an antenna when placed inside the ENZ channel. As a source, we consider a coaxial cable with and outer diameter . The outer conductor inner diameter is connected to the bottom plate of the narrow channel and the inner conductor to its upper plate. A hole with diameter is also opened in the bottom plate of the waveguide to connect the coaxial cable. For now, its position is assumed to be at the geometrical center of the channel, at the origin of the coordinate system. Fig. 5 reports the reflection coefficient, the input resistance and reactance, as seen by the coaxial cable in the two geometries of Fig. 2. In the Teflon geometry, we have considered a and , decoaxial cable with signed to have a characteristic impedance of about 150 , for the reasons explained in the following. In the previous section, we have shown how the ENZ channel is matched to the outer

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waveguide at the design frequency, and therefore we may expect that the coaxial cable “sees” the same value of local input impedance, since the ENZ channel is capable of “tunneling” the local value of impedance from the outer waveguide to the location of the cable. Since the coaxial line is locally feeding the channel, the input impedance that it effectively sees is given by the ratio of the tangential electric and magnetic fields in the outer . Moreover, since in the geometry of waveguide, i.e., Fig. 1 the cable is feeding two waveguide sections in parallel, this effective value is halved for practical purposes. For the geometry at hand, the expected input impedance provides:

(5) which is very close to the obtained value of input impedance in the channel from our full-wave numerical simulations (slightly larger than 150 ), and it justifies our choice of design for the coaxial cable. The slight difference between the extracted value and (5) is related to the sharp abruptions at the channel exits, which effectively introduce additional small loads, and also contribute to a slightly detuning of the tunneling frequency from . Although these additional abruption loads may be easily taken into account in the matching and their effect is rather small (see [24] for a thorough discussion), their contribution may be drastically reduced, if desired, by adding some short additional ENZ “buffers” at the exits of the channel, as shown in [24]. Our full-wave simulations of this geometry show that with the addition of few mm hollow sections at the abruptions before the Teflon waveguide sections, with same height and width as the outer waveguide, operating effectively as ENZ buffers as described in [24], the extracted value of impedance exactly coincides with (5), and the tunneling frequency is shifted back to the exact cutoff frequency . Such additional buffers are not considered in the following of this paper, since they somehow complicate the geometry and provide only small variations to these results, not particularly relevant to the following discussion. is lower, and therefore a higher In the hollow scenario, input resistance is expected. In this case, (5), ensures that with a 300 coaxial cable excellent matching may be achieved at the tunneling frequency, and this is confirmed by our full-wave simulations. These results show that it is indeed possible to achieve good matching between a coaxial cable and a rectangular waveguide by properly “squeezing” the feeding channel in a section of arbitrary length to which the cable is connected. We notice, as an aside, that the plots of Fig. 5 do not show matching at higher frequencies, near the first Fabry-Perot resonance that was noticed in Fig. 2 at high frequencies. This is because Fabry-Perot resonant tunneling does not provide a uniform field inside the channel, and therefore due to the reciprocity the radiation and matching of an antenna becomes strongly dependent on its position along the channel. In this case, the coaxial cable is placed at a null of the first Fabry-Perot resonance, explaining the absence of matching and radiation.

Fig. 5. Reflection coefficient and input impedance as measured by the coaxial cable for the two geometries of Fig. 2.

Fig. 6 shows the variation of the matching input resistance seen by the coaxial line, at the same feeding point, in terms of the height of the outer waveguide sections, while all other geometrical parameters are kept fixed. In particular, Fig. 6 refers to the , , case of the Teflon geometry ( , , , ). It is seen, as expected, that the input impedance (solid line in the figure) follows very closely the waveguide impedance (dashed line), with a small discrepancy associated with the perturbation caused by the presence of the inner conductor inside the channel. This offers a very easy matching design for this setup. As reported in the bottom panel in Fig. 6, the matching frequency is only very slightly detuned around the “ideal” ENZ frequency by the increase in , due to the parasitic reactance caused by the E-plane abruption. Even though the matching frequency

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Fig. 6. Input resistance and matching frequency for the Teflon waveguide of Fig. 2, versus a for a coaxial line with d : and d : .

= 0 5 mm

= 6 1 mm

is not significantly changed by a variation in , its bandwidth is strongly affected, as expected, i.e., being larger for (lower field enhancement in the channel) and being lower substantially reduced by an increase in . This is consistent with the discussions in the previous section. Fig. 7 reports corresponding plots for the hollow geometry (here the plots for the fractional bandwidth are not reported for sake of clarity, but the results are consistent with the Teflon case). In this geometry, the width of the outer waveguide is also an important parameter to consider in the matching, and it is accordingly varied in the figure. The other parameters remain the , , , same as in Fig. 2: , . , It is noticed that for small H-plane steps the outer waveguide sections are almost at cutoff at the ENZ frequency , and this implies a high characteristic impedance matched to the channel, which is even increased as seen by , the the coaxial line (solid line). By increasing input impedance gets lower and closer to the outer waveguide increases further, the input impedance. However, when impedance is not sensibly affected. The input impedance lines (upper panel) still follow the trend of the waveguide impedance (middle panel), but the abruptions at the E- and H-planes now affect more sensibly the input impedance. Still, the matching is easily achieved by considering design charts like those in Fig. 7. As discussed in the reciprocal problem analyzed in the previous section, the matching frequency is even less affected in the case of abruptions both in the E- and H-planes, as compared with

Fig. 7. Same as in Fig. 6, but for the hollow waveguide. In this case, we vary also w as a parameter.

only E-plane steps (case of Fig. 6) and the matching frequency is always very close to . The fractional bandwidth decreases and , consistent with with increasing the ratios the Teflon geometry. Fig. 8 analyzes the dependence of the input resistance and . In this case and in the following, reactance on variations in we concentrate on the Teflon geometry, which provides a more direct matching design, simply associated with the impedance of the outer waveguide section. Similar considerations may be equally applied to the hollow scenario. All the other parameters . are kept fixed, as in Fig. 2 and Fig. 6, with If for thin channels and high ratios the matching mechanism is excellent, opening up the channel and sensibly implies a detuning of the tunneling properties increasing of the channel, which may substantially affect the possibility of matching. In all the examples of Fig. 8 the inner conductor

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Fig. 8. Input impedance for the Teflon waveguide of Fig. 2, with various values of a for a coaxial line with d : and d : .

Fig. 9. Input impedance of the Teflon waveguide of Fig. 2 with various values : . sizes of the coaxial cable, keeping fixed the ratio d =d

of the coaxial cable is kept connected to the upper plate of the waveguide, since it has been verified that as soon as this connection is lost, as expected, the matching deteriorates rapidly. , the fractional bandwidth inIt is seen that, by increasing creases, consistent with the previous discussion, the tunneling resonance gets slightly detuned and the input resistance at the matching frequency remains constant (equal to , which is not changing here). This is completely consistent with the reciprocal tunneling problem through the ENZ channel [23], [24]. gets too large, the input reactance grows However, when increasingly inductive and the possibility of matching is lost. This is expected, since in the limit of a regular waveguide with , the matching using a simple coaxial no ENZ channel cable connected to the upper plate is expected to be poor and is dominated by the inner conductor inductance. This is also consistent with the reciprocal problem, for which ENZ tunneling cannot be achieved as soon as the longitudinal cross sectional area of the channel gets too large [20]–[24]. Fig. 8 fully confirms these predictions and implies, counterintuitively, that by squeezing the connecting channel we can substantially improve the matching to a coaxial line connected to it. In Fig. 9 we report, again for the Teflon geometry, the dependence of the input impedance on the coaxial design. We do not sensibly affect the input have found that variations of impedance seen by the coaxial line, and therefore this parameter may be used to tune the matching of the line to the desired value may indeed slightly affect the of . However, increasing

input impedance, since the aperture opened in the channel gets larger and larger. In Fig. 9, we have considered different sizes , of the coaxial cable, keeping fixed the ratio in order to provide an input impedance of 150 . It can be seen how larger coaxial cables experience a somewhat lower input impedance and slightly higher matching frequency. These properties may provide another degree of freedom in the matching design. We have verified that the hollow waveguide are charac. terized by a similar variation of the input impedance with Figs. 10 and 11 report the variation of input resistance with the position of the feeding line inside the channel for the Teflon geometry of Fig. 5. The frequency dependence of the input reactance, which is consistent with the standard dispersion near a resonance, and is related to the frequency dispersion of the input resistance, is not reported here for sake of brevity. Fig. 10, in particular, shows the variation of input resistance when the coaxial line is moved along the channel from the entrance of the channel to its center, along its central axis. As predicted in the previous section, the input impedance is not affected at all by the position of the line, consistent with the uniform field distribution of the electric field along the central axis inside the ENZ channel. This represents an important advantage of this matching technique, in addition to its simplicity: the independence on the position of the feeding line along the channel, due to its ‘ultrafast’ properties. Even when the line is connected at the entrance of the channel, its matching properties are surprisingly unchanged. As a side feature, it is evident in Fig. 10 how by moving the line towards the edge of the channel a second

= 0 5 mm

= 6 1 mm

= 12 2

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Fig. 10. Input resistance for the Teflon waveguide of Fig. 5, varying the position of the coaxial line along the channel at the central width. Fig. 12. Side view (left) and perspective view (right) for three different variations on the original Teflon geometry of Fig. 5, obtained by arbitrarily bending the ENZ matching channel.

independent of the position of the line along the channel, properties quite appealing for several applications. However, the special features of the ENZ channel are not limited to only the matching features highlighted in the previous section, but they may provide several other exciting possibilities for different applications. We outline some examples in the following. A. Independence Upon Arbitrary Bending and Abruptions

Fig. 11. Input resistance for the Teflon waveguide of Fig. 5, varying the position of the coaxial line along the channel width at the central length.

peak of input resistance occurs at higher frequencies. This is associated with the first Fabry-Perot resonance of the channel highlighted in Fig. 2. It is clear that for this resonance the possibility of matching would indeed be affected by the feed location. In Fig. 11 we vary the position of the coaxial line along the channel width, half way along the length of the channel. Since the electric field and voltage distribution vary sinusoidally along the transverse width of the channel, following the modal pattern, the matching features are affected by a variation in the position of the line along this axis, as expected. The input resistance decreases together with the electric field in the reciprocal transmission problem. This variation may be used as a further tool in matching a given coaxial line to the channel. At any given transverse location, the location of cable along the coaxial line of the channel remains irrelevant, consistent with Fig. 10. These considerations may be equally applied to the hollow geometry in Fig. 2. IV. ANOMALOUS MATCHING PROPERTIES AND OTHER GEOMETRIES OF INTEREST The anomalous matching properties of the ultranarrow ENZ channel may offer several interesting possibilities in matching a coaxial line to a rectangular waveguide. As outlined in the previous section, the matching design may be very simple, and

Following our extensive studies outlined in [24]–[26] for the reciprocal problem, here we explore how arbitrary bending in both E- and H-planes and arbitrary abruptions in the channel cross-section do not sensibly modify the matching properties of the channel. This is a very interesting property for this matching device, which may be useful in several applications. Indeed, once the matching is designed, turning, twisting or bending the ENZ channel for redirecting the output waveguides would not substantially change the overall matching properties. As an example, Fig. 12 reports some interesting variations on the previous geometry, obtained by arbitrarily rotating and bending the original Teflon setup of Fig. 5. Fig. 13 reports the corresponding plots for their matching properties. Using the same 150 coaxial line of the previous section, which was well matched to the straight channel of Fig. 2 (reported as the solid lines in Fig. 13), we have bent the channel in three different ways, as depicted in the geometries of Fig. 12: in Fig. 12(a) the ENZ channel is bent in three pieces with a 180 degree bend in the E-plane, feeding two parallel waveguides (dashed lines in Fig. 13). In the second case (Fig. 12(b), dash-dotted lines in Fig. 13), the channel is connected to a 90 degree bend in the E-plane and in the third geometry (Fig. 12(c), dotted lines in Fig. 13) the channel is directly plugged into the same parallel waveguides as in the first case, but without any connecting section. It can be clearly seen how the same coaxial cable, connected in the same way to the channel, is capable of providing a good matching at the same ENZ frequency, with a weak change in its overall matching properties. Of course, the bandwidth of operation may vary depending on how much the field is squeezed

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Fig. 13. Reflection coefficient and input impedance for the three geometries depicted in Fig. 12.

in the channel and its total length, but this does not sensibly affect the matching design. In particular, in the third scenario in which the channel travels all over the entrance face of the waveguide sections, the bandwidth is greatly enhanced, since the wave is much less squeezed at the entrance of the channel. This is consistent with adding transition channels, as considered in [20]–[24], to the entrance of the channel, which have been shown to be effective in reducing the reactance associated with the entrance and exit abruptions. Analogous results have also been obtained in the hollow geometries (not reported here for sake of brevity). B. Single-Ended Output Channel In all the previous setups, the matching ENZ channel has always been considered in geometries connecting to two distinct waveguide sections. Although this solution may be prac-

tical for several purposes, and we have shown in the previous section how there is large flexibility in bending and arbitrarily redirecting the output waveguides, it may be desirable for many applications to direct the total impinging power into one single output waveguide. Exploiting the fast-wave properties of the ENZ channel, it may be possible to overcome this limitation in some setups of interest. At the first look, to in order to direct all the impinging power towards one rectangular waveguide, one may consider simply closing one end of the channel with a perfect reflector. However, this approach is not successful, since on the one hand, the fast wave properties of the channel provide us with the possibility of closing the channel at any arbitrary distance from the coaxial line, but on the other hand due to the same properties using a conducting plate to close the channel would imply ‘shorting’ the coaxial line, therefore causing huge mismatch and very poor coupling around the ENZ frequency. It is evident that the ideal way to close the channel on one end is to use a magnetic conductor, which would provide zero phase-shift upon reflection. In Fig. 14 we have reported (solid line) the matching properties of the same identical channel of Fig. 5 when closed on one side by an ideal magnetic conductor. The structure is fed now by a coaxial line with double value of its characteristic , which provides a good matching at the impedance ENZ frequency, as seen in the middle panel of Fig. 14. Evidently, placing a magnetic conductor at any distance from the coaxial line provides specular reflection of the fast-wave excited by the antenna with no phase delay, causing the doubling of the input impedance and producing practically the same matching properties as in the double waveguide setup. This allows redirecting the whole power into one waveguide section. Since magnetic conductors do not exist naturally (although they can be designed as high-impedance surfaces), in practice it may be inconvenient to realize the required magnetic conductor at the design frequency. However for such narrow channels an open-ended waveguide should look somehow analogous to a magnetic conductor. The dotted lines in Fig. 14 indeed refers to the case in which the same channel is now open in free space, effectively removing one waveguide section from the original setup of Fig. 5. It can be clearly seen how the matching properties are practically unperturbed from the ideal magnetic conductor case. We have verified with full-wave simulations that, due to the ultranarrow thickness of the ENZ channel, only about 7% of the impinging power is radiated out from the aperture, whereas the remaining is tunneled through the ENZ channel towards the single output waveguide section. As an alternative, again exploiting the fast-wave properties of the channel around the ENZ frequency, we have designed another setup as shown in Fig. 15. In this case, the setup analyzed in Fig. 12 for the 180 degree bend has been connected with both ends to the same waveguide section. As evident, due to the anstisymmetric orientation of the electric field in the two branches of the channel, feeding the waveguide at the same transverse section would produce total cancellation of the modal excitation, and total reflection at the coaxial line. However, if one of the two feeding points is shifted by a length equal to , with being the effective wavelength of the mode in the outer waveguide section at the ENZ frequency, the excita-

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Fig. 15. A single-ended geometry with 180 degree-bent. Its matching properties are reported in Fig. 14, dashed line.

Fig. 16. Side view (top) and perspective view (bottom) of a double ENZ channel connects two thicker waveguide sections. Each channel is fed by a coaxial cable.

the coaxial line is achieved, and even in the open-ended scenario over 93% of the impinging power is indeed channeled through the ENZ channel into the tiny coaxial cable with no reflections at the various drastic abruptions that the mode encounters along its path. In some senses, this scenario is a corollary of the supercoupling phenomenon described in Section II, as applied to the matching between a rectangular waveguide and a coaxial cable with very different cross-sections. C. Presence of Multiple Antennas Along the Channels

Fig. 14. Reflection coefficient and input impedance for three single-ended geometries.

tions would sum in phase, producing constructive interference and phase matching. In this case, the input impedance is halved as compared to the original scenario, and therefore we have re, designed the coaxial cable feed with , which provides perfect matching at the same ENZ frequency, despite the somehow intricate design, the abruptions and the bending. This setup is particularly interesting, also considering the fact that once again the position of the line along the central line of the ENZ channel is totally irrelevant for what concerns the matching. It is also interesting to realize that these setups of Fig. 14 may provide analogous matched performance in the dual scenario, in which it is desirable to couple the impinging mode traveling in a thick waveguide into a tiny coaxial cable. Indeed, we have verified that at the matching frequency total energy squeezing through the channel and perfect matching at

The fact that in different locations along the ENZ channel the coaxial line may radiate with unchanged matched properties and same phase delay suggests that an array of in-phase antennas may possibly radiate along the channel in phase. However, the addition of other coaxial lines along the channel with the same input impedance strongly affects the overall matching performance of each antenna. Indeed, each coaxial cable placed along the line is well modeled by a shunt stub with its own characteristic impedance. It is evident that by adding these shunt elements the overall matching is affected and the matching design should properly take into account their presence. Using a simple transmission-line model, the matching may be tailored to the presence of multiple stubs/antennas, but part of the radiation from each active antenna is necessarily coupled into the neighboring elements. These considerations apply equally well to the reciprocal problem in the geometry of Fig. 1. Indeed, considering the excitation from the outer rectangular waveguide sections, as we have done in Section II, but now with the presence of the coaxial cable inside the channel, it is easy to realize that the tunneling is strongly affected by the presence of the antenna, which acts as a shunt stub that sensibly affects the overall matching. The presence of the antenna generates some reflection and

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would absorb some of the impinging power, as predicted by the associated transmission-line model. As an interesting alternative, we have considered the geometry depicted in Fig. 16, formed by two identical channels, both connected to the same outer waveguide sections, in a geometry very similar to that in Fig. 1, but with two identical channels connecting the waveguides. A distinct coaxial cable is connected to each one of the channels. As we have verified with full-wave simulations, the two coaxial cables are in general strongly coupled, despite being placed in different ENZ channels. Independent of their positions inside the channel, the coupling between the two antennas increase with their input impedance. In the limit of input impedance much larger than , here at the ENZ frequency we may realize two completely isolated connections: the first connection is between the outer waveguide sections, which would be totally coupled together, not experiencing any mismatch, since the parallel stubs introduced by the coaxial lines have a high impedance load. The second is the link established between the two coaxial lines, which would be ideally coupled in the limit of very high characteristic impedance, exchanging the whole impinging power between the two channels from one antenna to the other, with no leakage into the two outer waveguide sections. The waveguide sections would serve only as a link between the two ENZ channels and the phase delay between the channels remains close to zero. This configuration, with its very counterintuitive behavior, is another example of the anomalous electromagnetic properties of ENZ materials and channels at cutoff. In this Section, we have concentrated on Teflon waveguide geometries, but similar concepts may be applied to the hollow scenarios with analogous results. V. CONCLUSION In this paper, using full-wave numerical simulations we have applied the counterintuitive electromagnetic properties of ENZ materials to a specific design for matching and coupling coaxial cables to rectangular waveguides with very different cross sections. We have realized the metamaterial properties by simply designing ultranarrow hollow channels operating near their cutoff frequencies, introducing a novel matching technique inspired by metamaterials, but realizable using simple techniques. We believe that these findings may have interesting potential applications in matching coaxial cables to rectangular waveguides with novel properties, such as the independence of the matching design on the position of the line in the connecting channel, large degrees of freedom in its shape, form and geometry, and ultrafast phase propagation. These findings may provide an effective example of how the findings in metamaterial theory and techniques may indeed inspire and provide novel designs for practical utility in applied technology. REFERENCES [1] , G. V. Eleftheriades and K. G. Balmain, Eds., Negative Refraction Metamaterials: Fundamental Properties and Applications. Hoboken-Piscataway, NJ: IEEE Press-Wiley, 2005.

[2] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Hoboken-Piscataway: Wiley-IEEE Press, 2005. [3] , N. Engheta and R. W. Ziolkowski, Eds., Electromagnetic Metamaterials: Physics and Engineering Explorations. New York: Wiley, 2006. [4] R. Marqués, F. Martín, and M. Sorolla, Metamaterials With Negative Parameters: Theory, Design and Microwave Applications, ser. Wiley Series in Microwave and Optical Engineering. New York: Wiley, 2008. [5] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, no. 18, pp. 3966–3969, 2000. [6] W. Rotman, “Plasma simulation by artificial dielectrics and parallelplate media,” IRE Trans. Antennas Propag., vol. 10, no. 1, pp. 82–84, 1962. [7] I. J. Bahl and K. C. Gupta, “A leaky-wave antenna using an artificial dielectric medium,” IEEE Trans. Antennas Propag., vol. 22, pp. 119–122, 1974. [8] S. A. Kyriandou, R. E. Diaz, and N. G. Alexopoulos, “Radiation properties of microstrip elements in a dispersive substrate of permittivity less than unity,” in IEEE Antennas and Propag. Int. Symp. Digest, Atlanta, GA, Jun. 21–26, 1998, vol. 2, pp. 660–663. [9] S. Enoch, G. Tayeb, P. Sabornoux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett., vol. 89, no. 21, p. 213902, Nov. 18, 2002. [10] J. Pacheco, T. Gregorczyk, B. I. Wu, and J. A. Kong, “A wideband directive antenna using metamaterials,” in Proc. PIERS, Honolulu, HI, Oct. 13–16, 2003, abstract p. 479. [11] A. Alù, F. Bilotti, N. Engheta, and L. Vegni, “Metamaterial covers over a small aperture,” IEEE Trans. Antennas Propag., vol. AP-54, no. 6, pp. 1632–1643, Jun. 2006. [12] A. Alù, F. Bilotti, N. Engheta, and L. Vegni, “Sub-wavelength planar leaky-wave components with metamaterial bilayers,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pt. 2, pp. 882–891, Mar. 2007. [13] A. Alù, F. Bilotti, N. Engheta, and L. Vegni, “Theory and simulations of a conformal omnidirectional sub-wavelength metamaterial leakywave antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pt. 2, pp. 1698–1708, June 2007. [14] A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilonnear-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B, vol. 75, p. 155410, Apr. 11, 2007, (13 pages). [15] A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phy. Rev. E, vol. 72, p. 016623, Jul. 26, 2005. [16] A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: Mechanism, robustness, and physical insights,” Opt. Express, vol. 15, no. 6, pp. 3318–3332, Mar. 19, 2007. [17] A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express, vol. 15, no. 12, pp. 7578–7590, Jun. 5, 2007. [18] A. Alù and N. Engheta, “Multifrequency optical cloaking with layered plasmonic shells,” Phys. Rev. Lett., vol. 100, p. 113901, Mar. 18, 2008. [19] A. Alù and N. Engheta, “Plasmonic and metamaterial cloaking: Physical mechanisms and potentials,” J. Opt. A: Pure Appl. Opt., vol. 10, no. 9, p. 093002, Aug. 19, 2008, (17 pages). [20] M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett., vol. 97, p. 157403, 2006. [21] M. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using epsilon near-zero metamaterials,” Phys. Rev. B, vol. 76, p. 245109, 2007. [22] R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett., vol. 100, p. 023903, 2008. [23] B. Edwards, A. Alù, M. E. Young, M. G. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett., vol. 100, p. 033903, Jan. 25, 2008. [24] A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of "-near-zero (ENZ)-filled narrow channels,” Phy. Rev. E, vol. 78, p. 016604, Jul. 23, 2008. [25] A. Alù and N. Engheta, “Light squeezing through arbitrarily-shaped plasmonic channels and sharp bends,” Phy. Rev. B, vol. 78, p. 035440, Jul. 24, 2008.

ALÙ AND ENGHETA: COAXIAL-TO-WAVEGUIDE MATCHING WITH ENZ ULTRANARROW CHANNELS AND BENDS

[26] B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys., vol. 105, no. 4, p. 044905, Feb. 18, 2009. [27] A. Alù and N. Engheta, “Boosting molecular fluorescence with a plasmonic nanolauncher,” Phys. Rev. Lett., 2009, in press. [28] D. A. Powell, A. Alù, B. Edwards, A. Vakil, Y. S. Kivshar, and N. Engheta, “Nonlinear control of tunneling through an "-near-zero channel,” Phys. Rev. B, 2009, to be published. [29] A. Alù and N. Engheta, “Dielectric sensing in "-near-zero narrow waveguide channels,” Phy. Rev. B, vol. 78, p. 045102, Jul. 3, 2008, (5 pages). [30] R. Marques, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ringresonator-loaded metallic waveguides,” Phys. Rev. Lett., vol. 89, p. 183901, 2002. [31] J. D. Baena, L. Jelinek, R. Marqués, and F. Medina, “Near-perfect tunneling and amplification of evanescent electromagnetic waves in a waveguide filled by a metamaterial: Theory and experiments,” Phys. Rev. B, vol. 72, p. 075116, 2005. [32] S. Hrabar, J. Bartolic, and Z. Sipus, “Waveguide miniaturization using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag., vol. 53, pp. 110–119, 2005. [33] M. D. Deshpande, B. N. Das, and G. S. Sanyal, “Analysis of an end launcher for an X-band rectangular waveguide,” IEEE Trans. Microw. Thheory Tech., vol. 27, pp. 731–735, Aug. 1979. [34] S. M. Saad, “A more accurate analysis and design of coaxial-to-rectangular waveguide end launcher,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 2, pp. 129–134, Feb. 1990. [35] B. Z. Wang, “Full-wave analysis of coaxial to waveguide adapters by the FDTD method,” Int. J. Infrared Millimeter Waves, vol. 19, no. 8, pp. 1121–1130, 1998. [36] G. H. C. Kwan and N. K. Das, “Excitation of a parallel-plate dielectric waveguide using a coaxial probe—Basic characteristics and experiments,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 6, pp. 1609–1620, June 2002. [37] CST Microwave Studio 2008 [Online]. Available: www.cst.com

Andrea Alù (S’03–M’07) received the Laurea (M.S.) degree in electronic engineering, the M.S. degree in environmental engineering, and the Ph.D. degree in biomedical electronics, electromagnetics and telecommunications from the University of Roma Tre, Rome, Italy, in 2001, 2003, and 2007, respectively. From 2002 to 2008, he worked periodically at the University of Pennsylvania, Philadelphia, where he developed significant parts of his Ph.D. research. Currently, he is an Assistant Professor at the University of Texas at Austin. He is the coauthor of over 90 scientific papers in peer-reviewed journals and book chapters. His current research interests span over a broad range of areas, including metamaterials and plasmonics, electromangetics, optics and photonics, cloaking and transparency, nanocircuits and nanostructures modeling, miniaturized antennas and nanoantennas, RF antennas and circuits. Dr. Alù has been the recipient of several international awards and recognitions for his research studies, which include the L. B. Felsen Award for Excellence in Electrodynamics, the SUMMA Graduate Fellowship in Advanced Electromagnetics, three URSI Young Scientist Awards, and the Raj Mittra Travel Grant Young Researcher Award.

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Nader Engheta (S’80–M’82–SM’89–F’96) received the B.S. degree in electrical engineering from the University of Tehran, Tehran, Iran and the M.S. and Ph.D. degrees in electrical engineering both from the California Institute of Technology (Caltech), Pasadena. After spending one year as a Postdoctoral Research Fellow at Caltech and four years as a Senior Research Scientist at Kaman Sciences Corporation’s Dikewood Division in Santa Monica, CA, he joined the faculty of the University of Pennsylvania, Philadelphia, where he is currently the H. Nedwill Ramsey Professor of Electrical and Systems Engineering. He is also a member of the Mahoney Institute of Neurological Sciences, and holds an appointment in the Bioengineering Department at the University of Pennsylvania. He was the Graduate Group Chair of Electrical Engineering from July 1993 to June 1997. His current research interests and activities span over a broad range of areas including metamaterials and plasmonics, nanooptics and nanophotonics, nanocircuits and nanostructures modeling, bio-inspired/biomimetic polarization imaging and reverse engineering of polarization vision, miniaturized antennas and nanoantennas, hyperspectral sensing, biologically-based visualization and physics of sensing and display of polarization imagery, through-wall microwave imaging, fractional operators and fractional paradigm in electrodynamics. Prof. Engheta is a member of the American Association for the Advancement of Science (AAAS), Sigma Xi, Commissions B, D, and K of the U.S. National Committee (USNC) of the International Union of Radio Science (URSI), and a member of the Electromagnetics Academy. He is a Guggenheim Fellow, an IEEE Third Millennium Medalist, IEEE Fellow, American Physical Society Fellow, an Optical Society of America Fellow, and an American Association for Advancement of Science Fellow. He was elected as one of the Scientific American Magazine 50 Leaders in Science and Technology in 2006 for developing the concept of optical lumped nanocircuits. He was the recipient of the 2008 George H. Heilmeier Award for Excellence in Research from UPenn, the Fulbright Naples Chair Award, NSF Presidential Young Investigator award, the UPS Foundation Distinguished Educator term Chair, and several teaching awards including the Christian F. and Mary R. Lindback Foundation Award, the W. M. Keck Foundation’s 1995 Engineering Teaching Excellence Award, and two times recipient of S. Reid Warren, Jr. Award. He was an Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS (2002–2007), the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (1996–2001), and Radio Science (1991–1996). He was on the Editorial Board of the Journal of Electromagnetic Waves and Applications. He served as an IEEE Antennas and Propagation Society Distinguished Lecturer for the period 1997–99. He is the Chair of the Commission B of USNC-URSI for 2009–2011. He was the Chair (1989–91) and Vice-Chair (1988–89) of the joint chapter of the IEEE ANTENNAS AND PROPAGATION/MICROWAVE THEORY AND TECHNIQUES in the Philadelphia Section. He served as a member of the Administrative Committee (AdCom) of the IEEE Society of Antennas and Propagation from January 2003 till December 2005. He is on the Editorial board of the journal Metamaterials, and on the board of Journal Waves in Random and Complex Media. He is a Co-Guest Editor of the Metamaterials Special Issue of the IEEE Journal of Special Topics on Quantum Electronics. He has Guest Edited/Co-Edited several special issues, namely, the special issue of the Journal of Electromagnetic Waves and Applications on the topic of “Wave Interaction with Chiral and Complex Media” in 1992, the Journal of the Franklin Institute on the topic of “Antennas and Microwaves” (from the 13th Annual Benjamin Franklin Symposium) in 1995, Wave Motion on the topic of “Electrodynamics in Complex Environments” in 2001, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION on the topic of “Metamaterials” in 2003, and Solid State Communications on the topic of “Negative Refraction and Metamaterials for Optical Science and Engineering” in 2008. He co-edited the book “Metamaterials: Physics and Engineering Explorations” (Wiley-IEEE Press, 2006). He has been elected to be the Vice-Chair of the Gordon Research Conference on Plasmonics in 2010 and its chair in 2012.

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Periodic Leaky-Wave Antenna for Millimeter Wave Applications Based on Substrate Integrated Waveguide Feng Xu, Senior Member, IEEE, Ke Wu, Fellow, IEEE, and Xiupu Zhang, Senior Member, IEEE

Abstract—Substrate integrated waveguides (SIW) are built up of periodically arranged metallic via-holes or via-slots. The leakage loss of SIW structures increases with the distance between the viaholes or via-slots. An open periodic waveguide with a large via distance supports the propagation of leaky-wave modes and can thus be used for the design of a leaky-wave antenna. In this paper, this leakage loss is studied in detail and used to design a periodic leaky-wave antenna. The proposed concept represents an excellent choice for applications in the millimeter-wave band. Due to its versatility, the finite difference frequency domain method for periodic guided-wave or leaky-wave structures is used to analyze the characteristics of the proposed periodic leaky-wave antenna. Two modes ( 10 and 20 ) are investigated and their different leaky-wave properties are analyzed. Based on the proposed leaky-mode analysis method, a novel periodic leaky-wave antenna at 28–34 GHz is designed and fabricated.

TE

TE

Fig. 1. Periodic leaky-wave antenna of SIW version.

Index Terms—Finite difference frequency domain, periodic leaky-wave antenna, substrate integrated waveguide.

I. INTRODUCTION

UBSTRATE integrated waveguide (SIW) technology has been studied very extensively in recent years and has by now become a widely applied technique in planar microwave circuit design [1]–[4]. These waveguide-like structures are fabricated in planar form and are built up by periodically arranged metallic via-holes or via-slots [4] and take advantage of the well-known characteristics of conventional rectangular waveguides, namely, its high Q-factor and high power capacity, as well as its low losses. Though they have been studied for the use in antenna applications [5], [6], SIW structures have only been considered in the form of standard rectangular waveguides or cavities for slot antennas. In this paper, a novel concept is proposed that takes advantage of the increasing leakage loss for large via distances, which favors the forming of leaky-wave

S

Manuscript received November 20, 2008; revised April 07, 2009. First published June 30, 2009; current version published February 03, 2010. This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. F. Xu and K. Wu are with the Poly-Grames Research Center, Département de Génie Électrique, École Polytechnique de Montréal, Montréal, QC H3C 3A7, Canada (e-mail: [email protected]; [email protected] ). X. Zhang is with the Department of Electrical and Computer Engineering, Concordia University, Montréal, QC H3G 1M8, Canada (e-mail:xzhang@ece. concordia.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.2009.2026593

modes in the structure. As a result, a periodic leaky-wave antenna can be designed based on SIW technology. The operation principle of leaky-wave antennas has been well known for many years [7], [8]. There are two different kinds of leaky-wave antennas; one type is related to uniform guidedwave structures [9]–[11], while the second consists of an array of periodic guided-wave structures [12]–[15]. The first kind of leaky-wave antennas provides radiation into the forward quadrant and can yield scanning from broadside to forward end fire directions. The scanning range for periodic leaky-wave antennas reaches from backward end fire through broadside directions into a part of the forward quadrant. The dominant mode on the former type represents a fast wave, while the latter type is a slow wave structure. As a result, the dominant mode on periodic leaky-wave antennas does not radiate and radiation is achieved by using one of its space harmonics. A general SIWbased leaky-wave antenna architecture is shown in Fig. 1. Obviously, it belongs to the group of periodic leaky-wave antennas. The analysis and design of leaky-wave antennas mainly consists of the extraction of two parameters, namely the phase constant and the leakage constant . A number of numerical methods such as the transverse resonance method and the spectral-domain method have been used to extract the complex propagation constant. In this paper, the finite difference frequency domain (FDFD) method is selected due to its high versatility for analyzing the characteristics of the SIW leaky-wave antenna [16]–[18]. In recent years, this method has been successfully applied to extract the propagation characteristics of various types of SIW structures and is therefore directly applicable to the analysis of SIW leaky-wave antennas. For the analysis of this spe-

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cific type of structure, the domain decomposition technique has to be used because the simulation of SIW leaky-wave antenna is carried out both in the substrate region and the air region. The and in the SIW will be investigated in two modes order to get their leakage mode characteristics and to obtain the appropriate space harmonics. II. FDFD ALGORITHM When the FDFD method is used to resolve the characteristics of guided-wave structures, a nonsymmetrical standard eigenvalue problem is obtained if the equivalent resonant cavity model is used [17]. As an alternative, it is possible to use a standard FDFD method [16] by eliminating the longitudinal field components. In order to obtain more accurate results, in our case, a nonsymmetrical generalized eigenvalue problem is obtained [18],

Fig. 2. Transversal view of the SIW simulation domain.

radiate. The radiation of periodic leaky-wave antennas is originating from space harmonics introduced by periodicity,

(1)

(4)

where is the propagation constant, which may be a complex number depending on the type of problem. The vector is equal to for a wave propagation along the -direction. We introduce a shift-and-invert (SI) Arnoldi technique [19]–[21] in order to obtain a direct solution of the generalized eigenvalue problem in (1). The SI Arnoldi method can be used to compute a small number of eigenvalues close to a given shift and/or the associated eigenvectors of a large matrix pair. In the application of periodic guided-wave structures, we are interested in computing only some eigenvalues in the complex plane, which represent the complex propagation constants. In (1), for eigenvalues close to a shift , we obtain

where is the period, and , the fundamental space harmonic, represents the propagation constant of the dominant mode takes different for a periodic guided-wave structure. When values, the related space harmonic can be forward or backward in nature. The necessary condition for the radiation of a space harmonic is

(2) Thus

(3) where

(5) In a practical antenna design, usually only a single main radiated beam is needed and thus is selected. As a result, when space harmonic starts to rathe frequency increases, the diate from a backward end fire direction. With a further increase in frequency, the beam moves from the backward end fire direction into the back quadrant. For even higher frequencies, the beam moves toward broadside radiation, then traverses broadside direction, and moves into the forward quadrant. However, the range in the forward quadrant is usually limited by the emerbeam at the backward end fire direction or by gence of the a higher guided-wave mode [8]. The beam direction of periodic leaky-wave antennas can be calculated from

(6) As shown in Fig. 2, the entire simulation domain includes the substrate sub-domain and the air sub-domain in the case of leaky-wave antennas. As a result, the concept of domain decomposition is introduced in the simulation. III. PARAMETRIC ANALYSIS It is well known that the wave propagation for the dominant ) relative mode in periodic leaky-wave antennas is slow ( to free-space velocity, and the dominant mode itself does not

where is the maximum beam angle measured in broadside direction and

(7) The next section includes a detailed discussion of the radiation characteristics for the two leakage modes and in SIW structures.

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A. Leaky-Wave Mode Related to the

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Mode

For the analysis of this specific case, a magnetic wall is introduced in the center of the SIW as shown in Fig. 2. The propagation constant of the dominant mode can be approximated as

(8) where is the equivalent width of the waveguide and is larger than the inner width (as shown in Fig. 1) of an SIW [4], [22]. From (5) and (8) we obtain the condition, for which the dominant mode does not radiate as

Fig. 3. Comparison of the angles of maximum radiation for the SIW leakywave antenna for the TE mode, s = 4:8 mm and d = 0:8 mm.

(9) Obviously, if the condition is fulfilled at a certain low frequency, the dominate mode does not radiate at higher frequencies. The permittivity of the substrate is selected according to the low fre. quency of leaky-wave antennas to leaky-wave mode is obtained The phase constant for the from (7) and (8) as

(10) From (10), we can conclude that the period is a key factor in the design of leaky-wave antennas as it directly defines the antenna’s frequency range with a smaller period resulting in a higher operating frequency. Although the design of periodic leaky-wave antennas also depends on the leakage constant, we can use (9) and (10) to determine its frequency range and select an approximate value for the substrate permittivity. In a next step, the FDFD method will be used to extract the leakage constant and the phase constant of SIW leaky-wave antennas accurately. As shown in Fig. 1, the dimensions are se, rectangular metal slot lected to an SIW width dimensions and , and a distance . This yields a period length between the slots . An RT/duroid 5880 substrate with a , and a loss thickness of 0.787 mm, a permittivity tangent is selected in the simulation. The value of the phase constant is very important in determining the main beam direction, as well as for identifying single or multi beam operation. For the presented analysis we assume an infinitely extended substrate ( is infinite) in order to highly simplify the simulation model. The phase constants obtained from these approximate equations are very similar to the results obtained for the case of a finite substrate. However, the main beam direction needs to be multiplied with a factor according to the Snell’s law. Please note that if is infinite, the dominant mode is a fast wave. In this analysis, we only focus on the calculation of the first space harmonics. Figs. 3 and 4 show the angle of maximum radiation calculated from (6) and the normalized leakage constant. In Fig. 3, a

Fig. 4. Normalized leakage constant for the SIW leaky-wave antenna for the TE mode, s = 4:8 mm and d = 0:8 mm.

comparison of the results of the FDFD method and the approximate relation in (10) ( ) illustrates that the latter is a suitable approximation for evaluating the phase constant of SIW leaky-wave antennas. From Fig. 4, we observe a stop band that emerges for beam angles close to zero. and For modified dimensions (note that the overall periodical distance keeps unchanged), the angle of maximum radiation and the normalized leakage constant calculated with the FDFD method are shown in Figs. 5 and 6, respectively. As the period keeps unchanged, the phase constants are expected to remain almost constant, which is proved by comparison of the results in Figs. 3 and 5. On the contrary, the leakage constants in Figs. 4 and 6 are different. However, the influence of the stop band is very small in both cases. B. Leaky-Mode Related to the

Mode

For the analysis of the mode, an electric wall is introduced in the center of the SIW as shown in Fig. 2. The propagation constant of the dominant mode can be calculated approximately as

(11)

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Fig. 5. Angles of maximum radiation of the SIW leaky-wave antenna for TE mode, s = 4:0 mm and d = 1:6 mm.

Fig. 7. Comparison of the angles of maximum radiation of the SIW leaky-wave antenna for the TE mode, s = 4:8 mm and d = 0:8 mm.

Fig. 6. Normalized leakage constant of the SIW leaky-wave antenna for TE mode, s = 4:0 mm and d = 1:6 mm.

Fig. 8. Normalized leakage constant of the SIW leaky-wave antenna for the TE mode, s = 4:8 mm and d = 0:8 mm.

From (5) and (11), we obtain the condition, for which the dominant mode does not radiate as

(12) By comparing (9) and (12), we observe that the lowest frequency mode, for which the dominant mode does not raof the diate, is higher than that of the mode. Therefore, it is better to use a substrate with higher permittivity. The phase constant of the leaky-wave mode can be calculated from (7) and (11) as

(13) According to (13), the frequency for obtaining a certain beam angle in mode is higher than in mode . mode The FDFD method can also be used in the case of for extracting the leakage constant and phase constant of SIW leaky-wave antennas. The dimensions are selected to an SIW , rectangular metal slot dimensions width and , and a slot distance , which yields a period . Again we assume ) for the simulation. an infinitely extended substrate (

An RT/duroid 5880 substrate with a thickness of 0.787 mm is selected. Figs. 7 and 8 show the angle of maximum radiation and the normalized leakage constant, respectively. In Fig. 7, a compar) illustrate ison of the FDFD method and (13) ( once more that the approximation in (12) is suitable to evaluate the phase constant of SIW leaky-wave antennas. Furthermore, from Fig. 8, we also observe that the influence of the stop band is small. and ( For different dimensions keeps unchanged), the angle of maximum radiation and the normalized leakage constant obtained from the FDFD method are shown in Figs. 9 and 10, respectively. Again, for an unchanged period , the phase constants remain almost constant, which is proved by comparison between the results in Figs. 7 and 9. The leakage constants shown in Figs. 8 and 10 are again different. A comparison of Figs. 4 and 8, as well as Figs. 6 and 10, leads to the conclusion, that the radiation characteristics of the mode are better than for the mode. C. Finite Substrate In practical applications is finite, as shown in Fig. 1. This section describes an analysis of the impact of different values for on the characteristics of SIW leaky-wave antennas. Based on the analysis results in the previous section, a leakymode will be investigated. The wave antenna operating in dimensions are the same as for the case in Fig. 7, i.e., an SIW

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Fig. 9. Angles of maximum radiation of the SIW leaky-wave antenna for the TE mode, s = 4:0 mm and d = 1:6 mm.

Fig. 11. Normalized leakage constant of the SIW leaky-wave antenna for the TE mode, s = 4:8 mm, d = 0:8 mm.

Fig. 10. Normalized leakage constant of the SIW leaky-wave antenna for the TE mode, s = 4:0 mm and d = 1:6 mm.

Fig. 12. Angles of maximum radiation of the SIW leaky-wave antenna in free space radiation for the TE mode, s = 4:8 mm, d = 0:8 mm and c = 2:2 mm.

width

, rectangular metal slot dimensions and , a slot distance ,a , a loss tangent , and a permittivity . Three different values for substrate thickness are investigated, i.e., , 2.2 mm, and 2.3 mm. As shown in Fig. 2, the entire FDFD simulation domain includes the substrate sub-domain and the air sub-domain and the concept of domain decomposition is introduced into the simulation. Fig. 11 shows the normalized leakage constants for the three selected values obtained from the FDFD method. From Fig. 11, we observe that the antenna has the best leakage behavior for . The angle of maximum radiation for is shown in Fig. 12. Again, (13) is used for an approximate evaluation of the angle. Note, that for this case the wave number for free space radiation in (6) needs to be used. In addition, according to Snell’s law, the angle of maximum radiation needs to be mulat the interface between the subtiplied with a factor strate and free space. From Fig. 12, it is observed that multiple reflections and refractions occurring at the interface have only a small influence on the phase constant. However, their impact on the leakage constant is relative large. IV. MEASUREMENT mode leaky-wave Based on the analysis in Section III, a antenna is designed and fabricated. Although the leakage char-

acteristics in Fig. 11 are not as good as for the result presented in Fig. 8, an SIW leaky-wave antenna operating in the range of 28–34 GHz is designed in order to verify the conception and the design rules for SIW leaky-wave antenna proposed here. The dimensions are selected according to the values given in Fig. 11, and is selected to 2.2 mm. A. Feed The excitation of the mode is realized by combining modes with opposite phases. As shown in Fig. 13, a two microstrip line equally divides the input signal in two parts and mode in two narrow SIW branches. The travexcites the elling wave is a quasi-TEM mode on the microstrip line and a mode in the SIW. In order to realize the opposite phase modes, different lengths of micombination of the two crostrip and SIW are selected according to

(14) where as indicated in Fig. 13, and is the effective permittivity of the microstrip. Finally, the two SIW branches are combined into a wide SIW branch representing the leaky-wave mode. antenna in order to excite its required

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Fig. 13. Topology of TE

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mode feed. Fig. 16. Photograph of the manufactured the SIW TE

leaky-wave antenna.

Fig. 14. Field distribution of feed extracted from Ansoft HFSS.

Fig. 17. Measured and simulated return loss of the leaky-wave antenna.

Fig. 15.

S parameters of feed calculated with HFSS.

The dimensions of the feed are selected to , , , , , , , and . The width of the two narrow SIW branches is 4.8 mm. The simulation of the proposed structure is carried out with the help of Ansoft HFSS, offering a significant reduction in simulation time. Fig. 14 shows the calculated field distribution and mode in the wide confirms the development of the desired SIW branch at a central frequency of 31 GHz. In addition, a detailed -parameter simulation for the feed design in Figs. 13 and 14 is performed. Two ports, one for a quasi-TEM mode at the and modes input of the microstrip line, and one for at the end of the wide SIW branch are introduced. Fig. 15 shows the -parameter results obtained from HFSS, which prove a good performance in terms of input matching of the feed as well mode. as suppression of the B. Measurement of the SIW Leaky-Wave Antenna The dimensions of the presented leaky-wave antenna have been outlined in Section IV. A Rogers RT/Duroid 5880 wit and is used for fabrication. According to the simulated leakage constant calculated from the FDFD method, . The manufactured the antenna length is selected to prototype of the SIW periodic leaky-wave antenna is shown in Fig. 16 and the simulated and measured results for the return

Fig. 18. Measured and simulated H plane pattern at 31 GHz.

loss are shown in Fig. 17. The return loss is lower than over the entire frequency range of 28–34 GHz. The simulated and measured far-field patterns at the center frequency of 31 GHz in the -plane (the plane of the substrate) are shown in Fig. 18. As expected from the results obtained in . FurtherFig. 12, the antenna radiates at an angle of about more, the measured maximum cross polarization in the plane is ). The gain in simulation and reasonably low (less than measurement are 9.514 dB and 9.3 dB at 31 GHz, respectively. The -plane (the plane vertical to the propagation direction) pattern is shown in Fig. 19. A good agreement for the radiation patterns in Figs. 18 and 19 verifies the design procedure and analysis of the presented SIW leaky-wave antenna.

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Fig. 19. Measured and simulated E plane pattern at 31 GHz.

V. CONCLUSION The simulation and measurement results have validated the proposed design and analysis method of an SIW-based periodic leaky-wave antenna. The design of such an antenna is based on the leakage loss of SIW structures, which increases with distance between the via-holes or via-slots. An open periodic waveguide with a large via distance supports the propagation of leaky-wave modes and can therefore be used for the design of a leaky-wave antenna. The developed SIW-based structure is very suitable for applications in the millimeter-wave band. Important design characteristics have been identified, such as mode over the the better radiation properties of the mode. Moreover, the substrate permittivity and the periodical distance are key parameters for the performances such as the directivity, operating frequency and efficiency. Future work includes a more detailed investigation for solving the problem related to the observed stop band.

ACKNOWLEDGMENT The authors would like to thank Miss S. Winkler of the Poly-Grames Research Center, Département de Génie Électrique, École Polytechnique de Montréal, for her great assistance in the revision of the manuscript.

REFERENCES [1] A. Zeid and H. Baudrand, “Electromagnetic scattering by metallic holes and its applications in microwave circuit design,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 4, pp. 1198–1206, Apr. 2002. [2] D. Deslandes and K. Wu, “Single-substrate integration technique of planar circuits and waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 593–596, Feb. 2003. [3] C.-H. Tseng and T.-H. Chu, “Measurement of frequency-dependent equivalent width of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1431–1437, Apr. 2006.

[4] F. Xu, X. Jiang, and K. Wu, “Efficient and accurate design of substrate integrated waveguide circuits synthesized with metallic via-slot arrays,” IET Microw. Antennas Propag., vol. 2, no. 2, pp. 188–193, Mar. 2008. [5] L. Yan, W. Hong, G. Hua, J. Chen, K. Wu, and T. J. Cui, “Simulation and experiment on SIW slot array antennas,” IEEE Microwave and Wireless Compon. Lett., vol. 14, no. 9, pp. 446–448, Sep. 2004. [6] H. –C. Lu and T. –H. Chu, “Equivalent circuit of radiating longitudinal slots in substrate integrated waveguide,” in IEEE AP-S Int. Symp. Dig., 2004, pp. 2341–2344. [7] R. E. Collins and F. J. Zucker, Antenna Theory. New York: McGrawHill, 1969, ch. 19–20, pt. Part 2. [8] A. A. Oliner and R. C. Johnson, Leaky-Wave Antennas, Antenna Engineering Handbook, 3rd ed. New York: McGraw-Hill, 1993, ch. 10. [9] L. Goldstone and A. A. Oliner, “Leaky-wave antennas I: Rectangular waveguides,” IRE Trans. Antennas Propag., vol. 7, no. 4, pp. 307–319, Oct. 1959. [10] W. Menzel, “A new traveling-wave antenna in microstrip,” Arch. Elektron. Uebertragungstech, vol. 33, pp. 137–140, Apr. 1979. [11] A. A. Oliner, “Leakage from higher modes on microstrip line with application to antennas,” Radio Sci., vol. 22, no. 6, pp. 907–912, Nov. 1987. [12] M. Guglielmi and G. Boccalone, “A novel theory for dielectric-inset waveguide leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 39, no. 4, pp. 497–504, Apr. 1991. [13] J. A. Encinar, “Mode-matching and point-matching techniques applied to the analysis of metal-strip-loaded dielectric antennas,” IEEE Trans. Antennas Propag., vol. 38, no. 9, pp. 1405–1412, Sep. 1990. [14] A. Grbic and G. V. Eleftheriades, “Leaky CPW-based slot antenna arrays for millimeter-wave applications,” IEEE Trans. Antennas Propag., vol. 50, no. 11, pp. 1494–1504, Nov. 2002. [15] J. L. Gómez, F. D. Quesada, and A. A. Melcón, “Analysis and design of periodic leaky-wave antennas for millimeter waveband in hybid waveguide-planar technology,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2834–2842, Sep. 2005. [16] F. Xu, Y. Zhang, W. Hong, K. Wu, and T. J. Cui, “Finite-difference frequency-domain algorithm for modeling guided-wave properties of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 51, pp. 2221–2227, Nov. 2003. [17] F. Xu, K. Wu, and W. Hong, “Equivalent resonant cavity model of periodic guided-wave structures and its application in finite difference frequency domain algorithm,” IEEE Trans. Microw. Theory Tech., vol. 55, pp. 697–702, Apr. 2007. [18] F. Xu, L. Li, K. Wu, S. Delprat, and M. Chaker, “Parameter extraction of interdigital slow-wave coplanar waveguide circuits using finite difference frequency domain algorithm,” Int. J. RF Microw. Comput.Aided Eng., vol. 18, no. 3, pp. 250–259, Mar. 2008. [19] Y. Saad, Numerical Methods for Large Eigenvalue Problems. Manchester, U.K.: Manchester Univ. Press, 1992, Algorithms and Architectures for Advanced Scientific Computing. [20] Z. Jia and Y. Zhang, “A refined shift-and-invert Arnoldi algorithm for large unsymmetric generalized eigenproblems,” Comput. Mathematics Applicat., vol. 44, pp. 1117–1127, Oct.-Nov. 2002. [21] M. N. Kooper, H. A. ven der Vorst, S. Poedts, and J. P. Goedbloed, “Application of the implicitly updated Arnoldi method with a complex shift-and-invert strategy in MHD,” J. Computational Phys., vol. 118, pp. 320–32, May 1995. [22] F. Xu and K. Wu, “Guided-wave and leakage characteristics of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 66–73, Jan. 2005. Feng Xu (M’05–SM’09) was born in Jiangsu, China. He received the B.S. degree in radio engineering from Southeast University, Nanjing, China, in 1985, the M.S. degree in microwave and millimeter-wave theory and technology from the Nanjing Research Institute of Electronics and Technology, Nanjing, in 1998, and the Ph.D. degree in radio engineering from Southeast University, Nanjing, in 2002. From 1985 to 1996, he was with the Nanjing Research Institute of Electronics and Technology, where he conducted research in the areas of antenna and RF circuits design. Since 2002, he has been with the Poly-Grames Research Center, École Polytechnique, Montréal, QC, Canada, where he has been a Postdoctoral Researcher and is currently a Research Associate. His current research interests include numerical methods for electromagnetic field problem and advanced microwave and millimeter-wave circuits and components.

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Ke Wu (M’87–SM’92–F’01) received the B.Sc. degree (with distinction) in radio engineering from Nanjing Institute of Technology (now Southeast University), Nanjing, China, in 1982 and the D.E.A. and Ph.D. degrees in optics, optoelectronics, and microwave engineering (with distinction) from Institut National Polytechnique de Grenoble (INPG) and University of Grenoble, France, in 1984 and 1987, respectively. He is currently a Professor of electrical engineering, and Tier-I Canada Research Chair in RF and Millimeterwave Engineering at Ecole Polytechnique (University of Montreal). He has (co)-authored over 680 referred papers, a number of books/book chapters and patents. His current research interests involve substrate integrated circuits (SICs), antenna arrays, advanced CAD and modeling techniques, and development of low-cost RF and millimeter-wave transceivers. He is also interested in the modeling and design of microwave photonic circuits and systems. He serves on the Editorial Board of Microwave Journal, Microwave and Optical Technology Letters, and Wiley’s Encyclopedia of RF and Microwave Engineering. He is an Associate Editor of International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE). Dr. Wu also holds a number of Visiting (Guest) and Honorary Professorships at various universities including the first Cheung Kong Endowed Chair Professorship at Southeast University, the first Sir Yue-Kong Pao Chair Professorship at Ningbo University, and Honorary Professorship at Nanjing University of Science and Technology and City University of Hong Kong. He has been director of the Poly-Grames Research Center and has recently become the Founding Director of “Centre de recherche en électronique radiofréquence” (CREER) of Quebec. He is a member of Electromagnetics Academy, the Sigma Xi Honorary Society, and the URSI. He has held many positions in and has served on various international committees, including co-chair of the Technical Program Committee (TPC) for 1997 and 2008 Asia-Pacific Microwave Conferences (APMC), General Co-Chair of 1999 and 2000 SPIE’s Inter. Symposia on Terahertz and Gigahertz Electronics and Photonics, General Chair of 8th Inter. Microwave and Optical Technology (ISMOT’2001), TPC Chair of 2003 IEEE Radio and Wireless Conference (RAWCON’2003), General Co-Chair of RAWCON’2004, Co-Chair of 2005 APMC Inter. Steering Committee, General Chair of 2007 URSI Inter. Symp. on Signals, Systems and Electronics (ISSSE), and General Co-Chair of 2008 and 2009 Global Symposia on Millimeter-Waves, and Inter. Steering Committee Chair of 2008 Inter. Conference on Microwave and Millimeter-Wave Technology. In particular, he will be General Chair of 2012 IEEE MTT-S International Microwave

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Symposium (IMS). He has served on Editorial or Review Boards of various technical journals, including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He served on the Steering Committee for the 1997 joint IEEE AP-S/URSI Inter. Symp. and the TPC for the IEEE MTT-S Inter. Microwave Symp. He is currently chair of the joint IEEE chapters of MTTS/APS/LEOS in Montreal. He is an elected MTT-S AdCom member for 2006–2012 and was Chair of the IEEE MTT-S Transnational Committee. He is Chair of the newly formed IEEE MTT-S Member and Geographic Activities (MGA) Committee. He was the recipient of a URSI Young Scientist Award, IEE Oliver Lodge Premium Award, Asia-Pacific Microwave Prize, IEEE CCECE Best Paper Award, University Research Award “Prix Poly 1873 pour l’Excellence en Recherche” presented by the Ecole Polytechnique on the occasion of its 125th anniversary, Urgel-Archambault Prize (the highest honor) in the field of physical sciences, mathematics and engineering from ACFAS, and 2004 Fessenden Medal of IEEE Canada. In 2002, he became the first recipient of the IEEE MTT-S Outstanding Young Engineer Award. He is Fellow of the Canadian Academy of Engineering (CAE) and Fellow of the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities).

Xiupu Zhang (SM’07) received the B.Sc. degree from Harbin University of Science and Technology, Harbin, China, in 1983, the M.Sc. degree from Beijing University of Posts and Telecommunications, Beijing, China, in 1988, and the Ph.D. degree from the Technical University of Denmark, Lyngby Denmark, in 1996, all in electrical engineering. Following his Ph.D. study, he worked as a Research Fellow at Chalmers University of Technology, Goteborg, Sweden, for one and a half of years. Prior to joining Concordia University in June 2002, he had worked in fiber-optics industry in China, Canada, and the USA for about 10 years. Currently, he is a Professor in Department of Electrical and Computer Engineering, Concordia University, Quebec Canada. His current research interests include optical fiber transmission, radio-over-fiber systems, quantum dot semiconductors, THz generation and broadband optical sources. Dr. Zhang is a member of OSA.

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Multiband Handset Antenna With a Parallel Excitation of PIFA and Slot Radiators Jaume Anguera, Senior Member, IEEE, Iván Sanz, Josep Mumbrú, and Carles Puente, Member, IEEE

Abstract—A handset antenna technique combining a parallel excitation of a PIFA and a slot is presented. The number of frequency bands is given by the sum of bands given per each radiator which can be controlled independently. Component interaction (battery, display, and speaker) is analyzed to determine the best place to mitigate performance degradation. Finally, a concept featuring a small footprint (39 11 mm2 ) and low profile (2 mm) is proposed for multiband operation. Index Terms—Component interaction, handset antennas, multiband, planar inverted F antenna (PIFA), slot, specific absorption rate (SAR).

I. INTRODUCTION

O

NE of the decisive aspects of a portable radio device, such as for instance a hand-held telephone or a wireless device is its volume and size. From the consumer perception, the overall volume, mechanical design, ergonomics and aesthetics of the phone are decisive. There is an increased trend in making thinner phones that can better fit inside a shirt or jacket pocket or a bag or case. This need in making smaller, thinner phones enters into conflict with the trend of adding more features to the phone. On one hand, phones are increasingly adding components and features such as large color screens, digital cameras, digital music players, digital and analogue radio and multimedia ) and come with a broadcast receivers (FM/AM, DVB-H, wider range of form factors (bar phones, clamshell phones, flip-phones, slider phones, ). On the other hand, new cellular and wireless services are being added, which in some cases means that multiband capabilities are required (to feature several standards such as for instance GSM850, GSM900, GSM1800, GSM1900, UMTS) or that other connectivity components (for instance for Bluetooth, IEEE802, WiFi, WiMax, ZigBee, Ultrawideband). All these trends put an increasing pressure on the antenna specifications, which need to feature a small footprint, a thin mechanical profile, yet performing efficiently at several frequency bands [1]–[16]. Manuscript received December 08, 2008; revised August 03, 2009. First published December 08, 2009; current version published February 03, 2010. J. Anguera is with the Department of Electronics and Telecommunications, Universitat Ramon Llull-Barcelona, Barcelona 08022, Spain and also with Fractus, S.A., 08174 Barcelona, Spain (e-mail: [email protected]). I. Sanz and J. Mumbrú are with Fractus, S.A., 08174 Barcelona, Spain. C. Puente is with the Polytechnic University of Catalonia (UPC), Barcelona, Spain and also with Fractus, S.A., 08174 Barcelona, Spain. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2038183

Several techniques employing PIFA and slots have already appeared in the literature. In [17] authors present a slot on the ground plane to make the ground plane resonant at the same frequency than the PIFA. This way, a broadband behavior covering is obtained, from 800–1230 MHz approximately that is, an antenna that fulfills at least the standards GSM850 (824–894 MHz) and GSM900 (880–960 MHz). In [18] a similar approach using a resonating ground plane is shown (design ). covers from 750 to 1250 MHz approximately, In [19], a design using multiple slots on the ground plane is studied in order to achieve a multiband behavior. In spite of the good reflection coefficient results, the proposed structure is difficult to be integrated into a handset phone due to battery, displays, and speakers, among others that can short-out the slots effect. To overcome the component integration problem, in [20]–[23] a similar design is proposed using a slot underneath the antenna area. The slot has two objectives: on the one hand to tune the ground plane to resonate at lower bands (around 900 MHz) as in [17], [18] obtaining a broadband behavior: (GSM850-900); on the other hand, the slot is designed in such a way that operates as a parasitic antenna resonating at the upper band (1900 MHz). With a proper coupling between the slot and the PIFA [22], the bandwidth at the upper band is improved achieving GSM1800 (1710–1880 MHz), 1900 (1850–1990 MHz), and UMTS (1920–2170 MHz). Characteristic modes [24] have been used to give a good understanding on how the ground plane can be used to enhance the behavior of a handset antenna [25]–[27]. Other techniques have been proposed in [28]. In this case, the slot in not printed on the ground plane but embedded on the PIFA geometry. This technique creates an extra mode which enhances the bandwidth at the upper band covering from GSM900-1800 for the original design to GSM900-1800 and 1900 for the embedded-slot design. In [29] a multiband low profile handset designed only with slot antennas is analyzed. Slots are not only useful to antenna design but also for damping undesired modes for EMC purposes [30]. Finally, other solutions employing monopole antenna for multiband purposes can be found in [31]. The objective of the paper is to present a handset antenna technique that combines a PIFA and a slot suitable for slim-profile and multiband cell-phones [32]–[34]. Although PIFA is not low profile compared with a slot-type antenna, the ground plane underneath facilitates component integration as it is demonstrated in this paper. This paper is a detailed extension as well as new data (component interaction) of the author’s previous work presented in [34]. A similar concept based on a parallel excitation of two different antenna types (slot and monopole) can

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Fig. 3. (a) Single branch PIFA + Slot; (b) Dual-branch PIFA + Slot: Adding an extra resonance to the PIFA antenna.

II. ANTENNA CONCEPT Fig. 1. Sequence showing the antenna concept. (a) a slot on the ground plane is tuned at 1.9 GHz (band#2); (b) PIFA is tuned at 900 MHz (band#1); (c) parallel excitation of both antennas (PIFA + Slot). Ground plane is 100 40 mm for all cases.

2

Fig. 2. Simulated reflection coefficient for the sequences shown in Fig. 1.

be found in [35] which also demonstrates to be very useful for multiband performance. The paper is structured as follows: Section II explains the antenna concept. In Section III, simulation gives a physical insight into the antenna behavior. Component interaction (battery, display, and speaker) is analyzed in Section IV. Section V presents a low-profile design covering GSM900, 1800, 1900, UMTS, and S-DMB (2630–2655 MHz). Reflection coefficient, efficiency, radiation patterns, as well as SAR (specific absorption rate) results are shown. Finally, Section VI summarizes the work.

One of the techniques to obtain multiband behavior for handset antennas is to create several resonant paths [2], [16]. Parasitic elements or increasing height may be used to enlarge bandwidth. However both techniques increase antenna volume which is especially prohibitive for the new generation of slim phones. Some solutions remove the ground plane under the antenna area resulting in a monopole type antenna. For these particular cases, once the ground plane under the antenna has been removed, cell-phone components such a camera, vibrator or speaker may degrade the antenna behavior [22]. The antenna technique presented here overcomes the problem of the small bandwidth for low profile PIFA and facilitates component integration. An illustration on how the concept works is shown next. Fig. 1(a) depicts a slot on a ground plane having 100 40 mm . In this case, the slot is excited around 1900 MHz. The . obtained bandwidth covers GSM1800-UMTS at Fig. 1(b) shows a 900 MHz PIFA on the same ground plane. The feeding mechanism is in the same position used to excite the previous slot. The bandwidth is quite poor as the PIFA height is only 4 mm. Both designs are combined, that is, the PIFA and the slot share the same feeding mechanism [Fig. 1(c)]. It can be observed that the new antenna combines both reflection coefficients (Fig. 2). It is important to notice that bandwidth at 900 MHz has been improved. A rationale for this may be found since some ground plane has been removed under the PIFA area reducing its quality factor. Another justification may be explained as the currents follow a larger path due to the slot on the ground plane. The ground plane wave mode gets closer to 900 MHz reaching a better bandwidth [17]–[23]. For the at higher bands is combined solution, bandwidth similar as the single slot case. To increase the bandwidth at the second band, slot width may be increased [36].

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Fig. 4. Simulated reflection coefficient for the sequence shown in Fig. 3(a), (b).

Fig. 6. Electrical field above the slot area.

been tuned at 2600 MHz band which is centered at S-DMB approximately (Satellite Digital Media Broadcast); (Figs. 3, 4). For these two examples we can conclude that: a) . b) Bands due to the PIFA and the slot can be adjusted independently.

III. CURRENT AND FIELD SIMULATIONS

Fig. 5. Current distributions on the PIFA surface. (a) 900 MHz, (b) 1900 MHz, (c) 2600 MHz. The same dynamic range is used.

Current distribution on the PIFA and electrical field on the slot has been computed using the IE3D MoM package to give an extra physical insight into the behavior of this antenna. Fig. 5(a)–(c) shows the current distribution at 900, 1900, and 2600 MHz, respectively. It is remarkable that the PIFA is highly excited at 900 MHz [larger branch at Fig. 5(a)] and at 2600 MHz [short branch at Fig. 5(c)] whereas it is weakly excited at 1900 MHz. The PIFA modes for both resonances are fundamental ones, that is, maximum of current distribution is at the feeding/ short area and the minimum is at the open edge. and ) on To check the slot excitation, electrical field the slot area is computed at several frequencies Fig. 6. Notice that at 900 MHz the slot is weakly excited compared to 1720 and 2000 MHz. At these frequencies, the field distribution corresponds to a quarter wave mode: minimum and maximum at the shorted and open edge, respectively having the illumination . field IV. COMPONENT INTERACTION

Since the PIFA has only one branch, used for the low band, the space can be reused to create a second path, that is, a new resonant frequency [33]. In this case, a new electrical path has

This section analyses the effect on the antenna performance of three particular cell-phone components such as a speaker, a battery, and a display.

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Fig. 7. Simulation to evaluate the effect to floating or connecting the speaker to the PCB.

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be DC-connected to the GND, two RF chokes (100 nH) are introduced to achieve it and at the same time at RF frequencies the speaker is disconnected. This way, the negative effects of the speaker are mitigated. Since the results for floating and RF chokes cases are practically the same, the following experiments uses the speaker in floating conditions. The analysis carried out in Figs. 8–11 is explained next. a) A PIFA-Slot prototype based on Fig. 3(a) operating at GSM850, 900, 1800, 1900 bands has been designed (see slot in Fig. 9 and PIFA in Fig. 10). Groundplane is 100 mm 40 mm and PIFA is 4 mm height. are meab) Reflection coefficient and total efficiency sured using a plastic back-cover to emulate a more realistic scenario. Total efficiency is measured using 3D pattern integration with the Satimo Star-Gate 32 chamber at Fractus Lab. is calculated using (1) c) Radiation efficiency (1)

Fig. 8. Measured results without battery, speaker and display. Handset included both front and back plastic covers.

The characteristic of the components and their connection to the PCB or ground (GND) are described next. • Speaker: circular shape of diameter 13 mm. Floating. • Battery: width is the same as the ground mm. Material: externally shielded with metal. Adhesive surrounding the whole structure. It is GND connected using the ground pad connection. mm. • Display: width is the same as the ground A thin metallic layer covers the back side of the display which is facing the slot radiator. It is GND connected. The reason why the speaker has not been connected to the PCB is explained using simulations results (Fig. 7). The radiation efficiency is computed for four particular situations: without the speaker, with the speaker not connected, with the speaker connected to GND, and finally the speaker connected nH. When the speaker is connected to the GND using directly to the GND it degrades both lower and upper frequencies because the PIFA has more metallic part underneath and the slot is shielded. However, when the speaker is not connected, efficiency at lower frequencies is not degraded since the speaker is not an extension of the groundplane. The metal box of the speaker induces more ground effect to high-band slot and causes a poor efficiency. Since the speaker needs to

It is important to outline that a component may shift reflection coefficient with minor changes in which is true if the component introduces low losses or not degrades the antenna radiation; in other situations, a component may introduce losses or short out the antenna causing reflection coefficient to change and to drop dramatically as it is shown next. d) Aforementioned measurements are performed without components (Fig. 8) and with components at three different positions (Figs. 9–11). Note: for comparison purposes, 4 marks are included in all graphs indicating total efficiency at 824, 960, 1710, and 1990 MHz. Fig. 9 shows the speaker effect when it is placed above the slot area. For position 1 neither the reflection coefficient nor changes. Since the speaker is above the short-edge of the slot, the effect is negligible, meaning that the speaker may be integrated at this position without affecting antenna. However, as the speaker moves closer the open edge, there is a dramatic change at the higher band. This in reflection coefficient as well as means that the speaker reduces radiation from the slot. Lower bands are affected in a much lesser way. This result corroborates data obtained from the simulation: slot is weakly excited at the lower bands. and Fig. 10 depicts the evolution of reflection coefficient, for the following situations: (a) battery at 9 mm, (b) 5 mm, and (c) 0 mm from the PIFA inner edge. It is shown that the performance remains almost the same for b and c situations compared to non-component situation. However, at 0 mm, reflection coefficient at GSM850-900 is shifted to lower frequencies and drops degrading the antenna behavior. At at the same time GSM1800–1900 frequencies, antenna performance is slightly affected since the battery does not interfere with the slot area. Fig. 11 explains the display effect when it is placed above the slot. The effect is pretty much the same as the speaker: to block the radiation from the slot. It is interesting to outline that in spite of the acceptable reflection coefficient at GSM1800–1900 bands, is less than 30%. The lower bands are weakly affected.

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Fig. 9. Speaker effect.

From this experiment it is concluded that the slot should be kept free from the display, being this one a critical component. We can outline that the slot is an attractive solution in terms of a low profile antenna, but it is sensitive when handset components are close to the slot aperture. V. SLIM HANDSET ANTENNA FOR MULTIBAND BEHAVIOR This section illustrates a particular design for pentaband (GSM900, 1800, 1900, UMTS, and S-DMB) behavior using the geometry depicted in Fig. 3(b). Physical implementation is shown in Fig. 12 and Fig. 13. Slot uses a wide aperture to enhance bandwidth especially at the upper bands (DCS-UMTS frequencies) since the slot is the antenna operating at these

Fig. 10. Battery effect.

frequencies. Increasing the slot width may have a lateral effect on the PIFA at GSM900 frequencies since it has less ground plane underneath even the slot is not resonating at GSM900. This way, the PIFA has a partial ground plane which decreases its quality factor, that is, more bandwidth may be obtained. A similar explanation may be observed for a partial grounded microstrip patch. When a slot is placed underneath the microstrip patch, it reduces the quality factor, and therefore the bandwidth of the antenna is increased [37]. More research needs to be done to include GSM850 band which should be achieved by either increasing slot width, PIFA height, or using broad banding networks.

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Fig. 13. PIFA and slot dimensions in mm. F: feeding point; S: short. F and S are two metal parts having 2 mm width 2 mm (h).

2

Fig. 14. The handset prototype using the PIFA-slot of Fig. 12. Speaker at position 2 (see Fig. 9). Battery at 9 mm from the PIFA. Display covering 0% the slot area.

Fig. 15. Measured reflection coefficient for the antenna shown in Fig. 12. Fig. 11. Display effect.

Fig. 12. Slim PIFA-Slot antenna: 39 mm polymethacrylimide foam.

2 11 mm 2 2 mm (h). Substrate is

Following the guideline presented in Section IV, a battery, a speaker, and a display have been attached to the ground plane. In addition, a camera has also been placed near the short-edge of the slot (Fig. 14). A front and a back cover are also taken into consideration for all the experiments. Fig. 15 represents the reflection coefficient where it can be obdB). At 1710 MHz, which served a good matching ( is the starting frequency of GSM1800, matching may be further improved as there is enough room since the end part of UMTS dB. It should be outlined that the (2170 MHz) has almost PIFA height is only 2 mm; as shown in previous section, increasing to 4 mm would be useful to include also GSM850 band being still a low profile PIFA [38].

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TABLE I RADIATION ( ), TOTAL EFFICIENCY ( ), AND REFLECTION COEFFICIENT FOR THE ANTENNA HAVING ALL THE COMPONENTS SHOWN IN FIG. 14

Fig. 16. Measured radiation cuts at 900, 1800, 2100, and 2600 MHz. Measured maximum gain at each frequency is 1.3, 0.7, 2.7, and 0.85 dBi.

TABLE II MEASURED SAR VALUES FOR THE ANTENNA PROTOTYPE OF FIG. 14

imum transmit power is 33 dBm; however, a transmit channel of a time slot. This results in 24 dBm which is the uses only power of a continuous wave to test SAR. Similar procedure is done at GSM1800; in this case, maximum transmit power is 30 dBm. Thus, SAR is tested using 21 dBm. SAR passive testing is indicative of a preliminary measure since SAR is finally tested with an active device. However, it is interesting to test in a passive way to analyze if the antenna may pose a SAR problem. For example, from this passive data (Table II and Fig. 17) some conclusions can be obtained. a) At the low frequencies (900 MHz), the hot-spot (maximum SAR value) is located at the centre of the ground plane confirming again that the ground plane mode determines radiation. The slot is weakly excited, meaning that it is not an issue for SAR. b) At the higher frequencies, the hot-spot in mainly fixed by the slot on the ground plane since in this case the slot is excited, that is, SAR is more antenna dependent [17]. This is useful information since SAR can be dramatically reduced at higher bands by placing the antenna at the opposite short edge of the ground plane (180 rotation of the handset) [22]. VI. CONCLUSION

Fig. 17. Measured SAR distribution at right cheek position 900 MHz (left) and 1800 MHz (right).

Table I shows the measured and . Even the existence of several cell-phone components, can fulfil mobile service requirements. Radiation cuts have been measured at 900, 1800, 2100, and 2600 MHz (Fig. 16). Dipole-type radiation pattern can be observed at 900 MHz determined by the ground plane mode: omhaving linear polarization following nidirectional at y-axis. Radiation patterns at higher frequencies present a larger directivity due to the larger electrical size of the ground plane. Finally, specific absorption rate (SAR) in passive mode has been tested using Dasy-4 at Fractus-Lab. At GSM900 the max-

The concept based on a PIFA-slot has been shown to be useful to design multiband handset antennas where the number of frequency bands is given by the sum of the bands given by each radiator. Moreover, said bands can be controlled independently which adds an extra freedom design. Component interaction has been analyzed showing that: a) the speaker mainly affects the slot radiator (introduces mismatch and losses) but its negative effect can be minimized by placing the speaker near the short-edge of the slot, b) battery affects the PIFA causing a detuning and introduce losses, c) the display is a critical component which should keep the slot free. This means that for planar handset antennas such as monopoles or slots, component interaction should be carefully taken into account. Although PIFA type occupies more space, components can be placed at the other part of the ground plane with a minimum impact on the performance of the antenna. Thanks to the slot radiator, the PIFA volume can be reused to add more bands; for this research, an extra band centered at S-DBM has been added to finally design a pentaband prototype at GSM900, 1800, 1900, UMTS, and S-DMB. The total antenna volume results in only 39 11 2(h) mm . Results for total efficiency taking into account several components (battery, display, speaker, camera, and phone covers) are satisfactory and make this concept very attractive for the new generation of low-profile multiband handset phones.

ANGUERA et al.: MULTIBAND HANDSET ANTENNA WITH A PARALLEL EXCITATION OF PIFA AND SLOT RADIATORS

REFERENCES [1] T. Taga and K. Tsunekawa, “Performance analysis of a built-in planar inverted-F antenna for 800 MHz band portable radio units,” IEEE Trans. Sel. Areas Commun., vol. SAC-5, no. 5, pp. 921–929, Jun. 1987. [2] C. Puente, J. Romeu, C. Borga, and J. Anguera, “Multilevel antennas,” patent Appl. WO 01/22528, Sep. 20, 1999. [3] D. Manteuffel, A. Bahr, and I. Wolff, “Investigation on integrated antennas for GSM mobile phones,” presented at the Millennium Conf. on Antennas and Propag., Davos, Switzerland, Apr. 2000. [4] T. Y. Wu and K. L. Wong, “On the impedance bandwidth of a planar inverted-F antenna for mobile handsets,” Microw. Opt. Technol. Lett., vol. 32, pp. 249–251, Feb. 20, 2002. [5] K. L. Virga and Y. Rahmat-Samii, “Low-profile enhanced-bandwidth PIFA antennas for wireless communications packaging,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 10, pp. 1879–1888, Oct. 1997. [6] C. R. Rowell and R. D. Murch, “A compact PIFA suitable for dualfrequency 900/1800-MHz operation,” IEEE Trans. Antennas Propag., vol. 46, no. 4, pp. 596–598, Apr. 1998. [7] J. Ollikainen, M. Fischer, and P. Vainikainen, “Thin dual-resonant stacked shorted patch antenna for mobile communications,” Electron. Lett., vol. 35, no. 6, pp. 437–438, Mar. 18, 1999. [8] Y. X. Guo, M. Y. W. Chia, and Z. N. Chen, “Miniature built-in quadband antennas for mobile handsets,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 30–32, 2003. [9] B. Sanz-Izquierdo, J. Batchelor, and R. Langley, “Multiband printed PIFA antenna with ground plane capacitive resonator,” Electron. Lett., vol. 40, no. 22, pp. 1391–1392, Oct. 28, 2004. [10] Y. J. Cho, S. H. Hwang, and S. O. Park, “A dual-band internal antenna with a parasitic patch for mobile handsets and the consideration of the handset case and battery,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 429–432, 2005. [11] M. Martínez-Vázquez, O. Litschke, M. Geissler, D. Heberling, A. M. Martínez-González, and D. Sánchez-Hernández, “Integrated planar multiband antennas for personal communication handsets,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 389–391, Feb. 2006. [12] P. Lindberg and E. Öjefors, “A bandwidth enhancement technique for mobile handset antennas using wavetraps,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2226–2233, Aug. 2006. [13] B. N. Kim, S. O. Park, Y. S. Yoon, J. K. Oh, K. J. Lee, and G. Y. Koo, “Hexaband planar inverted-F antenna with novel feed structure for wireless terminals,” IEEE Microw. Wireless Compon. Lett, vol. 6, pp. 66–69, 2007. [14] B. Jung, J. S. Lee, M. J. Park, Y. S. Chung, F. J. Harackiewicz, and B. Lee, “TDMB-AMPS-GSM-DCS-PCS-SDMB internal antenna using parasitic element with switching circuit,” Electron. Lett., vol. 42, pp. 734–736, Jun. 22, 2006. [15] K. R. Boyle and P. J. Massey, “Nine band antenna system for mobile phones,” Electron. Lett., vol. 42, no. 5, pp. 265–266, Mar. 2006. [16] K. L. Wong, Planar Antennas for Wireless Communication, ser. Wiley Series in Microwave and Optical Engineering, K. Chang, Ed. New York: Wiley, 2003. [17] P. Vainikainen, J. Ollikainen, O. Kivekäs, and I. Kelander, “Resonator-based analysis of the combination of mobile handset antenna and chassis,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1433–1444, Oct. 2002. [18] R. Hossa, A. Byndas, and M. E. Bialkowski, “Improvement of compact terminal antenna performance by incorporating open-end slots in ground plane,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 283–285, Jun. 2004. [19] M. F. Abedin and M. Ali, “Modifying the ground plane and its effect on planar inverted-F antennas (PIFAs) for mobile phone handsets,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 226–229, 2003. [20] J. Anguera, I. Sanz, A. Sanz, A. Condes, D. Gala, C. Puente, and J. Soler, “Enhancing the performance of handset antennas by means of groundplane design,” presented at the IEEE Int. Workshop on Antenna Technology: Small Antennas and Novel Metamaterials (IWAT), New York, Mar. 2006. [21] J. Anguera, A. Cabedo, C. Picher, I. Sanz, M. Ribó, and C. Puente, “Multiband handset antennas by means of groundplane modification,” presented at the IEEE Antennas Propagation Society Int. Symp., Honolulu, HI, Jun. 2007. [22] A. Cabedo, J. Anguera, C. Picher, M. Ribö, and C. Puente, “Multiband handset antenna combining PIFA, slots, and ground plane modes,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2526–2533, Sep. 2009.

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[23] C. Picher, J. Anguera, A. Cabedo, C. Puente, and S. Kahng, “Multiband handset antenna using slots on the ground plane: Considerations to facilitate the integration of the feeding transmission line,” Progr. Electromagn. Res. C, vol. 7, pp. 95–109, 2009. [24] R. F. Harrington and J. R. Mautz, “Theory of characteristic modes for conducting bodies,” IEEE Trans. Antennas Propag., vol. 19, pp. 622–628, Sep. 1971. [25] E. Antonino, C. A. Suárez, M. Cabedo, and M. Ferrando, “Wideband antenna for mobile terminals based on the handset PCB resonance,” Microw. Opt. Technol. Lett., vol. 48, no. 7, pp. 1408–1411, Jul. 2006. [26] M. Cabedo, E. Antonino, A. Valero, and M. Ferrando, “The theory of characteristic modes revisited: A contribution to the design of antennas for modern applications,” IEEE Antennas Propag. Mag., vol. 49, no. 5, pp. 52–68, Oct. 2007. [27] M. Cabedo, E. Antonino, V. Rodrigo, and C. Suárez, “Análisis modal de un plano de masa radiante doblado y con una ranura para terminales móviles,” presented at the XXI National Symp. URSI’06, Oviedo, Spain, 2006, Sep.. [28] C. Di Nallo and A. Faraone, “Multiband internal antenna for mobile phones,” Electron. Lett., vol. 41, pp. 514–515, 2005. [29] C. Lin and K. L. Wong, “Printed monopole slot antenna for internal multiband mobile phone antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 3690–3697, Dec. 2007. [30] S. Kahng, “The rectangular power-bus with slits GA-optimized to damp resonances,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1892–1895, Jun. 2007. [31] S. Hong, W. Kim, H. Park, S. Kahng, and J. Choi, “Design of an internal multiresonant monopole antenna for GSM900/DCS1800/USPCS/S-DMB operation,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1437–1443, May 2008. [32] R. Quintero and C. Puente, “Multilevel and space-filling ground-planes for miniature and multiband antennas,” patent appl. WO 03/023900, Sep. 13, 2001. [33] J. Anguera and C. Puente, “Shaped groundplane for radio apparatus,” patent appl. WO 06/ 070017, Dec. 29, 2005. [34] J. Anguera, I. Sanz, J. Mumbrú, and C. Puente, “Multiband handset antenna behavior by combining PIFA and a slot radiators,” presented at the IEEE Antennas Propag. Society Int. Symp., Honolulu, HI, Jun. 2007. [35] C. Lin and K. L. Wong, “Internal hybrid antenna for multiband operation in the mobile phone,” Microw. Opt. Tech. Lett., vol. 50, no. 1, pp. 38–42, Jan. 2008. [36] S. Kumar, L. Shafai, and N. Jacob, “Investigation of wide-band microstrip slot antenna,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 865–872, Mar. 2004. [37] K. L. Wong, Compact and Broadband Microstrip Antennas, ser. Wiley Series in Microwave and Optical Engineering, K. Chang, Ed. New York: Wiley, 2002. [38] B. N. Kim, S. O. Park, Y. S. Yoon, J. K. Oh, K. J. Lee, and G. Y. Koo, “Hexaband planar inverted-F antenna with novel feed structure for wireless terminals,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 66–68, 2007. Jaume Anguera (S’99–M’03–SM’09) was born in Vinaròs, Spain, in 1972. He received the Technical Ingeniero degree in electronic systems and Ingeniero degree in electronic engineering from the Ramon Llull University (URL), Barcelona, Spain, in 1994 and 1997, respectively, and the Ingeniero and Ph.D. degrees in telecommunication engineering from the Polytechnic University of Catalonia (UPC), Barcelona, Spain, in 1998 and 2003, respectively. From 1998 to 2000, he joined the Electromagnetic and Photonic Engineering Group (EEF), Signal Theory and Communications Department, UPC, as a Researcher in microstrip fractal-shaped antennas. In 1999, he was a Senior Researcher at Sistemas Radiantes, Madrid, Spain, where he was involved in the design of a dual-frequency dual-polarized fractal-shaped microstrip patch array for mobile communications. In the same year, he became an Assistant Professor at the Department of Electronics and Telecommunications, Universitat Ramon Llull-Barcelona, where he is currently teaching antenna theory. Since 2000, he has been with Fractus, S.A., Barcelona, Spain, where he holds the position of R&D Manager. At Fractus, he leads projects on antennas for base station systems, antennas for automotion, handset and wireless antennas. His research interest are multiband and small antennas, microstrip patch arrays, feeding network architectures, broadband matching networks, array pattern synthesis with genetic algorithms, diversity antenna systems, electromagnetic dosimetry, and handset antennas.

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He is a Leading Engineer for the Innovation Antenna Group. From September 2003 to May 2004, he was with Fractus-Korea (Republic South of Korea) where he was managing projects for miniature and multiband antennas for handset and wireless applications. Since 2005, he has been leading research projects in the antenna field for handset and wireless applications in a frame of industry-university collaboration: Fractus company and the Department of Electronics and Communications, Universitat Ramon Llull-Barcelona, Spain. He holds more than 27 patents on fractal an other related antennas. He is the author/coauthor of more than 120 journal, international, and national conference papers and he has directed more than 50 bachelor and master thesis. Dr. Anguera was member of the Fractal team that in 1998 received the European Information Technology Grand Prize from the European Council for the Applied Science an Engineering and the European Commission for the fractal-shaped antenna application to cellular telephony. He was the 2003 Finalist for the Best Doctoral Thesis (Fractal and Broadband Techniques on Miniature, Multifrequency, and High-Directivity Microstrip Patch Antennas) on UMTS (this prize has been promoted by “Technology plan of UMTS promotion” given by Telefónica Móviles España). He was named one of the New faces of Engineering 2004 by the IEEE. In the same year, he received the Best Doctoral Thesis (Ph.D.) in “Network and BroadBand Services” (XXIV Prize Edition “Ingenieros de Telecomunicación”) organized by Colegio Oficial de Ingenieros de Telecomunicación (COIT) and ONO Company. He is reviewer for the IEEE TRANSACTIONS AND ANTENNAS AND PROPAGATION, IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, Progress in Electromagnetic Research (PIER), IEE Electronics Letters, and the ETRI Journal (Electronics and Telecommunications Research Institute, South Korea). His biography is listed in Who’s Who in the World, Who’s Who in Science and Engineering, Who’s Who in Emerging Leaders and in IBC (International Biographical Center, Cambridge-England).

Iván Sanz was born in Oviedo, Spain, in 1974. He received the Ingeniero degree in telecommunication engineering from the Polytechnic University of Catalonia (UPC), Barcelona, Spain, in 2008. He joined Fractus S.A., Barcelona, Spain, in 2005. He has been involved in several investigation projects about the application of the fractal technology in the design of miniature multiband antennas.

Josep Mumbrú was born in Barcelona, Spain, in 1971. He received the Ingeniero degree in telecommunication engineering (with specialization in communication systems) and the Ingeniero degree in electronics from the Polytechnic University of Catalonia (UPC), Barcelona, Spain, in 1995 and 1996, respectively, and the M.Sc. degree in electrical engineering and the Ph.D. degree in electrical engineering and social science from the California Institute of Technology (Caltech), Pasadena, in 1998 and 2002, respectively. From 1997 to 2002, he was Research and Teaching Assistant in the Optical Information Processing Group, Caltech. He conducted research work in the fields of holographic data storage for optical memories, holographic imaging systems and components for WDMA optical fiber network. He also collaborated with Holoplex Inc., in the development of next generation systems for optical data storage. Since 2002 he has been with Fractus S.A., Barcelona, where he holds the position of R&D and IPR Manager. At Fractus, he has contributed to the development of the fractal antenna technology and their applications to mobile communication and wireless connectivity devices, and to the growth and strengthening of Fractus patent portfolio.

Carles Puente (M’92) received the M.Sc. degree from the University of Illinois at Urbana-Champaign, in 1994 and the Ph.D. degree from Polytechnic University of Catalonia (UPC), Barcelona, Spain, in 1997. From 1994 to 1999, he worked with the faculty of Electromagnetic and Photonic Engineering, UPC, on pioneering developments of fractal technology applied to antennas and microwave devices. He is a co-founder of Fractus, S.A., Barcelona, Spain, and currently leads its Antenna Technology Research Team, with responsibility for the company’s intellectual property portfolio development and antenna development. He is also a Professor at UPC, Barcelona, Spain, where he started researching fractal-shaped antennas while a student in the late 1980s. He has authored more than 50 invention patents and over 90 scientific publications in fractal and related antenna technologies. Dr. Puente was awarded the Best Doctoral Thesis in Mobile Communications 1997 by the COIT, the European Information Society Technology Grand Prize from the European Commission in 1998, and the Premi Ciutat de Barcelona in 1999. He and his team at Fractus where awarded the Technology Pioneer distinction by the World Economic Forum in 2005.

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On the Design of a Compact Neural Network-Based DOA Estimation System Nelson Jorge G. Fonseca, Member, IEEE, Michael Coudyser, Jean-Jacques Laurin, Senior Member, IEEE, and Jean-Jules Brault, Associate Member, IEEE

Abstract—A system to measure the direction of arrival (DOA) of a signal within a 45-degree conical sector is demonstrated. The system is compact and uses only four circularly polarized patch elements. A printed beamforming network is used to create a set of partially overlapping beams allowing DOA estimation without ambiguity. Neural networks are used to first classify the antenna signals and then estimate the DOA. The proposed system was validated experimentally in C band and, in spite of the highly disturbed beams caused by the finite size of the antenna platform, it was shown that DOA estimation errors in the order of one degree were achievable under signal-to-noise ratios of 10 dB. Index Terms—Array feeding network, direction finding antenna, monopulse radar, neural network.

I. INTRODUCTION

D

IRECTION finding is used in a number of applications such as radio emitter tracking, jammer localization and mobile communications. In this case, the link budget between user terminals and a fixed base station can be enhanced if an adaptive system is implemented to optimize the alignment of the antenna beams. A recent example of this is the developing OFDMA systems (IEEE Standard 802.16a), which allow for adaptive antenna in the base station. In the future, one may foresee that this feature will be implemented in mobile terminal as well. Currently, some portable computers equipped with wireless connections use space diversity, i.e., one of the antennas located at different positions on the computer is selected based on the maximum received signal strength. Electronic beam steering control, coupled with a system to estimate the direction of arrival of the strongest signal would be a more flexible and potentially better alternative. Difficulties arise due to the fact that mobile terminals have inherently small electrical dimensions. Consequently, it is not possible to have

Manuscript received October 14, 2008; revised August 04, 2009. First published December 04, 2009; current version published February 03, 2010. N. J. G. Fonseca was with the Poly-Grames Research Centre, Montreal, QC H3C 3A7, Canada. He is now with the Antenna and Sub-Millimeter Wave Section, European Space Agency/ESTEC, 2200 AG Noordwijk, The Netherlands (e-mail: [email protected]). M. Coudyser was with the Department of Electrical Engineering, Ecole Polytechnique, Montreal, QC H3C 3A7, Canada. He is now with Ineo-Suez, Paris Cedex 92059, France (e-mail: [email protected]). J.-J. Laurin and J.-J. Brault are with the Department of Electrical Engineering, Ecole Polytechnique, Montreal, QC H3C 3A7, Canada (e-mail: jean-jacques. [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.2009.2037766

a large array to accurately determine the direction of arrival (DOA). On the other hand, high accuracy is not necessary in many cases because the limited antenna size does not allow narrow beamforming. Another requirement is that the DOA system has to be able to operate over a broad angular sector due to the random orientation of the mobile terminal. There is therefore an interest for a compact system that can estimate DOA within such wide range of directions. The system has to respond rapidly to make the adaptive function effective. In this paper we are proposing such a compact DOA system based on two orthogonal 2-element patch arrays. The system is inspired from the basic monopulse system in which sum and difference beams are formed by combining the signals from two antenna or sub-array ports. Monopulse systems can operate on phase, amplitude or combined phase and amplitude differences [1]. A monopulse system based on arbitrary complex ratios between the antenna outputs was proposed in [2]. A three-beam monopulse amplitude-based architecture was simulated in [3] and it was shown that an error on the DOA of less than 1 can be achieved for signal-to-noise ratios (SNR) of 26 dB or more. However, this accuracy was only obtained over angular sectors of about 10 and 14 in the cases of amplitude-only and complex signal processing respectively. A DOA system based on amplitude comparisons between the signals received by an omnidirectional antenna and an eight-beam directional antenna was implemented by Li et al. in [4]. DOA errors of about 2 were obtained over a 360 angular range and an 11 GHz frequency bandwidth. A similar approach called “power pattern cross-correlation”, in which a number of beams are realized with an electronically steerable parasitic array radiator, was later demonstrated by Taillefer et al. [5]. Again, very good performance, with an average DOA error of less than 1.5 was achieved over a full 360 sector. Although the results of [3]–[5] confirm the feasibility of achieving accurate DOA estimations with magnitude-only processing, in all cases a fitness function based on power measurements and recorded patterns need to be calculated over the whole angular sector investigated, which increases processing time. This is in addition to the time for switching between the beams. The system we are presenting in this paper is inspired from the monopulse system and uses only amplitude data. Instead of using only two inputs ( and ), three beam functions, as proposed in [3], are formed by a beamforming network (BFN) to prevent ambiguities. Processing of the three normalized amplitude values is accomplished by neural networks (NN) that determine the DOA without evaluating a fitness function over the whole range of possible directions. This results in fast

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processing but foremost in importance, it eliminates all needs for phase and magnitude calibration and it compensates to some extent for non-ideal hardware performance. The NN approach was successfully used previously in systems estimating only one angle (e.g., [6]–[8]). In [9], DOA estimations using a NN-based system and using a well-known subspace method (MUSIC) were compared. The accuracy of both approaches was similar but there was a substantial gain in processing speed with the NN system. The system presented here finds the DOA in a conical sector, and therefore estimates two angles. The feasibility of two-dimension (2D) DOA systems with an L-shaped array, has been demonstrated through numerical simulations using different estimation approaches. In [10], the matrix pencil method was used to estimate the DOA and polarization state of correlated incoming signals. A “propagator” method that avoids the need for eigenvalue decomposition was successfully demonstrated in [11] with two different arrangements of L-shaped array, whereas an approach based on the singular value decomposition of crosscorrelation matrices presented in [12] also demonstrated a high accuracy. These works assumed ideal elements and neglected mutual couplings. A scheme to compensate for mutual couplings in the array has been presented in [13] for the case of a circular 2-D DOA array. The system we are presenting in this paper uses an L-shaped array. However, mutual couplings and other non-ideal characteristics of the elements do not have to be taken into account explicitly, as they are present during the NN training phase. The system was designed with the objective of studying the feasibility of doing DOA with a compact antenna, and thus emphasis was put on angular resolution for the case of a single incoming signal. In a multipath environment where the same incoming signal arrives from multiple directions, with variable amplitudes and delays, the system would have to be augmented with high resolution time domain processing to allow for signal classification. In the case studied here, it was simply assumed that one of the directions would correspond to a signal having much stronger amplitude than the others, which can then be neglected. A monopulse system designed for more than one target has been investigated in [14] but such a case will not be considered here. The paper is organized as follows. Section II gives a general description of the proposed system whereas Section III describes the realized antennas and beamforming network in details. The implementation of the NN-based DOA system is presented in Section IV, which is followed by experimental validations and performance assessment in Section V.

II. DESCRIPTION OF THE DOA SYSTEM The proposed DOA system is based on the angle estimation between the direction of arrival of the considered signal and a reference axis. Combining two such one-axis systems in orthogonal directions, say and , makes possible the estimation of and defined in Fig. 1. These angles are related to angles

Fig. 1. Angles definitions for DOA estimation.

Fig. 2. General architecture of the proposed DOA system.

the spherical coordinate angles formulas:

and

through the following

(1) Let us first consider a basic monopulse system consisting of a two-element array connected to a circuit, typically a hybrid junction, generating sum and difference signals. If the magnitude and relative phase of these two output signals are known, it is possible to determine the angle of arrival of a signal with respect to the array axis. However, if only the magnitudes are known, there is an ambiguity, as both sum and difference magnitudes have symmetrical responses with respect to the array’s boresight. The one-axis system we are proposing, illustrated in Fig. 2, is also a simple 2-element linear array but the BFN has three beam ports generating signals for which the magnitudes provide sufficient information to determine the DOA without ambiguity. In a 2-beam-port system, removing the ambiguity would necessitate mechanical or electronic scanning, which is not the case here. Such use of NN processing in conjunction with a switched beam network has been presented in the 1D NN-DOA system reported in [15] for which estimation errors of less than 1 were achieved with seven beam ports. The three signals at the beam ports are transmitted to a RF receiver unit for frequency down-conversion and appropriate processing (see Fig. 2). In a practical system, various multiple access techniques and coding schemes could be implemented to recover the signal of interest. Once these signals are recovered, their magnitudes are processed by a neural network (NN), implemented in a digital processing unit. A controller responsible for switching between channels and automatic gain control is also shown in the figure. Since the purpose of this paper is to demonstrate the DOA estimation capability, no receiver unit has been implemented. In fact, our experimental demonstrations

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each output signal. This leads with the following constraints (see Appendix):

(4) The following parameters values are a possible solution for these constraints:

(5) Fig. 3. Schematic of the antenna array and BFN.

were made with an incident continuous wave of fixed frequency generated and received by a network analyzer. The tests were done in a shielded anechoic chamber. Thus, it can be assumed that only the desired incident wave is present and in this case, simple RF power level measurements are sufficient to produce the NN inputs. III. ANTENNA ARRAY AND BEAMFORMING NETWORK A. RF Design Description The topology of the proposed BFN for only one axis is shown in Fig. 3. The proposed concept is adapted from Nolen matrices, characterized by serial feeding [17], [18]. It is composed of directional couplers, delay lines and a power divider. The output signals are obtained with the following expressions (see Appendix for details)

(2)

are the complex signals received at antenna level where . and In the foregoing analysis, plane wave incidence and equal and will be assumed. If we consider amplitudes of signals isotropic elements, we can write the complex inputs at antenna relative to one axis as follows: level for a given 1D-DOA

(3) where is the wave number and the spacing between the two radiating elements. This assumption implies that the design is based on identical antennas operating in identical environments. This is of course not the case in practice. Removing the unpredictable effects of the antenna setup is one of the reasons justifying the use of a NN. The BFN design parameters are the phase of the delay lines and , as well as parameters and defining the coupling levels. These parameters are fixed by imposing that the two antennas contribute with equal amplitude weights to

The last parameter, , is adjusted so as to locate the curves symmetrically around . The corresponding value is . The magnitudes of the signals produced by the BFN, assuming isotropic elements, are presented in Fig. 4. These curves basically represent different array factors which were designed with the following rationale. Firstly, it can be seen and of are the same over that the signs of ). a wide range of angles (approximately Thus, knowing this sign allows us to remove the ambiguity , which is present in the case of monopulse on varies rapidly in this angular range. systems. Secondly, based on This sensitivity favors the estimation of the value of . The choice of the spacing between the two antennas has to ensure that these two characteristics will be maintained over the desired angular range of operation. For the implementation presented in this paper the objective was to op. erate in a 45-degree conical sector centered on By varying in simulations of the NN estimation performance, led to the best results it was found that a separation of in this sector. This distance was then used for the curves of Fig. 4 and in the rest of the paper. It is important to point out however that the system is robust with respect to variations of . For instance, performance degradation was barely noticeable , which suggests that the size of the with a separation of system could be reduced. However, mutual coupling between the elements was not taken into account in these simulations. In practice, coupling between nearby elements can affect the radiation pattern. For this reason, we preferred use the larger value of . B. Simulation and Measurement Results In the proposed system, the two angles and are estimated by two identical systems having their antenna axes rotated by 90 degrees with respect to each other, as displayed in Fig. 5. In addition to rotation, the two systems are also separated (vertically in the figure) in order to allow printed implementation of the two BFNs without crossing between traces. This arrangement is also preferred to limit mutual coupling between the radiating elements of the two axes. In theory, separating the phase centers of the two systems should have no effect since the two angles to estimate are independent. Compact aperture-coupled square patch elements were used. Circular polarization (CP) response is produced by using a cross

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Fig. 4. Magnitude of signals produced by the proposed BFN assuming isotropic elements separated by 0:58.

shaped slot of unequal arms lengths [19] coupled to a microstrip line. The DAO system prototype was designed for operation at 5.8 GHz. The substrate used for the antenna array was a laminate (GML 1032 by GIL Technologies) with and a thickness of 1.52 mm. The microstrip BFN was imple, mented on RT/Duroid 6002 by Rogers Corporation ( ) having a thickness of 0.51 mm. The complete structure therefore comprises three metal layers, i.e., microstrip lines, ground plane with cross slots and a patch layer. This imposes alignment constraints on the manufacturing process. The layout of the simulated antenna presented in Fig. 5(a) was simulated with the method of moments using Agilent’s ADS Momentum. As it can be seen, two of the directional couplers have a ring shape so as to minimize the number of sharp bends in the BFN design. A Wilkinson power divider with isolation resistor was used to reduce the effect of impedance mismatch between components. The delay lines were simply implemented with microstrip meanders. Finally, to minimize radiation from the slots in the back side, a solid metal plate [see Fig. 5(b)] isolated from the microstrip layer by a 6.35 mm thick foam spacer was added. Besides SNR considerations, the estimation of the DOA defunctions with respect pends on the relative levels of the to each other, and not on their absolute levels. Therefore, the NN’s input layer can be fed with data normalized with respect to one of the inputs. As mentioned in [2], there are many possible choices for normalization. Of course the normalization reference must be different from zero for all possible DOAs. Also, from the accuracy point of view, it is preferable to normalize with respect to a signal that is the least affected by noise. As seen in Fig. 4, each of the beam ports gives a null over the 180-degree angle interval. Thus, to avoid divisions by zero, the normalized NN’s input variables for each of the two one-axis arrays were defined as

(6)

Fig. 5. Layout of the simulated system showing (a) the four patch elements and the six beam ports. (b) Picture of the fabricated prototype.

Fig. 6. Normalized elevation patterns of the realized CP element in the plane.

xz

When the antenna elements are identical but not isotropic, term is multiplied by the element factor, which is then each cancelled in the normalization process. This suggests that the

FONSECA et al.: ON THE DESIGN OF A COMPACT NEURAL NETWORK-BASED DOA ESTIMATION SYSTEM

Fig. 7. Predicted and measured

X

patterns of the sub-array associated with the

x direction at 5.8 GHz.

Fig. 8. Predicted and measured

X

patterns of the sub-array associated with the

y direction at 5.8 GHz.

choice of antenna element can be arbitrary if DOA estimation is based on the normalized variables. In practice, we must however avoid nulls in the element patterns over the angle range of interest. The printed CP patch elements used here have degraded gain and axial ratio when the DOA has a low elevation above the plane, as illustrated in the measured and simulated radiation patterns shown in Fig. 6. The half-power beamwidths is approximately 90 degrees. For those reasons, the DOA interval considered was limited . A mean to determine if the DOA is to included in this range will be described in the next section. functions for the and sub-arrays are plotted in The Fig. 7 and Fig. 8 respectively. Theoretical curves are shown to( sub-array) and gether with measured ones done in the ( sub-array) planes. The theoretical curves are based on the normalized BFN outputs [see (6)] and it is assumed that the elements are identical and operate over an infinite ground plane. In general there is a good agreement between measurements and predictions in the 45 –135 range for both sub-arrays but rapid oscillations are clearly visible outside this range. These discrepancies can be due to scattering by the edges of the and to non idenfinite-size antenna ground plane tical element factors due to inter-element couplings, two effects which are not included in the simplified theoretical array factors. Although the symmetry of the fabricated antenna should result in similar responses for the two sub-arrays, unpredictable differences can be noticed, in particular outside the angular range of interest. As expected from (6), each signal saturates to a value of 1 over the angular range where it is the strongest among the three outputs.

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IV. NEURAL NETWORK

Previous work on compact DOA systems have shown that neural networks can be used to predict the DOA, in a one-dimension case, with a compact system including a small number of elements [7]. A similar approach was used here in the case of the proposed dual 1-D system. The chosen NN is a conventional multilayer perceptron network (MLP), because such a network is well adapted to regression problems [20]. A MLP is essentially based on simple mathematical operations at the neuron level, i.e., multiplication, addition and application of a smooth nonlinear sigmoidal function; hence, MLPs are particularly convenient for implementation on a DSP. In this work, we however did not implement our neural network in hardware. Instead, we used the NN Toolbox of Matlab (R12) to build and optimize our MLPs. Also, our DOA estimations were not done in real time. Instead, power measurements at the antenna ports were first recorded and this data was later processed off line with Matlab. The topology of the MLP is defined as a trade-off between the computational capacity of the NN and the expected risk minimization of the DOA estimate. In other words, an important issue that was addressed in the choice of the MLP topology is the number of perceptrons in the hidden layers. Usually, higher numbers of neurons lead to a better fit between training data and the expected NN estimation but they could also lead to overtraining. For instance, the NN may learn the noise superimposed to the specific training data used and may then become more sensitive to noise affecting the input data during the recall stage.

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Finally, to improve the performances in the presence of noise, an averaging filter was implemented at the output of the NNDE. This filter is particularly adapted to limit the impact of Gaussian noise, characterized by a mean equal to zero, which is representative of the noise at receiver level. B. NN for DOA Classification (NNDC)

Fig. 9. Validation error versus the number of neurons in the hidden layer.

For this study, we restricted the MLP topology to a single hidden layer. A. NN for DOA Estimation (NNDE) Two approaches were investigated to define the NN. In the first one, we have two independent NNs (2x1D), one by axis; in the second one, one NN is used to process simultaneously the inputs from the and axes BFNs (1x2D). The latter approach was finally selected because it led to better DOA estimations when used with measured data. This result could be anticipated as a consequence of the chosen normalization scheme. A oneaxis normalization suppresses one information by axis, while a 1x2D normalization only suppresses one information for the whole system. Then, the input signals in the 1x2D approach carry more information after normalization when compared to the 2x1D approach. The resulting MLP was optimized by cross-validation using the Levenberg-Marquardt [20] algorithm and with a training data set that will be described in Section V. It is composed of the following three layers. 1) The input layer has a number of neurons equal to the number of input signals, six in our system. 2) The hidden layer comprises 20 neurons. The regression error versus the number of neurons on the hidden layer is presented in Fig. 9. The error is defined as

where is the size of the training set and is the difference between the estimated and expected DOAs in direction. the sampled These results were obtained by training the NN with data measured within the range of interest, . Although Fig. 9 suggests that the error could still decrease slightly for larger number of neurons, 20 appeared as a good trade-off between accuracy and complexity. 3) The output layer has a number of linear neurons equal to the number of outputs, i.e., two in the 2D system.

In order to suppress ambiguities, a second MLP was implemented to determine if the DOA was within the conical sector of . It is referred to as the classification interest defined by MLP. The output of this MLP is equal to 1 if the DOA is within the range of interest and it is set to 0 otherwise. Depending on this output, the signals will be processed or not by the NNDE. An optimal number of 8 neurons was obtained in the hidden layer. The same dataset as for the NNDE was used. The complete DOA system is illustrated in Fig. 10. As shown, after normalization and filtering, the six power levels are fed to the input nodes of the NNDE and NNDC, both comprising only one hidden layer with 20 and 8 perceptrons, respectively. The NNDE uses two linear neurons (weighted summer) in the output layer while the NNDC uses a single neuron with a sigmoidal activation function. V. DIRECTION FINDING SYSTEM PERFORMANCE In this section, the performance of the proposed DOA system using measured RF power levels from the six beam ports is presented. The tests were performed in an anechoic chamber with the incident CP wave generated by a fixed helix antenna, located in the far field of the system under test. The system was rotated with an antenna positioner and measurements at the six BFN output ports [coaxial ports in Fig. 5(b)] were taken over an hemisphere with steps of 1 in and . A network analyzer (Agilent HP8510C) operating in continuous wave at 5.8 GHz was , between the used to measure the transmitted power, i.e., transmit antenna (port A) and each BFN port (port B). These parameters were then normalized according to (6). The NNDE was trained with a sub-set of 1000 directions and in the interval. It randomly chosen with was then tested on another random sub-set of 500 data directions. The uniform step measurement sweeps in and yield a higher density of points as . Therefore, the DOA estimates should be more accurate for smaller values of . Finally, in order to simulate measurement conditions with various noise levels, white Gaussian noise was added independently to each of the antenna output signals prior to the A/D conversion and normalization steps. A digital filter then computed the average of thirty values which was then processed by the NNDE to perform the DOA estimation. The experimental results for the 1x2D implementation of the NNDE with 20 hidden neurons are shown in Fig. 11. In part (a) it can be seen that the estimation error decreases rapidly and then varies slowly after only ten training iterations. Training was stopped after a predetermined value of 100 iterations. However, it can be noticed that near the end the regression error is still slowly decreasing. There is therefore a possibility to achieve

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Fig. 10. Complete DOA system showing the antennas, the beamforming network and the processing stages.

slightly better DOA estimations that those presented in the sequel. The system performance was first evaluated with no noise added to the measured data. It can be seen that the estimated and are within few degrees of the expected angles values. The root mean square (rms) error on the DOA is presented in Fig. 12. In this figure, the DOA error is defined as where the unit vectors are . This gives given by the angular distance between the exact and estimated DOAs. It can be seen that the error increases when we consider a larger angle for the conical sector of operation of the system. This was expected because, as mentioned previously, both the training and testing sets have a higher point density when decreases. It is also shown that if a 1-degree rms error can be tolerated, a conical sector with a half-angle slightly larger than 20 degrees can be used. We then added noise numerically on the measured data. DOA estimations for several noise levels are shown in Fig. 13. It appears that for a SNR of 9 dB, an rms error lower that 4 is achieved over a cone of 45-degree half-angle. For low values of , this error is less than 2 .

Fig. 11. Performance of the DOA system. (a) Regression error for the training and test data sets; (b) and (c) measured accuracy for the estimated  and  respectively.

The performance of the 1x2D NNDC system implemented with 8 hidden neurons is presented in Fig. 14 for a SNR of 10 dB. The solid curve is giving the cumulative probability of false positive error integrated with increasing from zero, whereas the dashed curve is the cumulative probability of false negative error integrated with decreasing from 90 degrees. As expected, this error is higher for DOA approaching the limit of the range

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Fig. 12. RMS DOA error as a function of the half angle of the considered conical sector.

Fig. 13. RMS error on DOA estimation with noise added to the measured data.

of interest, , which is the decision threshold. The results indicate that if the system is operated with a maximum value of 30 degrees, while being trained with a threshold of 45 degrees, the percentage of classification error will be very small (basically zero for the 500 test samples used). If the full 45-degree interval is used, DOA estimates will be made for DOAs that are actually outside the range about 1.7% of the time. As seen in Fig. 13, most of these erroneous DOAs are within 5 degrees of the classification threshold.

VI. CONCLUSION A compact system to determine the DOA of an incident wave with respect to two orthogonal axes was proposed and demonstrated experimentally. Only four patch antennas are used together with a six-beam port BFN, all occupying a planar area of approximately four squared wavelengths. The system implements a classifier that inhibits the response when the DOA is outside a limiting conical sector in which the beam port signals do not allow accurate and unambiguous DOA estimations. The error on the DOA varied from 0.8 to 1.8 for operating

Fig. 14. Classification error versus DOA with measured data and a SNR of 10 dB.

cone half-angles chosen within the 5 to 45 range. The system is robust to noise, as the SNR had to be decreased to 9 dB to double the error on the estimated DOA. These performances are comparable to those presented in [4] and [5] for the case of single angle estimations, with 8 and 6 beam port systems respectively. However, the use of a NN to establish a bijective relationship between the beam amplitude sextuplets and the two angles defining the DOA is leading to very fast estimation, with no need to compute a fitness function over the whole range of solutions. For a SNR level of 10 dB, the simulated 2D systems based on L-shaped arrays presented in [10]–[12] had DOA errors of approximately 1 , 0.25 and 0.5 respectively, with corresponding number of elements of 24, 10 and 10. This is better than the accuracy obtained experimentally with the 4-element system proposed in this paper. However, it is not possible to say if theses simulated models would perform well in a practical case where mutual couplings and antenna manufacturing errors are present. The 8-element circular array system simulated in [13] included mutual coupling effects and achieved an accuracy of approximately 2 (SNR not available), which is comparable to our measured results shown in Figs. 12 and 13. In conventional monopulse systems, the sum and difference patterns require careful control of the sidelobe level in order to achieve optimal DOA estimations [21]. This requirement could possibly be alleviated by using the approach presented here in which NN training compensated for antenna beam patterns that were significantly different than the expected ones. A first concept based on an antenna array composed of 3 antennas per axis and a BFN with 8 beam ports per axis was investigated in [16]. Testing of this system revealed that similar or even better DOA estimations were attainable with the system of reduced complexity presented here.

APPENDIX This appendix derives the requirements on the couplers and delay lines for the design of a BFN of desired characteristics.

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The solution is

(14) The full expressions for

and

are

Fig. 15. Ideal directional coupler and power divider.

First, we need to define an ideal 4-port directional coupler (see Fig. 15). A convenient form for the scattering matrix uses the parameter as follows:

(15) Imposing again that for each of these outputs the antenna signals and contribute with weights of equal magnitudes leads : to following conditions on

(7)

(16) Using (11) and (13) to simplify (16) gives

The coupling coefficient is then defined as

(17)

(8) The scattering matrix of the Wilkinson power divider is

The solution is

(18)

(9) Using these matrices leads us to the expressions in (2). Let us now detail the constraints summarized in (3). From the expression of in (2), we see that the two antennas contributions will be of equal amplitudes if

, the Finally, imposing symmetry with respect to and are equal, brings the following conangle for which dition on the “difference” pattern :

(19) (10) This expression can be simplified to

Using (11), (13) and : tion on

gives the following condi-

(20)

(11) for which the solution is

for which the solution is

. REFERENCES

(12) This solution corresponds to a directional coupler dividing the total input power in parts of 1/3 and 2/3. This solution satisfies also the condition that the three output signals have equal power will have 1/3 of the total available power since the signal at each antenna. From these remarks, we see that the second direction coupler has to be balanced, hence

(13)

[1] W. Hausz and R. A. Zachary, “Phase-amplitude monopulse system,” IRE Trans. Military Electron., vol. MIL-6, pp. 140–146, Apr. 1962. [2] W. Kederer and J. Detlefsen, “Direction of Arrival (DOA) determination based on monopulse concepts,” in Proc. Asia-Pacific Microw. Conf., Sydney, Australia, Dec. 2000, pp. 120–123. [3] W. Kederer and J. Detlefsen, “Comparison of amplitude matching and complex monopulse algorithms with respect to SNR,” AEÜ-Int. J. Electron. and Commun., vol. 57, no. 3, pp. 168–172, 2003. [4] J. Li, G. Fan, and Q. Mei, “A new method to find the direction of radar signal,” in Proc. CIE Int. Conf. Radar, Beijing, China, Oct. 1996, pp. 601–604. [5] E. Taillefer, A. Hirata, and T. Ohira, “Direction-of-arrival estimation using radiation power pattern with an ESPAR antenna,” IEEE Trans. Antennas Propag., vol. 53, pp. 678–684, Feb. 2005.

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[6] H. L. Southall, J. A. Simmers, and T. H. O’Donnel, “Direction finding in phased arrays with a neural network beamformer,” IEEE Trans. Antennas Propag., vol. 43, pp. 1369–1374, Dec. 1995. [7] E. Charpentier and J.-J. Laurin, “An implementation of a directionfinding antenna for mobile communications using a neural network,” IEEE Trans. Antennas Propag., vol. 47, pp. 1152–1159, Jul. 1999. [8] C. Bracco, S. Marcos, and M. Benidir, “Improving the resolution of a sensor array pattern by neural networks,” in Proc. IEEE Workshop on Neural Networks and Signal Processing, Ermioni, Greece, Sep. 1994, pp. 575–584. [9] El Zhooghby, C. G. Christodoulou, and M. Giorgiopoulos, “Performance of radial-basis function networks for direction of arrival estimation with antenna arrays,” IEEE Trans. Antennas Propag., vol. 45, pp. 1611–1617, Nov. 1997. [10] J. F. Fernández de Río and M. F. Cátedra-Pérez, “The matrix pencil method for two-dimensional direction of arrival estimation employing an L-shaped array,” IEEE Trans. Antennas Propag., vol. 45, pp. 1693–1694, Nov. 1997. [11] N. Tayem and H. M. Kwon, “L-shaped 2-dimensional arrival angle estimation with propagator method,” IEEE Trans. Antennas Propag., vol. 53, pp. 1622–1630, May 2005. [12] L. Gan, J.-F. Gu, and P. Wei, “Estimation of 2-D DOA for noncircular sources using simultaneous SVD technique,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 385–388, 2008. [13] T. T. Zhang, Y. L. Lu, and H. T. Hui, “Compensation for the mutual coupling effect in uniform circular arrays for 2D DOA estimations employing the maximum likelihood technique,” IEEE Trans. Aerosp. Electron. Syst., vol. 44, pp. 1215–1221, Jul. 2008. [14] S. M. Sherman, “Complex indicated angles applied to unresolved radar targets and multipath,” IEEE Trans. Aerosp. Electr., vol. 7, pp. 160–170, Jan. 1971. [15] K. A. Gotsis, K. Siakavara, and J. N. Sahalos, “On the direction of arrival (DoA) estimation for a switched-beam antenna system using neural networks,” IEEE Trans. Antennas Propag., vol. 57, pp. 1399–1411, May 2009. [16] J.-J. Laurin, M. Coudyser, J.-J. Brault, and Y. Xu, “A direction finding antenna based on neural networks for space applications,” in Proc. Int. Symp. Antennas (JINA), Nice, France, Nov. 2002, pp. 411–414. [17] J. Blass, “Multi-directional antennas: A new approach to stacked beams,” in IRE Conv. Record, 1960, vol. 8, pp. 48–51, Part 1. [18] N. J. G. Fonseca, “Study and design of a S-band 4 4 Nolen matrix for satellite digital multimedia broadcasting applications,” in Proc. Int. Symp. Antennas Techn. Appl. Electromag. (ANTEM-URSI), Montreal, QC, Canada, Aug. 2006, pp. 481–484. [19] C.-Y. Huang, “Designs for an aperture-coupled compact circularly polarized microstrip antenna,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 146, pp. 13–16, Feb. 1999. [20] S. Haykin, Neural Network, A Comprehensive Foundation, 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 1994. [21] K. W. Lo and T. B. Vu, “Improving performance of monopulse phased array in direction finding,” Proc. Inst. Elect. Eng., vol. 135, pt. H, pp. 391–394, Dec. 1988.

