IEEE Transactions on Antennas and Propagation [volume 59 number 9]

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SEPTEMBER 2011

VOLUME 59

NUMBER 9

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Size Reduced Multi-Band Printed Quadrifilar Helical Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Rabemanantsoa and A. Sharaiha Engineering the Input Impedance of Optical Nano Dipole Antennas: Materials, Geometry and Excitation Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. P. G. de Arquer, V. Volski, N. Verellen, G. A. E. Vandenbosch, and V. V. Moshchalkov Electromagnetic Resonances of a Straight Wire on an Earth-Air Interface . . . . . . . . . . . . . . J. M. Myers, S. S. Sandler, and T. T. Wu Gain Enhancement by Dielectric Horns in the Terahertz Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Andres-Garcia, E. Garcia-Muñoz, S. Bauerschmidt, S. Preu, S. Malzer, G. H. Döhler, L. J. Wang, and D. Segovia-Vargas Part I: A New Theory for Modeling Conductor-Backed Planar Slot Antenna Elements, in the Presence of a General Non-Planar Surrounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. L. McCabe and N. K. Das Part II: A Conductor-Backed Slot Antenna Element Surrounded by a Shorting-Post Cavity to Suppress Parallel-Plate Mode Excitation—Design Analysis and Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. L. McCabe and N. K. Das Modal Analysis of Dielectric-Filled Rectangular Waveguide With Transverse Slots . . . . . . . . . J. Liu, D. R. Jackson, and Y. Long Arrays Very Small Footprint 60 GHz Stacked Yagi Antenna Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. Kramer, T. Djerafi, and K. Wu Using Bayesian Inference for Linear Antenna Array Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.-Y. Chan and P. M. Goggans Performance of Planar Slotted Waveguide Arrays With Surface Distortion . . . . . . . . . . . L. Song, B. Duan, F. Zheng, and F. Zhang An Effective Synthesis of Planar Array Antennas for Producing Near-Field Contoured Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-T. Chou, N.-N. Wang, H.-H. Chou, and J.-H. Qiu Time-Harmonic Echo Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Capozzoli, C. Curcio, and A. Liseno Synthesis of Sub-Arrayed Time Modulated Linear Arrays Through a Multi-Stage Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Rocca, L. Poli, G. Oliveri, and A. Massa A Transmit-Receive Reflectarray Antenna for Direct Broadcast Satellite Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. A. Encinar, M. Arrebola, L. F. de la Fuente, and G. Toso An Active Annular Ring Frequency Selective Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. S. Taylor, E. A. Parker, and J. C. Batchelor Numerical Techniques Isotropic Spatial Filters for Suppression of Spurious Noise Waves in Sub-Gridded FDTD Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Zadehgol and A. C. Cangellaris High-Order Split-Step Unconditionally-Stable FDTD Methods and Numerical Analysis . . . . . . . . . . . . Y.-D. Kong and Q.-X. Chu Modeling of Sloped Interfaces on a Yee Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. M. Shyroki Incorporating the G-TFSF Concept into the Analytic Field Propagation TFSF Method . . . . . . . . . . . . J. B. Schneider and Z. Chen A New Boundary Closure Scheme for the Multiresolution Time-Domain (MRTD) Method . . . . . . . . . . . . . . . . . P. Yao and S. Zhao

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(Contents Continued on p. 3137)

(Contents Continued from Front Cover) The Magnetostatic Frill Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Simpson Generation of Excitation-Independent Characteristic Basis Functions for Three-Dimensional Homogeneous Dielectric Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Laviada, M. R. Pino, and F. Las-Heras Integral Equation Based Domain Decomposition Method for Solving Electromagnetic Wave Scattering From Non-Penetrable Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Z. Peng, X.-C. Wang, and J.-F. Lee Scattering and Absorption of Waves by Flat Material Strips Analyzed Using Generalized Boundary Conditions and Nystrom-Type Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. V. Shapoval, R. Sauleau, and A. I. Nosich Adaptive Aperture Partition in Shooting and Bouncing Ray Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. B. Tao, H. Lin, and H. J. Bao Wave Scattering and Propagation On the Relation Between Optimal Wideband Matching and Scattering of Spherical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Nordebo, A. Bernland, M. Gustafsson, C. Sohl, and G. Kristensson A New Solution for Characterizing Electromagnetic Scattering by a Gyroelectric Sphere . . . . . . . . . . . . J. L.-W. Li and W.-L. Ong Exact Complex Source Point Beam Expansions for Electromagnetic Fields . . . . . . . . . K. Tap, P. H. Pathak, and R. J. Burkholder Scattering by an Eccentrically Loaded Cylindrical Cavity With Multiple Slits . . . . . . . . S. Seran, J. P. Donohoe, and E. Topsakal Controllable Metamaterial-Loaded Waveguides Supporting Backward and Forward Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.-Y. Meng, Q. Wu, D. Erni, and L.-W. Li Anomalous Negative Group Velocity in Coupled Positive-Index/Negative-Index Guides Supporting Complex Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Mirzaei, R. Islam, and G. V. Eleftheriades

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COMMUNICATIONS

The Half-Width Microstrip Leaky Wave Antenna With the Periodic Short Circuits . . . . . . . Y. Li, Q. Xue, H.-Z. Tan, and Y. Long An Electronically Reconfigurable Microstrip Antenna With Switchable Slots for Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. S. Nishamol, V. P. Sarin, D. Tony, C. K. Aanandan, P. Mohanan, and K. Vasudevan A Novel Hexa-Band Antenna for Mobile Handsets Application . . . . . . . . . . . . . . . . . . . . . . . . . C.-M. Peng, I.-F. Chen, and C.-T. Chien Design of Single-Feed Dual-Frequency Patch Antenna for GPS and WLAN Applications . . . . . . . . . . . . . . S.-L. Ma and J.-S. Row A Methodology for the Design of Frequency and Environment Robust UHF RFID Tags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Chaabane, E. Perret, and S. Tedjini A Planar UWB Patch-Dipole Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. K. Toh, X. Qing, and Z. N. Chen Compact Dielectric Resonator Antennas With Ultrawide 60%–110% Bandwidth . . . . . . . . . . . Y. Ge, K. P. Esselle, and T. S. Bird Zeroth-Order Resonator Antennas Using Inductor-Loaded and Capacitor-Loaded CPWs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.-P. Lai, S.-C. Chiu, H.-J. Li, and S.-Y. Chen Corrugated-Conical Horn Analysis Using Aperture Field With Quadratic Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Densmore, Y. Rahmat-Samii, and G. Seck High-Strength, Metalized Fibers for Conformal Load Bearing Antenna Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. E. Morris, Y. Bayram, L. Zhang, Z. Wang, M. Shtein, and J. L. Volakis A Compact MIMO Array of Planar End-Fire Antennas for WLAN Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.-D. Capobianco, F. M. Pigozzo, A. Assalini, M. Midrio, S. Boscolo, and F. Sacchetto Axial Ratio Enhancement for Circularly-Polarized Millimeter-Wave Phased-Arrays Using a Sequential Rotation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. B. Smolders and U. Johannsen Gain-Enhanced 60-GHz LTCC Antenna Array With Open Air Cavities . . . . . . . . . . . . . . . . . . . . S. B. Yeap, Z. N. Chen, and X. Qing Weighted Thinned Linear Array Design With the Iterative FFT Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. P. du Plessis Robust Beamforming With Magnitude Response Constraints Using Iterative Second-Order Cone Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Liao, K. M. Tsui, and S. C. Chan Performance Improvement of a U-Slot Patch Antenna Using a Dual-Band Frequency Selective Surface With Modified Jerusalem Cross Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-Y. Chen and Y. Tao Numerical Reflection Coefficients of ID-FDTD Scheme for Planar Dielectric Interface . . . . . . . . . . . . . . . . . . . P. Deng and I.-S. Koh Multilevel Fast Multipole Algorithm-Based Direct Solution for Analysis of Electromagnetic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Z. Jiang, Y. Sheng, and S. Shen Application of Multiplicative Regularization to the Finite-Element Contrast Source Inversion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Zakaria and J. LoVetri Statistical Characterization of Medium Wave Spatial Variability Due to Urban Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U. Gil, I. Peña, D. Guerra, D. de la Vega, P. Angueira, and J. L. Ordiales

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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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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (ISSN 0018-926X) is published monthly by the Institute of Electrical and Electronics Engineers, Inc. Responsibility for the contents rests upon the authors and not upon the IEEE, the Society/Council, or its members. IEEE Corporate Office: 3 Park Avenue, 17th Floor, New York, NY 10016-5997. IEEE Operations Center: 445 Hoes Lane, Piscataway, NJ 08854-4141. NJ Telephone: +1 732 981 0060. Price/Publication Information: Individual copies: IEEE Members $20.00 (first copy only), nonmembers $107.00. (Note: Postage and handling charge not included.) Member and nonmember subscription prices available upon request. Copyright and Reprint Permissions: Abstracting is permitted with credit to the source. Libraries are permitted to photocopy for private use of patrons, provided the per-copy fee indicated in the code at the bottom of the first page is paid through the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923. For all other copying, reprint, or republication permission, write to Copyrights and Permissions Department, IEEE Publications Administration, 445 Hoes Lane, Piscataway, NJ 08854-4141. Copyright © 2011 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved. Periodicals Postage Paid at New York, NY and at additional mailing offices. Postmaster: Send address changes to IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE, 445 Hoes Lane, Piscataway, NJ 08854-4141. GST Registration No. 125634188. CPC Sales Agreement #40013087. Return undeliverable Canada addresses to: Pitney Bowes IMEX, P.O. Box 4332, Stanton Rd., Toronto, ON M5W 3J4, Canada. IEEE prohibits discrimination, harassment, and bullying. For more information visit http://www.ieee.org/nondiscrimination. Printed in U.S.A.

Digital Object Identifier 10.1109/TAP.2011.2166711

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 9, SEPTEMBER 2011

Size Reduced Multi-Band Printed Quadrifilar Helical Antenna Josh Rabemanantsoa and Ala Sharaiha, Senior Member, IEEE

Abstract—A simple and innovative method for designing a spiral folded printed quadrifilar helix antenna (S-FPQHA) for dual-band operations is presented. The axial length of a conventional PQHA is miniaturized of about 43% by meandering and turning the helix arms into the form of square spirals. Parametric studies are performed to explore the performance improvements. Based on the studies a dual band antenna working in L1/L5 GPS application is realized with a good gain and a good circular polarization. Measured results are presented to validate the concept. Index Terms—Circular polarization, dual-band antenna, folded antenna, GPS application, miniaturization, printed quadrifilar helical antenna.

I. INTRODUCTION ATELLITE communication systems have been widely used in recent years such as in broadcasting satellite systems or global positioning systems [1]. For these systems, circular polarization antennas are well suited for channel transmission due to their insensitivity to ionospheric polarization rotation. More precisely, for global positioning systems, the link between ground antenna receivers and satellites requires antennas that can provide right hand circular polarization and uniform pattern coverage over approximately the entire upper hemisphere. A very attractive candidate for all these applications is the resonant quadrifilar helical antenna (QHA) [2], [3] and more recently the conventional printed quadrifilar helical antenna (RefPQHA) [4] due to their performance in terms of circular polarization, good axial ratio, light weight, high dimensional stability, ease of fabrication and low cost. Practically, the PQHA is composed of four parallel arms printed on a thin dielectric substrate wrapped around a cylinder and fed in phase quadrature on each helix (see Fig. 1(a) and (c)). Although the Ref-PQHA structure is already small, further size reduction is necessary to satisfy the space limitations of GPS terminal. Several methods to reduce the axial length have already been developed and presented, using an integrated feeding network [5], introducing a dielectric rod inside the helix [6]–[8], employing variable pitch angles on the wires [9], winding the arms

S

Manuscript received May 03, 2010; revised December 17, 2010; accepted February 09, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Institute of Electronic and de Telecommunication of Rennes, Université de Rennes I, 35042 Rennes Cedex, France (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161436

Fig. 1. The unwrapped Ref-PQHA (a) and S-FPQHA (b) and 3D representation Ref-PQHA (c) and S-FPQHA (d).

according to a sinusoidal profile [10], [11], meandering the radiating elements [12]–[15], through the folding line technique or using coupled segment [16], [17], and via insertion of gaps [18]. The main limitation of these antennas is the narrow impedance bandwidth and therefore insufficiency to meet requirements of new antenna communication applications. However, recently some new antenna configurations have been reported which can provide dual band operations. These include integrated switching circuits [19] or using pin diodes [20], impedance matching in two frequency bands [21], arrangements of collocated antennas [2], arm loaded with lumped elements [22], a double layer structure [23], [24], and antennas with several resonators [25], [26]. In this paper, we present a novel design method to reduce the size of the Ref-PQHA of about 43% and to obtain dual band even multiband behavior with a good gain and circular polarization. The dual band operation is achieved by meandering and turning the helix arms into the form of a square spirals to obtain the Spiral Folded Printed Quadrifilar Helix Antenna called S-FPQHA. A parametric study of the spacing between each strip is achieved to show the influence of mutual coupling between arms on the impedance matching. The commercial simulation

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TABLE II PHYSICAL CHARACTERISTICS OF THE S-FPQHA

Le = arm length; Le = a + b + c + d + e + f + g + h

Fig. 2. Representation of meandering and turning helix arms.

TABLE I GEOMETRICAL PARAMETERS OF REF-PQHA AND S-FPQHA

Fig. 3. Comparison of the reflection coefficient versus frequency between RefPQHA and S-FPQHA at F = 1 to 2 GHz.

Lax = axial length; Le = arm length; R = radius;  = pitch angle; wa = arm width; mm = millimeter; N = number of turn; = degree. tool HFSS [27] has been used in this study. The results allow one to find an optimal design which has been realized and measured. We provide details of the proposed antenna design operating in L-band for L1/L5 GPS applications, as well as the experimental results.

The arm length (Le) of the S-FPQHA is equal to for an operating frequency of 1.26 GHz whereas the Ref-PQHA operating at the same frequency, where is is equal to the free space wavelength. These antennas were modeled using HFSS code [44]. In Fig. 3, where the comparison of the active reflection coefficient versus frequency of the S-FPQHA and the Ref-PQHA are presented, we can note a multi-resonance behavior corre. sponding to the arm length which is a multiple of

II. ANTENNA STRUCTURE The configuration of the Ref-PQHA and the proposed S-FPQHA is shown in Fig. 1, where the four helix-shaped radiating elements are printed on a thin dielectric substrate [Fig. 1(a) and (b)], wrapped around a cylindrical support, and mounted on a small ground plane [Fig. 1(c) and (d)]. Each helix element consists of meandering and turning the helix arms into the form of square spirals (see on Fig. 2). The antennas with folded arms tend to resonate at frequencies much lower than single antenna elements of equal length. By meandering and tuning the helix arms into the form of square spirals, we increase the effective length of each arm in order to shorten its overall element height. The physical characteristic of the S-FPQHA is given in Table I and Table II using the same radius R, arm width Wa, pitch angle and number of turns N as the Ref-PQHA.

The resonance frequency (see in Table III), can be easily calculated using the following relation:

where is the effective permittivity of the dielectric substrate and ( 1.15 for a thin substrate of relative permittivity of thickness [3]). For the S-FPQHA, we obtain a small frequency shift regarding this expression due to the mutual coupling in the folded arm. The radiation pattern of the S-FPQHA displays almost identical characteristics to the radiation pattern of the Ref-PQHA at 1.26 GHz (Fig. 4(a)). For the higher frequencies, the radiation pattern of S-FQPHA remains in axial mode shown

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TABLE III RESONANT FREQUENCY (GHz) OF REF-PQHA AND S-FPQHA

Fig. 4. Simulated radiation pattern versus frequency of Ref-PQHA and S-FPQHA at (a) F = 1:26 GHz, (b) F = 1:85 GHz and (c) F = 2:2 GHz.

TABLE IV SIMULATED RESULT OF S-FPQHA

in Fig. 4(b) and Fig. 4(c) whereas the Ref-PQHA radiation pattern tends to a normal mode. The performances of this antenna at different resonant frequencies are resumed in Table IV and compared to the RefPQHA operating at 1.26 GHz. III. PARAMETRIC STUDY In this section, we study the effect of the coupling in the folded arm by the variation of “DA” between the portions “c” and “g”, the variation of “DD” between the portions “g” and “e” and the variation of width “Wg”of “g”. We focus our analysis on the resonance frequencies F2 and F3. A. Effect of the Spacing Between the Portions “c” and “g” We consider “DA” as the spacing between the segments “c” and “g”. Several values of DA were used in simulations (

Fig. 5. Simulated reflection coefficient versus frequency of the S-FPQHA for different values of DA.

, 4, 6, 8, and 10 mm) while the other dimensions were fixed and . with values of In Fig. 5(a), we present the reflection coefficient versus frequency for different values of DA. We can note that the increase of “DA” causes a decrease of the resonance frequencies F2 and F3 mainly due to the lengthening of the total arm length. For DA greater than 8 mm the antenna is no more matched in the F3 band. As for the radiation pattern seen in Fig. 5(b), we observe that when DA increases the level of the LHCP, at the resonance frequencies in F3 band, decreases in the beamwidth from a maximum of 15 dB to a maximum of 18 dB for which corresponds to an axial ratio going from 3.3 dB to 2 dB respectively. Otherwise the co-polar remains stable. B. Effect of the Spacing Between the Portions “g” and “e” Here, we consider “DD” as the spacing between the segments , “g” and “e”. Several value of “DD” have been tested ( 6.5, 8.5, 10.5 and 12.5 mm) where the other dimensions are fixed and . with a The reflection coefficient versus frequency is shown in Fig. 6, for different values of DD. We can observe the dual-band behavior and a decreasing of the impedance variation in F3 band. For DD greater than 8 mm the antenna is no more matched in the F3 band. Moreover, the radiation pattern in the F3 band remains stable as well as the level of the LHCP.

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Fig. 8. The unwrapped spiral folded helix (a) and a picture of the prototype.

Fig. 6. Simulated reflection coefficient versus frequency of the S-FPQHA for different values of DD.

Fig. 9. Measured and simulated reflection coefficient versus frequency of S-FPQHA-L1L5.

Fig. 7. Simulated reflection coefficient versus frequency of the S-FPQHA for different values of Wg.

TABLE V PHYSICAL CHARACTERISTICS OF THE PROTOTYPE OF THE S-FPQHA

Le = arm length; Le = a + b + c + d + e + f + g + h C. Effect of the Width of Portion “g”: Wg To modify the mutual coupling between the portion c, g and e we choose to vary the width of the g portion. Several values of Wg have been tested ( , 4, 6, 8, 10 and 12.5 mm) where , the other dimensions are fixed with and . In Fig. 7, we show the reflection coefficient versus frequency. We note that the main effect of the increasing of Wg is in the F3 band where we can obtain a good impedance matching without affecting the radiation pattern. For Wg equal to 12 mm, we have

a good impedance matching in the bandwidth and the coupling loop in the Smith chart becomes more centered. IV. EXPERIMENTAL RESULTS In order to validate the previous method, a single prototype of S-FPQHA-working in L1/L5 bands for GPS applications had been realized. We increased the arms length of the previous structure, presented in Section II, to adjust the first resonant frequency so as to cover L5 band. The arm length (Le) of the S-FPQHA is now equal to 300 mm and the height (Lax) is equal to 77 mm. The other dimensions are then modified (see Table V) using the parametric study results to cover the L1 frequency band and to obtain a good impedance matching. This antenna is printed on a thin Neltec substrate of relative and a thickness of 0.127 mm. The final permittivity configuration of the antenna, assembled on a circular disk of 3 mm of thickness and 25 mm of radius used as a ground plane, is shown in Fig. 8. The circular disk is metalized only from the side of the connectors and isolated from the other side. In Fig. 9, the measured reflection coefficient versus frequency of the SFPQHA is presented and compared to the simulation results of the antenna with the circular disk and with a thin ground plane. As expected, we obtain multiband behavior. We , can note three resonant measured frequencies: , and . The bandwidth is about 2.7% to 4.7% for the first two frequencies and 3, 6% MHz for the third. We can also note that the use of a thin ground plane will shift and adjust the resonant frequencies in the GPS L1/L5 bands.

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Fig. 11. 2D Measured axial ratio versus frequency of S-FPQHA-L1L5 (a) F =

1:14 GHz, (b) F = 1:54 GHz.

Fig. 12. Measured maximum gain versus frequency of S-FPQHA-L1L5.

Fig. 10. Comparison of measured and simulated radiation patterns in circular polarization of S-FPQHA-L1L5 at (a) F = 1:14 GHz, (b) F = 1:54 GHz and (c) F = 1:84 GHz.

The far-field radiation patterns of magnitude and phase for linear polarizations were measured in an anechoic chamber and combined to give the circular polarization. For this, we use a classical commercial feeding system composed of three hybrid couplers such as a hybrid coupler (3 dB, 90 ) and two hybrid couplers (3 dB, 180 ). The measured radiation patterns in circular polarization compared to the simulated one are shown in Fig. 10.

We can note that the radiation patterns remain stable with frequency. There is an obvious discrepancy for the cross polarization (LHCP) component. This could be due on the one hand to fabrication errors to maintain arms symmetry which is important especially in high frequency and on the other hand to phase and amplitude errors of the commercial hybrid couplers used. and In Fig. 11 we show the 2D axial ratio at . We notice that the axial ratio is less than 2 dB in a large beamwidth (160 ) and in all the directions of the upper hemisphere. The measured maximum gain is greater than 1.5 dBic in the three frequency bands, as shown in Fig. 12. The performances of these antennas are resumed in Table VI.

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TABLE VI MEASURED RESULT OF S-FPQHA-L1L5

V. CONCLUSION In this paper, a new S-FPQHA design is presented. The resonant frequencies of the SFPQHA are analyzed and compared with the calculated resonant frequency of the Ref-PQHA. The parametric study shows how to optimize the antenna geometry to control the multiband behavior as well as the impedance matching and maintaining the antenna performances in term of gain, axial ratio and radiation pattern stability. A prototype of the S-FPQHA operating in L-band is realized. It provides a 43% size reduction for the axial length compared to a conventional PQHA, to satisfy space limitations of a mobile satellite terminal, with good performances. These properties make the antenna a good candidate to cover the dual frequency bands of global positioning systems.

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[14] Y. Letestu, A. Sharaiha, and P. Besnier, “A size reduced configurations of printed quadrifilar helix antenna,” in Proc. Antenna Technology: Small Antennas and Novel Metamaterials, 2005, pp. 326–328. [15] B. Bhandari, S. Gao, and T. Brown, “Meandered variable pitch angle printed quadrifilar Helix antenna,” in Proc. Loughborough Antennas and Propagation Conf., 2009, pp. 325–328. [16] A. Petros and S. Licul, ““Folded” quadrifilar helix antenna,” in Proc. Antennas and Propagation Society Int. Symp., Boston, 2001, vol. 4, pp. 569–572. [17] D. F. Filipovic, M. A. Tassoudji, and E. Ozaki, “A coupled-segment quadrifilar helical antenna,” IEEE MTT Wireless Applicat., pp. 43–46. [18] M. Amin and R. Cahill, “Compact quadrifilar helix antenna,” Electron. Lett., vol. 41, no. 12, pp. 672–674, Jun. 2005. [19] R. A. Sainati, J. J. Groppelli, R. C. Olesen, and A. J. Stanland, “A band switched resonant quadrifilar helix,” IEEE Trans. Antennas Propag, vol. 30, no. 5, pp. 1010–1013, Sep. 1982. [20] R. C. Olesen, R. A. Sainati, J. J. Groppelli, and A. J. Stanland, “Quadrifilar helix antenna tuning using pin diodes,” U.S. patent 4554554, Nov. 1985. [21] M. Hosseini, M. Hakkak, and P. Rezaei, “Design of dual-band quadrifilar helix antenna,” IEEE Antennas Wireless Propag. Lett., vol. 15, no. 12, pp. 39–42, Apr. 2005. [22] D. Lamensdorf and M. A. Smolinski, “Dual band quadrifilar helix antenna,” in Proc. IEEE Int. Symp. Antennas and Propagation, Jun. 2002, vol. 3, pp. 488–491. [23] A. Sharaiha and C. Terret, “Overlapping quadrifilar resonant helix antenna,” Electron. Lett., vol. 26, pp. 1090–1092, 1990. [24] D. Grybos and C. C. Hung, “Multi-Band concentric helical antenna,” U.S. patent 5986619, Nov. 1999. [25] K. Tanabe, “Method of producing a helical and the helical antenna apparatus,” U.S. patent 6072441, Jun. 2000. [26] M. Ohgren and S. Johansson, “Dual frequency quadrifilar helix antenna,” U.S. patent 6421028, Jul. 16, 2002. [27] High Frequency Structure Simulator v. 11.0 Ansoft Corp., 2008 [Online]. Available: www.ansoft.com

REFERENCES [1] J. V. Evans, “Satellite systems for personal communications,” IEEE Trans. Antennas Propag. Mag., vol. 39, pp. 7–20, 1997. [2] J. M. Tranquilla and S. R. Best, “A study of the quadrifilar helix antenna for global positioning system (GPS) applications,” IEEE Trans. Antennas Propag., vol. 38, pp. 1545–1550, 1990. [3] C. C. Kilgus, “Resonant quadrifilar helix design,” Microw. J., vol. 13, no. 12, pp. 49–54, Dec. 1970. [4] A. Sharaiha and C. Terret, “Analysis of quadrifilar resonant printed helical antenna for mobile communications,” IEE Proc. H Microw. Antennas Propag. , vol. 140, no. 4, pp. 269–273, 1993. [5] A. Sharaiha, C. Terret, and J. P. Blot, “Printed resonant helix antenna with integrated feeding network,” Electron. Lett., vol. 33, pp. 265–257, 1997. [6] O. Leinsten, J. C. Vardaxoglou, P. Mcevoy, R. Seager, and A. Wingfield, “Miniaturized dielectrically-loaded quadrifilar antenna for global positioning system (GPS),” Electron. Lett., vol. 37, no. 22, Oct. 2001. [7] B. Desplanches, A. Sharaiha, and C. Terret, “Parametrical study of printed quadrifilar helical antennas with central dielectric rods,” Microw. Opt. . Tech. Lett., vol. 20, pp. 932–933, 1999. [8] Y.-S. Wang and S.-J. Chung, “A miniature quadrifilar helix antenna for global positioning satellite reception,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3746–3751, 2009. [9] J. Louvigné and A. Sharaiha, “Synthesis of printed quadrifilar helical antenna,” Electronic Lett., vol. 37, pp. 271–272, 2001. [10] M. E. Ermutlu, “Modified quadrifilar helical antennas for mobile satellite communication,” in Proc. IEEE AP-S for Wireless Commun., 1998, pp. 141–144. [11] A. Takacs, N. J. G. Fonseca, H. Aubert, and X. Dollat, “Miniaturization of quadrifilar helix antenna for VHF band applications,” in Proc. Loughborough Antennas & Propagation Conf., 2009, pp. 597–600. [12] M. G. Ibambe, Y. Letestu, and A. Sharaiha, “Compact printed quadrifilar helical antenna,” Electron. Lett., vol. 43, no. 13, pp. 697–698, 2007. [13] D. K. C. D. Chew and S. R. Saunders, “Meander line technique for size reduction of quadrifilar helix antenna,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 109–111, 2002.

Josh Rabemanantsoa was born in Fianarantsoa, Madagascar. He received the Master degree in telecommunications from the University of Rennes 1, France, in 2005, where he is currently working toward the Ph.D. degree. His research interests include antennas for mobile communication systems

Ala Sharaiha (SM’09) received the Ph.D. and Habilitation à Diriger la Recherche (HDR) degrees in telecommunication from the University of Rennes 1, France, in 1990 and 2001, respectively. Currently, he is a Professor at the University of Rennes 1, Rennes, and a Researcher in the Antennas and High Frequency Group, Institute of Telecommunication of Rennes, where he is responsible of a research theme concerning the new concepts and architecture of antennas design. Main research activities include broadband and UWB antennas, miniaturization, printed spiral and helical antennas, antennas for mobile communications, etc. He conducted and involved in more than 15 development projects for private companies and participates in the European Network of Excellence ACE (Antenna Center of Excellence) in the small antenna WP. He is the author and coauthor of 35 international papers, 105 conference presentations and holds ten European patents. Prof. Sharaiha is a reviewer for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, the IET Letters and the IET Microwave Antennas Propagation. He was the conference Chairman of the 11th International Canadian Conference ANTEM (Antenna Technology and Applied Electro- Magnetics), held at Saint-Malo in France, 2005.

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Engineering the Input Impedance of Optical Nano Dipole Antennas: Materials, Geometry and Excitation Effect F. Pelayo García de Arquer, Vladimir Volski, Niels Verellen, Guy A. E. Vandenbosch, and Victor V. Moshchalkov

Abstract—An optical nano dipole antenna is analyzed by means of its input impedance as well as the matching properties of the antenna topology and material configuration. A comparison of this classical microwave driving method with plane wave excitation is accomplished, contrasting the resonances in the input impedance and optical cross sections for several setups, and analyzing the spectral response shape. It is found that for all structures analyzed, a simple linear expression can be defined characterizing the relation between total dipole length and resonant wavelength. The fact that this linear relationship remains valid for different excitation models, for most widely used antenna materials (Au, Ag, Cu, and Al) and even in the presence of substrates is important with respect to practical designs. To our knowledge, such an extensive study has not been performed before. Index Terms—Optical dipole antennas, plasmons, resonance, scattering.

I. INTRODUCTION

T

HE quest for miniaturization, integration, higher compactness, and speed in circuit devices, is leading towards ever higher and higher frequency regimes. This trend gave way from microwave to terahertz, and more lately to optical frequency regimes. However, metal dispersive properties at optical frequencies present a new challenge in the electromagnetic modeling of such devices, as a direct scaling from lower frequency bands is no longer possible and thus a proper review is required. In these regimes the interaction of free light with the electron bulk sea in the metal’s surface gives Manuscript received July 09, 2010; revised January 19, 2011; accepted January 31, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the Flemish government through the Methusalem framework and in part by the Fonds Wetenschappelijk Onderzoek Vlaanderen (FWO) through several research projects. F. P. G. de Arquer was with the Katholieke Universiteit Leuven, ESAT-TELEMIC, 3001 Leuven, Belgium. He is now with the Institute of Photonic Sciences, 08860 Barcelona, Spain (e-mail: [email protected]). V. Volski, and G. A. E. Vandenbosch are with the Katholieke Universiteit Leuven, ESAT-TELEMIC, 3001 Leuven, Belgium (e-mail: [email protected]; [email protected]). N. Verellen is with the Katholieke Universiteit Leuven, ESAT-TELEMIC; Institute for Nanoscale Physics and Chemistry, Nanoscale Superconductivity and Magnetism and Pulsed Fields Group; IMEC, 3001, Leuven, Belgium (e-mail: [email protected]). V. V. Moshchalkov is with the Institute for Nanoscale Physics and Chemistry, Nanoscale Superconductivity and Magnetism and Pulsed Fields Group, 3001 Leuven, 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.2011.2161544

rise to a kind of waves named plasmon polaritons. Because of their subwavelength and non diffraction limited nature [1], [2] they may play a key role in compact nano circuitry, allowing the possibility of handling light with structures much smaller than a wavelength. Since the systems supporting surface plasmons are also electrical conductors, the integration with photonic circuitry will be facilitated. This is viewed as a major breakthrough in electric-photonic nanocircuitry [3]–[7]. The necessity of interconnecting all of such circuit constitutive elements implies the analysis of waveguides and antennas, among others [8]–[20], resembling the state of the art in microwave technology decades ago. As an example of the failure in direct scaling from radio, microwave and terahertz regimes, the response of these antennas has been reported to considerably shift to lower frequencies because of the dielectric character of the metal [21]. The subwavelength focusing properties of surface plasmons, particularly applied in plasmonic nano dipole and monopole antennas make them a very productive topic [6]–[13]. More complex configurations as Yagi Uda dipole arrangements [17]–[19] and bowtie antennas [20] have also been studied. The strong field enhancement present in the vicinity of the antennas is also very useful in other areas, such as non linear optics and biosensing. Up to now, most research has focused on the scattering properties of these structures when excited by plane waves or Gaussian laser beams [9]–[20]. However, from a circuit point of view, the parameter to study is the input impedance of the . The input impedance concept is common antenna at microwave and lower frequency regimes, but is quite new in the study of plasmonics. The first insights in the calculation of optical input impedance for nano dipole antennas were obtained by Alù et al. in [22], where they analyzed the response and matching properties of a silver dipole nanoantenna embedded in vacuum when varying its length and gap material. More recently, Huang et al. [23] studied the impedance in a circuit consisting of two gold dipole nano antennas interconnected by a two wires transmission line, and then obtained the impedance from the standing wave relation (SWR) taken at a fixed point on this line. They analyzed also the matching properties varying antenna width and length. In both of these investigations however finite domain discretization technique solvers were used and only in the last one the substrate was taken into account. Because free standing nano structures are extremely difficult to fabricate, the effect of the substrate on the input impedance

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DE ARQUER et al.: ENGINEERING THE INPUT IMPEDANCE OF OPTICAL NANO DIPOLE ANTENNAS

should also be included in further antenna analysis and characterization. In this paper, the input impedance of a gap fed nano dipole antenna is analyzed for several metal and substrate materials, in a two semi infinite media environment that models the effect of the substrate, for varying antenna length and width. These results are compared with the behavior of radio frequency (RF) antennas, where the dispersive properties of the metals are negligible. A linear model of the resonant response of the antennas is established. This is very useful in designs as it can take advantage of all the knowledge in antenna theory acquired at lower frequencies. After this study a comparison of gap fed excitation with plane wave excitation is performed. The optical cross sections, corresponding to the plane wave excitation, are determined. All the calculations were performed with an in-house developed simulation tool, MAGMAS 3D [24]. The simulation procedure is based on a combination of surface and volume integral equations in a multilayered environment. These equations are solved using the method of moments (MoM). Surface currents are meshed using rectangles and triangles and volume currents are meshed using hexahedra. Additional details about the used techniques can be found in [25]–[28]. This tool was originally developed for the microwave regime and extended considerably to the optical regime [29]. The MoM technique [30] is known as a very powerful, robust, and fast technique in classical antenna problems. In the optical domain however, up to now it has been completely deprecated by the finite elements method (FEM) or finite-difference time-domain (FDTD) modeling technique, which is used in most commercial solvers there. The main intrinsic strength of a MoM solver is the much lower number of unknowns describing the problem. This argument actually still holds also in the optical regime, as only micro/nanostructures need to be discretized. II. THEORY AND VALIDATION The first great advantage of the MoM in comparison with the FEM and the FDTD method is that only the current carrying components (the dipoles) in Fig. 1 have to be discretized. The complex environment, which may include multilayered substrates, is taken into account via Green’s functions. The discretized components are modeled in terms of equivalent currents. These currents fill completely all metal/dielectric volumes and the boundary conditions are satisfied in an average sense inside these volumes, on surfaces or lines depending on the selected testing procedure. The result is that MoM is intrinsically faster than the two other methods. The second advantage of the MoM technique is that, if properly formulated, it is variationally stable since most of the output parameters are expressed in integral form over the equivalent currents. As a consequence even if the calculated currents differ considerably from the exact solution, integral parameters over both currents may remain very similar. Further, this will be illustrated by showing that even with a rather rough mesh high quality physical results may be obtained. A third advantage is that MoM does not heavily suffer from field singularities near sharp edges, since they are analytically incorporated inside the

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Fig. 1. (a) Schematic of the dipole antenna geometry and dimensions. (b) Schematic of the excitation setup and input impedance retrieval. An imposed current (red) is flowing over two passive patches (green) to the two dipole arms. (c) Plane wave illumination.

Green’s functions. For the FEM and the FDTD method special ) should be taken to describe correctly care ( these field singularities. The antenna design is based on the work of Muhlschlegel et al. [12] where the optical response of plasmonic nano antennas was studied. The geometry and defining parameters are depicted , and . in Fig. 1(a): This structure is modeled in MAGMAS sandwiched between two semi infinite media, one (the bottom) taken as substrate and the other as vacuum. The dipole arms are meshed with 5 5 8 hexahedrons, which is optimal for this structure and increases also the efficiency of MoM by the use of translation symmetry. The extinction cross section of a gold monomer calculated using MAGMAS 3D with different meshes is plotted in Fig. 2(a). The analysis of these data demonstrates clearly the stability of the MoM [30]. The results obtained even with a very rough mesh 6 1 1 (41.7 nm 40 nm 40 nm) provides already a very good estimation of the antenna resonance properties. The differences in extinction cross section calculated with the rough and fine meshes are almost negligibly small. On the adjacent figure Fig. 2(b) extinction cross sections are calculated using Lumerical [31], a commercial solver based on the FDTD method. As expected, in general the results obtained with the FDTD method depend considerably on the chosen mesh. A fine mesh should be used for reliable calculations of the monomer resonant wavelength. The calculated wavelengths as a function of the mesh cell size are plotted in Fig. 3. It should be also noted that in contrast to Fig. 2(a) (MoM) in Fig. 2(b) (FDTD) not only peak positions but also their levels depend clearly on the mesh. However, for both methods shown in Fig. 2, the convergence of the numerical results is not an ultimate proof of the accuracy. Comparison with the exact solution or experimental results still remains a more convincing validation. As a next step, several gold monomers with different lengths were fabricated on a glass substrate by means of electron beam lithography and ion-milling. The fabricated monomers have a height and width of about 60 nm and 110 nm, respectively. The maximal lengths of the monomers are 265 nm, 205 nm and 165 nm. A scanning electron microscopy (SEM) image of the

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2

2

Fig. 2. Extinction cross section of a 250 nm 40 nm 40 nm gold monomer on a substrate with n = 1:5 (a. MAGMAS 3D, b. Lumerical). Fig. 4. (a) Simulation and measurement of extinction cross section of gold monomers with different lengths on a 1.5 refractive index glass substrate, polarization along longitudinal axis (thick lines: measurements, solid lines: MAGMAS, dashed lines: Lumerical), (b) SEM image of a monomer with length 265 nm.

Fig. 3. Resonance wavelength of a gold monomer as a function of mesh cell size.

monomer with length 265 nm is shown in Fig. 4. As a consequence of the fabrication method the side walls of all particles have a slope of about 20 degrees. Extinction cross secwith 2 pitch were meations of arrays of 50 by 50 sured using a Fourier transform infrared spectrometer equipped with a microscope. They are shown in Fig. 4. The monomers were illuminated by a plane wave polarized along the longitudinal axis. Considering the fact that the distance between the dipoles in the array is considerably larger than a wavelength, in the case of normal incidence the response of a dipole in the array is almost identical to the response of a single dipole. Regarding the electromagnetic modeling, in the case of Lumerical (FDTD method), the space was meshed in 5 nm side cubes in a box surrounding the particle. The plane wave excitation within MAGMAS illuminates the sample from the substrate

side, whereas in Lumerical it is top illuminated. This difference does not affect the obtained cross sections. The MAGMAS mesh cell size is about 20 nm, i.e., 16 6 3 cells for 265 nm, 10 6 3 cells for 205 nm, and 9 6 3 cells for 165 nm. The used complex permittivity of gold was taken from [32]. The substrate was modeled with a constant 1.5 refractive index. In the case of the FDTD method the dielectric response of the metal was fitted using the multicoefficient model of Lumerical [31] that indeed offers a good fit to [32] (not shown here). Simulation values for both solvers were normalized and adjusted to match the experiment. This change of scale does not affect the interpretation of the results, because the important feature here is the extinction shape. The agreement between them is evident, although both simulations are red-shifted with respect to the measurements. These differences are partially due to the fact that rectangular prisms were used in both simulations resulting in larger monomer volumes in comparison with their real sizes in the experiment. III. GAP EXCITATION An essential parameter of any receiving or transmitting antenna is its input impedance [33]–[36]. This parameter is absolutely needed in order to be able to describe the coupling between an antenna and the connected circuits, being part of a receiver or transmitter, respectively. This is evident at RF and

DE ARQUER et al.: ENGINEERING THE INPUT IMPEDANCE OF OPTICAL NANO DIPOLE ANTENNAS

microwave frequencies, but stays valid also at IR and optical frequencies. There are different techniques to calculate the input impedance. In general the input impedance can be expressed in terms of incident and reflected waves, as in [23]. This approach requires the incorporation of the feeding line in the full wave calculations, in this way increasing the numerical size of the problem. However, a well-known simplification is the gap excitation [22]. It is well-known that, despite the absence of the feeding line in the gap technique, the antenna impedances calculated with both techniques are very similar. This can be explained physically. Since the conductors of the feeding transmission line in normal circumstances are very close to each other compared to the wavelength, the incident and reflected waves on the feeding line actually impose a voltage between the two parts of the antenna connected to the two conductors of the transmission line. This voltage can be considered a gap voltage. Since the currents on both conductors of the transmission line are opposite and located extremely close to each other, they have a negligible effect on the antenna. Omitting them in the radiation problem by consequence does not introduce large errors. The gap excitation can be considered a voltage source. It thus corresponds to a Thevenin equivalent. In this paper, the Norton equivalent is used. In order to retrieve the antenna input impedance the dipole is fed at its gap by imposing a theoretical current flowing through infinitely conducting connections to the dipole arms. The primary feed used is thus an imposed current source placed at the top (metal/vacuum) dipole interface covering the gap. This feeding scheme is meshed with 5 nm side square segments (Fig. 1(b)) in such a way that they are aligned with the dipole arm mesh. No boundary condition is imposed for this current. Instead, the impedance can be calculated by determining the resulting voltage over the gap. This way of working is equivalent with taking into account both conduction and displacement current in the impedance calculation. Only when the dipole is close to one of its natural resonances, the input impedance demonstrates resonant behavior due to a very strong response of the dipole to the gap excitation. The goal of this paper is to provide simple guidelines to estimate the position of the resonance for different widely used metals in the optical regime. It should be noted that although the resonance response is determined mainly by the natural dipole resonances, the exact resonance position depends slightly on the chosen excitation. Nevertheless qualitative differences in the response between different metals remain the same as demonstrated in this paper for the gap excitation and plane wave excitation as shown in Fig. 1(b, c). , and the metal is chosen among Au, The substrate is Ag, Cu, Al and perfect electric conductor (PEC). The complex permittivity of these materials was modeled with experimental from [37]. The data, Au, Ag taken from [32], Cu, Al and obtained input impedance (resistance and reactance ) is depicted in Fig. 5 for varying antenna arm length . For high off-resonance wavelengths , where the dipole is electrically ) and the resmall, the reactance is largely capacitive ( sistance relatively small when compared to resonance (where is maximum), independent of the antenna metal used. Depending on feeding line impedance, this may be highly incon-

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venient since it drastically enhances return losses, therefore requiring a large impedance matching network. As the wavelength gets closer to resonance, the reactance starts to behave induc) and the resistance starts to grow rapidly. The tively ( crosses zero from negative to positive values points where show series LC resonant behavior. For those wavelengths the input impedance is purely resistive, but still much smaller than , at what can be seen as a short circuit resonance. For gold this occurs at approximately 1225 nm for an arm length , yielding [Fig. 5(a)–(b)]. This is of . one order of magnitude smaller than This resonance is the counterpart of the well known for thin RF dipoles. Going down in wavelength, close to the maximum resistance resonance, the reactance crosses zero again and now starts to behave capacitively (parallel LC resonant behavior). In these points the impedance is only resistive but larger . For the same case at 920 nm is obthan tained. For longer wavelengths the reactance remains negative and the resistance decays until second order resonances appear at 565 nm). Be(in the same case cause the resistance peak corresponds to maximum scattering cross section or, equivalently, radiated power, dipole dimensions (or the working wavelength band) should be chosen as a trade off between radiation efficiency and impedance matching, both strongly relying on the feeding line properties [23]. For other antenna materials the results are qualitatively similar. The case of silver [Fig. 5(c)–(d)] is blueshifted with respect to gold, and presents also slightly smaller resistance values. For copper [Fig. 5(e)–(f)] the similarities with gold are remarkable. Although resistance values are a little bit higher for the second resonance (about 500 ), resonance positions only differ in a few nanometers, opening up interesting design choices depending on the environment. This fact is explained by the almost identical complex permittivity in the analyzed wavelengths for both materials. The aluminium case [Fig. 5(g)–(h)] presents a new phenomenon with respect to the previous metals. An additional at around 900 peak appears in the resistance for nm, and also the reactance is shifted to negative values in a way different from typical series and parallel LC resonances. This is explained by the interband transitions that approximately occur at this wavelength. The resistance is also much smaller, below at maximum. Note that for all cases the resistance 1.5 and the impedance amplitude show a significant decrease as dipole arms get shortened. This is caused by the bigger electrical size of the gap for each resonant frequency. The case of PEC [Fig. 5(i)–(j)] serves as a non dispersive reference, thus representing the RF down scaled equivalent. The overall response is qualitatively similar, but a very clear and important differ) for the real metals ence is noticed. Resonant positions ( are strongly redshifted with respect to the expected values in classic antenna theory. The first one is traditionally given by for thin dipoles. This phenomenon is properly explained by the dispersive properties of the metals at optical frequencies, which introduces an effective wavelength scaling in the form of (1) [21] (1)

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Fig. 5. Input impedance as a function of  when varying dipole arm length L in steps of 10 nm for different antenna materials over SiO . W = H = 40 nm and G = 30 nm are kept constant. Resistance and reactance, respectively, for (a)-(b) Au, (c)-(d) Ag, (e)-(f) Cu, (g)-(h) Al and (i)-(j) PEC. Horizontal semi transparent planes show zero crosses in reactance.

where is the plasma wavelength, and both and coefficients determining the offset and slope of the linear relation. correspond to the resonances of the ideal The values for non-dispersive PEC case given the same geometry. Apart from the wavelength shift each material will give rise to different ) in the antenna, that can be qualiOhmic losses ( tatively classified according to the imaginary part of the permit. Although (1) tivity as serves as a good description of the wavelength scaling, a more convenient form is given by (2), linearly relating both total anand the associated resonance with tenna length two parameters (slope) and (offset) that will depend in each case on the material and geometric properties of the setup. This formula is easy to use for making a practical design (2) Fig. 6 consists of slices of Fig. 5 for specific configurations. As discussed before, even if there is only one physical resonance, resonant frequency can be defined in different ways: 1. at zero values of the reactance going from negative to positive ( );

2. at zero values of the reactance going from positive to neg); ative ( ). 3. at the maximum of the real part ( For the topologies depicted there, they are clearly marked on , square, pentagram – ). The Fig. 6 (diamond – resonant wavelengths are calculated using (2) and Table I. A comparison of the real part resonance positions among the different materials is plotted in Fig. 7(a), when varying dipole arm length. Au antennas present bigger shifts than other materials as predicted in [21] for cylindrical nanorods embedded in a homogeneous medium. It will be shown however that the shift in the resonances depends not only on the properties of the materials, but also on the antenna excitation. For gold the resonance is approximately 330 nm redshifted with respect to PEC, while silver yields a softer displacement of 280 nm. Table I depicts the linear fit of the resonance positions as a function of total antenna for several material configurations. The length bigger gold redshift is clearly translated in the offset . The in the PEC case is further from the value two beslope cause of the substrate and the non negligible antenna width. For copper the results are very similar to gold. Aluminium presents the smallest displacement, only about 110 nm above PEC. Note

DE ARQUER et al.: ENGINEERING THE INPUT IMPEDANCE OF OPTICAL NANO DIPOLE ANTENNAS

Fig. 6. Calculated input impedance of the dipole (L = 100 nm, W = H = 40 nm, G = 30 nm) for several antenna materials. (a) real part, (b) imaginary part.

TABLE I LINEAR FIT OF INPUT IMPEDANCE RESONANCE POSITIONS FOR SEVERAL MATERIAL CONFIGURATIONS AS A FUNCTION OF L . W H AND

G = 30 nm

= = 40 nm

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Fig. 7. (a) Resonance wavelengths in the input impedance (for maximum R) for several antenna materials and substrate: varying L, W H L G. (b) Resonance wavelengths in the ,G .L input impedance (for maximum R) for Au over when varying antenna width W . L ,G and H .

40 nm

= 30 nm = 110 nm

SiO =2 + = 30 nm

=

=

SiO = 40 nm

width variation influence is studied for the Au over configuration fixing all other dimensions and maintaining the initial mesh. Fig. 7(b) shows the relation between the dipole width and the real part of the impedance resonances. The observed behavior coincides with classic antenna theory, as increasing the width means a reduction of the effective dipole length because of the bigger reactance at the rod ends, producing a blue shift of the resonances. IV. PLANE WAVE EXCITATION

that second order resonances appear also in the case of Au and Cu, although they are not studied here. The effect of considering a substrate with constant refractive permittivity [37] index instead of the frequency dependent for gold as antenna material. The difwas studied with ferences are not appreciable for resonances, very slightly redshifted, at least for the wavelengths under consideration. The

After the gap excitation analysis, the same topology is now studied when illuminated by a plane wave. The setup of this configuration in MAGMAS is shown in Fig. 1(c). A plane wave polarized along the dipole axis impinges the structure from the substrate at its bottom interface. From the scattered far fields, extinction cross sections are obtained by application of the Optical Theorem [38]. There is a clear difference between this excitation and the gap feeding. In the latter case, a source of current was placed in the middle of the dipole, in this way establishing a connection between them: the two nanorods constituting the antenna are coupled through the current flowing between them. The current shows a maximum in the gap and decays up to the rod ends (not shown here) in a quasi sinusoidal distribution along the longitudinal direction. With the plane wave excitation however, the rods are only capacitively coupled, resulting in an extinction peak considerably redshifted when compared to the expected longitudinal plasmon band of the individual rods [39],

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Q

=15 = 40 nm

= = 40 nm

Fig. 8. Full characterization of the normalized extinction spectra (a-d) as a function of L and G for Au over a n : substrate with W H . (a) Resonant positions corresponding to maximum extinction cross section, (b) maximum value of normalized extinction, (c) FWHM and (d) normalized extinction : substrate with H and G . The amplitude of average. (e) Resonant positions and (f) FWHM as a function of L and W for Au over a n the current distribution jJ j at resonance for the same configuration with W H and L is plotted for several gap sizes: (g) G , (h) 10 nm and 40 nm.

=15 = = 40 nm = 110 nm

TABLE II LINEAR FIT OF EXTINCTION CROSS SECTION RESONANCE WAVELENGTHS FOR PLANE WAVE ILLUMINATION AS A FUNCTION OF L . W H AND G . SUBSTRATE WITH REFRACTIVE INDEX n :

= 30 nm

= = 40 nm =15

but blue shifted with respect to gap feeding. A huge field enhancement appears now in the vicinity of the gap, where there is no current flowing at all. Each nanorod indeed presents at its ends zero current and a maximum value somewhere in the middle, depending on the relative position and the strength of the coupling [Fig. 8(g)–(i)]. To overcome the mismatch introduced by the gap for the plane wave illumination in the currents distribution, the use of one equivalent rod without gap was proposed in [21]. However, under this assumption the results are red-shifted a lot, about 500 nm, compared to the same length dipole with gap excitation. Resonances corresponding to the extinction peak values are analyzed for Au, Ag and PEC when varying dipole arm length as it was done for the gap excita, subtion, this time with constant refractive index, strate (Table II). The same shifts appear because of the material dispersive properties, but the resonant positions are overall

= 10 nm

= 1 nm

blue shifted compared with gap excitation. Differences are evident in the offset of the wavelength-dipole length relation, but almost negligible for the slopes, where for two significant digits the only difference appears for the PEC configuration. Far and near field resonance shifts under plane wave illumination were also reported in [41], [42] for single cylindrical nanorods but as a function of their radii. Now that the comparison of the two excitation methods has been accomplished, a full analysis of the effect of the gap for the plane wave excitation is performed. The gap size is crucial since it determines the coupling between the two dipole arms, in this way affecting the far field response of the structure and shifting the obtained resonances. Plasmon resonance spec) dependence on coupled nanorod antenna ditral position ( mensions has been studied extensively before [40]–[42]. However, little attention has been given to the spectral shape, i.e., the full width at half maximum (FWHM) of these resonances [42]. This parameter is of great importance for several applications ranging from localized surface plasmon resonance (LSPR) sensing to plasmonic lasing [43], where high quality resonators having sharp lineshapes are required. For that reason, in addition maps [Fig. 8(a) and (e)], for each to the and set a full characterization of the normalized extinc(taken as the extinction cross section divided tion spectra by the antenna horizontal area) is given, by means of analyzing , linewidth , and its peak

DE ARQUER et al.: ENGINEERING THE INPUT IMPEDANCE OF OPTICAL NANO DIPOLE ANTENNAS

average over value [Fig. 8(b), (c) and (f) and (d) respectively]. It is shown that smaller gap sizes imply red-shifted peak positions [Fig. 8(a)], and that the slope of the relation between this peak and arm length also increases when reducing its size. For a fixed , the redshift in the resonance as a function of the gap is not linear, which is more evident for larger values ). It tends asymptotiof (thus for higher aspect ratios cally to the no-gap resonant wavelength. If is fixed however, resonant positions behave almost linearly. The peak value of [Fig. 8(b)] shows a very strong dependence on the gap size as smaller gaps yield bigger amplitudes. For instance, keeping , varies from 1 to 20. In Fig. 8(c) FWHM space is plotted. Broader peaks appear for variation in longer dipoles. The sensitivity to seems to be different depending on the arm length. For , reducing the gap no size increases the width, whereas in the case clear relation between these two parameters can be observed. The average of the normalized extinction is shown in Fig. 8(d). As it happens for the peak value, reducing increases its overall amplitude. As FWHM reveals the same behavior, this does not mean that the extinction is more concentrated around a narrower and bigger peak, but that it is broadened, enhanced and redshifted when reducing the gap size. This behavior reveals a clear tendency in the extinction parameters to reach the no-gap is reduced. This can be explained based on the spectra as current distribution amplitude over the two individual nanorods, tending to a sinusoidal shape in the overall structure as the particles get closer. [Fig. 8(i)] the two particles interact, in this For way displacing the maximum current position of the individual nanorods towards the gap. This is more evident when the two nanorods get closer [Fig. 8(h) and (g)] and thus couple more the differences with a single nanorod strongly. For of double length are only appreciable in a small region around the gap, that will eventually vanish in the limit. The current, i.e., the current density integrated over the dipole cross section, is shown in Fig. 9 along the length of the dipole. The behavior of the current around the gap is clearly demonstrated. This confirms our physical expectation looking at the field distribution that there is a smooth transition between the monomer and a dipole with a very small gap. The same analysis performed for silver (not shown here) produced similar behavior, although blue shifted because of the different material dispersive properties. and length is analyzed Next, the influence of varying , fixing again for a Au antenna over a substrate with to 10 nm. Fig. 8 depicts the resonance peak position (e) and FWHM (f) dependence on these parameters. The blue shift ob) for the gap excitation served in the resonance positions ( with increasing dipole width is also noticed here, in agreement relation is howwith [21] and [40]. The slope of the ever not affected by , contrarily to the gap variation. For the FWHM [Fig. 8(f)], keeping fixed results in a monotonic increasing dependence on . For small aspect ratio antennas the sensitivity to almost disappears (e.g., the case of with ). More elongated antennas present a stronger dependence on that parameter. This is again in agreement with

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Fig. 9. The amplitude of the current distribution jJ j along the dipole integrated over its cross section at resonance for Au over a n : substrate with W H and L for several gap sizes.

= 40 nm

= 110 nm

=15

=

classic antenna theory, where a bigger radius yields an enhanced bandwidth. Here, however, due to the response of metals at optical frequencies, the FWHM is in the first instance determined by the damping and losses inside the metal, that directly depend on the total volume and the current distribution along the structure. Since the resonances studied here correspond to the first sustained mode, that exhibits a quasi-shifted sinusoidal current distribution, the bigger the volume the bigger the losses and therefore an enhanced bandwidth is obtained. On the other hand the spectral position of the resonances is mainly determined by the antenna length, and the remaining parameter that allows tuning the quality factor is the width . The damping strength inside the metal also depends on the metal used. Using materials with lower losses, like Ag, will result in smaller FWHM values. characterization obIt must be noted that the tained here, although presenting similar behavior if compared with the work of Muskens et al. for similar sizes [42], is scaled down by approximately a factor of two with respect to their results. The reason is that they calculated the cross sections from a 2D simplification of the antenna structure based on a Green’s function formalism instead of using a full wave 3D electromagnetic solver. This suggests that for nano dipole antennas, where the aspect ratios are upper-bounded by the considerable value of due to fabrication issues and due to the fact that cannot be chosen arbitrarily large in order to keep the dipole resonances ) in the optical regime, the antenna width must be taken ( into account to accurately characterize the spectral shape.

V. CONCLUSION AND REMARKS The nano dipole antenna topology was studied with a method of moments based software tool, comparing gap excitation and plane wave illumination. To the knowledge of the authors, for the first time, the input impedance of the antenna was calculated for several material configurations, varying lengths and widths, and including the substrate effects, this in contrast with previous work [22]. The calculation of the input impedance for gap feeding can be used to determine the parameters of the optimal feeding line [23]. It was shown that besides a displacement in the response of the antenna and an increase in the ohmic

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losses due to dispersive not PEC materials, the remaining aspects of the behavior are similar to the behavior of classical antennas at microwave and RF frequencies. Transfer of know-how from these areas is very feasible. A linear relation for the resonances in terms of the antenna length was introduced, as well as some basic design rules. The effect of the dispersive properties of some important metals was studied for the input impedance. Resembling [21], a shift comparison was observed for gap excitation. Gold produces the bigger redshift, which is a good characteristic for designing subwavelength structures, but because of its permittivity, also presents than silver and other metals. Copper higher absorption and shows a similar response than gold making both of them particularly suited for red and infrared wavelengths. Silver is less redshifted but has certain practical issues related to oxidation. Aluminium is the one closest to the ideal PEC case, although it presents larger losses. Concerning losses, in the range 500–1200 . Therefore, a trade-off among the desired working window, device size, and material characteristics in each environment must be accomplished in the design and engineering of the input impedance. The study of plane wave excitation reveals that the phenomenology of this configuration is not the same as for gap excitation, where the coupling between dipole arms is conductive instead of capacitive. This difference, in addition to the width influence on the far field, introduces a considerable blue shift in the cross section resonances. A strong dependence of the scattering properties on antenna gap and width values was shown under this illumination. The gap influence on the extinction amplitude and peak position can be understood in terms of input impedance as an increase and redshift in its values. It was shown that the antenna width is crucial to accurately determine the values of the cross sections. Further research, for example using different materials for the individual nanorods could also introduce interesting properties in the frequency response of the system. Also, radiating quantum dots can be studied. The intriguing world of nanocircuitry and the engineering of these devices represent an incredible challenge. It is shown that a proper design and the full understanding of each individual block and its connections undeniably requires a thorough study of the antenna input impedance. REFERENCES [1] J. R. Krenn, B. Lamprecht, H. Ditlbacher, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, “Non diffraction-limited light transport by gold nanowires,” Europhys. Lett., vol. 60, pp. 663–669, 2002. [2] N. F. Van Hulst, “Light in chains,” Nature, vol. 448, pp. 141–142, 2007. [3] N. Engheta, “Circuits elements at optical frequencies: Nanoinductors, nanocapacitors and nanoresistors,” Phys. Rev. Lett., vol. 95, p. 95504, 2005. [4] N. Engheta, “Circuits with light at nanoscales: Optical circuits inspired by metamaterials,” Science, vol. 317, pp. 1698–1702, 2007. [5] E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science, vol. 311, p. 5758, 2006. [6] P. Ginzburg, D. Arbel, and M. Orenstein, “Efficient coupling of nano-plasmonics to micro-photonic circuitry,” presented at the Conf. on Lasers and Electro Optics, Baltimore, MD, 2005. [7] W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett., vol. 9, pp. 4403–11, 2009. [8] Z. Chun-Lin, R. Xi-Feng, H. Yun-Feng, D. Kai-Min, and G. GuangCan, “FDTD studies of metallic cylinder arrays: Plasmon waveguide and Y-splitter,” Chin. Phys. Lett., vol. 25, pp. 559–562, 2008.

[9] T. H. Taminiau, F. B. Segerink, and N. F. van Hulst, “A monopole antenna at optical frequencies: Single-molecule near-field measurements,” IEEE Trans. Antennas Propag., vol. 55, pp. 3010–3017, 2007. [10] T. H. Taminiau, F. B. Segerink, R. J. Moerland, L. K. Kuipers, and N. F. van Hulst, “Near-field driving of an optical monopole antenna,” J. Opt. A: Pure Appl. Opt., vol. 9, pp. S315–S321, 2007. [11] T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nature Photon, vol. 2, pp. 234–237, 2008. [12] P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science, vol. 308, pp. 1607–1609, 2005. [13] J. R. Krenn, G. Schider, W. Rechberger, B. Lamprecht, A. Leitner, and J. C. Weeber, “Design of multipolar plasmon excitations in silver nanoparticles,” Appl. Phys. Lett., vol. 77, pp. 3379–3381, 2000. [14] P. Genuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopy mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett., vol. 101, p. 116805, 2008. [15] H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Expr., vol. 16, pp. 9144–9154, 2008. [16] H. Fischer and O. J. F. Martin, “Polarization sensitivity of optical resonant dipole antennas,” J. Euro. Opt. Soc. Rap. Public., vol. 3, p. 08018, 2008. [17] J. Li, A. Salandrino, and N. Engheta, “Shaping the beam of light in nanometer scales: A Yagi-Uda nanoantenna in optical domain,” Phys. Rev. B., vol. 76, p. 245403, 2007. [18] T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna,” Opt. Expr., vol. 16, p. 16858, 2008. [19] T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Single emitters coupled to plasmonic nano-antennas: Angular emission and collection efficiency,” New J. Phys, vol. 10, p. 105005, 2008. [20] N. Yu, E. Cubukcu, L. Diehl, D. Bour, S. Corzine, J. Zhu, and G. Höfler, “Bowtie plasmonic quantum cascade laser antenna,” Opt. Expr., vol. 15, p. 13272, 2007. [21] L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett., vol. 98, p. 266802, 2007. [22] A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nano antennas,” Phys. Rev. Lett., vol. 101, p. 043901, 2008. [23] J. Huang, T. Feichtner, P. Biagioni, and B. Hecht, “Impedance matching and emission properties of nanoantennas in an optical nanocircuit,” Nano Lett., vol. 9, p. 1897, 2009. [24] MAGMAS 3D Full Wave Solver, [Online]. Available: http://www.esat. kuleuven.be/telemic/antennas/magmas/ [25] G. A. E. Vandenbosch and A. R. Van de Capelle, “Mixed-potential integral expression formulation of the electric field in a stratified dielectric medium – application to the case of a probe current source,” IEEE Trans. Antennas Propag., vol. 40, pp. 806–817, Jul. 1992. [26] F. J. Demuynck, G. A. E. Vandenbosch, and A. R. Van de Capelle, “The expansion wave concept–Part I: Efficient calculation of spatial Green’s functions in a stratified dielectric medium,” IEEE Trans. Antennas Propag., vol. 46, pp. 397–406, Mar. 1998. [27] M. Vrancken and G. A. E. Vandenbosch, “Integral equation formulation and solution for extended vertical current sheets in multilayered planar structures,” Radio Sci., vol. 38, no. 2, pp. VIC17.1–VIC17.11, Apr. 2003. [28] Y. Schols and G. A. E. Vandenbosch, “Separation of horizontal and vertical dependencies in a surface/volume integral equation approach to model quasi 3-D structures in multilayered media,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1086–1094, Apr. 2007. [29] F. P. G. de Arquer, “Plasmonic antennas and scattering nanostructures,” Master’s thesis, Katholieke Universiteit Leuven, TELEMIC, Belgium, 2009. [30] R. F. Harrington, Field Computation by Moment Methods. New York: IEEE Press, 1993. [31] “Optical and Photonic Design and Engineering Software Products,” Lumerical Solutions, Inc. [Online]. Available: http://www.Lumerical.com [32] P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B, vol. 6, p. 4370, 1972. [33] S. R. Best and B. C. Kaanta, “A tutorial on the receiving and scattering properties of antennas,” IEEE Antennas Propag. Mag., vol. 51, no. 5, pp. 26–37, 2009.

DE ARQUER et al.: ENGINEERING THE INPUT IMPEDANCE OF OPTICAL NANO DIPOLE ANTENNAS

[34] A. Locatelli, C. de Angelis, D. Modotto, S. Boscolo, F. Sacchetto, M. Midrio, A.-D. Capobianco, F. M. Pigozzo, and C. G. Someda, “Modeling of enhanced field confinement and scattering by optical wire antennas,” Opt. Expr., vol. 17, p. 16792, 2009. [35] A. Alu and S. Maslovski, “Power relations and a consistent analytical model for receiving wire antennas,” IEEE Antennas Propag., vol. 58, no. 5, pp. 1436–1448, 2010. [36] A. Alu and N. Engheta, “Wireless at the nanoscale: Optical interconnects using matched nanoantennas,” Phys. Rev. Lett., vol. 104, no. 21, p. 213902, 2010. [37] E. D. Palik, Handbook of Optical Constants of Solids. New York: Academic Press, 1985. [38] R. G. Newton, “Optical Theorem and beyond,” Amer. J. Phys., vol. 44, p. 639, 1976. [39] A. M. Funston, C. Novo, T. J. Davis, and P. Mulvaney, “Plasmon coupling of gold nanorods at short distances and in different geometries,” Nano Lett., vol. 9, p. 1651, 2009. [40] J. Aizpurua, G. W. Bryant, L. J. Richter, F. J. G. de Abajo, B. K. Kelley, and T. Mallouk, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B., vol. 71, no. 2005, p. 235420, 2005. [41] G. W. Bryant, F. J. G. de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of plasmonic nanoantennas,” Nano Lett., vol. 8, pp. 631–636, 2008. [42] O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. G. Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Expr., vol. 15, p. 17736, 2007. [43] D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett., vol. 90, p. 027402, 2005. F. Pelayo García de Arquer received the M.Sc. degree in telecom engineering in 2009 from the Universidad de Oviedo, after doing his Master Thesis at the Katholieke Universiteit Leuven on the topic of “Plasmonic Antennas and Scattering Nanostructures” in Prof. Vandenbosch’s group. He was awarded a Fundación la Caixa grant to take the M.Sc. in photonics at the Universidad Politénica de Catalunya. In his second Master Thesis, “Plasmonically Enhanced Absorption in Metal-Semiconductor Nanocomposites” he worked with Prof. Konstantatos’ group at the Institute of Photonic Sciences, Barcelona, Spain, where he is working toward the Ph.D. degree.

Vladimir Volski graduated from the Moscow Power Engineering Institute, Moscow, Russia, in 1987 and received the degree of Candidate of Science (Ph.D.) in 1993. In 1987, he joined the Antennas and Propagation of Radio Waves Division, Moscow Power Engineering Institute, as a Researcher. Since January 1996, he has been a Postdoctoral Researcher at the ESAT-TELEMIC Division, Katholieke Universiteit Leuven, Leuven, Belgium. His main research interests include electromagnetic theory, computational electromagnetics, and modeling and measuring of electromagnetic radiation, including bio-electromagnetics.

Niels Verellen received the B.S. and M.S. degree in physics from the Katholieke Universiteit Leuven, Leuven, Belgium, in 2005 and 2007, respectively, where he is currently working toward the Ph.D. degree. His doctoral work is focused on optical confinement phenomena in plasmonic nanomaterials with predesigned electromagnetic properties.

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Guy A. E. Vandenbosch was born in Sint-Niklaas, Belgium, on May 4, 1962. He received the M.S. and Ph.D. degrees in electrical engineering from the Katholieke Universiteit Leuven, Leuven, Belgium, in 1985 and 1991, respectively. From 1985 to 1991, he was a Research and Teaching Assistant with the Telecommunications and Microwaves Section, Katholieke Universiteit Leuven, where he worked on the modeling of microstrip antennas with the integral equation technique. From 1991 to 1993, he held a postdoctoral research position at the Katholieke Universiteit Leuven. Since 1993, he has been a Lecturer, and since 2005, a Full Professor at the same university. He has taught, or currently teaches, courses on “electrical engineering, electronics, and electrical energy,” “wireless and mobile communications, part antennas,” digital steer- and measuring techniques in physics,” and “electromagnetic compatibility.” His research interests are in the area of electromagnetic theory, computational electromagnetics, planar antennas and circuits, electromagnetic radiation, electromagnetic compatibility, and bio-electromagnetics. His work has been published in ca. 120 papers in international journals and has been presented in ca. 200 papers at international conferences. Prof. Vandenbosch has convened and chaired numerous sessions at many conferences. He was Co-Chairman of the European Microwave Week 2004 in Amsterdam, and Chaired the TPC of the European Microwave Conference within this same Week. He was a member of the TPC of the European Microwave Conference in 2005, 2006, 2007, and 2008. He has been a member of the Management Committees of the consecutive European COST actions on antennas since 1993, where he is leading the working group on modeling and software for antennas. Within the ACE Network of Excellence of the EU (2004–2007), he was a member of the Executive Board and coordinated the activity on the creation of a European antenna software platform. He holds a certificate of the postacademic course in Electro-Magnetic Compatibility at the Technical University Eindhoven, The Netherlands. Since 2001, he has been President of SITEL, the Belgian Society of Engineers in Telecommunication and Electronics. Since 2008, he is a member of the board of FITCE Belgium, the Belgian branch of the Federation of Telecommunications Engineers of the European Union. During 1999 to 2004, he was Vice-Chairman, and from 2005 to 2009, Secretary of the IEEE Benelux Chapter on Antennas and Propagation. Currently he holds the position of Chairman of this Chapter. From 2002 to 2004, he was Secretary of the IEEE Benelux Chapter on EMC.

Victor Moshchalkov was born in Russia. He received the M.Sc., Ph.D., and “Habilitation” degrees in physics from the Lomonosov Moscow State University, Moscow, Russia, in 1975, 1978, and 1985, respectively. From 1978 to 1988, he was a Research Physicist, Assistant Professor, and Professor at the Lomonosov Moscow State University where, in 1988, he became Head of the Laboratory of High Temperature Superconductivity. Since 1986, he has held a number of positions as Guest Scientist or Guest Professor at Toronto University, Canada, TH Darmstadt, Marburg University, RWTH Aachen, Germany, Centre d’Etudes Nucléaires de Grenoble, France. In 1991, he joined the Katholieke Universiteit Leuven, Leuven, Belgium, as a Visiting Professor and where he became a Full Professor in 1993. He has over 780 publications in international peer reviewed journals and more than 9300 citations. Prof. Moshchalkov has received several awards: the Young Researcher award in 1986, the High Education Scientific Prize in 1988, the ISI Thomson Scientific Award “Top Cited Paper in Flanders” in 2000, the Dr. A.De LeeuwDamry-Bourlart Prize for Exact Sciences from the Flemish FWO in 2005, and the Methusalem Research Award in 2009. He was a finalist for the EU Descartes Research Prize in 2006. He is an American Physical Society Fellow since 2007. He was the Chairman of the ESF Program “Vortex Matter in Superconductors-VORTEX” from 1999 until 2004. He became the Director of the INPAC Institute for Nanoscale Physics and Chemistry (Center of Excellence at the K.U. Leuven) in 2005. Since 2007, he is also the Chairman of the ESF-NES program, which includes 60 teams from 15 European countries. He was an invited speaker at 96 international conferences and workshops, and he is Founder of the new series of International Conferences on “Vortex Matter in Nanostructured Superconductors.” He is a member of the International Advisory Committee of 34 international conferences. He has been, or is, Promoter of 48 Ph.D. theses (12 at the Moscow State University and 36 at the K.U. Leuven).

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Electromagnetic Resonances of a Straight Wire on an Earth-Air Interface John M. Myers, Sheldon S. Sandler, Life Member, IEEE, and Tai Tsun Wu

Abstract—Using a variational method, we recently determined an electromagnetic “signature” for characterizing a straight wire in free space. The signature consists of the first five resonant frequencies and their widths, more compactly expressed as the first five complex-valued resonant frequencies. Here we apply the variational method to the much more complicated case of determining the same signature for a straight wire or wire pair on a flat interface between a homogeneous earth and air. To calculate the resonances we obtain an integral equation for the current on a wire on the interface between two dielectric media. Complex-valued resonant frequencies are defined as those for which the homogeneous integral equation for the current in an equivalent thin strip on the interface has non-zero solutions. The variational method extracts good approximations to these complex-valued resonant frequencies, without having to solve the integral equation. A table of resonances is given for the case of a relative dielectric constant of the earth equal to 4 and for three values of the ratio of wire radius to wire half-length. Index Terms—Antenna theory, earth, integral equation, interface phenomena, resonance, variational methods.

I. INTRODUCTION

I

T has long been of interest to establish electromagnetic signatures by which to recognize objects. Here we study complex-valued resonant frequencies [1], [2] in a straight wire located on the surface of a flat earth, to serve as a “signature” for locating such a wire. An important property of the resonances is that the ratios of successive resonant frequencies are characteristic of such a wire and are approximately invariant with wire diameter and wire length, so long as the diameter is very much less than both the wire length and the wavelength. In the case of a thin wire in a homogeneous medium, the shape of its cross section has negligible effect on the electrical behavior of the wire, but some shapes are much more convenient than others for analysis. Although mostly interested in a wire of circular cross section, we draw on conformal mapping to find that for perfect conductors a circular wire of radius is essentially equivalent to a flat strip of width and negligible thickness [3, p. 59]. This equivalence to a strip holds also for a thin wire on the boundary between two different dielectric media; indeed a wire pair can be treated the same way, namely as equivalent to a suitably scaled strip. We analyze a flat and perfectly conducting strip of zero thickness. We assume the strip is Manuscript received March 19, 2010; revised December 21, 2010; accepted February 28, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the Army Research Office under Grant W911NF-07-1-0509 with Harvard University. The authors are with the Harvard School of Engineering and Applied Sciences, Cambridge, MA 02138 USA (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2161539

located in a plane interface, thought of as horizontal, between a medium below (earth) and a medium above (air). The problem is to find the first five resonant frequencies for electromagnetic radiation scattered by this thin strip as functions of the strip length, strip width, and the dielectric constant of earth and of air; the width of each resonance is also to be determined. Our approach, based on the theory of linear antennas, is to formulate an integral equation for the current in the wire induced by an incident electromagnetic field. For the much simpler case of a wire in a single homogeneous medium, two forms of integral equation for the current generated by an incident electromagnetic field were well developed over fifty years ago [4]. The formulation of the Hallén integral equation for the current in the wire induced by an incident vector potential evades non-integrable singularities that would infest the integral if the integral equation were formulated directly for an incident electric field. From this Hallén integral equation, in which the vector potential plays a central role, it follows that the current on a thin antenna is approximately sinusoidal, and the propagation constant for this approximately sinusoidal current is that of the surrounding medium. In the early 1970s resonant frequencies were defined as the complex values of frequency at which a homogeneous integral equation (in which the incident field is set to zero) has a non-zero solution for the current [1], [2]. For a straight wire thicker than that of interest to us, the singularity expansion method (SEM) yielded solutions for the case of a homogeneous medium [5]. Recently we recast the SEM method to express the resonant frequencies as solutions to an equation that sets to zero a functional of the current, with the nice property that this functional has zero first-order variation under variation of the current [6]. This insensitivity allows a relatively crude approximation to the current at resonance to be used in solving for an accurate approximation to the complex-valued resonant frequency. In contrast to the case of a wire in a single medium, for the present case of a wire on an interface, there is no useful definition of a vector potential, which complicates formulating the problem. When the wire involves two media with distinct propagation constants, the questions arise: 1) Is the current along the wire approximately sinusoidal? 2) If so, what is its propagation constant? Based on the known expression for the electric field emanating from a point current element located on and parallel with the interface [7], it turns out that integral equations of both the Pocklington type and the Hallén type can be formulated, but with a more complex kernel that introduces qualitatively new features, associated physically with the complexity of paths by which energy can propagate near an interface. Solving the integral equation in any form is difficult; however, we will find a good ap-

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MYERS et al.: ELECTROMAGNETIC RESONANCES OF A STRAIGHT WIRE ON AN EARTH-AIR INTERFACE

proximation to the resonant values of without having to solve the integral equation. As described below, the current near resonance is approximately sinusoidal, but with a complex-valued propagation constant, even when both media are lossless. Using the approximate current, the variational method developed for free space is applied to determine approximately the first five complex-valued resonant frequencies of a wire on an earth-air interface. The rest of the paper is organized as follows. 1) In Section II, we state the known expression of the electric field generated by a current element on the strip. This electric-field kernel, however, is non-integrable and so cannot serve directly as the kernel of an integral equation. 2) To obtain an integrable kernel, the trick is similar to that used in the case of a single medium: in Section III we determine a pair of complex-valued zeros in the Fourier transform of the electric-field kernel with respect to distance along the strip, based on an approximation valid for a thin strip. 3) Using the zeros determined in Section III, in Section IV, we derive the integral equation of the Pocklington type satisfied by the current induced in the strip, and show how the values of the zeros in the Fourier transform of the electric field supply the main ingredient in the answer to the two questions posed above. 4) In Section V, the complex-valued resonant frequencies are defined as the frequencies at which the homogeneous integral equation for the current in the strip has non-zero solutions. The integral equation is manipulated into a form in which the solutions for the resonances are insensitive to small errors in the current, allowing us to use the kernel of the integral equation, obtained after much labor, together with a simple approximation for the current to obtain approximate equations for the resonant frequencies of better accuracy than the approximation for the current. 5) In Section VI, the equations defining the kernel of the integral equation for the current are rearranged to facilitate numerical computation, and examples of calculated resonances are reported. 6) In Section VII we make some concluding remarks.

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Fig. 1. Schematic diagram of the antenna geometry: (a) side view, (b) top view.

Since the antenna is assumed to be thin, so that

is taken to be small,

(2) . For present purposes both which, of course, implies earth and air are assumed lossless, so that both and are taken to be real-valued and positive. (As in the usual case of the linear antenna in a homogeneous medium, once the lossless case is understood, the introduction of loss in the medium is fairly straightforward.) The formulation of the problem hinges on the electric field generated by a “horizontal” point dipole in the interface. Consider a delta-function current at the origin, (3) then, the -component of the electric field at a point the interface is given by (see (5.4.13) of [7])

II. FORMULATION We analyze a flat, perfectly conducting strip of length , , and zero thickness, located in a plane interface, width thought of as horizontal, between a medium below (earth) with and a medium above (air) with a a propagation constant propagation constant , as shown schematically in Fig. 1. Following the notation of [7], let the planar boundary between , and let the -axis air and earth be the -plane given by point down into the direction of the earth. The problem is to find the first five complex resonant frequencies for electromagnetic radiation scattered by this thin strip as functions of the , and . parameters As holds for earth and air, it will be assumed throughout this paper that (1)

on

(4) where

(5) In deriving (4), it has been assumed that the magnetic permeability for both region 1 and region 2 is given by ; that is, we do not deal here with magnetizable earth. The advantage of studying a strip antenna of negligible thickness is the invariance

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of (4) under interchange of regions 1 and 2, unlike the much more complex formula that would express a thick wire. At points off the strip, the electric field produced by the current density on the strip is expressed as an integral over times the current density as a function of and , can be thought of as an “electric-field kernel”; so that however at points on the strip the integral fails to converge. As in the case of a single-medium, for the wire on an interface the key property of the electric-field kernel that makes possible the elimination of a non-integrable singularity is a pair of zeros of the spatial Fourier transform of the electric-field kernel. where is In the single-medium case, the zeros are at the propagation constant of the single medium, so the Fourier which leads in transform of the kernel has a factor the Pocklington integro-differential equation to the differential . That method works also for a thin strip operator at an interface, except that finding the zeros in the Fourier transis a considerable task, to which we now turn. form of

TABLE I SAMPLE VALUES OF k =k AS A FUNCTION OF k a AND k =k

where

III. ZEROS IN THE FOURIER TRANSFORM OF From (4) it follows that the Fourier transform from is times

to

(10)

of and

(11)

(6) This depends also on on frequency through

and

, which in turn depend

can be evaluated exactly in As discussed in detail in [8], terms of Bessel functions. Then the small-argument approximation [9] for the Bessel functions together with an approximate described in [8] imply evaluation of

(7) where and are the relative dielectric constants of the two and are media. We assume lossless media, so that both real-valued and positive; taking region 2 to be air, . In marked contrast to the case of a single medium, however, the is zero turn out to be complex, not values of at which real. , the integral on the right-hand To find the zeros of side of (6) must be evaluated, which presents two difficulties: (a) this integral cannot be expressed exactly in terms of standard special functions, and (b) the integrand oscillates and decays only slowly, making it unsuitable for direct numerical evaluation. To overcome these difficulties, we develop approximations , and of the integral, accurate for a thin strip. Because taking to be positive without loss of generality, the condition for the strip to be thin is

(12)

(8) The approximation to be developed has the form of the first two terms of a series expansion. Although it neglects terms of the , where is the larger one of and , order of the approximation is highly accurate for small values of . To proceed, we rewrite (6) as (9)

This is the desired approximate expression from which to determine the zero of this as a function of when and are given subject to the conditions (8). For several values of and , values of at which as specified in (12) has a zero at are listed in Table I. Equation (12) determines a unique value for , and hence a pair of values for ; by we denote the complex value having a positive real part.

MYERS et al.: ELECTROMAGNETIC RESONANCES OF A STRAIGHT WIRE ON AN EARTH-AIR INTERFACE

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as a differential operator acting on the non-integrable a function that has only a logarithmic singularity, and then to bring the differential operator outside the integration. The needed differential operator is obtained by exploiting the in the Fourier transform , found above, zero-points into the form to put (15)

Fig. 2. A usual choice of the first (“proper”) sheet in the  -plane. With this and ~ (; y ) at  = k and  = k are a number of similar choices, the zeros of E on the first sheet.

0

, which has only a logarithmic where the function singularity, follows from defining its Fourier transform to be (16)

In this calculation the neglect of terms on the order of generates a relative error in of order . For the th resonant frequency, we bound by . For and , the ne, and under these conditions, which glected terms are hold for Table I, we expect a relative error in less than . To gain more physical insight than is visible in these calculated numbers, we keep only the leading term of order and neglect all terms of order 1 on the right-hand side of (12) to obtain the rough approximation:

From (16) together with (6) one obtains (15). Inverting the gives Fourier transform

(17) For comparison, in the case of a homogeneous medium char, one finds [8] acterized by

(13) Thus, in this rough approximation, the zero of given simply by the known formula [10], [11]

, called

, is

(14) A more accurate approximation yields an additional interesting result. Because the right-hand side of (12) is complex in the sense of having both a real part and an imaginary part, setting to zero can be expected to lead to a solution that is complex, and such a zero off the real axis leads to many new phenomena. Reference [8] displays the approximate value of the imaginary part of the zero, to show that both the real and the of in the -plane imaginary parts of the zero point , at least for small values of . That are positive when is not real even in the absence of dissipation leads to features that are not seen in the usual case of the linear antenna in is in the first (“proper”) sheet a uniform medium. Note that of the complex -plane as usually chosen—see Fig. 2—unlike the Sommerfeld pole for the linear antenna. IV. INTEGRAL EQUATION FOR THE CURRENT If the electric field kernel had only an integrable singularity, it would work immediately as the kernel for an integral equation for the current in the wire; however because can be shown to behave as as tends toward zero, something has to be done to evade this non-integrable singularity. As in the case of a homogeneous medium, the trick is to express

(18) is electroThe argument that a thin strip antenna of width magnetically equivalent to a circular one of radius (see [4, pp. 16–18]) works without change for the more general situation of two media, leading to the integro-differential equation (19) where, from (15) and (17), the kernel

of (19) is

(20) is the current on the antenna, and is the -component of the external electric field applied to the thin strip, evaluated at . Equation (19) together with the approximation (14) to the allows one to anzeros in the Fourier transform of swer the two questions posed in the introduction. For a wire in in the Pockfree space there appears a factor lington equation, where is the free-space propagation constant, and this factor contributes large terms of the form when is near for odd. Thus at resonance with odd, the current is roughly approximated by a constant times . Similarly for even the current is . For roughly approximated by a constant times the present case of a wire on an interface, the Pocklington-type

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equation (19) has the factor , with the result that near resonance the current is roughly approximated as in the free-space case except that of free space is replaced by the complex-value .

there is a non-zero solution Suppose that for some value to (21). As shown in [6], the variation with respect of is zero, that to (as a function of ) around this is:

V. COMPLEX-VALUED RESONANT FREQUENCIES As in [1], [2], we relate complex-valued resonant frequencies to solutions of the homogeneous integral equation, that is the on set to 0 integral equation (19) with (21) At first glance, one might expect the only solution to be . However, the key to defining resonance is to note that the kernel depends not only on position along the wire, but also on the angular frequency , on which and depend through (7). Therefore the solution to the integral equation (21) depends on , and for certain discrete special values of , denoted , the equation has non-zero solutions. These values are expected to be complex, so that, listed in increasing order of their real parts, we have

(22) Under the assumption made here that is real-valued, the parameter (where we choose instead of , with the speed of light in vacuum) to express the th complex resonant frequency. The issue is how to find the at which (21) has non-zero solutions.

Equation (21) is just the statement that , whence it follows that . But since the first variation of the left-hand side is zero, replacing by an approximation makes no first-order error in . Thus we will determine the expression as the solution, for a suitable approximating current , to (24) Now we take advantage of the relative insensitivity of (24) to choose a simple approximating current. The resonances partition into those for which is odd and the current is symmetric in and those for which is even and the current is anti-symmetric in . For a single medium, we discussed in [6] the “shifted-cosine” approximation to the current, and rejected it because it introduced spurious values for the resonant frequencies. As in that case, here for the case of two media we approximate the , and for resonant current for odd as even as . For computational convenience we carry an -derivative under the integral and integrate by parts to obtain for the equation for

(25)

A. Approximate Solution for Resonant Frequencies To determine the complex values , we use a variational technique which can be expressed symbolically as follows. and both depend Noting that for the th resonance, , we let denote the linear integro-differential on operator in (21), and let the current for the th resonance be so that (21) is abbreviated as denoted by

where the denotes an integral over the spatial variable, and we have omitted writing the spatial variables and to emphasize the dependence of and on the propagation constant . The problem of resonance is to determine for the such that has a solution for . What we require is only ; we do not seek non-zero an accurate solution to the current . As in the case of a wire in a homogeneous medium [6], the is based on considering the funcmethod for determining tional

, the sought value of , is now limited the dependence on and the parameter to the dependence of the kernel on . B. Resonances Define . With the chosen approximating curfor the case of symmetric currents, (25) for determining rents in which is odd becomes

(26) We define (27) (28) and carry out the differentiations in (26) to obtain for

(23)

odd: (29)

MYERS et al.: ELECTROMAGNETIC RESONANCES OF A STRAIGHT WIRE ON AN EARTH-AIR INTERFACE

which in a form more convenient for calculation becomes (30) Similarly for the resonances with antisymmetric currents so that is even, one finds

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Similar to the case of the wire in a homogeneous medium, when the strip width is much smaller than all the other dimensions in this scattering problem, we can approximate the kernel [defined in (20)] by defined in (17), so that we have

(31)

(37) We will use this approximation to determine resonances.

which implies (32) To find the complex-valued resonant frequencies, one holds fixed the geometrical and material parameters and while varying the angular frequency in order to find complex values of for which (30) for the symmetric resonances and (32) for the antisymmetric resonances have non-zero solutions. Varying in the complex plane implies also varying the via for ; propagation constants and the dependence of and on (complex-valued) varies as determined by (12). The next step, derived in Appendix A, is to reduce the double integrals to single integrals to obtain, for all resonances, regardless of whether is even or odd, (33) where we define

(34)

D. Accuracy Apart from the minor contribution of numerical integrations to error, the error in the complex-valued resonant frequencies comes from the neglected terms in the solution for the pole in the Fourier transform of the electric-field kernel (Section III), the approximation of the kernel by (37), and the inexactitude of the approximation to the resonant current. For thin wires with , by far the largest contribution to error in the complex-valued resonant frequency is the inexactitude of approximation to the current. In order to get an idea of the size of this error, we proceed in a two-phase cycle: first we provisionally accept the value of complex-valued resonant frequency , which we now rewrite as , determined as above and use it to obtain an improved approximation to the current; second we use the improved approximation to the current to refine the approximation of the resonant frequency, denoted . Although here we limit ourselves to a single round of the cycle, the iteration can be continued. 1) Phase I: We view the dielectric constant as fixed, so that is rigidly geared to . Making explicit the dependence of the kernel and the current on the parameter , we have that evaluated at the complex resonant frequency for odd and , the exact current solves the equation obtained by transforming (21) and imposing a normalization condition (38)

C. Scaling to For calculation purposes, we save effort by recognizing that the resonant frequencies scale with the wire half-length . Thus one can set to 1 and obtain resonant frequencies for other we have values of by dividing by . With and the task is to solve the equation for resonances, namely, regardless of whether is even or odd,

(for even the cosine is replaced by a sine). When the approxiis substituted for the exact current , mate current the equation no longer holds, and the discrepancy can be expressed by

(35) where with

and

we have

(39)

(36)

To get a better approximation of the current, , we allow additional terms in the approximating current, with coefficients evaluated at obtained by minimizing the resulting .

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2) Phase II: Insert the determined in Phase I into . Reduction in error is measured by the ratio (25) to compute

complex values we want with the rule of “no cross branch lines” which tells us how to deform branch lines so as to avoid such given in (37), we define crossings. To express the kernel

(40)

(46)

Result for Lowest Resonance in Free Space: For the lowest resonance in free space we carried out the above steps. To get a better approximation to the current we enlarge the space of possible currents to include not only the in the approximating current but also a term term , so the approximating current becomes

After dropping constants that cancel out in the equations for resonant frequencies, for real-valued frequencies the kernel of the integral equation can be taken to be

(41) where

and

are determined numerically by minimizing . For this case, with , we find and . The coefficient is just a normalization factor, while the coefficient gives the relative size of the correction to the approximating current; thus is that correction is about 1.5%. The error measure . The resulting value reduced by a factor of 47 relative to . for the free-space resonant frequency is The relative magnitude of the difference between this and the computed in [6] is a little complex value less than 0.1%. Because this first cycle of the iterative procedure reduces the error measure so strongly, we take 0.1% to be a reasonable estimate of the error in the complex-valued resonant . We expect more or less the frequency computed using same size of error for the wire on the interface, and we expect ; determining the error for less error for smaller values of higher resonances is left to future work.

(47) The kernel depends on the resonant frequency through its , and , where itself is defined by the dependence on to and given in Section III. relation of . In [8] we show that for The integrand has poles at a real-valued dielectric constant , over the range has a relatively large positive real part and a small positive imaginary part for the pole that contributes to contour integrals evaluated in [8]. To avoid direct integration over an oscillating integrand that drops off only slowly, we split the integrand into a part that drops off slowly but can be integrated exactly analytically and other parts that have to be integrated numerically but that converge faster. By use of

VI. NUMERICAL ANALYSIS OF RESONANCES approxWe numerically solved (35) through (37) with imated by (37) for a wire scaled to a half-length of . To do this we had to evade two obstacles. The first obstacle is that, for complex-valued resonant frequencies of interest, the integration path must be deformed to avoid crossing branch cuts. For keeping track of branch cuts, we note that for imaginary values , and , the kernel is real-valued. of the parameters For this reason, keeping track of branch cuts is made easier by changing variables to ’s and the corresponding ’s defined by

(48) one obtains (49) where (50)

(42) (43) (51) where

and (44)

For

and

defined in (5) we have (45)

For real values of ’s, the branch points and branch lines present no trouble. We determine the proper branches involved in various integrals by following a path from positive real ’s to whatever

(52)

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Fig. 3. Two cases of the kernel and a=h = 10 .

10

K(x) when k =k

= 2 and k

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h = 5; a=h =

The last relation follows because the integrand falls off fast smooth as , so enough as becomes large to make that in the integrand we replace by 0 without significant loss of accuracy. can be evaluated exactly [8]: The term Fig. 4. Plot of first five resonant values of k h for k =k = 2.

(53)

TABLE II COMPLEX VALUES OF k h AT RESONANCE n FOR THREE VALUES OF a=h

where . In [8] the integrals in and are transformed to single integrals and put into forms convenient for numerical evaluation. for the case and Fig. 3 shows the kernel , and for two values of and . Unlike the wire in a homogeneous medium, the kernels for the two differ noticeably even for values of . cases of For the case of earth having a dielectric constant 4 times that , the first five normalized of free space, leading to are listed in Table II complex-valued resonant frequencies , the ratio of wire and plotted in Fig. 4 for three cases of radius to wire half-length. (Recall that the propagation constant where is the frequency in Hertz and is in air is m/s. Note that an even resonance the speed of light number corresponds to anti-resonance for the impedance of a center-driven antenna.) The computations were carried out using MATLAB on a personal computer. Because several multiple integrals had to be evaluated many times, a running time of several hours was involved. VII. CONCLUSION The problem of determining the first few complex-valued resonant frequencies of a straight wire on a flat earth-air interface has been formulated and solved. The approach started from the electric-field kernel, which, however, contains a non-integrable singularity, evaded by finding the zero point in the Fourier transwe form of the electric field kernel. Using this zero point arrived at the integro-differential equation (21), in which the kernel, defined by (37), is a double integral over a slowly decreasing, oscillating integrand. This kernel was transformed into

a form suitable for numerical analysis. Although an exact determination of the resonant frequencies would require solving the extremely challenging integro-differential equation (21), a variational technique allowed a crude approximation to the resonant current to be combined with our hard-won kernel to determine approximations to the resonant frequencies accurate to better than 0.1%. Because of the crude approximating current, we expect the method presented here to offer an accuracy in resonant frequencies on the order of 0.1% only for the first five resonances and

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The resonant frequencies as determined here can enter a future analysis of backscattering by a wire on an earth-air interface. At frequencies near resonance, the backscattered field is approximately a product of a geometric factor (involving the incident field and the resonant current) and a frequency factor; the frequency factor near the th resonance is proportional to , the real and imaginary parts of which are plotted in Figs. 5, 6, and 7 for the same three cases of as reported in Table II. Also left to future analysis are the effects of: (1) departure of the wire from straightness; (2) departure of the earth-air interface from flatness; and (3) resistive loss in the wire. APPENDIX A AND INTEGRATIONS FOR Fig. 5. Plot of 1=(k h

0k

) for case of k =k = 2 and a=h = 10

.

Because depends only on the difference between and , it must be possible to rewrite the integrals in (27) and (28) as single integrals. We do this without using the fact that is an even function of its argument. Define (A1) Changing integration variables duces the relation

and

pro(A2)

With this relation one expresses

and

as (A3) (A4)

Fig. 6. Plot of 1=(k h

0k

so that we have ) for case of k =k = 2 and a=h = 10

.

(A5) (A6) For the reduction to single integrals we compute, using the integration region shown in Fig. 8,

Fig. 7. Plot of 1=(k h

0k

) for case of k =k = 2 and a=h = 10

.

for a ratio of wire diameter to wire length that is less than or . equal to

(A7)

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[10] B. L. Coleman, “Propagation of electromagnetic disturbances along a thin wire in a horizontally stratified medium,” Phil. Mag. Series 7, vol. 41, pp. 276–288, 1950. [11] J. R. Wait, “Theory of propagation along a thin wire parallel to an interface,” Radio Sci., vol. 7, pp. 675–679, 1972.

John M. Myers received the Ph.D. degree in applied physics from Harvard University, Cambridge, MA, in 1963. Author of papers on electromagnetic wave scattering, quantum computing, quantum cryptography, clock synchronization, and issues in testing implementations of quantum-mechanical designs. As Senior Research Scientist, Raytheon Company, analyzed radar cross sections and effects of glint on radar tracking. Past analyst Office of the Secretary of Defense. Past member Defense Science Board Task Force for Electronic Warfare. Former consultant to Draper Laboratory in synchronization of fault-tolerant computers. Former consultant to the MIT Laboratory for Computer Science (parallelizing a program for solving hydrodynamic equations). IBM Visiting Professor at Brown University in 1998, co-teaching a course on quantum computing. International Advisory Board, Measurement Science and Technology. Co-inventor of receiver for single-photon light pulses under Patent 5999285.

Fig. 8. Integration region on (x ; y )-plane.

The case

is worked out directly to show

(A8)

ACKNOWLEDGMENT This work was undertaken at the suggestion of J. Hersey, whom the authors thank for posing the problem and for many helpful discussions along the way. They also thank the referees for extraordinarily helpful suggestions.

Sheldon S. Sandler (SM’53–M’63–LM’02) received the Ph.D. degree in applied physics from Harvard University, Cambridge, MA, in 1962. Engages in research in applied electromagnetics, antennas, and remote sensing. He is at Harvard and also a Research Fellow in the Department of Archaeology at Boston University, Boston, MA, working on ground-penetrating radar applications to archaeology. He is Professor Emeritus in the Electrical and Computer Engineering Department, Northeastern University, Boston, MA. At Northeastern he worked on intelligent antennas, electromagnetic effects on biological systems and insect vision, and picture processing and reconstruction. At Geo-Centers Inc., Newton Centre, MA, he did research on topics related to geophysical exploration. He was a Visiting Professor at the E.T.H., Zurich, Switzerland, and at the University of Zurich; also he was a Visiting Scholar at the M.R.C. Laboratory for Molecular Biology, Cambridge, England. He was a Thomas A. Edison Fellow in the Department of Electrical Engineering, Yale University, New Haven, CT.

REFERENCES [1] C. E. Baum, “The singularity expansion method,” in Transient Electromagnetic Fields, L. B. Felsen, Ed. New York: Springer-Verlag, 1975. [2] C. E. Baum, “Emerging technology for transient and broad-band analysis and synthesis of antennas and scatterers,” Proc. IEEE, vol. 64, pp. 1598–1616, Nov. 1976. [3] H. Kober, Dictionary of Conformal Representations. New York: Dover, 1957. [4] R. W. P. King, The Theory of Linear Antennas. Cambridge, MA: Harvard Univ. Press, 1956. [5] F. M. Tesche, “On the analysis of scattering and antenna problems using the singularity expansion technique,” IEEE Trans. Antennas Propag., vol. AP-21, pp. 53–62, Jan. 1973. [6] J. M. Myers, S. S. Sandler, and T. T. Wu, “Electromagnetic resonances of a straight wire,” IEEE Trans. Antennas Propag., vol. 59, pp. 129–134, Jan. 2011. [7] R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves: Theory and Applications to Communications, Geophysical Exploration and Remote Sensing. New York: Springer-Verlag, 1992. [8] J. M. Myers, S. S. Sandler, and T. T. Wu, “Electromagnetic Resonances of a Wire on an Earth-Air Interface,” Tech. Rep., Army Research Office Grant W911NF-07-1-0509, Nov. 12, 2009 [Online]. Available: http://www.dtic.mil/ from the Defense Technical Information Center, by searching on ADA520611. [9] Higher Transcendental Functions, A. Erdélyi, Ed. New York: McGraw-Hill, 1953, vol. II.

Tai Tsun Wu received the Ph.D. degree in applied physics from Harvard University, Cambridge, MA, in 1956. He is the Gordon McKay Professor of Applied Physics and Professor of physics at Harvard University, and Scientific Associate at CERN, Geneva, Switzerland. Most recently, he was the Principal Investigator of the Harvard University study on Physics of Secure Quantum Communication, for BBN Technologies as part of the DARPA QuIST Program (August 2001 through May 2006). He is a co-author of the following six books: Scattering and Diffraction of Waves; The Two-Dimensional Ising Model; Antennas in Matter: Fundamentals, Theory, and Applications; Expanding Protons: Scattering at High Energies; The Ubiquitous Photon: Helicity Method for QED and QCD; and Lateral Electromagnetic Waves: Theory and Applications to Communications, Geophysical Explorations, and Remote Sensing. In addition, he has published 400 original papers in refereed scientific journals, conference proceedings, and chapters in books, many contributing to quantum field theory and classical electromagnetic theory. He was involved in designing a light-weight, low-frequency, broad-band antenna for Air Force One. Dr. Wu was co-recipient of the Dannie Heineman Prize for Mathematical Physics from the American Institute of Physics in 1999, and received the Alexander von Humboldt U.S. Senior Scientist award in 1985–1986.

In the special case which we have here, in which and simplify to

(A9) which are the single integrals that we wanted to obtain.

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Gain Enhancement by Dielectric Horns in the Terahertz Band Belen Andres-Garcia, Enrique Garcia-Muñoz, Sebastian Bauerschmidt, Sascha Preu, Stefan Malzer, Gottfried H. Döhler, Lijun Wang, and Daniel Segovia-Vargas, Member, IEEE

Abstract—A new geometry for the design of antennas in the Terahertz band is presented. The structure is based on a horn antenna etched in the substrate and fed with a planar printed antenna used for generation of terahertz radiation, designed for the 200 GHz to 3 THz range. For the proposed antenna, the energy distribution through the substrate is reduced towards an increase in the gain of the system, at least, 8 dB in a 1:10 bandwidth. The structure has been measured showing the expected behavior in the low band. Index Terms—High power terahertz, horn antennas, photomixers, planar antennas, spiral antennas, terahertz.

I. INTRODUCTION HE Terahertz (THz) band has been in a great expansion during the last decade due to its growing interest for spectroscopy, radioastronomy, imaging, etc. [1], [2]. The room temperature generation of THz radiation, especially high power signals, has been an issue for all the technologies in this frequency band. From electronics, THz generation has been achieved through microwave technology achieving a maximum power of 1 mW around 1 THz [3]. On the other side, optoelectronic devices have been developed for THz generation by achieving a maximum power in the range of a few photomixing of two laser beams in a semiconductor [9], [10]. Here, different devices based on a photoconductor with short carrier lifetime [6], especially designed UTC p-i-n-diodes [7], or n-i-pn-i-p superlattice devices [8] are used to absorb the optical beating signal and transform it into a an AC (THz) current. The current is fed into an antenna to radiate the THz signal. All these devices operate at room temperature and are widely tunable by tuning the optical frequency of one of the mixing lasers. The design of the antenna has to address the coverage of

T

Manuscript received December 02, 2010; revised January 21, 2011; accepted February 17, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported by CONSOLIDER TERASENSE CSD2008-00068 and CICYT NeoImag TEC2009-14525-C02-01. B. Andres-Garcia, E. Garcia-Muñoz, and D. Segovia-Vargas are with the Signal Theory and Department, Carlos III University of Madrid, 28911 Madrid, Spain (e-mail: [email protected]). S. Bauerschmidh, S. Malzer, and G. H. Döhler are with the Max Planck Institute for the Science of Light, 91058, Erlangen, Germany. S. Preu was with the Max Planck Institute for the Science of Light, 91058, Erlangen, Germany. He is now with the Materials and Physics Department, University of Santa Barbara, Santa Barbara, CA 93106 USA. L. J. Wang is with Physics Department, Tsinghua University, Beijing 100084, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161563

the desired frequency range and the impedance matching to the photomixer. The output signal is usually radiated by an antenna designed in the THz range [8], [10]. These feeding structures have strong restrictions in terms of antenna implementation. Typically, they are placed on a semi. conductor substrate with a high dielectric constant The design of antennas with good performance in terms of radiation efficiency and input impedance that could perfectly match the source (as in the microwave regime) results in a great challenge, especially for broadband radiation. During the last years, several techniques have been developed to optimize the radiated power, especially, at room temperature. These techniques have been based either on the antenna gain increase [11], or on a a reduction of the matching loses [11], [12]. As the photomixers are broadband devices, although their roll-off factor [8] strongly limits the output power, the main limiting factor, in terms of bandwidth performance, is usually the antenna. These radiating elements must be designed with the photomixers in planar technology, printed over a high dielectric constant material. In addition to the high dielectric constant, the thickness of the substrate must be large enough in wavelengths to allow the device heat dissipation. Previous conditions lead to a poor directivity due to the energy distribution in the semiconductor substrate. This forces the use of silicon lenses in order to improve the radiation pattern [12]. The joint effect of these drawbacks and the lack of high gain or large power generation devices for the THz range in planar technology is the driving force for new structure designs for the outcoupling of the generated THz radiation. One example of these new structures is the use of micro-machined horns, fed with printed antennas over membranes, working in the millimeter wave regime [13]. In this example, the micro-machined horns are metallic air-loaded horns. They are also based on the idea of focusing the THz radiation to increase the final system gain. From another point of view, the properties of antennas based on dielectric filled waveguides are well known. These structures reduce the energy distribution in the material, guide the power in a high efficient way, and feed the dielectric loaded horns or dielectric horns [14]–[17]. Dielectric filled-horns have been widely studied [14]–[17], as well as dielectric waveguides [18] in the previous years. These studies show good results in terms of cross-polarization levels, allowing a reduction in the size of the horns as well as the elimination of corrugations to achieve symmetric radiation patterns and good beam characteristics with a broad bandwidth. The use of semiconductor substrates can open new ways for implementing planar printed antennas in the THz range.

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Fig. 1. (a) Energy distribution on a high permittivity substrate, (b) proposed structure with the antenna on top.

A dielectric semiconductor horn antenna etched on a high permittivity and thick substrate is proposed to focus the radiation, reduce the energy distribution and enhance the gain of the device in the broadside direction in the THz band. A sketch of this process is shown in Fig. 1(a) and (b). In the first case a conventional system is depicted, where the radiation has a strong diffusion through the substrate. In order to overcome this issue, a horn antenna is etched into the semiconductor as shown in Fig. 1(b). The energy diffusion is reduced in the substrate, focalizing all the radiation into the broadside direction. The large thickness of the substrate and its high dielectric permittivity are now used in advantage to increasing the efficiency of the dielectric horn in the THz range. For the experimental realization, the antennas are fed by n-i-pn-i-p photomixers [12], which provide high power at room temperature. This new structure also allows the use of a dielectric lens in the mouth of the horn, with which higher gains can be achieved with higher directivity and also eliminate reflection losses due to the dielectric-air interface. The paper is organized as follows: in Section II theoretical results for the dielectric waveguide and horn are presented, while in Section III the results are particularized for the excitation of the system by an antenna. In Section IV the measurement results and fabrication process are shown. II. DIELECTRIC WAVEGUIDE AND HORN A. Propagating Modes on a Dielectric Waveguide First, the modes propagating in a dielectric waveguide at THz frequencies (the minimum operating frequency will be 100 GHz) are calculated. A modal chart is represented in Fig. 2. It corresponds to an asymptotic case of a dielectric waveguide directed along based on the assumption that the dimension in the axis is infinite while the direction has the thickness of the substrate, . The curves of Fig. 2 are the results of solving (propagation constant) and the characteristic (1) and (2) for (attenuation) for and modes respectively (even and odd) for a dielectric waveguide of InGaAs and according to [18] at 200 GHz

(1) (2)

= 12 9

: , Fig. 2. First four propagating modes for a dielectric waveguide of  and a  at 200 GHz. TE (dashed) modes odd and even and TM (solid) modes odd and even.

= 180 m

The first two modes that propagate in this waveguide are the and [18] modes, thus, the fundamental mode is the and a , the hybrid result of the combination of a [15], [18]. The Fourier transform of the fields for this mode in the dielectric waveguide aperture produces broadside radiation. and ) because they We neglect the two first modes ( do not correspond to any guided mode [19]. and modes For all the modes shown in Fig. 2, both (of the same order) have the same cut-off frequency. Thus all the modes propagating in the structure are combination of and , resulting in hybrid modes. In some references [15], [20], through mode matching techniques and after some approximations to the problem of solving the modes that propagate in a rectangular and circular dielectric waveguides, it is concluded that these structures allow hybrid . This modes propagation, with the fundamental mode agrees with other works on this field [14], [18] about substrate waveguides and horns partially filled with dielectrics. Due to the high dielectric constant, the single mode operation bandwidth is reduced in comparison with previous results [14], [16], and more power is coupled into higher order modes. This prevents the radiation from being highly directive [14] especially when the flare angle is large. Fig. 2 is used in this case for the design of the waveguides for Section III, for not allowing the propagation of higher order modes. Previous works on dielectric waveguides or loaded-dielectric waveguides have presented good results by reducing the cross polarization levels of unloaded horns thanks to the excitation higher modes [14]. In spite of the cross polarization of reduction, the presence of these higher order modes also limits the bandwidth when the power coupled to these modes is too high [14]. Here, the dielectric waveguide is used to propagate the mode to improve the energy coupled from the mode to the dielectric horn. If we discard the slab, this coupling is less efficient in terms of the energy coupled to the fundamental mode.

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TABLE I GAIN COMPARISON WITH DIPOLE EXCITATION

Fig. 3. Sketch of the system: horn ended with an infinite substrate layer.

B. Dielectric Horn The rectangular dielectric waveguide is open to form a dielectric horn as depicted in Fig. 3. In this figure, the antenna is printed on the left side of the dielectric slab, while in the right-hand side the substrate is infinite towards the direction. In order to study the propagation in the dielectric horn, one can make the same assumption that in conventional horns. When mode is propagating, the phase term for the spherical the wave front in the mouth of the horn is added. Finally, the radiated fields are obtained as there is no change in the medium because an infinite dielectric is assumed [15]. mode can According to previous studies [14]–[16], the propagate in dielectric filled horns if an air-gap between the substrate and the metal walls were present [16]. In this paper we assume an infinite gap so the equations are still valid in the limit. mode or a combination of modes Therefore, the can be excited in the horn. These modes will produce broadside radiation. Regarding the broadband behavior of the dielectric horn, the operation frequencies are from the cut-off frequency of the dielectric slab that feeds the horn up to the limit when higher order modes are propagating in the structure with comparable power. Within higher frequencies, more energy is coupled to these higher order modes and not only to the fundamental mode, and the radiation is no longer broadside [14]. This is also strongly limited by the dielectric permittivity [14] where the energy into higher order modes is strongly coupled producing a reduction in the operating bandwidth in the upper limit. III. WAVEGUIDE AND HORN EXCITATION BY A PRINTED ANTENNA For the excitation of the dielectric waveguide, a planar printed antenna is used, so the modes excited on each case can be identified by the set of modes that can propagate in the specific waveguide. The antennas employed in this paper are dipole and logspiral antennas. The proposed feeding antennas are patterned in one of the ends of a dielectric slab that feeds the substrate horn

as shown in Fig. 3. The dimension of the dielectric waveguide and is the same in both cases, that feeds the horn is and is chosen according Fig. 2. This size only allows the propagation of the first two modes that compose the fundamental mode. All the waveguides and horns considered in this paper have the same flare angle in both E and H plane. For the generation of THz radiation a planar structure is employed. For the proposed system, a n-i-pn-i-p superlattice phois used. The photomixer [12] on an InP substrate tomixer generates the current in the THz band that results from the beat of two monochromatic laser signals. All the simulations are performed with CST 2010 [21] assuming an infinite dielectric along the axis in the mouth of the horn. This assumption is made to neglect the reflections from the air-InGaAs interface that would cause bad quality radiation patterns. In Section IV, for practical implementation a high resistivity Silicon lens will be added to the system agreeing with this assumption. The polarization of the output field agrees with the one of the original antenna. A. Dipole Excitation For the first case a dipole excitation is studied. As it is linearly polarized, fewer modes than with the spiral feeding will be excited in the waveguide. The field excited by a dipole inside a dielectric horn is the fundamental mode. The half-wavelength dipole for the simulations is designed for a resonating frequency of 900 GHz, printed in the air-InGaAs interface of the dielectric. The input impedance of the antenna does not change, compared with the case of the antenna over InGaAs without the horn, as the dipole can be considered as printed in the same air-InGaAs interface as for previous studies [12]. The main difference in the performance of the antenna lies in the radiation pattern characteristics: the 3 dB beamwidth and the gain. Table I shows the result for the antenna gain in simulation, printed over a flat substrate and over the etched dielectric horn, using an infinite dielectric width. It can be seen that an increase in the gain of the antenna of 15 dB at 700 GHz results when a substrate pyramidal horn with an aperture of 737.2 in both E and H of InGaAs and a flare planes, a length of 900 angle of 28 This change in the radiation pattern is due to the focusing of the energy through the broadside direction reducing the diffusion of the energy into the substrate. Fig. 4 shows the behavior for the gain of the antenna in terms of the aperture angle and the length when the excitation is the half-wavelength dipole. It has a maximum for each aperture angle, and an absolute maximum, as in the conventional metallic

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TABLE II GAIN COMPARISON WITH SPIRAL EXCITATION

Fig. 4. Gain in terms of aperture angle and length of the horn at 900 GHz.

horns [18]. The maximum gain achieved is 22.5 dB with a single dipole feeding the dielectric slab. As long as the aperture angle is small enough (less than 35 ), there is a strong increase in the gain of the system, especially when the length of the horn . Nevertheless, when the increase in gain is more than is minimum, compared to a conventional system, this structure produces an increase of at least 4 dB in gain. B. Spiral Excitation In order to take advantage of the broadband characteristics of the dielectric horn and the photomixer used for THz generation, a spiral antenna printed on one side of a dielectric waveguide is used to feed the dielectric horn. This spiral is designed to work . The from 150 GHz to 3 THz [22], with a diameter of 150 size of the antenna should be the smallest possible in order to reduce the dielectric waveguide input aperture size, . In this way a highest working frequency will be achieved. The size of the waveguide has also been chosen according to Fig. 2, allowing and ). the propagation only of the first two modes ( The cut-off frequency of the waveguide is always below the dimensions of the antenna. Then, according to (3) and taking into account that the antenna is always larger in diameter than one half of the minimum wavelength in the dielectric, being the mode order, and , , , and the dielectric and vacuum permittivities and permeabilities respectively, and the aperture size of the waveguide, the lower cut-off frequency will be given by the size of the antenna

(3) It should be noted that the limiting factor for the upper frequency is the dielectric horn substrate due to the higher order modes propagating in the substrate. This is due to the combined effect of a high dielectric constant with a wide aperture shape [14]. In addition, the coupled energy to these higher order modes is large. For the case of the spiral, the modes excited in the structure are two modes, shifted 90 . Then, the pyramidal horn

Fig. 5. Gain for the spiral fed antenna in terms of aperture angle and length of the horn at 1 THz. The black dot represents the manufactured antenna for Section IV.

substrate will also radiate circular polarization as the isolated spiral. The axial ratio for broadside direction is 0.6 dB at 1 THz in simulation. Table II shows a comparison between the gains for the spiral system at different frequencies similar to the one shown for the dipole case. The length of the horn substrate has been set to while the aperture angle to 25 . There is an increase of 450 14.5 dB at 1 THz and more than 7 dB in the whole operation bandwidth. In this way, the broadband characteristics of photomixers, log-spirals and substrate horns are combined to build an ultra-wideband system. All these results are obtained with the simulation of the antenna over a semi-infinite substrate. The gain of the system for different aperture angles and horn lengths at 1 THz is shown in Fig. 5. The maximum gain is achieved when the flare angle is 28 . It can also be seen that is almost independent with the length of the horn. The best values to achieve the maximum gain at 1 THz are: aperture angle of . With these values, the working fre28 and length of 850 quency band is centered at 1 THz and the horn is working from 0.2 THz to 2 THz.

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TABLE III SIMULATED GAIN COMPARISON WITH SPIRAL EXCITATION FOR THE MANUFACTURED ANTENNA

Fig. 6. (a) Measurement setup and (b) SEM image of the dielectric horn antenna processed with a depth of 150  and a 54 deg. flare angle.

m

Fig. 7. Power spectrum of two different measurements for the antena+horn (solid and dots) for showing measuring tolerances. Reference antenna (dashed) in logarithmic scale.

IV. EXPERIMENTAL RESULTS The proposed structure has been manufactured to provide validation to the theoretical analysis. The excitation element employed used for THz generation is a spiral, printed over a dielectric slab which opens with a 54 . angle . The aperture of the horn on the and has a depth of 240 other side is approximately (a SEM picture of the device is shown in Fig. 6(a)). The semiconductor dielec. Due to the fabricatric substrate is InGaAs with an tion process and etching limitations [23] the resulting angle is , this point is indicated in Fig. 5 54 for a horn length of 150 with a black dot. The final dimensions of the horn are not optimal for the application at 1 THz. The dielectric slab was not etched, so the antenna was directly printed on the mouth of the horn, not being mode as optimal regarding the coupling of energy to the mentioned in Section II. The results were compared with a reference of the same antenna, printed on the same wafer but without etching the horn. The measurements were made with the set up shown in Fig. 6(b), using a hyperhemispherical 5 mm radius Si lens , two parabolic off-axis mirrors with with a slab of 1080

Fig. 8. Power spectrum of the antenna+horn normalized to the reference antenna in logarithmic scale.

and a Golay cell placed in the focus of the second parabolic mirror. We have measured the power spectrum of the device from 100 GHz up to 1.5 THz. The efficiency defined as is shown in Fig. 7 with the spectrum of the reference antenna. According to the simulations performed for the dimensions of the physical horn after the etching, it provides 4 dB increase of gain at 200 GHz. The simulated gain comparison for the manufactured antenna is shown in Table III for different frequencies. Despite the horn is not optimal regarding the size, the gain of the sistem is still above the reference antenna in the whole band. This agrees with the results obtained from the measurements, where there is an increase in the efficiency in the 200 GHz–400 mode can propagate in the manuGHz band, where the factured structure. The strong peak below 200 GHz is attributed to an artefact as the antenna used for these devices has a low frequency cut off at around 200 GHz (where the horn would work optimum). For frequencies higher than 400 GHz, the horn is extremely big so the performance is not optimum, and higher order modes

ANDRES-GARCIA et al.: GAIN ENHANCEMENT BY DIELECTRIC HORNS IN THE TERAHERTZ BAND

are propagating. In Fig. 8 we show the efficiency of the dielectric horn antenna normalized to the efficiency of the reference antenna in a logarithmic scale. This efficiency is higher than the reference antenna for nearly the whole band. Regarding measurement tolerances, in Fig. 7 are also shown two different measurements for the antenna under test (AUT), this is the horn antenna. For all the cases, the system is setup in an alignment when the maximum power is received, discarding alignment errors for all the measurements. The error obtained is below the difference for the horn-reference case, so the deviation is due to an improve on the efficiency for the horn antenna with respect to the reference. V. CONCLUSION A new substrate horn antenna in the THz band has been developed. A horn substrate with a planar printed antenna over a mode has been used. The dielectric slab propagating the energy coupling to this mode is maximized through the dielectric slab. A 1:10 bandwidth has been achieved with a broadband log-spiral antenna. The measurements agree with calculations despite the frequency shift associated with manufacturing issues. At this moment new devices are being processed with a central frequency of 1 THz with a different mask set.