Space - France), and in the Antenna Department, French Space Agency (CNES), Toulouse, France, before joining his current position at the Antenna and Sub-Millimeter Wave Section, European Space Agency (ESA), Noordwijk, Netherlands, in 2009. His interests cover the telecommunication antennas, beam forming network designs, and new enabling technologies such as metamaterials and membranes applied to antenna applications. He has authored or co-authored more than 60 papers in journals and conferences, including two CNES Technical Notes. He holds two patents and has eight patents pending. Mr. Fonseca is a member of the Electromagnetic and Microwave Circuit CNES Technical Competence Center board. He received the special prize of Toulouse City for abroad studies in 2003 and the Best Young Engineer Paper Award at the 29th ESA Workshop on Antennas in 2007. He was also co-recipient of the Best Application Paper Award at the 30th ESA Workshop on Antennas in 2008. He is currently serving as a Technical Reviewer for the Journal of Electromagnetic Waves and Applications - Progress in Electromagnetic Research (PIER), MIT and the IEEE Microwave and Wireless Components Letters (MWCL).

Nelson Jorge G. Fonseca (M’06–SM’09) was born in Ovar, Portugal, in 1979. He received the Electrical Engineering degree from Ecole Nationale Supérieure d’Electrotechnique, Electronique, Informatique, Hydraulique et Telecommunications (ENSEEIHT), Toulouse, France and the Master degree from the Ecole Polytechnique de Montreal, Quebec, Canada, both in 2003. He is currently working toward the Ph.D. degree at Université de Toulouse - Institut National Polytechnique de Toulouse, France. He worked as an Antenna Engineer successively in the Antenna Study Department, Alcatel Alénia Space (now Thalès Alénia

Jean-Jules Brault (A’01) received the B. Eng. degree in engineering physics, and the M.A.Sc. and Ph.D. degrees in electrical engineering from Ecole Polytechnique de Montreal, QC, Canada in 1980, 1983, and 1988, respectively. In 1990, he joined the staff of the Electrical Engineering Department, Ecole Polytechnique de Montreal, where he is presently an Associate Professor. His current interests are mainly in the area of machine learning and artificial neural networks applied to pattern recognition and classification in various areas. He also conducted research in hardware acceleration in bayesian networks.

2

Michael Coudyser graduated from Ecole Polytechnique of France in 1998 and received the M.A.Sc. degree from Ecole Polytechnique de Montreal, QC, Canada, in 2003, where he specialized in neuronal networks. On returning to France in 2003, he joined the PSA Peugeot Citroën Group where he served as Project Manager at the direction of innovation. Since 2007, he works for INEO (electricity group of GDF SUEZ), Paris, France, in the direction of development where he is in charge of the development of the photovoltaic market.

Jean-Jacques Laurin (S’87–M’91–SM’98) received the B.Eng. degree in engineering physics from Ecole Polytechnique de Montreal, Montreal, QC, Canada, and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 1983, 1986, and 1991, respectively. In 1991, he joined the Poly-Grames Research Centre, Ecole Polytechnique de Montreal, where he is currently a Professor. He was an Invited Professor at Ecole Polytechnique Fédérale de Lausanne (EPFL) from 1998 to 1999, and a Visiting Scientist at ESA/ESTEC in 2008. His research interests include antenna design and modeling, near-field antenna measurement techniques, microwave tomography, and electromagnetic compatibility.

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Design of a Beam Switching/Steering Butler Matrix for Phased Array System Chia-Chan Chang, Member, IEEE, Ruey-Hsuan Lee, Student Member, IEEE, and Ting-Yen Shih, Student Member, IEEE

Abstract—A compact broadband 8-way Butler matrix integrated with tunable phase shifters is proposed to provide full beam switching/steering capability. The newly designed multilayer stripline Butler matrix exhibits an average insertion loss of 1.1 dB with amplitude variation less than 2.2 dB and an average phase imbalance of less than 20.7 from 1.6 GHz to 2.8 GHz. The 100 mm2 , which corresponds to an circuit size is only 160 85% size reduction compared with a comparable conventional microstrip 8-way Butler matrix. The stripline tunable phase shifter is designed based on the asymmetric reflection-type configuration, where a Chebyshev matching network is utilized to convert the port impedance from 50 to 25 so that a phase tuning range in excess of 120 can be obtained from 1.6 GHz to 2.8 GHz. To demonstrate the beam switching/steering functionality, the proposed tunable Butler matrix is applied to a 1 8 antenna array system. The measured radiation patterns show that the beam can be fully steered within a spatial range of 108 . Index Terms—Antenna array, beam steering, beam switching, Butler matrix, reflection-type phase shifter.

I. INTRODUCTION

T

HE rapidly developing smart antenna systems require the employment of multiple antennas to create various beam patterns, such as switched beam or continuous beam, based on different beamforming technologies. The tunable phase shifters are often employed in continuous-beam systems [1], [2], whereas the Butler matrix has been widely adopted in switched-beam systems due to its simplicity and easy realization. Traditionally, the Butler matrix has been realized by microstrip lines [3]–[7], and therefore occupies a large area. Consequently, several multilayer Butler matrix configurations have been reported which emphasize size reduction [8]–[11]. CMOS More recently, the multilayer Butler matrix in 0.18-

Manuscript received November 26, 2008; revised June 10, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported in part by the National Science Council of Taiwan, R.O.C., under Grant NSC-97-2221-E-194-003-MY2. C.-C. Chang is with the Department of Electrical Engineering, Department of Communications Engineering, and Center for Telecommunication Research, National Chung Cheng University, 621 Taiwan, R.O.C. (e-mail: ccchang@ee. ccu.edu.tw). R.-H. Lee and T.-Y. Shih are with the Department of Electrical Engineering, National Chung Cheng University, 621 Taiwan, R.O.C. (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.2009.2037693

Fig. 1. (a) Conventional 8-way Butler matrix, (b) proposed switching/steering Butler matrix.

technology is also proposed, and is regarded as the most compact design to date [12], [13]. For a conventional -way Butler matrix, one can only gensets of phase distribution. Fig. 1(a) exhibits a convenerate tional 8-way Butler matrix with 8 radiation beams distributed spatial coverage. The relative progressive phase within and the corresponding beam directions , resulting shifts from different feeding ports, are listed in Table I. In some applications, high beam resolution may be required. Although increasing the order of can improve the beam resolution, the circuit size of the Butler matrix will also grow impractically large. Another beamforming approach is to use tunable phase shifters to generate the continuous beam. However,

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this requires a significant amount of phase tuning for the same spatial coverage, thus implying a great design challenge. In this work, an enhanced switching/steering Butler matrix combining both techniques is proposed for the first time. As shown in Fig. 1(b), the radiation beam is initially switched to a certain direction through the Butler matrix, and then slightly adjusted by the tunable phase shifters. The phase shifters are only responsible for a small steering range between two adjacent beams. By using this approach, the beam resolution is dramatically improved, while the Butler matrix still remains low order and only a small amount of phase tuning is needed. It not only helps in the size compactness, but also alleviates the design difficulty of the beamforming circuitry. To verify the above beamforming topology, an 8-way Butler matrix integrated with eight tunable phase shifters is introduced here. In addition to its enhanced functionality, this switching/steering Butler matrix is also intended to achieve wideband performance and ultra-compact circuit size. Therefore, multilayer stripline is chosen as the transmission medium in this design. This paper is organized as follows. Section II describes the circuit designs in detail. Experimental results of array system demonstration are presented and discussed in Section III, and a conclusion is drawn in Section IV.

TABLE I THE BEAM DISTRIBUTIONS OF AN 8-WAY BUTLER MATRIX (RELATIVE PROGRESSIVE PHASE SHIFT  , BEAM DIRECTION  )

1

II. DESIGN OF SWITCHING/STEERING STRIPLINE BUTLER MATRIX A. Stripline Butler Matrix This Butler matrix consists of three 25N printed circuit boards , ) to create a stripline configuration. ( The circuit layouts on the inner board are shown in Fig. 2(a), while the lay-ups of this multilayer structure with corresponding board thickness are shown in Fig. 2(b). For the inner board, the black lines (layer2) are patterned on the front side, while the gray ones (layer3) are laid on the backside. The top (layer1) and bottom (layer4) metals serve as ground planes to prevent outside interferences. To achieve broadband performance, several key elements, including 3-dB couplers, fixed phase shifters, and crossovers, are re-designed for this Butler matrix. Similar to the structure proposed in [8], the 3-dB coupler is designed based on the broadside coupling configuration, where two coupled lines are placed opposite each other on the inner board. However, owing to the choice of inner board thickness, it is found that 50- coupled lines cause the coupling coefficient to exceed 3-dB when they are fully overlapped. To reduce the value of the coupling, the coupled lines are laid with a slight offset so that both the 50impedance and the 3-dB coupling can be simultaneously retained. Over a frequency range of 1.6 to 2.8 GHz, the simulation , the amplitude results show that the phase balance is balance is 0.3 0.7 dB, the return loss is better than 22 dB, and the isolation is better than 20 dB, respectively. Conventionally, the fixed phase shifters incorporated in a Butler matrix employ a length difference to generate the requisite phase shift. However, the operating bandwidth is limited due to its narrowband nature. To design a broadband Butler matrix, the Schiffman phase shifter is utilized herein. In a standard Schiffman phase shifter, a coupled section and a

Fig. 2. The structure of the stripline Butler matrix (a) inner board layouts (b) the lay-ups of three PCBs with corresponding board thickness.

transmission line are used to give a differential phase shift. By properly selecting the length of the coupled section as well as the coupling coefficient, a nearly constant phase difference can be produced over a certain bandwidth [11]. In this 8-way Butler matrix, three different values of phase shifters including 22.5 , 45 and 67.5 are required. The simulations predict that a very is obtained in those three cases stable phase variation of from 1.6 to 2.8 GHz.

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Fig. 3. Measured power distributions of proposed 8-way stripline Butler matrix. Signal is fed at (a) port-1, (b) port-2, (c) port-3, (d) port-4.

Crossover is achieved when two lines are intersected on the opposite sides of the inner board. However, due to the thin thickness of the inner board, a strong coupling could be produced in the intersection region, with concomitant degradation in the isolation. To solve this problem, crossing lines with reduced line width are employed in the intersection to decrease the coupling capacitance [8]. Although the isolation is improved significantly, the return loss is also degraded by the impedance mismatch from the discontinuity. Therefore, four curved open stubs were added around the crossover region to further improve the impedance matching. The simulations show that an isolation better than 20 dB is obtained over 1.6–2.8 GHz. The circuit has been realized with those three PCBs stacked up carefully by the screws, and the overall size is about . To provide a quantitative estimate of the amount of size reduction, an 8-way microstrip Butler matrix, which is extended from [7] wherein a 4-way Butler matrix was reported, has also been designed for comparison purposes. The result shows that nearly 85% size reduction is achieved by this proposed circuit. The measured power distributions are presented in Fig. 3(a)–(d) when the input signal is fed from port-1 to port-4, respectively. Because of the symmetry of the Butler

matrix, the results with the signal fed from port-5 to port-8 are not shown here. Over the frequency range of 1.6 to 2.8 GHz, the average insertion loss is 10.1 dB (9 dB from the theoretical distribution loss) with amplitude imbalance less than 1.5 dB. Fig. 4(a)–(d) show the measured output phase distributions. By feeding the signal at port-1 to port-4 in sequence, the , , relative progressive phase shift of , are obtained, respectively. It is found that the layout deviation causes the phase errors to exceed the expectation. However, this problem is less crucial in the proposed switching/steering Butler matrix since the following tunable phase shifter can be used to calibrate this phase error. B. Tunable Phase Shifter As seen in Table I, there is a 45 phase difference in required progressive phase shift for each two adjacent beams. To progressive phase achieve full scanning, an additional shift is needed so that the radiation beam can be steered either upward or downward to the midway toward its neighboring beams. Therefore, the maximum phase tuning of each tunable phase shifter is 157.5 . In this work, the reflection-type phase shifter (RTPS) is utilized for the tunable phase shifter design. Fig. 5 illustrates the

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Fig. 4. Measured output phase distribution of proposed 8-way stripline Butler matrix. Signal is fed at (a) port-1, (b) port-2, (c) port-3, (d) port-4.

where

(2) to Assuming that the load reactance is varied from , the maximum phase tuning range can be determined Fig. 5. Schematic diagram of a reflection-type phase shifter.

schematic diagram of an RTPS, which is composed of a 3-dB quadrature coupler with port-2 and port-3 terminated with two identical reflection loads. Ideally, the reflection load is pure reactive (lossless) such that the output signal will become the duplicate of the input signal with a relative phase shift equal to the phase of the reflection coefficient of the load. Varactors are often employed in the reflection loads to provide the tunable reactance. Hence, the phase shift can be varied using different DC bias voltages [14]. The reflection coefficient from the load is given as

(1)

(3) It is apparent from (3) that the maximum phase shift can be is close to infinity. Unfortuachieved when the ratio of nately, a given commercial varactor usually provides a limited . In order to increase the ratio capacitance tuning range , the terminal port impedance is also reduced so of that the phase tuning range can be further enhanced [15]. In this case, the input/output port (port-1/4) impedance is kept as 50 , where the terminal port (port-2/3) impedance is chosen as 25 to create an asymmetrical coupler. To transfer the port impedance from 50 to 25 , a 4th-order Chebyshev

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Fig. 8. Schematic of the proposed stripline reflection-type phase shifter and implemented circuit photograph. Fig. 6. The structure of an asymmetric stripline 3-dB quadrature coupler.

Fig. 7. The measured impedance trajectory of the load (@2.45 GHz) (a) Varactor SMV 1763 only, (b) Varactor is connected in series with an inductor and a DC-blocking capacitor.

impedance matching network is utilized [16], where the maximum allowable reflection coefficient magnitude in the passband is chosen as 0.05 and the section number is chosen as 4. The calculated characteristic impedance of each section is indicated on Fig. 6. In order to integrate with the previous stripline Butler matrix, this tunable phase shifter is also designed based on a stripline medium. The design challenge of this asymmetrical broadside coupler is not only achieving the impedance transformation, but also maintaining a constant coupling value during the transition. Therefore, as shown in Fig. 6, the two coupled lines are slightly stretched out to keep the overlapped area uniform during the transformation so that a constant coupling coefficient can be maintained. Different loading circuitries of an RTPS have been studied in the literature to either maximize the relative phase tuning range [17]–[22] or widen the operation bandwidth [23]–[26]. The load in this work is composed by a series-resonated circuitry, where a varactor diode is connected in series to an inductor. The varactor chosen in this work is the Skywork SMV-1763 with and within 0–5 volts.

Fig. 9. The measurement results of an asymmetric stripline reflection-type phase shifter (a) S (b) maximum phase tuning range.

As revealed in Fig. 7(a), the measured impedance trajectory of a single varactor is closely along the periphery on the lower Smith chart. With an additional inductor (2.7 nH) added, the trajectory can be relocated symmetrically across the zero reactance

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axis of the Smith chart. As a result, a larger angle span of 93.4 is obtained, shown in Fig. 7(b). An additional capacitor (1.5 pF) serving as a DC block is also taken into consideration in the loading circuitry design. Fig. 8 shows the schematic drawing and implemented circuit photograph of the proposed stripline reflection-type phase shifter. To conserve the circuit area, the loading circuitries are placed on the top and bottom ground planes by connecting to the coupler through the vias. Additionally, a resistor of 12 provides the choke of the RF signal. Fig. 9(a), (b) exhibit the measured scattering parameters and phase shifts of this asymmetric stripline RTPS. Over the frequency range of 1.6–2.8 GHz, the average insertion loss is increased from 2 dB to 3 dB with a loss variation of 1 dB along with phase tuning. The loss variation is due to the asymmetric RTPS structure. The maximum phase shift is 142 at 2 GHz and remains greater than 120 within the entire band. It is found that the parasitic induced by the vias degrades the loading circuitry performance, and causes a degradation of the phase tuning range from the expectation. C. Circuit Integration Fig. 10 shows the photos of the proposed switching/steering Butler matrix, which consists of an 8-way stripline Butler matrix integrated with eight asymmetric tunable phase shifters. The overall circuit size is 175 mm 135 mm. Fig. 10. The photos of proposed switching/steering 8-way stripline Butler matrix with tunable phase shifters (a) inner board layouts and (b) top view of the integrated circuit.

Fig. 11. The planar monopole antenna with backside reflector (a) configuration and (b) measured E-plane (y-z plane) and H-plane (x-z plane) patterns.

III. ANTENNA ARRAY MEASUREMENT To demonstrate the beam switching/steering functionality, the proposed Butler matrix is connected with a 1 8 antenna array, where an H-plane broadband rectangular planar monopole antenna [27], [28] with backside reflector is chosen as the array element. The antenna configuration and measured antenna patterns are shown in Fig. 11. The measured array patterns at 2.45 GHz are illustrated in ” in each figure indicates Fig. 12. The statement “ the progressive phase difference of port -port (N), where is contributed from 8-way Butler matrix and is generated by the tunable phase shifters. Again, only half patterns (1L-4L) are shown because of the circuit symmetry. Table II summarizes the measured beam direction and corresponding array power gain. For each case, the main beam direction is initially contributed by the Butler matrix, and then driven to the maximum steerable angle using the tunable phase shifter. , and can be For the 1L case, the initial beam direction is to 2 . For the 2L case, the initial beam direcsteered from , and can be steered from to . In the 3L tion is to while the initial case, the steering range is from beam direction is . For the 4L case, the initial beam direc, and can be steered from to . However, tion is the power gain drops significantly in this case, especially when . This is mainly caused by the the beam is steered beyond gain drops from the antenna element. As seen in Fig. 11(b), the antenna gain in H-plane drops at least 2 dB when the beam is compared to the gain at 0 . Therefore, the steered beyond overall scanning range in this demonstration is considered as 108 . With different antenna elements, the spatial coverage can be further enhanced. In addition to the influence of the antenna

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Fig. 12. Measured antenna array patterns (a) 1L (fed from port-8), (b) 2L (fed from port-4), (c) 3L (fed from port-6), (d) 4L (fed from port-2).

TABLE II MEASURED BEAM DIRECTION AND RELATED ARRAY POWER GAIN

element pattern, note that the phase shifters under different bias conditions introduce different degrees of insertion loss, which also causes the gain variation and dissimilar patterns between the upward and downward tuning cases. Since the tunable phase shifter can be precisely controlled, the beam resolution is better than 2 .

IV. CONCLUSION In this paper, a new switching/steering 8-way Butler matrix integrated with eight tunable phase shifters is proposed for continuous beam scanning capability. The featured components of a Butler matrix, including 3-dB couplers, Schiffman phase shifters and crossovers, are designed using the multilayer stripline medium to achieve the broadband performance and compact size. Over the range 1.6 GHz to 2.8 GHz, the measured insertion loss is 1.1 dB with a loss variation of 2.2 dB. A stripline asymmetric reflection-type phase shifter is chosen as the tunable phase shifter in this work to provide the additional phase tuning. For a given varactor with a limited capacitance ratio, the relative phase shift range can be further increased when the quadrature coupler has the unequal port impedances. Therefore, a Chebyshev impedance transformer is employed to 25 . Over the to convert the port impedance from 50 bandwidth of 1.6 GHz to 2.8 GHz, the measured return loss is better than 10 dB, the maximum tunable phase shift is greater than 120 , and the average insertion loss is 2 dB to 3 dB with a loss variation of 1 dB along with phase tuning. The overall circuit size of this switching/steering Butler matrix is 175 mm 135 mm, which is extremely compact compared to the conventional microstrip Butler matrix. From the

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array measurement, a full scanning range of 108 with beam resolution better than 2 is demonstrated. Instead of using the Butler matrix or phase shifters alone, the proposed work takes advantages of both techniques. The apparent benefits are that we can reduce the circuit size (the low-order Butler matrix) and design difficulty (smaller phase tuning range of the phase shifter). This proposed work shows great potential for various applications, such as a tracking system where the target can be roughly detected by the switched beams and then precisely tracked using the steering beams. Although this work demonstrates an 8-way switching/steering Butler matrix, the same technique can also be applied to a 4-way Butler matrix in a similar manner, which is more influential since the increase of the beam resolution is more significant.

REFERENCES [1] T. Yu and G. M. Rebeiz, “A 24 GHz 4-channel phased-array receiver in 0.13  CMOS,” in IEEE Radio Freq. Integr. Circuits Symp. Dig., Apr. 2008, pp. 361–364. [2] K. J. Koh and G. M. Rebeiz, “An X- and Ku-band 8-element phasedarray receiver in 0.18-  SiGe BiCMOS technology,” IEEE J. SolidState Circuit, vol. 43, pp. 1360–1371, June 2008. [3] F. Cladwell, J. S. Kenney, and I. A. Ingram, “Design and implementation of a switched-beam smart antenna for an 802.11b wireless access point,” in Proc. Radio and Wireless Conf., Aug. 2002, pp. 55–58. [4] R. Comitangelo, D. Minervini, and B. Piovano, “Beam forming networks of optimum size and compactness for multibeam antennas at 900 MHz,” in Proc. Antennas Propag. Society Int. Symp., Jul. 1997, vol. 4, pp. 2127–2130. [5] T. Bechteler, B. Mayer, and R. Weigel, “A new high-temperature superconducting double-hybrid coupler with wide bandwidth,” in IEEE MTT-S Int., Microw. Symp. Dig., Jun. 1997, pp. 311–314. [6] H. Hayashi, D. A. Hitko, and C. G. Sodini, “Four-element planar Butler matrix using half-wavelength open stubs,” IEEE Microw. Wireless Compon. Lett., vol. 12, pp. 73–75, March 2002. [7] M. R. C. Rose, S. R. M. Shah, M. F. A. Kadir, D. Misman, M. Z. A. Aziz, and M. K. Suaidi, “The mitered and circular bend method of Butler matrix design for WLAN application,” in Proc. IEEE Asia-Pacific Applied Electron. Conf., Dec. 2007, pp. 1–6. [8] M. Bona, L. Manholm, J. P. Starski, and B. Svensson, “Low-loss compact Butler matrix for a microstrip antenna,” IEEE Trans. Microw. Theory Tech., vol. 50, pp. 2069–2075, Sep. 2002. [9] M. Nedil, T. A. Denidni, and L. Talbi, “Novel Butler matrix using CPW multilayer technology,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 499–507, Jan. 2006. [10] S.-F. Chang, J.-L. Chen, Y.-H. Jeng, and C.-T. Wu, “New high-directivity coupler design with coupled spurlines,” IEEE Microw. Wireless Compon. Lett., vol. 14, pp. 65–67, Feb. 2004. [11] J. L. R. Quirarte and J. P. Starski, “Synthesis of Schiffman phase shifters,” IEEE Trans. Microw. Theory Tech., vol. 39, pp. 1885–1889, Nov. 1991. [12] C.-C. Chang, T.-Y. Chin, J.-C. Wu, and S.-F. Chang, “Novel design of a 2.5-GHz fully integrated CMOS Butler matrix for smart-antenna systems,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 1757–1763, Aug. 2008. [13] T.-Y. Chin, S.-F. Chang, C.-C. Chang, and J.-C. Wu, “A 24-GHz CMOS Butler matrix MMIC for multi-beam smart antenna systems,” in IEEE Radio Freq. Integr. Circuits Symp., Dig., Jun. 2008, pp. 633–636. [14] R. N. Hardin, E. J. Downey, and J. Munushian, “Electronically variable phase shifter utilizing variable capacitance diodes,” Proc. IRE, vol. 48, no. 5, pp. 944–945, May 1960. [15] C.-S. Lin, S.-F. Chang, C.-C. Chang, and Y.-H. Shu, “Design of a reflection-type phase shifter with wide relative phase shift and constant insertion loss,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 9, pp. 1862–1868, Sept. 2007. [16] D. M. Pozar, Microwave Engineering, 2nd ed. , Canada: Wiley, 1998. [17] R. V. Garver, “Broadband binary 180 diode phase modulators,” IEEE Trans. Microw. Theory Tech., vol. 13, pp. 32–38, Jan. 1965.

m

m

[18] B. T. Henoch and P. Tamm, “A 360 reflection-type diode phase modulator,” IEEE Trans. Microw. Theory Tech., vol. 19, pp. 103–105, Jan. 1971. [19] Y. H. Liew, J. Joe, and M. S. Leong, “A novel 360 analog phase shifter with linear voltage phase relationship,” in Proc. IEEE Asia-Pacific Microwave Conf., Dec. 1999, pp. 17–20. [20] S. Shin, R. V. Snyder, and E. Niver, “360-degree linear analog phase shifter design using tunable short-circuit terminated combline filters,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2001, pp. 303–306. [21] F. Ellinger, R. Vogt, and W. Bächtold, “Compact reflective type phase shifter MMIC for C-band using a lumped element coupler,” IEEE Trans. Microw. Theory Tech., vol. 49, pp. 913–917, May 2001. [22] K. O. Sun, H. J. Kim, C. C. Yen, and D. Weide, “A scalable reflection type phase shifter with large phase variation,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 10, pp. 647–648, Oct. 2005. [23] R. V. Garver, “360 varactor linear phase modulator,” IEEE Trans. Microw. Theory Tech., vol. 17, pp. 137–147, Mar. 1969. [24] T. W. Yoo, J. H. Song, and M. S. Park, “Phase shifter with high phase shifts using defected ground structures,” Electron. Lett., vol. 41, no. 4, pp. 196–197, Feb. 2005. [25] S. Lucyszyn and I. D. Robertson, “Synthesis techniques for high performance octave bandwidth 180 analog phase shifters,” IEEE Trans. Microw. Theory Tech., vol. 40, pp. 731–740, Mar. 1992. [26] C. T. Rodenbeck, S.-G. Kim, W.-H. Tu, M. R. Coutant, S. Hong, M. Li, and K. Chang, “Ultra-wideband low-cost phased-array radars,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 3697–3703, Dec. 2005. [27] G. Kumar and K. P. Ray, Broadband Microstrip Antennas. Boston: Artech House, 2003. [28] T.-Y. Shih, C.-L. Li, and C.-S. Lai, “Design of an UWB fully planar quasi-elliptic monopole antenna,” presented at the ICEMAC, 2004. Chia-Chan Chang (S’99–M’04) received the B.S. degree in communication engineering from National Chiao-Tung University, Hsinchu, Taiwan, in 1995, the M.S. and Ph.D. degrees in electrical and computer engineering from the University of California at Davis (UCD), CA, in 2001 and 2003, respectively. She was a full-time Teaching Assistant with the Department of Electronics Engineering at National Chiao-Tung University from 1995 to 1997. In February 2004, she joined the faculty of the Department of Electrical Engineering at National Chung-Cheng University, Chiayi, Taiwan, as an Assistant Professor, becoming an Associate Professor in 2009. She also holds a joint-appointment with the Department of Communications Engineering. Her current research interests include phased antenna array technologies, microwave/millimeter-wave integrated circuit (IC) designs, and the application of radar systems.

Ruey-Hsuan Lee (S’09) was born in Taipei, Taiwan, R.O.C., in 1983. He received the B.S. degree in electrical engineering from National Chung Hsing University, Taiwan, R.O.C., in 2005 and the M.S. degree in electrical engineering from the National Chung Cheng University, Taiwan, R.O.C., in 2007, where he is currently working toward the Ph.D. degree. His research interests include development of beamforming circuits, phased array antenna system, and radar-based locating applications.

Ting-Yen Shih (S’06) received the B.S. degree in electrical engineering from Tamkang University, Taipei, Taiwan, R.O.C., in 2004 and the M.S. degree in electrical engineering from the National Chung Cheng University, Chiayi, Taiwan, R.O.C., in 2006. He completed his military service in 2008. Currently, Currently, he is an Assistant Research Fellow in Center of Telecommunication Research at National Chung-Cheng University. His research interests include the design and analysis of high frequency circuits and antennas, beamforming technologies, and the application of radar and sensor systems.

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Novel Composite Phase-Shifting Transmission-Line and Its Application in the Design of Antenna Array Xian Qi Lin, Member, IEEE, Di Bao, Hui Feng Ma, and Tie Jun Cui, Senior Member, IEEE

Abstract—A novel composite phase-shifting transmission line (TL) with designable characteristics is presented, which can be used to achieve arbitrary phase of the transmission coefficient at any required frequency with a certain length of the TL. An empirical formula is given of the relationship between the phase and physical length of the composite TL at a required frequency. A sample of 0 phase-shifting TL is designed in details, and is verified by the full-wave simulation. At the required frequency of 5 GHz, the amplitude of S21 is equal to 0 23 dB with a phase of 0 467 . The electric length is only 0 212 0 , which has been decreased by 68.5% compared to the conventional microstrip line. Using the proposed composite TL, an antenna array is designed with two radiation patches excited by the novel series feed-line. The detailed procedure of such design is presented. The lowest reflection coefficient is exactly achieved at the required frequency of 5 GHz. The maximum radiation is obtained at 0 = 0 , which indicates that the 0 phase-shifting TL works very well. The sample is also fabricated and good agreements between simulation and measurement results are obtained. Index Terms—Antenna array, composite phase-shifting transmission-line, microstrip line, series feed-line.

I. INTRODUCTION

P

HASE-SHIFTING transmission-line (TL) is widely used in the design of microwave components and antennas, such as phase shifters, 3-dB branch line couplers, balun, circular polarization feed-lines and series-fed antenna array [1]–[5]. In the conventional design, a linear phase response of the TL is utilized, and hence the phase shift is correlated to the electric length of TL, which leads to considerably large sizes and limited functions. In 2002, a novel left-handed (LH) TL was proposed simultaneously by three different research groups [6]–[8]. Negative permittivity and permeability were obtained in broader frequency bands simultaneously. Considering the Manuscript received October 20, 2008; revised May 04, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported in part by the National Science Foundation of China under Grants 60871016, 60671015, 60601002, 60621002, and 60901022, in part by the Natural Science Foundation of Jiangsu Province under Grant BK2008031, in part by the National Basic Research Program (973) of China under Grant 2004CB719802, in part by the 111 Project under Grant 111-2-05, and in part by the Foundation for Excellent Doctoral Dissertation of Southeast University. X. Q. Lin is with the with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China and also with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China. D. Bao, H. F. Ma, and T. J. Cui are with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, 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.2009.2037764

parasitical parameters of real structures of LH TLs, Itoh et al. proposed a concept of composite right/left handed (CRLH) TL. After that, different structures of CRLH TLs have been presented, and many novel microwave components and antennas, such as hybrids, couplers, filters, phase shifters and antenna arrays, have been designed using the unusual characteristics of CRLH TLs [9]–[15]. Compared to the conventional TLs, CRLH TLs have nonlinear phase responses and can achieve arbitrary phases of the transmission coefficient independent of the electric length at any required frequency [9], [16], [17]. However, the balance between left-handed and right-handed frequency-bands is highly sensitive to the detailed geometrical dimensions of CRLH TLs and we have to redesign the CRLH structures when the required phase or working frequency is changed. In another word, it is difficult to design arbitrary phase response just by adjusting the length of CRLH TLs as the linear conventional TLs done. In our earlier work, we proposed a novel compact CRLH structure [18], in which good balance is kept while the length of CRLH structure is changed. Using such a novel CRLH structure, a series of super-wide bandpass filters were designed and fabricated at different frequency bands by changing solely the length of the structure. Good agreements between simulation and experiment results have been achieved, and good performance in the passband and stopband has been observed with a relative 3-dB bandwidth larger than 70%. In this paper, we focus our attentions on the phase responses of such compact CRLH structure. A concept of designable composite phase-shifting TLs is presented in Section II. An empirical formula is given based on which the arbitrary phase can be easily designed at a required frequency of 5 GHz. A sample of 0 phase-shifting line is designed in details and is verified by the full-wave simulation. Using the proposed composite phaseshifting TLs, an antenna array is designed with two radiation patches excited by novel series feed-lines in Section III. The detailed procedure of such design is proposed, and the lowest reflection coefficient is exactly achieved at the required frequency of 5 GHz. The sample is also fabricated and good agreements between simulation and measurement results are obtained. We present conclusions in Section IV. II. DESIGN OF COMPOSITE PHASE-SHIFTING TRANSMISSION LINE The proposed novel designable composite phase-shifting TL is shown in Fig. 1, which consists of a center CRLH TL with non-linear phase response and two sections of microstrip lines with linear phase response. The CRLH TL is constructed by two unit cells of the CRLH structure which is the same as that in [18].

0018-926X/$26.00 © 2009 IEEE

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Fig. 1. The structure of composite phase-shifting TL.

Two sections of microstrip line are used for the connection to other components whose length can be adjusted. The composite phase-shifting TL is fabricated on a substrate of F4B-1/2, which is made of polytetra-fluoroethylene and glass fiber with a thickand a relative permittivity of . ness of The loss tangent of such substrate is 0.001. The feed-lines at . Four two ports are 2.2 mm wide, which are matched to , , peg holes are used to fix the sample. The parameters , indicate the distance between and two metallic vias, the length of interdigital capacitor, the total length of two microstirp lines, and the total length of composite phase-shifting TL, respectively. According to [19] and [20], in which the substrate integrated waveguide is synthesized using dielectric substrate with linear arrays of metallic vias, the sizes of via-walls can be chosen with the following relation

(1) is the free-space wavelength at the operating frewhere quency, is the diameter of vias and is the distance between and . two vias. Here, we choose By using the via-hole arrays, such novel structure is easy to be integrated with other components and side coupling is is increased by addigreatly reduced. More importantly, tional coupling from interdigital capacitor to two side vias-wall and the balance condition of at similar level of is easier to be preserved while single length of is and refer to the shunt stub inchanged, where ductor shorted to via walls, the series interdigital capacitor, the shunt capacitance and series inductance provided by the natural parasitics of interdigital capacitor and stub inductor, recan be seen spectively. The detailed values versus different in [18]. Here, we mainly focus our attentions on the phase responses of such compact structure. We first study the phase responses of the proposed CRLH TL. Since it is difficult to obtain analytical formula of such complicated structures, empirical design formula based on experiment or full-wave simulation results is adopted. In our work, the commercial software CST Microwave Studio 5 [21] is used to achieve the simulation results. De-embedding technology is

Fig. 2. The amplitude and phase of S plitude and (b) phase.

versus different values of l

(a) Am-

used and the phase reference plane is fixed at the interface of microstrip line and CRLH TL, so that only the phase response is studied. of the center CRLH TL versus different values of with Fig. 2 illustrates the amplitude and phase of . From Fig. 2(a) we clearly observe that good balance is kept while the length of CRLH TL is changed. We also list the main performances at the required frequency of 5 GHz in refers to the phase response of CRLH TL. Table I, where Non-linear phase response can be observed from Fig. 2(b) and Table I, which can be characterized by an empirical formula deand : scribing the relation between

(2) For the microstrip line, the phase response can be given as [1]

(3)

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TABLE I MAIN PERFORMANCES OF CRLH TLS AT THE REQUIRED FREQUENCY OF 5 GHz

where is the phase constant of wave propagation, is the effective permittivity of the microstrip line and can be obtained as [1] Fig. 3. Simulated S-parameters of the novel 0 phase-shifting line.

(4) in which and are the thickness and is the width of the relative permittivity of the substrate, and microstrip line. At the required frequency of 5 GHz, we have (5) Finally, we have the total phase response posite TL as

of the novel com-

(6) Using (2), (5) and (6), we can easily obtain an arbitrary phase response at the required frequency of 5 GHz. Take the 0 phaseshifting line for example, if we choose , we will and . It can be easily calcuhave . We late from (2) that the needed length is simulate the designed structure by the CST Microwave Studio 5. Fig. 3 illustrates the amplitude and phase of S-parameters, is equal from which we easily observe that the amplitude of with the phase of at the required freto quency of 5 GHz. The total length of the composite TL is , which is decreased by 68.5% compared to the conventional phase-shifting microstrip line. We also remark by the data shown in Fig. 2(b) that we can develop empirical formulas at other frequencies based on which phase shifting TLs with different physical lengths can be achieved III. ANTENNA ARRAY DESIGN USING NOVEL SERIES FEED-LINE There are two basic kinds of feeding modes in the antenna array design. One is series feeding, and the other is parallel feeding. For the parallel feeding, uniform feed-direction with the same length of feed-lines is required, while in the series mode 0 phase-shifting feed-line is needed. The conventional series feed-line between two radiation cells is obtained as

(7)

where is the light speed and is calculated by (4). At the required frequency of 5 GHz, we have . However, the distance between two radiation cells is limited by the condition [22]

(8) where is the maximum radiation angle. In the side radiation and ; while in the end radiaarray, we have tion array and . This implies that we have to curve the series feed-line in some array designs. [10] and [13] proposed two kinds of non-linear series feed-lines and novel antenna arrays were fabricated. However, both designs are sensitive to the detailed geometries and one has to redesign the structures when the required phase, the frequency, or the distance between radiation cells is changed. Moreover, an added discontinuity was presented in such novel non-linear series feed-lines which will influence the final radiation characteristics. Here, we design a new antenna array series fed using our novel composite phase-shifting TL. With the effect of two periodical metallic vias, the isolation for side coupling is much better than those arrays series fed by the conventional TLs or CRLH TLs. The schematic structure is illustrated in Fig. 4 and the detailed procedure is described as follows. • Choose the primary values. The substrate of F4B-1/2 with and a relative permittivity of a thickness of is chosen. The feed-line at input port . has a width of 2.2 mm, which is matching to • Design the 0 phase-shifting feed-line. In our antenna array design, rectangle radiation patch is selected. The smaller is, the larger the radiation resistance is. Here, the width and we choose geometrical parameters as . From (5) and (6), we obtain . Solving the (2), we finally obtain the needed length . as • Design the unit cell of radiation patch. We have selected . At the frequency of 5 GHz, the the width length of the patch can be obtained as [4] (9)

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Fig. 4. The structure of antenna array with two radiation patches excited by the novel series feed-line.

(10)

From (9) and (10), we obtain the primary value of . Using the parameter sweeping in the CST sim. ulation, we finally obtain a suitable length • Design the impedance matching TL. We present a quarter wavelength TL to match the characteristic impedance of input port and the input impedance of the antenna array. . The length can be obtained as The radiation resistance and the input impedance of one radiation patch are given by [4] (11) (12) is the distance between the feeding where point and the edge of width side, and is the width of the radiation patch. From (11), we obtain . Equation (12) gives the input impedance of radiation patch fed at the width side. However, we can use the equation to obtain the primary values of the antenna array fed at the length side. For one radiation patch, we , from which we can obtain have the required characteristics impedance of the matching TL . Limited by our as fabrication technology, we choose the width impedance with the characteristics matching TL as . impedance of • Full wave simulations in CST. Setting the geometry as , , , , , in the CST simulator, we finally obtain and the reflection coefficient and direction coefficient in the H-plane, as shown in Fig. 5(b) and (c), respectively. At the required frequency of 5 GHz, the minimum reflection coefwith frequency-band ficient is of 38.42 MHz and half-power radiation angle of 64.4 . The , which implies that maximum radiation angle is the 0 series feed line works very well. At the frequencies of 4.48 GHz and 5.904 GHz, we also have two peaks

Fig. 5. The measurement results of antenna array with two radiation patches excited by novel series feed-line (a) Photograph. (b) Amplitude of S . (c) Simulated direction coefficient in H-plane. (d) Measured normalized radiation pattern of power in H-plane.

of minimum reflection coefficient. However, the maximum and 25 , respectively. radiation angle are

LIN et al.: NOVEL COMPOSITE PHASE-SHIFTING TRANSMISSION-LINE AND ITS APPLICATION

• Fabrication and measurement of the sample. The photograph of the fabricated sample is shown in Fig. 5(a) and normalized radiation pattern with the measured in Fig. 5(b) and (d), respectively. We obtain the minimum at the frequency reflection coefficient of 5.068 GHz with the relative error of 1.36%. The measured gain is 5.04 dB with the half-power radiation angle of 58.6 . A very good agreement is achieved between the simulation and measurement results. We remind that the total physical length of can be changed and at the just by tuning different combinations of working frequency of 5 GHz for variable radiation parameters. We also remark that the proposed design procedure can be widely used in other antenna array designs at different frequencies to achieve compact sizes and lower coupling between the feed-lines and radiation patches. IV. CONCLUSION We present a novel designable composite phase-shifting TL. Arbitrary phase response is achieved by tuning the combination of different CRLH and microstrip lines at the required frequencies and even with required total length of the structure. The empirical formula is proposed based on several primary simulation results. Using the formula, we accurately design a 0 phase-shifting line. The electric length is only about , which has a reduction of 68.5% compared to the conventional microstrip line. Moreover, such a composite line has minimized side coupling and hence suffers little coupling effects when integrated with other components or antennas. In order to illustrate the useful and designable characteristics of our proposed composite phase-shifting TL. A compact antenna array with two radiation patches fed by the novel series feed-line is presented using the proposed composite line. Detailed design procedure is extracted and a very good agreement between simulation and measurement results is achieved. Compared to the conventional composite series line based on other structures, the proposed structure and design procedure prove to be a convenient way to design other antenna arrays at various frequencies with a fixed length of the series feed-line. REFERENCES [1] Microstrip Circuits. Beijing, China: Tsinghua Univ. and Post & Telecom Press, 1976. [2] I. Bahl and P. Bhartia, Microwave Solid State Circuit Design. New York: Wiley, 1998. [3] D. M. Pozar, Microwave Engineering, 3rd ed. New York: McGrawHill, 2003. [4] I. J. Bahl and P. Bhartia, Microstrip Antennas. Boston, MA: Artech House, 1980. [5] J. D. Kraus and J. M. Ronald, Antennas: For All Application, 3rd ed. New York: , 2002. [6] C. Caloz and T. Itoh, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip ‘LH line’,” in Proc. IEEE AP-S Int. Symp., San Antonio, TX, Jun. 2002, vol. 2, pp. 412–415. [7] A. A. Oliner, “A periodic-structure negative-refractive-index medium without resonant elements,” in IEEE AP-S/URSI Int. Symp. Dig., San Antonio, TX, Jun. 2002, p. 41.

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[8] A. K. Iyer and G. V. Eleftheriades, “Negative refractive index metamaterials supporting 2-D waves,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, Jun. 2002, pp. 1067–1070. [9] A. Lai, C. Caloz, and T. Itoh, “Composite right/left-handed transmission line metamaterials,” IEEE Microwave Mag., pp. 34–50, Sep. 2004. [10] Q. Zhu, Z. X. Zhang, and S. J. Xu, “Millimeter wave microstrip array design with CRLH-TL as feeding line,” in IEEE Antennas and Propagation Society Int. Sympo., Jun. 2004, vol. 3, pp. 3413–3416. [11] S.-G. Mao and Y.-Z. Chueh, “Broadband composite right/left-handed coplanar waveguide power splitters with arbitrary phase responses and balun and antenna applications,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 234–250, Jan. 2006. [12] X. Q. Lin, R. P. Liu, X. M. Yang, J. X. Chen, X. X. Ying, Q. Cheng, and T. J. Cui, “Arbitrarily dual-band components using simplified structures of conventional CRLH-TLs,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 7, pp. 2902–2909, Jul. 2006. [13] M. Gil, J. Bonache, J. Selga, J. García-García, and F. Martín, “Broadband resonant type metamaterial transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 17, pp. 97–99, Feb. 2007. [14] M. Gil, J. Bonache, J. García-García, J. Martel, and F. Martín, “Composite right/left handed (CRLH) metamaterial transmission lines based on complementary split rings resonators (CSRRs) and their applications to very wide band and compact filter design,” IEEE Trans. Microw. Theory Tech., vol. 55, pp. 1296–1304, Jun. 2007. [15] Y. S. Wang, M. F. Hsu, and S. J. Chung, “A compact slot antenna utilizing a right/left-handed transmission line feed,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 675–683, Mar. 2008. [16] N. Engheta, “Compact cavity resonators using metamaterials with negative permittivity and permeability,” in Proc. Int. Conf. Electromagnetics in Advanced Applications (ICEAA), Torino, Italy, Sep. 2001, pp. 739–742. [17] N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas Wireless Lett., vol. 1, pp. 10–13, 2002. [18] X. Q. Lin, H. F. Ma, D. Bao, and T. J. Cui, “Design and analysis of super-wide bandpass filters using a novel compact meta-structure,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 4, pp. 747–753, Apr. 2007. [19] L. Yan, W. Hong, K. Wu, and T. J. Cui, “Investigations on the propagation characteristics of SIW,” Proc. Inst. Elect. Eng. Microw., Antennas, Propag., vol. 152, no. 1, pp. 35–42, Feb. 2005. [20] Z. C. Hao, “Investigations on the Substrate of SIW,” Doctor, Southeast University, Nanjing, 2005. [21] CST Microwave Studio 5 User’s Manual. Darmstadt, Germany, CST Ltd, 2003. [22] M. G. Wang, S. W. Lv, and R. C. Liu, Analysis and Synthesis of Array Antennas. Chengdu: University of Electronic Science and Technology of China Press, 1989. Xian Qi Lin (M’09) was born in July 1980, in Zhejiang, China. He received the B.S. degree in electrical engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2003 and the Ph.D degree in information science and engineering from Southeast university, Nanjing, China, in 2008. In June 2008, he joined the School of Electrical Engineering, UESTC, and became an Associate Professor in August 2009. His research interests include micro/millimeter wave technology and circuits, metamaterial, and antennas.

Di Bao was born in Jiangsu Province, China, on May 16, 1983. She received the B.S. degree in radio engineering from Southeast University (SEU), Nanjing, China, in 2006, where she currently working toward the M.S. degree Her current research interests include nano-materials and applications of metamaterials in microwave circuit designs.

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Hui Feng Ma was born in Jiangsu Province, China, on December 10, 1981. He received the B.S. degree in electrical engineering from the Nanjing University of Science and Technology, Nanjing, China, in 2004, and is currently working toward the Ph.D. degree at Southeast University, Nanjing, China. His research interests include design and analysis of microwave circuits and metamaterials.

Tie Jun Cui (M’98–SM’00) was born in September 1965, in Hebei, China. He received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Xidian University, Xi’an, China, in 1987, 1990, and 1993, respectively. In March 1993, he joined the Department of Electromagnetic Engineering, Xidian University, and was promoted to an Associate Professor in November 1993. From 1995 to 1997 he was a Research Fellow with the Institut für Höchstfrequenztechnik und Elektronik (IHE) at the University of Karlsruhe, Ger-

many. In July 1997, he joined the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, first as a Postdoctoral Research Associate and then a Research Scientist. In September 2001, he became a Chang-Jiang Professor with the Department of Radio Engineering, Southeast University, Nanjing, China, under the Cheung Kong Scholar Program awarded by the Ministry of Education, China. He is currently an Associate Director of the State Key Laboratory of Millimeter Waves, Southeast University. He is the author of four book chapters, over 100 scientific journal articles, and has presented over 50 conference papers. His research interests include wave propagation, scattering, inverse scattering, land mine detection, geophysical subsurface sensing, fast algorithms, integrated circuit simulations, and metamaterials. Dr. Cui was awarded a Research Fellowship from the Alexander von Humboldt Foundation, Bonn, Germany, in 1995, received a Young Scientist Award from the International Union of Radio Science (URSI) in 1999, was awarded a Chang-Jiang Professor from the Ministry of Education, China, in 2001, received a National Science Foundation of China for Distinguished Young Scholars in 2002, and received a Teaching Award from Southeast University, China, in 2003. He is a member of URSI (Commission B), and a senior member of the Chinese Institute of Electronics (CIE). Currently he serves as an Associate Editor of the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, and is an Editorial Staff member of the IEEE Antennas and Propagation Magazine.

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Hermite-Rodriguez UWB Circular Arrays Gaetano Marrocco, Member, IEEE, and Giovanni Galletta

Abstract—Pulsed circular arrays are collecting growing interest in radar applications such as automotives and indoor navigations. This contribution presents the analytic derivation of the spacetime and energy patterns of pulsed circular arrays in terms of geometrical and electrical parameters as well as of the signal distortion produced by the antennas’ response. It is shown that the field emitted by circular arrays with many elements can be represented as a summation of a practically finite set of high-order Hermite-Rodriguez waveforms, while the energy pattern is a generalized Hypergeometric Function. The angular and temporal resolutions are finally related, through handy formulas, to the array size, the input signals and to the antenna types. Index Terms—Circular array, Hermite-Rodriguez functions, pulsed arrays, ultrawideband (UWB) antennas.

I. INTRODUCTION N the last decades there has been an increasing interest in ultrawideband (UWB) pulsed arrays [1]–[3] consisting in an arrangement of antennas having more than 25% bandwidth, which are sourced by baseband carrier-free input signals. Possible applications are for communications, radar, precise indoor positioning and tracking [4]–[6]. The properties of UWB pulses permit to design “spots” as opposed to “beams” in narrow band systems and hence a set of distributed radios can be used to communicate to a distant specific point in space. Pulsed linear arrays (PLA) have been theoretically investigated from different points of view for what concerns the general features [7]–[12], the dynamics of ultrasparse configurations [13] and the modal phenomena arising in large, at limit infinite, two-dimensional configurations [14], [15]. Since real antennas have a finite bandwidth, they produce a distortion of the input waveform so that the radiated signals depend on both the array geometry and beamforming network as well as on the antenna space-time features. These effects can not be simply kept separate as instead commonly done in the narrowband regime. A few synthesis techniques have been moreover presented with the underlying idea to control the individual waveforms exciting each antenna with the goal to achieve localized radiation [16], [17] or the compliance with a given mask having imposed some constraints over the beamforming network [18].

I

Manuscript received December 01, 2008; revised September 10, 2009. First published December 04, 2009; current version published February 03, 2010. G. Marrocco is with the Dipartimento di Informatica, Sistemi e Produzione, University of Roma “Tor Vergata”, Roma, Italy (e-mail: [email protected]). G. Galletta was with the Dipartimento di Informatica, Sistemi e Produzione, University of Roma “Tor Vergata”, Roma, Italy. He is now with Terasystem SpA, Rome 00143, 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.2009.2037695

Pulsed circular arrays (PCA) are particular attractive in precise radar applications such as indoor radio-navigation, automotive radars and homeland surveillance. PCA are recognized to can provide better angle of arrival estimates [19] than linear array and planar rectangular arrays, in light of the following two important advantages: i) the azimuth of PCA covers 360 in contrast to the 180 of PLAs; ii) the spot of the PCA is unchanged for scanning around the azimuth angle while that of the PLA broadens and distorts as the spot is steered from the boresight [20]. Moreover it is has been demonstrated that PCA recordings contain 3D information, which can be used to identify, separate, and reconstruct first-order ceiling reflections in indoor environments [21]. PCA configurations have been mainly studied in acoustics and optics for impulse-like input signals because of their relevance to sound and image formation. Many publications may be found about the possibility to achieve highly localized and non-diffracting waves (see the book [22] for an exhaustive review). Fewer contributions are instead available about the electromagnetic properties of PCAs. The study in [23] introduces the main formalism and gives many numerical examples for the only Gaussian input waveform with no specification of the single antenna features. The mathematical properties of pulsedriven antennas are instead considered in [16] with a particular emphasis to the formation of non diffracting beams and analytical examples for annular arrays of short dipoles are discussed with a detailed physical insight. In this paper the UWB radiation from circular arrays is addressed for higher-order Gaussian input stimuli and for canonical models of distorting antennas within the unitary framework of Hermite Rodriguez representation of signals. A new class of pulsed circular arrays is therefore originated: the Hermite-Rodriguez arrays, and the purpose of the work is to establish some fundamental limits in beam formation and to provide handy relationships among the global radiation features and the geometrical and electrical parameters useful for a preliminary array design. In particular, analytical formulas are derived in the case of array with many elements for both the space-time radiated field and the energy pattern. The proposed representation permits, moreover, to give a simple interpretation to the formation of the transient radiated patterns in terms of superposition of Hermite-Rodriguez waveforms. The paper is organized as follow: Section II briefly recalls the formalism of transient circular arrays and introduces the concept of ideal th order Hermite-Rodriguez array. Section III deals with the representation and UWB properties of circular arrays with many elements by means of exact expressions. Section IV shows some examples to better understand the role of the geometrical and physical parameters and, finally, Section IV gives some preliminary considerations of further issues arising for real systems.

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For a circular equi-spaced array of radius (Fig. 1) whose input currents are delayed to focus the main spot along direction, the term, may be expressed, according the formalism in [23], as (3) where arrays elements and

is the angular position of the .

A. Hermite-Rodriguez Arrays Uniform arrays are here considered, in the sense that all the antennas are supposed to be sourced by the same input waveform and specific delay. In particular, such waveforms belong to the family of high-order Gaussian pulses, also known as Hermite-Rodriguez functions [25] defined as Fig. 1. Geometry and reference system of a circular array of N elements.

(4) where “ ” tags the order of the function and the parameter is , the waveform’s width. Hence, it is assumed that . for any The function (Fig. 2) is a Gaussian pulse, gives a a doublet. All the functions of parameter monocycle and are up-bounded by a zeroth order Gaussian pulse of larger . parameter: The Fourier transform of (4) is (5)

Fig. 2. Hermite-Rodriguez functions of some order k .

II. CIRCULAR ARRAYS IN PULSED REGIME Time-dependent radiation from antennas is generally described by the transmitting-mode Time Domain (TD) effective which accounts for the antenna-source height [24] mismatch and depends on the time and on the observation . unitary vector A set of identical and equi-oriented antennas, each excited , is now considered. Folby an inward travelling current lowing usual far-field approximations [24], the total field radiated by the array, when the coupling effects are neglected, is (1) where is the speed of light and function

the vacuum impedance. The

(2) . is the TD array factor [9] with is a time delay to achieve spot steering along a given direction , and tags the th antenna position with respect to a local coordinate system.

and its spectrum peaks at . The HR functions exhibit a compact support in both time and frequency domains and can be generated by a cascade of differentiation modules. Such functions are generally accepted as typical input signals in real transmitters (see for instance [16], [26]), and more complex finite-support waveforms can be synthesized as a superposition of HR functions [18], [27]. A discussion on this issue is presented in Section V. The space-time dispersive effect of the radiating elements is taken into account by means of the following simple mathematical model of the antenna’s effective height, wherein the angular and time dependences are supposed decoupled such as (6) is a time delay and the operator , as already diswhere cussed in [16] and in [18], produces the th derivative of the input signals. It is worth noticing (see the review in [28] for an extensive gallery of the transient response of antennas) that the (ideal differentiator) is a good model for optimal case represents the UWB antennas (TEM horn, IRA), the case close resemresponse of a small dipole [24] and, finally, bles the impulse response of moderately large-band antennas such as a diamond dipole [29] and the Achimedean spiral. This representation could be also extended to more complex impulse responses by a linear combination of derivative operators. Nevertheless, the only canonical cases described by (6) will be here

MARROCCO AND GALLETTA: HERMITE-RODRIGUEZ UWB CIRCULAR ARRAYS

considered, while some more realistic models will be discussed in Section V. Thanks to the distributive property of the differentiator operator with respect to the convolution (e.g., ) and to the definition of HR functions, the overall field radiated by the circular array sourced by above input stimuli may be rewritten as

(7) is a constant with

where

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where

is the Gamma function. For example for to 5, while for further increasing . III. HR-ARRAYS WITH MANY ELEMENTS

of a uniform It is well known that the array factor circular array with a large number of elements, theoretically when approaches infinity (continuous uniform current over a circle), can be expressed in the frequency domain [23] by means of the zeroth order Bessel function (13)

. Denoting with (8) the th order Hermite-Rodriguez Array Factor, which embodies both the input waveform order and the time-dispersive , the array’s total field will effect of the antennas be finally proportional to (9) The array pattern can be globally characterized by the time duration, or time resolution of the main spot, and by its angular width or angular resolution. The angular resolution is associated to an energy indicator, norm removing the time dependence in (9). Hence e.g., to an the Energy Pattern of the array, also accounting for the temporal-dispersive effect of the antennas and of the input signals may be defined as (10) where is the norm with respect to time. The pattern multiplication principle still holds in the energy domain, so that it is possible to define a half-power beamwidth for the main spot in analogy to the frequency domain beamwidth. Calculation of needs to be generally performed numerically, but interesting analytical formulas can be retrieved in the assumption of large number of elements as discussed in the Section III.

is the spectrum of the input signals and where metric function independent on frequency

a trigono-

(14) for focusing along the vertical axis while when the main spot . is formed on the horizontal array plane, e.g., for Taking the Fourier transform of (13), the HR array factor in case of many elements can be approximated by the integral representation where

(15) Above integral can be considered as a windowed (Gabor) Fourier Transform of the Bessel function and it is solved by using the expression in (5) for the HR function and the series definition [30] of the Bessel function . After some tedious but straightforward manipulations the expression in (15) is finally written as

(16) In particular, for beam focusing along the array plane , the above expression becomes

B. Time Resolution The time resolution along the main beam is defined as the effective signal duration [24]

(17) (11) A. Pattern Formation For a uniform array, coincides with the duration of the th , since and it can be HR function expressed (see Appendix) as (12)

It is worth noting that along the main beam the 0th term in (17) is the only surviving one and hence , as expected. At other directions, the radiated field is given by the superposition of infinite high-order HR waveforms whose amplitude is angularly modulated by factor (or by for focusing on the the vertical direction). Such waveforms are therefore as more

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Fig. 3. Excitation coefficients B (17) for some choices of the order signal duration.

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and (R=c ) B of the HR waves in m and of size R with respect to the input

weighted as the observation direction departs from the main beam. However, the number of emitted waves is practically limited since the excitation coefficients exhibit a low-pass behavior as shown, for example in Fig. 3. The span of the practically excited waves enlarges along with the size of the array (with respect to the input pulse duration, ), and with the array order . In other words, only e.g., a limited set, among the infinite HR waves emitted from the array, will contribute to the propagation. For instance, in case and ideal antennas sourced with Gaussian pulses , the practically excited HR functions are less than 10. waveforms interfere and, Far from the main beam, the as sketched in Fig. 4(a), the internal out-of-phase oscillations of the multiplicity of signals tend to mutually cancel leaving a couplet of external residual waists. The resulting angle-time pattern exhibits a typical “X”-like shape, whose vertex gives the main spot and the external fringes have the meanings of images in the example of the side-lobes. The sequence of Fig. 4(b) provides evidence of the above discussed cancellation of internal oscillations and of the finite number of excited HR waves. This kind of pattern is typical of uniform current distributions on circles, in the context of the “X”-waves theory [22]. These concepts are further enlighten in the example Section.

Fig. 4. Formation of the TD radiation pattern of an Hermite-Rodriguez array timed to focus along the horizontal plane. a) Sketch of the far field waveforms, w (t), along the main beam and far from it (superposition of higher order HRs w (t)). b) Example of “X”-shaped pattern formation (m = 1, R = c ) for increasing truncation p = 0 . . . P of the series in (16).

in the Appendix. After some mathematical manipulation, (18) becomes

(19)

B. Angular Resolution A significant expression for the energy pattern in (10) is in the now found by starting from (15) and calculating frequency domain through the Parseval theorem and using the spectrum in (5): (18) The above integral has an exact solution in terms of the Generalized Hypergeometric Function [31], as described

having introduced the auxiliary variable . , appears for The peak value of (19), corresponding to and, accordingly, . anFig. 5 shows the plot of the normalized gular energy for some orders , versus the auxiliary variable . The width of the main-lobe reduces for increasing and the first side-lobe appears for while further lobes may be . recognized for The angular resolution of the th HR array factor, e.g., the is not less than half the angle wherein the energy pattern

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Fig. 5. Normalized energy pattern of the HR array E (^ r )=jg (^ r )j versus the auxiliary variable u =  =cT for some orders m. The horizontal line tags the 03 dB amplitude for the calculation of the u values, and hence of the angular resolution. Fig. 6. TD array factor F (^ r ;  ) and angular energy on the horizontal plane  = =2 timed to focus at (^ r = x ) for electric and geometric parameters:  = 0:1 ns, array radius R = 5c and increasing number of array elements N = f5; 10; 20g.

TABLE I HALF-POWER VALUES FOR THE HYPERGEOMETRIC SERIES

It is worth noticing that expression in (23) and (24) generalizes those in [23] to higher order pulses and multiple differentiating antennas. peak value, is calculated by enforcing (20) are given in Table I. The first values of The angular resolution of the array factor is finally retrieved for array focusing along the vertical and the horizontal planes. and the visible space is limited In the first case by (21) which is accordingly affected by the radius of the array and by , is the time duration of the pulse. The angular resolution, so that obtained by enforcing (22) For directive arrays this expression simplifies as follows: (23) Having fixed the array radius, the angular resolution can be therefore improved by increasing the order of the input signal (or of the antenna), at expense of the formation of energy side-lobes, as well as by reducing the input signal duration. A similar expression (24) can be found for an array focusing on the horizontal plane along a direction , where is an integer number.

IV. NUMERICAL EXAMPLES Space-time-patterns and energy diagrams are here shown with respect to some choices of the electrical and geometrical features of the HR array. An elementary pattern synthesis is also illustrated. In all the given examples only the signal distortion effect of the antennas is considered while, for generality, the angular is dropped (which means to consider spreading function antennas with isotropic radiation, at least over the observation plane). A. Radiation Versus Number of Antennas Fig. 6–8 shows the for the case of input pulses with Gaussian width and array radius , concerning spot focusing along the horizontal plane and along the vertical plane , respectively. The diagrams are specified for increasing numbers of elements . The resulting space-time functions may represent the radiation from ideal antennas sourced by Gaussian pulses. The presence of side partial interferences among a subset of emitters is clearly visible. The interference fringes, outside the main spot are quite different for the case of vertical and horizontal focusing. On the vertical plane, the phenomenology is rather similar to that of a linear array and side “interference rails” appears, due to the radiation of each emitters [32]. On the horizontal plane instead, the interference pattern is more comtrue distinct multiple side spots are visible at plicated and the left-side and at the right-side of the main spot. These are produced by the mutual interference originating by couplets of

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Fig. 7. Some cuts of F (1;  ) from the 2D representation in Fig. 6 for different numbers of elements and observation directions. Functions are normalized with respect to the maximum amplitude. The branches of the “X” shape are clearly visible.

r ;  ) and angular energy on the horizontal plane Fig. 9. TD array factor F (^ ^) for three different order m = f3; 4; 6g. = x  = 0 timed to focus at (^ r Other parameters as in the previous examples.

B. Radiation vs. Array Order

Fig. 8. TD array factor F (^ r ;  ) on the vertical plane  = 0 timed to focus ^) for electric and geometric parameters:  = 0:1 ns, array radius at (^ = z r R = 5c and increasing number of array elements N = f5; 10; 20g.

The previous numerical experiments are now repeated for higher order array factors with which may represent several combinations of antenna types and input represents the signal’s orders. Just for instance, the case radiation of broadband dipoles (diamond-like, ) sourced , as well as small dipoles by Gaussian pulses sourced by monocycles . The case with may with difreproduce the radiation of broadband dipoles ferentiated Gaussian input signals . Results are given in Fig. 9 where multiple time oscillations appear along the main spot with increasing order of the HR functions. When comcase in Fig. 6, it is possible paring Fig. 9 with the to note a narrowing in the main spot and a more “pixelated” pattern of side spots as also apparent in the energy diagram. The corresponding angular resolutions are for . C. Example of Synthesis

emitters placed at different distance from the axis orthogso that their delays are synonal to the focusing direction chronized at a particular direction. However, in both the configurations, as the density of elements increases, the positive pulse of a spot cancels the negative pulse of the adjacent one, only leaving the most external fringes (Fig. 7). The resulting pattern is the “X” shape, already discussed in the previous Section from a different point of view. The energy diagrams reveal that, while the angular resolution remains unchanged with respect to the increase in the number of elements, the side spots become instead sensibly lower. It is worth mentioning that extensive numerical simulations demonstrate that the representations (16) and (17) are able to reproduce the whole “X” phenomenology when just with a weak dependence on the array size, also . Such approximations are therefore for radius valid even when the inter-antenna distance (element density) is not particular small (in terms of the waveform duration).

Equations (10) and (22) or (23) help to choose the values of driving Gaussian pulses’ parameter and of the array radius to achieve the desired temporal and angular resolution, for the particular pulses’ order and antenna family. For instance, in case of focusing over the array plane, the array size is related to the spot’s resolution by the equation (25) where as derived from (10). Just for example, consider the synthesis of a narrow beam and . with required resolution In case of an array of ideal UWB antennas sourced by the needed array radius from (25) is Gaussian pulses , while a rather smaller radius is required and input monocycles in case ultra wideband dipoles are used, e.g., . It is interesting

MARROCCO AND GALLETTA: HERMITE-RODRIGUEZ UWB CIRCULAR ARRAYS

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derivative is replaced by a finite realization which broaden the input signal. The total field in (1) is therefore written as a triple summation (28) (29) The convolution between two HR functions of different widths is still an HR function [25] having order equal to the sum of the two orders and width, , given by the average quadratic summation of the two widths, e.g.,

(30) with Fig. 10. HR array factors and angular energies for two choices of antennas family and input signal waveforms such to give, in both the cases, the same temporal- and angular-resolution (T; 8) = (5 ns; 5 ). Upper row: m = 1 with a required radius R = 5:95 m. Lower row: m = 4 with a required radius R = 3:20 m.

to note that these sizes are in full agreement with those found in [32] for the case of linear arrays with the same resolution. The angle-time and energy patterns for the above examples are elements. It can be displayed in Fig. 10 for the case of observed that the highest order HR array factor exhibits a more case. localized “X” pattern than the V. EXTENSION TO REAL ARRAYS Real antennas and real input signals may differ from the canonical models given above. For instance the input signal could be a differently synthesized waveform, or the antennas’ impulse response could be not symmetric or decay less regular than a single HR function. Nevertheless the presented formulation can be still useful to manage the dominant behavior of real systems. At this purpose the complexity of the real word is parametrized again by the help of the HR functions which are a useful tool to represent compact-support signals [25]. In particular, both the input current and the time-dependence of the antennas’ impulse response are represented as a finite superposition of HR families having, for generality, different parameters , , respectively (26)

(27) where and are constant coefficients and the HF functions are defined as in (4) with proper Gaussian width as indicated in the subscript. is a time delay assumed to be identical for all the functions. The representation in (27) can be considered as a generalization of the ideal model in (6) wherein the th

. Denoting with ,

then

the

total field becomes

(31) where is the array factor as defined in (8) for an ideal HR array with input signals of width . The real array may be therefore interpreted as the superposition of th order HR arrays with weights . Along the main spot the signals are all synchronized, , and hence by reordering the terms, the radiated field is the superposition of HR functions of same width and different or. The strongest coders, e.g., efficient identifies the dominant HR contributing function to the radiation, in the sense of a weighted quadratic norm since [25]. The corterms can be considered as the dominant HR responding array factor whose phenomenology and resolutions gives a firstorder indication of the whole array features. The macroscopic difference with respect to the ideal case is that the width of the by transmitted signal will be now enlarged the presence of the antennas and hence the time and angular resolutions are expected to be degraded. It is worth finally recalling that all the given representations have been derived neglecting the antenna coupling. The formal analysis of the circular array coupling is beyond the scope of this paper but recalling a previous study of the authors [32] about the coupling in pulsed linear array, it is expected that the coupling should be a second-order phenomenon which is going to affect only slightly the shape of the main spot without sensible modification in its overall characteristics. Moreover, the inter-element distance may not be mandatory small to achieve a high resolution and the characteristic “X” waveform, and it is known that the coupling effect can be however greatly reduced by using input signals with small duration in comparison with the inter-antenna spacing (normalized by the light speed). The given results and guidelines are therefore considered of general applicability in the first step of real array design.

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VI. CONCLUSION The Hermite-Rodriguez formalism offers a compact representation of UWB arrays with high-order excitation also accounting, in a uniform way, for the distorting effect introduced by the antennas. The main results are now summarized. i) A circular array with many UWB antennas, sourced by high-order Gaussian pulses, produces a time-varying pattern which is the superposition of a practical finite set of Hermite-Rodriguez waveforms, angularly weighted, giving an “X”-shaped space-time pattern; ii) Hence the radiation is practically localized in angle, time and even in waveform domains. The energy pattern is a generalized Hypergeometric function; iii) The time resolution is dependent on both the input waveform’s order and duration and on the distortion’s order of the antennas. However the time resolution tends to satu; rate to the Gaussian parameter as , reduce) by iv) The angular resolution improves ( increasing the order of the HR array; v) The space-time localization globally improves with the increase of the array size; vi) The HR array has still some meaning in case of realistic systems with more complicated dispersive behavior since it gives information about the dominant contribute to the radiation; vii) The angle-time resolution of a real array is expected to be worse than in the case of a canonical HR array of same geometry. The presented simple mathematical dissertation permits not only to give some physical insight to the complex time-domain radiation mechanism, but also provides a first tool for an elementary array synthesis wherein both the angular- and the timeresolution are subjected to constraints. The array resolution and the strength of side spot may be controlled not only by the array size and by the number of elements, but also by acting on the order and on the duration of input pulses and by selecting the proper distorting effect of the antennas.

and by some mathematical manipulation, the numerator in (32) is rewritten as in (33), shown at the bottom of the page. The denominator is instead (34) The above integrals are of the types [33]

with

, and hence

(35)

(36) Finally (37)

B. Details About the Calculation of the Energy Pattern in (18) The integral in (18) is of the type

APPENDIX A. Details About the Calculation of

in (12)

The effective signal duration [24] of the formally defined as

th HR function is (38) (32)

It is now convenient to express the HF functions in , e.g., terms of the Hermite polynomials , with the scale factor . By using the definition

where fined as

is the Hypergeometric Function [30], [31], [34], de-

(39)

(33)

MARROCCO AND GALLETTA: HERMITE-RODRIGUEZ UWB CIRCULAR ARRAYS

The term

is the Pochhammer operator (40)

After substitution , tions

, ,

, , and the simplifica-

, the (19) is obtained. The convergence can be discussed by applying the definition of the function in (39) and hence

(41) where the following identities have been used:

The series in (41) is convergent according to the Leibniz rule since the terms have alternate sign and it is easy to show that and finally they are uniformly decreasing after a given . each term goes to zero as ACKNOWLEDGMENT The authors wish to thank Prof. R. Ziolkowski for having suggested interesting papers on X-waves. REFERENCES [1] C. E. Baum, “Transient arrays,” in Proc. Ultra-Wideband, Short-Pulse Electromagnetics 3, C. E. Baum, Ed. et al. New York: Plenum Press, 1997, pp. 129–138. [2] E. L. Mokole, “Behavior of ultrawideband-radar array antennas,” in Proc. IEEE Int. Symp. on Phased Array Systems and Technology, 1996, pp. 113–118. [3] F. Anderson, W. Christensen, L. Fullerton, and B. Kortegaard, “Ultrawideband beamforming in sparse arrays,” Proc. Inst. Elect. Eng., vol. 138, no. 4, pt. H, pp. 342–346, 1991. [4] K. Siwiak and D. McKeown, Ultra-Wideband Radio Technology. New York: Wiley, 2004. [5] M. Ghavami, L. B. Michael, and R. Kohno, Ultra Wideband: Signals and Systems in Communication Engineering. Chichester, U.K.: Wiley, 2004, pp. 31–37. [6] M. G. M. Hussain, “Principles of space-time array processing for ultrawide-band impulse radar and radio communications,” IEEE Trans. Veh. Technol., vol. 51, no. 3, pp. 393–403, 2002. [7] M. Malek and M. Hussain, “Antenna patterns of nonsinusoidal waves with the time variation of a Gaussian pulse: Part I,” IEEE Trans. Antennas Propag., vol. 30, no. 4, pp. 504–512, Nov. 1989. [8] M. Malek and M. Hussain, “Antenna patterns of nonsinusoidal waves with the time variation of a Gaussian pulse: Part II,” IEEE Trans. Antennas Propag., vol. 30, no. 4, pp. 513–522, Nov. 1989.

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[9] A. Shlivinski and E. Heyman, “A unified kinematic theory of transient arrays,” in Ultra-Wideband Short-Pulse Electromagnetics 5, P. D. Smith and S. R. Cloude, Eds. New York: Kluwer Academic/Plenum, 2000, pp. 327–334. [10] G. Franceschetti, J. Tatoian, and G. Gibbs, “Timed arrays in a nutshell,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 4073–4082, 2005. [11] W. Sorgel, C. Sturm, and W. Wiesbeck, “Impulse response of linear UWB antenna arrays and the application to beam steering,” in Proc. IEEE Int. Conf. on Ultra-Wideband, 2005, pp. 275–280. [12] Y. Kang and D. M. Pozar, “Optimization of pulse radiation from dipole arrays for maximum energy in a specified time interval,” IEEE Trans. Antennas. Propag., vol. 34, no. 12, pp. 1383–1390, 1986. [13] J. L. Schwartz and B. D. Steinberg, “Ultrasparse, ultrawideband arrays,” IEEE Trans. Ultrason., Ferroelect., Freq. Control, vol. 45, no. 2, pp. 376–393, Feb. 1998. [14] L. B. Felsen and F. Capolino, “Time-domain Green’s function for an infinite sequentially excited periodic line array of dipoles,” IEEE Trans. Antennas Propag., vol. 50, no. 1, pp. 31–41, Jan. 2002. [15] F. Capolino and L. B. Felsen, “Time-domain Green’s function for an infinite sequentially excited periodic planar array of dipoles,” IEEE Trans. Antennas Propag., vol. 51, no. 2, pp. 160–170, Feb. 2003. [16] R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” IEEE Trans. Antennas Propag., vol. 40, no. 8, pp. 888–905, 1992. [17] J. E. Hernandez, R. W. Ziolkowski, and S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,” J. Acoust. Soc., Am., vol. 92, no. 1, pp. 550–562, Jul. 1992. [18] M. Ciattaglia and G. Marrocco, “Time domain synthesis of pulsed arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 1928–1938, Jul. 2008. [19] H. Arslan, Z. N. Chen, and M. G. Di Benedetto, Ultra Wideband Wireless Communication. Hoboken, NJ: Wiley-Interscience, 2006, ch. 3. [20] C. Gentile, A. J. Braga, and A. Kik, “A comprehensive evaluation of joint range and angle estimation in ultra-wideband location systems for indoors,” in Proc. IEEE Int. Conf. on Commun., May 2008, pp. 4219–4225. [21] D. De Vries, L. Horchens, and P. Grond, “Extraction of 3D information from circular array measurements for auralization with wave field synthesis,” J. Appl. Signal Process., vol. 2007, no. 1, pp. 190–200, 2007. [22] , H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Eds., Localized Waves. Hoboken, NJ: Wiley, 2008. [23] M. G. M. Hussain, M. M. Al-Halabi, and A. A. Omar, “Antenna patterns of nonsinusoidal waves with the time variation of a Gaussian pulse: Part III,” IEEE Trans. Antennas Propag., vol. 31, no. 1, pp. 34–47, Feb. 1989. [24] A. Shlivinski, E. Heyman, and R. Kastner, “Antenna characterization in the time domain,” IEEE Trans. Antennas Propag., vol. 45, no. 7, pp. 1140–1149, 1997. [25] L. R. Lo Conte, R. Merletti, and G. V. Sandri, “Hermite expansion of compact support waveforms: Application to myoelectric signals,” IEEE Trans. Biomed. Eng., vol. 41, no. 12, pp. 1147–1159, Dec. 1994. [26] M. Ghavami, L. B. Michael, and R. Kohno, Ultra Wideband: Signals and Systems in Communication Engineering. London, U.K.: Wiley, 2004, pp. 31–37. [27] M. A. Yusoff, “Application of Hermite-Rodriguez functions to pulse shaping analog filter design,” in Proc. World Academy Sci., Eng. Technol., Dec. 2007, vol. 26, pp. 762–785. [28] D. Ghosh, A. De, M. C. Taylor, T. Sarkar, M. C. Wicks, and E. Mokole, “Transmission and reception by ultra-wideband (UWB) antennas,” IEEE Antennas Propag. Mag., vol. 48, no. 5, pp. 67–99, Oct. 2006. [29] H. Schantz, “Time domain array design,” in Ultra-Wideband Short-Pulse Electromagnetics 5. New York: Kluwer Academic/Plenum Publisher, 2002, pp. 385–392. [30] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. New York: Dover, 1964. [31] E. W. Weisstein, CRC Concise Encyclopedia of Mathematics. Boca Raton, FL: Chapman & Hall/CRC Press, 1999, pp. 872–875. [32] M. Ciattaglia and G. Marrocco, “Investigation on antenna coupling in time domain,” IEEE Trans. Antennas Propag., vol. 54, no. 3, pp. 835–843, March 2006. [33] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. San Diego, CA: Elsevier, 2007. [34] A. M. Mathai, Generalized Hypergeometric Functions With Applications in Statistics and Physical Sciences. Berlin: Springer-Verlag, 1973.

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Gaetano Marrocco was born in Teramo, Italy, on August 29, 1969. He received the Laurea degree in electronic engineering and the Ph.D. degree in applied electromagnetics from University of L’Aquila, Italy, in 1994 and 1998, respectively. He has been a Researcher at the University of Rome “Tor Vergata” since 1997 where he currently teaches antenna design and bioelectromagnetics. In summer 1994, he was at the University of Illinois at Urbana Champain as a Postdoctoral student. In autumn 1999, he was a Visiting Scientist at the Imperial College in London. His research is mainly directed to the modeling and design of broadband and ultrawideband antennas and arrays as well as of miniaturized antennas for RFID applications. He has been involved in several space, avionic, naval and vehicular programs of the European Space Agency, NATO, Italian Space Agency, and the Italian Navy. He holds two patents on broadband naval antennas and one patent on sensor RFID systems. Prof. Marrocco currently serves as an Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

Giovanni Galletta received the Laurea degree in telecommunications engineering from the University of Rome “Tor Vergata” in 2008. His main scientific interest concerns the design of antenna systems for telecommunications. He is currently employed at Terasystem SpA, Rome, Italy, where he works on IT Service Management.

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Analysis and Design of a Novel Dual-Band Array Antenna With a Low Profile for 2400/5800-MHz WLAN Systems Shanhong He and Jidong Xie, Member, IEEE

Abstract—A novel dual-band array antenna and its associated feed networks for 2400/5800-MHz wireless local area networks (WLAN) is proposed. The nested dual-band array is composed of a 2400-MHz band array formed by 16 (4 4) parallel-fed rectangular patches and a 5800-MHz band array formed by 64 (8 8) hybrid-fed double-sided printed dipoles. The measured and simulated results are presented and analyzed. The relative impedance bandwidths and the peak gains of the 2400-MHz band 1 5), 24 and 5800-MHz band arrays reach 8%, 13% (VSWR dBi and 18 dBi respectively. The dual-band array has regular radiation patterns, high polarization purity and antenna efficiency, and it is suitable for integration with other microwave circuits and can be applied in other dual-band system by scaling its size. The design consideration for side lobe level (SLL) suppression by amplitude weighting combining the quarter-wavelength transformer and a small adjustment in the antenna geometry is discussed as well. Index Terms—Dual-band, low profile, low side-lobe, planar array.

I. INTRODUCTION

W

ITH the development of WLAN, the wireless networks for WLAN will have to be compatible with multimode and several criterions such as IEEE 802.11b (2400–2483 MHz) and IEEE 802.11a (5150–5850 MHz). Thus, the array antenna can operate at two or more frequency band in WLAN system is demanded. At the same time, these arrays must satisfy the technical requirements such as high antenna efficiency, low crosspolarization level, being lightweight and low profile as well as the need to meet the additional economic constraints of low cost, reliability and easy in fabrication. Modern communication and radar requirements promote the development of the dual-band array. A dual-band base-station array for cellular communication system is proposed in [1]. It dB) and a specific half-power has low side-lobe levels ( beam-width in the horizontal plane at both frequency band, but it cannot achieve the antenna structure with low profile and light weight due to the corn reflector, the dielectric covered driven dipole and parasitic element. Compared with the array

Manuscript received December 04, 2008; revised July 21, 2009. First published December 04, 2009; current version published February 03, 2010. S. H. He is with School of Electrical Engineering and Information, Anhui University of Technology, Ma’anshan, Anhui Province 243002, China (e-mail: [email protected]). J. D. Xie is with Department of Communication Engineering, Nanjing University of Posts and Telecommunication, Nanjing, Jiangsu Province 210003, China. Digital Object Identifier 10.1109/TAP.2009.2037699

[1], the height of the dual-band array for 2/3 G base station [2] is reduced greatly, but when the operating frequency exceeds 5 GHz or higher frequency, many unexpected phenomena will occur due to its special structure. A C-and X-band array [3] using interlaced microstrip patches and printed slots has been applied in SAR system and the optimal performance has been achieved, but its relative bandwidth is too narrow to satisfy this of WLAN system. In addition, since two layers of solid teflonbased substrates are used, a relatively thick and heavy antenna structure is inevitable scaling to the lowest frequency of WLAN system. Contrasted to the array [3], the analogous dual-band array formed by the L-band slots and C-band microstrip patches with foam substrates for spacing borne SAR [4] has a very lightweight structure, but the large slots lead to some spurious radiation, which results in the decrease of gain and front-to-back ratio of radiation patterns. The low profile egg-crate array with “bunny-ear” elements [5] has not only an ultra-wide bandwidth but also good cross-polarization suppression, while the antenna height cannot be scaled to the lowest operation frequency of WLAN system by scaling its height to the given size within 12 mm. Although the shared-aperture L-band and X-band dual-polarized microstrip array [6] has numerous desirable features, the relative bandwidth of the upper frequency band is less than 2% , which cannot cover this of WLAN in terms of system. The dual-band dual-polarized nested Vivaldi slot array with mulltilevel ground plane was reported in [7]. The nested arrangement of the elements and the mulltilevel ground plane avoid the grating lobe and deterioration of the radiation patterns respectively, but the height of the array is approximately at the lowest frequency, which cannot satisfy the requirement of low profile. In this paper, the analysis and design of a novel dual-band array antenna for WLAN system by experiment and simulation utilizing the finite element method software called HFSS are addressed. This paper is organized as follows: the geometry of the antenna is described in Section II. In Section III, the design considerations and the measured and simulated results are presented followed by conclusion in Section IV. II. ANTENNA CONFIGURATION A dual-frequency array can be conceived in one of two fundamental ways: using dual-frequency elements to cover both bands simultaneously or using nested single-frequency radiating elements for each band. For the present design, an array using dual-frequency elements requires elements operate

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simultaneously at 2400-MHz and 5800-MHz band. Because of the approximately 2:1 frequency ratio, element spacing in the array is dictated by the requirement to avoid grating lobes at 5800-MHz band, so that element spacing will be on the order of at 5800-MHz band. This leads to an unnecessary increase in the number of elements at 2400-MHz band, and a larger associated feed network. In addition, if the elements are resonant at 2400-MHz band and the substrate dielectric constant is relatively low (preferable for good bandwidth), the size of the element may be large enough to preclude such close spacing (at least without strong mutual coupling effects) and/or crowding of the feed network. Other potential problems with specific dual-frequency elements include poor polarization purity, low antenna efficiency and complicated feeding requirements. Coupling within a dual-frequency structure often results in distorted radiation patterns and poor isolation between frequency bands. For these reasons, it was determined that the use of dual-frequency elements was not an appropriate choice, and this could be accomplished with 2400-MHz elements interleaved between groups of two 5800-MHz elements. In this work, the wide-band double-sided printed dipoles [8] are adopted in 5800-MHz band, with 2400-MHz rectangular radiating patches below the openings of the 5800-MHz band dipoles. As shown in Fig. 1(c), the substrate structure consists of five layers. The 2400-MHz microstrip line feed network and the ground plane are printed on the topside and downside of an inexpensive FR4 material named layer 1 with relative permitand thickness of 1 mm, respectively. Layer 2 is tivity a 6-mm-thick air layer serving not only a space for the ground plane and the 2400-MHz radiating patches but also the dielectric of the 2400-MHz radiating patches. Layer 3 is the metallic layer composed of the 2400-MHz radiating patches, which are fed by the feed network through the probes and supported by the plastic posts with diameter of 6 mm at both ends of radiating patches. Layer 4 is a 3-mm-thick air layer, which keeps the spacing between radiating elements of the 2400-MHz and 5800-MHz arrays. The 5800-MHz radiating elements and its feed network are etched on each side of Layer 5, which is a 0.78-mm-thick Teflon material supported by plastic posts. The design considerations for the dual-band array are described in detail in the following subsection. III. ANTENNA DESIGN CONSIDERATIONS AND MEASURED AND SIMULATED RESULTS A. Design of the 5800-MHz Array Antenna As discussed in [3], the key initial design for a dual-band array is the choice of radiating elements. Factors to consider are element bandwidth, feeding, and a size and shape that allow collocation of elements to cover both frequency bands within the same aperture plane. Two-dimensional array antenna can be constructed in many ways. However, the hybrid-fed arrays combining the series-fed and parallel-fed techniques are the more typical ones [8]. With a parallel-fed structure, the array can maintain a symmetrical structure and relative wide operating bandwidth, and the excitation coefficient of elements can be controlled with facility, but higher insertion loss exists in the feed network due to its

Fig. 1. Geometry of the (a) top viewer of the 5800-MHz array with dimension of 280 280 12 mm, (b) top viewer of the 2400-MHz array with dimension 280 12 mm and its feed network (c) side viewer of the dual-band of 280 array and the plastic posts are not presented for clarity of the diagram.

2 2

2 2

complexity and the lengthy transmission lines. In contrast to the parallel-fed arrays, the series-fed array has a simple physical construction, as a result, not only the metallic areas in layer 5, which can be treated as blocked pieces or parasitic radiators

HE AND XIE: ANALYSIS AND DESIGN OF A NOVEL DUAL-BAND ARRAY ANTENNA WITH A LOW PROFILE

of the 2400-MHz array, but also the insertion loss in the feed network are substantially reduced. For longer series-fed array, although the impedance response of the array tends to oscillate remains with frequency for wider bandwidth and the ratio and denotes the approximately constant with , where percentage bandwidth and the number of elements of the array, respectively, the longer series-fed array cannot be actually designed since the increase of cross-polarization level or grating lobes, even the beam splitting limits the operation of the longer series-fed arrays [10]. For these reasons, the hybrid-fed array is designed to push for the optimum performance. As shown in Fig. 1(a), the 5800-MHz array is composed of 64 (8 8) wide-band double-sided printed dipoles with each arm printed on the opposite sides of layer 5, and they are fed by a balanced twin-lead transmission line. Two adjacent elements along -axis (elements 1 and 2, 3 and 4, 5 and 6, 7 and 8) or two adjacent linear sub-arrays along -axis of a Cartesian reference frame shown in Fig. 1 constitute a series-fed sub-array, of which the at the cenelement spacing should be a guided wavelength tral frequency to make all the elements have uniform excitation phase. The relative permittivity of the substrate Teflon is 2.2, which is selected to achieve the element spacing without stretching or shrinking the twin lines and make the best of the mm. The thickness of given space of 280 mm 280 mm the substrate is 0.78 mm; hence, the dipole arms are only slightly out of phase, resulting in the insignificant effect on the radiation patterns and the low surface wave losses. The array is designed to be a symmetrical structure with a uniform element spacing of mm both in - plane and - plane. To provide a transition between the microstrip line and the balanced transmission line, the wide-band balun, as shown in Fig. 1(a), is designed; it is connected to the microstrip line with gradual transition into the balanced twin line. Information on the principle of operation of similar baluns can be found in [11]. The distance between the dipoles and the ground plane shall affect the input impedance of mm is the best comprothe dipoles, so the selection of mise between the input impedance and the low profile. of the dipole has been obOnce the input impedance tained, the excitation coefficients of the elements can be obtained through the transmission line theory described in the following Section III.C. Obviously, the excitation coefficients vary with frequency along -axis and -axis, nonetheless, it is easily seen from the symmetrical antenna structure that the main beam will continue to be pointed into the broadside direction. The measured far-field radiation patterns on - plane and - plane of the 5800-MHz array are plotted in Fig. 2(a)–(b), the array bandwidth is greater than 13% from 5150 to 5850 MHz in terms and dB. The simulated radiaof plane are compared with the measured tion patterns on ones in Fig. 2(a)–(b) for the purpose of assessing the reliability of HFSS, and the very close agreement between them qualifies HFSS as an accurate and effective tool for the following design and analysis of this dual-band array. In order to observe the effects of the 2400-MHz array on the 5800-MHz array eliminating the random manufacturing and measuring errors, the simulated radiation patterns on - plane of the 5800-MHz array and the dual-band array are compared with each other in Fig. 2(c), there are very little discrepancies

393

Fig. 2. Normalized radiation patterns of the (a) 5800-MHz array at 5150 MHz; (b) 5800-MHz array at 5850 MHz; (c) 5800-MHz array and dual-band array at 5500 MHz.

between them and the simulated cross-polarization level on broadside direction is less than dB, which illustrates that the effects can nearly be ignored. Many quite deep nulls testify for negligible phase error, which in turn is likely due to the

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Fig. 3. Measured VSWR of the (a) 5800-MHz array and (b) 2400-MHz array.

quite small mutual effects between the elements in the same frequency band and in the two sub-bands. The measured VSWR is illustrated in Fig. 3(a) and the peak gain and antenna efficiency is about 24 dBi and 50% over the bandwidth, respectively. B. Design of the 2400-MHz Array Antenna As shown in Fig. 1(b), (c), the 2400-MHz array consists of 16 (4 4) rectangular radiating patches. The patches are made of copper with thickness 1 mm and fed by a metallic strip (marked as a probe in Fig. 1) which originally is an internal part of the patch and is soldered to the conventional parallel-fed network at the position with an offset distance 6 mm away from the center of the patch along -axis. Due to the nested arrangement of the elements of the two sub-bands, the element spacings both on plane and - plane are twice of those of the 5800-MHz array. The dimension of the radiating patch plays a crucial role on the resonant frequency and it is optimized to be mm. The distance between the radiating patches and the ground plane is designed to 6 mm to achieve both the desired input impedance of the element and the low profile. Due to the uniform excitation coefficients both on - plane and - plane, the radiation patterns on - plane and plane will nearly resemble each other. The measured radiation pattern on - plane is plotted in Fig. 4(a), and the simulated co-polarization and cross-polarization patterns of the dual-band array, from which the more accurate cross-polarization level, SLLs and clearer nulls can be found without aforementioned errors, are presented in Fig. 4(a) to investigate the effects of the 5800-MHz array on the 2400-MHz array. Fig. 4(b) shows the induced surface current strength distribution on the 5800-MHz array and the excitation one on the 2400-MHz radiating patches by simulation, in which the brighter regions on the surface mean stronger current. It is observed that the current strength on some parts of the 5800-MHz array cannot be ignored compared with that on the 2400-MHz patches, which may partly account for the small difference between the radiation patterns of the 2400-MHz array and the dual-band array. The

Fig. 4. (a) Normalized radiation patterns of the 2400-MHz and dual-band arrays at 2450 MHz. (b) Simulated surface current strength distribution.

simulated cross-polarization level of the dual-band array on broadside direction exceeds dB, which is also attributed to the inevitable small effects, but it is not a serious concern. One can deduce that if only the parallel-fed method is adopted in the 5800-MHz array, the cross-polarization level will be increased and the antenna gain will be decreased with the increased blockage effect and parasitic effect. The measured VSWR is plotted in Fig. 3(b) and the peak gain and antenna efficiency is about 17.5 dBi and 50% at central frequency, respectively. C. Design Consideration for the SLL Suppression of the Dual-Band Array Antenna The amplitude weighting and non-uniformly element spacing of the array are commonly employed to reduce the SLL of array. In this designed array with uniform element spacings, only the amplitude weighting can be considered. For the parallel-fed 2400-MHz array, the amplitude weighting can be achieved by the conventional method with many unequal power dividers and it is not presented for brevity. The amplitude weighting has been successfully applied to the series-fed array by a simple adjustment in the geometry of each element [9], [10], [12]–[14], but a special radiating element geometry, of which some parts can be adjusted to control the transmission coefficient of the element is required. Therefore, the feasibility of amplitude weighting combining the quarter-wavelength (at central frequency) transformer and a small adjustment in the geometry of the antenna is introduced.

HE AND XIE: ANALYSIS AND DESIGN OF A NOVEL DUAL-BAND ARRAY ANTENNA WITH A LOW PROFILE

Fig. 5. (a) Configuration of the hybrid-fed linear array with four elements and (b) equivalent circuit mode of one half of the linear array.

Fig. 5(a), (b) depict the hybrid-fed linear array with four elements and the corresponding equivalent circuit mode of one half of the array, respectively. Each half of the array consists of two elements and four quarter-wavelength transformers with and besides the tercharacteristic impedances of minal transmission line with the length of half wavelength and the characteristic impedance equals to the input impedance of the element. As shown in Fig. 5(b), based on the transmission line theory, once the input impedance of the element has been obtained, the desired excitation coefficients of the elements can be achieved by selecting suitable characteristic impedance of the quarterat the wavelength transformers, and the input admittance position of the 2nd element towards the terminal element and of half of the array can be expressed as the input admittance

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transformer is placed between the junction and the input port to achieve the impedance matching. In the aforementioned design, some factors, such as the transmission-line loss, the mutual effects between elements in the same frequency band and in the two sub-bands, the discontinuity between the quarter-wavelength transformers and the fact that the hybrid-fed array is not a pure traveling-wave antenna, etc., are ignored. Therefore, the discrepancies between the simulated and the predicted SLLs are inevitable. To make the simulated SLLs more agree with the prescribed ones, a small adjustment in the geometry of the radiating elements and feed network by experiment or simulation, which is an analogous means proposed in [9], [10], [12]–[14] would be adopted based on the approximate range of the antenna geometrical parameters established by the transmission line theory. of the element For an instance, if the input impedance at the central frequency is 58 approximately, and the symmetrical array with 64 elements shown in Fig. 1(a) has tapered normalized amplitude distribution along -axis and they are

The predicted SLLs at the central frequency are below dB, according to these formulae from (1) to (3), one group of the characteristic impedance of the quarter-wavelength transformers shown in Fig. 5(a) are selected as follows:

Considering these ignored factors mentioned above, the lengths of the quarter-wavelength transformers and radiating elements are shortened slightly and the characteristic impedances of these quarter-wavelength transformers are adjusted as follows:

(1)

(2)

Finally, the currents of the 1st and the 2nd elements can be obtained and they are

(3) where is the voltage between and . Similarly, as shown in Fig. 5(a), the currents of the 3rd and the 4th elements of the hybrid-fed linear array can be obtained. The input admittance at the junction of the two series-fed subarrays is the summation of the two, and a quarter-wavelength

The simulated radiation patterns on - plane are shown in Fig. 6 and it can be seen that, despite the decrease of gain, dB at the central frequency, which the SLLs are below approximate to the prescribed ones very much. Although a few dB with increasing deviation from the central SLLs exceed frequency, they have been decreased greatly compared to those of the array without consideration of the SLLs suppression. Similarly, the SLLs of the radiation patterns on - plane can also be suppressed in the case of that all the linear sub-arrays along -axis are regarded as identical loads. IV. CONCLUSION In this paper, a simple, inexpensive, lightweight, low profile, dual-band array for the 2400/5800-MHz WLAN applications is designed and analyzed. By virtue of the arrangement of the elements, the feed method and the shape of the elements, the

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Fig. 6. Simulated radiation patterns on y -z plane of the shaped symmetrical lineal array antenna with eight elements.

high polarization purity and low insertion loss have been obtained and the mutual effects between the elements in the same frequency band and in the two sub-frequency bands have been minimized, which result in the regular radiation patterns, high antenna efficiency, maintenance of the original impedance characteristic and so on. The simulation result qualifies the effectiveness of the simple method of suppressing the SLLs by amplitude weighting combining the quarter-wavelength transformer and a small adjustment in the geometry of the antenna. The proposed dual-band array is also suitable for applications in other dual-band system by scaling its size and integration with other microwave circuits. ACKNOWLEDGMENT The authors would also like to thank the anonymous reviewers for their constructive comments. REFERENCES [1] Z. Zahairs, E. Vafiadis, and J. N. Sahalos, “On the design of a dual-band base station wire antenna,” IEEE Antennas Propag. Mag., vol. 42, no. 6, pp. 144–151, Dec. 2000. [2] X. Liu, S. He, H. Zhou, J. Xie, and H. Wang, “A novel low-profile dual band dual polarization broadband array antenna for 2/3 G base station,” in Proc. Inst. Elect. Eng. Int. Conf. on Wireless Mobile & Multimedia Networks , Nov. 2006, pp. 1335–1338. [3] R. Pokuls, J. Uher, and D. M. Pozar, “Dual-frequency and dual- polarization microstrip antennas for SAR applications,” IEEE Trans. Antennas Propag., vol. 46, no. 9, pp. 1289–1296, Sep. 1998.

[4] D. M. Pozar, D. H. Schaubert, S. D. Targonski, and M. Zawadski, “A dual-band dual-polarized array for spacing borne SAR,” in Proc. IEEE Int. Symp. Antennas Propaga., Atlanta, GA, Jul. 1998. [5] J. J. Lee, S. Livingston, and R. Koenig, “A low-profile wide-band (5:1) dual-polarized array,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 46–49, Feb. 2003. [6] D. M. Pozar and S. D. Targonski, “A shared-aperture dual-band dualpolarized microstrip array,” IEEE Trans. Antennas Propag., vol. 49, no. 2, pp. 150–157, Feb. 2001. [7] H. Loui, J. P. Weem, and Z. Popovic, “A dual-band dual-polarized nested vivaldi slot array with multilevel ground plane,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2168–2175, Sept. 2003. [8] B. G. Duffley, G. A. Morin, M. Mikavica, and Y. M. M. Antar, “A wide-band printed double-sided dipole array,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 628–631, Feb. 2004. [9] K. Y. Kim, Y. H. Chung, and Y. S. Choe, “Low side lobe series-fed planar array at 20 GHz,” in Proc. IEEE AP-S Int. Symp., Atlanta, GA, Jun. 21–26, 1998, pp. 1196–1199. [10] A. Vallecchi and G. B. Gentili, “Design of dual-polarization series-fed microstrip arrays with low losses and high polarization purity,” IEEE Trans. Antennas Propag., vol. 53, no. 5, pp. 1791–1798, May 2005. [11] M. Gans, D. Kajfez, and V. H. Rumsey, “Frequency independent baluns,” Proc. IEEE, pp. 647–648, Jun. 1965. [12] C. Niu, J. She, and Z. Feng, “Design and simulation of linear series-fed low side lobe microstrip antenna array,” presented at the Asia-Pacific Microw. Conf., 2007. [13] L. James, Drewniak, and P. E. Mayes, “The synthesis of patterns using a series-fed array of annular sector radiating line(ANSERLIN) elements: Low-profile circularly polarized radiators,” IEEE Trans. Antennas Propag., vol. 39, no. 2, pp. 184–189, Feb. 1991. [14] B. B. Jones, F. Y. M. Chow, and A. W. Seeto, “The synthesis of shaped patterns with series-fed microstrip patch arrays,” IEEE Trans. Antennas Propag., vol. 30, no. 6, pp. 1206–1212, Nov. 1982. Shanhong He was born in Hunan Province, China, on November 26, 1973. He received the B.S. degree from Xi’dian University, China, and the M.S. degree from Nanjing University of Science and Technology, China, in 1995 and 2003, respectively. From 1995 to 2004, he was with Nanjing Research Institute of Electronics Technology, where he conducted research on antenna design. Now he is working in the School of Electrical Engineering & Information, Anhui University of Technology as an Associate Professor. His current research interests include numerical calculation and design of microstrip antenna; reflect antennas; array antenna and ultrawidebandwidth antennas.

Jidong Xie (M’01) was born in Jiangsu Province, China, on November 08, 1958. He received the B.S. and M.S. degrees from Chinese University of Science and Technology, in 1982 and 1984 respectively. From 1984 to 2000, he was with Nanjing Research Institute of Electronics Technology, where he was engaged in the research and development of antenna system for Electronic engineering. Now he is working in the Department of Communication Engineering, Nanjing University of Posts and Telecommunications. As a Professor, he is engaged in numerical calculation of reflect antenna, design of antenna system for electronic engineering and development of satellite communication antenna system.

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Dual Polarization Interleaved Spiral Antenna Phased Array With an Octave Bandwidth Régis Guinvarc’h, Member, IEEE, and Randy L. Haupt, Fellow, IEEE

Abstract—A technique to design dual polarization spiral antenna phased arrays using mono polarization spirals is presented. The proposed technique consists of two interleaved subarrays, one for each polarization. The position of every spiral antenna is optimized through a genetic algorithm so that each array is nearly the size of the platform while having low sidelobes. Both resulting arrays have the same properties and are steerable. This method also helps to increase the bandwidth. An 80-element spiral array (spiral of width 0.25 m) with dual polarization is demonstrated, its beam can be steered 30 , over almost one octave (1.97:1). The physical rejection of the sidelobes (i.e., without weighting) is about 10 dB. Index Terms—Genetic algorithms, phased arrays, polarization, spiral antennas.

I. INTRODUCTION

B

ROADBAND, dual-polarized arrays have diverse applications from airborne reconnaissance to surveillance operations. Broadband arrays require broadband elements, such as a spiral antenna. The size of the spiral is proportional to its bandwidth. Thus, increasing the bandwidth forces the element spacing to get larger which in turn causes grating lobes to appear at smaller steering angles. Examples of mono polarization spiral arrays can be found in the literature. For instance, in [1], Nakano has developed a spiral array with a tilted beam for land mobile communications systems. Steyskal et al. [2] have developed a phased array with spiral elements of only one polarization. ARGUS [3] offers an example of applications for astronomy. Dual polarization spirals have been studied (for examples, see [4], [5]) but, to the knowledge of the authors, none have been put in an array, because they are too large. This paper is based upon the results presented in three previous papers. Haupt introduced optimized thinned arrays using a genetic algorithm (GA) [6]. Certain elements are “turned off” or connected to a matched load instead of the array feed network. In this way, a spatial taper is created across the array that results in low sidelobes. A follow on idea used the “turned off” elements in another array [7]. These interleaved arrays can be Manuscript received September 08, 2008; revised November 25, 2008. First published December 04, 2009; current version published February 03, 2010. R. Guinvarc’h is with SONDRA, Supelec, 91192 Gif-Sur-Yvette France (e-mail: [email protected]). R. L. Haupt is with the Applied Research Laboratory, The Pennsylvania State University, State College, PA 16804-0030 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.2009.2037767

optimized for low sidelobes and efficient use of the entire array aperture. In a third paper, Guinvarc’h extended this idea to the design of a dual polarized spiral array [8]. This paper presents an approach to optimizing the design of interleaved spiral antenna arrays simultaneously using relative sidelobe level (RSLL), axial ratio (AR), voltage standing wave ratio (VSWR), and gain as design goals. The spiral element must be optimized in conjunction with the array in order to achieve all the design goals. The optimum design is a dual polarization spiral antenna array with almost an octave bandwidth, steerable . over Section II briefly presents the characteristics of the center-fed two-arm spiral antenna, used as the array element in this paper. Section III will then review the interleaving technique developed and its extension to dual polarization applications. Section IV investigates the radiation properties of a dual polarization spiral array developed according to our method. It especially highlights the effect of this method on its radiation properties. A drawback is then identified and the elementary spiral is modified accordingly in Section V. The last part presents the results for an array constituted of these modified spirals with an octave bandwidth. II. ISOLATED ARCHIMEDEAN SPIRAL ELEMENT The 80-element array proposed in this paper is large enough to demonstrate the concept of a dual polarized, interleaved array while still being small enough to optimize. Since a method of moments solution is proposed, available memory and computer speed become important in deciding the type of element used in the simulation. A free space Archimedean spiral was chosen because it was simple enough to easily compute, but complex enough to be broadband and have a polarization that is easy to change. Placing a ground plane behind the spiral would limit its bandwidth. A more realistic cavity-backed spiral is computationally too difficult for an 80-element array. Calculating the far field pattern for the array proposed in this paper required 3650 MB of memory and 420 minutes over the bandwidth (on a 8 processors server). Thus, the Archimedean spiral is a compromise between a point source or dipole and the more complex cavity-backed spiral. The array presented in this paper is made of spirals with five turns and of 34 mm in diameter. Fig. 1 shows a right hand circularly polarized (RHCP) Archimedean spiral modeled using a commercial method of moment software package FEKO [9]. The spiral is made from a perfect electric conductor as wide as the gap between two strips (self-complementary). It is center fed with a 1 V source. The spiral is broken into triangles with a

0018-926X/$26.00 © 2009 IEEE

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Fig. 1. A two-arm Archimedean spiral.

maximum edge length of . The spiral is in free space and radiates out both sides. The parameters of interest are its axial ratio and its voltage standing wave ratio, which are shown on Fig. 2. The VSWR is below 2 starting at 2.8 GHz (theory also gives 2.8 GHz [10]), while the AR is under 3 dB from 3.5 GHz. III. TECHNIQUE DESCRIPTION The aim here is to find a way to interleave two subarrays, one per polarization. In 1994, Haupt [6] introduced the thinning of an array so that the maximum relative sidelobe level of the thinned array is kept as low as possible. The thinned array has or turned off elements that are either turned on . For a uniform linear array, the RSLL is defined as (outside the main beam):

Fig. 2. Axial Ratio (Left) and VSWR (Right) of a 2-arm center fed Archimedean spiral, that has five turns and a diameter of 34 mm. (a) Axial Ratio. (b) VSWR at 250 ohms.

(1) Fig. 3. Example of a dual polarization spiral array, 1001100110.

where Thanks to the antisymmetry, both subarrays have the same radiation patterns. Because of the binary nature of the problem, this is very well suited to a genetic algorithm optimization [6], where the cost function is the maximum RSLL. , the spiral is RHCP while For this application, when the spiral is LHCP. For the case of the spiral, when this is very well suited as both types of elements (left or right handed) have the same dimensions. Fig. 3 shows a 10-element array corresponding to a chromosome of [1001100110].

number of elements in the array; element on/off (1,0); element spacing; with

angle from endfire direction; steering phase; element pattern; peak of main beam.

Only an even number of elements is considered, because the array consists of two equal sized interleaved arrays. In [7], an array or chromosome had weights represented by

(2) (3)

IV. INTERLEAVED ARRAY OF SPIRALS The array design goals are a scan with an AR less than . The 3 dB, a VSWR less than 2 and a RSLL under targeted lower frequency is 3 GHz and the targeted upper frequency is 6 GHz. The array under study is an 80-element array, with 40 RHCP spirals and 40 LHCP spirals. A first study [8] of only the array factor has shown that 80 elements is a good

GUINVARC’H AND HAUPT: DUAL POLARIZATION INTERLEAVED SPIRAL ANTENNA PHASED ARRAY

trade-off between the computation cost (both memory and time) and the effectiveness of the technique. Equation (1) does not take the coupling into account, so the method of moments program, FEKO [9], is used to model the array instead. Since FEKO takes considerably longer to calculate the antenna pattern, a two step approach is used for the optimization. First, the spiral element is optimized for AR and VSWR using FEKO. Then, the array is optimized for the RSLL only (through the array factor). At the end of this process, we then do a FEKO run to look at the ”true” RSLL and at the AR and the VSWR of the array of spirals. The cost function for the array optimization returns the minimum RSLL over the bandwidth and scan angles of the array. The isolated spiral radiation pattern of Section II is used. The distance center-to-center in the array is chosen to be 38 mm (trade-off between the size of the elements and the inter-element coupling). A chromosome is made of 2N bits (one per antenna), but, thanks to the symmetry, only N bits are to be optimized. The population is made of 4N chromosomes. At each iteration, we keep the first half of the chromosomes (ranked by RSLL). The second half is then recreated by swapping the first half. Finally, two random mutations (of one bit) are introduced. The resulting chromosome is [1010011110110100101010 11011100000110001100111001111100010010101011010010 00011010]. Fig. 4(a) is a plot of the axial ratio of the array for three steering angles. The AR shows little variation as a function of steering angle and is under 3 dB above 4 GHz for all steering angles. Fig. 4(b) shows the VSWR of element 20 (the worst case of all elements) when matched to 250 ohms for three steering angles. The VSWR rapidly decreases until 2.5 GHz. It exhibits a small resonance at 3.25 GHz when the beam is steered to 20 or 30 . Afterwards, it has very few variations. Here, a bandwidth where the VSWR is less than 2 can be defined for frequencies from 2.75 GHz to greater than 10 GHz (except around 3.25 GHz, but this exception only occurs for element 20). Comparing Fig. 4 with Fig. 2 shows only slight differences. The AR bandwidth (AR less than 3 dB) in Fig. 4(a) is 500 MHz less than in Fig. 2(a), probably because of the coupling. The AR and not the VSWR of the spiral elements limits the array bandwidth. Fig. 5 shows the maximum RSLL for an optimized 80-element array, with 40 RHCP spirals and 40 LHCP spirals. These spirals have a diameter of 34 mm, the distance center-to-center is 38 mm. This separation distance leads to grating lobes starting around 3.94 GHz for an array with a uniform spacing of 38 mm. This maximum RSLL is shown for various steering angles, up until 7 GHz to 30 . At broadside, the RSLL stays under (maximum frequency of the simulation). For the other steering until 5.25 GHz, the frequency angles, it remains under at which the grating lobe appears at a 30 steering angle. Whatever the steering angle, the RSLL is quite flat. This is obviously due to the choice of the fitness function. Fig. 6(a) is a plot of the array pattern at broadside for 3 GHz, and in Fig. 6(b) the pattern is steered to 210 at 5 GHz. For the

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Fig. 4. Characteristics of an 80-spiral antenna array. These spirals are standard 2-arms center fed Archimedean spirals, they have five turns, for a diameter of 34 mm. The distance center-to-center in the array is 38 mm. (a) Axial ratio. (b) VSWR at 250 ohms.

first case, the 3-dB beamwidth is 1.7 while it is 1.1 for the second case (thanks to the higher frequency). Assuming the bandwidth is defined for a RSLL less than , an AR less than 3 dB and a VSWR less than 2, all of to , this array has an apthis for steering angles from proximate bandwidth of 4–5.25 GHz, or 27%. This bandwidth is valid for both polarizations. Fig. 7 shows the AR, VSWR, and RSLL bandwidths. This figure clearly shows that the AR establishes the lowest frequency in the bandwidth, while the RSLL sets the highest frequency. The VSWR bandwidth extends beyond these limits. Thus, to increase the bandwidth, two possibilities remain: • extend the lower limit of the AR; • extend the upper limit of the RSLL, this means decreasing the element spacing in the array, thus decreasing the size of the spiral.

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Fig. 7. Bandwidth of the standard array, following the three criteria.

Fig. 8. Bandwidth of a simple interleaved array of spiral with alternating polarization, following the three criteria.

Fig. 5. Maximum RSLL for a 80-element array, spirals are 34 mm wide; interelement spacing is 38 mm.

The corresponding figure for a simple interleaved array of spirals with alternating polarization (0 and 1 alternating for the 80 elements of the array) is provided in Fig. 8 as a reference. Obviously, for the latter array, the grating lobes limit appears well before the spirals start to radiate (and this is even worse for the AR limit) because of the double interelement spacing. Therefore, this array does not function, with respect to our definition of the bandwidth. V. A SMALLER SPIRAL ELEMENT A. Functioning of a Spiral

Fig. 6. Samples of the radiation patterns for a standard array. (a) At 3 GHz and from broadside. at broadside. (b) At 5 GHz and steered at

+30

If we want to reduce the size of the spiral to improve the RSLL, the AR will get worse. Several authors have explained how a center-fed Archimedean spiral works [11], [12]. A wave starting at the center travels through various radiation zones. If the spiral is large enough, the wave dampens enough before reaching the ends. Thus, it will not be reflected and will affect neither the VSWR nor the AR. The first radiation zone is . So, to excite this first radiroughly located at a radius of ation zone, the diameter of the spiral has to be greater than . In order to maintain a good AR and a good VSWR, it is necessary to reduce the reflections from the ends of the spirals. One solution is to increase the length of a spiral arm, using some meandering techniques [13]. An alternative way is based on some work by Wu and King [14], where they have added resistive loads in order to thermally dissipate the excess current, i.e., the current that would otherwise be reflected from the ends. Some methods to realize it are summarized in [15]. We have added some distributed resistive loads to the spiral in order to increase its bandwidth without increasing its size. These 10 loads start at half a turn from the edge and have a linearly increasing values up to 125 ohms. Fig. 9(a) shows the comparison of the VSWR (matched at 250 ohms) of a standard Archimedean spiral and of a distributively resistively loaded (DRL) one. The result is an improvement of the VSWR-bandwidth over the whole frequency range studied. The improvement to the AR is shown in Fig. 9(b). Instead of having a 3-dB AR starting at 3.5 GHz, we now have an AR under 2 dB from less than 1.8 GHz to over 10

GUINVARC’H AND HAUPT: DUAL POLARIZATION INTERLEAVED SPIRAL ANTENNA PHASED ARRAY

Fig. 10. Comparison for 4 different spirals of (Left) their their S . (a) S at 250 ohms. (b) S .

Fig. 9. Comparison of a standard Archimedean spiral and of a distributively resistively loaded one. (a) VSWR at 250 ohms. (b) Axial ratio at broadside. (c) Gain.

GHz. This means that, even if we use a smaller spiral, we will still have better results than with the standard Archimedean one. The drawback of this technique, cf. Fig. 9(c), is a loss of gain at lower frequencies. However, above 3 GHz, the maximum loss is only of 1 dB. To further asses the DRL spiral, we compare in Fig. 10 two smaller (28 mm) DRL spirals with two standard Archimedean spirals without any loading: one with a 34 mm

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S

and of (Right)

diameter, the other with a 28 mm diameter. Fig. 10(a) is a repfor these four cases. Above resentation of the return loss 4 GHz, the 4 spirals have the same behavior; this is no surprise as the current stays in the center of the spirals at high frequencies. Below 4 GHz, there are two groups: the first with the two unloaded spirals and the second with the DRL spirals. The unloaded spirals have a similar behaviour, but the smaller one has a low frequency of 3.85 GHz while the larger one has a low frefrom quency of 3.45 GHz. For the DRL, both are under less than 2.5 to 7.8 GHz. In Section IV, we hypothesized that the degraded AR in the array (compared to that of a spiral alone) was due to mutual coupling. In this section, a second co-polarized spiral is used to look of the four spirals. The distance between the edges at the [cf. Fig. 10(b)] of the pair of spirals is always 4 mm. The is actually improved by the addition of the resistive loads. The longer the loads, the lower the coupling, at the lowest frequency, because there is less current on the ends of the spirals. We can therefore expect a less degraded AR in the array when using

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Fig. 12. Maximum RSLL for a 80-element array, spirals are 28 mm wide; interelement spacing is 34 mm.

Fig. 11. A 80-loaded smaller spiral antenna array, with DRL spiral of 28 mm diameter. (a) VSWR at 250 ohms. (b) Axial ratio.

such spirals. Besides this, it is worth noting that the is actually maximum around 4 GHz. At high frequencies, the agreement between all three 28 mm spirals is due to the very low magnitude of the current on the outer part of the spirals. VI. AN INTERLEAVED SPIRAL ARRAY WITH INCREASED BANDWIDTH A new array is built using DRL spirals as elements. Both co-polarized and cross-polarized spirals have identical resistive tapers. A smaller number of turns is used in order to speed up the simulations. A proper choice of the loading helps to maintain a good AR and a VSWR over the bandwidth. The GA optimization process was re-run, using the new radiation pattern, and based on a lower inter-element spacing. Fig. 11 shows the AR and the VSWR for this new DRL-array, with a 34 mm inter-element spacing. As usual, the VSWR, shown on Fig. 11(a) and matched to 250 ohms, is under 2 for the whole frequency range, except around 4.4 GHz for some of the elements. The AR is now under 3 dB

starting at 3 GHz. So, considering the AR and the VSWR, this smaller array is at least as good as the standard one. The plot of the RSLL shown on Fig. 12 translates to a bandwidth from 2.7 to 5.9 GHz (compared to a maximum of 5.25 GHz previously). It is worth noting that the sidelobe levels are quite flat, for all frequencies and steering angles. The gain is over 14 dB over the bandwidth. The VSWR of Fig. 11(a) has a small increase around 4.4 GHz and the AR of Fig. 11(b) around 3.5 GHz. Regarding the VSWR, it is worth remembering that the inter-element spacing is 34 mm, corresponding to a frequency of 4.4 GHz, thus explaining the increase in the VSWR. It is therefore interesting to look more thoroughly at the coupling/current on the spirals. After comparing the current distribution on the spirals at various frequencies, we did not notice anything of interest at 3.5 GHz or 4.4 GHz. We then simulated a new array where only the co-polarized spirals are loaded. The VSWR is roughly the same. However, a 4 dB peak in the AR occurs at 3.5 GHz. Fig. 13 shows the current on the spirals; the same logarithmic scale has been used for Fig. 13(a)–(c). Fig. 13(a) shows the current at 3.2 GHz with no loads on the cross-polarized spirals. There is no strong current on these cross polarized spirals (loading these spirals does not change the coupling). But, at 3.5 GHz, where the increase in AR occurs, the results are clearly different. When the cross spirals are not loaded, some current can be seen on the cross spirals, therefore explaining the increase of the AR [cf Fig. 13(b)]. When loaded, there is no more current on these cross spirals [cf. Fig. 13(c)] and a good AR result. This motivates us to try to lower the coupling in order to improve the Axial Ratio. Fig. 14 shows the maximum RSLL of the array factor, compared to that obtained from FEKO, cf. Fig. 12. Differences clearly appear, both at low frequencies and at high frequencies , for all steering angles. At high frequencies, when the radiation pattern is computed with the array factor formula, the grating lobes start after 6.5 GHz, while it is after 5.9 GHz when computed with the MoM code,

GUINVARC’H AND HAUPT: DUAL POLARIZATION INTERLEAVED SPIRAL ANTENNA PHASED ARRAY

Fig. 13. Zoom on the current on the spirals. (a) 3.2 GHz, no loads on the cross polarized spirals. (b) 3.5 GHz, only co-polarized spirals loaded. (c) 3.5 GHz, all spirals loaded.

Fig. 14. Maximum RSLL for a 80-element array without coupling (array factor), spirals are 28 mm wide; inter-element spacing is 34 mm.

the main differences between the two cases being the coupling. Reducing the mutual coupling therefore improves the RSLL. VII. CONCLUSION This paper has demonstrated a technique of developing dual polarization wideband phased array composed of mono-polarization spirals. The effect of the technique on the elementary spiral parameters (axial ratio, VSWR, gain) has been highlighted. It has been shown that the coupling reduces the AR bandwidth, while it has almost no influence on the VSWR bandwidth. A modified spiral with distributed loads has been introduced to extend the bandwidth. An array of these DRL spirals has been shown to have almost an octave bandwidth, where the bandwidth is defined for an AR less than 3 dB, a , all of VSWR less than 2 and a sidelobe level less than this for a steering angle . It should certainly be interesting to look at other elementary spiral designs, such as in [16]. REFERENCES [1] H. Nakano, Y. Shinma, and J. Yamauchi, “A monofilar spiral antenna and its array above a ground plane formation of a circularly polarized tilted fan beam,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1506–1511, Oct. 1997. [2] H. Steyskal, J. Ramprecht, and H. Holter, “Spiral elements for broadband phased array,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2558–2562, Aug. 2005.