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[12] S. Preu, F. H. Renner, S. Malzer, G. H. Döhler, L. J. Wang, M. Hanson, A. C. Gossard, T. L. J. Wilkinson, and R. E. Brown, “Efficient terahertz emission from ballistic transport enhanced n-i-p-n-i-p superlattice photomixers,” Appl. Phys. Lett., vol. 90, no. 21, pp. 212115–212115-3, May 2007. [13] G. M. Rebeiz, L. P. B. Katehi, W. Y. Ali-Ahmad, G. V. Eleftheriades, and C. C. Ling, “Integrated horn antennas for millimeter-wave applications,” IEEE Trans. Antennas Propag. Mag., vol. 34, no. 1, pp. 7–16, Feb. 1992. [14] A. D. Olver, P. J. B. Clarricoats, and K. Raghavan, “Dielectric cone loaded horn antennas,” IEE Proc., vol. 135, no. 3, pt. H, pp. 158–162, June 1988. [15] S. K. Palit and W. Perris, “Dielectric-loaded pyramidal horns,” J. Elect. Electron. Eng., vol. 16, no. 2, pp. 139–145, 1996, Australia. [16] E. Lier, “A dielectric hybrid mode antenna feed: A simple alternative to the corrugated horn,” IEEE Trans. Antennas Propag., vol. Ap-34, no. 1, pp. 21–29, Jan. 1986. [17] G. V. Eleftheriades, W. Y. Ali-Ahmad, L. P. B. Katehi, and G. M. Rebeiz, “Millimeter-wave integrated-horn antennas: Part I theory,” IEEE Trans. Antennas Propag., vol. 39, no. 11, pp. 1575–1581, Nov. 1991. [18] C. A. Balanis, Advanced Engineering Electromagnetics. Englewood Cliffs, NJ: Prentice Hall, 1989. [19] I. A. Eshrah, A. B. Yakovlev, A. A. Kishk, A. W. Glisson, and G. W. Hanson, “The TE00 waveguide mode—The complete story,” IEEE Antennas Propag. Mag., vol. 46, no. 5, pp. 36–41, Oct. 2004. [20] E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J., Sept. 1969. [21] CST Microwave Studio, 2010. [22] J. L. Volakis, Antenna Engineering Handbook. New York: McGraw Hill, 2007. [23] P. Elias, I. Kostic, J. Soltys, and S. Hasenöhrl, “Wet-etch bulk micromachining of (100) InP substrates,” J. Micromech. Microeng., vol. 14, pp. 1205–1214, 2004.

REFERENCES [1] P. H. Siegel, “Terahertz technology,” IEEE Trans. Microwave Theory Tech., vol. 50, no. 3, pp. 910–928, Mar. 2002. [2] M. Tonouchi, “Cutting-edge terahertz technology,” Nature Photon., vol. 1, pp. 97–105, 2007. [3] B. Thomas, A. Maestrini, J. Gill, C. Lee, R. Lin, I. Mehdi, and P. de Maagt, “A broadband 835–900 GHz fundamental balanced mixer based on monolithic GaAs membrane Schottky diodes,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 7, pp. 1917–1924, Jul. 2010. [4] I. C. Mayorga, P. Muñoz Pradas, E. A. Michael, M. Mikulics, A. Schmitz, P. van der Wal, C. Kaseman, R. Güsten, K. Jacobs, M. Marso, H. Luth, and P. Kordos, “Terahertz photonic mixers as local oscillators for hot electron bolometer and superconductor-insulator-superconductor astronomical receivers,” J. Appl. Phys., vol. 100, no. 4, pp. 043116–043116-4, Aug. 2006. [5] A. Stohr, A. Malcoci, A. Sauerwald, I. C. Mayorga, R. Güsten, and D. S. Jäger, “Utra-wide-band travelling wave photodetectors for photonic local oscillators,” J. Lightw. Technol., vol. 21, no. 12, pp. 3062–3070, Dec. 2003. [6] E. R. Brown, “THz generation by photomixing in ultrafast photoconductors,” Int. J. High Speed Electron. Syst., vol. 13, no. 497, pp. 147–195, 2003. [7] H. Ito, F. Nakajima, T. Futura, and T. Ishibashi, “Continuous THzwave generation using antenna-integrated uni-traveling-carrier photodiodes,” Semicond. Sci. Technol., vol. 20, no. 191, pp. 141–150, 2005. [8] G. H. Döhler, F. Renner, O. Klar, M. Eckardt, A. Schwanhuer, S. Malzer, D. Driscoll, M. Hanson, A. C. Gossard, G. Loata, T. Löffler, and H. Roskos, “THz-photomixer based on quasi-ballistic transport,” Semicond. Sci. Technol., vol. 20, pp. 178–190, 2005. [9] I. C. Mayorga, P. Muñoz Pradas, E. A. Michael, M. Mikulics, A. Schmitz, P. van der Wal, C. Kaseman, R. Güsten, K. Jacobs, M. Marso, H. Luth, and P. Kordos, “Terahertz photonic mixers as local oscillators for hot electron bolometer and superconductor-insulator-superconductor astronomical receivers,” J. Appl. Phys., vol. 100, no. 4, pp. 043116–043116-4, Aug. 2006. [10] A. Stohr, A. Malcoci, A. Sauerwald, I. C. Mayorga, R. Güsten, and D. S. Jäger, “Utra-wide-band travelling wave photodetectors for photonic local oscillators,” J. Lightw. Technol., vol. 21, no. 12, pp. 3062–3070, Dec. 2003. [11] I. S. Gregory, W. R. Tribe, C. Baker, B. E. Cole, M. J. Evans, L. Spencer, M. Pepper, and M. Missous, “Continuous-wave terahertz system with a 60 dB dynamic range,” Appl. Phys. Lett., vol. 86, no. 20, pp. 204104–204104-3, May 2005.

Belen Andres-Garcia (S’08) was born in Ourense, Spain, in 1984. She received the Engineer degree in telecommunications from Carlos III University, Madrid, Spain, in 2008, where she is currently pursuing the Ph.D. degree in communications. She authored and coauthored 15 journal articles and conference contributions. Her research interests include terahertz antennas, quasioptical systems and antenna arrays. Miss Andrés-Garcéia received the Best Master Thesis Dissertation award from the COIT/AEIT (Official Association of Spanish Telecommunication Engineers) in 2008.

Enrique Garcia-Muñoz is an Associate Professor at the Universidad Carlos III de Madrid, Spain. He has managed or participated in several national and European research projects on areas such as antennas and array design. He has coauthored more than 50 papers in international journals and conferences and holds three patents. His current research interests include terahertz antennas, array design, truncation in antenna arrays and radioastronomy instrumentation.

Sebastian Bauerschmidt was born in Fürth, Germany, in 1983. He received the Diploma degree in physics in 2010 from the Max Planck Institute for the Science of Light, Erlangen, Germany. He is currently pursuing the Ph.D. degree at the University of Erlangen-Nuremberg. His research is focused on the development of THz photomixers as well as their application in spectroscopy and experiments with whispering gallery mode resonators.

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Sascha Preu was born in Weissenburg, Germany, in 1980. He received the Diploma degree in physics from the Max Planck Institute for the Science of Light, Erlangen, Germany, in 2005 and the Ph.D. degree in physics (summa cum laude) from the Univ. of Erlangen-Nuremberg, in 2009. He is currently with the Materials and Physics Department, University of Santa Barbara, California. He authored and coauthored 17 journal articles and conference contributions. His research interests focus on the development of semiconductor-based THz sources and detectors. He also works on applications of THz radiation, in particular the characterization of novel THz optics such as whispering gallery mode resonators.

Stefan Malzer received the Ph.D. degree in semiconductor physics from the University of ErlangenNürnberg, Germany, in 1991. He has been a staff member at the Institute of condensed Matter at the University of Erlangen. There he is involved in III-V-optoelectronic research projects on quantum dots and high speed opto-electronics. Currently, his research focus goes into THz-generation by photomixing in semiconductor nanostructures. He is coauthor of 80 papers in international journals and conference proceedings and holds one patent.

Gottfried H. Döhler received the Ph.D. degree from the Technical University in Munich (Germany) in 1968. He has worked with the Max-Planck-Institute in Stuttgart and with HP-Labs in Palo Alto, before taking a position as Professor at the University of Erlangen-Nürnberg (Germany). After retiring he joined a Max-Planck Research Group, which became the Max Planck Institute for the Science of Light in 2009. Most of his research has been devoted to experimental and theoretical investigations of transport and optical properties of superlattices, nanostructures, nanodevices and quantum dots. Most recently he has worked on femtosecond transport in nanostructures and its application to THz emitters. He has (co)-authored more than 400 publications including many reviews, book chapters, and patents. Dr. Döhler has been a chairman or committee member in numerous international conferences and has been serving as a referee for many funding agencies and scientific journals. The Walter-Schottky-Prize of the German Physical Society was awarded to him for his work on n-i-p-i doping superlattices in 1984, a

Sarojini Damodaran Fellowship by the Tata Institute of Fundamental Research (TIFR, Mumbai, India) in 2004 and a Chair of Excellence by the University Carlos III in Madrid in 2010.

Lijun Wang was born in Beijing in 1966. He received the B.S. degree in physics from the University of Science and Technology of China, in 1986 and the Ph.D. degree in physics from the University of Rochester, in 1992. He was a Research Associate at Duke University from 1992 to 1994. From 1994 to 1996, he was a Senior Scientist at the Research Institute of General Atomics Corp. From 1996 to 2004, he was a Research Scientist and subsequently a Senior Member of the Technical Staff and Department Head at the NEC Research Institute (NEC Laboratories America) in Princeton, NJ. From 2004 to 2008, he was a Director of the Max-Planck Research Group Institute for Optics, Information, and Photonics. He is also a Chair Professor of Experimental Physics (C4) at the University of Erlangen-Nuremberg, in Erlangen, Germany. In 2006, he was appointed adjunct Professor of Engineering. Since January 2009, he is a Max-Planck Fellow (part time) at the Max-Planck Institute for the Science of Light (MPL Erlangen). He joined the Physics Department and the Department of Precision Instrumentation at Tsinghua University in 2010, to start a new Joint Institute of Metrological Science installed jointly by Tsinghua University and the National Institute of Metrology of China. Dr. Wang’s research work at the NEC Institute was widely reviewed. His work on anomalous light pulse propagation, published in Nature in 2000 was selected as Top 10 Physics News in 2000 by Science News Magazine.

Daniel Segovia-Vargas (M’98) was born in Madrid, Spain, in 1968. He received the Telecommunication Engineering degree and the Ph.D. degree from the Polytechnic University of Madrid, in 1993 and in 1998, respectively. From 1993 to 1998, he was an Assistant Professor at Valladolid University. Since 1999, he is an Associate Professor at Carlos III University in Madrid where he is in charge of the Microwaves and Antenna courses. Since 2004, he is the Leader of the Radiofrequency Group, Dept. of Signal Theory and Communications, University Carlos III of Madrid. He has authored and coauthored over 110 technical conference, letters and journal papers. His research areas are printed antennas and active radiators and arrays, broadband antennas, LH metamaterials, terahertz antennas and passive circuits. He has also been expert of the European Projects Cost260, Cost284 and COST IC0603.

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Part I: A New Theory for Modeling Conductor-Backed Planar Slot Antenna Elements, in the Presence of a General Non-Planar Surrounding Brian L. McCabe, Member, IEEE, and Nirod K. Das, Member, IEEE

Abstract—We have developed a new analytical technique to model conductor-backed slot antenna elements in the presence of a general non-planar surrounding. The basic approach provides a new analytical framework to rigorously combine independent modeling of a central planar element and a non-planar surrounding. This would lead to an efficient modeling and design of “locally planar” structures. The theory is based on a complex spectral-plane analysis, which incorporates general scattering from its surrounding through a contour-deformation and harmonic decomposition analysis. The accuracy and versatility of the new method is demonstrated by applying it to selected source and scattering structures. The results are validated by comparing with independent data. Index Terms—Cavity-backed slot, hybrid method, moment method, quasi-planar elements.

I. INTRODUCTION ONDUCTOR backing behind a slot antenna or a slot-coupled microstrip patch antenna is of particular interest for various practical considerations [1], [2]. It may also be useful to introduce metal shorting posts in the conductor-backing region to suppress any unwanted parallel-plate radiation. Other added structures around the slot element, such as a metal or dielectric cavity backing of different shapes, a packaging enclosure for any circuit in the conductor-backing region, or a non-planar dielectric guide to couple signal to the slot element, may also be used. Such geometries represent a general class of practical configurations where planar elements are used together with non-planar surrounding. Geometries that are “strictly planar” can be efficiently analyzed using a spectral-domain technique [3]–[6]. Similar spectral models may also be extended to layered structures on cylindrical surfaces [7]. However, such planar models would not be applicable to structures that are only “locally planar,” with a general non-planar surrounding. The local planar element and the non-planar surrounding could have distinct functional roles, and they may have been modeled independently using different computational tools or techniques. It would be useful to develop modeling approaches which can

C

Manuscript received December 09, 2008; revised December 24, 2010; accepted December 28, 2010. Date of publication July 14, 2011; date of current version September 02, 2011. B. L. McCabe is with Sikorsky Aircraft, Stratford, CT 06614 USA. N. K. Das is with the Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, Six Metrotech Center, Brooklyn, NY 11201 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2161433

combine results from independent modeling of the planar and non-planar parts, in order to efficiently analyze and design composite structures. Hybrid approaches combining different analysis techniques and physical environments have been investigated in the past [8], [9]. In this part of the paper, we present a new analysis to model a general class of geometries discussed above. The approach would allow new computational features that are not feasible using conventional numerical techniques and computational tools available today. The potential new features include: (1) Planar elements embedded in a non-planar surrounding can be modeled essentially like open planar structures, with significant computational efficiency. (2) A planar antenna or circuit configuration and an external environment, such as a packaging or a cavity-backing structure, may be independently modeled using different computation tools or techniques, as optimally suited for the particular part. The results for the separate parts can then be rigorously combined in the final analysis. (3) Any repositioning or reorientation of the parts can be handled with minimal re-computation, without having to model the total structure again. (4) Experimental scattering signature for the surrounding may also be rigorously combined with the field modeling of a total structure. (5) Powerful topological flexibility, such as adding, removing, and rearranging parts, can be achieved with significant computational efficiency and formulational simplicity. This would open new possibilities for modular design strategies and optimization methods. The analysis is based on a new theory, established in the complex space-spectral domain. Contour deformation techniques on a complex plane are useful for simplification of radiation integrals [10]. The new method incorporates a suitable mixture of both “incoming” as well as “outgoing” cylindrical waves, by using a non-conventional contour deformation on the complex plane. This is in contrast to only outgoing waves, normally included in a conventional spectral-domain treatment of planar structures [11]–[14]. Similar non-conventional contour deformation was used in [15], [16] for modeling leaky-wave phenomena, involving fields that are exponentially increasing away from a source. Here, for the first time, we use such contour-deformation calculus as the foundation to treat a different class of basic problems involving “incoming waves.” The detailed theoretical derivations are presented in Section II. Though the analysis is formulated for a slot element of interest, it would similarly apply to multilayered, printed strip structures in a parallel-plate and packaging environment. Further, scattering in a radiating, open environment

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Fig. 1. Geometry of a conductor-backed slot antenna element surrounded by a general non-planar structure. The slot element in general can be of any arbitrary shape and size.

can also be equivalently modeled by placing a metal enclosure above the source sufficiently spaced apart. Accordingly, the present work potentially covers a much larger class of problems, carrying broader theoretical and computational significance. The basic analysis presented in this part of the paper considers the external scatterer to be general in nature, for which a complete “scattering signature” is assumed to be defined and available. For simple scattering structures, such as a rectangular metallic cavity or waveguide structure, or a cylindrical dielectric or metallic cavity, a complete scattering matrix can be derived in purely analytical form. We demonstrate the validity of the method in Section III, by comparing results for rectangular cavity-backed or waveguide-coupled structures with independent data. More complex scattering is introduced when the environment is of arbitrary geometry or material distribution. For example, the environment may consist of metal shorting posts and/or an arbitrary-shaped packaging structure. In such cases, the scattering matrix may have to be computed or analytically derived using a more involved process. Once available, the scattering matrix can then be reused for different source configurations operating in the given environment. Such specific modeling of a shorting-post structure, investigating basic physical performance and design trends of the structure, is presented in Part-II [17] of the paper. II. ANALYSIS Fig. 1 shows the geometry of a slot element printed on the top wall of a planar, parallel-plate structure, enclosed by a general non-planar surrounding. We specifically show a slot-ring, or an alternate rectangular slot element, which are commonly used. Any other slot configuration may replace the rectangular or ring element in Fig. 1 for general application. , The electric fields , or its equivalent magnetic currents in a general slot element may be modeled by a superposition of with unknown amplitudes known basis distributions (1)

where is the normal to the surface of the slot element, directed toward the medium where the fields are to be modeled. The unknown amplitudes can then be solved using a Galerkin moment-method procedure, if a matrix of mutual admittances between different basis distributions can be calculated [2], [11]. The moment-method procedure for printed, layered structures is considered well known, and will not be covered here. Accordingly, it will suffice to derive only the self or mutual admittance (generically represented as ) between two arbitrary and ,) source distributions (generically represented as placed in a general scattering environment. The second source distribution would serve as a testing function in a Galerkin’s moment-method procedure. The new theory is founded as a spectral-domain analysis, where all field and related parameters are expressed in the spaof tial Fourier domain. For reference, the Fourier transform the source distribution is defined as follows:

(2) “ ” is used as a standard notation for any Fourier transform and are the transform coordinates in quantity. the rectangular and polar coordinates, respectively. may be decomposed into The self or mutual admittance two independent parts (3) where and are, respectively, the admittances as seen into the air and parallel-plate sides. and are the magnetic fields in the air and dielectric sides, respectively, produced by . the source current is unaffected by the parallel-plate medium and the scattering environment. It sees a planar medium, which may be one semi-infinite, free-space medium as shown in Fig. 1, or may consist of additional dielectric substrates as covering layers for other designs. Therefore, it can be treated in the transform domain using suitable spectral Green’s functions [3]–[6]. Treatment of such planar elements in the spectral domain with a single- [11] or multi-layered [2], [18] arrangement is considered well known, and will not be elaborated further. For the rest to be known. of this work we will assume as seen into the parOn the other hand, the admittance allel-plate medium is influenced by the presence of a scattering structure between the parallel plates. “sees” a non-planar environment, and therefore can not be treated directly in the specis discussed in the foltral domain. A new formulation for lowing sections. A. Open Parallel-Plate Structure, No Scattering Environment First, assume that the scattering structure is removed and the planar medium between the parallel plates is infinitely extended in transverse directions. The slot element in this case would see

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an ideal planar medium, and therefore the resulting admittance may be treated using a planar analysis similar to that used discussed earlier. The admittance can be expressed for in the transform domain using suitable Green’s functions in the parallel-plate medium. We will use polar forms of the transforms (4)

The superscript “ ” refers to computation at diametrically oppo. This would site spectral coordinates be equivalent to a complex conjugation operation for a real distribution and real spectral coordinates. The basic spectral inteis expressed in (4) in a standard polar form, having gral for and an anone-sided radial integration for . An alternate form using a gular integration over double-sided radial integral , but with an, will also be useful for analytic gular integration over plane treatment on the complex

(5)

The transformation from the single-sided polar integration of (4) to the double-sided form of (5) would require appropriate analytic symmetry of the integrands. The main integrand in (4), evaluated at a diametrically opposite , is projected in (5) to the negative radial spectral angle, axis, . This assumes basic analytic symmetry for the field and source transforms, where and . However, the additional in (4) has an odd symmetry about the origin, polar variable and can not be similarly projected from (4) to (5). This problem can be analytically overcome in suitable ways. We may introin (5) to substitute duce an alternate function for the in (4). The branch plane and branch cut for each of in the new function are to be selected the square-root terms along the differently, such that the total product is equal to positive real axis, but is along the negative real axis. The resulting even symmetry of the function would allow the desired projection from (4) to (5). This arrangement essentially substiby an analytic version of the real magnitude function tutes . A similar technique for symmetric projection through manipulation of branch cuts is also introduced in Section II-C. integration path in (5) needs to be properly detoured The around any pole or singularity of the integrand, occurring along or near the real axis. These poles refer to the propagating guided modes of the parallel-plate structure. The conventional integration contour is illustrated in Fig. 2(a). This contour may be seen as an analytic continuation of the original Fourier integration along the real axis, as the poles approach the real axis with a small imaginary part (negative and positive imaginary parts, respectively for the positive and negative real-axis poles). This is physically meaningful, when the material medium realistically

Fig. 2. Different contours used in the complex-plane treatment of incoming or outgoing waves.

includes a small, non-zero quantity of loss. The choice of the spectral contour is known to model fields that are spatially “outgoing” in nature. This means that the resulting fields are propagating or decaying away from the central source, having a type of variation, with , . Now, the admittance as seen into the parallel-plate side and , contributed due can be decomposed into two parts, potential and potential of the total field, to respectively

(6)

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(7) and Such independent decomposition into the parts is also valid for the admittance of the total structure, in , the presence of a general scatterer. That is, where and are contributed from the total and fields in the immediate vicinity of the slot source, in the and do not see a presence of the scatterer. Because strictly open, planar environment, one may not explicitly treat them in the spectral form. However, it may be possible to relate and in a simple manner, without significant them to and computational cost. One may recognize that include the effects of the incoming waves scattered from the surrounding environment, in addition to the outgoing waves propagating outward from the source. This is in contrast with and , which include only the outgoing waves. Therefore, and , it would be useful to in order to properly model see if one could treat arbitrary incoming fields in the spectral form, by proper contour deformation on the complex spectral plane. This is studied in the following section. and account for any interaction with a general scattering body, and are expected to be in general different from the and , for the transversally open corresponding quantities, problem. There may be simple special cases, where only the field is affected (for thin vertical shorting posts) or only the field is affected (for small circular conducting loops plane.) In other special cases both the and on the fields could be affected, but the two fields may be decoupled from each other (for a cylindrical metal or dielectric cavity.) and admittances and In these special cases, the including the scattering are related to the respective parts and for the open environment, without any cross relation between the two parts. For formulational simplicity and brevity, in the following analysis we will explicitly treat scattering of field, decoupled from the part. In this case, only the we would have (8) However, as will be evident from the analysis, the treatment of the scattering can be duplicated for scattering of the wave. Any coupling between the and fields can also be similarly treated using a suitable cross-scattering matrix for the incoming waves. We will address the issue more specifically in Section II-D. 1) Complex-Plane Treatment for Incoming Waves: As discussed, the conventional contour used in the last section produces only outgoing waves. In principle, one could select any deformation of the contour on the complex plane, while using the same spectral integrand. This would still ensure that the Maxwell’s equations and all source boundary conditions are satisfied everywhere in a local sense, although on a purely mathematical basis. The resulting solutions may not be physically

valid, however, by not conforming to one or more obvious physical realities. They may result in waves that are inward propagating, bringing energy into the source instead of taking away from it, or produce negative radiation resistance for a passive device, or produce infinite fields at infinite distances leading to infinite total power produced from a finite-powered source. The mathematical validity of such possible contours, in the sense that the resulting solutions locally satisfy the Maxwell’s equations and source boundary conditions, may still carry intrinsic analytical significance. This could be exploited for special modeling. , as shown Let us consider a deformed integration contour in Fig. 2(b), which is oriented on the opposite side of all the spectral-plane singularities, with respect to the conventional is traversing below a contour . If the integration path particular singularity, then the new contour traverses above it, and vice versa. It is implicitly assumed that the contour is in analytic continuation with , with respect to any branch-point that may occur in a particular problem. Using , instead of the conventional contour , the new contour would result in switching the “outgoing” nature of all fields type of variation) This into “incoming” type (having conclusion is mathematically established in Section II-C. One may wish to switch only specific guided-wave modes from outgoing to incoming type, while maintaining conventional in Fig. 2(a) outgoing forms of the other waves. The contour is one such integration contour, which switches only the dominant propagating mode to its incoming form. The contour in Fig. 2(c) represents a more general situation, where only the th guided mode associated with the singularity at is switched to the incoming form. One may similarly think of many other possibilities where more than one mode is switched to incoming waves, leaving the others as outgoing. We know that such new contours, and the resulting incoming waves, are not physically valid for the basic geometry of a slot element in an open parallel-plate structure. They could, however, be useful to model interaction with any incoming waves, which may exist in a modified problem. For example, the incoming waves may have been injected into the slot element from another external source, or from an external scatterer which feeds part of the outgoing wave generated from the slot source back towards the same source. For now, we may keep aside any specific physical interpretation or source of such incoming waves. Let us assume that the slot element would produce all . incoming waves, and express the resulting slot admittance Mathematically, we simply need to change the integration conto . Using residue calculus, this amounts to tour from adding residue contributions around all singularities, to the integration along

(9) (10)

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total integrand. The harmonic decomposition of the residue of , combined with the harmonic decomposition of the second , will provide the compart, at each guided-mode pole plete decomposition of the admittance. part in (6), at Let us first decompose the residue of the is one specific th guided-mode pole. As shown in (12), expressed in terms of source transform and Green’s function components

(14)

(11)

(12)

(15)

and are the spectral Green’s functions for the , produced due to an - and z-directed magnetic potential -directed magnetic current source, respectively. The singularities (also referred to as poles) in the above integrands are introduced directly due to poles in the potential transform function , or equivalently in the potential Green’s functions and . Proper symmetry condition for the residue function is assumed in the derivation of (11). The symmetry condition can be established from basic functional characteris, and tics and analytic symmetry of the potential transform suitable residue mathematics

The symmetry condition in (14) is equivalent to that of the potential residue in (13). Now, we need similar decomposition of the other part of the spectral integrand in (6), (11)

(13) 2) -Harmonic Decomposition of Residue: The residue of the potential transform at a given pole represents the level of excitation of the corresponding guided mode. Further, the spectral dependence of the residue function would represent the spatial dependence of the guided-wave excitation. With this in mind, it would be useful to decompose the residue into spectral harmonics in . The resulting harmonic coefficients can then be related to the amplitudes of excitation of the cylindrical harmonics of the associated guided mode. In Section II-C we will examine such relationships between the space and spectral harmonics of the potential function. Here, we will evaluate contributions of the individual spectral harmonics to the admittance seen by a slot antenna. It may be noted that the cylindrical guided modes are often expressed directly using the potential functions [19], with which various field quantities can be related using suitable field-potential relationships. Therefore, we will use the potential form, not the field form, of the admittance integral in (6) for the harmonic decomposition. Further, the spectral integrand in (6) may be seen to consist of two distinct parts: the potential transform , and the remaining part that multiplies with to form the

(16) (17)

It may be noted that the above harmonic decomposition into and involve only single-dimensional integrations, which are relatively simple in computational terms. This is in contrast with the two-dimensional integrations needed to and of the open structure. compute the admittances Fast Fourier transform (FFT) techniques may be used for and parallel computation of the Fourier coefficients , (truncated at sufficiently large index ,) for a given pole index . For many practical problems the scatterers are not placed in tight proximity to the central source. Therefore, the propagating guided modes and only a few additional evanescent modes may be sufficient to describe coupling to the scatterer. In such practical situations, the formulation of the total structure would need Fourier decomposition for only a handful of guided-wave . Accordingly, the computation cost involved in the poles above harmonic decomposition (14)–(17) is typically a fraction and of the open of that to evaluate the admittances structure. Further, the integrands used in the decomposition of and are parts of the same integrand formulated for the admittance computation. Therefore, the computation of and may not carry any new formulational overhead. The above considerations are important in order to appreciate and quantify the computational merits of the new method.

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We can combine the harmonic decompositions of the two parts in (14), (15) and (16), (17), to obtain a decomposition of the th mode residue in (11)

the following form: (22) where

is selected such that it has only one singularity at . Further, the residue of at is equal to that of the th harmonic of the total potential transform.

(18) in (10), we can now Using the above decomposition of relate to . It may be recalled that is the admittance when all outgoing guided modes excited by the source are switched to incoming waves

(19) B. Admittance Including Scattering as As we know, the complex plane integrations along are both mathematically valid, in the sense that well as the corresponding fields satisfy the Maxwell’s equations and source-boundary conditions. A more general form of complex-plane integration, which would also be mathematically valid, may be performed as a partitioned superposition of the integrations along and (20) where is an arbitrary complex number. Such partitioning into two integration contours, instead of only one ( with only outwith only incoming waves,) is equivalent going waves, or to having a mixture of both incoming and outgoing waves. The Maxwell’s equations and the source boundary conditions are still satisfied. This mixture may represent scattering from an external environment, where a part of the outgoing waves is reflected back to the source in the form of incoming propaga, which includes both tion. The corresponding admittance the outgoing and incoming waves, can be expressed accordingly

(23) The total potential transform and each of its parts in (22) satisfy a basic symmetry condition

(24) After one decomposes the total potential transform into the , the remaining part does not consingular functions tain any modal singularity. The residues of the singular parts represent a set of cylindrical guided modes, which completely describe the fields outside the source region. The remaining is not associated with any guided mode, non-singular part and therefore would only relate to the non-guided fields inside does not have any the source region. Incidentally, because modal singularity, its integration along the standard contour , or any detoured contour , would provide the same result. Once the potential transform is decomposed into (22), the that relates to the potential transform, as shown admittance in (6), can also be decomposed into respective parts (25)

(26) is the admittance associated with the potential , when the guided mode is outgoing, with the integration path , when selected along . The corresponding admittance the mode is incoming, is evaluated by switching the integration path from to

(21) (27) The above partitioning into incoming and outgoing parts by a single parameter may not still be the most general form, and may not be that useful. In the most general form, the complex integrand may first be expressed as a sum of several parts. and , as described in The partitioned integration along (20), can be performed for different parts of the integrand, each having a different partition parameter (at this point unknown.) as a sum of parts, in With this idea in mind, let us express

consists of only one sinBecause the potential function , any integration along the path is anagularity at (see the complex lytically equivalent to that along the path contours of Fig. 2(b) and (c)). may be related to through its residue contribution , using (23), (16) (28)

MCCABE AND DAS: PART I: A NEW THEORY FOR MODELING CONDUCTOR-BACKED PLANAR SLOT ANTENNA ELEMENTS

(29) Now, we will use the partitioned integration of (21) along both and , with a different partitioning factor for a dif. The resulting admittance would include ferent mode a general combination of both incoming and outgoing modes, and therefore, would be applicable in an arbitrary scattering environment

(30) The above general expression may be rearranged using (25), to the conventional admittance without (28), relating the any external scattering

(31) It may sometimes be analytically useful to express following alternate form:

in the

(32) (33) If we include only the propagating modes (poles on the real would be the principal integration axis) in our calculation, along the real axis, excluding any residue contribution for may be called the around the poles. With this reference, “principal-value” contribution for the admittance . C. Fields and Potential in a Parallel-Plate Medium In the last section, the incoming and outgoing waves are assumed to carry arbitrary amplitudes, purely for analytical considerations. The incoming and outgoing wave amplitudes must be physically related through the scattering characteristics of a specific external environment. The arbitrary partitioning conand used in (21), (30) represent relative stants strengths of the incoming and outgoing waves, respectively. Accordingly, the partitioning constants must also be physically related to the external scattering characteristics. In order to establish such relationship, it would be necessary to examine the potential function in the space domain under different conditions.

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To this end, we need to examine the Fourier inverse of the potential transform using different integration contours: leading to only incoming waves, and comto only outgoing waves, binations of and , with partitioning parameters and , to both incoming and outgoing waves. In the following, we will consider the three situations separately. 1) Open Structure, Outgoing Wave Formulation: The decomposition of the potential transform in (22) would result in corresponding decomposition of the potential function in space. The correspondence is established through Fourier inversion (34)

(35) We know a priori that outside the source region, the potential can be expressed rigorously as a superposition of cylindrical wave functions, with suitable coefficients for each mode. As it would be established in the following derivation, the complex in (35) would result in only outgoing waves. The contour wave functions for the outgoing waves take the form of Hankel functions of the second kind (36) We need to find the coefficients of the modal expansion in (36) from the Fourier inversion integral (35). One approach is , and to evaluate the integral (35) at large distances relate the resulting potential function (34) to the large-distance of (35) is esapproximation of (36). The behavior at tablished by performing integration around stationary-phase [10, Ch. 4], and then using residue calpoints at integration culus for the

(37) The square-root function in the above expression refers to the principal positive value for real . Using the symmetry condition (24) for the potential transform, the contribution from the second part of the integrand can be made to be equal to the axis. This is possible by first part computed at the negative choosing an appropriate branch-cut for the square-root function in order to extend its validity to the negative axis, or to the complane in general. The required branch cut is shown as plex in Fig. 2(b), in the first quadrant. Essentially, this amounts to having the square-root function, when evaluated along the negative real axis, to be negative imaginary. Under this condition, the single-sided integral of (37) can be transformed to a

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double-sided integral of only the first part, computed along the contour (see Fig. 2(a) and (b))

(38) With the branch cut in the first quadrant, which is above the real axis, we may analytically close the integration path along in the lower side of the complex plane (see Fig. 2(b)). The inte, goes to zero along . gration of (38), with the factor Therefore, closing the path along would not contribute to . Now residue calthe total loop integration along culus can be used to express the integration of (38) in terms of . The residue of its residue around the singularity at is defined in (23)

2) Open Structure, Incoming Wave Formulation: The derivation in the last section may be repeated by replacing the intein the residue treatment of (38). This gration contour by would capture the residue at , instead of that at . Therefore, the resulting wave would be incoming propagation factor. in nature, as we expected, with a This is in contrast to the outgoing wave of (39) obtained in the propagation factor. Accordingly, last section, with a Hankel functions of the first kind need to be substituted in (36), (41), instead of the second kind

(43)

(39) As we have expected, the resulting wave function in the above propagaequation is clearly outgoing in nature, with a tion factor. Now, using (39) in (34) we get,

(44)

(45) (40) As explained in Section II-B, a potential function in (34) does not contribute to the modal expansion outside the central source region. Therefore, the total potential far from the source of is expressed in (40) using only the modal functions (39). The cylindrical modal expansion in (36) may also be approximated for large radial distance, by using asymptotic expressions of the Hankel functions for large arguments [20]

(41) Now, equating (40) with (41), we can relate the modal coeffito the residue coefficients cients (42) As expected, the cylindrical modal decomposition in space is directly equivalent to harmonic decomposition of the residue in the Fourier domain.

The symmetry condition of (13), (14) is used in deriving (44), to that at . Notice that we relating the residue at have here incoming waves of the form as a result of , as we discussed earlier. These the contour deformation to incoming waves exhibit exponential growth with , because is strictly less than zero (assuming non-zero material loss). Therefore, their Fourier transforms would not exist for real , making a conventional inversion contour invalid for treatment of the incoming waves. Similar exponential growth away from the source is also seen in leaky waves [15], [16], although the leaky waves are fundamentally distinct phenomena from the incoming waves. Equating (45) with (43), we can relate the modal coefficients to the residue coefficients (46) Notice that the coefficients of the incoming modal formulation are negative of those of the outgoing wave formulation in (42). 3) Scattering Environment, Including Both Incoming and Outgoing Waves: Incorporating a scattering environment, with both outgoing and incoming waves, requires partitioning of

MCCABE AND DAS: PART I: A NEW THEORY FOR MODELING CONDUCTOR-BACKED PLANAR SLOT ANTENNA ELEMENTS

the integration contour into and , with a partitioning for the th mode. The spatial distribution of coefficient in the scattering environment is first estabthe potential lished using the partitioned contour. The associated partitioning can then be related to the reflection matrix of coefficients the scattering environment. in (22), (35) As may be recalled from Section II-B, the does not contain any pole. Therefore, evaluating an inverse along or , or a path partiFourier integral of the and , would be equal to each other. In tioned between , , and would other words, be equal to each other. Further, the would not contribute to any guided-mode field, because it is not associated with any guided-mode pole. Therefore, in the medium outside the source region, where the total field is completely described as a sum would not contribute to of only guided-mode fields, the the modal decomposition. With the above considerations, the in the scattering environment, outside the source potential region, may be expressed as follows:

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Some matrix algebra in (49) would relate the partitioned modal multiplied by and , to the coefficients, unpartitioned coefficients

(50) where is a unit diagonal matrix. The reflection matrix in (48) (or the modified version in (49), (50)) relates the amplitudes of the incoming Hankel function to those of the outgoing Hankel functions. It may sometimes be convenient to express the total wave of (47) as a superposition of outgoing Hankel functions and Bessel functions of first and ). It would then be meaningful to define a new kind ( that relates the coefficients of the Bessel reflection matrix functions to those of the outgoing Hankel functions

(51)

(52) (47) D. Slot Admittance Using External Scattering Matrix Now, we may define a reflection matrix for the scattering medium, which relates the amplitudes of the outgoing and incoming Hankel functions for each mode index . We may assume here, for convenience, that the scattering environment does not result in cross-coupling between modes having different mode indices . The assumption is valid for scatterers that have uniform cross section for all (all geometries of interest in this study). If this is not the case, one would use a more general form of the reflection matrix that accounts for the cross-coupling between all modes

in the In (30)–(32) we have expressed the admittance without presence of scattering in terms of the admittance scattering, the residue coefficients , and the arbi. Now that the residue and partitrary partitioning factors tioning coefficients are definitively related to a reflection matrix for the scattering environment, we can use the relationship to in terms of the reflection maexpress the slot admittance trix. The reflection matrix is properly defined in the last section, which can be separately computed for any given scattering structure around the antenna. Let us start with the admittance expression (31), and rewrite column matrices using two

(48) Here we have expressed a truncated reflection matrix with , using the modal coefficients . The are related to the residue coefficients in (42) with an . It would be useful to define a modiadditional factor fied matrix that relates directly to the residue coefficients, or equivalently to

(53) is the maximum number of -harmonics used for the residue decomposition, and stands for transpose operation. Using the relationship (50), (42) in (53), we get

(54)

(49)

(55)

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Using (52), (32), one can obtain equivalent expressions for in other useful forms

and , representing coupling between the harmonic compoand fields. nents of the III. RESULTS AND VALIDATION (56)

(57) , Equations (54)–(57) are final usable forms for modeling expressed in terms of a scattering matrix of the surrounding. , and then calculation of the total The above modeling of admittance using (3), (8), involve three computational parts: and of the original (a) computation of the admittances open structure, (b) computation of the scattering matrix in the or , and (c) computation of harmonic deform of , , and in order to implement the composition coefficients intercoupling between the source and scatterers. As we have discussed in Section II-A-2, the last item (c) usually involves only a fraction of the cost for computing the admittances of the open structure. For many practical scattering structures for which analytical results exist, or are relatively simple to model numerically, the item (b) can also be computationally inexpensive. A rectangular or a cylindrical conducting package, or a cylindrical dielectric cavity, or a set of thin vertical shorting posts, for example, falls into this class. In such designs, the computational cost in the modeling of the total structure, including the scattering, would mainly involve the computation of the item (a). Accordingly, the total computational cost with scattering is comparable to that for the open structure. This is an important development. Many practical applications may use a more complex scattering environment, such as a specific packaging box of a particular shape and size. But the same scattering environment may be used over and over again for different source configurations placed inside. In such cases, the modeling of the scatterer may be relatively difficult, but its scattering signature may have to be computed only once. The signature can then be used over repeatedly for all circuit elements or source distributions placed inside the scattering structure at arbitrary positions. Accordingly, any complexity in the external scattering environment does not effectively add to the total computational complexity. Further, the scattering structure may be analyzed independent of the source element, using a separate computational method as deemed efficient for the particular geometry. This is also a significant computational aspect of the theory. As mentioned before in (8), for analytical simplicity we have fields, without any so far treated the scattering of only the component. When there is scatcross-coupling with the waves, the same steps need be followed to tering of the . The final expressions for would be analogous derive to those for in (54), (57). The scattering parameters ’s would now refer to the reflection of the electric potential , instead of the magnetic potential . In more general cases with and waves, the same cross-coupling between the basic procedure would also apply. Except, suitable cross-reflection terms would be added in the final expressions of the

We will apply the new theory developed in the last section to compute self and mutual impedances between conductorbacked slot elements, placed at different locations inside a conducting rectangular cavity. The validity and accuracy of the new theory would be established by comparing results from an independent computation. A. Geometries for the Validation Study Consider a small rectangular slot or slot-ring element, printed on the top wall of a parallel-plate structure, enclosed on four sides by a conducting square cavity. Assume that the length of the rectangular slot, or the reference diameter of the slot-ring element passing through its feed point (reference -axis in Fig. 1), is aligned along one of the principal planes of the square cavity. However, the reference orientation for the slot-ring element is not technically critical, considering a uniform cylindrical symmetry of the particular element. The source distribution for the rectangular slot or the slot-ring element is approximately described by a single basis function, with known Fourier transform (see Appendix A). The self admittance for the slot element, or the mutual admittance between two such identical elements displaced with respect to each other, are modeled using the present theory as well as an independent computation described in the following section. The self and mutual admittances are measured across terminals located at the center of a rectangular element, or at the reference feed point of a slot-ring element. This assumes an ideal delta-gap input source. The Bessel reflection matrix of the square cavity, as needed in the new theory, is derived analytically in Appendix B. For the specific geometries studied in this section, the compuand , together with tation of the harmonic parameters the final processing to incorporate scattering effects of the surrounding, involved only a fractional second of processing time on a 3 GHz Pentium Processor, as compared with about 20 seconds for the admittance computation without any scattering. Accordingly, planar structures with and without a scattering environment are seen to be computationally equivalent problems. B. Independent Computation We select the particular geometries for a validation study, because an independent, rigorous solution for the square cavity environment is also available using periodic spectral Green’s functions. The alternate computational method may be explained here in principle. The known source distribution of (59), (61), located at a given position with respect to the cavity, and the various potential and field components produced by the source, can be expressed using appropriate Fourier-series expansions in the transverse plane. Each Fourier component of the various fields can be related to the corresponding Fourier component of the known source through periodic spectral Green’s functions for the particular parallel-plate structure. The total fields are then obtained by adding the individual Fourier parts. Once all the field components are determined, the complex power, and hence the equivalent self or mutual admittance, are derived by

MCCABE AND DAS: PART I: A NEW THEORY FOR MODELING CONDUCTOR-BACKED PLANAR SLOT ANTENNA ELEMENTS

integrating the Poynting vector over the source or the coupled mode, respectively. The above computation will have to be repeated for admittance calculations with variable locations for the source or coupled mode. Clearly, this is computationally intensive. In contrast, the admittance calculations using the new method required only an incrementally more computation time, when the source or coupled element was repeatedly relocated inside the cavity. The new theory and the above alternate approach using periodic spectral Green’s functions are fundamentally different formulations. The new theory treats the slot and the cavity surrounding as two separate problems, connected with each other analytically through Bessel harmonics. On the other hand, the alternate approach treats the slot and surrounding together using Fourier series decomposition and periodic spectral Green’s functions. A slot element is excited by a delta-gap feed across its input, and we model the input admittance seen at the same slot input, or the mutual admittance seen at the terminals of an identical second slot element, with variable locations for the source and the coupled elements. It may be noted that the total admittance consists of two parts, (i) admittance as seen into the airside above, and (ii) that into the cavity side below, the slot. The first part sees a uniform, semi-infinite air medium, and is computed in the same way for both the new method and the independent computation. The second part, which sees the surrounding cavity medium, is modeled separately using the two different approaches, to which the first part is added in order to obtain the total solution. C. Rectangular Slot Element The imaginary parts of the self admittance from the two solutions are compared in Fig. 3(a) for different cavity sizes, and in Fig. 3(b) for different slot locations along the principal and diagonal planes with respect to the cavity. On the other hand, Fig. 3(c) shows the mutual admittance between the slot element of Fig. 3(b) and an identical parallel source element placed at the cavity center. In Fig. 3(a), various resonance modes which couple to the particular source excitation, are clearly seen. And, the self and mutual coupling trends in Fig. 3(b) and (c) respectively relate to the field-squared and field patterns of the mode(s) (12 and 21 modes), as seen by the source and coupled slot. For the two cases, Fig. 3(a) and (c), the results from the new theory and independent computation compare well with each other over the entire range of computation. In contrast, the agreement in Fig. 3(b) is valid only over a limited region. This we used in the is because the particular reflection matrix computation (see Appendix B) can be shown to be strictly valid only over limited source and field regions. As may be verified from Fig. 3(b), the validity is restricted to all source regions (smallest circle enclosing the source) confined inside the largest that can be placed circle inside the square cavity. In contrast, in Fig. 3(a) and (c) the source is located at the center of the cavity, in which case the range of validity for the observed field, as sensed by the coupled slot, would cover the entire square box. This is a limitation of the scattering matrix we derived in Appendix B, not of the basic analytical technique itself. Therefore, it could be possible to extend the range of validity in Fig. 3(b) by using different

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scattering matrices that are valid for different ranges of the source and field locations, or by developing new scattering matrices that can be valid for source and field locations in the entire cavity region. This is, however, not the focus of the present research. In any event, considering that the two methods used in the computation of Fig. 3(a)–(c) are fundamentally and formulationally distinct approaches, the good comparisons in Fig. 3(a)–(c) clearly demonstrate the validity of the new theory under diverse conditions. In Fig. 3(a)–(c) the slot element is coupled to a square cavity, where all four sidewalls are closed by perfect conductors. Similar computation is also possible when the slot element is coupled to a rectangular waveguide, where only two sidewalls are covered by perfect conductors but the other two sides are left open. The reflection matrix in this case is established using the same procedure as for a closed cavity, by simply moving two opposing walls of the cavity far apart to infinity and excluding the corresponding image terms from the calculation in Appendix B. Results for such a problem, showing the equivalent normalized , as seen in series with the resonant resistance coupling waveguide, are presented in Fig. 3(d). The results are compared with independent theory and experiment available in [21]. Close comparison of independent data in Fig. 3(d) further expands the validity and accuracy of the new theory to more diverse environment. It may be mentioned here, that the Bessel reflection matrix used in the new theory changes with only the size of the coupling cavity or waveguide, but is independent of the particular source, source location, or the parallel-plate distance. Therefore, the Bessel reflection matrix of the cavity or waveguide may first be computed as a function of different cavity sizes, using the analytical approach in Appendix B. This matrix can then be reused, when the slot location, size, orientation angle and the parallel-plate dimensions are changed, without having to compute the matrix again. For example, the same reflection matrix is used for all data points in Fig. 3(b) and (c), because they share the same cavity size. Similarly, for a given frequency, calculations for all inclination angles in Fig. 3(d) use the same reflection matrix. If the results in Fig. 3(a) or (d) need to be recomputed for a different slot location, size, or a different parallel-plate distance, the same reflection matrix can be reused. As a convenient alternative to analytically derive the Bessel reflection matrix, one may also numerically “extract” the reflection matrix by equating the results from an independent computation with the new theory, for one set of slot and parallel-plate dimensions, as a function of different cavity sizes. This data may be similarly reused for variations of the slot and parallel-plate dimensions, as well as slot locations and orientations. This is an important development, which would allow us to significantly save expensive computation time, often consumed in modeling the effects of surrounding structures. No such possibility exists for the independent computational method we have implemented, or for any other computational method in common use today. D. Slot Ring Element The source structure for the results in Fig. 3 is a rectangular slot element. We now replace the rectangular slot element by a

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Fig. 3. (a) Parallel 0 plate cavity height = 0 05 , cavity length = width. (b) Parallel 0 plate cavity height = 0 05 , cavity length = width = 1 11 (close to [1, 2] resonance). defines the angle of the plane of displacement of the slot-center with respect to the -axis (axis along :

:



:







x

slot length.) (c) The imaginary part of mutual admittance for the same slot element as in (b), with an identical source element positioned at the center of the cavity. (d) Parallel plate height = waveguide height =1:016 cm, cavity width = waveguide width = 2:286 cm, cavity length . (a–c) Slot length L = 0:1 , slot width W = 0:01 , relative dielectric constant of the cavity medium =  = 2:2. I m(Y ) seen into the open air medium = 0:0114 1= . Re(Y ) = Re(Y ) = 2:7 10 1= . (d) L  =2, W = 0:1 cm, waveguide or parallel-plate medium  = 1:0.  =  =  is the wavelength in the parallel-plate medium.

p

0 0

!1

2

slot ring of radius and width , and compute the self and mutual admittance using the present theory as well as the alternate method using a periodic Fourier decomposition approach. Assuming that the ring diameter is sufficiently small compared to wavelength, the source distribution in this case may be expressed using a single basis function, uniform in , with a known Fourier transform (see Appendix A). The results are presented in Fig. 4(a) for the self admittance at different source locations, and in Fig. 4(b) for mutual admittance seen between the same slot in Fig. 4(a) and an identical source at the cavity center. The results show similar trends and agreements between the new theory and the independent model, as in the case of a rectangular slot. Similar agreements were also seen for different cavity size as in Fig. 3(a), but are not shown here. A small slot ring element produces cylindrical waves in the parallel plate medium . This having approximately uniform angular variation is fundamentally different from that produced by a rectangular

'

slot element, which produces cylindrical waves with odd har. The present results in Fig. 4(a), (b), tomonics gether with those in Fig. 3(a)–(d), demonstrate the validity of the new theory to diverse scattering conditions, with different source types, source positions and cavity dimensions. IV. CONCLUSION We have presented the basic formulation of a new technique for modeling general configurations of planar elements operating in a non-planar surrounding. Combined efficiently with a standard planar analysis model, such as [2], [18], the new method may provide a more powerful framework for a wider class of problems, involving “locally planar elements” operating in a non-planar neighborhood. The technical validity and accuracy of the new theory was demonstrated by comparing results with independent computation and experiment. Considering the formulational simplicity and significant computational

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(59) The corresponding expressions for a small slot ring element, electric field (see Fig. 1), are: with a radial

(60)

(61) For an identical source distribution, but with its original center , the transform needs to be location (0,0) displaced to . The upper or multiplied by a phase factor lower sign in the double-sign notations used in the above expressions are needed for modeling in the medium above or below the source, respectively. APPENDIX B MODAL REFLECTION IN A METALLIC RECTANGULAR CAVITY ( , , are integers) The modal reflection coefficients for a metallic rectangular cavity of length ( -dimension) and width ( -dimension) can be analytically derived. A source procylindrical mode is placed at the ducing an outgoing produced by center of the cavity. The potential function this source is expressed using an outgoing Hankel function (62)

Parallel 0 plate cavity height = 0 05 , cavity length = width = 1 57 (close to [1, 3] or [3, 1] resonance). defines the angle

Fig. 4. (a)

:

:







of the plane of displacement of the center of the slot ring, with respect to the x-axis (parallel to one cavity side). (b) Imaginary part of mutual admittance for the same slot element as in (a), with an identical slot-ring element positioned at the center of the cavity. Ring radius R = 0:1 , slot width W = 0:01 , relative dielectric constant of the cavity medium =  = 2:2.  =  is the wavelength in the parallel-plate medium. I m(Y ) seen into the  = open air medium = +0:0136 1= . Re(Y ) = Re(Y ) = 3:8 10 1= .

p

This serves as the incident wave that is reflected or scattered by the four sidewalls of the cavity. The reflected potential is expressed using double-infinite imaging across the four sidewalls

2

(63) efficiency and flexibility offered by the technique, it should find promising applications in a class of related geometries including cavity-backed and packaging designs. APPENDIX A SOURCE FUNCTIONS FOR SLOT ELEMENTS For a rectangular slot element, with a -directed electric field (see Fig. 1), the equivalent magnetic current distribution and its Fourier transform are expressed as follows:

(58)

(64) Note that the amplitudes as well as the harmonic index of different image elements in (63) change their relative signs. This accounts for proper imaging conditions for the present case of a wave and perfectly conducting cavity walls. Further, each image is placed at a different location, shifted from the center by the position vector . A shifted form of a cylindrical harmonic function can be decomposed into a spectrum of harmonics with respect to the center [19], [20]

(65)

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This decomposition can be applied to each image element of (63). This would result in expressing the total reflected wave as a superposition of cylindrical Bessel harmonics with respect to the center of the cavity. Using (65) in (63), and collecting only the th harmonic, we get the final expression for the reflection with respect to the cavity center coefficient

(66) A reflection matrix defined with respect to the center of the cavity may be transformed with respect to the center of a source element by using shifting relations similar to (65). Alternately, one may define the reflection matrix with respect to a fixed reference point, and the source and coupling elements are described with respect to the same fixed reference by using shifting properties of the Fourier transformation. REFERENCES [1] D. M. Pozar, “A reciprocity method of analysis of printed slots and slot-coupled microstrip antennas,” IEEE Trans. Antennas Propag., vol. AP-34, no. 12, pp. 1439–1446, Dec. 1986. [2] N. K. Das and D. M. Pozar, “Multiport scattering analysis of multilayered printed antennas fed by multiple feed ports, Part I: Theory; Part II: Applications,” IEEE Trans. Antennas Propag., vol. AP-40, no. 5, pp. 469–491, May 1992. [3] T. Itoh, “Spectral domain immitance approach for dispersion characteristics of generalized printed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-28, no. 7, pp. 733–736, Jul. 1980. [4] C. M. Krowne, “Green’s function in the spectral domain for biaxial and uniaxial anisotropic planar dielectric structures,” IEEE Trans. Antennas Propag., vol. 32, no. 12, pp. 1273–1281, Dec. 1984. [5] C. M. Krowne, “Numerical spectral matrix method for propagation in general layered media: Application to isotropic and anisotropic substrates,” IEEE Trans. Microw. Theory Tech., vol. 35, no. 12, pp. 1399–1407, Dec. 1987. [6] N. K. Das and D. M. Pozar, “A generalized spectral-domain Green’s function for multilayer dielectric substrates with applications to multilayer transmission lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 3, pp. 326–335, Mar. 1987. [7] K. E. Golden and G. E. Stewart, “Self and mutual admittance for axial rectangular slots on a cylinder in the presence of an inhomogeneous plasma layer,” IEEE Trans. Antennas Propag., vol. 19, no. 2, pp. 296–299, Mar. 1971. [8] E. H. Newman, “An overview of the hybrid MM/Green’s function method in electromagnetics,” Proc. IEEE, vol. 76, no. 3, pp. 272–282, Mar. 1988. [9] G. A. Thiele, “Overview of selected hybrid methods in radiating system analysis,” Proc. IEEE, vol. 80, no. 1, pp. 66–78, Jan. 1992. [10] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs: Prentice Hall, 1973. [11] D. M. Pozar, “Input impedance and mutual coupling of rectangular microstrip antennas,” IEEE Trans. Antennas Propag., vol. AP-30, no. 6, pp. 1991–1996, Nov. 1982. [12] M. C. Bailey and M. D. Deshpande, “Integral equation formulation of microstrip antennas,” IEEE Trans. Antennas Propag., vol. AP-30, no. 4, pp. 651–656, Jul. 1982. [13] D. R. Jackson and N. G. Alexopoulos, “Analysis of planar strip geometries in substrate-superstrate configurations,” IEEE Trans. Antennas Propag., vol. AP-34, no. 12, pp. 1430–1438, Dec. 1986.

[14] J. R. Mosig, “General integral equation formulation of microstrip antennas and scatterers,” Proc. IEE, Part-H, vol. 132, no. 7, pp. 424–432, 1985. [15] N. K. Das and D. M. Pozar, “Full-wave spectral-domain computation of material, radiation and guided wave losses in infinite multilayered printed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-39, no. 1, pp. 54–63, Jan. 1991. [16] D. Nghiem, J. T. Williams, and D. R. Jackson, “Leakage of the dominant mode on stripline with a small air gap,” IEEE Trans. Microw. Theory Tech., vol. MTT-43, no. 11, pp. 2549–2556, Nov. 1995. [17] B. McCabe and N. K. Das, “Part II: A conductor-backed slot antenna element with a shorting-post cavity to suppress parallel-plate mode excitation — Design analysis and experiment,” IEEE Trans. Antennas Propag., vol. 59, no. 9, pp. 3185–3193, Sep. 2011. [18] N. K. Das, “A Study of multilayered printed antenna structures,” Ph.D. dissertation, Dept. Elect. Comput. Engrg., Univ. Massachusetts, Amherst, Sep. 1987. [19] R. F. Harrington, Time Harmonic Electromagnetic Fields. Hoboken/ Piscataway, NJ: Wiley/IEEE Press, 2001, ch. 5. [20] Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, Eds. New York: Dover Publications, 1970, ch. 9. [21] , H. Jasik, Ed., Antenna Engineering Handbook. New York: McGraw-Hill, 1961, ch. 9. Brian L. McCabe (M’04) received the BS and MS degrees in electrical engineering from Carnegie Mellon and Rensselaer Polytechnic Institute, in 1984 and 1994 respectively, and the Ph.D. degree in electrical engineering from Polytechnic University (Now Polytechnic Institute of New York University), in 2004 for research on hybrid methods in computational electromagnetics. He is currently a Technical Fellow with Sikorsky Aircraft, Stratford, CT.

Nirod K. Das (S’87–M’88) was born in Puri, Orissa state, India, on February 27, 1963. He received the B.Tech (Hons.) degree in electronics and electrical communication engineering from the Indian Institute of Technology (IIT), Kharagpur, India, in 1985 and the M.S. and Ph.D. degrees in electrical engineering from the University of Massachusetts at Amherst, in 1987 and 1989, respectively. From 1985 to 1989, he was with the Department of Electrical and Computer Engineering, University of Massachusetts at Amherst, first as a Graduate Research Assistant and then, after receiving the Ph.D. degree, as a Postdoctoral Research Associate. In 1990, he joined the Department of Electrical Engineering at Polytechnic University of New York (now Polytechnic Institute of New York University) where he currently is an Associate Professor since 1997. He co-edited Next Generation of MMIC Devices and Systems (New York: Plenum, 1997). He also authored a computer-aided design (CAD)/instructional tool, i.e., PCAAMT, for microwave multilayer printed transmission lines, and another simulation tool, i.e., UNIFY, for unified modeling of multilayer printed antennas and arrays. His research interests have been in the general areas of electromagnetics, antennas, and microwave and millimeter wave integrated circuits. His recent research activities include numerical-analytical methods for electromagnetics and advanced materials for microwave circuits and antennas. Dr. Das is a member of the IEEE Antennas and Propagation Society (IEEE AP-S), the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), and the New York Academy of Sciences. He served in the Technical Program Committee of the IEEE MTT-S International Symposia from 1997–2003, and currently serves on the Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the Co-Chair of the 1996 International WRI Symposium, Brooklyn, New York. For his doctoral research work on multilayer printed antennas he received a Student Paper Award (Third Prize) in 1990 from the US National Council of the International Scientific Radio Union (URSI), and the R.W.P King Paper Award (below 35 age-group) in 1993 from the IEEE AP-S.

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Part II: A Conductor-Backed Slot Antenna Element Surrounded by a Shorting-Post Cavity to Suppress Parallel-Plate Mode Excitation—Design Analysis and Experiment Brian L. McCabe, Member, IEEE, and Nirod K. Das, Member, IEEE

Abstract—We investigate the basic physical characteristics of a conductor-backed slot element, surrounded by conducting shorting posts, designed to suppress the undesired parallel-plate mode. The shorting posts are assumed to be thin and are placed periodically along a circular boundary in the form of a “shorting-post cavity.” For analytical simplicity, thin cylindrical conducting posts are modeled as conducting strips of suitable equivalent width. The scattering matrix of the structure is modeled using cylindrical wave functions, by enforcing periodic boundary conditions on the conducting strips. The scattering matrix is then used, together with the general analysis presented in Part-I of the paper, to model the impedance and radiation characteristics of the total structure. Basic design trends of the shorting-post structure are investigated. Design data are presented, relating desired levels of parallel-plate mode suppression with the physical size, the number and the placement of the shorting posts. Reactive loadings associated with the designs are examined. Selected results are validated by measurements and independent computations. Index Terms—Cavity-backed slot, mode suppression, parallelplate guide, shorting post.

I. INTRODUCTION ONDUCTOR backing behind a slot antenna or a slot-coupled microstrip patch antenna would be useful for blocking any unwanted back radiation [1], [2]. It would also be desirable, as a metal backplane for mechanical support or thermal dissipation, behind slot-based microwave circuits such as a slot-coupled double-sided microstrip circuit [3] or a slotline/coplanar waveguide circuit [4], [5]. Conductor-backed slot elements may also be useful for multilayer integration [2], where circuit layers are stacked in multiple levels, with parallel metal planes placed between the layers for electrical isolation. In this case, the slotted elements are used to interconnect signals between adjacent circuit layers, at selected locations across a common conducting plane. All these designs involve slot elements, operating in an environment with parallel conducting plates. Although such designs

C

Manuscript received December 09, 2008; revised December 06, 2010; accepted December 28, 2010. Date of publication July 14, 2011; date of current version September 02, 2011. B. L. McCabe is with Sikorsky Aircraft, Stratford, CT 06614 USA. N. K. Das is with the Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, Brooklyn, NY 11201 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2161434

seem attractive and geometrically straightforward, they unfortunately present some fundamental problems. A slot element would excite guided modes between the parallel conducting plates. The dominant guided mode excited, referred to as the parallel-plate mode, can be prohibitively large for a typically small separation between the parallel plates. In order to successfully implement the conductor-backed antenna or the multilayer circuit mentioned, the excitation of the parallel-plate mode must be eliminated or significantly suppressed. One may use metal shorting posts placed around a coupling slot, which could suppress the parallel-plate mode by effectively “shorting-out” the electric field of the mode. The method would be practically useful, if one could achieve a significant suppression of the parallel-plate mode, using a reasonably small number of shorting posts. Shorting posts have been used to control leaky waves in conductor-backed slotlines [6], and also to emulate a closed metal waveguide for convenient integration with printed circuits [7]. We present a theoretical and experimental study of conductor-backed slot antenna elements, that are surrounded by shorting posts arranged in the form of a cylindrical “shorting-post cavity.” The structure is designed to effectively suppress any excitation of the undesired parallel-plate mode. The theoretical study is based on a new analysis presented in Part-I of the paper. Design data for different parameters of the shorting posts, which are placed around either a rectangular slot or a slot-ring element, are presented. The scattering matrix of the shorting posts, necessary for the analysis in part-I, is derived by injecting a cylindrical wave upon the shorting posts. The injected wave is described by an outgoing Hankel wave function, that is incident upon the shorting posts from the center of the post cavity. The resulting scattered fields in the central region are expressed as a superposition of Bessel wave functions, whose coefficients are derived by requiring that the electric field is zero along the metal posts. Practical assumptions and simplifications result in an analytically simple yet accurate procedure for computing the scattering matrix. The new theory of part-I, together with the scattering matrix derived for the shorting posts, provides a computationally efficient model for impedance and radiation characteristics of the entire structure. Once the scattering matrix is computed for a particular shorting-post configuration, the general theory of part-I allows the same matrix to be re-used for different source configurations, including different positions and orientations

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Fig. 1. Geometry of a conductor-backed slot antenna surrounded by a periodic arrangement of “shorting posts” for suppression of the parallel-plate mode. The slot element maybe a circular slot ring or a rectangular slot element.

of the sources. This makes the technique ideal for design optimization. The approach allows modeling of the source and the scattering characteristics separately, and then efficiently integrates the results to model different possible variations in an entire structure. Significant computation time is saved in this process, when compared to a conventional process of computation for each specific structure that needs to be repeated for the different variations.

Let us consider a modal incident : magnetic potential function

field, described by its

(1)

II. ANALYSIS Fig. 1 shows the geometry of a slot antenna element, which is printed on the top wall of a parallel-plate structure and surrounded by metal shorting posts. The slot element—a rectangular slot antenna or a circular slot-ring antenna—is excited by an ideal delta-gap voltage source of unit amplitude. The rectangular slot is excited across the center of the slot, whereas the ring . For analytical conelement is excited across the slot at venience, we replace the shorting posts by shorting metal strips . In this sense, in the remainder of an equivalent strip width of the paper they will be interchangeably referred to as shorting posts or shorting strips. A. Scattering Model for Circularly Periodic Shorting Posts The electromagnetic wave, generated by a central slot element, is incident upon the surrounding shorting posts. It then is partly reflected toward the source region and partly passes between the shorting posts escaping as radiation into the external region. Here we are interested only in the reflected fields, which can be rigorously modeled using a reflection matrix. The matrix represents the amplitudes of different cylindrical modes reflected toward the source, produced when any given cylindrical modal field is incident from the central source region. We assume that the thin shorting posts do not affect any incident field. Therefore, we will only study the reflection of fields.

(2) In general, the scattered fields produced by the incident wave would consist of all -harmonic modes (defined by the ). The scattering would not mode index ; produce any cross coupling between modes, with different -dependences (different mode index ). In other words, the scattered fields will have the same -dependence as the incident fields. Furthermore, the scattered fields are produced by the currents induced on the shorting posts, which are dominantly along the direction. Therefore the scattered fields are assumed fields, which can be modeled using a magnetic to be potential . or, The modal composition of the magnetic potential equivalently, of the -component of the reflected electric field , in the source region defines a suitable reflection matrix. can be expressed as a superposition of incoming Hankel functions . If this is done, a reflection matrix will represent the coefficients of the modes. can also be expressed as a superposition of Alternatively, , represented by a reflection Bessel function modes matrix . The superscript refers to the Bessel function representation of the reflected modal fields. We choose the second or , option here. Either form of the scattering matrix, can be used in the modeling of part-I, and either form can be

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deduced from the other when necessary

(8)

(3) 1) Induced Currents on the Shorting Posts: We assume that all shorting posts have the same dimensions. For analytical convenience, we will model each shorting post of radius as an . This is known to be equivalent conducting strip of width a valid approach, if suitable equivalence between the width and the radius is maintained [8], [9]. Also for analytical convenience, each strip is assumed to lie on a common (see Fig. 1). However, for cylindrical boundary of radius , the a sufficiently narrow strip of practical interest cylindrical profile would be essentially equivalent to a planar one. Further, we model each element of the strip array by re. placing it with its unknown induced current The induced current exhibits the same -variation as the incident mode. In addition, the induced currents in neighboring shorting posts exhibit the same relative phasing as the incident mode. Based on the above reasoning, we express the current distribuover the entire strip array as follows: tion

(9) By equating (9) with (3), the reflection matrix elements are all expressed in terms of the sole unknown quantity .

(10) 3) Boundary Condition on the Shorting Posts: If the shorting posts are perfect conductors, as we assume, the -component of the total fields from (1), (3) are zero over each strip element. We choose to enforce the boundary condition at the center of the . Due to the intrinsic periodicity of the reference strip total field, the boundary condition will be satisfied for the other shorting strips as well. Enforcing the boundary condition yields an independent equation, which can be solved for the unknown current coefficient

(4)

(5) The current distribution on each strip over its transverse dior, equivalently, over its angular width mension may be assumed to be a uniform function multiplied or modulated by the phase variation of the incident wave. This would be a valid assumption for typical narrow strips. This of the reference strip as the leaves the current amplitude only unknown to be solved for. The uniform function may be conveniently normalized so that the associated current Ampere. This is magnitude, integrated over each strip, is equivalent to having a total of one Ampere flowing through the elements of the strip array

(11) (12)

is known, the reflection coefficients can be obOnce tained from (10). This completes the modeling of the reflection matrix

(6) 2) Fourier Decomposition of Strip Currents and Fields: We in (4) and (5) will decompose the total current distribution into Fourier harmonics over . The field produced by each harmonic current can then be modeled using cylindrical wave functions [10]

(7)

(13) III. BASIC CHARACTERISTICS AND DESIGN STUDY We are now in a position to use the above scattering matrix with the general theory of part-I, to model the slot antenna element of Fig. 1. We study both the imaginary and the real parts of the input admittance. The imaginary part provides information about any reactive loading that might occur due to

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the shorting posts. On the other hand, the real part of the adseen into the parallel-plate medium, as mittance seen into the air medium, procompared to that vides useful design data for radiation efficiency , defined as . As mentioned before, the placement of the shorting posts emulates a cylindrical cavity. The radial distance of a shorting post from the source center is equal to the cavity radius or half the (see Fig. 1). In what follows we will intercavity diameter changeably use terms like “shorting location,” “shorting point,” “shorting distance,” “cavity radius” and “cavity diameter,” to refer to the radial position of the shorting posts.

A. Rectangular Slot Element Fig. 2 presents the radiation efficiency for a small rectangular slot element shown in Fig. 1. The small rectangular slot element represents a basic source of parallel-plate radiation, with a angular variation. It is interesting to observe that the radiation efficiency with shorting posts can be either better , depending or worse than that without shorting posts on the placement of the shorting posts. The shorting posts emulate an ideal cylindrical metal cavity, in the limit in which approaches . The radiation efficiency would get particularly low if a given post-cavity structure operates close to one of the resonance conditions of its limiting ideal cavity. For mode) for an example, the first resonance condition (the angular ideal cylindrical metal cavity, with a variation, is theoretically calculated to occur when the cavity [10]. Consistent with the above resodiameter nance principle, the radiation efficiency in Fig. 2 is at a minimum, when the cavity diameter is close to the theoretical res. The next resonant diameter in Fig. 2 onant value . Close to a resonant diameter, the minis at about imum efficiency in Fig. 2 is seen to reduce further by increasing . This interesting trend might be counter-intuitive. It means that any attempt to suppress the parallel-plate mode by placing shorting posts close to a resonant location would not only be ineffective the result would also get worse if one tried to improve the situation by increasing the number of shorting posts. is reAnother subtle detail may be noticed in Fig. 2. As duced, the resonant location with the minimum radiation efficiency shifts closer to the source, away from the ideal resonance . This may be viewed as a perturbalocation tion of the ideal resonance condition due to the spacing of the shorting posts and their reactive effects. Before implementing a particular design, its resonant conditions should be predicted. Placing the shorting posts near any resonance location should be carefully avoided. As the results of Fig. 2 show, placing the shorting posts approximately midway between two consecutive resonance locations raises the efficiency level to a local maximum. However, the best possible efficiency is achieved by placing the shorting posts as close to the central source as is feasible. For example, if the shorting dis, tance is slightly larger than half of the slot length and if 4, 8, 16 and 32 shorting posts are used, the best efficiency one could achieve according to Fig. 2 is about 0.78,

= 0:1  , slot width W = 0:01  , parallel-plate = 0:2  , parallel-plate medium  = 2:2, equiva= 0:02  .

Fig. 2. Slot length L distance (cavity height) d lent shorting-strip width W

0.97, 1.00 and 1.00, respectively. This is significant improvement compared to an efficiency of only 0.32 without shorting . The efficiency can be improved further by conposts trolling various parameters of the source and of the parallel-plate structure, such as the slot length, the parallel-plate distance and the dielectric constant of the parallel-plate medium. It may be useful to define a modal suppression factor , associated with a shorting-post design. This is the factor by which the power in the parallel-plate mode is reduced after the shorting posts are introduced (14) The is the real part of the admittance contributed by the parallel-plate mode power before any shorting post is placed; it therefore depends only on the source and parallel-plate pais the real part of the admittance after rameters. The the shorting posts are placed; it is expected to be proportional . The ratio of proportionality , and accordingly to , should in general be independent of the source and parallel-plate parameters, and should depend mainly on the design of the shorting posts, at least to first-order. This assumption should be valid except if the shorting points are too close to the source or if there is more than one propagating mode excited by the central antenna. Fig. 3 plots the suppression factor for the parameters of Fig. 2. A negative means that the excitation is stronger than without any shorting post. As Fig. 3 shows, a suppression factor greater than 80% can be achieved by placing four or more shorting posts at a radial distance close to half of the . If one prefers not to place the shorting slot length posts that close to the antenna (in order to avoid reactive loading effects on the source), the next best shorting distance is at about from the center, where a suppression of 65% can be shorting posts. As discussed achieved by using at least above, the suppression factor of Fig. 3 is mainly determined by the shorting-post design, and therefore can be applicable for a wide range of practical slot and parallel-plate parameters.

MCCABE AND DAS: PART II: A CONDUCTOR-BACKED SLOT ANTENNA ELEMENT SURROUNDED BY A SHORTING-POST CAVITY

Fig. 3. The modal suppression factor  for the geometry of Fig. 2. The data are generally independent of the slot size (in the range of small-slot approximation), and independent of the parallel-plate distance (assuming that no higher order mode is propagating or, if evanescent, does not couple to the shorting points.) The data are also independent of the material medium  between the parallel plates, as long as the parallel-plate distance and the shorting-strip width are proportionately scaled with respect to the wavelength  in the particular medium.

Some improvement in can be achieved by increasing the width of the shorting strip or the equivalent shorting-post diameter. The relative effect is illustrated in Fig. 3 which includes an , with a larger strip width additional graph for . Shorting strips that are too narrow or, equivalently, shorting posts that are too thin cannot be that effective. On the other hand, while it may be desirable to have thicker shorting posts in order to increase the suppression factor, they may not be easy to fabricate. Thus a proper balance is needed when selecting the size of the shorting posts. As discussed, the mode-suppression factor is determined by the design of the shorting posts. On the other hand, the actual values of the radiation and guided-wave power, excited by the original structure without the shorting posts, are determined by the parameters of the source and the parallel-plate structure. The combined effects of these two factors determine the radiation efficiency achieved by a complete design. In order to get a general idea of the radiation efficiency of complete designs, we considered some useful sets of design parameters, including the dielectric constant, the thickness of the parallel-plate structure, and the number and size of the shorting posts. The maximum values of the radiation efficiency that can be achieved with these designs by placing the shorting posts at a radial distance of about half the are plotted in Fig. 4. As expected, the slot-length radiation efficiency is higher if a thicker substrate and a lower dielectric constant is used for the parallel-plate medium, or if more and thicker shorting posts are used. While suppressing the power delivered to the parallel-plate mode, the shorting posts may also affect the reactive power in the device. This would appear as additional reactive loading across the antenna input, which may significantly affect the impedance matching and the bandwidth of the antenna. Fig. 5 shows the input susceptance as a function of the cavity diameter of shorting posts and the same set of for different numbers

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Fig. 4. Radiation efficiency of a small, rectangular slot antenna, printed on the top wall of a parallel-plate structure, with surrounding shorting posts circularly :  , slot width W arranged around the source. Slot length L :  , R slightly larger than L = .

0 01

2

= 01

=

physical parameters as in Fig. 2. Significant reactive loading is seen in Fig. 5 at the same resonant shorting locations at which Fig. 2 shows poor radiation efficiency. And minimal reactive loading is observed at the shorting locations at which the radiation efficiency shows peak performance—a desirable feature of the design. If the shorting points are very close to the source, appreciable reactive loading takes place although it may not be as large as at the resonant points. Reactive loading close to the source may not be desirable, particularly if one wants to place the shorting posts close to the source in order to achieve the best possible radiation efficiency. Since this additional loading could affect the impedance performance of the central antenna, it needs be taken into account when adjusting the resonant length of the antenna or tuning the frequency. If reactive loading needs to be avoided completely, one may place the shorting posts a bit farther from the source, thereby compromising the radiation efficiency to some extent. As seen in Fig. 5, one may find a location close to the source, with , where no reactive loading occurs. cavity diameter The design data in Figs. 2–5 for a small slot element are expected to be practically valid as well for reasonably long slots. We have computed the radiation efficiency for a con. This is ductor-backed slot antenna of length about half a wavelength in the effective surrounding medium, associated with a real input impedance. The results, presented in Fig. 6, may be compared with those in Fig. 2, which shows very similar efficiency levels and characteristic features. This suggests that the data set of Figs. 2–5 may be of general use up to about half a wavelength. for designs with slot length The same data set may still be usable for longer slots, but a note of caution must be added. Longer slot elements would strongly excite multiple cylindrical modes, all with odd angular harmonics (if a center-fed slot, symmetric slot-field excitation is assumed). The different modes may resonate each with a slightly different cavity diameter, because the zeros of the respective Bessel functions are closely located. This effect may

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Fig. 5. Physical parameters are the same as those in Fig. 2. Slot length L :  , slot width W :  , parallel-plate distance (cavity height) d : , equivalent shorting-strip width W :  , parallel-plate medium  : .

01 02 0 02

= 0 01

=22

= 04 = 0 01

( ) = 0 = 22 = 07

= = =

Fig. 6. Slot length L :  (Im Y , without any shorting :  , parallel-plate distance (cavity height) posts,) slot width W : , equivalent shorting-strip width d :  , parallel-plate medium  :  . Efficiency for slot length L :  , with N , all W other parameters as above, is also plotted for comparison, showing a distinct new resonance not seen with smaller slots.

= 02 = 0 02

= 32

just start to show in Fig. 6 for the slot length . In this case a small but noticeable “dip” or “bend” appears in the vicinity of the second resonance with cavity diameter . Data for a longer slot with length are also presented in Fig. 6, where the resonance for cavity diamis much more pronounced, as expected. Lengths eter of this size may not often be used in practice. Whenever they are used, all such additional resonances must be identified and avoided for reliable operation. Accordingly, a wider range of cavity diameters may become potentially unusable. B. Uniform Slot Ring The design data in Figs. 2–6 apply for sources that excite . Simcylindrical waves with odd angular harmonics ilar trends can be expected for sources that excite modes con-

= 0 1  , slot width W = 0:01  , parallel= 0:2  , parallel-plate medium  = 2:2, = 0:02  .

: Fig. 7. Slot-ring radius R plate distance (cavity height) d equivalent shorting-strip width W

sisting of even angular harmonics , but with certain basic differences. We now present selected results for a uniform , indicating the similarities and essenslot-ring element tial differences. Fig. 7 shows the radiation efficiency as a function of the shorting distance from the source center for different numbers of shorting posts. The results exhibit trends with respect to that are similar to those for a rectangular slot (Fig. 2). The dependence of the characteristic of a slot-ring element on the is also expected to exhibit trends simshorting-strip width ilar to those characteristic of a rectangular element. That is, inwould increase the maximum radiation efficreasing or ciency or the suppression factor, all other parameters pertaining to the source and the medium remaining the same. However, the shorting distances at which the radiation efficiency is at a minimum in Fig. 2 approximately correspond to those at a maximum in Fig. 7, and vice versa. This is because the resonance (or even , in gendiameters of a cylindrical cavity for (or odd , in general) eral) alternates with those for [10]. A data set for the maximum radiation efficiency that can be achieved by placing the shorting posts close to a particular slotring element is plotted in Fig. 8. The data are presented for the same combinations of parameters pertaining to the shorting ) and the medium between the parallel plates posts ( and as in Fig. 4. The trends with respect to the parallel-plate height in Fig. 8 are similar to those in Fig. 4. There are other similar trends in Figs. 4 and 8: improved efficiency with an increasing number of shorting points, or with thicker shorting posts, or with a lower dielectric constant for the parallel-plate medium, all other parameters remaining unchanged. However, the relative efficiency levels for different parameter sets in Fig. 4 do not always follow those in Fig. 8. For example, in Fig. 8 the relative efficiency levels for the parameter combinations 4 and 6 are switched compared to Fig. 4. We do not present data for the suppression factor and the reactive loading effect in a uniform slot-ring element. They are

MCCABE AND DAS: PART II: A CONDUCTOR-BACKED SLOT ANTENNA ELEMENT SURROUNDED BY A SHORTING-POST CAVITY

Fig. 8. Radiation efficiency of a slot-ring antenna, printed on the top wall of a parallel-plate structure, with surrounding shorting posts circularly arranged :  , slot width W :  ,R around the source. Slot-ring radius R slightly larger than R .

=01

= 0 01

similar to those for a rectangular element shown in Figs. 3 and 5, respectively, with the following basic difference. The resonance locations for the and for the susceptance of a uniform slot-ring element would alternate with those for a rectangular slot (Figs. 3 and 5). These slot-ring resonances correspond to the locations where the radiation efficiency in Fig. 7 is at a minimum. Further, as discussed before, these locations refer to the resonance harmonic. conditions of a cylindrical cavity for the IV. VALIDATION AND EXPERIMENT Fig. 9 shows results, for both a slot-ring as well as a rectangular slot element, obtained with our theoretical model as compared with a conventional full computation based on the moment-method [11], [12]. The moment-method computation models each of the post currents and the slot-field amplitude independent unknowns. The post currents are as assumed to be uniform along , and the slot fields are expressed using a piecewise sinusoid distribution for a rectangular element and a uniform annular distribution for a slot-ring element unknown quantities are then solved for [13]. The by establishing a moment matrix of mutual couplings. The couplings between the different elements in the parallel-plate environment were calculated using the theory of [11], [12]. Clearly, the moment-method procedure is computationally intensive, lacks meaningful physical and analytical insight, and therefore can be prohibitive for general design analysis and optimization. However, such a computation may still be useful for the one-time modeling of a specific geometry, and it is used here for independent validation of our new design model. Any small deviation between the two sets of results in Fig. 9 may be attributed to computational inaccuracy, particularly since the probes in the moment computation are assumed to be cylindrical in shape, whereas those in the new theory are modeled as equivalent current strips. In any case, the close trends and the agreement of the results in Fig. 9 are reasonable confirmation of the validity of our design model.

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Fig. 9. Validation of the design model by an independent computation. Uni: ,W : , post-cavity radius R : . form slot ring: R Rectangular slot: L : ,W : , post-cavity radius R : . : ,d : : ,N , ,W r : : W. , r

= 0 5 cm = 0 1 cm = 1 0 cm = 0 1 cm 4 0 cm Frequency = 1 5 GHz = 0 16 cm = 16 = 05 cm = 0 65

= 3 0 cm = = 0 24 cm

Fig. 10 shows the experimental results for the mode suppression factor of a conductor-backed rectangular slot element with different numbers of shorting posts. The slot element was excited by a microstrip line above the parallel-plate medium [1]. For practical convenience, the parallel-plate medium was se. The total equivalent series lected to be air or free space , as impedance , or the corresponding admittance seen by the microstrip feed line, was first measured without conductor backing. From this measurement the independent parts and , contributed by the two sides of the slot element, were obtained by multiplying the measured value of with a suitable factor obtained theoretically. The parallel-plate backing was then introduced . The total equivalent admitwithout shorting posts was measured in tance the presence of conductor backing, from which the value of already obtained was subtracted to yield . Different numbers of shorting posts were then introduced, and were similarly derived from the measurethe values of ment of the total admittance. The measured value of the mode suppression factor is obtained from definition (14) using the and . We need only the real parts of real parts of the admittances, which are equal to the respective admittances we measured at a resonant frequency of operation. In Fig. 10 the experimental results for are compared with , computed results from the new theory for the six values 2, 4, 8, 16, and 32, showing similar general trends and levels of suppression. The parallel-plate structure in the experiment was finite in the lateral dimension (a circular conducting disc about the slot center, extending somewhat beyond the boundary of the shorting posts), in contrast to the infinite structure used in the theoretical calculations. Although the general trends for the cases with finite and infinite parallel-plate structures are expected to be similar, the actual levels of excitation of the parallel-plate mode and the suppression factors can be different. This may explain the differences between the theoretical and

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Fig. 10. Experimental and theoretical results on a rectangular number of shorting posts = 1 0, = 2 5 mm, slot length = 0 1 cm, frequency = 3 7 GHz, diameter = 2 = 5 5 cm.

 W

: d : R

:

:

N

:

L W

for the effect of the ). slot element ( = 3 2 cm, slot width = 0 3 cm, post-cavity

: :

n; odd

experimental results in Fig. 10. The relative differences in the suppression levels can be significant for smaller , and the differences should approach negligible values for very large , as seen in Fig. 10. We would like to perform a validation experiment for a uniform slot-ring element. A practical slot-ring element operating near resonance, and excited from a single feed point across the slot, generally produce different combinations of even and/or odd harmonics , depending on the frequency and ring length. This would complicate the desired validation experiment for . However, we know that a uniform slot-ring mode is electrically equivalent the uniform slot-ring mode to a vertical current probe across the parallel plates, placed at the center of the ring, regardless of the frequency of operation. Therefore, a vertical probe feed from a coaxial input, with shorting posts placed around it, should exhibit a mode excitation or suppression behavior similar to that studied in Section III for a uniform slot ring. With this in mind, we select a coaxial probe-feed arrangement for our experimental study. This is done with the specific goal of validating our theoretical results, although such a vertical probe is clearly not a planar element we particularly address in the present study. Fig. 11 compares the measured and theoretical values of the input resistance seen by the coaxial feed, as a function of frequency. The input resistance is equivalent to the power coupled to the parallel-plate mode from a unit input current. The total number of shorting posts is chosen to be 16. The frequency dependence of the mode power, as measured in Fig. 11, is equivalent to the dependence on the electrical distance of the post location from the center, as studied in Section III, Fig. 7. The theoretical values for the mode power in Fig. 11 are obtained by , [10]), multiplying the parallel-plate mode power ( excited by a vertical coaxial input without any shorting post, by computed using the new theory. is the factor the mode suppression factor discussed in Section III. Consistently with the theoretical results in Fig. 11, the measured mode is relatively quite small for most electrical power with

Fig. 11. Experimental and theoretical results for the effect of the electrical disof the shorting posts on a uniform slot-ring mode ( = 0). Post tance = 0 75 mm, = 16, = 0 16 cm, cavity radius = 4 2 cm, = 2 33.



R= R :

:

N

r

:

d

n

:

distances, except close to the resonant values , 1.62 . These resonant distances are in agreement with those calculated in Fig. 7. The measured and theoretical data for the power levels in Fig. 11 differ only close to the resonant distances. Such a difference at a high-Q resonance is typically due to departures of practical situations from the ideal conditions commonly assumed in a theoretical calculation. This is particularly true for the present experiment, considering that the parallel-plate structure used is of finite lateral extent (somewhat larger than the shorting-post cavity), whereas in the theoretical calculations it is assumed to be of infinite extent. An input source, surrounded by a shorting-post cavity operating close to a resonance, excites a significant level of the parallel-plate mode. The mode, propagating outside the shorting-post cavity, is strongly reflected from the open end of the parallel-plate structure. The strong reflected power then couples back to the input source, through the resonant shorting posts, at a proportionately higher level. This resonant feedback process results in a higher deviation of the input excitation at the resonant frequency, as seen in Fig. 11. The deviation in Fig. 11 around the second-order resonance point is seen to be relatively more pronounced than . The that around the first-order resonance point reason for this is that, for a given number of shorting posts , the relative excitation of the parallel-plate mode is generally higher at larger shorting distances (smaller mode suppression, Fig. 7). This results in a deviation that is generally larger, due to the reflection from the open end of the parallel-plate structure, as explained above. The general agreement of our theoretical results in Figs. 9–11 with measurements and independent computations demonstrates the validity of the theoretical design model, and establishes confidence in the theory for diverse parameters of the source and the scattering structures. V. CONCLUSION We have performed the first study of the effects of metal shorting posts, arranged in the form of a cylindrical “shorting-

MCCABE AND DAS: PART II: A CONDUCTOR-BACKED SLOT ANTENNA ELEMENT SURROUNDED BY A SHORTING-POST CAVITY

post cavity,” on the parallel-plate mode excited by a conductorbacked slot element. A new modeling approach was used for the study to generate a complete set of design data for the mode-suppression factor, the radiation efficiency and the level of reactive loading on the slot element. Selected results were validated by measurements and independent computations. Data were presented for a range of useful parameters for the shorting posts and the parallel-plate medium, as well as for two basic types of sources. These data sets should find useful application in the design of conductor-backed slot antennas, slot-coupled printed antennas, and slot transitions across metal planes in a multilevel circuit design. As the general results indicate, with proper design of the parallel-plate structure and placement of the shorting posts, one can control and suppress the excitation of the parallel-plate mode, to any desired practicable level.

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[12] N. K. Das and D. M. Pozar, “A generalized CAD model for printed antennas and arrays with arbitrary multilayer geometries,” in Computer Physics Communication, Thematic Issue on Computational Electromagnetics, L. Safai, Ed. Amsterdam, The Netherlands: North-Holland Publications, 1991. [13] B. McCabe and N. K. Das, “Part I: A new theory for modeling conductor-backed planar slot antenna elements, in the presence of a general non-planar surrounding,” IEEE Trans. Antennas Propag, vol. 59, no. 9, pp. 3171–3184, Sep. 2011. Brian L. McCabe (M’04) received the BS and MS degrees in electrical engineering from Carnegie Mellon and Rensselaer Polytechnic Institute, in 1984 and 1994 respectively, and the Ph.D. degree in electrical engineering from Polytechnic University (Now Polytechnic Institute of New York University), in 2004 for research on hybrid methods in computational electromagnetics. He is currently a Technical Fellow with Sikorsky Aircraft, Stratford, CT.

REFERENCES [1] D. M. Pozar, “A reciprocity method of analysis of printed slots and slot-coupled microstrip antennas,” IEEE Trans. Antennas Propag., vol. AP-34, no. 12, pp. 1439–1446, Dec. 1986. [2] N. K. Das and D. M. Pozar, “Multiport scattering analysis of multilayered printed antennas fed by multiple feed ports, part I: Theory; part II: Applications,” IEEE Trans. Antennas Propag., vol. AP-40, no. 5, pp. 469–491, May 1992. [3] N. K. Das, “Generalized multiport reciprocity analysis of surface-tosurface transitions between multiple printed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-41, no. 7, pp. 1164–1167, Jul. 1993. [4] N. K. Das, “Power leakage, characteristic impedance and mode-coupling behavior of finite-length leaky printed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-44, no. 4, pp. 526–536, Apr. 1996. [5] M. Tsuji, H. Shigesawa, and A. Oliner, “New interesting leakage behavior on coplanar waveguides of finite and infinite widths,” IEEE Trans. Microw. Theory Tech., vol. MTT-39, no. 12, pp. 2130–2137, Dec. 1991. [6] N. K. Das, “Methods of suppression or avoidance of parallel-plate leakage from conductor-backed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-44, no. 2, pp. 169–181, Feb. 1996. [7] 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. [8] N. Marcuvitz, Waveguide Handbook. London, U.K.: Inst. Elect. Eng., 1986, ch. 5.11. [9] G. K. C. Kwan and N. K. Das, “Coaxial probe to parallel-plate dielectric waveguide transition: Analysis and experiment,” IEEE Trans. Microw. Theory Tech., vol. MTT-50, no. 6, pp. 1609–1620, Jun. 2002. [10] R. F. Harrington, Time Harmonic Electromagnetic Fields. Hoboken/ Piscataway, NJ: Wiley/IEEE Press, 2001, ch. 5. [11] N. K. Das, “A study of multilayered printed antenna structures,” Ph.D. dissertation, Dept. Elect. Comput. Engrg., Univ. Massachusetts, Amherst, 1987.

Nirod K. Das (S’87–M’88) was born in Puri, Orissa state, India, on February 27, 1963. He received the B.Tech (Hons.) degree in electronics and electrical communication engineering from the Indian Institute of Technology (IIT), Kharagpur, India, in 1985 and the M.S. and Ph.D. degrees in electrical engineering from the University of Massachusetts at Amherst, in 1987 and 1989, respectively. From 1985 to 1989, he was with the Department of Electrical and Computer Engineering, University of Massachusetts at Amherst, first as a Graduate Research Assistant and then, after receiving the Ph.D. degree, as a Postdoctoral Research Associate. In 1990, he joined the Department of Electrical Engineering at Polytechnic University of New York (now Polytechnic Institute of New York University) where he currently is an Associate Professor since 1997. He co-edited Next Generation of MMIC Devices and Systems (New York: Plenum, 1997). He also authored a computer-aided design (CAD)/instructional tool, i.e., PCAAMT, for microwave multilayer printed transmission lines, and another simulation tool, i.e., UNIFY, for unified modeling of multilayer printed antennas and arrays. His research interests have been in the general areas of electromagnetics, antennas, and microwave and millimeter wave integrated circuits. His recent research activities include numerical-analytical methods for electromagnetics and advanced materials for microwave circuits and antennas. Dr. Das is a member of the IEEE Antennas and Propagation Society (IEEE AP-S), the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), and the New York Academy of Sciences. He served in the Technical Program Committee of the IEEE MTT-S International Symposia from 1997–2003, and currently serves on the Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the Co-Chair of the 1996 International WRI Symposium, Brooklyn, New York. For his doctoral research work on multilayer printed antennas he received a Student Paper Award (Third Prize) in 1990 from the US National Council of the International Scientific Radio Union (URSI), and the R.W.P King Paper Award (below 35 age-group) in 1993 from the IEEE AP-S.

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Modal Analysis of Dielectric-Filled Rectangular Waveguide With Transverse Slots Juhua Liu, David R. Jackson, Fellow, IEEE, and Yunliang Long, Senior Member, IEEE

Abstract—The dispersion relations and modes for a dielectricfilled rectangular waveguide with transverse slots are investigated. The differences between the air-filled and the dielectric-filled rectangular waveguides with transverse slots are analyzed. The dielectric-filled rectangular waveguide with transverse slots supports a leaky waveguide mode, a surface-wave mode and a proper waveguide mode. The complex propagation wavenumber is calculated by enforcing an aperture magnetic field integral equation using either a space-domain approach or a spectral-domain approach. The physical significance of the solutions is explained. The theoretical results agree very well with the results from HFSS simulation. Index Terms—Leaky wave, leaky wave antenna, rectangular waveguide, slot array, surface wave. Fig. 1. The geometry of the dielectric-filled rectangular waveguide with transverse slots.

I. INTRODUCTION ECENTLY, a leaky-wave antenna based on a substrate integrated waveguide (SIW) with transverse slots in the broad wall of the SIW was proposed [1]. The SIW leaky-wave antenna has beam scanning from near broadside (though not exactly at broadside) to forward endfire. However, the nature of the waves propagating on the SIW with transverse slots was not analyzed theoretically. Usually, the SIW can be approximated as a rectangular waveguide [2]. Therefore, for understanding the modes propagating on an SIW with transverse slots, the transverse slotted rectangular waveguide (Fig. 1) is used. The purpose of the present paper is to explore the types of modes that propagate on the structure of Fig. 1, and to examine their roles in forming the radiated beam for this type of leaky-wave antenna. Several classical leaky-wave antennas based on rectangular waveguide use traveling-wave slots or slits [3]–[5] as the radiation sources, but with these antennas it is difficult to achieve endfire radiation due to the polarization of the radiating equivalent magnetic currents. The rectangular waveguide with transverse slots [6]–[10] has an orthogonal polarization and can scan

R

Manuscript received August 31, 2010; revised November 24, 2010; accepted January 17, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported by the Key Project of Guangdong Province (2009A080207006), the Research Program of Guangzhou (2010Y1C401) and NSFC-Guangdong (U0935002). J. Liu is with the Department of Electronics and Communication Engineering, Sun Yat-sen University, Guangzhou, China (e-mail: liujuhua_2000@hotmail. com). D. R. Jackson is with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204-4005 USA (e-mail: [email protected]). Y. Long is with the Department of Electronics and Communication Engineering, Sun Yat-sen University, Guangzhou, China (e-mail: [email protected]. edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161444

from near broadside to endfire if the waveguide is filled with a dielectric material. The structure shown in Fig. 1 makes a practical quasi-uniform leaky-wave antenna (radiating from the fundamental space harmonic [5]). In [8], Hyneman gave a comprehensive analysis of the same structure when the waveguide was air filled, and found that a leaky mode and a surface-wave mode can propagate on the structure. But the transverse slotted rectangular waveguide with dielectric filling has not been analyzed so far. The filling makes an important difference since it allows for the beam to scan to endfire, since the phase constant of the guided wave can then reach or exceed the wavenumber of free space as frequency increases. The dielectric filling also allows the structure to approximately model the corresponding SIW structure. This paper focuses on the analyses of the dielectric-filled leaky-wave antenna structure, though results are also represented for the air-filled case for comparison. The complex propagation wavenumber is calculated by using a magnetic field integral equation. The fields are expanded using either a space-domain approach or a spectral-domain approach, and both approaches have been implemented for validation purposes and found to give the same results. The theoretical results also agree very well with the results from HFSS simulation, as will be shown. Three different types of modes that exist on this structure are found for the first time: (1) a leaky waveguide mode, which rerectangular waveguide mode insembles the field of the side the waveguide, but which is leaky, with fields in the exterior air region that are improper (exponentially increasing away from the structure); (2) a proper waveguide mode, which also rectangular waveguide mode resembles the field of the inside the waveguide, but has a proper (exponentially decaying)

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LIU et al.: MODAL ANALYSIS OF DIELECTRIC-FILLED RECTANGULAR WAVEGUIDE WITH TRANSVERSE SLOTS

field distribution in the exterior region; and (3) a proper surface-wave type of mode that has most of the fields in the exterior region, and is only loosely bound to the structure. A transition between these modes occurs as the beam scans to endfire, and an examination of this modal transition will explain the radiation characteristics of this antenna near endfire. The modal transition is different from those that have been found previously for other types of leaky-wave structures [11]–[13]. It is seen here that at lower frequencies (before endfire is reached) the leaky waveguide mode and the surface-wave mode are both physical. As the frequency increases so that the beam reaches endfire, the leaky waveguide mode loses physical significance, while simultaneously the physical surface-wave mode evolves into the physical proper waveguide mode. Near endfire, it is not the leaky mode alone that is responsible for the radiation, but rather a combination of the leaky mode, surface-wave mode, and proper waveguide mode. One interesting aspect of the theoretical analysis concerns the correct choice of branch when analyzing the leaky-wave solution. In the spectral-domain approach, this relates to the corwavenumber plane. rect path of integration in the complex In the space-domain approach, this relates to the correct choice of square root branch for the transverse wavenumber. Another aspect is that when using the space-domain approach to find the conjugate solution, care must be taken in the correct calculation of the Hankel function, as discussed in [14]. II. SPACE-DOMAIN APPROACH In this section, the space-domain approach presented in Hyneman’s paper [8] is extended to analyze the dielectric-filled rectangular waveguide with transverse slots. In this approach the fields inside the waveguide are expanded in terms of waveguide modes, and the fields outside the waveguide are expanded using a line-source Green’s function involving a Hankel function.

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Suppose the fundamental propagation wavenumber of the , where is the phase constant guided mode is is the attenuation constant. Using Floquet’s theorem, and the fields are then expanded inside and outside the waveguide. Using the space-domain approach, the magnetic field outside the waveguide is represented as

(2) where (3) and (4) and is the wavenumber in free space. Inside the waveguide the fields are expanded in terms of waveguide modes and the magnetic field is represented as

A. Transcendental Equation for the Wavenumber The geometry of the dielectric-filled rectangular waveguide with transverse slots is shown in Fig. 1. The relative permittivity and permeability of the dielectric inside the rectangular waveguide are and . The width and height of the waveguide are and , respectively. The length and width of the slot are and . The period (distance between slots) is . To simplify the analysis, it is assumed that the waveguide is infinitely long and the top wall is extended into an infinite ground plane. The metal and dielectric are assumed lossless in this investigation, and the thickness of the waveguide wall is assumed infinitesimal. is rather small compared to the Suppose the slot width wavelength and the slot length. Then it can be assumed that the tangential electric field on the slot has only a component. As in Hyneman’s analysis, an approximation to the transverse slot field distribution is assumed [8], namely

(5) where

and is the wavenumber in the dielectric, . The unknown complex propagation wavenumber is calculated by enforcing an aperture magnetic field integral equation (MFIE), which imposes continuity of the magnetic field on the aperture of the number zero slot. Using a Galerkin testing, we have

(6) (1) In view of the symmetrical structure, all fields are then TE waves with respect to the axis.

where is the area of the zeroth slot (the periodic field representation will automatically then enforce the MFIE at all other slots).

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When the period of the slots is very small compared to the space harmonic of the leaky wavelength in free space, the waveguide mode is chosen to be an improper wave provided [5] and all other space harmonics are chosen to be proper waves. The surface-wave mode and the proper waveguide mode solutions have all the space harmonics chosen as proper waves. C. Conjugate Solution

Fig. 2. The complex plane for k

.

Substitution of (1), (2) and (5) into (6) yields a transcendental that is equation for the unknown complex wavenumber

When is a solution of (7), the conjugate of is also a solution. This follows from taking the conjugate of (7). However, finding the conjugate solution requires care in properly evaluating the conjugate of the Hankel function in (9). If a space haris in the third or fourth monic is a proper wave, namely, quadrant in Fig. 2, the conjugate of the Hankel function is

(10)

(7) where

(In the argument of the Hankel function in (9), the term is being neglected for simplicity.) If is a solution of (7) for a proper mode (all the space harmonics are proper waves), it can is also a solution by using be proven that the conjugate of (9) and (10). The conjugate solution will also be a proper mode, for each harmonic is the negative conjugate of the where corresponding radial wavenumber for the original mode. For an improper space harmonic (the zeroth space harmonic appears in the first quadrant. of the leaky waveguide mode), , For the conjugate solution, the transverse wavenumber, will appear in the second quadrant. The conjugate of is

(8) When the material inside waveguide is air, (7) reduces exactly to (14) in [8]. Performing the integration in (8) by parts [6] and simplifying the double integral into a single integral as in [3], (8) becomes

(11) As noted in [14], when

is in the second quadrant,

(12) (9) Equation (9) above yields the same numerical results as (15) in [8], but evidently in the simplification of the double integration of (8), a mistake was made in the final expression (15) given in [8]. The correct expression for the simplified double integration in (8) is that shown in (9). B. Branch Equation (4) shows that the transverse wavenumber in the exterior region for each Floquet wave (space harmonic) has two different possible values. The different values correspond to the proper and improper choices as shown in Fig. 2. An improper choice causes the wave to be exponentially increasing with distance from the axis while the proper choice causes the wave to be exponentially decaying with radial distance from the axis.

Using (11) and (12), the conjugate of

is then

(13) Clearly, this is a different result than (10). The implication of this is that for the improper space harmonic the Hankel function in (9) cannot be interpreted as the usual Hankel function as commonly defined in the four quadrants of the complex plane, with a branch cut along the negative real axis separating the second and third quadrants. If the usual Hankel function definition is used in the second quadrant, then property (10) will not hold, and the conjugate solution cannot be found from (7). Instead, the Hankel function for an argument in the second quadrant must be interpreted as the analytic continuation of the Hankel function from the third quadrant into the second quadrant, which means that the Hankel function for an argument in the second quadrant is

LIU et al.: MODAL ANALYSIS OF DIELECTRIC-FILLED RECTANGULAR WAVEGUIDE WITH TRANSVERSE SLOTS

being evaluated on a different sheet than the customary one, as noted in [14].

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where

III. SPECTRAL-DOMAIN APPROACH In this section, the fields outside and inside the waveguide are expanded using the spectral-domain approach. The magnetic field integral equation is again enforced for determining the propagation wavenumber.

A simplification of (17) will yield the exact same expression as in (5).

A. Exterior Field

C. Transcendental Equation

From (1), the Fourier transform of the electric field in the zeroth slot is

Inserting the exterior magnetic field from (15) and the interior magnetic field from (5) into the integral equation (6) for the zeroth slot, the transcendental equation for the propagation wavenumber is obtained as

(14) Since the structure is an infinite periodic structure in with a period , using the spectral-domain approach, the final expression for the magnetic field outside the waveguide is derived as

(18) Comparing (7) and (18), it can be proved numerically that the in (9) equals (15) where (19) (16) in (9) is Equation (19) can thus validate (numerically) that correct, and in so doing we note that (15) in [8] has an error in it.

B. Interior Field Since the tangential electric field is continuous, taking into account the boundary condition at the bottom of the waveguide, and accounting for the periodicity in and considering the mirror effects of the rectangular waveguide sidewalls, the interior magnetic field produced by all of the slots is given by

(17)

D. Integration Path In the calculation of the field above the waveguide, i.e., in (15), (18) and (19), there is an integration in the spectral wavenumber . The integration path should be selected for each space harmonic according to the proper/improper nature of the space harmonic. There are two possible integration paths is chosen as shown in Fig. 3, where the integration path for an improper wave (i.e., the zeroth harmonic of the leaky is chosen for a proper wave. mode) and the integration path The property of the complex plane and the different integration paths for leaky-wave antennas has been fully discussed in [15] and the reader is referred there for more details. It is interesting discussed here is equivalent to that the integration path that is discussed in the improper choice of branch for

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Fig. 3. The integration paths in the spectral-domain approach.

Fig. 5. Normalized attenuation constant for air-filled rectangular waveguide with transverse slots.

Fig. 4. Normalized phase constant for air-filled rectangular waveguide with transverse slots.

Section II.B, and the integration path . proper choice of branch for

is equivalent to the

IV. NUMERICAL RESULTS FOR AIR-FILLED WAVEGUIDE In this section results for the air-filled rectangular waveguide with transverse slots are shown. The dimensions of the rectanmm, mm, gular waveguide are and . The parameters of the slots are mm, mm, mm. A. Theoretical Results The propagation wavenumber is calculated by using the space-domain approach and the spectral-domain approach. The normalized phase and attenuation constants are shown in Figs. 4 and 5. Results for the corresponding closed rectangular waveguide are also shown for comparison. Figs. 4 and 5 show that the results from the space-domain approach are exactly the same as the ones from the spectral-domain approach. This remains the case for subsequent figures, though only the results from the space-domain approach are shown for simplicity. can After calculating the wavenumber, the magnetic field be calculated in the space-domain approach by using (2) and (5), or in the spectral-domain approach by using (15) and (17). Fig. 6 shows the magnetic field profiles of a typical leaky mode (radiating mode with a complex wavenumber) and bound mode

Fig. 6. Magnetic field along the line x = 0, z waveguide with transverse slots at 22 GHz.

= p=2 for air-filled rectangular

(unattenuated mode) as a function of along the line . It can be seen that the leaky mode (c) has most of the power stored inside the waveguide and eventually grows as a function of outside the waveguide, so this mode is classified as an improper leaky waveguide mode. On the other hand, the bound mode (d) has most of the power stored outside but close to the surface of the waveguide, and the field decays vertically outside the waveguide, so this mode is classified as a proper surface-wave type of mode. From Figs. 4 and 5, it is found that only the wavenumber of the leaky waveguide mode is very close to that of the closed rectangular waveguide (the phase constants are essentially coincident above cutoff of the closed waveguide, while the attenuation constants are almost coincident below cutoff). Hence, only the leaky waveguide mode represents a perturbation of the closed rectangular waveguide. When the slot size is extremely small, the surface-wave mode will disappear and the leaky waveguide mode in closed rectangular wavemode will become the guide. The surface-wave mode is loosely bound to the structure,

LIU et al.: MODAL ANALYSIS OF DIELECTRIC-FILLED RECTANGULAR WAVEGUIDE WITH TRANSVERSE SLOTS

Fig. 7. Steepest-descent plane for air-filled rectangular waveguide with transverse slots. The shaded region has full physical significance, while the region that is not shaded has no physical significance. The solid lines, dashed lines and dash-dotted lines are corresponding to the results in Fig. 4, while the dotted lines denote the conjugate solutions.

owing its existence to the presence of the slots in the ground plane. Fig. 7 shows the results plotted in the steepest-descent plane, defined by together with (20) where the sign of the square root is chosen according to the proper/improper nature of the wave. The transformation (20) is based on (16). The steepest-descent plane is useful because the proper and improper sheets of the plane get mapped into a single sheet of the plane for this kind of problem (due to the mapping in (20)). It is also convenient in explaining the physical meaning of the proper and improper solutions using the steepest-descent plane. Note that in connection with the steepest-descent plane the complex plane is only being used in the interpretation of the results, in accordance with (20), and not in the numerical calculation of the wavenumber solution. plane used in the calculation of the Hankel function (The in (2) for the wavenumber solution has an infinite number of sheets, but this is irrelevant for the graphical display of the wavenumber results using (20).) The physical properties of this plane in the context of periodic structures have been discussed further in [13]. Fig. 7 also shows the conjugate solution with a dotted line style. When searching for the conjugate solution of the leaky waveguide mode, care must be taken in evaluating the Hankel function as discussed in Section II.C. (Here, the conjugate solution from the spacedomain method has been validated with the spectral-domain method, although only the result from the space-domain method is shown.) Fig. 7 shows that the leaky waveguide modal solution is always in the physical fast-wave region (the shaded region to the left of the ESDP curve, which is the extreme steepest-de[13]), so this scent path that passes through the point mode has physical significance at all frequencies.

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In summary, with respect to phase constant shown in Fig. 4, the behavior of the modes in the different regions are as follows, labeled to match the corresponding labels in Fig. 4. GHz: The leaky waveguide mode is below the a) cutoff frequency, and is an improper mode. GHz: The leaky waveguide mode is b) cutoff frequency, is a wave with a negabelow the tive phase velocity , and is a proper mode. GHz: The leaky waveguide mode is an improper c) mode that remains in the physical fast-wave region. GHz: There exists a proper surface-wave mode d) with a real wavenumber. GHz: There exists a proper surface-wave mode e) with a real wavenumber, which is a backward wave [16] (the group velocity and power flow are in the negative direction, for a positive phase constant). For this mode it space harmonic that is is the wavenumber of the one (which is plotted for the plotted instead of the other curves). GHz: The two surface-wave modes have f) merged to form a single proper complex mode that is surface-wave like in nature. This mode is in the stop-band and has a constant value of and a nonzero attenuation constant, even though it does not radiate (the attenuation corresponds to evanescent decay). The frequency at which the stop-band begins is an upper cutoff frequency of the proper mode. Usually, when the slot length is equal to the waveguide width , the cutoff frequency is [8], where is the speed of light. approximately V. NUMERICAL RESULTS FOR DIELECTRIC-FILLED WAVEGUIDE This section presents numerical results for the dielectric-filled rectangular waveguide with transverse slots. The and . The dimensions of dielectric filling has the rectangular waveguide are mm and mm. mm, mm, and The parameters of the slots are mm. A. Theoretical Results Figs. 8 and 9 show the normalized phase and attenuation constants vs. frequency. The results are calculated using the space-domain approach and these results were found to be exactly the same as the ones from the spectral-domain approach. Results for the corresponding closed rectangular waveguide are also shown for comparison. With respect to the phase constant shown in Fig. 8, the following modes were found in the corresponding frequency regions that are labeled in Fig. 8. GHz: An improper leaky waveguide mode exa) ists in the cutoff region, below the cutoff of the mode. GHz: The leaky waveguide mode is b) still within the cutoff region, but is a wave with a negative phase constant (see the inset in Fig. 8) and is a proper wave. GHz: The improper leaky waveguide c) mode is in the physical (fast-wave) radiation region.

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Fig. 8. Normalized phase constant for dielectric-filled rectangular waveguide with transverse slots. Fig. 10. The magnetic field along the line x rectangular waveguide with transverse slots.

Fig. 9. Normalized attenuation constant for dielectric-filled rectangular waveguide with transverse slots.

d)

GHz: The improper leaky waveguide mode is a slow wave with . In this region the leaky mode is said to be in the “spectral gap” region and loses physical significance. GHz: A proper surface-wave mode exists, with e) a real wavenumber. GHz: A proper waveguide mode exf) ists, with a real wavenumber. GHz: A proper backward wave exists with a g) real wavenumber. This wave is actually the space harmonic of the same modal solution as in (f), but carrying power in the negative direction. GHz: The two proper waveguide modes have h) merged to form a single proper waveguide mode with a complex wavenumber. This mode is in the stop-band, having a constant value of and evanescent decay but no radiation. can again be calculated with the spaceThe magnetic field domain approach by using (2) and (5), or with the spectral-domain approach by using (15) and (17). Fig. 10 shows the magnetic field for solutions (c), (e) and (f). Fig. 10 shows that both

= 0; z = p=2 for dielectric-filled

solutions (c) and (f) have most of their power stored inside the waveguide. Moreover, Fig. 8 shows that solutions (c) and (f) have wavenumbers that are close to the wavenumber of the closed rectangular waveguide. Hence, solutions (c) and (f) represent perturbations of the closed rectangular waveguide. Solution (c) is referred to as the leaky waveguide mode while solution (f) is referred to as the proper waveguide mode. Solution (e) corresponds to a surface-wave mode since most of the power is stored in the external region outside the waveguide, as shown in Fig. 10. When the slots are extremely small, either the leaky waveguide mode or the proper waveguide mode becomes mode of the closed rectangular waveguide (depending the on the frequency), and the surface-wave mode disappears. The leaky waveguide mode in Fig. 10 is improper, though the vertical exponential increase is not observed on this scale. The leaky waveguide mode for the case of dielectric filling has an upper cutoff frequency at 10.1 GHz, corresponding to . When the frequency goes to this upper cutoff frequency, the antenna scans to endfire. As the frequency increases beyond this, the phase constant will enter the nonphysical region (it crosses the ESDP in the steepest-descent plane), as shown in Fig. 11. The leaky mode will lose its physical significance (somewhat gradually) when the frequency becomes larger than the upper cutoff frequency. The conjugate solutions that are calculated with the space-domain method (Section II.C) and have been validated with the spectral-domain method are also shown in Fig. 11 (dotted lines). The conjugate results are mathematically valid although they have no physical meaning. The proper waveguide mode (solution (f)) evolves from the surface-wave mode that exists at low frequency (solution (e)), as shown in Fig. 8. This transition occurs at exactly the same frequency where the leaky mode loses its physical significance (10.1 GHz). The transition is very sharp; below 10.1 GHz the proper mode is surface-wave like and above 10.1 GHz the proper mode is waveguide like. When the frequency continues increasing, the proper waveguide mode will enter the stop-band where it is complex with a large attenuation constant. The

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Fig. 12. Top view of the geometry (not to scale) for the simulation in HFSS. The slotted waveguide simulated in HFSS is a practical waveguide that has a top wall with a width a mm, and no top ground plane. The lengths are L mm and L mm, and the total length of the waveguide is 450 mm. The walls of the waveguide are assumed to be perfect conductor with a zero thickness.

= 300

= 12 = 60

Fig. 11. Steepest-descent plane for dielectric-filled rectangular waveguide with transverse slots. The shaded region has full physical significance, while the region that is not shaded has no physical significance. The solid lines, dashed lines and dash-dotted lines correspond to the results in Fig. 8, while the dotted lines denote the conjugate solutions.

upper cutoff frequency of the proper mode (beginning of the stopband) depends on the slot size, but not only on just the slot length . When the slot width becomes smaller, even though , the upper cutoff frequency the slot length is kept fixed at increases but is not given approximately by as it is for the air-filled case [8]. When the frequency is less than the upper cutoff frequency of leaky waveguide mode, both the leaky waveguide mode and the surface-wave mode have full physical significance and mode of the closed both waves can be excited by the rectangular waveguide. But when the frequency is larger than this upper cutoff frequency, the leaky waveguide mode loses its physical significance and the proper waveguide mode becomes the only mode with physical significance. So there is a physical continuity between the leaky waveguide mode and the proper waveguide mode as the frequency scans through endfire, although the two solutions do not connect mathematically. B. HFSS Simulation In addition to the theoretical calculations (from the space-domain method and the spectral-domain method, which give perfect agreement) the wavenumber can also be extracted numerically from simulation by HFSS. To do this, the structure shown in Fig. 12 is simulated. Note that the structure simulated in Fig. 12 has no top ground plane, and is not the structure assumed in Fig. 1 (which has an infinite top ground plane). As a matter of fact, we have also simulated the structure shown in Fig. 1 with HFSS, and an excellent agreement between the results from HFSS and the theoretical results is obtained, although not shown here. By comparing the HFSS results and the theoretical results in this section, it is established that the top ground plane has relatively little effect on the results. In Fig. 12, the slots at both ends are tapered to suppress the reflected wave. The near field in the region surrounding the uniform slots is used to extract the wavenumber of the propagating waves on the structure. In particular, it is assumed that there are

Fig. 13. Electric field plotted along the centerline inside the transverse-slotted waveguide at a frequency of 9.9 GHz from HFSS. The fields of the numerically extracted leaky and proper modes are shown, along with the sum of the two fields. The total field calculated by HFSS agrees well with the sum of the leaky and proper modal fields.

two traveling waves propagating along the waveguide, so that field for the

(21) where one wave propagates with wavenumber and the other . By numerically fitting the sum of the with wavenumber two travelling waves in (21) to the electric field obtained from and as well as the amplitudes HFSS, the wavenumbers and can be extracted. If it is known that one wave has a real wavenumber, this can be assumed in (21) to simplify the numerical extraction. As shown in Fig. 13, at a frequency of 9.9 GHz, it is assumed that there are two modes: a leaky mode (dotted line) with a comand a bound (nonradiating) mode (dashed plex wavenumber line) with a real wavenumber propagating on the structure. In Fig. 13, the field that is composed of the leaky mode and the bound mode (solid line) fits the total field from HFSS very well. Fig. 13 confirms that both the leaky mode and the surface-wave mode exist on the structure. The surface-wave mode is more difficult to excite as the slots become smaller since the mode vanishes in this limit. The normalized phase constant and attenuation constant from HFSS are shown in Figs. 14 and 15, and are compared with the

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mode (surface-wave mode or proper waveguide mode) play an important role in the radiation properties. VI. CONCLUSION

Fig. 14. Normalized phase constant from HFSS and theory for dielectric-filled rectangular waveguide with transverse slots.

This paper has investigated the modal properties of a dielectric-filled rectangular waveguide with transverse slots. This structure makes a practical leaky-wave antenna that has the capability of scanning down to endfire. The difference in the propagation characteristics between an air-filled waveguide and a dielectric-filled waveguide was shown. The propagation wavenumbers were calculated theoretically using two different methods, and also compared with numerical results from HFSS simulation. The dielectric-filled rectangular waveguide with transverse slots supports three different modes, depending on the frequency: a leaky waveguide mode, a surface-wave mode, and a proper waveguide mode. Both the leaky waveguide mode and the proper waveguide mode are perturbations of waveguide mode. For the frequency region where the the phase constant of the leaky waveguide mode is less than the wavenumber of free space (the leaky region), both the leaky waveguide mode and the surface-wave mode exist. As frequency increases and the beam scans to endfire the leaky waveguide mode loses physical significance. At the same time that the leaky waveguide mode loses physical significance, the surface-wave mode transitions into the proper waveguide mode. Hence, there is a physical continuity in the perturbed waveguide mode, changing from a leaky improper mode to a proper mode at the transition (endfire) frequency. ACKNOWLEDGMENT The authors wish to acknowledge V. R. Komanduri from the University of Houston for assistance during this research.

Fig. 15. Normalized attenuation constant from HFSS and theory for dielectricfilled rectangular waveguide with transverse slots.

theoretical results. Figs. 14 and 15 show that the theoretical results agree very well with the results from HFSS. Since the leaky and proper waveguide modes have most of their power stored inside the waveguide while the surface-wave mode has most of its power stored outside (but close to) the waveguide, it is best to extract the wavenumber of the leaky or proper waveguide mode by probing the near field inside the waveguide, and to extract the wavenumber of the surface-wave mode by sampling the near field outside the waveguide. This allows the wavenumber of each mode to be found as accurately as possible. Although the present paper has focused on the propagation properties of the guided modes and not the radiation (antenna) characteristics of the structure, the radiation properties have been studied in [17]. Results show that the leaky waveguide mode is the dominant radiation mechanism of the antenna when the radiated beam is not too close to endfire. As the beam approaches endfire both the leaky mode and the proper

REFERENCES [1] J. Liu, V. R. Komanduri, and D. R. Jackson, “Substrate integrated waveguide (SIW) leaky-wave antenna with transverse slots,” presented at the USNC/URSI National Radio Science, Charleston, SC, Jun. 2009. [2] F. Xu and K. Wu, “Guided-wave and leakage characteristics of substrate integrated waveguide,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 1, pp. 66–73, Jan. 2005. [3] V. H. Rumsey, “Traveling wave slot antennas,” J. Appl. Phys., vol. 24, pp. 1358–1365, Nov. 1953. [4] L. O. Goldstone and A. A. Oliner, “Leaky-wave antennas—Part I: Rectangular waveguides,” IRE Trans. Antennas Propag., vol. AP-7, pp. 307–319, Oct. 1959. [5] D. R. Jackson and A. A. Oliner, “Leaky-wave antennas,” in Modern Antenna Handbook, C. A. Balanis, Ed. Hoboken, NJ: Wiley, 2008, ch. 7. [6] R. S. Elliott, “Serrated waveguide—Part I: Theory,” IRE Trans. Antennas Propag., vol. 5, pp. 270–275, Jul. 1957. [7] K. C. Kelly and R. S. Elliott, “Serrated waveguide—Part II: Experiment,” IRE Trans. Antennas Propag., vol. 5, pp. 276–283, Jul. 1957. [8] R. F. Hyneman, “Closely-spaced transverse slots in rectangular waveguide,” IRE Trans. Antennas Propag., vol. 7, pp. 335–342, Oct. 1959. [9] J. P. Renault, “Leaky-Wave Radiation from a Periodically Slotted Waveguide,” U.S. Air Force, Bedford, Mass, Res. Rep. PIBMRI-1151-63 for Air Force Cambridge Res. Lab., May 1963. [10] S. L. Berdnik, V. A. Katrich, and V. A. Lyaschenko, “Closely spaced transverse slots in rectangular waveguide,” in Proc. Int. Conf. on Antenna Theory and Techniques, Sep. 2003, pp. 273–275.