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[3] S. Ellingson, G. Hampson, and R. Childers, “Argus: An l-band all-sky astronomical surveillance system,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 294–302, Feb. 2008. [4] “Dual Polarized Ambidextrous Multiple Deformed Aperture Spiral Antennas,” U.S. Patent 5,227,807, Jul. 13, 1993. [5] N. Stutzke and D. Filipovic, “Broadband two-arm dual-mode dual-polarized spiral antenna,” in Proc. IEEE Antennas and Propag. Society Int. Symp., 2005, vol. 1B, pp. 414–417. [6] R. Haupt, “Thinned array using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994. [7] R. Haupt, “Interleaved thinned linear arrays,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2858–2864, Sep. 2005. [8] R. Guinvarc’h, “Dual polarization wide-band interleaved spiral antenna array,” presented at the IET Int. Radar Conf., Edinburgh, U.K., 2007. [9] Feko [Online]. Available: www.feko.info [10] J. Kraus and R. Marhefka, Antennas for All Applications. New York: McGraw Hill, 2002. [11] U. R. Kraft, “Polarization properties of spiral antennas: A tutorial,” Electromagnetics, pp. 259–284, 1994. [12] R. G. Corzine and J. A. Mosko, Four-Arm Spiral Antennas. Boston, MA: Artech House, 1990. [13] H. Nakano, “A meander spiral antenna,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Jun. 2004, vol. 3, pp. 2243–2246. [14] T. T. Wu and R. King, “The cylindrical antenna with nonreflecting resistive loading,” IEEE Trans. Antennas Propag., pp. 369–373, May 1965. [15] J. Maloney and G. Smith, “A study of transient radiation from the Wuking resistive monopole-fdtd analysis and experimental measurements,” IEEE Trans. Antennas Propag., vol. 41, no. 5, pp. 668–676, May 1993. [16] M. Lee, B. Kramer, C.-C. Chen, and J. Volakis, “Distributed lumped loads and lossy transmission line model for wideband spiral antenna miniaturization and characterization,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2671–2678, Oct. 2007. Régis Guinvarc’h (S’02–M’04) received the engineering degree and the M.S. degree 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 Institut National des Sciences Appliquées (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 the Supelec ONERA NUS DSTA Research Alliance (SONDRA), Gif-Sur-Yvette, France, where he is working on antennas and HF surface wave radar.

Randy L. Haupt (M’82–SM’90–F’00) received the B.S. degree in electrical engineering from the U.S. Air Force Academy, U.S. Academy, CO, the M.S. degree in engineering management from Western New England College, Springfield, MA, in 1981, the M.S. degree in electrical engineering 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 a Professor of electrical engineering at the U.S. Air Force Academy and Professor and Chair of Electrical Engineering at the University of Nevada-Reno. In 1997, he retired as a Lt. Col. in the U.S. Air Force. He was a Project Engineer for the OTH-B radar and a Research Antenna Engineer for Rome Air Development Center. From 1999 to 2003, he was Professor and Department Head of Electrical and Computer Engineering at Utah State University, Logan. He is currently a Senior Scientist at the Applied Research Laboratory, Pennsylvania State University, State College. He has many journal articles, conference publications, and book chapters on antennas, radar cross section and numerical methods and is coauthor of the book Practical Genetic Algorithms, 2nd edition (New York: Wiley, May 2004). He has eight patents in antenna technology. Dr. Haupt is a member of Tau Beta Pi, Eta Kappa Nu, International Scientific Radio Union (URSI) Commission B, and the Electromagnetics Academy. He was the Federal Engineer of the Year in 1993.

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A Novel Geometrical Technique for Determining Optimal Array Antenna Lattice Configuration Srinivasa Rao Zinka, Il-Bong Jeong, Jong-Hoon Chun, and Jeong-Phill Kim, Member, IEEE

Abstract—We present a new 2D geometrical technique for determining optimal element arrangement for planar, phased array antennas with specified scan limits. This geometrical technique is not limited to conical or pyramidal scanning, but can be extended to any scan type that can be represented with an analytical equation. In addition, simple equations are given for two very important scanning types, conical and pyramidal. These equations provide deeper understanding and simpler graphical solutions than other pure graphical techniques. This paper discusses optimal array arrangement from the viewpoint of general lattice, which itself includes a hexagonal lattice as its subset. An important practical system, where this technique was found to be useful, is the -face phased array antenna providing scanning throughout a hemisphere. Simple equations are given for determining the maximum off-axis scan and tilt angles of each face with respect to the zenith. Finally, the lattice arrangement of each face is decided by the new design technique.

and tilt angle . After deciding and , of faces the scan sectors are transformed from the earth’s coordinates to the array coordinates . Then, the presented geometrical technique can be used to analyze the sine space of each individual array face.

Index Terms—Conical scan, grating lobes, hemispherical coverage, n-face array, phased arrays, pyramidal scan.

(1)

I. INTRODUCTION OMPREHENSIVE analysis of grating lobe appearance in planar, phased array antennas with different possible scanning specifications is presented. Although the concept of grating lobes and optimal planar array arrangement is well understood graphically [1]–[5], a complete analytical technique has not yet been performed. The basic concept behind the geometrical design technique presented in this paper is that almost doall types of scan specifications can be mapped onto main as a single ellipse or a combination of multiple ellipses. In addition, the technique discussed in this paper is generalized to include all types of planar array lattices and provides simple expressions for instructive graphical plots. One main application of this new design technique is determining the optimal lattice configurations for -face phased arrays to cover an entire hemisphere. A brief comparison between two different hemispherical sectorizations is given in Section IV. Although there are many parameters to be taken is into consideration, the maximum off-axis scan angle usually chosen as the main criterion for deciding the number

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Manuscript received March 19, 2009; revised May 27, 2009. First published July 07, 2009; current version published February 03, 2010. This work was supported by Samsung Thales Co., Ltd., Kangnam-gu, Seoul, under Contract STC-C-07-032. S. R. Zinka, I. B. Jeong, and J. P. Kim are with the School of Electrical and Electronic Engineering, Chung-Ang University, Seoul 156-756, Korea (e-mail: [email protected]; [email protected]). J. H. Chun is with the Research and Development Center, Samsung Thales, Gyeonggido, 449-885, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2026920

II. BASIC THEORY It can be shown [6] that for a general two-dimensional lattice structure described in Fig. 1(a), grating lobes occur in the domain (sine space) at

. where and Since by definition, , radiating far fields are confined to the circular , often known as visible space. The disk space, described as invisible space, is related remaining to the stored energy in the near field, which is analogous to the phenomenon of evanescent modes in a waveguide. Usually, an antenna engineer intends to avoid the appearance of all the , main lobe case) within grating lobes (except for the the visible space. Fig. 1(b) shows the visible space and the grating lobe spaces domain. Assuming that placed according to (1) in the represents the domain of the specified main lobe scan positions, – represent contours of all the possible the closed loops nearest grating lobe scan positions. From Fig. 1(b), it can be observed that all the grating lobe contours are just touching the and . Thus, for the given visible space circle, except for , the array lattice arrangement shown in scan specification Fig. 1(a) is not optimal. In this paper, optimal array arrangement is defined as the configuration that maximizes the array’s unit . In other words, an optimal configuration mincell area imizes the number of elements needed in a given array aperture. To achieve this optimal array lattice configuration, Fig. 1(b) should be modified according to the following description. 1) All the left-hand side grating lobe contours should be moved upward and the right-hand side contours downward. 2) After an optimal skew with respect to the axis, both left-hand side and right-hand side grating lobe contours should be moved horizontally toward each other, so that all would just touch the visible space circle.

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and a much easier graphical solution. To explain this technique, two very important and general scan options are considered in Section III. III. MAPPING OF GENERAL SCAN SPECIFICATIONS ONTO THE DOMAIN Two of the most important scan types are the conical and pyramidal scans. A complete analysis of these two scan options is presented in this section. The conical sector to be scanned is assumed to be an elliptical cone. Even though an elliptical conical scan does not have much practical significance, the theory is nevertheless the same as for the circular conical scan. The second scan option, the pyramidal scan, is described as scanning a polyhedral sector that has a polygon as its base and triangles with a common vertex (origin) for faces. Any practical scan specification can be expressed as either a conical or a pyramidal scan or as a combination of the two. A. Conical Scanning Fig. 2(a) shows an elliptical conical scan region with scan and in the and directions, respectively. It limits is further assumed that the conical region is symmetric with respect to the axis. Without loss of generality, a plane with an elliptical boundary and perpendicular to axis can plane. Any point , lying on the be mapped onto the boundary of this elliptical surface as shown in Fig. 2(a), is given by

(2) where

Fig. 1. (a) A general array lattice and (b) corresponding grating lobe locations in sine space.

3) If is symmetric with respect to both the and the axes, then the contours , , and will be at the same distance from the axis for the optimal array lattice configuration. This optimal array element arrangement is a . hexagonal array lattice However, it should be mentioned that a hexagonal lattice may is asymmetric with respect to be the optimal lattice even if the or axes. To substantiate this statement, one example of trapezoidal scanning is provided in Section IV. From the above domain for grating discussion, it is clear that to analyze the should be evaluated for a given set of scan lobe appearance, is obtained using pure graphical specifications. In [3]–[5], techniques. Although pure graphical techniques are simple and straightforward, they do not provide much information about the scanning procedure. Thus, in this paper a new geometrical technique is presented, which provides in-depth understanding

(3) It is shown in Appendix A that mapping of this elliptical conplane is given by tour onto the

(4) The mapped region in the plane, , given by (4), is shown in Fig. 2(b). The elliptical boundary of this region makes it difficult to obtain an analytical solution for the optimal array configuration. As mentioned before, elliptical scanning is rarely needed, unlike circular conical scanning. For circular conical scanning, and (4) represents a circle

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Fig. 3. Sine space corresponding to circular conical scan.

From (6), for a given general lattice configuration, the maximum , is given by off-axis scan angle,

(7) Thus, given the input parameters from (7). Fig. 2. Conical scanning: (a) geometry of the elliptical conical sector and (b) mapped scan region in sine space.

An analytical solution for circular conical scanning can be derived from Fig. 3. Centers of the grating lobe contours – are at distances – , respectively, from the origin, where – are given as

(6)

,

can be evaluated

B. Pyramidal Scanning Conical scanning, described in the previous subsection, is simple and gives an analytical solution for the optimal array configuration. However, in many practical applications the scan region is not conical, but pyramidal. To take advantage of the irregular shape of the pyramidal scan region, a new mapping technique is necessary. In this subsection, first the rectangular pyramidal scanning procedure is described, after which the results obtained for the rectangular pyramidal scan are extended for a more general pyramidal scan with a polygonal base. Fig. 4(a) shows a rectangular pyramidal scan region, which is symmetric with respect to the axis and with scan limits and in and directions, respectively. Again without loss and perpendicular of generality, a rectangular surface at plane. Only the region corto the axis is mapped onto the is mapped and the remaining map responding to can be constructed from the symmetry. Any point P lying on the boundary of the rectangular surface can be represented by (8), have the same shown at the bottom of the page, where ,

if (8) if if (9) if

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ZINKA et al.: A NOVEL GEOMETRICAL TECHNIQUE FOR DETERMINING OPTIMAL ARRAY ANTENNA

Fig. 4. Rectangular pyramidal scanning: (a) geometry of the rectangular pyramidal sector and (b) mapped scan region in sine space.

meaning as defined by (3). Following a similar procedure given in Appendix A, mapping of the rectangular pyramidal sector domain is given by (9). It is interesting to note onto the that the corresponding contour represents an intersection of two mutually orthogonal ellipses centered at the origin, as shown in Fig. 4(b). A complete description of rectangular pyramidal – is shown scanning with the closest grating lobe regions in Fig. 5(a). Because of the symmetry of the grating lobe lattice, – , as shown in Fig. 5(b). It can it is sufficient to analyze be observed that the optimal value of the array lattice parameter , , is solely decided by the ellipse

(10) The other array lattice parameter, , is controlled by two ellipses ( and ) and two critical points ( and ). In addition, the primary scanning region, , is symmetric with respect and the axes. Thus, the optimal array lattice to both the for rectangular scanning is a hexagonal lattice. With the conceptual insight provided by the rectangular pyramidal scanning procedure, it is intuitive that the same basic concept can be applied to any general pyramidal scanning. One

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Fig. 5. Rectangular pyramidal scanning.

simple example is shown in Fig. 6(a). The region to be scanned is a pyramidal sector with a triangular base coinciding with the . The lines joining the origin to the closest point on plane each side of the base are at the angles , and with respect to the axis. In addition, the lines joining the shifted origin to these closest points are at angles , and with respect axis. For better visualization, only information related to the to face 3 is shown in Fig. 6. From (9), it is clear that the mapping or contour onto the domain is an ellipse. If a of the contour is rotated along the axis, its corresponding mapdomain. This pheping also rotates by the same angle in nomenon can be explained by the basic idea behind mapping. domain onto the Mapping of a region from the domain is equivalent to projection of the corresponding contour onto the plane [1]. on the sphere Such projection does not change its shape, but just rotates with the corresponding rotation of the contour. Thus, as shown in Fig. 6(b), mapping of the triangular pyramidal sector is nothing more than a combination of three ellipses, with each ellipse rotated by a proper angle from the axis. This general technique can be used to map any arbitrary pyramidal region with a polygdomain. The application of this techonal base onto the nique for covering a hemispherical region is demonstrated in Section IV.

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Fig. 6. Triangular pyramidal scanning.

IV. APPLICATION OF SCANNING THEORY FOR COVERING A HEMISPHERE One of the main applications of phased array antennas is hemispherical scan coverage. Phased arrays providing beam scanning throughout a hemisphere require three or more faces, as shown in Fig. 7. Many criteria, such as the scan reflection coefficient and the realized gain at maximum scan angle, should be taken into consideration before choosing the number of faces. In this paper, no effort was made to determine the number of faces. Instead, the authors only attempt to explain the new mapping technique for a given number of array faces. It should be noted here that a hemisphere can be divided in several ways. Three simple sectorizations are shown in Figs. 7(b) and 8. A 3-face array and its far-field triangular pyramidal sectors are shown in Fig. 7. It is possible to reduce the maximum off-axis scan angle by increasing the number of radiating side faces. Fig. 8 shows two sectorizations, each having a sector perpendicular to the zenith. These two sectorizations are exactly the same from the viewpoint of the and the tilt angle of maximum off-axis requirement each face with respect to the zenith . The main difference is the path joining points and . Actually, there is an infinite number of possible paths joining these two points, but the more and . A plane going through path obvious ones are

Fig. 7. Three-face array for hemishpere scan coverage.

would intersect the axis at the origin, whereas a plane would be parallel to the plane going through path and intersect the axis at point , as shown in Fig. 9. These sectorizations are named after their respective authors [2]–[4]. The sectorization shown in Fig. 8(a) is optimal, and preferable, for the following reasons. 1) In the Kmetzo-Corey sectorization, the top face has to scan more regions, thus decreasing its corresponding unit cell area. 2) Knittel’s sectorization is much simpler with respect to each set of the individual array coordinates. Thus, to explain the theory proposed in this paper, Knittel’s sectorization was adapted. Before proceeding, simple equations and are given in Section IV-A. for evaluating A. Tilt Angle and Maximum Off-Axis Scan Requirement For an -face array with no face perpendicular to the zenith triangular pyramidal sectors), the tilt angle and max(i.e., imum off-axis scan requirements are given by

(11) where

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Fig. 9. A more detailed representation of the two sectorizations. Fig. 8. Comparison between different types of sectorizations.

B. Scan Specifications With Respect to Array Coordinates Similarly, and for an -face array with one face trapezoidal pyrabeing perpendicular to the zenith (i.e., midal sectors and one regular polygonal pyramidal sector) are given by

and are determined, scan specifications with Once respect to the earth coordinate system should be transformed to an array coordinate system. An equivalent (but not exactly same) representation of a trapezoidal pyramidal scan (Fig. 9(a)) is shown in Fig. 10(a). Scan specifications with respect to the array coordinate system are derived in Appendix B and they are given by

(12) where . Expressions given by (12) can be derived either by coordinate transformations [4] or by simply using the following conditions.

(14) where

(13) where points , and are depicted in Fig. 9(a). In addition, without loss of generality, the radius of the far-field sphere can be assumed as 1.

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Fig. 11. 2D top view of (z = 1)-plane [Fig. 10(a)].

C. Example

Fig. 10. Trapezoidal pyramidal scanning with respect to array coordinates.

Only scan specifications corresponding to contour are given in (14). Remaining parameters can be evaluated intuitively from Fig. 10(a). Mapping of this trapezoidal pyramidal space is done using the theory described sector onto the in the previous section and shown in Fig. 10(b). Following a similar procedure, scan specifications corresponding to a triangular pyramidal scan [Fig. 7(b)] are given by

(16) where

(17)

To demonstrate the new geometrical design technique, a 5-face array was considered. The sectorization for a 5-face array is shown in Fig. 9(a). The tilt angle of each side face with and the maximum off-axis scan angle respect to the zenith were determined from (12). Their values were 74.46 and 47.06 , respectively. From Fig. 9(a), it is evident that the array lattice configuration of the side face is different to that of the top face. Thus, mapping of each face is done separately as shown in Fig. 11. In addition, in Table I, these results were compared with the results obtained from the circular conical scan assumption (i.e., the irregular shape of the scan region is not taken advantage of). By taking the irregular shape of the scan region into consideration, unit cell areas obtained for the top and side faces were, respectively, 10% and 8% larger than is asymmetric for the circular conical scan. Even though with respect to the axis for the side face, from Fig. 11(a) it is clear that the optimal array configuration is a hexagonal lattice. Similar analysis can be performed for any arbitrary multiface array with specified conical or pyramidal scan sectors. V. CONCLUSION A 2D geometrical technique has been developed for determining the optimal array configuration of planar-phased array antennas. The basic theory for mapping different types of scan regions onto sine space has been presented. To demonstrate the proposed technique, a multiface array antenna for hemispherical coverage was considered. Simple equations were given to and of each array face. In addition, the theoevaluate retical analysis for transforming scan specifications from earth coordinates to array coordinates has been presented. Finally, the technique presented in this paper was applied to analyze a 5-face array antenna for the optimal array geometries. The geometrical technique presented in this paper provides deeper understanding and simpler graphical solutions than other pure graphical techniques. In addition, this geometrical technique is not limited to conical or pyramidal scanning, but can be extended to any scan type that can be represented with an analytical equation.

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From (2) or by simply observing Fig. 2(a), it is evident that

(20) and from the fundamental definition of sine space

(21) Substituting (20) into expressions given by (21) results in

(22) A similar procedure can be used to derive mapping equations for a pyramidal scan region given by (9). APPENDIX B SCAN SPECIFICATIONS WITH RESPECT TO ARRAY COORDINATES A 2D top view of Fig. 10(a) is shown in Fig. 11. Applying can be written as simple geometry, line (23) Fig. 12. Hemisphere scan coverage by 5-face array: (a) sine space corresponding to the side faces and (b) sine space corresponding to the top face.

Similarly, a line going through the origin and perpendicular to is given by

TABLE I LATTICE CONFIGURATIONS OBTAINED BY DIFFERENT SCAN SECTOR ASSUMPTIONS

(24) where

(25) APPENDIX A MAPPING OF CONICAL SCAN REGION ONTO SINE SPACE The locus of any point on the contour of the elliptical scan region as shown in Fig. 2(a) is given by (2). This expression was obtained using the following polar representation of ellipse. If an ellipse is expressed as

The point of intersection of lines and from the origin, where is given by

is at a distance

(26) and from Fig. 10(a), it is evident that

(18) (27) then its polar form can be written as

Therefore, from (26) and (27), (19)

is given by

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From (24) and (25),

is given by

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Finally, two more important equations for graphical analysis are

(30) Similar equations can be derived for triangular pyramidal scan and are given by (16) and (17). REFERENCES [1] W. H. V. Aulock, “Properties of phased arrays,” Proceedings of the IRE, pp. 1715–1727, 1960. [2] G. H. Knittel, “Choosing the number of faces of a phased-array antenna for hemisphere scan coverage,” IEEE Trans. Antennas Propag., vol. AP-13, pp. 878–882, 1965. [3] J. L. Kmetzo, “An analytical approach to the coverage of a hemisphere by n planar phased arrays,” IEEE Trans. Antennas Propag., vol. AP-15, pp. 367–371, 1967. [4] L. E. Corey, “A graphical technique for determining optimal array antenna geometry,” IEEE Trans. Antennas Propag., vol. AP-33, pp. 719–726, 1985. [5] A. Jablon and A. Agarwal, “Optimal number of array faces for active phased array radars,” in Proc. IEEE Antennas Propagat. Symp., Monterey, CA, 2004, pp. 4096–4099. [6] A. K. Bhattacharyya, Phased Array Antennas, Floquet Analysis, Synthesis, BFNs, and Active Array Systems. Hoboken, NJ: Wiley, 2006.

Srinivasa Rao Zinka was born in Guntur, India, on April 25, 1984. He received B.Tech. degree in electronics and communication engineering from Jawaharlal Nehru Technological University, Hyderabad, India, in 2005 and the M.Tech. degree in electrical engineering from the Indian Institute of Technology, Kanpur, in 2007. He is currently pursuing his Ph.D. degree at Chung-Ang University, Seoul. His current research interests are in the areas of phased array antennas, frequency selective surfaces, microwave circuits and application of optimization techniques for shaped beam synthesis. He is also involved in developing a comprehensive array design and analysis software.

Il-Bong Jeong was born in Gyeonggi, Korea, on February 20, 1982. He received the B.S. degree in electronic engineering from Chung-Ang University, Seoul, Korea, in 2008, where he is working toward the M.S. degree. His recent research interests are in the areas of phased array antennas and microwave circuits. He is especially interested in optimized beam pattern synthesis and coupling compensation techniques for phased arrays. In addition, currently he is involved in developing phased array antennas, where an aperture-coupled microstrip patch is used as a radiator.

Jong-Hoon Chun was born in Busan, Korea, on March 5, 1959. He received the B.S. degree in electronic engineering from Kyeongbook National University, Daegu, Korea, in 1982, the M.S. degree in electrical engineering from Hanyang University, Seoul, Korea, in 1985, and the and Ph.D. degree in electrical engineering from KAIST, Daejeon, Korea, in 2000. From 1985 to 2002, he was a Research Engineer in the Research and Development Center, LG Innotek, Korea, where he was involved with the design of systems, transmitters, and receivers for various kinds of radars. Since 2004, he has been involved with radar system & subsystem design in the Research and Development Center, Samsung Thales, Korea. Dr. Chun is a member of the IEEE Societies of Microwave Theory and Techniques and Aerospace and Electronic Systems.

Jeong-Phill Kim was born in Jeju, Korea, on November 2, 1964. He received the B.S. degree in electronic engineering from Seoul National University, Seoul, Korea, in 1988, and the M.S. and Ph.D. degrees in electrical engineering from Pohang University of Science and Technology, Pohang, Korea, in 1990 and 1998, respectively. From 1990 to 2001, he was a Research Engineer in the Research and Development Center, LG Innotek, Korea, where he was involved with the design of antennas, transmitters, and receivers for various kinds of radar system. Since 2001, he has been a faculty member with the School of Electrical and Electronic Engineering, Chung-Ang University, Seoul, Korea. He has made contribution to the development of material constants measurement setup for dielectric resonator. He also established efficient network models of microstrip-to-slotline transition, slot-coupled microstrip lines, microstrip-fed slot antenna, aperture-coupled microstrip patch antenna, and aperture-coupled cavity-fed microstrip patch antenna and coupler. Using these efficient network models, he has developed various kinds of novel slot-coupled microstrip circuits such as out-of-phase power dividers, multi-slot couplers, magic-T’s, and filters. In addition, he was involved in developing phased array antennas, where an aperture-coupled microstrip patch and a tapered slot were used as a radiator, and a microstrip meander-line on a ferrite substrate and coplanar waveguide on a thin film ferroelectric as a phase shifter. He also developed FDTD codes to simulate microwave circuits and antennas. In addition, he was involved in developing a phased array antenna with microstrip patch radiator and PIN-diode phase shifter, and calibrating this antenna using the REV (rotating-element electric field vector) and MTE (measurement of two elements) methods. As well as, he designed an antenna with two fixed beams for direction finding application. His recent research interests include microstrip circuits and antennas, dielectric resonator antennas, mutual coupling phenomena in phased array antenna, numerical modeling and analysis, microwave measurements, and wireless communication and sensor systems such as repeater and random noise radar. Dr. Kim is a member of the IEEE Societies of Microwave Theory and Techniques, Antennas and Propagation, and Aerospace and Electronic Systems.

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A Numerical Methodology for Efficient Evaluation of 2D Sommerfeld Integrals in the Dielectric Half-Space Problem Amit Hochman, Member, IEEE, and Yehuda Leviatan, Fellow, IEEE

Abstract—The analysis of 2D scattering in the presence of a dielectric half-space by integral-equation formulations involves repeated evaluation of Sommerfeld integrals. Deformation of the contour to the steepest-descent path results in a well-behaved integrand, that can be readily integrated. A well-known drawback of this method is that an analytical expression for the path is available only for evaluation of the reflected fields, but not for the evaluation of the transmitted fields. A simple scheme for numerical determination of the steepest-descent path, valid for both cases, is presented. The computational cost of the numerical determination is comparable to that of evaluating the analytical expression for the steepest-descent path for reflected fields. When necessary, contributions from branch-cut integrals and a second saddle point are taken into account. Certain ranges of the input parameters, which result in integrands that vary rapidly in the neighborhood of the saddle point, require special treatment. Alternative paths and specialized Gaussian quadrature rules for these cases are also proposed. An implementation of the proposed numerically determined steepest-descent path (ND-SDP) method is freely available for download. Index Terms—Green functions, integral equations, method of moments (MoM), nonhomogeneous media, Sommerfeld integrals.

I. INTRODUCTION

T

HE determination of the fields of an elementary source radiating in plane-stratified media is a canonical problem in electromagnetics. Even though some variants of this problem have been the subject of research since the beginning of the 20th century [1], [2], they are still of interest today. Comprehensive references are [3]–[5], and reviews of computational aspects can be found in [6]–[8]. Nowadays, interest is largely motivated by integral-equation formulations for scattering and propagation problems, as they entail repeated evaluation of the fields of elementary sources that constitute the Green’s function. The starting point for the evaluation of the various Green’s functions is an integral representation of the fields (or potentials), of the Sommerfeld integral (SI) type. Although the literature on SI evaluation is vast and the procedures are varied, most methods include some or all of the following steps. Manuscript received November 05, 2008; revised May 14, 2009. First published December 04, 2009; current version published February 03, 2010. This research was supported in part by the Israel Science Foundation. The authors are with the Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel (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.2009.2037761

• Contour Deformation: The integration contour is deformed from the real axis to a contour on the complex plane. The purpose of this step is to obtain a more well-behaved integrand by avoiding pole and branch-point singularities and possibly also minimizing phase variation along the path. Some possible paths are given in [9]–[11]. • Singularity Subtraction: Singular terms of the integrand are subtracted and then added back after analytical integration. This step has been used together with contour deformation [12], [13], or as an alternative to it [14]. • Numerical Integration: The value of the integral is estimated from a finite number of samples of the integrand. When this is done by a quadrature rule, the estimate is a linear combination of the samples of the integrand. In a more sophisticated scheme, the integrand (or some part of it) is approximated by a superposition of complex exponentials and this approximation is then integrated analytically [15]. This so-called Discrete Complex Image Method (DCIM), which has found widespread use [16], [17], is closely related to the continuous complex image method [18]. In a similar technique [19], the integrand is approximated by a superposition of rational functions, and the resulting approximation is then integrated analytically. Among the possible integration contours, the steepest-descent path (SDP) passing through a saddle point is considered, in some respects, the optimal choice [20]. Another aspect of using the SDP is that if, in the process of deforming the original path to the SDP, a branch point is intercepted, a path surrounding the intercepted branch point must be added. Although this entails some book-keeping, it also highlights an appealing feature of the method, namely, that the integral is obtained as a sum of distinct, physically meaningful, contributions. From a computational point of view, as simple quadrature is used, this method can potentially outperform the popular DCIM which involves finding the complex images by more computationally intensive methods such as Prony’s method [12], or the matrix pencil method [21]. Moreover, evaluating the Green’s function along the SDP is essential for the fast inhomogeneous plane wave algorithm [22] which can be used to solve electromagnetically large layered-media computational complexity. problems in When the observation point and the source point are in the same medium, an analytical expression for the SDP is available, and it has been used extensively [6], [23], [24]. In contrast, when the source and observation points are not in the same medium, an analytical expression for the SDP is not available. One op-

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tion, in this latter case, is to determine the path numerically, but this was deemed too computationally expensive, or otherwise impractical [23], [25]. In this paper, a simple and efficient scheme for numerical determination of the SDP is presented. Using this scheme, path determination is not slower than evaluating the analytical expression for the SDP, available for the reflected fields. The numerically-determined SDP (ND-SDP) can thus complement earlier works which used only the analytically known SDP. An example of such a work is that of Cui and Chew [23] in which the electric line-source case was studied. In [23], the analytically obtained SDP was used for the reflected fields, whereas for the transmitted fields, it was assumed that the observation point was close enough to the interface so that an analytically determined approximation of the SDP could be used. In the present work, the SDP is determined numerically both for the reflected and transmitted fields of electric and magnetic linesources. As far as the determination of the SDP is concerned, the reflected and transmitted fields can be treated equally, by use of the same code. The handling of intercepted branch points, however, should be different. For example, when a branch point is intercepted in the transmitted field case, a contribution from a second saddle point must be taken into account. Following earlier works that used the SDP [6], [23], the numerical integration employed throughout is Gaussian quadrature. An attractive feature of Gaussian quadrature is that it has only one parameter that must be set by the user, namely, the number of integration points. The integration points and the weights are then determined by the rule. We preserve this feature by deriving all the values of the parameters in the numerical scheme from the number of integration points. For certain combinations of source and observation point coordinates and material parameters, the integrand varies rapidly on the SDP and it becomes difficult to integrate it efficiently. This problem is well-known (see for example the introduction of [8]), and it is not particular to numerically determined SDPs. The parameters for which these difficulties are encountered correspond to three different physical cases, as follows. • Critical angle case: Occurs in the calculation of the reflected fields, when the source is in the dense medium, and the angle of specular reflection is close to the critical angle. This case was considered in [23], where the authors resorted to a uniform asymptotic expansion. • Grazing angle case: Occurs when the source and observation points are both near the interface, but far apart laterally, on a wavelength scale. • Quasi-static case: Occurs when the source and observation points are both near the interface, and close to each other laterally, on a wavelength scale. The difficulties encountered in this case were discussed in [26]. In all three cases the problem is mathematically similar and can be explained as follows. When evaluated on the SDP, the exponential factor of the integrand resembles a Gaussian curve, and its standard deviation can be used to define a neighborhood of the saddle point from which the dominant contribution to the integral comes. If the other factors, such as the amplitude and the reflection or transmission coefficients, vary rapidly in this neighborhood, the integration becomes difficult. As the most

rapid variation of these other factors is near the branch points, the difficulties arise when the saddle point is close to one of the branch points, on a standard-deviation scale. This can happen in two different ways. One is when the saddle point approaches the branch point, which occurs when the angle of specular reflection approaches the critical angle or becomes a grazing angle. The other, which occurs in the quasi-static case, is when the standard deviation increases to the extent that a branch point is included in the neighborhood of the saddle point. Assuming the number of Gaussian integration points is kept constant, the accuracy will deteriorate in these cases. Although each of the three cases is handled differently, all solutions are based on integration along an alternative path by use of a Gaussian quadrature rule that is tailored to the behavior of the integrand on the path. In the critical- and grazingangle cases, the Gaussian quadrature rules used are the generalized Hermite and Laguerre rules, which are known from the classical theory of orthogonal polynomials [27]. The Gaussian quadrature rules used in the quasi-static case, on the other hand, are derived by so-called discretization methods, which can generate these rules for quite arbitrary weight functions [28]. Appropriate weight functions are determined according to the behavior of the integrand on an alternative path, and their corresponding quadrature rules are derived. The rules are specific to the field component being calculated and to the material parameters. Once the rules are derived (a process which takes on the order of a second) they can be applied to all relevant source and observation point pairs. The techniques described in this paper have been combined in a MATLAB computer program that yields accurate results for a wide range of input parameters in very short execution time. One of the advantages of the ND-SDP method is that it can be easily vectorized. This means that all operations can be done by functions that operate on vectors of the input parameters, yielding a considerable reduction in function-call overhead. The program, which operates in this vectorized fashion, is freely available for download [29]. The remainder of this paper is organized as follows. The various integrals that are to be solved are summarily formulated in Section II. Next, the numerical determination of the SDP is explained in Section III, and the inclusion of contributions from intercepted branch points is discussed in Section IV. The cases for which integration along the SDP yields poor results: the critical-angle case, the grazing-angle case, and the quasi-static case are considered in Sections V–VII. Numerical results are shown in Section VIII, and the paper is summarized in Section IX. II. PROBLEM STATEMENT AND FORMULATION The configuration to be considered is that of a unit-amplitude electric line-source (ELS) or a magnetic one (MLS) situated in one of two contiguous dielectric half-spaces, as shown and it in Fig. 1. The coordinates of the line-source are is oriented in the direction, i.e., parallel to the interface. It is , as the case can be assumed throughout that treated by symmetry considerations. The relative permittivities and regions are denoted by and , of the respectively, and for both regions the permeability is taken to be that of free-space, . The free-space permittivity is denoted

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TABLE I EXPRESSIONS FOR I AND A (k ) IN (3)

Fig. 1. Two contiguous dielectric half-spaces, with a line-source at (x ; y ).

TABLE II EXPRESSIONS FOR I AND A (k ) (8)

by

. All fields and sources are assumed to follow a harmonic time dependence, which is suppressed. The fields in this configuration are well-known, and they can be expressed as follows: (1a) (1b) (2a) (2b)

for for for for

where and are the fields of the line-source in a homogeneous material of relative permittivity . They do not require an integral representation and can be evaluated in closed form and , (see, for example, [30, p. 224]). The reflected fields, can be expressed in terms of integrals of the following form:

in which is the wave number in the region. For future reference, we will use to denote , and to denote . The integration in (3) stays on the real axis except near the branch path where it is indented into the first and third points at quadrants. The correspondence between the integral and the various reflected field components is shown, together with the , in Table I. appropriate amplitude functions and , Like the reflected fields, the transmitted fields, can be expressed in terms of similar integrals. We have

(3)

(8)

where the complex phase function

is given by

where the complex phase function

is given by

(4) and

is related to region, i.e.,

through the separation equation in the

(5) in which is the wave number in the region, with being the speed of light in free-space. In (3), the is given by reflection coefficient for ELS

(6a)

for MLS

(6b)

is related to via (5), and is related to where region, i.e., the separation equation in the

through

(7)

(9) and the transmission coefficient

is given by

for ELS

(10a)

for MLS

(10b)

In Table II, the correspondence between the integral and the various transmitted field components is shown, together with the . Field components not appropriate amplitude functions shown in Tables I–II are zero. To unify a number of formulas in the sequel, pertaining to the reflected and transmitted fields, it is convenient to introduce a “unified” complex phase function, , which is equal to for the reflected fields, and to for the transmitted fields. This function can be written succinctly as

(11)

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and

are given by

(12) for the reflected fields, and by

traced path is as long as required by the Gaussian quadrature rule that is used to evaluate the integral, once the path is determined. To facilitate efficient integration, the piecewise-linear path is then approximated by a low-order polynomial that is fitted to the path vertices, in a least-squares sense. In this way we obtain a parametric representation of an approximate SDP that can be written in terms of a real parametric variable , as . We then have,

(13) for the transmitted fields. Lastly, to ensure proper behavior at infinity, we require

(14) The integrals in (3) and (8) have no known analytical expression and they are notoriously difficult to evaluate numerically. This is because the integrands are rapidly oscillating and slowly decaying. As already mentioned, the approach proposed in this paper consists of deforming the integration contour from the original path to the SDP. As we assume that the half-spaces are lossless and of positive permittivity, no poles are intercepted in the course of the deformation. However, branch points of the and may be intercepted. In integrand at this case, a path surrounding the intercepted branch point must be added. It should be remarked that in the integrand of , the appear only in the reflection coefbranch points at ficient but not in the complex phase function . On the other hand, in the integrand of , the two branch points appear both in the transmission coefficient and in the complex phase function . It is due to this essential difference that the calculation and analysis of the transmitted fields is in general more complicated. III. NUMERICALLY DETERMINED STEEPEST-DESCENT PATHS With the aid of the Cauchy-Riemann equations, the SDP can be shown to coincide with the level curve of the function that passes through the saddle point. Denoting value of the saddle point by , this curve is given the implicitly by

(15) For the reflected field, it is possible to manipulate (15) to obtain an explicit expression for , the imaginary part of , in terms of its real part. This, however, is not possible for the transis more complicated. mitted field, because the form of entails the solution of a nonIn this case, determining linear equation for each point of the curve. The observation that allows fast determination of the SDP is that even when there is no analytical solution to (15), there is always an analytical expression for the direction of steepest descent at any point in the plane. This can be used to approximate the SDP complex by a piecewise-linear path, obtained by stepping in the direction of steepest descent with a constant step size , beginning at the saddle point [see Fig. 2(a)]. This process continues until the

(16)

where is any of the integrands of (3) or (8). To this transformed integrand, Gaussian quadrature, which requires a real integration variable, can be readily applied. Clearly, the parametric representation should be smooth, otherwise the multiplication by its derivative will destroy the smoothness of the transformed integrand. This is why a low-order polynomial representation is preferred over a piecewise-linear one. Once the polynomial is found, the integration points are distributed in the range of according to the Gaussian quadrature rule, and the derivative of the path at these points, also required for the integration, is evaluated. An illustration of this scheme is depicted in Fig. 2(b). The parameters of the path determination scheme are set as follows. First, the number of integration points, denoted , is set by the user, according to the desired trade-off between accuracy and computation time. This trade-off is investigated in some detail in Section VIII. We take the number of vertices of the piecewise-linear path equal to . The step size is taken to be equal to the average Gaussian spacing between integration points. This ensures that the length of the piecewise-linear path fits the Gaussian quadrature rule. Also, since the step size of the Gaussian quadrature rule is scaled to fit the variation of the integrand, this choice usually provides an adequate step size for the calculation of the path. Lastly, we take a small number for the order of the polynomial: we have experimented with orders in the range of 5–25, and as long as the order is sufficiently ) the exact value does not seem smaller than (less than to matter much. Although this procedure may appear to involve significant is overhead, we show below how it can be expedited if kept constant for many pairs of source and observation points. Keeping constant also allows all operations to be vectorized, thus reducing function-call overhead considerably. Clearly, to be practical the method must therefore yield accurate results for a broad range of parameters, using a constant number of integration points. On the other hand, it is assumed that operations that are done once for a given (regardless of the number of pairs of source and observation points) may take longer without affecting the overall efficiency. A. Saddle Points The starting points for the SDP computation are the saddle points that are obtained from the condition that the derivative of the complex phase function vanish at the saddle point. When

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B. Numerical Determination of the SDP In the first stage of the algorithm, points on an approximate SDP are determined by stepping in the direction of steepest descent, beginning at the saddle point. Let denote a column complex values along the yet to be detervector of the mined SDP, where, for convenience, is assumed even. At the saddle-point, the angle between the SDP and the real-axis [4, Ch. 4.1b]. Hence, the th element of , , is calcuis and , lated by first setting the two middle elements, to and , respectively. The rest of the are then calculated by the iterative formulas

(18) (19) where

is a unit-amplitude complex number, given by

(20) which, when interpreted as a unit vector in the complex plane, points in the direction of steepest descent. Here, denotes the derivative of the unified complex phase function, , . Explicitly, is given by evaluated at the point Fig. 2. Illustration of the numerical scheme for the determination of the SDP. In (a), the uniformly distributed points are obtained by tracing the SDP, starting at the saddle point. In (b), a low-order polynomial is fit to the uniformly spaced points. Then, the integration points are distributed on the path defined by the polynomial according to the Gaussian quadrature rule.

m

calculating the reflected fields, the saddle point is determined readily as

(21) In (21), the signs of the radicals will differ from the proper choice in (14) whenever the path crosses a branch cut into an improper Riemann-sheet. When calculating the reflected fields, the computational cost of the numerical determination of the SDP can be compared with the computational cost of evaluating the analytical formula for the SDP in the reflected field case

(17) (22) where is the angle shown in Fig. 1 between the normal to the interface and the ray that reaches the observation point by specular reflection. When calculating the transmitted fields, the de. termination of the saddle point leads to a quartic equation in The four solutions of the quartic equation lead to eight candidate , not all of which are zeros of the derivative of solutions for the complex phase function on the relevant Riemann-sheet. As discussed in [31], two such solutions are relevant. In the lossless case, there is one solution on the portion of the real axis between the origin and the branch point of the thin medium, and one complex solution. To determine which of the eight candidate solutions are the two relevant ones, we evaluate the derivative on the relevant Riemann-sheet for all candidate solutions. The absolute value of the derivative is then a measure of the error of the solution, and it is seldom exactly zero for any of the candidate solutions. We take as the relevant solutions the real solution with smallest error, and the complex solution with smallest error.

where . Clearly, the cost of evaluating is roughly the same either way. It must be noted, however, that (22) gives the exact SDP, whereas (18), (19) only yield an approximation to it. Nevertheless, the error of the approximation, which is for each step, does not alter the general behavior of the integrand on the path appreciably. points on the approximate SDP path are deterAfter the mined, an analytical parametric representation of the path is obtained and the points are distributed according to a Gaussian quadrature rule (to be specified in Section III-D). C. Parametric Representation of the Path An analytical parametric representation of the numerically determined path is obtained by fitting a low-order complex poly-

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nomial, , to the points , calculated by (18), (19). For an th order polynomial , given by

TABLE III ANALYTICAL VS. NUMERICAL DETERMINATION OF THE SDP

(23) we denote by an -element column vector of its (yet to be determined) complex coefficients. The values of which correspond to the points are set, arbitrarily, to , i.e., they are uniformly distributed in the [ 1, 1] interval. The are then obtained by solving the following linear system in a least-squares error sense:

(24) where denotes the Vandermonde matrix whose th element is . As is well-known, this solution of (24) is given by

(25) where denotes the pseudo-inverse of , which can be . When the obtained from a singular-value decomposition of Green’s function has to be evaluated many times for different pairs of source and observation points, the various path vectors can be arranged in a matrix , and the corresponding vectors . Since of polynomial coefficients can be arranged in a matrix and depend only on , the matrix is given by

(26) where it is assumed that is kept constant for all paths. In this way, the matrices and are computed only once, regardless of the number of source and observation point pairs. Other once, the calcuthan the short time required to calculate lation of each coefficient of a polynomial requires just complex multiplications, and the calculation of all the coefficients complex multiplications. In of a polynomial requires is complex, whereas is real, the number fact, since only . of (real) multiplications is Once the polynomials have been obtained, they must be evaluated in order to redistribute the points along the path. The new , are given by , where denotes the points, Vandermonde matrix that corresponds to the points distributed according to the Gaussian quadrature rule. The th element , with the Gaussian quadraof this matrix is given by ture rule points scaled to fit the [ 1, 1] interval. Similarly, the points, , are given by derivatives of the paths at the , where holds the coefficients of the difis the corresponding Vanderferentiated polynomials, and by omitting its last monde matrix which is obtained from matrix, the and (highest-power) column. Similarly to the matrices are computed only once, regardless of the number of source and observation-point pairs. Other than the short time required to calculate these matrices once and for all, the calculaadditional multiplications, tion of each path requires

multiplicaand the evaluation of the derivatives requires tions. It is worth mentioning that when an analytical formula for the SDP is used, the derivative of the path must still be calculated by (20), (21), with the of the exact path. The computational cost of this calculation equals the computational cost of the calculation used to trace out the path numerically. It turns out then, that to compare the times of analytical vs. numerical determination of the SDP, one should compare the time it takes to calculate and given , with the time it takes to evaluate (22). , asThe computational complexity of both calculations is , and the calsuming to be a small constant. Typically, and given requires multiplications culation of per point. Since (22) includes a rather costly square-root operation, equivalent to about 85 multiplications in our system, its evaluation is not faster than the numerical determination. This comparison is summarized in Table III. D. Gaussian Quadrature Rule for Approximating the SDP Integral Bearing in mind that along the SDP the integrand has a Gaussian envelope, the most natural quadrature rule to apply is the Hermite-Gaussian rule. An -point quadrature rule of this type is known to yield an exact result for integrands that multiplied by a Gaussian are polynomials of degree weight function [32]. In this work, however, two half-range points were Hermite-Gaussian rules [28, p. 88], each with used for each half of the SDP. The lower order of each half adds robustness, which is particularly useful when the saddle point approaches a branch point. As the width of the Gaussian changes from one observation point to the next, the integration points and weights must be scaled according to the standard deviation of the Gaussian. The required standard deviation, , is readily determined from the second derivative of the complex at the saddle point. We have phase function

(27)

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Fig. 4. The real part of the complex phase function of the reflected fields,  , overlaid with contours of constant Re( ). The thick line shows the path used by the algorithm, and the circle marker marks the saddle point. The branch point of the y > 0 region is intercepted, necessitating the addition of a BCI. In (a), the source is in the optically thin medium. In (b), the source is in the optically dense medium. Fig. 3. The real part of the z component of the electric field due to a unit amplitude ELS. In regions I and II no branch points are intercepted. In (a), the source is in the optically thin medium. In (b), the source is in the optically dense medium. In region III, the branch point associated with the y > 0 region is intercepted, and in region IV the complex saddle point is intercepted.

IV. INTERCEPTED BRANCH POINTS In the course of deforming the original path to the SDP, the branch points may be intercepted. If this happens, an integral around the intercepted branch point must be added to the SDP contribution. A. Reflected Fields When calculating the reflected fields, an intercepted branch point leads to the well-known lateral wave contribution [4, p. 508]. In the lossless case assumed here, the branch point is intercepted whenever

(28) where and are the relative permittivities of the optically thin and the optically dense media, respectively. It follows that the branch point is intercepted whenever the angle of the spec, ular reflection is beyond an angle

which, in the case of a wave propagating in a dense medium and impinging on an interface with a thin medium, is the well-known critical angle. Note that although this critical angle does not have the total internal reflection physical interpretation when the source is in the thin medium, condition (28) applies regardless of whether the source is in the dense medium or the thin medium. In the latter case, the lateral wave contribution is evanescent [3, p. 263]. The region in physical space which corresponds to the branch point being intercepted is shown in Fig. 3, where it is marked as region III. The lines delimiting the regions have been plotted on top of an image of the component of the electric field due to a unit amplitude ELS, as calculated by use of the ND-SDP method. The case of the source being in the thin medium is shown in physical space in Fig. 3(a), and in the complex plane in Fig. 4(a). The complementary case, when the source is in the dense medium is shown in physical space in Fig. 3(b), plane in Fig. 4(b). and in the complex The branch-cut integral (BCI) that must be added to the SDP contribution is calculated along paths shown in Fig. 4. The BCI path surrounds the branch cut, which is taken along the steepestdescent path that originates at the branch point. In practice, the two halves of the path are taken infinitesimally close to the branch cut. This path has been used in [20], [23] and also in the asymptotic evaluation of SIs [3], [4]. As the two halves of

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the BCI path are on the same curve, it is convenient to form a new integrand that is the difference between the integrands on each half of the path. The integral is then a sum of the integral of this difference integrand and the contribution of an infinitesimal circle near the branch point. This last contribution turns out to be zero as can be seen by inspecting Tables I and II, and noting that the singularities at the branch points are at most of . Turning to the integration of the difference integrand, we first obtain a parametric representation of the path of steepest descent away from the branch point, by the same method used in , Section III. Denoting this parametric representation by it can be easily shown that the difference integrand will decay , while at exponentially with when singularity. In accorthe branch point itself, it will have a dance with this dependence on , we adopt a generalized Laguerre-Gaussian quadrature rule [32, p. 96] for which the cor. Taking the sinresponding weight function is gularity into account in this way is a very simple method to deal with the integration of the singular integrand in an accurate and robust manner.

B. Transmitted Fields As mentioned in Section III-A, when calculating the transmitted fields, there will always be a real saddle-point between the origin and the branch point associated with the thin medium. A SDP passing through this saddle point furnishes one contribution to the integral. This path will intercept the branch-point associated with the dense medium, whenever the path intercepts a second, complex, saddle point [31]. When this happens, a second SDP, that passes through the complex saddle point and surrounds the dense branch point must be added. The physical significance of this contribution has been considered for line-source and beam excitations in [31], [33]. As explained in these references, this contribution has the ray-optical interpretation shown in Fig. 5. When the source is in the dense medium and the observation point is close to the interface, the contribution of the complex saddle-point corresponds to a ray originating at the source and reaching a point on the interface close to the observation point. This ray then proceeds to the observation point as an evanescent wave. Similarly, when the source is in the thin medium and it is close to the interface, an evanescent component of the plane wave spectrum of the line source reaches the interface and then proceeds to the observation point as a propagating wave. The conditions for the inclusion of the complex saddle point and the contribution are as follows. Denoting by values of the real and complex saddle points, respectively, the complex saddle point is intercepted whenever . This condition is demonstrated in Fig. 6, where its correctness can be readily observed. The region in physical space which corresponds to the complex saddle point being intercepted is marked as region IV in Fig. 3. The lines delimiting region IV, in which the complex saddle point is intercepted, bear close resemblance to those of Fig. 1 in [31], where this region was sketched.

Fig. 5. Ray-optic interpretation of the contributions due to two saddle points. The real-line saddle point corresponds to the usual refracted ray, and the complex saddle point corresponds to a ray which is evanescent in the thin medium and propagating in the dense medium. In (a), the source is in the dense medium, whereas in (b), the source is in the thin medium.

V. CRITICAL ANGLE CASE The saddle point for the reflected field is given by (17), from which it is evident that the saddle point for the reflected fields plane, between is found on a segment of the real axis in the rethe origin and the branch point associated with the gion. When this region is the denser of the two, the saddle point region may and the branch point associated with the coincide, as shown in Fig. 7. This happens when is equal to the critical angle defined by . Although of the reflected field this branch point does not appear in , which case, it does appear in the reflection coefficient is given by (6). As can be readily verified from (6), the reis apflection coefficient near the branch point at . If the conventional Hermiteproximately Gaussian quadrature rule is used, the rapid variation of the integrand near the branch point degrades the accuracy of the integration. This problem can be dealt with by use of a generalized Hermite-Gaussian quadrature rule [28, p. 32–33] for which the . To obtain an integrand with weight function is this dependence, we first subtract and subsequently add a SI with a unit reflection coefficient. This SI can be evaluated analytically, as it is just the field of an image source which has the same amplitude as the original source. After subtracting the image source, if the saddle point and the branch point are close, but not exactly coincident, we shift the entire path laterally so that it is centered on the branch point. As long as the shift is

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the size of the neighborhood being some effective radius of the saddle point region. We found, empirically, that a good effective radius is , the standard deviation of the Gaussian envelope of the integrand, given by (27). To summarize the procedure, we represent the fields reflected at the critical angle by a sum of the fields due to an image source and a correction term which can be integrated readily, on a slightly shifted path, by the generalized Hermite-Gaussian quadrature rule.

VI. GRAZING ANGLE CASE

Fig. 6. The real part of the complex phase function of the transmitted fields,  , overlaid with contours of constant Re( ). In (a), the complex saddle point is not intercepted, whereas in (b) it is. The thick line shows the contour found by the algorithm, and the circle markers mark the saddle points.

Fig. 7. Path for the critical angle case. The value shown is the real part of the complex phase function of the reflected fields,  , overlaid with contours of constant Re( ). The saddle point, marked by the circle marker, coincides with the branch point of the y > 0 region.

small, the behavior of the integrand on the path will not be altered significantly, except in the neighborhood of the branch point, where it will have a square-root singularity. The generalized Hermite-Gaussian quadrature rule can now be applied to the integrand on the shifted path. A reasonable criterion for the use of this technique is that the branch point be contained in a neighborhood of the saddle point,

, or equivalently, , the When saddle point(s) approach the branch point(s), and this makes the integration more difficult. In the evaluation of the reflected field, . In the saddle point approaches the branch point at the evaluation of the transmitted field, two saddle points are relevant [31]. The real transmitted-field saddle-point approaches , and the complex transmitted-field the branch point at . As a saddle-point approaches the branch point at saddle point approaches a branch point, the integrand along the corresponding SDP becomes more difficult to integrate due to the proximity of the singularity. Also, the path bends more tightly around the branch point, and the contours of steepest-descent become aligned with the imaginary axis. Consequently, the integration along the SDP takes the form of a branch-cut integration around a vertical branch-cut. Branch cut integrals are common in the SI literature. However, the method usually employed for their evaluation, namely, integration along the path of steepest descent beginning from the branch point [3], [4], [20], [23], can only be applied in the reflected field case and only for , marked by the BCI around the branch point at in Fig. 8(a). It cannot be applied to the other BCIs because in these cases the complex phase function is singular at the branch point to be surrounded, and therefore no path of steepest-descent passing through the branch point exists. To overcome the above mentioned difficulty, we propose to calculate each BCI on a path that is made-up of two infinitesimally close, straight parallel lines, and an infinitesimal circle around the branch point. These keyhole-like paths are shown in ) for the reflected fields, and in Fig. 8(b) Fig. 8(a) (marked for the transmitted fields. The integrand along these paths becomes singular at the branch point when the radius of the circle and the distance between the straight lines vanish. The order of the singularity, however, is determined solely by which field component is being calculated, and it does not depend on the distance of the saddle point from the branch point. It can therefore be dealt with, in a very robust manner, by including it in the weight function of the Gaussian quadrature rule. The orientations of the keyhole paths are determined by the following considerations. As can be observed in Fig. 8, the contours of steepest descent tend to become parallel away from the branch points. The limiting values of the slopes of the contours can be readily derived by noting that in this as limit, and , where and are either 1 or 1 depending on the Riemann-sheet. For example, to calculate the asymptotic slopes on the proper Riemann-sheet,

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, where the regions for which the asymptotic slopes are , and are indicated in Fig. 8(b). These slopes too have simple geometrical interpretations that are shown in the inset of Fig. 8(b). If the slope of the branch cut and the path were taken equal to one of the asymptotic slopes, the integrand would oscillate slowly on one half of the path, but then it would oscillate rapidly on the other half. Since the accuracy of the integration is determined by the fastest variation along the path, we choose the slope of the branch cut so as to minimize the oscillation rate of the integrand. This corresponds to choosing the branch cut and the path on the bisector of the angle between two lines having asymptotic slopes. Lastly, choosing the keyhole path in plane guarantees that the conthe lower half of the complex tribution of the segments used to connect the two BCI paths to the original path and to each other at infinity is zero. A. Gaussian Quadrature Rule As the two halves of the keyhole paths coincide with the branch cut, it is convenient to form a new integrand that is the difference between the integrand evaluated on each side of the branch cut. The integral is then a sum of the integral of this difference integrand and the contribution of the infinitesimal circle, which turns out to be zero. To evaluate the integral, we express by the following parathe keyhole paths beginning at metric representations

Fig. 8. BCI paths for, (a) the reflected, and (b) the transmitted fields, in the grazing angle case. In the insets, geometrical interpretations of the asymptotic slopes given by (30) and (31) are shown. The source point is labeled S , its image is labeled S , the observation point is labeled P , and the point of specular reflection is labeled R. In (a), the slopes of the lines SR and RP with respect to the interface are equal to the asymptotic slopes a and a , respectively. In (b) the slopes of the lines S R, S P , and RP , are equal to a , a , and a , respectively.

both and are taken to be . The asymptotic slopes, , are then given, with the notation of (12), (13), by

(29) By specializing (29) to the case of the reflected fields, we find that the asymptotic slopes are given by

(30a) (30b) The asymptotic slopes have a simple geometrical interpretation shown in the inset of Fig. 8(a). Similarly, by specializing (29) to the case of the transmitted fields, we find that the asymptotic slopes are given by

(31a) (31b) (31c)

(32) where the angle between the positive real axis and the path is denoted by . The integration with respect to can now be effected by Gaussian quadrature. The choice of the Gaussian quadrature rule is dictated by the behavior of the difference integrand along the keyhole path. This , integrand decays exponentially with when or singularity. while at the branch point itself, it has a In accordance with this dependence on , we adopt a generalized Laguerre-Gaussian quadrature rule for which the corresponding . weight function is In this weight function, the exponent of must be set according to the singularity of the difference integrand at the branch point. By use of Tables I and II it can be determined that the exponent should always be set to 0.5 except when calculating the or components of the reflected electric or magnetic fields; in these cases, the exponent should be set to 0.5. B. Criterion for the Use of the Keyhole Paths As the specular reflection angle approaches , the integration along the keyhole paths becomes more accurate. This is because the asymptotic slopes and the slope of the path all become aligned, thus making the integrand slowly oscillating on the path. When the source and observation points recede from the interface, the accuracy of the integration along the keyhole path deteriorates, but then the usual SDPs can be used because the saddle points move away from the branch points. A criterion we found to be simple and reliable is to use the SDP whenever , the distance of the saddle point from the branch point,

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is greater than the standard deviation of the Gaussian envelope of the integrand , given by (27), and to use the keyhole path otherwise.

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, can be written as the The integral to be evaluated, , and an sum of two terms: an integral of a regular function, integral of a singular term, , as follows:

(35) C. High Contrast (36)

When , the evaluation of the fields of a ELS does not present any special difficulties. The MLS case, however, is complicated by the presence of a leaky pole near the branch . This follows from examination of the expoint at pressions for the reflection and transmission coefficients given is never zero in the ELS case, in (6). The denominator of has a pole at , where but in the MLS case

(33) is the free-space wave number. This pole is leaky, as and it occurs only in the two improper Riemann sheets defined by , and , . The leaky pole is never intercepted by the keyhole paths, but the accuracy of the integration will deteriorate if the pole is close to a path. From (33), it follows that this pole is constrained to the segment of the real axis between the origin and the branch . As the contrast increases, the pole approaches point at . A keyhole path surrounding this the branch point at branch point will penetrate into the improper sheet that contains the leaky pole, and consequently, accuracy will be poor if this path is used. To overcome this problem, we use a singularity subtraction technique similar to that employed in the modified saddle-point technique [34, pp. 615–620]. In this well-known technique, the singular term is subtracted from the integrand, and then added back after its integral along the SDP is written in terms of the . In this work, the path of integration is error function, the keyhole path and not the SDP, but nevertheless, the integral along the keyhole path can also be written in terms of the error function. the difference integrand evaluated on Let us denote by the keyhole path given by (32), as function of the parametric has the following properties.: variable . The function , where from (32), • It has a simple pole at

(34) Here is the angle of the path that surrounds the branch point associated with the thin medium. , with an attenuation • It decays exponentially when constant denoted by . it is , the sign depending on the • When component being calculated, as explained in the last paragraph of Section VI-A.

(37) (38) , and it behaves The integrand in (36) is regular at when and . asymptotically the same as Hence, the generalized Gauss-Legendre rule specified in Section VI-A can be used for its integration with confidence. The second term, , can be evaluated by use of the error function. If the exponent of in (37) is 0.5, we have

(39) and if the exponent of

is

0.5, we have

(40) In (39), (40), the principal branch of the square-root should be used. We consider a case to be one of high-contrast if and only if . The reason that the keyhole paths can is that the integrand along be used if these paths decays exponentially, with an attenuation constant . So if , which is approximately the integrand will decay rapidly on the keyhole paths compared to the variation due to the nearby branch point. D. Low Contrast , the critical angle approaches so the critWhen ical angle case and the grazing angle case merge. This situation is of considerable interest, among other things, as an approximation to continuous stratification (see [35, p. 25–32] and references therein). The keyhole paths and Gaussian quadrature rules described in this section are inadequate in the lowcontrast case. The Gaussian quadrature rule takes into account the singularity that the path surrounds, but not the other singularity, which, when the contrast is low, is nearby. To solve this problem, instead of using two keyhole paths, one around each branch point, we use one path that surrounds both of them at a prudential distance. The path, shown in Fig. 9, is made up of and two straight lines which begin a semicircle of radius on the real line and their slopes are the asymptotic slopes of (29)–(31). The integration along the straight lines is performed,

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the reflection coefficient by

and the transmission coefficient by plitude functions can all be written as

for ELS

(42a)

for MLS

(42b) . The am-

(43) Fig. 9. Path for the low contrast case. The value shown is the real part of the complex phase function  , overlaid with contours of constant Re( ).

similarly to the integration on the keyhole paths, by a (nongeneralized) Laguerre-Gaussian quadrature rule. The integration along the semicircle is performed by a Legendre-Gaussian quadrature rule. The number of integration points is divided equally between the semicircle and the straight lines. , is chosen as follows. On The radius of the semicircle, the one hand, the radius should be large enough so as to avoid the branch points. On the other hand, if the radius is too large, the along the semicircular number of oscillations of path will be proportionately large, and this will require a finer integration along the semicircle. In the grazing angle case, we can , make the approximation from which it can be deduced that letting ensures that only one period of fits into the semicircle. Still, it may not appear evident that this radius increases, the semicircle is large enough. Indeed, as shrinks towards the branch points. It turns out, however, that if this occurs, it is possible to revert to the keyhole paths proposed earlier. We therefore consider a case to be one of low . Like in the contrast if and only if high-contrast case, the reason that the keyhole paths can be used is that the integrand along these paths if decays exponentially, with an attenuation constant which is ap. So if , the integrand proximately will decay rapidly on the keyhole paths compared to the variation due to the nearby branch point.

where

is either 1, , or . Clearly, when , the variation of with becomes negligible, whereas that of the other factors, , and is unaffected. Naturally, a Gaussian quadrature rule that is tailored to the variation of the exponential factor is inadequate in this situation. It is more reasonable to use a Gaussian quadrature rule that is and to assume that the exponentailored to the variation of tial factor is slowly varying. If decays to zero quickly as , the main contribution to the integral will come from a small region in the plane, and, as the exponential factor will not vary significantly in this region, a small number of integration points will yield good accuracy. In some cases, however, the does not decay to zero, or it decays very slowly, function . To see this, note that is the product of or as by the appropriate . The amplitude function decays, if at all, no faster than , while, in some cases, and do not decay at all. For example, in the ELS case, when , but then . In the MLS nor decay to zero. This problem can be case, neither remedied by subtracting from or their limiting value , as is done in the DCIM [12]. The subtracted term as can be added after analytical integration as it corresponds to the field of a line-source placed either at the image point (for the reflected field) or at the source point (for the transmitted field). , deAfter the subtraction, the remaining factor, denoted . cays at least as fast as A. Integration Path

VII. THE QUASI-STATIC CASE It is well-known that in the quasi-static case the exponential in (3) varies more slowly than the other factor factors, denoted collectively by . As detailed in Tables I is the product of and II and in (6) and (10), the factor a reflection or transmission coefficient, and an amplitude function that depends on the field component being evaluated and on whether the source is an ELS or an MLS. The relationship between the various factors is more clear when they are written . The in terms of a normalized integration variable, exponential factor is then given by

(41)

The path is chosen according to the following guidelines. • The contributions due to segments connecting the ends of the path with the ends of the original path should be zero. This implies that the path should start and end on the proper plane. sheet, in the lower half of the complex • The path should steer away as much as possible from the branch points, and it should not intercept them. • The path should not venture too high-up into the upper plane, as the integrand can grow half of the complex . exponentially when These guidelines lead us to choose a parabolic path [shown in and , at Fig. 10(a)] that crosses the real line at ensures that the path does 45 angles. Passing through , while passing not pass close to the branch points at through ensures that it does not pass close to the . The 45 angles are a compromise branch point at

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B. Gaussian Quadrature Rule In Fig. 10(b) a typical quasi-static case integrand is shown on the proposed path, as a function of a together with parametric variable . Once the path and the permittivities have , which remains the same for all been set, the function source and observation points, is used as the weight function of the Gaussian quadrature rule. To obtain this rule, we use a discretization method [28]. In discretization methods, a positive weight func, is replaced by a discrete weight function tion, say , where the sampling points are distributed on the support of , and is the Dirac delta function. Deriving a Gaussian quadrature rule for is much more simple than deriving such a rule for because the difficult step in the derivation is the evaluation of the , and for moments of the weight function, this is trivial. It can be shown, that for any weight function that has finite moments of all orders, the integration points and weights obtained in this way converge to those of the Gaussian , as [28, p. 91]. In this work, quadrature rule for we used Gautchi’s implementation of this technique, which is freely available [36]. depends on the Lastly, it is important to note that while permittivities, it does not depend on the source or observation points. Therefore, for given permittivities, the same integration path and Gaussian quadrature rule can be used for all source and observation points belonging to the quasi-static case. Fig. 10. Integrand and path in the quasi-static case. In (a), the value shown is . the amplitude of the transmitted E integrand in dB above its value at k In (b), the real and imaginary parts of the integrand are shown, together with the weight function jF  j, on the proposed path. The plots are normalized by absolute value of the integral.

~( )

=0

C. Criterion for the Quasi-Static Case

between going too high into the upper half plane and passing too near the branch points. The equation for this path is given by

We use the quasi-static case method when the variation of is slow enough so that its period in the plane . is larger than the effective width of the envelope function , we note that the oscillations To find the period of of this function are most rapid in the direction perpendicular to . In that of steepest descent, and they are fastest when this limit, the smallest oscillation period, , is given by

(44)

(47)

To derive the Gaussian quadrature rule, it is convenient to replace this equation by a parametric representation of the path such that a finite range of the parametric variable, , spans it entirely. To this end, a transformation similar to the one in [10] is used. We have

Since the envelope function starts to decay as fast as when , we use the quasi-static method whenever . Rephrased in a more physically meaningful form, this , where condition reads is the wavelength in the denser medium.

(45)

VIII. NUMERICAL RESULTS

(46)

Sommerfeld integral evaluation techniques have to cope with a wide range of problem parameters. These are: the coordinates of the source and observation points, the material parameters, the type of source, and the field component being calculated. The behavior of the integrands, and the physics underlying this behavior, change significantly with the problem parameters. It therefore seems inevitable that only an amalgam of various techniques can efficiently yield accurate results across the parameter

which, when substituted into (44), yields

A Gaussian quadrature rule can now be derived for the integrand on the finite range of .

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domain. In this section, the performance of the techniques described in this paper, which we collectively term the ND-SDP method, will be charted. Throughout this section, we will show how the relative error of the numerical integration, , varies with the parameters of the problem and the algorithm. To calculate this error, we need the exact value of the integral, which, of course, is unavailable. To approximate the exact value, we use adaptive Gauss-Kronrod integration along a path which follows the real axis and is slightly indented into the first and third quadrants. This path, taken from [10], has the following parametric representation:

(48) where (49a) (49b) (50) The adaptive integration is stopped when it has converged to machine precision. In Fig. 11(a), the relative error in the computation of the fields is shown on a logarithmic reflected and transmitted , , scale. These fields are due to an ELS at being the free-space wavelength. The various paths used in each region are also indicated. Some of the paths, such as those labeled 1, 4, and 5, consist of a single curve, whereas the rest of the paths consist of two curves. The integrand is evaluated times along each curve, so the evaluation in regions 1, 4 and 5, is two times faster than the evaluation in the other regions. . A plot similar to the one shown in In this example Fig. 11(a) is shown in Fig. 11(b). The only difference between these figures is that in the latter the ELS is closer to the inter, and the range of the observation coordiface nates is smaller. This is done to show the calculation errors in the quasi-static case (region 2) and in the low-contrast grazing angle case (region 4). As can be readily observed in Fig. 11, , the largest error is less than 1% (it is 0.0058), with and in most regions the error is much smaller. Even smaller errors can be obtained by increasing . This can be seen in on is plotted, Fig. 12(a), where the dependence of for representative observation points taken from each region in Fig. 11(a). Similarly, for representative points in Fig. 11(b), the vs. is shown in Fig. 12(b). For corresponding plot of all regions, the plots imply convergence, approximately at an exponential rate. For some regions, like 1–4 in Fig. 11(a), the convergence is extremely fast, the results reaching machine precision with 20–40 samples. In other regions, like 5 in Fig. 11(b), the numerical integration converges more slowly. However, as can be seen in Fig. 12(b) for the point from region 5, converrelative error is achieved with as few gence with less than as 25 samples. An important advantage of using Gaussian quadrature, as opposed to more elaborate integration techniques such as those used in the DCIM, is that the computation time scales linearly with the number of integrand samples. This can be observed in Fig. 13, where the computation times on a 3 GHz PC for the

Fig. 11. Relative error on a logarithmic scale, log 1, as function of observation coordinates x and y . The fields computed are the transmitted and reflected E fields due to an ELS at x = 0 and: (a) y = , (b) y = =100. The relative permittivities are " = 3 and " = 1.

0

0

representative points of Fig. 11(a) are shown. Times shown are for the evaluation of a single integral with the time required for initialization (less than 1 sec) deducted. As can be inferred from these plots, fairly high accuracy can be obtained in less than 1 msec per evaluation by using, say, 40 sampling points. At the other end, very high accuracies (approaching machine precision) can be obtained by investing just a few milliseconds per evaluation. The numerical results presented so far indicate that the varies considerably with the accuracy obtained for a given problem parameters. To obtain a more comprehensive picture of this variation, we tested the algorithm on a large number (100000) of cases with randomly selected parameters, and . The range of parameters is given in the table in Fig. 14, together with a histogram of the relative errors. As expected, the range of relative errors is very large, spanning about 14 orders of magnitude. Like in Fig. 11, the largest relative error is less than 0.006. The various paths used in the algorithm generally yield more accurate results as the distance between the source and observation points increases. Hence, increasing the range of the coordinates would lower the errors. A. An Adaptive Integration Scheme distribution shown in Fig. 14 is The median of the . As , this small error implies that remarkably high accuracy can be attained at a modest computational

HOCHMAN AND LEVIATAN: A NUMERICAL METHODOLOGY FOR EFFICIENT EVALUATION OF 2D SIs

Fig. 13. Computation time vs. Fig. 12(a), on a 3 GHz PC.

1

427

m

for the representative points used in

m

Fig. 12. Relative error, , on a logarithmic scale, vs. , for representative points taken from: (a) regions 1–9 in Fig. 11(a), and (b) regions 1–5 in Fig. 11(b).

cost. Nevertheless, the large variation in the relative errors for may appear disadvantageous. If some minimum a constant accuracy is required, has to be taken large, even though for many cases a much smaller number would suffice. A simple adaptively, increasing it solution to this problem is to vary until the results have converged to a given tolerance. It should be noted, that if many integrals are to be computed, should be kept constant for as many cases as possible, as changing requires recalculation of the matrices and defined in (25), (26). In our implementation, we start by computing all inand . All integrals for which the diftegrals with ference between these two calculations is greater than the toler. We continue multiplying ance are recalculated with by 2 until all integrals have converged to the specified tolerance. It may seem natural to use a Gauss-Kronrod-type extension rule in order to reuse the integrand samples when increasing . Unfortunately, however, as proved in [37], such extensions of the Gauss-Laguerre and Gauss-Hermite rules do not exist. The results of applying the adaptive scheme to the same problem of Fig. 11(a), with a tolerance of , are shown in Fig. 15. In

Fig. 14. Histogram of relative errors obtained by testing the algorithm on 100000 integrals, with the problem parameters chosen randomly. The range of parameters is given in the table.

Fig. 15(a), the relative error is shown, on a logarithmic scale, and as can be readily observed, the largest error is indeed below at which the althe tolerance. In Fig. 15(b), the value of gorithm stopped for each observation point is shown. In a vast majority of observation points, the integrals converged in the or .A first or second steps, i.e., with few points, however, required . It should be noted that, for each point, the integrals must be calculated with all

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Fig. 16. Scattering width of a half-buried dielectric cylinder of radius r = 0:5 = k u . and relative permittivity " = 5. As the incidence is normal, k The scattering width is normalized to the free-space wavelength .

0

Fig. 15. In (a), the relative error, on a logarithmic scale, obtained with by adaptively modifying . In (b), the value of log at which the adaptive algorithm stopped. All problem parameters are the same as those of Fig. 11(a).

m

m

values preceding the value shown. Therefore, the number of integrand evaluations for each point is given by

ventional MoM techniques is that the current sources are elementary sources and they are placed at a small distance from the media boundaries and not right on them. In this way, the integration of the Green’s function, required when the basis functions are continuous current sources, is obviated. The SMT has been used to solve a wide range of electromagnetic problems [39]–[44] and is a viable alternative to more conventional MoM techniques. It has been used previously, in [10], to analyze scattering from cylinders in the presence of a dielectric half-space. Here, we followed the formulation given in [10], but replaced the SI evaluation routine with the present one, and reproduced all the numerical results given there with high-fidelity. One of these results is shown in Fig. 16. The problem analyzed is that of scattering by a circular dielectric cylinder, which is half-buried in a dielectric half-space. The cylinder is illuminated by a TM plane wave, the electric field of which is given by

(51) (52) This means that a plot of the total number of integrand evaluations for each point would look approximately the same as the . The plot of Fig. 15(b), shifted upwards by one unit of average computation time per point in this example is 9.5 msec, which is far less than it would take to compute all points with . the maximum , i.e., B. Solution of a Simple Scattering Problem The research into efficient evaluation of SIs is chiefly motivated by integral-equation formulations of scattering problems. Below, we show an example of the application of the SI evaluation techniques presented in this paper to the solution of a simple scattering problem. This problem has been solved in the past, and by comparing our results with the previous data we are able to partly validate our code. The solution of the scattering problem is effected by use of the source-model technique (SMT) [38], which belongs to the method of moments (MoM) class of techniques. What sets the SMT apart from more con-

, and are unit vectors, and is the wave where , its relative pervector. The radius of the cylinder is , and its axis coincides with the axis. The mittivity is region above the cylinder is air and the region below . the cylinder is characterized by a relative permittivity The polar plot in Fig. 16 shows the (angle-dependent) scattering , defined by width,

(53) where denotes the scattered electric field, and denotes the azimuthal angle. In [10], the polar plot was validated by comparing it with the results of [45], in which this problem was also studied, and good agreement was observed. To solve the scattering problem, we used 80 basis functions: 40 electric line sources for the fields outside of the cylinder, and

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TABLE IV COMPARISON OF COMPUTATION TIMES ON A 3 GHz PC

40 electric line sources for the fields inside of it. For the fields inside of the cylinder, the 40 line sources are assumed to radiate in a homogeneous medium of relative permittivity , and therefore these fields can be evaluated analytically. On the other hand, for the fields outside of the cylinder, the sources are assumed to radiate in the presence of the half-space, and therefore they are evaluated using the ND-SDP method. The continuity conditions were enforced on 80 testing points distributed uniformly on the boundary of the cylinder, leading to a 160 80 impedance matrix . The resulting matrix equation was solved in a least-squares sense and the 2-norm of the error in the continuity conditions, normalized to the incident field, was 0.004. After the amplitudes of the sources were calculated, the by computing SI evaluation routine was used to obtain the scattered electric field at 1000 points uniformly distributed . The adaptive scheme described in on a circle of radius Section VIII-A was used with a tolerance of 0.001. We also redid this computation with the older SI evaluation technique of [10], except that instead of using a trapezoidal rule with a fixed number of points as used there, we applied adaptive Gauss-Kronrod numerical integration with the same 0.001 tolerance. The computation times for both methods are shown in Table IV. A significant speed-up with respect to the method of [10] is evident, especially in the far-field calculation. Although it may seem natural use asymptotic methods for the far-fields, if the entire range of is of interest these methods usually require uniform asymptotic expansions and can become fairly complicated. Two additional examples of the solution of scattering problems are shown Fig. 17. The electric field of the previous example, this time near the cylinder, is shown in Fig. 17(a). The continuity of the field across the media boundaries, which graphically appears perfect, attests to the validity of the solution. Lastly, as shown in Fig. 17(b), scattering from a groove in a dielectric half-space may be analyzed by modeling the groove as a half-buried cylinder with a permittivity equal to that of the region. Like in the previous example, the continuity of the field appears to be perfect. In fact, a normalized error of 0.0055 in the continuity conditions resulted by using 60 basis image functions. The total computation time for the shown in Fig. 17(b) was about 8.5 minutes. IX. SUMMARY A numerical methodology for efficient evaluation of halfspace SIs is described in this paper. The approach is based on a simple and efficient method for numerical determination of the SDPs, and is termed the ND-SDP method. The idea is to determine the path numerically by stepping in the direction of

Fig. 17. The real part of the z -directed electric field near: (a) a half-buried dielectric cylinder, and (b) a semi-circular groove. In (a), all parameters are as in Fig. 16, whereas in (b), all parameters are also as in Fig. 16, except the incident k : u : u , and the relative permittivity wave vector, which is k . of the cylinder, which is "

= (0 8 00 6 ) =1

steepest descent, starting at the saddle point. A fast method for accurate evaluation of the derivative of the path, necessary for the numerical integration, is also described. The time required for computation of the path and its derivative scales linearly with the number of integration points and is comparable to the time an analytical evaluation of these quantities requires, when such analytical expression exists. On the SDP, Gaussian quadrature can be used to obtain accurate results with just a few samples of the integrand. When a branch point is intercepted, a second path is added giving rise to a lateral wave contribution (in the reflected field case), or an evanescent wave contribution due to a second, complex, saddle-point (in the transmitted field case). The numerical determination scheme is quite general and could be used to determine the SDPs of other spectral integrals. Other sources, such as dipoles and beams, and more general layered media configurations could be considered. In these cases, the path determination scheme would not have to be altered significantly, but the existence of other singularities would have to be taken into consideration. A number of special cases, for which straightforward integration of SIs along the SDP yields poor results, were also considered. The special cases were: the critical angle case, the grazing angle case, and the quasi-static case. Alternative paths and specialized Gaussian quadrature rules were proposed for

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these cases. The trade-off between accuracy and computational resources was investigated by testing the integration algorithms on a wide range of problem parameters. Also, the code was validated by comparison with published data and was shown to outperform the SI evaluation method of [10] by a significant margin. Lastly, some representative scattering problems, involving dielectric cylinders near a dielectric-half-space, were solved by use of the ND-SDP code. ACKNOWLEDGMENT Part of this work was performed while Y. Leviatan was a Visiting Professor at the University of Illinois at Urbana-Champaign. He would like to thank Provost L. Katehi and Prof. A. Cangellaris for their kind hospitality and for the valuable discussions he had with them. REFERENCES [1] A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys., vol. 333, no. 4, pp. 665–736, 1909. [2] J. Zenneck, “Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Ann. Phys., vol. 328, no. 10, pp. 846–866, 1907. [3] L. Brekhovskikh, Waves in Layered Media. New York: Academic Press, 1960. [4] L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, 1973. [5] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: Wiley, 1995. [6] Y. Rahmat-Samii, R. Mittra, and P. Parhami, “Evaluation of Sommerfeld integrals for lossy half-space problems,” Electromagnetics, vol. 1, no. 1, pp. 1–28, 1981. [7] K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 508–519, 1997. [8] V. I. Okhmatovski and A. C. Cangellaris, “A new technique for the derivation of closed-form electromagnetic Green’s functions for unbounded planar layered media,” IEEE Trans. Antennas Propag., vol. 50, no. 7, pp. 1005–1016, Jul. 2002. [9] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part II: Implementation and results for contiguous half-spaces,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 345–352, 1990. [10] Y. Leviatan and Y. Meyouhas, “Analysis of electromagnetic scattering from buried cylinders using a multifilament current model,” Rad. Sci., vol. 25, pp. 1231–1244, 1990. [11] M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E, vol. 62, no. 4, pp. 5797–5807, 2000. [12] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed-form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 588–592, 1991. [13] B. Wu and L. Tsang, “Fast computation of layered medium Green’s functions of multilayers and lossy media using fast all-modes method and numerical modified steepest descent path method,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 6, pp. 1446–1454, 2008. [14] F. J. Demuynck, G. A. E. Vandenbosch, and A. R. Van de Capelle, “The expansion wave concept. I. Efficient calculation of spatial Green’s functions in a stratified dielectric medium,” IEEE Trans. Antennas Propag., vol. 46, no. 3, pp. 397–406, 1998. [15] D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” Proc. Inst. Elect. Eng., vol. 135, no. 5, pt. H, pp. 297–303, 1988. [16] J. Bernal, F. Mesa, and F. Medina, “2-D analysis of leakage in printedcircuit lines using discrete complex-images technique,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 8, pp. 1895–1900, 2002. [17] E. A. Soliman and G. A. E. Vandenbosch, “Green’s functions of filament sources embedded in stratified dielectric media,” Progr. Electromagn. Res., vol. 62, pp. 21–40, 2006. [18] I. Lindell and E. Alanen, “Exact image theory for the Sommerfeld half-space problem—Part I: Vertical magnetic dipole,” IEEE Trans. Antennas Propag., vol. 32, no. 2, pp. 126–133, 1984.

[19] V. I. Okhmatovski and A. C. Cangellaris, “Evaluation of layered media Green’s functions via rational function fitting,” IEEE Microwave Compon. Lett., vol. 14, no. 1, pp. 22–24, 2004. [20] K. A. Michalski, “On the efficient evaluation of integrals arising in the Sommerfeld halfspace problem,” Proc. Inst. Elect. Eng., vol. 132, pt. H, pp. 312–318, 1985. [21] M. Yuan and T. K. Sarkar, “Computation of the Sommerfeld integral tails using the matrix pencil method,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1358–1362, 2006. [22] B. Hu and W. C. Chew, “Fast inhomogeneous plane wave algorithm for scattering from objects above the multilayered medium,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 5, pp. 1028–1038, 2001. [23] T. J. Cui and W. C. Chew, “Efficient evaluation of sommerfeld integrals for TM wave scattering by buried objects,” J. Electromag. Waves Appl., vol. 12, pp. 607–657, 1998. [24] T. J. Cui and W. C. Chew, “Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 2, pp. 887–900, 1999. [25] K. A. Michalski, “Evaluation of Sommerfeld integrals arising in the ground stake antenna problem,” Proc. Inst. Elect. Eng., vol. 134, pt. H, pp. 93–97, 1987. [26] T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Remote Sens., vol. 41, no. 9, pp. 2026–2036, 2003. [27] G. Szegö, Orthogonal Polynomials, 3rd ed. Washington, DC: American Mathematical Society, 1967. [28] W. Gautschi, Orthogonal Polynomials: Computation and Approximation. Oxford, U.K.: Oxford University press, 2004. [29] [Online]. Available: http://www.webee.technion.ac.il/people/leviatan/ ndsdp/index.htm [30] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [31] H. L. Bertoni, L. B. Felsen, and J. W. Ra, “Evanescent fields produced by totally reflected beams,” IEEE Trans. Antennas Propag., vol. 21, no. 5, pp. 730–732, 1973. [32] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration. New York: Wiley, 1975. [33] J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math., vol. 24, no. 3, pp. 396–413, 1973. [34] A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering. Englewood Cliffs, NJ: Prentice-Hall, 1991. [35] L. M. Brekhovskikh and O. A. Godin, Acoustics of Layered Media II: Point Sources and Bounded Beams. Germany: Springer-Verlag, 1990. [36] W. Gautschi, “Orthogonal polynomials (in Matlab),” J. Comput. Appl. Math., vol. 178, no. 1–2, pp. 215–234, 2005. [37] D. K. Kahaner and G. Monegato, “Nonexistence of extended GaussLaguerre and Gauss-Hermite quadrature rules with positive weights,” ZAMP, vol. 29, no. 6, pp. 983–986, 1978. [38] Y. Leviatan and A. Boag, “Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model,” IEEE Trans. Antennas Propag., vol. 35, no. 10, pp. 1119–1127, 1987. [39] Y. Leviatan, A. Boag, and A. Boag, “Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies-theory and numerical solution,” IEEE Trans. Antennas Propag., vol. 36, no. 12, pp. 1722–1734, Dec. 1988. [40] Y. Leviatan and A. Boag, “Analysis of TE scattering from dielectric cylinders using a multifilament magnetic current model,” IEEE Trans. Antennas Propag., vol. 36, no. 7, pp. 1026–1031, 1988. [41] D. I. Kaklamani and H. T. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag., vol. 44, no. 3, pp. 48–64, Jun. 2002. [42] A. Ludwig and Y. Leviatan, “Analysis of bandgap characteristics of two-dimensional periodic structures by using the source-model technique,” J. Opt. Soc. Am. A, vol. 20, no. 8, pp. 1553–1562, Aug. 2003. [43] A. Hochman and Y. Leviatan, “Analysis of strictly bound modes in photonic crystal fibers by use of a source-model technique,” J. Opt. Soc. Am. A, vol. 21, no. 6, pp. 1073–1081, June 2004. [44] A. Hochman and Y. Leviatan, “Efficient and spurious-free integralequation-based optical waveguide mode solver,” Opt. Express, vol. 15, pp. 14 431–14 453, 2007. [45] E. Marx, “Scattering by an arbitrary cylinder at a plane interface: Broadside incidence,” IEEE Trans. Antennas Propag., vol. 37, no. 5, pp. 619–628, 1989.

HOCHMAN AND LEVIATAN: A NUMERICAL METHODOLOGY FOR EFFICIENT EVALUATION OF 2D SIs

Amit Hochman (S’08–M’09) received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering in 1996, 2005, and 2009, respectively, all from the Technion-Israel Institute of Technology, Haifa. He spent a large part of his army service (1997–2002) at RAFAEL, working on the simulation of adaptive antennas and communication systems. He is currently a Postdoctoral Fellow at the Massachusetts Institute of Technology, Cambridge, where he is a member of the Computational Prototyping Group at the Research Laboratory of Electronics. His research interests center around the development of efficient computational modeling schemes, primarily for optical devices with wavelength-scale features.

Yehuda Leviatan (S’81–M’82–SM’88–F’98) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion-Israel Institute of Technology, Haifa, in 1977 and 1979, respectively, and the Ph.D. degree in electrical engineering from Syracuse University, Syracuse, NY, in 1982. He spent the 1982/83 academic year as an Assistant Professor at Syracuse University and subsequently joined the Department of Electrical Engineering at the Technion, where at present he is a Professor and the incumbent of the Joseph and Sadie

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Riesman Chair in Electrical Engineering. During his tenure at the Technion he held short-term visiting positions at Cornell University, the Swiss Federal Institute of Technology (ETH) in Zurich, the Catholic University of America, the University d’Aix-Marseilles III, the University of Washington, Bell Laboratories, the University of Michigan, Purdue University, and the University of Illinois Urbana-Champaign. During 1989–1991, while on sabbatical leave from the Technion, he was with the Department of Electrical Engineering and Computer Science at The George Washington University as a Distinguished Visiting Professor. Professor Leviatan’s research interests include computational methods applied to antennas, electromagnetic wave scattering, and microwave as well as optical guiding structures. He has published more than 100 journal papers and is listed among ISI’s Highly Cited Researchers. He also presented numerous papers at international symposia He is a recipient of the 2001 Henry Taub Prize for Excellence in Research. He is a member of Commission B of the International Union of Radio Science.

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Surface Current Source Reconstruction for Given Radiated Electromagnetic Fields Mehrbod Mohajer, Student Member, IEEE, Safieddin Safavi-Naeini, Member, IEEE, and Sujeet K. Chaudhuri, Senior Member, IEEE

Abstract—A general analytic approach is presented for reconstructing: 1) the minimum energy source enclosed by a sphere, and 2) the surface current distribution on a sphere from the knowledge of the radiated fields. The surface current source is derived by adding proper non-radiating sources to the minimum energy source. In contrast to the minimum energy volumetric distribution, the surface current derived in this paper is practically realizable. Finally, we present a closed form formula for the reconstructed spherical surface current source. We will show that this spherical surface current is indeed the unique solution of the inverse source problem for square-integrable surface electric current on a sphere in a homogenous medium. Index Terms—Current density, electromagnetic fields, inverse problems, wave functions.

I. INTRODUCTION

I

NVERSE source problems address the source reconstruction from the knowledge of the field outside the source region [1], [2]. It is often desirable to find a proper current source which produces a prescribed radiation field. In contrast to the forward problem of determining the radiation field of a given current source, there is no unique solution (current source) for the inverse problem [3], [4], because non-radiating currents can be added to the source without affecting the radiated fields [2]. However, purely radiating source so called minimum energy source is uniquely determined inside the given current source support region for a given radiated electromagnetic field. The closed form expression for the minimum energy volume current source within a sphere with a specific radius has been presented in [1]. It has been shown that any square-integrable source can be decomposed into the minimum energy source and non-radiating source which together will generate the same radiation fields as the minimum energy source produces [2]. Hence, the non-radiating part can be determined in such a way that a desired constraint such as a prescribed reactive power condition [5] can be imposed. As compared to volume current, the surface current is more feasible to implement. Although the equivalent surface sources

Manuscript received December 01, 2008; revised May 04, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and Research In Motion (RIM). The authors are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2009.2037696

and are one possible theoretical solution to inverse source problems, it is difficult in practice to implement both electric and magnetic current sources in the same place. To implement only one type of these equivalent currents, the enclosed volume would require to be filled by either PEC or PMC. Even with this requirement, it is not practical to implement either of these mathematical source functions. Hence, the main purpose here is to obtain pure electric current surface sources in homogeneous medium (without any PEC or PMC) that support the prescribed electromagnetic fields. In this paper, we add non-radiating sources and consider a new constraint of confinement of the source current to a surface rather than a volume. To achieve this, we propose a vector wave function-based approach to derive the required surface current source on a spherical shell for the predefined electromagnetic fields. We also determine the required non-radiating source which should be added to the minimum energy volume source so that the sought for source distribution becomes of surface type. We also show that the derived surface current source is the unique solution on a spherical shell for homogeneous media. The paper is organized as follows. In Section II, the vector wave functions in a general curvilinear coordinate system are reviewed. Then the forward problem of electromagnetic radiation is introduced in Section III. In Section IV, we solve the inverse source problem using the vector wave functions, and in Section V, a closed form formula for surface current source on a spherical shell is derived using the proposed vector wave function approach. Issue of uniqueness is discussed in Section VI and concluding remarks are presented in Section VII. II. VECTOR WAVE FUNCTIONS IN CURVILINEAR COORDINATES , and with the Let us consider curvilinear coordinates and the scale factors , respecunit vectors and vector potentials, tively [6]. The field vectors in a source free homogeneous and isotropic medium obey the vector Helmholtz equation [7]

(1) It has been shown that following independent vector wave functions (VWFs) satisfy (1) [6], [7]:

0018-926X/$26.00 © 2009 IEEE

(2)

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where is any arbitrary constant unit vector, and the scalar function is a solution of scalar Helmholtz equation

dinate system. The solution of homogeneous scalar Helmholtz equation in spherical coordinate system is as follows [7]:

(3)

(6)

The important properties of these three vectors are [7]

(4) which means that vector is curl-less, and and vector functions are divergence-less. It is advantageous to decompose the vector solution of equation (1) into longitudinal and transverse parts. Notice that the longitudinal and transverse vector functions are defined as zero-curl and zero-divergence vectors, respectively [6]. Hence, the electromagnetic fields can be represented as a linear combination of (longitudinal part), , and (transverse parts) which are the vector Eigen-functions of (1). The representation (2) is based on the constant unit vector . This representation can be extended to the cases where the unit vector , which is perpendicular to a constant coordinate surface in a curvilinear coordinate system, is not fixed. If the unit vector then and should be redefined as follows [6]: is (5) is the unit vector normal to curved surface , where and is a scalar function to be determined so that and functions satisfy vector Helmholtz equation. The choice of leads to be longitudinal, and and vectors to be transverse and satisfy (1) with respect to . It is shown in [6] that if and only if: ; 1is independent of ; 23- is either 1 or ; 4. Only six separable coordinate systems, Cartesian, three cylindrical ones, spherical and conical, out of eleven well-known separable coordinate systems, satisfy all these conditions. For other coordinate systems, for instance, spheroidal coordinate system, either a constant unit vector or position vector is used to construct VWFs. However, the constructed VWFs are not orthogonal [6]. III. VECTOR GREEN’S FUNCTION AND FORWARD PROBLEM Since the free space can be thought of as a spherical waveguide, the radiated fields can be represented in spherical coor-

where and stand for even and odd modes, is the associated Legendre Polynomial, and is an appro, or priate spherical function namely for , and 4, respectively. Based on the representation (5), the following spherical vector wave functions (SVWFs) establish the orthogonal basis functions for field expansion:

(7) The free space dyadic Green’s function is expanded in terms of SVWFs [8], shown in (8) at the bottom of the page. Using the dyadic Green’s function, the electric field radiated by an electric current source enclosed by a sphere of radius can be represented in terms of SVWFs as [6]–[8]

(9) where

(10) Equations (9) and (10) are well-known solutions to the forward problem, in which the radiated fields are determined for a , the far field can be approxgiven current source. As imated by replacing by in (9).

(8)

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Hence, the coefficients calculated by (10) are identical for far fields as well. By this way, the far field can be back-propagated to near field region, and similar equation can be written for coefficients obtained from far field spherical harmonics.

ficients in (11). Replacing expansion (11) in (10), we will have

IV. THE INVERSE SOURCE PROBLEM Unlike the forward problem, the inverse source problem is defined as constructing the current source localized within a limited space to generate a predefined radiated field. Assuming the current source is square-integrable, we propose to expand the current source in terms of the prescribed vector functions and then determine the coefficients of each vector function such that it generates the desired radiation fields. This expansion is also consistent with the Spectral Theorem. According to Spectral Theorem, if in which is a self-adjoint operator, can be expanded in terms of vector Eigen functions of operator . This expansion is also called Eigen-decomposition. Since the aforementioned , and are the vector Eigen-functions of vector Helmholtz equation, it is possible to simply expand in terms of VWFs in an appropriate coordinate system. Therefore, when it is desired to find the current source inside the arbitrary , the current source should be expanded in terms surface of VWFs based on in (5). However, among the six separable coordinate systems satisfying the required conditions for representation (5), only spherical coordinate systems have finite constant coordinate surfaces. Therefore, the spherical vector wave functions are used to expand the current source within a limited space. The source coefficient calculations are described below. Equations (9) and (10) show that the projections of the electric and are proportional to the projections of field on

(12) , and are given in [7]. where the integrals Notice that if VWFs are not orthogonal, the right-hand side of the relationship (12) will have to include infinitely many terms [9], [10]. As expected, the current source supporting the given electrical field is not unique. This is obvious from (12), and because infinite numbers of combinations of coefficients can give the same

coefficient, hence

generates the same radiation field. Note that the electric current source on

and

, respectively.

This observation suggests that the source can be expanded in , and vector functions, which terms of are finite at the origin. Therefore,

cients can be uniquely calculated from

coefficoefficients

. The non-uniqueness of the current source can also be observed from the Green’s function representation in (8), wherein the non-radiating sources can be added to radiating sources without affecting the radiated field [non-radiating sources are orthogonal to both and

is represented as

, and produce no field coefficient in (10)]. Therefore,

(11) Now, assuming that

and

are the desired electric

field expansion coefficients in (9) representing an electric field generated by an electric current source within a sphere of radius , it is desired to determine and coef-

any combination of the non-radiating sources with the radiating ones can generate the same radiation field. The non-radiating sources have been discussed in [2] for the inverse source problems. Chu theorem [7], [11] provides another approach to the source non-uniqueness problem when it is desired to construct the source from the far field knowledge. According to this theorem, for a given size and bandwidth of the source (antenna), instead of infinitely many spherical wave functions in (9), the electrical field contains only a finite number of SVWFs in the far zone. So if a source, which produces

MOHAJER et al.: SURFACE CURRENT SOURCE RECONSTRUCTION FOR GIVEN RADIATED ELECTROMAGNETIC FIELDS

a SVWF with an order higher than , is added to the radiating current source, the high order electromagnetic fields will be stored reactively around the source region and do not have any effect on the far field. Hence, in practice, we have a finite number of coefficients in (12) for far field radiation. Among the infinitely many possible solutions for , there is a particular minimum energy source solution inside the given volume. To find this solution, let us assume that the source is norm of current source is then calsquare-integrable. The culated as

(13) As mentioned before,

coefficients will determine

s, but we have the liberty to choose coefficients in such a way that the

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is a general approach and can deal with different types of constraints, and 2) both longitudinal and transverse parts are included which makes the source solution more physically meaningful. is a volume current distribution containing only In (15), and vector functions which are divergence-less in the interior of the sphere excluding the boundary surface. We know a current source is not divergence-less unless either charge denor . Hence, the current distribution in sity is zero (15) is not practically implementable. Thus, as mentioned in [2], without any non-rait is not possible to physically realize diating sources. The (11) suggests that by adding vector functions to current source it is possible to construct a non-zero divergence , which obviously is no longer a minimum energy source. In addition to having a non-zero divergence, the source expression now contains more coefficients which can be used to apply other desired constraints to the current source. Hence, the source expansion in terms of vector Eigen functions provides a general representation to which any arbitrary constraints can be applied. If the minimum energy constraint was applied, the previously derived results would be obtained. Alternatively, the new approach admits other desired constraints.

and

V. THE INVERSE SURFACE SOURCE PROBLEM

norm is minimized

(14) Solving the above simple minimization problem, we will find out that coefficients should be zero and in order to obtain the minimum

Since in the most practical cases, it is preferable to have surface current density rather than volume current source, in this Section, we find a surface current distribution, which can generate a predefined radiated field. For this purpose, vectors are added in such a way that the normal component of current becomes zero. Notice that it is source at the surface desirable to add a minimum number of vector functions to keep the source energy as small as possible, because adding any non-radiating source only changes the total energy of the current source and will not affect the radiated field. , only In a separable curvilinear coordinate system and vector functions have non-zero components along the direction

energy source. Consequently, the minimum energy current source will be represented as follows

(16)

(15) The same result has been derived using both a mathematical theory for so-called ill-posed inverse source problems [1], [2] and the Lagrangian optimization [5] to obtain the minimum energy volume current distribution confined in a sphere. The main advantages of the new proposed derivation are: 1) the proposed method

Since have [6]

is the scalar Helmholtz equation solution, we should

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or

(17) Knowing is a unit constant and , as it was required for representation (5), and substituting (17) into (16), we obtain the components of and vector functions on surface as follows

(20) where has been defined in (8), and the integrals have been precoefficients are uniquely detersented in [7]. Again the mined from and

(18) Therefore, components of both and vector functions over the surface . Thus, are in the form of it is possible to add vectors to the minimum energy volume current source in such a way that the total component of current source be canceled out. In the spherical coordinate system which has orthogonal is expanded in terms of VWFs, the surface current source SVWFs on the surface of spherical shell with radius

, whereas there are infinite choices for

coefficients. To have

coefficients in radiated

fields, those radii for which should be avoided to . Furthermore, should be non-zero ensure non-zero to have the liberty of choosing coefficients. Therefore, those radii for which avoided. As before,

should also be gives the minimum energy source, but

in general, the minimum energy source is not a surface current distribution on the spherical shell. Therefore the minimum energy condition should be relaxed. Using (18) for spherical coordinate system and (20), we have following two equations:

(21) (19)

Substituting (19) in (10), we will obtain the following equation:

The first equation is a result of (20), and the second one is imposed to cancel the radial component of the current source. Notice that the radial differential equation in the spherical coordinate has been used to convert the component of in (16) to the simple form in (21). Solving (21), we get the following surface current distribution on a spherical shell with the radius of which generates the predefined electromagnetic field specified in (9)

MOHAJER et al.: SURFACE CURRENT SOURCE RECONSTRUCTION FOR GIVEN RADIATED ELECTROMAGNETIC FIELDS

As it can be seen, vector functions on a

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consists of spherical surface.

and Therefore, ,

and

basis functions form the -, -, and -components of non-radiating current source, respectively. VI. DISCUSSIONS ON THE UNIQUENESS

(22) is the ratio of the coefficient of nth order to the cowhere efficient of nth order vector function. If was zero for all s, the minimum energy spherical surface current source would be obtained. However, since the radii for which are avoided, would not be zero, and generally speaking, it is impossible to obtain minimum energy spherical surface current source for arbitrarily given radiated electromagnetic fields. Alcontains all three SVWFs, the amplitudes of vector though functions have been adjusted so that the total current distribution only has - and -components. Note that for certain radii, we have , which means the radial component of is zero and it is imvector function. possible to cancel the radial component of Hence, to obtain the surface current distribution on the sphere, those radii for which the radial component of are equal to zero must be avoided. As we emphasized before, a finite number of SVWFs appear and at far field. Thus, should be satisfied for only a finite . Hence, these conditions are number of modes not too restrictive. The surface current source presented in (22) is not minimum energy as it consists of a non-radiating part. The non-radiating current source, which has been added to the minimum energy source in order to achieve the surface current distribution, is

(23)

It is interesting to note that the coefficients of surface source in (22) have been determined uniquely. This shows that the inverse surface source problem in homogenous medium admits a unique pure electric current surface solution. To elaborate on the details of the uniqueness in this formulation, we will prove the uniqueness of the solution for the defined problem using proof by contradiction. Let us assume the electric current surface-source in homogeneous media accepts other solutions than what is presented in (22), and all of them generate the same radiated electromagnetic fields. Hence, one should be able to determine different non-radiating spherical surface sources giving non-unique solutions. The theory of non-radiating surface sources has been presented in [12] and [13] for scalar Helmholtz equation. Assuming interior and a boundary surface with the volumes and exterior of this surface, respectively, it has been proven that the singlet and doublet components of any secondary source on generated by a primary source contained in are non-radiating , and vice versa. This theory surface sources into the region can be generalized to the vectorial surface sources by using the generalized Green’s Function theory. By applying vector Green’s theorems to homogeneous medium Green’s Function radiated by the for electric field and the electric field source located in , it is straightforward to obtain

(24) where and are respectively equivalent to singlet and doublet surface sources in within and are non-ra[12] that together radiate field . Obviously, these sources are the electric and diating into magnetic secondary surface sources on due to the primary , and consequently are non-radiating into . source within Therefore, any non-radiating surface source relative to can be mathematically constructed by considering the secondary due to any arbitrary primary source within resources on [12], [13]. Since the primary source can be chosen argion bitrarily, infinitely many non-radiating surface sources can be constructed as shown above, which according to (24) generate zero field and therefore, generally speaking, there is no unique solution to inverse surface source problem. Now let us consider how the above argument applies to our problem, which is to find a purely electric surface source to support a given electromagnetic field. Coming back to our proof, we should be able to find non-radiating electric surface sources if

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our problem does not admit a unique solution. In order to obwhich are tain purely electric surface sources non-radiating into outside the sphere with radius of , the other , should be zero. Hence, component of surface source, only those primary sources are acceptable that create equal to zero on the sphere. If either or surface source equals to zero, the fields generated by non-radiating surface source inside the sphere will be resonant, otherwise the field inside the sphere will be zero because of zero or surface boundary condition. Since simulsource alone cannot be non-radiating into both and taneously [12], [13], it is not possible to have zero field inside the sphere. On the other hand, we have already avoided the resonant fields inside the sphere by avoiding those radii for which and . This proves the fact that there is no purely electric non-radiating surface source that can be added to (22) in order to create a new electric surface current which supports the given electromagnetic fields. Note that the non-radiating electric source in (23), used to create surface source in (22), is not a surface source by itself. Non-radiating electric source in (23) was used to convert the radiating source required for generating the prescribed electromagnetic fields into a surface source, and no non-radiating electric surface source needs to be added to obtained solution in (22). Therefore, there is a unique square-integrable spherical surface electric source in a homogenous medium which generates the given electromagnetic fields. Notice that if a combination of electric and magnetic surface sources is desired, infinite number of non-radiating surface sources exist to support the same electromagnetic fields. However, as mentioned before, implementation of both electric and magnetic surface sources in the same place has practical difficulties. Finally, the equivalent electric and magnetic sources can provide alternative solutions for inverse surface source problem. As compared to equivalent sources, the advantage of the proposed surface current is the fact that it can be implemented by exciting a conducting sphere. Since spherical surface current distribution contains surface resonant modes, it can be excited by placing point sources at appropriate locations. In contrast to the spherical surface source in (22) which is practically feasible, the equivalent sources cannot be excited as the resonant modes of spherical conductor. If the interior region of the equivalent source sphere is filled with PEC or PMC, either magnetic or electric equivalent current will radiate in the presence of PEC or PMC sphere, respectively. In both cases, implementing the equivalent sources in presence of PEC or PMC is practically difficult as compared to the proposed derived electric source realization. Hence, the surface electric source in (22), which is uniquely determined in a homogeneous medium, is more convenient for practical realization.

VII. CONCLUSION The minimum energy volume current distribution has been re-derived for a given electromagnetic radiated fields using

general vector wave function approach. By adding a non-radiating source to the minimum energy source, we have proposed a VWF approach to obtain the spherical surface current distribution which is a more practical source derived from the knowledge of radiated fields. In addition to surface source constraint, VWF representation is also physically more meaningful and can admit other constraints. For instance, based on Chu theorem, it is possible to determine the radiating and non-radiating parts of surface source, and consequently, avoid the non-radiating modes in the current source solution. By this way, norm non-radiating source will be the minimum possible added to minimum energy source. Hence, the proposed surface current source not only is practically implementable to generate the prescribed EM fields, but also provides a physical insight for designing the source with desired requirements. Finally, it has been proven that there is no purely electric non-radiating surface source on a mathematical sphere, and the inverse surface source problem admits a unique solution for purely electric spherical surface current distribution in a homogenous medium. The derived electric source can be practically implemented as surface resonant modes of a spherical conductor whereas the equivalent sources cannot. We also have presented the mathematical representation of added non-radiating source to the minimum energy source to have surface current distribution.

REFERENCES [1] E. A. Marengo and A. J. Devaney, “The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution,” IEEE Trans. Antennas Propag., vol. 47, no. 2, pp. 410–412, Feb. 1999. [2] E. A. Marengo and R. W. Ziolkowski, “Nonradiating and minimumenergy sources and their fields: Generalized source inversion theory and applications,” IEEE Trans. Antennas Propag., vol. 48, no. 10, pp. 1553–1562, Oct. 2000. [3] A. J. Devaney and G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag., vol. 30, no. 5, pp. 1034–1037, Sep. 1982. [4] N. Bleistein and J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys., vol. 18, no. 2, pp. 194–201, 1977. [5] E. A. Marengo, A. J. Devaney, and F. K. Gruber, “Inverse source problem with reactive power constraint,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1586–1595, June 2004. [6] P. M. Morse and H. Feshbach, Methods of Theoretical Physics.. New York: McGraw-Hill, 1953. [7] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [8] C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory. San Francisco, CA: Intext Educational Publishers, 1971. [9] J. C.-E. Sten and E. A. Marengo, “Inverse source problem in the spheroidal geometry: Vector formulation,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 961–969, Apr. 2008. [10] J. C.-E. Sten and E. A. Marengo, “Transformation formulas for spherical and spheroidal multipole fields,” Int. J. Electron. Commun. (AEÜ), vol. 61, pp. 262–269, 2007. [11] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, 1948. [12] A. J. Devaney, “Nonradiating surface sources,” J. Opt. Society Amer. A, vol. 21, no. 11, pp. 2216–2222, Nov. 2004. [13] E. A. Marengo, “Observations on “Nonradiating surface sources”: Comments,” J. Opt. Society Amer. A, vol. 23, no. 1, pp. 142–145, Jan. 2006.

MOHAJER et al.: SURFACE CURRENT SOURCE RECONSTRUCTION FOR GIVEN RADIATED ELECTROMAGNETIC FIELDS

Mehrbod Mohajer (S’09) was born in Tehran, Iran, in 1980. He received B.Sc. degree (with honors) from KNT University of Technology, Tehran and the M.Sc. degree (with honors) from Amirkabir University of Technology (PolyTechnic of Tehran), Tehran, in 2002 and 2005, respectively, both in electrical engineering. He is currently working toward the Ph.D. degree at the University of Waterloo, Waterloo, ON, Canada. From 2003 to 2006, he was with Iran Telecommunication Research Center (ITRC), Tehran, as a Researcher. His research interests include antenna modeling, design, and analysis for advanced intelligent wireless systems, electromagnetic theory, RF/microwave circuit design, and wireless communication systems.

Safieddin Safavi-Naeini (M’79) received the B.Sc. degree from the University of Tehran, Tehran, Iran, in 1974 and the M.Sc. and Ph.D. degrees from University of Illinois at Urbana-Champaign, in 1975 and 1979, respectively, all in electrical engineering. He joined the University of Waterloo in 1996, where he is now a Professor in the Department of Electrical and Computer Engineering and holds the RIM/NSERC Industrial Research Chair in Intelligent Radio/Antenna and Photonics. He is also the Director of a newly established Center for Intelligent Antenna and Radio System (CIARS). His research activities deal with RF/microwave technologies, smart integrated antennas and radio systems, mmW/THz integrated technologies, nano-EM and photonics, EM in health science and pharmaceutical engineering, antenna, wireless communications and sensor systems and networks, new EM materials, bio-electromagnetics, and computational methods. He has published more than 80 journal papers and 200 conference papers in international conferences. He has led several international collaborative research programs with research institutes in Germany, Finland, Japan, China, Sweden, and the USA.

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Sujeet K. Chaudhuri (SM’85) was born in Kolkata, India, on August 25, 1949. He received the B.E. degree (with honors) in electronics engineering from Birla Institute of Technology and Science (BITS), Pilani, India, in 1970, the M.Tech. degree in electrical communication engineering from the Indian Institute of Technology, Delhi, India, in 1972, and the M.A.Sc. degree in microwave engineering and the Ph.D. degree in electromagnetic theory from the University of Manitoba, Canada, in 1973 and 1977, respectively. In 1977, he joined the University of Waterloo where he is currently a Professor in the Electrical and Computer Engineering Department and was the Chair of the Electrical and Computer Engineering Department from 1993 to 1998, 2007 to 2008, and the Dean of the Engineering Faculty from 1998 to 2003. He has also held a Visiting Associate Professor’s position in the EECS Department, University of Illinois at Chicago, during 1981 and 1984, a visiting Professorship at the National University of Singapore from 1990 to 1991, and the Erskine Fellowship at the University of Canterbury, New Zealand, in 1998. In 2004, 2005, and 2009 he visited the Korea Advanced Institute of Science and Technology (KAIST) and POSTECH as a BK-21 International Fellow. He has been involved in Contract Research and Consulting Work with several Canadian and U.S. industries and government research organizations. In 2004, in recognition of his sustained outstanding scholarship and academic leadership, he was installed as the O’Donovan Research Chair of RF/Microwaves and Photonics at the University of Waterloo. Current research interests are in guided-wave/electro-optic structures, planar microwave structures, dielectric resonators, optical and EM imaging, fiber/RF based broadband network and the emerging technologies based on the EBG/PBG-nanostructures. Dr. Chaudhuri is a member of URSI Commission B, and Sigma Xi.

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Microwave Imaging in Layered Media: 3-D Image Reconstruction From Experimental Data Chun Yu, Senior Member, IEEE, Mengqing Yuan, Yangjun Zhang, John Stang, Rhett T. George, Gary A. Ybarra, Senior Member, IEEE, William T. Joines, Fellow, IEEE, and Qing Huo Liu, Fellow, IEEE

Abstract—A prototype microwave imaging system for imaging 3-D targets in layered media is developed to validate the capability of microwave imaging with experimental data and with 3-D nonlinear inverse scattering algorithms. In this experimental prototype, the transmitting and receiving antennas are placed in a rectangular tub containing a fluid. Two plastic slabs are placed in parallel in the fluid to form a five-layer medium. The microwave scattering data are acquired by mechanically scanning a single transmitting antenna and a single receiving antenna, thus avoiding the mutual coupling that occurs when an array is used. The collected 3-D experimental data in the fluid are processed by full 3-D nonlinear inverse scattering algorithms to unravel the complicated multiple scattering effects and produce 3-D digital images of the dielectric constant and conductivity of the imaging domain. The image reconstruction is focused on the position and dimensions of the unknown scatterers. Different dielectric and metallic objects have been imaged effectively at 1.64 GHz. Index Terms—Experimental data reconstruction, inverse scattering, layered medium, microwave imaging, nonlinear inverse scattering algorithm, object detection.

I. INTRODUCTION HERE continues to be a growing interest in developing inverse scattering methods and microwave imaging systems for object detection with electromagnetic waves. In particular, applications such as subsurface sensing, through-wall imaging, and medical diagnostics call for microwave imaging of objects in a layered medium environment. The main objective of this work is to develop an imaging system and test a 3-D nonlinear inverse scattering algorithm for targets in a layered medium. A number of methods have been proposed for inverse scattering problems (e.g., [1]–[20]), including Born-type iterative methods [1]–[5], and contrast-source inversion [10], [11]. Several effective methods have also been developed to solve inverse

T

Manuscript received August 30, 2008; revised September 08, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported by NIH through grant 5R01CA102768-02. C. Yu, M. Yuan, J. Stang, R. T. George, G. A. Ybarra, W. T. Joines, and Q. H. Liu are with the Department of Electrical and Computer Engineering, Duke University, Box 90291, Durham, NC 27708-0291 (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; gary@ee. duke.edu; [email protected]; [email protected]). Y. Zhang was with the Department of Electrical and Computer Engineering, Duke University, Box 90291, Durham, NC 27708-0291, on leave from the Department of Electronics and Informatics, Ryukoku University, Seta, Ohtsu 5202194, Japan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037770

scattering problems in a layered medium (e.g., [16], [21]). The layered-medium model is important in inversion because the target to be reconstructed is likely buried below one or more overburden layers, and for such scenarios a layered medium model is necessary to represent a more realistic background than a homogeneous background. Recently, a 3-D data set has been produced by Geffrin and Sabouroux, and six groups of researchers have tested this data set with 3-D objects in a homogeneous background [22]. However, to our knowledge, measured data sets for layered media have not been available to test inverse solvers. In our previous work on layered-medium problems, we have mainly focused on algorithm development and synthetic data inversion. In [23], a 2-D contrast source inversion (CSI)-based imaging technique for a layered medium is developed to image objects in a room with walls. In [24] and [25], a fast inverse scattering algorithm with the diagonal tensor approximation (DTA) is proposed to solve inverse scattering problems in a layered medium, which is suitable for low to moderate-contrast objects. To solve inverse scattering problems from high contrast objects, a stabilized biconjugate gradient FFT (BCGS-FFT) algorithm is developed for the fast solution of the volume integral equation [26]–[28], and the algorithm has also been applied to the inversion for 3-D objects buried in a layered medium [16], [29]. Other relevant algorithms and numerical results have been shown by other groups for either 2-D or 3-D layered media (or half space), for example, see [30]–[32]. Recently, based on developed forward and inverse algorithms, the image reconstruction from microwave experimental data for 2-D and 3-D objects has been conducted and good reconstructed results have been obtained [33]–[35]. However, our previous efforts for experimental data inversion are limited to homogeneous backgrounds. This paper presents a 3-D microwave imaging system prototype for imaging targets buried in a layered medium. The experimental prototype includes a transmitting antenna and a receiving antenna that are placed in a rectangular tub containing a fluid. Unlike the experimental setup in free space, the experimental setup with a lossy fluid reduces reflections from the surrounding environment. (For an example of utilizing the reflection to enhance the microwave tomography performance, the reader is referred to [21], [36].) Two plastic slabs are placed in parallel in the fluid to form a five-layer medium. The scattered field from the object and the medium are measured in the frequency domain at several discrete frequencies; reconstruction is shown below for a single frequency. The collected 3-D experimental data from the layered-medium background are processed using a 3-D nonlinear inverse scattering algorithm. 3-D digital

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Fig. 2. An example of the geometry of the layered medium: the transmitting : and the receiving dipole dipole antenna moves on the plane of z antenna scans on the plane of z : to form multiple transmitter-receiver locations.

= 04 6 cm = 4 6 cm

Fig. 1. Experimental setup of the layered-medium microwave imaging system.

images of the dielectric constant and conductivity in the layered-medium background have been obtained with the proposed system prototype and nonlinear inverse scattering algorithm. This paper is organized as follows: Section II presents the experimental setup and 3-D experimental data acquisition. Section III summarizes the inverse scattering algorithm. Several microwave imaging examples are presented in Section IV. Finally, conclusions are provided in Section V. II. LAYERED-MEDIUM MWI EXPERIMENTAL SETUP Fig. 1 shows an experimental setup of the layered-medium microwave imaging (MWI) system. It consists of two dipole antennas (one transmitter and one receiver), two automatic posi), tioners, a rectangular tub (dimensions and an HP 8753E Vector Network Analyzer connecting to the source signal (Port 1) and receiver (Port 2). The tub is filled with either water or another fluid. Two plastic slabs are placed in parallel in the fluid to form a five-layer medium. The automatic positioners move the transmitting and receiving antennas in the fluid through a series of locations with high precision to produce multiview scattered field data. The transmitting and receiving antennas are linearly polarized dipole antennas made of 3-mm diameter semi-rigid coaxial cables. For details of the geometry of the dipole antennas, the reader is referred to [35]. An example of the geometry of the layered-medium with ) transmitting antenna locations (on the plane of ) and receiving antenna locations (on the plane of is shown in Fig. 2, where the interfaces of the layered medium , , , are located at and , respectively. The parameters of the layered , medium are , , and , respectively. Nine transmitter locations are arranged in a square array separated by 3-cm between the adjacent locations. Multiple receiving locations are obtained by scanning the receiving antenna on five lines (increments of 1 cm in the direction) with 11-points along each line (increments of 0.8 cm in the direction), resulting in a total number of receiver points of

. The object to be reconstructed is located in an imaging domain in layer 3. For inversion, the imaging domain will be discretized into many small cells (voxels) to reconstruct the unknown permittivity and conductivity distributions. Microwave signals at the receiving antenna are measured in through the HP vector netthe form of scattering parameter transmitter locations and work analyzer. If there are receiver locations, the number of complex data points for (corresponding to the th receiver signal due to the th trans. From this set of data, the mitter excitation) is in the imaging domain complex permittivity distribution can be found using an image reconstruction algorithm. III. IMAGE RECONSTRUCTION ALGORITHM Assume that measurements are taken on some surface in layer and the inhomogeneous object is within the domain in layer . The electrical properties of the layers , , and the object are characterized by the complex permittivity , , and , respectively, where the complex permittivity for the inhomo, is the relgeneous object is expressed as ative dielectric constant of the object, the permittivity of free space, and the conductivity of the object. The integral equations governing the layered-medium scattering problem [37] can be written as

(1) for the measured scattered field in layer

, and

(2) for the unknown electric field inside the object, where the is the contrast function defined as (3)

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and is the transmitter location, is an electric dyadic Green’s function at the observation point in layer due to a unit electric current source at the point in layer , and is an auxiliary potential dyadic Green’s function in which both and are in layer . The wavenumber in . layer is given by The integral (2) can be solved using forward solvers. The fullwave stabilized biconjugate-gradient FFT (BCGS-FFT) method and the approximation method such as the diagonal tensor approximation (DTA) have been implemented as efficient forward solvers [25]–[27]. In this paper, a hybrid technique [28] is used for the solver that combines the diagonal tensor approximation (DTA) and the stabilized biconjugate-gradient FFT (BCGSFFT) method where the DTA acts as a preconditioner for the BCGS-FFT method. Such a hybrid method has the advantage that, for low to moderate contrasts, the DTA alone will give accurate results without going through the BCGS-FFT iterations; for high contrasts, the DTA gives a good preconditioner so that the BCGS-FFT iterations will converge rapidly. Thus, both lowand high-contrast problems can be solved efficiently and accurately by this hybrid method. In solving the inverse scattering problem, the total number of . Suppose the imaging data points collected is domain is discretized into small voxels. By using the trapezoidal rule, the data equation with the unknown contrast function can be discretized as follows

(4) where is an -dimensional data column vector whose elements are the measured scattered electric field collected by the is the volume of each voxel, and receiver, denote the indices for the receiver and transmitter, respectively. For measurements and discretized voxels, (4) can be written compactly as (5) represents the simulated scattering fields, is an where -dimensional column vector of the contrast function , and is an matrix whose elements are given by (6) , and . where in the disSince the total field within the objects (and torted Born iterative method) is an unknown function of the material contrast function , (5) is a nonlinear equation in . Moreover, the limited amount of available information from measurement makes the problem ill-posed. The Born iterative method [2] or the distorted Born iterative method [1], [3]–[5] can be used to solve the above nonlinear inverse scattering equation. In this work, we utilize a two-step inversion algorithm [29]. In the first step of the inversion, we use an algorithm combining the diagonal tensor approximation and the Born iterative method to

obtain an initial reconstruction for a fast estimate of the image. In the second step of the inversion we use an algorithm combining the stabilized biconjugate-gradient FFT and the distorted Born iterative method to provide more accurate and fast inversion. With Tikhonov regularization technique [37], the normalized can be defined as cost functionals at iteration

(7) for the Born iterative method and

(8) for the distorted Born iterative method, where is the regularization parameter; subscripts and indicate the evaluation of the norms on the measurement surface and imaging domain , respectively; is the matrix evaluated in the th iteration. In the distorted Born iterative method, denotes the error between the measured scattered field and the predicted scattered field using the th iteration background Green’s function and is the correction of during the th iteration. can be obtained by the minimization The solution for or of above cost functionals. In image reconstructions, should be chosen to be much smaller than 1 to keep the first term of the cost functionals as the dominant term. In this work, has been chosen to be 0.05. Our experiments show that the reconstructed results are not sensitive when different values are used between 0.01 and 0.05. To determine whether the inversion has converged, we define the relative data residual error RES as

(9)

and are the th measured and simulated scatwhere tered field, respectively. In the first-step of inversion, the stopping criteria are related to the relative data residual error and a maximum iteration number: When the data error is less than a small number (e.g. 0.1%), or when the iteration exceeds the maximum iteration number (e.g. 10), the first step of the inversion is terminated and the inversion is switched into the second step of the inversion using the stabilized biconjugate-gradient FFT and the distorted Born iterative method. Finally the iteration in the second-step of the inversion stops with a given number of iterations (e.g. 20). More discussions on regularization can be found in [21], [37], [38]. IV. MICROWAVE IMAGING EXAMPLES A. Imaging From Synthetic Data and Layered-Medium Effects To illustrate the importance of the correct background model, we present an imaging example of a dielectric sphere from synthetic data assuming an inexact background model, i.e. a

YU et al.: MICROWAVE IMAGING IN LAYERED MEDIA: 3-D IMAGE RECONSTRUCTION FROM EXPERIMENTAL DATA

homogeneous background (water) to approximate a five-layer medium background depicted in Fig. 2. In this imaging example, a 1-cm diameter lossless sphere with dielectric constant 20 is placed at the center of the imaging domain (also the origin). To produce multiple source excitations, the point dipole transmitter is used as a source in our inversion algorithm. Nine transmitter locations are on the plane with , , where , ; the receiver having the same point dipole model is , placed at 55 locations on the opposite side with , and , where , , and , . Only the field component is used for image reconstruction, so the number of measured data . The imaging domain of dimension of is is divided into , resulting in a total number of complex unknowns to be reconstructed. The operating frequency is 1.64 GHz, which is the resonance frequency of the dipole antenna. Fig. 3(a) and (c) show the reconstructed permittivity in 3-D and 2-D plots with the stabilized biconjugate gradient-distorted Born iterative method using an inexact homogeneous background model for inversion. Fig. 3(b) and (d) show the reconstructed permittivity in 3-D and 2-D plots using the exact layered-medium model in inversion. The effects of the inexact layered-medium model (i.e., with small errors in permittivity and geometrical parameters in the layer medium) have also been studied. Our observations has shown that the small error will cause changes of the reconstructed image, even though the object can be seen. We observe from above figures that the reconstruction with the homogeneous background gives very poor results as shown in Fig. 3(a) and (c), while a good reconstruction of the position and size is obtained by using the exact layered-medium model. To study the sensitivity of the inversion, we change the layermedium parameters from their true values. In Fig. 3(e) we give (or 20% variation from the true the estimated value), and in Fig. 3(f) we change the thickness of layers 2 and 4 from 2.4 mm to 2.9 mm (or 20% variation from the true value); layer 3 changes from 6.52 cm to 6.42 cm thick as positions and are fixed. Under these changes, the images in Fig. 3(e) and (f) are degraded but are still reasonable. B. One Dielectric Sphere in the Layered Medium We use the microwave imaging setup shown in Fig. 1 to collect the scattered data. The large container is filled with water , ; the wavelength in water at (measured 1.64 GHz is about 2 cm) in order to avoid the environmental reflections caused by outer boundaries of the container. Dielectric spheres made of clay of various diameters have been used. The clay material has been measured in air (not in water, which may change the value) to have a dielectric constant of approximately 5 and lossless. Since scattered data sets are measured by linearly polarized resonant dipole antennas, a calibration procedure is used in the image reconstruction algorithm with an infinitesimal electric dipole having the same polarization as the finite-length dipole. The radiation field patterns from a short dipole and a resonant

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Fig. 3. The reconstructed relative permittivity for a 1 cm dielectric sphere in a five-layer medium background with the stabilized biconjugate gradient FFT-distorted Born iterative method (a) using an inexact homogeneous background model for inversion, and (b) using an accurate layered-medium model for inplane of the reconstructed relative version. The 2-D cross-section on the z permittivity profile (c) using an inexact homogeneous background model, and (d) using an accurate five-layer medium model. (e)   : (or 20% variation from the true value), and (f) the thickness of layers 2 and 4 are changed from 2.4 mm to 2.9 mm (or 20% variation from the true value); layer 3 changes from 6.52 cm to 6.42 cm thick.

=0

=

= 4 14

dipole are very similar within the viewing angles of the given antenna array. The difference between the measured data and simulated data can be calibrated out with a scale factor obtained by a simple normalization procedure. In the following imaging examples, we use a normalization factor based on the ratio between the measured background field and the simulated background field at the same receiver location. In this example, we demonstrate microwave imaging from experimental data by utilizing a clay sphere of diameter 1 cm placed at the center of the imaging domain (also the origin). The geometry of the sphere is the same as the last example of imaging from synthetic data, except that the clay sphere used in the experiment has a dielectric constant of about 5. Measurements are performed at several discrete frequencies, but only single-frequency reconstructed results are presented here. For this example, the imaging domain of dimension of is divided into , so there are complex unknowns to be reconstructed, while the number of . measured data is

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Fig. 4. Measured jS j from 9 transmitters and 55 receivers for one clay sphere in the layered medium: the magnitude jS j for the incident field and the total field (upper) and the magnitude jS j for the incident field and the scattered field (lower).

Fig. 4 presents the measured for the incident fields, total field and scattered field from a combination of 9 transmitters and is obtained by subtracting 55 receivers. The scattered field from the total field . It is observed that the incident field is approximately 70 to 90 the magnitude of the scattered dB, or about 20 to 40 dB lower than the incident field (about 50 dB). The signal-to-noise ratio in the measurement and data acquisition system (where the signal refers to the total field) should be better than the ratio of the total field to the scattered field (e.g. 20 dB). Figs. 5 and 6 show the reconstructed permittivity and conductivity, respectively, at 1.64 GHz using the diagonal tensor approximation-Born iterative method and the stabilized biconjugate gradient-distorted Born iterative method, respectively. It is observed that the reconstructed images indicate accurate location of the object, whether using the first step inversion with the diagonal tensor approximation-Born iterative method, or using the two-step inversion with the stabilized biconjugate gradient FFT-distorted Born iterative method. The value of the relative permittivity is about 60 from the two-step inversion, substantially higher than the actual value 5, because of the small electrical size of the object, limited aperture for the 3-D reconstruction, and single-frequency reconstruction. A more visual and direct observation of the above reconstructed 3-D images can be seen from the iso-surfaces in Figs. 7(a) and (b) for the reconstructed permittivity and conductivity, for permittivity respectively, for the object ( for conductivity). The image near and the center of the imaging domain is seen clearly. Also, we observe that the object image becomes elongated along the direction because the limited measurement locations are taken only on the transmitter plane and receiver plane both parallel plane. The multi-frequency reconstruction can imto the prove the imaging results. The frequency hopping approach has been applied to the collected multi-frequency data. However, due to small bandwidth of the transmitting and receiving antennas which results in measured data to contain more noise at

Fig. 5. The reconstructed permittivity profile in 3-D plots for a 1 cm clay sphere in the layered medium from measured data using the diagonal tensor approximation-Born iterative method (a) and the stabilized biconjugate gradient FFT-displane of the torted Born iterative method (b); 2-D cross section on the z reconstructed permittivity profile using the diagonal tensor approximation-Born iterative method (c) and the stabilized biconjugate gradient FFT-distorted Born iterative method (d).

=0

Fig. 6. The reconstructed conductivity profile in 3-D plots for a 1 cm clay sphere in the layered medium from measured data using the diagonal tensor approximation-Born iterative method (a) and the stabilized biconjugate gradient plane FFT-distorted Born iterative method (b); 2-D cross section on the z of the reconstructed conductivity profile using the diagonal tensor approximation-Born iterative method (c) and the stabilized biconjugate gradient FFT-distorted Born iterative method (d).

=0

low and high ends of the frequency band, only a small improvement of images is observed. Due to space limitation, multi-frequency inversion results are not presented here. To further demonstrate the capability of the reconstruction algorithm for experimental data, Fig. 8 shows the convergence curve of the data error as a function of the iteration number in the stabilized biconjugate gradient FFT-distorted Born iterative method. The inversion is terminated at the 20th iteration step.

YU et al.: MICROWAVE IMAGING IN LAYERED MEDIA: 3-D IMAGE RECONSTRUCTION FROM EXPERIMENTAL DATA

Fig. 9. Microwave imaging of three clay spheres in the layered medium. The three spheres of 1-cm diameter are contained in an imaging domain of and located on the plane of z with d and d : . : ,z The interfaces of the five-layer medium are located at at z ,z : , and z : . Experimental data are collected : using 9 transmitter locations on the plane of z : and 99 receiving : . antenna locations on the plane of z

3 cm 01 55cm

Fig. 7. The iso-surface of the reconstructed permittivity (a) and conductivity (b) for a 1 cm clay sphere in the layered-medium from measured data using the stabilized biconjugate gradient FFT-distorted Born iterative method with isovalue for permittivity and and isovalue for conduc: = tivity.

= 65

445

= 1 55 cm

6262 =0 = 3 cm = 2 5 cm = 01 78 = = 1 78 cm = 03 1 cm = 3 1 cm

= 13S m

Fig. 8. Convergence curve of the data error as a function of iteration number in the stabilized biconjugate gradient FFT-distorted Born iterative method for a 1-cm clay sphere in the layered medium.

It is observed that, due to the good inverse result from the diagonal tensor approximation-Born iterative method as an initial solution (data error 35%) for inversion, the data error of the inversion using the stabilized biconjugate gradient FFT-distorted Born iterative method converges quickly to 6% after 20 iterations for the real measurement data. C. Three Dielectric Spheres in the Layered Medium This example utilizes three clay spheres of diameter 1 cm placed on the plane of with and , as shown in Fig. 9. Multiple source excitations are produced by placing the dipole transmitting antenna at nine locations on the with , , where , plane . Experimental data for are collected for and each transmitter location by scanning the receiving antenna au, tomatically at 99 locations on the plane of with and , with , , and where , . Note that, compared to the last example, both the transmitting array aperture and the receiving array aperture have been enlarged in this setup. Due to the extension of the array aperture, more structure of the object profile can be viewed by electromagnetic waves. The measured data at 1.64 GHz is used for image reconstruction. An imaging domain is divided into . of size The total number of complex unknowns to be reconstructed is , while the number of measured data is .

Fig. 10. The reconstructed permittivity (a) and conductivity (b) profiles for three clay spheres in the layered medium in 3-D plots using the diagonal tensor approximation-Born iterative method. The reconstructed permittivity (c) and conductivity (d) profiles in 2-D cross section at z for the diagonal tensor approximation-Born iterative method.

=0

The reconstructed results of the permittivity and conductivity using the presented inversion method for the objects in the fivelayer medium are shown in Fig. 10, including a 3-D permittivity image (a), a 3-D conductivity image (b), a 2-D cross section permittivity image (c), and a 2-D cross section conductivity image (d). Also, the 3-D iso-surface of the reconstructed permittivity and conductivity is presented in Fig. 11, with for permittivity and and for conductivity. It is seen that the position and the shape of the three spheres are clearly determined. From the iso-surface results displayed in Fig. 11, we can observe that the resolution of the images in the direction is improved from the last example, likely due to the extension of the transmitter and receiver array apertures. D. Two Metallic Spheres in the Layered Medium In the above experiments, dielectric spheres with permittivity value smaller than the background have been imaged. To demonstrate the performance of the system and algorithms for objects having a higher complex permittivity value than the background, we show here an example for two metallic spheres in Fig. 12. The setup is the same as in the last experiment in

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Fig. 11. The iso-surface of the reconstructed permittivity (a) and conductivity (b) for three 1 cm clay spheres in the layered medium from measured data with isovalue for permittivity and and isovalue : = for conductivity.

= 64

= 16S m

Fig. 12. Microwave imaging of two metal spheres in the layered medium. The two spheres of 0.8-cm diameter are contained in an imaging domain of , separated center-to-center by 3 cm and located on the plane of z in the y direction. The interfaces of the five-layer medium are located at at z : ,z : : : ,z , and z . Experimental : data are collected using 9 transmitter locations on the plane of z : . and 99 receiving antenna locations on the plane of z

3 cm cm 01 78 = 01 55 cm

=0 = 1 55 cm

6262 = = 1 78 cm = 03 1 cm = 3 1 cm

Fig. 14. Reconstructed permittivity (a) and conductivity (b) images from the experimental data for two metallic spheres in 3-D plot. Permittivity (c) and conductivity (d) of reconstructed images from the experimental data for two metallic spheres in 2-D cross section at z .

=0

Fig. 15. Reconstructed permittivity (a) and conductivity (b) images from two spheres with large contrast to the background medium.

j j

Fig. 13. Measured S from 9 transmitters and 99 receivers for two metallic for the incident and total spheres in the layered medium: the magnitude S for the incident and scattered fields fields (upper) and the magnitude S (lower).

j j

j j

Fig. 9, except that the two metallic spheres with 0.8-cm diameter are separated by 3 cm center to center in the -direction of the plane. from 9 transmitters and 99 Fig. 13 shows the measured receivers for two metallic spheres in the layered medium. The for the incident and upper figure displays the magnitude total fields, and the lower figure presents a comparison of the between the incident field and the scattered magnitude field. It is observed that the magnitude of the scattered field is about 15 dB lower than the incident field for this case with metallic objects.

The reconstruction results for the permittivity and conductivity are shown in Fig. 14. The results clearly demonstrate the presence of the two metallic objects either from the permittivity image or from the conductivity image. As expected, Fig. 14 also shows that the resolution in the direction is lower than in the other two directions because the scattered field information is plane. collected only on the Our inversion method has been previously demonstrated to perform well for large contrasts [16], [21], [29]. Here we again show a synthetic example to demonstrate that the inversion result will be better if a larger contrast exists. In this example , ) sepatwo spheres (1-cm diameter, rated by 2 cm between their centers are imaged in the five-layer medium with the same configuration as Case A (Fig. 3). The reconstructed images in Fig. 15 confirm that the inversion results are good, especially for the relative permittivity. In the above examples, we note that the resolution of the reconstructed images is quite high, as we can clearly resolve objects smaller than 1 cm (i.e., smaller than half a wavelength) in a layered medium. Typical CPU time for the reconstruction of one case is about 1.5 hours on an Intel Q6600 computer. We have also done some reconstructions with multi-frequency inversion

YU et al.: MICROWAVE IMAGING IN LAYERED MEDIA: 3-D IMAGE RECONSTRUCTION FROM EXPERIMENTAL DATA

(using the frequency-hopping method), displaying only a little further improvement because the bandwidth of this antenna is small. V. CONCLUSION We have developed a layered-medium microwave tomographic imaging system prototype to test 3-D microwave imaging from experimental data when objects are buried in a multilayered medium. Such a system and data sets for 3-D objects in a layered medium are not known to exist previously. The 3-D inverse scattering algorithm used in this work combines the diagonal tensor approximation (DTA), the stabilized biconjugate gradient FFT algorithm (BCGS), the Born iterative method (BIM), and the distorted Born iterative method (DBIM). In this system, two plastic slabs are placed in water to form a five-layer medium and two linear-polarized dipole antennas are used to transmit and receive electromagnetic waves. The collected 3-D microwave scattered data in a multilayered medium background are inverted by the presented inverse scattering method. The focus of the image reconstructions is on the position and dimensions of the scatterers. Several imaging examples have testified the presented system and inversion method. Future work will focus on more realistic microwave imaging scenarios and biomedical applications. REFERENCES [1] W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iteration method,” IEEE Trans. Med. Imag., vol. 9, pp. 218–225, 1990. [2] Y. 1M. Wang and W. C. Chew, “An iterative solution of the two-dimensional electromagnetic inverse scattering problem,” Int. J. Imaging Systems Tech., vol. 1, pp. 100–108, 1989. [3] A. Roger, “A Newton-Kantorovich algorithm applied to an electromagnetic inverse problem,” IEEE Trans. Antennas Propag., vol. 29, pp. 232–238, 1981. [4] A. Franchois and C. Pichot, “Microwave imaging-complex permittivity reconstruction with a Levenberg-Marquardt method,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 203–215, 1997. [5] R. F. Remis and P. M. van den Berg, “On the equivalence of the Newton-Kantorovich and distorted Born method,” Inverse Problems, vol. 16, pp. L1–L4, 2000. [6] M. Slaney, A. C. Kak, and L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Tech., vol. 32, pp. 860–874, 1984. [7] S. Caorsi, G. L. Gragnani, M. Pastorino, and M. Sartore, “Electromagnetic imaging of infinite dielectric cylinders using a modified Born approximation and including a priori information on the unknown cross sections,” IEE Proc. Microw. Antennas Propag., vol. 131, 1994. [8] Z. Q. Zhang and Q. H. Liu, “Two nonlinear inverse methods for electromagnetic induction measurements,” IEEE Trans. Geosci. Remote Sensing, vol. 39, no. 6, pp. 1331–1339, 2001. [9] Q. H. Liu, Z. Q. Zhang, T. Wang, G. Ybarra, L. W. Nolte, J. A. Bryan, and W. T. Joines, “Active microwave imaging I: 2-D forward and inverse scattering methods,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 123–133, Jan. 2002. [10] P. M. van den Berg and R. E. Kleinman, “A contrast source inversion method,” Inverse Problems, vol. 13, pp. 1607–1620, 1997. [11] P. M. van den Berg, A. L. van Broehoven, and A. Abubakar, “Extended contrast source inversion,” Inverse Problems, vol. 15, pp. 1325–1344, 1999. [12] A. Abubakar and P. M. van den Berg, “Three-dimensional nonlinear inversion in cross-well electrode logging.,” Radio Sci., vol. 33, pp. 989–1004, Jul.–Aug. 1998. [13] A. Abubakar, P. M. van den Berg, and S. Y. Semenov, “Two- and threedimensional algorithms for microwave imaging and inverse scattering,” Progr. Electromagn. Res., vol. 37, pp. 57–79, 2002. [14] F.-C. Chen and W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett., vol. 72, no. 23, p. 3080, 1998.

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[15] M. Pastorino, S. Caorsi, and A. Massa, “A global optimization technique for microwave nondestructive evaluation,” IEEE Trans. Instrum. Meas., vol. 51, no. 4, pp. 666–673, 2002. [16] F. Li, Q. H. Liu, and L.-P. Song, “Three-dimensional reconstruction of objects buried in layered media using Born and distorted Born iterative methods,” IEEE Geosci. Remote Sensing Lett., vol. 1, no. 2, pp. 107–111, 2004. [17] L.-P. Song and Q. H. Liu, “GPR landmine imaging: 2D seismic migration and 3D inverse scattering in layered media,” Radio Sci., vol. 40, p. RS1S90, 2004, 10.1029/2004RS003087. [18] Z. Q. Zhang and Q. H. Liu, “3-D nonlinear image reconstruction for microwave biomedical imaging,” IEEE Trans. Biomed. Eng., vol. 51, pp. 544–548, Mar. 2004. [19] J. D. Zaeytijd, A. Franchois, C. Eyraud, and J. M. Geffrin, “Full-wave three-dimensional microwave imaging with a regularized Gauss-Newton method—Theory and experiment,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3279–3292, 2007. [20] C. Estatico, M. Pastorino, and A. Randazzo, “An inexact-Newton method for short-range microwave imaging within the second-order born approximation,” IEEE Trans. Geosci. Remote Sensing, vol. 43, pp. 2593–2605, Nov. 2005. [21] L. P. Song, Q. H. Liu, F. Li, and Z. Q. Zhang, “Reconstruction of threedimensional objects in layered media: Numerical experiments,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1556–1561, April 2005. [22] J. M. Geffrin and P. Sabouroux, “Continuing with the Fresnel database: Experimental setup and improvements in 3D scattering measurements,” Inverse Prob., vol. 25, p. 024001, 2009, (See also the papers in this Special Section, Guest Editors A. Litman and L. Crocco.). [23] L. P. Song, C. Yu, and Q. H. Liu, “Through-wall imaging (TWI) by radar: 2-D tomographic results and analyses,” IEEE Trans. Geosci. Remote Sensing, vol. 43, pp. 2793–2798, Dec. 2005. [24] L. P. Song and Q. H. Liu, “Fast three-dimensional electromagnetic nonlinear inversion in layered media with a novel scattering approximation,” Inverse Problems, vol. 20, no. 6, pp. S171–S194, 2004. [25] L. P. Song and Q. H. Liu, “A new approximation to three dimensional electromagnetic scattering,” IEEE Geosci. Remote Sensing Lett., vol. 2, no. 2, pp. 238–242, April 2005. [26] M. Xu and Q. H. Liu, “The BCGS-FFT method for electromagnetic scattering from inhomogeneous objects in a planarly layered-medium,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 77–80, 2002. [27] X. Millard and Q. H. Liu, “Fast volume integral equation solver for electromagnetic scattering from large inhomogeneous objects in planarly layered-media,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2393–2401, 2003. [28] B. Wei, E. Simsek, and Q. H. Liu, “Improved diagonal tensor approximation (DTA) and hybrid DTA/BCGS-FFT method for accurate simulation of 3-D inhomogeneous objects in layered media,” Waves Random Complex Media, vol. 17, no. 1, pp. 55–66, Feb. 2007. [29] B. Wei, E. Simsek, C. Yu, and Q. H. Liu, “Three-dimensional electromagnetic nonlinear inversion in layered media by a hybrid diagonal tensor approximation—Stabilized biconjugate gradient fast Fourier transform method,” Waves Random Complex Media, vol. 17, no. 2, pp. 129–147, May 2007. [30] A. Baussard, E. L. Miller, and D. Lesselier, “Adaptive multiscale reconstruction of buried objects,” Inverse Problems, vol. 20, pp. S1–S15, 2004. [31] N. V. Budko and R. F. Remis, “Electromagnetic inversion using a reduced-order three-dimensional homogeneous model,” Inverse Problems, vol. 20, pp. S17–S26, 2004. [32] T. J. Cui, Y. Qin, G. L. Wang, and W. C. Chew, “Low-frequency detection of two-dimensional buried objects using high-order extended Born approximations,” Inverse Problems, vol. 20, pp. S41–S62, 2004. [33] C. Yu, L. P. Song, and Q. H. Liu, “Inversion of multi-frequency experimental data for imaging complex objects by a DTA-CSI method,” Inverse Problems, vol. 21, pp. S165–S178, 2005. [34] C. Yu, M. Yuan, and Q. H. Liu, “Reconstruction of 3-D objects with multi-frequency experimental data using a fast DBIM-BCGS method,” Inverse Problems, vol. 25, p. 024007, 2009, 10.1088/0266-5611/25/2/ 024007. [35] C. Yu, M. Yuan, J. Stang, E. Bresslour, R. T. George, G. A. Ybarra, W. T. Joines, and Q. H. Liu, “Active microwave imaging II: 3-D system prototype and imaging reconstruction from experimental data,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 4, pp. 991–1000, 2008. [36] C. Gilmore and J. LoVetri, “Enhancement of microwave tomography through the use of electrically conducting enclosures,” Inverse Problems, vol. 24, p. 035008, 2008, 10.1088/0266-5611/24/3/035008, (21pp).

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[37] W. C. Chew, Waves and Fields in Inhomogeneous Media. Piscataway, NJ: IEEE Press, 1995. [38] 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.

Chun Yu (SM’07) received the Ph.D. degree in electrical engineering from Shanghai University, Shanghai, China, in 1998. From 1982 to 1992, he was an RF Design and Research Engineer with the China Research Institute of Radiowave Propagation, Xingxiang, China. From 1997 to 2001, he was a faculty member and an Associate Professor with the Department of Communication Engineering, Shanghai University. In March 2001, he joined the Electromagnetics Laboratory, Department of Electrical and Computer Engineering, University of Kentucky, initially as a Postdoctoral Research Fellow and then as a Research Scientist. Since December 2004, he has been a Research Associate with the Department of Electrical and Computer Engineering, Duke University, Durham, NC. His research interests include computational electromagnetics, electromagnetic scattering and wave propagation, inverse scattering, microwave and biomedical imaging, and antenna analysis and design.

Mengqing Yuan, photograph and biography not available at the time of publication.

Yangjun Zhang (M’00) received the Masters degree from Shanghai Institute of Technology, Shanghai, China, in 1992 and the Ph.D. degree in electrical engineering from Shizuoka University, Japan, in 2000. He then worked at Shizuoka University as a Research Associate. In 2003, he moved to Ryukoku University, Japan, where, since 2008, he has been an Associate Professor. His research interests include microwave moisture sensor, planar antenna, microwave resonators. Prof. Zhang is a member of IEICE.

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

Rhett T. George, photograph and biography not available at the time of publication.

Gary A. Ybarra, photograph and biography not available at the time of publication.

William T. Joines (M’61–SM’94–LSM’97–F’08) was born in Granite Falls, NC. He received the B.S.E.E. degree (with high honors) from North Carolina State University, Raleigh, and the M.S. and Ph.D. degrees in electrical engineering from Duke University, Durham, NC. He was a Member of Technical Staff at Bell Telephone Laboratories, doing research and development of microwave components and systems for military applications. He is now a Professor of electrical and computer engineering at Duke University. His research and teaching interests are in the area of electromagnetic wave interactions with structures and materials, mainly at microwave and optical frequencies. He is the author of more than 200 technical papers on electromagnetic wave theory and applications. He has received 12 U.S. patents. Prof. Joines received the Scientific and Technical Achievement Award from the Environmental Protection Agency in 1982, 1985, and 1990.

Qing Huo Liu (S’88–M’89–SM’94–F’05) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1989. His research interests include computational electromagnetics and acoustics, inverse problems, geophysical subsurface sensing, biomedical imaging, electronic packaging, and the simulation of photonic and nano devices. He has published over 400 papers in refereed journals and conference proceedings. He was with the Electromagnetics Laboratory at the University of Illinois at Urbana-Champaign as a Research Assistant from September 1986 to December 1988, and as a Postdoctoral Research Associate from January 1989 to February 1990. He was a Research Scientist and Program Leader with Schlumberger-Doll Research, Ridgefield, CT, from 1990 to 1995. From 1996 to May 1999, he was an Associate Professor with New Mexico State University. Since June 1999, he has been with Duke University where he is now a Professor of electrical and computer engineering. Dr. Liu is a Fellow of the IEEE, a Fellow of the Acoustical Society of America, a member of Phi Kappa Phi, Tau Beta Pi, a full member of U.S. National Committee of URSI Commissions B and F. Currently he serves as the Deputy Editor in Chief of Electromagnetic Waves and Applications, Deputy Editor in Chief of Progress in Electromagnetics Research, an Associate Editor for Medical Physics, for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, and for Radio Science. He received the 1996 Presidential Early Career Award for Scientists and Engineers (PECASE) from the White House, the 1996 Early Career Research Award from the Environmental Protection Agency, and the 1997 CAREER Award from the National Science Foundation.

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Viable Three-Dimensional Medical Microwave Tomography: Theory and Numerical Experiments Qianqian Fang, Member, IEEE, Paul M. Meaney, Member, IEEE, and Keith D. Paulsen, Member, IEEE

Abstract—Three-dimensional microwave tomography represents a potentially very important advance over 2D techniques because it eliminates associated approximations which may lead to more accurate images. However, with the significant increase in problem size, computational efficiency is critical to making 3D microwave imaging viable in practice. In this paper, we present two 3D image reconstruction methods utilizing 3D scalar and vector field modeling strategies, respectively. Finite element (FE) and finite-difference time-domain (FDTD) algorithms are used to model the electromagnetic field interactions in human tissue in 3D. Image reconstruction techniques previously developed for the 2D problem, such as the dual-mesh scheme, iterative block solver, and adjoint Jacobian method are extended directly to 3D reconstructions. Speed improvements achieved by setting an initial field distribution and utilizing an alternating-direction implicit (ADI) FDTD are explored for 3D vector field modeling. The proposed algorithms are tested with simulated data and correctly recovered the position, size and electrical properties of the target. The adjoint formulation and the FDTD method utilizing initial field estimates are found to be significantly more effective in reducing the computation time. Finally, these results also demonstrate that cross-plane measurements are critical for reconstructing 3D profiles of the target. Index Terms—Adjoint method, alternating-direction implicit finite-difference time-domain (ADI-FDTD), finite-difference timedomain (FDTD), microwave tomography.

I. INTRODUCTION

T

HE electrical properties of normal and cancerous tissues are significantly different across microwave frequencies [1]–[3]. To exploit this apparent contrast, substantial effort has been invested in the development of microwave imaging [4]–[8]. Compared to traditional X-rays, microwave energy

Manuscript received November 15, 2008; revised May 10, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported in part by the National Institutes of Health (NIH) through the National Cancer Institute under Grant P01-CA80139. Q. Fang is with Martinos Center for Biomedical Imaging, Massachusetts General Hospital (MGH), Charlestown, MA 02129 USA and also with Harvard Medical School, Boston, MA 02115 USA (e-mail: qianqian.fang.th05@alum. dartmouth.org). P. M. Meaney is with Thayer School of Engineering, Dartmouth College, Hanover, NH 03755 USA (e-mail: [email protected]). K. D. Paulsen is with Thayer School of Engineering, Dartmouth College, Hanover, NH 03755 USA and also with the Radiobiology and Bioengineering Research Program, Norris Cotton Cancer Center, Dartmouth-Hitchcock Medical Center, Lebanon, NH 03756 USA (e-mail: pkeith.d.paulsen@dartmouth. edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037691

is advantageous in several important respects, for example, it does not involve ionization, its associated imaging hardware is relatively low in cost and its absorption and scattering is altered by physiological processes of interest in tissue. These features make microwave methods an intriguing medical imaging option for situations where frequent scanning is required, such as in breast cancer screening or therapeutic monitoring. Within the microwave imaging arena, frequency-domain based tomography approaches [4]–[7] have been investigated along with time-domain techniques based on synthetic aperture radar [8]–[10] for breast cancer detection. In physical realizations of microwave imaging systems, the fields radiate into 3D space. Nonetheless, initial image reconstruction techniques reported in the literature were largely developed for 2D cases in order to achieve higher computational resolution at reasonable compute speeds and to reduce the amount of measurement data required [9], [11]–[14]. Approximations are necessary in order to utilize 2D methods, such as the tissues of interest being cylindrical in structure, and in most instances, the fields being confined to the transverse-magnetic (TM) mode. Because of these assumptions, the recovered 2D images often exhibit artifacts or distortions directly related to the approximations [15]. With the advent of increased computational power (including parallel computing using graphical processing units [16]–[18]) along with various algorithmic improvements (such as the adjoint approach [19]), 3D image reconstruction with clinically relevant spatial resolution and commensurate computational field and tissue property sampling is now within reach [20], [21]. In parallel to advances in image reconstruction, 3D microwave data acquisition systems have been developed by several research groups. A preliminary study of whole body imaging of a canine conducted by Semenov et al. [22] demonstrated, to a limited degree, the feasibility of 3D microwave tomography. However, both the data acquisition and image reconstruction times were unsatisfactory for realistic utility. More recently, a 3D scanning microwave imaging system reported by Yu et al. [23] has demonstrated respectable spatial localization for a simple target; however, the recovered microwave property contrast was relatively low. These algorithm and hardware advances set the stage for practical 3D microwave tomography. In this paper, we concentrate our efforts on applying computational innovations to accelerate the field solution as the forward modeling problem appears to be the bottleneck for accurate image reconstructions in 3D. We also explore the image quality improvements attained by adding cross-plane measurement data to co-planar transceiving antenna array configurations. Two 3D image reconstruction methods are

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evaluated based on 3D scalar and vector field models, respectively. Our previously published 2D [11], [19] and quasi-3D hybrid methods [24] along with the new 3D approaches outlined in this paper represent a spectrum of algorithms with increasing levels of complexity which have enabled us to explore trade-offs between model accuracy and computational efficiency. Several strategies developed for 2D reconstructions have been improved and incorporated into the new 3D algorithms including an iterative block solver [25] and an adjoint method for constructing the Jacobian matrix [24]. For the 3D vector field method, an optimized FDTD algorithm with a uniaxial perfectly matched layer (UPML) technique [26] has been developed to obtain more accurate forward models within an acceptable computational time. These algorithms were tested with simulated data and evaluated in terms of their computational efficiency and accuracy relative to previously developed methods. Four antenna array configurations that return combinations of in- and cross-plane field data were considered and the cross-plane measurements were found to be critical for recovering the 3D profiles of embedded heterogeneities. The paper includes a computational methods section (Section II) which describes the formulations of the two 3D reconstruction algorithms—(1) the FE-based 3D scalar forward/3D inverse reconstruction (scalar-3D) and (2) the FDTD-based 3D vector forward/ 3D inverse reconstruction (vector-3D). Key elements, such as the 3D dual-mesh, the nodal adjoint method and optimization of the 3D vector field solver, are discussed in detail and algorithmic options are evaluated from a computational complexity perspective. The results section (Section III) contains reconstructions from synthetic data which evaluate the performance of the 3D algorithms with respect to our existing 2D methods. In particular, a parametric study of 4 imaging array configurations is presented and the computational costs experienced in practice across the suite of imaging algorithms is reported. We conclude the paper with a summary discussion in Section IV.

II. COMPUTATIONAL METHODS The 3D reconstruction algorithms described in this paper exploit non-linear optimization based on a regularized Gauss-Newton method [27] and dual-mesh scheme [28]. We first present the dual-mesh configurations for the 3D scalar-field and vector-field reconstructions. A nodal adjoint method is subsequently derived in general form as a fast approximation to the original adjoint approach [24], which can be used not only in the two 3D algorithms presented in this paper, but also in the 2D and semi-3D methods described previously [11], [19], [24]. This development is followed by a brief discussion of the computational efficiency of the 3D FDTD algorithm. In the last subsection, enhanced finite-difference time-domain (FDTD) methods utilizing an alternating-direction implicit (ADI) update scheme and additional accelerations afforded by incorporating initial field distributions are discussed, and their overall efficiency improvements are compared.

Fig. 1. Forward and reconstruction mesh orientations for (a) the scalar-3D and (b) the vector-3D methods.

A. 3D Dual-Mesh The dual-mesh scheme is a simple and flexible approach to control the problem sizes of the forward and inverse computations comprising the image reconstruction by utilizing independent discretizations of the field and parameter representations [28]. In terms of the imaging system configuration at Dartmouth [6], the dual-meshes associated with the scalar-3D and the vector-3D methods are illustrated in Fig. 1(a) and (b). The forward (field) mesh for the scalar-3D method consists of a cylindrical domain (tetrahedrons) concentrically aligned with a circular monopole antenna array that extends radially beyond the antennas. The field mesh for the 3D vector reconstruction is a cubic-shaped 3D Yee-lattice [29] surrounded by several layers of UPML cells. The reconstruction (parameter) grids for both methods are identical in this instance, consisting of a 3D cylindrical domain centered within both the antenna array and the respective field meshes. The field and parameter meshes can be constructed with different and variable nodal densities. For a given target, the bilateral mappings between the field and parameter meshes can be precomputed and stored resulting in minimal increases in computational costs. B. Nodal Adjoint Method The adjoint method for Gauss-Newton parameter estimation is critical to achieving computational time reduction relative to the traditional sensitivity equation approach [30]. The Jacobian matrix for the dual-mesh configuration using the adjoint formulation [24] can be written in terms of a summation over field mesh elements

(1) where represents the measurement index between the th denotes the region within which the source and th detector; basis function of the th parameter node is non-zero, indicates summation over the field mesh elements which are located within . is a square matrix with each coefficient defined by

(2)

FANG et al.: VIABLE 3-D MEDICAL MICROWAVE TOMOGRAPHY: THEORY AND NUMERICAL EXPERIMENTS

where and are the basis functions over the field and paand rameter mesh elements, respectively, are the local node indices and is the total node number for a single field mesh element (for linear elein 2D and 4 in 3D). is the spatial domain ments, occupied by the th mesh element and is a 3D position vector. and are the fields at the vertices, , of the selected mesh element due to source antennas at and , respectively. Equation (1) is referred to as the element-based form of the adjoint formula. For cases where the boundaries of the field mesh elements do not precisely match those of the parameter mesh elements, the evaluation of (2) becomes more difficult because it involves integrations over partial elements of the field mesh. A nodal adjoint method is introduced to simplify the integration for a given dual-mesh pair under the assumption that the average size of the field mesh elements is significantly smaller than that of the parameter mesh elements (discussed at the end of the subsection). where , the parameter basis funcWithin domain can be expanded as a linear combination of the field tion basis functions

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Fig. 2. Plot of the maximum relative error of the nodal adjoint Jacobian as a function of parameter/field element area ratio. Sample parameter (solid lines) and field (dashed lines) mesh pairs are shown as insets.

, where is the volume of the th field element (in 2D, is the identity matrix. area of the element) and is an can be further approximated by The weighting matrix . By substituting back into (7) and then (4), the reorganized equation can be written as

(3) (8)

Inserting (3) into (1), yields

(4) where

is an

matrix defined as

.. .

.. .

..

(5)

.. .

.

and denotes volume integration over . Note that the result in cross-multiplinonzero off-diagonal elements in cation terms of the fields at different nodes when expanding (4). , To simplify the evaluation of (4), the weighting matrix, is approximated by summing each column (or row) and adding the off-diagonal elements to the diagonal while simultaneously zeroing the off-diagonal terms such that

.. .

.. .

..

.

.. . (6)

It is not difficult to prove that

(7)

where indicates summation over the field mesh nodes which fall inside and signifies summation over the field mesh elements that share the th node. The term is a scalar expression associated with the th , and referred to as the node which can be symbolized as effective volume of node . The nodal adjoint formula (8) and allows the Jacobian matrix to be easily computed: are the nodal electrical field values obtained directly and require only simple from the field problem; algebraic operations and can be calculated on-the-fly. This is important for forward techniques which dynamically generate their meshes, such as FDTD and some adaptive methods [31]. Note that the reconfiguration of the weighting matrix is only valid when the field mesh elements are substantially smaller than the parameter mesh elements such that the field values at their nodal vertices are approximately equal. To validate this derivation, we compute the Jacobian matrices using the nodal adjoint formula over a series of dual-meshes with different parameter/field element area ratios. The maximum relative error between the nodal adjoint and the true adjoint Jacobian is plotted as a function of the ratio of the averaged parameter and field element sizes (Fig. 2). From this plot, it is reasonable to conclude that when the forward element is small compared to the parameter element, the nodal adjoint Jacobian is a good approximation to an accurate Jacobian matrix (less than 2% difference for a 10:1 ratio in parameter to field element size). Given the derivation above, the nodal adjoint formulation for the vector-3D method is straightforward. In the 3D FDTD for all interior field nodes is grid, the effective volume

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identical, and is equal to the volume of a single voxel, i.e., . The nodal adjoint formula in this case is correspondingly written as

(9)

C. Computational Complexity Comparison In this subsection, the computational complexity of the 3D FDTD and 3D FE methods are compared by summing the total floating-point operations (FLOPs) for obtaining one field solution. The purpose of this comparison is to estimate how the computational complexity scales with increasing mesh densities and how the FE and FDTD methods perform in their generic is used forms. A uniform grid with to assess both approaches (each cube in the FE mesh is split into 6 tetrahedral elements). The total node number for both . After assembling the FE matrix for the weak meshes is form of the simplified scalar model, the size of the matrix is . The minimum half-bandwidth for the finite element when numbering the nodes sequentially in each approach is layer. If a boundary element matrix is incorporated to account for the far-field boundary conditions as in the hybrid method (FE/BE) [11], the minimum half-bandwidth increases to which is essentially the total number of the boundary nodes. Solving this matrix equation with a Cholesky factorization algorithm [32], the total FLOP count for the FE/BE hybrid approach while that for the FE method with abis (here, we sorbing boundary conditions is ignored the computations for assembling the FE/BE matrices). With the strategy described in [19], the total FLOP count for obtaining a 3D FDTD steady-state solution can be decomposed into a two-term expression

Fig. 3. Comparison of the total floating-point operation counts between the 3D FE/BE and 3D FDTD methods for different mesh sizes.

number of time steps required to reach steady state can be estimated as

(12) Consequently, the total FLOP count for the 3D FDTD method (with UPML for lossy medium) is given by (13) A plot of the total FLOP counts for the two methods over is shown in Fig. 3 where , a range of and are used in the calculations. From the graph, the 3D FDTD appears to be more efficient than the generic 3D FE method with increasing mesh sizes. Note that the implementation of iterative solvers can significantly reduce the computational expense associated with the FE or FE/BE field equations. In Section III, we tabulate the forward field computation time for a 3D FE method with an iterative solver and the 3D FDTD method. D. Computational Acceleration for the 3D FDTD Vector Field

(10) is the number of time steps needed to reach where is estimated as the time-steps required steady state ( for the radiated wave to travel round-trip through the domain is since the background medium is highly lossy), and can the number of operations within a single time step. be easily computed by counting the algebraic operations in and components which is the update equations for all approximately using a UPML medium [33]. , the Assuming the mesh is isotropic, i.e. Courant-Friedrichs-Lewy (CFL) number [34] is given by

(11) and the If the wave speed in the background medium is , the maximum wave speed among all inhomogeneities is

From the FLOP count analysis in Section II-C, the total FLOP number for the FDTD method is proportional to the number of time steps required to reach steady-state. We have found that is related to the initial the steady-state time step number field distribution: if the FDTD time-stepping starts from a null field distribution (i.e. all components are zero), it takes considerably longer to reach steady state than from a field distribution that resembles the final solution. A simple 2D forward problem is computed to illustrate this finding. A 2.5 cm 2.5 cm square dielectric object is located at the center of the antenna array, where the object properties and and those of the background are are 25 and 1.0 S/m, respectively. Utilizing polar coordinates, and and located at ( the transmitter operating at , ), the amplitudes of the receivers at and are recorded and plotted versus the number of time steps in Fig. 4 in comparison to the responses computed from the initial values of a similar field distribution, i.e. previously

FANG et al.: VIABLE 3-D MEDICAL MICROWAVE TOMOGRAPHY: THEORY AND NUMERICAL EXPERIMENTS

Fig. 4. Amplitudes at different time-steps for receivers located at (a)  and (b)  .

= 180

= 90

453

persion error becomes the limiting factor. A detailed study of the impact of the dispersion error from various ’s in the ADI FDTD is given by Zhao [36]. The unconditional stability of the ADI FDTD allows for simultaneous use with the iterative FDTD field initialization approach introduced in the previous subsection. To estimate the computational efficiency, the total FLOP count is calculated for this method. Assuming the 3D grid size , the floating-point operations per is iteration for the ADI approach can be written as (14)

computed fields due to the presence of a similarly sized object , . In both cases, the time step that has is set to to ensure stability. From the plot, it is evident that the second approach leads to significantly fewer time steps to achieve steady-state. The sharp oscillations in the solid lines are referred to as “spurious modes” induced by the jump in dielectric properties. To exploit this result, we have derived an iterative FDTD approach which utilizes the field distributions from the previous parameter estimate iteration to reduce the field modeling time. Implementation of this scheme is straightforward. Extra memory is required to store all field components and the accumulated elapsed time at the end of each iteration for each source. At the subsequent iteration, the fields are initialized by the stored values from the previous iteration of the corresponding source and continues the FDTD time-stepping. We demonstrate in Section III-A that it is possible to reduce the steady-state time step number by 1/2 to 2/3 by supplying initial field estimates which do not compromise either convergence or image quality. One must be careful when selecting the time step, , which must be a fixed number in this situation to avoid spurious waves. based on As a result, we can not determine the optimal the CFL condition per iteration, rather, a minimum permittivity, , should be estimated, and then a uniform determined for all iterations from (11). Overall, the acceleration provided by using an initial field distribution makes the iterative FDTD approach significantly faster. E. ADI FDTD With Lossy UPML As discussed above, a constant time-step in the iterative FDTD scheme may result in some computational redundancy during the first few iterations. To avoid this, we implemented an unconditionally stable FDTD scheme, the alternating direction implicit (ADI) FDTD method, for forward field modeling of lossy media. In the update equations of the ADI FDTD method, the target time step fields appear on both sides of the update equation; thus, this method yields an implicit difference update scheme. Based on this principle, the ADI form of the UPML update equation for a lossy medium is not difficult to derive. We have used a symbolic software package, Mathematica, to perform the derivation and the full formulation can be found in Appendix A in [35]. is not conWith this ADI technique, the time step size strained by the CFL stability condition (11). Instead, the dis-

where the number “2” results from the two sub-steps of the ADI FDTD, “177” is the FLOP count required to assemble the right-hand-side for the implicit update equation at each sub-step (optimized by Mathematica), “5” is the number of back substitutions needed to solve the system of tri-diagonal equations and “66” is related to the contributions from the remaining update equations. The total number of time steps required to reach steady-state for the ADI FDTD method can be written as (15) and are the steady time step and CFL where number defined in (12) and (11), respectively. Combining (14) and (15), the total FLOP count for the ADI FDTD with lossy UPML ABC is

(16) Based on (16) and (13), the CFL number for the ADI FDTD approach should be at least 6 times that used in the traditional FDTD in order to achieve faster computations. III. RESULTS AND DISCUSSION In this section, we present image reconstructions from simulated data to assess the performance of the proposed methods under ideal conditions. The antenna array configurations were not chosen to produce optimal images in each case, but rather to examine the influences of different array perturbations. The computational efficiency of the algorithm enhancements, for example, when incorporating initial field estimates and the ADI FDTD, are studied with these simulations. Finally, five dualmesh based algorithms are compared including the 2D, semi-3D and 3D methods, with benchmark reconstructions to profile their computational complexity with respect to increasing accuracy in the forward field modeling. In order to perform fair comparisons across these approaches, we applied a set of common parameters for all experiments unless otherwise noted. For instance, the background medium was and at a 0.9% saline solution having 900 MHz. The cylindrical reconstruction meshes for the two 3D methods were identical, comprised of 1660 nodes and 7808 tetrahedral elements. In this case, the origin of the Cartesian

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coordinate system was located at the center of the reconstruction mesh with the -axis aligned along the cylinder. A cirand comcular antenna array located on a radius prised of 16 equally spaced antennas was placed in the central plane. Each antenna in the array was modelled as an infinitely small -oriented dipole. For cases where multiple layers of antennas were used, schematic diagrams are provided to illustrate the positions of the array elements. For each iteration of the Gauss-Newton reconstruction, a Tikhonov regularization was imposed with the regularization parameter computed by the empirical method discussed in [37]. All reconstructions commenced from an initial property parameter estimate equal to the homogeneous background medium. A. Measurement Configuration Study The imaging target was a sphere ( , ) with , , ) and center location ( . For the scalar-3D reconstructions, the field radius mesh was a cylinder consisting of 56,636 nodes and 312,453 and extended tetrahedral elements. It had a radius to . For the vector-3D vertically from reconstructions, the interior grid was comprised of (in the , and directions, respectively) and was surrounded by 5 layers of a UPML (the final node size of the ). The FDTD cells were cubes with data array was a uniform node spacing of mm. The simulated measurement data was generated using an FDTD 3D vector solution over a much finer field mesh (40 nodes per wavelength with respect to the background medium compared with 15 nodes per wavelength in the reconstruction problem) components were extracted at the receiver sites. and the Four antenna array configurations (Fig. 5) were investigated. plane. Data For scheme A, all antennas resided in the was collected at the nine antenna sites opposing each transmitter for the total of 16 antennas (16 transmitters by 9 receivers). In schemes B and C, the antennas resided in two planes 1.5 cm plane and signals were transabove and below the central mitted from all 32 antennas. In scheme C, the signals were received by the 18 opposing antennas (nine in each plane), while for scheme B, the signals were only received by the opposing nine antennas in the same plane as the transmitters. The amount of measurement data for schemes B and C was 288 (32 transmitters by 9 receivers) and 576 (32 transmitters by 18 receivers), respectively. For scheme D, the antennas resided in 3 planes, plane and two 1.5 cm above and below one in the central it, respectively. Signals were transmitted only by antennas in the central plane and received at the 27 opposing antennas for a total of 432 pieces of measurement data (16 transmitters by 27 receivers). The reconstructed 3D dielectric profiles for both scalar (scheme A only) and vector methods are shown in Fig. 6. cross-sectional images are less effected by Because the the antenna array configurations evaluated here, we only include cross-sectional images in Fig. 6 and omit them for sample the rest of the results. Several important observations can be made here as follows. 1) The permittivity images have fewer artifacts than their conductivity counterparts, similarly to that observed in [24].

Fig. 5. Source configurations for 3D simulated reconstructions: (a) scheme A, (b) scheme B, (c) scheme C, and (d) scheme D. In each diagram, the star represents a transmitter and the triangles represent the corresponding receivers for that specific transmitter (Open circles represent non-receiving antennas for that transmitter).

2) The images reconstructed utilizing the scheme A antenna configuration from both the scalar (scalar-3D) and vector-3D methods have pronounced artifacts above and below the recovered object, particularly in the conductivity images. However, the vector-3D algorithm artifacts are noticeably reduced and the estimated object position appears to be more accurate in both the horizontal and vertical planes. 3) The permittivity contours for the single-layer receiving array (scheme B) are relatively accurate in the plane where the array is located. However, artifacts occur above and below the object making the permittivity appear elongated in the -direction whereas the conductivity images contain elevated zones above and below the actual target location. 4) The images acquired from the two-layer [scheme C—Fig. 7(b)] and three-layer [scheme D—Fig. 7(c)] receiving antenna configurations recover the target very well in terms of its shape, location and dielectric properties. These results demonstrate that more measurements, especially out-of-plane and cross-plane data, improve the quality of the 3D reconstructions. 5) Although the number of measurements is doubled in scheme B compared with scheme A, the artifacts along the -axis remain visible until the cross-plane measurements are included in Fig. 7(b) and (c). In Fig. 7(d), we compressed scheme B antenna array spacing from 3 cm to 2 cm. The vertical plane images from the two cases are quite distinct. The target reconstructed from the data acquired with the 3 cm array spacing appears elongated in the permittivity images, whereas it is recovered with very nearly the correct size, location and property values when the 2 cm array is used but with pronounced elevated property zones above and below. The corresponding conductivity images exhibit elevated property artifacts above and below the target in both cases. We also studied the impact of plane number on 3D image recovery for scheme A when the array was spaced in 1 cm increments by reconstructing the same target with the vector-3D algorithm utilizing three and five planes of data. Overall, a progression of image quality improvement occurs with an increase in the number of planes of data. It is interesting to note that

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Fig. 7. Vertical cross-sectional relative permittivity (left column) and conductivity (right column) images of the reconstructed dielectric profiles using the vector-3D algorithm and (a) scheme B, (b) scheme C, (c) scheme D and (d) scheme B with 2 cm separation.

Fig. 6. Cross-sectional relative permittivity (top row) and conductivity (bottom row) images of the reconstructed dielectric profiles using the scheme A antenna configuration (scalar-3D algorithm) for the (a) scalar-3D and (b) vector-3D algorithms. Circles show the exact location of the embedded heterogeneity in each cross-section.

For the single array, we found the semi-3D reconstruction produced better results than the scalar-3D approach. The relative residual, the residual normalized by that of the first iteration, was reduced to 13% after 10 iterations compared with 23% for the latter case. This finding can be understood by the low axial sensitivity of the given source configuration and the degradation of the field solution accuracy in the scalar 3D model. However, for the two-plane arrays, opposite results were observed, i.e., the scalar-3D reconstruction reduced the relative residual to 16% which was 3 times lower than the residual (44%) for the semi-3D reconstruction. This indicates that the scalar-3D reconstruction is advantageous when the measurements support the recovery of 3D profiles. In this case, the improvements associated with better geometric modeling outperformed the approximations introduced by the 3D scalar field model. C. Computation Time Improvements From Initial Field Estimation

the images from three planes of data which utilize cross-plane recordings [scheme D, Fig. 7(c)] are still better than five sets of in-plane measurements, especially with respect to the artifacts above and below the target. B. Comparison Between Semi-3D and Scalar-3D Reconstructions Additional numerical simulations were performed to compare semi 3D and scalar 3D reconstructions. In these experiments, a cylindrical reconstruction mesh was constructed by vertically ( -axis) extending a 2D horizontal circular mesh. Only the 2D profile is updated at each iteration in the semi-3D reconstruction, while the scalar-3D reconstruction calculates updates for all parameter nodes.

Utilizing simulated data from scheme A, the computational time reduction achieved with the initial field estimation approach was investigated. We first set the minimum dielectric property value to 1/5th that of the background and used to compute the time step for all iterations. The values of all field vectors and the accumulated time-steps were recorded starting from the second iteration. Additionally, we reduced the steady-state time step number estimate from (12) by factors of 2 and 3. For these simulations, the reconstructed images (not shown) demonstrate no obvious degradation. The relative errors in these reduced computation time reconstructions are plotted in Fig. 8 as a function of iteration number compared with those from the unenhanced version which confirms the benefits of the technique.

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TABLE I COMPARISONS BETWEEN DUAL-MESH BASED RECONSTRUCTIONS

y sensitivity equation method used to construct the Jacobian matrix; z used the initial field acceleration technique (factor = 2); total node number including those in the PML layers; 4 CPU’s operating in parallel; without multiple-RHS matrix solver. Note: the forward field computation for vector-3D utilizes single precision floating-point operations, which is two times faster than using double-precision.

the field problem and computing the additional vector field components.

IV. CONCLUSION

Fig. 8. Relative error plot of the reconstructions with and without the initial field estimates.

D. Computational Cost for All Dual-Mesh Based Algorithms Using the previous reconstructions as a benchmark, we tested five dual-mesh methods, i.e. the scalar-2D method [11], FDTD-2D method [19], semi-3D method [24], scalar-3D and vector-3D methods, and summarized the problem size and computational times in Table I. All computations were performed on an Alpha ES40 workstation with 4 600 MHz CPUs. From the table, we observe predictable trends when the problem size increases and when the reconstruction transitions from 2D to 3D. Independently of the increasing levels of computational complexity, implementation of the iterative block solver and initial field estimates seems capable of maintaining the 3D model computation time to be within acceptable limits even for the full vector approach. The adjoint techniques also deserve special mention because of the significant computational time reductions which make all of these approaches viable. Not surprisingly, the 2D algorithms demonstrate substantial speed advantages over their 3D counterparts. For the 3D reconstructions, the scalar technique based on the FE method together with the iterative block solver provides an efficient approach for modeling 3D field distributions with the understanding that the underlying scalar model imposes various approximations (and concomitantly important limitations). The 3D FDTD algorithm used in the vector-3D method is promising because of 1) the accuracy in field modelling, 2) the advantages being able to exploit parallel computing and 3) the flexibility in accommodating various optimizations as discussed in Sections II-E and II-D. From columns 5–7 of Table I, it is evident that the 3D FDTD method can compute the full vector field solution within 9 seconds, which is less than twice that required for the scalar technique, even when utilizing a mesh that is three times larger for

We have developed two 3D image reconstruction approaches including a 3D scalar field/3D reconstruction technique based on the FE method and a 3D vector field/3D reconstruction algorithm based on the FDTD method. The adjoint scheme devised in [24] was extended to a nodal-based approximation which significantly simplified the Jacobian matrix calculation and also led to an associated reduction in computation time. Additional enhancements in the 3D FDTD algorithm with respect to the image reconstruction problem were investigated including the use of initial field estimates and the ADI FDTD method. Despite that both proposed methods are capable of modeling the scattering field of arbitrary 3D structures, utilizing the initial field estimates can achieve significant acceleration only when the background medium is lossy. Reconstructions were performed to validate the proposed algorithms utilizing synthetic data. In most cases, the target objects were successfully recovered in both location and dielectric property values with the permittivity images exhibiting fewer artifacts than their conductivity companions. We compared this series of algorithms within the dual-mesh and iterative reconstruction framework. The 2D algorithms were superior in speed due to their considerably smaller problem size, while the 3D algorithms were generally superior in terms of image quality. Within the 2D methods, the 2D FDTD technique is promising and may facilitate quasi-real-time imaging applications because of its fast computation time. The 3D reconstructions are also promising and showed progressive improvements in terms of artifact reduction as the amount of measurement data was increased. The most significant improvement in image quality appeared to result from the use of cross-plane data in this regard. The investigations into these 3D image reconstruction algorithms are still preliminary and substantial work remains in order to make them viable in practice. At the same time, with the rapid increase in computing power, particularly with the availability of graphics processing unit-based (GPU) parallel computing, the use of vector field techniques such as the FDTD method becomes increasingly important for producing accurate field representations and consequently improved reconstructed image quality.

FANG et al.: VIABLE 3-D MEDICAL MICROWAVE TOMOGRAPHY: THEORY AND NUMERICAL EXPERIMENTS

ACKNOWLEDGMENT The authors would like to thank G. Zhu (McGill University, Canada) for the discussions on the nodal adjoint formulation. REFERENCES [1] K. R. Foster and J. L. Schepp, “Dielectric properties of tumor and normal tissues at radio through microwave frequencies,” J. Microw. Power, vol. 16, pp. 107–119, 1981. [2] R. Pethig, “Dielectric properties of biological materials: Biophysical and medical applications,” IEEE Trans. Elec. Insulation, vol. 19, pp. 453–474, Oct. 1984. [3] W. T. Joines, Y. Zhang, C. Li, and R. L. Jirtle, “The measured electrical properties of normal and malignant human tissues from 50 to 900 MHz,” Med. Phys., vol. 21, pp. 547–550, 1994. [4] L. E. Larsen and J. H. Jacobi, “Microwave scattering parameter imagery of an isolated canine kidney,” Med Phys., vol. 6, pp. 394–403, 1979. [5] S. Y. Semenov, A. E. Bulyshev, R. H. S. A. E. Souvorov, Y. E. Sizov, V. Y. Borisov, I. M. K. V. G. Posukh, A. G. Nazarov, and G. P. Tatsis, “Microwave tomography: Theoretical and experimental investigation of the iteration reconstruction algorithm,” IEEE Trans. Microw. Theory Tech., vol. 46, pp. 133–141, 1998. [6] P. M. Meaney, M. W. Fanning, D. Li, S. P. Poplack, and K. D. Paulsen, “A clinical prototype for active microwave imaging of the breast,” IEEE Trans. Microw. Theory Tech., vol. 48, pp. 1841–1853, 2000. [7] Q. H. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. Nolte, and W. T. Joines, “Active microwave imaging. I. 2-D forward and inverse scattering methods,” IEEE Trans. Microw. Theory Tech., vol. 50, pp. 123–133, 2002. [8] E. C. Fear, X. Li, S. C. Hagness, and M. A. Stuchly, “Confocal microwave imaging for breast cancer detection: Localization of tumors in three dimensions,” IEEE Trans. Biomed. Eng., vol. 49, pp. 812–822, 2002. [9] S. C. Hagness, A. Taflove, and J. E. Bridges, “Two-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Fixed-focus and antenna-array sensors,” IEEE Trans. Biomed. Eng., vol. 45, pp. 1470–1479, Dec. 1998. [10] E. C. Fear, J. Sill, and M. A. Stuchly, “Experimental feasibility study of confocal microwave imaging for breast tumor detection,” IEEE Trans. Microw. Theory Tech., vol. 51, pp. 887–892, 2003. [11] P. M. Meaney, K. D. Paulsen, and T. P. Ryan, “Two-dimensional hybrid element image reconstruction for TM illumination,” IEEE Trans. Antennas Propag., vol. 43, pp. 239–247, 1995. [12] W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging, vol. 9, no. 2, pp. 218–225, Jun. 1990. [13] S. Caorsi, G. L. Gragnani, and M. Pastorino, “Two-dimensional microwave imaging by a numerical inverse scattering solution,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 8, pp. 981–989, Aug. 1990. [14] S. Y. Semenov, R. H. Svenson, A. E. Boulyshev, A. E. Souvorov, V. Y. Borisov, Y. Sizov, A. N. Starostin, K. R. Dezern, G. P. Tatsis, and V. Y. Baranov, “Microwave tomography: Two-dimensional system for biological imaging,” IEEE Trans. Biomed. Eng., vol. 43, no. 9, pp. 869–877, Sep. 1996. [15] P. M. Meaney, K. D. Paulsen, S. Geimer, S. Haider, and M. W. Fanning, “Quantification of 3D field effects during 2D microwave imaging,” IEEE Trans. Biomed. Eng., vol. 49, pp. 708–720, 2002. [16] G. S. Baron, C. D. Sarris, and E. Fiume, “Fast and accurate time-domain simulations with commodity graphics hardware,” in Proc. Antennas and Propag. Society Int. Symp., 2005, vol. 4A, pp. 193–196. [17] S. Adams, J. Payne, and R. Boppana, “Finite Difference Time Domain (FDTD) simulations using graphics processors,” in Proc. DoD High Performance Computing Modernization Program Users Group Conf., 2007, pp. 334–338. [18] GPU Gems 3, H. Nguyen, Ed. Boston, MA: Addison-Wesley, 2007. [19] Q. Fang, P. M. Meaney, and K. D. Paulsen, “Microwave image reconstruction of tissue property dispersion characteristics utilizing multiple frequency information,” IEEE Trans. Microw. Theory Tech., vol. 52, pp. 1866–1875, Aug. 2004. [20] Z. Q. Zhang and Q. H. Liu, “3-D nonlinear image reconstruction for microwave biomedical imaging,” IEEE Trans. Biomed. Eng., vol. 51, pp. 544–548, 2004. [21] D. W. Winters, D. B. V. Veen, and S. C. Hagness, “Three-dimensional microwave breast imaging: Dispersive dielectric property estimation using patient-specific basis functions,” IEEE Trans Med. Imaging, vol. 28, no. 7, pp. 969–981, Jul. 2009.

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[22] S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, V. G. Posukh, and A. Pavlovsky, “Threedimensional microwave tomography: Initial experimental imaging of animals,” IEEE Trans. Biomed. Eng., vol. 49, pp. 55–63, 2002. [23] C. Yu, M. Yuan, J. Stang, E. Bresslour, R. T. George, G. A. Ybarra, W. T. Joines, and Q. H. Liu, “Active microwave imaging II: 3D system prototype and image reconstruction from experimental data,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 991–1000, 2008. [24] Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3D fields coupled to 2D parameter estimation,” IEEE Trans. Med. Imaging, vol. 23, pp. 475–484, Apr. 2004. [25] W. E. Boyse and A. A. Seidl, “A block QMR method for computing multiple simultaneous solutions to complex symmetric systems,” SIAM J. Sci. Comput., vol. 17, pp. 263–274, 1996. [26] S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag., vol. 44, no. 12, pp. 1630–1639, Dec. 1996. [27] N. Joachimowicz, C. Pichot, and J. P. Hugonin, “Inverse scattering: An iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag., vol. 39, pp. 1742–1752, 1991. [28] K. D. Paulsen, P. M. Meaney, M. J. Moskowitz, and J. M. Sullivan, Jr., “A dual mesh scheme for finite element based reconstruction algorithms,” IEEE Trans. Med. Imaging, vol. 14, pp. 504–514, 1995. [29] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302–307, Mar. 1966. [30] C.-T. Liauh, R. G. Hills, and R. B. Roemer, “Comparison of the adjoint and influence coefficient methods for solving the inverse hyoperthermia problem,” J. Biomech. Eng., vol. 115, pp. 63–71, 1993. [31] S. Burger, R. Klose, A. Schadle, F. Schmidt, and L. Zschiedrich, Adaptive FEM Solver for the Computation of Electromagnetic Eigenmodes in 3D Photonic Crystal Structures. Berlin Heidelberg: Springer, 2006. [32] G. H. Golub and C. H. van Loan, Matrix Computations. Baltimore, MD: The Johns Hopkins Univ. Press, 1991. [33] Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, Ed. Norwood, MA: Artech House, 1998. [34] , A. Taflove, Ed., Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 1995. [35] Q. Fang, “Computational Methods for Microw. Medical Imaging,” Ph.D. dissertation, Dartmouth College, Hanover, NH, Dec. 2004. [36] A. P. Zhao, “Analysis of the numerical dispersion of the 2D alternatingdirection implicit FDTD method,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 4, pp. 1156–1164, Apr. 2002. [37] A. Franchois and C. Pichot, “Microwave imaging-complex permittivity reconstruction with a Levenberg-Marquardt method,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 203–215, Feb. 1997.

Qianqian Fang (S’03–M’05) was born in Anyang, Henan, China, in 1976. He received the B.Eng. degree in electrical engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, and the Ph.D. degree in biomedical engineering from Thayer School of Engineering, Dartmouth College, Hanover, NH, in 2005. From 1997 to 2000, he was a Research Assistant with the Microwave and Computational Electromagnetics Labs, UESTC, working with millimeter microwave switch and finite difference-time domain (FDTD) simulations for EM pulse well-logging. From 2000 to 2005, he was a Research Assistant in the Microwave Imaging Group at Dartmouth College, developing tomographic microwave imaging for breast cancer detection, particularly focusing on fast three-dimensional forward and reconstruction algorithms. From 2005 to 2009, he was a Postdoctoral Fellow at the Martinos Center for Biomedical Imaging, Massachusetts General Hospital (MGH), performing studies in near-infrared tomographic breast imaging and brain functional imaging. He is currently an Instructor at MGH and Harvard Medical School. His current research interests include multi-modality imaging, translational near-infrared breast cancer imaging, compression-induced tissue dynamics and massively parallel computing using graphics processing units (GPU) for medical imaging applications.

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Paul M. Meaney (M’92) received A.B. degrees in computer science and electrical engineering from Brown University, Providence, RI, in 1982, the M.S. degree in electrical engineering from the University of Massachusetts, Boston, in 1985, and the Ph.D. degree in biomedical engineering from Dartmouth College, Hanover, NH, in 1995. He was a Postdoctoral Fellow at Dartmouth College, from 1995 to 1996, and an NSF-NATO Postdoctoral Fellow at the Royal Marsden Hospital in Sutton, U.K., from 1996 to 1997. He is currently a Research Professor at Dartmouth College. His interests include developing microwave imaging for biomedical applications especially breast imaging and hyperthermia monitoring, along with elastography and various applications of thermal modeling.

Keith D. Paulsen (S’85–M’86) received the B.S. degree in biomedical engineering from Duke University, Durham, NC, in 1981 and the M.S. and Ph.D. degrees in biomedical engineering from Dartmouth College, Hanover, NH, in 1984 and 1986, respectively. From 1986 to 1988, he was an Assistant Professor in the Electromagnetics Group within the Department of Electrical and Computer Engineering, University of Arizona, Tucson. He is currently a Professor at the Thayer School of Engineering, Dartmouth College and the Director of the Radiobiology and Bioengineering Research Program for the Norris Cotton Cancer Center within the Dartmouth-Hitchcock Medical Center, Lebanon, NH. His research interests include computational methods with particular emphasis on biomedical problems in cancer therapy and imaging, and model-guided surgery.

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Far Field Subwavelength Source Resolution Using Phase Conjugating Lens Assisted With Evanescent-to-Propagating Spectrum Conversion Oleksandr Malyuskin, Member, IEEE, and Vincent Fusco, Fellow, IEEE

Abstract—The imaging properties of a phase conjugating lens operating in the far field zone of the imaged source and augmented with scatterers positioned in the source near field region are theoretically studied in this paper. The phase conjugating lens consists of a double sided 2D assembly of straight wire elements, individually interconnected through phase conjugation operators. The scattering elements are straight wire segments which are loaded with lumped impedance loads at their centers. We analytically and numerically analyze all stages of the imaging process—i) evanescent-to-propagating spectrum conversion; ii) focusing properties of infinite or finite sized phase conjugating lens; iii) source reconstruction upon propagating-to-evanescent spectrum conversion. We show that the resolution that can be achieved depends critically on the separation distance between the imaged source and scattering arrangement, as well as on the topology of the scatterers used. Imaged focal widths of up to one-seventh wavelength are demonstrated. The results obtained indicate the possibility of such an arrangement as a potential practical means for realising using conventional materials devices for fine feature extraction by electromagnetic lensing at distances remotely located from the source objects under investigation. Index Terms—Array, diffraction limit, evanescent field, microwave imaging, phase conjugation, subwavelength resolution.

I. INTRODUCTION IGH-PERFORMANCE antenna devices employing phase conjugation (PC) phenomenon find application in modern self-tracking wireless communication systems [1], [2]. Recently it has been shown that surfaces performing PC of the field tangential components can lens through the process of negative refraction [3], [4]. The quality of imaging in these situations can be characterised by a resolution criterion normally defined as the width of the image field distribution at the 3 dB level across a focal spot under consideration. Imaging below the diffraction limit, i.e., with resolution less than half-wavelength is essential in many applications, such as electromagnetic sounding of a 3D structures for nanophysics, biological and medical diagnostics, subsurface remote sensing in geophysics, UWB radar etc., [5], [6].

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Manuscript received March 02, 2009; revised May 18, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported in part by the Strengthening all Island programme and in part by the U.K. Engineering and Physical Research Council under Grants EP/D045835/1, EP/E01707X/1. The authors are with the Institute of Electronics Communications and Information Technology, Queens University Belfast, Queens Island, Belfast BT3 9DT, Northern Ireland (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2037713

In the last years several techniques have been used for subwavelength imaging including left-handed metamaterial superlens [7], [8], wire or layered medium lens [9], [10], patterned screens and plates [11], [12]. The common feature of these structures operation is that the image is formed in the vicinity of the lens, some auxiliary lensing device has to be used to transfer the near field information to the far field zone [13]. The realizability constraints and operational effectiveness for these lensing devices are reported in [7]–[12] and for brevity are not repeated here. Phase conjugation techniques have been used in optics [14] and acoustics [15] for source localization. It is very important whether phase conjugation lensing operates in homogeneous or inhomogeneous environment. It has been shown [16], [17] that a “perfect” phase conjugating or time reversal mirror (lens) composed of monopole transducers cannot produce a source image with subwavelength resolution in a homogeneous space no matter how closely this lens is positioned to the source. The reason for this is that a “perfect” PC surface consisting of monopoles generates an imaginary part of the free-space Green’s function which oscillates at the half-wavelength scale, i.e., is diffraction limited [17]. However a PC lens composed of dipole transceivers [4], [17] can produce a subwavelength image in a homogeneous space when it is operated in the near field of the source, [4], [17]. Here the generated PC field is dominantly determined by the spatial derivative of the free space dyadic Green’s function [18] which results in a subwavelength oscillation of the field. In general if the PC lens is positioned in the far field zone of a source in a homogeneous space the resolution of this lens is always diffraction limited because the evanescent spectrum of the source carrying subwavelength information does not reach the conjugating surface. The lensing scenario can operate differently in an inhomogeneous medium—in this case scattering from heterogeneities can result in evanescent-to-propagating mode conversion. Therefore a portion of evanescent spectrum of the source can reach the PC surface after being converted into propagating waves. These propagating waves are negatively refracted by the PC surface. If identical scatterers are positioned in front of both the original source and image position the conversion of evanescent to propagating waves on the source side is reversed on the image side due to the scattering from these heterogeneities [19]. Experimental verification of this technique reveals the formation of a of a wavelength in an entirely closed focal spot as small as reverberation chamber of size equipped with a digital time reversal mirror and microstructured scattering media [20].

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This arrangement is of very limited value for applications which must operate in an open environment, e.g., a wireless communication system. The process of evanescent-to-propagating modes conversion and its reciprocal, propagating-to-evanescent, counterpart can be mathematically described by the symmetry properties between a propagating plane wave or superposition of propagating waves scattered on a heterogeneity and a generated evanescent wave or evanescent spectrum in a completely enclosed time reversal volume [19]. The relations in [19] establish the reciprocity between a propagating plane wave and an evanescent wave or spectrum and vice versa on the spatial plane containing the scattering potential, not the original source plane. Therefore a part of the evanescent spectrum will be inevitably lost even in the case when the near-to-far field converters are located very close to the source. However in the later case image resolution can be significantly improved. In general there are three major factors determining image resolution in lossless scattering medium (i) Evanescent-to-propagating conversion efficiency of the scattering arrangement. Here we define conversion efficiency as the ratio of surface integral of the scattered field to the surface integral of the source field over the same closed surface in the far field zone; (ii) the nature of the surrounding space—closed or open; (iii) the reflection/transmission properties and extent of the PC surface. The last two factors are important in assuring that as much as possible of the propagating wave energy is processed by the PC for re-assembly at the focal spot. This paper concentrates on the far-field imaging properties of a PC lens composed of a double periodic wire arrays, of finite or infinite extent which are illuminated by a Hertzian dipole source and operated in free space. The dipole source is operated, first without, and then with, arrangements of wire scatterers placed in to it. In the absence of the scatterers close proximity it is found that the focal spot produced by a PC lens in the far field region is about one wavelength in both the transversal and longitudinal ranges, so the image is indeed diffraction limited. We then show that with the scattering arrangement positioned both in the near field of the source and symmetric position in the focus of PC lens we can obtain automatic subwavelength source reconstruction. The use of a PC lens means that this can be achieved without the need for computer processing involving time reversal algorithms and digital hardware, [21]. In this paper we also present a mathematical formulation and simulation results that illustrate the degree of resolution that can potentially be achieved in the far field.

II. FAR FIELD TRANSMITTED BY A PC LENS DUE TO DIPOLE EXCITATION

Fig. 1. Geometry of the problem. Double sided PC lens. Focal spot occurs at z 2z , z is a distance from the origin of the coordinate system to the lens plane.



not large enough for the single plane wave approximation to be used. The lens is formed by two arrays separated by an isolation plane [4], e.g., a perfectly electric conducting (PEC) screen or electromagnetic (EM) absorbing layer. Below we compare the performance of an infinite PC double periodic wire frequency selective surface (PCFSS) and finite array consisting of 25, (5 5), wire elements. It is assumed that the PC device accurately conjugates the phase of a received signal wave and re-transmits it without any distortion. This is a good approximation to practically realized PC devices [1]. To calculate the transmitted field of an infinite PCFSS we represent the field of a dipole in a form of a continuous superposition of plane waves incident at all angles ,

(1) and verwhere the amplitudes of horizontal components are defined as a scalar tical product of a dipole moment vector with partial plane wave polarization vectors , . The scattering problem is solved for each partial wave in the wave packet (1) using a rigorous thin-wire kernel integral equation approach. The total transmitted field in the far field zone is reconstructed via continuous superposition of phase conjugated partial plane waves multiplied by the transof the PCFSS mission function

A. Phase Conjugating Surface Description In this section we consider the far field focussing properties of a wire based PC lens illuminated by a monochromatic, , Hertzian electric dipole with dipole moment , Fig. 1, positioned at a distance larger than one wavelength but

(2)

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Integration in (2) over the angle is restricted to the range since the contribution of the evanescent waves is negligibly small at the distances larger than due to exponential decay of the evanescent waves. The PC field transmitted through a finite array can be represented as a spatial convolution of the dyadic free-space Green’s induced by a phase confunction [18] with current densities jugated voltage sources [4] in the th wire

(3)

where

is the unit vector along the th wire direction.

B. Numerical Results In this section we compare the focusing properties of an infinite PCFSS and a finite PC wire array. In order to calculate the transmitted field of a finite PC array we use the FEKO 5.4 Suite [22], while for the infinite case in-house MOM software based on the equations above was used. The PC lens is illuminated by a -oriented Hertzian dipole with dipole moment set to be 1 mA-m radiating at frequency 1.0 GHz. Infinite double periodic PCFSS consists of the -oriented wire segments of a , wire radius , and lattice dimenlength , positioned at , stands sions for the wavelength, from the dipole source. The finite array is composed of wires with identical dimensions and separation between them. The number of elements in the finite array is 25, so the size by , the distance from the source is . of the array is The lattice dimensions are chosen to be to prevent possible length so grating lobes arising. The wire elements are of by . When unloaded, that they can fit into a unit cell of these elements operate below resonance with the consequence that large currents do not exist on them. To ensure the resonance condition and therefore maximize wire element currents, hence transmitted field the wire elements are centre loaded with 20 nH inductive loads. The results in Fig. 2(a) for the transmitted field of the infinite PCFSS show that the characteristic focal spot size in the longitudinal, , direction, and in the transis verse, , direction. Fig. 2(b) demonstrates that the focal spot in the direction is about one wavelength. Since the PCFSS is infinite in both and dimensions, the focal spot footprint does not depend on the distance from the exciting source (see, e.g., [23]). The reason for this is that the angular plane wave spectrum from the source propagating towards the surface reaches it without loss. The focussing properties of a finite array are described in Fig. 3. A primary feature of the transmitted field in Fig. 3 is in both and planes. the characteristic focal width is This spot enlargement, with respect to the case of infinite surface, Fig. 2, is due to the loss of angular spectrum of the source that reaches the surface [4]. In addition it can be seen in both Figs. 2 and 3 that the focal spot maximum is shifted towards and by for infinite and finite the PC surface by PCFSS cases respectively. This can be explained by the fact that

Fig. 2. Transmitted normalized field magnitude cited by a dipole source.

j

E

j

for infinite PCFSS ex-

in each case the array operates at slightly above the natural resonance frequency of the individual wire elements. Therefore it can be seen from (2) that the complex transmission coefficient with positive phase will shift the focal spot maximum position, , towards the array. In the case defined by the condition of the finite array case the shift is more pronounced than for the infinite array. Primarily this is due to inter element array mutual coupling impedance variation, [24]. In this Section we studied the focusing properties of infinite and finite PC arrays—in both cases the focal width is diffraction . In Section III, due its limited and its characteristic size is practical importance, we will discuss finite array subwavelength imaging in the far field. III. SUBWAVELENGTH FOCUSING IN THE FAR FIELD A. Formulation Subwavelength focusing in a heterogeneous medium using far field time reversal or phase conjugating mirrors has been experimentally examined in [20], [25], [26]. In these experiments a reverberation chamber equipped with scattering medium and digital time reversal mirrors positioned in the far field zone of the antenna have been used to demonstrate wave refocusing with subwavelength resolution. A scattering medium

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Fig. 4. Problem geometry. The 2D PCFSS is placed in the far-field zone of the z . In the half-space z > z identical counterpart scatterers sources at z are placed in the mirror positions.

=

Fig. 3. Transmitted normalized field magnitude jE

j

for finite PCFSS.

surrounded the EM or acoustical source facilitated the evanescent-to-propagating field conversion thus a substantial portion of the evanescent spectrum is converted into propagating waves. Due to the closed nature of a reverberation chamber all propagating waves after time reversal will be reassembled at the initial source position and partially converted back into the evanescent spectrum. In this case the resolution is limited only by near-to-far (evanescent field component) conversion efficiency. For practical implementation of a super-resolution far-field imaging system in an open (free space) environment we need to know two things: (i) can we convert by simple means evanescent energy into propagating energy and then back to evanescent energy again? (ii) can super-resolution be achieved by a suitably excited planar 2D phase conjugating surface positioned in free space? The second point is necessary since for a variety of wireless and sub-wavelength imaging applications imaging within an enclosing 3D system may be of little practical value. We now demonstrate the far-field subwavelength imaging properties of a PC lens augmented with wire scatterers assigned for simplicity this is done along the transversal x direction only. The dipole sources and the scatterers are aligned along axis and the characteristic resolution in this direction is equivalent to about the length of the scatterer, i.e., it is not necessarily

subwavelength. The PC lens wire segments are loaded with inductive loads of 20 nH to make them resonant. By using resonant scatterers we ensure large currents on these wire elements and therefore high PC conversion efficiency. Another additional feature of the proposed setup is again related to the loading of the scatterers. For the enclosed 3D cavity it does not matter whether the scatterers act as directors or reflectors—the propagating energy will reach the PC mirror after multiply reflections in any case. For free space operation it is essential that the scatterers direct the energy towards the PC surface, not reflect it away from the surface. To achieve directive behavior of a scatterer or arrangement of scatterers placed in the vicinity ) one can use of the source (typically at the distance parallel LC lumped inclusions. These loads modify the input impedance of the scattering element and therefore change the near field interference between the dipole source and driven scattering element. Thus the radiation pattern which is a function of frequency, wire scatterer length and radii, as well as source-scatterer separation can be flexibly adjusted to ensure maximum directivity, minimization of angular spectrum loss, towards the PC lens. The geometry of the PC lens augmented with scatterers is shown in Fig. 4. On the image side a receiving antenna or a probe is positioned to measure imaged field distribution. The EM field emanating form a dipole can be represented in a spatial, [18], or a spectral form (4) using the expansion of the free space Greens function into a spectrum of cylindrical waves

(5)

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where and is a propagating spectrum is the evanescent part of given by the first term while the field determined by the second term in (5). In (4) Helmholtz is defined as . From (5) vector operator on the adjacent to dipole it follows that the currents source wire elements will be predominantly determined by the near field of a dipole (its evanescent spectrum) since for . The scattered field

(6) will therefore contain the converted into propagating spectrum near field information. From (4), (6) it follows that in the far of the system dipole plus field zone the total field , so direct extraction of scatterers oscillates on a scale of the subwavelength information, without numerical processing (e.g., resorting to time reversal algorithms) is not possible. Therefore in order to extract encoded subwavelength information the total transmitted field has to be re-scattered using an identical scattering arrangement as was used for the initial evanescent-to-propagating spectrum conversion. In this process the PCFSS lens ensures constructive reassembly of all the propagating waves, reaching the surface. Thus at the image position wave fronts arrive with the correct phases and amplitudes [26]. Propagating-to-evanescent spectrum conversion will now be described by the scattered field on the image side of the PC lens

Fig. 5. a) Geometry of the imaging setup b) Normalized magnitude of the field y -component, jEy j, at z z : . Solid line: PC lens augmented with a pair of scatterers. Dotted line: PC lens without scatterers. Dashed line: field distribution of an isolated dipole source at distance = in free space.

=

+08

10

and transmitting modes. Although not pursued here both of these can in principle can be compensated for by using tuneable lumped loads [23]. B. Numerical Results

(7) on the wires positioned in where the current distributions the focal plane are due to the transmitted PC field (8) where is a transmission operator of a PC lens. Image quality is defined by the following factors: i) near-to-far field conversion efficiency; ii) loss of the propagating spectrum in free space, i.e., not all propagating waves reaching the surface; iii) distortions caused by a PC lens. Near-to-far conversion efficiency depends on the ability of a scattering arrangement to have large currents induced upon it in order to maximize the conversion of evanescent to propagating waves. Obviously the amount of conversion will depend on the separation between the source and scatterers, since the reciprocity between propagating and evanescent waves is established on the scattering plane only [19]. Therefore it is desirable to position the scatterers as close to a source as possible. The second factor depends on the electrical size of the PC lens. The open nature of free space should result in a spreading of the image with respect to the case of a totally enclosed cavity. The primary reasons for this are: (i) aberrations introduced by a PC lens associated with angular dispersion of an FSS array [27], (ii) the difference of currents on wire elements in the receiving

Numerical simulations have been carried out for the case of finite 5 5 elements PC array described in Section II. For all simulations we have chosen the separation between the source and the array to be equal to one wavelength. This is done in order to minimize possible angular spectrum loss with the view to achieving best possible resolution. Fig. 5 shows the image formation due to a dipole located at the origin of coordinate system by a pair of half-wavelength scat. The scatterers are centre loaded with LC terers with radii parallel load with 1 pF capacitance and 30 nH inductance to achieve a pattern with high directivity towards the PC surface. axis; the source-side scatThe scatterers are located on the in direction terer is displaced with respect to the source to (towards the PC surface); the image-side scatterer is displaced from the focal position set at towards by the PC surface. The field distribution is calculated in the image . It can be seen that the resolution in this plane at case is about a quarter-wavelength at the half-maximum amplitude (3 dB power) points. Also we note that the field distribution in the image plane without image side scatterer is diffraction . limited. Here characteristic width is In Fig. 6 we compare the field distribution in the image plane and then for two cases: the source-side scatterer is placed away from the source. The image-side scatterer is disthen from the image plane. placed accordingly at The parameters of the scatterers are as above, the scatterers are axis (cf. Fig. 5(a)). Fig. 6 demonstrates that positioned on

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=z + 20. Dotted

Fig. 6. Normalized magnitude of the field y-component, jE j, at z : . Solid line: separation between the scatterer and the source is = line: separation between the scatterer and the source is = .

08

10

Fig. 8. Normalized magnitude jE j of the transmitted field for imaging with 2 scatterers; a) Near field distribution in the vicinity of the scatterer in the image plane. b) Field distribution at z :  without, dotted line, and with z scatterer, solid line.

=

+08

Fig. 7. Imaging setup geometry in case of displaced scatterer.

the characteristic focal width that can be achieved for the separation is approximately . Next we show what happens if the source-side (and imageside) scatterer is displaced in the lateral, , direction. This can happen in imaging process when the location of the source is unknown and the volume is being scanned by a pair of scatterers. Figs. 7 and 8 illustrate the situation when a source side scatterer in the positive direction and in is displaced to direction (towards the PC surface) with respect to the source, , i.e., the scatterer centre is located at the point Fig. 7. The image scatterer is positioned accordingly with its , 0.0, ). centre located at ( From Fig. 8 it is clear that the PC lens augmented with scatterers (marked with cross) tends to produce the image field distribution with a maximum pointing towards the original source position (marked with circle). Residual displacement of the field distribution with respect to the origin is due to inefficiency of a single PEC scatterer as a near field antenna. Even so, in this case which is below the characteristic width of the curve is the diffraction limit. This result shows that symmetry of the scatterer arrangement on the source side and image side is essential. Next we examine the scattering arrangement where several scatterers are positioned across the range Fig. 9(a) and along the range Fig. 9(b). Counterpart scatterers are located in the mirror position in the image half space. The scatterer lengths and loading are the same as described above and the scatterer . radii are Fig. 10 shows the imaging properties of the PC lens augmented with the across-range and along-range arrangement. The

Fig. 9. Arrangement of scatterers across (a) and along the range (b).

across-range arrangement consists of three scatterers separated so the overall size of the arrangement in the direcby tion is (including wire diameters). The central scatterer is axis as shown in Fig. 9(a). The along-range located on the arrangement is composed of two scatterers of the same radii separation, Fig. 9(b). The resolution for both cases and is about . Consider next the situation when the across-range scatterers are displaced in the -direction Fig. 11. Fig. 12 shows the field distribution at the image plane for the in case when a 3-element scatterers array is displaced by

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Fig. 10. Across-range imaging by three scatterers, solid line, and along-range, dashed line. Dotted line—field distribution of an isolated dipole source at the distance of =10 in free space.

Fig. 11. Imaging setup geometry for the case of three displaced scatterers.

direction and in direction with respect to the origin of coordinate system. The counterpart array is displaced accordingly in the image half-space. It can be seen in Fig. 12(a) that this arrangement results in axis is at the 3 dB level the field distribution centred on the reproducing the original source location. However the spot size , i.e., subwavelength resolution is lost. The near field is distribution in Fig. 12(b) confirms that the near field generated by the array tends to produce a focal spot corresponding to the original source location. From Fig. 12 it can be seen that the subwavelength imaging ability of the PC lens assisted with a three element scattering arrangement is limited in the direction by to approximately a range while scanning at distance from the source in -direction. To increase the transversal range over which subwavelength resolution can be achieved it is necessary to i) scan at a closer-tosource distance or ii) improve the near field array evanescent to propagating conversion performance by combining the scatterers in the across- and along-range directions. Next we illustrate the possibility of imaging two sources closely located in the transversal , direction. This is equivalent to what is done by the classical 1951 USAF optical resolution test chart [28] designed to characterize the resolution property of any lens. The two dipoles case is necessary and practically sufficient to yield the characteristic resolution for a lens since any complex source can be decomposed into superposition of dipole and other higher order terms (higher order terms are not studied here for simplicity’s sake) [29]. In the situation

Fig. 12. Normalized magnitude jE j of the transmitted field for imaging with three across-range scatterers. a). Field distribution in the vicinity of the three elements array in the image plane, at z = z + 0:8. b) Near field distribution of the three across-range elements array.

of two dipole sources the additional coupling between scattering elements assigned to each dipole can smear the focal field distribution. Fig. 13(b) shows imaging of two dipoles in the cross range by a PC lens augmented separated by with along-range scatterer dispositions as shown in Fig. 13(a). The scatterers have the same parameters as above, i.e., radii, and separation between their centres. The field distribution produced by a PC lens in Fig. 13(b) displays two peaks corresponding to the dipoles locations, however the magnitude variation is much less than 3 dB. Mutual coupling between scattering assemblies and source dipoles plays some part in this. However in fact the near field array composed of aligned scatterers, Fig. 13(a) produces a directive beam pattern towards the PC lens which is not optimally shaped, such that the surface of the PC lens is not uniformly illuminated, Fig. 15. Modification of the scatterer arrangement to that shown in Fig. 14(a) helps to alleviate this problem and as a result improves the resolution, Fig. 14(b). Finally we illustrate imaging of a complex source consisting of three Hertzian dipoles separated by one third wavelengths and

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Fig. 15. Imaging of two dipoles separated by =4 in the x range. Field distribution in the source half-space between the scattering arrangement and the PC array. Solid line: scattering arrangement as shown in Fig. 13(a); dotted line: scattering arrangement as shown in Fig. 14(a).

Fig. 13. Imaging of two dipoles separated by =4 in the x range. a) PC lens plan view b). Field distribution in the source plane z = 0:1 and the image plane z = z + 0:8. Dotted line: field distribution of two closely spaced dipole sources at the distance of =10 in free space. Solid line: image field distribution.

Fig. 14. Imaging of two dipoles separated by =4 in the x range. a) PC lens top view b). Field distribution in the source plane z = 0:1 and the image plane z = z + 0:8. Dotted line: field distribution of two closely spaced dipole sources at the distance of =10 in free space. Solid line: image field distribution.

having different dipole moments of 1.6 mA-m (middle dipole), 1.2 mA-m (left dipole) and 0.8 mA-m (right dipole). The scatradii. terers are each one quarter-wavelength long and have , ensure directivity LC parallel loads with towards the surface. Fig. 16 demonstrates the field amplitude

and phase distribution at the source plane at and image plane at . It can be seen that the field amplitude variation is preserved at the image side however the left lobe of the image field has larger amplitude than at the source plane. This is believed to be caused by mutual coupling between the dipole sources and scattering elements and energy leaking from the middle and left dipole source. This distortion is exacerbated in an electrically small PC lens since the scattered field can be collected only over a limited angular range. The consequence of this is that the lens tends to smear sharp amplitude variations. It is also interesting to note that the image field phase variation shown in Fig. 16(c) is very close to the phase variation of the source despite some deterioration caused by the finite size of the array. The phase variation is largely preserved due to the autocorrecting properties [30] of a phase conjugating lens, to the best of our knowledge to date this effect has not been previously reported. These results show that proposed arrangement should be capable of imaging complex amplitude modulated sources. Figs. 13–16 demonstrate multiple sources subwavelength resolution which in terms of classical optical or photo-lithographical resolution terminology means that the modulation transfer function for the structure reported here is at least 18% in contrast to 9% for a diffraction limited lens [31]. It is important to note that not only the information about the location of dipoles but also the information about their amplitudes and phase variation can be transferred to the image plane and subsequently extracted with subwavelength resolution. IV. CONCLUSION In this work we have studied the imaging properties of infinite and finite phase conjugating lens operating in their far field regions. We have shown that the source near field information contained in its evanescent spectrum can be converted into a propagating spectrum by way of scattering. These propagating waves carrying both subwavelength and non-subwavelength information are refocused by a PC lens. In the case of a small finite size PC array with suitable scattering pairs allows . sub-wavelength one dimensional source resolution of up to These results indicate the possibility of such an arrangement

MALYUSKIN AND FUSCO: FAR FIELD SUBWAVELENGTH SOURCE RESOLUTION USING PHASE CONJUGATING LENS

Fig. 16. Imaging of three dipoles fed with different amplitude signals and separated by =3 in the x range. a) PC lens plan view. b) Field amplitude distribution in the source z = 0:05 and image z = z + 0:8 planes. c) Phase of the source and image fields. Dotted line—: field distribution of two closely spaced dipole sources at the distance of =20 in free space. Solid line—their image field distribution.

as a potential realizable means for constructing using conventional materials devices for fine feature extraction by electromagnetic lensing at distances remotely located from the source objects under investigation, or for advanced multiple-input multiple-output (MIMO) applications on restricted size platforms where multiple signals on an identical frequency carrier could be concurrently resolved in a smaller than currently possible aperture. ACKNOWLEDGMENT The authors acknowledge the financial support of the UK Engineering and Physical Science Council. REFERENCES [1] V. Fusco, C. B. Soo, and N. Buchanan, “Analysis and characterization of PLL-based retrodirective-Array,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 730–738, Feb. 2005.

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[2] G. S. Shiroma, R. Miyamoto, J. Roque, J. Cardenas, and W. Shiroma, “A high-directivity combined self-beam/null-steering array for secure point-to-point communications,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 5, pp. 838–844, May 2007. [3] S. Maslovski and S. Tretyakov, “Phase conjugation and perfect lensing,” J. Appl. Phys., vol. 94, no. 7, pp. 4241–4243, Oct. 2003. [4] O. Malyuskin, V. Fusco, and A. Schuchinsky, “Phase conjugating wire FSS lens,” IEEE Trans. Antennas Propag., vol. 54, pp. 1399–1404, May 2006. [5] K. P. Gaikovich, “Subsurface near-field scanning tomography,” Phys. Rev. Lett., vol. 98, pp. 183902–183902, 2007. [6] I. Aliferis, T. Savelyev, M. J. Yedlin, J.-Y. Dauvignac, A. Yarovoy, C. Pichot, and L. Ligfhart, “Comparison of the diffraction stack and time-reversal imaging algorithms applied to short-range UWB scattering data,” presented at the IEEE Int. Conf. Ultra-Wideband (ICUWB 2007), Singapore, Sep. 24–26, 2007. [7] V. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Uspekhi, vol. 92, no. 3, pp. 517–526, 1967. [8] N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett., vol. 88, no. 20, pp. 207403–207403, May 2002. [9] P. A. Belov, Y. Hao, and S. Sudhakaran, “Subwavelength microwave imaging using an array of parallel conducting wires as a lens,” Phys. Rev. B, vol. 73, pp. 033108–033108, 2006. [10] P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B, vol. 73, pp. 113110–113110, 2006. [11] G. Eleftheriades and A. Wong, “Holography-inspired screens for sub-wavelength focusing in the near field,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 4, pp. 236–238, Apr. 2008. [12] R. Merlin, “Radiationless electromagnetic interference: Evanescent-field lenses and perfect focusing,” Science, vol. 317, no. 5840, pp. 927–929, 2007. [13] M. Tsang and D. Psaltis, “Theory of resonantly enhanced near-field imaging,” Optics Express, vol. 15, no. 19, pp. 11959–11970, 2007. [14] Z. Yaqoob, D. Psaltis, M. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nature Photon., vol. 2, pp. 110–115, 2008. [15] M. Fink, “Time reversal of ultrasonic fields,” IEEE Trans. Ultrason., Ferroelect., Freq. Control, vol. 39, no. 5, pp. 555–566, Sep. 1992. [16] M. Nieto-Vesperinas and E. Wolf, “Phase conjugation and symmetries with wave fields in free space containing evanescent components,” J. Opt. Soc. Amer., vol. 2, no. 9, pp. 1429–1434, 1985. [17] J. de Rosny and M. Fink, “Focusing properties of near-field time reversal,” Phys. Rev. A, vol. 76, pp. 065801–065801, 2007. [18] H. Chen, Theory of Electromagnetic Waves. New York: McGrawHill, 1983. [19] R. Carminati, J. J. Sáenz, J.-J. Greffet, and M. Nieto-Vesperinas, “Reciprocty, unitarity and time-reversal symmetry of the S matrix of fields containing evanescent components,” Phys. Rev. A, vol. 62, pp. 012712–012712, 2000. [20] G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science, vol. 315, pp. 1120–1120, 2007. [21] F. Simonetti, M. Fleming, and E. Marengo, “Illustration of the role of multiply scattering in subwavelength imaging from far-field measurements,” J. Opt. Soc. Am. A, vol. 25, no. 2, pp. 292–303, Feb. 2008. [22] FEKO Suite [Online]. Available: www.feko.info [23] O. Malyuskin and V. Fusco, “Negative refraction lensing and signal modulation using a tuneable phase conjugating frequency selective surface,” presented at the IEEE Antennas Propagat. Int. Symp. San Diego, Jul. 2008. [24] B. Munk, Finite Antenna Arrays and FSS. New York: Wiley, 2003. [25] B. Henty and D. Stancil, “Multipath-enabled super-resolution for rf and microwave communication using phase-conjugate arrays,” Phys. Rev. Lett., vol. 93, pp. 243904–243904, Dec. 2004. [26] A. Derode, P. Roux, and M. Fink, “Robust acoustic time reversal with high-order multiple scattering,” Phys. Rev. Lett., vol. 75, pp. 4206–4209, 1995. [27] B. Munk, “Frequency selective surfaces,” in Theory and Design. New York: Wiley, 2000. [28] USAF 1951 Resolution Test [Online]. Available: http://www.aigimaging.com/Individual-Resolution-Targets.html [29] J. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941.

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[30] , R. A. Fisher, Ed., Optical Phase Conjugation. New York: Academic, 1983. [31] N. Koren, Understanding Image Sharpness and MTF [Online]. Available: www.normankoren.com/Tutorials/MTF.html

Oleksandr Malyuskin (M’04) received the M.Sc. degree in radiophysics and electronics and the Ph.D. degree in electrical engineering from Kharkov National University, Ukraine, in 1997 and 2001, respectively. He joined the Institute of Electronics, Communications and Information Technology, Queens University Belfast, in March 2004 as a Post Doctoral Research Fellow involved in the development of novel composite materials for advanced EM applications. His research interests include analytic and numerical methods in electromagnetic wave theory, characterization and application of complex and nonlinear materials, antenna arrays and time reversal techniques.

Vincent Fusco (S’82–M82–SM’96–F’04) received the Bachelors degree (1st class honors) in electrical and electronic engineering, the Ph.D. degree in microwave electronics, and the D.Sc. degree, for his work on advanced front end architectures with enhanced functionality, from The Queens University of Belfast (QUB), Belfast, Northern Ireland, in 1979, 1982, and 2000, respectively. His research interests include nonlinear microwave circuit design, and active and passive antenna techniques. He is the Research Director of the High Frequency Laboratories, Queens University of Belfast, and is also Director of the International Centre for Research for System on Chip and Advanced MicroWireless Integration, SoCaM. He has published over 420 scientific papers in major journals and international conferences, and is the author of two textbooks. He holds several patents on active and retrodirective antennas and has contributed invited chapters to books in the fields of active antenna design and EM field computation. Prof. Fusco is a Fellow of the Royal Academy of Engineering and a Member of the Royal Irish Academy. In 1986, he was awarded a British Telecommunications Fellowship and 1997 he was awarded the NI Engineering Federation Trophy for outstanding industrially relevant research.

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An E-J Collocated 3-D FDTD Model of Electromagnetic Wave Propagation in Magnetized Cold Plasma Yaxin Yu, Student Member, IEEE, and Jamesina J. Simpson, Member, IEEE

Abstract—A new three-dimensional finite-difference time-domain (FDTD) numerical model is proposed herein to simulate electromagnetic wave propagation in an anisotropic magnetized cold plasma medium. Plasma effects contributed by electrons, positive, and negative ions are considered in this model. The current density vectors are collocated at the positions of the electric field vectors, and the complete FDTD algorithm consists of three regular updating equations for the magnetic field intensity components, as well as 12 tightly coupled differential equations for updating the electric field components and current densities. This model has the capability to simulate wave behavior in magnetized cold plasma for an applied magnetic field with arbitrary direction and magnitude. We validate the FDTD algorithm by calculating Faraday rotation of a linearly polarized plane wave. Additional numerical examples of electromagnetic wave propagation in plasma are also provided, all of which demonstrate very good agreement with plasma theory. Index Terms—Earth, electromagnetic wave propagation, finitedifference time-domain (FDTD) method, ionosphere, magnetized cold plasma.

I. INTRODUCTION VER the past two decades, the finite-difference time-domain (FDTD) [1], [2] method has been extended to modeling electromagnetic (EM) wave propagation and interactions with cold plasmas. First, FDTD algorithms were developed for modeling nonmagnetized (therefore isotropic) cold plasma [3]–[9], or dispersive media FDTD algorithms were employed for isotropic plasma studies [10], [11]. A systematic analysis of these FDTD techniques has been published by S. A. Cummer [12]. An important aspect of plasma-related research is to study radio wave propagation through the ionosphere and to study lightning-related ionospheric phenomena. Many FDTD models have been developed to address these problems by treating the ionosphere as a simple nonmagnetized isotropic medium [13]–[15]. However, for accurate broadband investigations, effects introduced by the Earth’s magnetic field on the ionospheric plasma cannot be ignored. Thus, a magnetized (anisotropic) cold plasma ionospheric medium must be employed in the FDTD simulations.

O

Manuscript received January 03, 2009; revised April 23, 2009. First published December 04, 2009; current version published February 03, 2010. The authors are with the Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM 87106 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2037706

A few two-dimensional (2-D) and three-dimensional (3-D) FDTD models have been published that include the magnetic field effect on the ionospheric plasma. Cummer [16] proposed a 2-D cylindrical-coordinate FDTD model to study EM wave propagation in the Earth-ionosphere waveguide. Thèvenot et al. [17] reported another 2-D spherical-coordinate FDTD model to simulate VLF-LF propagation in the ionosphere. More recently, Hu and Cummer [18], [19] extended their full wave 2-D cylindrical coordinate FDTD model to explore lightninggenerated EM wave behavior in the ionosphere and to test the sprite initiation theory. Unlike the H-J collocation method [20], the stability condition of Hu and Cummer’s model involving the E-J collocation method is independent of medium properties and remains the same as for free space [12] (at the Courant stability limit [2]). This is a very important characteristic of their plasma algorithm. For some other FDTD plasma algorithms (see for example [21]–[23]), the time-stepping increment is linked to the plasma parameters, resulting in a strict time-step orders of magnitude smaller than that permitted by the Courant limit [2] when modeling the ionosphere [24], [25]. Further, some plasma algorithms, such as that of [21], produce nonphysically spurious electrostatic waves (of numerical origin) due to the spatially non-collocated status of electric fields and current densities [26] and thus exhibit seemingly incurable late-time instabilities when used to model ionospheric plasma, or they are only first-order accurate as for that proposed in [27], or their implementation requires a great amount of additional memory even for spatially unchanging plasma parameters as in [12], [28]. In this paper, we expand Hu and Cummer’s 2-D E-J collocation FDTD model [18] to the fully 3-D case. As such, similar accuracy attained previously by Hu and Cummer for their 2-D model at both high altitudes and over long distances when compared to experiments and mode theory is expected here for the newly developed 3-D model. However, the 3-D model described in this paper provides additional capabilities, such as modeling of Faraday rotation and the inclusion of fully 3-D spatial variations in the magnetization and characteristics of the cold plasma. These capabilities are essential for future global 3-D studies of EM propagation and in many other research areas. In Section II, the governing equations for the 3-D magnetized cold plasma are derived, as well as the resulting FDTD timestepping algorithm. In Section III, the plasma FDTD model is validated, and Section IV illustrates some additional numerical examples of EM propagation in plasma. Finally, Section V concludes and describes ongoing work.

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II. METHODOLOGY A. Governing Equations In the derivation of the 3-D FDTD magnetized cold plasma algorithm, wave propagation effects introduced by electrons, positive ions, and negative ions are included for generality. Here, we consider a cold plasma characterized by a free space permittivity and a free space permeability that is biased by an applied magnetic field . The magnetized cold plasma governing equations are cast in terms of Maxwell’s equations coupled to current equations derived from the Lorentz equation of motion. The Lorentz current equations consist of three auxiliary partial differential equations that model the response of each charged particle species to the electric field and the applied . The resulting whole governing equation set is given by

Notice that the cyclotron frequency is a function of the applied magnetic field. Thus, the cross-product terms in (3)–(5) make the plasma anisotropic so that the wave behavior depends on its propagation direction relative to the direction of the magnetic field. Without these cross product terms, the whole system of (1)–(6) simply becomes the governing equation set of a nonmagnetized (isotropic) cold plasma. By substituting (7)–(14) into (1)–(6) and separating out each Cartesian expression, the whole governing differential equation set is expanded into 15 scalar equations (15) (16) (17)

(1) (2)

(18)

(3)

(19)

(4)

(20)

(5)

(21)

(6) (22) Here the subscript denotes the charged particle species in the plasma ( , and as electrons, positive ions, and negative ions, respectively). , and are the collision frequencies of each and are the current densities of each species, and , is the source current density. Cartesian species, respectively. coordinate expressions of these current densities are as below

(23) (24) (25) (26)

(7) (8) (9) (10) The total induced charged-particle current density is then the combination of all three of the individual current densities as and forms the total current shown in (6) and the sum of , and are the plasma density as in (2). Further, frequencies of each species, respectively. By construction (11) as the charge, number density, and mass of with , and each particle species. In addition , and are the cyclotron frequencies of each species given by with Cartesian coordinate expressions (12) (13) (14)

(27) (28) (29) B. FDTD Discretization Scheme In this Section, the 15 scalar equations of (15)–(29) are applied to the FDTD mesh. Here, we adopt the E-J collocation method as described in [12]. This locates the , , and components at the same positions of , , and , respectively. The Yee cell describing the spatial positioning of the electric, magnetic, and current density field vector components is shown in Fig. 1. For central differencing of the space derivatives and at , in (15)–(29), we define a sample and at , and at , at , at and at . The 3-D magnetized plasma FDTD updating equations of the -fields derived from (15)–(17) are identical to those of the standard Yee algorithm [2]. Equation (30) provides a sample

YU AND SIMPSON: E-J COLLOCATED 3-D FDTD MODEL OF ELECTROMAGNETIC WAVE PROPAGATION

Fig. 1. Yee cell for spatial positioning of the field components.

updating algorithm of the component as derived from (15) and the spatial scheme of Fig. 1. Then, referring to Fig. 1 and using the so-called semi-implicit approximation [2], the 12 - and -related equations of (18)–(29) can be discretized first at the -component position . The 12 resulting tightly-coupled discretization equations can then be written in matrix form as expressed in

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(31), shown at the bottom of the page. The corresponding coefficient of each field component is then grouped into the three , and , coefficient matrixes which are detailed in Appendix for reference. Where and are scaled and current values of the original magnetic field intensity density . These scaled field values are actual quantities used for updating and during the iteration. This scaling is required to avoid instability and inaccuracies that would result from the large iteration coefficients [18]. is not scaled. Working towards an explicit expression, both sides of (31) to be transformed into are multiplied by the inverse matrix (32), shown at the bottom of the following page, which yields on the left-hand all of the field components at time-step side of (32) and all of the field values calculated at previous time steps on the right-hand side of (32). This results in (32) to be an explicit system suitable for FDTD implementation. In space domain, all of these field quantities are now assumed to be located , we therefore name (32) at -component position as the -equation. The same process can then be repeated at and -component pothe -component position to obtain the -equation and -equation, sition respectively. These three explicit matrix equations are very similar except different spatial positions of the field quantities and derivatives. We next notice that all of the non- field-components in the -equation (32) have no field values defined at -component location according to our leapfrog scheme in Fig. 1,

(30)

(31)

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which makes the direct implementing of the non-x field-quantities in first column of the -equation (32) at their pre-defined Yee-locations impossible. The same situation happens to the non- field-quantities of the -equation and the non- fieldquantities of the -equation, similarly. and are Notice that the coefficient matrix analogous for the -, - and -equations, we then pick up the 4 linear -component updating equations from the -equation ) the ( 4 linear -component updating equations from the -equation ) and ( the 4 -component updating equations from the -equation ) to ( be recombined to obtain a new explicit equation (33), which is the eventual iteration set used to implement the whole system. The field quantities in the first column of (33) are now all positioned at their pre-defined spatial locations, which allow them to be linearly direct-implemented. To update the

-components, all of the field quantities and derivatives at the right-hand side of (33) need to be evaluated at just as shown in (32). Similarly, for updating the -components and -components in the first column of (33), these quantities and , then need to be calculated at respectively. This makes the spatial indices impossible to be explicitly expressed for the field quantities and derivatives on the right-hand side of (33) and they are therefore only denoted in time domain. The iteration coefficients needed for implementation are then and , which only depend on the matrix elements in the plasma properties and the modeling parameters. For time-invariant homogeneous plasma, these coefficients are only needed to be calculated once. For inhomogeneous medium such as ionospheric plasma, these coefficients vary with height and position around the Earth, and additional calculations and storage are therefore required to account for the location-dependence of the parameters. However, these additional coefficients may

(32)

(33)

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be efficiently pre-calculated and stored before time-stepping to be used during the iteration. It is important to note that for this method, spatial-averaging is required for all of the spatially non-collocated state variables and derivatives to maintain the algorithm second-order through (33), accuracy. For example, to update all of the field quantities and derivatives at the right-hand side as mentioned of (33) need to be evaluated at above. However, according to our leapfrog scheme in Fig. 1, there are no pre-defined field values at this precise location , , , , , , , , , , for , , and . For the spatially non-collocated and components above, four neighboring diagonal field values are in order to find the field thus averaged about value at that position. As an example, the spatial averaging is illustrated in Fig. 2 and expressed as (34). For the of spatially non-collocated derivatives listed above, eight cubic diare then utilized agonal field values surrounding to evaluate the corresponding derivative at that point. As an example, the spatial averaging of derivative is illustrated in Fig. 3 and expressed as (35). All of the other spatially non-collocated field quantities are treated in a similar manner where spatial-averaging is needed. The iteration process of the whole system is realized by first updating the -field components through the three regular discretized FDTD equations from (15)–(17), scaling the -fields, then updating the - and -field components through (33), de-scaling the -fields, and finally repeating this process in the next iteration loop. C. Stability and Accuracy Analysis of the Scheme Unlike the H-J collocation method [20] and the algorithm in [21], the most appreciable advantage of this E-J collocation method is that the stability condition is independent of plasma properties, which is a crucial characteristic when modeling the ionosphere. The maximum stable Courant number in unmagnetized cold plasma for the E-J collocation method is unity just as

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Fig. 2. The illustration of the spatial averaging of J .

 Fig. 3. The illustration of the spatial averaging of (1H

=1z )

.

for free space [12]. For magnetized plasma medium, the effect of the anisotropy on the stability of the FDTD scheme is less predictable due to the complexity of the algorithm. It has been

(34)

(35)

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found empirically that the stability condition in the unmagnetized case is still valid for the magnetized case [18], which has been verified by our numerical results as well as the numerical experiments presented in [17]. Due to the semi-implicit differencing of the current density and must hold theoterms in (21)–(29), retically to maintain accuracy. However, as shown in the dispersion analysis of [12], the accuracy of this E-J collocation method is comparable to other differencing methods even for in unmagnetized cold plasma. Additionally, for both must hold the unmagnetized and magnetized cases, to approximate the analytical solutions of the original differential equations (21)–(29) since the homogeneous solutions of the semi-implicit differencing equations of (21)-(29) have a growth . Therefore, per time-step factor of for the magnetized case, a suitable spatial grid-cell size must be carefully chosen to satisfy both the stability condition and accuracy requirements. Criteria for choosing this grid-cell size may be obtained through dispersion analysis, however it is very complex due to the anisotropy and will not be shown here. In general, we see the E-J collocation method as having a major advantage over other algorithms, because the grid parameters may be chosen based on the Courant stability limit, and only further reduced to the level that provides an acceptable level of accuracy, as done in [18] for wave propagation in the ionosphere (as opposed to being forced to use a time step dependent on the plasma’s parameters [24] when implementing the method of [21], which results in a time step three orders of magnitude smaller than that imposed by the Courant limit [25]). III. VALIDATION OF THE ALGORITHM Having a fully 3-D cold plasma model, we choose to validate our FDTD algorithm by testing the Faraday rotation effect in a lossless electron plasma without ions. According to plasma theory [29], a linearly polarized plane wave propagating in a direction parallel to the direction of the applied magnetic field will be decomposed to a right-hand (RH) and a left-hand (LH) circularly polarized wave with different phase velocities. This causes the plane of polarization of the linearly polarized wave to rotate as the wave propagates through the plasma. The rotation can be written as angle per unit distance (36) where is the total rotation angle over a distance . and are the propagation constants for the LH- and RH- polarized wave, respectively. By construction (37) (38) and are the plasma frequency and cyclotron freHere, quency of the electrons, respectively. is the frequency of the linearly polarized plane wave. We test the Faraday rotation effect by sending an initially -polarized unit sinusoidal plane wave into the plasma. The

Fig. 4. The comparison of the simulated and analytical results of the Faraday rotation angle per meter.

wave propagation and the applied magnetic field are both along -coordinate. The simulation parameters are (39) (40) (41) (42) Each simulation is run for 1750 time steps and repeated for magnetic field values ranging from 1.0 to 1.7 Tesla. The electric field and are recorded at several distances away components from the source plane wave. The FDTD-calculated Faraday rofor each magnetic field value tation angle per unit distance is then given by (43) In Fig. 4, the simulation results of (43) are compared with the analytical results of (36). The FDTD simulation results are seen to be in very good agreement with the analytical results with an average error of 0.0031%. The average error is defined as (44) where and are the simulated and analytical Faraday rotation angles per unit distance, respectively, and, n is the sam. pling numbers. In our case To demonstrate the Faraday rotation effect, tracings of the total electric field vectors at different recording points along the case are illustrated in direction of propagation for the Fig. 5. These tracings show the rotation of the linearly polarized and wave, and they are obtained by plotting the recorded values over one cycle using the electric field magnitudes as the - coordinates. The initial plane of polarization of the linearly polarized wave is along -coordinate. As the wave propagates through the plasma, it starts rotating as shown in Fig. 5 with a constant rotation angle per unit distance stated in (36). Due to

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Fig. 5. Demonstration of the Faraday rotation effect.

the different phase velocities of the RH- and LH-circularly polarized wave as mentioned above, the traces of the total electric field vectors become more elliptical with the increasing propagation distance in the plasma. IV. NUMERICAL DEMONSTRATIONS OF ELECTROMAGNETIC WAVE PROPAGATION IN THE PLASMA In this Section, the 3-D FDTD cold plasma model is used to further demonstrate EM propagation characteristics in a lossless plasma medium. We again generate an -polarized Gaussian-pulsed plane wave propagating along the externally applied magnetic field. The Gaussian pulse is described by (45) We first model the plasma as having an electron density of 1.0 without ions under an applied 0.06T magnetic field. Thus the simulation is characterized by (46) (47) (48) (49) (50) (51) Fig. 6 illustrates the time and frequency domain waveforms of at the recording point located 40 cells away from the sourced plane wave. Clearly shown in the frequency domain waveform as in Fig. 6(b) is the slow whistler mode below the electron cyclotron frequency, and a resonance at the cyclotron frequency. The slow whistler mode is observed in the time-domain results in Fig. 6(a) as a low frequency oscillation arriving at about 600 time-steps. Above the stop band extending between the electron cyclotron frequency and the LHC cutoff frequency (for our case

Fig. 6. Waveforms for a Gaussian-pulsed plane wave propagating in the plasma with an applied magnetic field (a) Time Domain and (b) Frequency Domain.

of ), the LH- and RH-circularly polarized modes are observed with distinct cutoffs at and , respectively. These numerical results of our 3-D FDTD model agree very well with plasma theory. We next repeat the above experiment, but without the applied magnetic field. The solid lines in Fig. 7 illustrate the results of this second case having only electrons. The cutoff at the electron plasma frequency is clearly shown in the frequency domain waveform of Fig. 7(b). Further a long tail oscillating at the electron plasma frequency is observed in the time domain waveform of Fig. 7(a). This results from the very slow group velocities near the cutoff frequency. with As a final test, we introduce a positive ion species of atomic mass 32 into our model to study the impact of having a plasma comprised of both ions and electrons. The ion density is and all other parameters are kept the same as in 1.0 and the electron-only case above. The ion plasma frequency the total plasma frequency are then given by

(52) (53)

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observed in Fig. 7(b). Again, these numerical results agree with plasma theory very well. V. CONCLUSION AND ONGOING WORK We report a 3-D FDTD model of EM wave propagation in anisotropic magnetized cold plasma. This model is based upon the 2-D cylindrical FDTD model developed by Hu and Cummer [18]. In this work, we expand their 2-D, tightly coupled E-J collocation plasma method to the fully 3-D case. As a result, taking into account all three particle species (electrons, positive ions, and negative ions), our whole 3-D iteration system consists of 15 linear equations with 15 state variables. We use an equivalent set of explicit iteration equations to derive the FDTD iteration coefficients for these 15 linear equations (rather than deriving them analytically). Unlike for a 2-D plasma code, using our newly developed 3-D FDTD anisotropic plasma model, we are able to model such effects as Faraday rotation and complete 3-D spatial variations of the magnetized cold plasma. Our simulation results for Faraday rotation and EM propagation characteristics agree very well with plasma theory. Ongoing work includes extending the 3-D plasma algorithm developed here to the latitude-longitude [30] and geodesic [31] global 3-D FDTD models of the Earth-ionosphere waveguide. These global models could greatly improve simulation capabilities and results for a wide variety of applications, such as those described in [32] or [33]. APPENDIX Iteration Matrix A: see the matrix at the bottom of the page. Fig. 7. Waveforms for a Gaussian-pulsed plane wave propagating in the plasma without external magnetic field (a) Time Domain (b) Frequency Domain.