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[11] H. Shigesawa, M. Tsuji, and A. A. Oliner, “The nature of the spectral gap between bound and leaky solutions when dielectric loss is present in printed-circuit lines,” Radio Sci., vol. 28, no. 6, pp. 1235–1243, Nov./ Dec. 1993. [12] P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound-wave and leaky-wave ranges for a partially dielectric loaded open guiding structure,” IEEE Trans. Microwave Theory Tech., vol. 38, pp. 1831–1836, Dec. 1990. [13] S. Majumder, D. R. Jackson, A. A. Oliner, and M. Guglielmi, “The nature of the spectral gap for leaky-waves on a periodic strip-grating structure,” IEEE Trans. Microwave Theory Tech., vol. 45, no. 12, pp. 2296–2307, Dec. 1997. [14] J. R. James, “Comment: Leaky-waves on a dielectric rod,” Electon. Lett., vol. 5, pp. 252–254, May 1969. [15] P. Baccarelli, C. Di Nallo, S. Paulotto, and D. R. Jackson, “A full wave numerical approach for modal analysis of 1D periodic microstrip structures,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1350–1362, Apr. 2006. [16] A. A. Oliner, “Radiating Periodic Structures: Analysis in Terms of k vs. Diagrams,” Rep. SC-5-63 (41pp.), Jun. 4, 1963, Polytechnic Institute of Brooklyn (now Polytechnic Institute of New York University (NYU Poly)), short-course series, Microwave Field and Network Techniques. [17] J. Liu, D. R. Jackson, and Y. Long, “Substrate integrated waveguide (SIW) leaky-wave antenna with transverse slots,” IEEE Trans. Antennas Propag., submitted for publication.

Juhua Liu was born in Heyuan, Guangdong, China, in September, 1981. He received the B.S. degree from Sun Yat-sen University, China, in 2004, where he is currently working toward the Ph.D. degree. From 2008 to 2009, he was a Visiting Scholar in the Department of Electrical and Computer Engineering, University of Houston, Houston, TX. His present research interests include microstrip antennas, substrate integrated waveguide antennas, leaky-wave antennas, periodic structures, and computational electromagnetics. Mr. Liu was awarded a scholarship from the China Scholarship Council (CSC) to study in the United States.

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David R. Jackson (S’83–M’84–SM’95–F’99) was born in St. Louis, MO, on March 28, 1957. He received the B.S.E.E. and M.S.E.E. degrees from the University of Missouri, Columbia, in 1979 and 1981, respectively, and the Ph.D. degree in electrical engineering from the University of California, Los Angeles, in 1985. From 1985 to 1991 he was an Assistant Professor, from 1991 to 1998 an Associate Professor, and since 1998 a Professor in the Department of Electrical and Computer Engineering, University of Houston, Houston, TX. His present research interests include microstrip antennas and circuits, leaky-wave antennas, leakage and radiation effects in microwave integrated circuits, periodic structures, and electromagnetic compatibility and interference. He is a Fellow of the IEEE and is presently serving as the Chair of the Distinguished Lecturer Committee of the IEEE AP-S Society, and as a Member-atLarge for U.S. Commission B of URSI (the International Union of Radio Science). He also serves as the Chair of the MTT-15 (Microwave Field Theory) Technical Committee and is on the Editorial Board for the IEEE Transactions on Microwave Theory and Techniques. Previously, he was the Chair of the Transnational Committee for the IEEE AP-S Society, the Chapter Activities Coordinator for the AP-S Society, a Distinguished Lecturer for the AP-S Society, a member of the AdCom for the AP-S Society, and an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He has also served as the Chair of U.S. Commission B of URSI. He has also served as an Associate Editor for the journal Radio Science and the International Journal of RF and Microwave Computer-Aided Engineering.

Yunliang Long (M’01–SM’02) was born in Chongqing, China, in June, 1963. He received the B.Sc., M.Eng., and Ph.D. degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 1983, 1989, and 1992, respectively. From 1992 to 1994, he was a Postdoctoral Research Fellow, then employed as an Associate Professor, with the Department of Electronics, Sun Yat-sen University, Guangzhou, China. From 1998 to 1999, he was a Visiting Scholar in IHF, RWTH University of Aachen, Germany. From 2000 to 2001, he was a Research Fellow with the Department of Electronics Engineering, City University of Hong Kong, China. Currently, he is a Professor and the Head of the Department of Electronics and Communication Engineering, Sun Yat-sen University, China. He has authored and coauthored over 130 academic papers. His research interests include antennas and propagation theory, EM theory in inhomogeneous lossy medium, computational electromagnetics, and wireless communication applications. Prof. Long is a member of the Committee of the Microwave Society of CIE, and is on the editorial board of the Chinese Journal of Radio Science. He is Vice Chairman of the Guangzhou Electronic Industrial Association. His name is listed in Who’s Who in the World.

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Very Small Footprint 60 GHz Stacked Yagi Antenna Array Olivier Kramer, Tarek Djerafi, and Ke Wu, Fellow, IEEE

Abstract—Millimeter wave applications such as short-range high-speed wireless links require modular, compact-size and high-directivity antennas. In this paper, high-gain compact stacked multilayered Yagi designs are proposed and demonstrated in the V-band. This novel design shows for the first time an antenna array of Yagi elements in millimeter wave stacked structure. To demonstrate the proposed concepts and design features, a 4 4 antenna array is created having excellent gain performance as well as very small footprint. A single element stacked Yagi antenna fed with microstrip is studied in order to obtain the desired performance. An analysis is performed to define the structure limitations. Measured results of the fabricated antenna prototypes are in good agreement with simulated results The measured Yagi antenna attains 11 dBi gain over 4.2% bandwidth with a size of 6.5 6.5 3.4 mm3 . A 4 4 array of Yagi antenna using an SIW (Substrate Integrated Waveguide) feeding technique is conceived. Both simulated and measured results match with each other very well. The 4 4 array has a size of 28 24 2.4 mm3 , and reaches a measured gain of 18 dBi over 7% bandwidth. An alternate configuration of the array using angled Yagi antenna elements allows a significant improvement of the side lobe level (SLL) with a low impact on the gain performances. The proposed antennas are excellent candidates for integrated low-cost millimeter-wave and even terahertz systems. The small foot print, the antenna design flexibility as well as its easy adaptation to automatic fabrication processes are good assets for making short range portable imaging systems. Index Terms—Array, circular patch, feeding network, microstrip antenna, millimeter-wave and terahertz, SIW, SLL, stacked antenna, Yagi-Uda.

I. INTRODUCTION

T

HE recent trend on the development of millimeter-wave frequencies systems has led to many innovative techniques with their successful demonstrations in different applications. Among those applications, the unlicensed bands around 60 GHz and above provides an opportunity for high-data-rate wireless communications and sensing applications with reduced energy per bit [1]; 77 GHz automotive radar [2]; and 94 GHz imagers and radiometers [3], where a lower profile array Manuscript received November 25, 2010; revised January 25, 2011; accepted February 23, 2011. Date of publication July 14, 2011; date of current version September 02, 2011. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and in part by Regroupement strategique of FQRNT. The authors are with the Département de Génie Électrique, Poly-Grames Research Center, École Polytechnique de Montréal, Montréal, QC H3T 1J4, Canada (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161562

antenna can achieve high gain. Millimeter-wave front-ends necessitate antenna systems with small size, low-power consumption and power-efficiency requirements [4], which should be integrated together with circuits to avoid unnecessary transmission line loss. A low side lobe level (SLL) is another important characteristic parameter that must be controlled to minimize the interferences [5]. This is particularly in line with the emerging worldwide discussion on green information and communication technology (Green ICT) policy and its implementations. The physical gain saturation of planar antennas and more specifically planar antenna array is defined by Hall [6], and it is limited to circa 35 dBi for a large number of elements with a significant decrease in efficiency [7]. The stacked Yagi-Uda antenna can overcome this limit by using the third dimension. The classical Yagi antenna has been widely successful thanks to its simplicity and customizable high gain [8]–[12]. It consists of basically three-elements: a half-wavelength driver dipole, a longer reflector backing the driver and a director on the other side. Presented in a previous paper [13] is a stacked Yagi antenna working at 5.8 GHz for the purpose of proof-of-concept, stacking together the reflector, the driver, and the directors printed on substrate. However, the usual spacing is quite large at 5.8 GHz between parasitic elements of and imposes air gaps as the substrate would be too thick. This integration issue is naturally solved at higher frequencies like millimeter-wave and beyond where a completely integrated stacked structure can be achieved owe to wavelength comparable with dielectric substrate thickness. Such structures are suitable for multilayer processing techniques including PCB, LTCC and photoimageable thick-film process, which have become more mature in integrated circuit design, fabrication and integration [14], [15]. First of all, a V-band single antenna element is designed and demonstrated, using six elements fed by microstrip line. An analysis of the dielectric loss as well as the effect of coupling in the structure is performed in order to design the 4 4 planar antenna array. The array makes use of an SIW (Substrate Integrated Waveguide) network in order to feed a 4 4 array by coupling the radiating elements through rectangular slots. The SIW is used to reduce or even suppress the radiation that generally appears in microstrip feeding structure. Hence, it can be connected to a demodulation circuit without interferences. An alternate configuration of the array is also presented. It is composed of Yagi elements oriented in different directions with geometrical offset with respect to each other, allowing for a significant reduction of SLL.

0018-926X/$26.00 © 2011 IEEE

KRAMER et al.: VERY SMALL FOOTPRINT 60 GHZ STACKED YAGI ANTENNA ARRAY

Fig. 1. Proposed circular-patch based Yagi antenna.

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Fig. 3. Gain versus number of element.

TABLE I STACKED YAGI ANTENNA

Fig. 2. Side view of the circular- patch based Yagi antenna.

II. STACKED YAGI ANTENNA The general configuration of the stacked Yagi antenna is shown in Figs. 1 and 2. It is made of one circular patch driver, and four parasitic elements. Of course, other forms of radiating elements can be used instead of circular patch. The antenna respects the same design rules that are used in the conventional dipole based Yagi antenna, except that it is made on stacked planar substrates. The circular patch element offers flexibility of design and optimization without complexity since it requires optimizing one parameter (radius) only and allows the possibility of a dual polarization. By choosing the elements’ diameters, the spacing between them, the substrate thickness as well as the dielectric constant, the antenna’s performances such as gain, front-to-back ratio, beam width and center frequency, can be modified. The elements are printed on Duroïd 5880 substrates and have a relative permittivity of 2.2, which reduces spacing between parasitic elements. The spacing is within the standard range and . In this case, the spacing is which is between corresponding to a thickness of 0.508 mm. The radiating element configuration consists of two substrate layers with 0.254 mm of thickness. The circular patch is etched on the top of Layer 2, the feed lines etched on top of Layer 1 and the ground plane at its bottom. The patch is fed by coupling, allowing an increased bandwidth. The proposed stacked dipole Yagi antenna is simulated using Ansoft HFSS v12.0. Fig. 3 shows the gain as a function of the number of elements in the stacked Yagi structure. It can be noticed that the gain

rises steadily when the number of elements increases up to 8–9 elements, then begins to get saturated. Stutzman made a similar curve for classical Yagi antennas and a gain of 12.5 dBi was reached for 10 elements [10]. In our configuration, the gain for the same number of elements is 2 dB more. In addition, the saturation occurs after a higher number of elements. This is due to the fact that the circular patch has more gain than its dipole counterpart, the reflector (here the ground plane of the patch) is four times larger than the radiating elements, and the fact that the air-substrate interface is placed around the antenna confines the electromagnetic fields within the structure. The 0.2 to 0.5 dB of gain drop caused by the dielectric loss shows that it does not have a significant effect on the antenna performances. This is in particular interesting for millimeter-wave integrated antennacircuit platform based on semiconductor substrates. The optimized dimensions of the prototype antenna are tabulated in Table I. The total height of the designed structure is 3.4 mm. The surface size of the stacked substrates is 6.5 6.5 mm . The picture of the fabricated prototype is shown in Fig. 4. S-parameters are measured using Anritsu 37397C Vector Network Analyzer (VNA) to characterize the matching condition of the proposed structure over the V-band frequency range. Simulated and measured S-parameters versus frequency are presented in Fig. 5. We can notice a very small shift in frequency from the designed target but it is still very close to 59 GHz, which may be caused by our fabrication tolerance and uncertainty of madB) covers terial parameters. The antenna bandwidth (

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Fig. 4. Photography of the circular patch-based Yagi antenna.

Fig. 5. Simulated and Measured S-parameters of the circular patch-based Yagi antenna.

frequencies from 57.5 to 60 GHz or almost 4.2% at 60 GHz. The radiation pattern is obtained by using a measurement system composed of a rotating standard horn antenna (Quinstar, Model n qwh-vprr00) and a network analyzer. The gain is calculated using the Friis equation. Fig. 6 presents calculated and measured co-polar (E-plane) radiation patterns. From these results, we can observe a good agreement between the measured and simulated radiation patterns. The pattern is directive and its 3 dB beam width is approximately 55 for both measured and simulated E-plan patterns. The side lobes are found higher in the measurement and are the direct result of surface waves diffracting at the border. Ripples appear because of the multiple reflections of the measurement process and the test fixture used to feed the antenna. The maximum antenna gain is 12 dBi in simulation and 11 dBi in measurement. This difference of 1 dB is attributed to the loss of the connectors and the measurement process. This shows a very good agreement in performances between measurements and simulations. III. 4

4 STACKED YAGI ANTENNA ARRAY USING SIW FEEDING MECHANISM

The antenna array uses SIWs to feed the different elements, which are very suitable for millimeter wave frequencies thanks to their integrability, low loss, negligible radiation [16], and hence a more suitable solution than the previous microstrip or other planar feeding techniques. In order to couple the circular

Fig. 6. Measured and simulated gain of the circular-patch based Yagi antenna. (a) H-plane. (b) E-plane.

[19] patch, driver slots are made on the top layer of the SIW feeding network. Micromachining technique laser perforation was employed. These techniques are amenable to arbitrarily shaped perforations. As suggested in [17], long and large rectangular vias are used for lower leakage and better definition of the SIW side walls. Those vias (or slots) are electroplated. Rounded corners increase the overall mechanical stability, allowing a better metallization, and they often cannot be completely avoided in the fabrication process because of the laser diameter. The SIW is operated between the first and second stop bands. Because of the feeding structure, this configuration does not allow the use of a reflector. The proposed feeding is shown in Fig. 7. The geometry of each antenna is very similar to the previous geometry as described in the above section. The antenna is optimized to achieve the best possible gain and good bandwidth. Each antenna element reaches a simulated gain of 10.5 dBi, which conforms to the theory without resorting to a dedicated reflector. Each branch possesses four elements. The branches are short-circuited at their extremities. Each element is fed by slots located where the gradient of electric field is the maximum in the in the waveguide, corresponding to a spacing of

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Fig. 7. SIW feeding network with SIW-microstrip transition (a) Power divider with couplings slots. The 1 4 network is a combination of one T junction and two Y junctions [19]. (b) Coupling slots.

2

Fig. 8. Isolated mutual coupling between two branches center with and without air gap.

substrate [18]. This configuration allows for an easy feeding of each element. Since the distance between elements is less than half a wavelength in the air and also since the energy distribution is not uniform, the gain does not get doubled when the number of elements is doubled. A gain of more than 14 dBi is reached for each branch with very low side lobe level characteristics. Four branches are subsequently added in order to make up a 4 4 array. Fig. 8 shows the coupling between two of those branches. When dielectric substrate between the elements is used, the isolation is low and reaches a maximum of 30 dB. Air slots are created through various techniques such as partial substrate removal between branches in order to reduce the coupling between them. This improves significantly . This the isolation with a distance of more than 3 mm coupling affects the gain performance of the complete array and changes the impedance of each branch [10]. This change of impedance could be very time consuming as it requires optimizing again each branch. In addition, metalized slots are inserted around the patches to avoid the waves to leak out or spread horizontally, which can use the substrate as a waveguide through parallel-plate slab mode and radiate out at the edges of the structure. The proposed configuration is a good structure for beam steering. A matrix (Butler, Blass, Nolen, etc.) can be used to scan a single dimension. A two dimensions scan (as in

Fig. 9. Representation of the entire array antenna including the transition.

TABLE II

4

2 4 ARRAY OF STACKED YAGI ANTENNA

[20]) would require a phase shifter controlling each elements of the array. The discussed antenna array is shown in Fig. 9. In this structure, the feeding structure uses Rogers 6002 with 0.127 mm thickness for the two first layers because of its better properties with our laser micro-machining used during the fabrication process and Rogers RF/Duroïd 5880 substrate with 0.762 mm thickness for the upper layers. This leads to a total design height of 2.41 mm. The surface size of the upper layers is 28 mm 24 mm. The SIW slots are 1.41 mm long, 0.65 mm wide, with a spacing of 0.254 mm between them. The SIW is 2.2 mm wide. The resulting array parameters are listed in Table II. The fabricated prototype is illustrated in Fig. 10.

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2

Fig. 13. Side view of: (a) Standard 4 4 stacked Yagi antenna array (b) 4 stacked Yagi antenna array with slanted elements. Fig. 10. Photography of the 4

24

2 4 stacked Yagi antenna array.

Fig. 11. Measured and simulated S-parameters of 4 array.

2 4 stacked Yagi antenna

2

Fig. 14. General view of 4 4 stacked Yagi antenna array with slanted elements, the taped is added as SIW- microstrip transition.

be better than 22 dB over the entire band of frequency. The radiation pattern is obtained by using the same measurement technique as done for the first design example. Fig. 12 presents the simulated and measured radiation patterns for the array antenna for each plane. The measured radiation pattern agrees well with the simulated ones. The radiation pattern is very directive. The dB) is 60 degrees, which matches beam width measured (at the simulation. The side lobes are higher in measurements. The respective maximum gain for the simulation and the measurement are 19 dBi and 18 dBi. This gain and SLL discrepancies may be caused by the feed line losses, patch losses, feed structure radiation, surface wave generation, mutual coupling, design and manufacturing tolerance errors [6].

Fig. 12. Measured and simulated radiation patterns of 4 tenna array.

2 4 stacked Yagi an-

Fig. 11 illustrates the calculated and measured -parameters of the designed prototype. The resonant frequency for both measurement and simulation matches with each other very well. The bandwidth spans from 56.7 GHz to 63 GHz or 10% at 60 GHz. A noticeable small difference between the measured and simulated results can be attributed to the fabrication tolerances and additional losses. The measured cross-polarization level is found to

IV. LOW SIDE LOBE LEVEL (SLL) 4 4 ARRAY OF STACKED YAGI ANTENNA WITH SLANTED ELEMENTS An interesting property of the Yagi antenna is that the directors can control the direction of the main beam. Figs. 13 and 14 show the structure that deploys this characteristic to reduce which the side lobe levels. The optimized parameter was is the shift on the X axis of each Yagi director in respect to the one below. This allowed having a single parameter of optimization. The ratio of three was chosen arbitrarily and allows having the two inner branches not to close from each other and from the outer branches, throughout the optimization. Fig. 15 parameters on the radiation pattern shows the effect of the

KRAMER et al.: VERY SMALL FOOTPRINT 60 GHZ STACKED YAGI ANTENNA ARRAY

Fig. 15. Effect of the parameter

1x on the E-plane radiation pattern.

Fig. 17. Measured and simulated pattern of stacked Yagi Antenna 4 with slanted elements.

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2 4 Array

an improvement of almost 6 dB in both simulation and measurement. The H-plane pattern remains almost unchanged compared to a standard array element, except for a slight decrease in SSL.

Fig. 16. Photograph of the stacked Yagi Antenna 4 ements prototype with a V connector.

2 4 Array with slanted el-

(E-plane). By increasing the parameter , we can notice that the first side lobe reduces but the other side lobes increase. The optimal configuration is when the side lobes are at the same level and the gain is not affected too much. The reduction of SLL is explored, but it would be possible to have better results with an analytical model and optimizing the ratio. The optimized value mm corresponds of slanted angle of the branches and . This allows a different pattern for each branch. Since the gain of each element has an opening (at dB) of 60 , the vertical maximal gain is not much affected, but the side lobes are in a different angle position. Hence, the structure has a simulated gain of 18.5 dBi but corrects the high side dB down to lobe level of the previous design from dB. By controlling the angle of an individual Yagi element, the structure is similar to conformal array where arrays follow the shape of an object and therefore have each element pointing in a different direction [21]. Fig. 16 is the picture of the fabricated low SLL array prototype. The measured radiation pattern at 60 GHz is shown in Fig. 17 that is compared to the simulated one. Both measurement and simulation are in good agreement. The gain of the main lobe is measured at 17.7 dBi with a 3 dB beamwidth of 15 degrees on the E-plane. There is a difference of 15.6 dB between the main beam and side lobes observed, which suggests

V. CONCLUSION A complete study and demonstration of a new stacked multilayer array using the Yagi concept were carried out in this work. The proposed design is based on several substrate stacked directly on each other, resulting in a very small footprint of the entire geometry. This configuration was demonstrated using two types of feeding techniques, showing the outstanding flexibility of the concept and suggesting its easy integration into millimeter-wave front-ends. The building elements and the 4 4 antenna array are analyzed, designed and measured. This offers the conception of a high gain antenna (19 dBi was reached) with compact size and minimal footprint. The measured results of S-parameters show a good bandwidth, a gain of 18 dBi, which are in very good agreement with the simulated results. The prototyped 4 4 array has a size of 28 24 2.4 mm (including the distributed network feeding). Such a very small footprint leads to a far field around 16 cm at 60 GHz that allows a very close scanning unlike standard horn antenna. Finally, our design using differently oriented elements highlights its potential in beam forming. This capability allows for instance the placing of zero in radiation pattern, therefore solving multipath issues. A large choice of feeding techniques and interesting properties of Yagi antennas provides this design with the possibilities of dual-polarization, circular polarization (using circularly polarized patch or sequential feeding, with low mutual coupling), beam forming (using network such as Butler, Blass or Nolen) and higher array scanning angle, and therefore makes it a viable alternative solution for many applications. In order to use the complete band available at 60 GHz, it is possible to further improve the bandwidth of the radiating element, using: aperture coupled patch, vias, T and L fed, etc. The platform could also achieve dual mode of operation, which is one of the desirable specifications of an automotive radar antenna. By changing electrically the length of the Yagi directive elements, dual band or configurable Yagi antennas can be created. The structure is also suitable

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for automatic fabrication process and it would be easy to fabricate by using many micro-fabrication processes including LTCC and photoimageable thick-film process. Obviously, the proposed scheme presents an excellent light-weight, low-cost, high-performance antenna technology for millimeter-wave and terahertz systems such as imaging applications. ACKNOWLEDGMENT The author wishes to thank J. Gauthier, T. Antonescu and S. Dubé of Poly-Grames Research Center, École Polytechnique de Montreal, in the fabrication and mounting of prototypes. REFERENCES [1] D. Pepe and D. Zito, “60-GHz transceivers for wireless HD uncompressed video communication in nano-era CMOS technology,” in Proc. 15th IEEE Mediterranean Electrotechnical Conf., Apr. 2010, pp. 1237–1240. [2] C. Wagner, R. Feger, A. Stelzer, and H. Jaeger, “A phased-array radar transmitter based on 77-GHz cascadable transceivers,” in Proc. IEEE MTT-S Int. Microwave Symp. Digest, 2009, pp. 73–76. [3] D. S. Goshi, Y. Liu, K. Mai, L. Bui, and Y. Shih, “Recent advances in 94 Ghz FMCW imaging radar development,” in Proc. IEEE MTT-S Int. Microwave Symp. Digest, 2009, pp. 77–80. [4] E. G. Hoare and R. Hill, “System requirements for automotive radar antennas,” in Proc. IEEE Colloq. Antennas for Automotives, Mar. 2000, pp. 1/1–1/11. [5] F. D. L. Peters, B. Boukari, S. O. Tatu, and T. A. Denidni, “77 Ghz millimeter wave antenna array with Wilkinson divider feeding network,” Progr. Electromagn. Res. Lett., vol. 9, pp. 193–199, 2009. [6] P. S. Hall and C. M. Hall, “Coplanar corporate feed effects in microstrip patch array design,” in Proc. Inst. Elect. Eng., June 1988, vol. 135, pp. 180–186, pt. H. [7] J. Ashkenazy, P. Perlmutter, and D. Treves, “A modular approach for the design of microstrip array antennas,” IEEE Trans. Antennas Propag., vol. AP-31, pp. 190–193, Jan. 1983. [8] H. Yagi, “Beam transmission of ultra short waves,” in Proc. IRE, June 1928, vol. 16, pp. 715–741. [9] S. Uda, “Radiotelegraphy and radiotelephony on half-meter waves,” in Proc. IRE, 1930, vol. 18, pp. 1047–1063. [10] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed. New York: Wiley, 1998. [11] Q. Yongxi and T. Itoh, “Active integrated antennas using planar quasiYagi radiators,” in Pro. Int. Microwave and Millimeter Wave Technology Conf., Sep. 2000, pp. 1–4. [12] L. M. Alsliety and D. Aloi, “A low profile microstrip Yagi dipole antenna for wireless communications in the 5 Ghz band,” in Proc. IEEE Int. Conf. on Electro/Information Technology, 2006, pp. 525–528. [13] O. Kramer, T. Djerafi, and K. Wu, “Vertically multilayer-stacked Yagi antenna with single and dual polarizations,” IEEE Trans. Antennas Propag., vol. 58, pp. 1022–1030, 2010. [14] M. S. Aftanasar, P. R. Young, I. D. Robertson, J. Minalgiene, and S. Lucyszyn, “Photoimageable thick-film millimetre-wave metal-pipe rectangular waveguides,” Electron. Lett., vol. 37, no. 18, pp. 1122–1123, 2001. [15] D. Stephens, P. R. Young, and I. D. Robertson, “Millimeter-wave substrate integrated waveguides and filters in photoimageable thick-film technology,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 12, pp. 3832–3838, 2005. [16] M. Bozzi, L. Perregrini, and K. Wu, “Modeling of conductor, dielectric, and radiation losses in substrate integrated waveguide by the boundary integral-resonant mode expansion,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 3153–3161, Dec. 2008. [17] A. Patrovsky, M. Daigle, and K. Wu, “Millimeter-wave wideband transition from CPW to substrate integrated waveguide on electrically thick high-permittivity substrates,” in Proc. 37th Eur. Microw. Conf., Munich, Germany, Oct. 8–12, 2007, pp. 138–141.

[18] F. Kolak and C. Eswarappa, “A low profile 77 GHz three beam antenna for automotive radar,” in Proc. IEEE MTT-S Dig., 2001, vol. 2, pp. 1107–1110. [19] S. Germain, D. Deslandes, and K. Wu, “Development of substrate integrated waveguide power dividers,” in Proc. Canadian Conf. on, Electrical and Computer Engineering, 2003, vol. 3, pp. 1921–1924. [20] W. F. Moulder, W. Khalil, and J. L. Volakis, “60-Ghz two-dimensionally scanning array employing wideband planar switched beam network,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 818–821, 2010. [21] L. Josefsson and P. Persson, Conformal Array Antenna Theory & Design. Piscataway, NJ: IEEE Press/Wiley, 2006.

Kramer Olivier was born in Lyon, France, in 1984. He received the B.Sc. degree (with distinction) from the Ecole Nationale Supérieure d’Ingénieurs des Systèmes Avancés et Réseaux of the Institut National Polytechnique de Grenoble (ESISAR-INPG), France, in 2007 and the M.A.Sc. degree in electrical engineering from Ecole Polytechnique, Montreal, QC, Canada in 2010. His current interests involve millimeter-wave multilayer structures.

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

Dr. 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), 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 a Professor of electrical engineering, and Tier-I Canada Research Chair in RF and Millimeter-wave Engineering at Ecole Polytechnique (University of Montreal). He also holds a number of visiting (guest) and honorary professorships at various universities including the first Cheung Kong Endowed Chair Professorship at Southeast University, the first Sir Yue-Kong Pao Chair Professorship at Ningbo University, and Honorary Professorship at Nanjing University of Science and Technology and City University of Hong Kong. He has been the Director of the Poly-Grames Research Center and the Founding Director of the “Centre de recherche en électronique radiofréquence” (CREER) of Quebec. He has (co)-authored over 700 refereed 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 the International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE).

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Using Bayesian Inference for Linear Antenna Array Design Chung-Yong Chan, Member, IEEE, and Paul M. Goggans, Member, IEEE

Abstract—Based on the observation that design and inference are both generalized inverse problems, we devise a new approach that uses the Bayesian inference framework for the automated design of linear antenna arrays. Compared to the optimization-based techniques that are widely used for automated antenna design, this newly-developed method has a prominent advantage, which is the capability to determine automatically the number of antenna elements required to satisfy design requirements and specifications. Three broadside array design problems, which include the nullcontrolled pattern, the sector beam pattern, and the Chebyshev pattern, along with an end-fire array design problem, are presented as examples. The obtained results demonstrate the advantages of using the Bayesian inference framework for the design of linear antenna arrays. Index Terms—Bayesian methods, design automation, inference algorithms, linear antenna arrays.

I. INTRODUCTION HEN analytic or textbook array synthesis methods are inadequate for achieving a desired radiation pattern, linear antenna array design is commonly automated by treating the design problem as an optimization problem. In this case, the goal of the optimization approach is to reduce some measure of the error between the desired and obtained array factor (usually defined by a fitness function) to an acceptable level. While many optimization techniques such as Taguchi’s method [1], particle swarm optimization (PSO) [2]–[5], genetic algorithms [6]–[8] and simulated annealing [9] have achieved notable success, the optimization approach has a significant drawback in that the array synthesis is performed with a fixed and predetermined number of array elements. As a result, the designer using synthesis by optimization is left to choose the number of elements in the final array design via ad hoc methods of her own devising. In the absence of a systematic procedure for determining the number of elements, the designer typically chooses a larger number of elements than is actually needed since an array with too many elements can easily achieve the desired pattern. In applications such as space-deployed arrays, designing arrays using the minimum necessary number of elements is a critical requirement. Even in less-critical

W

Manuscript received May 25, 2010; revised August 10, 2010; accepted January 24, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Department of Electrical Engineering, The University of Mississippi, University, MS 38677 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2161437

applications, it is desirable to design arrays having no more elements than necessary, as this simplifies array construction and reduces weight as well as cost. For these reasons, it is very desirable to have the required number of elements determined automatically during the design process, and the design method presented here has this capability. Because inference and design are both generalized (ill-posed) inverse problems, tools and methods developed for Bayesian parameter estimation and model comparison can be adapted and used for the solution of design problems. Here the developed method uses the Bayesian inference framework for the design of linear antenna arrays [10]. In the Bayesian inference framework, the solution to a design problem is a probability distribution function (PDF) called the posterior. For the array design problem, the posterior PDF is a function of the array design parameters. The design parameters include the number of elements, the location of each element, and the driving current for each element. The posterior—which comes from Bayes’ theorem—arises as a result of the assignment of a probability distribution function to the acceptable error between the desired and achieved pattern and the assignment of PDFs expressing constraints on the array’s design parameters. The posterior cannot be derived in a closed form; however, a Monte Carlo approximation of it can be obtained. In this approximation, a reasonable number of samples drawn from the posterior are used to represent the posterior. Samples can be drawn using sampling techniques such as Markov chain Monte Carlo (MCMC) methods [11], [12]. Each sample in the Monte Carlo approximation is a potential design solution having specific values for the design parameters. It is thus possible for different potential design solutions to have different values for the number of elements. The “answer” to a Bayesian inference problem is the posterior distribution; hence, application of the inference framework for design concludes with a number of potential designs rather than a single final design. To obtain the final design, an additional decision step is required and it is up to the designer to choose a single final design from all the candidate designs based upon judgement or additional criteria.1 For instance, if minimizing the number of elements is paramount, the designer may decide to select a candidate design that uses the fewest number of array elements as the final design. This paper is organized as follows: Starting with a description of the parametric model, Section II presents the basic concepts of the Bayesian inference framework. The application of Bayesian inference to four representative examples is discussed in Section III. Section IV concludes with some final thoughts. 1In

an actual inference problem, decision theory would be employed.

0018-926X/$26.00 © 2011 IEEE

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B. Bayes’ Theorem The foundation of Bayesian inference is Bayes’ theorem. In the context of linear antenna array design, Bayes’ theorem can be expressed in the following form:

Fig. 1. Linear antenna array of

N isotropic radiators positioned on the z -axis. (4)

II. BAYESIAN INFERENCE FRAMEWORK Bayesian inference uses probability theory as extended logic for making scientific inference [13]. The Bayesian method provides a systematic and logically consistent way for conducting inference, and has been widely utilized in scientific and engineering applications. For details of the theory and application of Bayesian inference beyond the minimum presented here that are required for the exposition of the present method, readers can refer to a number of recent and well written books [13]–[17]. A. The Array Model, Design Requirements, and Design Goals The inference framework presented here requires the use of a parametric model. For the problem of designing linear antenna arrays, the model is of the form

(1) where denotes the far-field observation angle defined in Fig. 1. isotropic radiators positioned on the For a linear array of -axis, the array factor can be expressed as

are the parameters of where the elements, and denotes the relevant background information about the design problem of interest. The vector of positive , indicates the revalued constants, quired degree of compliance between the desired and achieved pattern at the designated angles. The value at each angle is assigned by the designer with the knowledge that a smaller value produces a greater degree of compliance and vice versa. A probability distribution function can be in the discrete, continuous, or mixed form. A discrete PDF has parameters that can only take a discrete number of values while a continuous PDF has parameters defined over a continuous interval. A mixed PDF has a combination of both the discrete and continuous PDFs. To simplify the notation, the symbol is used to represent all three types of PDFs in this article. Because every PDF is conditioned on the background information, an additional simplification to the notation is made by suppressing the symbol in the conditioning of all PDFs that follow. Every term in (4) has a formal name. The term is called the prior probability function because it is only conditioned on the background information. Applying the product rule2, it can be expanded to

(2) (5) where and is the free space wavelength at the time-harmonic operating frequency. The element parameters and are the position and complex driving current of the th antenna element. The examples presented here are symmetric arrays having pairs of radiators. For this case, the array factor can be simplified to

(3) and . where There are a number of ways to specify design requirements. In the work presented here, upper and lower bounds in decibels at a finite set of designated angles which are denoted respectively by and are used, where is the total number of designated angles. Given specific design requirements, the ultimate goal is to obtain a final design that achieves the best compromise between design complexity and and design performance. In this context, the values for , , which correspond to the final design, are to be determined.

is the prior probability function for the number where of elements, and all other terms are the prior probability density functions for the parameters of the elements. These prior distributions are assigned by the designer based on the background information, which could include the maximum acceptable number of elements and the possible or acceptable range of values for the element parameters. The is called the likelihood function, term and its assignment will be discussed in Section III. Having obtained the likelihood function, the posterior distribution can be approximated computationally using a sampling technique. C. The Likelihood To determine the likelihood, we begin by assigning a PDF for the error between the desired and realized pattern. At every 2Because the background information does not include the pattern requireand ments, there is no basis for asserting any dependence between or ... , and . Hence, ... , and between are treated as independent to avoid asserting information we do not possess in our supposed background state of knowledge.

z

I ;z ;I ;z ; ;I

z

N X N; I ; z ; I ; z ; ; I

CHAN AND GOGGANS: USING BAYESIAN INFERENCE FOR LINEAR ANTENNA ARRAY DESIGN

angle for , the error is heuristically assigned the following Gaussian PDF:

where is the number of elements of the th sample, is the Dirac delta function and for otherwise.

(6)

(11)

Equation (10) can be used to approximate the expected values of functions under the posterior distribution. Using this property yields the following:

where for for otherwise.

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(7)

As mentioned earlier, the value for each is assigned by the designer. The assigned value indicates the width of the error dis, a designer can tribution for . Through the assignment of specify, based on design preferences, a different degree of compliance between the desired and achieved pattern for different regions. Because of logical independence, applying the product rule yields the following likelihood function:

(8) where the term inside the braces is minus the total weighted squared error. It is interesting to observe the relationship between the Gaussian likelihood function defined in (8) and a fitness function such as might be used in an optimization framework. The fitness function

(9) is equal to the total squared error for the case where for all . D. Sampling of the Posterior The posterior distribution in (4) can be approximated by drawing a reasonable number of samples from it. In the work presented here, a program called BayeSys3 [18], [19] which samples from employs the MCMC method is used to draw the posterior distribution. In the context of antenna design, each sample is a potential solution to the design problem of interest. The drawn samples form an approximation to the posterior distribution so that

(12) and

(13) which are the expected value of, and the posterior probability for the number of elements respectively.4 Equation (13) states that the number of drawn samples with elements is approximately equal to the total number of samples drawn times the posterior . The Bayesian framework probability for , embodies the principle of parsimony and quantitatively implements Ockham’s razor to render less-probable array designs with more elements than required to meet pattern requirements.5 The Ockham’s razor implicit in the Bayesian inference framework ensures that a design with a fewer number of elements is preferred to one with more elements, if both designs yield the same degree of compliance between the desired and achieved pattern. This property acting through the posterior probability , allows automatic for the number of elements, determination of the number of elements. E. Selecting the Final Design The Bayesian inference portion of the design process ends with the drawing of the design candidates from the posterior design candidates drawn from the postedistribution. The rior during the inference portion of the design process all come where the values of the posterior are from regions of close to or equal to the maximum; hence, all design candidates satisfy the pattern requirements with low or no error.6 Because all the design candidates satisfy the pattern requirements to a large extent, the designer must choose a single final design design candidates based upon additional criteria. from the The array designer will determine the additional criteria based on the specifics of the application for which the array is being designed and will, presumably, be built.

B B

(10) 3The

public-domain C-language program BayeSys is available from the following site: http://www.inference.phy.cam.ac.uk/bayesys/.

4Equation (13) follows from p(N j ; ;  ) = hN 0 N i where N is some value of N . 5The selection of the number of array elements is equivalent to model selection. The mechanism by which Ocham’s razor operates in Bayesian model selection is discussed at length in, among other places, [13, Ch. 3], [14, Ch. 4], and [15, Ch. 28]. 6This statement assumes that the array model used is adequate and that the prior distributions assigned to array parameters do not adversely constrain their values.

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TABLE I ESTIMATED POSTERIOR PROBABILITY FOR THE NUMBER OF ELEMENT PAIRS FOR THE CHEBYSHEV PATTERN DESIGN. IN THIS CASE, hN i :

= 7 25

Fig. 2. Achieved Chebyshev pattern using the inference framework. B U and

B L are denoted by dashed lines.

The array design examples presented here are academic in nature and intended only to illustrate the design method. Lacking any real application for the design examples to inform our choice of additional criteria, we have simply chosen additional criteria that seem in some sense reasonable and that illustrate the kinds of additional criteria that can be employed. For all the examples presented here, the final design is chosen using a multistep process. The process begins by first selecting the design candidates that have zero or the least weighted-squared-error. Next, we narrow down the remaining design candidates to those designs that use the fewest antenna elements. Of all these designs, the design that has the smallest aperture is selected as the final design. We make no claim that this method of selecting the final design is optimal in any way, but it seems a reasonable approach for the purpose of illustrating the design method and it may be useful in some array design applications.

1) Chebyshev Pattern Design: A recently published paper [20] discusses the importance of designing linear arrays using as few elements as possible. This article presents a method which can reproduce the array factor of an existing linear array design using a linear array with a reduced number of elements. For results comparison, the broadside Chebyshev pattern which is Example 1 of [20] is used. The design specifications for the Chebyshev pattern are indicated by the dashed lines in Fig. 2. The deand the sired direction of maximum radiation is at . The beam side lobe levels are restricted to be less than and must be less than 16.0 and at least widths at 6.3 respectively. To strictly enforce the beam widths at and , the sample points and are replaced by new sample points and respectively. For this design problem, we made the following assignments:

and for for

III. RESULTS To illustrate the application of Bayesian inference in the design of linear antenna arrays, we present three examples of broadside array which are presented in [1], [2], [4], [20] and an end-fire array design problem inspired by [21, Ch. 6]. In equally spaced angles are all four design problems, used. The posterior distribution for each case is approximated samples from it. The design goal is by drawing to determine the number of elements and the values for the element parameters of the final design. A. Broadside Array With Real-Valued Currents For a symmetric broadside array that has real-valued currents only where and is a real-valued parameter, the parametric model is of the form

(14) Since the array factor of a broadside array is symmetric about , the 101 predefined angles need to cover only the range of .

Note that the prior assigned to constrains the array aperture to be less that 9.5 wavelengths and that the prior assigned to allows the real valued element currents to be positive or negaare set tive. The maximum and minimum possible values of to allow a single pair of elements to achieve the required 0 dB , power pattern at broadside. The error distribution widths, are set to very small values, particularly in critical regions of , because we desire designs with no or very small error. The prior for the number of element pairs is assigned a broad uniform distribution to bracket all potential values. The obtained results are summarized in Tables I and II, and Fig. 2. The 1st and 3rd columns of Table II list the parameter values of the antenna elements for the final design. These parameter values are used to plot the radiation pattern at the 101 predefined angles. The graph of Fig. 2 demonstrates that the design pattern specifications are fully met. Table II indicates that the final designs produced by the method in [20] and the inference framework are comparable to each other. In addition to having the same number of elements, 12, the values of the location and normalized current amplitude of each element are comparable as well. While the method in [20] manages to reconstruct the original Chebyshev pattern using fewer antenna elements, the

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TABLE II CURRENT AMPLITUDES AND LOCATIONS OF THE ANTENNA ELEMENTS OBTAINED BY THE INFERENCE FRAMEWORK AND MATRIX PENCIL METHOD FOR THE CHEBYSHEV PATTERN

TABLE III ESTIMATED POSTERIOR PROBABILITY FOR THE NUMBER OF ELEMENT PAIRS : FOR THE SECTOR BEAM PATTERN DESIGN. IN THIS CASE, hN i

= 6 65

Fig. 3. Achieved sector beam pattern using the inference framework. B U and B L are denoted by dashed lines.

inference framework manages to produce a number of linear antenna array designs that realize the desired pattern specifications using the same number of elements, though as many as 50 pairs of antenna elements are allowed in a design. The method in [20] has the advantage of low computational cost; however, since it relies on a pre-existing array design satisfying the pattern specifications, it lacks generality. In addition, the method may fail in shaped beam pattern design problems such as the following sector beam pattern design problem. 2) Sector Beam Pattern Design: The design requirements for the sector beam pattern are indicated by the dashed lines in Fig. 3. There are basically two regions in the pattern. The first region, which falls in the range of , contains ripples that have to be smaller than 0.5 dB. The second region, and which is in the range of , has desired side lobe levels that are less than . For this design problem, we made the following assignments:

and

TABLE IV CURRENT AMPLITUDES AND POSITIONS OF THE ANTENNA ELEMENTS FOR THE FINAL DESIGN OF THE SECTOR BEAM PATTERN

antenna elements, which is two pairs fewer than the 7 pairs obtained in [1]. A notable difference between these two results is that the final design produced by the inference framework manages to meet the design requirements without using the phase of array-element driving-currents as design parameters. Instead, the real-valued phasor driving currents are allowed to have negative values. This along with the continuously variable positions of the array elements is sufficient to achieve the required pattern. 3) Null Controlled Pattern Design: The array pattern to be synthesized is required to contain nulls at specified directions. Fig. 4 illustrates the desired antenna pattern where the dashed lines indicate the upper and lower bounds of the design requirements. The main beam is at with a minimum 3 dB beam width of 7.2 . The side lobe levels are desired to be less and the beamwidth at is specified to be than less than 20.9 . In addition, a null is required in the two and . For regions of this design problem, we made the following assignments:

for for The obtained results are summarized in Tables III and IV, and Fig. 3. Table IV displays the parameter values of the antenna elements for the final design. Using these values, the radiation pattern at the 101 predefined angles is plotted in Fig. 3. This plot shows that all 101 points of the achieved pattern fall within the design specifications. The final design uses 5 pairs of

and for for

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Fig. 4. Achieved null controlled pattern using the inference framework. B U and B L are denoted by dashed lines.

Fig. 5. Achieved end-fire pattern using the inference framework. B U and B L are denoted by dashed lines.

TABLE V ESTIMATED POSTERIOR PROBABILITY FOR THE NUMBER OF ELEMENT PAIRS : FOR THE NULL CONTROLLED PATTERN DESIGN. IN THIS CASE, hN i

The number of equally spaced angles used remains at 101; how. Fig. 5 illustrates the ever, the range is now design requirements for the end-fire array design problem of interest. The maximum radiation is desired to be at the direction of 180 with a minimum 3 dB beam width of 36.0 . Addition. ally, the side lobe levels are constrained to be less than This desired antenna pattern is inspired by the Hansen-Woodyard end-fire array example in [21, Ch. 6]. The goal here is to obtain a result roughly comparable to the Hansen-Woodyard end-fire array using the model for an ordinary end-fire array.7 For this design problem, we made the following assignments:

= 7 11

TABLE VI CURRENT AMPLITUDES AND LOCATIONS OF THE ANTENNA ELEMENTS FOR THE FINAL DESIGN FOR THE NULL CONTROLLED PATTERN

and The obtained results are summarized in Tables V and VI, and Fig. 4. The parameter values of the elements for the final design are displayed in Table VI. Using these parameter values, the radiation pattern at the 101 predefined angles is plotted. The graph of Fig. 4 shows that the design pattern specifications are fully satisfied. The final design produced by the inference framework uses only 7 pairs of antenna elements, which is fewer than the 10 pairs used in [1]. The 30% savings in antenna elements used illustrates the tendency for a designer to use more elements than necessary when the required number of elements is not determined automatically during the design process. B. End-Fire Array To further demonstrate the use of inference in linear array design, an end-fire array design problem where the direction of maximum radiation is at 180 is presented here. For a symmetric array with complex currents and , the parametric model can be expressed as

(15)

for for for In addition, is set to so that a power pattern of 0 dB is obtained at 180 . Table VIII displays the parameter values of the elements for the final design. The achieved antenna pattern for the end-fire array design at the 101 predefined angles is plotted in Fig. 5. The plot shows that the design specifications have been fully satisfied. The 6 element pairs of the designed ordinary end-fire array is in line with the 10 elements given by [21] for the HansenWoodyard end-fire array. These results demonstrate that the inference approach can be successfully applied to an end-fire array design problem. IV. CONCLUSION The approach of using Bayesian inference for linear antenna array design has been successfully applied to solve four design problems. In all four cases, the inference approach has successfully determined the number of elements required and produced designs that completely satisfy the given design requirements. 7To obtain a truly comparable result would require the use of the general array model given by (1) and (2). This is possible with the present method but beyond the scope of this paper.

CHAN AND GOGGANS: USING BAYESIAN INFERENCE FOR LINEAR ANTENNA ARRAY DESIGN

TABLE VII ESTIMATED POSTERIOR PROBABILITY FOR THE NUMBER OF ELEMENT PAIRS FOR THE END-FIRE PATTERN DESIGN. IN THIS CASE, h i

N = 7:05

TABLE VIII CURRENT AMPLITUDES AND LOCATIONS OF THE ANTENNA ELEMENTS FOR THE FINAL DESIGN FOR THE END-FIRE PATTERN

The importance of using a design method that automatically determines the required number of elements increases as the complexity of the required pattern, the number of element design parameters, and the constraints on the array parameters increases. Because of its unique ability to solve these problems and to produce designs that balance design performance and design complexity, with additional development, use of the Bayesian inference framework for the automated design of antenna arrays is likely to become an important antenna design tool. REFERENCES [1] W.-C. Weng, F. Yang, and A. Z. Elsherbeni, “Linear antenna array synthesis using Taguchi’s method: A novel optimization technique in electromagnetics,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 723–730, Mar. 2007. [2] M. M. Khodier and C. G. Christodoulou, “Linear array geometry synthesis with minimum sidelobe level and null control using particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 53, pp. 2674–2679, Aug. 2005. [3] N. Jin and Y. Rahmat-Samii, “A novel design methodology for aperiodic arrays using particle swarm optimization,” in Proc. Nat. Radio Sci. Meeting Dig., Boulder, CO, Jan. 2006, pp. 69–69. [4] D. Gies and Y. Rahmat-Samii, “Particle swarm optimization for reconfigurable phased-differential array design,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 168–175, Aug. 2003. [5] N. Jin and Y. Rahmat-Samii, “Advances in particle swarm optimization for antenna designs: Real-number, binary, single-objective and multiobjective implementations,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 556–567, Mar. 2007. [6] D. W. Boeringer, D. H. Werner, and D. W. Machuga, “A simultaneous parameter adaptation scheme for genetic algorithms with application to phased array synthesis,” IEEE Trans. Antennas Propag., vol. 53, pp. 356–371, Jan. 2005. [7] F. J. Ares-Pena, A. Rodriguez-Gonzalez, E. Villanueva-Lopez, and S. R. Rengarajan, “Genetic algorithms in the design and optimization of antenna array patterns,” IEEE Trans. Antennas Propag., vol. 47, pp. 506–510, Mar. 1999. [8] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithm for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, pp. 771–779, Mar. 2004. [9] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing,” IEEE Trans. Signal Process., vol. 44, no. 1, pp. 119–123, Jan. 1996.

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[10] P. M. Goggans and C. Y. Chan, “Antenna array design as inference,” in Proc. 28th Int. Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 2008, vol. 1073, AIP, pp. 294–300. [11] W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Markov Chain Monte Carlo in Practice. London, U.K./Boca Raton, FL: Chapman and Hall/CRC, 1996. [12] P. J. Green, “Reversible jump Markov chain Monte Carlo computation and Bayesian model determination,” Biometrika, vol. 82, no. 4, pp. 711–732, Dec. 1995. [13] P. Gregory, Bayesian Logical Data Analysis for the Physical Sciences. Cambridge, U.K.: Cambridge Univ. Press, 2005. [14] D. S. Sivia and J. Skilling, Data Analysis-A Bayesian Tutorial, 2nd ed. Oxford, U.K.: Oxford Science, 2006. [15] D. J. C. Mackay, Information Theory, Inference, and Learning Algorithms. Cambridge, U.K.: Cambridge Univ. Press, 2003. [16] E. T. Jaynes, Probability Theory: The Logic of Science. Cambridge, U.K.: Cambridge Univ. Press, 2003. [17] C. P. Robert, The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation, 2nd ed. New York: Springer Verlag, 2007. [18] J. Skilling, BayeSys and MassInf Maximum Entropy Data Consultants Ltd, 2004 [Online]. Available: http://www.inference.phy.cam.ac.uk/bayesys/ [19] J. Skilling, “Using the Hilbert curve,” in Proc. 23rd Int. Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 2004, vol. 707, AIP, pp. 388–405. [20] Y. Liu, Z. Nie, and Q. H. Liu, “Reducing the number of elements in a linear antenna array by the matrix pencil method,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2955–2962, Sep. 2008. [21] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1996. Chung-Yong Chan (M’10) was born in Kuala Lumpur, Malaysia, in 1979. He received the B.S. degree in electrical engineering and the M.S. and Ph.D. degrees in engineering science with an emphasis in electrical engineering all from The University of Mississippi, Oxford, in 2000, 2003, and 2010, respectively. His research interests include the use of Bayesian inference in engineering design applications such as antenna array and digital filter design, and Bayesian analysis in various inference problems such as landmine detection and sound energy decay analysis in room acoustics. Dr. Chan was an Organizing Committee Member of the 29th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2009) and Co-Editor of the MaxEnt 2009 Proceedings. He is a member of the Phi Kappa Phi Honor Society.

Paul M. Goggans (S’78–M’89) was born in Opelika, AL, in 1954. He received the B.S. and M.S. degrees in electrical engineering and the Ph.D. degree all from Auburn University, Auburn, AL, in 1976, 1978, and 1990, respectively . From 1979 to 1985, he was employed by Sandia National Laboratories, Albuquerque, NM, in the Radar Signal Analysis Division. From 1985 to 1990, he was with Auburn University as an Instructor while working toward the Ph.D. degree. In 1990, he was appointed Assistant Professor and, in 1994, he was promoted to Associate Professor in the Department of Electrical Engineering, University of Mississippi, Oxford. His research interests include the application of Bayesian inference to engineering model-comparison, parameter-estimation, and design problems, Markov chain Monte Carlo methods, and computational electromagnetics. Dr. Goggans was General Chair of the 29th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2009) and Co-Editor of the MaxEnt 2009 Proceedings. He is a member of the Audio Engineering Society and the IEEE Antennas and Propagation Society.

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Performance of Planar Slotted Waveguide Arrays With Surface Distortion Liwei Song, Baoyan Duan, Fei Zheng, and Fushun Zhang

Abstract—This paper deals with the prediction of the effects of surface distortion on the radiation characteristics of arrays of slot antennas. The deformation of an object caused by exterior loads is calculated by a finite element method. The coupled structural electromagnetic (CSE) model of slotted waveguide arrays (SWA) is developed by the analysis of the positional difference of radiating slots in the aperture field caused by structural displacement. The slot voltages in this model are either obtained from finite-element simulations or by near-field measurements. Then the far fields can be calculated according to the coupled model. Two prototype antennas were manufactured and measured, and their deformation accomplished by a specific device with some fixed points and forcing points. Finally, measurements and simulation results are presented for the two prototypes. Index Terms—Finite element method, slotted waveguide arrays, slot voltages, structural deformation.

I. INTRODUCTION INCE a SWA antenna has the advantage of a compact configuration, stable mechanical characteristics, low loss and perfect efficiency, it is widely used in communication and radar systems. The SWA technology can satisfy the requirement of high performance and high survivability, however, whose electromagnetic (EM) performance and process period have a direct influence on the performance [1]–[5]. Generally the whole SWA consists of some waveguides with slot elements, which are subject to exterior loads such as temperature load and vibration load. For example, structural distortion is generated when the SWA operates under an airborne, missile-borne, vehicle-borne and space-borne environment. The SWA structure is the carrier and boundary condition of the electromagnetic signal transmission, with its displacement field directly influencing the amplitude and phase distribution of the EM field. Subsequently, the antenna’s shape and size and the flatness of the array plane have a direct influence on the EM performances of the SWA such as sidelobe level (SLL), beam pointing and gain [6], [7].

S

Manuscript received November 27, 2009; revised February 10, 2011; accepted March 09, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported by the Key Project of the Chinese National Programs for Fundamental Research and Development (973 program: 61358), the Key Project of the Chinese National Natural Science Foundation (No: 51035006), and the Fundamental Research Funds for the Central Universities (No: JY10000904019). L. Song, B. Duan, and F. Zheng are with the Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi’an, Shaanxi 710071, China (e-mail: [email protected]; [email protected]; [email protected]). F. Zhang is with the National Key Laboratory of Antenna and Microwave Technology, Xidian University, Xi’an, Shaanxi 710071, China. Digital Object Identifier 10.1109/TAP.2011.2161537

Because the performance of an SWA radar system is directly influenced by the degradation of EM performances of the SWA, some researchers explored the different influencing factors on antenna EM performances. In terms of general array antennas, the prime interest has been in the determination of the boresight gain loss, and for this, Ruze’s [8] simple formula has been very effectively used. Wang [9] investigated the influence of the random error of each radiating element on the performance of a phased array antenna based on the probability method. But, both Ruze and Wang assumed that the structural error met the pregiven distribution and did not analyze the practical structural error through the finite element analysis of the antenna structure. Then Adelman and Padula [10] implemented an integrated thermal-structural-electromagnetic optimization design by a cascade-coupled method. Liu and Hollaway [11] realized a coupled structural-electromagnetic design of reflector antenna by a multidisciplinary model with two optimization objects. However, Adelman and Padula [10] and Liu and Hollaway [11] did not analyze whether any quantitative relation exists between structural parameters and EM performances. For this limitation, Wang [12], [13] presented a coupled structural-electromagnetic-thermal model and analyzed a quantitative relation among them. However, this study aims only at phased array antennas, and does not give any guidance for the SWA antennas. Considering the effect of vibrating objects on the EM performances, Kleinmann [14] studied the scattering by linearly vibrating objects. However, Kleinmann assumed that the vibrating objects are rigid and do not deformed by exterior loads. For conformal antennas, Schippers [15] and Knott [16] analyzed the effects of the distortions and vibrations of the conformal antenna on the EM performance. But, these antennas are different from the SWA antennas. An antenna is a mechatronic system whose structural design should conform to the requirements for structural intensity and stiffness. However, the most important use of antenna design is that EM performances should meet the requirements. Therefore, it is a beneficial way of achieving the optimal synthesis design by studying the CSE problems and investigating the relation between structural distortion and the EM performances of the SWA. Wang [17], [18] studied the coupling relation between the structural displacement field and the EM field of reflector and phased array antennas, presented a coupled structural-electromagnetic model that directly described the antenna EM performances as a function of structural and temperature design variables. But, it is solely applicable to these antennas. With radar antennas developing for high-frequency band, high gain, low SLLs, high performances, ultra-wide band and high precision [19], [20], the CSE problems of the SWA become increasingly serious. Therefore, based on the previous studies of

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tion can be calculated. For this purpose, a piecewise sinusoidal approximation for the slot electric fields is used, which simplifies the expression of (1) for the radiated pattern to

(2) where

Fig. 1. Geometry of the slotted waveguide arrays.

(3) and

authors, the paper investigates the relation between the structural displacement and EM fields, and discusses the CSE solution method of SWA antennas.

are the length and width of the th radiating slot. III. STRUCTURAL DISTORTION AND SLOT VOLTAGES

A. Antenna Structural Distortion II. THE CSE MODEL OF SWA The geometry of a SWA is shown in Fig. 1 with reference to the particular case of a traveling wave planar array. At a glance, we distinguish two regions: the internal one, formed by the waveguide rows, placed side by side, and the external one, consisting of the front plane containing the radiating slots. The EM and structural performances of the SWA vary according to the exterior loads on its structure: temperature distribution, vibration, and so on. So, we will present the expression of the EM performances with the structural deformation of the SWA. First, during the solution of the internal problem, the effect of the deformation of the waveguide on the EM performances is neglected. Although the deformation causes internal reflections in the waveguides, the insensitivity of the radiation pattern of the radiating slots to these reflections allows for neglecting this effect in practical configurations [21]. Then, the solution of the external problem is the net radiation pattern, which is calculated with a simple (vectorial) superposition of the radiation patterns of the radiating slots, taking the location of each slot into consideration. When the antenna was distorted by exterior loads, the external plane containing the radiating slots became a curved surface. of the th radiating slot can The displacement be determined by the analysis of the structural finite element. So the effect of the displacement of each slot is to introduce additional phase factors, the radiation pattern can be introduced as

(1) where are the slot voltage and the pattern of the th are the standard spherical coordinate variradiating slot, are the initial locations of the th radiating slot. ables, Once the slot voltage and the displacement of each radiating slot are known, the radiation pattern of SWA with surface distor-

In analyzing the structural performance of a SWA, it is necessary to accurately characterize the surface containing the radiating slots. Any deviation of the radiating slots from its ideal plane will cause the EM performances to degrade. So the antenna’s structural static analysis is the evaluation of the displacement of radiating slots. Although a SWA is made of the material of aluminum alloy, the distortion will be appearing due to exterior loads on the structure. So, to simulate the structural deformation in an actual load environment, a device is made to force the SWA to deformation. This device can provide the different shapes of the distortion for the SWA, which is shown in Fig. 2. The four vertexes of the device are used to fix up the SWA, and the nuts to force the back of the SWA to the anticipated distortion. The surface containing the radiating slots is measured with API to acquire its displacement of discrete points on the surface. However, the measured discrete points are too few to determine the position of the radiating slots, so the finite element analysis is used to work out the displacement of the surface precisely. First, the finite element model of the SWA is used in Fig. 3 with the shell element described to be SHELL63 in ANSYS. Then, the displacement of those nuts and the corresponding restricted points are added to the finite element model. So, the structural analysis can obtain the nodal displacements on the surface. Because the displacements of the nuts are not measured with API, that may lead to the wrong displacements of nuts to be added to the finite element model of the SWA. So an approach is given to resolve this problem. With the help of the structural analysis and API, the differences between the simulated and measured of the nodal displacements on the surface can be obtained. By constantly changing the displacements of the nuts, the differences will be drop off. Then the displacements of the radiating slots can be obtained from the structural analysis. B. Slot Voltages The slot voltages of the SWA may be given by two kinds of methods. In the first method, the finite element solution of the internal region formed by the waveguide obtains the electric

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Fig. 4. The slot antenna formed by 80 radiating slots with rectangular waveguide in the Ku-band.