The dashed lines of Fig. 7 illustrate the results of this third test plasma case including both ions and electrons. The oscillating frequency of the time-domain tail has clearly increased from of the electron-only plasma case to as shown in Fig. 7(a) and to as the frequency-domain cutoff has also shifted from

Iteration Matrix B: see the first matrix at the top of the following page. Iteration Matrix C: see the second matrix at the top of the following page. ACKNOWLEDGMENT The authors gratefully acknowledge Prof. S. Cummer of Duke University and Dr. W. Hu for technical discussions relating to their 2-D FDTD plasma model. The computing

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support for this work was provided by the University of New Mexico High Performance Computing Center (HPCC). REFERENCES [1] K. Yee, “A numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, pp. 302–307, 1966. [2] A. Taflove and S. C. Hagness, Computational Electromagnetics: Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [3] R. J. Luebbers, F. Hunsberger, and K. S. Kunz, “A frequency-dependent finite-difference time-domain formulation for transient propagation in a plasma,” IEEE Trans. Antennas Propag., vol. 39, pp. 29–34, Jan. 1991. [4] L. J. Nickisch and P. M. Franke, “Finite-difference time-domain solution of Maxwell’s equations for the dispersive ionosphere,” IEEE Antennas Propag. Mag., vol. 34, pp. 33–39, Oct. 1992. [5] J. L. Young, “A full finite difference time domain implementation for radio wave propagation in a plasma,” Radio Sci., vol. 29, pp. 1513–1522, 1994. [6] J. L. Young, “A higher order FDTD method for EM propagation in a collisionless cold plasma,” IEEE Trans. Antennas Propag., vol. 44, pp. 1283–1289, Sep. 1996. [7] Q. Chen, M. Katsurai, and P. H. Aoyagi, “An FDTD formulation for dispersive media using a current density,” IEEE Trans. Antennas Propag., vol. 46, pp. 1739–1746, Oct. 1998. [8] J. Zhonghe et al., “Propagation of electromagnetic TM (S-polarization) mode in two-dimensional atmospheric plasma,” Plasma Sci. Tech., vol. 8, pp. 297–299, May 2006. [9] G. Bin, W. Xiaogang, and Z. Yu, “FDTD numerical simulation of absorption of microwaves in an unmagnetized atmosphere plasma,” Plasma Sci. Tech., vol. 8, pp. 558–560, Sep. 2006.

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[10] D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propag., vol. 44, pp. 792–797, Jun. 1996. [11] D. M. Sullivan, “Z-transform theory and the FDTD method,” IEEE Trans. Antennas Propag., vol. 44, pp. 28–34, Jan. 1996. [12] S. A. Cummer, “An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy,” IEEE Trans. Antennas Propag., vol. 45, pp. 392–400, 1997. [13] M. Cho and M. J. Rycroft, “Computer simulation of the electric field structure and optical emission from cloud-top to the ionosphere,” J. Atoms. Terr. Phys., vol. 60, pp. 871–888, 1998. [14] V. P. Pasko, U. S. Inan, T. F. Bell, and S. C. Reising, “Mechanism of ELF radiation from sprites,” Geophys. Res. Lett., vol. 25, pp. 3493–3496, 1998. [15] G. Veronis, V. P. Pasko, and U. S. Inan, “Characteristics of mesospheric optical emissions produced by lightning discharges,” J. Geophys. Res., vol. 104, pp. 12 645–12 656, 1999. [16] S. A. Cummer, “Modeling electromagnetic propagation in the earthionosphere waveguide,” IEEE Trans. Antennas Propag., vol. 48, pp. 1420–1429, 2000. [17] M. Thèvenot, J. P. Bérenger, T. Monedière, and F. Jecko, “A FDTD scheme for the computation of VLF-LF propagation in the anisotropic earth-ionosphere waveguide,” Ann. Télécommun., vol. 54, pp. 297–310, 1999. [18] W. Hu and S. A. Cummer, “An FDTD model for low and high altitude lightning-generated EM fields,” IEEE Trans. Antennas Propag., vol. 54, pp. 1513–1522, May 2006. [19] W. Hu, S. A. Cummer, and W. A. Lyons, “Testing sprite initiation theory using lightning measurements and modeled electromagnetic fields,” J. Geophys. Res., vol. 112, pp. 12645–12656, 2007. [20] J. L. Young, A. Kittichartphayak, Y. M. Kwok, and D. Sullivan, “On the dispersion errors related to (FD)2TD type schemes,” IEEE Trans. Microw. Theory Tech., vol. 43, pp. 1902–1909, 1995.

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[21] J. H. Lee and D. K. Kalluri, “Three dimensional FDTD simulation of electromagnetic wave transformation in a dynamic inhomogeneous magnetized plasma,” IEEE Trans. Antennas Propag., vol. 47, pp. 1148–1151, 1999. [22] L. Xu and N. Yuan, “FDTD formulations for scattering from 3-D anisotropic magnetized plasma objects,” IEEE Antennas Wirless Propag. Lett., vol. 5, pp. 335–338, 2006. [23] S. Liu and S. Liu, “Runge-Kutta exponential time differencing FDTD method for anisotropic magnetized plasma,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 306–309, 2008. [24] J. A. Payne, U. S. Inan, F. R. Foust, T. W. Chevalier, and T. F. Bell, “HF modulated ionospheric currents,” Geophys. Res. Lett., vol. 34, L23101, 2007. [25] Personal Communication Apr. 21, 2009. [26] T. W. Chevalier, U. S. Inan, and T. F. Bell, “Terminal impedance and antenna current distribution of a VLF electric dipole in the inner magnetosphere,” IEEE Trans. Antennas Propag., vol. 56, pp. 2454–2468, Aug. 2008. [27] F. Hunsberger, R. Luebbers, and K. Kunz, “Finite-Difference time-domain analysis of gyrotropic media-I: Magnetized plasma,” IEEE Trans. Antennas Propag., vol. 40, pp. 1489–1495, Dec. 1992. [28] S. J. H. Huang and F. Li, “FDTD simulation of electromagnetic propagation in magnetized plasma using z-transforms,” International Journal of Infrared Millimeter Waves, vol. 25, no. 5, pp. 815–825, May 2004. [29] F. C. Francis, Introduction to Plasma Physics and Controlled Fusion, 2nd ed. New York and London: Plenum Press, 1984. [30] J. J. Simpson and A. Taflove, “Three-dimensional FDTD modeling of impulsive ELF antipodal propagation and Schumann resonance of the earth-sphere,” IEEE Trans. Antennas Propag., vol. 52, pp. 443–451, Feb. 2004. [31] J. J. Simpson, R. P. Heikes, and A. Taflove, “FDTD modeling of a novel ELF radar for major oil deposits using a three-dimensional geodesic grid of the earth-ionosphere waveguide,” IEEE Trans. Antennas Propag., vol. 54, pp. 1734–1741, Jun. 2006. [32] J. J. Simpson and A. Taflove, “A review of progress in FDTD Maxwell’s equations modeling of impulsive sub-ionospheric propagation below 300 kHz,” IEEE Trans. Antennas Propag.: Special Issue on Electromagn. Wave Propa. Complex Environments: A Tribute to Leopold Benno Felsen, vol. 55, no. 6, pp. 1582–1590, Jun. 2007.

[33] J. J. Simpson, “Current and future applications of full-vector 3-D Maxwell’s equations FDTD global earth-ionosphere waveguide models,” Surveys Geophys., vol. 30, no. 2, pp. 105–130, 2009. Yaxin Yu received the B.S. degree in physics from Northwest University, China, in 2000, the M.S. degree in optics from Nankai University, China, in 2003, and the M.S. degree in electrical engineering from the University of New Mexico, Albuquerque, in 2006, where he is currently working toward the Ph.D. degree. His research interests include semiconductor optoelectronics, especially the III–V compound semiconductor materials and devices, and the finite-difference time-domain (FDTD) solution of Maxwell’s equations. His current research focuses on FDTD simulation of electromagnetic wave propagation in ionosphere.

Jamesina J. Simpson (S’01–M’07) received the B.S. and Ph.D. degrees from Northwestern University, Evanston, IL, in 2003 and 2007, respectively. She joined the Electrical and Computer Engineering Department, University of New Mexico, Albuquerque, as an Assistant Professor in August 2007. Her research focuses on the finite-difference time-domain (FDTD) solution of Maxwell’s equations. To date, her work has spanned applications ranging from geophysically induced electromagnetic propagation and phenomena in the Earth-ionosphere system, to electromagnetic compatibility issues arising in compact portable electronic devices and to optical interactions with living tissues. Dr. Simpson is a member of Tau Beta Pi and received the National Science Foundation Graduate Research Fellowship, Walter P. Murphy Fellowship, IEEE AP-S Graduate Research Award, and IEEE MTT-S Graduate Fellowship to support her graduate studies. She was also awarded the 2007 Best Ph.D. Dissertation Award from the Northwestern Electrical Engineering and Computer Science Department.

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Calculation of the Impedance Matrix Inner Integral to Prescribed Precision John S. Asvestas, Life Fellow, IEEE, Stephen Paul Yankovich, and Oliver Eric Allen, Senior Member, IEEE

Abstract—We present a new method for evaluating the inner integral of the impedance matrix element in the traditional Rao-Wilton-Glisson formulation of the method of moments for perfect conductors. In this method we replace the original integrand (modified by a constant phase factor) by its Taylor series and keep enough terms to guarantee a number of significant digits in the integration outcome. We develop criteria that relate the number of Taylor terms to the number of required significant digits. We integrate the leading Taylor terms analytically and the rest through iteration formulas. We show that the iteration formulas converge for all observation points within a sphere with a radius of half-a-wavelength and center the triangle’s centroid. We compare results of our method with existing ones and find them in excellent agreement. We also outline a procedure for using cubatures outside the region of convergence. Index Terms—Boundary-integral equations, cubatures, Gordon-Bilow transformation, impedance-matrix, method of moments (MoM), numerical integration, significant digits, Taylor’s theorem with a remainder.

I. INTRODUCTION INCE the introduction of the Rao-Wilton-Glisson functions [1] in the method of moments (MoM), a considerable volume of research has appeared addressing the computation of the impedance matrix (IM) elements. An excellent review on the subject and the various approaches used in evaluating the surface integrals of the elements, as well as bibliographical information, is found in the article by Khayat and Wilton [2]. Bibliographies appear also in Taylor [3], and in Ylä-Oijala and Taskinen [4]. Recent publications include [5] and [6], with their titles accurately describing the focus of the work. In none of the methods to-date is it possible to ascertain the accuracy with which the surface integrals that comprise the matrix element are computed. Till recently, the quantities of interest in electromagnetic simulations were the scattered or radiated far fields. From these, we could compute the scattering cross section of a target or the radiation pattern of an antenna. This allowed for a considerable degree of error in computing the elements of the IM because of

S

Manuscript received December 10, 2008; revised July 24, 2009. First published December 04, 2009; current version published February 03, 2010. J. S. Asvestas and O. E. Allen are with the NAVAIR, Radar and Antenna Systems Division, Code 4.5.5, Patuxent River, MD 20670 USA (e-mail: john. [email protected], [email protected]). S. P. Yankovich (deceased) was with the NAVAIR, Radar and Antenna Systems Division, Code 4.5.5, Patuxent River, MD 20670 USA. Digital Object Identifier 10.1109/TAP.2009.2037703

the error smoothing effect of the near- to far-field transformation (integration). Moreover, due to computer hardware limitations, the size of the IM was small enough so that round-off error did not have a severe effect on accuracy. In recent years, the MoM has been applied to problems where near-field information is required (e.g., input impedance of an antenna). This necessitates a more accurate computation of the elements of the IM than when only the far fields are of interest. Additionally, advances in computer hardware allow us to address problems that result in a system with millions of unknowns. Thus, the effect of the round-off error becomes more pronounced. Both reasons lead us to the conclusion that the more accurate the representation of the IM is, the better the quality of the solution. This is the central theme of our paper: the computation of the inner surface integral of an IM element to prescribed precision when the platform of interest is perfectly conducting. Our method relies on approximating the exponential function present in the integrand by a polynomial of the same argument. Before doing so, we extract from the exponential a constant phase term related to the observation point (OP). This minimizes the variation of the exponential over the integration triangle. The relative error involved in replacing the exponential by a polynomial is computable. This allows us to compute the degree of the polynomial required to produce agreement with the exponential to a specified number of significant digits (SD) (Section II). The resulting integral can now be evaluated term-by-term. The evaluation process is iterative: the first one or two terms are evaluated explicitly and the rest through two recursion formulas (Section III). We show that these formulas do not result in converging sequences for all permissible values of the OP but that they both converge for OPs within half a wavelength from the triangle’s centroid. Thus, the proposed method is valid for all OPs in a sphere centered at the centroid of the integration triangle and of radius less than half-a-wavelength (Section IV). We validate the present method by comparing with results in [2] and find the agreement to be excellent (Section V). In Section VI, we outline a strategy for using the present method and cubatures1 to obtain a specified precision for the OP anywhere in space. In conclusion, we mention that we have prepared a report [7] where we present the contents of this paper in detail. The second part of this report is especially useful because it provides all the details we have omitted in Section VI. 1By cubature we mean a numerical method for evaluating a surface integral; we reserve the term quadrature for a numerical method for evaluating a line integral.

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where is the exterior unit normal to the side of a triangle, lying on the plane of the triangle and is the projection of the OP onto the triangle’s plane. The splitting of the integral as in the first line above appears also in [8] for the OP on the triangle. The conclusion in (4), however, is original and valid for all positions of the OP. We proceed one step further and measure all distances in wavelengths. We note that

(5) when we move from measuring length in meters to length in wavelengths. With this in mind, we get from (4)

Fig. 1. The integration triangle.

II. STATEMENT OF THE PROBLEM AND APPROACH In the Rao-Wilton-Glisson [1] formulation of the MoM, we need evaluate the integrals

(1) where

(2) with

the integration point and

the OP

(3) and the region of integration being the triangle in Fig. 1. This triangle lies on the -plane and its centroid is the origin of coordinates. Boldface letters denote vectors. The same letters in italics denote the magnitudes of these vectors while a caret over a letter denotes a unit vector. The vector is the position vector to the -the vertex of the triangle, as shown in Fig. 1. We note that

(6) We note that both integrals are independent of the index , and that, although the first one is a vector integral, it is in essence a vector sum of three scalar integrals, each defined over a side of the triangle. The objective here is to calculate the integrals in (6) to a pre-assigned number of SD. Neither of these integrals can be evaluated analytically. The integrands, however, can be replaced by polynomials that we know how to integrate and that can approximate the actual integrands arbitrarily closely. The obvious way to proceed is to expand the exponential function in a Taylor and truncate after the required number of series about terms. This works well when is small, so that the number of terms for agreement to the required number of SD is also small. This means that the OP must be near the integration triangle. The OP, however, can be any point in space; hence, as the OP recedes from the triangle, more and more terms are required in the Taylor sum and there comes a point when this method becomes counter-productive due to the large amount of time required to compute this sum. We attempt to get around this difficulty by multiplying and dividing (6) by a common factor; we thus write

(7) We can show that [7]

(8)

(4)

where is the length of the longest side of the triangle. Though this is not a strict bound, it is a bound that holds for all OPs; thus, no matter how far away the OP is from the integration triangle, the number of terms in the expansion will be the same as for a point near the triangle. Moreover, for an actual grid, we can search among all triangles for the longest side and

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use that value in (8). In this way, we do not have to test all triangles separately, achieving substantial computational savings. We proceed now to expand the exponential in the integrand of (7) in a Maclaurin series and keep the first terms

radians or 24 deargument is approximately equal to grees. This means that the trigonometric functions in the original integrals hardly exhibit an oscillatory behavior. The argument for the cosine function is slightly more complicated and is driven by the fact that the integral of the leading term of the first sum in (9) is equal to zero. Thus, the first contributor to the integral is the second term in the cosine expansion. We then proceed to rewrite the relevant integral in the form

(9)

(14)

In determining the number of required terms, , to produce a certain accuracy we use Taylor’s Theorem with a Remainder of correct SD, then ([9, p. 113]). If we require a number we proceed as follows to determine . The series for the sinc function is

and obtain the following condition for determining

(15)

III. EVALUATION OF INTEGRALS

(10)

Since the leading term of the first sum in (9) is zero, we can rewrite this expression in the form

is the remainder of the series and is bounded by the where first omitted term

(16)

(11) We can then write for the relative error

where

(17) We employ the Gordon-Bilow transformation to convert the surface integral to a line integral [7], [10], [11]. The result is (12) (18) The number of terms is determined by requiring that the relative error is smaller than 5 divided by the number 10 raised to the number of SD plus one, or

where

(13) (19) In this discussion we have assumed that the argument of the sine function is small; in practice, the longest side of a triangle does not exceed one tenth of a wavelength and, hence, the largest

where is the th side of the triangle and is the exterior unit normal to it, lying on the triangle’s plane (see Fig. 2), while is the distance between the integration point on the triangle

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Fig. 2. Local rectangular coordinates and various vectors.

Fig. 4. Geometrical meaning of the quantities in (24).

by dropping the normal from the OP to the th side We obtain by taking its projection onto the triangle’s of the triangle and plane (see Fig. 4). From (21)

Fig. 3. Definition of

$: $ =

j

r0



j

with

 = r hz^. 0

(25) and the projection of the OP Fig. 3). In this notation

onto the triangle’s plane (see

with

(20) (26)

where

(21)

(27)

From (19), we obtain the iteration

(22) (23) Thus, we only need evaluate (19) for ration for evaluating (21), we define

with the sign when (Region 1) and (Region 2). These the—sign when two regions of space are defined by two infinite planes, both perpendicular to the th side and each containing one endpoint of the side. The region between the two planes (Region 3) is , . In this region defined by

, and (21). In prepa-

(24)

(28)

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In all three regions we have the recursion formula

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Substitution in (23) leads to the characteristic equation

(34)

(29) whose roots are where, see (30) at the bottom of the page. can be found in [7]. The The details of the evaluation of resulting expressions are

(35) We thus have the two solutions

(36) According to [13, Theorem 7.2.9.6], a necessary and sufficient condition for the two solutions to satisfy the stability condition

(37)

(31) and the lower for

the upper sign for ; moreover

and, if equality holds, then the root must be is that simple. This implies that

(38) The two roots are equal when tions are stable provided must have

, and, in this case the solu. Thus, for stable solutions we

(39) (32) We have also derived expressions for small values of can be found in [7].

. They

IV. REGIONS OF CONVERGENCE OF RECURSION FORMULAS

We plot this relationship in Fig. 5. The stability analysis for (29) is motivated by comments in [14, p. 142]. The homogeneous part of the difference equation is

(40)

According to [12, p. 371], the stability of the solutions of the difference equations (23) and (29) depends on the homogeneous parts of these equations. For (23) we seek homogeneous solutions of the form

The coefficient here is no longer a constant but depends on the independent variable. We can, however, proceed as follows in our test for stability. We let

(33)

(41)

(30)

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By construction,

is non-negative. But

(48)

Fig. 5. Relation between jhj and  for stability. The stable region is the one below the curve and bounded by the two axes. The point (1, 0) is excluded.

and treat

as a constant. The characteristic equation is (42)

with solutions

since the OP and any of the three vertices of the triangle are inside the sphere. From (47) and (48) we conclude that (44) is satisfied. Thus, our method does not work in the entire observation space. It is guaranteed, however, to work when the OP is less than half a wavelength away from the triangle’s centroid. We do not know whether the integrals can be evaluated in a way that removes this obstacle. This is, however, an academic question since outside this sphere we can use cubatures to obtain the same precision in less time. At least, this has been our experience in doing numerical experiments for 7-SD precision. A more meaningful question we may ask is whether it is necessary to introduce the exponential factor as we did in (7). If we break the integrands in (6) and (7) into sines and cosines and expand these in a Maclaurin series, we find that at observation distances of half a wavelength from the centroid, we need at least twice as many terms in (6) as in (7) for a specified precision ranging from 4 to 15 SD. Details appear in [7] where a simple argument shows that the unaltered series is competitive with the current method only for the OP at the centroid of the triangle. V. VALIDATION We compare here results of our method with those of Khayat and Wilton [2]. They compute the second (scalar) integral in (6), i.e., the integral of the free-space Green’s function

(43) Since , for the solutions and be stable, we need to be less or equal to one

to

(44) Conditions (39) and (44) do not imply one another. We can show, however, that both are satisfied within a sphere centered at the centroid of the triangle and radius equal to 0.5. The proof is simple: we note that (45) from which we get that

(46) so that (39) is satisfied. We also note that

(47)

(49) The integration triangle is shown in Fig. 6. It is an isosceles right triangle whose equal legs have length 1 m. The wavelength is equal to 10 m. The results of the comparison are shown in Tables I and II. The four OPs lie on the right angle’s bisector. Their coordinates are given with respect to an origin located at the vertex of the right angle, as in [2]. According to (13) and (15), the number of terms required for fifteen SD is fourteen (total). The Khayat and Wilton results come from Table II in [2], with a reported accuracy of 14 SD. In Table I we exhibit the real part of (49) while in Table II the imaginary. In Table III, we display the remaining OPs computed in [2] to 14 SD. In this case, we move away from the OP of Case 1 along the normal to the triangle and compute the integral at three OPs. The integrand now is not singular as in the first four cases but the proximity of the OP to the integration triangle makes it behave as if it were. We find excellent agreement in general, except for the real part of Case 5 where we note a significant difference. We placed a fine grid around this point and found our result to lie on this curve. In an exchange of correspondence, the authors of [2] verified the correctness of our result using both the original approach and a modified one [15]. This concludes our validation. We have also used cubatures to test our method and we obtained good agreement [7].

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TABLE I COMPARISON OF PRESENT APPROACH TO RESULTS IN [2]. REAL PART OF THE INTEGRAL IN (49). OP STANDS FOR OBSERVATION POINT. IT IS INSIDE THE TRIANGLE AND ITS COORDINATES ARE GIVEN WITH RESPECT TO AN ORIGIN CENTERED AT THE RIGHT-ANGLE VERTEX

TABLE II COMPARISON OF IMAGINARY PART OF (49). EVERYTHING ELSE AS IN TABLE I

TABLE III OBSERVATION POINT NEAR INTEGRATION TRIANGLE. OBSERVATION POINT COORDINATES (0.1, 0.1, z )

Fig. 6. The Khayat-Wilton triangle.

VI. CUBATURES Two questions we can ask is (a) whether, for OPs within half a wavelength from the centroid, there is a cubature that can provide the same accuracy faster than the method above, and (b) what size cubature we should use in the rest of space. We have addressed both questions in Part 2 of [7] and we summarize the results here. Although the procedure we use is very general (in the sense that every parameter is a variable), we use 7 SD to demonstrate it in [7]. We chose this number because it is almost the arithmetic mean between 0 SD and 15 SD (double precision) and corresponds to single precision (32 bits) in the IEEE-754 floating-point standard [16]. All statements below refer to this precision.

As a first step we timed the cubatures against the present approach and found that cubatures with more than 21 points are slower than our method. We next addressed the issue of whether cubatures of 21 or fewer points can be employed for OPs within the sphere with center the centroid and radius of half a wavelength. We designed an elaborate experiment involving 25 different triangles and 10,000 OPs (lying on a hemisphere above the triangle). We computed 7 complex numbers, the scalar integral in (49) and the three vector integrals in (1). We called a failure if, for a given OP, one of the complex numbers is accurate to fewer than 7 SD. We then defined a failure rate for a cubature over a single triangle in a rather complex way that is explained in detail in [7]. We also defined a minimum (smallest number of failures over a triangle) and maximum (greatest number of failures over a triangle) failure rate, and an average failure rate (failure rate averaged over the 25 triangles). For OPs inside the half-wavelength sphere but not on the triangle, we ran our method along with those cubatures that are faster than it. We determined where those cubatures provide the required accuracy by comparing their answer to that of our method. Based on this, we defined a new sphere, concentric with the old one but of radius less than that of the old one. For OPs outside the new sphere, we ordered all cubatures according to increasing size (number of points employed in cubature). We then defined a convergence criterion for the sequence of cubatures [7]. For a judiciously selected set of OPs and for all triangles, we ran the sequence of cubatures. Using the results, we defined another sphere, concentric with the previous one but of a larger radius. In the space between the two spheres, we chose the smallest cubature that has converged and we did the same outside it. We show a typical example of this strategy in Fig. 7. For a maximum failure rate of 1% or less, we see that

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[4] P. Ylä-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n RW G functions,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1837–1846, 2003. [5] S. Järvenpää, M. Taskinen, and P. Ylä-Oijala, “Singularity subtraction technique for high-Order polynomial vector basis functions on planar triangles,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 42–49, 2006. [6] H. Zhou, W. Hong, and G. Hua, “An accurate approach for the calculation of MoM matrix elements,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1185–1191, 2006. [7] J. S. Asvestas, S. Yankovich, and O. E. Allen, Calculation of Impedance Matrix Inner Integral to Prescribed Precision Patuxent River, MD, 2008, NAVAIR Report No. NAWCADPAX/TR-2008/227. [8] D. R. Wilton et al., “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag., vol. AP-32, no. 3, pp. 276–281, 1984. [9] T. M. Apostol, Mathematical Analysis, 2nd, Ed. Reading, MA: Addison Wesley, 1974. [10] W. A. Gordon and H. J. Bilow, “Reduction of surface integrals to contour integrals,” IEEE Trans. Antennas Propag., vol. 50, no. 3, pp. 308–311, 2002. [11] J. S. Asvestas and H. J. Bilow, “Line-integral approach to computing impedance matrix elements,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2767–2772, 2007. [12] F. Dahlquist and Å. Björck, Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall, 1974. [13] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis. New York: Springer-Verlag, 1980. [14] W. H. Press et al., Numerical Recipes. Cambridge: Cambridge University Press, 1986. [15] M. A. Khayat, D. R. Wilton, and P. W. Fink, “An improved transformation and optimized sampling scheme for numerical evaluation of singular and near-singular potentials,” IEEE Antennas and Wireless Propag. Letters., to be published. [16] IEEE Standard for Floating-Point Arithmetic, (IEEE Std 754-2008), 2008.

2

Fig. 7. A possible zoning strategy for 7 SD based on a less than 1% MaxFR.

the present method is faster than any cubature for all OPs that are within three tenths of a wavelength from the triangle’s centroid; that for OPs between three tenths of a wavelength and two wavelengths from centroid, a 21-point cubature is the minimum cubature that will provide 7 SD; and that for OPs farther than two wavelengths, a 15-point cubature is sufficient to provide this accuracy.

VII. CONCLUSION We have introduced a new method for computing the inner integral of the impedance elements in the Rao-Wilton-Glisson [1] formulation of the method of moments in electromagnetics (Section II). The distinguishing feature of this method is that it can compute the integral to a prescribed precision. We know of no other method that can do this. The method is valid for all OPs that lie within a sphere with center the triangle’s centroid and radius of one half of a wavelength (Section IV). This restriction is due to the fact that the two iteration formulas we use do not converge everywhere in the observation space but have a common domain of convergence in the interior of this sphere. There are also points outside this sphere where we have convergence; they are of no consequence, however, since we can use cubatures that are faster than our method and yield a prescribed accuracy outside this sphere.

REFERENCES [1] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, no. 3, pp. 409–418, 1982. [2] M. A. Khayat and C. R. Wilton, “Numerical evaluation of singular and near-singular potential integrals,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3180–3190, 2005. [3] D. J. Taylor, “Accurate and efficient numerical integration of weakly singular integrals in Galerkin EFIE solution,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1630–1637, 2003.

John S. Asvestas was born in Athens, Greece. He received the B.S.E., M.S.E., and Ph.D. degrees in electrical engineering from The University of Michigan, Ann Arbor, in 1963, 1965, and 1968, respectively. He has worked at The University of Michigan Radiation Laboratory, the Technical University of Denmark Applied Mathematical Physics Laboratory, the University of Delaware Mathematics Department, the Radar Systems Group of Hughes Aircraft Company (currently Raytheon), and the Corporate Research Center of Grumman Corporation (currently Northrop/Grumman). He presently works in the Radar and Antenna Systems Division, NAVAIR, Patuxent River, MD. His main interest is in analytical and computational electromagnetics.

Stephen Paul Yankovich (deceased) received the B.S. degree in mathematics from Drexel University, Warminster, PA. in 1980. Following graduation, he worked at Burroughs Corporation, Philadelphia, PA, as a Mathematical Programmer from 1980 to 1984. Beginning in 1984, he worked at the Naval Air Development Center, Warminster, supporting research in acoustic anti-submarine warfare. In 1996, he relocated to the Naval Air Warfare Center, Patuxent River, Maryland. His most recent work focused on mathematical analysis of computational electromagnetics problems, underwater acoustic and hydrodynamic propagation modeling, and radar signal processing. Mr. Yankovich passed away unexpectedly in June 2009. Mr. Yankovich received both the NAVAIR Commanders Award and Dr. Delores M. Etter Top Navy Scientists and Engineers of the Year Award in 2009.

ASVESTAS et al.: CALCULATION OF THE IMPEDANCE MATRIX INNER INTEGRAL TO PRESCRIBED PRECISION

Oliver Eric Allen received the B.S.I.E. degree from the University of Wisconsin-Madison, in 1982, the B.E.E. degree from Johns Hopkins University, Baltimore, MD, in 1989, the M.S.E.E. degree from the University of Colorado-Boulder, in 1992, and the D.Sc.E.E. degree from the George Washington University, Washington, DC, in 2004. From 1982 to 1985, he was detailed to the Fleet Analysis Center, Corona, CA, the Naval Air Rework Facility, Norfolk, VA, and the Naval Electronics Systems Engineering Activity, St. Inigoes, MD, while he was assigned to the Naval Material Command, Washington, DC. From 1985 to 1987, he was with the Naval Air Test Center, Anti-Submarine Aircraft Test Directorate, Patuxent River, MD. In 1987, he transferred to the Electromagnetic Environmental Effects Division of the Naval Air Warfare Center. During

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the 1989–1990 academic year, he conducted research at the National Institute of Standards and Technology, Boulder, CO, while attending the University of Colorado. In 1997 he served a two-year assignment as the Science and Technology Advisor to the Commander of the U.S. Seventh Fleet in Yokosuka, Japan. From 1999 to 2003, he was detailed to the Office of Naval Research, Ocean, Atmosphere and Space Department in Arlington, VA, directing research in nonacoustic anti-submarine warfare. He is currently the Chief Radar Systems Engineer at the Naval Air Systems Command, Patuxent River, MD. He is a senior member of the IEEE, a member of Tau Beta Pi Engineering Honor Society and a NAVAIR Associate Fellow. Dr. Allen has received the Meritorious Civilian Service Medal, the Fleet/Force Rapid Technology Insertion Award, the NAVAIR Commanders Award and Dr. Delores M. Etter Top Navy Scientists and Engineer of the Year Award.

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Method of Moments Solution of Electromagnetic Scattering Problems Involving Arbitrarily-Shaped Conducting/Dielectric Bodies Using Triangular Patches and Pulse Basis Functions Anne I. Mackenzie, Member, IEEE, Sadasiva M. Rao, Fellow, IEEE, and Michael E. Baginski, Senior Member, IEEE

Abstract—We present a new method of moments solution procedure for calculating electromagnetic scattering and radiation from conductor/dielectric composite structures. The solution is obtained using triangular patch modeling and a recently developed pair of mutually orthogonal pulse basis functions to represent the equiva. The pulse basis functions are defined with lent currents and respect to the edges in the triangulated model and partially spread over the triangles connected to the edge. The orthogonality of the basis functions allows the development of stable solutions for all formulations and configurations investigated. A set of numerical results is presented that illustrates the efficacy of the present approach. Index Terms—Basis functions, conducting bodies, dielectric bodies, electromagnetic scattering, scattering.

I. INTRODUCTION ECENTLY there has been an increased interest in calculating the electromagnetic scattering and radiation associated with composite structures consisting of dielectric and conducting materials. This is a direct result of the rapid growth of the use of composites for aircraft design and conformal antenna fabrication in the military and civilian sectors. Traditionally, the electromagnetic characterization of these types of configurations has relied on surface integral equation techniques and has been solved using the familiar method of moments (MoM) algorithm. MoM solutions are ideal for such problems since the problem formulation and the solution may be confined to the object’s surface [1], [2]. In [3], [4], a solution for the composite problem involving both conducting and dielectric objects was developed using the Electric Field Integral Equation (EFIE). There the authors use the triangular patch modeling scheme and express , the equivalent electric current, using the well known RWG basis functions [5] and , the equivalent magnetic current, as functions, where represents the unit outward vector normal to the surface of the object. Since only

R

Manuscript received April 13, 2009; revised July 24, 2009. First published December 04, 2009; current version published February 03, 2010. A. Mackenzie is with the Electromagnetics & Sensors Branch, NASA Langley Research Center, Hampton, VA 23681-0001 USA (e-mail: [email protected]). S. Rao and M. Baginski are with the Electrical & Computer Engineering Department, Auburn University, Auburn, AL, 36849-5201. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037839

the EFIE formulation is used, the algorithm presented in [3], [4] generates erroneous solutions near the object’s characteristic (resonant) frequencies and, hence, is limited to low-frequency problems. To alleviate the instabilities that occur at or near the characteristic frequencies, a new algorithm was developed [6], [7] in which the conducting bodies are handled using either the EFIE or the Combined Field Integral Equation (CFIE), depending on whether the body is open or closed, and the dielectric bodies are handled using a Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation. Although the formulation presented in [6], [7] is general, the algorithm generates good solutions only when the conductors and dielectric bodies do not share a common surface. This major drawback makes the work of [6], [7] of limited value for many practical problems. The main difficulty of the work presented in [6], [7] can be attributed to the use of identical basis functions for and . When testing functions and basis functions are all vectors having the same direction, the curl operation in the integral equations results in zero-valued or very small diagonal matrix terms [8]. This results in an unstable numerical solution. Sheng et al. [9] have shown numerous examples of incorrect dielectric RCS’s that were calculated using identical RWG basis functions for and and the testing functions. Further, Sheng et al. [9] have proposed a new testing scheme to overcome the problem, but the scheme is numerically intensive. Wu [10] and Djordevic and Notaros [11] have demonstrated other types of basis functions to solve dielectric and composite problems by surface integral, MoM approaches. Wu used harmonic basis functions for bodies of revolution; Djordevic and Notaros used higher order polynomial basis functions for arbitarily-shaped bodies. In this work, we present the method of moments solution using triangular patch modeling and recently developed mutually orthogonal pulse basis function pairs to represent the equivalent currents and [8]. These basis functions allow the development of stable solutions for all configurations. A set of numerical results is presented that illustrates the efficacy of the present approach. II. SURFACE INTEGRAL EQUATION FORMULATION Consider an arbitrarily-shaped conductor along with an arbitrarily-shaped dielectric body as shown in Fig. 1. Although the bodies are shown distinct, this need not be the general case. If

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Fig. 3. Internal problem. Fig. 1. Composite body problem.

The interior equivalent situation is shown in Fig. 3. Since this case concerns the interior region, the material parameters must be kept the same as those in the interior region of the original problem. However, the external region has no restrictions. The medium outside is given the same material parameters as the region inside. By enforcing the continuity of tangential fields across the dielectric interface , we have

Fig. 2. External problem.

the conductor and the dielectric body are joined together, then we treat them as two bodies with an infinitesimally thin layer separating them. and denote the surfaces of a conducting/dielectric Let scatterer illuminated by an incident electromagnetic plane wave. The regions exterior and interior to the dielectric body have constitutive parameters ( , ) and ( , ), respectively. Furthermore, we assume that the dielectric is a closed body so that a unique outward normal vector can be defined unambiguously. By invoking the equivalence principle [12], we formulate two equivalent problems, one valid for regions external and one internal to the dielectric material. In the equivalent problem external to the dielectric region, the dielectric surface is replaced by a fictitious mathematical surface and all regions are filled by the homogeneous material having the constitutive parameters of the external region as shown in ) is the sum of the incident Fig. 2. The total external field ( field ( ) and the scattered field ( ) from the object. Also, since the equivalence is valid only for the external region, the internal field may be conveniently set to zero. Two equivaand are allowed to flow on the surface to lent currents ensure the continuity of tangential electric and magnetic fields and , where is the given by outward unit vector normal to the surface . By enforcing the boundary conditions for the electric and magnetic fields at the conducting interface , we derive the following equations:

(5) (6) where the subscript denotes the dielectric medium where the scattered fields are evaluated. Adding (1) and (2), we develop a CFIE for the conducting bodies, given by

(7) where is a constant between 0 and 1 and is the wave impedance of the external medium. We also note that when the conducting body is open. However, for the dielectric region, one can develop several different formulations by combining (3)–(6) in an appropriate manner. By choosing (3) and (5) or (4) and (6) only, one develops either (EFIE) or (HFIE) formulations. Either of these two formulations along with (7) can be solved to obtain , , and using MoM as presented in the following sections. Unfortunately, such formulations are prone to erroneous solutions near characteristic frequencies associated with the space occupied by the dielectric body, commonly referred as the internal resonance problem [13]. To alleviate the internal resonance problem, one may develop a CFIE formulation analogous to (7) as

(1) (2) In (1) and (2), the subscript denotes the medium in which the scattered fields are computed. Similarly, by enforcing the boundary conditions of the electric and magnetic fields across the dielectric interface , we derive the following equations:

(3) (4)

(8) (9) In a similar way, several other combined field formulations such as PMCHWT and Müller’s formulation [14] can be created and may be solved using a MoM solution as presented in the following sections.

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Fig. 5. Definition of testing functions.

Fig. 4. Triangle pair to define basis functions.

III. DESCRIPTION OF BASIS AND TESTING FUNCTIONS IN MOM PROCEDURE In this work, a given composite structure is approximated by planar triangular surface patches. Triangular patches easily conform to any surface or boundary and permit an easy description of the entire region for numerical analysis. The grid density can be adjusted to allow greater densities for portions of the surface where higher resolution is desired. Next, using the pair of mutually orthogonal, pulse-type basis functions defined in [8], we approximate the currents , , and as

IV. NUMERICAL SOLUTION PROCEDURE Defining , we can test (7)–(9) using the testing vectors defined previously to obtain

(13)

(14) (15) (10) and represent the number of edges in the trianIn (10), gulated model of the conducting and the dielectric object, respectively. Thus, the total number of unknowns in the MoM . Further, referring to Fig. 4, we procedure is and in (10) as follows: describe the Let and represent two triangles connected to the edge of the triangulated surface model. The edges of each triedge are referred as free edges. Thus, angle other than the we have

for . We note that, from (13)–(15), one needs to evaluate the quan, , , and tities for an basis function when using the MoM procedure to develop the overall solution scheme. Once these quantities are evaluated accurately, it is a trivial matter to generate the solution to any desired formulation. We also note that these quantities are evaluated thoroughly in [8] and, in the following, we only outline the important steps. Also, we drop the subscripts on , , and for the sake of clarity. A. Evaluation of

otherwise

otherwise

(11)

We note that

(16) (17)

(12)

defines a unit vector along the edge, , a unit where , and , the dovector normal to the plane of the triangle is the region whose perimeter main of the basis functions. is drawn by connecting the mid-points of the free edges to the and to the nodes of edge as shown in centroids of triangles is 2/3 of the total triangular patch the shaded area in Fig. 4. area. and For the testing scheme, we define vectors associated with edge , as shown in Fig. 5. The vector extends from the centroid of triangle to the edge midextends from the edge midpoint to the centroid of point; . Vector extends from the beginning to the end of triangle edge . The vector is used to test the electric field , while is used to test the magnetic field .

and

where

(18) (19) (20) (21) (22) (23)

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In (18)–(21), and , 2 are the permeability and permittivity of the surrounding medium, denotes the source surface, and is the wave number for each region. The superscripts on and indicate the location of the field points associated with each quantity; these points lie at the ends of the testing vectors shown in Fig. 5. The vectors and are position vectors to observation and source points, respectively, from a global coordinate origin. , , The mathematical and numerical steps to evaluate , and are discussed in detail in [8] and hence are not repeated here. B. Evaluation of

and

We note that

Fig. 6. Geometries for which bistatic RCS was calculated, including a) two spheres, b) a disk/cone, and c) a missile. a) = Two spheres. b) Disk/cone. c) Missile.

(24)

(25)

where the symbol

identifies the principal values of the inte-

Fig. 7. Geometry of a dielectric cube capped by conducting plates.

grals. Again, (24) and (25) are evaluated using the procedures described in [8]. V. NUMERICAL EXAMPLES The scattering solution using an orthogonal pair of pulse basis vectors is demonstrated by calculating the bistatic RCS for four composite cases: a dielectric sphere close to a PEC sphere, a dielectric cone capped with a PEC disk, a missile composed of a dielectric nose cone and a PEC cylinder, and a cube capped with PEC plates at opposite ends. The geometries are shown in Figs. 6 and 7; the EFIE solution method was employed for all four cases. In addition, the two-spheres problem was also solved by using the HFIE. We choose to present the numerical results for EFIE and HFIE because obtaining accurate solutions for these formulations is a prerequisite for obtaining accurate results for any other formulation for a closed body. As mentioned earlier, it is relatively a simple matter to combine EFIE and HFIE to obtain solutions for other formulations as well. In the first problem, a dielectric sphere of radius 0.2 and and a PEC sphere of radius 0.3 are situated on the -axis. There is a gap of 0.1 between them, and a plane wave direction impinges on the dielectric sphere traveling in the first. The dielectric sphere mesh has 324 edges, or , while the conducting sphere mesh has 750 edges, or . The bistatic RCS results are shown for the pulse basis MoM EFIE and HFIE solutions and compared to a body of revolution (BOR) MoM solution in Fig. 8. The vertical axis represents RCS normalized by the region 1 wavelength.

Fig. 8. Bistatic RCS for two non-touching spheres, one dielectric,  one PEC.

= 4, and

In the second problem, a dielectric cone has , , and . The circular end of the cone is covered by a PEC disk. Because the EFIE solution was chosen and the disk was PEC, it was allowable to model the disk as an open body in contact with the closed dielectric cone in the figure, thus reducing the size of the required PEC mesh. The results are and . For an HFIE solution, the shown for disk would be modeled as a closed body having a larger mesh. direction, The plane wave was assumed to be traveling in the

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Fig. 9. Bistatic RCS for a composite disk/cone, cone 

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= 2, PEC disk.

Fig. 10. Bistatic RCS for a composite missile, nose cone  cylinder, incident wave approaching the nose of the missile.

= 7 5, PEC :

and the bistatic RCS is shown in Fig. 9 for the pulse basis MoM solution, again compared to a BOR MoM solution. In the third problem, an air-to-air missile shape was selected and having a curved dielectric nose cone with . The PEC cylinder has and . , while . A plane wave was assumed to be traveling in the direction, toward the nose. The bistatic RCS is shown for the pulse basis MoM EFIE solution in Fig. 10. Additional results are shown in Fig. 11 for the case direction, toward where the incident wave is traveling in the the tail. In the fourth problem, a dielectric cube having and is sandwiched between two PEC plates and the structure is irradiated from below as shown in Fig. 7. Two pulse basis EFIE results are shown for the structure, one obtained by using a coarse mesh and one using a finer mesh. For and for each plate, . the coarse mesh, and for each plate, . For the finer mesh, The normalized bistatic RCS in dB is shown in Figs. 12 and 13 and compared with combination volume integral equation (VIE) and surface integral equation (SIE) results from Sarkar et al. [3]. Their formulation used 192 unknowns for the dielectric volume currents and 32 unknowns for the PEC plate currents.

Fig. 11. Bistatic RCS for a composite missile, nose cone cylinder, incident wave approaching the tail of the missile.

Fig. 12. Bistatic RCS at  . with PEC plates, 

=4

Fig. 13. Bistatic RCS at  . with PEC plates, 

=4



= 7 5, PEC :

=0

for a dielectric cube of length 0.1  capped

= 90

for a dielectric cube of length 0.1  capped

The graphical results show very good agreement between RCS plots calculated with orthogonal pulse basis vectors and their corresponding BOR plots. In the example of the dielectric cube with PEC plates at top and bottom, the pulse basis and in VIE/SIE methods similarly indicate a deep null at

MACKENZIE et al.: METHOD OF MOMENTS SOLUTION OF ELECTROMAGNETIC SCATTERING PROBLEMS

the RCS curve (Fig. 12). Compared to the finely meshed pulse basis RCS curve (Fig. 13), the coarsely meshed pulse basis more closely resembles the VIE/SIE RCS curve at results of Sarkar et al., who also used a coarse mesh for the PEC plates. Finer meshes show convergence to an almost flat RCS curve. VI. CONCLUSION We have demonstrated the solution of PEC/dielectric composite scattering problems by using a pair of orthogonally placed pulse basis vectors. These basis functions, which model equivalent electric and magnetic surface currents, allow for the correct implementation of the EFIE and HFIE for dielectric bodies. In addition, the electric current pulse basis vector allows the implementation of the EFIE and HFIE for PEC bodies. For the example composite EFIE and HFIE bodies, the basis functions combined with the testing schemes generate accurate and stable solutions. By arithmetically combining the EFIE and the HFIE, the CFIE may be used to guarantee unique solutions for composite scattering problems. REFERENCES [1] R. Mittra, Ed., Computer Techniques for Electromagnetics. New York, Pergammon, 1973. [2] R. F. Harrington, Field Computation By Method of Moments. New York: Macmillan, 1968. [3] T. K. Sarkar, S. M. Rao, and A. R. Djordjevic, “Electromagnetic scattering and radiation from finite microstrip structures,” IEEE Trans. Microw. Theory Tech., vol. 38, pp. 1568–1575, Nov. 1990. [4] S. M. Rao, T. K. Sarkar, P. Midya, and A. R. Djordjevic, “Electromagnetic scattering from finite conducting and dielectric structures: Surface/surface formulation,” IEEE Trans. Antennas Propag., vol. 39, pp. 1034–1037, Jul. 1991. [5] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, pp. 409–418, May 1982. [6] S. M. Rao, C. C. Cha, R. L. Cravey, and D. L. Wilkes, “Electromagnetic scattering from arbitrary shaped conducting bodies coated with lossy materials of arbitrary thickness,” IEEE Trans. Antennas Propag., vol. 39, pp. 627–631, May 1991. [7] E. Arvas, A. Rahhal-Arabi, A. Sadigh, and S. M. Rao, “Scattering from multiple conducting and dielectric bodies of arbitrary shape – Surface formulation,” Antennas Propag. Magazine, vol. 33, pp. 29–36, Apr. 1991. [8] A. I. Mackenzie, S. M. Rao, and M. E. Baginski, “Electromagnetic scattering from arbitrarily shaped dielectric bodies using paired pulse vector basis functions and method of moments,” IEEE Trans. Antennas Propag., vol. 57, pp. 2076–2083, Jul. 2009. [9] X. Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag., vol. 46, pp. 1718–1726, Nov. 1998. [10] T. K. Wu, “Radar cross section of arbitarily shaped bodies of revolution,” Proc. IEEE, vol. 77, pp. 735–740, May 1989.

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[11] M. Djordevic and B. M. Notaros, “Double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers,” IEEE Trans. Antennas Propag., vol. 52, pp. 2118–2129, Aug. 2004. [12] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [13] J. R. Mautz and R. F. Harrington, “H-field, E-field, and combined field solutions for conducting bodies of revolution,” A. E.Ü., vol. 32, no. 4, pp. 157–164, Apr. 1978. [14] C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin: Springer, 1969. Anne I. Mackenzie (M’87) received the B.S.E.E. degree from Purdue University, West Lafayette, IN, in 1986, the M.S.E.E. degree from Virginia Polytechnical Institute and State University, Blackburg, in 1992, and the Ph.D. degree in electrical engineering from Auburn University, Auburn, AL, in 2008. Since 1987, she has worked at the NASA Langley Research Center, Hampton, VA, in the Electromagnetics and Sensors Branch. Her work areas have included weather radar, radiometry, materials measurements, and numerical methods. Dr. Mackenzie currently serves on the Electromagnetics Code Consortium Government Executive Committee.

Sadasiva M. Rao (M’83–SM’90–F’00) received the Ph.D. degree with a specialization in electromagnetic theory from the University of Mississippi, Oxford, in 1980. Since 1980, he has been a faculty member at various institutions and, at present, works as a Full Professor at the Department of Electrical and Computer Engineering, Auburn University, Auburn, AL. He has published extensively in various international journals/conferences. His research interests are in the area of numerical methods applied to electrically large structures and highly complex, arbitrarily-shaped scattering bodies such as phased array antennas, missiles, and aircraft.

Michael E. Baginski (M’87–SM’95) received the B.S., M.S., and Ph.D. degrees, all in electrical engineering, from Pennsylvania State University, University Park. He is currently an Associate Professor of Electrical Engineering at Auburn University, Auburn, AL, where he has resided since the completion of his doctorate. His research interests include analytic and numerical solutions to transient electromagnetic problems, transient heat flow and solid state structural analysis using finite element routines, EMI and EMC characterization of MCM’s and PCB’s, simulation of rapid thermal expansion of metals under the action of large electric currents, S-parameter permittivity extraction routines, synthetic aperture radar (SAR) design and data processing routines, and the use of Genetic Algorithms for antenna optimization. Dr. Baginski is a member of Eta Kappa Nu, Sigma Xi, the New York Academy of Sciences, and the IEEE Education and Electromagnetic Compatibility Societies. He is also a member of Who’s Who in Science and Engineering and Who’s Who Among America’s Teachers.

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GPU-Based Shooting and Bouncing Ray Method for Fast RCS Prediction Yubo Tao, Hai Lin, and Hujun Bao

Abstract—The shooting and bouncing ray (SBR) method is highly effective in the radar cross section (RCS) prediction. For electrically large and complex targets, computing scattered fields is still time-consuming in many applications like range profile and ISAR simulation. In this paper, we propose a GPU-based SBR that is fully implemented on the graphics processing unit (GPU). Based on the stackless kd-tree traversal algorithm, the ray tube tracing can rapidly evaluate the exit position in a single pass on the GPU. We also present a technique for fast electromagnetic computing that allows the geometric optics (GO) and Physical optics (PO) integral to be carried out on the GPU efficiently during the ray tube tracing. Numerical experiments demonstrate that the GPU-based SBR can significantly improve the computational efficiency of the RCS prediction, about 30 times faster, while providing the same accuracy as the CPU-based SBR. Index Terms—Compute unified device architecture (CUDA), graphics processing unit (GPU), kd-tree, radar cross section (RCS), ray tracing, shooting and bouncing ray (SBR).

I. INTRODUCTION

T

HE shooting and bouncing ray (SBR) [1], [2] method is a popular and effective technique for the Radar Cross Section (RCS) prediction of arbitrarily shaped targets. This is because the ray tube makes the SBR clear in the concept of physics and also makes it easy to be implemented. More importantly, besides the first-order scattered fields, the SBR provides more accurate results by including the scattered fields arising from multiple bounces. The procedure of the SBR involves two steps: ray tube tracing and electromagnetic computing, as illustrated in Fig. 1. The incident plane wave is modeled as a dense grid of ray tubes, which are shot toward the target. Each corner ray of ray tubes is recursively traced to obtain the exit position. The exit position and field of the central ray are also evaluated via ray tracing, in which the reflected field is calculated according to the law of geometrical optics (GO) [3]. Finally, the physical optics (PO) integral is preformed to obtain the scattered field of this ray tube based on the pre-calculated exit positions and field. All scattered fields of ray tubes are summed to produce the scattered field of the target.

Manuscript received November 29, 2008; revised September 11, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported in part by the National Hi-Tech Research and Development Program of China under Grant 2002AA135020. The authors are with the State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310058, 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.2009.2037694

Fig. 1. The illustration of the SBR method for calculating the RCS of the trihedral corner reflector.

The SBR method is more effective than other numerical methods, such as the method of moments (MoM), for the high-frequency RCS prediction. However, both the ray tube tracing and electromagnetic computing are very time-consuming for electrically large and complex targets [4]. The total number of ray tubes depends on the electrical size of the target, since the density of ray tubes on the virtual aperture perpendicular to the incident direction should be greater than about ten rays per wavelength in view of the convergence of results. This requirement enormously increases the computational amount of ray tube tracing and the PO integral for electrically large targets. Moreover, if the target is described in terms of triangles, the number of intersection tests for each ray without any acceleration is proportional to the number of triangles, which further aggravates the computational burden of ray tube tracing. Due to such two compute-intensive steps, the SBR method is still not fast enough for applications such as range profile and ISAR simulation of real targets. In order to reduce the computation time, various acceleration techniques have been proposed. Sundararajan and Niamat [4] presented the ray-box intersection algorithm in FPGA to concurrently determine whether rays hit or miss the bounding box of the target. Suk et al. [5] introduced the multiresolution grid algorithm to reduce the initial number of ray tubes. Jin et al. [6] utilized the octree, recursively subdividing the box into eight children boxes using three axis-perpendicular planes, to decrease the number of intersection tests. Bang et al. [7] extended this work with a combination of the grid division and space division algorithms. As the kd-tree, recursively subdividing the box into two uneven boxes using one axis-perpendicular plane, has been proved as the best general-purpose acceleration structure for ray

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Fig. 2. The procedure of GPU-based SBR. The gray polygon is the projection of the target on the virtual aperture. The first step is to recursively trace rays on the grid (the virtual aperture). The intersected rays are shown as solid dots on the left. The next step is checking the validity of ray tubes, and only valid ray tubes need to trace the central ray and calculate the scattered field. Valid ray tubes are marked with the dot on the center of ray tubes in the middle. The final step is to reduce scattered fields of valid ray tubes to the scattered field of the target. All these steps are performed in a multithreaded manner on CUDA GPU Computing environment.

tracing of static scenes in computer graphics [8], Tao et al. [9] suggested utilizing the kd-tree to accelerate the ray tube tracing of the SBR. Over the past few years, with the rapid development of graphics hardware, especially the programmability of graphics processing units (GPUs), commodity graphics hardware provides large memory bandwidth and high computing power in general-purpose processing, which is known as GPGPU (general-purpose processing on the GPU) [10]. Compute unified device architecture (CUDA) developed by NVIDIA offers an effective way to directly access the massively parallel computing resources on the GPU and is specialized for computationally demanding, highly parallel tasks [11]. Many have reported success in performing general-purpose parallel computation on CUDA, such as molecular dynamics simulations [12] and fast multipole methods [13]. In this paper, we are interested in utilizing the GPU to accelerate both ray tube tracing and electromagnetic computing of the SBR. It is obvious that ray tracing is well suitable for parallel processing due to the independence of rays. Carr et al. [14] first implemented the ray-triangle intersection on the GPU in 2002, while Purcell et al. [15] presented a GPU ray tracing algorithm in multiple passes using a uniform grid as the acceleration structure in the same year. Due to the lack of stack support on the GPU, Foley and Sugerman [16] introduced two kd-tree traversal algorithms on the GPU, kd-restart and kd-backtrack, which both eliminate the need of a stack during the kd-tree traversal; subsequently, they extended the kd-restart algorithm from multiple passes to a single pass using branching and looping abilities of the GPU [17]. Recently, Popov et al. [18] developed a stackless kd-tree traversal implementation using CUDA, and the kd-tree augmented with ropes reduces the redundant traversal steps of interior nodes. The GPU ray tracing in computer graphics focuses mainly on ray casting (primitive rays). However, ray tube tracing in the SBR requires taking into account multiple bounces, and is more concerned about the exit position and field for the next electromagnetic computing. It is necessary, therefore, to adapt the existing GPU ray tracing to satisfy the requirement of ray tube tracing.

Graphics hardware has been employed in computational electromagnetics as early as 1993. Graphical electromagnetic computing (GRECO) [19], [20] method is the first proposal of using graphics hardware to accelerate computations of the first-order scattered fields of visible surfaces and wedges of the target. The identification of surfaces and wedges visible from the incident direction can be rapidly obtained through the Z-Buffer of workstation graphics hardware. The electromagnetic computing part of GRECO has been moved to graphics hardware by using the programmable GPU, which greatly improved the computational efficiency of the RCS prediction [21], [22]. If only the first-order scattered field is considered, the proposed GPU-based SBR is similar to the GRECO method, using ray tracing instead of rasterization to identify visible surfaces. However, we further take into account the calculation of the multiple-order scattered fields on graphics hardware. Inman and Elsherbeni [23] discussed the GPU implementation of FDTD and obtained a speedup factor of 40 in 2D case and 14 in 3D case. Recently, Peng and Nie [24] proposed the GPU accelerated method of moments and achieved an acceleration ratio about 20. In summary, we present a GPU-based SBR, in which ray tube tracing and electromagnetic computing are fully implemented on CUDA GPU Computing environment. Ray tube tracing is based on the stackless kd-tree traversal algorithm, which is modified to evaluate the exit position and field quickly. Electromagnetic computing is integrated into the process of central ray tracing, including the evaluation of the reflected field using the GO and the scattered field using the PO integral. The proposed approach can significantly accelerate the RCS prediction for electrically large and complex targets. II. GPU-BASED SBR CUDA GPU Computing environment can be thought of as programming massively parallel processors. A 32-thread warp operates in the Single Instruction Multiple Data (SIMD) fashion, i.e., 32 threads execute the same instruction on different data simultaneously. A thread block is composed of several warps, and these warps run in the single program multiple data

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(SPMD) fashion. In addition, CUDA also processes multiple thread blocks in the SPMD fashion at one time. As ray tubes are evaluated independently, without access to others, the SBR can be easily restructured into the multi-threaded fashion. As illustrated in Fig. 2, the procedure of GPU-based SBR is divided into three steps, and each step executes one kernel (program) on CUDA in a multi-threaded manner while synchronizing these threads of each kernel on the CPU. When the grid of ray tubes on the virtual aperture is determined, the first step is to recursively trace the corner rays of ray tubes in parallel to obtain the exit positions. Corner rays shared by neighbor ray tubes need to be traced only once. In the second step, each thread deals with one ray tube. It firstly checks the validity of the ray tube, then traces the central ray of the valid ray tube recursively and calculates the reflected field during the central ray tracing, finally performs the PO integral to obtain the scattered field of the ray tube. The scattered field of the target is given through the parallel reduction of scattered fields of ray tubes on CUDA. The details of these steps are discussed in Sections II-A through II-C. A. Corner Ray Tracing Given the incident direction of the electromagnetic wave, we can construct the virtual aperture perpendicular to the incident direction and divide it into a dense grid of ray tubes according to the criterion of ten rays per wavelength. The grid should be large enough to cover at least the projected area of the target. Since ray tracing is required for the corner and central rays of ray tubes, we describe the implementation of ray tracing on the GPU in detail, which is based on the stackless kd-tree traversal algorithm [18]. The kd-tree is a variation of the binary space partitioning tree and it is constructed by recursively employing the axis-perpendicular plane to split the target space into uneven axis-aligned boxes. The choice of the splitting plane is based on the raytracing cost estimation model, in which the cost consists of the traversal time of interior nodes and the ray-triangles intersection time of leaf nodes. The best known heuristic is the greedy Surface Area Heuristic (SAH) [25] that minimizes the cost for the node individually to construct the approximately optimal kd-tree. Triangles of the spitted node are then associated with one child node they overlap in space, and if the triangle is across the splitting plane, it should be associated with both children nodes. These two children nodes are then processed recursively until the termination condition is satisfied, such as the number of triangles of the node less than the user-defined number and no benefit to further split the node. While the octree simply puts the splitting positions at the middle point of the extend in each direction, the kd-tree takes into account the triangle distribution in the target space to find the optimal splitting axis and position. As a result, the kd-tree can provide faster ray tracing than the octree for general scenes. A simple 2D kd-tree is shown in Fig. 3(a). The detail of fast kd-tree construction is well described in Pharr and Humphreys’ book [26]. The kd-tree can be augmented with ropes in leaf nodes. The rope on each face of leaf nodes is a pointer to the adjacent leaf node, the smallest interior node including all adjacent leaf

Fig. 3. 2D kd-tree. (a) A simple 2D kd-tree. Interior nodes are labeled as their splitting planes and leaf nodes are labeled in their boxes. (b) A graph representation of the same 2D kd-tree. Each leaf node has four ropes on the face and these ropes directly link the adjacent leaf node, the smallest interior node or a nil node.

nodes or a nil node for leaf nodes on the border. During the traversal, the ray passing through a leaf node can directly move onto the adjacent node through its exit rope avoiding the requirement of the stack to keep to-be-visited nodes, and subsequently this manner removes the unnecessary traversal steps of interior nodes. The graph representation of the same 2D kd-tree of Fig. 3(a) is illustrated in Fig. 3(b), and four ropes of the leaf are also shown. The rope is constructed during the crenode ation of kd-tree offline on the CPU, and the algorithm for rope optimization can be found in [18]. Once the kd-tree with ropes is constructed, the rays for all incident and reflected directions could be traced efficiently. Ray tracing in the SBR is slightly different from ray tracing in computer graphics. This is mainly because ray tracing in the SBR puts more emphasis on the exit position and field rather than intermediate radiance contributions. Since ray tracing in computer graphics often finds one intersection in one pass, we recast the algorithm to find all intersections in a single pass (Algorithm I), and the results are stored in the device memory on the GPU for the next electromagnetic computing. Ray tracing in the SBR starts with the root node of the kd-tree with ropes, and determines the entry position through the intersection of the ray and the bounding box of the target. At the interior node, one of the two child nodes is selected to continue the traversal according to the relative position between the entry position and the splitting plane. If the entry position is on the left of the splitting plane, the left child needs to be traversed next. Otherwise, the traversal continues to the right child. Following the above rules, the kd-tree is recursively traversed down until a leaf node is encountered. At the leaf node, we first determine the exit face of the ray , ) range, on the leaf node’s boundary box and the ( which defines the part of the ray that is inside the leaf node. Then the ray is iteratively tested for intersection with triangles . Actually, of the leaf node to find the nearest intersection, the nearest intersection found may not be inside the current leaf with to determine the node. However, we can compare exact leaf node where the intersection is located. If the ray does not intersect any triangle of this leaf node or the , nearest intersection is not inside this leaf node the traversal continues to the adjacent node through the rope on the exit face. If the adjacent node is the nil node, the ray goes off the target, and then the last intersection position and the triangle

TAO et al.: GPU-BASED SHOOTING AND BOUNCING RAY METHOD FOR FAST RCS PREDICTION

Algorithm I Single-pass Ray Tracing

while

and

// process interior nodes while do if

do

then

end if end while // intersection test with the triangles in the leaf node

if

in

do and

From the description above, the proposed single-pass ray tracing not only utilizes the ropes to reduce the number of interior-node traversal steps for the primary ray, but also directly starts at the leaf node containing the origin of the ray to further eliminate interior-node traversal steps for the reflected ray. An illustration of the proposed ray tracing in 2D is shown in Fig. 4. In this way, each corner ray of ray tubes is recursively traced in parallel on CUDA, and the exit position and the corresponding triangle ID are evaluated and stored in the device memory on the GPU for the following electromagnetic computing.

B. Central Ray Tracing and Electromagnetic Computing

else

for

497

then

end if end for // check intersection and then if // generate the reflected ray

With the knowledge of the exit position and the triangle ID of each corner of ray tubes, the scattered field of ray tubes could be calculated in a multi-threaded manner on CUDA. This procedure involves three parts. The first part is to check the validity of the ray tube. If any one corner ray of the ray tube does not intersect the target, this ray tube is invalid. Invalid ray tubes need not trace the central ray and calculate the scattered field. Besides this simple rule, other stricter rules are required to judge whether the ray tube diverges in the recursive corner ray tracing. However, it is difficult to determine the divergence of ray tubes accurately due to the discrete sampling. The last intersected triangle IDs of four corner rays are also examined in our implementation. If the last intersected triangles are not the same triangle, the ray tube is highly divergent in the ray tube tracing and is discard as the invalid ray tube. Each valid ray tube constructs the central ray, and the central ray is recursively traced in a similar fashion like the corner ray. The primary difference is the field tracing. At each intersection, the GO is applied to calculate the reflected field through the field before the intersection and the geometric information of the target as follows:

(1) else // follow the rope of the exit face

where

,

, , and

end if end while return ID, if any, are stored for the next electromagnetic computing. Otherwise, if the nearest intersection along the ray is inside this leaf node, the origin and direction of the ray are replaced with the intersection position and the reflected direction respectively, and the traversal continues to this leaf node with the reflected ray. If the number of intersections is larger than the maximum order of the reflection, the traversal terminates, and then the last intersection position and the triangle ID are also stored for the next electromagnetic computing.

, . The vector

is the propagation direction before the intersection, is the propagation direction after the intersection, and is the normal and the reflected of the intersection. The incident field is . The detail formulas about field is the reflection coefficients are explained in [1], [3]. Therefore, after the central ray tracing, both the exit position and field of the central ray could be obtained. The exit ray tube is modeled as a four-sided polygon calculated from the corner ray tracing. The scattered filed of the ray tube can be approximated by the PO integral as follows:

(2)

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where is the observation direction. The can be of the four-sided polygon expressed as the exit field

(3) As pointed out in [2], the coefficients and in the EH formulation(0.5) provide a better result. Under the assumption that the field on the exit ray tube has the same amplitude and a linear phase variation with the exit field of the central ray, the PO integral can be approximated in a more computable form as shown in [2]. As a result, the scattered field for both the vertical and horizontal polarization can be calculated, and the final results are 12 floating-point numbers. In order to reduce the number of outputs, the complex results of vv, vh, hv, and hh polarization are produced, and these 8 floating-point numbers are stored in the device memory on the GPU. C. Field Reduction When scattered fields of ray tubes are available, the scattered field of the target can be easily obtained by summing up these scattered fields. Although current graphics hardware provides high memory bandwidth between the CPU and GPU, the summing on the CPU are not very effective due to the low memory access on the CPU as shown in [22]. The scan primitives such as the prefix-sum algorithm for GPU computing have been well studied recently [27], [28] and the parallel reduction on CUDA can be implemented in a similar way. The expected complexity . Therefore, of this parallel reduction for elements is the reduction of scattered fields is implemented through the scan primitives on CUDA, and the final result, only 8 float numbers, are read back from the device memory to the CPU. III. IMPLEMENTATION DETAILS In order to verify the accuracy and efficiency of the proposed GPU-based SBR, the original SBR and the CPU kd-tree accelerated SBR were also implemented for comparison, and several numerical examples were tested. These experiments were performed on an NVIDIA GeForce 8800 GTX and an Intel Core 2 Duo 3.0 GHz CPU. Our implementation ran atop Windows XP with the CUDA Toolkit 1.1. As all future NVIDIA GPUs will support CUDA, the proposed GPU-based SBR is scalable across future generations. CUDA provides a simple and general C language interface to the hardware functionality on GeForce 8800, so it is possible that the GPU can be directly utilized as a data-parallel computing device, eliminating the special mapping between the computation of GPGPU applications and the graphics APIs. The kernel such as the ray tracing function is executed as a large number of threads simultaneously on the GPU. GeForce 8800 has 16 multiprocessors, each with 8 scalar processors, and each multiprocessor can process multiple thread blocks concurrently. The 32-thread warp is executed in SIMD and is the scheduled unit in the multiprocessor. The number of thread

Fig. 4. Recursive ray tracing. (a) The ray is recursively traced in the 2D target space and has two intersections with the target. (b) The traversal path of the ray is shown as bold lines. The traversal begins with the root node s , and proceeds down through the interior node s and the leaf node n . The ray intersects the triangle in the leaf node n , and the first reflected ray is generated. The first reflected ray does not have intersection with the triangle in leaf node n , follows the rope of the exit face to the interior node s , and moves on to the leaf node n . The first reflected ray passes through the leaf node n without intersection, and continues following the rope of the exit face to the leaf node n . An intersection is found between the first reflected ray and the triangle in leaf node n , and the second reflected ray is generated. The second reflected ray continues to be traced, as the nil node is encountered on the exit face, ray tracing terminates.

blocks and threads per block is specified by the programmer, and each thread has a unique thread ID and block ID to identify the unique data assigned to each thread. Therefore, each corner ray and tube can be specified through thread ID and block ID. The geometry data of the target are packed and transferred into the texture memory in the device. The texture memory is analogous to a read-only 1D/2D array for random access. As each multiprocessor provides a small texture cache, it speeds up data access from the texture memory than from the global memory. The kd-tree node and its associated triangle IDs are also packed and transferred into the texture memory at the stage of preprocessing, as these data structures are only dependent on the target and not changed during the RCS prediction. Leaf nodes require additional memory for the information about the ropes and bounding boxes. As there is no cache support for the global memory and the access latency of the global memory is much higher, it is very necessary to follow the right access pattern to obtain maximum memory bandwidth, especially the coalescing rule. The coalescing rule states that if each thread of a half-warp reads or writes the global memory with contiguous, aligned addresses, these operations can be coalesced into a single contiguous, aligned memory access. The detail specification can be found in the CUDA programming guide [11]. Therefore, the block size in our implementation is (16, 4), as the half-warp size is 16, and we also take into account the limited register number in each multiprocessor and the coherence of ray tracing. The output of the corner ray tracing can be coalesced into a single contiguous, aligned memory access, as threads in the warp execute the same write instruction in a sequence. The access of these data in the electromagnetic computing can also be coalesced for the same reason. The memory access in parallel field reduction is highly optimized using the on-chip shared memory, and this improves the performance by eliminating memory traffic to the device memory and avoiding the bank conflicts. The shared memory is another special feature of CUDA. Each block can obtain part of the on-chip shared

TAO et al.: GPU-BASED SHOOTING AND BOUNCING RAY METHOD FOR FAST RCS PREDICTION

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Fig. 5. Four test targets: (a) generic missile, (b) ship , (c) airplane A, and (d) airplane B.

TABLE I THE GEOMETRY SIZE AND MEMORY REQUIREMENT OF THE FOUR TARGETS (KB). THE BASIC KD-TREE MEMORY CORRESPONDS TO THE INFORMATION OF THE KD-TREE NODES AND THE ASSOCIATED TRIANGLE IDS OF LEAF NODES. THE ROPES KD-TREE MEMORY IS THE STORAGE REQUIREMENT OF THE ROPES AND THE BOUNDING BOX OF LEAF NODES

memory and its associated threads can access this shared memory in one clock circle if there is no band conflicts [11]. The grid size on the virtual aperture is proportional to the projected area of the target and the frequency. For example, the maximum grid size in our experiments is 7364 1502. Each ray needs to record the last intersected position and each ray tube also requires 8 float number for the scattered field. Therefore, the required memory would be larger than 500 M for this example. As is known to us, the amount of available memory on the GPU is very limited, for example, NVIDIA GeForce 8800 has only 768 M device memory. For electrically large targets, the large amount memory requirement limits the scalability of the GPU-based SBR. We resolve this problem by partitioning the grid into several sub-grids. The GPU-based SBR is applied to each sub-grid and the final scattered field is calculated by summing up scattered fields of sub-grids. In our implementation, the sub-grid size is 1024 1024, which corresponds to about 48 M device memory. IV. RESULTS AND DISCUSSION Several different types of targets were tested to evaluate the proposed GPU-based SBR. As shown in Fig. 5, there are a generic missile, a simple ship, and two airplanes. The geometry size and triangle number of the four targets are listed in Table I. As can be seen from Table I, the four targets vary in the geometry size and are modeled using different triangle numbers. The missile and ship have simple sharps, while structures of two airplanes are much more complex. The monostatic RCS of the four targets were computed from 0 to 360 in 361 equal-spaced incident directions at 10 GHz frequency. The incident direction for the four targets is also illustrated in Fig. 5. As can be observed from Fig. 5, the incident axis, while the direction for the ship is rotated around the others are rotated around the axis. At most fifth-order reflection was considered for complex structures of the four targets.

The ray tube size of the grid on the virtual aperture is 3 mm . The memory requirements of the four targets in our implement are available in Table I. The memory requirements of triangles for the ray-triangle intersection test on the GPU are relatively small, at most 1257.5 kB memory in the four targets. The leaf numbers of the kd-tree depend on the geometrical structure. The simple structure only has a small number of leaf nodes, while a large number of leaf nodes are generated for the complex structure. The kd-tree memory requirements of the four targets are also listed in Table I. The basic memory requirement includes the information of the kd-tree nodes and the associated triangle IDs of leaf nodes. Due to the stackless kd-tree traversal algorithm, additional memory is required for the information about the ropes and bounding box of leaf nodes. Fortunately, these memory requirements are not very large compared with the basic kd-tree memory requirements, as shown in Table I. One advantage of the stackless kd-tree traversal algorithm is that it can effectively reduce the number of interior-node traversal steps through the ropes. Additionally, the reflected ray, which directly starts at the leaf node containing the origin of the ray, further eliminates interior-node traversal steps. Table II describes the average interior-node traversal steps of the primary ray and the reflected ray. As seen clearly from Table II, the interior-node traversal steps of stackless ray tracing are significantly reduced compared with that of standard ray tracing, especially the reflected ray. The total computation time of all incident angles are shown in Table III using the original SBR, the CPU kd-tree accelerated SBR, and the proposed GPU-based SBR. As demonstrated in [18], the kd-tree accelerates the ray tracing in the SBR and the computation time is extremely reduced compared with the original method. The reason is that most rays could find the intersection in the first leaf nodes visited [8], and it eliminates the unnecessary ray-triangle intersection tests. In our experiments, the

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Fig. 6. The comparison of our GPU-based SBR result and the MLFMM result for the trihedral corner reflector. (a) HH-polarization result for the incident plane  at 3 GHz, (b) VV-polarization result for the incident plane  at 6 GHz.

= 60

= 45

Fig. 7. The comparison of our GPU-based SBR result and the MLFMM result for the ship at 10 GHz. (a) VV-polarization result, (b) HH-polarization result.

TABLE II THE AVERAGE INTERIOR-NODE TRAVERSAL STEPS (PER RAY) OF THE FOUR TARGETS FOR STANDARD RAY TRACING AND STACKLESS RAY TRACING

TABLE III THE COMPUTATION TIME OF THE FOUR TARGETS OF THE ORIGINAL SBR, CPU KD-TREE ACCELERATED SBR, AND PROPOSED GPU-BASED SBR (SEC)

average number of ray-triangle intersection tests of the primary ray is 16.55, 8.75, 7.16, and 5.88 for the four targets, respectively. With the reduced ray-triangle intersection tests, the proposed GPU-based SBR method is 3 orders of magnitude faster than the original method. Due to high computing power on the GPU and the stackless kd-tree traversal algorithm, the proposed GPU-based SBR is about 30 times faster than the CPU kd-tree

accelerated SBR. It also can be observed from Table III that the speedup factor increases with the geometry size and complexity. This further verifies that the proposed GPU-based SBR are highly effective for electrically large and complex targets. The trihedral corner reflector is a typical benchmark target for verifying the high frequency multiple-bounce scattering [2], [29]. The trihedral corner reflector used in this paper is constructed of three right-angled triangles with the side length 1 m. Two different incident parameters are used to evaluate the accuracy of the proposed GPU-based SBR: (a) from 0 to 90 on plane with an angular resolution of 1 at 3 GHz; the plane with an angular res(b) from 0 to 90 on the olution of 1 at 6 GHz. As the valid checking of ray tubes in the second step of the GPU-based SBR is very strict, the discarding of divergent ray tubes would affect the accuracy of the RCS prediction. In order to reduce the impact of divergent ray tubes, we divide the grid on the virtual aperture into denser ray to obtain more accurate results. As the CPU-based tubes SBR result is almost the same as the GPU-based SBR result, we only compare the GPU-based SBR result and the MLFMM result. The monostatic RCS results of the HH-polarization using the parameter (a) and the VV-polarization using the parameter (b) are shown in Fig. 6. The comparison shows a good agreement between the GPU-based SBR result and the MLFMM result. The computation time of the MLFMM are approximately

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Fig. 8. The comparison of our GPU-based SBR + TW-ILDC result and the MLFMM result for the ship at 10 GHz. (a) VV-polarization result, (b) HH-polarization result.

3.75 and 27.47 minutes per-angle at 3 GHz and 6 GHz, respectively. In contrast to this, the total computation time of all incident angles and polarization types are 8.73 and 32.17 seconds in the GPU-based SBR. The ship illustrated in Fig. 5(b) as well as the measured data is also widely used to validate the accuracy of the SBR [5]–[7]. Since some of the geometric details are unknown, a new ship is modeled based on the available geometric parameters and a MLFMM result is used to verify the accuracy. Fig. 7 shows the monostatic RCS comparison from 0 to 360 in 361 equalspaced incident directions at 10 GHz. A good agreement is observed between the two results, and the deviation may be partly due to the edge-diffraction effect [30]. To verify this, the edgediffraction effect of the ship is computed based on truncatedwedge incremental-length diffraction coefficients (TW-ILDC) [31] on the CPU, and the diffraction fields are added to the result of the GPU-based SBR. The SBR + TW-ILDC result is compared with the MLFMM result in Fig. 8. The SBR TW-ILDC result is more accurate than the SBR result, especially in the incident angles after 180 . This is due to the fact that the relative impact of the edge-diffraction effect, as the secondary dominant scattering mechanism, can not be ignored in the incident angles after 180 , where there is only the first-order scattered field. V. CONCLUSION It has been shown that thanks to the rapid development of graphics hardware and the stackless kd-tree traversal algorithm, ray tube tracing and electromagnetic computing of the SBR are fully implemented on CUDA GPU Computing environment. Ray tube tracing, based on the stackless kd-tree traversal algorithm, can quickly evaluate the exit position and field, and electromagnetic computing is integrated into the process of central ray tracing, including the calculation of the reflected field using the GO and the scattered field using the PO integral. Numerical results show excellent agreement with the exact solution, and demonstrate that the GPU-based SBR method can greatly reduce the computation time. Furthermore, the proposed GPU-based single-pass ray tracing can also be adapted to other computational electromagnetic methods, such as statistic ray tracing for RCS prediction [29] and radio propagation [32].

ACKNOWLEDGMENT The authors would like to thank Prof. T. Cui from South East University for providing the MLFMM method used in this paper. REFERENCES [1] H. Ling, R. C. Chow, and S. W. Lee, “Shooting and bouncing rays: Calculating the RCS of an arbitrarily shaped cavity,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 194–205, 1989. [2] J. Baldauf, S. W. Lee, L. Lin, S. K. Jeng, S. M. Scarborough, and C. L. Yu, “High frequency scattering from trihedral corner reflectors and other benchmark targets: SBR vs. experiments,” IEEE Trans. Antennas Propag., vol. 39, no. 9, pp. 1345–1351, 1991. [3] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [4] P. Sundararajan and M. Y. Niamat, “FPGA implementation of the ray tracing algorithm used in the XPATCH software,” in Proc. IEEE MWSCAS’01, Dayton, OH, Aug. 2001, vol. 1, pp. 446–449. [5] S. H. Suk, T. I. Seo, H. S. Park, and H. T. Kim, “Multiresolution grid algorithm in the SBR and its application to the RCS calculation,” Microw. Opt. Technol. Lett, vol. 29, no. 6, pp. 394–397, 2001. [6] K. S. Jin, T. I. Suh, S. H. Suk, B. C. Kim, and H. T. Kim, “Fast ray tracing using a space-division algorithm for RCS prediction,” J. Electromagn. Waves Applicat, vol. 20, no. 1, pp. 119–126, 2006. [7] J. K. Bang, B. C. Kim, S. H. Suk, K. S. Jin, and H. T. Kim, “Time consumption reduction of ray tracing for RCS prediction using efficient grid division and space division algorithms,” J. Electromagn. Waves Applicat., vol. 21, no. 6, pp. 829–840, 2007. [8] V. Havran, “Heuristic Ray Shooting Algorithms,” Ph.D. dissertation, Univ. Czech Technical, Prague, 2000. [9] Y.-B. Tao, H. Lin, and H.-J. Bao, “Kd-tree based fast ray tracing for RCS prediction,” Progress Electromagn. Res. (PIER), vol. 81, pp. 329–341, 2008. [10] J. D. Owens, D. Luebke, N. Govindaraju, M. Harris, J. Krüger, A. E. Lefohn, and T. J. Purcell, “A survey of general-purpose computation on graphics hardware,” Comput. Graphics Forum, vol. 26, no. 1, pp. 80–113, 2007. [11] NVIDIA CUDA Compute Unified Device Architecture Programming Guid 1.1. Internet Draft NVIDIA CORPORATION, 2008 [Online]. Available: http://developer.nvidia.com/object /cuda_get.html [12] J. A. Anderson, C. D. Lorenz, and A. Travesset, “General purpose molecular dynamics simulations fully implemented on graphics processing units,” J. Comp. Phys., vol. 227, no. 10, pp. 5342–5359, 2008. [13] N. A. Gumerov and R. Duraiswami, “Fast multipole methods on graphics processors,” J. Comp. Phys., vol. 227, no. 18, pp. 8290–8313, 2008. [14] N. A. Carr, J. D. Hall, and J. C. Hart, “The ray engine,” in Proc. Graphics Hardware’02, Sep. 2002, pp. 37–46. [15] T. J. Purcell, I. Buck, W. R. Mark, and P. Hanrahan, “Ray tracing on programmable graphics hardware,” ACM Trans. Graph., vol. 21, no. 3, pp. 703–712, 2002.

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[16] T. Foley and J. Sugerman, “Kd-tree acceleration structures for a GPU raytracer,” in Proc. Graphics Hardware’05, Jul. 2005, pp. 15–22. [17] D. R. Horn, J. Sugermann, M. Houston, and P. Hanrahan, “Interactive k-d tree GPU raytracing,” in Proc. Interactive 3D Graphics’07, Aug. 2007, pp. 167–174. [18] S. Popov, J. Günther, H.-P. Seidel, and P. Slusallek, “Stackless kd-tree traversal for high performance GPU ray tracing,” Comput. Graphics Forum, vol. 26, no. 3, pp. 415–424, 2007. [19] J. M. Rius, M. Ferrando, and L. Jofre, “High frequency RCS of complex radar targets in real time,” IEEE Trans. Antennas Propag., vol. 41, no. 9, pp. 1308–1318, 1993. [20] J. M. Rius, M. Ferrando, and L. Jofre, “GRECO: Graphical electromagnetic computing for RCS prediction in real time,” IEEE Antennas Propag. Mag., vol. 35, no. 2, pp. 7–17, 1993. [21] Z.-L. Yang, L. Jin, and W.-Q. Li, “Accelerated GRECO based on GPU,” in Proc. Radar’06, Oct. 2006, pp. 1–4. [22] Y.-B. Tao, H. Lin, and H.-J. Bao, “From CPU to GPU: GPU-based electromagnetic computing (GPUECO),” Progress Electromagn. Res. (PIER), vol. 81, pp. 1–19, 2008. [23] M. J. Inman and A. Z. Elsherbeni, “Programming video cards for computational electromagnetics applications,” IEEE Antennas Propag. Mag., vol. 47, no. 6, pp. 71–78, 2005. [24] S.-X. Peng and Z.-P. Nie, “Acceleration of the method of moments calculations by using graphics processing units,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 2130–2133, 2008. [25] J. D. Macdonald and K. S. Booth, “Heuristics for ray tracing using space subdivision,” Proc. Graphics Interface’89, pp. 152–163, Jun. 1989. [26] M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation. New York: Morgan Kaufmann, 2004. [27] S. Sengupta, M. Harris, Y. Zhang, and J. D. Owens, “Scan primitives for GPU computing,” in Proc. Graphics Hardware’07, Aug. 2007, pp. 97–106. [28] M. Harris, J. Owens, S. Sengupta, Y. Zhang, and A. Davidson, CUDA Data Parallel Primitives Library 2008 [Online]. Available: http://www. gpgpu.org/developer/cudpp/ [29] F. Weinmann, “Ray tracing with PO/PTD for RCS modeling of large complex objects,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1797–1806, 2006. [30] R. G. Koujoumijan and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, vol. 62, pp. 1448–1461, 1974. [31] P. M. Johansen, “Uniform physical theory of diffraction equivalent edge currents for truncated wedge strips,” IEEE Trans. Antennas Propag., vol. 44, no. 7, pp. 989–995, 1996.

[32] T. Fügen, J. Maurer, T. Kayser, and W. Wiesbeck, “Capability of 3D ray tracing for defining parameter sets for the specification of future mobile communications systems,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3125–3137, 2006. Yubo Tao received the B.S. and Ph.D. degree in computer science and technology from Zhejiang University, Hangzhou, China, in 2003 and 2009, respectively. He is currently a Postdoctoral Researcher in the State Key Laboratory of CAD&CG of Zhejiang University. His research interests include computational electromagnetics, GPU programming, and scientific visualization.

Hai Lin received the Ph.D. degree in computer science from Zhejiang University, Hangzhou, China. Currently, he is a Professor of Visual Computing in the State Key Lab. of CAD&CG, Zhejiang University. He was a Visiting Professor in the Department of Computing and Information Systems, University of Bedfordshire, U.K. His research interests include computer graphics, scientific visualization, volume rendering and computational electromagnetics.

Hujun Bao received the Bachelor and Ph.D. degrees in applied mathematics from Zhejiang University, Hangzhou, China, in 1987 and 1993. He is Currently the Director of the State Key Laboratory of CAD&CG of Zhejiang University. He is also the Principal Investigator of the Virtual Reality Project sponsored by Ministry of Science and Technology of China. His research interests include realistic image synthesis, realtime rendering technique, digital geometry processing, field-based surface modeling, virtual reality and video processing.

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A Body Area Propagation Model Derived From Fundamental Principles: Analytical Analysis and Comparison With Measurements Andrew Fort, Member, IEEE, Farshad Keshmiri, Student Member, IEEE, Gemma Roqueta Crusats, Christophe Craeye, Member, IEEE, and Claude Oestges

Abstract—Using wireless sensors worn on the body to monitor health information is a promising new application. To realize transceivers targeted for these applications, it is essential to understand the body area propagation channel. Several numerical, simulated, and measured body area propagation studies have recently been conducted. While many of these studies are useful for evaluating communication systems, they are not compared against or justified by more fundamental physical models derived from basic principles. This type of comparison is necessary to provide better physical insights into expected propagation trends and to justify modeling choices. To address this problem, we have developed a simple and generic body area propagation model derived directly from Maxwell’s equations revealing basic propagation trends away, inside, around, and along the body. We have verified the resulting analytical model by comparing it with measurements in an anechoic chamber. This paper develops an analytical model of the body, describes the expected body area pathloss trends predicted by Maxwell’s equations, and compares it with measurements of the electric field close to the body. Index Terms—Body area networks, propagation model.

I. INTRODUCTION

W

IRELESS bio-medical sensors are a promising new application made possible by recent advances in ultralow power technology [1]. Each sensor measures parameters of interest and sends the data in short bursts to a central device, such as a PDA. Both the sensors and the PDA are worn directly on the body. Examples include sensors to observe brain activity for recording or warning against seizure events, or sensors to examine heart activity for diagnosis and automatic emergency calls. The large diversity and potential of these applications makes it an exciting new research direction. A computationally simple and generic body area propagation model is required to develop efficient low power radio systems for use near the human body. Measurements [2]–[5] and

Manuscript received November 25, 2007; revised February 13, 2009. First published June 23, 2009; current version published February 03, 2010. A. Fort is with IMEC, B-3001 Leuven, Belgium and also with Vrije Universiteit Brussel (VUB), Dept. ELEC, B-1050 Brussels, Belgium (e-mail: [email protected]). F. Keshmiri, C. Craeye, and C. Oestges are with Université Catholique de Louvain (UCL), B-1348 Louvain-La-Neuve, Belgium (e-mail: [email protected]; [email protected]; [email protected]). G. R. Crusats is with Universitat Politècnica de Catalunya (UPC), 31 08034 Barcelona, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2025786

finite difference time domain (FDTD) simulations [6]–[8] have successfully described very specific communication scenarios. Using measured or simulated results, complete statistical ultrawideband and narrowband models have been developed in [9]–[11] and standardized by the IEEE [12]. These approaches are entirely appropriate for body area propagation modeling and have already proven effective for evaluating body area communication system proposals [13], [14]. However, measurements do not directly consider the physical propagation mechanism, forcing researchers to rely on some ad-hoc modeling approaches that are not always motivated by fundamental principles. On the other hand, realistic FDTD simulations that numerically evaluate the solution to Maxwell’s equations of small antennas worn close to the sophisticated curved inhomogeneous human bodies are very time-consuming. Another approach is to use the uniform theory of diffraction (UTD) [15]. UTD is an extension to ray tracing allowing the propagation channel to be described in terms of the sum of rays diffracting around and reflecting from body parts [8]. While this approach offers considerable physical insight, it typically relies on a high-frequency asymptotic approximation which is not valid when the wavelength is of the same order of magnitude as the diffracting object, or for tangentially polarized antennas placed very close to the human body [15]. This scenario is particularly important for body worn antennas where it is desirable to have low-profile antennas for user comfort. To help understand propagation near the body, we have implemented a computationally simple but generic model of body area propagation derived directly from Maxwell’s equations. We assume the antenna can be modeled as a tangentially polarized point source, and the body can be modeled as a lossy cylinder, then evaluate the resulting electromagnetic fields. This approach is related to models developed for antennas printed on coated cylinders [16]–[19]. Unlike UTD, this exact computation is valid regardless of the distance of the antenna from the body and can be used for tangentially polarized antennas. The physical meaning of the main steps of this derivation is provided. We begin with the solution for an infinite line source in the vicinity of an infinite lossy cylinder. We then numerically calculate the inverse Fourier transform of the line source to obtain the fields due to a point source, representing the antenna, near a lossy cylinder, representing the body. Using this model, we have evaluated the electric fields for propagation around, along, away, and inside a lossy cylinder to estimate the major propagation trends near a human body, and we have proposed simple

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pathloss laws which can be used in practice. Finally, we have validated this approach by comparing the fields predicted by this simple model with actual measurements around the human body in an anechoic chamber. The resulting approach can therefore be used to rapidly approximate average pathloss near a body, or to physically justify the average pathloss trends of more realistic statistical models extracted from measurements and FDTD simulations. This article summarizes our approach to analytical body area propagation modeling. Section II begins by outlining the various steps in our analytical developments. Section III reviews the well-known solution for a tangentially polarized line-source near a lossy conductor. To obtain a more realistic model of a small antenna close to the body, Section IV proposes a numerical integration technique to transform the line source into a point source and addresses other practical issues required for computer implementation. Section V demonstrates, from fundamental principles, the major pathloss trends expected for propagation near a curved lossy body and identifies some simplified pathloss laws which are more convenient to use in practice. Section VI then compares the total model with measurements taken near a body in an anechoic chamber. Finally, Section VII summarizes the major conclusions of this study.

Fig. 1. Proposed two-step procedure for body area modeling. First we obtain the solution for a line-source near a lossy cylinder. Second, we transform the solution to a point source by taking its inverse Fourier transform.

II. GENERIC APPROACH TO BODY AREA MODELING Fig. 1 summarizes our proposed approach to body area modeling. We model the antennas as a point source with some polarity, and the human body as an infinite lossy cylinder with arbitrary material parameters. A point source is a reasonable approximation of an antenna if the antenna size is small compared with the wavelength. This is normally the case for compact body-worn devices. While this approximation is not appropriate for analyzing near-field propagation trends of realistic antennas, it can form the basis of future studies employing the method of moments. A lossy cylinder is a reasonable first order approximation of a human body allowing us to take into account many propagation phenomena, including diffraction around a curved lossy surface, reflections off the body, and penetration into the body. All of these effects are expected to play a role in body area propagation, though the relative importance of each effect will depend on such factors as the frequency, polarity, radius of curvature, and tissue properties. Thus, our chosen geometry will allow us to explore many important far-field body area propagation phenomena, while remaining analytically tractable so that a solution can be derived directly from fundamental principles. In order to derive the electric fields around and inside a lossy cylinder resulting from a point source, we follow a two step procedure as indicated in Fig. 1. First, we solve the problem of a linear current source located outside a lossy cylinder. The solution to this problem can be obtained by representing the displaced cylindrical waves of the line source by a superposition of un-displaced cylindrical waves originating from the cylinder axis. This allows us to easily enforce the boundary conditions around the surface of the lossy cylinder. Section III describes the

Fig. 2. Geometry and coordinate system for our analysis.

details of this step. Second, we convert the line source to a point source by performing an inverse Fourier transform of the current . The transform source along the vertical spectral coordinate must be obtained numerically using a contour integral to avoid a singularity in the line source solution. Section IV describes the details of this step. Throughout this article, we focus on the special case of a tangentially polarized point-source which is representative of low-profile antennas. Low profile antennas are more desirable for comfortable, low-cost body-worn devices. However, the general approach could also be used to investigate antennas polarized normal to the body having more desirable pathloss characteristics. We recommend this as future work.

III. LINE SOURCE Fig. 2 provides a more precise diagram of the geometry emphasizing the major variables used throughout our analysis. An infinite cylinder of radius is oriented along the -axis at the origin of the coordinate system. An infinite linear current source . We let the current is located at cylindrical coordinates . The exponent inon this line vary along the z-axis as dicates the current source is a traveling wave in the -direction with a propagation constant . This choice of current will allow us to convert it to a point source by means of an inverse Fourier transform in Section IV.

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We begin the body area modeling developments by writing down the well-known vector Helmholtz equations which are most convenient for solving this problem

(1) is the wavenumber, is the angular frewhere quency, is the material permeability, is material permittivity, and is the current density. We want to determine the electric field and magnetic field that satisfy (1) for the geometry shown in Fig. 2. We know that the general solution to this equation can be written as the sum of the solution to the homogeneous equation or , and any particular solution to (1). The solution to the homogeneous equation will correspond physically to the scattered field by the cylinder and will be discussed in Section III-A. The particular solution will correspond to the incident field from the line source propagating through free space without the presence of the lossy cylinder and will be discussed in Section III-B. Section III-C describes the total solution obtained by summing the scattered and incident fields and enforcing the boundary conditions along the surface of the cylinder.

Fig. 3. The left side shows the wavefronts of an undisplaced cylindrical wave, while the right side shows the wavefronts of a displaced cylindrical wave that would be generated by a line source located at coordinates ( ;  ).

B. Incident Field The incident field can be determined by noting that the field everywhere, except on the current source itself, must take the form of the homogeneous solution in (2) since there are no current or charge sources in free-space. Thus, the component of the incident field will take the form

(5)

A. Scattered Field The z-component of the homogeneous solution to the Helmholtz equations for the electric and magnetic fields is obtained by separation of variables [16]

or

(6) (2) where outside also have

as a delayed Unit step function set A to zero and to zero inside the cylinder. We

(3) Following the same reasoning, a similar expression can be derived for the magnetic field

(4) and represent the Hankel function of the second kind with order and the Bessel function of the first kind with order respectively. The constants , , , and will be determined in Section III-C from the boundary conditions imposed by the geometry. We will show later how the other components of the field can be obtained from the vertical components.

Physically, we expect the fields generated by the line-source to consist of displaced outgoing cylindrical waves originating (see Fig. 3). This expression indifrom the line source at cates we can also express those fields by a sum of un-displaced cylindrical harmonics originating from the cylinder axis. The reason why we want to express the incident field solution as a sum of waves originating from the cylinder axis is that they have the same geometry as the lossy cylinder. This will make it easier to apply the boundary conditions of the complete solution at the surface of the cylinder in the following section. and by apWe now need to determine the constants plying the boundary conditions at the current source. The current source for the geometry shown in Fig. 2 can be expressed mathematically as

(7) Note that, without loss of generality, we have assumed that the source is located at and , which will simplify our notation. We have also written this expression such that it . This requires the term (for is valid for all integers ) since our cylindrical geometry is periodic with

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every radians. The vector indicates a vertical polarization. For the horizontal polarization case, we will simply replace with and denote the current density and its amplitude as and respectively. The current source representation in (7) will make it difficult to apply the boundary conditions since it has the form while the incident field in (10) is expressed in terms of the sum . We will thereof cylindrical harmonics of the form fore transform our line source into a similar sum of cylindrical harmonics by applying the following Fourier transform pair

The expressions for the magnetic field have the same form, so we can also write the -component of the magnetic field

(12) We can determine the other components of the electromagnetic field from the vertical components using the following relations which are valid in any homogeneous medium, as proven from Maxwell’s equations [16]:

(8) (13)

resulting in the following expression: with (9) which now has the same mathematical form as (5). The source is , indicating now defined for all and is periodic every a cylindrical current sheet. Thus, by using the Fourier Transform relation of (8), we have represented the original line source as a sum of cylindrical current sheets centered at the origin. This will allow us to easily apply the boundary conditions on the surface of the current sheet. Since all expressions are now written using cylindrical harwe will simply assume this monics having the form term from now on without explicitly writing it every time. Furthermore, we introduce the following notation to represent a particular harmonic of propagation:

(14) The vectors and represent the and components of the and represent the electromagnetic field, while the scalars components. Applying this relation to (11) and (12) allows us to express the components as follows:

(15)

and

(10) represents the harmonic of the inIn this notation, cident field from (5). While we have not explicitly written the , it is still there and we must remember to factor multiply the signal by or whenever we take its derivative with respect to or . We know that the Hankel function from (10) ap. Since we cannot have infinite proaches infinity as fields in practice, the portion of our geometry which includes the origin can only consist of Bessel functions. Similarly, the region outside our cylindrical sheet can only consist of Hankel functions. This allows us to re-write the incident field inside and outside the cylindrical current sheet as

(11)

(16)

Equations (11), (12) must satisfy the following tangential boundary conditions at the surface of the current sheets :

(17) (18) (19) The subscripts 1 and 2 indicate fields just inside and outside the current sheet respectively. The resulting solutions for a vertically and horizontally polarized current source are discussed

FORT et al.: A BODY AREA PROPAGATION MODEL DERIVED FROM FUNDAMENTAL PRINCIPLES

below. By adding together weighted combinations of these components, we can generate the solution for a point source having arbitrary tangential polarity. 1) Vertical Polarization: For the vertical polarity, the current source is polarized in the direction, as indicated in (9), resulting in the following solution:

(20) (21) (22) (23) 2) Horizontal Polarization: The developments for the horizontally polarized current source proceed in the same manner as the vertically polarized current source except now the current sheets have a polarization instead of the polarization. This results in the following solution:

(24) (25) (26) (27) For both the horizontal and vertical polarization cases, we have made use of the following well-known identity to write the result in this simple form [16]:

507

and

(31) harmonic and we implicitly asSubscript indicates the sume the term . As in the case of the incident field, we know the solution in the region inside the cylinder can only consist of Bessel functions while the region outside the cylinder can only consist of Hankel functions. The scattered and incident fields inside the cylinder have the same form so we combine these terms into , which represents the total field, and assign it the constants and . However, the scattered and incident fields outside the cylinder have a different form and cannot be combined together. We can also use the same approach to determine the from the vertical components by applying (13) to (30) and (31). , , , and by applying We can now solve for the tangential boundary conditions of (17)–(19) to the surface . In our case, these boundary conditions of the cylinder at state that the tangential fields just inside the cylinder equal the tangential fields just outside the cylinder. We can write this as follows: (32) (33) , represent the scattered and incident tangential where represents field components just outside the cylinder, and the total field just inside the cylinder. Similar expressions can be written for the magnetic field. Finally, we can use our previous definitions of these various field components to re-write the second line of (33) in matrix format as follows:

(28)

C. Total Field We have now developed the solution for the homogeneous equation in Section III-A, and a particular solution to the nonhomogeneous equation in Section III-B. The general solution is the sum of these two solutions

(29) As indicated at the end of Section III-A, we still need to solve for , , , and from (2)–(4) to determine the unknowns the component of the scattered solution. We can re-write the field using the notation developed in Section III-B as

(30)

(34)

and repreOn the right side of the equations, the terms sent the incident fields for vertically and horizontally polarized and respectively. Their soline-sources with amplitudes lution was already given in Section III-B. The left side of the equation is obtained from (30) and (31) using (13) to describe the components from the components. We have introduced , , to express the the terms equation more compactly. Similarly, we have also introduced the notation and , where and refer to the material just inside and outside the cylinder respectively.

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We could solve (34) and express the constants , , , in closed form. However, the resulting expressions are and complicated and do not provide any additional insight. Therefore, we will simply solve the equation numerically when we implement the model on a computer in Section IV. IV. POINT SOURCE In the previous section, we developed the solution for a line source near a lossy cylinder. Our goal now is to convert the solution for a line-source into the solution for a point source. This can be accomplished using the following Fourier transform equivalence:

(35) In our case, we can see that the left hand side of the equation could represent a point source. We could also interpret the right hand side of the equation to be a sum of line sources each having a current that is a traveling wave with a vertical propagation constant . This, is the type of current source we considered in the previous section. From the linearity of Maxwell’s equations, we know that the combined field due to several sources is equal to the sum of the fields of each individual source. Therefore, we can write the following:

=

Fig. 4. A parabolic contour around the singularity at k k is a good numerical method for evaluating the inverse Fourier transform when converting to a point source.

then the Hankel functions become difficult to evaluate since they . An exact numerical analincrease exponentially with ysis of this problem is beyond the scope of this article. Instead, we have simply plotted the integrand and chosen the contour integration parameters to produce a smooth result. Typically, C is chosen to be a positive number less than 0.01. Finally, we can only evaluate a finite number of terms to calculate the summation over . This can be problematic since routines for computing Hankel functions may exhibit numerical and small complex or real aroverflow problems for large guments. Fortunately, this numerical aspect of the problem has already been investigated. A simple recursive formula for obtaining the largest calculable for Hankel functions has been derived in [20] and can be used to approximate an appropriate number of terms in the summation. Typically, less than 200 terms are required for an accurate solution. V. BODY AREA PATHLOSS TRENDS

(36) represents the field due to a point source. The vector It is too difficult to perform the integration of (36) analytically, so we must resort to a numerical integration by computer. Unfortunately, this is complicated by the presence of a pole in . This pole can be seen the line source solution when as indicated in (3). Thus, by noting when which will cause the various Hankel functions in our solution to approach infinity. There may be other poles resulting from the solution of (34). To avoid these singularities in the line-source solution, we can perform a contour integral in . We have the complex plane around the singularity at found that a parabolic contour, defined as in Fig. 4, together with Simpson’s rule, provides a practical numerical integration technique that rapidly converges to a good approximation of the solution. This contour integral can be expressed as

(37) We must carefully choose the parameter defining the height of the parabolic contour in Fig. 4. If is too small, then the contour gets too close to the singularity and the integral becomes difficult to evaluate numerically. On the other hand, if is too large,

We have implemented the analytical model of a point source near a lossy cylinder described in Section IV. We are now able to use these expressions to calculate basic pathloss trends around an arbitrary cylinder much faster than with FDTD. Section V-A investigates propagation trends away from and into the lossy cylinder. Section V-B investigates propagation trends along and around the surface of a lossy cylinder. We will compare these results with actual measurements around a human body in Section VI. A. Propagation Away and Into a Lossy Cylinder Fig. 5 shows the electric field magnitude expressed in dBV/m as a function of observation radius for a point source just outside a lossy cylinder1. It shows the field values for both a 400 MHz and a 2.45 GHz source. In both cases, the point source is located . The lossy cylinder is centered at the origin at which roughly corresponds and has a radius of to the radius of a typical human torso. It is indicated on the figure with dashed lines. The source is therefore just one centimeter away from the cylinder on the right side. We set the material properties of the cylinder according to the high-water content human tissues in Table I. The electric field away from the cylinder on the right side of the figure decays proportionally with distance as expected in free space. The electric field into the cylinder between the dashed lines is roughly a straight line on our log-linear

20 10(E ) and represents the magnitude of the electric field in

1dBV/m is log dB relative to 1 V/m

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Fig. 5. Electric field as a function of observation radius for a point source just : outside a lossy cylinder. A 1 Amp point source is located at  on the right side of the figure. The lossy cylinder is shown with dashed lines.

= 0 16 meter

TABLE I ELECTRIC PROPERTIES OF HUMAN TISSUES WITH HIGH WATER CONTENT SUCH AS MUSCLE AND SKIN [21]

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Fig. 6. Electric field as a function of distance through free-space, along the length of the cylinder, and around the cylinder. The lossy cylinder radius is 0.15 m, and both the observation and source radii are 1 centimeter away from the cylinder. All results are for a frequency of 2.45 GHz and a 1 Amp current.

electric field. Nevertheless, we can expect this field component to decay rapidly close to the body surface resulting in higher pathloss. Note that a normally polarized electric field would not decay close to the surface of a conductor in the same manner indicating that antenna polarization will have a fundamental impact on body area propagation and hence communication performance. B. Propagation on the Surface of a Lossy Cylinder

scale indicating it decays exponentially with distance . This is also not surprising since the waves are propagating through a lossy medium and therefore decay exponentially [22]. As expected, the electric field decays more rapidly at 2.45 GHz than at 400 MHz since the skin depth is proportional to the square root of the wavelength [22]. This effect is compounded by the fact that body tissues of high water content tend to be better conductors at higher frequency. As expected, the field values on the opposite side of the cylinder are much lower since they are more shadowed by the cylinder. Fig. 5 shows that there is a local minimum of the electric field inside the cylinder. The field decays roughly exponentially on either side of this minimum. This behavior suggests that, at high frequencies, the fields on the opposite side of the body will result from diffraction around the surface of the body rather than penetration through the body. This occurs because the body is a relatively good conductor. These results have also been verified with FDTD simulations on more sophisticated models of the body incorporating heterogeneous tissues [23]. Finally, the left side of Fig. 5 shows how tangential fields decay rapidly close to the surface of a lossy conductor. For example, the fields on the left edge of the figure are actually higher than the fields next to the cylinder even though they are physically much further away from the antenna. If bodies were perfect conductors, the electric field would vanish on the surface completely. The human body is actually an imperfect conductor, so there will be some tangential component to the

Fig. 6 shows the electric field magnitude expressed in dB V/m as a function of distance for a 2.45 GHz point source. As be. The fore, the lossy cylinder has a radius of point source is located just 1 centimeter away from the body at . Three curves are presented. The top curve shows the electric field versus distance for a point source in free space. The middle curve shows the electric field versus distance along the front surface of the cylinder. The bottom curve shows the electric field versus distance around the surface of the cylinder. The top two curves are obtained using a horizontally polarized point source and calculating the electric field along the direction. For the top curve, we set the material properties of the lossy cylinder to the properties of free space. For the middle curve, we set the lossy cylinder properties according to the high water content tissues of Table I. The bottom curve was obtained by using a vertically polarized point source and varying the angle of observation . In all cases, the observation radius . is set to the radius of the source We can see from Fig. 6 that the two curves along and around the surface of the cylinder (shown with circles and dots respectively) are shifted down compared with the free space curve. As explained in the previous section, this is because we expect the tangential fields to be small close to a lossy conductor compared with fields in free space. It is also clear that we can expect the pathloss versus distance trend on a body to be very sensitive to the trajectory we consider. The electric field will decay significantly more rapidly when considering propagation around a

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body into the shadow region compared with propagation along the front of a body. The middle curve, representing propagation along the front surface of the cylinder, decays somewhat faster than in free while we have found that space. In free space, with m between 3–3.5 provides a reasonable approximation for different frequencies and radii. The excellent fit of this power law relationship is shown by the dashed line. It is difficult to provide a physical interpretation of this trend other than to say it is a consequence of Maxwell’s Equations. We can suggest that higher pathloss trends are normally expected near conducting surfaces due to material losses. However, given the close proximity of the source and lossy surface, other contributions such as surface waves may also influence results. The bottom curve representing propagation around the cylinder is approximately a straight line on our log-linear scale indicating it decays roughly exponentially with distance . The excellent fit of an exponential decay law is shown by the dashed line. The exponential decay is not obvious from our developments in Sections III, IV since the model is expressed in terms of Hankel and Bessel functions. However, high-frequency asymptotic approximations for diffraction around curved conducting surfaces based on the uniform theory of diffraction (UTD) also tend to decay exponentially with distance [15]. In this case, waves guided by a curved surface are often called creeping waves. Unfortunately, we do not know of any suitable UTD approximations valid for tangentially polarized sources close to a conducting surface2. This is because the source and surface are very close with respect to the wavelength we consider. UTD approximations are only valid when the geometry does not vary significantly over an interval on the order of a wavelength [15]. Nevertheless, we propose that the exponential decay our model predicts can be interpreted as creeping waves diffracting around curved surfaces and suggest as future work finding suitable approximations to show this analytically for our geometry. For creeping wave propagation, the rapid attenuation around the body is related to the continuous shedding of energy tangent to the cylinder as a wave rotates around the surface into the shadowed region [15]. Thus, energy is radiated away from the body during the diffraction process which results in a faster pathloss versus distance trend close to the surface of the body. The fluctuation of the curve near the back of the body can be interpreted as the interference of clockwise and counter-clockwise creeping waves. Fig. 7 shows the pathloss versus distance trend for propagation around the same 0.15 m radius cylinder at different frequencies. The pathloss is shown in dB relative to the pathloss at a reference distance of 0.1 meters. In general, the rate of exponential decay increases with frequency around a lossy cylinder which is also consistent with creeping wave diffraction [15]. We also see that the curves tend to flatten out on the opposite side of the body with respect to a perfect exponential decay, especially at lower frequencies. This is likely due to clockwise and counter-clockwise diffracting waves interfering with each other on opposite sides of the cylinder. 2On the other hand, UTD approximations valid for normally polarized sources located on perfectly conducting surfaces are well known [15] and have even been employed in body area propagation studies [8]

Fig. 7. Propagation around a lossy cylinder at different frequencies.

VI. COMPARISON WITH MEASUREMENTS Obviously, the geometry considered in the previous sections does not perfectly represent an antenna and body. Thus, we have compared the pathloss trends predicted by our simplified lossy-cylinder model with measurements of the electric field close to the human body using actual antennas. Section VI-A describes our measurement setup, while Section VI-B compares our model and measurement results. A. Measurement Setup An HP8753ES vector network analyzer (VNA) is used to measure the S21 parameter between two antennas placed at various positions on a human body in an anechoic chamber. The two antennas are connected to the VNA using 6 meter lowloss coaxial cables. Measurements are taken at 915 MHz and 2.45 GHz. We focus on these frequencies since they represent unlicensed ISM bands available internationally [24], and are also used by recent narrowband standardization efforts such as Zigbee [25] and Bluetooth [26]. The same small, low-profile Skycross SMT-8TO25-MA [27] antennas are used for all measurements. The antennas are 50.5 by 28 by 8 mm in size and weigh only 4.2 grams. These antennas were chosen since they are close to the size and profile requirements typical of comfortable body worn sensor devices [1]. Furthermore, they have a wide bandwidth which minimizes degradation resulting from the antenna being de-tuned when placed near the body [28]. The distance between the body and the antenna can significantly influence the pathloss and needs to be carefully controlled [3]. We control this separation by putting a 5 mm dielectric between the body and the antenna. In the same manner as [9], the antenna is taped to this dielectric and held against the body using tight elastics so they can not move while a measurement is being made. In all cases, the antennas are mounted so they are vertically polarized parallel to the body surface. paWe analyze the antenna matching by measuring the rameter in free-space and close to the body. In free space, the

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Fig. 8. Experimental setup: measurement locations around body.

parameter is below 10 dB across the band of interest indicating the measurement setup itself does not introduce signifpaicant matching loss. When mounted on the body, the rameter can vary depending on the placement of the antenna on but the body. It remains good at 915 MHz . in some cases becomes marginal at 2.4 GHz It is possible that the coaxial cable and connector may radiate energy influence the pathloss versus distance trends. However, the display on our VNA remains stable when we move the cables indicating they do not radiate enough energy to appreciably alter our results. Fig. 8(a) shows where the antennas are placed around the torso. All channel parameters are extracted from measurements performed in 3 planes separated by approximately 15 cm along the vertical axis (see left diagram). The right diagram shows where the antennas are placed for each plane. The receiver positions are marked with circles, while the transmitter is marked with a box around the circle. The transmitter is always placed on the front, and the receiver is placed at distances of 10–45 cm in steps of 5 cm measured around the perimeter of the body. Fig. 8(b) shows where the antennas are placed for communication along the torso. The transmitter is worn at approximately shoulder height at one of two different positions. The receiver is placed directly below the transmitter at seven positions separated by 10 cm covering the range from the shoulder to the knees. To gather more measurement points, we repeat the procedure on the back of the body. B. Measurement and Model Comparison Fig. 9 compares the pathloss versus distance measured around the body and along the body for 2.45 GHz and 915 MHz frequencies with the analytical model developed in Sections III, IV. The vertical axis represents the pathloss. The horizontal axis is the distance traveled by the wave around or along the surface of the body. The circles indicate individual measurements, while the solid line was calculated using our physical model derived from Maxwell’s equations. The torso used in this experiment has a radius of between 13–15 cm depending on the height at which the measurement is made. Thus, we use a cylinder radius and placed the antenna and observation of away from the lossy cylinder point at in our analytical model. Furthermore, we set the material properties to the high-water content tissues of Table I. Since a point source does not exhibit the same near-field losses as the antenna used in our experiment, the analytical results are normalized to minimize the mean squared error.

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The analytical model of a lossy cylinder matches the average trends of the measurements around a real body remarkably well considering the gross simplification of the body shape and antenna. As predicted by Maxwell’s Equations, propagation around the body results in a more rapid attenuation versus distance trend than propagation along the body. However, there is a large variance in the pathloss measured at a particular distance around the torso consistent with other measured and simulated results [8]. This variance can be attributed to several physical factors including random interaction of the antenna and body at different locations, reflections off the arms, and variations in the local curvature or tissue properties. Since our analytical model is based on a uniform lossy cylinder and does not incorporate antenna losses, we can not expect it to take into account these fluctuations. This effect is better analyzed using statistical models based on either measurements or FDTD simulations. Some researchers have proposed modeling these contributions using a log-normal random variable since many of the effects are multiplicative [8], [10], [11]. Comparing the top two figures with the bottom two figures, it is clear that the pathloss within 10–20 centimeters of the antenna is significantly lower at 915 MHz than at 2.45 GHz. This indicates that the losses for these particular antennas are lower at 915 MHz than at 2.45 GHz. Part of this can be explained by the higher matching losses at 2.45 GHz compared with 915 MHz. Furthermore, at least in free-space, the antennas are constant gain (non-constant aperture) between 800 MHz–2.45 GHz resulting in an attenuation of higher frequencies. However, the antenna properties are also affected by the nearby body and propagation is along the body surface rather than through free-space. Finally, similar pathloss trends for tangentially polarized antennas have been reported in the literature consistent with our theoretical derivations [2], [10], [11]. However, pathloss measurements in office building rather than in an anechoic chamber tend to flatten out on the back of the body in comparison with Fig. 9 due to the presence of nearby scatterers [2], [9], [11], and can also exhibit small-scale fading due to the interference of diffracting components near the body with reflections from nearby scatterers [11]. For narrowband systems, Ricean and Rayleigh small-scale fading models can be used depending on the amount of shadowing [11]. For ultrawideband systems, modified Saleh-Valenzuela models can be used to incorporate later arriving multipath clusters [9], [29]. VII. CONCLUSIONS AND FUTURE WORK We have proposed a simplified physical body area propagation model derived directly from Maxwell’s equations. The model assumes the body can be approximated as an infinite lossy cylinder and the antenna can be approximated as a point source just outside the cylinder. A complete derivation starting from Maxwell’s equations was developed together with a practical numerical method for evaluating it on a computer. We then used this model to identify the following approximate body area propagation trends which can be used to justify more sophisticated empirical models: away from the body; • into the body; • • around the body;

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Fig. 9. Comparison of the analytical model for propagation around/along a cylinder and measurements around/along a real human body in an anechoic chamber. (a) 2450 MHz, around torso; (b) 2450 MHz, along front of torso; (c) 915 MHz, around torso; (d) 915 MHz, along front of torso.

• along the body with –3.5 is inversely proportional to the distance The electric field away from the body in free space. However, the field has a much more rapid exponential decay into and around the body. The exponential decay into the body is expected in a lossy medium. The exponential decay around the body is typical of creeping wave propagation around conducting surfaces. Along the front of the body, the pathloss versus distance decays more rapidly than in free space but does not follow the same rapid exponential trends as propagation around the body. The decay rates ( , , and ) depend on several factors including the tissue properties, frequency, polarity, and proximity to the body. While they cannot be evaluated directly, they can be estimated easily from our analytical model much faster than with numerical simulation methods such as FDTD. In general, normal polarization, lower frequency, and increasing the body-antenna separation will result in more desirable pathloss trends from an electromagnetic perspective. Unfortunately, these trends are less desirable from an application perspective where we desire small, low-profile antennas worn directly against the body. Thus, emerging body area communication systems will need to find a good compromise between these conflicting constraints. For the first time, we have shown how the average pathloss trends of small tangentially polarized antennas measured in an anechoic chamber closely follow the trends predicted from Maxwell’s Equations for propagation around and along a lossy cylinder. As predicted, pathloss due to diffraction around the body is significantly higher than pathloss along the body. There is a large variance in the pathloss measured at a particular distance around the torso. This variance can be attributed to

several physical factors including random interaction of the antenna and body at different locations, reflections off the arms, and variations in the local curvature or tissue properties. Our modeling approach is very generic and we can recommend several future uses and extensions. Frequency dispersion and pathloss trends can be investigated for ultrawideband propagation systems by evaluating the fields across several different frequencies. Furthermore, by using different Fourier Transform relations, the line-source solution can be converted to different current density patterns more representative of practical antennas such as finite length dipoles. Finally, the model can be used as a Green’s function in an integral equation and solved using the method of moments to predict the fields near a lossy cylinder generated by practical three-dimensional antennas. This approach is computationally faster than FDTD at the cost of a simplified body model. However, considering that all bodies are different, an exact calculation of electric field values near a particular body model may not provide significantly more insight than a cylindrical approximation. Furthermore, if more precise field values are necessary, FDTD methods can be used to fine-tune or verify antenna parameters obtained more easily with the method of moments. REFERENCES [1] B. Gyselinckx, C. V. Hoof, and S. Donnay, “Body area networks, the ascent of autonomous wireless microsystems,” presented at the Int. Symp. on Hardware Technol. Drivers of Ambient Intelligence, 2004. [2] T. Zasowski, F. Althaus, M. Stager, A. Wittneben, and G. Trõster, “UWB for noninvasive wireless body area networks: Channel measurements and results,” in Proc. IEEE Conf. on Ultra Wideband Systems and Technol., 2003, pp. 285–289.

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[3] A. Fort, C. Desset, J. Ryckaert, P. De Doncker, L. Van Biesen, and P. Wambacq, “Characterization of the ultra wideband body area propagation channel,” in Int. Conf. on Ultra-Wideband (ICU) Proc., Zurich, Sep. 2005, pp. 22–27. [4] P. Hall, M. Ricci, and T. Hee, “Measurements of on-body propagation characteristics,” in Int. Conf. on Microw. and Millimeter Wave Technol., 2002, pp. 770–772. [5] A. Alomainy, Y. Hao, X. Hu, C. G. Parini, and P. S. Hall, “UWB on-body radio propagation and system modelling for wireless bodycentric networks,” IEE Commun. Proc., vol. 153, no. 1, pp. 107–114, Feb. 2006. [6] W. Scanlon and N. Evans, “Numerical analysis of bodyworn UHF antenna systems,” Electron. Commun. Engineering J., vol. 13, no. 2, pp. 53–64, 2001. [7] J. Ryckaert, P. DeDoncker, S. Donnay, A. Delehoye, and R. Meys, “Channel model for wireless communication around the human body,” Electron. Lett., vol. 40, no. 9, pp. 543–544, Apr. . [8] Y. Zhao, Y. Hao, A. Alomainy, and C. Parini, “UWB on-body radio channel modeling using ray theory and subband FDTD method,” , vol. 54, no. 4, pp. 1827–1835, Jun. 2006. [9] A. Fort, J. Ryckaert, C. Desset, P. De Doncker, and L. Van Biesen, “Ultra wideband channel model for communication around the human body,” IEEE J. Sel. Areas Commun., 2006. [10] A. Fort, C. Desset, P. De Doncker, and L. Van Biesen, “An ultra wideband body area propagation channel model: From statistics to implementation,” IEEE Trans. Microw. Theory Tech., 2006. [11] A. Fort, P. Wambacq, C. Desset, and L. V. Biesen, “An indoor body area channel model for narrowband communications,” IET Microw. Antennas Propag., to be published. [12] A. F. Molisch, D. Cassioli, C.-C. Chong, S. Emami, A. Fort, B. Kannan, J. Karedal, J. Kunisch, 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. 3143–3150, Nov. 2006. [13] A. Fort, C. Desset, P. Wambacq, and L. Van Biesen, “Body area RAKE receiver communication,” presented at the IEEE Conf. on Commun. (ICC), Turkey, 2006. [14] A. Fort, M. Chen, C. Desset, P. Wambacq, and L. Van Biesen, “Impact of sampling jitter on mostly-digital architectures for UWB bio-medical applications,” presented at the IEEE Conf. on Commun. (ICC), Glascow, 2007. [15] D. A. Mcnamara, C. Pistorius, and J. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction. Boston, MA: Artech House, 1991. [16] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: IEEE Press, 1995. [17] V. B. Erturk and R. G. Rojas, “Efficient computation of surface fields excited on a dielectric coated circular cylinder,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 10, pp. 1507–1516, 2000. [18] J. Sun, C. F. Wang, L. W. Li, and M. S. Leong, “Further improvement for fast computation of mixed potential Green’s functions for cylindrically stratified media,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 3026–3036, 2004. [19] S. Raffaelli, Z. Sipus, and P. S. Kildal, “Analysis and measurements of conformal patch array antennas on multilayer circular cylinder,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1105–1113, 2005. [20] C. F. du Toit, “A procedure for determining the largest computable order of bessel functions of the second kind and hankel functions,” IEEE Trans. Antennas Propag., vol. 41, no. 12, pp. 1741–1742, Dec. 1993. [21] C. C. Johnson and A. W. Guy, “Nonionizing electromagnetic wave effects in biological materials and sytems,” Proc. IEEE, vol. 60, no. 6, pp. 692–718, Jun. 1972. [22] D. K. Cheng, Field and Wave Electromagnetics, 2nd ed. Reading, MA: Adisson-Wesley Publishing Company, 1992. [23] A. Fort, C. Desset, J. Ryckaert, P. DeDoncker, L. Van Biesen, and S. Donnay, “Ultra wide-band body area channel model,” in Int. Conf. Commun. (ICC) Proc., Seoul, Korea, May 2005, vol. 4, pp. 2840–2844. [24] International Telecommunications Union—Radiocommunications (ITU-R), Radio Regulations, Section 5.138 and 5.150 [Online]. Available: http://www.itu.int/home [25] IEEE, IEEE 802.15.4, Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (LR-WPANs) October 2003. [26] The Bluetooth Standard Version 1b [Online]. Available: http://www. bluetooth.com [27] Skycross [Online]. Available: http://www.skycross.com

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[28] W. G. Scanlon, N. E. Evans, and M. Rollins, “Antenna-body interaction effects in a 418 MHz radio telemeter for infant use,” in 18th Annu. Int. Conf. of the IEEE Eng. in Medicine and Biology Society, 1996, pp. 278–279. [29] A. Saleh and R. A. Valenzuela, “A statistical model for indoor multipath propagation,” IEEE J. Sel. Areas Commun., vol. 5, no. 2, pp. 128–137, Feb. 1987. Andrew Fort (S’04–M’07) was born in Ottawa, ON, Canada, in 1975. He received the Bachelor’s degree in computer and electrical engineering from the University of Victoria, Canada, in 1998 and the Ph.D. degree in applied sciences (with greatest distinction) from the Free University of Brussels, Belgium, in 2007. Between 1998 and 2000, he worked with IVL Technologies, Canada, researching real-time DSP algorithms for pitch recognition of the human voice. He joined IMEC, Belgium, in January 2000 and spent seven years researching and developing of a wide range of communication systems, including satellite, wireless local-area networks, and ultralow power radios. He is currently a wireless specialist for low-power body area communication at the Cochlear Technology Center in Belgium.

Farshad Keshmiri (S’08) was born in Iran in 1982. He received the B.S. degree in electrical engineering and the M.S. degree in communication engineering from the Iran University of Science and Technology (IUST), Tehran, in 2004 and 2007, respectively. He is currently working toward the Ph.D. degree at the Ecole Polytechnique de Louvain, Université Catholique de Louvain (UCL), Belgium. While at IUST, he concentrated on miniaturization of microwave antennas using electromagnetic bandgap (EBG) structures and implementation of some high gain EBG antennas. Since October 2007, he has been with the Laboratory of the Ecole Polytechnique de Louvain, UCL, where he has been working on the design of antennas devoting to the body area networks. His research interests include computational electromagnetics, body area networks, metamaterials and electromagnetic band gap antennas.

Gemma Roqueta Crusats was born in Girona, Spain, in 1983. She received the Telecommunication Engineer degree from Universitat Politècnica de Catalunya, Spain, in 2007, where she is working toward the Ph.D. degree. In 2006, she was involved in body area networks research as a Visiting Student at the Université Catholique de Louvain, Belgium. Since September 2007, she is involved in developing microwave based non-destructive testing methods for monitoring and predicting civil infrastructure condition. Her research interest include indoor and outdoor propagation, wideband microwave imaging with spiralometric discrimination and ultrawideband antennas.

Christophe Craeye (M’08) was born in Belgium in 1971. He received the Electrical Engineer and Bachelor in Philosophy degrees in 1994 and the Ph.D. degree in applied sciences in 1998 from the Université Catholique de Louvain (UCL), Belgium. From 1994 to 1999, he was a Teaching Assistant at UCL and carried out research on the radar signature of the sea surface perturbed by rain, in collaboration with the rain-sea interaction facility of NASA, Wallops Island, VA, and with the European Space Agency. From 1999 to 2001, was Postdoctoral Researcher at the Eindhoven University of Technology, The Netherlands. His research there has been carried out in the framework of the square kilometer array (SKA) radio telescope project, and consisted of studying a technology based on

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phased-arrays traditionally used for broadband radar applications. In this framework, he was a Visiting Researcher at the University of Massachusetts in Fall 1999, and worked with the Netherlands Institute for Research in Astronomy (ASTRON) in 2001. Since 2002, he is an Associate Professor (Chargé de Cours) at UCL. His research interests are finite antenna arrays, multiple antenna systems and numerical methods for fields in periodic media, including metamaterials. Proc. Craeye participates in the COST IC0603 Action on Antennas for Sensor applications funded by the European Commission. He is currently an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

Claude Oestges received the M.Sc. degree and the Ph.D. degree in applied science from the Université Catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 1996 and 2000, respectively. From 1996 to 2000, he was an Assistant Lecturer in the Microwave Laboratory, UCL. From January to December 2001, he was a Postdoctoral Scholar in the Smart Antennas Research Group (Information Systems Laboratory) of Stanford University, CA. From October 2001 to September 2005, he was a Postdoctoral Fellow at the Belgian National Science Founda-

tion FNRS (FRS-FNRS—Fonds de la Recherche Scientifique—FNRS), associated with the Microwave Laboratory UCL. He is presently a FRS-FNRS Research Associate and Assistant Professor at UCL. His research interests cover wireless and satellite communications, with a specific focus on the propagation channel and its impact on system performance. He is leading sub-working groups in COST 2100 “Pervasive mobile and ambient wireless communications” as well as in the Network of Excellence NEWCOM++. He is the author or coauthor of one book and more than 100 papers in international journals and conference proceedings. Prof. Oestges was the recipient of the IEE Marconi Premium Award in 2001 and the IEEE Vehicular Technology Society Neal Shepherd Award in 2004. He currently serves as Associate Editor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY and the EURASIP Journal on Wireless Communications and Networking.

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Exploration of Whole Human Body and UWB Radiation Interaction by Efficient and Accurate Two-Debye-Pole Tissue Models Masafumi Fujii, Ryo Fujii, Reo Yotsuki, Tuya Wuren, Toshio Takai, and Iwata Sakagami, Member, IEEE Abstract—We have developed a computationally efficient finite-difference time-domain (FDTD) model of a whole human body based on accurate 2-pole Debye dispersion dielectric tissue properties. Comprehensive FDTD analyses of the interaction between a whole human body and ultrawideband (UWB) radiation are carried out by including the proposed frequency dependent tissue models. The 2-pole Debye models have been obtained for 50 individual human tissues from Gabriel’s Cole-Cole data by the least squares fitting technique over the frequency range from 100 MHz to 6 GHz. A whole human body composed of the 2-pole Debye models is exposed to spread spectrum radiation. Local energy absorption in a human body is compared between the proposed model and the conventional model of frequency-independent permittivity and conductivity. Resonance states are then investigated in the human body exposed to electromagnetic nano-second pulse radiation. For the extraction of the frequency contents from the highly damped FDTD time signals, a spectrum analysis technique based on an auto-regressive (AR) model has been applied. Pulse propagation in the vicinity of the human body is also characterized by the proposed model for the wireless body area network (WBAN) application that has been proposed recently for computer assisted medical diagnostics and rehabilitation. Index Terms—Biophysics, electromagnetic propagation, electromagnetic propagation in dispersive media, electromagnetic radiation effects, finite-difference time-domain (FDTD) methods.

I. INTRODUCTION number of human body models have been generated for the numerical analysis of electromagnetic effects caused by the radiation from various digital wireless communication devices [1]–[3]. However, because the electromagnetic properties of human tissues depend largely on frequency, the actual effects on the human body caused by ultrawideband (UWB) spread spectrum signals may differ greatly from those obtained for the conventional frequency-independent model. Precise frequency dependent permittivity and conductivity values for human body tissues have been compiled and made available by Gabriel [4]. These have been fitted by relatively

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Manuscript received October 16, 2008; revised January 07, 2009. First published June 10, 2009; current version published February 03, 2010. M. Fujii, R. Yotsuki, and I. Sakagami are with the Department of Electrical and Electronic Engineering, University of Toyama, Toyama-shi 930-8555 Japan (e-mail: [email protected]). T. Wuren was with the Department of Electrical and Electronic Engineering, University of Toyama, Toyama-shi 930-8555 Japan. She is now with Toyohashi University of Technology, Aichi 440-8580, Japan. T. Takai is with Nippon Paper Industries, Ltd, Shizuoka 417-8610, Japan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2024968

complicated 4-pole Cole-Cole equations. Frequency dependent electromagnetic effects in the microwave range are efficiently analyzed by the finite-difference time-domain (FDTD) method through the auxiliary differential equation (ADE) technique [5], [6]. In our study [7] we have applied the least squares fitting technique to reduce Gabriel’s 4-pole Cole-Cole equations to simpler 2-pole Debye equations with a constant loss term. By adopting two poles for every tissue property, the large variation in the complex permittivity values at a low frequency regime of around 100 MHz is accurately fitted; as a consequence, the model provides good approximation over the wide frequency range from 100 MHz to 6 GHz. The frequency dependency of human tissues is implemented in the FDTD analysis more effectively by the Debye dispersion than by the Cole-Cole model, and the 2-pole Debye model provides almost perfectly accurate tissue properties in this particular frequency range compared either to the conventional frequency independent model or to a single-pole Debye model [8]. The previously reported frequency dependent tissue model in [9] is similar but less accurate; it contains no constant loss term, having only three parameters for each tissue property, and the frequency range is limited up to 1.5 GHz. Our present FDTD model maintains both simplicity and accuracy for a wider frequency range, and it has enabled for the first time the frequency dependent time-domain analysis of a whole human body under UWB radiation. The frequency dependent time domain analysis is a powerful tool to investigate eigenvalue problems, especially for very large scale problems where the common solution of matrix inversion is computationally too extensive. What is proposed in this paper is not only a matter of switching from a single-frequency model to a multi-frequency model of a human body, but it realizes the totally different possibility of solving effectively new and large scale problems. As for UWB radiation, the frequency spectrum is broadened and thus the frequency content is very weak. However, the radiation yields cumulative effects, and therefore, it is important to evaluate the energy absorption by time integration of fields even for UWB signals. Assessment of the safety of radiation is commonly done by using specific absorption (SA); SA is obtained by simply dividing the energy absorption (EA) by volume density. However, the accurate value of volume density for each human tissue type is not available to the authors, and thus we have used energy absorption rather than the specific absorption. The energy absorption is, however, an effective measure to evaluate the loss effects on any object, and it is a more fundamental physical quantity. Also for the same reason, only a local peak value of energy absorption, not an averaged value over a certain

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volume, is evaluated in this study; taking an average may dilute an important effect. To follow the official standard of safety assessment will be a future subject. In this paper, we propose an accurate and computationally efficient frequency dependent dielectric model of a whole human body, which will be useful for exploring the UWB radiation effects on us. We present a successful FDTD analysis of these radiation effects for particular applications as follows. However, lack of physiological knowledge restricts this study only to the evaluation from a standpoint of physical interaction, not being extended to the assessment of the safety of electromagnetic radiation. We have proposed an efficient time-domain calculation technique of energy absorption in general lossy media [7], and in this paper the method has been verified by an analytical solution of plane wave propagation in such media. We have exposed the human body model to spread spectrum radiation at 1 GHz center frequency modulated by typical binary phase shift keying (BPSK), whose frequency spectrum spreads broadly. The geometrical feature of the human body is read from a voxel whole-body model that has been generated from realistic nuclear magnetic resonance computer tomography (NMR-CT) data by Nagaoka et al. [3]. Computation was performed with our efficient in-house code that has been implemented to run on parallel cluster computers. Next we apply our frequency-dependent tissue models to the analysis of resonance states of the human body that is exposed to UWB radiation. Since the human body is composed of lossy media in an electromagnetically open environment, the resonance states are highly damped and radiative, thus the usual Fourier spectrum analysis may be insufficient in predicting the possible resonance states of human bodies. In order to suffice our requirement we have adopted a spectrum analysis technique based on an auto-regressive (AR) signal analysis referred to as the ’Sompi’ method [10]. Moreover, since our model is versatile with respect to pulse waveforms and geometrical configuration, we demonstrate the analysis of pulse propagation for a wireless body area network (WBAN), where digitally modulated signals propagate in the vicinity of the human body. This application has been recently proposed for the purpose of computer assisted rehabilitation [11]. Through these examples we clearly show the advantage of the large-scale frequency-dependent whole human body model, which may still be applied to various other problems regarding the interaction between a human body and digitally modulated electromagnetic radiation. II. TWO-POLE DEBYE DISPERSION MODEL OF HUMAN BODY In order to investigate the interaction between electromagnetic radiation and a human body at microwave frequencies, we first approximate Gabriel’s permittivity and conductivity data for each human tissue type [4] from 100 MHz to 6 GHz with a complex relative permittivity of 2-pole Debye dispersion and a conductive loss term

(1)

and are defined as the real part and where , respectively, with denotes electhe imaginary part of is permittivity at infinite frequency. tric conductivity and Notations and for are the strengths and the reis the laxation times of the Debye dispersion, respectively; electric conductivity at DC, the permittivity of vacuum, and the angular frequency. By applying the Newton method of root finding to the nonlinear least squares fitting of the Gabriel’s Cole-Cole data to (1), we have obtained the 6 parameters of (1) for all the 46 tissues in [4], as well as for other averaged tissue properties used in Nagaoka et al.’s body indices [3]; due to the limited space, only a part of them is listed in Table I, and the fitting results for representative tissues are plotted in Fig. 1; it is clearly demonstrated that the 2-pole Debye models exhibit good approximation to Gabriel’s data for the frequency range of our interest. To obtain the fitting parameters, initial values had to be chosen carefully; besides, the standard least squares procedure worked well. For the implementation of the model in the FDTD algorithm, we employ the ADE technique in terms of polarization [5], [6], which is flexible enough to include as many polarization terms as needed. Ampère’s law in the frequency domain is written for magnetic field and electric flux density as (2) where denotes a frequency domain function. The electric flux density is expressed in terms of electric field and polarizations for Debye dispersion and conductive loss as (3) For (3) the electric conductivity is usually included by the so-called semi-implicit formulation of the FDTD method, however, in this paper it is formulated as one of the polarization terms, which eventually leads to more consistent implementation. The polarization terms are given by (4) (5) denotes the Debye dispersion where electric susceptibility represented by the 2nd term of (1). All these frequency dependent terms are implemented into the ADE formalism for the FDTD calculation. In the present formalism and are available for every time step of the FDTD calculation, which allows simple calculation of energy absorption as shown in the next section. III. CALCULATION OF ENERGY ABSORPTION IN TIME DOMAIN We adopt the local energy absorption time domain by

calculated in the

(6)

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in this formulation includes all the terms of Debye polarization and the loss as in (3). Note that (6) gives an exact value of absorbed energy in general lossy media as shown in the following; and by applying Parseval’s theorem and considering that are real functions, (6) can be written in the frequency domain as

(7) where * denotes complex conjugate. We consider that together with (1), as well as the fact that the time functions of must satisfy causality, namely the real part and are an even function and an odd the imaginary part of function, respectively. Then (7) reduces to the usual integral of the energy absorption in the frequency domain,

(8) has vanished when it In (8) the contribution of the real part to . For accurate calculahas been integrated from tion of the time integral of (6), it is noted that the time derivative of is approximated by a backward difference, and is obtained by averaging the fields at the current and the previous time steps, which allows the evaluation of the two quantities exactly at the same time. Specific absorption is then obtained by by volume density. dividing calculation by comparing the FDTD result We verify the of damped oscillation in a lossy medium with an analytical solution. Plane wave propagation in the -direction in a medium of relative dielectric constant and conductivity is represented by its electric field as a function of distance and time

(9) where by

is an amplitude, and the propagation constant is given

(10) where is the speed of light. From this field distribution is is then obtained by integrating the squared field in time, and divided by the time duration of integration to obtain the energy , absorption rate

(11) Fig. 1. Representative Gabriel’s human tissue data and least squares fitted 2-pole Debye model.

where denotes total current density, i.e., the sum of conduction and displacement current densities. The flux density

For verification, the energy absorption rate was calculated with FDTD for plane wave propagation in a lossy medium and . For the FDTD analysis, the disof cretization in the -direction was 1 mm, and plane wave at 10 GHz was excited at the center of the 500 mm-long region for time duration of 20 ns. The calculation region was terminated

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TABLE I LEAST SQUARES FIT PARAMETERS OF 2-POLE DEBYE MODEL FOR HUMAN TISSUES. REPRESENTATIVE 15 SETS OUT OF 50 ARE SHOWN DUE TO THE LIMITED SPACE.

by 20 layer PML. The results of the distribution are compared in Fig. 2(a); it is found that the FDTD and the analytical results agree well. A snapshot of the field distribution at time 20 ns is shown in Fig. 2(b); one can see the field attenuates as it propagates away from the center, and the time integral of Fig. 2(b) yields exponentially decaying smooth variation, which of Fig. 2(a). The verificacorresponds to the distribution of tion in this section is valid also for lower frequencies because the EA algorithm is verified for an arbitrary material loss, and the implementation of polarization has been verified in our previous research of various nonlinear and dispersive materials e.g., [12], [13]; therefore, once complex permittivity and conductivity are . given, the computation will give correct distribution of IV. FDTD ANALYSIS OF ENERGY ABSORPTION UNDER SPREAD SPECTRUM RADIATION The present 2-pole Debye human body model was then exposed to spread spectrum radiation, and the energy absorption in the model was calculated. The time signal used in this study is generated tentatively by BPSK as shown in Fig. 3(a), and its frequency spectrum is shown in Fig. 3(b). The center frequency was chosen to be 1 GHz. In these figures it is noted that a spread spectrum signal has considerably large frequency content at lower frequency range. The spread spectrum signal was launched from a nearly-plane-wave source, which is a uniformly distributed electromagnetic source in front of the human body model, however bound by the perfectly matched layer (PML) absorbing boundaries. Therefore, the field propagates like a plane wave at first, but gradually spreads as it propagates. The computational in width, in thickness region is in height, and is surrounded by perfectly and matched layer (PML) absorber. The polarization of the source was chosen in the -direction, or in the direction of height of the human model. The FDTD calculation was carried out for the maximum time duration of 50 ns. The discretization was chosen

Fig. 2. Verification of EA calculation for plane wave propagation at 10 GHz in a simple lossy medium of EAR. " and  : = . (a) comparison of EAR for a plane wave in a lossy medium obtained by FDTD of (6) and by analytical form (11). (b) Snapshot of field at time 20 ns.

=2 E

= 02S m

to be , and in order to reduce the computational load without significant degradation in accuracy. The analysis code has been implemented for

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Fig. 4. EA distribution at the center of the body in the y -direction for (a) the single frequency model and for (b) the 2-pole Debye model. For quantitative evaluation see Fig. 5.

Fig. 3. (a) Tentative BPSK time signal and (b) its frequency spectrum.

efficient parallel computation; the memory requirement was approximately 12 GB, and the CPU time was typically 24 hours with 12 processors. distribution in Fig. 4 for the single We plot the results of frequency model (electrical constants at 1 GHz) and the 2-pole Debye model. The distributions do not appear much different. However, when they are quantitatively compared in detail, we have found at most 10% differences between them; the typical near the lower arms results are shown in Fig. 5(a) and (b) for and the knees, respectively. Further investigations have revealed is generally weaker for the 2-pole Debye model than that for the single frequency model. However, this tendency is not becomes greater for always the case, and at some locations the 2-pole Debye model than for the single frequency model. For accurate evaluation of the effect of the spread spectrum radiation, precise frequency dependent models must be used. V. ANALYSIS OF ELECTROMAGNETIC RESONANCE IN HUMAN BODY BY PULSE EXCITATION As the wireless digital communication like UWB and spread spectrum signals have a wide range of frequency spectrum, the analysis of resonating states in a human body under such radiation is only done with a frequency dependent model of the tissues. In particular as clearly indicated in the frequency dependent and of Fig. 1, the property of the human tissue changes drastically around 100 MHz to 1 GHz, and therefore the response to digital signal radiation also changes. Since the human body exposed to electromagnetic radiation is an open circumstance, the resonance states are highly damped

Fig. 5. Comparison of EA between the 2-pole Debye model and the single frequency model on two lines. (a) Near the lower arm at depth y = 160 mm and height z = 941 mm. The insets are expansions of the two large peaks. (b) Near the knee at depth y = 160 mm and height z = 350 mm.

and/or radiative. Therefore, the usual Fourier transform is not an effective technique to investigate the frequency responses of a human body. In order to alleviate the difficulty we have adopted the so-called ’Sompi’ spectrum analysis method [10], which is a versatile spectrum analysis method based on the auto-regressive (AR) signal analysis. Although describing the detail goes beyond the scope of this paper, the Sompi method resolves the resonance frequency, the attenuation rate, the initial amplitude and the phase of each damped oscillation from many signals superimposed; the spectral decomposition is done as the usual AR analysis but with solving a characteristic eigenvalue equation with respect to the complex frequency of damped oscillation. The Sompi method yields a discrete line spectrum, rather than a continuous spectrum like the Fourier transform or the

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Fig. 7. (a) The excitation pulse form. (b) Fourier spectrum of the excitation pulse. Figures (c) and (e) show the late part of the E field [V/m] of the FDTD time data (solid line) and fitting results by the Sompi analysis (’+’). For clarity the vertical axis is adjusted for each signal. Note that for spectral analysis the initial part of the excitation has been removed, and the fitting starts after 3.5 ns. Figures (d) and (f) are frequency spectra from 100 MHz to 6 GHz for the Fourier (solid line) and the Sompi (vertical lines with ’+’) spectrum analysis methods. The magnitudes of the spectra are normalized. (a) excitation. (b) excitation. (c) ankle. (d) ankle. (e) hand. (f) hand.

Fig. 6.

EA distribution by a pulse plane wave. The scale is in arbitrary units.

well-known maximum entropy method (MEM). These capabilities of the Sompi method are highly suitable for our purpose of

detecting resonance frequencies, damping factors, and strengths of the oscillation in the lossy and radiative circumstances. First, we exposed the human body model to the nearly-planewave pulse radiation coming from the front of the body with -directed (directed in height) polarization. In Fig. 6 we show distribution by the pulse radiation; in these figures, it the is clear that strong is observed in parts of the leg and arm bones. Therefore, we have investigated the frequency spectra of the time signals obtained at these points. For comparison we have performed both the Fourier and the Sompi spectrum analyses, the resulting time signals and the frequency spectra are shown in Fig. 7. The excitation pulse used in this analysis is a one-period sinusoidal pulse of Fig. 7(a), and the frequency content of the pulse is shown in Fig. 7(b); the input spectra have major contents around 1 to 2 GHz, without a DC content. This frequency range was chosen because major wireless devices currently use this frequency range, and we anticipated some resonances. Of course the center frequency can be raised as long as the major frequency content fits within the effective range of our model below 6 GHz. Time signals are then detected at the ankle and the hand. The initial part of the excitation was removed, and

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Fig. 8. Analysis region. The source (ankle) and the detection (inner-side wrist, front waist and head above ear) points are indicated by white dots. Unit in mm.

the time signals after 3.5 ns are used for the spectrum analysis. Time signals of Fig. 7(c) and (e) can be fitted by the sum of many damped oscillations obtained by the Sompi analysis, and both the frequency and the strength of each damped oscillation are plotted in the frequency spectrum of Fig. 7(d) and (f). The fittings of the time data seem to be very accurate, and the spectra show a certain consistency between the Fourier and the Sompi methods. For these cases the AR order of the Sompi analysis was chosen to be 50, which yields 25 spectral lines in the positive frequency region, but if the AR order reduces to around 20, the fitting of the time data starts to show discrepancy. It has been shown by the authors of [10] that when a time signal contains any Gaussian noise, the prediction of the initial amplitude degrades faster than that of the frequency. The present results may contain some uncertainties in the heights of the line spectra. From these results, it is obvious that some resonances exist in the human body, depending on the size of the part of the body. In particular at around 0.5 GHz relatively strong spectrum has been observed for the ankle and the hand of both male and female models (not shown). Even for the strongly damped signals, the Sompi method resolves major frequency contents in the FDTD time data; in Fig. 7(d) the largest frequency content appears at 0.36 GHz, and in Fig. 7(f) at 0.56 GHz; the second largest frequency content is over 10 dB down for each case. The resonance signals were in general much stronger for vertical polarization (directed in height) than horizontal polarization (directed in width, not shown). VI. PULSE PROPAGATION IN THE VICINITY OF HUMAN BODY FOR WBAN APPLICATION One of the practical applications of wireless communication in the vicinity of a human body is the so-called wireless body area network (WBAN) [11]. This has been proposed mainly for the purpose of collecting and analyzing biomedical information of patients under rehabilitation, and thus allowing real-time consultation by a doctor. By removing many electrical cables that can restrict patients’ freedom of movement, this application in fact improves largely the patients’ quality of life under treatment. In such applications, sources and detectors of wireless digital signals locate on various parts of the body. This application has been proposed only recently; in order for it to be widely used, the quality of the wireless signal transmission needs to

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be stable and fast. To ensure these requirements, good understanding of digital pulse wave propagation near or on the surface of the human body is essential. In this section, we assume a compact wearable communication device similar to those in [11], and suppose it is attached to a patient’s ankle as an electromagnetic digital pulse source. In Fig. 8 we show the geometrical configuration for the analysis, and the source and the detection points by white dots. The pulse source is a short dipole of length 8 mm placed on the outside surface of the ankle with the polarization (direction of the elecpolarization) or tangential tric field) either perpendicular ( ( polarization) to the surface of the ankle. The pulse signal is then detected at some locations of the FDTD electric field node on the surface of the body; as in Fig. 8 the ankle as a source point, the inner-side wrist, the front waist, and the head right above the ear as detection points. In order to properly observe the wave propagation outside the body, we have expanded the space by 200 mm in each axis ( , and ) for both negative and positive directions. The analysis region is then surrounded by PML; the analysis region has dimensions of in width, in height, and in depth. The whole space is discretized by a cube of 4 mm. The source signal is a single period pulse same as shown in Fig. 7(a). The center frequency of the pulse was 1 GHz. For the computation, 8 processors are used in parallel, and approximately 16 GB of memory is needed in total. The CPU time for each core was typically 3 hours for 15 ns time evolution. To clarify the effects of the dielectric property distribution over the whole body, we tested three cases, where (i) the 2-pole Debye dispersion is considered with the exact spatial distribution of the body tissues, (ii) the 2-pole Debye dispersion is considered but with the whole dielectric property represented by the averaged skin property (Table I), and (iii) the exact spatial distribution of the dielectric property of the body tissues is considered but with an approximation by frequency-independent dielectric constants defined at 1 GHz. The detected time signals are shown -polarization. The maximum ampliin Fig. 9 for the case of tude of the electric field detected 2 cm away from the source was 21.1 V/m. It is clearly seen in these results that the amplitude of the detected signal is largely different for the three cases: the electric fields for the 2-pole Debye skin model (ii) are larger by a factor of 1.5 to 3.2 than those for the 2-pole Debye exact model (i), and the electric field for the approximate frequency-independent model (iii) is larger by a factor of 1.1 to 1.7 than those of (i). These results implicate that the wave penetration into the body influences substantially the absorption of the field energy. For the analysis of pulse propagation of WBAN, the frequency dependent dielectric property must be taken into consideration in order to accurately evaluate the signal amplitude. Then for the 2-pole Debye exact model of (i), we launched from the same source a pulse with the other polarization of point as (at the ankle). The fields detected at the same points polarization are compared in Fig. 10; for (a) and (c), as field is perpendicular to the surface at the detection the field is perpendicular to the surface; in points, and for (b) the these results the perpendicular fields are an order of magnitude larger than the tangential fields. This tendency holds for the signals detected at other parts of the body. It is interesting to note

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Fig. 9. Pulse waveforms detected at wrist, waist and head for E polarization. (a) wrist. (b) front waist. (c) head.

that although the dielectric property of the body is highly lossy and complicated, the dielectric waveguide properties hold in the vicinity of the body surface. To illustrate these results, the electric field distribution is , and plotted in Fig. 11 for the field on the plane of . In both figures, the background in Fig. 12 for is plotted for the cross section of the dielectric constant body in (a) to recognize the outline of the human body in the fields from the following plots. Plots (b) and (c) show

Fig. 10. Pulse waveforms detected at wrist, waist and head. Comparison between E and E polarizations for the 2-pole Debye exact model of (i). (a) wrist. (b) front waist. (c) head.

source, and (d) and (e) show fields from the source. Along the outline of the body we see that the field propagates upwards, and it is clearly observed that the field near the surface of the body is much stronger for the perpendicular polarization (b) and (c) than for the tangential polarization (d) and (e). This tendency of the field distribution is often seen for dielectric waveguides, and these results are consistent with those of the time data in Fig. 10.

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= 360mm

Fig. 11. Electric field distribution on the plane of y at time 3 ns and 5 ns. (a) The distribution of " for the guide to the outline of the body. (b),(c) the snapshots of E field (electric field component in the direction of the width) with x-polarized source, and (d),(e), the E field with y -polarized source. The radiation source locates on the surface of the ankle. Color is in linear scale of V/m.

VII. CONCLUSION We have developed 2-pole Debye dispersion models of human tissues, and have applied the models to the analysis of the interaction between a whole human body and UWB radiation. The numerical analysis was performed efficiently with a parallel cluster computer. It has been found that the 2-pole Debye dispersion model gives generally weaker energy

Fig. 12. Electric field distribution on the plane of y ment of figures as in Fig. 11.

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= 408 mm. Same arrange-

absorption than the single-frequency model, while for some becomes greater for the 2-pole locations in the body Debye model. The frequency dependent tissue models have allowed the analysis of electromagnetic resonances in the human body. By combining with some spectrum analysis techniques, the frequency spectrum suggested some insights into the phenomena caused by UWB radiation. In particular, strong absorption of energy is observed in relatively small parts of the body such as the hand and the ankle. The analysis of pulse propagation in the vicinity of the human body has clearly indicated interactions between the pulse field and the surface of

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the body. The pulse field is an order of magnitude stronger for the polarization perpendicular to the surface of the body than for the polarization tangential to it; the field strength is also strongly affected by the penetration of the field into the body. It is again emphasized that large scale eigenvalue problems such as resonance detection in a whole human body can be solved effectively with a frequency dependent time domain analysis like those presented in this paper. ACKNOWLEDGMENT Authors acknowledge the National Institute of Information and Communications Technology (NICT), Japan, for providing them with the human body geometrical data.

Masafumi Fujii received the B.E. and M.E. degrees in electrical and electronic engineering from Kobe University, Hyogo, Japan in 1989 and 1991, respectively, and the Ph.D. degree from the University of Victoria, British Columbia, Canada, in 1999. From 1991 to 1998, he was with Sumitomo Metal Industries, Ltd., Japan, where he was involved in numerical analysis and measurement of microwave devices and materials. He was a Postdoctoral Research Fellow in the Department of Electrical and Computer Engineering, University of Victoria, from 1999 to 2000, and a Humboldt Research Fellow in the Institute for High-Frequency Engineering, University of Technology Munich, Germany, from 2001 to 2003. He is currently an Associate Professor in the Department of Electric, Electronic and System Engineering, University of Toyama, Japan. He spent his sabbatical in 2005 as a Visiting Professor at the Institute of High-frequency and Quantum Electronics, University of Karlsruhe, Germany. He coauthored chapter 9 in Computational Electrodynamics – The Finite-Difference Time-Domain Method, 3rd ed., Artech House, 2005. His research interest is in the area of numerical modeling of electromagnetic and optical fields in various materials of nonlinear, dispersive, and negative index properties, and applications of wavelets to scientific computation.

REFERENCES [1] M. Okoniewski and M. A. Stuchly, “A study of the handset antenna and human body interaction,” IEEE Trans. Microw. Theory Tech., vol. 44, pp. 1855–1864, Oct. 1996. [2] J. Wang and O. Fujiwara, “FDTD analysis of dosimetry in human head model for a helical antenna portable telephone,” IEICE Trans. Commun., vol. E83-B, pp. 549–554, 2000. [3] T. Nagaoka et al., “Development of realistic high-resolution wholebody voxel models of Japanese adult male and female of average height and weight, and application of models to radio-frequency electromagnetic-field dosimetry,” Phys. Medicine Biol., vol. 49, pp. 1–15, 2004. [4] C. Gabriel, “Compilation of the Dielectric Properties of Body Tissues at RF and Microwave Frequencies” Brooks Air Force, 1996, Tech. Rep. AL/OE-TR-1996-0037. [5] M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “Highorder FDTD and auxiliary differential equation formulation of optical pulse propagation in 2D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron., vol. 40, pp. 175–182, Feb. 2004. [6] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Boston, MA: Artech House, 2005, ch. 9, pp. 361–383. [7] T. Wuren, T. Takai, M. Fujii, and I. Sakagami, “Effective 2-Debye-pole FDTD model of electromagnetic interaction between whole human body and UWB radiation,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 7, pp. 483–485, Jul. 2007. [8] E. Zastrow, S. K. Davis, and S. Hagness, “Safety assessment of breast cancer detection via ultrawideband microwave radar operating in pulsed-radiation mode,” Microw. Opt. Technol. Lett., vol. 49, no. 1, pp. 221–225, Jan. 2007. [9] O. P. Gandhi and C. M. Furse, “Currents induced in the human body for exposure to ultrawideband electromagnetic pulses,” IEEE Trans. Electromagn. Compat., vol. 39, pp. 174–180, May 1997. [10] T. Matsuura, Y. Imanishi, M. Imanari, and M. Kumazawa, “Application of a new method of high-resolution spectral analysis, ’Sompi’, for free induction decay of nuclear magnetic resonance,” Appl. Spectroscopy, vol. 44, pp. 618–626, 1990. [11] E. Jovanov, A. Milenkovic, C. Otto, and P. C. de Groen, “A wireless body area network of intelligent motion sensors for computer assisted physical rehabilitation,” J. Neuro Eng. Rehab., vol. 2, no. 6, pp. Doi:10. 1186/1743-0003-2–6, 2005. [12] M. Fujii, N. Omaki, M. Tahara, I. Sakagami, C. Poulton, W. Freude, and P. Russer, “Optimization of nonlinear dispersive APML ABC for the FDTD analysis of optical solitons,” IEEE J. Quantum Electron., vol. 41, pp. 448–454, Mar. 2005. [13] M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP/InP racetrack micro-resonators,” IEEE Photon. Technol. Lett., vol. 18, pp. 361–363, Feb. 2006.

Ryo Fujii received the B.E. degree in electric, electronic and system engineering from University of Toyama, Japan, in 2007, where he is currently working toward the M.E. degree. His Master’s degree research theme is on the numerical modeling of electromagnetic fields in frequency dispersive complex media.

Reo Yotsuki received the B.E. degree in electric, electronic and system engineering from University of Toyama, Japan, in 2007, where he is currently working toward the M.E. degree. His Master’s degree research theme is on the numerical modeling of optical and electromagnetic fields in metamaterials.

Tuya Wuren received the B.E. degree in electronic and science engineering from Liao Ning University, China, in 1994 and the D.E. degree in electrical and electronic engineering from Muroran Institute of Technology, Japan, in 2005. From 1994 to 1998, she was a Technician at Cai Hong Jituan TV Factory, Inner Mongolia, China. From 2005 to 2007, she was a Postdoctoral Researcher at the University of Toyama, Japan. She is currently a Research Assistant at Toyohashi University of Technology, Japan. Her current research interest is in the area of microwave oscillators and microwave planar filters.

Toshio Takai received the B.E. and M.E. degrees in electric, electronic and system engineering from University of Toyama, Japan, in 2004 and in 2006, respectively. His thesis theme was on the numerical modeling of interaction between a human body and electromagnetic waves. He is currently with Nippon Paper Industries, Ltd.

Iwata Sakagami (M’81) received the B.S., M.S., and Ph.D. degrees in electronic engineering from Hokkaido University, Sapporo, Japan, in 1972, 1977 and 1980, respectively. After working at Mitubishi Electric Corp., he completed the doctoral program and became a Research Associate at the Institute of Applied Electricity, Hokkaido University. He became an Associate Professor at Kushiro National Technical College in 1987, and at Muroran Institute of Technology in 1988. He has been a Professor at University of Toyama, Japan, since 2000. He has been engaged in research on size-reduction techniques of distributed constant circuits and their time domain responses. Dr. Sakagami is a member of IEICE.

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Efficient Numerical Modal Solutions for RF Propagation in Lossy Circular Waveguides Ronald W. Moses, Jr. and D. Michael Cai

Abstract—The propagation of electromagnetic waves is reconsidered in the context of cylindrical tunnels through material of uniform electrical properties. In the absence of internal conductors, the wave solutions reduce to well-known analytic expressions, requiring the solution of a single transcendental equation in complex phase space. Any given tunnel configuration has a wide range of solutions in phase and frequency space. Identifying these solutions can be a significant numerical challenge. Both the NewtonRaphson method and the winding number technique have been described in previous publications, and each can be prone to missing solutions. This paper reviews these two solution techniques and develops a combination of the two to provide a fast and efficient procedure for locating the roots in well-defined regions of phase space. Index Terms—Electromagnetic, propagation, tunnels, waveguides.

I. INTRODUCTION

T

HE propagation of radio waves in lossy waveguides and tunnels has been researched extensively for many years, as can be seen in the detailed book by Wait [1]. The mathematics used to model waveguides is essentially the same as that needed to model radio-frequency (RF) propagation in simple tunnels. The presence or lack of conductors in a waveguide or tunnel is a key driver in the nature of the solutions for a particular application, Delogne [2]. When conductors pass through a waveguide or tunnel, the simplest propagation modes are surface-guided waves following the conductor, typically enabling long-range transmission. A tunnel with a core conductor can act like a coaxial cable, propagating waves at a nearly constant speed, regardless of frequency. A tunnel or waveguide without internal conductors is subject to very different wave patterns, resulting in a much more complex propagation analysis. Holloway et al. [3]. presented a comprehensive study of RF propagation in circular structures embedded in lossy surroundings. The work of Holloway et al. is the basis for this paper, where we discuss application of their computational techniques and present refinements gleaned from our work. II. THE MODEL When a tunnel has no discrete conductors inside, there is no material to channel the axial current as mentioned above. NevManuscript received November 06, 2008; revised April 26, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported by the DOE Nonproliferation and Verification R&D Office (NA-22). The authors are with the Space Data Systems Group, Los Alamos National Laboratory, Los Alamos, NM 87545 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.2009.2037758

Fig. 1. The computed electric field lines of a transverse-magnetic (TM) wave propagating along a circular tunnel are shown in (a), while the transverse-magnetic field lines are illustrated in (b). In (a), only the E field lines of half a wave are shown. Each E field line starts on the wall and returns to the wall, while confined to a  z plane. In (b), the corresponding orthogonal B field lines are given. Note that while there is an actual current along a central conductor in a coaxial cable, the corresponding mechanism of the TM waves is just the displacement electric flux along the axis, in the core of the tunnel, as illustrated in (a). The time-varying displacement flux of (a) induces the magnetic field of (b).

0

ertheless, RF waves can propagate in manners similar to classic waveguides. We introduce a schematic of a relatively simple transverse-magnetic (TM) wave in a round tunnel in Fig. 1. If one had the luxury of highly conducting walls, standard waveguide theory could be used. A good reference to this is the textbook by Panofsky and Phillips [4]. Here, the theoretical basics of waveguides are presented clearly, especially the concept of the cutoff frequency. An intuitive perspective on this may be gained by thinking of the waveguide as a resonant cavity for transverse standing waves, where the tangential electric field at the wall approaches zero. One can expect that no standing waves will be contained with wavelengths greater than about twice the . Hence, greatest transverse dimension of the waveguide, no waves will be contained in the guide with frequencies lower , where is the speed of light. Applithan cations close to the cutoff frequency are not particularly useful, as dispersion and attenuation can be significant in that regime. In addition to these rudimentary caveats, real tunnels cannot be expected to have highly conducting walls, so a significantly more refined theory is needed for practical modeling of actual tunnels. The step from modeling classic waveguides to consideration of more realistic tunnel conditions is done well in the paper by Holloway et al. [3]. Here, a Fourier-Bessel expansion is developed for electromagnetic waves in a circular tunnel, buried in a material of uniform electrical properties. This provides a good start for understanding wave propagation in tunnels without conductors inside. Although this is a relatively idealized problem, it still requires sophisticated numerical analysis. We have written a code in Mathematica to automate much of the solution process. We report on this code below.

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Following Holloway et al., a cylindrical coordinate system is , with a frequency, , and defined in terms of an azimuthal mode number, , chosen. In addition to the radius of the tunnel, , the electrical properties of the surrounding material are expressed in terms of a complex permittivity

Holloway et al. [3] describe the “winding number” approach to locating solutions of (3). By taking the following contour integral around a region in space

(5) (1) where is the real part of the dielectric constant, and is the effective resistivity of the wall. Since there may not be a true time-independent resistivity applicable to RF frequencies, one often drops the concept of a separate resistivity and considers a dielectric constant, , with both real and imaginary components. For this paper, the magnetic permeability of the wall is taken . The electric to be the same as that of free space, and magnetic fields and have azimuthal dependence that is sinusoidal in terms of . The axial and time dependences are exponential, (2) with having both real and imaginary components, signifying the axial attenuation and propagation respectively. Inside the tunnel, the fields have a radial dependence that goes as , while outside they go as the Hankel the Bessel function . The radial wave numbers and are for function inside and outside the tunnel respectively. With frequency and azimuthal mode number specified, one must solve for , , and . Solutions are obtained when the electromagnetic fields are correctly matched at the boundary between the tunnel and the surrounding earth or structure. This is known as meeting the Fresnel conditions. In the Holloway paper, the Fresnel conditions are met by solving a transcendental equation cast in the form

one can indeed find the number of solutions, , within the contour, . Holloway et al. use an algorithm described by Singaraju et al. [5]. and modified by Tijhuis and van der Weiden [6] to find the roots. Although we find this approach interesting, it is our opinion that it is unduly complex. Here, we report on a technique that combines a simple numerical search with the winding number method as a check, to give an effective tool for accurately finding large numbers of roots. Unfortunately, there are often singularities of (3) close to the solutions. Every singularity enclosed within a contour of inteto (5). If the presence of singularities was not gration adds recognized, one could miss counting solutions. For the parameters that we considered, there are typically singularities of (3) running along the real axis of space. Hence, in establishing the numerical integration of (5), one would encounter the singularities if he were to integrate along the positive real axis defined as the lower bound of the acceptable region of (4). When the lower limit of integration is raised above the real axis, it is easy to be too close to the singularities, leading to poor numerical results. Or, one may raise the line of integration too much and pass above some of the zeros, missing solutions. Consequently, we have chosen to develop a search algorithm to find roots satisfying the restrictions of (4). Then we use the winding number approach as a valuable check on the results. III. NUMERICAL METHOD The region of acceptable solutions is well-defined in as any expression satisfying the relation

space

(3) (6) ([3, Eq. 14]). This is a transcendental equawhere tion involving numerous quotients of Bessel and Hankel functions and their derivatives. The solutions, , are complex numbers representing a wide range of radial modes inside the tunnel. Not all solutions of (1) are physically acceptable for our applications, for example we are only interested in waves starting inside and at one end of the tunnel, then propagating along and out of the tunnel into the surrounding earth. Meanwhile, solutions of (3) can propagate both outward and inward with respect to the tunnel. Holloway et al. [3] provide detailed criteria restricting the use of space to regions that allow only solutions viable for our applications; see [3, Fig. 5]. Acceptable solutions occur only in the first quadrant of space, inside the hyperbola defined by the equation

(4) [3, Eq. (22)].

Modal solutions do exist outside these limits, but they are not useful to the present application. For example, modes above the limit of (6) represent waves that start outside the tunnel and propagate inward. Although (6) forms a semi-infinite plane, experience can indicate local regions where applicable modes are found. Our first step in solving for the roots and singularities of (3) is to choose upper and lower bounds for to go along with (6) and form a region of computation (ROC). Next, a rectangular grid of points is placed on the ROC, where the vertical is uniform. The horizontal spacing in spacing in is linearly increasing out from . This is achieved over the ROC. A by choosing uniform spacing of sample grid is shown in Fig. 2 with only every tenth horizontal point shown. The variable spacing is used, because singularitend to be nearly uniformly distributed ties and zeros of rather than in , then this spacing leads to better in numerical coverage of the ROC. Once the computational grid is

MOSES AND CAI: EFFICIENT NUMERICAL MODAL SOLUTIONS FOR RF PROPAGATION IN LOSSY CIRCULAR WAVEGUIDES

Fig. 2. A computational grid of points in w space used for the subsequent computation of TM modes. For visual clarity, only every tenth point is shown in the Re[w ] direction.

chosen, is computed for each point on the grid and stored in a matrix. Next, interior points of the ROC grid are searched for the aband compared with the values at solute value eight adjacent points. When the central point of a nine-point rectangle is either a relative minimum or maximum of the nine values, its location is stored in a separate table as a locator of a root or singularity respectively. When this search is complete, the tables of locators serve as starting points for Newton-Raphson numerical solutions for the zeros and singularities. This numerical method is usually very effective in finding , but they can be missed if the roots and singularities of grid spacing is too coarse. For example, when placing a grid , it is not guaranteed on a region of steep gradient of that each root or singularity will have a nine point rectangle on . As grids are made that grid with a central extremum of successively finer, such an extremum is ultimately a certainty. We use the winding number method to check our computations and make changes to the grid if necessary. This check would be straight forward if it weren’t for the fact that there are usu, ally a set of singularities lying on the positive real axis of and there is often a set of roots relatively close to the same axis. You want to choose a path of integration in the ROC that encloses the likely region of solutions, without passing too close to roots or singularities and impairing the numerical line integration process. There could be numerous approaches to this integration; here it is treated as follows. First, it is easier to perform line integraspace than on the hytions around the rectangular ROC in perbolic contours of interest in space. Therefore, we replace (5) with the following line integral to determine the winding number:

(7) which is the mathematical equivalent of (5). We start at the lower left-hand corner of the ROC in space as seen in Fig. 2. A straight line is drawn to each of the root solutions in our table, and the minimum angle between those lines and the positive real axis where the singularities lie is computed. We do a numerical line integral for (7) from the lower left corner of the ROC along

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Fig. 3. The local minima and maxima of jG[w ]j are found on the computational grid for 100 MHz TM waves in a 2 m radius tunnel, where the walls have  = (10 0 i) . The Newton-Raphson technique is used to precisely locate the roots (small dots) and singularities (large dots on the real axis). A line integration is taken around the observed roots (solid line) to test whether or not all roots have been found.

this line to the right-hand side of the ROC. We then complete the line integral of (7), looping around the remaining edges of the ROC. An example of this procedure is shown in Fig. 3. If the winding number does not match the number of numericallydetermined roots minus singularities inside the line integral, one can assume the search grid is not fine enough, and a solution has been missed. In our experience, no zeros have been seen off the . We have encountered a few cases of positive real axis of missed roots, that have been corrected by selection of a finer mesh. If one finds the density of modes required to numerically locate all the modes in the ROC, it is possible to subdivide the ROC with horizontal line integrals and use the winding number to eliminate regions from computation that are free of roots and singularities. IV. COMPUTATIONAL RESULTS As a first example, consider a 100 MHz TM wave in a 2 m for (1), and radius tunnel, where the walls have takes the place of the explicit resistivity the imaginary term term in (1). The grid in Fig. 2 is consistent with the upper and lower boundaries imposed by (6). The numerical procedure of the preceding section was used to find the roots and singularities in this ROC. These are displayed in Fig. 3 as small and large dots respectively, with six singularities on the positive real axis and the roots as six dots above that axis. Also shown of in Fig. 3 is the path of integration for the line integral used to check the winding number for the roots in the ROC. Indeed, six roots are found by both numerical techniques. One can see how the lower segments of the line integration must be above the real axis to avoid hitting the singularities. It must also be noted that the ROC is bounded on the left by a negative value . Although we have not seen roots in the negative half of space, there are circumstances where singularities are seen at the origin. Hence, it is prudent to avoid crossing the origin with the line integration for the winding number. The results in Fig. 3 are reformatted in space and plotted in Fig. 4 for wave motion in the positive axial direction. Here, the ROC is bounded by the axes, the limiting hyperbolic curve and our arbitrarily determined edges taken from the left and right sides of Fig. 3. The lower segment of the line integration from

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Fig. 6. Isometric plots of the normalized radial electric displacement field, D , as a function of z and  for the (a) left-most and (b) second modes in Fig. 5.

Fig. 4. The computational grid, roots, singularities, and line integration of w space in Fig. 3 are displayed in w space. Large dots are singularities, and small dots are roots.

Fig. 7. The roots (small dots) and singularities (large dots) for 1 GHz TM waves i  , are plotted in the w in a 2 m radius tunnel, with walls having  plane (a). The corresponding solutions for are shown in (b).

= (10 0 )

. By contrast, the group velocity speed of light, at which information is propagated is

(8)

Fig. 5. The axial attenuation rate, , and wave number, , for the solutions in Figs. 3 and 4. These are the real and imaginary components of .

Fig. 3 can be seen close to the axes and between the roots and singularities. The rationale for choosing the horizontal limits in Fig. 3 can be understood better by looking at the placement of the solutions for as shown in Fig. 5. The left-most solution in this plot has , while the next an attenuation rate of point has a value of . This means that at 100 MHz in this tunnel the longest attenuation length is about 5 m, and the rest are much shorter than 1 m. The higher modes are essentially evanescent waves, providing no meaningful propagation. This comparison of the first and second modes is seen in Fig. 6 where the radial electric displacement field, , is plotted as a function of and . The lowest mode is seen to propagate along the axis of the tunnel as it dissipates radially away from the tunnel. The second mode is concentrated near the left-most plane of the system and shows little propagation along the tunnel. Operating at 100 MHz, one is only marginally above the cutoff frequency of about 37 MHz, so it is no surprise that wave propagation is not very effective. One may also note that the axial phase velocity, , is for the lowest mode, is higher than the

When the group velocity is determined numerically for the , significantly less than . lowest mode, it is Assuming the dielectric and magnetic properties of the tunnel walls are unchanged, the range of propagation greatly increases as the wave frequency is raised. When the RF frequency is 1 GHz, the plots of and change to those shown in Fig. 7, and . This the lowest mode has an attenuation rate of attenuation length of about 70 m is of much greater interest for potential transmission applications. Also, all of the modes have ; hence, the axial phase velocity and the group velocity are now approximately the speed of light. The transverse electric (TE) modes are similar to the TM modes, with the electric and magnetic fields being interchanged. The same tunnel conditions as above were used to compute the first six modes for a 100 MHz TE wave, and the results are shown for and in Fig. 8. The local energy dissipation per , and unit volume in the surrounding material is given by the azimuthal electric field of the TE modes is not the same as electric field of the TM modes. Hence, the TE and TM the modes are similar in appearances, but their different dissipation rates lead to somewhat different solutions for and . In the present example, the lowest mode has an attenuation length of 16 m, and that becomes 0.6 m in the next mode. The TE modal solutions for a frequency of 1 GHz are shown in Fig. 9. Here, the attenuation lengths are 730 m for the first mode, and they gradually come down to 25 m for the sixth mode, indicating a much

MOSES AND CAI: EFFICIENT NUMERICAL MODAL SOLUTIONS FOR RF PROPAGATION IN LOSSY CIRCULAR WAVEGUIDES

Fig. 8. The roots (small dots) and singularities (large dots) for 100 MHz TE i  , are plotted in waves in a 2 m radius tunnel, with walls having  the w plane (a). The corresponding solutions for are shown in (b).

= (10 0 )

Fig. 9. The roots (small dots), singularities (large dots), and values are given in (a) and (b) respectively for 1 GHz TE waves in the same tunnel as above.

Fig. 10. The roots (small dots), singularities (large dots), and values are shown in (a) and (b) respectively for 100 MHz hybrid waves in the same tunnel as above.

greater capability of transmission for TE modes in the present example. Solutions that have azimuthal dependence are called hybrid modes. These are the roots of (3) that have angular dependence with being nonzero. Snitzer [7] has categorized the two and according different types of hybrid modes as to the dominant Bessel functions that contribute to the transmode, , has transverse fields. The lowest order verse fields formed by orthogonal, relatively straight, line patis characterized by more circular patterns. terns, while See Snitzer [7, p. 497] for details. Once again, our numerical method was applied to a 2 m radius , for 100 MHz waves, now having tunnel with . Considering the same ROC as used before, the computed results for and are shown in Fig. 10. These results contain both the and modes as described by Snitzer and Holloway et al. The left-most solution in Fig. 10(a) and (b) is mode and the next solution is the mode. The the first of these roots has an attenuation length of 10 m, and the second has 2.7 m. Beyond that, the attenuation lengths go from 60 cm down, once again like evanescent waves. When the frequency is increased to 1 GHz, the modal structure becomes considerably more complex as shown in Fig. 11.

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Fig. 11. The roots (small dots), singularities (large dots), and values are shown in (a) and (b) respectively for 1 GHz hybrid waves in the same tunnel as above. Note that there are now 15 roots in these plots, seen in three separate arcs in the plot. This indicates the variety of hybrid modes one may see at higher frequencies with more admissible solutions.

. The attenNow we see 15 modes for the same range of uation lengths come from 331 m for the left-most mode, down to 15 m for the right-most. Correspondingly, the phase velocity . The winding goes from very close to up to number test is valuable in computing such a complex mode structure, because it indicates missed modes when the computational grid is not sufficiently fine. This was the case while developing Fig. 11, so the grid spacing was reduced until the tabulated results matched the winding number. Clearly, waves with azimuthal dependence cover a very large parameter space. Modes with the lowest radial and azimuthal wave numbers tend to propagate the farthest along the tunnel. It may be necessary to assemble a great many modes to model a wave close to its source, but only a relatively small number of its modes are likely to propagate any significant distance. V. CONCLUSION We have described and demonstrated a simple and thorough approach to solving the transcendental equations associated with RF propagation in circular tunnels and waveguides surrounded by material of uniform electrical properties. Holloway et al. [3] and others have promoted the winding number method of locating roots over the Newton-Raphson search method alone. We demonstrate what may be a simpler approach, whereby phase space is covered with a grid of computational , is test points upon which the transcendental expression, are identified computed. Local minima and maxima of as certain indicators of roots and singularities respectively. Both roots and singularities are then computed using the Newton-Raphson method. The line integration of the winding number method is taken around the region of observed roots, excluding the singularities. If the winding number of roots exceeds the number of roots found numerically, the computational grid density is increased until a match is realized. We do not attempt to present a wide-ranging study of mode structures. The numerical technique is illustrated with a set of examples having TM, TE, and hybrid modes in a 2 m radius tunnel carrying 100 MHz and 1 GHz waves. ACKNOWLEDGMENT The authors would like to express their appreciation to Dr. S. Knox of Los Alamos National Laboratory for encouraging this effort

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REFERENCES [1] J. R. Wait, Electromagnetic Wave Theory. New York: Harper and Row, 1985. [2] P. Delogne, “EM propagation in tunnels,” IEEE Trans. Antennas Propag., vol. 39, pp. 401–406, 1991. [3] C. L. Holloway, D. A. Hill, R. A. Dalke, and G. A. Hufford, “Radio wave propagation characteristics in lossy circular waveguides such as tunnels, mine shafts, and boreholes,” IEEE Trans. Antennas Propag., vol. 48, pp. 1354–1366, 2000. [4] W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism. Reading, MA: Addison-Wesley, 1962. [5] B. K. Singaraju, D. V. Giri, and C. E. Baum, “Further developments in the application of contour integration to the evaluation of the zeros of analytic functions and relevant computer programs,” Air Force Weapons Lab, Mathematics Note, vol. 42, Mar. 1976. [6] A. G. Tijhuis and R. M. van der Weiden, “SEM approach to transient scattering by a lossy, radially inhomogeneous dielectric circular cylinder,” Wave Motion, vol. 8, pp. 43–63, 1986. [7] E. Snitzer, “Cylindrical dielectric waveguide modes,” J. Opt. Soc. Amer., vol. 51, no. 5, pp. 491–498, 1961. Ronald W. Moses, Jr. was born and raised in Ames, Iowa. He received the B.S. degree in physics from Iowa State University, Ames, in 1963 and the Ph.D. in physics from the University of Wisconsin, Madison, in 1968. As an undergraduate at Iowa State, he started work in particle optics by computing proton orbits in a small cyclotron, with the support of an NSF Undergraduate Research Grant. At Wisconsin, his research was supported by a NASA Fellowship, and he worked in the field of electron optics as related to the electron microscope. After receiving his Ph.D. he went on to hold an NSF Postdoctoral Fellowship at the Cavendish Laboratory, Cambridge University, U.K., followed by research positions at the Technische Hochschule Darmstadt, Germany, and the University of Chicago. He returned to the University of Wisconsin in 1973 where he began energy-related work with interests in magnetic

energy storage and fusion power. He became a staff member at Los Alamos National Laboratory in 1976. There, he worked on the theory of magnetic confinement fusion during his first 14 years at Los Alamos National Laboratory (LANL), Los Alamos, NM. After the fusion program was greatly scaled back at LANL, he went on to work on a variety of additional subjects in the fields of: plasma physics, ionospheric physics, electromagnetic detection of buried objects, synthetic aperture radar, pulsed power, particle optics, fluid dynamics, ductile failure of metals, and energy production and storage. Over the last 17 years much of his work has emphasized RF propagation in a multitude of configurations and in a variety of media: underground, air, and the ionosphere. He retired from LANL on July 1, 2005, and is now an independent consultant, currently working with LANL and Sandia National Laboratories, Livermore. He is co-founder of a small consulting company, Applied Science Enterprises, LLC. Dr. Moses is a member of American Physical Society and the American Geophysical Union.

D. Michael Cai received the Ph.D. degree in biomedical engineering from the University of Minnesota, Minneapolis, in 1995. He continued his postdoctoral research on biophysics at the School of Medicine, Johns Hopkins University, Baltimore, MD. From 1997 to 2001, he was with the Center for Adaptive System Application, Inc., Los Alamos, NM, and HNC Software (now a subsidiary of Fair Isaac Corporation) at San Diego, CA, where he developed several new algorithms and operations for financial, insurance, and pharmaceutical companies. He is currently a Project Leader in the Space Data System Group, Los Alamos National Laboratory, Los Alamos. His research interests include statistical learning, modeling, simulation, data mining, data fusion, distributed sensor networks, and high performance computing. Dr. Cai is a member of the American Geophysical Union (AGU) and the International Society for Optical Engineering (SPIE). He was a member of the organization committee and a session chair for SPIE Defense & Security Symposium.

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Acceleration of Ray-Based Radar Cross Section Predictions Using Monostatic-Bistatic Equivalence Hermann Buddendick, Student Member, IEEE, and Thomas F. Eibert, Senior Member, IEEE

Abstract—An approach is presented to simulate the monostatic scattering properties of complex shaped realistic objects in a very efficient way. To achieve this, the calculation of the radar cross section (RCS) in the high frequency regime based on the well known shooting and bouncing rays (SBR) technique is considerably accelerated by the use of the monostatic bistatic equivalence principle. Instead of performing independent simulations for all required aspect angles, the concept is based on the idea of additionally exploiting bistatic information for some neighboring aspect angles. This information is obtained relatively cheaply during the SBR process and it can be favorably exploited under certain conditions, mainly that the bistatic angle is small and the object is sufficiently smooth. In this case, the results of the geometrical ray tracing, which consumes a large part of the computational resources for complex shaped objects is reused multiple times with only low additional computational resources. The basic principles and benefits of the methodology are discussed as well as its limitations and drawbacks. Different generic simulation examples are used to show the general applicability of the method and to examine the degradation of the results depending on the applied bistatic angle. Based on these experiences, a passenger car model is simulated at 10 GHz and a considerable reduction of the computational effort by a factor of 32 is estimated for the complete multiaspect simulation problem. Index Terms—Electromagnetic scattering, physical optics, radar cross section (RCS), ray tracing.

I. INTRODUCTION

HE radar cross section (RCS) is an important characterization of the electromagnetic scattering properties of large and complex objects and in many areas it is desired to have a good estimation of it for various aspect angles. It is common practice to use this quantity in the fields of radar system design and dimensioning, as well as for radar algorithm development and validation [1]. Unfortunately, it is computationally very demanding to perform the required simulations for complex shaped objects like vehicles or airplanes at very high frequencies, even if suitable high frequency approximations such as the widely used geometrical optics (GO) and physical optics (PO) are used.

T

Manuscript received November 24, 2008; revised March 04, 2009. First published December 04, 2009; current version published February 03, 2010. H. Buddendick is with the Institute of Radio Frequency Technology, Universität Stuttgart, 70550 Stuttgart, Germany (e-mail: [email protected]. de). T. F. Eibert is with the Lehrstuhl für Hochfrequenztechnik, Technische Universität München, 80290 München, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037692

In this work, we focus on the acceleration of a ray-based asymptotic simulation approach, namely the shooting and bouncing rays (SBR) technique, first introduced in [2] and now widely used for RCS predictions of arbitrarily shaped objects [3]–[8]. Based on a well known relationship for the scattering properties at different aspect angles (often referred to as monostatic bistatic equivalence principle, see e.g., [9]) we present an approach which enables us to accelerate considerably these ray-based multiaspect RCS simulations. Basically, in the SBR approach rays representing the incident fields in a GO manner are used to determine the resulting equivalent surface currents and finally the resulting field contributions at the given observation points are derived by a PO integration. This is explained in Section II with special focus on those aspects that are important for the application of the bistatic monostatic equivalence principle, which itself is introduced in the first part of the following Section III. In this context, previous work and how it relates to the present study is also discussed. The second part of Section III demonstrates how, with a proper phase correction, contributions related to deviating illumination and observation angles are utilized for an approximation of the desired monostatic RCS. In this way, from each determined ray, contributions to several neighboring monostatic receivers are obtained, and therefore the total number of rays required for a proper sampling of the objects can be reduced significantly. As the overall computational cost for a multiaspect problem clearly depends on the number of aspect angles to be considered, the angular sampling is explained in Section IV and the important relation between sampling rate and the efficiency of the bistatic acceleration is stressed. In Section V some simple simulation examples are presented to demonstrate the applicability of the approach as well as its limitations. Furthermore, a simulation example of a passenger car is given to show the possible acceleration for a more realistic simulation scenario. II. THE SHOOTING AND BOUNCING RAYS TECHNIQUE For very high frequencies, scattering problems for large objects cannot be treated using exact numerical techniques like the method of moments (MoM) due to the arising computational complexity. To analyze the scattering properties of electrically large and complex objects asymptotic, high frequency methods are to be applied. In this paper we use an approach which is based on an SBR algorithm to determine the local incident fields on the scatterer and the PO to determine the scattered field contributions. In the SBR approach, see for example [2], a dense grid of rays is launched in the direction of the object and the rays are traced with one or several bounces (reflections) until

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they exit the scenario, so that the associated fields at the interaction points determine the equivalent surface currents. Nowadays the concept of SBR is widely applied to compute the scattering fields of large objects in scientific as well as in commercial codes. Some examples can be found in [4], [7], [8]. Considering the computational complexity, it is important to see that the SBR algorithm splits into two parts: • Ray tracing is used to find the fields on the objects surface according to the GO principle; • PO integration determines the scattered fields based on the surface currents. Usually the ray tracing part of the algorithm consumes most of the computational resources if detailed object models are considered. For typical applications the complexity of the geometric model can easily be in the order of a few hundred thousand triangles. This clearly indicates the need for a very efficient implementation, which, in the presented case, is based for example on a uniform spatial subdivision [10], [11]. that are directed towards the object The number of rays must be large enough to capture all relevant object details sufficiently for the coherent superposition of the field contributions. Starting from a regular grid of ray directions ( – and –direction) a random offset is added to each ray. However, each grid cell still contains one ray and finally a uniform distribution is achieved. By this, aliasing effects at surface borders or wedges are avoided while a minimum ray density is maintained in the whole angle range [8]. Due to this stochastic ray launching process, the results of two independent simulation runs with identical parameter settings differ from each other and include a stochastic component as well, called SBR noise here. With an increasing number of rays this deviation decreases and the simulation results converge. Note that the negative impact of the SBR noise is dominant for weak scattering fields. Whereas strong scattering fields are obtained due to in-phase superposition of contributions from large object features and are therefore stable in a statistical sense for a lower number of rays. In many cases, this enables to capture the main scattering characteristics efficiently with a relatively low number of rays and a more resolving simulation setting is required to predict weak scattering fields. For simulating complex shaped objects, it is feasible to determine the required number of rays through observing the SBR noise level in relation to the expected simulation accuracy and to adjust the ray density if required. For each interaction point of the ray on the object, the PO scattering contributions are determined. A local coordinate system according to Fig. 1 is used and the electric surface currents are derived from the field components in the following form:

Fig. 1. Local coordinate system used for the PO integration around the interaction point of a ray on a plane surface element.

from a PEC surface element for a given observation point can be expressed as distance

at

(3) where is the free-space wave number, and are the surface integrals of the electric currents depending on the observation direction. The subscripts denote the corresponding polarization. The integration is carried out numerically as a sum of contributions of all rays. For the following considerations and without any loss in generality, the observation points are restricted to be within the horizontal plane in the global coordinate system (only -dependence). Applying far-field approximations, the scattered field can thus be written as

(4) being the wave vector directed towards the obwith the position vector of the hit point of the servation point and –th ray on the surface. Note that depends on the direction of the incident field . III. PROPOSED ACCELERATION APPROACH A. Monostatic Bistatic Equivalence Considering the scattering properties of an object and a geometrical setup as shown in Fig. 2, a simple and well known relationship between bistatic and monostatic RCS values exists [9] and can be expressed in the following form:

(1) (2)

(5)

with the free space wave impedance and the vectors as well as the electric and magnetic field components being defined in Fig. 1. Following the notation in [12], the far-field scattering

and being the monostatic and bistatic RCS, respecwith the angle of incidence and the bistatic angle. tively, and This means that the bistatic RCS equals the monostatic RCS

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Fig. 3. Symmetric distribution of temporary bistatic receivers around the monostatic receivers and final monostatic aggregation.

Fig. 2. Equivalence of bistatic and monostatic RCS.

observed at the bisecting angle (see Fig. 2), for a different carrier frequency . The underlying assumptions for this identity are the consideration of very high frequencies and a sufficiently smooth surface of the scattering object. Details on the monostatic bistatic equivalence including an analytical discussion is given by Kell in [9] or, in a slightly modified form, by Siegel et al. in [13]. A discussion of the two different formulations of the equivalence and a validation is given in [15]. In the present case, Kell’s scattering center concept can be more intuitively adapted to the interaction points of individual rays and is used in the following. Several interesting applications of the aforementioned principle can be found. Some applications are in the context of RCS measurements, e.g., monostatic RCS values derived from bistatic measurement setups, or, in the reverse direction, bistatic RCS derived from monostatic measurements [9]. In the field of simulation techniques the equivalence has been successfully used to accelerate time domain (e.g., FVTD, FDTD), or MoM codes in a post-processing manner, see for example [16] for an overview. According to the authors knowledge no publication on the use of the monostatic bistatic equivalence on a direct ray-by-ray basis exists. Another interesting use of this principle is related to the possibility of generating multifrequency data from multiaspect (bistatic) data. This can be advantageously utilized in the field of Synthetic Aperture Radar (SAR) imaging as described for example in [17]. As these methods are based on a coherent signal processing, a coherent scattering model has to be used as well. Similarly, in the presented SBR approach rays are added up coherently and consequently a coherent scatterer model is used to analyze the monostatic and bistatic relation. This results in a simple but important phase correction factor to be considered for bistatic contributions as derived in the following section. B. Bistatic Acceleration of the SBR Algorithm Now, with the monostatic bistatic equivalence in hand, the rays that are launched during a standard (monostatic) SBR simulation can be reused to provide information for some of the neighboring observation points due to the equivalence principle.

To capture this bistatic information, additional bistatic receivers (Rx) are used for each monostatic transmitter (Tx). In each simulation one Tx is located at one of the observation points and spacing in an angular the Rx are equally distributed with around the Tx. This is depicted in angular domain sector in Fig. 3. In this representation the Tx/Rx constellation of one simulation builds a diagonal as highlighted for one example. In this example, all rays launched from the Tx contribute to five Rx, one being the monostatic Rx and four being bistatic Rx. In general, the number of monostatic observation angles within a , which is called bistatic reuse factor here, is desector fined as

(6) According to (5), the bistatic contributions produce monostatic information at the bisecting observation points. For example, a ray contribution captured at the closest bistatic Rx, which is separated from the Tx by , is considered for the monostatic RCS of the bisecting observation point, which is separated . In this way, all bistatic Rx forming a column in Fig. 3 by contribute to the corresponding observation point (monostatic aggregation, see highlighted area in Fig. 3). Regarding a given observation point and the number of rays launched for this direction, this monostatic aggregation increases the number of rays effectively evaluated for this direc, i.e., the sum of all rays tion by the bistatic reuse factor captured by the receivers in one column. The resulting scattered field, or equivalently the RCS, for an individual observation direceivers, each with ray rection is a modified average of contributions. Modified in a sense that the monostatic bistatic equivalence principle is considered as shown below. Consequently, for a fixed number of rays the SBR noise level is reduced for the cost of the additional bistatic calculations. Therefore, the number of rays per incidence direction can be reduced significantly by this bistatic scheme to achieve an equivalent SBR noise level. This results in the acceleration of the proposed approach. In detail, this will also be shown with the example in Section V-A. Next, the consequences of the monostatic bistatic equivalence on the monostatic aggregation are discussed. Therefore, a point

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scatterer on the surface of the object is assumed as depicted in of Fig. 4. The length of the actual bistatic path of the path to the bisecting the ray differs from the length monostatic observation point to which the ray finally shall contribute according to the bistatic equivalence principle. In (5), this results in a frequency shift and considering individual rays the path length difference can be conveniently handled by a phase correction. Using simple geometrical relations, for a far-field the path length difference deobservation distance with and the observation point is pending on the coordinates of found to be

(7) or, alternatively depending only on the quantities directly accessible in the simulation:

(8) Following this model a simple phase correction corresponding to (9) is applied to all field contributions on a ray-by-ray basis. By extending (4) the scattered field including the bistatic contributions can now be written as

(10) with . Here, the sum over is the monostatic aggregation. Equation (10) reveals that the number of contributions evaluated for the scattered field in a given observation direction exceeds the number of distinct rays launched for this . direction by a factor of C. Expected Performance and Limitations Obviously, the use of the bistatic monostatic equivalence principle limits the presented approach to high frequency applications and to sufficiently smooth objects. The first restriction is not very stringent, as this is the main focus of this acceleration approach anyway, and it scales well with its necessity and efficiency, as it will be shown later. The standard SBR algorithm as well as the accelerated algorithm can handle multiple bounces, however, the phase correction as explained above degrades in case of multiple bounces (e.g., double reflections). This is also closely related to the smoothness of the object and therefore not a new restriction. Here, the smoothness of the object is partly related to a similar illumination of the object for the bistatic and

Fig. 4. The actual bistatic path differs from the monostatic path for which the contribution is to be considered and the phase has to be corrected accordingly.

the monostatic cases. Exposed geometrical features causing self-shadowing on the object itself lead to a degradation in the quality of the contribution of rays with larger bistatic angles. Thus, systematic errors are introduced and consequently this leads to a limitation of the maximum usable bistatic angle. The example in Section V shows how the smoothness of the object affects the results. is the factor by which the number of effectively evaluated rays exceeds the number of distinct rays. Consequently, it can be assumed that the number of rays required to achieve results comparable with pure monostatic simulations could be . For details on this, different simulation examlowered by ples are studied in Section V. As for most realistic scenarios the required CPU time is proportional to the number of rays, the acceleration that can be expected should be in the range of the bistatic reuse factor. Unfortunately, the improvement that can be realized is somewhat lower. An additional bistatic overhead in both, memory and computational cost, due to the temporary bistatic receivers decreases the performance. Generally, by considering only the main part of the simulation, the computational expense can be approximated by (11) is the number of receivers and and are the exwhere ecution times for the geometrical ray tracing and the calculation of the field contributions of a single ray, respectively. The execution time for the accelerated approach is yielded for and considering the reduced number of rays by substituting with . Now, the acceleration can be expressed as the ratio of the required simulation times, resulting in

(12) with being the ratio of ray tracing and complex field computation times. For dominating ray tracing effort the accel, but in general it is below this maximum eration approaches . The factor clearly deby approximately a factor of pends on the objects geometrical representation. For the actual

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TABLE I EXAMPLES FOR THE REQUIRED ANGULAR SAMPLING INTERVAL.

performance of the complex example shown in Section V-C with and the acceleration is thus reduced by 1.6 due to the bistatic overhead. Furthermore, as each observation point contains (bistatic) information originating from neighboring transmitters, the simuat both ends lated aspect range should be extended by to capture the bistatic information symmetrically. Considering this overhead is a simulation with a desired aspect range of . For example, a 180 aspect range simulation with a results in a 5.6% overhead. Hence, bistatic sector for many practical scenarios with large observation ranges this has only a minor impact. IV. ANGULAR SAMPLING From the preceding sections we conclude that the efficiency that is expected from the bistatic acceleration depends on the according to (6). As the maximum bistatic reuse factor is determined only by the geometry of the bistatic angle object and is therefore fixed, the resulting efficiency depends on the applied (monostatic) sampling rate. Often it is desired to have a full representation of the backscattering fields of an object for a certain aspect range, e.g., in the azimuthal plane. In this case the required observation angles are to be determined according to the rules of the sampling theorem. Assuming far field conditions an angular sampling interval of (13) is required with being the carrier wavelength and the maximum object radius in the considered observation plane. Equation (13) is based on the resulting phase shift of observed point scatterers rotating in the observation plane. Even though the context is somewhat different, the derivation can be carried out similar to that in [18] and is skipped here for brevity. Some for a bistatic sector of sampling intervals and the resulting are given in Table I. Note that at frequencies as high as 80 GHz, objects of the size of a passenger car require a large amount of observation directions according to the sampling theorem. On the other hand the increasing bistatic reuse factor offers a compensation for this enormous computational burden of such problems at very high frequencies. V. SIMULATION RESULTS A. Faceted Sphere First, a simple object in form of a faceted sphere is used to demonstrate the basic principle of the bistatic acceleration ap-

Fig. 5. Normalized RCS of the faceted sphere computed with a very low number of rays. The improved convergence can be seen depending on the amount of bistatic information used. Additionally, the SBR reference result is shown as solid curve.

proach. The surface of a sphere with radius and the dimension , i.e., large enough to be well within the optical region [19], is approximated by 726 metallic triangles. As the surface is composed of plane facets the monostatic RCS is not independent of the aspect angle as it would be for an ideal sphere. The normalized monostatic RCS for an arbitrarily chosen 45 aspect range is depicted as solid curve in Fig. 5, where the RCS of the corresponding ideal sphere is . This SBR reference solution has been obtained with a high number of rays ensuring good convergence (here five million). With a decreasing number of rays per aspect angle the SBR simulation results will oscillate more and more around the reference curve. To demonstrate the effect of using bistatic information the following investigation is considered: a 45 aspect range is simulated using a very low number of rays, namely one thousand. Consequently, the desired RCS reference cannot be recognized due to the strong random oscillations. Now a simulation run with the same low number of rays is carried out using bistatic ray con). A 0.5 -aspect sampling rate has been tributions ( used resulting in a bistatic reuse factor of . During the monostatic aggregation (see Fig. 3) intermediate results for difcan be generated. The resulting plot of the ferent levels of normalized RCS depending on the monostatic aspect angle and on the maximum bistatic angle is depicted in Fig. 5. It is observed that the RCS converges towards the SBR reference result which is also shown as solid curve in this graph. It should be noted that the (unrealistic) low number of rays prohibits a better convergence in this case, but has been chosen for the purpose of a clearer illustration of the principal behavior. It is not surprising that a minimum ray density is required to capture the objects surface properly. This can be attributed to the fact that inaccurate modeling of the incident fields cannot be recovered by the bistatic evaluation. As it has already been stated in Section II the SBR results generally contain SBR noise. This noise can be reduced and hence SBR convergence is achieved by increasing the number of rays resulting in a more accurate representation of the surface currents on the object. One way of quantifying the noise level is to compare the (noisy) simulation of interest with a noiseless,

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Fig. 6. Noise of the SBR simulation studied in terms of standard deviations. For two different sphere sizes the standard deviations are compared for the accelerated and the standard SBR/PO approach.

Fig. 7. Generic object representing a smooth body (cylinder) with blades causing bistatic shadowing and double reflections.

Fig. 8. RCS of the generic cylinder model with different levels of bistatic informeans standard SBR/PO without any bistatic information). mation (

=0

i.e., converged result and determine the corresponding standard deviation. In this way, the noise is determined for simulations with and without the proposed acceleration method to demonstrate its impact. Fig. 6 depicts how the noise reduces as the ray density increases. This is shown for two different sphere sizes, with and without bistatic information being used. The bistatic improvement can now be defined as the factor by which the number of rays can be reduced while still a comparable SBR noise level is achieved (see Fig. 6). According to (12), the computational cost for the overall simulation can be assumed to be proportional to the number of rays and consequently the bistatic improvement is also the factor by which the required computation time can be

reduced. In this way, a bistatic improvement factor in the range of 4 to 10 can be estimated for the larger sphere, which is quite (see below). close to the bistatic reuse factor It can also be seen from Fig. 6 that for low standard deviations, or equivalently for high numbers of rays, the efficiency of the bistatic contributions tends to degrade. This can be explained by the fundamental bistatic limitations for grazing incidence angles1. These error contributions are expected to be rather constant, but for a high number of rays with already good accuracy and thus low standard deviations they become more and more significant and they limit the bistatic benefit. For realistic targets with a large dynamic range of the aspect dependent RCS, this is however not a severe limitation. Of course, considering the two sphere sizes, the standard deviations for a fixed absolute number of rays are higher for the larger sphere as the ray density is significantly lower in this case. The curves for both spheres match quite well if they are corrected for the ray density. Since, the results presented for the small sphere include a considerable amount of monostatic oversampling for a more stringent analysis of the proposed approach it is desirable to consider monostatic Nyquist sampling (see Section IV). Note, that additional aspect angles due to monostatic oversampling can be considered as an effective increase in the number of rays and this effect might not be clearly separated from the effect caused by the bistatic acceleration. This separation is achieved for a given and if the electrical size of the object is increased. is determined by the shape of the object (but not by its size) and the tolerated bistatic inaccuracies, whereas a given is required for the bistatic acceleration approach to be efficient. Therefore the larger sphere in combination with 1 –sampling and has been used resulting in a bistatic reuse factor . In this way, the bistatic improvement seen in Fig. 6 of can directly be contributed to the proposed acceleration scheme. B. Generic Cylindrical Object To determine the effect of the bistatic approximation in case of a non-smooth object including self-shadowing features, a small metal cylinder with four radial blades is considered as depicted in Fig. 7. With the given relative dimensions of the blades, the resulting shadows on the object are much more penalizing for the presented approach than what is expected from a realistic object like a passenger car. Besides these shadowing effects multiple reflections occur between the blades and the cylinder. Therefore, we conclude that this object qualifies well for a test . case and to derive an upper bound for Due to symmetry only a 45 sector must be considered for the monostatic RCS. Fig. 8 shows in bold the reference RCS obtained with the straightforward SBR without bistatic inforand with a sufficiently high number of mation, i.e., rays guaranteeing good convergence. This can be considered as reference of what can be achieved with the SBR approach. It should be added that the object is fairly small for the SBR and that no diffraction effects are included. Concentrating on the effect of the bistatic approximation, it is is seen that the degradation for a bistatic angle 1In an overall view only a small fraction of the rays hit the surface close to grazing.

BUDDENDICK AND EIBERT: ACCELERATION OF RAY-BASED RCS PREDICTIONS USING MONOSTATIC-BISTATIC EQUIVALENCE

Fig. 9. Simulated monostatic RCS of a passenger car at 10 GHz using the : and . bistatic acceleration approach with a resolution 

1 =02

= 10

still very limited. Only a bistatic angle will cause a considerable impact due to bistatic shadowing. It should be noted that Fig. 8 is obtained with a high degree of monostatic oversampling (see Section IV) for visualization purposes. Of course, this is not a sufficient setup for a comparison of the computational efficiency. The improvement, that is observed using the bistatic approximation, could result in part from averaging the oversampling redundancy. To avoid this, a simulation with the generic object being up-scaled by a factor of 10 compared to Fig. 7 has been studied. This results in faster RCS oscillations and an increased required sampling rate. For and with varying numbers of rays, the deviadifferent tions of the RCS from the converged SBR reference are determined. By comparing the required number of rays for a given level of deviation from the reference (e.g., standard deviation) the bistatic acceleration is estimated. This is the same procedure as described in more detail for the example of the faceted sphere (leading to Fig. 6) and is skipped here for brevity. As a result, an achievable acceleration factor in the range for of 10–20 is observed. C. Complex Passenger Car Model As a final example the monostatic RCS of a complex shaped object, in this case a passenger car, is shown in Fig. 9. The geometric model of the car consists of approximately 100000 metal triangles and a frequency of 10 GHz has been assumed. A maxis observed for broadside aspects imum RCS of ). According to the sampling theorem the observation ( . directions have been computed with a spacing of For the simulations the PO solver has been used and, according to the experience with the previous simulation examples, a maxhas been considered. This imum bistatic angle of , which leads to a results in a bistatic reuse factor of considerable acceleration in this case. The SBR noise, which has been explained earlier, is clearly limiting the achievable accuracy. For the simple objects in the previous examples the standard deviation between a result of interest and a converged SBR reference result has been determined to quantify the SBR noise level. For this complex object computation time prohibits a reference simulation and a more convenient way should be followed to determine the noise level. This can be done by comparing two stochastically independent

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simulations with identical settings, again in terms of the standard deviation. By this, the influence of the bistatic acceleration approach can be assessed by considering the computational cost (number of rays) and accuracy in terms of SBR noise. For a detailed analysis, a 40 aspect range of the monostatic RCS is shown in Fig. 10 for different simulation settings. The first graph on top shows two independent standard SBR simulation runs with only 500000 rays per aspect angle. No bistatic information has been used for these results. It can be seen that the main scattering characteristics can be observed in both results, e.g., a local maximum around 10 . Nevertheless, the two results differ significantly in detail and the standard deviation in this case is almost 2 dB. This SBR noise could be reduced further by increasing the number of rays for the cost of increased computational complexity. The second graph of Fig. 10 shows again two independent simulation runs with identical settings, now using bistatic infor each aspect angle. Comparing formation with these results with the graph on top, it can be found that the same main scattering characteristics can be observed but the deviations between the simulation runs are reduced significantly (standard deviation of 0.37 dB), indicating a reduced SBR noise level. From this point of view bistatic information can be used to reduce the SBR noise, which on the other hand would require a considerably increased number of rays and computational resources. For the third graph at the bottom of Fig. 10 the number of rays is reduced by a factor of 10 and additional bistatic in) is used. Compared to the graph in the formation ( middle, the SBR noise slightly increases (standard deviation of 0.54 dB) due to the reduced number of rays. But compared to the graph on top, still a significantly improved standard deviation can be observed along with a significantly reduced computational effort. As mentioned above the effective acceleration factor will be , mainly due to the somewhat below the bistatic reuse factor additional overhead for capturing the bistatic field data. In this case, a simulation with 5 million rays and bistatic assistance ) took on average 4.5 minutes per aspect angle.2 For ( comparison, some aspect angles have been simulated without bistatic information but with an increased number of rays (250 million), which leads to an identical number of effectively evaluated rays. For this configuration an average computation time of 2.4 hours has been observed. This leads to an acceleration by a factor of approximately 32 for this example in the current implementation. For higher frequencies an even higher acceleration factor is expected as outlined in the previous sections. It should be noted that the bistatic overhead can be further reduced through a more advanced implementation which would increase the overall acceleration closer to the bistatic . This has not been considered up to now for reuse factor the sake of reusability of software modules from the standard bistatic case. VI. CONCLUSION Nowadays, an important part of high frequency electromagnetic scattering analysis is based on asymptotic SBR and PO 2AMD Athlon 64 X2 4200+ processor operated by a Windows XP system (single core usage).

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information, which is collected with very low additional computational cost, is reused to improve the convergence of the stochastic SBR algorithm, which means that less rays must be used for a required accuracy. In this way the effort for the monostatic sampling of the object of interest is partly replaced by the use of bistatic side information. To maximize the acceleration, large values for the utilized bistatic angle are desired. From the investigation of a canonical object with some strong self-shadowing features it was concan be used in a simulation of cluded that a value of a passenger car without significant degradation in the RCS refor a complete multiaspect sults. In this case a simulation is shown. It should be noted that in our acceleration approach the rays are still launched from all observation directions, only the number of rays is lowered equally for all directions. Bistatic information is used for all directions to compensate this and the ratio of monostatic to bistatic information is constant for all observation directions. Therefore, the inaccuracies introduced by this approach are expected to be distributed over all observation directions. Consequently, the resulting quality and accuracy of all aspect angles is affected in a similar way. This is in clear contrast to other simulation techniques, where the equivalence principle is used in a post-processing manner to interpolate observation points leading to a reduced accuracy of the interpolated points compared to the directly simulated ones. Note that the presented acceleration technique has an advantageous scaling property as it is expected to be most efficient for very high frequencies, where, at the same time, it is needed the most to reduce computational complexity. REFERENCES

Fig. 10. Detailed view on a part of the monostatic RCS of the passenger car in for different simulation parameters (note the number of rays as well as each subfigure). Each graph shows two curves obtained from two independent simulation runs (one as bold curve and one in dashed style). No bistatic information has been used for the graph on top. 10 bistatic angle reduces the stochastic influence significantly (see standard deviation for the graph in the middle) and this even holds for further reduced computational expense (bottom).

techniques. In this paper it is not the focus to legitimize this approach but rather a methodology is proposed to accelerate these simulations significantly. It is a ray-based method applying the principle of monostatic bistatic equivalence. Additional bistatic

[1] M. I. Skolnik, Introduction to Radar Systems, 3rd. ed. New York: McGraw-Hill, 2001. [2] H. Ling, R. C. Chou, and S. W. Lee, “Shooting and bouncing rays: Calculating the RCS of an arbitrarily shaped cavity,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 194–205, Feb. 1989. [3] D. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, and Y. Chu, “Xpatch: A high frequency electromagnetic scattering prediction code and environment for three-dimensional objects,” IEEE Trans. Antennas Propag., vol. 36, pp. 65–69, 1994. [4] D. Andersh, J. Moore, R. Bhalla, and J. Hughes, “Xpatch 4: The next generation in high frequency electromagnetic modeling and simulation software,” in Proc. IEEE Int. Radar Conf., 2000, pp. 844–849. [5] J. Pérez and M. F. Catedra, “Application of physical optics to the RCS computation of bodies modeled with nurbs surfaces,” IEEE Trans. Antennas Propag., vol. 42, pp. 1404–1411, Oct. 1994. [6] F. Saez de Adana, I. G. Diego, O. G. Blanco, P. Lozano, and M. F. Catedra, “Method based on physical optics for the computation of the radar cross section including diffraction and double effects of metallic and absorbing bodies modeled with parametric surfaces,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3295–3303, Dec. 2004. [7] P. Galloway and T. Welsh, “Extensions to the shooting bouncing ray algorithm for scattering from diffuse and grating structures,” in Proc. Int. Radar Symp., IRS, Berlin, Sep. 2005. [8] F. Weinmann, “Ray tracing with PO/PTD for RCS modeling of large complex objects,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1797–1806, Jun. 2006. [9] R. E. Kell, “On the derivation of bistatic RCS from monostatic measurements,” Proc. IEEE, vol. 53, pp. 983–988, Aug. 1965. [10] A. Fujimoto, T. Tanaka, and K. Iwata, “ARTS: Accelerated ray tracing system,” IEEE Comput. Graph. Appl., vol. 6, no. 4, Apr. 1986. [11] A. Glassner, An Introduction to Ray Tracing. London: Academic Press, 1989. [12] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989.

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[13] K. M. Siegel et al., Methods of Radar Cross-Section Analysis. New York: Academic Press, 1968. [14] W. A. Pierson, C. S. Liang, and R. W. Clay, “The effect of coupling on monostatic-bistatic equivalence,” Proc. IEEE, vol. 59, pp. 84–86, Jan. 1971. [15] R. L. Eigel, P. J. Collins, A. J. Terzuoli, G. Nesti, and J. Fortuny, “Bistatic scattering characterization of complex objects,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 5, pp. 2078–2092, Sep. 2000. [16] J. Schuh, A. C. Woo, and M. P. Simon, “The monostatic/bistatic approximation,” IEEE Antennas and Propagation Magazine, vol. 36, no. 4, 1994. [17] D. L. Mensa, High Resolution Radar Cross Section Imaging. Boston: Artech House, 1991. [18] T. Vaupel and T. F. Eibert, “Comparison and application of near-field ISAR imaging techniques for far-field radar cross section determination,” IEEE Trans. Antennas Propag., vol. 54, no. 1, Jan. 2006. [19] E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section, 2nd ed. Norwood, MA: Artech House, 1993. Hermann Buddendick (S’08) received the Dipl.-Ing. degree in electrical engineering from the Universität Stuttgart, Stuttgart, Germany, in 2001. From 2001 to 2002, he was with the Alcatel Corporate Research Center, Stuttgart, Germany where he worked as a Research Engineer and was involved in mobile network system simulations within the mobile communication research department. From 2002 to 2006, he was with AWE Communications GmbH, Böblingen, Germany, where he developed radio network planning and simulation software. Since 2006

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he has been a Research Assistant at the Institute of Radio Frequency Technology, Universität Stuttgart, Stuttgart. His main research activities are in the field of asymptotic computational electromagnetics in particular radio channel modeling and electromagnetic scattering analysis.

Thomas F. Eibert (S’93–M’97–SM’09) received the Dipl.-Ing.(FH) degree from Fachhochschule Nürnberg, Nürnberg, Germany, the Dipl.-Ing. degree from Ruhr-Universität Bochum, Bochum, Germany, and the Dr.-Ing. degree from Bergische Universität Wuppertal, Wuppertal, Germany, in 1989, 1992, and 1997, all in electrical engineering. From 1997 to 1998, he was with the Radiation Laboratory, EECS Department of the University of Michigan, Ann Arbor, from 1998 to 2002, he was with Deutsche Telekom, Darmstadt, Germany, and from 2002 to 2005, he was with the Institute for High-frequency Physics and Radar Techniques of FGAN e.V., Wachtberg, Germany, where he was head of the department Antennas and Scattering. From 2005 to 2008, he was a Professor for radio frequency technology at Universität Stuttgart, Stuttgart, Germany. Since October 2008, he has been a Professor for high-frequency engineering at the Technische Universität München, Munich, Germany. His major areas of interest are numerical electromagnetics, wave propagation, measurement techniques for antennas and scattering as well as all kinds of antenna and microwave circuit technologies for sensors and communications.

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Scattering by Polygonal Cross-Section Dielectric Cylinders at Oblique Incidence Mario Lucido, Member, IEEE, Gaetano Panariello, Member, IEEE, and Fulvio Schettino, Member, IEEE

Abstract—We analyze the scattering by polygonal cross-section lossless dielectric cylinders illuminated by an obliquely incident plane wave. The problem is formulated in terms of a system of surface integral equations opportunely devised so as to be valid for objects with edges and incident angles including the total reflection limit angle. By means of Galerkin’s method in the spectral domain with analytically Fourier-transformable expansion functions factorizing the correct edge behaviors and continuity conditions of the unknowns, convergence of exponential type is achieved and the coefficients of the scattering matrix are reduced to single integrals that can be efficiently evaluated. Numerical results for both near field and far field parameters are presented, showing the quick convergence of the method even when applied to composite cylindrical objects. Index Terms—Dielectric cylinders, electromagnetic scattering, Galerkin’s method, spectral domain analysis.

I. INTRODUCTION

S

CATTERING by dielectric objects has been widely studied for its relevance in a great number of applications, e.g., target identifications or prediction of poor reception areas caused by obstacles. In particular, for cylindrical objects, different approaches have been developed in the past decades depending on the complexity of the structures, the range of frequencies of interest and the behavior of the source along the cylinder axis [1]–[20]. Particularly troublesome configurations in scattering problems include the presence of wedges. In fact, unlike metallic objects, for dielectric wedges closed form solutions of Maxwell’s equations are available in the static case for the general sectorial medium [21]–[23], but only for some configurations of isorefractive two-dimensional media [24]–[28] in time-varying case. Obviously, such occurrence has prevented the development of asymptotic solutions directly from exact solutions. Therefore an effort has been devoted to devise approximate solutions even for the canonical two-dimensional wedge [29]–[32]. As for metallic objects, a key point in the analysis of dielectric objects is the need to consider the correct edge behaviors and continuity properties of the fields in order to characterize the functional space to which the solution has to belong [29]–[34]. Meixner [35] derived the edge behavior of the fields on a wedge by expanding them in power series of the distance from the wedge itself. This approach in general is not consistent Manuscript received April 16, 2009; revised July 09, 2009. First published December 08, 2009; current version published February 03, 2010. The authors are with the D.A.E.I.M.I., University of Cassino, 03043 Cassino, Italy (e-mail: [email protected], [email protected], [email protected]). Digital Object Identifier 10.1109/TAP.2009.2038181

for dielectric wedges [36]. A more general consistent representation of the fields was introduced by Makarov and Osipov [37]. Anyway the dominant edge behavior predicted by both theories coincides with the static edge behavior found by Greenberg and Karp [21], [22] for the general sectorial medium. In the literature dedicated to the propagation and scattering by perfectly conducting and dielectric cylinders studied by means of surface integral operator formulations it has been widely observed that a fast convergence (of exponential type) can be achieved by means of the application of Galerkin’s method with a suitable choice of the expansion functions. For dielectric cylinders with quite general smooth contour cross-sections, the angular exponents reveal to be a good choice for the expansion functions because the integral operator can be decomposed into a main part with an explicit Fourier representation and a remaining part with smooth kernel [38], [39]. However, the convergence rate is slower for structures with sharper corners and in presence of wedges different expansion functions have to be introduced. In the analysis of the scattering by polygonal cross-section perfectly conducting cylinders the goal can be obtained by using expansion functions factorizing the correct edge behavior of the unknowns [40]–[47]. Moreover it has been demonstrated that Galerkin’s method in the spectral domain with basis functions analytically Fourier-transformable leads to single integrals which can be efficiently numerically evaluated [40]–[47]. In this contribution the scattering by polygonal cross-section lossless dielectric cylinders when a plane wave impinges obliquely with respect to the cylinders axis is analyzed by applying Galerkin’s method in the spectral domain to a system of surface integral equations opportunely devised so as to be valid for objects with edges and incident angles including the total reflection limit angle. In particular, by using basis functions analytically Fourier-transformable exhibiting the correct edge behavior of the unknowns, it is shown that all the positive results listed above are preserved in the case of general configurations consisting of more than one dielectric and/or conducting cylinder. The paper is organized as follows. In Section II, for incident angles different from limit angle, by means of the equivalence principle a system of integral-differential equations for the equivalent surface electric and magnetic current densities in the spectral domain is obtained. In Section III the obtained integral-differential operator is transformed into an integral operator by using a procedure based on the asymptotic behavior of the unknowns in the spectral domain related to the edge behavior and continuity conditions at edges of the fields. Section IV is devoted to the generalization of the integral operator in order to include the limit angle. Moreover the obtained integral operator

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(1b) (1c) (1d)

Fig. 1. Geometry of the problem.

reveals to be numerically stable even for incidence angles very close to limit angle. In Section V Galerkin’s method is applied to the system of integral equations by expanding the unknowns on each face of the cylinder with analytically Fourier-transformable functions factorizing the correct behavior of the fields at edges. Section VI is devoted to the presentation of the numerical results. In particular a converging test showing the exponential convergence of the method and some comparisons with the literature and a commercial software are presented. Two mathematical appendices conclude the paper. II. FORMULATION IN TERMS OF A SYSTEM OF INTEGRAL-DIFFERENTIAL EQUATIONS In this section a system of surface integral-differential equations governing the problem of scattering by a dielectric cylinder is devised. For the sake of simplicity an homogeneous and isotropic lossless dielectric cylinder with a polygonal cross section (medium 2) dipped in an homogeneous and isotropic lossless medium (medium 1) is considered (Fig. 1), but the obtained equations can be immediately generalized to more than one dielectric and/or conducting cylinder. is introduced so that the axis A coordinate system coincides with the cylinder axis. The sides of the polygonal cross-section are numbered clockwise and a local coordinate ) is introduced on the th side with the origin system ( with respect at the centre of the side itself in position to the main coordinate system and oriented in the outward is the angle of the direction of the axis direction, the dimension of the th side. with respect to the axis and A plane wave impinges onto the cylinder with an azimuth angle with respect to the plane and a zenith angle with respect to the axis (Fig. 1) so all fields in both where media have a behavior with of the kind being the wave number of the external medium (medium 1). In order to efficiently formulate the problem fictitious electric and magnetic surface currents are introduced on each face of the cylinder. It is worth noting that such unknowns lead to a not well-balanced system of equations because of the different nature of the involved quantities [48]. In order to get a better balanced system, normalized quantities are used instead of the original ones defined as (1a)

where is the normalized scattered field in medium 1, is the normalized total field in medium 2, is the normalized incident field, and possible choices for and which preserve the continuity of the fields are [48] and where is the the magnetic permeability of the dielectric permittivity and involved media. By means of the equivalence theorem, the incident field plus the field generated by the superposition of the sources vanishes in medium 2 while the field generated by the superposition of the sources vanishes in medium 1. Therefore the fields in each medium can be obtained from the vector potentials in homogeneous medium. and . Moreover , where is the wave As a first task it is assumed , while the case in which number of the medium will be examined in Section IV. In such a case, normalized magnetic and electric vector potentials generated by the sources on the th side in medium can be written as

(2a)

(2b) where

, and is the Hankel function of second kind and order zero. As reported in [45] it is possible to show that

(3a)

(3b) where and are the Fourier transforms of the sources on the th side with respect

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and continuity conditions of the currents at edges are strictly related to their asymptotic behaviors in the spectral domain and, then, to the possibility to transform the integral-differential operator into an integral operator. It is well-known that a closed form solution for the general sectorial medium (see Fig. 2) exists only for the static case [21]–[23]. In the time-varying case, following Meixner’s theory [35], the transverse components of the electromagnetic field in the vicinity of the edge are expanded in series like (7) where , while the are organized in ascending order and can be evaluated by boundary conditions and Maxwell’s equations as shown later. The representation of the longitudinal components can be obtained from transverse components by means of Maxwell’s equations. In fact the following relations can be written

Fig. 2. General sectorial medium.

to

defined as

(8a) (4a)

(8b)

(4b)

thus leading to a representation of the kind (7) also for the derivative of the longitudinal components of the fields with respect to . Moreover, by integrating both members of (8) with respect to , it is obvious that the longitudinal components of the fields are finite and continuous for . From Maxwell’s equations it is possible to obtain a recurrent system of ordinary differential equations for the coefficients . For the only term remaining in (7) is the one for and is solution of the harmonic oscillator equation. The boundary and continuity conditions for the tangential components of the fields at the discontinuity surfaces can thus be imposed directly on

The total electromagnetic field generated in each medium by the superposition of the sources can be obtained by using the well-known relations between fields and potentials obtaining

(5a)

(5b) where is the angular frequency and the functional dependences are omitted. Now substituting (3) in (5) and imposing the continuity of the tangential components of the fields on the cylinder surface, the following system of integral-differential equations is obtained (6a) (6b) with

and

.

III. EDGE BEHAVIOR AND DERIVATION OF A SYSTEM OF INTEGRAL EQUATIONS In the analysis of scattering by objects with wedges the edge behavior of the fields plays a key role because the fast evaluation of the solution depends on the choice of the functional space to which the solution has to belong. Moreover the edge behaviors

(9) obtaining an homogeneous system of algebraic equations for the parameters and . The requirement that the above system be solvable leads to transcendental equations for the parameters . However Bach Andersen and Solodukhov [36] demonstrated that Meixner’s expansion is consistent for perfectly conducting wedges but not always for penetrable wedges with could not exist or for which a solution for it could not be unique. To overcome this problem Makarov and Osipov [37] introduced the following more general representation in place of (7)

(10) It is interesting to note that the only term remaining in (10) for is the one for , and is again solution of the harmonic oscillator equation and has consequently the

LUCIDO et al.: SCATTERING BY POLYGONAL CROSS-SECTION DIELECTRIC CYLINDERS AT OBLIQUE INCIDENCE

expression (9). For this reason the dominant behavior for is the same in both the proposed solutions and it coincides with that obtained in the static case. For the problem in Fig. 1, by means of relations (1), (8) and (10), the dominant behavior of the electromagnetic field at edges in terms of equivalent electric and magnetic surface currents is given by

In this way the first integral in (15) can be inverted with the derivative, whilst it can be analytically demonstrated that the second term is zero as a direct consequence of the continuity of the currents at each edge [47], obtaining

(11) is the minimum solution of the transcendental equawhere tion [35]

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(16) where (17a)

(12) such that , and is one of the two angles in corre. In addition, the spondence to the wedge at abscissa transverse currents must be continuous even on the wedges. Once the edge behavior of the longitudinal and transverse components of the electric current (analogously for the magnetic current) on the th side is stated, it is possible to evaluate the asymptotic behavior of their Fourier transforms. In fact, by means of Watson’s lemma [49], it can be obtained

(17b) and . Therefore with the following expressions can be obtained for the normalized and directions fields on the

(13a) (13b) are suitable parameters depending on the problem, for and . Then the decaying of the integrand in the expression of the potentials is such that it is always possible to invert one derivative and the integral in (5). In this way, the superposition of the double derivative of the magnetic potential (analogously for the electric potential) can be written as where the

(18) where the elements of the dyadic operator are defined as (19a)

(19b)

(14) (19c) For the analytical evaluation of the second derivative it is possible to proceed as in [47] accelerating the asymptotic behavior of the integrand in (14)

(19d)

(15)

(19e)

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is very close to . All these problems can be overcome observing that it is possible to write

(19f)

(19g)

(19h)

(21)

(19i)

(analogous considerations can be done on the quantity ). It can be demonstrated (see Appendix A) that at the second member of (21) the integrals in the first term are finite for and the last term is an thus leading to the effectiveness of (21) for . , the last sum in (21) can be written as For

(19j) (19k) (19l) (19m) (19n) (19o) (19p)

(22)

(19q) and (20a)

where and indicate the Fourier transform , and by of the surface currents on the th side for simple algebraic manipulation as

(20b)

IV. INCIDENCE AT AND ABOUT LIMIT ANGLE If the refractive index of the internal medium is less than that of the external medium, for each frequency incident directions exist and vice versa. In such cases it is not for which possible to use the expressions (3) for the potentials in the internal medium. However, being the fields continuous functions and the frequency, their expressions for can of by means of be obtained apart from the ones for a limit operation. Unlikely in (18) it is not possible to invert limit operation and integration because the integral operators and diverge for . Furthermore, such pathology can be reflected on the difficulty in the accurate evaluation of the operators themselves even when

(23) Being the currents continuous functions of the incidence angle, the problem is regularized even for by introducing new unknowns on each side defined as (24a) (24b)

LUCIDO et al.: SCATTERING BY POLYGONAL CROSS-SECTION DIELECTRIC CYLINDERS AT OBLIQUE INCIDENCE

V. GALERKIN’S METHOD, EXPANSION FUNCTIONS AND NUMERICAL EVALUATION OF THE SCATTERING MATRIX COEFFICIENTS From (11), the longitudinal components and the derivative of the transverse components of the electric current (analogously for the magnetic current) can be written as

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is the complete Beta has been used and function. The Fourier transform in the complex plane of (29) on the generic side is analytical and is [46]

(25a) (25b)

(31)

and well-behaved functions in . It is with worth noting that due to the Weierstrass approximation theorem any function continuous in a compact interval can be expanded by a uniformly convergent series of polynomials. Moreover it has been shown in [40]–[47] that, in a Galerkin scheme, fast convergence can be achieved by expanding the induced currents with functions factorizing the behavior of the fields at edges. However a proper expansion for the transverse currents can be obtained by integrating the expansions of the derivative itself and imposing the continuity of the transverse currents across the wedges. To this end, suitable expansions for the longitudinal and transverse electric currents (analogously for the magnetic currents) were introduced in [46] and are reported here for the sake of completeness

The Fourier transforms in the complex plane on the generic side of all the other expansion functions are also analytical [45]

(26a)

(26b) where

(32) In (31) and (32) is the confluent hypergeometric function of first kind, also known as Kummer function of first kind [50]. So it can be concluded that the Fourier transforms of all the expansion functions are analytical. By substituting the Fourier transforms of the expansions into (18) and the expressions (18) into (6), and projecting the equations involving the longitudinal/transverse components of the fields onto the expansion functions of the longitudinal/transverse components of the currents respectively, a linear system of algebraic equations is obtained. The coefficients’ matrix of such a system has many symindicates the generic coefficient metries. In fact, if identify the of the scattering matrix, where identify the sides, generic operator, identify the expansion and projection functions respectively, by means of reciprocity it is possible to show that (see Appendix B) (33)

(27) and from (19) and (33) that is the Jacobi polynomial of order and parameters which, suitably chosen, factorize the desired edge behavior of the unknown function, is a suitable normalization quanis the unitary rectangular window, while tity, and

(28)

(29) where the definition of the incomplete Beta function [50]

(34a) (34b) (34c) thus leading to the numerical evaluation of elements instead of when each component of the curexpansion functions. rents on each side is represented with Moreover following the same line of reasoning as in [46] it is possible to demonstrate that all the elements of the scattering matrix can be reduced to single integrals. The numerical computation of such integrals is not time-consuming since either the integrands are exponentially decaying, or the integrals can be analytically accelerated by using the procedure developed in [46]. VI. NUMERICAL RESULTS

(30)

This section is devoted to the validation of the presented method by means of the analysis of the convergence and the

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Fig. 3. Normalized truncation error for the complex cylindrical structure dipped in vacuum sketched in the inset for oblique TM and TE incidence.  = ;  = ; " : ; ;" : ; ; AB DE EA ; BC  = ; CD = ; F G GH = ; AF = ; ABC = ; B CD = ; HF  = ; AF = ; C DE = ; DEA E AB F GH = ; GHF HF G = .

= 6 = 4 = = 3 = p2 3 3 2 ^ = 6 ^ 4

= 25p = 2  = = 32 = = 5 24 = 4 = ^ = ^ =

0 75  = 1 = 2 = = ^ = 3 ^ = 2 ^ = ^ =

comparisons with results presented in the literature and with the commercial software CST Microwave Studio. All the simulations have been performed on a laptop with a Core2 Duo 2.4 GHz, with 2 GB RAM, running Windows XP. The following normalized truncation error is introduced

Fig. 4. Normalized surface currents on the cylinders in the inset of Fig. 3 for . oblique TM and TE incidence. N

=9

(35) where is the usual euclidean norm and is the vector of all the expansion coefficients of the normalized currents on terms on each side. In order all the sides evaluated with to demonstrate the quick convergence and versatility of the method, in Fig. 3 the normalized truncation error is plotted for the complex cylindrical structure sketched in the inset for oblique TM and TE incidence with respect to the cylinder axis. The electromagnetic properties of the media and the incident angle with respect to the cylinder axis are chosen so that is equal to the wave number of the innermost medium. It is worth noting the exponential decaying of the error. In Figs 4(a) and (b) some components of the resulting normalized surface currents are plotted by using 9 expansion functions for each (in the following currents on each side so that the same truncation criterion will be used). In Table I the convergence of the longitudinal fields in a point of the boundary of a rectangular dielectric cylinder dipped in vacuum for both oblique and orthogonal TM and TE incidence are reported in order to enable the reader to more clearly appreciate the accuracy and efficiency of the proposed method. In Fig. 5 the normalized surface magnetic current on the circumference of a square dielectric cylinder dipped in vacuum for . The agreement orthogonal TE incidence is plotted for with the results reported in [19] by means of the domain-product technique, and CST Microwave Studio is very good. It is interesting to note that the electric field is zero at the edges and which is consistent with the static analysis of the problem [51].

Fig. 5. Normalized surface magnetic current on the square dielectric cylinder dipped in vacuum sketched in the inset for orthogonal TE incidence, " : and d : = . Solid lines: this method; dotted lines: CST Microwave Studio (about 2500000 meshcells); circles: data from [19]. N .

= 1 75

=25

=5

In Fig. 6 the normalized surface electric and magnetic currents on the circumference of a rectangular dielectric cylinder dipped . in vacuum for orthogonal TM incidence are plotted for Also in such a case the obtained results agree very well with the ones reported in [11] obtained applying the boundary element method to the extended integral equation formulation, as well as with the adopted commercial software. Figs. 7 and 8 show the normalized scattered electric field on the circumference of a square dielectric cylinder dipped in vacuum for orthogonal TM

LUCIDO et al.: SCATTERING BY POLYGONAL CROSS-SECTION DIELECTRIC CYLINDERS AT OBLIQUE INCIDENCE

Fig. 6. Normalized surface electric and magnetic currents on the rectangular dielectric cylinder dipped in vacuum sketched in the inset for orthogonal TM ;a =  and b a. Solid lines: this method; incidence, " dotted lines: CST Microwave Studio (about 500000 meshcells); circles: data . from [11]. N

= 2 = 3 ( p2) =8

=2

Fig. 7. Normalized scattered electric field on the circumference of the square dielectric cylinder dipped in vacuum sketched in the inset for orthogonal TM : and d : = . Solid lines: this method; dotted lines: incidence, " CST Microwave Studio (about 500000 meshcells); circles: data from [7]. N .

= 2 89

9

=38

=

incidence alone and backed by a bent perfectly conducting strip respectively. The agreement with the results reported in [7] by means of the boundary element method and [19], and CST Microwave Studio is very good. In Fig. 9 the bistatic radar cross sections of a square dielectric cylinder dipped in vacuum for orthogonal TM and TE incidence are compared with the data presented in obtained with [17] by modelling the scatterer with circular cylinders, and the ones obtained with CST Microwave Studio with good agreement. In Fig. 10 the normalized scattered far-field of a hollow square dielectric cylinder dipped in vacuum for orthogonal TM . Also in this case the agreement incidence is plotted for with the data reported in [7] and the commercial software is very good. To conclude, in Fig. 11 the bistatic radar cross sections of a dielectric slab dipped in vacuum for oblique TE incidence and different widths are compared with the data obtained in [9]

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Fig. 8. Normalized scattered electric field on the circumference of the square dielectric cylinder backed by a bent perfectly conducting strip dipped in vacuum : and d sketched in the inset for orthogonal TM incidence, " : = . Solid lines: this method; dotted lines: CST Microwave Studio (about 150000 meshcells); circles: data from [19]. N .

= 2 89

=38

=9

Fig. 9. Bistatic radar cross sections of the square dielectric cylinder dipped in vacuum sketched in the inset for orthogonal TM and TE incidence, " and d = . Solid lines: this method; dotted lines: CST Microwave Studio (about . 100000 meshcells); circles: data from [17]. N

= 2

=4

=6

by means of the conjugate gradient method and FFT with good agreement. VII. CONCLUSION In this work the analysis of the scattering by polygonal crosssection lossless dielectric cylinders illuminated by an obliquely incident plane wave has been performed. Galerkin’s method in the spectral domain with analytically Fourier transformable expansion functions factorizing the correct edge behaviors and continuity conditions of the unknowns, has been applied to a system of surface integral equations opportunely devised so as to be valid for objects with edges and incident angles including the total reflection limit angle. In this way, as shown by the reported numerical results, fast convergence of exponential type has been achieved and the scattering matrix coefficients have

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TABLE I CONVERGENCE TESTS ON THE NORMALIZED LONGITUDINAL COMPONENTS OF THE FIELDS AT THE CENTRE OF THE LARGER ILLUMINATED SIDE OF A RECTANGULAR DIELECTRIC CYLINDER DIPPED IN VACUUM FOR OBLIQUE AND ORTHOGONAL TM AND TE INCIDENCES. a b  (DIMENSIONS OF = AND " THE SIDES), 

= 3

=2 = =2

Fig. 10. Normalized scattered far-field of the hollow square dielectric cylinder dipped in vacuum sketched in the inset for orthogonal TM incidence, " = . Solid lines: this method; dotted lines: CST : ;a : = and b . Microwave Studio (about 2000000 meshcells); circles: data from [7]. N

= =8

=3

2 89 = 3 8

rent). To this end it is sufficient to observe that the function

(A2) is integrable. In fact, it is continuous for

Fig. 11. Bistatic radar cross section of the dielectric slab dipped in vacuum : ;" sketched in the inset for oblique TE incidence with  and b for : . Lines: this method; symbols: data from [9]. N ; ; a : ; ;  respectively.

2

= Arccos(00 3) = = 5 6 12

= 0 15 =03 3

been efficiently evaluated, even when the method has been applied to the analysis of composite cylindrical objects with polygonal cross-section. APPENDIX A As

a

first

task

it

can

be

demonstrated

that

(A1) where , is finite (analogous considerations can be done for the integral involving the magnetic cur-

exist and the Fourier transform of a function with is finite being decays asymptotically at least as compact support, and . It can now be demonstrated that

(A3) and are the mean values of Observing that the normalized transverse electric and magnetic currents on the th side respectively, and using the definitions (1), it is possible to write

(A4) where is the discontinuity curve in the transverse plane between the two media oriented in clockwise sense with respect to

LUCIDO et al.: SCATTERING BY POLYGONAL CROSS-SECTION DIELECTRIC CYLINDERS AT OBLIQUE INCIDENCE

the positive axis, and is the unit vector tangent to the curve . The scalar Stokes’ theorem

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(B2b) in an homogeneous and isotropic medium has the following expression

(A5) (B3a) where is a continuous and piecewise continuously differentiable function, is a regular surface delimited by the contour and is the normal to defined in such a way that and respect the “right hand rule”, leads to

where (B3b) Then, from Maxwell’s equations, it is possible to demonstrate that the field generated by the sources (B4a) (B4b)

(A6) is the cross-section of the internal medium. where By using the well-know relation between the transverse and longitudinal components of the electromagnetic fields

is (B5a) (B5b) For such a case, the reciprocity theorem states

(A7) it is possible to conclude that (B6)

(A8) where the integral at the second member is always finite. APPENDIX B By means of reciprocity, it can be demonstrated that the scattering matrix obtained by using Galerkin’s method has the following property (B1) As a first task it is important to note that the coefficient is the reaction of the field generated by the surface source represented by the expansion function of order of the transverse/longitudinal component of the electric/magnetic current (depending on the value of ) on the th side, onto the surface source represented by the expansion function of of the transverse/longitudinal component of the elecorder tric/magnetic current (depending on the value of ) on the th is the reaction obtained side. On the other hand reversing the role of the sources. Being even if the involved sources are both transverse or longitudinal and odd otherwise, (B1) states that the reactions at the first and second member are equal if the involved sources are both transverse or longitudinal and opposite otherwise. To demonstrate the last sentence and then (B1), note that the electromagnetic field generated by the sources (B2a)

where is the plane orthogonal to the axis. Equation (B1) is finally proved by substituting in (B6) surface sources represented by the expansion function of order of the transverse/longitudinal component of the electric/magnetic curof rent on the th side and the expansion function of order the transverse/longitudinal component of the electric/magnetic current on the th side. REFERENCES [1] J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary crosssection shape,” IEEE Trans. Antennas Propag., vol. 13, pp. 334–341, May 1965. [2] J. H. Richmond, “TE-wave scattering by a dielectric cylinder of arbitrary cross-section shape,” IEEE Trans. Antennas Propag., vol. 14, pp. 460–464, Jul. 1966. [3] V. V. Solodukhov and E. N. Vasile’v, “Diffraction of a plane electromagnetic wave by a dielectric cylinder of arbitrary cross section,” Sov. Phys.-Tech. Phys., vol. 15, pp. 32–36, Jul. 1970. [4] S.-K. Chang and K. K. Mei, “Application of the unimoment method to electromagnetic scattering of dielectric cylinders,” IEEE Trans. Antennas Propag., vol. 24, pp. 35–42, Jan. 1976. [5] N. Morita, “Surface integral representations for electromagnetic scattering from dielectric cylinders,” IEEE Trans. Antennas Propag., vol. 26, pp. 261–266, Mar. 1978. [6] R. Kastner and R. Mittra, “A spectral-iteration technique for analyzing scattering from arbitrary bodies—Part 1: cylindrical scatterers with E-wave incidence,” IEEE Trans. Antennas Propag., vol. 31, pp. 499–506, May 1983. [7] K. Yashiro and S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag., vol. 33, pp. 383–389, Apr. 1985. [8] E. Arvas, S. M. Rao, and T. K. Sarkar, “E-field solution of TM-scattering from multiple perfectly conducting and lossy dielectric cylinders of arbitrary cross-section,” Proc. Inst. Elect. Eng., vol. 133, pp. 115–121, Apr. 1986. [9] C. C. Su, “Calculation of electromagnetic scattering from a dielectric cylinder using the conjugate gradient method and FFT,” IEEE Trans. Antennas Propag., vol. 35, pp. 1418–1425, Dec. 1987.