Fig. 2. The device that forces the antenna to deformation. (a) The front face of the device; (b) the back plate of the device.

Fig. 3. The finite element model of antenna with shell elements.

field distribution of each slot of the antenna. The peak values are used as the slot voltages for the radiating slots [21]. And For the second method, it utilizes the near-field measurement data to determine slot voltages over a fictitious plane that encompasses the antenna [22]–[24]. These two methods are applied separately to the two SWAs with operating frequency bands of Ku and X. IV. NUMERICAL RESULTS We have applied the expression of (2) in Section II in the analysis of two SWAs. We have obtained the simulated results that are very close to the measured ones. We have first applied the model in a SWA formed by the 80 radiating slots operating in Ku band, shown in Fig. 4. The slot

Fig. 5. The radiation pattern of a slotted array of 80 slots under the ideal antenna. (a) Azimuth plane; (b) elevation plane.

voltages of the SWA have been obtained by the finite element method. For simulating the antenna distortion, we have used the device in Fig. 2 to force the SWA to deformation in a straight way. Then, the expression of (2) can be solved. We first compare the calculated results with measured ones of the realistic antenna with the zero distortion, shown in Fig. 5. One may see that the calculated and measured patterns in the azimuth and elevation

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Fig. 7. The slot antenna formed by 1172 radiating slots with rectangular waveguide in the X-band.

Fig. 6. The radiation pattern of a slotted array of 80 slots under the distorted antenna. (a) Azimuth plane; (b) elevation plane.

TABLE I THE 80 SLOTS ANTENNA PERFORMANCE IN THE UNDISTORTED AND DISTORTED CASES

Fig. 8. The radiation pattern of a slotted array of 1172 slots under the ideal antenna. (a) Azimuth plane; (b) elevation plane.

planes are in a good agreement. So, the slot voltages from the finite element solution are considered as a correct prediction. Then, the slot voltages are used to calculate the distorted SWA obtained by the forcing device according to the actual loading cases, the performances of the SWA with surface distortion can be obtained. So, Fig. 6 shows the pattern of the same antenna

under the distorted antenna by the forcing device. By comparing the pattern results of the deformed antenna with the ideal one, it is seen that the distortion has caused a reduction in the gain and a rise in the near-SLL in the elevation plane. The antenna performances in the undistorted and two distorted cases are listed in Table I. When the maximum of the displacements of the distorted antenna is 1.5 mm, the maximum difference between the simulated and measured results is 0.4 dB of the maximum SLL in the azimuth plane.

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is carried out by the structural model of the shell and beam elements. Fig. 10 shows the pattern of the distorted SWA. After the SWA has been deformed, the prediction of the mathematical model also has a very good agreement, except for very few positions in the pattern. These deviations are due to the hypothesis of changeless slot voltages and the approximate pattern of each radiating slot element. V. CONCLUSION

Fig. 9. The device that forces the antenna to deformation.

From the results of our study, we see that mechanical errors in a slotted waveguide array generally produce a phase error in the signals of each radiating slot due to the changes of its position in an ideal plane. Structural distortion as a result of loads is predictable by the structural analysis with the finite element method. The effect of the errors can be estimated by the direct calculations. It is demonstrated that the impact of the distortion manifests itself primarily as reduction in the gain and a rise in the SLL. ACKNOWLEDGMENT The authors would like to express their gratitude to the staff of the Research Institute on Mechatronics, Xidian University, China, for their assistance in the completion of this paper. REFERENCES

Fig. 10. The radiation pattern of a slotted array of 1172 slots under the distorted antenna. (a) Azimuth plane; (b) elevation plane.

We have also applied the method to a large SWA formed by 1172 of radiating slots in the X-band shown in Fig. 7. Since the dimension of the antenna is very large, vast computer resources are needed to obtain the slot voltages by the finite-element method. So, we have measured the near-field data to calculate the slot voltages by the method in [22]. Fig. 8 shows a comparison between the measured and computed radiation patterns of the ideal SWA. As shown in Fig. 8, a good agreement is found between the calculated and measured patterns, which prove that the slot voltages are correct. So the slot voltages can be used to calculate the distorted SWA. Then, the forcing device shown in Fig. 9 is made to impose deformation on the SWA, and the structural analysis

[1] R. S. Elliott, “An improved design procedure for small arrays of shunt slots,” IEEE Trans. Antennas Propag., vol. 31, no. 1, pp. 48–53, 1983. [2] H. Y. Yee, “The design of large waveguide arrays of shunt slots,” IEEE Trans. Antennas Propag., vol. 40, pp. 775–781, 1983. [3] J. C. Coetzee, J. Joubert, and D. A. McNamara, “Off-center-frequency analysis of a complete planar slotted-waveguide array consisting of subarrays,” IEEE Trans. Antennas Propag., vol. 48, no. 11, pp. 1746–1755, 2000. [4] A. Morini, T. Rozzi, and G. Venanzoni, “On the analysis of slotted waveguide arrays,” IEEE Trans. Antennas Propag., vol. 54, pp. 2016–2021, 2006. [5] L. Sikora and J. Womack, “The art and scicence of manufacturing waveguide slot-array antennas,” Microw. J., pp. 157–162, 1988. [6] P. N. Richardson and H. Y. Yee, “Design and analysis of slotted waveguide antenna arrays,” Microw. J., vol. 31, no. 6, pp. 109–125, 1988. [7] N. V. Larsen and O. Breinbjerg, “Modelling the impact of ground planes on antenna radiation using the method of auxiliary sources,” IET Microw. Antennas Propag., vol. 1, no. 2, pp. 472–479, 2007. [8] J. Ruze, “Pattern degradation of space fed phased arrays M.I.T. Lincoln Laboratory, Lexington, MA, Project report SBR-1, Dec. 1979. [9] H. S. C. Wang, “Performance of phased-array antennas with mechanical errors,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 2, pp. 535–545, April 1992. [10] H. M. Aderman and S. L. Padula, “Integrated thermal-structural-electromagnetic design optimization of large space antenna reflectors,” NASA-TM-87713, 1986. [11] J. S. Liu and L. Hollaway, “Integrated structure-electromagnetic optimization of large reflector antenna systems,” Struct. Multidiscip. Optim., vol. 16, no. 1, pp. 29–36, 1998. [12] C. S. Wang, B. Y. Duan, F. S. Zhang, and M. B. Zhu, “Coupled structural-electromagnetic-thermal modelling and analysis of active phased array antennas,” IET Microw. Antennas Propag., vol. 4, no. 2, pp. 247–257, 2010. [13] C. S. Wang, B. Y. Duan, F. S. Zhang, and M. B. Zhu, “Analysis of performance of active phased array antennas with distorted plane error,” Int. J. Electron., vol. 96, no. 5, pp. 549–559, 2009. [14] R. E. Kleinman and R. B. Mack, “Scattering by vibrating objects,” IEEE Trans. Antennas Propag., vol. 27, no. 3, pp. 344–352, 1979. [15] H. Schippers, J. H. van Tongeren, P. Knott, T. Deloues, P. Lacomme, and M. R. Scherbarth, “Vibrating antennas and compensation techniques, research in NATO/RTO/SET 087/RTG 50,” in Proc. IEEE Aerospace Conf., Mar. 2007, pp. 1–13. [16] P. Knott, “Deformation and vibration of conformal antenna arrays and compensation techniques,” NATO/OTAN, RTO-MP-AVT-141.

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[17] B. Y. Duan and Y. H. Qi, “Study on optimization of mechanical and electronic synthesis for the antenna structural systems,” Mechatronics, vol. 4, no. 6, pp. 553–564, 1994. [18] C. S. Wang, H. Bao, F. S. Zhang, and X. G. Feng, “Analysis of electrical performances of planar active phased array antennas with distorted array plane,” J. Syst. Eng. Electron., vol. 20, no. 4, pp. 726–731, 2009. [19] N. Wang, Z. H. Xue, and S. M. Yang, “Characters of time domain radiated field of ultra wide band ultra low side lobe phased array antenna,” (in Chinese) Acta Electron. Sin., vol. 34, no. 9, pp. 1605–1609, 2006. [20] Q. Q. Jiang, “Development trend of active phased array radar technology,” (in Chinese) Technol. Found. Nat. Def., vol. 2, no. 4, pp. 9–11, 2005. [21] P. T. Benko, B. Ladanyi-Turoczy, and J. Pavo, “A coupled analytical-finite element technique for the calculation of radiation from tilted rectangular waveguide slot antennas,” IEEE Trans. Magn., vol. 44, no. 6, pp. 1666–1669, Jun. 2008. [22] P. Petre and T. K. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach,” IEEE Trans. Antennas Propag., vol. 40, no. 11, pp. 1348–1356, Nov. 1992. [23] P. Petre and T. K. Sarkar, “Planar near-field to far field transformation using an array of dipole probes,” IEEE Trans. Antennas Propag., vol. 42, no. 4, pp. 534–537, Apr. 1994. [24] A. Taaghol and T. K. Sarkar, “Near-field to near/far-field transformation for arbitrary near-field geometry, utilizing an equivalent magnetic current,” IEEE Trans. Electromagn. Compat., vol. 38, no. 3, pp. 536–542, Aug. 1996. Liwei Song was born in Inner Mongolia, China, on December 6, 1982. He received the B.S. degree in applied physics and the Ph.D. degree in mechanical engineering from Xidian University, Xi’an, China, in 2004 and 2010, respectively. Currently, he is a teaching assistant in the Research Institute on Mechatronics, Xidian University. His major research interests include coupled structural-electromagnetic analysis, electromechanical synthesis design and the distorted compensation method of electronic equipment.

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Baoyan Duan received the B.S., M.S., and Ph.D. degrees in mechanical engineering from Xidian University, Xi’an, China, in 1981, 1984, and 1989, respectively. Previously, he was a Postdoctoral Fellow at the University of Liverpool, Liverpool, U.K., and a Research Fellow at Hokkaido University, Japan, from 1991 to 1994. He was a Guest Professor at Cornell University, Ithaca, NY, in 2000. Currently, he is the President of Xidian University and a Full Professor in the School of Electromechanical Engineering. His research interests are antenna structures and servo systems, engineering structural optimization design, mechatronics, and CAD/CAE.

Fei Zheng received the B.S., M.S., and Ph.D. degrees in mechanical engineering from Xidian University, Xi’an, China, in 1989, 1992, and 1995, respectively. From 1999 to 2000, he was a Visiting Scholar at the Department of Information Engineering, University of Zurich, Switzerland. He is now a Full Professor in the Research Institute on Mechatronics, Xidian University. His research interests include modeling and simulation of electronic equipment structure, design and analysis of space deployable structure, and multidisciplinary integration analysis of electronic equipment.

Fushun Zhang received the B.S., M.S., and Ph.D. degrees in electromagnetism and microwave technology from Xidian University, Xi’an, China, in 1982, 1993, and 1997, respectively. Currently, he is a Full Professor in the School of Electronic Engineering, Xidian University, China. His research interests are antenna theory, engineering and measurement.

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An Effective Synthesis of Planar Array Antennas for Producing Near-Field Contoured Patterns Hsi-Tseng Chou, Senior Member, IEEE, Nan-Nan Wang, Hsi-Hsir Chou, and Jing-Hui Qiu

Abstract—We present an efficient procedure for the synthesis of planar array antennas to radiate contoured patterns in the near zone of array aperture, as desired in RFID applications. It first grids the target area, and then utilizes a global basis set to span the array’s excitations. Each basis’s excitation of the array radiates a focused spot beam in the near zone and serves as a local basis function for representing the radiation field. The unknown coefficients of the array’s basis set are then found via a minimum least square error (MLSE) technique. It is found that only a small number of basis functions are sufficient to synthesize the radiation pattern, and make the present approach very fast. Index Terms—Antenna arrays, near field focus, pattern synthesis.

I. INTRODUCTION N efficient procedure is developed to synthesize the radiation pattern of a planar antenna array in the near zone of array aperture. This work is motivated by the increasing interest in the applications of RFID [1], [2], vital life-detection systems [3], and noncontact microwave detection systems [4]–[7], where the objects under detection may be located in the near zone of the antenna. In this case, an optimum focused field pattern helps not only to reduce the interferences resulting from the scattering due to the presence of neighborhood structures, but also save the system power. In the past, pattern synthesis techniques have been widely investigated in the far-zone of the array for applications such as satellite communications, where it is used to compensate the power deficiency in an angular space [8]–[13]. In the nearfield applications, most of the works focused on developing antenna arrays or reflectors that produce a point-focused spot beam [4]–[6], [14]–[18]. The efforts to synthesize the pattern in the near zone were very few. In particular, [19] treated the pattern synthesis along the axis of a source aperture, which is not applicable to solve realistic problems. The works most related to the current interest are presented in [20]–[24]. In particular,

A

Manuscript received September 22, 2010; revised January 13, 2011; accepted February 17, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the Ministry of Economics Affairs and National Science Council, Taiwan. H.-T. Chou is with the Department of Communication Engineering and Communication Research Center, Yuan-Ze University, Chung-Li 320, Taiwan (e-mail: [email protected]). N.-N. Wang and J.-H. Qiu are with the Department of Microwave Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected], [email protected]). H.-H. Chou is with the Communication Research Center, Yuan-Ze University, Chung-Li 320, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161554

[20]–[22] represent a brute-force approach that employs a direct synthesis over sampled field points in a test area using linear least square approach (LLSA). The goal is to produce a field distribution similar to a plane-wave for the purpose of RF testing and RCS measurement, which specifies the field points on the surface of test volume. This approach is relatively time-consuming and not physically appealing. Also [23], [24] further utilize potential integral solution of source currents, Nyquist sampling of near-field data, and LLSA to synthesize near-field patterns for several array configurations. The drawback of the above conventional approaches is that they use the radiations of array elements to synthesize the desired radiation pattern, and are therefore computationally very cumbersome because the array needs to be synthesized element-by-element. Instead, this paper utilizes a global basis set to span the array excitation as previously proposed in [25]. Each basis’s excitation radiates a near-field focused spot beam in the target area, and serves as local basis function to synthesize the contoured pattern in the target area. The unknown coefficients of array’s basis set are then found via a minimum least square error (MLSE) technique. It is found that only a small number of basis functions are sufficient to resemble the radiation pattern, and make the present approach very fast because the fields in the region external to this target area are usually required to be low, which can be achieved via the low near field side lobes of the local basis functions. It is noted that the actual element excitations can be found by multiplying the coefficients with an impedance matrix that accounts for the mutual coupling effects between elements as addressed in the method of moment. The paper is organized as follows. Section II describes the optimization procedure. Section III presents an approach based on the similarity between local basis functions to accelerate the numerical computation of these local field basis functions. Section IV demonstrates the validity of this approach by synthesizing a field distribution over a rectangular target area, and investigates the characteristics of field radiations in the near zone. Accuracy and efficiency are also investigated. Finally a short discussion is presented in Section V as the conclusion. II. SYNTHESIS APPROACH A. Mathematic Modeling of Antenna Elemental Pattern Fig. 1 illustrates a periodic array of identical antenna elements with periods, and , along the and coordinates, respectively. Its th element is located at , and , expressed by has a radiation field,

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(1)

CHOU et al.: AN EFFECTIVE SYNTHESIS OF PLANAR ARRAY ANTENNAS FOR PRODUCING NEAR-FIELD CONTOURED PATTERNS

N

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2 M

Fig. 1. Illustration of a near-field focus problem. The (2 + 1) (2 + 1) antenna array radiates near fields to provide a coverage at a target area, where related position vectors are also indicated.

where , with and being their representations in a spherical coordinate system. , is the radiation pattern of an When element located at the origin of the coordinate system which is time dependence is at the center of the planar array. An assumed in (1) and suppressed throughout in this paper. Also this assumption of identical radiation pattern for each element remains valid for an infinite array or a standing along element, where the unequal mutual coupling effects on each element due to the truncation of finite array are ignored in the following development. For near-field applications, where the target area of interest is located in front of the array, it is sufficient to approximate the element’s radiation pattern by a cosine taper for a broad main-lobe pattern radiated from a small element. In particular, for a circular polarization can be approximated by [26] (2) where (3) The shapes of patterns depend on the element’s actual radiations, and are controlled by indices that are determined by matching (3) with the element’s actual radiation patterns obtained from either numerical simulations or experimental meain (2) is used to indicate surements. Also the parameter left- and right-hand circular polarizations, respectively. The assumption of identical element patterns in (1) remains valid for most types of array elements, such as dipoles and microstrip patches that operate at a single fundamental mode for a size less than half a wavelength. The field is assumed to be radiated from a source current that equivalently represents the array element via a numerical method. B. Synthesis Procedure for Near-Field Optimum Patterns of The objective is to find a set of complex amplitudes array excitations, so that the superposition of elements’ radia. Thus, tions will assemble the desired field pattern, (4)

Fig. 2. Schemes of pattern synthesis through using the sets of global and local basis functions, respectively in (a) and (b) to illustrate their characteristics in patterns synthesis. (a) Global basis expansion. (b) Local basis expansion.

where the left-hand side denotes the synthesized field while the is right-hand side indicates the desired field pattern. When , a direct a pattern of near-zone focused field at approach selects to enable , which are radiating fields with equal phases at created by the propagation from all antenna elements. It however results in a focused single spot pattern, and is not able to form a contoured pattern in the target area. In a general synthesis are treated as unknown variables that need to procedure, be optimized through a synthesis procedure until an acceptable pattern in (4) is obtained. has a broad pattern and usuIn a conventional approach, ally serves as a global basis function in the pattern synthesis. It exhibits disadvantages, as illustrated in Fig. 2(a), because, in unknowns need to this case, not only be synthesized simultaneously, but also a large number of field points need to be used in the synthesis due to the difficulty to identify dominating components of the fields. The computation in the analysis and synthesis of array patterns will be cumbersome if the array is extremely large. As an alternative, this paper presents a procedure based on the creation of a new set of local radiation basis functions (defined later in (7)) to describe the desired pattern, as illustrated , have radiation in Fig. 2(b). These new field basis functions, patterns with much narrower beamwidths in comparison with in (1). This procedure expects to perform better in practice because it only needs to consider the very few basis functions that are confined within the desired coverage area. The use of the basis functions outside of the coverage area is unnecessary since in that region the only requirement is to make the power level

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as low as possible, and this goal can be achieved very easily by removing almost all the local radiation basis functions external to this region. , are produced by inThe local radiation basis functions, troducing the following transformation

where is the distance of target area. Let the target area is sam, and (7) becomes pled by

(5)

(8b)

where is the number of local basis functions, and the new coefficients. Substituting (5) into (4) gives

are now

which exhibits a discrete Fourier transform (DFT) when (8c)

(6) A criterion to distribute the local basis functions can be adopted, where a half of distance in (8c) will work fine. Thus for basis functions within the target area can be consequently determined.

where

(7)

Equation (7) may be identified as a field pattern radiated from an “entire” array with a global array excitation distribution to . In (7), produce radiation field focused at , and is used to provide an amplitude tapering in order to control the beamwidth of each basis function and the so that they may properly overlap to provide a shapes of good expansion without causing too many ripples in the pattern. Those position vectors are also shown in Fig. 1. , (7) results When [27], in a largest directivity or the narrowest beam spot at where and “ ” indicate the desired polarization and a complex represents in a spherical conjugate, and coordinate system. In this synthesis approach, one first optimizes the coefficients, , using (6) and then finds the actual element excitation co, by (5). One may simply employ the standard efficients, so that the desired radiMLSE technique [25] to solve for ation coverage beam can be formed. As mentioned earlier, only a small number of narrow spot beams are sufficient to synthesize a realistic contoured pattern. This number is simply a small fraction of the total number of elements in the entire array. It is noted that the actual element excitations can be found by multiplying the coefficients with an impedance matrix that accounts for the mutual coupling effects between elements as addressed in the method of moment.

III. AN EFFICIENT COMPUTATION ALGORITHM FOR The computation of in (7) can be accelerated if one properly does the following selections: (9) and (10) Let the focused region under synthesis be sampled by (11) Substituting (9)–(11) into (7) and using (1) give

(12) where

and in (12) are mainly a function of given by

. and

, which are

(13)

C. Determination of Local Basis Function Arrangement

It can be shown that

The arrangement of may determine , and is planed using the following algorithm. Since is a narrow beam fo, one thus approximates in (1) by cused at

, where and are integers. Similar rules also apply to . The computation is accelerated by using the similarity between adjacent basis functions as illustrated in Fig. 3. In particular, Fig. 3(a) and , shows the similarity between while Fig. 3(b) shows the similarity between and . As illustrated in Fig. 3(a), the fields that

(8a)

and that

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It is noted that when this concept is similarly applied to (15) and (17) as also illustrated in Fig. 3(a) and (b), they become

(18) and

(19) These rules in (14), (16), (18) and (19) are applied iteratively to save a lot of computation. IV. NUMERICAL EXAMPLES This section presents examples to demonstrate the implementation of this technique and exhibit the characteristics of synthesized contoured near-field patterns.

Fig. 3. Similarity of array radiations that contribute to  ( and R r ). Part (a) shows the similarity between E ( r ), and between C ( r ) and C  ( r ) and E (b) shows the similarity between E R (r ) and R ( r ).

E , C (r ) E (r ) and A. Examinations of Elemental Patterns and Basis Functions ( r ). Part One first examines the validity of the cosine tapers to model ( r ), and

contribute to at and radiate from antenna elements grouped by the closed solid-line loop are identical to the at and radiate fields that contribute to from antenna elements grouped by the closed dashed-line loop. has been found, can be Thus once obtained by shifting the array downward by , which gives , then subtracting the contributions from the bottom row elements, which does not exist in the original array, and adding the contribution from the top row elements that has been missed after the array shift. As a result, (12) become

(14) where (15) Similar identity applies in Fig. 3(b). The following relation also holds:

(16) where (17)

the radiation patterns of array element. In this case, 4 array sizes of 3 3, 5 5, 9 9 and 13 13 microstrip patch antennas are considered, where the antennas were printed on a FR4 substrate with period between elements (the element a size is to be shown in Fig. 12). Fig. 4(a)–(c), show the HFSS (a widely used commercial software) [33] simulated radiation patterns of array elements located at the center, corner and edge’s pattern is also shown in the figcentral elements, where a ures for comparison. It is observed that the radiation patterns of array elements do not change significantly within a reasonable angular range where the target area is located. Next, one considers an application scenario shown in Fig. 5 where an array of 17 17 half-wavelength spaced antennas is used for a typical RFID reader. In this case, the indices of cosine tapers for the elements’ pat. The target area is 0.5 0.5 located terns are at 1.5 m away from the array aperture. The design goal is to obtain a uniform near-field distribution in the target area while retaining low power level in the region external to this area. To demonstrate the validity of the developed approach, three are examined, as shown in Table I. They functions of exhibit various advantages and characteristics in the numerical using functions (i) raimplementations. In particular, diates same fields as from a conventional near-field focus an. Functions tenna array [17] with an equal-phase focusing at (ii) and (iii) are used for tapering the array excitation distributions to reduce the edge effects of finitely truncated array aperture, and make the array radiations in the region external to the target area decaying more smoothly to low levels, as having been demonstrated in the asymptotic theory of finite array diffraction

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Fig. 5. A scenario of 2.4 GHz RFID applications for a department store management. TABLE I VARIOUS FUNCTIONS OF f (n; m) USED IN THE EXAMINATION

. These asymptotic solutions [32] applicable to find will be more efficient than the algorithm shown in Section III since only a few rays, regardless the number of array elements, are sufficient to compute the array’s radiations, and will be reported in the future. On the other hand, the asymptotic evaluusing functions (i) and (ii) will suffer from ation of the singularities in the focused region because the equal-phase results in zero derivatives, which makes a radiation at location of ray-caustics in the asymptotic evaluation [28]–[30]. Fig. 6(a)–(c) show the distributions of central, edge in the target area and corner basis functions of in Table I, respectively. Here for the various basis functions are used to synthesize the near-field distribution in this target area. As expected, the employment of taper functions for array excitations reduces the levels of side-lobes and increases slightly the beamwidths. Fig. 4. Far-field patterns radiated from a single array element, which is located at the center, corner and edge center of the array in the presence of various array sizes. A cos  is also plotted for comparison. Radiation pattern of (a) a central element, (b) a corner element, (c) an edge element.

[28], [29]. It is noted that the asymptotic evaluation of radiation from a finite array decomposes the total radiation field into a superposition of ray-fields contributed from Floquet modes of an equivalent infinite array, edge-diffracted Floquet waves, and corner-diffracted fields [29], [30]. The reduction of excitations along the edge boundary will decrease the diffraction effects. In an implementation of accelerating numerical computations, function (iii) have a Gaussian taper, which gives a complex phase, and makes the asymptotic evaluation techniques [31],

B. Examination of Near-Zone Field Pattern Synthesis Fig. 7 shows the normalized synthesized near-field patterns in the target area. In particular, Fig. 7(a) shows the comparison of synthesized patterns using the excitation tapering functions in Table I. It is observed that all tapering functions result in similar and good field distributions within the target area, but the global cosine function (ii) and narrow Gaussian function (iii) give lower side-lobe levels in the region external to the target area. Fig. 7(b) shows the contoured pattern using a conventional near-field focus approach [17], which is also identical to the central beam using the function (i) from Table I. Fig. 7(c) and (d) shows the synthesized contoured patterns using functions (i) and (iii) in Table I. It is observed

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 ( Fig. 6. Central, edge and corner elements of E r ) for various f (n; m) in Table I. (a) Center basis function (p = 0; q = 0). (b) Edge basis function (p = 2; q = 0). (c) Corner basis function (p = 2; q = 2).

0

0

0

that the conventional near-field focus approach focuses fields at the center of the target area, which decay very quickly when the observation point moves away from the center. Significant side-lobe variations also appear in this target area. On the other hand, the synthesized contoured patterns exhibit very smooth and nearly uniform distributions within the target area, as shown in Fig. 7(c) and (d). It is also observed that the function (i) (i.e., no tapering on the array excitation) results in higher

Fig. 7. Normalized synthesized pattern in the target zone. (a) The pattern at x = 0 cut, (b) the pattern of a conventional near-field focused beam, (c) and (d) the synthesized pattern using function (i) and (iii), respectively.

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Fig. 8. Comparison of near-field distributions along the propagation between conventional near-field focused beam and synthesize contoured beam (using Function (iii) for f (n; m)). (unit: dB). (a) Conventional beam. (b) Synthesized beam.

side-lobes in the vicinity of the target area in comparison with the case that uses a narrow Gaussian function (iii) to taper the array excitation. Further investigation of the characteristics of array’s radiations along the propagation is illustrated by Fig. 8. It is achieved by considering the contoured patterns on the - plane, which is orthogonal to the array aperture, and comparing it for the conventional near-field focus approach and the current synthesis technique, as shown in Fig. 8(a) and (b), respectively. It is observed in Fig. 8(a) that even though the array elements’ radiations have equal phases at the center of the target area, which is located 1.5 m away from the array aperture, they do not yield a maximum field value at the focus point. Using the conventional approach, the maximum field strength occurs at 1.2 m, and away from this point the fields decay rapidly, and cause ripples and nulls that might degrade the communication quality. On the other hand, Fig. 8(b) shows that the presented technique results in a good field distribution in the target area. It is noted that the function (6) from Table I is used to taper the array excitation in order to reduce the side lobes in the region ex-

Fig. 9. Array excitations and their corresponding far field patterns. (a) Magnitude of array excitation. (b) Phase of array excitations. (c) Far field pattern.

ternal to the target area. Numerical examinations have exhibited that all 3 functions from Table I give almost identical contoured patterns on this plane within the region shown in Fig. 8(b). It is observed that the field distribution remains smooth and near uniform along the propagation after . As shown in Fig. 8(b), the array’s radiations quickly focus at 0.2 m, where the maximum field strength occurs, and then start to defocus. , the defocused fields tend to become smooth Beyond and form the required patterns.

CHOU et al.: AN EFFECTIVE SYNTHESIS OF PLANAR ARRAY ANTENNAS FOR PRODUCING NEAR-FIELD CONTOURED PATTERNS

Fig. 10. Synthesis of a non-uniform near-field pattern in a 0.4 area, where both desired and synthesized patterns are shown.

2 0.7 m

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target

Finally the magnitudes and phases of array excitations on the plane are shown central row, and the far field patterns at in Fig. 9(a)–(c), where a reference far field pattern obtained from a conventional near-field focus array is also shown for a comparison. It is observed that all three functions from Table I result in similar excitation weightings in the central region of the array, where the global Gaussian taper (i.e., Function (iii) in Table I) gives lower values of weighting for edge elements. The far-field patterns in Fig. 9(c) show that the cosine tapers (i.e., Function (3) and (4)) result in lower side-lobe levels and broader beamwidth in the far field. To further demonstrate the validity and flexibility of the present technique, the array is also used to synthesize a rectangular area of 0.4 0.7 with a linear amplitude taper. This scenario occurs in a situation where the array is placed with a tilted angle with respect to a horizontal plane as illustrated in Fig. 5. In this case, the distances between the array aperture and the surface of the transportation belt are linearly increased. The linear radiation pattern assists to achieve a uniform distribution over the horizontal surface. The desired and synthesized patterns are shown in Fig. 10, where a good agreement has been achieved. C. Examination of Computational Efficiency and Accuracy The accuracy and efficiency of this method are also investigated and shown in Fig. 11(a) and (b), respectively. In particular, Fig. 11(a) shows the averaged error between the synthesized and desired patterns in the case of Fig. 7, which was plotted with respect to the number of local basis functions (i.e., in (5)). The error is defined by the difference between co-polarized components as shown below:

(20) where is the number of sampling points over near-fields in the target area. In this case, 0.1 sampling distance of field distribution is used to obtain the result in Fig. 11(a). In this case, , the synthesis achieves a good result of accuracy. when

Fig. 11. Demonstration of accuracy and efficiency of the presented technique in synthesizing a near-field pattern shown in Fig. 7. labeled in the horizontal axis of (a) indicates the separation distance between local basis functions. (a) Percentage of error (b) CPU time.

1

The criterion in part C of Section II gives , which is shown to be reliable in this examination. The efficiency of this technique is examined in comparison of a steepest decent method (SDM) [9], [13], which is a reliable and fast convergent technique that is widely used in the antenna synthesis problems to find a minimum of a cost function. The cost function is usually defined based on the differences of field strengths between the computed values and the desired ones. The formulations in the SDM iterative procedure require one to find the derivatives of cost function in order to determine the new excitation weightings. In this case, the derivatives of a cost function used in SDM can be expressed in closed-forms (instead of using numerical computations in most of applications) for array synthesis, and is considered as one of most efficient techniques. On the other hand, to demonstrate the efficiency and robustness of the developed technique, we have removed the implementation of algorithm in Section III in this comparison. Thus all values of field distribution are found by element-by-element summation of individual element’s contribution. The result in terms of CPU time is shown in Fig. 11(b), where both cases were run on a same notebook with CPU Intel

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Fig. 12. Synthesized and numerically HFSS-simulated near-field patterns of a microstrip patch array with 13 13 elements. This figure shows the normalized patterns, which are found by the presented approach and that by HFSS.

2

Core 2 Quad Q8300 @2.50 GHz. It is noted that the variable, , in Fig. 11(b) is the dimension of a square array with elements. It is observed that the proposed approach appears very efficient as the number of element increases. Especially SDM recomputational CPU time while the proposed quires an technique tends to be when is large. Finally we present a realistic application of this technique to synthesize the NF radiation pattern of a microstrip patch array antenna, which has 13 13 elements and is printed on a FR4 substrate with a period between elements. The elements are trimmed on two opposite corners to radiate CP fields, whose excitations are computed using the presented technique. The radiation patterns computed by using the presented technique and HFSS, respectively are shown in Fig. 12. This case demonstrates the applicability of this technique to a realistic antenna array with strong mutual couplings between elements. As shown in Fig. 12, agreement in the comparison between the synthesized pattern using the presented technique and HFF simulated pattern has been found to be extremely good. Furthermore, the HFSS simulated result exhibits a lower sidelobe level. V. CONCLUSION This paper presents an efficient approach to synthesize the near-field pattern of an antenna array. The method utilizes a global basis set to represent the excitation amplitude of the array with an additional phase impression to generate focused spot beam in the near zone. It is combined with a standard synthesis MLSE algorithm to generate a contoured pattern. An efficient algorithm to compute the near-fields radiated from the array is also developed, which makes this approach very effective for practical applications. Numerical examples are presented to demonstrate the efficiency and validity of this new synthesis procedure. REFERENCES [1] K. Finkenzeller, RFID Handbook, 2nd ed. New York: Wiley, 2003. [2] 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, Dec. 2005.

[3] J. Lin and C. Li, “Wireless non-contact detection of heartbeat and respiration using low-power microwave radar sensor,” in Proc. Asia-Pacific Microwave Conf., 2007, pp. 1–4. [4] M. Bogosanovic and A. G. Williamson, “Antenna array with beam focused in near-field zone,” Electron. Lett., vol. 39, no. 9, pp. 704–705, May 2003. [5] M. Bogosanovic and A. G. Williamson, “Microstrip antenna array with a beam focused in the near-field zone for application in noncontact microwave industrial inspection,” IEEE Trans. Instrum. Meas., vol. 56, no. 6, pp. 2186–2195, Dec. 2007. [6] K. D. Stephan, J. B. Mead, D. M. Pozar, L. Wang, and J. A. Pearce, “A near field focused microstrip array for a radiometric temperature sensor,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1199–1203, Apr. 2007. [7] T. W. R. East, “A self-steering array for the SHARP microwave-powered aircraft,” IEEE Trans. Antennas Propag., vol. 40, no. 12, pp. 1565–1567, Dec. 1992. [8] O. M. Bucci, N. Fiorentino, and T. Isernia, “An efficient approach to 2-D arrays mast constrained power pattern synthesis,” in Proc. IEEE AP-S Int. Symp. Digest, Montreal, Canada, Jul. 13–18, 1997, vol. 4, pp. 2252–2255. [9] A. R. Cherrette, S.-W. Lee, and R. J. Acosta, “A method for producing a shaped contour radiation pattern using a single shaped reflector and a single feed,” IEEE Trans. Antennas Propag., vol. 37, no. 6, pp. 698–705, Jun. 1989. [10] 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. 27–39, Jan. 1995. [11] G. T. Poulton, “Power pattern synthesis using the method of successive projections,” in Proc. IEEE AP-S Int. Symp. Digest, 1986, vol. 2, pp. 667–670. [12] H.-T. Chou, P. H. Pathak, and R. J. Burkholder, “Feed array synthesis for reflector antennas in contoured beam applications via an efficient and novel Gaussian beam technique,” Radio Sci., vol. 36, no. 6, pp. 1341–1351, 2001. [13] H.-T. Chou and P. H. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc., Microw. Antennas Propag., vol. 151, no. 1, pp. 13–20, Feb. 2004. [14] Z.-M. Liu and R. R. Hillegas, “A 3-patch near field antenna for conveyor bottom read in RFID sortation application,” in Proc. IEEE AP-S Int. Symp. Digest, Albuquerque, NM, Jul. 9–14, 2006, pp. 1043–1046. [15] L. Shafai, A. A. Kishk, and A. Sebak, “Near field focusing of apertures and reflectors antennas,” in Proc. Conf. on Communications, Power and Computing, Winnipeg, MB, May 22–23, 1997, pp. 246–251. [16] S. Karimkashi and A. A. Kishk, “A new Fresnel zone antenna with beam focused in the fresnel region,” in Proc. XXIXth General Assembly of the Int. Union of Radio Science, Chicago, IL, Aug. 7–16, 2008. [17] A. Buffi, A. A. Serra, P. Nepa, H.-T. Chou, and G. Manara, “A focused planar microstrip array for 2.4 GHz RFID readers,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1536–1544, May 2010. [18] J. W. Sherman, “Properties of focused apertures in the Fresnel region,” IRE Trans. Antennas Propag., vol. 10, no. 4, pp. 399–408, Jul. 1962. [19] W. J. Graham, “Analysis and synthesis of axial field pattern of focused apertures,” IEEE Trans. Antennas Propag., vol. 31, no. 4, pp. 665–668, Jul. 1983. [20] D. A. Hill, “A numerical method for near-field array synthesis,” IEEE Trans. Electromagn. Comp., vol. EMC-27, no. 4, pp. 201–211, Nov. 1985. [21] D. A. Hill and G. H. Koepke, “A near-field array of Yagi-Uda antennas for electromagnetic-susceptibility testing,” IEEE Trans. Electromagn. Comp., vol. EMC-28, pp. 170–178, Jan. 1986. [22] B. Stupfel and S. Vermersch, “Plane-Wave synthesis by an antenna-array and RCS determination: Theoretical approach and numerical simulations,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 3086–3095, Nov. 2004. [23] M. S. Narasimhan and B. Philips, “Synthesis of near-field patterns of arrays,” IEEE Trans. Antennas Propag., vol. 35, no. 2, pp. 212–218, Feb. 1987. [24] M. S. Narasimhan and B. Philips, “Synthesis of near-field patterns of a nonuniformly spaced array,” IEEE Trans. Antennas Propag., vol. 35, no. 11, pp. 1189–1198, Nov. 1987. [25] H.-T. Chou, Y.-T. Hsaio, P. H. Pathak, P. Nepa, and P. Janpugdee, “A fast DFT planar array synthesis tool for generating contoured beams,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 287–290, 2004.

CHOU et al.: AN EFFECTIVE SYNTHESIS OF PLANAR ARRAY ANTENNAS FOR PRODUCING NEAR-FIELD CONTOURED PATTERNS

[26] Y. Rahmat-Samii and S.-W. Lee, “Directivity of planar array feeds for satellite reflector applications,” IEEE Trans. Antennas Propag., vol. AP-31, no. 3, pp. 463–470, May 1983. [27] H.-T. Chou, Y.-Y. Lin, and S. Sun, “Applications of pattern-matched beam forming algorithm in the design of adaptive array antennas,” J. Chinese Inst. Elect. Engrg., vol. 9, no. 2, pp. 175–180, May 2002. [28] Ö. A. Çivi, P. H. Pathak, H.-T. Chou, and P. Nepa, “Extension to a hybrid UTDMoM approach for the efficient analysis of radiation/scattering from tapered array distributions,” in Proc. IEEE AP-S Int. Symp. Digest, Salt Lake City, UT, Jul. 2000, pp. 70–73. [29] F. Mariottini, F. Capolino, S. Maci, and L. B. Felsen, “Asymptotic high-frequency Green’s function for a large rectangular planar periodic phased array of dipoles with weakly tapered excitation in two dimensions,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 608–620, 2005. [30] Ö. A. Civi, P. H. Pathak, H.-T. Chou, and P. Nepa, “A hybrid uniform geometrical theory of diffraction-moment method for efficient analysis of electromagnetic radiation/scattering from large finite planar arrays,” Radio Sci., vol. 35, no. 2, pp. 607–620, 2000. [31] Y. Kim, “On a uniform geometrical theory of diffraction based complex source beam diffraction by a curved wedge with applications to reflector antenna analysis,” Ph.D. dissertation, Electrosci. Lab., Ohio State Univ., , 2009. [32] H.-T. Chou and P. H. Pathak, “Uniform asymptotic solution for electromagnetic reflection and diffraction of an arbitrary Gaussian beam by a smooth surface with an edge,” Radio Sci., vol. 32, no. 4, pp. 1319–1336, 1997. [33] High Frequency Structure Simulator (HSFF) User Manual-v12. Pittsburgh, PA: Ansoft Corporation, 2009.

Hsi-Tseng Chou (S’96–M’97–SM’01) was born in Taiwan, in 1966. He received his B.S. degree in electrical engineering from National Taiwan University in 1988, and his M.S. and Ph.D. degrees in also electrical engineering from Ohio State University (OSU) in 1993 and 1996, respectively. He joined Yuan-Ze University (YZU), Taiwan, in August 1998, and is currently a professor in the Department of Communications Engineering. His research interests include wireless communication network, antenna design, antenna measurement, electromagnetic scattering, asymptotic high frequency techniques such as Uniform Geometrical Theory of Diffraction (UTD), novel Gaussian Beam techniques, and UTD type solution for periodic structures. Dr. Chou has received two awards from Taiwanese Ministry of Education and Ministry of Economic Affairs in 2003 and 2008, respectively to recognize his distinguished contributions in promoting academic researches for industrial applications, which were the highest honors these two ministries have given to university professors to recognize their industrial contributions. He has published more than 320 journal and conference papers. Dr. Chou is a senior member of IEEE AP-S and an elected member of URSI International Radio Science US commission B.

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Nan-Nan Wang was born in Harbin, China, in 1982. She received the B.S. degree from Dalian Nationalities University, China, in 2005 and the M.S. degree from Harbin Institute of Technology (HIT), China, in 2007, all in electrical engineering. She is currently working toward the Ph.D. degree at Harbin Institute of Technology. She worked one year as an exchange student in the Department of Communication Engineering, Yuan-Ze University, Taiwan. Currently, she is mainly engaged in the researches of millimeter wave imaging technology, microwave antenna and the synthesis of array antennas to produce near-field focused contoured patterns.

Hsi-Hsir Chou was born in Chang Hua, Taiwan, in 1975. He received the Ph.D. degree in Engineering from Cambridge University, U.K. in 2008. He was involved in collaborating with ALPS UK Co. Ltd. and Dow Corning Co. Ltd. in the development of patented free-space optical interconnection technologies, ferroelectric liquid crystal devices and carbon nanotube dielectric devices during his Ph.D. program at Cambridge University from 2004 to 2008. He joined the Department of Engineering Science, Oxford University, U.K. in July 2008 as a Postdoctoral Researcher in the development of high-speed visible light communication technologies sponsored by Samsung Electronics Co. Ltd., Korea, before he returned Taiwan to join the Communication Research Center, Yuan Ze University, Taiwan as a Researcher in May, 2009. His current research interests include free-space optical interconnection technologies, ferroelectric liquid crystal devices, carbon nanotube dielectric devices and antenna design. Dr. Chou is a lifetime member of Trinity College, Cambridge and a Fellow of Cambridge Overseas Society since 2005.

Jing-hui Qiu was born in Heilongjiang Province, China in 1960. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from Harbin Institute of Technology (HIT), China, in 1982, 1987 and 2008, respectively. He jointed HIT in 1987, and is currently a Professor and the Head of the Department of Microwave Engineering, School of Electronics and Information Engineering. His research interests include the developments of ultra-wideband antenna, pulsed antenna, high-power EM pulse technologies, airborne antenna, passive millimeter wave imaging technologies and communication technologies for the applications of near-space high speed aircrafts. He has published more than 200 journal and conference papers. Prof. Qiu received a number of awards to recognize his distinguished contributions in promoting academic researches for industrial applications including the Science and Technology Progress Awards from Department of Space and from Aerospace Industry Corporation of China, respectively.

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Time-Harmonic Echo Generation Amedeo Capozzoli, Member, IEEE, Claudio Curcio, and Angelo Liseno

Abstract—We propose an approach to the design of array-based, 2D echo generators. The radiating elements are located on non-uniform grids and the quiet zone (QZ) design specifications are enforced at non-uniformly spaced sampling locations. The approach, based on a singular values optimization process, allows dimensioning the size of the echo generator to meet the QZ specifications, defining the number and locations of the QZ sampling points and of the radiators to control the ill-conditioning when searching for the excitation coefficients, dramatically reducing the number of radiating elements, and finding the excitations by a Singular Value Decomposition approach. The performance of the method in terms of QZ field behavior and robustness against realization errors is numerically assessed. An experimental validation of the technique is also presented. Index Terms—Antenna arrays, antenna measurements, compact range, near-fields, radar testing, singular value decomposition.

I. INTRODUCTION ANY applications require the generation of “canonical” waves in prescribed regions of space known as quiet-zones (QZs). These include antenna characterization [1], the testing of electronic equipments (e.g., integrated aircraft electronic systems) subject to electromagnetic stimuli [2], medical treatments [3], the electromagnetic exposure of biological specimens [4], system testing of radars under their actual working conditions by synthesizing the waveforms produced by reflecting targets [5] and tomography [6]. When the canonical wave of interest is a plane wave, reflector- [7], lens- [8] or hologram-based [9] compact antenna test ranges (CATRs) represent a commonly employed solution. However, CATRs exhibit some drawbacks. Indeed, for reflector-based CATRs, sufficiently precise manufacturing, mitigation of the diffraction phenomena, and accurate alignment procedures are needed, and, especially at low frequencies, large QZs require large reflecting surfaces due to diffraction effects. Lens-based CATRs are quite expensive due to the costly workmanship of the lens surface and the homogeneity needs of the employed lens material. Furthermore, large, cumbersome antennas are difficult to be characterized in a CATR, due to the movements (rotations) required during the testing. Finally, CATR solutions lack in flexibility since they do not easily enable real-time changes of the radiated wavefront, or simply dealing with disconnected and/or arbitrarily shaped QZs [2].

M

Manuscript received August 02, 2010; revised January 03, 2011; accepted February 18, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Dipartimento di Ingegneria Biomedica, Elettronica e delle Telecomunicazioni, Universitá di Napoli Federico II, I 80125 Napoli, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161548

These limitations have then suggested the development of plane wave synthesizers (PWS’), i.e., of arrays purposely designed to generate prescribed waves in their near-field regions [1]. Although exhibiting more hardware complexity, PWS’ should permit to significantly improve the flexibility by allowing, for example, circular geometries [10] and disconnected QZs (i.e., QZs made of disconnected subregions) by sets of disconnected panels of elementary radiators [2]. Furthermore, they are very helpful whenever the testing of large objects under test (OUTs) is required, by avoiding likewise large QZs [1], [2]. Due to the higher hardware complexity as compared to CATRs, the design of a PWS is not an easy task, requiring not only the synthesis of the excitation coefficients of the radiating elements, but also dealing with key points as the minimum required size of the radiating panels and the minimum number of radiating elements per panel. Different design techniques for standard PWS’ have been developed, often assuming the array of radiators located on a regular, Cartesian grid [1], [2], [11] and enforcing the design specifications on regular QZ lattices. Most recently, methods to face the problem ill-conditioning have also been developed [12] and approaches for the design of generalized PWS’ (GPWS’), i.e., PWS’ made of disconnected radiating panels with optimum size and involving disconnected QZ fields, reducing the number of elements in each panel by arranging them on a convenient non-regular lattice, and enforcing the design specifications on non-regular grids of a quiet plane (QP, typically the edge plane of the QZ) have been presented for a 1D geometry [13]. For fixed QZ regions, the strategy in [13] allows the determination of the overall number and size of radiating panels, along with their positions and orientations, and the number, the positions, and the excitation coefficients of the radiating elements located within the panels. By a singular value optimization approach, similar to that proposed for the non-redundant sampling of near-fields and very near-fields in [14], [15], the QP grid and the radiator locations are properly chosen to control the ill-conditioning [16], also associated to the (point-matching) enforcement of the QP constraints. Following the definition of the size of the radiating panels, the definition of both the QP and radiator lattices are preliminary to the determination, performed by a singular value decomposition (SVD), of the array excitation coefficients. By improving the ill-conditioning, the field level outside the QP can also be controlled and kept low to improve the QZ depth. A proper strategy is furthermore employed by the algorithm to provide reliability of the results and to control the computational burden, especially concerning the step performing the optimization of the QP grid and element positions. Following [13], [17], the purpose of this paper is to propose a technique for the synthesis of 2D time-harmonic echo generators (superposition of plane waves), i.e., of systems for which the canonical waves of interest are not only plane waves but, more generally, radar echoes.

0018-926X/$26.00 © 2011 IEEE

CAPOZZOLI et al.: TIME-HARMONIC ECHO GENERATION

Fig. 1. Geometry of the problem.

The strategy in [13], [17] is developed in more detail and the performance is numerically and experimentally evaluated, by focusing the attention on the case of a single radiating panel and QP domain. The generation of radar echoes has over time drawn much attention in many research directions indeed, and mainly in the developments of apparatuses for the functional testing of radars [18], for the performance evaluations of direction finding [19], target detection and identification [20], and of countermeasure radars [21], as well as in the development of near-field techniques for radar synthetic environment simulators [5]. It is worth stressing that, in all these circumstances, the signals of interest correspond to non-impulsive plane wave spectra, as it especially occurs in the case of radar countermeasures. On referring to the classical problem of generating plane waves, it is shown how the approach followed in this paper improves the performance of standard PWS’ involving uniform lattices, in terms of QZ field behavior, of robustness against uncertainties and of dramatic reduction of the number of involved radiators. Furthermore, this paper emphasizes how one of the purposes of the synthesis procedure of echo generators, namely reducing the number of radiators as compared to standard approaches, is fulfilled. Indeed, such a reduction means significant benefits, especially for electrically very large structures. In this paper, an extensive numerical and experimental analysis assesses the performance of the approach for producing echoes of the scattering from objects with canonical shapes, for testing purposes. Obviously, in this case the echo can be represented by means of plane waves propagating in a cone with axis in the observation direction. For the sake of brevity, the method presented here is for timeharmonic fields, but can be properly extended to the case of time-domain waveforms. II. THE PROBLEM Let us consider a time harmonic echo generator composed by rectangular an array of radiators, arranged over a plane, and with the th element panel, located in the and excitation coefficient having coordinates (see Fig. 1) [17]. The purpose of the array is to radiate a prescribed wave in a half space, termed prescribed and finite 3D region of the the QZ. Throughout the paper we suppose the QZ being a rectangular box, starting at (see Fig. 1).

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In the case of CATRs and GPWSs, the wave of interest is a portion of a plane wave propagating along the positive direction [1], and the QZ field is required to have uniform amplitude and phase over portions of planes parallel to the scattering/radiating structure. In practice, both the amplitude and phase are required to be uniform with tolerances of dB for the amplitude and for the phase [1], [7]. In the case of echo generators considered in this paper, the wave of interest is not necessarily a plane wave and will be required, as a design specification, to deviate from the desired one with the same above mentioned tolerances of dB for the amplitude and for the phase. For a fixed QZ, the aim of the design procedure is to choose the panel size, the number of radiators, their locations and their excitation coefficients to meet the design specifications. Having chosen the dimensions of the radiating panel (see the following Section), the synthesis procedure attempts to satisfy the design specifications on the QZ with the least number of radiators. However, it can be seen that the constraints can be enforced only on the first edge of the QZ itself, namely, only on a rectangular region of the plane , henceforth finite termed the quiet plane (QP), which will at the same time also control the field behavior outside of it on [22]. III. DIMENSIONING THE RADIATING PANEL of the radiating panel, To determine the overall size we resort to a continuous, scalar modelling of the problem, by regarding the panel as a continuous aperture and the sized QP domain as not discretized. the aperture Under this assumption, denoting by field, the link between the field radiated over the sized QP region and is provided by

(1) where is the obliquity factor [23], is the angle between the normal to the aperture and the unit vector , with and , and is the wavenumber. Assuming non-superdirective apertures, the aperture field can be represented by the prolate spheroidal wave functions (PSWFs) as [24], [25] (2) where products”

is the th, 1D PSWF with “space-bandwidth are expansion coefficients, denoting the is the wavelength. Obviously, the integer part, and and , having series in (2) has been limited to the indices considered only the contributions to the visible region. In the framework of continuous domains, the synthesis of and the the radiating aperture amounts to determining ’s, given the QP field , which represents the design specification in this case. , the

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Given and , the problem of determining the ’s is a linear, inverse and ill-conditioned one, and can be solved by an SVD approach, i.e., by truncating the singular values falling below a prescribed threshold [16], [24]. of singular values above the threshold1 deThe number pends on the geometry of the continuous problem, i.e., on and , and represents the number of source parameters that can be effectively exploited to synthesize a field matching the design specifications on the QP. represents On the other hand, the number of visible source parameters exploitable in the case when the design specifications are set over the whole QP and is then also the maximum number of source parameters available. is significantly smaller than , then the Now, if apertures are uselessly large, with detrimental effects on the system encumbrance and complexity and on the computational of the QP region of interest burden. Indeed, the size does not allow exploiting all the available number of source pais always smaller than , rameters. Accordingly, since then the panel size is chosen so as to ensure that the conditions be as close as possible to 1. IV. THE DESIGN PROCEDURE Once the size of the radiating panel has been defined, the most convenient number of radiating elements and QP sampling points along with their respective locations should be determined. In order to provide details about the design procedure, we first observe that the constraints are expressed through a point matching procedure with the desired field [17] on a sampling points of the QP at the coordinates set of (see Fig. 1). The numbers and and the locaof such points are also properly selected by the design tions procedure. Furthermore, the synthesis algorithm is also responsible to avoid “superdirectivity-like” effects, as high field values or abrupt field decays outside the QP. Indeed, the former would affect the field downstream from the QZ, thus reducing its effective size. The latter would correspond to having many closely spaced radiating elements, which would render the realization of the echo generator unstable. To provide more details about the synthesis process, the model employed for the field radiated by the echo generator in its near-field region is determined first. Then, the scheme for determining the element excitations, which is based on a SVD and for fixed raapproach [16], is described for fixed and diator and matching point lattices. The optimization procedure for determining the most convenient above-mentioned lattices is then discussed. Finally, the rationale for the determination of and is provided. A. The Field Radiated by the Echo Generator The radiated field

at

can be written as (3)

1For the cases of our interest, a step-like behavior of the singular values occurs, so that weakly depends on the chosen threshold.

N

where

is the element factor, is the wavenumber and are the spherical coordinates of in a (see Fig. 1). Equation (3) reference system with center at holds true if is located in the far-zone region of each of the involved radiators. In Section V.A, it will be shown that this approximation does not affect the accuracy of the synthesized echo generator. As in the above, for the sake of simplicity, a scalar problem is addressed so that the specifications concern the -component of the field given by (4) where is the homologous component of . Furthermore, to match the addressed experimental setup, as the element factor, that of an open-ended waveguide (OEWG) is considered, so that is the -component of . Finally, it is worth observing that is weakly dependent on the particular kind of adopted antenna, being elementary. Otherwise, it would affect the conditioning of the radiation operator (see below). B. The SVD-Based Determination of the Array Excitation Coefficients By enforcing the design specifications at the sampling , the excitation coefficients should satisfy the folpoints lowing linear system (5) where is the vector containing all the element excitations, is the vector containing the field sample values prescribed at the and points

.. .

..

.

.. .

(6)

where . The excitation coefficients can be determined by means of a SVD approach to solve (5). Once the ’s satisfying (5) are determined, the field radiated by the synthesized array should be analyzed to verify that the QP design specifications are met. of the QP region of However, for a prescribed size interest, the possibility of meeting the specifications is affected by , • the choice of a (sufficiently large) panel size • the choice of the number and locations of the radiators, and the locations of the • the choice of the number QP sampling points. On the other hand, the number and locations of both the radiators and the sampling points affect the ill-conditioning of the problem of inverting and therefore also the ability of controlsized region of the QP. This, in ling the field outside the turn, is responsible of unsatisfactory behavior of the field within

CAPOZZOLI et al.: TIME-HARMONIC ECHO GENERATION

TABLE I COMPARING UNIFORM AND NON-UNIFORM (ACCORDING TO [26], [27]) QP SAMPLINGS

the rest of the QP [12], [13]. Furthermore, large condition numbers of undermine the robustness of the synthesized QP field. can lead to an unmanFinally, improper choices of and/or ageable computational burden. In the remaining part of this Section, the choice of the number and locations of both the radiators and the QP sampling points is addressed. C. Choosing the Number and Locations of the Radiators and QP Sampling Points Let us now consider the choice of the number of radiators. In order to keep low, is chosen to be equal to the 2D PSWFs needed to represent the field in the number of aperture field model of the foregoing Section. of QP sampling points. To Let us now choose the number consistently solve the synthesis problem, we should set . At the same time, to avoid any unwanted increase of the should be as small as possible, to also ill-conditioning of . control the computational effort. Consequently, we set We clarify that, since the panel size is chosen, according to the procedure in Section III, to be the smallest capable to radiate a field having enough degrees of freedom on the near-field region of the QP (in order to meet the design specifications), by the criterion here provided and are essentially chosen as the minimum radiator and QP sampling point numbers compatible with the source and QP dimensions. We also stress that, if identified with the number of degrees of freedom would vanish for increasing , thus of the QP field, then becoming less than . However, such a circumstance cannot occur since, by the procedure in Section III, the panel size is chosen so that the number of degrees of freedom of the source is close to the number of radiated degrees of freedom on the QP. By changing , the panel should be resized to meet such a condition. It should be mentioned that an alternative first possibility for defining the radiating panel and QP grids would be to employ sampling steps [1], [2]. However, this sampling uniform, strategy is highly redundant, leading to high condition numbers for . In [12], it has been shown how a non-uniform sampling, according to the theory in [26], [27], allows reducing the degree of ill-conditioning of the problem and the overall computational burden. To better emphasize this point, let us consider, sized radiating panel and a QP region for example, a located at . Table I illustrates the advantages achieved when using the non-uniform sampling , using in [26], [27] for the QP grid and having assumed uniformly spaced radiators. The significantly better condition

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number of can be appreciated, along with the strong reduction of the computational cost of the SVD calculation2 Now, following the choices of and , proper locations for the radiating elements and QP sampling points need to be deis fined to control the ill-conditioning. By noting that the ratio between the maximum and minimum singular values of , then a convenient QP grid can be sought for that will ensure the singular value dynamics of to be as flat as possible [13]–[15]. This is accomplished here by maximizing the functional (7) which entails evaluating the “area” subtended by the normalized , where is the th singular value of singular values [13]–[15], and . It will be shown in the next Section that this approach can be further improved upon the condition number that is reported in Table I. D. The Representation Function An iterative procedure for the singular value optimization (7) can be very demanding from a computational point of view, as [28] at it requires the evaluation of the singular values of each iteration step. Moreover, although the sum of the singular defines a convex function [29], the functional values of that needs to be optimized is found to be multimodal due to the nonlinearity of the relation between the matrix entries and the radiator and QP sampling point locations. Thus the solution can be susceptible to being trapped in a local maximum. To reduce the computational burden and strengthen the singular value optimization against the traps, proper mapping functions transforming uniform 2D lattices into non-uniform ones and by a few parameters, are employed to represent analogously to what was proposed in [30], [31]. In this way, the of both the radiators and QP points are reprelocations sented as (8) with (9)

(10) where defines a uniform lattice of is a properly defined set of basis functions (e.g., Legendre polynomials, as employed in the numerical analysis) and and are proper expansion coefficients. The representation in (8), (9) and (10) is able to significantly reduce the number of the parameters to be sought for during the maximization of , as the actual unknowns of the procedure 2Table I refers to a processing performed on an Intel Dual Core, 1.80 GHz with 2 Gb of RAM.

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Fig. 2. QP sampling grid. Dots: uniform positions with =2 spacings. Circles: optimized positions.