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[10] R. G. Rojas, “Scattering by inhomogeneous dielectric/ferrite cylinder of arbitrary cross-section shape—Oblique incidence case,” IEEE Trans. Antennas Propag., vol. 36, pp. 238–246, Feb. 1988. [11] I. Toyoda, M. Matsuhara, and N. Kumagai, “Extended integral equation formulation for scattering problems from a cylindrical scatterer,” IEEE Trans. Antennas Propag., vol. 36, pp. 1580–1586, Nov. 1988. [12] S. Eisler and Y. Leviatan, “Analysis of electromagnetic scattering from metallic and penetrable cylinders with edges using a multifilament current model,” Proc. Inst. Elect. Eng., vol. 136, pp. 431–438, Dec. 1989. [13] R. G. Riechers, “The application of Lanczos S-expansion method to the solution of TM scattering from a dielectric cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag., vol. 38, pp. 1204–1212, Aug. 1990. [14] K. Sarabandi and T. B. A. Senior, “Low-frequency scattering from cylindrical structures at oblique incidence,” IEEE Trans. Geosci. Remote Sensing, vol. 28, pp. 879–885, Sep. 1990. [15] E. Michielssen, A. F. Peterson, and R. Mittra, “Oblique scattering from inhomogeneous cylinders using a coupled integral equation formulation with triangular cells,” IEEE Trans. Antennas Propag., vol. 39, pp. 485–490, Apr. 1991. [16] A. C. Cangellaris and R. Lee, “Finite element analysis of electromagnetic scattering from inhomogeneous cylinders at oblique incidence,” IEEE Trans. Antennas Propag., vol. 39, pp. 645–650, May 1991. [17] A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag., vol. 40, pp. 96–99, Jan. 1992. [18] H. Tosun, “Novel differential formulation of electromagnetic scattering by dielectric cylinders of arbitrary cross-section,” Proc. Inst. Elect. Eng. Microw., Antennas Propag., vol. 141, pp. 189–195, June 1994. [19] V. P. Chumachenko, “Domain-product technique solution for the problem of electromagnetic scattering from multiangular composite cylinders,” IEEE Trans. Antennas Propag., vol. 51, pp. 2845–2851, Oct. 2003. [20] J. L. Tsalamengas, “Exponentially converging Nyström methods applied to integral-integrodifferential equations of oblique scattering/hybrid wave propagation in presence of composite dielectric cylinders of arbitrary cross section,” IEEE Trans. Antennas Propag., vol. 55, pp. 3239–3250, Nov. 2007. [21] G. A. Greenberg, “On a method of solving the fundamental problem of electrostatics and allied problems,” Zh. Exp. Teor. Fiz. (Russian), vol. 8, pt. 1, pp. 221–252, Mar. 1938, and vol. 9, 2, pp. 725-728, Jun. 1939. [22] S. N. Karp, “The effect of discontinuities of dielectric constant on electrostatic fields near conductors,” Inst. Math. Sci. Dec. 1954, New York Univ., Res. Rep. EM-71. [23] R. W. Scharstein, “Mellin transform solution for the static line-source excitation of a dielectric wedge,” IEEE Trans. Antennas Propag., vol. 41, pp. 1675–1679, Dec. 1993. [24] P. L. E. Uslenghi, “Exact scattering by isorefractive bodies,” IEEE Trans. Antennas Propag., vol. 45, pp. 1382–1385, Sep. 1997. [25] V. G. Daniele and P. L. E. Uslenghi, “Closed-form solution for a line source at the edge of an isorefractive wedge,” IEEE Trans. Antennas Propag., vol. 47, pp. 764–765, Apr. 1999. [26] P. L. E. Uslenghi, “Exact geometrical optics solution for an isorefractive wedge structure,” IEEE Trans. Antennas Propag., vol. 48, pp. 335–336, Feb. 2000. [27] P. L. E. Uslenghi, “Exact geometrical optics scattering from a tri-sector isorefractive wedge structure,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 94–95, 2004. [28] P. L. E. Uslenghi, “Exact geometrical optics scattering from a rightangle isorefractive wedge structure,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 127–128, 2004. [29] A. D. Rawlins, “Diffraction by a dielectric wedge,” J. Inst. Math. Appl., vol. 19, pp. 261–279, 1977. [30] S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge—Part I: Physical optics approximation,” IEEE Trans. Antennas Propag., vol. 39, pp. 1272–1281, Sep. 1991. [31] S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge—Part II: correction to physical optics solution,” IEEE Trans. Antennas Propag., vol. 39, pp. 1282–1292, Sep. 1991. [32] S. Y. Kim, “Hidden rays of diffraction,” IEEE Trans. Antennas Propag., vol. 55, pp. 892–906, Mar. 2007. [33] G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightw. Technol., vol. 20, pp. 1219–1231, Jul. 2002.

[34] T. Lu and D. Yevick, “Comparative evaluation of a novel series approximation for electromagnetic fields at dielectric corners with boundary element method applications,” J. Lightw. Technol., vol. 22, pp. 1426–1432, May 2004. [35] J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag., vol. 20, pp. 442–446, Jul. 1972. [36] J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag., vol. 26, pp. 598–602, Jul. 1978. [37] G. I. Makarov and A. V. Osipov, “Structure of Meixner’s series,” Radiophys. Quantum Electron., vol. 29, pp. 544–549, Jun. 1986. [38] S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Highly efficient full-vectorial integral equation solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Sel. Topics Quantum Electron., vol. 8, pp. 1225–1232, Nov./Dec. 2002. [39] S. V. Boriskina, P. Sewell, T. M. Benson, and A. I. Nosich, “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Society Amer. A, vol. 21, pp. 393–402, Mar. 2004. [40] K. Eswaran, “On the solutions of a class of dual integral equations occurring in diffraction problems,” Proc. Royal Society London, ser. A, pp. 399–427, 1990. [41] E. I. Veliev and V. V. Veremey, “Numerical-analytical approach for the solution to the wave scattering by polygonal cylinders and flat strip structures,” in Analytical and Numerical Methods in Electromagnetic Wave Theory, M. Hashimoto, M. Idemen, and O. A. Tretyakov, Eds. Tokyo, Japan: Science House, 1993. [42] R. Araneo, S. Celozzi, G. Panariello, F. Schettino, and L. Verolino, “Analysis of microstrip antennas by means of regularization via Neumann series,” in Review of Radio Science 1999–2002, W. R. Stone, Ed. Piscataway, NJ/New York: IEEE Press, Wiley Interscience, 2002, pp. 111–124. [43] M. Lucido, G. Panariello, and F. Schettino, “Analytically regularized evaluation of the scattering by perfectly conducting cylinders,” Microw. Opt. Technol. Lett., vol. 41, pp. 410–414, 2004. [44] M. Lucido, G. Panariello, and F. Schettino, “Accurate and efficient analysis of stripline structures,” Microw. Opt. Technol. Lett., vol. 43, pp. 14–21, 2004. [45] M. Lucido, G. Panariello, and F. Schettino, “Analysis of the electromagnetic scattering by perfectly conducting convex polygonal cylinders,” IEEE Trans. Antennas Propag., vol. 54, pp. 1223–1231, Apr. 2006. [46] M. Lucido, G. Panariello, and F. Schettino, “Electromagnetic scattering by multiple perfectly conducting arbitrary polygonal cylinders,” IEEE Trans. Antennas Propag., vol. 56, pp. 425–436, Feb. 2008. [47] M. Lucido, G. Panariello, and F. Schettino, “TE scattering by arbitrarily connected conducting strips,” IEEE Trans. Antennas Propag., vol. 57, pp. 2212–2216, Jul. 2009. [48] M. Taskinen and P. Ylä-Oijala, “Current and charge integral equation formulation,” IEEE Trans. Antennas Propag., vol. 54, pp. 58–67, Jan. 2006. [49] D. S. Jones, The Theory of Electromagnetism. New York: Pergamon Press, 1964. [50] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. The Netherlands: Verlag Harri Deutsch, 1984. [51] J. Van Bladel, Singular Electromagnetic Fields and Sources. New York: IEEE Press, 1991.

Mario Lucido (M’04) was born in Naples, Italy, in 1972. He received the Laurea degree (summa cum laude) in electronic engineering from the University of Napoli “Federico II”, Naples, Italy, in 2000 and the Ph.D. degree in electric and telecommunication engineering from the University of Cassino, Cassino, Italy, in 2004. Since April 2005, he has been a Researcher at the University of Cassino. His research interests include scattering problems, microwave circuits and microstrip antennas. Dr. Lucido received the Giorgio Barzilai Prize for the Best Young Scientist paper at the Italian National Congress on Electromagnetics in 2006.

LUCIDO et al.: SCATTERING BY POLYGONAL CROSS-SECTION DIELECTRIC CYLINDERS AT OBLIQUE INCIDENCE

Gaetano Panariello (M’04) was born in Herculaneum, Italy, in 1956. He graduated summa cum laude in electronic engineering and received the Ph.D. degree from the University “Federico II” of Naples, Naples, Italy, in 1980 and 1989, respectively. He was a Staff Engineer in the Antenna Division, Elettronica SpA, until 1984 when he joined the Electromagnetic Research Group, University “Federico II” of Naples, where from 1990 to 1992, he was a Research Assistant, and from 1992 to 2000, he was an Associate Professor in the Department of Electronic and Telecommunication Engineering, teaching optics, microwave circuit design, and radiowave propagation. In 2000, he joined the Department of Automation, Electromagnetics, Information Engineering and Industrial Mathematics (DAEIMI), University of Cassino, Cassino, Italy, as Full Professor of Electromagnetic Fields. His research interests include inverse problems, nonlinear electromagnetic, analytical and numerical methods, and microwave circuit design. Prof. Panariello is a Member of the Scientific Committee of the National Inter-University Consortium for Telecommunication (CNIT) and of the Scientific Committee of the Italian Electromagnetic Society (SIEM).

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Fulvio Schettino (M’02) was born in Naples, Italy, in 1971. He received the Laurea degree (summa cum laude) in electronic engineering and the Ph.D. degree in electronics and computer science from the University Federico II, Naples, in 1997 and 2001, respectively. Since June 2001, he has been a Researcher at the University of Cassino, Cassino, Italy. His main research activities concern analytical and numerical techniques for antenna and circuits analysis and adaptive antennas. Dr. Schettino received the Giorgio Barzilai Prize for the Best Young Scientist paper at the Italian National Congress on Electromagnetics in 2006.

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Backscattering From a Two Dimensional Rectangular Crack Using FIE Mehdi Bozorgi, Ahad Tavakoli, Giovanni Monegato, Seyed H. Hesamedin Sadeghi, Senior Member, IEEE, and Rouzbeh Moini, Senior Member, IEEE Abstract—The (numerical) solution of a rectangular crack in a perfectly conducting surface is appropriate for non-destructive testing (NDT) applications to model faults. The paper presents a direct modeling technique for determining the H and E-polarized backscattering signatures of a two-dimensional crack in a metallic surface that is suitable for inverse scattering problem. The governing field integral equations (FIE) with logarithmic and hyper singular kernels is first discretized and solved by a collocation method based on Chebyshev polynomials. By using ad hoc quadrature rules, the integral equation is then reduced to a linear system of algebraic equations. This approach does not have the size or frequency limitations of the regular techniques such as modal expansion and quasi-static manners. The results are in good agreement with the entirely numerical and non-reversible solution of a finite element method. Index Terms—Backscattering, Chebyshev polynomials, integral equation, rectangular crack.

I. INTRODUCTION

C

RACKS are the cause of many defects in machines and structures such as pressure vessels or piping of chemical and power plants. Tests on welded structures demonstrate that the initial defects are mainly surface-breaking cracks of approximately semi-elliptical or rectangular shape. These small cracks gradually grow and link together, forming a long surface crack [1]. There are several electromagnetic techniques that can be used to detect surface cracks in metals, including the potential drop technique [2], the eddy current technique [3], the AC field measurement technique [4] and microwaves technique [5]–[7]. Since in many structures such as blast furnaces, use of ordinary NDT techniques is not possible or easy, far field electromagnetic scattering measurement is recommended [8]. Harrington and Mautz suggested a general MoM solution for aperture problems [9]. For a narrow and arbitrary shaped gap, Senior and Sarabandi presented the solution of the integral equation using a point matching MoM approach with a pulse basis function [10]. A quasi-static and higher order boundary conditions solution for the equivalent magnetic current distribution over a dielectric filled narrow groove is also given by Barkeshli and Volakis [11], [12]. Whites and Asvestas utilized impedance boundary condition to obtain scattering fields of a crack [13],

Manuscript received August 18, 2008; revised May 10, 2009. First published November 06, 2009; current version published February 03, 2010. M. Bozorgi, A. Tavakoli, S. H. H. Sadeghi, and R. Moini are with the Electrical Engineering Department, Amirkabir University of Technology, Tehran 15914, Iran (e-mail: [email protected]; [email protected]; [email protected]; and [email protected]). G. Monegato is with the Department of Mathematics, Politecnico di Torino, I10129 Torino, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2036194

[14]. The mode expansion scattering solution for wide cracks is provided by Morgan [15]. Finite Element and hybrid technique solutions for scattering from a crack have also been reported in the literature [16]–[19]. Many studies attempt to directly solve integral equations with non-oscillatory or moderately oscillatory kernels [20]. These methods partially mesh the object surface and thus provide highly efficient solutions. Most integral equations do not have a closed form solution but, they can often be discretized to be solved numerically. Usually, integral equations in electromagnetices are solved by the method of moments (MoM). Proper management of the kernel singularities is an important challenge in implementation of MoM. For Dirichlet boundary conditions, the kernel of integral equation has singularities where and even of the of the forms Cauchy-type [21]. For Neumann boundary conditions where the kernel of the integral equation contains a second order derivative, a hyper singularity is encountered. In this paper, we present the H and E-polarized backscattering signatures of a two dimensional crack in a metallic surface. To avoid the problems involved in the treatment of kernel singularities in the MoM, we resort to the hypersingular integrals introduced by Hadamard introduced for solving Cauchy’s problem of hyperbolic partial differential equations [22]. There are approximate methods of evaluating the hypersingular integrals. One method is based on the direct numerical computation of the finite part integrals by a variety of quadrature techniques that avoid problem restrictions. The paper is organized as follows. In Section II, the governing field integral equation (FIE) for the equivalent magnetic current distribution over the aperture is formulated. The H-polarization integral equation contains a logarithmic singularity that is solved by removing the singularity. For E-polarization, a hypersingular integral equation is derived and consequently solved by a collocation method. The extractions of singular terms of the kernels are given in Section III. In Section IV, the descritization of the first kind Fredholm’s integral equation and its numerical solution are discussed while the far-field backscatter due to the equivalent magnetic current is calculated. The comparison of the results with fully numerical and approximate methods is presented in Section V, demonstrating the validity and efficiency of the proposed method for predicting the H and E-polarized backscattering signatures of long cracks with narrow and wide openings. II. FORMULATION OF THE PROBLEM Assume a plane wave with its z-component of magnetic (H-polarization), electric field (E-polarization), expressed

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Considering the Green’s functions inside and outside of the crack, a Fredlholm’s integral equation of the first kind for (H-pol.) and (E-pol.) are obtained. The equivalent magnetic currents are the solution of the following singular integral equations:

(4)

Fig. 1. Geometry of a two dimensional dielectric-filled rectangular groove illuminated by a plane wave.

in (1), shown at the bottom of the page, where and are respectively, the amplitudes of the incident electric and is the free space wave number magnetic fields. is the incidence angle. For the sake of simplicity, we and and with respect to and normalize the values of correspondingly. With reference to Fig. 1, the wave illuminates the two-dimensional rectangular crack filled with . The crack has perfect conducting walls on three sides and an open aperture on one side. By invoking the equivalence principle on that generates the aperture, the equivalent magnetic current the scattered waves in Fig. 1 at is

and are defined in (5) and (6), shown at the where is the free-space intrinsic admitbottom of the page, where , and is the zero tance, order Hankel function of the second kind. The crack is modeled as a rectangular cavity with the Green’s functions given [11] at

(7) and

(2) The total field is the sum of the incident field and the scattered field . By imposing the continuity of the total tangential on the both side of the apermagnetic and electric fields ture (Region 1: free space, Region 2: crack) we have (3), shown at the bottom of the page.

(8) where . Note that the superscript and in (7) and (8) as well as the rest of the manuscript denote Hand E-polarizations, respectively.

(1)

(3)

(5) and (6)

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III. KERNELS ANALYSIS In order to solve the integral equations expressed in (4), the behavior of their kernels must be examined. This is done by and breaking up the expressions in (5) into free-space components crack-dependent, (9) we can write

(10) and (11) is the first-order Hankel function of the second where kind. It is noted that there is a second-order hypersingularity in in (4) as well as a logarithmic singularity. The Hankel . functions and the infinite series are both singular at In both cases the corresponding integral (4) turns out to be spaces [25], [27]. However, well-posed in proper weighted for an efficient and accurate numerical solution of the integral equations, all singular components of the kernels must be extracted and integrated exactly. Then, the integral equations can be solved by ad hoc collocation methods. A. Kernel By referring to Appendix A, in (10) consists of both hyper and logarithmic singularities where the kernels

can be arranged as (12), shown at the bottom of the page, and are smooth functions defined in where Appendix A. B. Kernel To study the singularities in , it is necessary to explore the infinite sinusoidal series. Figs. 2(a) and (b) show variand for ations of series coefficients H- and E-polarizations in (7) and (8) as functions of the mode number . Propagating modes cause the oscillation, while the monotonic behavior is due to evanescent modes. The series rapidly converges as increases. In contrast, the series coeffifor E-polarization diverges with increasing . cient As a result, the truncated mode number is defined as the value of where the series coefficients start moving into the evanescent mode region. If we only consider the excited modes for , and truncate the series at , a large error is created due to the missing singularities existing in the (complete) series [24], [27]. Thus has singular components at and a nonsinand can be divided into a singular part, , i.e., gular part, (13) where, see (14) and (15), shown at the bottom of the page, where is the sum of the first propagating modes, while is the sum of the remaining infinite evanescent modes. in (15), we use In order to remove the singularities at the procedure given in Appendix B. Expanding the series coefand leads to (16), shown ficients and are given in at the bottom of the page, where Appendix B. It is noted that, for both H and E- polarizations, the first term of the series in (16) is an oscillatory nonsingular

(12)

(14) (15)

(16)

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Thus, the integral equations can be solved by an ad hoc collocation method. IV. SOLUTION OF THE INTEGRAL EQUATIONS Following the treatment of the kernels in H- and E-polarized integrals in terms of appropriate singular and nonsingular terms, we now focus on their numerical solutions. A. H-Polarized Several numerical methods have been proposed and studied for the solution of a weakly singular integral equation of type (4), with the kernel given by (5). For example, for a first rough approximation of the solution we could use a collocation method based on a piecewise constant approximant associated with a uniform partition of the domain of integration. In this small elements of equal case, the domain is divided into . Thus denoting by the length midpoints of these intervals, we first collocate the equation at these abscissas and rewrite it in the form (18) and are defined in (5) and (6). We then rewhere place by the above defined piecewise constant approximant and replace the integral by the sum of the integrals over all seg. By ments, except that defined on the interval containing approximating the first set of integrals by the midpoint rule we obtain

Fig. 2. Behavior of series coefficients (7) and (8) when p is increasing for " j; w  and d :  (a) k = k d (H-polarization), (b) k = k d (E-polarization).

(19)

function while the second term contains hyper and logarithmic . singularities at Finally, (16) is rewritten in (17), shown at the bottom of the in page. A useful representation for the coefficients (17) is given in Appendix B. Notice that in the E-polarization case the integral in (4) is defined in the finite part sense [25].

The integral in (19) is weakly singular over the th interval at . After rebecause of the behavior of moving the singularity by the approach described below, the original integral equation is reduced to a linear system , where and is defined by (20), shown at the bottom of the page. In (20) for the case

= 40 =4 tan( )

= 05

tan( )

(17)

(20)

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case, the Hankel function has a logarithmic singularity that can be removed by the method of [21] as follows:

to reduce the integral equation to the interval . By multiplying and dividing the integrands by and defining the new unknown function (24)

(21) As seen in (21), the resulting integral is nonsingular. Consequently by small argument approximation of the Hankel funcis calculated using (22), shown at tion, the matrix element and , the bottom of the page, where are, respectively, the Euler’s and Napier’s constants. Due to the slow convergence of the above method, one has to take a moderate/large number, , of elements for obtaining a reasonable accuracy. Thus, the order of the final linear system integrals should be is not small and therefore, the associated computed where each one requires the evaluation of the corresponding series expansions of the smooth kernel components. For more accurate results, we could use piecewise polynomial approximations of higher local degree. Instead, we propose to use a more sophisticated and efficient method, requiring the solution of a final linear system having a much smaller order, which also turns out to be fairly well-conditioned, thus reducing the computation time significantly. The method is based on the polynomial collocation approach, using the Chebyshev polynomial expansion. Its computational , where is the degree of the complexity is of order polynomial approximant. Indeed, to determine the coefficients of this approximant, we have to solve a linear system of order , where the computation of each matrix element requires arithmetic operations, with . This computational complexity is more than other methods such as the collocation method that is based on piecewise constant or linear polynomial approximants, such as the one we have described where is the order of the corabove. This would be responding linear system. However, we must observe that using . our method, a given accuracy is obtained by taking To apply our method, we arrange preliminarily (18) in the form

at this point, we replace

by the polynomial of degree (25)

are unknown coeffiwhere are the th degree cients that must be calculated and Chebyshev polynomial of the first kind. We recall that the latter satisfies the well-known three-term recurrence . , i.e., Thus, the collocation points are the zeros of . By substituting (24) in (23), for any we have

(26) Finally, we approximate the integrals in (26) by the -point quadrature rules that are indicated in Appendix C and obtain the following linear system of equations: (27) where ments of

and the eleare given by

(28) (23) and are defined in Appendix C. Finally, Also, the weights is obtained as follows: the magnetic current where is the sum of the are defined in Appendix A and B. We set

and and

(29)

(22)

BOZORGI et al.: BACKSCATTERING FROM A TWO DIMENSIONAL RECTANGULAR CRACK USING FIE

B. E-Polarized We propose to solve E-polarized field by a polynomial collocation method very similar to that described in the previous section for the case of H-polarized field. To this end, we write

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, i.e., Thus, collocation points take the zeros of . By substituting (34) as in (30), we obtain a system of linear equations for any follows:

(30)

is sum of and as dewhere means that the first fined in Appendixes A and B in (30) integral is defined in the finite part [25]. The boundary condition at the edges of the groove enforces and to be zero. After introducing variand , we set ables

The integrals in (35) are approximated by -point quadrature rules as described in Appendix C. Finally, the matrix form of (30) is given below

(31)

(36)

where is an unknown continuous function defined in the . Moreover, the integral (30) can be written in interval the following operator form:

(35)

where ments of

and the eleare given by

(32) where

(37) The weights Appendix C.

and

in (37) are defined in the

C. Far Field Scattering (33)

The far field two-dimensional E- and H-polarization backscattered fields are given by [9], [14]

It can be shown that (32) has a unique solution in a proper weighted Sobolev type space and that can be solved by a collocation method based on Chebyshev orthogonal polynomials of by a the second kind [26], [27]. Therefore we approximate , i.e., polynomial of degree (38) (34) and where are the unknown coefficients that must be calculated and is the th degree Chebyshev polynomial of the second kind. Note that the latter satisfies the well-known three-term recurrence .

(39)

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w = 2:1

M d = 0:2

Fig. 3. Comparison between H-polarized magnetic current j j of DIES and FEM for a crack width of and depth of filled by various materials.

(" = 1)

M of DIES and 5. Echowidth of an empty and a dielectric filled crack of w =  and w =  d = 0:2 ' = 15; 30; 60 Fig. d = 0:25 at various incidence angles  . (a) H-polarized, (b) E-polarized.

Fig. 4. Comparison between E-polarized magnetic current j of FEM, for an empty crack and at and 90 degrees.

j

From which the corresponding echowidths are determined as follows:

(40)

(41)

V. THEORETICAL RESULTS To demonstrate the validity of the solution technique, we first consider the special case of a two-dimensional crack. We then present results for a long slot of rectangular shape as well as combination of multiple long rectangular slots. The latter case serves to show the generality of the proposed model as a useful mean for predicting the perturbation of a real fatigue crack.

The comparisons are made with respective results using the approximate quasi-static method (Q-SM) [11], the mode expansion method (MEM) [15] and a fully numerical 2-D finite element method (FEM) that utilizes absorbing boundary condition (ABC) with triangular nodal meshing [28]. A. Two-Dimensional Crack To study the validity of the proposed method, Fig. 3 illustrate variations of H-polarized magnetic current density for a 2-D crack (Fig. 1) of and when the crack opening is filled with a dielectric ( and ). The results shown in Fig. 3 are obtained by 100 division points using the conventional piecewise constant function method and only 20 points by Chebychev collocation method. It is also noted that collocation method based Chebyshev polynomials is more efficient than the conventional method as it requires a smaller number of discretization points for achieving a desired degree of accuracy. A comparison of the results in this figure demonstrates the accuracy of the DIES

BOZORGI et al.: BACKSCATTERING FROM A TWO DIMENSIONAL RECTANGULAR CRACK USING FIE

method. We then study the E-polarization results. Fig. 4 depicts the equivalent magnetic current distributions on an air-filled and . crack at different incidence angles where As seen in these figures, the results are obtained using the DIES method and the finite element method. An examination of the results shown in Figs. 3 and 4 further validates the accuracy of DIES method. To investigate the effect of incident angle on H and E-polarization echowidths and validation of results at various incidence angles, we have considered an air and a dielectric filled cracks. The results are shown in Fig. 5(a) and (b) demonstrates that, as opposed to the case of E-polarization, the value of echowidth is more influenced by the incidence angle in the case of H-polarization. The effect of crack width on H and E-polarization echowidths is, respectively, shown in Figs. 6(a) and (b) where an air and filled cracks of various widths a lossy dielectric are considered. As can be seen in these figures, when the crack opening is filled with a lossy dielectric, the corresponding echowidth tends to decrease. This is due to the energy loss in the crack opening. Taking the FEM as the reference, it can be deduced from the results shown in Figs. 6 that the DIES is accurate for both cases of wide and narrow cracks, whereas the Q-SM and MEM are found to be appropriate for narrow and wide cracks, respectively. To examine the computation efficiency of the proposed method, we have measured the simulation time of each method used in Figs. 6 for air filled cracks, Table I. An inspection of the results in Table I demonstrates that DIES is faster than all numerical methods, including MEM and FEM. This is more accentuated in the case of FEM for narrow cracks as more mesh points are required. One of the advantages of the DIES is solving of integral equations with minimal mesh points at optimum accuracy and thus, with less memory requirements. In NDT/NDE applications, a serious difficulty is encountered when using full numerical methods such as FEM and MoM. For small cracks in large objects, the number of meshes must increase significantly and consequently, the memory requirements and cpu time drastically increase. It should be however noted that in DIES method as the value (Fig. 1) increases the propagating modes increase of w and becomes increasingly oscillatory. and the magnetic current In other words, the computation of the associated integrals conin the truncated sum requires taining the highly oscillating and a large number of quadrature nodes. Thus we need large to make the desired accuracy. For the matrix, the compuarithmetic tation of each element of the matrix requires operations, with (see Appendix C). In our computa. To avoid unnecessary repetitions, we have taken tion of the same computations, it is more convenient to use the following data bank for both cases of E- and H-polarized fields. and ; • Two -components vectors to store matrices to store • Two ; • A

-components vector to store the moments , needed to compute the weights for a given value of ;

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Fig. 6. H-polarized echowidth of an empty and a dielectric filled crack at normal incidence and depth of 0:2 at various crack widths. (a) H-polarized, (b) E-polarized.

TABLE I THE SIMULATION TIMES FOR THE VARIOUS APPLIED METHODS FOR GRAPH " = 1 IN FIG. 6 (PENTIUM 4, 2 GHz AND 1 G RAM)

• A

matrix to store . From a numerical point of view, to illustrate a quantitative well conditioned matrix of the corresponding linear systems of (27) and (36), a 2-norm condition number (Cond) or spectral norm of Table I is calculated as shown in Table II. Table II verifies

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TABLE II 2-NORM CONDITION NUMBERS (COND) OF LINEAR SYSTEMS (27) AND (36) IN FIG. 6 FOR GRAPH "

=1

that the systems (27) and (36) are fairly well-conditioned; in the used in practice it is reasonable sense that for the values of . Obvismall. In Table II, For instance, taking ously, the 2-norm condition number for -polarization is larger than -polarization. Fig. 7. Geometry of a 3-D empty rectangular long crack illuminated by a plane wave.

B. Long Crack ) at normal inciFor electrically long cracks (i.e., dence angels [11], three dimensional radar cross section (RCS) could be approximated by two dimensional echowidths [10], [29], [30]. Here, we place the magnetic current is obtained from two dimensional DIES at the phase center of the long crack, so the integration is carried out over the length of the cavity, using the three-dimensional radiation integrals [31]. Fig. 7 depicts a general setup for gathering scatter waves that works like a radar system, including a transmitter and a receiver feeler. In this case, the empty long crack is illuminated by a plane wave described below

(42) and cross-polarIn this case, The co-polarization ization bistatic cross-sections are obtained using both the proposed method and the well-known High Frequency Simulation Software (HFSS) for the crack depicted in Fig. 7 at and when and . The results for two cases of E- and H-polarizations are shown in Figs. 8(a) and (b), respectively. A comparison of the results given in these figures substantiates the validity of this procedure and demonstrates that the 2-D magnetic currents are accurately calculated. Moreover, the results are also acceptable for negligible deviation of the crack orientation from the z-axis [11]. C. Multiple Short Cracks A typical connection of three neighboring short cracks in fatigue growth is shown in Fig. 9 forming a so-called Z-shaped crack. The total equivalent magnetic current on the Z-shaped crack is approximated as follows:

(43)

Fig. 8. The long crack Bistatic radar cross section of Fig. 7 at incidence angle  ; and  for the various observation angles  and  (w= : ; L= : ; d= : and " ). (a) H-polarized  ; , (b) E-polarized  ;  .

(

= 45 = 90

)

= 90 =05

=0 =25 (

= 0 25 )

=1

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Fig. 9. Representation of a 3-D empty S-shaped rectangular crack.

where and denote the H and E-polarized 2-D is The magnetic currents obtained using DIES method and angle of crack direction with respect to z-axis (Fig. 7). Integrating along the crack length and using the far-field radiation integrals [33], the bistatic radar cross sections are calculated. To validate the accuracy of this procedure for these configurations, we compare the bistatic radar cross section results with HFSS simulations. Assume a plane wave of 3 millimeters . With wavelength is incident on a Z-shaped crack at mm mm, mm reference to Fig. 9, mm and mm . Figs. 10(a) and (b) show, respectively, the H and E-polarized bistatic radar cross section results using this procedure and the HFSS. Comparisons of the figures demonstrate that these results closely follow their HFSS counterparts except at grazing angles. Placing the magnetic current at the phase center of a long crack leads to an accuracy suitable for straight cracks but not as proper enough for Z-shaped ones. When the observation angle increases, the accuracy diminishes particularly for cross-polarizations. There reasons are first, the assumption of constant magnetic current along the Z-shaped crack is a coarse approximation. Second, the edge effects are prevailing more for Z-shaped grooves. The simulation times for the Z-shaped crack of Fig. 9 are 8.23 and 7.93 minutes for FEM and 4.78 and 4.68 seconds for this procedure for E- and H-polarizations respectively. It is noting that the solution of the problem described in this paper can be utilized to devise appropriate algorithms for predicting crack parameters (i.e., length, depth and orientation). Solution of such an inverse problem can be done using the phenomenological [32], [3] or non-phenomenological methods [33], [34]. In practice, the far field scattering measurement and modeling of electromagnetic waves is an innovative approach in NDT/NDE applications. Thus, invertible solutions to many complex problems in this category must be derived. The disadvantages of such approach include omission of some near field information as well as measurement complexity due to

Fig. 10. The Z-shaped crack bistatic radar cross section of Fig. 9 at incidence and  for the various observation angles  and  angle  (w : mm, d : mm, L : mm  ;L mm  ;L : mm  and " ). (a) H-polarized , (b) E-polarized  ;  .  ;

= 45 90 = 18 ( = 15 ) ) (

= 90 = 0 75 = 75 ( (

= 75 = 0) )

( = 0) =1

= =6

noise, clutter, and narrow beam focusing. In addition, both measurements and simulation algorithms contain complicated processing. VI. CONCLUSION For inverse scattering problems, methods that convert the integral equations to efficient and accurate linear system of equations are desirable. The advantage of the proposed direct integral equation solver (DIES) method is the numerical computation of the singular integral equations by a direct application of quadrature formulas requiring little analytical pre-work. Because of the strong singularity in many integral equations, especially for E-polarization, we cannot utilize the standard descretizing methods. The integrals are approximated by ad hoc

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quadrature rules which lead to a linear system of equations. We associated slow convergence and the divergence in ordinary methods to the singular terms that were extracted from the infinity harmonic series and accelerated the simulation process. The echowidth calculation of an empty and a dielectric filled groove were compared with the quasi-static low frequency approximation, the high frequency mode expansion and the numerical finite element results. Contrary to other methods, the DIES is accurate for both cases of narrow and wide cracks. The proposed method is very efficient and is in good agreement with the time consuming, totally numerical and non- reversible FEM. Consequently, if we can manage singularity in DIES, they are suitable methods for inverse scattering applications such as NDT where no size limitation on crack and good accuracy is in demand.

series coefficients in (15) we have: (B.3) (B.4) and where tion of (B.3) and (B.4) in (15) leads to

. Substitu-

(B.5) and APPENDIX A THE SINGULARITY TERMS OF THE The singular components (10) are extracted by recalling [23]

(A.1)

(B.6)

(A.2)

and It is clear that are singular. To take out the singular terms, we use the following expansions:

(A.3) (A.4)

(B.7) (B.8)

(A.5) where the Euler constant.

and

is

. Since , the above expanwhere sions produce sufficiently accurate results. Obviously, and create hyper and logarithmic singularities. By considering (B.2)–(B.8) we set (B.9)

APPENDIX B THE SINGULARITY TERMS OF THE To extract the singular components of note that for sufficiently large , we have

in (11), we

(B.1) This relation holds as long as (B.2)

(B.10)

However, depends on the wave number and the crack’s difollowing [24] for the mension. Once is chosen, for

(B.11)

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(C.6)

(B.12)

is the th degree Chebyshev polynomial of the first where kind. These are product type quadrature rules, constructed by and [26]. The interpolation of a function at zeros of smooth ordinary integrals are approximated by the well known Gauss-Chebyshev quadrature formula (see [23])

(C.7)

REFERENCES

(B.13) The series in converge very fast.

and

are the smooth terms that

APPENDIX C THE QUADRATURE RULES In (23) and (33), a hypersingular, a logarithmic singular and an ordinary integral are present. An expression for the hypersingular integrals is in the form of (C.1) The quadrature rules for approximating the logarithmic kernels integrals are shown in

(C.2) where

(C.3) where

being the Neumann symbol and

(C.4) so (C.5)

[1] W. D. Dover and R. Collins, “Recent advances in the. detection and sizing of cracks using alternating current field,” British J. Nondestructive Testing, Nov. 1980. [2] M. D. Halliday and C. J. Beevers, The dc Electrical Method for Crack Length Measurement. Warley, U.K.: Engineering Materials Advisory Service Ltd., 1980, pp. 85–112. [3] Y. Li, L. Udpa, and S. S. Udpa, “Three-dimensional defect reconstruction from eddy-current NDE signals using a genetic local search algorithm,” IEEE Trans. Magn., vol. 40, no. 2, pp. 410–417, Mar. 2004. [4] R. Khalaj-Amineh, M. Ravan, S. H. H. Sadeghi, and R. Moini, “Removal of probe lift-off effects on crack detection and sizing in metals by the AC field measurement technique,” IEEE Trans. Magn., vol. 44, no. 8, pp. 2066–2073, Aug. 2008. [5] C. Huber, H. Abiri, S. Ganchev, and R. Zoughi, “Modeling of surface hairline-crack detection in metals under coatings using an open-ended rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 11, pp. 2049–2057, Nov. 1997. [6] F. Mazlumi, S. H. H. Sadeghi, and R. Moini, “Interaction of rectangular open-ended waveguides with surface tilted long cracks in metals,” IEEE Trans. Instrum. Meas., vol. 55, no. 6, pp. 2191–2197, Dec. 2006. [7] F. Mazlumi, S. H. H. Sadeghi, and R. Moini, “Interaction of an openended rectangular waveguide probe with an arbitrary-shape surface crack in a lossy conductor,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3706–3711, Oct. 2006. [8] N. Ida, Microwave NDT. Ohio, 1992, vol. 10, Dept. Elect. Eng., University of Akron. [9] R. F. Harrington and J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag., vol. AP-24, pp. 870–873, Nov. 1976. [10] B. A. Senior, K. Sarabandi, and J. R. Natzke, “Scattering by a narrow gap,” IEEE Trans. Antennas Propag., vol. 38, no. 7, pp. 1102–1111, Jul. 1990. [11] K. Barkashli and J. L. Volakis, “Scattering from narrow rectangular filled grooves,” IEEE Trans. Antennas Propag., vol. AP-39, no. 10, pp. 804–810, Jun. 1991. [12] K. Barkashli and J. L. Volakis, “TE scattering by a two-dimensinal groove in a ground plane using higher order boundary conditions,” IEEE Trans. Antennas Propag., vol. AP-38, no. 10, pp. 1421–1428, Sep. 1990. [13] K. W. Whites, E. Michielssen, and R. Mittra, “Approximating the scattering by a material filled 2-D trough in an infinite plane using the impedance boundary condition,” IEEE Trans. Antennas Propag., vol. 41, no. 2, pp. 146–153, Feb. 1993. [14] J. S. Asvestas, “Scattering by an indentation satisfying a dyadic impedance boundary condition,” IEEE Trans. Antennas Propag., vol. 45, no. 1, pp. 28–33, Jan. 1997. [15] M. A. Morgan, “Mode expansion solution for scattering by a material filled rectangular groove,” Progr. Electromagn.Res., vol. PIER 18, 1998. [16] S. K. Jeng, “Scattering from a cavity- backed slit on a ground plane TE case,” IEEE Trans. Antennas Propag., vol. 16, pp. 1523–1529, Oct. 1990. [17] T. Van and A. W. Wood, “Finite elements analysis of electromagnetic scattering from a cavity,” IEEE Trans. Antennas Propag., vol. 51, no. 1, pp. 130–136, Jan. 2003. [18] S. S. Bindiganavale and J. L. Volakis, “A hybrid FE-FMM technique for electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 45, no. 1, pp. 180–182, Dec. 1997.

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[19] Y. Shifman and Y. Leviatan, “Scattering by a groove in a conducting plane a PO-MoM hybrid formulation and wavelet analysis,” IEEE Trans. Antennas Propag., vol. 49, no. 12, pp. 130–136, Dec. 2001. [20] Y. Chen, “Fast direct solver for the Lippmann-Schwinger integral equation,” Adv. Comput. Math., vol. 16, pp. 175–190, 2002. [21] S. Amari and J. Bornemann, “Efficient numerical computation of singular integral with application to electromagnetic,” IEEE Trans. Antennas Propag., vol. 43, no. 11, pp. 1343–1348, Nov. 1995. [22] J. Hadmard, Le Probleme de Cauchy et les Equations aux Derivees Particelles Lineares Hyperboliques. Paris, France: Hermann, 1932. [23] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, ser. Applied Mathematics series. Washington, DC: National Bureau of Standards, 1972, vol. 55, Department of Commerce. [24] G. Monegato, R. Orta, and R. Tascone, “A fast method for the solution of a hypersingular integral equation arising in a waveguide scattering problem,” Int. J. Numer. Methods Eng., vol. 67, no. 2, pp. 272–297, Jul. 2006. [25] G. Monegato, “Definitions, properties and applications of finite part integrals,” J. Comput. Appl. Math., vol. 229, no. 2, pp. 425–439, Jul. 2009. [26] M. R. Capobianco, G. Criscuolo, and P. Junghanns, “A fast algorithm for Prandtl’s integro-differential equation,” J. Comput. Appl. Math., vol. 77, pp. 103–128, Jan. 1997. [27] G. Monegato, “Numerical evaluation of hypersingular integrals,” J. Comput. Appl. Math., vol. 50, pp. 9–31, May 1994. [28] A. Polycarpou, Introduction to the Finite Element Method in Electromagnetics (Synthesis Lectures on Computational Electromagnetics, Lec#4), 1st ed. San Rafael, CA: Morgan & Claypool, Jun. 2006. [29] A. K. Dominek, H. T. Shamansky, and A. Wang, “Scattering from three- dimensional cracks,” IEEE Trans. Antennas Propag., vol. 37, no. 5, pp. 586–591, May 1989. [30] J. J. Kim and O. B. Kesler, “Efficient 3-D RCS prediction technique for steps and grooves on an arbitrary shaped surface,” in IEEE Antennas and Propagation Society Int. Symp. AP-S. Dig., Jun. 1994, vol. 2, pp. 688–691. [31] C. A. Balanis, Antenna Theory, Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005. [32] Z. Chen, G. Preda, O. Mihalach, and K. Miyak, “Reconstruction of crack shapes from the MFLT signals by using a rapid forward solver and an optimization approach,” IEEE Trans. Magn., vol. 38, no. 2, pp. 1025–1028, Mar. 2002. [33] M. Ravan, S. H. H. Sadeghi, and R. Moini, “Neural network approach for determination of fatigue crack depth profile in a metal, using AC field measurement data,” IET J. Sci., Meas. Technol., vol. 2, no. 1, pp. 32–38, Jan. 2008. [34] R. Hasanzadeh, A. H. Rezaie, S. H. H. Sadeghi, M. H. Moradi, and M. Ahmadi, “A density-based fuzzy clustering technique for non-destructive detection of defects in materials,” J. NDT&E Int., vol. 40, no. 4, pp. 337–346, Jun. 2007.

Mehdi Bozorgi was born in Isfahan, Iran, on September 4, 1977. He received the B.S. degree in electrical engineering from Kashan University, Tehran, Iran, and the M.S. degree in electrical engineering from the Amirkabir University of technology, Tehray, where he is currently working toward the Ph.D. degree. His research field of interest is scattering of electromagnetic waves, electromagnetic nondestructive testing and optics.

Ahad Tavakoli was born in Tehran, Iran, on March 8, 1959. He received the B.S. and M.S. degrees from the University of Kansas, Lawrence, and the Ph.D. degree from the University of Michigan, Ann Arbor, all in electrical engineering, in 1982, 1984, and 1991, respectively. In 1991, he joined the Amirkabir University of Technology, Tehran, Iran, where he is currently a Professor in the Department of Electrical Engineering. His research interests include EMC, scattering of electromagnetic waves and microstrip antennas.

Giovanni Monegato was born in Turin, Italy, on January 22, 1949. He is a Professor of Numerical Analysis in the Department of Mathematics, Politecnico di Torino, Italy, since 1980. His current research interests include numerical analysis, orthogonal polynomials, approximation theory, numerical integration and numerical methods for integral equations.

Seyed H. Hesamedin Sadeghi (M’92–SM’05) received the B.S. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1980, the M.S. degree from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1984, and the Ph.D. degree from the University of Essex, Essex, U.K., in 1991. From 1980 to 1983, he worked within the electrical power industry in Iran. In 1984, he was a Research Assistant with the University of Lancaster, Lancaster, U.K. In 1991, he joined the University of Essex. In 1992, he became a Research Assistant Professor with Vanderbilt University, Nashville, TN. From 1996 to 1997 and from 2005 to 2006, he was a Visiting Professor with the University of Wisconsin-Milwaukee. He is currently a Professor of electrical engineering with the Amirkabir University of Technology, Tehran, Iran. His current research interests include electromagnetic nondestructive evaluation of materials and electromagnetic compatibility issues in power engineering.

Rouzbeh Moini (M’93–SM’05) was born in Tehran, Iran, in 1963. He received the B.S., M.S., and Ph.D. degrees in electronics from Limoges University, Limoges, France. In 1988, he joined the Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran, where he is currently a Professor of telecommunications. From 1995 to 1996, he was a Visiting Professor with the University of Florida, Gainesville. His main research interests are numerical methods in electromagnetic compatibility, and antenna theory. Dr. Moini was the recipient of the 1995 Islamic Development Bank (IDB) Merit Scholarship Award.

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Orbital Angular Momentum in Radio—A System Study Siavoush Mohaghegh Mohammadi, Lars K. S. Daldorff, Jan E. S. Bergman, Roger L. Karlsson, Member, IEEE, Bo Thidé, Kamyar Forozesh, Student Member, IEEE, Tobia D. Carozzi, and Brett Isham

Abstract—Recent discoveries concerning rotating (helical) phase fronts and orbital angular momentum (OAM) of laser beams are applied to radio frequencies and comprehensive simulations of a radio OAM system are performed. We find that with the use of vector field-sensing electric and magnetic triaxial antennas, it is possible to unambiguously estimate the OAM in radio beams by local measurements at a single point, assuming ideal (noiseless) conditions and that the beam axis is known. Furthermore, we show that conventional antenna pattern optimization methods can be applied to OAM-generating circular arrays to enhance their directivity. Index Terms—Directivity optimization, microwave orbital angular momentum, orbital angular momentum (OAM), radio orbital angular momentum, rotating phase fronts.

I. INTRODUCTION S described in the standard electrodynamics literature, an electromagnetic (EM) system will not only radiate energy (linear momentum) but also angular momentum (AM) into the far zone [1], [2]. Already during the early 20th century, predictions of how the polarization of light, i.e., the spin part of the angular momentum (SAM), could transfer angular momentum to a mechanical system were made. In 1935, Beth [3] demonstrated experimentally that this was indeed possible. However, it is only recently that the orbital part of the electromagnetic angular momentum (OAM) has found practical use [4], [5]. Hitherto, the applications have mostly been within the optical regime, but the

A

basic physical properties of the EM fields can be translated from optics to radio; in 2007 the first radio OAM simulations were performed [6]. In the present paper, we make a comprehensive system simulation of OAM carried by radio beams generated by a circular antenna array. It should be emphasized that it is sufficent, but not necessary, to use antenna arrays to generate OAM. Optical methods used at radio frequencies will also give rise to OAM-carrying radio beams [7]. The simulations presented here has been performed in two steps. First, we address issues concerning the generation and directivity of OAM radio beams, while the second part of the paper concerns the measurement of OAM at radio frequencies. It is important to remember that angular momentum (AM) behaves similarly to other conserved electromagnetic quantities such as energy and linear momentum. The AM modes, and hence also the OAM modes, are therefore carried by radio beams just as other conserved electromagnetic quantities are [1], [2]. In this paper, the EM field data are always referring to the three-dimensional (3D) electric and magnetic vector fields, with complete phase information. Furthermore we only consider integer AM, OAM and SAM modes. II. THEORETICAL BACKGROUND The angular momentum of (a spectral component of) the electromagnetic field

(1) Manuscript received December 08, 2008; revised August 25, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported in part by the Swedish Research Council (VR), the Swedish Governmental Agency for Innovation Systems (VINNOVA) and in part by the Swedish Space Board (SNSB). S. M. Mohammadi, J. E. S. Bergman, and B. Thidé are with the Swedish Institute of Space Physics, SE-751 21 Uppsala, Sweden (e-mail: [email protected]; [email protected]; [email protected]). L. K. S. Daldorff was with the Department of Physics and Astronomy, Uppsala University, SE-751 20 Uppsala, Sweden. He is now with the Finnish Meteorological Institute, Helsinki FI-00101, Finland (e-mail: [email protected]). R. L. Karlsson is with the Department of Physics and Astronomy, Uppsala University, SE-751 20 Uppsala, Sweden and also with the Space Research Institute, Austrian Academy of Sciences, A-8042 Graz, Austria (e-mail: [email protected]). K. Forozesh is with the Department of Physics and Astronomy, Uppsala University, SE-751 20 Uppsala, Sweden (e-mail: [email protected]). T. D. Carozzi is with the Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, Scotland, U.K. (e-mail: [email protected]. uk). B. Isham is with the Department of Electrical and Computer Engineering, Interamerican University of Puerto Rico, Bayamón PR 00957-6257, USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2037701

can in a beam geometry be decomposed into a polarizationdependent intrinsic rotation (SAM) and an extrinsic rotation (OAM) [8]. This decomposition of angular momentum is often referred to as the Humblet decomposition [9], i.e., (2) where (3) (4) in Note the occurrence of the OAM operator (3). Polarization is the classical manifestation of the quantum mechanical concept spin and we will refer to the intrinsic, polarization dependent, rotation as spin angular momentum (SAM) with mode number denoted . The extrinsic rotation is referred

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to as orbital angular momentum (OAM) with mode number . The angular momentum is the composition of OAM and SAM . The angular mosuch that the AM mode number mentum mode number can be explicitly calculated from the formula [1]

(5) where

(6) As shown in [6], an OAM-carrying radio beam can be generated equidistant elements, elecby a circular antenna array with , trically phased such that the th element has phase is the angle of the element position; where where denotes the radius of the array. For generality, we assume that each antenna element consists of three orthogonal and collocated small electric dipole elements (tripole) of length . The current distribution in the th element is then . Depending on the phasing of the dipoles, the constant current density vector, , is in general complex. The vector potential for an array of tripoles can then be written

(7) for phases and where we have used for amplitudes, i.e., the standard infinitesimal in (7) cordipole approximation [10]. The vector potential responds to electric dipole radiation (see, e.g., [1]) and the sum, , is the array factor. If is suffiwhich we denote by ciently large the array factor can be approximated by an integral over the angle

(8) , in the final exNote the azimuthal phase dependency, pression of the vector potential. When the OAM operator, , in (3) operates on the azimuthal phase dependence becomes clearly visible, since (9) A general characteristic of an OAM beam is that the fields have rotating phase fronts [10]; see Fig. 1. This contrasts with

Fig. 1. The rotational phase front of an OAM radio beam as obtained in our numerical simulations. Panels (a) and (b) are visualizations of the phase of an electric field component for a beam of OAM mode l and l , respec from the transmitting tively. In both cases the beams are intersected at z array, and the window is  wide in both x and y direction. A change in color from blue to red, yellow, green, and back to blue again corresponds to a change in phase of  .

20

=1 = 25

=2

2

polarized beams, which have rotating electric field vectors. Generally, the polarization-dependent angular momentum (SAM) can be found by measuring the 3D generalized Stokes param[11]–[13], whereas the OAM part must be calculated eter from explicitly, either through the Humblet decomposition or by taking the difference between the total angular momentum and the “spin” (proportional to ). III. ANTENNA SIMULATION SETUP Electromagnetic orbital angular momentum can be described in both classical and quantum physics language. Even though antennas are not usually described in quantum language, the field characteristics of photon laser experiments [5] can be reproduced in the radio regime. The OAM-carrying fields can be generated in a simple way by a phased circular antenna array, array elements, distributed equidistantly around where the the perimeter of the circle, are phased such that the phase dif, where is the ference between each element is desired OAM state [6]. The array defines the plane with the axis normal to this plane; see Fig. 2. In the simulations, far zone approximations of the fields cannot be made before the actual calculations are performed since, trivially (10) showing that the OAM transported to the far zone originates in second order terms normally associated with near- and intermediate-zone fields that are (usually) neglected in the far zone [14], [15]. This does not imply that the OAM of an EM field is restricted to the near or intermediate zone; on the contrary, as can be seen from Fig. 3, when in the near zone, pure angular momentum modes cannot be estimated reliably from (1), due to strong near-field and components. Since the OAM mode is defined by its rotating phase front, it is a highly phase-dependent phenomenon and hence much smaller phase errors are tolerable than for an ordinary radiating system, where the standard tol[16]. The rotating erated far-field phase error is typically phase front created by the radiating antennas should be calculated as exactly as possible in order to achieve the necessary precision. Once the actual calculation is performed, one can apply the far zone approximation without losing information about the angular momentum mode.

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IV. DIRECTIVITY The phase configuration of the OAM-generating circular array gives rise to coherent OAM-carrying radio beams with characteristic on-axis phase singularities and hence amplitude nulls at the centers of the beams. As shown by Chireix [19], who considered the radiation patterns of phased circular arrays but not the angular momentum of the radio beams generated by them, the array factor is given by (11) Fig. 2. Configuration of a typical OAM-generating antenna array used in the present work. In this case electrically short dipoles are equidistantly spaced along the perimeter in a circular geometry, phased such that .

N = 12

 = 2l=N

Fig. 3. OAM radiation patterns (a) for beams carrying OAM mode corre(red, solid), (blue, dash-dot) sponding to (green, dashed), (black, dotted); (b) the calculated behavior of the same beams as a and function of distance from the antenna array. The fall-off for the higher modes visible in (b) is due to the wide null in the simulation box (volume). The beam itself still carries the correct mode but outside the simulation box. Due to low power density at the beam null, the simulation volume must be large enough to capture a part of the main beam. The beams were created with and electrically short dipoles placed in a circular array of diameter operating at a frequency of (10 m vacuum wavelength).

l=4

l=1

l=2

l=3

l

l

f = 29:98 MHz

N = 12 D=

We have produced our raw data (EM fields) by using the Maxwell equation solver NEC2 and the geometry editor of 4NEC2 [17]. However, our results are reproducible with any sufficiently accurate electromagnetic simulation tool. The angular momentum can be calculated by averaging the OAM on cross-sections of the radio beam with the cross-sectional planes perpendicular to the beam-axis [18]. In this paper, the calculations were performed in rectangular 3D volumes, sliced such that the axis becomes the normal to the intersecting measurement planes. In each plane, the energy, Poynting flux, OAM, and the “spin” (i.e., wave polarization, SAM) are calculated from the 3D electric and magnetic fields.

where is normalized to unity, is the diameter of the circular array, are Bessel functions, is the angle between the normal to the plane of the circle and the direction from the ring array to the field point, and the azimuth of the same point [20]. From (11) the phase singularity in the beam center can be found. Note that the array factor in (8) corresponds well to Chireix expression presented in (11). In contrast to the highly collimated and cylindrical OAM laser beams [5], the spatial extent of the null region of the radio beams widens fast with the vertical distance from the transmitting array due to the conical beam intensity structure; see Fig. 3. Clearly, this makes it inappropriate to place the receiving antennas in the null at the center of the transmitted beam. A convenient way to produce OAM modes is to use an array of electrically short dipoles. These small dipoles are geometrically simple and hence easier to work with in the simulation dipoles may also be used. tool, but for practical purposes The basic configuration of the array of short dipoles can be seen elements. in Fig. 2 for Radiation patterns of four OAM modes generated with an anelements are shown in Fig. 3. The tenna array with width of the null along the axis can be reduced with standard methods such as increasing the directivity of the individual elements, or widening the diameter of the circular array. Another way to generate more-collimated OAM-carrying radio beams is to superimpose different OAM modes [6]; this technique will not be discussed in the present paper. A. Number of Elements of the Array The number of array elements has an additional influence on an OAM-generating antenna array compared to a regular antenna array: it determines the largest mode the array can genwhere is the erate. Theory predicts number of array elements [6]. In an ideal circular OAM dipole array, the short dipoles are continuously distributed along the circle. In the discrete approximation used here, OAM modes with too large values of ) will not generate a pure rotating phase front and hence no perfect OAM mode. The beam will be distorted and will not produce a stable OAM mode number over the whole conical intensity maximum profile. The effect of a low number of antenna elements on the radiation pattern can be seen in Fig. 4, and its general effect on the gain maximum in Table I. When the elements are not short dipoles but, for instance, dipoles, not even is sufficient to yield a satisfactory

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Fig. 4. For a low number of elements in the circular array, such as the case which is shown in the upper part of the figure, the radiation pattern . The diameter of the array is for all exhibits ripples which vanish for , and the elements are excited at 29.98 MHz (10 m vacuum wavelength).

N=4 N



N 6

TABLE I THE TABLE SHOWS THAT THE ANGLE OF MAXIMUM GAIN, IN THE TABLE, DOES NOT VARY WITH THE NUMBER OF ELEMENTS FOR AN OAM BEAM, AS LONG AS MORE THAN FIVE ANTENNA ELEMENTS ARE USED. THE DIAMETER OF THE ARRAY IS for All , AND THE ELEMENTS ARE EXCITED AT 29.98 MHz (10 m VACUUM WAVELENGTH)





l=1

N

=2

Fig. 5. OAM modes generated with six dipoles excited at 2.4 GHz in a . Here one observes the difficulty of being too circular array with close to the limit when one uses antennas which are not electrically short. For each mode, (red, solid) and (green, dashed), the radiation pattern is shown in panel (a), whereas the OAM mode number as a function of vertical distance to the array is shown in panel (b).

D = 1:5 jlj > N=2 l=1

result. Then the measured OAM mode number will be close to the transmitted value but will never quite reach it; see Fig. 5. It is reasonable to assume that these limiting cases are not as ; this will robust against noise as when is far away from be investigated in a subsequent paper. B. Aperture of the Array To minimize the null in the direction along the axis of the array, the diameter (aperture), , of the circular array can be increased. We have studied a twelve-element dipole array phased mode. By increasing the diameter of the to create an array, the angle of maximum gain is reduced, as seen in Fig. 6 and Table II. The radiation pattern in Fig. 7 illustrates the cylindrical shape of the beam for the case in Fig. 6(d) for a circle with . The sidelobes also increase with , but they are at much wider angles than the main lobe. They will not be detected by a measurement close to the beam axis, and will have a small effect on the simulations as long as they are weaker than the center lobe; see Fig. 6. Increasing sidelobes mean a loss of energy, but for our OAM system simulation this has no significant importance. V. SIZE OF THE MEASUREMENT REGION We shall now show that we do not need to measure the entire beam profile in order to get a reliable OAM measurement. It

l=2

is, however, important to measure at the correct positions with respect to the beam axis. A. Optimal Measurement Position The estimate of the OAM number reflects the radiation intensity in a very direct way, as can be seen in Fig. 8. The asymmetries of the radiation pattern in and Fig. 8(b) are reflected in the surface plot of the OAM mode number. A measurement at the beam null will give a very weak signal and hence not a reliable OAM mode estimate, while a measurement in the major lobe will result in a strong signal and an accurate OAM mode estimate. In other words, when antenna arrays are used, a good OAM measurement position corresponds to a point of high beam intensity (Poynting vector or linear momentum intensity). Any asymmetry of the radiation pattern will be reflected in the OAM mode measurement. The angular momentum intensity can be compared to the radiation intensity measurement by plotting the intensity of angular momentum, , together with , the radiation intensity; this was done in Fig. 9 for the case , and . It can be seen that the angular momentum intensity has a more collimated structure than the radiation intensity. That is, the boundary between the beam maximum and the beam null is sharper for the angular momentum intensity than for the radiation intensity. This makes us claim that the angular momentum intensity is a better indication of where the angular momentum is measurable than the conventional radiation (linear momentum) intensity.

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Fig. 7. Plot showing the 3D radiation intensity of a beam carrying OAM mode , generated by a circular array of identical electrically short dipoles and with a diameter . Notice the almost cylindrical shape of the main lobe.

l=1

D = 4

N = 12

B. Minimum Measurement Area Another crucial question is over how large an area the fields must be measured for a fixed number of sample points and an arbitrarily-chosen fixed position in space. We have performed simulations for a fixed number of four measurement points, while we have varied the distance between the points from to ; see Fig. 8. The figure shows that the system does not exhibit any radical changes for different measurement areas. The only substantial variations of the OAM number occur in the beam null, where there in any case is no usable signal. Fig. 8(c) shows that there is a small improvement in the estimation of the radio beams OAM mode for larger “measurement” cross-sections, however it is not of significant order.

D

Fig. 6. Variation of the power density with the diameter of the circular array for an element array. When the diameter of the array is changed the pattern becomes more collimated and the side lobes increase. In all four cases, the beams carry the OAM mode number . The elements were operating at ; (b) ; 29.98 MHz (10 m vacuum wavelength). (a) ; (c) (d) .

N = 12

l=1 D=

D = 4

D = 2

D = 3

TABLE II THE TABLE SHOWS THE VARIATION OF THE ANGLE OF MAXIMUM GAIN, , WHEN THE DIAMETER, , OF THE CIRCULAR ARRAY IS VARIED. CF. FIG. 6(D)

G

D



C. Minimum Number of Measurement Points Another important question is how many measurement points are needed within a fixed area in order to measure an OAM mode , where is the measurement error of integer within . As can be seen in the table in the desired OAM number Fig. 10, the number of points can be rather small and still one can estimate the OAM mode number. From a field perspective, if the 3D electric and magnetic fields are measured and the position of the beam axis is known (or the exact position and orientation of the source is known), the OAM mode number is a locally measurable quantity, i.e. can be measured in a small surrounding of a single point. This is in contrast to laser experiments where the entire beam profile is usually measured in order to estimate the OAM mode number [4]. Nevertheless, the OAM of an electromagnetic beam is as much a property of the fields themselves [21] as in the polarization or Poynting flux, and this suggests that it should be as stable or unstable as the other electromagnetic quantities. D. OAM and Polarization So far we have not taken into account the polarization of the field, i.e., the spin angular momentum, SAM, which is a frequently-utilized rotational/topological quantity. Can the two different rotational modes, OAM and SAM, be distinguished and, if so, how? In fact, SAM (polarization) modes and OAM modes can be separated. Instead of using a circular array of , an array of crossed dipoles may dipoles to generate

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=2

Fig. 8. OAM measurement simulations for a beam carrying OAM mode number l , generated by 12 short electric dipoles in a circular array with diameter D . Panel (a) shows l estimated in a plane at a distance of  from the circular array and perpendicular to the z axis, i.e., the beam axis. Panel (b) shows the 3D radiation intensity. In (a) each OAM number is averaged over four grid points, which in are separated by : . This means that the sample area is only :  . Panel (c) shows the dependence of the calculated OAM mode number on the position of the measurement area when the position is passing from the beam maximum through the beam null and to the beam maximum on the other side of the null. The colors correspond to the side of the sample square, i.e., the distance between two adjacent sample points, as follows: blue, : , red, :  green, :  cyan, :  magenta, :  yellow, :  and black : . The radiation intensity directly affects the estimate of the OAM mode number. At positions of low field intensity, e.g., in the beam null, the OAM mode number deviates significantly from l .

= 0 01

50

1 00

01

0 75

0 50

0 40

0 30

0 20

0 10

=2

and SAM , and a second beam carrying OAM and SAM , as seen in Fig. 11. In both rying OAM cases the total angular momentum mode number . In this case the AM mode number cannot be separated into and without determining by either calculating the polarization from the generalized Stokes parameter [11]–[13] or by dividing the signal into vertical or horizontal polarization. In order to determine the polarization from the generalized Stokes parameters, we consider the component of the 3D Stokes parameter [11]–[13] Fig. 9. Panel (a) shows the radiation intensity and panel (b) the angular momentum intensity for a radio beam carrying OAM proportional to l . The beam nulls are along the z axis and the angular momentum intensity has a more collimated structure than the radiation intensity. In this case N and D .

=1 = 12

=

(12) which can be normalized such that

(13) which, for the special case of a circular polarized beam, equals . VI. CONCLUSION AND DISCUSSION

Fig. 10. The table shows how the deviation of the integrated OAM number depends on the number of grid points in the meafrom the ideal case l to in a fixed measurement area. The number of grid points varies from surement area of size  . The plot illustrates how the grid points were distributed (cross), (stars) and (dots). for the first three cases,

( = 2) 1

1

2

10

3

be used to generate a perfect circularly polarized field with the same but also with an additional SAM mode number . If one only considers , some of the modes will become degenerate. Consider for instance two different beams, one car-

Our simulations show that standard antennas arranged in circular arrays can be used to generate beams that carry orbital angular momentum. Hence, directivity and other antenna properties can be calculated using standard methods, and we can use existing optimization techniques to incorporate and study OAM in radio beams. This is of course interesting if one would like to incorporate OAM radio modes into existing radio technology. We have also demonstrated that only a local measurement is needed in order to detect OAM in a radio beam, provided that the beam axis is known. In our simulations, neither the number of measurement points nor the measurement area affect significantly the (noiseless) measured OAM mode of the field. In practice, however, real-world measurement noise is likely to favor larger numbers of points and larger measurement areas.

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Fig. 11. Two different beams, corresponding to two degenerate cases of angular when (a) OAM mode number l , SAM momentum mode number j and (b) l ,s . If only the angular momentum mode number s J " r E B V is measured, the two cases are indistinguishable. However, they can be resolved if the generalized Stokes parameter V is used to determine s.

=

= 01 2Ref 2 gd

=0 = 01 = 1

=1

In the optics literature dealing with OAM [22], it is shown that paraxial Laguerre-Gauss (LG) modes carry OAM and therefore have non-planar, rotating (helical) phase fronts. Although the OAM antenna array system does not generate pure LG beams, a rotation in the phase front is clearly visible even in radio beams which carry OAM; see Fig. 1. This suggests that the radio beams generated in this way have similarities with LG beams.

ACKNOWLEDGMENT The authors thank L. Gustafsson for useful comments and discussions.

[6] B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. E. S. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utlilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett., vol. 99, no. 8, p. 087701, Aug. 22, 2007. [7] G. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre wave frequencies by use of a spiral phaseplate,” Optics Commun., vol. 127, pp. 183–188, 1996. [8] S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quant. Semiclass. Opt., vol. 4, pp. S7–S16, 2002. [9] J. Humblet, “Sur le moment d’impulsion d’une onde électromagnétique,” Physica, vol. X, no. 7, pp. 585–603, 1943. [10] J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett., vol. 81, no. 22, pp. 4828–4830, 1998. [11] T. Carozzi, R. L. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E, vol. 61, pp. 2024–2028, 2000. [12] J. E. S. Bergman, S. M. Mohammadi, T. D. Carozzi, L. K. S. Daldorff, B. Thidé, R. L. Karlsson, and M. Eriksson, “Conservation laws in generalized Riemann-Silberstein electrodynamics,” arXiv vol. 0803.2383, 2008 [Online]. Available: http://arxiv.org/abs/0803.2383 [13] C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach. New York: Wiley, 1998. [14] V. M. Abraham, “Der Drehimpuls des Lichtes,” Physik. Zeitschr., vol. XV, pp. 914–918, 1914. [15] B. Thidé, Electromagnetic Field Theory. Uppsala, Sweden: Upsilon Books, Sep. 2007 [Online]. Available: http://www.plasma.uu.se/CED/ Book [16] C. A. Balanis, Antenna Theory, 3rd ed. New York: Wiley, 2005. [17] A. Voors, 4nec2 [Online]. Available: http://home.ict.nl/~arivoors/ [18] M. V. Berry, “Paraxial beams of spinning light,” in Singular Optics, M. S. Soskin, Ed. Philadelphia, OA: SPIE, Aug. 1998, vol. 3487, Int. Society for Opt. Eng., pp. 6–11. [19] H. Chireix, “Antennas á Rayonnement Zénital Réducit,” L’Onde élec., vol. 15, pp. 440–456, 1936. [20] H. L. Knudsen, “The field radiated by a ring quasi-array of an infinite number of tangential or radial dipoles,” Proc. I.R.E., vol. 45, no. 11, pp. 781–789, Jun. 1953. [21] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature, vol. 412, pp. 313–316, 2001. [22] L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum. Bristol, U.K.: IOP, 2003.

Siavoush Mohaghegh Mohammadi received the M.Sc. degree in engineering physics from Uppsala University, Uppsala, Sweden, in the spring of 2008. During 2008, he was hired as a Research Assistant at the Swedish Institute of Space Physics, Uppsala. Currently, he is a Guest Researcher at the Interamerican University of Puerto Rico, Bayamón, Puerto Rico, USA.

REFERENCES [1] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1998, ch. 7, Problem 7.29. [2] J. Schwinger, L. L. DeRaad, Jr., K. A. Milton, and W. Tsai, Classical Electrodynamics. Reading, MA: Perseus Books, 1998. [3] R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev., vol. 50, no. 2, pp. 115–125, 1936. [4] G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express, vol. 12, no. 25, pp. 5448–5456, Nov. 21, 2004. [5] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Optical angular momentum of light and the transformation of Laguerre-Gauss laser modes,” Phys. Rev. A, vol. 45, no. 11, pp. 8185–8189, 1992.

Lars K. S. Daldorff received the M.Sc. degree in astronomy from the University of Oslo, Norway, in 2001, on the topic of numerical simulations and the Ph.D. degree in space physics, focusing on numerical simulations, from Uppsala University, Uppsala, Sweden, in 2009. As a Research Assistant at the University of Oslo, he worked on streaming-data radio sensor networks for space science in collaboration with Uppsala University, Växjö University, the Swedish Institute of Space Physics, and IBM. Currently, he holds a Postdoctoral Research position at the Finnish Meteorological Institute, Helsinki, Finland.

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Jan E. S. Bergman received the Ph.D. degree in space physics from Uppsala University, Uppsala, Sweden, in 2000. After his dissertation, he was a Consultant in a private company working in the field of telecommunications where he developed digital pre-distortion techniques for 3G base-station multicarrier power amplifiers (MCPA), and later he co-founded a university spin-off telecom company where he worked as a research manager. Upon returning to academia in 2004, he spent one year as a postdoc at the Space Research Centre of the Polish Academy of Sciences in Warsaw, Poland, before being employed by his home institution, in November 2004, as a Project Manager for a nanosatellite payload encompassing miniaturized software defined radio (SDR) receivers. Currently, he is a Senior Scientist and, since 2006, a permanent staff member at the Swedish Institute of Space Physics (IRF), Uppsala. He serves as Co-I for radio payloads on several satellites, notably Compass-2, which was launched in 2006 and carried an SDR developed in house. He has recently been engaged as PI and Co-PI in two international space borne low-frequency radio astronomy proposals but also works with nanosecond radio pulse detection of ultrahigh energy neutrinos in the Antarctic ice. He has around 25 published papers and has given 10 invited talks since his return to IRF. He teaches microsatellite technology at the masters program in engineering physics and is a vice chairman of the education programme committee for the electronics and information technology undergraduate programs at Uppsala University.

Roger L. Karlsson received the M.Sc. degree in engineering physics and the Ph.D. degree in space physics from Uppsala University, Uppsala, Sweden, in 1997 and 2005, respectively. In 2000, he co-founded the company Red Snake Radio Technology AB. He has previously been at ScandiNova Systems AB, Uppsala. Presently, he is Postdoctoral Researcher at the Space Research Institute of the Austrian Academy of Sciences, Graz, Austria, and a Lecturer at the Department of Physics and Astronomy at Uppsala University. His main research interests are antenna theory, antenna calibration, and solar radio emissions.

Bo Thidé received the Ph.D. degree in theoretical physics from Uppsala University, Uppsala, Sweden, in 1979. In 1980, he joined the Swedish Institute of Space Physics, Uppsala, where he was appointed Senior Scientist in 1983, Associate Professor in 1984, Director of Science 1991, and Professor in Space Physics in 2000. In 1981, while conducting an experiment at the EISCAT Heating Ionospheric HF Interaction Facility near Tromsø, Norway, he discovered that the ionospheric plasma, when perturbed by a strong radio beam into a turbulent state, produces systematic secondary radio emissions. This discovery laid the foundation for the so called SEE diagnostic technique now in use at all ionospheric HF interaction facilities in the world and also led to the Edlund Prize from the Royal Swedish Academy of Sciences. He later developed, together with Associate Professor Bengt Lundborg, an analytic method for calculating the full three-dimensional wave pattern, and associated quantities such as angular momentum, of a radio wave propagating in an inhomogeneous, magnetized, collisional plasma. In 2005, he and his project members at the Swedish Institute of Physics in Uppsala demonstrated that a radio beam generated by a circular phased arrays can carry

both spin angular momentum and orbital angular momentum. He has also authored and coauthored articles in applied mathematics, plasma physics, and particle physics. In 1985 and 1986, he was a Visiting Scientist at the Arecibo Observatory, Puerto Rico. Currently, his main research interests are radio wave propagation and interactions, development of innovative methods for studies of space based on fundamental physics, cosmic electrodynamics, computers and computing in physics, and theory of science. He is the author of the textbook Electromagnetic Field Theory, that is available on the Internet.

Kamyar Forozesh completed his Master’s thesis at KDDI R&D Laboratories, Tokyo, Japan. His research was on advanced modulation formats for fiber-optic applications such as optical-OFDM. He is currently working toward the Ph.D. degree at Uppsala University, Uppsala, Sweden. He is also working in the Department of Physics and Astronomy, Uppsala University. Mr. Forozesh’s Master’s thesis was acknowledged by the IEEE Laser and Electro-Optical Society (LEOS).

Tobia D. Carozzi received the M.Sc. degree in engineering physics, specializing in radiation sciences, from Uppsala University, Uppsala, Sweden, in 1993 and the Ph.D. degree in space physics from the Swedish Institute of Space Physics, Uppsala, in 2002. He has more than 10 years of experience working with space research in general and radio science in particular. In 2000, he co-founded Red Snake Radio Technology AB, where he was a Senior Researcher. He holds several international patents and won several innovation awards. He is currently a Research Fellow at the University of Sussex, U.K., where he works with several recent novel developments in theoretical electromagnetism.

Brett Isham received the B.S. degree in electrical engineering from Stanford University, Stanford, CA, in 1980 and the Ph.D. degree in space plasma physics from Cornell University, Ithaca, NY, in 1991. He worked as a Postdoctoral Researcher in 1991 at the National Astronomy and Ionosphere Center in Ithaca, New York; from 1991 to 1994 at the Swedish Institute of Space Physics in Kiruna, Sweden; and from 1994 to 1997 at the Arecibo Observatory, Arecibo, Puerto Rico. From 1997 to 1998, he worked as a Staff Scientist at the Arecibo Observatory; from 1998 to 2000 as an Associate Professor of physics at Interamerican University of Puerto Rico, Bayamón; from 2000 to 2004 as a Scientific Advisor at the European Incoherent Scatter (EISCAT) Scientific Association in Tromsø, Norway; in 1994 as a Scientific Advisor at the EISCAT Svalbard Radar in Longyearbyen, Norway; from 2004 to 2007 and from 2007 to the present as an Associate Professor and Professor, respectively, of electrical engineering at Interamerican University of Puerto Rico, Bayamón. His main research interests are artificial and natural Langmuir turbulence in the ionosphere, remote sensing of plasma and atmospheric parameters, and radio wave transmission, propagation, and detection.

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On-Body Diversity Channel Characterization Imdad Khan, Yuriy I. Nechayev, Member, IEEE, and Peter S. Hall, Fellow, IEEE

Abstract—Statistical analysis of the on-body diversity combined and branch signals are presented. The distribution of short-term fading envelopes of the branch and combined signals show that the fading environment for the on-body channels is Rician. The longterm fading best fits the Log-Normal distribution. The average best fit parameters are presented for short-term and long-term fading envelopes of the branch and combined signals. The Doppler spectra are also presented. Three different types of antennas were used and two on-body channels were characterized at 2.45, 5.8, and 10 GHz. Index Terms—Channel characterization, diversity, doppler frequency shift, fading channels, on-body channels.

I. INTRODUCTION

B

ODY-CENTRIC wireless communications has established itself as an important branch of personal communication systems. There is a significant breadth of potential to the work on body-centric communications. By way of example, we quote the remit of IEEE 802.15.6 (task group for BAN) [1] as: “developing a communication standard optimized for low power devices and operation on, in or around the human body (but not limited to humans) to serve a variety of applications including medical, consumer electronics, personal entertainment and other”. There is also a large, but little reported, interest from the defence community, with emphasis on equipping the future soldier with personal wireless connected on-body equipment. The medical sensor network area seems to be perhaps the biggest potential market with many new products coming on to the market, some of which, in current and future generations, use on-body channels. The ever increasing use of wireless devices and future high data rate requirement in the above mentioned and other application areas like entertainment, security and personal identification, fashion, and personalized communications, etc. drives the research to establish more reliable and efficient link between the devices mounted on the body, and demands the use of multiple antennas for the on-body and off-body channels. The very significant fading that occurs in these sorts of channels suggests that, to minimize power needs, diversity may well be used. Channel characterization will then be an important issue. Antenna diversity is a well Manuscript received November 07, 2008; revised May 13, 2009. First published December 04, 2009; current version published February 03, 2010. I. Khan was with the University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. He is now with COMSATS Institute of Information Technology, Abbottabad, Pakistan (e-mail: [email protected]). Y. I. Nechayev and P. S. Hall are with the University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037759

known technique to overcome fading and provide a power efficient link. Signals from two or more uncorrelated branches are combined in different ways to achieve the diversity combined signal. Mobile radio channels are usually characterized by Rayleigh fading, which assumes a random variation in the amplitude of the fading envelope and uniform distribution of the phase in nonline-of-sight (NLOS) scenario with no dominant component. Both the diversity branches in a mobile diversity system are assumed to be independent and identically distributed. Rayleigh fading distribution is not valid for the on-body diversity channels as most of the commonly used on-body channels have a strong LOS link. The fading in this case is mainly caused by the movement of the scatterers surrounding the antennas and also by the environment. The diversity branches can be independent but the distributions of the branch signals may not be identical. The scattering due to the motion of the body may change the fading distribution at the two branches [2]. Much attention has been paid to the characterization of antennas for on-body channels, the effect of human body presence on the communication link, antenna position on the body, use of phantoms, and the link budget of the on-body channels [2]–[5]. The diversity performance analysis of the on-body channels in terms of diversity gains and correlation between the branches is presented in [6]–[9]. The absence of any statistical model for the on-body channels requires an extensive measurement campaign to determine the channel statistics. It is important to know the fading distribution and perform the statistical analysis of the on-body channels. It is also of interest to know the distribution of the diversity combined signals, as diversity gain is commonly calculated from the distributions of the branch and combined signals. To the best of the authors’ knowledge, no attempt has so far been made to characterize the on-body diversity channels with random movements and to find a distribution for the short-term and long-term fading caused by the body movements and the environment surrounding the body-area network (BAN). This paper presents the statistics and the fading distributions of the on-body diversity channels with antennas mounted on a real human body. The subject performed random movements in a typical indoor laboratory environment. Two channels, which show importance in the current application areas, were selected for the measurement, namely, the belt-wrist and belt-head channels. Three types of antennas were used and the channels were characterized at three frequencies, i.e., 2.45, 5.8, and 10 GHz. The description of the environment and the antennas used, along with the measurement setup, is presented in Section II. Section III of the paper describes the data processing and the results. Spectral analysis of the signals is also presented in this section. Section IV concludes the results.

0018-926X/$26.00 © 2009 IEEE

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Fig. 1. (a) Space diversity monopole antenna with spacing d and the two orientations (b) and (c).

Fig. 2. (a) The diversity printed-IFA and (b) the transmitting printed-IFA with y^ for belt-head and a^ = z^ for belt-wrist channel. Black lines are the metallic conductors. Metallic ground planes are in grey and the substrate size is indicated by the thin line.

a^ =

0

0

Fig. 4. Body-coordinate system.

B. Measurement Setup

Fig. 3. (a) Top view of the diversity PIFA and (b) the transmitting PIFA with

a^ = y^ for belt-wrist and a^ = 0z^ for belt-head channel.

II. MEASUREMENT PROCEDURE A. Antennas Used in Measurement Due to the better link performance of the monopole antennas on the body [5], diversity measurements were first performed monopole antennas were with monopoles. Two thin-wire placed on the same ground plane to achieve space diversity, as shown in Fig. 1. The high-profile shape of monopole antennas makes them impractical for use in wearable devices. Therefore, some low profile realistic antennas were also used. These were the printed inverted-F antenna (printed-IFA) and planar inverted-F antenna (PIFA), as shown in Figs. 2 and 3, respectively. The transmitting antenna in each case was a single element of the same type. Assume that is a unit vector pointed towards the direction of the feeding line (as shown in Figs. 2 and 3), and are the unit vectors corresponding to the x, y, and z axes of the body-coordinate system shown in Fig. 4. The diversity printed-IFA and diversity PIFA were used in four orientations in orienfor both the channels tested. For both channels, in orientation 3. In orientation 2, for tation 1 and for belt-wrist channel, whereas, in orienbelt-head and tation 4, for belt-head and for the belt-wrist. Further details of all the antennas can be found in [8], [9]. The orientation of Tx antennas is shown in Figs. 2 and 3.

For each on-body channel, the transmitting antenna was placed at the waist (belt) position on the left side of the body about 100 mm away from the body centre line and the receiving antennas were placed at the right side of the head and right wrist, thus forming two on-body channels named belt-head and belt-wrist, respectively. The distance between the body and the antennas mounted on the body was kept to about 7–10 mm including the clothing. Measurements were done in an indoor environment, which was an L-shaped laboratory room containing equipment, tables, chairs, and computers, thus providing a rich multipath propagation environment. The dimensions of the two arms of the room were 5 m 2 m and 7.5 m 1.5 m. For the 2.45 GHz measurements, the two ports of the receiving diversity antenna were connected to the two ports of a vector network analyzer (VNA) calibrated in tuned receiver mode. The transmitting antenna was connected to a signal generator which was generating the desired 2.45 GHz unmodulated carrier signal. The sampling time was set to 10 ms and a total of 16010 points were collected for each diversity branch. For 5.8 GHz and 10 GHz measurements, the two receiving antennas were connected through an RF switch to the receiving port, whereas, the transmitting antenna was connected to the transmitting port of the VNA with response through calibration. The switching time of the RF switch was 40 s, which was much lower than the coherence time of the channels [8]. The sampling time was set to 21.2 ms for 5.8 GHz and 9.2 ms for 10 GHz measurements. A total of 8000 points were collected for each branch at these two frequencies. The sampling time in each case was selected to ensure that all the variations caused by any fast movement of the body are captured. This was achieved by choosing the sampling frequency higher than twice the maximum Doppler shift, , where . Here is the free space wavelength and is the average velocity of motion, which was assumed to be 1.5 m/s. More details and the coherence time are given in Section III.D. Each of

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TABLE I MOVEMENTS DONE FOR EACH CHANNEL

Fig. 5. Autocorrelation function of the short-term fading envelope for belthead channel with PIFA at 2.45 GHz before resampling.

collected sample contained the amplitude and phase information. During the measurements, a sequence of pseudo-random activities were performed and repeated for all cases. Each activity set was composed of 10 postures. The activity sets for both channels are given in Table I. Further details of the measurement setup can be found in [8], [9]. III. CHANNEL CHARACTERIZATION A. Data Processing The movements were divided into two groups. The first group was a set of postures which cause a dynamic change in the path length and also involves movement of the body and hence the antennas in the environment. This group consists of sweeps 3, 4, 6, 8, 9, and 10 for belt-head and sweeps 1, and 5–10 for belt-wrist channel (see Table I). The second group was a set of static postures during which the channel is static in terms of the path length and involves no movement of the antennas in the environment. This group consists of the rest of the sweeps for each channel. It is usually assumed that long-term fading is a multiplicative factor to the received signal envelope [10], [11]

distribution fitting. Fig. 5 shows, as an example, the autocorrelation among the data sample points for one data set before resampling was done. The sample autocorrelation reduced to below 0.5 after resampling for all the data sets. The resampling for short-term fading data was done by sampling every 4th point for 2.45 GHz data and every 2nd point for 5.8 and 10 GHz data in order to make the total number of points approximately the same (2300 approx. for belt-head and 2600 approx. for belt-wrist channel) for all the cases, with a difference of few samples resulting from different window sizes. The long-term fading was extracted by local averaging and resampling every 10th sample for 5.8 and 10 GHz and every 20th sample for 2.45 GHz using sliding window of the same size as was used for short-term fading extraction in each case. B. Diversity Combining The three schemes for diversity combining i.e., selection combining (SC), equal gain combining (EGC), and maximal ratio combining (MRC) were applied. The diversity combined signals were achieved by using the expressions given in [12]. The expression for MRC is

(3) (1) (2) where is the received signal envelope, is is the local the short-term fading envelope, and root-mean-square (RMS) value of the envelope constituting the long-term fading. The averaging window size 2 is critical in separating the short-term and long-term fading. The window was selected such that there were sufficient short-term fading oscillations (approx. 4 to 6) inside the window and yet small enough compared to the time scale of the long-term variation. The window size used for various channels at various frequencies falls within the 5 to 20 interval, where is the free space wavelength. To extract the short-term and long-term fading, each sweep data was demeaned using (1) and (2) with sliding window. The demeaned data for the corresponding individual sweeps in each group of movements were concatenated to constitute the short-term fading data set for each channel. The data was then resampled to reduce the correlation among the adjacent samples and get independent data samples for

where and represent the received branch signal envelopes. The results are shown for MRC only as it is the optimal among the three. The long-term and short-term fading envelopes for the combined signal were separated by the same method described above. C. Distribution Fitting Each data set was fitted to seven distributions, namely, log-normal, Nakagami, Rician, Rayleigh, Weibull, gamma, and normal. The goodness of the fit was verified using Kolmogorov-Smirnov (KS) test. It was observed that the second group of movements, which involves static postures, does not fit any of the distribution and thus only the results for the first group of movements are presented. Out of the seven distributions tested for the first group of movements, the short-term fading envelope of the branch and combined signal (MRC) for both the on-body channels fitted to four prominent distributions, namely, Rician, Weibull, normal, and Nakagami, with significance levels above 5%. Among the four distributions, Rician was the best fit for most of the cases with highest p-values, usually above 50%. Normal was the

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TABLE II SHORT-TERM FADING PARAMETERS OF THE RICIAN BRANCH AND COMBINED SIGNALS FOR THE BELT-HEAD CHANNEL

TABLE III SHORT-TERM FADING PARAMETERS OF THE RICIAN BRANCH AND COMBINED SIGNALS FOR THE BELT-WRIST CHANNEL

Fig. 6. Graphs showing the No. of times short-term fading data sets of branch and combined signals fitted the four dominant distributions with p-values higher than 5% and the best fit (highest p-value) among the 58 cases for (a) belt-head and (b) belt-wrist channel.

second best but the significance level values were lower for the majority of cases. Fig. 6 shows the number of times each distribution had significance level above 5% and the number of times each one was best fit (highest p-value) for the 58 cases tested altogether for different orientations and repetitions of each of the three antennas at the three frequencies. It is evident that Rician distribution is the best fit for almost 85% of the cases for the short-term fading envelope. The Rician probability density function (PDF) is given by [13]

(4) is the random variable representing the short-term where is the modified Bessel function of fading envelope the first kind and 0th order, and and are the two parameters characterizing the strongest ray power, , and the average , respectively. The Rician K-factor, which scattering power, represents the ratio of the strongest ray power to the average . The total scattered power can be calculated as power can be calculated as . Due to the deor 0 dB for meaning method used, as described above, all the short-term fading envelopes. Thus, the Rician distribution is completely characterized by the K-factor. The average , and combined signal K-factor values for the branch signal,

, are shown in Tables II and III for the belt-head (MRC), and belt-wrist channels, respectively. The averaging was done with dB values of the K-factor over different orientations and repetitions of the measurements for each antenna type and frequency. The tables also show the maximum absolute deviation , with different orientations and repetitions, and (in dB), , between average values the maximum difference (in dB), ) of the of the two diversity branches. Only one spacing (i.e., monopole diversity antenna was used at 2.45 GHz with no repvalues do not etition and no other orientation hence the exist. The same seven distributions were fitted to the long-term fading envelopes of the branch and combined signals. The p-values for all the distributions were low compared to the p-values for the short-term fading and among those only log-normal and gamma were good fits and the rest of the distributions had p-values close to zero. The same significance level of 5% was set for the goodness of fit. Fig. 7 shows the number of times log-normal and gamma had p-values of 5% or more out of the 58 cases for each channel and the number of times each distribution was the best fit. It is clear from the figures that log-normal is the best fit for the long-term fading envelope for both the on-body channels. The PDF for the log-normal distribution is given by [13]

(5) and are the mean and the standard deviation of the natural logarithm of , where is a random variable representing the long-term fading envelope . The average parameters for

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TABLE V LONG-TERM FADING PARAMETERS OF THE LOG-NORMAL COMBINED SIGNALS FOR THE BELT-HEAD CHANNEL

TABLE VI LONG-TERM FADING PARAMETERS OF THE LOG-NORMAL BRANCH SIGNALS FOR BELT-WRIST CHANNEL

Fig. 7. Graphs showing the No. of times the long-term fading data sets of combined signal fitted the two dominant distributions with p-values higher than 5% and the best fit (highest p-value) among the 58 cases for (a) belt-head and (b) belt-wrist channel.

TABLE VII LONG-TERM FADING PARAMETERS OF THE LOG-NORMAL COMBINED SIGNALS FOR THE BELT-WRIST CHANNEL

TABLE IV LONG-TERM FADING PARAMETERS OF THE LOG-NORMAL BRANCH SIGNALS FOR BELT-HEAD CHANNEL

the log-normal long-term fading are shown in Tables IV and V for the belt-head and in Tables VI and VII for belt-wrist channels, along with the maximum absolute deviation from the average value with different orientations and repetitions and their difference between the two diversity branches. Parameters and are presented here in dB to make them consistent with the rest of the paper. It can be noted from Tables II and III that the K-factor is not as high as expected in a short range LOS communication. This may be due to the fact that the direct ray propagates along the surface of the body as creeping wave and is attenuated much more rapidly compared to free space propagation. Also, the movement of the body causes significant scattering along with the scattering from the environment. Thus, the scattering

ray power is comparable to the direct ray power, resulting in relatively low K-factor. This phenomenon is reflected more clearly in the case of printed-IFA, which has negative K-factor values i.e., direct ray power is less than the power in the scattering rays. Due to the structure of the printed-IFA, the direct ray is polarized tangentially to the surface of the body and is attenuated much more than the perpendicularly polarized ray in case of monopole antennas. Thus, monopole antennas have the strongest direct ray followed by PIFA, which also has a strong perpendicularly polarized component. Furthermore, the parameter in Tables IV–VII, which represents the mean received power, is the highest for the monopole antenna, followed by the PIFA with the standard deviation, , lower than that of the printed-IFA. This signifies better link budget for the monopole antennas and worse for printed-IFA among the three antenna types tested. Although the monopole outperforms other antenna types, the structure of the monopole antenna

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is not suitable for body-worn devices. For BAN applications, PIFA seems to be the best choice among the three antennas due to its moderate performance and low profile structure. The comparison of the parameters at the three frequencies signifies higher losses at the higher frequencies, as expected. It can be seen from Tables IV and VI that both, the pathloss and the standard deviation, are higher at high frequencies. No trend was observed for the K-factor with increasing frequency. The maximum deviation in the parameters with various repetitions and orientations is acceptable in most cases but is slightly higher for the printed-IFA. The reason may be the fact that the direct ray is very weak and the power is mostly in the multipath components, which may vary significantly with the movement of the body and the environment. As the movements cannot be replicated exactly and also the antenna position on the body can be slightly offset in various repeated measurements, this can affect the link performance. The maximum deviation in parameters with various orientations of the antennas, in general, is within the repeatability error [8], [9]. Thus, it can be concluded that the orientation of the antenna does not significantly change the statistics apart from the repeatability error. The difference between the two branch signals, which effectively shows the power imbalance, is noticeable for monopole antennas at 2.45 GHz. In this case, the separation between the antennas is larger compared to the separation for other two frequencies (for the ) and the body shadowing can produce same spacing say a difference in the received power at the two branch antennas. This difference signifies the fact that the two branch signals may not be identically distributed. The smaller spacing between the antennas at the higher frequencies results in approximately the same amount of shadowing for both branches and hence, lower power imbalance is observed. For the other two antennas, the difference between branches is small. It can be observed from the Tables II and III that the Rician K-factor increases significantly for the combined signals compared to the K-factors of the corresponding branch signals. This clearly suggests an improvement of the coherent component over the scattering power with diversity. This improvement is almost similar for all the antennas and also at the three frequencies, which shows that use of diversity offers similar improvement at the three frequencies and the three antenna types. This was reported in [8] and [9] as well. This improvement can also be seen by comparing the and values in Tables IV and V for branch and combined signals, respectively, for belt-head and, similarly, in Tables VII and VIII for belt-wrist channels. D. Spectral Analysis The transmitter and receiver antennas mounted on the body move relative to each other as well as in the environment due to the body movement, and hence, a Doppler frequency shift is introduced referred to here as the body Doppler shift. The Doppler shift, , can be calculated as [13]:

(6) where is the velocity of motion, is the angle between the direction of arrival of the signal and the direction of receiver

movement, and is the wavelength. For on-body channels, and are variable, resulting in many Doppler components. The coherence time, , for the channels can be estimated from the , as [13] maximum Doppler shift,

(7) The Doppler spectrum of the received branch signal envelope for each channel was calculated by taking the Fourier transform of the autocorrelation function of the signal envelope. The branch and the corresponding diversity combined signals (obtained from the undemeaned branch signals) have identical spectra. Hamming window was used to reduce the side lobes. The average envelope Doppler spectrum at a certain frequency was obtained as the average of the spectra for the three antennas and all their orientations and repetitions at that frequency. Only the walking movement was selected for the spectral analysis. Figs. 8 and 9 show the Doppler spectra of the two channels at the three frequencies. The side lobe level for Hamming window dB, so any frequency point with an ordinate value below is dB may represent a side lobe and hence should be ignored. , can be approximated from The maximum Doppler shift, these plots. The frequency where the slope of the curve changes [14]. The slope of the curves and drops down rapidly is in Figs. 8 and 9 changes slowly, as opposed to the ideal case, due to the presence of side lobes and the noise in the system. point can be identified for the 2.45 GHz case but The not for the other two frequencies due to the limitations of the measurement system on the sampling time and the noise level. The maximum Doppler shift at 2.45 GHz, as observed from the plots, was about 4 Hz for belt-head channel, and about 10 Hz for the belt-wrist channel. The belt-wrist channel involves faster movements of the Rx due to wrist motion and hence has higher Doppler shift. The scenario for the belt-head channel is similar to what is explained in [15] with fixed Tx and Rx antennas and moving scaterrers. The belt-wrist scenario is different, as it also involves the motion of Tx and Rx with respect to each other as well as in the environment. In addition, both these scenarios are different from the mobile moving in the environment with a constant speed of motion. The several peaks after the first minimum position in Figs. 8 and 9 can be attributed to various periodic activities during the movements. The first peak after the minimum represents a periodic movement with a time period of about 6s for both the channels. This coincides with the period of walking up and down the room. This peak cannot be observed for the 10 GHz frequency as the resolution is not high enough due to the limitation of the total sweep time. The speed of motion for the 2.45 and 5.8 GHz measurements was approximately the same and hence the peaks are approximately in the same position. For 10 GHz, the motion was slightly faster and hence the peaks are at slightly higher frequencies. There is a bigger peak at about 0.8 Hz for the belt-wrist channel, which is not present in the belt-head spectrum. This peak corresponds to the periodic movement of the Rx antenna on the wrist relative to the Tx with approximately 1.5 s period, for 2.45 and 5.8 GHz measurements. This peak is at about 1 Hz

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Fig. 8. Average Doppler spectrum for the belt-head channel.

Fig. 10. Average Doppler Spectrum for the belt-head channel for walking posture using complex signal.

Fig. 9. Average Doppler spectrum for the belt-wrist channel.

Fig. 11. Average Doppler Spectrum for the belt-wrist channel for walking posture using complex signal.

for 10 GHz, corresponding to faster oscillation of the arm. The presence of a dominant component introduces two peaks in the , where is the angle of arrival of spectrum at the dominant component [16]. There is a dominant component present for both the channels and hence peaks will be introduced in the spectrum but are undetected due to random speed of motion and random . The Doppler spectrum with complex signals was determined by taking the FFT of the autocorrelation function of the complex received signal. Figs. 10 and 11 show a portion of the average Doppler spectra calculated for the two channels using complex signals. The spectra are not the typical U-shaped curves but are similar in shape to the spectra presented in [15]. The spectrum is asymmetric about the center frequency, as shown in [16] for mobile channels. This asymmetry may be due to the variable speed of motion of the receiver and transmitter relative to each other and also in the environment. Also, the scatterers around the antennas move randomly due to the random movements. In the measurement environment (a laboratory room) most scatterers (equipment and furniture) were positioned in the horizontal plane. Besides, the scatterers on each side of the test subject were closer to him, as the walking path of the subject had

to be clear of obstructions. Combined with the fact that a large proportion of the directions of arrival in the horizontal plane are shielded by the body on which the antennas are mounted, it is clear that the scattering is non-isotropic. IV. CONCLUSION Statistical and spectral analyses of two on-body diversity channels are presented at three frequencies using three different types of antennas. The short-term and long-term fading envelopes of the diversity branch and combined signals were fitted to seven distributions and the best fit was evaluated for each case. The short-term fading branch and combined signals best fit the Rician distribution. Long-term fading best fits the log-normal distribution. The average best-fit parameters were presented for each case along with the maximum deviation from the average values with different orientations and repetitions for each antenna at each frequency. The maximum difference of the parameters between the two branch signals was also presented, which suggested that the two branch signals may not be identically distributed. The monopole antenna gives better performance for the on-body channels followed by the PIFA, as the mean path gain for the monopole antennas

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is the highest among the three types. For printed-IFA, the direct ray is polarized parallel to the surface of the body and is, therefore, attenuated much more than a perpendicularly polarized ray. The Rician K-factors for printed-IFA were very small, which suggests that for this antenna, the local off-body scattering is dominant over the direct ray. The K-factor values were compared for the branch and combined signals, which clearly indicated that there is an improvement due to diversity reception. The average Doppler spectra are presented for both channels and the maximum body Doppler shift is estimated.

REFERENCES [1] , [Online]. Available: http://IEEE802.org/15/pub/TG6.html [2] Y. I. Nechayev and P. S. Hall, “Multipath fading of on-body propagation channels,” presented at the IEEE Int. AP-S Symp.—USNC/URSI National Radio Science Meeting San Diego, CA, 2008. [3] , P. S. Hall and Y. Hao, Eds., Antennas and Propagation for Body-Centric Wireless Communications. London, U.K.: Artech House, 2006. [4] Y. I. Nechayev, P. S. Hall, C. C. Constantinou, Y. Hao, A. Alomainy, R. Dubrovka, and C. Parini, “Antennas and propagation for on-body communication systems,” presented at the 11th Int Symp. on Antenna Tech and Applied Electromagnetics—ANTEM, France, 2005. [5] M. R. Kamarudin, Y. I. Nechayev, and P. S. Hall, “Performance of antennas in the on-body environment,” in Proc. IEEE Antennas and Propagation Society Int. Symp., July 3–8, 2005, vol. 3A, pp. 475–478. [6] A. A. Serra, P. Nepa, G. Manara, and P. S. Hall, “Diversity measurements for on-body communication systems,” IEEE Antenna Wireless Propag. Lett., vol. 6, no. 1, pp. 361–363, 200. [7] I. Khan, L. Yu, Y. I. Nechayev, and P. S. Hall, “Space and pattern diversity for on-body communication channels in an indoor environment at 2.45 GHz,” presented at the 2nd Eur. Conf. on Antennas and Propagation (EuCAP), Edinburgh, U.K., Nov. 11–16, 2007. [8] I. Khan and P. S. Hall, “Multiple antenna reception at 5.8 and 10 GHz for body-centric wireless communication channels,” IEEE Trans. Antennas Propag., vol. 57, no. 1, Jan. 2009. [9] 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, Apr. 2009. [10] , R. Steele, Ed., Mobile Radio Communications. London, New York: Pentech Press, 1994. [11] N. L. Scott and R. G. Vaughn, “The effect of demeaning on signal envelope correlation analysis,” presented at the 4th Int. Symp. on Personal, Indoor and Mobile Radio Communications, Yokohama, Japan, Sep. 1993. [12] A. M. D. Turkmani, A. A. Arowojolu, P. A. Jefford, and C. J. Kellett, “An experimental evaluation of the performance of two-branch space and polarization diversity schemes at 1800 MHz,” IEEE Trans. Veh. Technol., vol. 44, no. 2, May 1995. [13] R. Prasad, Universal Wireless Personal Communications. Boston, MA: Artech House, 1998. [14] W. C. Y. Lee, Mobile Communications Engineering. New York: McGraw-Hill, 1982, ch. 6, p. 202. [15] A. Domazetovic, L. J. Greenstein, N. B. Mandayam, and I. Seskar, “Estimating the Doppler spectrum of a short-range fixed wireless channel,” IEEE Commun. Lett., vol. 7, no. 5, May 2003.

[16] J. D. Parsons and J. G. Gardiner, Mobile Communication Systems. New York: Halsted Press, p. 35 and 54. Imdad Khan received the M.S. degree in electrical engineering from NWFP University of Engineering and Technology, Peshawar, Pakistan, in 2003 and Ph.D. degree from University of Birmingham, Birmingham, U.K., in 2009. He is currently working with COMSATS Institute of Information Technology, Abbottabad, Pakistan. His major field of research is diversity and MIMO for on-body communication channels.

Yuriy I. Nechayev (S’00–M’03) received the Diploma of Specialist in Physics (with honors) from the Kharkiv State University, Ukraine, in 1996 and the Ph.D. degree in electronic and electrical engineering from the University of Birmingham, Birmingham, U.K., in 2004. Since 2003, he has been with the University of Birmingham as a Research Associate, and later, a Research Fellow, working on the problems of on-body propagation channel. He has coauthored a book chapter, an IEEE magazine article, and a number of technical papers on radio propagation in urban environments and around human body. His research interests include radiowave propagation modeling and measurements, propagation in random media, wave scattering, and electromagnetics.

Peter S. Hall (M’88–SM’93–F’01) received the Ph.D. degree in antenna measurements from Sheffield University, Sheffield, U.K. After graduating, he spent three years with Marconi Space and Defence Systems. He then joined The Royal Military College of Science as a Senior Research Scientist, progressing to Reader in Electromagnetics. He joined the University of Birmingham in 1994. Currently, he is a Professor of communications engineering, Leader of the Antennas and Applied Electromagnetics Laboratory, and Head of the Devices and Systems Research Centre in the Department of Electronic, Electrical and Computer Engineering, University of Birmingham, Birmingham, U.K. He has researched extensively in the areas of microwave antennas and associated components and antenna measurements. He has published five books, over 250 learned papers and taken various patents. Prof. Hall’s publications have earned six Institution of Electrical Engineers (IEE) premium awards, including the 1990 IEE Rayleigh Book Award for the Handbook of Microstrip Antennas. He is a Fellow of the IEE and the IEEE and a past IEEE Distinguished Lecturer. He is a member of the IEEE AP-S Fellow Evaluation Committee. He chaired the Organizing Committee of the 1997 IEE International Conference on Antennas and Propagation and has been associated with the organization of many other international conferences. He was Honorary Editor of IEE Proceedings Part H from 1991 to 1995 and is currently on the editorial board of Microwave and Optical Technology Letters. He is a member of the Executive Board of the EC Antenna Network of Excellence.

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Communications Switchable Frequency Selective Surface for Reconfigurable Electromagnetic Architecture of Buildings Ghaffer I. Kiani, Kenneth L. Ford, Lars G. Olsson, Karu P. Esselle, and Chinthana J. Panagamuwa

Abstract—A frequency selective surface (FSS) that is electronically switchable between reflective and transparent states is tested. It can be used to provide a spatial filter solution to reconfigure the electromagnetic architecture of buildings. The FSS measurements show that the frequency response of the filter does not change significantly when the wave polarization changes or the angle of incidence changes up to 45 from normal. The FSS is based on square loop aperture geometry, with each unit cell having four PIN diodes across the aperture at 90 degree intervals. Experiments demonstrated that almost 10 dB additional transmission loss can be introduced on average at the resonance frequency, for both polarizations, by switching PIN diodes to ON from OFF state. Index Terms—Active frequency selective surface (FSS), electromagnetic architecture, frequency selective surface (FSS), oblique incidence, PIN, security, stability, switchable. Fig. 1. The front close-up view of the switchable FSS prototype.

I. INTRODUCTION In large buildings and offices, frequency re-use methods are required to enhance the spectral efficiency and capacity of wireless communication systems. This observation has led to the concept of electromagnetic architecture of buildings [1], [2]. Passive bandstop frequency selective surfaces (FSSs) can be used to enhance the electromagnetic architecture of a building, and hence to improve spectral efficiency and system capacity, but switchable FSSs can provide a better reconfigurable solution. If switchable FSSs are placed in strategic locations of a building, they can be reconfigured remotely and rapidly, which is not possible with passive FSSs [1]. This paper describes an electronically-switchable FSS, with a highly stable frequency response, useful in such applications. Recently, a considerable amount of research is carried out in the field of switchable FSS to achieve a reconfigurable frequency response for different applications [2]–[12]. Among different methods to obtain a variable FSS frequency response, the PIN diodes are mostly used to switch an FSS between ON and OFF states. Most research has been car-

Manuscript received June 02, 2008; revised September 15, 2009. First published December 04, 2009; current version published February 03, 2010. G. I. Kiani was with the Department of Physics and Engineering, Macquarie University, Sydney NSW 2109, Australia. He is now with the CSIRO Information and Communication Technologies Centre (ICT), Epping NSW 1710, Australia (e-mail: [email protected]). K. L. Ford is with the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K. (e-mail: [email protected]). L. G. Olsson is with the Department of Electrical and Information Technology, Lund University, Lund SE-221 00, Sweden (e-mail: lars.olsson@eit. lth.se). K. P. Esselle is with the Department of Physics and Engineering, Macquarie University, Sydney NSW 2109, Australia (e-mail: [email protected]; karu. [email protected]). C. J. Panagamuwa is with the Department of Electronic Engineering, Lougborough University, Loughborough, Leicestershire LE11 3TU, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2037772

ried out on bandstop FSSs, such as arrays of metallic dipoles, which have a switching device (e.g., diode) placed between the arms of the dipoles. Conversely, their bandpass version consisting of an array of slots is not suitable for the inclusion of active elements as the bias applied to the devices will short across the metal surface. Recently, an attempt was made to design an active band pass FSS using two FSS layers [8]. In this design, one of the FSS layers is a standard bandstop active FSS which is placed in front of a passive bandpass FSS with a thin layer of PCB. This design is quite complex and require accurate design tools and high manufacturing accuracy. In order to alleviate these constraints, a single layer switchable FSS is presented in this paper. Its frequency response does not change significantly with polarization (TE and TM) and angle of incidence (up to 645 from normal) and can be used to electronically reconfigure electromagnetic architecture of buildings. Switchable FSS Prototype: The FSS tested here is based on the theoretical design given in [11]. These theoretical results have predicted a frequency response that does not change significantly with polarization (TE and TM) and angle of incidence up to 60 . An FSS with 25 2 15 elements was fabricated based on the theoretical design. The overall size of FSS prototype is 45 cm 2 30 cm. The thickness of the FR4 substrate was 1.6 mm. Figs. 1 and 2 show the front and rear close-up views of the switchable FSS prototype, respectively. There are four PIN diodes in each unit cell. Positive dc biasing is applied from the front side of FSS. The diagonal negative dc bias lines on the reverse side are joined together on the border of the FSS prototype. It can be seen in reference [12] that crossed shape bias lines produced stable frequency response at oblique angles only for TE polarization (while unstable for TM), therefore diagonal biased lines were preferred in this particular design. II. MEASUREMENT SETUP Transmission: Fig. 3 shows the measurement setup for the switchable FSS prototype. HyperLog 7060 log periodic antennas from Aar-

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Fig. 3. The transmission measurement setup showing the switchable FSS fixed in a metallic frame between two log-periodic antennas.

Fig. 2. The rear close-up view of the switchable FSS prototype.

iona were used for the measurements [13]. These antennas operate between 700 MHz to 6 GHz and were well suited for the experimentation. The antennas were connected to a vector network analyzer (Rohde & Schwarz ZVC, 20 KHz to 8 GHz). The measurement procedure for each set-up included a reference measurement without the FSS in the window. The instrument was then calibrated for transmission over the frequency range. This means that the subsequent measurement was relative to the transmission through the aperture. In this way any diffraction around the edges of the finite metal sheet were included. Furthermore, the influence of reflections from the floor was studied by putting in absorbing material, which did not make any detectable difference in the measurement. Personnel moving around in the room more than a meter and a half away from the set-up did not affect the result, which lead to the conclusion that other reflections did not cause any major errors. To measure the oblique incidence performance of FSS prototype, the antennas were kept stationary while the position of FSS aperture was changed for each angle of incidence. Straight lines were marked on the floor for each angle of incidence by considering the initial position of the FSS aperture (in metal frame) as a reference (normal incidence). The metal frame having FSS aperture was carefully rotated to align it exactly to the marked lines on the floor (for each angle of incidence, see Fig. 3). This experimental arrangement was found to have sufficient positioning accuracy to provide repeatable measured results. To measure both TE and TM polarizations, the antennas were rotated around their axes by 90 . Reflection: Reflectivity measurements were carried out using a standard NRL (Naval Research Laboratory) arch [14], which is housed within an anechoic chamber. The measurement system is illustrated in Fig. 4 and consists of two wide band horns, covering 2–18 GHz, connected to an HP8510C automatic vector network analyzer (VNA). Rohde and Schwarz HF906 double ridged waveguide horn antennas were used [15]. The switchable FSS sample under test was supported on a low density expanded polystyrene table, which was surrounded by 12 inch pyramidal absorbers. The system was calibrated using a response/isolation technique (vector error correction). The isolation measurement was carried out by first removing the polystyrene table and sample so that the horns were illuminating the chamber absorber directly. For the response measurement the table was replaced and a metal plate was positioned so that its center was on the axis of rotation of the

Fig. 4. The block diagram showing the reflection measurement setup.

arch arms which carry the wide band horns. This resulted in a calibration which gave a dynamic range of better than 055 dB across the entire frequency range and this was deemed sufficient to assess the performance of the FSS. To improve the measurements further, the time domain gating feature within the VNA was used to reduce any erroneous scattering. In order to carry out oblique incidence measurements the supporting arms of the arch were rotated to the appropriate angle using accurate predetermined fixings and the horns were rotated through 90 each depending on the polarization needed. The previously described calibration was repeated for each angle and polarization. PIN Diodes Biasing: A total of 1500 Philips (BAP51-03) [16] diodes were used in the fabrication of FSS prototype which is relatively costly even when cheaper diodes were used. However, the cost may be further reduced by using technique described in references [2], [7]. In a practical situation it is envisaged that an active FSS would be used as a small aperture within a conducting wall, rather than applying the FSS over the entire wall or room. This would reduce the costs of the FSS and this approach was investigated successfully in [7]. To switch these PIN diodes from OFF to ON state, a forward voltage of 2

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Fig. 5. Measured transmission and reflection characteristics of switchable FSS for perpendicular polarization (TE) when the diodes are in OFF state.

V was applied to the FSS prototype in both transmission and reflection measurements. The biasing of the diodes is accomplished such that all the diodes are electrically in parallel. To forward bias them the power supply is set to deliver a current of 2 A, which is divided among the diodes giving each a bias current of about 1.3 mA. A further increase in current did not affect the attenuation. No voltage was applied in the reverse bias case. III. MEASURED RESULTS A. Perpendicular (TE) Polarization PIN Diodes OFF: Fig. 5 shows the measured transmission and reflection characteristics at 0 , 30 and 45 incidence angles for perpendicular polarization (TE incidence) when the PIN diodes are in OFF state. The resonance occurred at 3.2 GHz for 0 , with a 017.5 dB reflection. The insertion loss at this frequency for 0 is 2.6 dB. At 30 and 45 incidence angles, the resonance frequency slightly shifts downward to 3.1 and 3.05 GHz, respectively, both with a reflection coefficient of 014 dB. The insertion losses for these incident angles are 2.6 and 2.6 dB, respectively. Comparing the theoretical results in [11], the measured resonance frequency is shifted upwards by about 0.6 GHz. The average transmission loss in theory is 0.7 dB as opposed to 2.6 dB in measured results. There are three main causes for the higher values of measured transmission loss: dielectric losses, diode losses and the reflection losses due to physical presence of diodes and the soldering material on FSS surface. The shift in resonance frequency from 2.45 GHz (theoretical) to 3.2 GHz (measured) may be due to: (a) lower value of dielectric constant of the substrate used for the fabrication of active FSS; (b) inaccuracies in the parameters values in the diode model; (c) extra inductance added by the dc interconnecting lines on the border of finite FSS (rear side); and (d) the soldering material used in FSS fabrication. The other reason is the extra inductance added by the continuous diagonal bias line as opposed to Fig. 2 in [11], in which the bias line has a small discontinuity. PIN Diodes ON: Fig. 6 depicts the measured transmission and reflection coefficients for perpendicular (TE incidence) polarization when the diodes are in ON state. For 0 , 30 and 45 incidence angles, the reflection coefficient is close to 1 dB while the transmission loss is 11.5, 13, and 14.6 dB, respectively. Therefore, at 0 , 30 and 45 incidence angles, the transmission loss can be switched by 8.9, 10.4, and 12.0

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Fig. 6. Measured transmission and reflection characteristics of switchable FSS for perpendicular polarization (TE) when the diodes are in ON state.

dB, respectively, by switching PIN diodes from OFF to ON state. As far as the amplitude variation is concerned, although the transmission response varies over the measurement frequency range, the actual used bandwidth is much smaller than this (IEEE802.11b). The frequency variation over this bandwidth is much smaller and so would not be an issue in the application it is intended for. B. Parallel (TM) Polarization PIN Diodes OFF: Fig. 7 shows the measured transmission and reflection characteristics at normal and oblique incidence angles for parallel polarization (TM incidence). The resonance and insertion loss at normal incidence are the same as in TE case, as expected from the symmetry of the unit cell. The change in the resonance frequency is also the same as in TE case while the reflection coefficients at 30 and 45 are 021 dB and 028 dB, respectively. The transmission losses for these angles of incidence are 2.4 and 2.3 dB (due to Brewster effect), respectively. PIN Diodes ON: Fig. 8 depicts the measured transmission and reflection coefficients for perpendicular (TM incidence) polarization. For 0 , 30 and 45 incidence angles, the reflection coefficient is close to 1 dB while the transmission loss is 14, 13.5, and 12.2 dB, respectively. Therefore, at 0 , 30 and 45 incidence angles, the transmission loss can be changed by 11.4, 11.1, and 9.9 dB, respectively, by switching PIN diodes from OFF to ON state. The use of any FSS design to control wireless coverage in a building environment depends on many factors, such as transmitting source power, receiver sensitivity, room layout, distance between sources and building construction materials. It is not proposed that entire rooms be coated with FSS. This is discussed in [8] in which a simple path loss analysis shows that >30 dB transmission loss may be required for an FSS wall, which this FSS design would not provide. The potential in FSS designs is in the use as a small aperture embedded in a shielded room, as discussed and demonstrated in [8]. In this case and with full recognition of the factors mentioned above, this FSS design offers a practical solution. IV. CONCLUSION In this paper, experimental results for a single-layer switchable FSS are presented. PIN diodes along with square apertures are used.

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REFERENCES

Fig. 7. Measured transmission and reflection characteristics of switchable FSS for parallel (TM) polarization when the diodes are in OFF state.

Fig. 8. Measured transmission and reflection characteristics of switchable FSS for parallel polarization (TM) when the diodes are in ON state.

The prototype FSS has shown a stable transmission response for both TE and TM polarizations. Experimentally, an average additional transmission loss of 10 dB is achieved for both polarizations at normal and oblique incidence, by switching PIN diodes between forward and reverse bias. Further isolation may be obtained by using a dielectric having low loss tangent, better quality PIN diodes and dual layer FSS architecture. Beside other applications, it may find use in electronically reconfiguring electromagnetic architecture of buildings. ACKNOWLEDGMENT The authors thank Prof. R. Langley, Department of Electronic and Electrical Engineering, University of Sheffield, U.K., for his invitation to carry out this collaborative work on active FSS. Also, many thanks to Prof. A. Karlsson, Department of Electrical and Information Technology, Lund University, Sweden, for providing lab facilities for transmission measurements.

[1] [Online]. Available: http://www.ee.kent.ac.uk/research/theme_ project. aspx?pid=84(Accessed in Oct. 2009) [2] E. A. Parker et al., “Minimal size FSS for long wavelength operation,” Electron. Lett., vol. 44, no. 6, pp. 394–395, 2008. [3] T. K. Chang, R. J. Langley, and E. A. Parker, “An active square loop frequency selective surface,” IEEE Microw. Guided Wave Lett., vol. 3, no. 10, pp. 387–388, Oct. 1993. [4] B. Philips, E. A. Parker, and R. J. Langley, “Active FSS in an experimental horn antenna switchable between two beamwidths,” IEE Electron. Lett., vol. 31, no. 1, pp. 1–2, Jan. 1995. [5] T. K. Chang, R. J. Langley, and E. A. Parker, “Active frequency selective surfaces,” Proc. Inst. Elect. Eng., vol. 143, pt. H, pp. 62–66, Feb. 1996. [6] B. M. Cahill and E. A. Parker, “Field switching in an enclosure with active FSS screen,” IEE Electron. Lett., vol. 37, no. 4, pp. 244–245, Feb. 2001. [7] A. Tennant and B. Chambers, “A single layer tunable microwave absorber using an active FSS,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 1, Jan. 2004. [8] K. Mitchell, A. Keen, L. Davenport, C. Smartt, P. Leask, R. Larson, and J. Davies, “Research to demonstrate the ability of close-coupled frequency selective structures to enhance the spectral efficiency of radio systems in buildings,” Ofcom Project Rep. AY4462B, 2004. [9] G. I. Kiani, K. P. Esselle, A. R. Weily, and K. L. Ford, “Active frequency selective surface using PIN diodes,” presented at the IEEE Antennas and Propag. Int. Symp., Jun. 9–15, 2007. [10] G. I. Kiani, K. P. Esselle, A. R. Weily, and K. L. Ford, “Active frequency selective surface design for WLAN,” presented at the 10th Australian Symp. on Antennas, Sydney, Australia, Feb. 14–15, 2007. [11] G. Kiani, K. L. Ford, K. P. Esselle, and A. R. Weily, “Oblique incidence performance of an active square loop frequency selective surface,” presented at the 2nd Eur. Conf. on Antennas and Propag., Edinburgh, U.K., Nov. 11–16, 2007. [12] G. I. Kiani, K. L. Ford, K. P. Esselle, A. R. Weily, C. Panagamuwa, and J. C. Batchelor, “Single-layer bandpass active frequency selective surfaces,” Microw. Opt. Technol. Lett., vol. 50, no. 8, pp. 2149–2151, Aug. 2008. [13] [Online]. Available: http://www.elektrosmog.de/Gutachten/HyperLOG 7000-E.pdf(Accessed in Oct. 2009) [14] F. C. Smith, B. Chambers, and J. C. Bennett, “Caliberation techniques for free space reflection coefficients measurements,” Proc. Inst. Elect. Eng., vol. 139, no. 5, pp. 247–253, Sep. 1992. [15] [Online]. Available: http://www2.rohde-schwarz.com/product/HF906. html(Accessed in Oct. 2009) [16] [Online]. Available: http://www.nxp.com/acrobat_download/datasheets/BAP51-03_4.pdf(Accessed in Oct. 2009)

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Performance Improvement for a Varactor-Loaded Reflectarray Element L. Boccia, G. Amendola, and G. Di Massa

Abstract—The application of varactor diodes as phase-shifting devices in reflectarrays has received considerable attention because it represents a low-cost and low-complexity solution for the creation of scanning antennas with continuous phase tuning. The main issues related to the design of varactor-loaded reflectarrays are the difficulty in the correct characterization of the unit cell, and the limited phase range achievable by changing the polarization voltage of the device. The contribution presented in this communication attempts to cover both these aspects. In particular, a simple model for the analysis of a varactor-loaded microstrip reflectarray cell is presented, and then employed to increase the patch’s phase response to 360 . The result is then validated experimentally, with the results showing good agreement between simulated and measured data. Index Terms—Microstrip, reflectarray, reflector antenna, varactor.

I. INTRODUCTION Microstrip reflectarray antennas are arrays of patches printed onto a grounded dielectric slab and illuminated by a primary source. Due to their planar form, reflectarrays are easy to manufacture and install. Furthermore, they offer notable flexibility in the radiation pattern, which can be controlled both in direction and shape by simply adjusting the geometry of the patches printed onto the reflecting surface. In the last few years reflectarrays have received increasing attention, and a number of different successful configurations have been proposed. For example, designs of reflectarrays with improved bandwidth [1]–[3] or with integration of amplifiers into the radiating elements [4] have been proposed. There have also been several examples of electronically reconfigurable reflectarrays. The dynamic phase control in the radiating elements was achieved either by employing low-loss phase shifters or MEMS [5]–[7] or by placing miniature motors under the patches [8]. Although these techniques were experimentally demonstrated with good results, they significantly increase the complexity and the cost of the scanning systems, especially when large reflectarrays with a high scanning resolution are considered. Recently, as an alternative solution, it has been demonstrated [9] that the phase of each reflectarray element can be reconfigured by loading a patch with a varactor diode onto the radiating edge. A voltage-controlled tuning varactor can be used to introduce a variable capacitive reactance which modifies the electrical length of the patch. By controlling the capacitance of the diode, the resonant frequency of the antenna can be varied within a specified range as is done in the case of frequency-agile antennas. This capability is beneficial for reflectarrays because the small shift in the resonant frequency introduced by the tuning diodes changes the phase of the field reflected by the single element, thus resulting in dynamic phase control. Unfortunately, early studies [9], [10] conducted on this configuration showed a phase range that was insufficient, limiting its practical applicability. However, in the last few years improvements have been made in varactor-based reflectarray elements, and two Manuscript received December 03, 2008; revised May 21, 2009. First published December 04, 2009; current version published February 03, 2010. The authors are with the Dipartimento di Elettronica, Informatica e Sistemistica Rende, University of Calabria, CS 87036, Italy (e-mail: luigi.boccia@ unical.it). 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.2009.2037697

Fig. 1. Varactor-loaded reflectarray unit cell.

alternative arrangements with a wider phase response have been proposed in the literature. In the first [11] two surface mounted diodes are series-connected to the two halves of a patch. Similarly, in the other [12] a printed dipole is used in combination with a varactor diode. Although both these configurations provide nearly full phase control, the risk of perturbation in the radiation pattern may arise at higher frequencies since varactors and bias circuitry are mounted directly on the radiating aperture. This potential limitation has been overcome in recent studies [13], [14] where a resonant patch is aperture-coupled to a transmission line loaded with two varactors. In this communication, a configuration based on a single shunt varactor, originally proposed in [9], will be reexamined in order to boost its performance as far as full phase control. A hybrid model for a varactor-loaded reflectarray unit cell will also be presented, along with an analysis technique based on the representation of the passive circuit by means of an S -matrix derived from a full-wave analysis and integrated with the diode equivalent circuit. This model will then be employed to show how the phase range of a reflectarray cell loaded with a shunt-connected varactor diode can be improved by adjusting the diode position and the patch dimensions, in the end permitting full phase control.

II. VARACTOR-LOADED REFLECTARRAY MODEL A reflectarray consisting of rectangular patches of width W , length and uniform inter-element spacing g is considered, as shown in Fig. 1. The array is printed onto a grounded dielectric slab of thickness h and with dielectric constant "r . Each element is loaded with a shunt-connected varactor diode placed at a distance d from the radiating edge. DC bias to the active device is driven by means of a metalized via-hole that extends through the backside of the ground plane. Isolation between the RF currents on the patch and the bias circuitry is achieved by locating the bias terminal at the center of the non-radiating edge, where the patch currents are virtually zero. The behavior of the proposed reflectarray configuration is studied by considering a single cell in an infinite array environment. This approach, indeed, is particularly well suited for varactor-loaded reflectarrays, which are usually large and made up of identical elements. In addition, the single cell is modeled so as to differentiate the simulation of the passive part and the active part of the circuit. The rectangular patch is analyzed with commercial full-wave software [15] that is able to simulate the actual antenna geometry, including bias circuitry, and

L,

0018-926X/$26.00 © 2009 IEEE

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Fig. 4. Phase of the field scattered by a reflectarray unit cell versus frequency ,W and d . for various junction capacitances, C . L

= 19mm

Fig. 2. Simulation setup used to extract the two-port S matrix of the reflectarray unit cell in an infinite-array scenario. Arbitrary incidence angles can be set by controlling the master-slave phase-shift walls.

Fig. 3. Equivalent circuit for varactor-loaded reflectarray cell.

supports Floquet harmonics, which are necessary in order to enforce periodic boundary conditions. The simulation setup, shown in Fig. 2, includes two ports: one port that supports Floquet modes and which models the incident plane wave, and a lumped port located at the same position as the diode. Periodic boundary conditions are enforced through the use of phase-shift walls implemented through master-slave couples [15] that can be configured to represent different angles of incidence. In order to better take into account the current-crowding effect caused by the presence of the varactor on the patch surface, the width of this lumped port has to be made equal to the diameter of the diode. The solution of the full-wave analysis can then be cast in the form of a two-port S matrix cascaded to the diode-equivalent circuit, as shown in Fig. 3. The phase of the field scattered by the varactor-loaded reflectarray cell will thus coincide with the phase of the reflection coefficient 0in , which can effectively be evaluated for various junction capacities by assuming Z0 to be equal to the free-space impedance. III. PHASE RESPONSE OF THE REFLECTARRAY UNIT CELL For this work, a Microsemi GC15006–89 varactor diode was used. It is a silicon epitaxial mesa device, designed to provide wideband linear tuning. Its junction capacity, Cvar , varies in the range between 2.2 and 0.2 pF when it is reverse-biased between 0 and 30 V. The equivalent

= 22mm

=0

circuit, as provided by the manufacturer, includes package effects that are modeled by a parallel parasitic capacitance of C = 0:15 pF and a series inductance of L = 0:2 nH. Diode losses are taken into account by inserting into the circuit a series resistor, R, of 0.07 , whose value is calculated by the device Q factor, taken at 4 V of reverse bias. The package geometry is very important for the proposed configuration, as the diode has to be mounted through a dielectric of the same height. In general, the presence of a varactor will distort the patch currents. However, if the diode size is not too large with respect to the patch length, the impact on the radiating behavior of the single cell is negligible as was demonstrated in an earlier study [10]. In the present case, the configuration of the varactor package is cylindrical, with a radius of 1.27 mm and a height of 0.762 mm. A substrate having the same height as the varactor and with a relative permittivity of "r = 2:33 has been employed. To investigate the relationships between the design parameters L, W , and d and the unit-cell phase response, several simulations were conducted at an operating frequency of 5 GHz. In a first assessment, the patch length, L, was fixed at 19 mm, the nominal resonant length of a rectangular patch antenna, while the width, W , was set at 22 mm, and the cell size, g , was 45 mm. For a passive patch, the phase of the field scattered by the antenna is zero at the resonant frequency while it varies by 360 as frequency is changed over a certain range. As is shown in Fig. 4, when a varactor is placed across a radiating edge, i.e., d = 0, it provokes a reduction of the patch resonant frequency which tends to decrease as the junction capacitance increases [16]. Accordingly, once the operating frequency is fixed, e.g., at 5 GHz, then only limited phase agility can be achieved when the varactor-loaded patch length is set as in the passive case. This behavior can be observed in Fig. 5, where the phase response of the varactor-loaded reflectarray cell as a function of the junction capacitance is presented at a fixed frequency of 5 GHz. However, this limitation can be avoided in two ways. Firstly, wider phase ranges can be achieved over the diode tuning span by reducing the patch length, L. Indeed, this raises the active reflectarray cell resonance so that the slope of the reflection phase reaches a maximum when the junction capacitance is approximately half of its maximum value. Secondly, a similar effect can be achieved by changing the diode position, d, from the radiating border to the center of the patch. This effect is shown in Fig. 6, where the phase response versus junction capacitance, Cvar , is reported for various varactor positions, d, at a fixed patch length, L. As can be seen, when the diode is moved to a more central position, the resonance of the varactor-tuned patch gradually rises, thus increasing the available phase range up to about 354 . As will be shown in Section IV, these two techniques can be applied in unison. Indeed, even though it is theoretically possible to raise the varactor-loaded patch resonant frequency by just reducing the patch’s length, the exclusive use of this technique might not be effective in applications at higher frequencies. In fact, in these cases the diode

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Fig. 5. Phase of the field scattered by a varactor loaded microstrip antenna versus junction capacitance, C , for various patch lengths, L, at a frequency of 5 GHz and with d .

=0

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Fig. 7. Experimental setup: varactor-loaded reflectarray unit cell in place at the end of a WR-187 rectangular waveguide.

Fig. 6. Phase of the field scattered by a varactor loaded reflectarray unit cell versus junction capacitance, C , for various diode positions, d, at a frequency of 5 GHz.

package might be excessively bulky with respect to the microstrip antenna size, thus significantly perturbing the patch fields and making the integration difficult.

IV. MODEL VALIDATION The varactor-loaded microstrip antenna model proposed for the analysis was experimentally validated through the creation of an active reflectarray cell at a frequency of 5 GHz. The parametric simulations presented in the previous section were used as a starting point. Both the diode position and the patch length were tuned to maximize the available phase range. The optimal patch behavior was obtained at L = 17 mm and d = 5 mm. A varactor-loaded reflectarray unit cell was then fabricated on top of an Arlon Diclad substrate having the characteristics reported above. Experimental measurements were then carried out following the approach proposed in [16] The prototype was placed at the end of a section of standard WR187 rectangular waveguide excited by a T E10 mode. The test environment, shown in Fig. 7, is therefore very similar to an infinite array scenario with a vertical inter-element spacing equal to the waveguide height, which is 22.25 mm. However, it is worth noticing that the T E10 travelling in the waveguide simulates illumination at an off-broadside direction [17]. In particular, for the WR187 waveguide at 5 GHz the wave-propagation angle is  = 39:12 in the H-plane. This condition was replicated in the simulation environment by opportunely setting the phase delay relationships of the master-slave boundaries. As a preliminary result, the simulated phase response of the active reflectarray cell is shown in Fig. 8 at different angles of incidence in the E- and H- plane. As can

Fig. 8. Simulated phase response of the reflectarray cell for various angles of incidence. a) E-plane variations; b) H-plane variations.

be observed, the effects of off-axis incidence are negligible unless very large reflectarrays are considered. The response of the varactor-loaded reflectarray unit cell was studied taking into account both the phase and the amplitude of the scattered field at different frequencies. The simulated and measured phase response are shown in Fig. 9(a). Both simulated and measured phase data were de-embedded on the ground plane of the patch, while the reverse voltage was converted to the corresponding junction capacitance using the conversion formula provided by the diode manufacturer. As can be observed, the experimental results are in excellent agreement with the simulations and the available tuning range is almost 360 at 5 GHz. The simulated and measured amplitude response of the reflectarray unit-cell are shown in Fig. 9(b). As can be seen, the amplitude of the

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field scattered by the varactor-loaded patch is not uniform when the junction capacitance, Cvar , is varied, experiencing a maximum drop of about 2.5 dB. There are two main causes of this non-uniformity. The first one is due to the amplitude response of the patch [18], which depends on the resonant frequency of the antenna, which in turn changes when the junction capacitance, Cvar , is varied. This phenomenon is also observed in passive reflectarrays, where the amplitude response of each element is related to its geometry, which may vary across the array [19] The second cause is the varactor losses, which have been taken into account by including in the model the resistor R, whose value has been calculated from the diode quality factor measured by the manufacturer at 4 V of reverse bias. However, it is worth pointing out that diode losses estimated through this procedure do not perfectly match the physical phenomena of the active device. Firstly, the diode Q provided by the manufacturer is usually measured using a lower frequency test signal. Secondly, R is not constant with reverse bias but depends on the thickness of the depletion zone [20] In principle, R increases as one approaches 0 V bias, and decreases at the higher safe reverse-bias voltage. These effects may be part of the causes of the differences between the simulated and measured data presented in Fig. 9(b). However, it should be noted that the non-uniform amplitude response of the field scattered by the varactor-loaded unit cell does not limit the applicability of the proposed configuration. Indeed, the same problem is present also in passive reflectarrays but it has been demonstrated [21] that the gain for even small reflectarrays is dictated by the array factor, and the small fluctuation in the element amplitude has a limited effect on the main lobe shape. On the other hand, amplitude losses have a negative effect on side lobe levels and on side lobes. Another important aspect of reflectarray design is the evaluation of the operational bandwidth. In general, for very large apertures reflectarray bandwidth is limited by the non-constant feed-element path delays over the reflector surface, while for reflectarrays of smaller size the main constraint comes from the element frequency response [22] A first estimation of the varactor-loaded scattering behavior at different frequencies is given in Fig. 9, where both measured and simulated scattered fields are reported at 4.8, 5.0, and 5.2 GHz. As can be observed, the phase error generated by frequency variation is not constant but depends on the varactor capacitance. Hence, it is possible to define the active reflectarray bandwidth as the frequency range for which the phase error does not exceed a certain limit for all possible varactor capacitances. This evaluation can be done graphically as shown in Fig. 10, where the scattered field phases versus frequency are presented for different junction capacitances taken within the varactor tuning span. The different phase responses are normalized at the center band frequency, 5 GHz. Normally the maximum acceptable phase error would depend on the reflectarray configuration and on the desired side lobes constraints. When a maximum phase error of 630 is considered, the active patch element bandwidth does not exceed 0.6% which is consistent with a microstrip patch of the same size realized on the same substrate. Larger bandwidths can be achieved by increasing the substrate thickness, as was done in [16] and [13], where bandwidths of 1.3% and 2.2% were achieved at a similar frequency on substrates of thickness 1.5 mm and 3 mm, respectively. V. CONCLUSION In this communication a study of varactor-loaded reflectarray elements has been presented. In particular, the effect of the diode position and patch size on the reflection phase response has been analyzed numerically and experimentally demonstrating that a patch antenna loaded with a shunt-connected varactor diode can be designed so as to attain full phase control, thus enabling the design of varactor-loaded reflectarrays with full-beam scanning.

Fig. 9. Simulated field and measured field scattered by a varactor-loaded re,W and d . (a) phase flectarray unit cell of L response; (b) amplitude response. Markers indicate measured data.

= 17 mm

= 22 mm

= 5 mm

Fig. 10. Simulated scattered field phases versus frequency for different junction capacitances taken in the varactor tuning span, i.e., from 0.2 pF to 2.2 pF.

The proposed study has been conducted by employing a hybrid model that combines full-wave analysis and circuit analysis. One of the main advantages of the proposed approach is that the actual geometry of the unit cell can be analyzed with full-wave software, and the results of this simulation—cast in the form of S-parameters—can be integrated with the active device-equivalent circuit. Simulations for various diode reverse biases can therefore be obtained by merely performing circuit analysis.

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REFERENCES

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Collocated Microstrip Antennas for MIMO Systems With a Low Mutual Coupling Using Mode Confinement

[1] J. Encinar and J. Zornoza, “Broadband design of three-layer printed reflectarrays,” IEEE Trans. Antennas Propag., vol. 51, pp. 1662–1664, 2003. [2] F. Venneri, S. Costanzo, and G. D. Massa, “Wideband aperture-coupled reflectarrays with reduced inter-element spacing,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2008, pp. 1–4. [3] S. Costanzo, F. Venneri, and G. D. Massa, “Parametric analysis of bandwidth features for aperture-coupled reflectarrays,” in Proc. 2nd Eur. Conf. on Antennas Propag., 2007, pp. 1–4. [4] M. Bialkowski, A. Robinson, and H. Song, “Design, development, and testing of X-band amplifying reflectarrays,” IEEE Trans. Antennas Propag., vol. 50, pp. 1065–1076, 2002. [5] R. Gilbert, Dipole Tunable Reconfigurable Reflector Array U.S. Patent Patent US 2001/0050650. [6] J. Perruisseau-Carrier and A. Skrivervik, “Monolithic MEMS-based reflectarray cell digitally reconfigurable over a 360 phase range,” IEEE Antennas and Wireless Propag. Lett., vol. 7, pp. 138–141, 2008. [7] H. Legay, B. Pinte, M. Charrier, A. Ziaei, E. Girard, and R. Gillard, “A steerable reflectarray antenna with MEMS controls,” in Proc. IEEE Int. Symp. on Phased Array Systems and Technol., 2003, pp. 494–499. [8] J. Huang, “Capabilities of printed reflectarray antennas,” in Proc. IEEE Int. Symp. on Phased Array Systems and Technol., 1996, pp. 131–134. [9] L. Boccia, F. Venneri, G. Amendola, and G. D. Massa, “Experimental investigation of a varactor loaded reflectarray antenna,” in Proc. IEEE Int. Microw. Symp. Digest, 2002, pp. 69–71. [10] F. Venneri, L. Boccia, G. Angiulli, G. Amendola, and G. D. Massa, “Analysis and design of passive and active microstrip reflectarrays,” Int. J. RF Microw. Comput.-Aided Engineering, vol. 13, pp. 370–377, 2003. [11] S. Hum, M. Okoniewski, and R. Davies, “Realizing an electronically tunable reflectarray using varactor diode-tuned elements,” IEEE Microw. Wireless Compon. Lett. , vol. 15, pp. 422–424, 2005. [12] O. Vendik and M. Parnes, “A phase shifter with one tunable component for a reflectarray antenna,” IEEE Antennas Propag. Mag., vol. 50, pp. 53–65, 2008. [13] M. Riel and J. Laurin, “Design of an electronically beam scanning reflectarray using aperture-coupled elements,” IEEE Trans. Antennas Propag., vol. 55, pp. 1260–1266, 2007. [14] M. Riel and J. Laurin, “Design of a C-band reflectarray element with full phase tuning range using varactor diodes,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2005, vol. 3A, pp. 622–625. [15] Ansoft HFSS Ansoft Corporation. [16] S. Hum, M. Okoniewski, and R. Davies, “Modeling and design of electronically tunable reflectarrays,” IEEE Trans. Antennas Propag., vol. 55, pp. 2200–2210, 2007. [17] P. Hannan and M. Balfour, “Simulation of a phased-array antenna in waveguide,” IEEE Trans. Antennas Propag., vol. 13, pp. 342–353, 1965. [18] H. Rajagopalan and Y. Rahmat-Samii, “Loss quantification for microstrip reflectarray: Issue of high fields and currents,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2008, pp. 1–4. [19] F. Venneri, G. Angiulli, and G. D. Massa, “Design of microstrip reflect array using data from isolated patch analysis,” Microw. Opt. Technol. Lett., vol. 34, pp. 411–414, 2002. [20] I. Gutierrez, J. Meléndez, and E. Hernández, Design and Characterization of Integrated Varactors for RF Applications. Hoboken, NJ: Wiley, 2007. [21] K. M. Shum, Q. Xue, C. H. Chan, and K. M. Luk, “Investigation of microstrip reflectarray using a photonic bandgap structure,” Microw. Opt. Technol. Lett., vol. 28, pp. 114–116, 2001. [22] D. Pozar, “Bandwidth of reflectarrays,” Electron. Lett., vol. 39, pp. 1490–1491, 2003.

J. Sarrazin, Y. Mahé, S. Avrillon, and S. Toutain

Abstract—Collocated antennas for mutiple input multiple output (MIMO) systems are presented. The structure is composed of a dual-polarized microstrip square patch combined to a dual-polarized microstrip square ring. Consequently, four different radiation patterns are available simultaneously. With a length of about one guided wavelength, the proposed structure is well suited to reduce the MIMO terminal size. The structure operates in the 5.25 GHz band and exhibits a mutual coupling between the four input ports less than 23 5 dB in simulation and 20 dB in measurement. MIMO channel capacity measurements have been performed and demonstrate that the proposed antenna system can replace a classical system based on space diversity. Index Terms—Channel capacity measurements, collocated antennas, mutiple input multiple output (MIMO) system, radiation pattern diversity.

I. INTRODUCTION

Mutiple input multiple output (MIMO) systems can drastically improve wireless communication capacity and robustness by exploiting multipath effects. Performances of these multiple antenna systems largely depend on the correlation between received signals. To maximize the capacity, the correlation must be as low as possible. Spatial diversity is commonly used to decrease this correlation. However, this requires a space between antennas of about less than 0:5 up to several  (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 are also investigated in order to co-localize antennas still keeping the decorrelation capabilities [1]. Thus, the antenna system presented in this communication is composed of two collocated dual-polarized antennas: a microstrip square patch and a microstrip square ring. This system is able to produce four uncorrelated signals by using radiation pattern diversity [2]. Furthermore, the structure is planar and contains a ground plane which is well-suited to be integrated in a wireless terminal. Metal filled via holes have been introduced in the design in order to confine patch and ring resonance modes under their respective structures. Consequently, the mutual coupling between the four input ports has been decreased. The communication has been organized as follows. The study of the proposed antenna system is presented in Section II. The fabrication and measurement results are detailed in Section III. Finally, MIMO channel measurements have been conducted and capacity results are given in Section IV in order to compare performances of the proposed system with classical antennas based on space diversity. Concluding remarks are given in Section V. Manuscript received November 12, 2008; revised May 19, 2009. First published December 04, 2009; current version published February 03, 2010. J. Sarrazin, Y. Mahé, and S. Toutain are with the IREENA laboratory, University of Nantes, 44000 Nantes, France (e-mail: [email protected]). S. Avrillon is with the IETR laboratory, University of Rennes I, 35042 Rennes Cedex, France (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037690

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TABLE I PARAMETER VALUES OF THE ANTENNA STRUCTURE

Fig. 1. Microstrip antenna configuration.

II. ANTENNA DESIGN A. Antenna Overview The antenna system, shown in Fig. 1, is composed of a dual-polarized microstrip square patch and a dual-polarized microstrip square ring. L-slots have been etched on the patch to reduce its dimensions [3]. Consequently, the patch can be located in the ring’s center in order to obtain a compact structure. Because of slot locations, linear polarizations of the patch antenna are at 645 . Labels F1 to F4 indicate the coaxial feeding points. F1 and F2 feed respectively the TM10 and TM01 modes of the patch and F3 and F4 feed respectively the TM21 and TM12 modes of the ring. Thanks to the dual-polarization behavior of the structures and since the ring radiates differently from the patch, four different radiation patterns are available simultaneously. So pattern diversity is achieved between the four feeding ports. The whole structure acts as four independent antennas. Use these collocated antennas instead of a system based on classical space diversity enables to reduce MIMO terminal size. However, a drawback of such a structure is the strong coupling between patch and ring modes, especially between F2 and F3 (S32 ) and between F1 and F4 (S41 ). On a transmitter, the energy coupled between input ports is not radiated and so the radiation efficiency decreases. That is why metallic vias have been inserted between the ring and the patch. By enclosing resonant modes inside their respective structures, it is possible to decrease the coupling between them [4]. Four other metallic vias take place on the square-ring to set shortcircuit locations (according to TM12 and TM21 modes) in order to help finding the 50 feeding points. B. Simulation The structure has been simulated with Ansoft HFSS. The dimensions are given in Table I (according to the Fig. 1) for a substrate with a relative permittivity "r = 2:55 and a thickness h = 1:58 mm. The distance between each via is lvia = 2 mm which corresponds approximately to g =18 (with g , the guided wavelength). Their influence is studied by comparing the performances of the structure with and without vias. By enclosing patch and ring modes respectively, vias decrease the coupling between them. Their effects on the ring return loss

Fig. 2. Simulated S-parameters (a) without decoupling via and (b) with decoupling vias.

can be neglected whereas they shift slightly the patch resonance frequency. This effect is compensated by a small variations of the slot length in order to keep the patch resonance frequency identical in both cases (lm = 5:22 mm for the structure with vias and lm = 5:38 mm for the structure without via). Simulated results are represented in Fig. 2. S-parameters are given for the structure without decoupling via [Fig. 2(a)] and with decoupling vias [Fig. 2(b)]. In both cases, return loss results show that the two patch modes and the two ring modes resonate around 5.25 GHz with a 010 dB bandwidth of respectively 1f = 1:6% and 1f = 1%. Without via, strongest couplings occur between input ports F2 and F3 (S32 ) and F1 and F4 (S41 ) with levels about 018 dB. In Fig. 2(b), metallic vias have been inserted in order to enclose resonant modes under their respective structure. Therefore, the coupling decreases

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Fig. 5. Measured radiation patterns of patch modes.

Fig. 3. Picture of the prototype with the four input ports.

Fig. 6. Measured radiation patterns of ring modes.

Fig. 4. Measured S-parameters of the built antenna with decoupling vias.

down to 023:5 dB. So a 5.5 dB improvement has been obtained thanks to vias. III. FABRICATION AND MEASUREMENTS To prove the concept, the collocated antenna system has been built on a Teflon substrate ("r = 2:55) with the same dimensions as in simulation (see Table I). The prototype includes the decoupling vias and a picture is shown in Fig. 3. Measured S-parameters are shown in Fig. 4. Ring modes resonate at 5.22 GHz (S33 and S44 ) and patch modes at 5.2 GHz (S11 and S22 ). The coupling is a little bit higher than the simulation results but still lower than 020 dB. Measured radiation patterns for the four modes are given in Figs. 5 and 6. Co-polarization and cross-polarization planes are given according to the 645 polarization of the square patch. Patch modes are nearly orthogonal between them as well as ring modes. Furthermore, patch and ring radiation patterns have different shapes which means that pattern diversity is achieved. In order to quantify the pattern diversity capability, channel capacity measurements have been conducted as it is described in Section IV. IV. MIMO CHANNEL CAPACITY RESULTS To determine the channel capacity with the proposed collocated antennas, we performed an experimental 4 2 4 MIMO channel measurement as it is described in [5]. By using a vector network analyzer,

Fig. 7. Room of measurements.

channel responses have been measured between the multiple transmission antennas and the multiple reception antennas. Since the analyzer has only two input ports, two switches 4-ways (one for TX and one for RX) have been used to feed successively each antenna. Non fed antennas are terminated by matched loads to not disturb the behavior of the system. Measurements have been performed in our laboratory which is a rich scattering environment. To ensure the channel is stationary, data were obtained while nobody was in the room. The measurement bandwidth represents a typical WiFi channel: 22 MHz around 5.22 GHz (with 101 frequency sample points).

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Furthermore, simulated capacities of independent and identically distributed (i.i.d.) SISO, MIMO 3 2 3 and MIMO 4 2 4 channels are also given. For the LOS environment [Fig. 8(a)], we note that the capacity of the four monopoles spaced by a wavelength slightly lead to a higher capacity than those spaced by a half-wavelength. Approximately the same difference can be observed between the half-wavelength spaced monopoles and the collocated system. However, capacities provided by the three tested antenna systems are quite equivalent and are between the i.i.d. MIMO 3 2 3 and 4 2 4 channel results. In the NLOS environment [Fig. 8(b)], results are quite similar. Capacities of the three tested antenna systems are also very close together. For both cases, the collocated antenna system provides performances which are close to those available with monopoles. Furthermore, the proposed system is smaller than four spaced monopoles and is more suitable for integration on a terminal. Consequently, we can conclude that proposed collocated antennas are well-suited in order to reduce MIMO communication system sizes without decreasing their performances. V. CONCLUSION A collocated antenna system has been proposed in order to reduce sizes of MIMO devices. By combining microstrip square-ring and square-patch antennas on their both polarizations, radiation diversity is achieved. This allows obtaining a compact structure which is able to transmit or receive four uncorrelated signals. MIMO channel measurements have demonstrated the potential of such a system to be used instead of classical space diversity antennas.

REFERENCES

Fig. 8. Measured capacities at 5.22 GHz for (a) LOS environment and (b) NLOS environment.

For all the results, the MIMO channel is averaged over 20 snapshots in order to reduce the measurement noise. The channel is then normalized with the Frobenius norm and the capacity is determined with the following expression:

C = log2 det IN + Nr H (H )H t with r the signal-to-noise ratio at the reception, Nt the number of transmission antennas, H the MIMO channel matrix ((1)H denotes the Hermitian transposition) and INr the identity matrix. Measurements have been conducted for 10 locations of the receiver and 2 of the transmitter for a LOS and a NLOS environment, as shown in Fig. 7. The distance between TX and RX is about 3.5 meters for the LOS case and 4.5 meters for the NLOS case. Capacity results are determined from an average of these 20 relative TX/RX locations and from the 101 frequency sample points (so a total of 2020 different MIMO channel matrices). Results for LOS and NLOS conditions are given in Fig. 8(a) and 8(b), respectively. For both environments, the transmitter is constituted of four monopoles spaced by a wavelength. The receiver is successively constituted of four monopoles spaced by a half-wavelength, four monopoles spaced by a wavelength and the proposed collocated antenna system. Capacity results of these three receivers are compared.

[1] 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. [2] J. Sarrazin, Y. Mahé, S. Avrillon, and S. Toutain, “Four co-located antennas for MIMO systems with a low mutual coupling using mode confinement,” presented at the IEEE Antennas and Propagation Symp., Jul. 2008. [3] G. S. Row, S. H. Yeh, and K. L. Wong, “Compact dual-polarized microstrip antennas,” Microw. Opt. Technol. Lett., vol. 27, pp. 284–287, Nov. 2000. [4] J.-M. Kim and J. Yook, “A parallel-plate-mode suppressed meander slot antenna with plated-through-holes,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 118–120, 2005. [5] H. Nishimoto, Y. Ogawa, T. Nishimura, and T. Ohgane, “Measurement-based performance evaluation of MIMO spatial multiplexing in a multipath-rich indoor environment,” IEEE Trans. Antennas Propag., vol. 55, no. 12, Dec. 2007.

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A Multiband Quasi-Yagi Type Antenna Sung-Jung Wu, Cheng-Hung Kang, Keng-Hsien Chen, and Jenn-Hwan Tarng

Abstract—A new design of multiband quasi-Yagi type antenna for 700 MHz band, GSM900, DCS1800, GPS, and Bluetooth/WLAN applications is presented. In contrast to conventional quasi-Yagi antenna design, our proposed antenna realizes the multiband performance by interaction between the section extended ground and the derived driver element which is a branch of driver dipole element. The parametric studies of the proposed antenna are discussed to explore the antenna operating mechanism. The performances of the antenna are demonstrated along with measured and simulated results. Index Terms—700 MHz band, antenna, multiband antenna, quasi-Yagi antenna.

I. INTRODUCTION With the growing demand for wireless communication, wireless applications such as 2G/3G cellular phone, global positioning system (GPS), wireless local area network (WLAN), and Bluetooth have been developed at a fast pace during the past decade. On July 31, 2007, the Federal Communications Commission (FCC) revised the 700 MHz band plan and service rules to establish a nationwide, interoperable public safety broadband network for the benefit of state and local public safety consumer [1]. The 700 MHz band, ranging from 698 to 806 MHz, is expected to be used for commercial multimedia broadcasting, digital television, and local public safety network in regulated channels. Several studies have reported applications and technologies of the multiband antenna design, including the internal quad-band handset antenna of compact size [2], single and double layer multiband planar inverted-F antenna (PIFA) [3], multi-resonant monopole antenna [4], and multiband multiple ring monopole antenna [5]. Different from the multiband antenna, broadband antennas have been designed for high speed transmission rates and high capacity broadband services. The broadband sleeve monopole antenna [6] and metal-plate monopole antenna [7] were designed for DTVs and portable multimedia players, respectively. The quasi-Yagi antenna has been widely used in wireless communication because of its broadband characteristics and good radiation performance [8], [9]. This antenna has various transitions, is inexpensive, has a low profile, is easy to implement, and can readily be integrated with other circuit components. For example, the quasi-Yagi antennas were designed with microstrip-to-CPS transitions in order to connect to other components [10], [11] and a 100 GHz quasi-Yagi antenna was achieved using a silicon process [12]. In this paper, we present a new design for the multiband quasi-Yagi antenna for the 700 MHz (698–806 MHz), GSM900 (880–960 MHz), DCS1800 (1720–1880 MHz), GPS (1575 MHz), and Bluetooth/WLAN (2400–2484 MHz) applications. The proposed antenna shows the three section operating bands (measured return loss > 7) Manuscript received May 01, 2009; revised October 07, 2009. Current version published February 03, 2010. This work was supported by the National Science Council, R.O.C., under Grants NSC 98-2221-E-009-051 and NSC 98-2219-E009-001. The authors are with Department of Communication Engineering, National Chiao Tung University, Hsin-Chu, Taiwan, R.O.C.(e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2041522

Fig. 1. Configuration of the proposed antenna. (a) Top view. (b) Cross-sectional view.

which cover the above-mentioned applications and provide acceptable absolute gains (0–4.4 dBi). The initial design of the proposed antenna begins with a conventional quasi-Yagi antenna. Furthermore, the interaction between the section extended ground plane and the branch of driver dipole element is employed for multiband performance. The organization of this paper is followed. In Section II, we present the geometry and design concept of the proposed antenna as well as important parameters for the design process. The measured and simulated radiation patterns are given in Section III. Finally, we present the conclusions in Section IV. II. ANTENNA CONFIGURATION AND DESIGN THEORY Fig. 1 shows the proposed multiband quasi-Yagi type antenna. The initial design begins with a conventional quasi-Yagi antenna. The antenna consists of a microstrip-to-CPS transition, a driver dipole element, a parasitic director element, a derived driver element, and a truncated ground plane as the reflector element of the proposed antenna. The derived driver element is a branch of the driver dipole element. The driver dipole element is wound to reduce the antenna size. The parasitic director element on the top plane simultaneously directs the wave propagation toward the end-fire direction, and acts as an impedance matching element. The antenna was fabricated on a 1 mm FR4 substrate of size of 130 mm 2 95 mm. The final antenna parameters are optimized by using the commercial electromagnetic solver HFSS 9.2:

;

;

;

;

W1 = W2 = 21 W3 = 4 W4 = W8 = 2 W5 = 20 W6 = 17 W7 = 15 W9 = 110 W10 = 12 L1 = 55 L2 = 20 L3 = 15 L4 = 9 L5 = 3 L6 = 7 L7 = 28 H = 1, where all units are in

; ;

;

;

;

;

;

;

;

;

;

mm. The measured return loss agrees with the simulated one as shown in Fig. 2. There are three section operating bandwidths, which are on the 10 dB return loss condition. It is clear that the derived driver element not only provides impedance matching around the GPS and DCS1800 bands but also improves the performance for nearby the 700 MHz and GSM900 bands. Here, the microstrip-to-CPS transition forms a folding

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Fig. 2. Measured and simulated return loss of proposed antenna.

Fig. 4. Simulated return loss for various derived driver element positions.

Fig. 3. Simulated return loss of various lengths of the section extended ground.

Fig. 5. Simulated return loss for various distances between the driver dipole element and the parasitic director element.

microstrip line and exhibits broadband impedance transforming properties. The transition is designed to excite the odd-mode at the CPS line by providing 180 phase delay and is capable of transforming an unbalanced input signal to a balanced signal at the driver dipole element. In order to maximize the antenna impedance bandwidth, the optimum length of L1 is around one-eighth the length of the lowest operating frequency. To further improve the front-to-back ratio and the impedance matching of the proposed antenna, a part of the ground plane is extended toward the radiator. Fig. 3 shows the simulated return loss of various lengths of the section extended ground. With this arrangement, the bandwidth of each band becomes large when L3 changes from 0 to 15 mm. However, bandwidth of 700 MHz and GSM900 bands decrease as L3 = 18 mm. Fig. 4 shows the simulated return loss for various derived driver element positions. The derived driver element connects to different locations on the driver dipole element, i.e., Points A, B, C, and D. For the GPS and DCS1800 bands, the bandwidth changes at various locations. The derived driver element not only influences the lowest operating frequency slightly but also strongly relates to the bandwidth of the GPS and DCS bands. Fig. 5 shows the simulated return loss for various distances between the driver dipole element and the parasitic director element. At the

700 MHz and GSM900 bands, the distance becomes shorter, which strengthens the coupling and helps the wave propagate along the end-fire direction, ensuring a good end-fire pattern. However, the coupling also affects the wide impedance bandwidth. Although the bandwidths of the proposed antenna do not accurately cover the operating frequency of the wireless standards on 10 dB returnloss condition, the parameter studies are still useful for providing the design guideline. According to the Fig. 2 to 5, the lowest operating frequency is strongly related to the length of the driver dipole element. Furthermore, the bandwidths of each band are related to the capacitive interaction between the derived driver element and extended ground plane. Thus, by adjusting the position of the derived driver element and the extended ground plane, the bandwidths of the proposed antenna may satisfy bandwidth of wireless standards. III. RADIATED PATTERN

The antenna radiation patterns are measured in a 7.0 m 2 3.6 m 2 3.0 m anechoic chamber with an Agilent E362B network analyzer and the NSI2000 far-field measurement software. The xy- and the yz-plane radiation for each band are illustrated in Fig. 6. The agreement between the simulations and measurements is fairly good in most of the results. The discrepancies between simulated and measured cross-polarization

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TABLE I LIST OF THE MEASURED BAND CHARACTERISTICS

Referring to Fig. 6(a) and (b), the co-polarized patterns in xy-plane demonstrate end-fire radiation patterns. The measured cross-polarized gains are around 20 dB lower than the co-polarized one in the xy-plane. It is not a surprising result because the operating mechanism of the proposed antenna has the characteristics of the quasi-Yagi radiated. Referring from Fig. 6(c) to (d), the cross-polarized level becomes larger than the one in Fig. 6(a) and (b). The radiated patterns do not present the end-fired forms because the operating mechanism of the proposed antenna is realized by capacitive interaction between the extended ground plane and the derived driver element. Referring to Fig. 6(e), the maximum gain in the xy-plane is at = 180 . The measured gain and bandwidths of the proposed antenna are shown in Table I. Three measured section bandwidths on the 10 dB return loss condition are 700–937 MHz, 1508–1860 MHz, and 2382–2435 MHz, respectively. If the 7 dB return loss condition is adjusted, the operating bandwidths of the proposed antenna cover the bandwidth of 700 MHz band, GSM900, DCS1800, GPS, and Bluetooth/WLAN standards. The measured maximum gains are 0.3–4.4 dBi within operating bands. This is in good agreement with the simulated results.

IV. CONCLUSION In this paper, a new multiband quasi-Yagi type antenna has been proposed for 700 MHz band, GSM900, DCS1800, GPS, and Bluetooth/ WLAN applications. The antenna configuration and design methodology have been discussed. The set of parametric studies of the proposed antenna provides brief guidelines for the antenna designer. Simultaneously, the measured return loss and radiation patterns agree with the simulated ones. The proposed antenna is expected to find applications in wireless communication.

REFERENCES

Fig. 6. Simulated and Measured radiation patterns (a) at 750 MHz. (b) at 900 MHz. (c) at 1575 MHz. (d) at 1800 MHz. (e) at 2400 MHz. (Unit: dBi)

level in xy-plane can be mostly attributed to the measured uncertainty such as the connecting coaxial cable and the absorber used in measured arrangement.

[1] [Online]. Available: http://www.fcc.gov/pshs/public-safety-spectrum/ 700-MHz/ [2] Y.-X. Guo, I. Ang, and M. Y. W. Chia, “Compact internal multiband antennas for mobile handsets,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 143–146, 2003. [3] B. Sanz-Izquierdo, J. C. Batchelor, R. J. Langley, and M. I. Sobhy, “Single and double layer planar multiband PIFAs,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1416–1422, May 2006. [4] S. Hong, W. Kim, H. Park, S. Kahng, and J. Choi, “Design of an internal multiresonant monopole antenna for GSM900/DCS1800/USPCS/S-DMB operation,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1437–1443, May 2008. [5] C. T. P. Song, P. S. Hall, and H. Ghafouri-Shiraz, “Multiband multiple ring monopole antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 4, pp. 722–729, Apr. 2003. [6] H.-D. Chen, “Compact broadband microstrip-line-fed sleeve monopole antenna for DTV application and ground plane effect,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 497–500, 2008.

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[7] Y.-S. Yu, D.-H. Seo, S.-G. Jeon, and J.-H. Choi, “Design of an internal DTV antenna for portable multimedia player,” in Proc. Asia-Pacific Microwave Conf., Dec. 12–15, 2006, pp. 1601–1603. [8] N. Kaneda, W. R. Deal, Q. Yongxi, R. Waterhouse, and T. Itoh, “A broadband planar quasi-Yagi antenna,” IEEE Trans. Antennas Propag., vol. 50, no. 8, pp. 1158–1160, Aug. 2002. [9] S.-Y. Chen and P. Hsu, “Broadband microstrip-fed modified quasiYagi antenna,” in IEEE/ACES Int. Conf. on Wireless Commun. and Appl. Comput. Electromagn., Apr. 3–7, 2005, pp. 208–211. [10] D.-S. Woo, Y.-G. Kim, K. W. Kim, and Y.-K. Cho, “A simplified design of quasi-Yagi antennas using the new microstrip-to-CPS transitions,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Jun. 9–15, 2007, pp. 781–784. [11] R.-C. Hua, C.-W. Wang, and T.-G. Ma, “A planar quasi-Yagi antenna with a new microstrip-to-CPS balun by artificial transmission lines,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Jun. 9–15, 2007, pp. 2305–2308. [12] M. Sun and Y. P. Zhang, “100-GHz quasi-Yagi antenna in silicon technology,” IEEE Electron Device Lett., vol. 28, no. 5, pp. 455–457, May 2007.

A Novel Wideband and Compact Microstrip Grid Array Antenna Xing Chen, Guosheng Wang, and Kama Huang

Abstract—A wideband and compact microstrip grid array antenna is presented. The antenna is printed on a dielectric substrate, backed by a metal board, and directly fed from a 50 coaxial cable. It adopts elliptical radiation elements to enhance its impedance and gain bandwidths, and sinusoid lines to reduce its size. Structural parameters of a proposed antenna with 7 radiation elements were optimized by a parallel genetic algorithm (GA) on a cluster system. A prototype antenna was fabricated and tested. Results of simulation and measurement agree well and show the antenna bandwidth of 25% exhibits encouraging properties, e.g., and a 3 dB gain-drop bandwidth of 16.3%, both of which are much wider than that of conventional grid array antennas, as well as a maximum gain of approximately 13.7 dBi with an area-reduction factor of 47%.



10 dB

Index Terms—Compact, microstrip grid array antenna, wideband.

I. INTRODUCTION The grid array antenna [1]–[11] is a low-profile flat and linear or circular polarization antenna. This antenna possesses advantages such as high gain, narrow beam, and low side lobes, but has not received much attention and found popular applications since it was first presented by Kraus[1]. Literatures investigating it are limited, e.g., Conti et al. [2] reported a microstrip version of the array, and H. Nakano [3]–[11] analyzed its radiation characteristics and introduced some extensions. A grid array antenna is composed of many grid cells and backed by a conducting ground plane. Grid cells are formed by two kinds of lines. One acts as radiation elements, and another acts as transmission lines Manuscript received July 17, 2008; revised March 22, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported by a grant from the National High Technology Research and Development Program of China (863 Program, No. 2007AA01Z279). The authors are with the College of Electronic and Information Engineering, Sichuan University, Chengdu 610064, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2037769

that provide a phase delay of 360 at resonance to ensure the in-phase excitation of all the radiation elements. A grid array antenna may be categorized as a wire grid array antenna [1], [4]–[6], [8], [9], [11] or a microstrip grid array antenna [2], [3], [7], [10], [12] by that its grid cells are formed by metal wires or microstrip lines. In comparison with the former, the latter is easier to be fabricated because it allows the use of simple, low cost, and accurate microstrip fabrication techniques [2]. To make a grid array antenna compact, [6] uses meander lines rather than straight lines to construct the grid cells, and so that reduces the size of the antenna at a degree of 38%, but the gain of the antenna is also dropped by approximately 2 dB. There are still some problems existing in the grid array antenna. Firstly, the input impedance of the grid array antenna, as revealed in [3], [7], sometimes is much higher than 50 , which means an impedance transformer is necessary if the antenna is fed from a commonly used 50 coaxial line. Secondly, the frequency bandwidth of the grid array antenna is narrow. For example, the grid array antenna in [6] has VSWR < 2 bandwidth of only 2.6% and a 3 dB gain-drop bandwidth of less than 7%; the microstrip grid array antenna designed in [7] has VSWR < 2 bandwidth of approximately 13%. To the knowledge of the authors, it is the largest impedance bandwidth reported in previous literatures; in [4], [8] and [9], several grid array antennas are presented, but their 3 dB gain-drop bandwidth are only approximately 3.3%, 9%, and 5.6%, respectively. In our previous work [12], the genetic algorithm (GA) was employed for optimizing a microstrip grid array antenna, but the antenna’s bandwidth is only 4.5%. Today, antennas with properties of small-size, low-cost, high gain and wide frequency band are desired in the wireless communication and other applications. This communication presents a new microstrip grid array antenna, which is designed to be directly fed from a 50

coaxial line without an impedance transformer. To enhance its frequency bandwidth and make it compact, it adopts elliptical radiation elements and sinusoid transmission lines to form its grid cells. Traditionally, engineers work with their intuition, exact or approximate analysis, and simulation to determine antenna’s structures and parameters. The method of designing antennas is mainly by hand, so is time- and labor-intensive, and unlikely produce truly optimal results. In recent years, when the high speed computer became available, more and more engineers employ optimization algorithms in the antenna design as a powerful computer aided design technology, which is able to speed the antenna design and allow for greater design complexity. Hence, this work use the GA in conjunction with the finite different time domain (FDTD) method to optimize structural parameters of the proposed antenna on a cluster system for high radiation gain, wide frequency bandwidth and large area-reduction. Section II introduces the configuration of the new antenna. The antenna optimization based on the parallel GA is presented in Section III. Simulated and measured properties of the designed antenna are given in Section IV. Conclusions are stated in Section V. II. ANTENNA CONFIGURATION Fig. 1 illustrates a top view and a side view of the proposed microstrip grid array antenna, whose grid array is printed on a dielectric substrate of relative permittivity "r and thickness h1 , and backed by a metal board. Grid cells of the antenna are composed of Y -directed elliptical radiation elements and X -directed sinusoid transmission lines. In comparison with conventional rectangular microstrip lines or straight metal wires used as radiation elements in previous literatures [6], [7], [9], [10], elliptical elements are able to enhance the frequency bandwidth of the antenna due to their smooth and broad configuration.

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Fig. 2. The fabricated prototype antenna.

Fig. 1. The configuration of the microstrip grid array antenna.

For the elliptical elements, the lengths of their major and minor axis are La and Lb respectively. Acting as transmission lines, sinusoid lines with width Wt , side length Lt , and amplitude Am are employed for connecting adjacent elliptical radiation elements. The number of periodicities in a section of sinusoid lines is Np . The use of sinusoid lines leads to antenna-size reduction. In this work, an area-reduction factor is defined as

 = LwL0 Lt w

(1)

where  is the area-reduction factor, Lt and Lw are the side length and wire length of a section of sinusoid lines. An air space of thickness h2 is between the substrate and the metal board. The grid array antenna is fed directly from a 50 coaxial line, whose inner conductor penetrates both the metal board and the dielectric substrate, and then connects with the grid array at point A. In this work, the microstrip grid array antenna consists of 7 radiation elements and works at a center frequency of 2.45 GHz. A printed circuit board (PCB) with relative permittivity "r = 2:65 and thickness h1 = 1 mm is chosen as its dielectric substrate, and an aluminum board is selected as its metal ground. III. ANTENNA OPTIMIZATION The genetic algorithm (GA) [13] is a powerful and efficient mathematical technique for the solution of problems involving the maximization (or the minimization) of a given function. They are based on stochastic methods on the model of the natural evolution process: the improvement of a population of parameters along successive generations. A complete set of initial parameters constitutes an individual. The population is a collection of individuals. Each generation is built from the parent generation by applying some selecting operators, e.g., mutation, selection and crossover. After a large number of generations the best individuals will approximate the optimum parameters.

The GA has been successfully applied in the optimization of various antennas, such as microstrip dipole antennas [14], wire antennas [15], [16], patch antennas [17], and antenna arrays [18], [19]. In this work, the GA is employed for optimizing structural parameters of the proposed microstrip grid array antenna. The radiation properties of the antenna are obtained by the full-wave EM simulation using the finite-difference time-domain (FDTD) method. A procedure of the antenna optimization based on the GA usually invokes hundreds or even thousands EM simulations, hence is computationally intensive. To greatly reduce the design time, the computation of the GA-based antenna optimization is parallelized in a master-slave model and implemented on a Beowulf cluster system [14], [20], [21]. The Beowulf cluster system is composed of 32 processors interconnected by a fast 1000 Mb/s Ethernet and uses the message passing interface (MPI) library. One processor, named the master processor, carries out the GA optimization while other processors, called slave processors, execute full-wave EM simulations using FDTD, which is a standard FDTD in this work. To simplify the antenna optimization, Np is fixed to be 5. According to results of FDTD simulations, larger conducting ground plane can slightly enhance the antenna’s gain. As a trade-off between the size and the gain of the antenna, the dimensions of the conducting ground plane is set to be that of the grid area plus a margin of 10 mm in both X and Y direction. There are total 6 parameters needed to be optimized, i.e., h2 , La , Lb , Am , Lt and Wt . According to results in previous works [7], [12], parameters h2 , La , Lb and Wt are usually selected to be about 0:05   0:08 , 0.5 , 0:2   0:4  and 0:03   0:04  respectively, and the wire length of a section of a sinusoid transmission lines should about be 0:9   1:0 , where  is the free-space wavelength at the working frequency. After given a considerable margin for the GA-based optimization, the 6 parameters h2 , La , Lb , Wt , Am and Lt are confined within the range of 4:9 mm  11 mm(0:04   0:09 ), 4:9 mm  73 mm (0:4   0:6 ), 24 mm  55 mm (0:2   0:45 ), 2:4 mm  6:1 mm (0:02   0:05 ), 4:9 mm  18 mm (0:04   0:15 ) and 37 mm  98 mm (0:3   0:8 ), respectively. The fitness function plays a key role in the GA optimization because it guides the direction of the GA optimization. In this work, the fitness function should take the impedance bandwidth, gain bandwidth,

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Fig. 3. The measured and simulated reflection coefficient of the prototype antenna.

Fig. 4. Gains of the designed antenna against frequencies.

maximum gain, good impedance match and area-reduction factor into account simultaneously, hence it is defined as

F

= C1 3 ImpBW + C2 3 GainBW +C3 3 MaxGain + C4 3 MaxS 11 + C5 3 

(2)

where F is the fitness value, ImpBW and GainBW are the impedance bandwidth for S11 < 010 dB and gain bandwidth for 3 dB gaindrop criterion respectively, MaxGain refers to the maximum radiation gain obtained by the proposed antenna at the center frequency of 2.45 GHz, MaxS11 denotes the maximum S11 over a preset frequency band ranging from 2.35 to 2.55 GHz, and  is the area-reduction factor defined in (1). Values of the gain and S11 are in dB. C1 , C2 , C3 , C4 and C5 are weight factors, which are determined by experience and are set to be 0.5, 0.5, 0.02, 00:01 and 0.2 respectively in this work. A GA-based optimization is executed. In the optimization, the GA employs tournament selection with elitism, single-point crossover with probability Pc = 0:5, and jump mutation with probability Pm = 0:2, and it uses 100 generations, 120 chromosomes, and 80 individuals in a population.

Fig. 5. Measured and simulated radiation patterns on the XZ and YZ Plane.

IV. RESULTS AND DISCUSSION Structural parameters of the proposed microstrip grid array antenna generated by the GA-based optimization are as follow (unit: mm): h2 = 6:42, La = 65:6, Lb = 45:5, Am = 8:4, Lt = 55 and Wt = 3:6. This GA-based antenna optimization procedure cost about 110 hours on our cluster system, and will cost at much more time (at least 1000 hours in this case) without the parallel computation. From the definition (1), the area-reduction factor achieved by the proposed

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antenna is 47%. A prototype antenna as shown in Fig. 2 has been fabricated. The reflection coefficient of the prototype antenna was measured by an Agilent E8362B Network Analyzer. Fig. 3 compares the measured and simulated reflection coefficient. One can observe they are in good agreement that validates the reliability of the GA-based antenna optimization. The antenna has achieved a very wide frequency bandwidth for S11 < 010, approximately 25% (from 2.34 to 2.98 GHz), which is much wider than that of grid array antennas introduced in previous literatures [1], [6], [7], [12]. The measured maximum in the preset frequency band ranging from 2.35 to 2.55 GHz is 010:5 dB, which estimates one of the optimization objectives, good impedance match, is also accomplished. Gains of the designed antenna against frequencies are given in Fig. 4, which shows the maximum gain of the antenna is 13.7 dBi, and the 3 dB gain-drop bandwidth is 16.3% (from 2.25 to 2.65 GHz), which is also much wider than that of grid array antennas designed in previous literatures [4], [6], [8], [9]. Measured and simulated radiation patterns at different frequency points over the frequency band ranging from 2.25 to 2.65 GHz are presented in Fig. 5. From the figure, we can observe the measured and simulated radiation patterns agree very well, and the proposed antenna maintains directional patterns over the frequency band though the patterns become worse with the frequency deviating from the center frequency. Over the whole frequency band and in both XZ and YZ plane, the measured side-lobes are more than 10 dB below the main lobe, and the antenna radiates in the linear polarization with the cross-polarization level less than 010 dB. V. CONCLUSION A new microstrip grid array antenna is presented. It adopts elliptical radiation elements to enlarge the frequency bandwidth and sinusoid lines to reduce its dimension. After its structural parameters were optimized by the GA in parallel on a cluster system, a prototype antenna was fabricated and measured. Results of the measurement are compared with that of the simulations and shown they are in good agreement. The optimized antenna can directly be fed from 50 coaxial cable, obtains a maximum gain of 13.7 dBi with the area-reduction factor of 47%, provides the S11 < 010 dB impedance bandwidth and 3 dB gain-drop bandwidth up to 25% and 16.3% respectively, which are much wider than those of conventional grid array antennas. The properties of wideband, compact, high gain and simple feed will make the antenna to be useful in a variety of applications.

REFERENCES [1] J. D. Kraus and R. J. Marhefka, Antennas, 3rd ed. New York: McGraw-Hill, 2003, pp. 578–581. [2] R. Conti, J. Toth, T. Dowling, and J. Weiss, “The wire grid microstrip antenna,” IEEE Trans. Antennas Propag., vol. AP-29, pp. 157–166, 1981. [3] H. Nakano, I. Oshima, H. Mimaki, K. Hirose, and J. Yamauchi, “Center fed grid array antennas,” in Proc. IEEE AP-S Int. Symp., 1995, pp. 2010–2013. [4] H. Nakano, T. Kawano, and J. Yamauchi, “A cross-mesh array antenna,” in Proc. 11th lnt. Conf. on Antennas and Propag., Apr. 2001, pp. 77–20. [5] H. Nakano, H. Osada, H. Mimaki, Y. Iitsuka, and J. Yamauchi, “A modified grid array antenna radiating a circularly polarized wave,” in Proc. IEEE Int. Symp. on Microwave, Antenna, Propag. and EMC Technol. for Wireless Commun., Aug. 2007, pp. 527–530. [6] H. Nakano, T. Kawano, and J. Yamauchi, “Meander-line grid-array antenna,” Proc. Inst. Elect. Eng. Microw Antennas Propag, vol. 145, no. 4, Aug. 1998.

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[7] H. Nakano, H. Osada, and J. Yamauchi, “Strip-type grid array antenna with a two-layer rear-space structure,” in Proc. 7th ISAPE, Guilin, China, Oct. 2006, pp. 58–61. [8] T. Kawano and H. Nakano, “A grid array antenna with C-figured elements,” Electron. Commun. Jpn, vol. 85, no. 1, pt. 1, pp. 58–68, 2002. [9] H. Nakano, T. Kawano, H. Mimaki, and J. Yamauchi, “Analysis of a printed grid array antenna by a fast MoM calculation technique,” in Proc. 11th Int. Conf. on Antennas and Propag., Apr. 2001, pp. 17–20. [10] H. Nakano, I. Oshima, H. Mimaki, K. Hirose, and J. Yamauchi, “Numerical analysis of a grid array antenna,” in Proc. ICCS’94, Singapore, 1994, pp. 700–704. [11] H. Nakano, T. Kawano, Y. Kozono, and J. Yamauchi, “A fast MoM calculation technique using sinusoidal basis and testing functions for a wire on a dielectric substrate and its application to meander loop and grid array antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3300–3307, Oct. 2005. [12] C. Xing, C. Kain, and H. Kama, “A microstrip grid array antenna optimized by a parallel genetic algorithm,” Microw. Opt. Technol. Lett., vol. 50, no. 11, pp. 2976–2978, Nov. 2008. [13] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, 3rd ed. Berlin, Heidelberg, Germany: Springer-Verlag, 1996. [14] K. Chen, X. Chen, and K. Huang, “A novel microstrip dipole antenna with wideband and end-fire properties,” J. Electromagn. Waves Appl., vol. 21, no. 12, pp. 1679–1688, 2007. [15] D. S. Linden, “Automated Design and Optimization of Wire Antennas Using Genetic Algorithms,” Ph.D. dissertation, Massachusetts Institute of Technology (MIT), Cambridge, 1997. [16] E. A. Jones and W. T. Joines, “Design of Yagi-Uda antennas using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 45, no. 9, pp. 1386–1392, Sep. 1997. [17] F. J. Villegas, T. Cwik, Y. Rahmat-Samii, and M. Manteghi, “A parallel electromagnetic genetic-algorithm optimization (EGO) application for patch antenna design,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2424–2435, Sep. 2004. [18] M. Donelli, S. Caorsi, F. D. Natale, M. Pastorino, and A. Massa, “Linear antenna synthesis with a hybrid genetic algorithm,” Progr. Electromagn. Res., vol. PIER 49, pp. 1–22, 2004. [19] I. S. Misra and R. S. Chakrabarty, “Design, analysis and optimization of V-dipole and its three-element Yagi-Uda array,” Progr. Electromagn. Res., vol. PIER 66, pp. 137–156, 2006. [20] X. Chen, K. Huang, and X. Xu, “Automated design of a three dimensional fishbone antenna using parallel genetic algorithm and NEC,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 425–428, 2005. [21] X. Chen and K. Huang, “Microwave imaging buried inhomogeneous objects using genetic parallel algorithm combined with FDTD method,” Progr. Electromagn. Res., vol. PIER 53, pp. 283–298, 2005.

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Modular Broadband Phased-Arrays Based on a Nonuniform Distribution of Elements Along the Peano-Gosper Space-Filling Curve T. G. Spence, D. H. Werner, and J. N. Carvajal

Abstract—The Peano-Gosper space-filling curve provides an excellent framework for designing broadband planar antenna arrays with highly modular architectures. Uniformly distributing elements along the curve leads to an element distribution with a triangular lattice that has an irregular fractal boundary contour. This boundary contour allows for a modular subarray configuration and better sidelobe suppression than conventional triangular lattice arrays with a regular boundary contour. While they have a greater bandwidth than square-lattice distributions, arrays based on a triangular lattice still possess a rather limited bandwidth for beam steering applications due to the formation of grating lobes. In this communication it will be shown that the beam steering capabilities of the Peano-Gosper array can be enhanced by introducing perturbations into the basic recursive array generation scheme. With the proper implementation, the perturbed arrays retain the attractive features of modularity and recursive beamforming that are associated with the standard Peano-Gosper array. Examples will be presented for several stages of Peano-Gosper arrays that were designed for 2:1 broadband performance while scanning within a 30 conical volume. Full-wave simulations will be used to examine the effects of mutual coupling on these aperiodic array layouts. Index Terms—Broadband arrays, modular arrays, fractal arrays, PeanoGosper arrays, genetic algorithm.

I. INTRODUCTION Arrays based on a triangular lattice offer important advantages over square lattice arrays, namely a slightly larger bandwidth and fewer elements within a given aperture [1]. These attributes make them an attractive choice for planar array designs. Recently, it was shown that the Peano-Gosper (PG) space-filling curve [2] could be effectively exploited in the design of triangular lattice arrays with additional desirable characteristics [3], [4]. Uniformly distributing elements along the length of the PG curve leads to a triangular lattice element distribution that has an irregular fractal boundary contour [see Fig. 1(a)]. This boundary contour provides enhanced sidelobe suppression over conventional triangular lattice arrays with regular boundary contours. In addition, it allows for a convenient modular architecture whereby a number of subarrays could be tiled together. This modularity offers the potential to support simultaneous multibeam and multifrequency operation as well as simplified subarray construction. While they have a greater bandwidth than traditional square lattice distributions, arrays based on a triangular lattice exhibit a rather limited bandwidth for beam steering applications [3]. This issue can be demonstrated by observing, for example, the radiation pattern of a 344-element stage-3 PG array shown in Fig. 1(b) with a minimum element spacing of  and its main beam directed 30 from broadside. Clearly, in this example there are two grating lobes in addition to the main beam within the visible region of the array. Similar characteristics are seen for other azimuthal scan angles. In this communication we will introduce a design technique that mitigates this issue and is capable of extending the bandwidth and beam steering capabilities of PG array distributions. Additionally, these Manuscript received October 30, 2008; revised March 20, 2009. First published December 04, 2009; current version published February 03, 2010. The authors are with the Department of Electrical Engineering and Applied Research Laboratory, The Pennsylvania State University, University Park, PA 16802 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2037763

distributions retain the desirable attributes of highly modular architectures and the ability to utilize a recursive pattern formulation for fast array factor calculations. The primary limiting factor on the bandwidth and beam steering capabilities of PG arrays is the triangular-lattice that makes up their interior. The design technique that will be discussed here is based on altering this interior lattice. This has the effect of disrupting the unwanted constructive interference that accounts for the formation of grating lobes, thus extending the useable bandwidth of the arrays. The design technique accomplishes this by introducing perturbations to the locations of the antenna elements that are uniformly distributed along the curve. For large array layouts this would require the adjustment of many element positions, ranging from dozens up to potentially thousands. Rather than adjusting the location of every element in the array, the design technique simply perturbs the locations of the interior elements along a stage-1 generating curve and then uses these locations to recursively generate higher stages of arrays (see Fig. 2). The effect of the iterative construction is that the perturbation of an element location along the stage-1 curve considerably alters the layout of higher stages. Consequently, adjustment of several elements along the curve can lead to a wide range of planar array geometries and associated radiation characteristics. Traditional perturbation techniques require adjusting the position of every element in an array based on an initial periodic lattice [5]–[7]. For large linear arrays and planar arrays this process becomes very challenging due to various issues such as minimum spacings between elements and increased design complexity due to a large number of design parameters. Consequently, alternate perturbation schemes have been developed to address these design challenges [8], [9]. These schemes have made use of iteratively-constructed geometries for efficient element perturbation in the design of broadband arrays. The novelty of the recursive-perturbation technique presented here lies in its ability to generate large broadband planar arrays based on the displacement of only a small number of elements along a simple 1-D generator curve. Dealing with a small set of element positions along a curve significantly lessens issues involving design complexity and element spacings. It also provides for a tractable design problem that is easily handled by an optimizer-based approach. II. RECURSIVE ARRAY FACTOR FORMULATION The recursive array factor formulation that was developed in [3] is generalized here to make it applicable to both standard and perturbed PG arrays. With the perturbed stage-1 generator array as the fundamental building block, stages of the PG array are simply comprised of a collection of three different orientations of generator arrays (see Fig. 2). The recursive formulation uses pattern multiplication of these orientations of the generator arrays and the underlying hierarchical architecture of the PG curve. It provides a significant reduction in the analysis time required to evaluate the radiation patterns of these arrays; for instance, in the case of the stage-4 array it is nearly thirty times faster. The array factor of a stage-P perturbed PG array at a particular far-field location (; ') is conveniently expressed using matrix multiplication as

A

B

AFP (; ') = AP BP C

(1)

where P and P are associated with the array factor of the generator arrays and the PG curve structure, respectively. The three orientations of the generator component are calculated using

AP = [A1;P

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A2;P A3;P ]

(2)

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Fig. 2. (a) Example of a stage-1 generating array with perturbed element locations. The solid line denotes the PG curve and the dashed line denotes the initiator curve. The polar coordinates of the locations of the perturbation elements are referenced to the center of the initiator, as shown for the 7th element. (b) Stage-2 and (c) stage-3 arrays that were generated using the element locations of the perturbed stage-1 generating array.

F = f f = q

P;q t;i

P;q t;i

(7)

(323)

exp[jk r sin  cos(' 0 ' 0 ' q

2

n

n

i

n N

Fig. 1. (a) Geometry of a stage-3 Peano-Gosper array and (b) a contour plot of its normalized radiation pattern with d =  and its main beam directed toward ' = 0 ;  = 30 . It is plotted against the axes u and v with a maximum extent of u + v < 1, where u = sin  cos ' and v = sin  sin '. The array is comprised of isotropic radiators with uniform current amplitude excitations.

A

i;P

=

8

I exp[jkr sin  cos(' 0 ' 0 ' m

m=1

m

m

i

+ (P 0 1)) + j ] (3) = 0kr sin o cos (4) 2 ['o 0 ' 0 ' + (P 0 1)] 2  ' = (i 0 1) 3 (5) where rm (in wavelengths) and 'm correspond to the location of the i;m;P

i;m;P

m

m

i

i

mth generator element from the center of the initiator (as shown in Fig. 2) and Im is its associated current amplitude excitation. The amplitude excitations have a value of one for the endpoint elements and a value of two for all others. The phase shifts i;m;P can be used to direct the main beam to a particular far-field location (o ; 'o ). The component of the array factor associated with the PG generating array structure p is calculated using

B

BP =

P01

F

q

q =1

(6)

+ (P 0 q 0 1)) + j ] (8) = 0k r sin o cos[' 0 ' 0 ' + (P 0 q 0 1)] (9) N = [N ](323) f1; 3; 5; 6g f2g f4; 7g = f4; 7g f1; 3; 5; 6g (10) f2g f2g f4; 7g f1; 3; 5; 6g C = [ 1 0 0 ]T (11) where rn (in wavelengths) and 'n correspond to the location of the

i;n;q;P

i;n;q;P

q

n

o

n

i

t;i

center of the nth segment of the stage-1 curve (Fig. 2). A table of these locations can be found in [3] or they can be readily calculated based on the geometry of the PG curve. The angle, , corresponds to the acute angle between the first segment p of the stage-1 curve and the initiator, which is equal to arctan( 3=5). The scale factor,  , is defined as the ratio of the length of thepinitiator to the length of a segment in the stage-1 curve and is equal to 3=(2sin ). The values of  and  can be readily derived through a geometrical construction by overlaying the stage-1 curve on a triangular lattice of elements. Elements of the macorrespond to groupings of segments with similar orientations trix along the stage-1 curve (see Fig. 2) [3]. These groupings are taken into account in the array factor summation of (8); for instance, evaluation of (8) for N1;1 sums the contribution of an array element located at the center the first, third, fifth and sixth segments. The current distribution that results from the formulation of (1)–(11) consists of unity amplitude

N

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excitations on the elements at the two end points of the Peano-Gosper curve and excitations with a magnitude of two on all interior elements [3]. This particular almost uniform current distribution is preserved for the perturbed Peano-Gosper fractile arrays considered here and shall be referred to as uniform throughout this communication. III. DESIGN PROCESS & EXAMPLES The design of a perturbed PG array requires the proper selection of six perturbation element locations subject to design constraints and objectives. A convenient way to represent the perturbations is by translational offsets from their original locations along the length of the stage-1 PG curve. This representation allows the entire geometry of a high-order perturbed PG array to be completely specified by only six parameters. For practical considerations, it is assumed that the arrays in this communication have a minimum element spacing, dmin , of =2 at their lowest operational frequency fo (with corresponding wavelength ). The objective of the following design process is to optimize the perturbed array configurations for improved scanning performance at a frequency f = 2fo , with dmin = . It is important to note that in order for the perturbation technique to be applied to an array it is necessary to start with a PG curve that has a segment length that is greater than the targeted spacing. This allows the elements to be perturbed from their initial locations while not violating the targeted minimum element spacing constraint. For simplicity, a starting segment length of 2  was selected for all of the designs. A few stages of perturbed Peano-Gosper arrays were designed for enhanced beam steering within a 30 conical volume (centered at broadside) at f = 2fo . This was accomplished by minimizing the peak lobe that appears within the visible region of the array if its main beam was steered throughout this volume. A simple way to determine this peak lobe is by evaluating the radiation pattern of the array within the region bounded by u2 + v 2 < 1 + sin(T ), where T is the targeted elevation scan angle from broadside [10]. The peak sidelobe level within this expanded visible region corresponds to the peak level that appears if the main beam of the array is steered throughout the targeted scan volume. The design process was carried out for stage-2, stage-3, and stage-4 arrays using a genetic algorithm [11]. In each case, the perturbation process provided a significant level of sidelobe suppression over the standard PG array. Table 6.1 lists some of the relevant geometrical and radiation characteristics of the optimized stages of the perturbed PG arrays. The following discussion will examine some selected aspects of these designs. The layout of the perturbed stage-3 PG array is shown in Fig. 3. By comparing this with that of Fig. 1(a), it is apparent that the adjustment of the six perturbation locations led to a highly altered element distribution. The best way to illustrate the enhanced scanning performance of the optimized stage-3 array is through the examination of its radiation pattern during beam steering, as shown in Fig. 3(b). Clearly, there are no grating lobes contained within the visible region of the array at this particular scan angle. This is in direct contrast to the standard stage-3 PG array that has several grating lobes present at the same scan angle, as shown in Fig. 1(b). The stage-4 optimization produced a design with a sidelobe suppression of 010:2 dB while scanning within the targeted 30 conical volume at f = 2fo . If the array were limited to only broadside operation it would have a 3:1 bandwidth, which corresponds to the ratio of the upper operating frequency to the lowest intended operating frequency over which sidelobes remain below 010 dB. In addition, it maintains a sidelobe level below 07 dB up to f = 6:4fo . This is in contrast to the standard PG array that has grating lobes in its pattern past f = 2:25fo . A plot of the sidelobe performance of the optimized design is shown in Fig. 4. As it can be seen in this plot, the enhanced scanning capabilities and improved sidelobe suppression at higher frequencies come at

Fig. 3. (a) Geometry of the perturbed stage-3 PG array and (b) a contour plot of its normalized radiation pattern at f f with the main beam steered to ' ; .

=0

= 30

=2

the expense of higher sidelobes at lower frequencies. For instance, the perturbed stage-4 array has a sidelobe level of 010:7 dB for broadside operation at f = fo ; this level is 5.42 dB higher than that of the standard stage-4 array. From a subarray standpoint, the stage-4 array could be operated as a single array or as a collection of seven stage-3 modular subarrays as shown in Fig. 5. In either case, the level of sidelobe suppression that is exhibited during beam steering would be comparable to the values reported in Table I. It was expected that mutual coupling would not play a significant role in the performance of perturbed PG arrays due to their relatively large average element spacings. This was investigated through the use of full-wave analysis (method of moments [12]) of an entire optimized 50-element stage-2 array. The antenna element that was selected for this investigation was a probe-fed microstrip patch designed for operation at 1 GHz. The antenna is based on a substrate with "r = 4 and a thickness of 8 mm. Radiation pattern cuts of the array are shown in Fig. 6. Cuts are also shown for an array that was analyzed using pattern multiplication of a full-wave simulation of a single antenna element (in isolation) with the array factor of the stage-2 array. Excellent agreement is exhibited between these two models over both cuts as well as over their entire visible region. This indicates that mutual coupling

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603

TABLE I CHARACTERISTICS OF THE OPTIMIZED STAGES OF THE PERTURBED PG ARRAYS. THE PEAK SIDELOBE LEVEL (SLL) CORRESPONDS TO THE PEAK LOBE LEVEL f . THE LAST THREE COLUMNS PERTAIN TO OPERATION AT f f . THE THAT APPEARS WHILE BEAM STEERING THROUGH A 30 CONICAL VOLUME AT f AVERAGE ELEMENT SPACING CORRESPONDS TO THE AVERAGE OF THE SPACINGS BETWEEN SUCCESSIVE ELEMENTS ALONG THE PG CURVE

=2

=

Fig. 4. Sidelobe performance of the perturbed stage-4 PG array, the standard stage-4 PG array, and a 1793-element square lattice periodic array during broadside operation.

= 90

Fig. 6. Radiation pattern cuts at ' of the perturbed stage-2 PG array at GHz with its main beam directed to (a) broadside and (b) ' ; f . The azimuthal cut angle corresponds to E-plane orientation of the microstrip patch elements. The centers of the array elements have a spacing of at least of 15 cm.

=1 30

= 90

=

IV. SUMMARY

Fig. 5. Construction of a stage-4 array via a tiling of seven optimized stage-3 perturbed Peano-Gosper fractile subarrays. The element locations are superimposed on the stage-4 Peano-Gosper curve and each stage-3 array is contained within a Gosper island fractile (fractal tile).

does not have a significant impact on the radiation properties of the perturbed patch antenna array layout.

This communication introduced an effective design technique for generating modular, wideband phased arrays that are based on perturbed Peano-Gosper array geometries. Unlike traditional perturbation techniques that require independently adjusting the position of every element in an array, this technique is based on perturbing only six element locations along a stage-1 Peano-Gosper generator curve and then using these locations to iteratively construct larger arrays. This allows for a tractable perturbation process that is easily combined with an optimization algorithm. Stages of perturbed PG arrays were designed for enhanced beam steering capabilities over a 2:1 bandwidth. In particular, the stage-4 design has a peak sidelobe of 010:2 dB at f = 2fo

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while steering throughout a 30 conical volume. Full-wave simulations were used to analyze the effects of mutual coupling on the performance of perturbed PG array layouts. It was shown that these effects do not have a significant impact on the radiation properties of a perturbed PG array that is comprised of microstrip patch elements.

REFERENCES [1] E. Sharp, “A triangular arrangement of planar-array elements that reduces the number needed,” IRE Trans. Antennas Propag., vol. 9, no. 2, pp. 126–129, Mar. 1961. [2] B. B. Mandelbrot, Fractals: Form, Chance, and Dimension. New York: W. H. Freeman and Company, 1977. [3] D. H. Werner, W. Kuhirun, and P. L. Werner, “The Peano-Gosper fractal array,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 2063–2072, Aug. 2003. [4] D. H. Werner, W. Kuhirun, and P. L. Werner, “Fractile arrays: A new class of tiled arrays with fractal boundaries,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 2008–2018, Aug. 2004. [5] A. Tennant, M. M. Dawoud, and A. P. Anderson, “Array pattern nulling by element position perturbations using a genetic algorithm,” IEE Electron. Lett., vol. 30, no. 3, pp. 174–176, Feb. 1994. [6] M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag., vol. 50, no. 12, pp. 1732–1742, Dec. 2002. [7] 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. [8] J. S. Petko and D. H. Werner, “An autopolyploidy-based genetic algorithm for enhanced evolution of linear polyfractal arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 583–593, Mar. 2007. [9] T. G. Spence and D. H. Werner, “Design of broadband planar arrays based on the optimization of aperiodic tilings,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 76–86, Jan. 2008. [10] Y. T. Lo and S. W. Lee, “Affine transformation and its application to antenna arrays,” IEEE Trans. Antennas Propag., vol. 13, no. 6, pp. 890–896, Nov. 1965. [11] R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics. Hoboken/Piscataway, NJ: Wiley-IEEE Press, 2007. [12] FEKO. EM Software and Systems-S.A., [Online]. Available: http://www.emssusa.com

Reducing the Number of Elements in the Synthesis of Shaped-Beam Patterns by the Forward-Backward Matrix Pencil Method Yanhui Liu, Qing Huo Liu, and Zaiping Nie

Abstract—The matrix pencil method (MPM) has been used to reduce the number of elements in the linear antenna array with a pencil-beam pattern. This work extends the MPM-based synthesis method to the synthesis of shaped-beam patterns by using the forward-backward matrix pencil method (FBMPM). The FBMPM-based synthesis method places a necessary restriction on the poles which correspond to element positions, and consequently obtains more accurate synthesis results, particularly for the synthesis of asymmetric patterns. Numerical examples show the effectiveness and advantages of the proposed method in the reduction of the number of elements for shaped-beam patterns. Index Terms—Array synthesis, sparse antenna array, forward-backward matrix pencil method (FBMPM), shaped-beam pattern.

I. INTRODUCTION The problem of synthesizing shaped-beam patterns has received much attention over the past sixty years, for example, in [1]–[6]. Most of synthesis techniques for the shaped beam pattern have been developed only with uniformly spaced linear arrays, such as in [1]–[4]. Despite the success of these techniques, the synthesis of uniformly spaced arrays sometimes requires a large number of elements to produce a desired pattern characteristics. Naturally, using nonuniform element spacings can reduce the total number of elements, which is very useful in some applications, particularly for the situation where the weight of antenna systems is extremely limited. This work focuses on the problem of reconstructing the shaped-beam pattern with as few elements as possible. The synthesis of a completely nonuniform array is usually required. This is a highly nonlinear inverse problem involving finding the solution of many unknowns (excitation amplitude, phase and position for each element). Many iterative synthesis techniques cannot guarantee a global optimum for all the variables unless exceptionally good starting values are supplied. Some stochastic optimization algorithms capable of finding the global optimal solutions may be appropriate, but they can be time-consuming since even the number of elements required is unknown. In addition, to our knowledge, most of existing nonuniform array geometry synthesis techniques deal only with the case of pencil-beam patterns, and it is not clear whether they can be directly applicable to the synthesis of shaped-beam patterns. Manuscript received January 27, 2009; revised May 05, 2009. First published December 04, 2009; current version published February 03, 2010. This work was supported in part by the U.S. National Science Foundation under Grant CCF-0621862, in part by the 111 project of China under Contract No. B07046, and in part by the Joint Ph.D. Fellowship Program of the China Scholarship Council. Y. Liu is with the School of Electronic Engineering, University of Electronic Science and Technology of China, Sichuan 610054, China and also with the Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA. Q. H. Liu is with the Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA (e-mail: [email protected]). Z. Nie is with the School of Electronic Engineering, University of Electronic Science and Technology of China, Sichuan 610054, China. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037709

0018-926X/$26.00 © 2009 IEEE

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Recently, we presented a non-iterative synthesis technique based on the matrix pencil method (MPM) [7]. This MPM-based synthesis technique organizes the desired pattern data in a form of Hankel matrix and performs the singular value decomposition (SVD) of this matrix to obtain an optimal lower-rank matrix that corresponds to fewer antenna elements. The MPM is then utilized to reconstruct new element locations and excitations. This synthesis method has shown its ability of reducing the number of elements for many linear arrays with pencil-beam patterns. However, as we mentioned in [7], applying this method to synthesize a shaped-beam pattern may encounter a problem that the estimated poles usually do not lie on the unit circle, which results in unrealized imaginary parts of the element positions. Note that this is also observed by Miller and Goodman in the Prony-based synthesis method [8]. In this paper, we will show that this problem can be partially solved by using the forward-back matrix pencil method (FBMPM) [9], [10] which places a necessary constraint on the distribution of poles. This constraint is not sufficient but very useful for limiting all the poles on the unit circle. In the following sections of this paper, we will introduce the FBMPM-based synthesis method and give results for reconstructing some typical shaped patterns with fewer elements.

II. PATTERN RECONSTRUCTION METHOD Let a linear array be composed of far field array factor is given by

M identical antenna elements. The

M

FM (u) = Riejw u j p0 ;u

i=1



w

(1)

d =

1 = cos( ) and i = 2 i . The above expreswhere = sion is in the form of a sum of exponentials. The problem of reducing the number of antenna elements is to use as few exponentials as possible to approximate the original pattern M ( ) within a desired tolerance.

F u

A. MPM-Based Synthesis and Its Limitation In the MPM-based synthesis method, the desired pattern function is uniformly sampled in the space of from = 1 to = +1. , where = + 1 ... . We have Let n = 1 = M ( ) = M ( 1) = M i=1 i in , where i = jw 1 . The sampling condition can be found in [7]. Then the MPM-based synthesis method organizes the pattern data into a Hankel matrix

u u 0 u n 0N; 0N ; ;N Rz z e

u n n=N f n F n

Y = [y0; y1 ; . . . ; yL ]

(2)

where l = [ l l+1 . . . 2N 0L+l ]T and l = M ( is ). called the pencil parameter, and the subscript denotes the transpose of a matrix. By performing the singular value decomposition (SVD) of and discarding some small singular values, we can obtain an optimal ). The matrix Q lower-rank approximation of , say Q ( corresponds to an approximate pattern that can be produced by fewer elements. The positions of new elements can be obtained by solving a generalized eigenvalue problem given by (11) or its equivalent problem given by (12), both of [7]. Assume that i0 s are the estimated eigenvalues. The element positions are given by

y

Y

y ;y ; ;y

y f l0N L T

Y

Y QM

Y

z

0 di0 = j 2ln1zi :

(3)

605

d

As in the Prony-based synthesis method [8], the source locations i0 calculated in this way may turn out to be complex if i0 = 1. Only the real parts of i0 are physically realizable. Through numerous synthesis examples, we found that the magnitudes of imaginary parts of i0 were always very insignificant for the synthesis of broadside pencil-beam patterns [7]. Therefore taking only the real parts of i0 in [7] and [8] does not degrade the accuracy of synthesis results in this case. However, this is not true for the synthesis of shaped-beam patterns where the estimated poles of i0 are frequently not very close to the unit circle. We shall show this problem can be overcome or at the least partially solved by the FBMPM-based synthesis.

jz j 6

d

d

d

z

B. FBMPM-Based Synthesis Method Now, we consider the FBMPM [9], [10] for solving the problem mentioned above. To do so, we organize the sampled pattern data into a Hankel-Toeplitz matrix that is given by

Yfb = yyL30 yyL3 01 1 11 11 11 yyL03

(4)

L

where 3 denotes complex conjugate, the pencil parameter is chosen 2 such that [10]. Consider the matrix pencil

M L N 0M

Yffb 0 zYlfb (5) where Yffb is obtained from Yfb by deleting the first column, and Ylfb is obtained from Yfb by deleting the last column. It can be proven that if fzi0 ; v1 g is a pair of generalized eigenvalue and eigenvector of this 3 matrix pencil, f(1=zi0 ) ; v2 g must be another pair of generalized eigenvalue and eigenvector, where v2 (k) = v13 (L0k). That is, all the eigen3 values (or poles) must be obtained as a pair of fzi0 ; (1=zi0 ) g. Although this is only a necessary condition for guaranteeing that jzi0 j = 1, this 0

z

constraint is actually locally sufficient for i in the neighborhood of the true value. Using this constraint can improve significantly the estimation accuracy of the poles, which is very important for extending the MPM-based synthesis method to deal with asymmetric shaped-beam patterns. Remember our final goal is to reduce the number of elements for a desired pattern. Therefore, actually we do not directly solve the eigenvalue problem of (5). Instead, we first perform the SVD of fb and then find an optimal lower-rank approximation of this matrix by retaining only largest singular values, just as we did in the MPM-based fb . It can be synthesis method. Denote the lower-rank matrix by Q fb shown that Q is not a Hankel-Toeplitz matrix anymore, but still maintains exactly the same vector structure as that of (4) in terms of l . Consider the following matrix pencil:

Y

Q

Y

Y

y

fb 0 zYQ;l fb YQ;f

(6)

fb (resp., Q;l fb ) is obtained from Qfb by deleting the first where Q;f column (resp., deleting the last column). This matrix pencil maintains a similar matrix structure as (5), and the generalized eigenvalues of this pencil would have the same constraint as the eigenvalues of (5). This constraint is very useful for avoiding the poles moving off the unite circle, which will be validated by the synthesis experiments in Section III. Once the eigenvalues (or the poles) are obtained correctly, the positions and excitations of new elements can be immediately calculated through (14)–(18) of [7].

Y

Y

Y

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Fig. 1. (a) The pattern synthesized by [6] with 20 elements and the patterns reconstructed by the MPM- and FBMPM-based methods, both with 15 elements. (b) Distributions of the poles.

Fig. 2. (a) The pattern synthesized by [4] with 16 elements and the patterns reconstructed by the MPM- and FBMPM-based methods, both with 13 elements. (b) Distributions of the poles.

III. NUMERICAL RESULTS

Now, only one question remains. How do we determine the value of

Q or what is the minimum number of elements required for a satisfactory approximation of the desired pattern? In [7], it was found that the criterion for choosing Q based on index q P 2 i=q+1 i q 2 i=1 i

Q^ = min q;

0 of dimension (2N-1) such that

P

VK ( x) =  xT P x:

Q

(5)

Q

Given a real symmetric matrix , the matrix inequality > 0 holds if > 0. Correspondingly and only if for every nonzero vector ; T < 0 if and only if T < 0 for every > 0. Applying (3) and (5) in (4) one obtains the following matrix inequality

Q

y Qy

y y Qy y

WL(1^; P; q) = PK~ L (1^) + K~ LT (1^)P + 2qP < 0: (6) For a given matrix P, finding 1 ^ that maximizes q is an eigenvalue (EVP) problem that is convex [6] and can be efficiently solved. Addi^ , finding the symmetric matrix P > 0 satistionally, for a given 1

fying (6) that maximizes q is a generalized eigenvalue problem (GEVP) that is quasi-convex and can also be efficiently solved using a bisection algorithm [6]. IV. STABILITY OPTIMIZATION ALGORITHM The array factor can be written in the following form allowing for a perturbation in the excitation vector [3], [4]:

x

AF() = Vo uH 1N + uH [ IN jVo IN ] x where u() = [ej(nkd sin()0 ) ]H for n = 1; . . . ; N. IN

(7)

is the identity matrix of dimension N. Similarly to [1], [3], [4], the main beam is first steered to an angle o from broadside by assuming uniform amplitude excitation Vo and a progressive phase shift on = nkd sin(o ). The distance between the antenna elements is d. In a second step, constraints in the array factor at m ; m = 1; . . . ; M, with = + = [ r (1 ) i (2 ) 1 1 1 (M ) ]H are imposed by setting AF(m ) 2 (em + jem )[01; 1], which, using (7), leads to

U C jS u

u

u

0Vo C1N 0 er  C 0Vo S x 0Vo S1N 0 ei S VoC r 0V N+e  0VooC1 i S1N + e

or, in compact form

fl  Fx  fh

It is straightforward to express (9) in terms of

fl  F 1^  fh

(3)

^) of dimension 2N-1, defined described by the square matrix ~ (  in the Appendix, whose eigenvalues determine the stability of o . ~ ( ^) does not have a linear dependence on ^ because of the matrix inversion involved in its derivation and the exponential de^ , which can be verified by looking at pendence on the phases ' its definition in the Appendix. However, based on the fundamental assumption of small and  in (1), one may consider instead the ^) of ~ ( ^). 1st order Taylor expansion ~ L (  ^) is the maximum real part of The spectral abscissa of ~ L (  ^) [5]. Alternatively, the decay rate q of the eigenvalues of ~ L (  ~ L ( ^) is the negative of the abscissa of ~ L ( ^) [6]. A positive decay rate corresponds to a stable o . Furthermore, maximizing the

K1

609

(8)

(9)

1^ (10)

fl ; fh , and F can be easily obtained from fl ; fh , and F and (2). The optimization problem can therefore be written as

s:t: (i) (ii) (iii) (iv)

min L(t; q) = (t 0 q) G0x^ 1g 0 G0x^ 1G^ 1^ 2 + k1^k2  t fml  F 1^  fmh WL(1^; P; q) = PK~ L(1^) + K~ LT (1^)P + 2qP < 0 P>0 ^ ;P;t;q) (1

(11)

x

where (i) enforces that the magnitude of the perturbation k^ k2 + k ^k2 is minimized. Problem (11) is not convex due to (iii), and

1

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Fig. 1. Coupled oscillator system schematic.

therefore it is not straightforward to solve. An algorithm to solve (11) is proposed as follows. Step 1: Let e > 0 the algorithm termination tolerance, and k = 0 the iteration number. Set  ^ 0 2 arg min(1^ ;t) L(t; 0) = t subject to (i) and (ii). This is the original problem without a stability related constraint. q0 is the initial decay rate ^0 ) and t0 the initial pervalue corresponding to ~ L (  turbation vector. Step 2: Repeat f P1: Find k+1 2 arg min(P;q) L(0; q) = 0q subject ^; ; q) < 0 and (iv). This is a generalto (iii) L (  ized eigenvalue minimization (GEVP) problem. q1;k+1 is the decay rate value of this step. P2: Find  ^ k+1 2 arg min(1^ ;t;q) L(t; q) = t 0 q subject to (i), (ii), (iii) using as input k+1 of the previous step. q2;k+1 is the decay rate value of this step. tk+1 is the perturbation vector of this step. This is an eigenvalue minimization (EVP) problem. P3: k = k + 1g until q2;k+1 0 q1;k+1 < e. The optimization problems P1 and P2 can be efficiently solved in Matlab using an appropriate toolbox. In this work the freely available toolbox SeDuMi [7] was used. In addition, the free graphic environment Yalmip [8] was used as an interface to SeDuMi, to formulate the optimization problems.

1

K 1

P W 1 P 1

Fig. 2. Decay rate versus iteration for main beam direction  location  .

= 064

= 20

and null

P

V. A DESIGN EXAMPLE The optimization problem (11) is demonstrated for a linear N = 9 element array. The array elements are assumed to be cou-

pled using resistive loaded transmission line networks (Fig. 1). The = 1 with Zo = 100 Ohm, and transmission line section is the loading resistors are R = 270 Ohms. The antenna spacing is o =2(kd = ). A 5.5 GHz voltage controlled oscillator (VCO) was simulated in commercial HB software, using the available nonlinear models for the NE3210S01 NEC HJ-FET and the MACOM MA46H070 varactor. The free-running steady state of each element was (Vo ; !o ; o ) = (0:47 V; 25:491 GHz; 1 V). The main beam is steered to o = 20o and a null is introduced at n = 064o . The solution of the original algorithm which does not contain the stability constraint (Step 1), corresponds to a perturbation vector tmino = 0:8307 and a decay rate qmaxo = 0:0126(106 s01 ). The proposed algorithm (Step 2) required 176 iterations and resulted in tmin = 0:8371 and decay rate qmax = 0:0533(106 s01 ). The decay rate was improved by 322% while the perturbation vector was increased by only 0.77%. The convergence of the algorithm is shown by ploting the decay rate versus the iteration number in Fig. 2. The radiation patterns of the two solutions are shown in Fig. 3, where one can see that the two solutions lead to very small differences in the array factor. A nonlinear harmonic balance simulator (Agilent ADS) was used to verify the algorithm. In order to do so the phase perturbation vectors obtained from Step 1 and Step 2, were imposed in the harmonic balance simulator and the amplitudes and controls of the oscillator elements were optimized. This is in agreement with the fact that typically the amplitude variations that are introduced are very small.

Fig. 3. Array factor comparison for the original solution of Step 1, and the final solution optimized for stability (Step 2). The harmonic balance solution corresponding to the solution of Step 2 is also included. The main beam direction is  and the null location at  .

= 20

= 064

The result of the harmonic balance simulator is also included in Fig. 3, where it is seen that it practically overlaps the solution of Step 2. The envelope transient simulator of Agilent ADS was then used to provide an indication of the stability of the solutions of Step 1 and Step 2. First, the phases of the oscillators corresponding to the two solutions were imposed in the harmonic balance simulator and the amplitudes and controls of the oscillators corresponding to the two steady states were obtained. In a second step, this solution was used to initialize the envelope transient simulator of Agilent ADS, and the oscillator array is left to evolve in time. The result is shown in Fig. 4, where one can see that both solutions, after a small transient, evolve to the nearest stable steady state. One can see that the transient of the solution of Step 2 is much smaller than the transient of Step 1, indicating that the decay rate of the solution of Step 2 is larger. Additional tests consisted of steering the main beam to o = 20o and placing the desired null at different angles. The location of the null was varied along  with a step of 2o , except for angles that correspond to directions inside the main lobe. First, the validity of the linear approximation ~ L of the matrix ~ used to determine the stability of the solutions is examined. The decay rate of the two matrices for the solutions corresponding to the original algorithm of Step 1 is shown in

K

K

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Fig. 4. Envelope transient simulation of the solutions corresponding to the algorithm of Step 1 and Step 2. Evolution of the phase of oscillator 4 versus time. and the null location is  . The main beam direction is 

= 20

= 064

K~

K~

Fig. 5. Accuracy of the linear approximation of for the solution of the original problem of Step 1. Decay rate for different null angles (main beam ). 

= 20

Fig. 5. The array factor assuming a uniform excitation with a progressive phase shift is shown in the figure background (not to scale) in order to better visualize the effect of the null condition in the optimization. One can see that the linear approximation is very good for most null locations. In Figs. 6 and 7 the original algorithm (Step 1) and the proposed one (Step 2) are compared in terms of the magnitude of the perturbation vector and the decay rate respectively. The perturbation vector tmin = ^k2 + k1^k2 is not zero at null locations that kxk2 + k1k2 = kx correspond to an existing null in the uniform pattern (Fig. 6). This is because the perturbation vector includes the control vector perturbation that is required to generate the constant phase shift distribution in order to steer the main beam. The minimum perturbation vector is zero only when the main beam is pointing at broadside. Furthermore, the perturbation vector takes locally maximum values when the null coincides with a sidelobe of the uniform array factor. As it was also observed in [3], [4], there is a null location (n = 041:15o ), where both algorithms fail to converge. This, in fact is an inherent limitation of the perturbation method of [2], and can be demonstrated by

611

Fig. 6. Perturbation vector magnitude t for different null angle values.

Fig. 7. Decay rate for different null angle values.

analytically minimizing x = [ 1VT 1'T ]T subject to (9) [3]. Following the procedure of [3], one finds that xT x goes to infinity for

sin(n ) 0 sin(o ) = 0=kd

(12)

When kd =  , and o = 20o , one gets n = 041:15o . One can verify from Fig. 6 that the solution of the proposed algorithm is very close to the solution of the original convex optimization algorithm of Step 1, which is used as the starting point. One can also identify three points at approximately n = 072o ; 060o ; 2o where the algorithm failed to converge. The proposed alternate minimization algorithm is not convex and it may converge to a stationary point that is not a global minimum of the optimization problem. Moreover, due to the non smooth nature of the eigenvalue functions, the algorithm may fail to converge to a stationary point [9]. Finally, in Fig. 7 one can see that the solution of the proposed algorithm corresponds to a higher decay rate than the original algorithm and thus to a more stable solution. In some cases the improvement is minimal, in others however such as the ones between n = 072o and o n = 060 the improvement is significant.

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Ec = [ EVc EPc ] = @@!Y IN + 8H Yc! 8 [ jIN 0V^ ] Ac = [ AcV AcP ] ^ N = @@YV V^ + 8H Yc 8 + 1Y j8H Yc 8V^ 0 jdiag 8H Yc 8V1 8 = diag [exp(j1')] ; V^ = Vo IN + diag[1V]; 1Y = diag @@ YV 1V + @@Y 1 :

APPENDIX

1' ]T , corresponding to

The array steady state x = [ 1V a control vector 1 is given by (1). Note that in (1) the frequency of the array synchronized steady state is set equal to the free-running frequency !o . This is possible by allowing all oscillators to be tuned [3]. G and g are defined as follows [3], [4]: T

T

equations of (A.1) are reduced by one by subtracting the equation that corresponds to the phase of oscillator j (i.e., row N+j) from every other phase equation. The equation that corresponds to the phase of oscillator j can then be eliminated. Consequently, it is also possible to eliminate column (N + j) of (A.1) because it is multiplied by zero. ~ = 'ji6=j 0 j 1N01 ;  x~ = [  VT '~ T ]T one has Defining  '

~_ = K~ (xo ) x~ x G1 = Vo @@ YV IN + 8Ho Yc 8o ~ (xo ) dimensions have been reduced by 1. K~ (xo ) where the matrix K G2 = jVo 8Ho Yc 8o 0 diag 8Ho Yc 8o 1N has the same eigenvalues with K(xo ) without the zero eigenvalue. T The solution xo = [ 1VoT 1' T o ] is stable if all the eigenvalues G3 = Vo @@Y IN ~ of K(xo ) have negative real parts. Using (2), one obtains xo as a funcGc = [ G1 G2 G3 ] ^ , and therefore K~ = K~ (1^) is a square matrix of dimension tion of 1 2N 0 1, with N 0 1 parameters. gc = 0Vo 8Ho Yc 8o 1N and G = [ Re(Gc ) Im(Gc ) ]T ; g = [ Re(gc ) Im(gc )]T . REFERENCES Y is the oscillator nonlinear admittance at !o and Yc is the coupling

network Y-matrix. The partial derivatives of Y that are used in the formulation can be easily computed from a harmonic balance simulation of the free-running oscillator element [3]. The coupling matrix Yc in addition to contribution from a coupling network (e.g., a transmission line based network, Fig. 1), may include radiation coupling that can be calculated from an EM simulation. 8o is a diagonal matrix with the phase exponents expfjon g in its main diagonal. A steady state solution with a progressive phase distribution along the elements is given by setting on = nkd sin(o ), where o corresponds to the desired main beam direction from broadside. = The stability of the array steady state solution xo [ 1VoT 1'To ]T of (1), is examined by perturbing the steady state amplitudes and phases by  x = [  VT  ' T ]T , leading to [3]

E(xo ) x_ = A(xo ) x )  x_ = K(xo ) x

(A.1)

K = E01 A; E = [ RefEc gT ImfEc gT ]T ; A = [ RefAc gT ImfAc gT ]T , where, see the equation at the top of the page, Yc! is the frequency derivative of the coupling matrix. In the

with

case of typically used broadband coupling networks, one may assume

Yc!  0. This assumption is used in this work. Since the analysis is based on a perturbation of the oscillator free-running state, one has V^  Vo IN (to zero order). As a result, E is diagonal without any zeros in its main diagonal and, therefore, invertible. The existence of

K is then guaranteed. The stability of the solution xo is determined by the 2N eigenvalues of K.

Due to the arbitrary time reference of the array steady state, one of the eigenvalues of K is always zero. One can easily verify that A and subsequently K is unchanged to phase perturbations that are common to all oscillators. It is possible to reduce the dimension of the system by 1, eliminating the zero eigenvalue, by forming phase differences as follows. Selecting arbitrary element j as a reference, the N phase

[1] T. Heath, “Simultaneous beam steering and null formation with coupled, nonlinear oscillator arrays,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 2031–2035, Jun. 2005. [2] H. Steyskal, “Simple method for pattern nulling by phase perturbation,” IEEE Trans. Antennas Propag., vol. AP-31, no. 1, pp. 163–166, Jan. 1983. [3] A. Georgiadis, A. Collado, and A. Suarez, “Pattern nulling in coupled oscillator antenna arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1267–1274, May 2007. [4] A. Georgiadis and K. Slavakis, “A convex optimization method for constrained beam-steering in planar (2D) coupled oscillator antenna arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2925–2928, Oct. 2007. [5] A. S. Lewis and M. L. Overton, “Eigenvalue optimization,” in Acta Numerica. Cambridge, U.K.: Cambridge Univ. Press, 1996, pp. 149–190. [6] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, “Linear matrix inequalities in system and control theory,” in SIAM Studies in Applied Mathematics. Philadelphia, PA: SIAM, 1994, vol. 15. [7] J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11–12, pp. 625–653, Aug. 1999. [8] J. Lofberg, “YALMIP: A toolbox for modeling and optimization in MATLAB,” in Proc. IEEE Int. Symp. on Computer Aided Control Syst. Design, Sep. 2004, pp. 284–289. [9] K. C. Goh, L. Turan, M. G. Safonov, G. P. Papavassilopoulos, and J. H. Ly, “Biaffine matrix inequality properties and computational methods,” in Proc. American Control Conf., Jul. 1994, vol. 1, pp. 850–855.

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Wireless Transmission in Tunnels With Non-Circular Cross Section Samir F. Mahmoud

Abstract—We study wireless communication via the free modes in a uniform tunnel with non-circular cross section. The dominant modes propagating in such tunnel are considered as perturbed versions of corresponding modes in the well studied circular tunnels. Of particular practical interest are tunnels whose cross section comprise an incomplete circle with flattened base. It is shown that in these tunnels, a vertically polarized mode is more attenuated than its horizontally polarized counterpart. We also investigate the approximation of a non-circular tunnel with an equivalent rectangular tunnel regarding the attenuation of the dominant modes. Comparison is made with experimental attenuation measurements available in the literature. Index Terms—Perturbation theory, propagation in tunnels, wireless communication.

I. INTRODUCTION Wireless communication in road and mine tunnels have recently received much attention [1]–[3]. A typical straight tunnel with cross sectional linear dimensions of few meters can act as a waveguide to electromagnetic waves at UHF and the upper VHF bands. In this range, a tunnel is wide enough to support free propagation of electromagnetic waves. In addition, the tunnel walls act as a good dielectric with small loss tangent at these frequencies. So the electromagnetic wave losses are caused mainly by radiation or refraction through the walls with little or negligible ohmic losses as deduced by Glaser [4]. Tunnels with regular cross sections such as the circular or rectangular shape are amenable to analytical analysis that lead to full characterization of their main modes of propagation [2], [3]. However, most existing tunnels do not have regular cross sections and their study may require exhaustive numerical methods [5]. In this communication we consider cylindrical tunnels whose cross-section comprise an incomplete circle with a flat base as depicted in Fig. 1. Perturbation theory is used to predict the attenuation and phase velocity of the dominant modes from those in a perfectly circular tunnel. The latter has an analytical solution for its modal parameters.

Fig. 1. A tunnel with a perturbed circular cross section. The circular sector has a radius a and the flat base has a width L.

where "r (possibly complex) is the relative permittivity of the tunnel wall and 0 is the free space wave impedance. In a tunnel with perfectly circular cross-section, the dominant modes, with the least attenuation, are the TE01 and the HE11 modes [1]. Obviously, the HE11 mode has two possible polarizations where E is vertically or horizontally oriented. In a tunnel with a circular cross section, these two versions of the HE11 mode are degenerate; they have the same propagation constant. Detailed study of the field distribution of the dominant modes; TE01 and the HE11 modes, in an electrically large circular tunnel shows that the field of these modes is quite weak near the tunnel walls [3], [6]. Namely the transverse fields at the tunnel wall are inversely proportional to k0 a. This observation justifies the use of simple perturbation theory to predict the modal propagation in wall perturbed tunnel as the one shown in Fig. 1. In the next section we present the perturbation analysis. III. MODAL PROPAGATION PARAMETERS IN PERTURBED TUNNEL We consider a cylindrical circular tunnel of radius “a” and cross section S0 surrounded by a homogeneous earth of relative permittivity "r . ~ 0; H ~ 0 ) exp( 0 z ) Let us denote the vector fields of a given mode by (E where 0 is the longitudinal (along +z) propagation constant. Simi~ H ~ )exp(+ z ) be the vector fields of the corresponding larly, let (E; mode in the perturbed tunnel of area S (Fig. 2). Note, however, that the mode is a backward mode; travelling in the ( z) direction. Both circular and the perturbed tunnels have the same wall constant impedance Z and admittance Y . Now, we use Maxwell’s equations that must be satisfied by both modal fields to get the reciprocity relation [7] ~0 H ~ E ~ H ~ 0 ) = 0. Integrating over the infinitesimal volume :(E between z and z + dz in the perturbed tunnel, after some manipulations we obtain

0

0

r 2 0 2

II. MODAL PROPAGATION IN A CIRCULAR TUNNEL When the applied radio frequency is sufficiently high such that the tunnel radius a is greater than the free space wavelength , the following condition applies to the dominant, or low order modes of the k0 , where k0 is the free space wavenumber, and k is the tunnel: k radial wavenumber. Under this condition, the modal rays are incident at grazing angles to the tunnel walls. Therefore these walls are accurately modeled by constant surface impedance and admittance defined as follows [3], [6]



p Z = E =Hz =  = "r 0 1 0

and

Y

p"r 0 1

= 0 H =Ez = 001 "r =

(1)

Manuscript received December 16, 2008; revised June 16, 2009. First published December 04, 2009; current version published February 03, 2010. The author is with the Electrical Engineering Department, Kuwait University, Safat 13060, Kuwait (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037704

0 C (E~ xH~ 0 E~ xH~ ):a^ndC

0 = ~ ~ ~ ~ ^z dS S (E xH 0 E xH ):a 0

1

0

0

0

1

(2)

0

^n is a unit vector where C1 is the flat part of the cross-section contour, a ^y ) and a^z is a unit axial along the outward normal to the wall (= a vector. The integration in the denominator is taken over the cross section of the perturbed tunnel. In order to evaluate the numerator of (2), we use the constant wall impedance and admittance satisfied by the perturbed fields on the flat surface: Ex = ZHz and Hx = Y Ez . Z and Y are given in (1). Using these relations, (2) reduces to

0

0

0

0

= 02

L=2 x=0

[E0xHz + Ez H0x + Y Ez E0z 0 ZHz H0z ]dx : ~ ~ ~ ~ ^z dS S (E0 xH 0 ExH0 ):a

(3)

The integration in the numerator is taken over the flat surface of the perturbed tunnel. So far, the above result is rigorous, but cannot be used

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Fig. 2. A circular tunnel and a perturbed circular tunnel with a flat base. The walls are characterized by constant Z and Y.

Fig. 4. Percentage increase of attenuation of the perturbed HE with vertical and horizontal polarization in the noncircular tunnel versus L=a.

"

1 GHz) and two values of r (6 and 12) in Fig. 4. It can be seen that the ) is fairly weak on and r . Since the attenuadependence of (1 tion in an electrically large circular tunnel is inversely proportional to 2 [6], so will be the attenuation in the perturbed flat based tunnel.

=

f

"

f

V. AN EQUIVALENT RECTANGULAR TUNNEL MODEL Fig. 3. Attenuation of the perturbed HE (curves 1 and 2) and TE modes (curve 3) in a flat based tunnel versus L=a. Note that the HE mode has vertical and horizontal polarizations (curves 1 and 2 respectively).

as such since the perturbed fields are not known. As a first approximation we can equate these fields to the backward mode fields in the unperturbed (circular) tunnel. So we set: z  0z , z  0 0z in the numerator. In the denominator, the fields involved are the transverse fields (to z). So we use the approximations: x^z  0 x^z and x^z  0 0 x^z . Therefore (3) is approximated by

H

H~ a

H~ a

0 0 =

L=2 x=0

H E E E~ a E~ a

E0x H0z 0 E0z H0x 0 Y E02z 0 ZH02z dx : ~ ~ ^z dS S (E0 xH0 ):a

L

L=a

f L=a

L=a

L=a

L

L=a

=

w

f

jHEnm = Z=20k+2 a3Y 0 xnm01 2 0

(5)

01 where n th zero of the Bessel function n01 ( ). This form is the 01 mula is based on the condition: 0  n m . For the rectangular tunnel with width “ ” and height “ ” the attenuation of the HEnm mode (with vertical polarization) is [2], [3]

x

m w

ka h

x

J

2 2 n2 Y 0 0 jHEnm = 2k2 m wZ= + 3 h3 0

a

We consider a perturbed circular tunnel of radius = 2 meters with a flat base of width . The surrounding earth has a relative permittivity r = 6. For an applied frequency = 500 MHz, the modal attenuation factor, computed by (4), is plotted in Fig. 3 for the perturbed TE01 and . Note that HE11 modes as a function of the = 0 corresponds to a full circle and = 2 corresponds to a half circle. Generally the . As stated earlier, the HE11 mode has attenuation increases with two versions depending whether the polarization is horizontal (along x) or vertical (along y). Obviously the two modes are degenerate in a increases, this deperfectly circular tunnel ( = 0). However as generacy breaks down in the perturbed tunnel. It is remarkable to see that the attenuation of the horizontally polarized HE11 mode becomes less then that of the vertically polarized mode in the perturbed tunnel. This has been observed in real measurements made in rectangular tunnels [8] and in arched tunnels with flat base [9]. It is interesting to study the effect of changing the frequency or the wall permittivity on the mode attenuation in the perturbed flat based tunnel. So we plot the percentage increase of the attenuation relative to ) for two values of (500 MHz and that in the circular tunnel (1

"

h

(4)

Thus, for a given mode in the circular tunnel, (4) can be used to get the propagation constant of the perturbed mode in the corresponding perturbed tunnel. Of particular interest is the attenuation factor of the various modes for which numerical examples will be given next. IV. NUMERICAL EXAMPLES

We have seen that the attenuation of the perturbed HE11 mode in the perturbed circular tunnel (with flat base) depends on the mode polarization; namely the vertically polarized HE11 mode is more attenuated than the horizontally polarized mode (see Figs. 3, 4). The same observation is true for the HE11 mode in a rectangular tunnel whose height “ ” is less than its width “ ” [2], [3]. This raises the question whether we can model the flat based tunnel of Fig. 1 by a rectangular tunnel. We will investigate this possibility in this section. To this end let us start by comparing the attenuation of the HE11 mode in tunnels with circular and a square cross sections. For the circular tunnel we have [3]

x

(6)

Specializing (5) and (6) to the HE11 mode in circular and square tunnel ( = and = = 1) and equating the attenuation rates we obtain

w h

m n

w = h = (42 =2:40482 )1=3 a = 1:897a

(7)

which means that the area of the equivalent square tunnel is equal to 1.145 times the area of the circular tunnel. This contrasts the work of Dudley et al. [2] who adopted an equal area of tunnels. It is important to note that this equivalence is valid only for the HE11 mode in both tunnels; for other modes the attenuation in the circular and the square tunnels are generally not equal. Now let us turn attention to the perturbed circular tunnel with flat base (Fig. 1) for which we attempt to find an equivalent rectangular tunnel. We base this equivalence on equal attenuation of the HE11 mode (or perturbed form) in both tunnels. Let us maintain the ratio of areas as obtained from the square and circular tunnels; namely we fix the ratio of the equivalent rectangular area to the perturbed tunnel area

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Fig. 5. Attenuation of the HE mode in the perturbed tunnel of Fig. 1 using perturbation analysis and rectangular equivalent tunnel.

to 1.145. Meanwhile we choose the ratio h=w equal to the perturbed tunnel height to its diameter. So, we write

wh = 1:145 ( 0 )a2 + (La=2)cos  and

h=w = (1 + cos )=2

Fig. 6. Measured field down the Massif Central Tunnel in South France [2] at 450 and 900 MHz.

(8)

where  = Arc sin(L=2a). Equation (8) defines the rectangular tunnel that is equivalent to the perturbed circular tunnel regarding the HE11 mode. In order to check the validity of this equivalence, we compare the estimated attenuation of the HE11 mode in the perturbed circular tunnel as obtained by perturbation analysis and by the equivalent rectangular tunnel in Fig. 5. There is a reasonably close agreement between both methods of estimation for values of L=a between zero and 1.82. VI. SOME EXPERIMENTAL DATA In this section we assess the perturbation analysis and the rectangular tunnel model as means of predicting the attenuation of the dominant modes in the perturbrd circular tunnel. We do so by comparing the predicted attenuation with measured results in real tunnels as reported in the literature. So, we present measurements in the Massif Central tunnel in France and in one of the National Japanese Railways tunnels.

Fig. 7. Attenuation of the HE (like) mode in the Massif Central Tunnel by the perturbation analysis and the rectangular tunnel model versus measured values.

A. The Massif Central Road Tunnel

B. Japanese Tohoku National Railways

Measurements of the electric field down the Massif Central road tunnel south Central France have been taken by a research group in Lille University, France and the results are reported by Dudley et al. [2]. The Massif Central tunnel has a flat-based circular shape as in Fig. 1 with radius a = 4:3 m and L = 7:8 m; that is L=a = 1:81. The relative permittivity of the wall "r = 5 and the conductivity  = 0:01 S=m. The transmit and receive antennas were vertically polarized and the field measured down the tunnel at the frequencies 450 and 900 MHz are given in Fig. 6. For the lower frequency, the field shows fast oscillatory behavior in the near zone, but at far distances from the source (greater than 1800 m), the field exhibits almost a constant rate of attenuation, which is that of the dominant HE11 (like) mode. We estimate the attenuation of this mode as 27.2 dB/km. At the 900 MHz frequency, there are two interfering modes that are observed in the range of 1500-2500 m. One of these two modes must be the dominant HE11 mode. Some analysis is needed in this range that lead to an estimation of the attenuation of the HE11 mode, which we find as 6.8 dB/km. The predicted attenuation rates of the dominant mode by the present perturbation analysis and by the rectangular tunnel model are plotted versus the frequency in Fig. 7. The measured attenuation at 450 and 900 MHz are also plotted for comparison. It is seen that there is generally good agreement between measured and predicted attenuation.

Chiba et al. [9] have provided field measurements in one of the National Japanese Railways tunnel located in Tohoku. The tunnel cross section is circular with a flat base as that depicted in Fig. 1. The radius a = 4:8 m, L = 8:8 m (L=a = 1:83), the wall "r = 5:5 and  = 0:03 S=m. Field measurements were taken down the tunnel for different frequencies and polarizations. The attenuation of the dominant HE11 mode was then measured for both horizontal and vertical polarization at the frequencies 150, 470, 900, 1700, and 4000 MHz. We plot the predicted attenuation of the horizontally polarized HE11 mode in this same tunnel using both the present perturbation analysis and the rectangular tunnel model in Fig. 8. On top of these curves, the measured attenuation are shown as discrete dots at the above selected frequencies. The predicted attenuation shows the expected inverse f 2 dependence. The measured attenuation follows well the predicted attenuation except at the highest two frequencies (1700 and 4000 MHz) for which it is higher than predicted. This may be explained, however, on account of wall roughness or micro-bending of the tunnel walls that affect the higher frequencies in particular [8]. VII. CONCLUSION We have presented a perturbation analysis for evaluating the propagation character of the dominant modes in uniform tunnels with cross

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Numerical Evaluation of the Scattering of Brillouin Precursors From Targets Inside Water Reza Safian

Abstract—In a dispersive medium excited by a wideband pulse, the appearance of the steady-state part of the propagated signal is preceded by oscillations known as precursors. Precursor fields in lossy Debye media have been shown to present a sub-exponential attenuation rate, thus becoming good candidates for applications requiring field penetration into such media. This communication aims to study the performance of a pulse consisting of two mutually delayed precursors for the detection of dielectric objects inside triply distilled water. Finite-difference time-domain (FDTD) simulations are employed to evaluate the strength of the scattered field obtained by illuminating the targets with the precursor-based pulse excitation and to compare it with other conventional alternatives, namely, Gaussian, rectangular and sinusoidal pulses. Fig. 8. Attenuation of the HE mode in the Japanese National Railway tunnel by the perturbation analysis and the rectangular tunnel model versus measured values (as reported in [8]).

Index Terms—Dispersive media, finite-difference time-domain (FDTD) methods, underwater communication, wave propagation.

I. INTRODUCTION sections that deviate from the circular shape. Simple forms are thus obtained for the propagation parameters of the dominant modes in a perturbed circular tunnel with a flat base. It has been shown that the HE11 mode with vertical polarization suffers more attenuation than the same mode with horizontal polarization. This result has suggested that we seek an equivalent rectangular tunnel that would match this behavior. Comparisons are made with measured attenuation of the dominant mode in the Massif Central tunnel of south central France and in Tuhoku National Railways tunnel in Japan. These comparisons support the validity of the perturbation analysis as well as the proposed equivalent rectangular tunnel model as means of predicting the attenuation of the dominant modes in the perturbed circular tunnel.

REFERENCES [1] D. G. Dudley and S. F. Mahmoud, “Linear source in a circular tunnel,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2034–2048, Jul. 2006. [2] D. G. Dudley, M. Lienard, S. F. Mahmoud, and P. Degauque, “Wireless propagation in tunnels,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 11–26, Apr. 2007. [3] S. F. Mahmoud, “Modal propagation of high frequency electromagnetic waves in straight and curved tunnels within the Earth,” J. Electromagn. Waves Applicat., vol. 19, pp. 1611–1627, 2005. [4] J. I. Glaser, “Attenuation and guidance of modes in hollow dielectric waveguides,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, pp. 173–174, Mar. 1969. [5] J. Pingenot, R. N. Rieben, D. A. White, and D. G. Dudley, “Full wave analysis of RF signal attenuation in a lossy rough surface cave using high order time domain vector finite element method,” J. Electromagn. Wave Applicat., vol. 20, no. 12, pp. 1695–1705, 2006. [6] S. F. Mahmoud, Electromagnetic Waveguides, Theory and Applications, ser. IEE Electromagnetics Series 32. London, U.K.: Peter Peregrinus, 1991, sec. 3.4. [7] R. F. Harrington, Time Harmonic Electromagnetic Fields. New York: McGraw Hill, 1961, sec. 3-8. [8] A. G. Emslie and R. L. Lagace, “Theory of the propagation of UHF radio waves in coal mine tunnels,” IEEE Trans. Antennas Propag., vol. AP-23, no. 2, pp. 192–205, 1975. [9] J. Chiba, T. Inaba, Y. Kuwamoto, O. Banno, and R. Sato, “Radio communication in tunnels,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 6, pp. 439–443, 1978.

The propagation of ultrawideband electromagnetic pulses through a causal, temporally dispersive dielectric has been a topic of research interest for many years [1]–[3]. A focal point of this research is the reshaping that a pulse undergoes due to the effects of dispersion and attenuation. These result in the evolution of the so-called precursor fields, which precede the main part of a pulse propagating in a causal dielectric. For Debye-type dielectrics in particular, a rectangular pulse excitation with temporal support T , modulated by a carrier signal of fre1=fc , is reshaped into a pair of Brillouin prequency fc , where T cursors [3], as it propagates within the medium. Interestingly, the peak amplitude of the Brillouin precursors decays as the inverse square root of the propagation distance, a clearly sub-exponential rate. Based on this observation, it was recently proposed that a pulse excitation consisting of a pair of appropriately defined Brillouin precursors (henceforth referred to as “double Brillouin pulse”) is “near optimal” in the sense that it produces a propagated field which attenuates with the low attenuation rate of the Brillouin precursor [4]. Numerical results attesting to the validity of this claim were provided in [4]. This development inspires the possibility of employing precursortype shape pulses for the detection of objects inside lossy dielectrics, such as moist soil, human tissue and water itself. This communication provides the first study of Brillouin precursor scattering from dielectric objects inside a Rocard-Powles-Debye dielectric with properties that correspond to those of triply distilled water. Comparisons of the scattered field strength obtained for different positions of the targets inside the dielectric with a double Brillouin pulse, Gaussian, rectangular and sinusoidal excitations of the same initial energy illustrate the advantages of using of the double Brillouin waveform.



Manuscript received January 28, 2009; revised April 15, 2009. First published December 04, 2009; current version published February 03, 2010. The author is with Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 84156, Iran (e-mail: [email protected]. ir). 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.2009.2037710

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Fig. 1. The geometry under study. The dashed line shows the interface between water and free space. The letter “R” is for receiver and “T” is for transmitter.

II. PROBLEM STATEMENT AND COMPUTATIONAL MODELING Fig. 1(a) shows the geometry under study. A transmitter (T)—receiver (R) pair is located above the air-water interface, similar to the ground penetrating radar geometry of [5]. The transmitter is a stationary source, located at a distance z D from a square target of side d, in free space. The receiver moves parallel to the interface at the same . z -coordinate as the transmitter. The target is dielectric of r The complex index of refraction of the medium follows the RocardPowles-Debye model, given by [4]

1+

= 10

(s 0 1 ) ( ) = 1 + (1 0 i! )(1 0 i!f )

n !

1=2

:

(1)

C = 4 62 100 s

= 76 2 = 8 44 100 s

(

= 0)

=

= 1 GHz =

=

= 500

40

The complex wave number is given by

~( ) = (!) + i(!) = ! n(!) c

k !

In (1),  is the relaxation time with an associated friction time f . The are: values of the parameters in (1) for triply-distilled water at 25 12 14  : , s : , : 2 , f : 2 [4]. It is assumed that the carrier frequency is away from the resonant frequency for this model. Two dimensional FDTD method on the xz -plane is used to simulate the propagation and scattering of the double Brillouin pulse. The excitation is a point source, which radiates omnidirectionally and has polarization. It is in the middle transverse electric Ey ; Hx ; Hz 6 of the computational domain along the x-axis, 0 above the interc=fc is the free space wavelength where face along the z -axis (0 fc ). We have used the space and time discretization paz 0 = , t sx =c, where s : is the rameters: x Courant stability number. Both free space and the dispersive medium are semi-infinite, truncated by 10 cell perfectly matched layers (PML) along the x and z directions. In the following, this computational dox , lz z ) is excited with double Brillouin as main (lx well as other pulses of equal energy, in order to evaluate and compare the scattered field from the target in each case.

1 = 21

Fig. 2. (a) The time distribution of the double Brillouin pulse (b) the frequency distribution of the double Brillouin pulse.

( ) = ~( ) ( ) = ~( )

where, !