Fig. 3. Radiator positions. Dots: uniform positions with =2 spacings. Circles: optimized positions.

reduce to a few coefficients and [30], [31]. The number of searched for parameters can be further reduced by accounting for the reflection symmetry of the radiator locations and QP grid around the origin. In particular, can be chosen as odd with respect to and even with • respect to ; can be chosen as even with respect to and odd with • respect to . V. NUMERICAL RESULTS We now evaluate the performance of the approach by deeply sized array of analyzing a test case consisting of a 441 radiating elements, generating a sized QP region . For this size of the QP, the panel dimensions have at been chosen according to the criterion sketched in Section 3. Figs. 2 and 3 illustrate the radiator positions and the QP sampling point locations for the synthesized echo generator following the optimization of . In addition, Fig. 4 compares the

Fig. 4. Singular values behavior. Circles: uniform =2 sampling points and radiator positions. Continuous line: optimized sampling points and radiator positions.

Fig. 5. Computation times estimated for processing performed by a Matlab code run on an Intel Dual Core, 1.80 GHz with 2 Gb of RAM, equipped with a Windows XP operating system.

behavior of the singular values for uniform and optimized sampling point and radiator positions illustrated in Figs. 2 and 3. As can be seen, the singular value dynamics is flatter, leading to a better problem conditioning. In other words, Fig. 4 shows that the radiator and QP sampling point locations affect the number of significant singular values of , so that improper locations could prevent exploiting all the degrees of freedom of the radiating panel. However, once the radiator and QP sampling point locations have been properly chosen, the robustness of the echo generator is not significantly affected by positioning and excitation error, as shown by the numerical analysis below reported. It is also worth showing the computational advantages of the proposed strategy. To this end, Fig. 5 illustrates the processing time needed, for a panel size equal to that considered in this paper, to synthesize the echo generator for different QP area sizes. For the uniform sampling and non-uniform sampling

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Fig. 6. Plane wave orthogonally impinging to the QP: amplitude of the synthesized QP field.

cases, the processing times have been estimated from the theoretical asymptotic computational complexity of the SVD routine. On the other hand, for the non-uniform sampling case with use of the mapping functions, the processing times are actually measured times. As can be seen, the case of non-uniform sampling aided by the mapping functions (9) and (10) is able to reQP area by two orders of duce the processing time for a magnitude, with respect to the case of non-uniform sampling when no mapping functions are exploited and four orders of sampling. magnitude with respect to the case of uniform, In the next Section, the synthesis problem of a single plane wave with its propagation direction orthogonal to the QP will be considered first and the robustness of the designed echo generator against errors on the radiator locations and excitation coefficients will be assessed. Subsequently, the synthesis of tilted plane waves will be dealt with. Such an analysis is of interest to foresee the performance of the echo generator, since a generic wavefront can be represented by a proper superposition of plane waves with propagation vectors distributed in a cone with its axis in the observation direction. In other words, determining the set of incidence directions that can be synthesized entails determining the capability of the echo generator to synthesize arbitrary wavefronts. Finally, the results concerning a more realistic wavefront will be shown. In particular, synthesizing the field scattered by two perfectly conducting cylinders will be addressed. A. Synthesis of a Plane Wave Orthogonally Impinging to the QP Figs. 6 and 7 illustrate the amplitude and phase of the QP field radiated by the synthesized echo generator. It can be verified that such a radiated field satisfies the design specifications. In order to point out the accuracy of the assumptions for (3), Fig. 8 compares the cuts, along the -axis, of the field as evaluated by (3) and the “exact” one radiated on the QP. As can be seen, the lines concerning the “exact” and that evaluated by (3) lie on top of each other. A similar result holds true also for the phase. Following the design of the echo generator, assessing its robustness against possible implementation errors is in order. In the following, errors in either the radiator locations or in the radiator excitation coefficients are individually considered along with their effects on the synthesized QP field.

Fig. 7. Plane wave orthogonally impinging to the QP: phase of the synthesized QP field.

Fig. 8. Cuts, along the x-axis, of the “exact” field (circles) and that evaluated by (3) (solid line).

1) Errors in the Radiator Locations: The errors in the radiator locations have been modeled according to random , variables uniformly distributed in with being a parameter determining the extent of the error. For each value considered for , a number of positioning error realizations, numbered , have been drawn. For each instance, the maximum and minimum field amplitude and phase variations and respectively have been recorded, as compared to their corresponding spatial mean values over the QP region of interest. From the recorded values, the average values (over the realizations) of the maximum field amplitude and phase variations and , respectively, have been calculated and the frequency of occurrence of meeting the design specifications evaluated. Table II summarizes the results of such an analysis. , the probability of missing the deIt is noted that, for sign specifications is close to 100%, almost completely due to exceedingly large field amplitude oscillations. However, these oscillation averages are about 0.61 dB for this case, which can

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TABLE II ROBUSTNESS OF THE ECHO GENERATOR AGAINST POSITIONING ERRORS

TABLE III ROBUSTNESS OF THE ECHO GENERATOR AGAINST ERRORS ON THE EXCITATION COEFFICIENTS

still be acceptable due to the large errors considered for the radiator positions. Indeed, considering for example cm corresponds to m, which is (X-band), an error of more than the typical positional uncertainties [32] exhibited by a commercial implementation of a synthetic echo generator [1]. 2) Errors in the Radiator Excitation Coefficients: The errors on the array excitation coefficients have been modeled as additive noise governed by Gaussian random variables. Different Signal to Noise Ratios (SNRs) have been considered between 30 dB and 40 dB, and the rationale of the analysis has been the same as that in Subsection V.A.1. Table III summarizes the results. dB, the design specifications are practically When always missed due to large field oscillations. However, considerations similar to those in Subsection V.A.1 also hold true, since the average maximum field amplitude variations stays within dB.

Fig. 9. Maximum field amplitude deviation for different values of z. Continuous line: specifications. Circles: proposed approach. Stars: =2 panel and QP grids. Maximum field amplitude deviation limits to 3:7 the longitudinal extent of the QZ for the =2 panel.

B. Volumetric Extent of the QZ In order to evaluate the volumetric extent of the synthehas been sized QZ, the field radiated over planes calculated. Figs. 9 and 10 show, for different values of , the field amplitude and phase deviations given by and respectively, portions of the plane . As can be within seen, the longitudinal extent of the QZ is limited by the field . It is worth pointing out phase deviation and is equal to that the echo generator designed by a variant of the algorithm uniformly spaced radiators and QP grids has employing worse performance in terms of longitudinal extent of the QZ.

Fig. 10. Maximum field phase deviation for different values of z. Continuous line: specifications. Stars: proposed approach. Maximum field phase deviation limits to 10 the longitudinal extent of the QZ for the proposed approach.

They are limited by the field amplitude deviations and, as illustrated by Fig. 9, the longitudinal extent of the QZ equals for this case.

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Fig. 11. Field amplitude in the y

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= 0 plane. Fig. 13. Tilted plane wave generation: cuts along the x-axis of the desired (circles) and synthesized (continuous line) field phases over the QP. The two lines lie virtually on top of each other.

Fig. 12. Field phase in the y

= 0 plane.

Finally, to illustrate the QZ field behavior, Figs. 11 and plane, 12 report the field amplitude and phase in the respectively.

Fig. 14. Tilted plane wave generation: cuts along the x- (continuous line) and y - (circles) axes of the field phase deviations from specifications over the QP.

C. Synthesis of a “Tilted” Plane Wave We consider now the generation of a plane wave with tilted angle of incidence on the QP. In particular, a tilt angle of along the -axis has been considered. Fig. 13 illustrates the good agreement between the behavior of the synthesized tilted phase as compared to the desired one along the -axis. Fig. 14 depicts the field phase deviations from the specifications along the - and -axes. As can be seen, the field phase deviations are less than the 5 requirement. It has been verified that the design specifications are met all over the QP. D. Synthesis of the Field Scattered by Two Conducting Cylinders We now report on the results concerning the generation of the field scattered by two, infinitely long, perfectly conducting cylinders having circular cross section and radius equal to , with reciprocal center spacing of (see Fig. 15).

The cylinders’ axes are assumed to be parallel to the -axis, they are illuminated by an orthogonally impinging plane wave, while . Under these the reciprocal distance to the QP is circumstances, the spatial spectrum of the field impinging on the QP contains plane waves with maximum impinging angles of up to approximately 4 . It has been verified that the synthesized echo generator is capable of generating the scattered field within the required accuracy. Figs. 16 and 17 compare the behavior of the synthesized versus desired scattered field amplitude and phase respectively, for a cut along the -axis. VI. EXPERIMENTAL RESULTS The experimental validation of the above design procedure has been performed by a “synthetic” implementation of the involved radiating array [1] which, as compared to a “real” one with all radiating elements contemporary present, disregards the

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Fig. 15. Geometry relevant to the generation of the field scattered by two perfectly conducting cylinders.

Fig. 17. Scattered field generation: cuts along the x-axis of the desired (circles) and synthesized (continuous line) field phases over the QP. The vertical line denotes the QP boundary.

Ingegneria Biomedica, Elettronica e delle Telecomunicazioni of the Università di Napoli Federico II. The 5 m 8 m 3.5 m chamber is certified to operate in the 900 MHz–40 GHz band, dB at 1.8 GHz and dB with a nominal reflectivity of at 26 GHz. The positioning system consists of a planar scanner equipped with a horizontal and a vertical slide, each 1.8 m long, driven by a control unit, and of a turntable installed over a manual slide orthogonal to the planar scanner. A 40 MHz–20 GHz Anritsu 37247C Vector Network Analyzer has been used for the far-field acquisitions. Lastly, the working frequency was 10 GHz, and a standard, X-band open ended waveguide probe, model MI-6970-WR90, has been employed as radiating element. B. Validation by the Antenna Pattern Comparison (APC) Method Fig. 16. Scattered field generation: cuts along the x-axis of the desired (circles) and synthesized (continuous line) field amplitudes over the QP. The vertical line denotes the QP boundary.

mutual coupling. A single radiator occupies, one after the other, all the established array positions, while its radiated QP field is acquired by a probe. Then, all the radiated QP fields are coherently summed according to the weights determined by the array excitation coefficients. In the following, after a brief description of the measurement setup, the experimental validation is performed using the antenna pattern comparison (APC) method [33], [34]. It entails using the designed echo generator to determine the far-field of a reference antenna whose far-field is already known by direct measurements. A. The Measurement Setup The measurements have been performed in the anechoic chamber designed and realized by MI Technologies for the Microwave and Millimeter-Wave Lab of the Dipartimento di

In order to reduce the measurement time, an echo generator, smaller than that considered for the foregoing numerical analysis, has been used for validation purposes. The considered QP was , while the radiating array comprised 49 elements, panel positioned at . With the occupying a aim of speeding up the implementation of the synthetic array, the element positions have been arranged in a “serpentine” sequence to minimize the movements of the positioner from one element to the next (see Fig. 18). To implement the APC method, a Narda Microwave Corporation standard gain pyramidal horn, model 640, has been used as Antenna Under Test (AUT). A reference H-plane far-field pattern of the considered AUT was already obtained in a different experiment by direct far-field measurements [33]. The AUT was installed on the turntable, so that it could be rotated within the H-plane. Starting from the “optimal” radiator locations and from their corresponding excitation coefficients, the measurement process proceeded by rotating the AUT in the within the interval [0 , 90 ], with generic angular position a sampling step of 6 , while the radiator was moved along all the positions in Fig. 18. The measurements were then processed

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Fig. 18. “Serpentine” raster-type scanning of the radiating elements.

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Fig. 20. Comparison between the far-field pattern obtained by the synthetic tilting of the plane wave (dots) and that following direct far-field measurements (dashed line).

one, when the synthetic tilting of the plane waves is applied. As . can be seen, good agreement can be achieved when VII. CONCLUSIONS AND FUTURE DEVELOPMENTS

Fig. 19. Comparison between the far-field pattern obtained by the echo generator (continuous line) and that following direct far-field measurements (dashed line).

using the array excitation coefficients to attain the AUT far-field pattern. Fig. 19 compares the far-field pattern obtained by the echo generator against the direct far-field measurements, showing good agreement. It is worth mentioning that, to reduce the measurement time needed to determine the AUT far-field pattern, the angular rotations of the AUT could be “synthetically” obtained (up to a limited extent) by synthesizing a plane wave propagating in a tilted needs direction. In this way, only a measurement set for to be collected. Obviously, when the QP is electrically small, as for the considered test case, the degrees of freedom of the field are “few”, and so the propagating directions of the plane waves that can actually be synthesized and that are not normal to the radiating array, are “few” as well. Fig. 20 refers to the ansized array of 121 elements, other example with a sized QP at also radiating at 10 GHz, and generating a . It compares the AUT far-field pattern to the reference

We have presented an approach for the synthesis of arraybased, 2D echo generators. The method allows the determination of the radiating panel size and exploits a singular value optimization approach to determine the number and the positions of the radiators and the QP grid points, where the design constraints are enforced. Furthermore, it employs a SVD procedure to determine the excitation coefficients of the radiating elements located within the panels. A proper strategy is also employed to provide reliable results and to control the computational burden. The procedure has been numerically and experimentally validated for the particular case of a single radiating panel and QP region, showing satisfactory results in terms of synthesized QP field amplitude and phase variations, robustness and computational cost. The method can be further improved through better circumvention of the local maxima trapping problem by adopting multistage approaches based on global optimization techniques [35]. It can be easily extended to account for a full vector, two-dimensional geometry and to cases involving multiple panels and disconnected QZs. Furthermore, the method is also of interest to applications in acoustics [36] and the capability of choosing the radiator positions provides further degrees of freedom that can be employed to improve the frequency-behavior of the echo generator in the band of interest, as long as time-domain echo generation is of interest. Also, constraints on the radiator and/or QP sampling point locations can be enforced [30], [31]. Finally, the perturbation theory of linear operators [37], [38] can be fruitfully exploited to improve the numerical effectiveness of the technique. REFERENCES [1] A. W. Rudge, K. Milne, A. D. Olver, and P. Knight, The Handbook of Antenna Design. London: Peter Peregrinus, 1982.

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[2] C. C. Courtney, D. E. Voss, R. Haupt, and L. LeDuc, “The theory and architecture of a plane-wave generator,” in Proc. 24th AMTA Symp., Cleveland, OH, Nov. 3–8, 2002, pp. 353–358. [3] M. E. Kowalski and J. M. Jin, “Model-based optimization of phased arrays for electromagnetic hyperthermia,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1964–1977, Aug. 2004. [4] Z. Ji, S. C. Hagness, J. H. Booske, S. Mathur, and M. L. Meltz, “FDTD analysis of gigahertz TEM cell for ultra-wideband pulse exposure studies of biological specimens,” IEEE Trans. Biomed. Eng., vol. 53, no. 5, pp. 780–789, May 2006. [5] M. Ciattaglia, A. De Luca, L. Infante, S. Mosca, and M. Albani, “Near field techniques for radar synthetic environment simulator,” in Proc. Int. Conf. on Electromagn. in Adv. Appl., Turin, Italy, Sep. 17–21, 2007, pp. 780–783. [6] R. Aitmehdi, “Scattered data transformation from cylindrical to plane wave illumination for quantitative diffraction tomography of cylindrical objects,” in IEE Proc. Pt. H, Jun. 1990, vol. 137, no. 3, pp. 160–162. [7] A. D. Olver, “Compact antenna test ranges,” in Proc. 7th Int. Conf. on Antennas Propag. (ICAP), York, UK, Apr. 15–18, 1991, pp. 99–108. [8] W. Menzel and B. Huder, “Compact range for millimetre-wave frequencies using a dielectric lens,” Electron. Lett., vol. 20, no. 19, pp. 768–769, Sep. 1984. [9] J. Hakli, T. Koskinen, A. Lonnqvist, J. Saily, V. Viikari, J. Mallat, J. Ala-Laurinaho, J. Tuovinen, and A. V. Räisänen, “Testing of a 1.5-m reflector antenna at 322 GHz in a CATR based on a hologram,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3142–3150, Oct. 2005. [10] D. A. Hill, “A circular array for plane-wave synthesis,” IEEE Trans. Electromagn. Compat., vol. 30, no. 1, pp. 3–8, Feb. 1988. [11] R. Haupt, “Generating a plane wave with a linear array of line sources,” IEEE Trans. Antennas Propag., vol. 51, no. 2, pp. 273–278, Feb. 2003. [12] A. Capozzoli and G. D’Elia, “On the plane wave synthesis in the nearfield zone,” in Proc. Int. Conf. on Antenna Tech., Ahmedabad, India, Feb. 23–24, 2005, pp. 273–277. [13] A. Capozzoli, C. Curcio, G. D’Elia, A. Liseno, and P. Vinetti, “A novel approach to the design of generalized plane-wave synthesizers,” presented at the 3rd Eur. Conf. on Antennas Propag., Berlin, Germany, Mar. 23–27, 2009, CD ROM. [14] P. Vinetti, “A non-invasive, near-field and very near-field phaseless antenna characterization system,” Ph.D. dissertation, Universitá di Napoli Federico II, Italy, Nov. 2008. [15] A. Capozzoli, C. Curcio, A. Liseno, and P. Vinetti, “Field sampling and field reconstruction: A new perspective,” Radio Sci. vol. 45, p. RS6004, 31 pages, 2010, DOI: 10.1029/2009RS004298. [16] M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging. Bristol, U.K.: Institute of Physics Publishing, 1998. [17] A. Capozzoli, C. Curcio, G. D’Elia, A. Liseno, G. Ianniello, and P. Vinetti, “Effective 2D generalised plane wave synthesizers: Experimental validation,” presented at the Proc. 30th Annual Antenna Measurement Tech. Association Symp., Salt Lake City, UT, Nov. 1–6, 2009, CD ROM. [18] K. G. Mills, T. Maxwell, E. C. Bergsagel, and R. K. Richardson, “Electromagnetic target generator,” U.S. Patent no. 5 892 479, Apr. 6, 1999. [19] K. S. Kim, “Tracking radar signal generator,” U.S. Patent no. 5 870 055, Feb. 9, 1999. [20] K. A. J. Warren, “Target detecting apparatus equipped with testing device for simulating targets at different ranges,” U.S. Patent no. 4 319 247, Mar. 9, 1982. [21] D. J. Grone, “Counter-based simulated target generator,” U.S. Patent no. 4 737 792, Apr. 12, 1988. [22] A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “On the sampling of electromagnetic fields,” presented at the URSI-B Int. Symp. on Electromagn. Theory, Berlin, Germany, Aug. 16–19, 2010. [23] M. Born and E. Wolf, Principles of Optics. Cambridge, U.K.: Cambridge Univ. Press, 1999. [24] B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progr. Opt., E. Wolf, Ed. Amsterdam: North-Holland, 1971, vol. 9, pp. 311–407. [25] A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Phaseless antenna characterization by effective aperture field and data representations,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 215–230, Jan. 2009.

[26] O. M. Bucci and G. D’Elia, “Advanced sampling techniques in electromagnetics,” in Review of Radio Sci. 1993–1996. London, U.K.: Oxford Univ. Press, 1996, pp. 177–204. [27] O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples,” IEEE Trans. Antennas Propag., vol. 46, no. 3, pp. 351–359, Mar. 1998. [28] R. L. Burden and J. D. Faires, Numerical Analysis. Belmont, CA: Thomson Brooks/Cole, 2005. [29] M. L. Overton and R. S. Womersley, “Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices,” Math. Progr., vol. 62, no. 2, pp. 321–357, Nov. 1993. [30] A. Capozzoli, C. Curcio, G. D’Elia, A. Liseno, and P. Vinetti, “FFT & equivalently tapered aperiodic arrays,” presented at the URSI General Assembly, Chicago, IL, Aug. 7–17, 2008, CD ROM. [31] A. Capozzoli, C. Curcio, G. D’Elia, A. Liseno, and P. Vinetti, “FFT & aperiodic arrays with phase-only control and constraints due to superdirectivity, mutual coupling and overall size,” presented at the 30th ESA Workshop on Antennas for Earth Observ., Science, Telecommun. and Navig. Space Missions, Noordwijk, The Netherlands, May 27–30, 2008, CD ROM. [32] “Planar NF/cylindrical NF/far field antenna measurement system,” Proposal CAR-0704-08, Revision A Feb. 2005, MI Technologies. [33] A. Capozzoli, C. Curcio, and G. D’Elia, “The indoor test range at the University of Naples Federico II: Characterization and antenna testing,” Atti Della Fondazione Ronchi, vol. LXII, no. 2, pp. 163–188, Mar.–Apr. 2007. [34] V. Viikari, J. Hakli, J. Ala-Laurinaho, J. Mallat, and A. V. Räisänen, “A feed scanning based APC technique for compact antenna test ranges,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3160–3165, Oct. 2005. [35] A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Power pattern synthesis of multifeed reconfigurable reflectarrays,” presented at the 29th ESA Antenna Workshop on Multiple Beams and Reconfigurable Antennas, Nordwijk, The Netherlands, Apr. 18–20, 2007, CD ROM. [36] D. De Vries and M. M. Boone, “Wave field synthesis and analysis using array technology,” in Proc. IEEE Workshop on Signal Proc. to Audio and Acoust., New York, Oct. 17–20, 1999, pp. 15–18. [37] T. Kato, Perturbation Theory for Linear Operators. Berlin-Heidelberg: Springer Verlag, 1995. [38] A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Gradient-based, singular value optimization in near-field measurements,” presented at the Int. Symp. on Antennas Propag., Toronto, Canada, Jul. 11–17, 2010, CD ROM.

Amedeo Capozzoli (M’00) graduated (summa cum laude) in electronic engineering and received the Ph.D. degree in electronic engineering and computer science from the University of Naples Federico II, in 1994 and 2000, respectively. Since January 2005, he has been an Associate Professor of electromagnetic fields at the University of Naples Federico II. His research interests include synthesis and diagnosis of radiating systems, inverse-scattering techniques, advanced measurement techniques, and the restoring of aberrations due to propagation through random media. Prof. Capozzoli was awarded the 1996 Telecom Italia Prize for the best degree thesis in electronic engineering discussed at the University of Naples Federico II in the Academic Year 1994–1995. In November 1999, he won the open competition for the post of Researcher at the University of Naples Federico II. In September 2002, he was awarded the Barzilai Prize for young scientists at the XIV Riunione Nazionale di Elettromagnetismo. In April 2003, he won the open competition for the post of Associate Professor at Politecnico di Milano. He was awarded the 2009 Antenna Measurement Techniques Association (AMTA) Best Technical Paper Award.

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Claudio Curcio received the Laurea degree (summa cum laude) in electronic engineering and the Ph.D. degree in electronic and telecommunication engineering from the Università di Napoli Federico II, Naples, Italy, in 2002 and 2005, respectively. From 2006 to 2007, he held a postdoctoral position at the University of Naples Federico II. He is currently a Researcher at the Università di Napoli Federico II. His main fields of interest are antenna measurements, phaseless near-field/far-field transformation techniques, optical beamforming techniques for array antennas, and reflectarray synthesis. Dr. Curcio was the recipient of the Optimus Award at the SIMAGINE 2002 “Worldwide GSM & Java Card Developer Contest.” He was awarded the 2009 Antenna Measurement Techniques Association (AMTA) Best Technical Paper Award.

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Angelo Liseno was born in Italy in 1974. He received the Laurea (summa cum laude) and Ph.D. degrees in electrical engineering from the Seconda Università di Napoli, Napoli, Italy, in 1998 and 2001, respectively. From 2001 to 2002, he held a postdoctoral position at the Seconda Università di Napoli. From 2003 to 2004, he was a Research Scientist with the Institut für Hochfrequenztechnik und Radarsysteme, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Oberpfaffenhofen, Germany. Since 2005, he has been a Researcher with the Dipartimento di Ingegneria Biomedica, Elettronica e delle Telecomunicazioni, Università di Napoli Federico II, Napoli. His main fields of interest are phaseless near-field/far-field transformation techniques, antenna synthesis, remote sensing, and inverse scattering. Dr. Liseno was awarded the 2009 Antenna Measurement Techniques Association (AMTA) Best Technical Paper Award.

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Synthesis of Sub-Arrayed Time Modulated Linear Arrays Through a Multi-Stage Approach Paolo Rocca, Member, IEEE, Lorenzo Poli, Student Member, IEEE, Giacomo Oliveri, Member, IEEE, and Andrea Massa, Member, IEEE Abstract—The synthesis of sub-arrayed time-modulated linear arrays is addressed in this paper by means of a multi-stage approach. An efficient algorithm based on an excitation matching strategy is used at the first stage to define the “best compromise” sub-array configuration. Successively, the digital pulse sequence controlling a set of RF switches devoted to enforce the time-modulation at the sub-array level is optimized by means of a stochastic global optimizer. Selected results are reported and compared with state-of-the-art solutions to show potentialities/advantages and limitations/disadvantages of the proposed approach. Index Terms—Antenna arrays, array synthesis, linear arrays, sub-arrayed antennas, time-modulated arrays.

I. INTRODUCTION

T

HE simplification of the antenna architecture is an issue of great interest in array synthesis. Indeed, it has a key role on the manufacturing costs and on the power requirements of the beam forming network (BFN) thus enabling/preventing the use of arrays in several practical applications. Furthermore, it is worthwhileto pointout thatthepossibilityofreconfiguringtheradiation pattern just modifying a reduced number of control elements enables the synthesis of antennas with enhanced performances as well as the proliferation of innovative wireless services. On one hand, the use of sub-arrays among suitable alternatives is a well-assessed solution to reduce the hardware complexity of the antenna system. As a matter of fact, several papers have been published on this subject in the reference literature (see [1]–[12] and the references quoted therein). On the other hand, a renewed interest has been recently shown towards time-modulated arrays able to arbitrarily shape the radiation patterns by using on-off switches [13]–[25]. In such a framework, the design of array architectures characterized by simplicity and reconfigurability is addressed in this paper by means of a hybrid approach. Unlike other approaches where the elements of the array are individually modulated [13]–[24], the synthesis is concerned with sub-arrayed time-modulated devices where a set of radio-frequency (RF) switches is used, at the sub-array level, to control the excitations of the array aperture. Accordingly, the problem Manuscript received August 05, 2009; revised December 03, 2010; accepted March 09, 2011. Date of publication July 14, 2011; date of current version September 02, 2011. The authors are with the Department of Information Engineering and Computer Science, University of Trento, 38123 Trento, Italy (e-mail: paolo. [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161535

deals with the definition of two sets of unknowns, describing the sub-array configuration and the on-off pulse sequence, such that (a) the antenna radiates a desired pattern at the working (carrier) frequency and (b) the power losses and the interferences caused by the harmonic radiations are reduced as much as possible. Instead of simultaneously optimizing the whole set of unknowns using a computationally-expensive global search algorithm, the two set of unknowns are here determined by solving three different and simpler problems. More specifically, the aggregation of the elements into sub-arrays is determined by means of a customized [to the synthesis of single-beam time-modulated linear arrays (TMLAs)] version of the border element method (BEM) [12], [26] because of its computational effectiveness. As a matter of fact, although sub-arraying problems (e.g., the compromise between sum and difference patterns [4] or the synthesis of a single-beam array antenna [9]) have been effectively solved by means of genetic algorithms (GAs), the differential evolution (DE) [5], [10], and the simulated annealing (SA) [8], global optimization strategies generally require a huge amount of computational resources since the number of admissible sub-array configurations grows exponentially with the number of array elements. Although the BEM for TMLAs also gives an estimate of the pulse sequence modulating the static array excitations, the losses in the sideband radiation (SR) [27] could severely affect the performance and the reliability of the antenna. As a matter of fact, the shift of the power radiated by the antenna to SR due to the periodic on-off commutation of the RF switches causes a reduction of the directivity and the gain of the array at the working frequency [28]–[31]. Therefore, once the sub-array configuration has been determined, the time sequence controlling the sub-array switches is optimized by means of a particle swarm optimizer (PSO) [32]. Unlike [13], [14], [25], [33], [35] where the energy wasted in the SR is indirectly reduced through the minimization of the sideband levels (SBLs)1 with non-negligible computational costs2, the closed-form relationship yielded in [27] for quantifying the total power losses is here exploited. Finally, in order to uniformly spread the power content of the SR within the whole visible angular range, the SBLs are minimized acting on the switch-on instants [24], [36]. The paper is organized as follows. The problem is mathematically formulated in Section II and the three-stage synthesis approach is described (Section II-A). A set of numerical results 1Namely, the peak level of the harmonic patterns with respect to the maximum value of the field radiated at the carrier frequency. 2As a matter of fact, the minimization of SR through the reduction of SBLs needs the generation of the corresponding harmonic patterns (ideally infinite). The process turns out being cumbersome and it is often restricted to a few (i.e., the first ones) harmonic radiations.

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is reported and discussed in Section III to point out potentialities and limitations of the proposed method. Comparisons with state-of-the-art solutions are also presented. Finally, some concluding remarks are drawn (Section IV). II. MATHEMATICAL FORMULATION Let us consider a linear array of elements equally-spaced ) RF switches are used to moduby . A set of (with late the “static” sub-arrays of the antenna through periodic pulse sequences (Fig. 1). Dealing with isotropic radiators, the array pattern is given by [28]

Fig. 1. Sketch of the antenna system.

pattern related to the SR (1) carrier angular frequency, is the set of static excitations, is the free-space wavenumber, being the is the angle measured from the direction wavelength, is the Kronecker delta orthogonal to the array axis, and if , otherwise). The set function (i.e., , describes of integer values, the grouping of the array elements into sub-arrays where means that the th element belongs to the th sub-array is used to identify those elements which are not and is time-modulated. For instance, . the aggregation vector for the array of Fig. 1 where , model the on-off The periodic functions states of the RF switches enforcing the time-modulation at for and the sub-array level: elsewhere, and being the switch-on (short circuit state) and the switch-off (open circuit state) instants, respectively. is a periodic function, it can be expressed in terms Since of the corresponding Fourier series [27] where

is

the

(2) where are

is the modulation period and the Fourier coefficients

(3) By substituting (2) into (1), the radiation pattern turns out being composed by an infinite number of harmonic components [27]. The pattern at the carrier frequency is then given by (4) where the coefficients , (where ) are computed through (3) by setting . Moreover, the summation of the harmonic terms at multiple of , determines the array the modulating angular frequency,

(5) A. Sub-Arrayed TMLAs Synthesis In order to radiate a desired pattern at the carrier frequency and to minimize the SR by means of a sub-arrayed TMLA architecture, the unknowns to be optimized are the sub, and the pulse array configuration, . sequence, Since we are dealing with the generation of a single beam, , [9] such the static excitations are set to (4) directly depends on the switch-on that the pattern at , , . Because of the times, use of sub-arrays, the values of the switch-on times are a stepwise approximation of reference coefficients (i.e., the weights derived from known analytical procedures—Chebyshev, Taylor, or Bayliss [37]—and exactly synthesizing a reference pattern). Therefore, the problem at hand consists in searching the “best” stepwise approximation of a (normalized) reference distribution of coefficients using values. Moreover, the problem related to the generation of undesired sideband radiations must be carefully taken into account because of the trade-off between the possibility of arbitrarily and the amount and shaping the radiation pattern at “distribution” of the losses in the SR. To address these issues, the following three-stage procedure is proposed. It is aimed at synthesizing a desired pattern at the carrier frequency (Stage 1 and Stage 2) while keeping low the power in the SR (Stage 2) by setting the time-pulse durations . Successively, the SBLs are minimized by spreading the power of the harmonic radiations as uniformly as possible (Stage 3) thanks to the setup of the switch-on times of the modulation pulses. As far as the optimization of Stage 2 and Stage 3 is concerned, the PSO is used. Although potentially the two sets of unknowns and can be jointly optimized, the motivation of splitting the problem in two sub-problems is twofold. On one hand, changes of do not influence the amount of power in the SR in case of conventional array antenna syntheses. Therefore, the two sets can be considered as independent and are employed

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to minimize two different metrics (i.e., the SR and the SBLs, respectively). On the other hand, since the dimension of the solution space grows exponentially with the number of parameters to be optimized [34], reducing the number of unknowns greatly improves the computational efficiency of the approach also limiting the probability of being trapped into local minima of the functional at hand. • Stage 1—Definition of the Sub-array Architecture— The first stage exploits the guidelines of the excitation matching procedure presented in [26] and here customized to the synthesis of time-modulated arrays. The elements , of a set of (real) excitations, affording a desired/reference pattern, are sorted on a line sub-sets to define a trial successively subdivided into . The sub-array configuration, positions of the cut points along the line are iteratively ( being the iteration index) updated by means of the Border Element Method [26] applied to minimize the mismatch between the reference distribution of coefficients and its stepwise approximation

by means of the PSO starting from the solution obtained and ). Moreover, at the previous stage (i.e., , , and are real weights and is the total power radiated by the array [27]. It is worth to notice that the last two terms in (9), although not related to the sideband radiation, act like constraints to keep the pattern at the carrier frequency, which is also a function of the set (4), close to the reference one. The values of the switch-on times synthesized at this stage are denoted by as ; • Stage 3—Side-Band Level Control— Although the optimization of (9) accounts the minimization of the power losses at the undesired harmonics , it does not allow to control the angular power distribution of the SR. Towards this end, additional degrees of or , ) in defining the freedom (i.e., have been profitably exploited while pulse sequence (i.e., ) and keeping unaltered (i.e., and ). As a matter of fact, the harmonic coefficients (3) of the th modulating , can be expressed as function,

(6) (10) , In (6), the normalized switch-on times, , are analytically computed [26] for each trial , as follows sub-array configuration, (7) , , is the number of array elements where belonging to the th sub-array. The iterative process is stopped after an heuristically-determined maximum number of iterations, , or when the stationary condition [26] on the value of given by , and being user-defined parameters, is satisfied. and Let us indicate with the values determined at the convergence; • Stage 2—Sideband Radiation Control— Since the power losses due to SR amount to [27]

(8)

, Thus, once the switch-on times are fixed, only depends on the array pattern [12], [36]. Accordingly, the SBLs at the harmonic frequencies can be optimized by solving the following minimization problem (11) where value, and SBL

,

, is a fixed integer , is computed as follows

(12) being the pattern radiated at the th harmonic an. gular frequency, As regards the optimizer, a suitable version of the PSO [38], [39] is used because of (a) the simplicity of its implementation, (b) the ability to deal with real unknowns [32], and (c) the stability (to the parameter setting) and the effectiveness demonstrated in several and similar optimization problems [40]. III. NUMERICAL RESULTS

where if and , otherwise, the reduction of the SR is forced by minimizing

(9)

In order to show the potentialities and the limitations of the proposed approach, two benchmark experiments are considered. More specifically, they concern with the synthesis of a difference pattern and a sum pattern previously dealt with in [14] and [18], respectively. In the reference examples [14], [18], the antenna elements were individually time-modulated and the pulse sequence controlling the radiated power pattern was defined through a stochastic optimizer (namely, the SA). As for the proposed method, 10 simulations have been carried out for each experiment any time the PSO was used. The simulation whose performance was closest to the average performance value has

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TABLE I SUB-ARRAY CONFIGURATIONS

M = 15;G = 4)—SA pulse sequence

Fig. 2. Difference Pattern Synthesis ( [18].

been considered as representative result and reported within the paper. However, the statistics (i.e., minimum, maximum, average, and variance values) of representative pattern features are reported to be more definite about the performance of the PSO-based procedure. As far as the first experiment is concerned, the array is comelements uniformly-spaced by . posed by Starting from such an array geometry, the difference pattern has been synthesized in [18] by considering a symmetric , array with uniform and unitary static excitations (i.e., ) just modulating the 8 central elements (Table I, second row) according to the pulse sequence shown in Fig. 2. In order to perform a fair comparison, a symmetric sub-arrayed TMLA has been designed too and the synthesis process has been constrained to half structure defining aggregations (i.e., 4 switches) over elements. Concerning the set of reference excitations, , a distribution affording a modified-Zolotarev difference pattern [41] with and has been chosen. Such a choice has been motivated by the need to generate a pattern with the same sidelobe level of the solution achieved in [18] with a reduced dynamic range ratio to simplify the hardware realization [42]. The results at the convergence of the BEM (Stage 1) are summarized in Fig. 3 and Table I. The power patterns at the carrier and at the first two harmonics frequency as well as the synthesized pulse sequence are shown in Fig. 3(a) and (b)3, respectively. As it can be observed [Fig. 3(b)], , , the switch-on instants are fixed to

 

Q n ; ;N

3In Figs. 3(b), 5(b), 7(b), and 9(b) there are + 1 different pulse duration = 1, = 1 . . . , are not time-modulated values. The elements with = 0). and therefore they do not belong to any sub-array of the BFN (i.e.,

c

M

h

;G

h

Fig. 3. Difference Pattern Synthesis ( = 15 = 4)—Stage 1. Plots of (a) the normalized power patterns generated at = 0 and = 1, 2 with the BEM and (b) the pulse sequence, .



since, at this stage, they are not degrees of freedom of the optimization. The pattern indexes in Table II confirm that the BEM presents a sidelobe level and a main compromise at lobe beamwidth (BW) close to those of the reference pattern versus SLL and (SLL versus BW ), but the BW power losses are significantly higher than those of the SA solu). tion (i.e., To reduce the SR, the cost function in (9) has been then minimized (Stage 2) by means of the PSO. As far as the PSO control parameters are concerned, the following setup has been chosen [30], [39]: , , . A swarm of and particles has been used and the iterative process has been . Moreover, the simulations have been stopped at

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M

;G = 4)—P

TABLE II DIFFERENCE PATTERN SYNTHESIS ( = 15

M = 15;G = 4)—S

ATTERN INDEXES

TABLE III

DIFFERENCE PATTERN SYNTHESIS (

h

M

TATISTICS OF THE

PATTERN FEATURES OF THE PSO SIMULATIONS

;G

Fig. 4. Difference Pattern Synthesis ( = 15 = 4)—Stage 2. Normalized power patterns at = 0 for the SA [18] and the BEM - PSO.

carried out on a 3 GHz PC with 1 GB of RAM with an average CPU-time of about 0.576 [sec] for each iteration. Besides the representative solution discussed in the following, the , and maximum SBL for the statistics of the SLL, BW, solutions achieved by the repeated PSO simulations are given in Table III. Fig. 4 shows the plots of the patterns at the carrier frequency for both the PSO-based approach and that in [18]. The losses due to SR turn out to be three times lowered from down to , despite the pattern features synthesized at the Stage 1 are still kept unaltered (Table II). It is also worthwhile to notice [Fig. 5(a)] that, although not directly and involved in the optimization process, the SBLs at have been reduced of about 8 dB and 6 dB, respectively. For completeness, the SBLs of the first harmonic terms have been reported in Fig. 6 and compared with those coming from the BEM and the ones obtained in [18], as well. As regards the architecture complexity, the BFN is now further are switched-off and simplified since 3 elements over —Table I, forth row). Moreover, can be discarded (i.e., is enough for each half of the array only a sub-array , the third stage of the proposed [Fig. 5(b), Table I]. Since

M

h

;G

h

Fig. 5. Difference Pattern Synthesis ( = 15 = 4)—Stage 2. Plots of (a) the normalized power patterns generated at = 0 and = 1, 2 with the BEM - PSO and (b) the BEM - PSO pulse sequence.

approach has not been performed. As a matter of fact, only the switch-on instant has to be set and its value has no influence on the radiated patterns. The second experiment concerns with the synthesis of a sum elements. The number of pattern and an antenna of

ROCCA et al.: SYNTHESIS OF SUB-ARRAYED TIME MODULATED LINEAR ARRAYS THROUGH A MULTI-STAGE APPROACH

h

;

M

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;G

Fig. 6. Difference Pattern Synthesis ( = 15 = 4). Sideband levels SBL , 2 [1 25], for the reference solution [14] and the solutions at the end of the first stage (BEM) and the second stage (BEM - PSO).

sub-arrays has been chosen equal to (Table I) and the , , reference excitations have been set to being the th normalized switch-on time computed through the SA [14]. Fig. 7(a) shows the pattern generated by the pulse sequence (Stage 1) in Fig. 7(b) and the reference one synthesized in [14]. As it can be observed, although the asymmetric distribution of the pulse distribution of Fig. 7(b), which is caused by , the asymmetries of the reference pulse distribution , the generated sum pattern is symmetric and directed at broadside [Fig. 7(a)] because the equivalent element excita, ) of the array factor tions (i.e., AF are real. As for the compromise pattern, it almost perfectly matches the reference beam as also confirmed by the values and BW in Table IV. The sidelobe level inof SLL versus SLL creases of just 0.3 dB ( ), while the main beamwidth is slightly narrower ( versus BW ). The directivity, computed according to [29], has been slightly improved versus ). Moreover, a ( smaller amount of power is wasted in the SR ( versus ) and the value of SBL is reduced of 0.7 dB. It is also worth noting that the BEM solution reduces the number of switches to be used (Table IV). It is three times smaller than that required by the TMLA architecture obtained in [14] with a non-negligible advantage in terms of costs and complexity of the BFN. In order to further minimize the SR (Stage agents has been then used to optimize 2), a PSO with the switch-on times at the sub-array level. The same PSO parameters of the previous experiment have been used and itthe minimization of (9) has been terminated after erations in 33.3 [sec]. The obtained results are summarized in decreases (Table IV) Table IV and in Fig. 8. As expected, only slightly modifies [Fig. 8 versus and the radiation at Fig. 7(a)]. Successively (Stage 3), the optimization of the switch-on instants, , is taken into account to minimize the SBLs. The number of harmonics under analysis has been limited to since (i) the energy wasted in SR is relevant only at the first harmonics and (ii) a smaller value of saves computational time.

N

h

;Q

Fig. 7. Sum Pattern Synthesis ( = 30 = 3)—Stage 1. Plots of the normalized power patterns generated at = 0 with the SA [14] and the BEM (a). Pulse sequence determined at the end of the BEM optimization (b).

h

N

;Q

Fig. 8. Sum Pattern Synthesis ( = 30 = 3)—Stage 2. Normalized power patterns at = 0, 1, 2 synthesized with the BEM - PSO.

The patterns radiated in correspondence with the BEM - PSO solution obtained at the convergence after 28.04 [sec] as well as the optimized pulse sequence are reported in Fig. 9(a) and (b), respectively. It is worth noting how the power content of the SR is uniformly spread in the visible range thus lowering the SBL value (Table IV). As a matter of fact, it turns out that SBL with a non-negligible improvement (taking into account the small number of , and that only degrees of freedom in the optimization, ) compared to the solution at the the 29th element has

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TABLE IV

N = 30; Q = 3)—SWITCHES NUMBER AND PATTERN INDEXES

SUM PATTERN SYNTHESIS (

TABLE V

N = 30;Q = 3)—STATISTICS OF THE PATTERN FEATURES OF THE PSO SIMULATIONS

SUM PATTERN SYNTHESIS (

N

;Q

Fig. 10. Sum Pattern Synthesis ( = 30 , = 3). Sideband levels SBL 2 [1 25], for the [14] and the solutions at the end of the first stage (BEM), the second stage (BEM - PSO), and the third stage (BEM - PSO ).

h

;

SBLs are reported in Fig. 10 with the corresponding values from [14]. For the sake of completeness, the statistics of the pattern features for the whole set of PSO solutions synthesized at the Stage 2 and the Stage 3 are given in Table V.

N

;Q

Fig. 9. Sum Pattern Synthesis ( = 30 = 3)–Stage 3. Plots of (a) the normalized power patterns generated at = 0 and = 1, 2 with the BEM PSO and (b) the corresponding pulse sequence.

second stage (i.e., SBL with in [14] (i.e., SBL

h

h

) and the one ). Finally, the obtained

IV. CONCLUSIONS In this paper, an innovative approach to the synthesis of time-modulated sub-arrayed antennas has been presented and assessed. The potentialities of the sub-arraying strategy in simplifying the BFN architecture as well as the advantages offered by the time-modulation technique of reconfiguring the pattern just acting on a digital pulse sequence have been combined. At the first stage, the effectiveness and computational efficiency of the BEM have been exploited to determine the “best compromise” sub-array configuration. Successively, the PSO has been used to optimize the pulse sequence for minimizing both the power losses in the SR and the maximum levels of the harmonic

ROCCA et al.: SYNTHESIS OF SUB-ARRAYED TIME MODULATED LINEAR ARRAYS THROUGH A MULTI-STAGE APPROACH

radiations. The obtained results have proved the reliability of the proposed technique in designing simple and efficient array architectures. REFERENCES [1] R. L. Haupt, “Reducing grating lobes due to subarray amplitude tapering,” IEEE Trans. Antennas Propag., vol. 33, no. 8, pp. 846–850, Aug. 1985. [2] A. P. Goffer, M. Kam, and P. Herczfeld, “Design of phased arrays in terms of random subarrays,” IEEE Trans. Antennas Propag., vol. 42, no. 6, pp. 820–826, Jun. 1994. [3] U. R. O. Nickel, “Monopulse estimation with subarray output adaptive beamforming and low sidelobe sum and difference beams,” in Proc. IEEE Int. Symp. Phased Array Syst. Tech., Boston, MA, Oct. 1996, pp. 283–288. [4] P. Lopez, J. A. Rodriguez, F. Ares, and E. Moreno, “Subarray weighting for difference patterns of monopulse antennas: Joint optimization of subarray configurations and weights,” IEEE Trans. Antennas Propag., vol. 49, no. 11, pp. 1606–1608, Nov. 2001. [5] S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, “Optimization of the difference patterns for monopulse antennas by a hybrid real/integercoded differential evolution method,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 372–376, Jan. 2005. [6] N. Toyama, “Aperiodic array consisting of subarrays for use in small mobile earth stations,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 2004–2010, Jun. 2005. [7] R. J. Mailloux, S. G. Santarelli, and T. M. Roberts, “Wideband arrays using irregular (polyomino) shaped subarrays,” Electron. Lett., vol. 42, no. 18, pp. 1019–1020, 2006. [8] M. D’Urso and T. Isernia, “Solving some array synthesis problems by means of an effective hybrid approach,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 750–759, Mar. 2007. [9] R. L. Haupt, “Optimized weighting of uniform subarrays of unequal size,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1207–1210, Apr. 2007. [10] Y. Chen, S. Yang, and Z. Nie, “The application of a modified differential evolution strategy to some array pattern synthesis problems,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 1919–1927, Jul. 2008. [11] R. J. Mailloux, S. G. Santarelli, T. M. Roberts, and D. Luu, “Irregular polyomino-shaped subarrays for space-based active arrays,” Int. J. Antennas Propag., pp. 9–9, 2009. [12] L. Manica, P. Rocca, and A. Massa, “Design of subarrayed linear and planar array antennas with SLL control based on an excitation matching approach,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1684–1691, Jun. 2009. [13] S. Yang, Y. B. Gan, and P. K. Tan, “A new technique for power-pattern synthesis in time-modulated linear arrays,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 285–287, 2003. [14] J. Fondevila, J. C. Brégains, F. Ares, and E. Moreno, “Optimizing uniformly excited linear arrays through time modulation,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 298–301, 2004. [15] S. Yang, Y. B. Gan, A. Qing, and P. K. Tan, “Design of a uniform amplitude time modulated linear array with optimized time sequences,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2337–2339, Jul. 2005. [16] S. Yang and Z. Nie, “Time modulated planar arrays with square lattices and circular boundaries,” Int. J. Numer. Model, vol. 18, pp. 469–480, 2005. [17] S. Yang, Z. Nie, and F. Yang, “Synthesis of low sidelobe planar antenna arrays with time modulation,” in Proc. APMC 2005 Asia-Pacific Microw. Conf., Suzhou, China, Dec. 4–7, 2005, vol. 3, pp. 3–3. [18] J. Fondevila, J. C. Brégains, F. Ares, and E. Moreno, “Application of time modulation in the synthesis of sum and difference patterns by using linear arrays,” Microw. Opt. Technol. Lett., vol. 48, no. 5, pp. 829–832, May 2006. [19] Y. Chen, S. Yang, and Z. Nie, “Synthesis of satellite footprint patterns from time-modulated planar arrays with very low dynamic range ratios,” Int. J. Numer. Model., vol. 21, pp. 493–506, 2008. [20] Y. Chen, S. Yang, and Z. Nie, “Synthesis of optimal sum and difference patterns from time-modulated hexagonal planar arrays,” Int. J. Infrared Milli. Waves, vol. 29, pp. 933–945, 2008. [21] A. Tennant and B. Chambers, “Time-switched array analysis of phaseswitched screens,” IEEE Trans. Antennas Propag., vol. 57, no. 3, pp. 808–812, Mar. 2009. [22] G. Li, S. Yang, and Z. Nie, “A study on the application of time modulated antenna arrays to airborne pulsed Doppler radar,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1578–1582, May 2009.

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[23] P. Rocca, L. Poli, G. Oliveri, and A. Massa, “Synthesis of time-modulated planar arrays with controlled harmonic radiations,” J. Electromagn. Waves Appl., vol. 24, pp. 827–838, 2010. [24] L. Poli, P. Rocca, L. Manica, and A. Massa, “Handling sideband radiations in time-modulated arrays through particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1408–1411, Apr. 2010. [25] P. Rocca, L. Manica, L. Poli, and A. Massa, “Synthesis of compromise sum-difference arrays through time-modulation,” IET Radar Sonar Navig., vol. 3, no. 6, pp. 630–637, Dec. 2009. [26] L. Manica, P. Rocca, A. Martini, and A. Massa, “An innovative approach based on a tree-searching algorithm for the optimal matching of independently optimum sum and difference excitations,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 58–66, Jan. 2008. [27] J. C. Brégains, J. Fondevila, G. Franceschetti, and F. Ares, “Signal radiation and power losses of time-modulated arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1799–1804, Jun. 2008. [28] W. H. Kummer, A. T. Villeneuve, T. S. Fong, and F. G. Terrio, “Ultra-low sidelobes from time-modulated arrays,” IEEE Trans. Antennas Propag., vol. 11, no. 6, pp. 633–639, Jun. 1963. [29] S. Yang, Y. B. Gan, and P. K. Tan, “Evaluation of directivity and gain for time-modulated linear antenna arrays,” Microw. Opt. Technol. Lett., vol. 42, no. 2, pp. 167–171, Jul. 2004. [30] L. Manica, P. Rocca, L. Poli, and A. Massa, “Almost time-independent performance in time-modulated linear arrays,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 843–846, 2009. [31] L. Poli, P. Rocca, G. Oliveri, and A. Massa, “Harmonic beamforming in time-modulated linear arrays,” IEEE Trans. Antennas Propag.. [32] J. Kennedy, R. C. Eberhart, and Y. Shi, Swarm Intelligence. San Francisco, CA: Morgan Kaufmann, 2001. [33] S. Yang, Y. B. Gan, and A. Qing, “Sideband suppression in time-modulated linear arrays by the differential evolution algorithm,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 173–175, 2002. [34] A. S. Nemirovsky and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization. New York: Wiley, 1983. [35] S. Yang, Y. B. Gan, A. Qing, and P. K. Tan, “Design of a uniform amplitude time modulated linear array with optimized time sequences,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2337–2339, Jul. 2005. [36] P. Rocca and A. Massa, “Innovative approaches for optimized performance in time-modulated linear arrays,” presented at the IEEE AP-S Int. Symp., Charleston, SC, Jun. 1–5, 2009. [37] R. S. Elliott, Antenna Theory and Design. New York: Wiley-Interscience, IEEE Press, 2003. [38] A. Massa, R. Azaro, M. Donelli, M. Benedetti, and P. Rocca, “Complex synthesis of antenna structures through evolutionary-optimization techniques,” presented at the 10th Int. Conf. on Electromagnetics in Advance Appl. (ICEA 07), Torino, Italy, Sep. 17–21, 2007. [39] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Prob., vol. 25, pp. 1–41, 2009. [40] M. Donelli and A. Massa, “Computational approach based on a particle swarm optimizer for microwave imaging of two-dimensional dielectric scatterers,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1761–1776, May 2005.  —Distributions for difference patterns,” [41] D. A. McNamara, “Discrete n Electron. Lett., vol. 22, no. 6, pp. 303–304, Jun. 1986. [42] R. J. Mailloux, Phased Array Antenna Handbook, 2nd ed. Boston, MA: Artech House, 2005.

Paolo Rocca (M’08) received the M.S. degree in telecommunications engineering and the Ph.D. degree in information and communication technologies from the University of Trento, Trento, Italy, in 2005 and 2008, respectively. He is currently an Assistant Professor at the Department of Information Engineering and Computer Science and a member of the ELEDIA Research Group, University of Trento. He has been a visiting student at the Pennsylvania State University and at the University “Mediterranea” of Reggio Calabria. His main interests are in the framework of antenna array synthesis and design, electromagnetic inverse scattering, and optimization techniques for electromagnetics. Dr. Rocca received the best Ph.D. thesis award from the IEEE-GRS Central Italy Chapter.

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Lorenzo Poli (S’08) received the M.S. degree in telecommunication engineering from the University of Trento, Italy, in 2008, where he is currently working toward the Ph.D. degree. He is currently a member of the ELEDIA Research Group, University of Trento. His main interests are in the framework of antenna array synthesis and electromagnetic inverse scattering.

Giacomo Oliveri (M’09) received the M.S. degree in telecommunication engineering and the Ph.D. degree in space science and engineering from the University of Genoa, Italy, in 2005 and 2009, respectively. He is currently a member of the ELEDIA Research Group, University of Trento. His research is mainly focused on antenna arrays, electromagnetic propagation in complex environments and numerical methods for electromagnetic problems.

Andrea Massa (M’96) received the “Laurea” degree in electronic engineering and Ph.D. degree in electronics and computer science from the University of Genoa, Genoa, Italy, in 1992 and 1996, respectively. From 1997 to 1999, he was an Assistant Professor of electromagnetic fields at the Department of Biophysical and Electronic Engineering (University of Genoa) teaching the university course of Electromagnetic Fields 1. From 2001 to 2004, he was an Associate Professor at the University of Trento. Since 2005, he has been a Full Professor of electromagnetic fields at the University of Trento, where he currently teaches electromagnetic fields, inverse scattering techniques, antennas and wireless communications, and optimization techniques. At present, he is the Director of the ELEDIALab, University of Trento and Deputy Dean of the Faculty of Engineering. His research work since 1992 has been principally on electromagnetic direct and inverse scattering, microwave imaging, optimization techniques, wave propagation in presence of nonlinear media, wireless communications and applications of electromagnetic fields to telecommunications, medicine and biology. Prof. Massa is a member of the PIERS Technical Committee, the Inter-University Research Center for Interactions Between Electromagnetic Fields and Biological Systems (ICEmB) and the Italian representative in the general assembly of the European Microwave Association (EuMA).

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A Transmit-Receive Reflectarray Antenna for Direct Broadcast Satellite Applications Jose A. Encinar, Fellow, IEEE, Manuel Arrebola, Member, IEEE, Luis F. de la Fuente, and Giovanni Toso, Senior Member, IEEE

Abstract—A 1.2-meter reflectarray antenna has been designed to accomplish the requirements of a Direct Broadcast Satellite (DBS) mission, which provides a South America transmit-receive coverage in Ku band. The antenna has been designed by applying first of all a pattern synthesis technique to obtain the required phase distribution on the reflectarray at several frequencies in transmit and receive bands. Then, the patch dimensions have been optimized in a configuration made of three stacked layers of varying-sized patches in order to provide the required phase distribution at transmit and receive frequencies. An antenna demonstrator has been manufactured and tested. The measured patterns are in good agreement with the simulations and they are close to fulfill the coverage requirements in both transmit and receive bands. Index Terms—DBS antennas, reflectarray, satellite antennas, transmit-receive antennas.

I. INTRODUCTION HE design requirements of spacecraft antennas for satellite broadcast and telecommunication missions are becoming extremely stringent. In particular they may include highly shaped contoured beams to efficiently illuminate the required geographical area, dual polarizations with very low levels of cross polarization, co-polar isolation in other geographical regions, in order to avoid interference, and transmit-receive (Tx-Rx) operation. Even if in some missions the stringent requirements recommend using two separate antennas for the transmit (Tx) and for the receive (Rx) link [1], the use of a single transmit-receive antenna is very attractive, because of the reduction in volume, mass and costs. Currently, shaped reflectors are satisfactorily used in many missions [2] to provide all the requirements of coverage, cross-polarization and isolation in both transmit and receive frequency bands, which are separated more than 20% in Ku band. However,

T

Manuscript received September 17, 2010; revised January 10, 2011; accepted January 15, 2011. Date of publication July 07, 2011; date of current version September 02, 2011. This work was supported in part by the European Space Agency (ESA) under contract ESTEC/19571/06/NL/JA and in part by the Spanish Ministry of Science and Innovation, under the project: CICYT TEC 2007-63650/TCM and TEC2010-17567. J. A. Encinar is with Universidad Politécnica de Madrid, E-28040 Madrid, Spain (e-mail: [email protected]). M. Arrebola is with Universidad de Oviedo, E-33203 Gijón, Spain (e-mail: [email protected]). L. F. de la Fuente is with EADS-CASA Espacio, E-28022 Madrid, Spain, (e-mail: [email protected]). G. Toso is with the European Space Agency ESA-ESTEC, 2200 AG Noordwijk, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161449

the main disadvantage of reflector antennas is the manufacturing of a specific mold for the shaped reflector, which depends on the antenna requirements and therefore cannot be reused for other missions, with an associated impact on the cost of the antenna and the manufacturing time. On the other hand, reflectarray antennas do not require any mold to be manufactured, because they are made using a flat panel. In addition, for a specified coverage only the dimensions of printed patches or lines are varied [3] and not the structural panel. Asa consequence, the mechanical models and tests can be reused for different missions. Contoured beam reflectarrays made of one [4] and three [5], [6] layers of varying-sized patches have been successfully used for Direct Broadcast Satellite (DBS) applications. Although the narrow bandwidth is a severe limitation for reflectarray antennas [7], [8], it was demonstrated in previous works [5], [9] that the bandwidth in large reflectarrays can be improved by stacking three layers of rectangular patches and optimizing the patch dimensions to compensate the spatial phase delay in the working frequency band. Using this technique, a 1-meter reflectarray breadboard has been designed in a 10% bandwidth for a DBS transmit antenna, providing a European coverage in H-polarization and a North-America coverage in V-polarization [6]. The reflectarray antenna has been manufactured and tested using space-proven technologies. The measured radiation patterns demonstrated that the coverage requirements are practically fulfilled in a 10% bandwidth for both polarizations. This bandwidth is sufficient for either a transmit or receive antenna, but for a transmit-receive antenna the design should be extended to dual frequency operation. Several concepts have been proposed for reflectarray antennas operating at two different frequencies [10]–[14]. A reflectarray was proposed for operation at 5 GHz and 20 GHz [10] using in the same unit cell four printed patches working at 20 GHz on top of one element designed for 5 GHz. Recently, a reflectarray was proposed to work at X and Ka bands using the Ka-band reflectarray with an FSS acting as ground plane above the X-band reflectarray [11]. Another alternative is to place the low-frequency elements on the top of the high-frequency elements, as demonstrated in [12], [13] for operation at X (or C) and Ka bands. In [12] the X-band elements are crossed dipoles and the Ka-band elements are square patches. The reflectarray reported in [13] was proposed for deep-space telecommunications using circular polarization in both bands, where the phasing in circular polarization was achieved by sequential rotation of printed circular rings with two oppositely located small gaps. However, previous concepts are only valid when the two frequency bands are significantly separated,

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which is not usually the case for transmit-receive antennas. For closer frequency bands, a reflectarray based on two stacked patches with variable size was designed to produce a focused beam at 6.5 GHz and 10.6 GHz [14], but this was a simple design with no attempt to fulfill any requirement in terms of bandwidth. Another reflectarray element based on two stacked layers, with two concentric split rings printed on each layer, has been proposed for dual-frequency operation in two close frequencies (7.4 GHz and 8.9 GHz) [15]. In this case, the gap of the concentric rings in one layer is placed at the orthogonal position with respect to the other layer, in order to use each layer as phase shifter for only one polarization. Therefore, the reflectarray element operates in different polarization at each frequency. A reflectarray demonstrator for a Tx-Rx ground terminal DBS antenna in Ku-band using a single layer of printed elements was reported in [16]. In this case, the reflectarray was designed to operate in one linear polarization in reception (11.4 GHz–12.8 GHz) and in the orthogonal polarization in transmission (13.7 GHz–14.5 GHz). Therefore, the phasing at each frequency was decoupled by using different polarization, and it was implemented by varying the dimensions of double square open loops (for Rx) interleaved with double-cross open loops (for Tx) on the same grounded substrate. The measured radiation patterns show satisfactory gain in the two specified frequency bands. The use of a single substrate and a single polarization for Tx and Rx is important to produce low-cost terminal DBS antennas, but it is not valid for spacecraft DBS antennas, where dual polarization should be used with very stringent requirements in the contoured beam, in cross-polarization and in co-polar isolation. In this work, a reflectarray demonstrator is designed to accomplish the requirements of a real Ku-band telecommunication mission, which consists of a Tx (11.7–12.2 GHz)-Rx (13.75–14.25 GHz) antenna on the top floor of the Amazonas spacecraft (located at a longitude of 61 W in the geostationary orbit). The antenna should provide the same coverage to South America (PAN-S) in both linear polarizations for Tx and Rx frequency bands. A reflectarray configuration based on three stacked layers with printed patches of varying size [5] has been selected because it provides better electrical performance than reflectarrays using other type of elements, as shown in [3]. The technique previously developed for the designing of reflectarrays based on three layers of varying-sized patches [5] has been extended in several aspects. First of all, the technique for the optimization of the patch dimensions to fulfill the requirements in a 10% bandwidth has been extended to two separate bands (Tx and Rx). Second, the pattern synthesis based on the Intersection Approach [17] has been modified in order to enforce co-polar isolation requirements, in the present mission with respect to a European coverage (EUR). Finally, the incident field on each reflectarray element is computed by using the near field radiated by the feed-horn. This technique was proposed and validated in [18] for a more accurate analysis, and here is implemented in the design of the reflectarray antenna. A 1.2-meter antenna breadboard has been designed, manufactured and tested, including mechanical and thermo-elastic constraints

Fig. 1. Coverage of the Amazonas satellite.

for space applications. The antenna performance is close to fulfilling the very stringent requirements of the selected mission in both polarizations and in both frequency bands. II. COVERAGE REQUIREMENTS AND ANTENNA DEFINITION The requirements of a real DBS spacecraft antenna providing service to South America, the “South Pan-American mission (PAN-S)” on the Amazonas satellite, have been selected to design the Tx-Rx reflectarray antenna. The satellite also supports three more DBS missions: North Pan-American (PAN-N), Europe (EUR) and Brazil (BRA) missions, being PAN-S, PAN-N and EUR coverage regions shown in Fig. 1 using the normalized angular coordinates referred to the satellite coordinate system. A. Antenna Requirements The antenna operates in dual-linear polarization, vertical (V) and horizontal (H). The gain and cross-polar requirements, cross-polar discrimination (XPD) for Tx and cross-polar isolation (XPI) for Rx, for each coverage zone are given in Table I in dB. These requirements shall be met including a margin of 0.1 for the antenna pointing error. The mission also includes a requirement of 30 dB co-polar isolation with EUR mission in the same satellite. Since the gain requirement of the EUR antenna is higher than 30 dBi in all the European zones, the co-polar isolation will be fulfilled if the radiation levels of the PAN-S antenna are lower than 0 dBi in Europe zones. This is the requirement that will be enforced in the design process. B. Antenna Definition The antenna actually used on board the satellite for PAN-S mission is a dual-reflector antenna made of a 1.5-meter main shaped reflector and a 50-cm subreflector. In the present work, a single-offset flat reflectarray of smaller dimensions (1.2-meter) is considered as a possible alternative solution. The reflectarray is elliptical with axes 1248 mm 1196 mm, containing 96 rows by 92 columns (6944 cells of 13 mm 13 mm). The phase

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TABLE I GAIN AND CROSS-POL. REQUIREMENTS FOR V- AND H-POLARIZATION

Fig. 3. Reflectarray element based on three-staked varying-sized patches. (a) Lateral view, (b) top view.

Fig. 4. Electrical sandwich configuration.

TABLE II DEFINITION OF SANDWICH LAYERS

Fig. 2. Reflectarray antenna including the coordinate systems.

centre of the feed, which is initially modeled by a funcin mm referred to tion, is placed at coordinates the reflectarray coordinate system, see Fig. 2. It was checked that for the defined period and for the angles of incidence on the reflectarray elements, which vary from 0 to 34 , there is no grating lobe generation. The reflectarray elements consist of three layers of varying-sized rectangular copper patches sandwiched in a multi-layer structure, see Fig. 3. The electrical sandwich containing the three layers of patches and the ground aluminum plane is shown in Fig. 4. To provide stiffness to the reflectarray panel, the electrical sandwich is backed by a 30-mm structural sandwich, not shown in the figure. The materials and the sandwich configuration have been selected to reduce the ohmic losses and to improve the thermal and mechanical properties. The details of the sandwich layers are given in Table II. The 18- m copper patches are printed on 127- m , which is Metclad MY1 217 bonded to Quartz honeycomb by 76- m prepreg Quartz-fiber layers AQ525/EX1515 . The slight anisotropy of the honeycomb is taken into account in the deaxis oriented along the ribbon sign, being the , and the axis across the ribbon .

C. Characterization of Reflectarray Elements The thickness of the honeycomb, the period and the initial ratio between the patch dimensions in each layer have been chosen in order to produce a linear phase variation (broadband reflectarray element), small losses and low cross-polarization. A parametric study was carried out considering stacked patches with different relative sizes and the following ratios were , chosen: because they provide a sufficient linear phase response in both Tx and Rx frequencies. Fig. 5 shows the phase and ohmic losses as a function of the patch size considering the previous dimension ratios at central frequencies in Tx and Rx bands. The curves are obtained for different angles of incidence by

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Fig. 5. Ohmic losses and phase-shift computed by SD-MoM at 11.95 GHz (a) and 14.00 GHz (b) for different angles of incidence.

using the spectral-domain method of moments (SD-MoM) [19] assuming a periodic structure. Note that the effect of the angle of incidence can produce variations in the phase-shift up to 100 in both bands, and therefore it should be taken into account in the design process. In both frequency bands, the dissipative losses are smaller than 0.25 dB and the phase range is around 600 degrees, as shown in Fig. 5. The levels of the cross-polarization components generated on the reflectarray elements have been computed for oblique incidence, being at dB below the co-polar components for least . Note that previous size-ratio in each layer is only used in the first design step at 11.95 GHz, which is used as starting point for optimizations. However, in the optimizations all the patch dimensions are optimized independently and the previous size ratio is not maintained any more. III. ANTENNA DESIGN To accomplish the very stringent coverage requirements shown in Fig. 1 and Table I, a phase-only pattern synthesis is required, because the amplitude of the incident field is fixed by the radiation pattern of the feed-horn. A direct optimization process, in which all the element dimensions are simultaneously optimized in an iterative process to achieve the required contoured pattern, is computationally unaffordable, because it

would require the analysis of the 6944 elements to compute the radiation patterns at every iteration of the synthesis. In this work, an efficient design process based on the technique described in [5] has been used. The time efficiency is based on carrying out the following two steps separately: first, the pattern synthesis to obtain the required phase distribution on the reflectarray; and second, the optimization of each reflectarray element. This avoids the need to carry out an electromagnetic simulation of all the elements in each iteration. The design process applied in this work includes some additional features with respect to that reported in [5], such as additional requirements for co-polar isolation, optimization in two separate bands and the use of the near field of the feed-horn in the design. These features are implemented according to the following steps. First, a pattern synthesis technique based on the Intersection Approach [17] is applied at a single frequency (central frequency of the Tx band) to obtain the phase distribution on the reflectarray surface that provides the required shaped pattern. At this stage, the requirements of co-polar isolation with other missions in the same satellite are also included. Second, the patch dimensions are adjusted, element by element, to match the phase distribution at the selected frequency by iteratively calling an analysis routine based on the SD-MoM [19] considering local periodicity and accounting for the angle of incidence and the polarization of the incident filed. Third, the required phase distribution for the given contoured beam is obtained at several frequencies in each band (Tx and Rx) by using a multi-frequency pattern synthesis [20]. Finally, the patch dimensions are optimized element by element to match simultaneously the phase distribution at central and extreme frequencies in both Tx and Rx bands for each polarization. A conical corrugated horn has been designed, manufactured and tested. The feed-horn (length: 268 mm and diameter: 116 mm), provides a directivity going from 19.9 dBi (at 11.7 GHz) to 21.6 dBi (at 14.25 GHz) and cross-polar levels below dB. Although the reflectarray elements are placed relatively far from the feed, the near field radiated by the feed-horn has been computed using a full-wave commercial software [21], and it has been used as incident field on the reflectarray elements for a more accurate design of the reflectarray. A. Pattern Synthesis Including Co-Polar Isolation The pattern synthesis based on the Intersection Approach [17] model for the horn and was initially carried out using a then it was improved by including the near field radiated by the feed and the requirements for co-polar isolation with EUR mission, see Fig. 1. The isolation with Europe is implemented in the pattern synthesis by adding the defined European zones in the mask of requirements and enforcing a level of radiation in these zones below 0 dBi. After applying the Intersection Approach at 11.95 GHz with the isolation requirements, the resulting radiation patterns fulfill the gain requirements in South America with an excess margin of 1 dB, for possible uncertainties and losses. Fig. 6(a) shows the radiation patterns in dBi for H-polarization -direction). The radiation levels in the Eu(electric field in rope regions are below the required level of 0 dBi. Similar radiation patterns are obtained for V-polarization (electric field in direction). Although in the present antenna configuration,

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the reflectarray as a function of the frequency. As a result, it is easier to realise the phases at different frequencies by optimizing the patch dimensions. The resulting phase distribution provides a radiation pattern at 14 GHz that fulfils the requirements of coverage and isolation with Europe. B. Optimization of Patch Dimensions for Tx and Rx Bands

Fig. 6. Co-polar radiation patterns for H-polarization obtained from pattern synthesis at 11.95 GHz and enforcing co-polar isolation with Europe, superimposed to the gain requirements in the reflectarray coordinate system. For comparison, the radiation patterns are obtained using the near field (a) and the far-field model (b) of the horn.

the model of the feed is acceptable because the reflectarray elements are already in the far field region of the horn, the field radiated by the horn obtained from full-wave simulations using a commercial software [21] has been used. To evaluate the effect of the feed model in the antenna performance, the radiamodel tion patterns have also been computed using the for the same phase distribution obtained from the synthesis and the results are slightly different as shown in Fig. 6(b). The starting phase distribution for the pattern synthesis at 14 GHz (Rx) is obtained by applying a linear variation with frequency to the phase distribution synthesized at 11.95 GHz, which consists in multiplying the phase-delay distribution (not truncated to a range of 360 ) by the frequency ratio as described in [9]. Then, the Intersection Approach is applied including the near field of the horn at 14 GHz. Since the initial phase distribution already provides a contoured beam close to the prescribed coverage, a small adjustment in the phase should be sufficient to fulfil the requirements; for this reason, the variation of phase . This limitation also in the pattern synthesis is limited to ensures a smooth variation of the phase distribution required on

As a first step, the patch dimensions have been adjusted, keeping the same ratio between layers , to produce the phase distribution at 11.95 GHz obtained from pattern synthesis. For the reflectarray designed at 11.95 GHz the radiation patterns are compliant with the requirements at the design frequency but they suffer severe distortions in the Rx band. The required phase distributions at central and extreme frequencies in the Tx and Rx bands are obtained by applying a linear variation with frequency [9] to the phase distributions obtained from pattern synthesis at the central frequency in each band (11.95 GHz and 14.00 GHz). These phase distributions are used for the first iteration in the optimization, in which the patch dimensions are optimized to match at every element the required phase distribution at the defined frequencies. The optimization routine uses a Fletcher Powell algorithm that adjust all the patch dimensions to minimize an error function, defined as the sum of squared phase errors at the six frequencies (three in Tx band and three in Rx band) for the two polarizations. Since the optimization to minimize the error function involves six variables, three patch dimensions for each polarization, and the phasing performance for the two polarizations is practically uncoupled, as shown in previous works [6] and [22], the optimization is carried out independently for each polarization, i.e., the dimensions -direction are optimized to achieve the required phase in -direction, in distribution for the electric field polarized in order to reduce the CPU time. The error functions for - and -polarization, which respectively correspond to V-and H-polarization in the present application, are defined at a generic reflectarray element by,

(1) (2) where is a sub-index to indicate each frequency, are weighting coefficients, which can be different for each frequency, and are, respectively, the computed for and objective phase-shift on element at frequency polarization. The dimensions and where first optimized by minimizing the error function (1) at each element . Then, these dimensions where used for the optimization of and using (2). Note that during the optimization for one polarization, the orthogonal dimensions of the patches are kept unchanged. After individual optimizations for each polarization, all the patch dimensions can be optimized in the same step for a further refinement to account for the slight influence of the orthogonal dimensions of the patches. This process has

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TABLE IV COMPLIANCE OF THE REQUIREMENTS FOR V-POL. AT 14.25 GHz

Fig. 7. Co-polar radiation pattern (H-polarization) for the optimized reflectarray at 11.70 GHz (a) and 14.25 GHz (b).

ments are not fully accomplished. The compliance of the requirements is improved by further optimizations in an iterative process, as follows: 1) the phase distribution produced by the optimized reflectarray are computed by the SD-MoM analysis technique at the defined frequencies (11.70, 11.95, 12.20, 13.75, 14.00 and 14.25 GHz) and they are used as starting point in a new pattern synthesis; 2) the Intersection Approach is applied again at each frequency until the radiation patterns obtained from the new synthesized phases are compliant with the requirements; and 3) the synthesized phase distributions at the defined frequencies are used in the next iteration for optimizing the path dimensions to minimize the error functions (1) and (2). An improvement was observed in the radiation patterns after the new optimization of the patches, but still the requirements were not met at 100%. For this reason, the process was repeated several times, i.e., the phases produced by the optimized reflectarray were used as a new starting point for a next pattern synthesis, and the new phase distribution were used as the objective for the next optimization of the patch dimensions. The design was concluded after four iterations in the optimization of the patches. C. Analysis of Reflectarray Demonstrator

TABLE III COMPLIANCE OF THE REQUIREMENTS FOR V-POL. AT 11.7 GHz

been done in previous works [6] and [22], demonstrating that it does not produce a significant improvement in the antenna performance, particularly if the optimization process is repeated several times as explained in the next paragraph. For this reason, in the present work the optimizations are carried out independently for each polarization, which allow to account in a , simple manner for the anisotropy of the honeycomb ( for polarization ad for polarization). After a first optimization for and polarizations, the resulting reflectarray shows a significant improvement in the coverage performance in Tx and Rx frequencies, but the require-

The radiation patterns have been computed at several frequencies in Tx and Rx for the optimized reflectarray and the antenna performance has been evaluated. The computed co-polar radiation patterns are shown in Fig. 7 at extreme frequencies (11.70 GHz and 14.25 GHz) for H-polarization (electric field in -direction), where contoured lines are represented for the required gain levels increased in 0.3 dB. The corresponding patterns for V-polarization are not shown because they are very similar to those shown in Fig. 7 for H-polarization. The antenna performance and the compliance with the requirements for V-polarization are summarized in Tables III and IV at 11.70 GHz and 14.25 GHz, respectively. To account for additional uncertainties and losses, the compliance in these tables is evaluated by subtracting 0.3 dB to the minimum value of gain in each zone obtained from simulations, which already includes the ohmic losses in the reflectarray (estimated as 0.25 dB) and the spillover losses. The tables also provide the percentage of the surface in each zone that fulfills the requirements. Note that at the lowest frequency in Tx (11.7 GHz) the minimum gain requirement is practically accomplished at 100% and the XPD is also very satisfactory. For the highest frequency in Rx (14.25 GHz), in the worst case (zone SD) the minimum gain is accomplished in more than 90% of the surface. The XPI is not fully accomplished, particularly in zone SD where the required gain

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is lower. Co-polar isolation with Europe is well accomplished in Tx band, being the levels of radiation in the Europe zones below dBi at 11.7 GHz for both polarizations. For the Rx band, the co-polar isolation is worse, particularly at the highest frequency, showing levels up to 5 dBi in some particular locations of the European zones. It has been checked that the reason for this increase of the radiation in some areas out of the prescribed coverage is associated to the remaining phase errors after several iterations of the patch optimizations. The phase errors have been evaluated for the six frequencies in Tx and Rx bands and they are below 25 degrees in more than 90% of the reflectarray elements for the worst frequency case. The phase errors, in spite of their small values, produce some spurious radiation that worsen the antenna performance in cross-polarization and co-polar isolation with other regions, particularly at higher frequencies (the worst case occurs at 14.25 GHz in the present application).

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Fig. 8. Tx-Rx reflectarray demonstrator.

IV. MANUFACTURE OF REFLECTARRAY DEMONSTRATOR The reflectarray panel was manufactured by a co-curing process, in which all the layers in the sandwich are cured in a single step in order to reduce manufacturing time and costs. First, the arrays of copper patches are manufactured from a copper-clad Metclad layer by using a conventional photo-etching process. The different layers of the sandwich are piled on a curing tool, which consist of a flat aluminum plate. The stacking is started with the third layer of Metclad, which acts as radome to protect the copper patches, then the Quartz fiber layer (previously prepared) and the Quartz honeycomb are piled, and so on until all the layers detailed in Table II are piled up. The last layer for the electrical sandwich corresponds to the aluminum ground plane that was previously treated in order to have better adherence with the rest of the elements. After the aluminum ground plane, a backside structural sandwich was piled to provide stiffness to the reflectarray panel. The structural sandwich consists of three layers of Quartz fiber, a 30-mm Quartz honeycomb core and another three layers of Quartz fiber. The complete reflectarray panel, including electrical and structural sandwiches has been cured in one step at the EADS-CASA Espacio facilities. The demonstrator is assembled using a supporting structure made of aluminum which provides the correct position and alignment of the feed-horn and the reflectarray panel. The completed reflectarray demonstrator installed in the anechoic chamber is shown in Fig. 8. V. MEASUREMENTS Several mechanical, thermal and electrical tests have been carried out on the antenna demonstrator, including modal test, thermal cycling, thermoelastic and radiofrequency (RF) tests, in order to check the antenna performance in space environment. The measurements in RF include the losses and phase-shift of individual reflectarray elements and the co-polar and cross-polar radiation patterns in Tx and Rx bands. A. Measurements of Losses and Phase-Shift in Waveguide Small samples have been cut from the panel manufactured for the reflectarray demonstrator to measure the losses and

Fig. 9. Samples for measurement in WGS.

phase-shift in waveguide simulator (WGS). Each sample consists of two periods of the three-layer reflectarray placed in a quarter wavelength section of WR90 waveguide (dimensions 22.86 mm 10.16 mm). The reflection coefficient is measured in an Agilent PNA network analyzer N5230A and also computed using the analysis routine based on SD-MoM, taking into account the propagation constant in the waveguide. Three different types of samples have been manufactured: 1) samples mm, mm), with large square patches ( mm, 2) samples with small patches ( mm) and 3) samples without patches, see Fig. 9. Fig. 10 shows the measured and simulated values of losses and phase-shift for the three samples. To check the repeatability, three sets of samples with the same dimensions have been measured and the results are practically identical. As shown in Fig. 10, very good agreement is obtained for losses and phase-shift, with differences in phase smaller than 10 degrees in all the cases. The ohmic losses are small, from 0.1 dB up to 0.35 dB (average value in frequency for large patches), excluding the abrupt variations near the resonance at 10.85 GHz. This resonance is related to the waveguide cavity formed by the ground plane of the samples and the short-circuit of the coaxial-transition, but it depends on the reactance of the stacked patches. In fact, the resonance appeared in the three samples with large patches at the same frequency (10.85 GHz) and for the same value of ). The measured losses are slightly higher phase (around than the predictions, probably because the loss tangent of the materials is higher than the nominal value. These values of losses are compatible with the objective of 0.25 dB losses in

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Fig. 11. Measured radiation patterns at 11.7 GHz for H-polarization. Contoured lines in dBi for co-polar (a) and cross-polar (b) patterns.

Fig. 10. Comparison of measured and computed reflection coefficient (losses and phase-shift) for three samples in WGS: without patches (a), with small (b) and large (c) patches.

the breadboard, because the breadboard contains patches above and below resonance. B. Measured Radiation Patterns Co-polar and cross-polar radiation patterns and gain have been measured for both polarizations in a planar near field system at the facilities of Universidad Politecnica de Madrid. To evaluate the compliance with the coverage requirements,

the measured patterns in gain are represented as contoured lines and are superimposed to the mask of requirements, in the reflectarray coordinate system shown in Fig. 2. The measured gain contours for H-polarization corresponding to the levels required in each zone, see Table I, at the extreme frequencies in Tx and Rx are shown in Figs. 11(a) and 12(a) respectively. As shown in the figures, the requirements of minimum gain are practically fulfilled at both frequencies in all the zones except in zone SD, where the minimum measured gain is around 18 dBi at 11.7 GHz and around 17 dBi at 14.25 GHz. Similar results are obtained for V-polarization and for the other frequencies, being the gain requirements better accomplished in the Rx band compared to the Tx band. The cross-polarization levels have also been measured according to the third Ludwig’s definition [23]. The contoured patterns at 11.70 GHz and 14.25 GHz are respectively shown in Figs. 11(b) and 12(b) for H-polarization. In general the level of cross-polarization is low in all the coverage, with a maximum of dBi at 11.7 GHz and 0.6 dBi at 14.25 GHz. Similar results are obtained for V-polarization and for the rest of frequencies, being the XPD requirements accomplished in more than 90% of the coverage in Tx band, however XPI requirements in Rx band are not satisfactorily accomplished.

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Fig. 13. Measured Z coordinate in mm of the reflectarray surface after mechanical and thermal tests.

mm including the mechanflatness within a variation of ical tolerances and the distortions associated to the space environment. VI. CONCLUSION

Fig. 12. Measured radiation patterns at 14.25 GHz for H-polarization. Contoured lines in dBi for co-polar (a) and cross-polar (b) patterns.

The ohmic losses of the reflectarray demonstrator are obtained as the difference between measurements of gain and directivity, and they vary from 0.26 dB to 0.3 dB in Tx band and from 0.36 dB to 0.44 in Rx band. These values are slightly higher than those obtained from simulations, similarly as the results shown in Fig. 10(c) for the elements in WGS. When comparing the gain contours with those obtained by simulations, see Fig. 7, there are some differences in the beam shaping at both frequencies, showing some distortions particularly in the Brazil area. As a result of this distortion, the gain requirements are fulfilled in a lower percentage of the zones than in the simulated results shown in Tables III and IV. A similar effect is observed for the orthogonal polarization. The reason for these discrepancies is probably a deformation in the reflectarray surface produced after the thermal cycling. The reflectarray surface has been measured after the mechanical and thermal tests coordinate in mm measured in a grid of 50 and the result ( mm 50 mm) is shown in Fig. 13, which reveal a peak value of 1.711-mm. It has been checked that these deviations in the surface can produce the distortions observed in the radiation patterns. These results suggest that the mechanical design of the reflectarray should be modified, in order to maintain the surface

A reflectarray demonstrator based on three staked layers of varying-sized patches has been designed, manufactured and tested in order to satisfy the electrical requirements of a real Tx-Rx antenna for DBS applications. The coverage requirements of this mission are very stringent, because they include two separated regions of high gain (28.8 dBi) and several regions of lower gain. In addition, the antenna must work in two separate frequency bands in Tx and Rx. In the simulations, the coverage gain requirements are accomplished always in more than 90% of the region, but the cross-polar and the co-polar isolation are not fully accomplished. The results presented in this paper demonstrate that a reflectarray could be designed to fulfill all the requirements typical of a Tx-Rx DBS antenna provided that the electrical requirements (complexity of the coverage, cross-polarization and co-polar isolation requirements) are slightly relaxed compared to the one considered in this paper. The reflectarray demonstrator has shown the viability of this technology with very promising capabilities for Tx-Rx space antennas for telecommunications and broadcasting. REFERENCES [1] K. Shogen, H. Nishida, and N. Toyama, “Single shaped reflector antennas for broadcasting satellites,” IEEE Trans. Antennas Propag., vol. 40, no. 2, pp. 178–187, Feb. 1992. [2] R. A. Pearson, Y. Kalatidazeh, B. G. Driscoll, G. Y. Philippou, B. Claydon, and D. J. Brain, “Application of contoured beam shaped reflector antennas to mission requirements,” in Proc. 8th Int. Conf. on Antennas and Propagation, Edinburgh, U.K., 1993, pp. 9–13. [3] J. Huang and J. A. Encinar, Reflectarray Antennas. Piscataway/Hoboken, NJ: IEEE Press/Wiley, 2008, pp. 143–169. [4] D. M. Pozar, S. D. Targonski, and R. Pokuls, “A shaped-beam microstrip patch reflectarray,” IEEE Trans. Antennas Propag., vol. 47, no. 7, pp. 1167–1173, July 1999. [5] J. A. Encinar and J. A. Zornoza, “Three-layer printed reflectarrays for contoured beam space applications,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1138–1148, May 2004.

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[6] J. A. Encinar et al., “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2828–2837, Oct. 2006. [7] J. Huang, “Bandwidth study of microstrip reflectarray and a novel phased reflectarray concept,” in Proc. IEEE AP-S/URSI Symp., Newport Beach, California, Jun. 1995, pp. 582–585. [8] D. M. Pozar, “Bandwidth of reflectarrays,” Electronics Letters, vol. 39, no. 21, pp. 1490–1491, Oct. 16, 2003. [9] J. A. Encinar and J. A. Zornoza, “Broadband design of three-layer printed reflectarrays,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1662–1664, Jul. 2003. [10] D. I. Wu, R. C. Hall, and J. Huang, “Dual-frequency microstrip reflectarray,” in Proc. IEEE AP-S/URSI Symp., Newport Beach, CA, Jun. 1995, pp. 2128–2131. [11] M. R. Chaharmir, J. Shaker, and H. Legay, “Dual-band Ka/X reflectarray with broadband loop elements,” IET Microwaves, Antennas and Propagation, vol. 4, no. 2, pp. 225–231, Feb. 2010. [12] M. Zawadzki and J. Huang, “A dual-band reflectarray for X- and Ka-bands,” in Proc. PIERS Symp., Honolulu, Hawaii, Oct. 2003. [13] C. Han, C. Rodenbeck, J. Huang, and K. Chang, “A C/Ka dual-frequency dual-layer circularly polarized reflectarray antenna with microstrip ring elements,” IEEE Trans. Antennas Propag., vol. 52, pp. 2871–2876, Nov. 2004. [14] J. A. Encinar, “Design of a dual-frequency reflectarray using microstrip stacked patches of variable size,” Electron. Lett., vol. 32, no. 12, pp. 1049–1050, June 1996. [15] N. Misran, R. Cahill, and V. F. Fusco, “Concentric split ring element for dual frequency,” Electron. Lett., vol. 39, no. 25, pp. 1776–1777, Dec. 2003. [16] M. R. Chaharmir, J. Shaker, N. Gagnon, and D. Lee, “Design of broadband, single layer dual-band large reflectarray using multi open loop elements,” IEEE Trans. Antennas Propag., vol. 58, no. 9, pp. 2875–2883, Sept. 2010. [17] O. M. Bucci, G. Franceschetti, G. Mazzarella, and G. Panariello, “Intersection approach to array pattern synthesis,” IEE Proc., vol. 137, no. 6, pp. 349–357, Dec. 1990, Pt. H. [18] M. Arrebola, Y. Álvarez, J. A. Encinar, and F. Las-Heras, “Accurate analysis of printed reflectarrays considering the near field of the primary feed,” IET Microw. Antennas Propag., vol. 3, no. 2, pp. 187–194, Mar. 2009. [19] C. Wan and J. A. Encinar, “Efficient computation of generalized scattering matrix for analyzing multilayered periodic structures,” IEEE Trans. Antennas Propag., vol. 43, pp. 1233–1242, Nov. 1995. [20] J. A. Zornoza, M. Arrebola, and J. A. Encinar, “Multi-frequency pattern synthesis for contoured beam reflectarrays,” in Proc 26th ESA Antenna Workshop, Estec, Noordwijk, The Netherlands, 2003, pp. 337–342. [21] CHAMP user’s manual for software package for analysis of corrugated and/or smooth wall horn with circular cross section, TICRA, Denmark. [22] J. A. Encinar, M. Arrebola, M. Dejus, and C. Jouve, “Design of a 1-metre reflectarray for DBS application with 15% bandwidth,” in Proc. Eur. Conf. on Antennas and Propagation (EuCAP), Nice, France, Nov. 2006, pp. 1–5. [23] A. C. Ludwig, “The definition of cross polarization,” IEEE Trans. Antennas Propag., vol. 21, no. 1, pp. 116–119, Jan. 1973. Josá A. Encinar (S’81–M’86–SM’09–FM’10) was born in Madrid, Spain. He received the Electrical Engineer and Ph.D. degrees, both from Universidad Politécnica de Madrid (UPM), in 1979 and 1985, respectively. Since January 1980, he has been with the Applied Electromagnetism and Microwaves Group, UPM, as a Teaching and Research Assistant from 1980 to 1982, as an Assistant Professor from 1983 to 1986, and as Associate Professor from 1986 to 1991. From February to October of 1987, he was with the Polytechnic University, Brooklyn, NY, as a Postdoctoral Fellow of the NATO Science Program. Since 1991, he has been a Professor of the Electromagnetism and Circuit Theory Department, UPM. He was a Visiting Professor with the Laboratory of Electromagnetics and Acoustics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland, in 1996 and with the Institute of Electronics, Communication and Information Technology (ECIT), Queen’s University Belfast, U.K., in 2006. His research interests include numerical techniques for

the analysis of multi-layer periodic structures, design of frequency selective surfaces, printed arrays and reflectarrays. He has published more than 150 journal and conference papers, and he is holder of three patents on array and reflectarray antennas. Prof. Encinar was a co-recipient of the 2005 H. A. Wheeler Applications Prize Paper Award and the 2007 S. A. Schelkunoff Transactions Prize Paper Award, given by IEEE Antennas and Propagation Society. He has been a member of the Technical Programme Committee of several International Conferences (European Conference on Antennas and Propagation, ESA Antenna Workshops, Loughborough Antennas & Propagation Conference).

Manuel Arrebola (S’99–M’07) was born in Lucena (Córdoba), Spain. He received the Ingeniero de Telecomunicación degree from the Universidad de Málaga (UMA), Málaga, Spain, in 2002 and the Ph.D. degree from the Universidad Politécnica de Madrid (UPM), Madrid, Spain, in 2008. From 2003 to 2007, he was with the Electromagnetism and Circuit Theory Department, UPM, as a Research Assistant. From August to December 2005, he was with the Microwave Techniques Department, Universität Ulm, Ulm, Germany, as a Visiting Scholar. In December 2007, he joined the Electrical Engineering Department, Universidad de Oviedo, Gijón, Spain, where he is an Associate Professor. His current research interests include analysis and design techniques of printed reflectarrays both in single and dual-reflector configurations and planar antennas. Dr. Arrebola was co-recipient of the 2007 S.A. Schelkunoff Transactions Prize Paper Award, given by IEEE Antennas and Propagation Society.

Luis F. De La Fuente was born in Madrid, Spain. He received the M.S. degree in telecommunication from the Polytechnic University in Madrid. Previously, he was working for the Radiation Department of the Polytechnic University, where he focused on RF measurements in anechoic chamber. Since 1993, he is working at EADS CASA Espacio where he has been involved on the design of shaped single/dual reflector antennae for spacecraft in Ku and Ka bands (Hispasat 1C/1D, Amazonas I and II, Chinasat, Astra-3B, etc.). Since 2006, he is involved in passive and active DRA and reflectarray antennas. Also, he is Technical Coordinator of the DRA-ELSA (DRA-electronically steerable antenna) for the AG1 Small-Geo Satellite from ESA/Hispasat. This project includes the development of sub-arrays, filters, ASICs, MMICs, Hybrids and BFNs.

Giovanni Toso (S’93–M’00–SM’07) was born in La Spezia (Italy) on May 3, 1967. He received the Laurea degree (summa cum laude) and the Ph.D. in electrical engineering from the University of Florence, Florence, Italy, in 1992 and 1995, respectively. In 1996, he was Visiting Scientist at the Laboratoire d’Optique Electromagnátique, University of Aix-Marseille III, Marseille, France. From 1997 to 1999, he was a Postdoctoral student at the University of Florence. In 1999, he was a Visiting Scientist at the University of California, Los Angeles (UCLA). In the same year he received a scholarship from Thales Alenia Space (Rome, Italy) and was appointed Researcher in a Radioastronomy Observatory of the Italian National Council of Researches (CNR). Since 2000, he is with the Antenna and Submillimeter Section, European Space and Technology Centre, European Space Agency, ESA ESTEC, Noordwijk, The Netherlands. His research interests are mainly in the field of array antennas for satellite applications and, in particular, in non regular arrays, reflectarrays and multibeam antennas. In 2009, he was coeditor of the Special Issue on Active Antennas for Satellite Applications in the International Journal of Antennas and Propagation.

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An Active Annular Ring Frequency Selective Surface Paul S. Taylor, Edward A. Parker, and John C. Batchelor, Senior Member, IEEE

Abstract—Offering good performance in terms of all polarizations affected and good angular stability, the ring element is a popular choice in frequency selective surface (FSS) designs. This paper introduces a topology for two-state switching of a ring based FSS. The two states offered by the surface enable it to be either transparent or reflective at the frequency of interest. A design targeted at the 2.45 GHz WLAN band, and intended for the control of the electromagnetic architecture of buildings (EAoB), is realized both by simulation and measurement, the results of which are presented and evaluated. Index Terms—Active frequency selective surface, annular ring, electromagnetic architecture, frequency selective surface (FSS).

I. INTRODUCTION S more devices become wireless and the demand for such is on the increase, particularly in the built environment, then the Electromagnetic Architecture of these spaces needs to be considered. Modern construction materials offer improvements in building thermal efficiency and UV protection. This is often achieved by a metallic coating or loading and can be greatly detrimental to the building EA. Another consequence of the convenience of wireless connectivity is security, particularly where unbounded signals might radiate beyond their intended boundaries. Containing these signals would greatly enhance security. Modifying the electromagnetic architecture of buildings (EAoB) and therefore controlling their spectral efficiency and security can be achieved by the application of frequency selective surfaces (FSS) [1]–[5]. A passive FSS although suitable for some applications might be considered to be restrictive, as it offers no flexibility once installed, whereas an Active FSS (AFSS) allows the potential for some element of control by changing the behavior of the surface. This allows for reconfiguration in the event of time or frequency dependant propagation requirements, or the actual physical movement of boundaries such as dividing walls or temporary partitions. Many element types are used in FSS designs, from simple dipoles to complex fractal and convoluted structures. The propagation characteristics of the built environment can be complex, with signals arriving at any angle of incidence or polarization, due to diffraction, reflection and scattering, and the element type selected needs to be appropriate. An added complication is that these environments are often dynamic with the movement of equipment, furnishings and people continually changing the propagation characteristics of the space. Singly polarized

A

Manuscript received September 22, 2010; revised January 11, 2011; accepted February 26, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Department of Electronics and Digital Arts, University of Kent Canterbury CT2 7NT, U.K. (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2161555

elements such as dipoles only offer frequency selectivity in the plane of the element; dual polarization is achievable with crossed dipoles and related structures but, as with most elements, their performance suffers at oblique angles due to the grating responses which are angle of incidence dependent [6], [7]. A popular choice of rotationally symmetrical geometries offering good stability to angle of incidence is the loop family of elements, and particularly the annular ring [8]–[11]. These features make it a good choice for, but not restricted to, applications in the built environment. Achieving two-state switching of a dipole based surface utilizing semiconductor switches, such as PIN diodes is a recognized technique, and has also been applied to other element types [12]–[16]. The term two-state means that the surface, for a patch element design, can be configured to a reflective or transparent state at the frequency of interest by application or removal of a control signal, usually a dc bias. This paper presents a novel technique targeted at the WLAN band of 2.45 GHz, for two-state switching of a ring based AFSS design, whilst still maintaining appropriate performance for the applications previously outlined. Section II demonstrates how two-state operation of the design is realized by exploiting the resonances [17]–[20] that are achievable with split-ring elements. Simulations using CST Microwave Studio (CST MWS) are used to verify the basic operation of the design. Section III looks at the implementation of the PIN diode switching elements and deals with the transparent distribution, from an RF point of view, of the dc control signal. Section IV details the construction and practical measurements of an actual functional prototype surface at angles of incidence up to 45 with a linearly polarized source at rotation angles of 0 , 22.5 , 45 and 90 . Simulations using CST MWS are included for comparison, and the results are discussed. The paper closes with concluding remarks that summarize the design and measurements. II. TWO-STATE ANNULAR RING FSS DESIGN A. Theory of Operation Fig. 1(a) illustrates the performance of a two-State dipole FSS where the surface behaves as a conventional FSS array in its reflective state with the elements open and in a transparent state by connecting the rows of dipole ends together, usually with semiconductor switches such as PIN diodes. This results in an inductive surface with a high-pass filter response. Providing the high-pass band is low enough in frequency then negligible loss is experienced at in the transparent state, as shown in Fig. 1(a). Initial experiments have shown that applying a similar technique to an annular ring FSS resulted in a lowering of the surface’s resonant frequency, but with the response being rather broad and lossy, and also falling within the original stop-band as

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Fig. 2. A unit cell of the experimental surface.

Fig. 1. Development of a two-state ring surface from the dipole structure of (a), the shorted ring version of (b) and the final open-circuited structure of (c).

shown in Fig. 1(b). Another approach, and the method adopted here, was to remove the fundamental resonance by introducing discontinuities in the elements. This is achieved by open-circuiting the rings into four sections, with the breaks being orthogonally positioned at 45 , 135 , 225 and 315 respectively. This results in a transparent surface that is no longer resonant at its original design frequency. Reconnecting these breaks in the conductors returns the surface back to its reflective state. Fig. 1(c) demonstrates the basic principle of operation. It is worth noting that for the transparent state the surface still exhibits a resonance, which is at approximately twice the fundamental design frequency, with each unit cell being made up of elements at this frequency. This resonance is considfour ered far enough removed from the target band as not to be problematic. B. Experimental Structure Shown in Fig. 2 is a unit cell for a design frequency of 2.45 GHz, its dimensions are: mm, mm,

Fig. 3. (a) ON and (b) OFF state simulation results for TE —- and TM polarizations at normal incidence for the structure of Fig. 2.

0000

mm, and mm. These dimensions result in a closely spaced array on a square lattice. The close spacing is advantageous in terms of both angular stability and also distribution of the biasing control signal. The design and associated simulations include a 0.17 mm thick polyester substrate with and . All metallic elements are of 0.015 mm copper. Initial simulations were carried out using this structure, where ideal switches were assumed, that is switches were either open or closed at points , and introduced no additional strays or losses to the surface. The dimension is dictated more by the component package used rather than a critical dimension. Simulation results for this structure at normal incidence are given in Fig. 3, and show a pronounced stop-band at 2.45 GHz for the ON state and a 1 dB transmission loss for the OFF state, with the OFF state resonance at approximately 5 GHz.

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Fig. 4. (a) PIN diode structure, with its forward bias equivalent circuit in (b) and reversed biased in (c). Fig. 5. Inductor impedance jZj response.

III. PIN DIODE SWITCHING AND CONTROL A. PIN Diode Switch PIN diode switching is an established and reasonably efficient technique in RF and Microwave circuits, and is used here. Fig. 4 shows a PIN diode structure and simplified equivalent circuits for when the diode is forward and reverse biased. In the forward bias condition the diode presents a resistance in series with the package inductance . The reverse bias condition the circuit becomes a parallel combination of and in series with . is actually a combination of and its package parasitic. the device junction capacitance At frequencies above about 10 MHz and up to several GHz the equivalent circuit is a good approximation to a resistance whose value is controlled by a dc or low frequency control current. The HMSP3862 [21] from Avago has been used in this application. The device is actually a series connected pair of diodes in a SOT-23 package with an OFF capacitance of 0.3 pF and an ON resistance of 1.5 . This results in a device with 0.15 pF and 3 for a series connected pair. The surface topology required four devices per unit cell. B. Control Signal Distribution and Isolation In AFSS designs any additional control or bias lines if not correctly isolated from the resonant surface will impact upon its operation and performance. Suitably chosen inductors achieve the required isolation. Inductors when used as RF chokes present a high impedance at the design frequency whilst allowing a dc path for the PIN diode control signal. For choke applications, owing to the presence of stray reactances, the as the minimum self blocking impedance rises above resonant frequency (SRF) of the device is approached [22]. It is acceptable, and even advantageous, to exploit this feature as a greater impedance is presented by the device and consequently improved isolation is achieved. The SIMID 0603 series of inductors from EPCOS [23] offer a suitable component. With an SRF of 2.5 GHz, the 56 nH inductor was the selected device. Normally this value of inductance would present an impedance of approximately 860 at 2.45 GHz, but as we are operating to just below its SRF a greater value is achieved. Presented in Fig. 5 is an SRF measurement

Fig. 6. Schematic representation of a 2 2 2 array, detailing the surface topology and component locations. L = 56 nH inductor, D = HMSP3862 PIN diode.

performed on an EPCOS 56 nH inductor using a HP8722ES vector network analyzer (VNA). It clearly shows the SRF peak, and the device presenting an impedance of approximately k at the design frequency of 2.45 GHz. Providing we have good isolation of the control signal, in the interest of efficiency and also economy, where practical it makes sense to use the FSS elements themselves as current carrying conductors for the control signal. The topology is such that providing the correct polarities of the PIN diodes are observed then the biasing can be applied in either a row or column format. The latter is adopted here. Conveniently, the spacing between adjacent elements supports an 0603 inductor with no additional tracking required. Shown in Fig. 6 is a schematic representation of a 2 2 array.

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Fig. 7. ON state simulation results for TE incidence at angles of, 0 , 30 1 1 1 11 and 45 0 0 0. 15

0 0 00

00,

Fig. 8. OFF state simulation results for TE incidence at angles of, 0 15 0 0 00, 30 1 1 1 11 and 45 0 0 0.

00

,

IV. SIMULATION AND EXPERIMENTAL RESULTS A. Simulation Results With the component values and their associated strays known, these were incorporated in the structure shown in Fig. 2. The values for the PIN diode were: 0.15 pF for the OFF state capacitance and 3 for the ON state resistance and an impedance of 4.5 k for the bias line inductor was also included. Figs. 7 and 8 show the simulated results for both ON and OFF states of the surface illuminated with a linearly polarized source at rotation angles of 0 , 22.5 , 45 and 90 at incidence angles of 0 –45 in 15 increments. The rotation angle is the angle between the y axis and the E vector in the plane of the array. They clearly show a good stop-band at the design frequency and good angular stability, with increasing angles of incidence at all polarization angles. It is evident from the results that the OFF state capacitance of the PIN diodes has the effect of lowering the 5 GHz surface secondary resonance by approximately 1 GHz when compared with that of an ideal switch as previously shown in Fig. 3(b). The OFF state simulation results of Fig. 8(a)–(d) show an additional unwanted narrow-band response at approximately 2.8 GHz. Although this null was quite deep in cases, its bandwidth was sufficiently narrow that it was not expected to be observable in practice. Its origin is a weak resonance corresponding to a current distribution mode approximating that for an un-segmented ring-simulations show that its exact frequency is moderately sensitive to the gap width . Note that there are slight differences between Fig. 7(a) and (d), and also differences between Fig. 8(a) and (d), though in both cases the E vector is parallel to a side of the lattice square. This is related to the attached chokes along y. B. Prototype FSS Fabrication The surface tested was a 5 5 array on an axial lattice fabricated using standard printed circuit board (PCB) photographic and wet-etch techniques, which resulted in a 25 element surface

Fig. 9. Close-up view of the constructed prototype test surface.

of 200 200 mm requiring a total of 100 diodes and 30 inductors. For mechanical stability the test surface was backed with a . Shown in 12 mm thick sheet of polystyrene foam Fig. 9 is the constructed test surface. C. Prototype FSS Measurements A dc bias voltage, , of 17 volts, current limited to a total forward current of 200 mA was required for the prototype surface. The supply current is divided equally over five columns, resulting in approximately 40 mA per column. In the OFF state, no bias was applied. The measurement set-up consisted of a plane-wave chamber equally divided by a microwave absorber loaded rotatable screen, thus allowing for angle of incidence transmission measurements. The screen has a centrally located adjustable

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TABLE I SUMMARY OF SIMULATED AND MEASURED RESULTS

Fig. 10. ON state measurement results for TE incidence at angles of, 0 , 30 1 1 1 11 and 45 0 0 0. 15

0 0 00

00,

aperture that accepts the surface under test. A pair of Rohde and Schwarz HL050 broadband log-periodic antennas and a Hewlett Packard 8722ES VNA were used for the transmission system. To ensure consistency in the measurements a transmission calibration was carried out with an open aperture before each measurement. Figs. 10 and 11 show the results for the prototype surface. For the ON state the stop-bands are at approximately dB, and good stability for in2.45 GHz, with a rejection of creasing angles of incidence. For the OFF state, with increasing angles of incidence a loss of between 1 and 3.5 dB is experienced at 2.45 GHz. As anticipated the transmission null at 2.8 GHz is much reduced when compared with the simulation results. D. Discussion of Results Table I contains a summary of the simulated and experimental results including the dB bandwidths. Comparing the results shows good centre frequency stability near 2.4 GHz for all polarisations at the angles of incidence measured, with any minor differences being attributed to tolerance of manufacture

Fig. 11. OFF state measurement results for TE incidence at angles of, 0 15 0 0 00, 30 1 1 1 11 and 45 0 0 0.

00

,

of the test surface. The centre frequency rejection was dB for the test surface, which is comparable with non-active surfaces. The OFF state insertion loss varied over a range of 1–3.5 and , but was much redB, with the worst case being duced when compared with the simulations of Fig. 8(a)–(d). This was expected, as very narrow FSS responses are often attenuated [24]—the main contributor to loss at 45 being the minor resonance at approximately 2.8 GHz encroaching into the pass-band. As previously shown, the OFF capacitance of the PIN diodes significantly lowers the overall frequency response. A PIN diode with a lower value of capacitance would effectively increase the frequency of the unwanted response and consequently reduce these OFF state losses. Applying a reverse, as opposed to a zero bias offered no improvement in the surface OFF state performance. dB bandwidth was 1060–420 MHz, dependent upon The polarization and angle of incidence. Significantly it was lower than the simulated results, which would suggest that the ON resistance of the PIN diodes was less than the 1.5 per device quoted, resulting in a higher Q surface and hence narrower bandwidth. With a PIN diode effectively being a current controlled

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variable resistor, this is a feature of the device that might be used to give the surface an amount of bandwidth control.

V. CONCLUSION This paper has presented a novel method of two-state switching of a ring FSS structure, both through computer simulation and practical measurements. A prototype 5 5 two-state ring FSS structure has been designed and fabricated, targeted at the WLAN band of 2.45 GHz. The prototype surface was constructed using readily available materials and components and measured. Performance was in good agreement with computer simulations for both its ON and OFF states. Good stability for angle of incidence, at least up to 45 , and at all rotation angles was also demonstrated. Although the rings were in this case set on a square lattice for symmetry, the same biasing arrangement is certainly feasible for modified lattice geometries [25], giving different reflection bandwidths—an issue of secondary importance here—and others with higher grating lobe onset frequency. In the interest of energy conservation, the ON state current and hence the power consumption, may be reduced below the 40 mA in Section IV.C, but below about 10mA the signal attenuation in the operating band centred at 2.45 GHz would be reduced. Furthermore, the power requirements are lower for small finite size FSS [26]. The surface presented here could be of interest to applications in both the built and other environments. One application is communications control between adjoining rooms in a building, by the simple operation of a switch, or a more intelligent control system.

REFERENCES [1] M. Philippakis, C. Martel, D. Kemp, R. Allan, M. Clift, S. Massey, S. Appleton, W. Damerell, C. Burton, and E. A. Parker, “Application of FSS structures to selectively control the propagation of signals into and out of buildings,” 2004 [Online]. Available: http://stakeholders.ofcom.org.uk/market-data-research/spectrum-research/fss-structures, Ofcom ref. AY4464A, (Accessed in Dec. 2010) [2] M. Hook and K. Ward, “A Project to demonstrate the ability of frequency selective surfaces and structures to enhance the spectral efficiency of radio systems when used within buildings,” Ofcom ref. AY4462A, 2004. [3] T. Parker, J. Batchelor, J.-B. Robertson, B. Sanz-Izquierdo, and I. Ekpo, “Frequency selective surfaces for long wavelength use in buildings,” in Proc. IET Seminar on Electromagnetic Propagation in Structures and Buildings, Dec. 4–4, 2008, pp. 1–22. [4] J.-V. Rodriguez, M. Gustafsson, F. Tufvesson, A. Karlsson, and L. Juan-Lldcer, “Frequency-selective wallpaper for reducing interference while increasing MIMO capacity in indoor environments,” in Proc. 2nd Eur. Conf. on Antennas and Propagation , Nov. 11–16, 2007, pp. 1–6. [5] G. H.-H. Sung, K. W. Sowerby, M. J. Neve, and A. G. Williamson, “A frequency-selective wall for interference reduction in wireless indoor environments,” IEEE Antennas Propag. Mag., vol. 48, no. 5, pp. 29–37, Oct. 2006. [6] J. C. Vardaxoglou, Frequency Selective Surfaces: Analysis and Design. Taunton, England: Research Studies Press, 1997, ISBN 0-86380-196-X. [7] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000, pp. 26–62, ISBN 0-471-37047-9. [8] E. A. Parker and S. M. A. Hamdy, “Rings as elements for frequency selective surfaces,” Electron. Lett., vol. 17, no. 17, pp. 612–614, Aug. 20, 1981.

[9] T. K. Wu, K. Woo, and S. W. Lee, “Multi-ring element FSS for multiband applications,” in Proc. Antennas and Propagation Society Int. Symp. Digest. Held in Conjunction With URSI Radio Science Meeting and Nuclear EMP Meeting., Jul. 18–25, 1992, vol. 4, pp. 1775–1778, IEEE. [10] E. A. Parker, S. M. A. Hamdy, and R. J. Langley, “Arrays of concentric rings as frequency selective surfaces,” Electron. Lett., vol. 17, no. 23, pp. 880–881, Nov. 12, 1981. [11] E. A. Parker and J. C. Vardaxoglou, “Plane-wave illumination of concentric-ring frequency-selective surfaces,” in Inst. Elect. Eng. Proc.-H Microwaves, Antennas and Propagation, June 1985, vol. 132, no. 3, pp. 176–180. [12] P. Edenhofer and A. Alpaslan, “Active frequency selective surfaces for antenna applications electronically to control phase distribution and reflective/transmissive amplification,” in Proc. IEEE/ACES Int. Conf. on Wireless Communications and Applied Computational Electromagnetics, Apr. 3–7, 2005, pp. 237–240. [13] B. M. Cahill and E. A. Parker, “Field switching in an enclosure with active FSS screen,” Electron. Lett., vol. 37, no. 4, pp. 244–245, Feb. 15, 2001. [14] A. Tennant and B. Chambers, “A single-layer tuneable microwave absorber using an active FSS,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 1, pp. 46–47, Jan. 2004. [15] T. K. Chang, R. J. Langley, and E. A. Parker, “Active frequency-selective surfaces,” in Inst. Elect. Eng. Proc.—Microw., Antennas Propag., Feb. 1996, vol. 143, no. 1, p. 62. [16] G. I. Kiani, K. L. Ford, L. G. Olsson, K. P. Esselle, and C. J. Panagamuwa, “Switchable frequency selective surface for reconfigurable electromagnetic architecture of buildings,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 581–584, Feb. 2010. [17] A. D. Chuprin, E. A. Parker, and J. C. Batchelor, “Resonant frequencies of open and closed loop frequency selective surface arrays,” Electron. Lett., vol. 36, no. 19, pp. 1601–1603, Sept. 14, 2000. [18] S. B. Savia and E. A. Parker, “Current distribution across curved ring element FSS,” in Proc. Inst. Elect. Eng. National Conf. on Antennas and Propagation, Aug. 1999, pp. 332–335. [19] A. E. Martynyuk and J. I. M. Lopez, “Frequency-selective surfaces based on shorted ring slots,” Electron. Lett., vol. 37, no. 5, pp. 268–269, Mar. 1, 2001. [20] R. Dickie, R. Cahill, H. S. Gamble, V. F. Fusco, P. G. Huggard, B. P. Moyna, M. L. Oldfield, N. Grant, and P. de Maagt, “Polarisation independent bandpass FSS,” Electron. Lett., vol. 43, no. 19, pp. 1013–1015, Sept. 13, 2007. [21] [Online]. Available: http://www.avagotech.com/docs/AV02-0293EN (Accessed in Sept. 2010) [22] [Online]. Available: http://www.inductors.ru/pdf/doc671_Selecting_ RF_Inductors.pdf(Accessed in Sept. 2010) [23] [Online]. Available: http://www.epcos.com/inf/30/db/ind_2008/b824 96c.pdf(Accessed in Sept. 2010) [24] P. Callaghan and E. A. Parker, “Loss-bandwidth product for frequency selective surfaces,” Electron. Lett., vol. 28, no. 4, p. 365, Feb. 13, 1992. [25] S. M. A. Hamdy and E. A. Parker, “Influence of lattice geometry on transmission of electromagnetic waves through arrays of crossed dipoles,” in Inst. Elect. Eng. Proc.-H Microw., Optics and Antennas, Feb. 1982, vol. 129, no. 1, p. 7. [26] E. A. Parker, J.-B. Robertson, B. Sanz-Izquierdo, and J. C. Batchelor, “Minimal size FSS for long wavelength operation,” Electron. Lett., vol. 44, no. 6, pp. 394–395, Mar. 13, 2008.

Paul S. Taylor received the B.Eng.(Hons.) from the University of Greenwich, U.K. Since 2008, he has been working toward the Ph.D. degree at the University of Kent, Canterbury, U.K. He has many years industrial experience as a Research & Development Engineer. His main focus has been VHF transceiver design, particularly marine band, working with such companies as Simrad and Navico. More recently he worked for Thales as part of the radio team at the Channel Tunnel. His main research interests are frequency selective surfaces, particularly active, and their applications in the built environment.

TAYLOR et al.: AN ACTIVE ANNULAR RING FREQUENCY SELECTIVE SURFACE

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Edward (Ted) A. Parker graduated from St. Catharine’s College, Cambridge University, U.K., with an M.A. degree in physics and Ph.D. degree in radio astronomy. He established the Antennas Group in the Electronics Laboratory, University of Kent, U.K. The early work of that group focused on reflector antenna design, later on frequency selective surfaces and patch antennas. He was appointed Reader at the University of Kent in 1977, and since 1987 he has been Professor of Radio Communications, now

John C. Batchelor (S’93–M’95–SM’07) received the B.Sc. and Ph.D. degrees from the University of Kent, Canterbury, U.K., in 1991 and 1995, respectively. In 1997, he became a Lecturer with the Electronics Department, University of Kent, and a Senior Lecturer in 2006. He now heads the Antennas Group and was appointed to Reader in 2010. His research interests include compact printed antennas, low frequency FSS, wearable antennas and RFID tag design.

Professor Emeritus. Prof. Parker is a member of the IET.

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Isotropic Spatial Filters for Suppression of Spurious Noise Waves in Sub-Gridded FDTD Simulation Ata Zadehgol, Student Member, IEEE, and Andreas C. Cangellaris, Fellow, IEEE

Abstract—A class of isotropic spatial filters is proposed to suppress spurious noise waves due to sub-gridding in finite difference time domain simulations. The proposed filters are suitable for both two-dimensional and three-dimensional applications. A simple procedure is introduced for the a-priori determination of the required filter order and the frequency with which the filter is applied based on the maximum temporal frequency bandwidth of the electromagnetic source. The proposed filters are easy to implement in the context of the standard Yee algorithm and are applicable for computational domains that involve quite arbitrary material and structural inhomogeneities. Index Terms—Finite difference time domain (FDTD), spatial filter, sub-gridding.

I. INTRODUCTION HE finite difference time domain (FDTD) method with the standard Yee algorithm [1] has been the most popular method in the electromagnetics community for the computer-aided analysis of field interaction in complex structures and media. While the Yee algorithm can be applied to uniform grids accurately and efficiently, the well known Courant stability criteria places upper limits on the maximum temporal discretization based on the minimum spatial discretization. As such, when the need arises to simulate structures with disparate spatial scales to obtain the steady-state field solutions, the very small temporal discretization dictates the need for vastly more iterations in the uniform FDTD grid, making it computationally costly. In such cases, and in order to overcome the computational cost imposed by the use of a uniform FDTD grid, various methods have been proposed, including sub-gridding, use of sub-cells, and macro-modeling. In the case of the popular approach of sub-gridding, the grid is divided into fine and coarse spatial domains as necessary to comprehend the finer spatial variations of the structures to be modeled. This requires a variable-length spatial discretization for regions that require higher spatial sampling in order to resolve the finer structural details. In principle, sub-gridding should be quite robust, since it continues to use Yee’s simple update scheme in both fine and coarse grids; however, in practice, sub-gridding introduces

T

Manuscript received September 03, 2010; revised February 13, 2011; accepted February 17, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Department of Electrical and Computer Engineering, University of Illinois, Urbana-Champaign, IL 61801 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161551

various side effects that can impact solution accuracy. For example, any abrupt change in grid size gives rise to spurious reflections at the grid interface, due to the numerical wave impedance mismatch. This necessitates the use of proper schemes to mitigate or suppress such spurious effects in order to ensure solution accuracy [2]–[12]. While some of these schemes consider direct changes to the standard Yee update equations or suggest new temporal/spatial interpolation techniques, a few others propose the application of filter operators to eliminate those spurious waves that dominate the spurious degradation in solution accuracy [6], [8]–[10]. For example, the novel work [9], [10] uses 1D digital filters (1D convolution) for spatial decimation and interpolation of the fields at the fine-coarse interface to overcome aliasing, while resorting to phase compensation (scaling of space metric) to suppress the spurious reflections due to impedance-mismatch at the fine-coarse interface. In this work, our goal is to develop a general method to suppress spurious waves in FDTD simulations due to spurious reflections caused by the numerical impedance mismatch at finecoarse grid interface. In particular, we propose the use of a class of spatial filters that: a) are straightforward to design and implement within the existing Yee style FDTD explicit timestepping scheme [13]; b) their implementation does not require complicated spatial/temporal interpolation in the field update equations; c) are able to accommodate broadband electromagnetic sources; d) exhibit spatial isotropy in their suppression of the spurious numerical reflections while preserving the bulk of power spectral density (PSD) of the signal of interest. The paper is organized as follows. In Section II we review the reflection properties of the interface between two finite difference grids of different grid size. We use the results of this analysis to inform the discussion that follows on the development of the proposed spatial filters. In Section III, making use of ideas from digital image/signal processing [14]–[16], we establish a systematic method for the development of spatial filters in two and three dimensions via the McClellan transformation [17]. Furthermore, we provide a design procedure for a-priori determination of the required filter order and the required frequency of application of the filter based on the maximum temporal frequency bandwidth of the electromagnetic source. This is followed by a series of numerical experiments in Section IV that demonstrate the effectiveness of the proposed filters. Finally, Section V concludes with a brief summary and closing remarks. II. REFLECTION PROPERTIES OF THE GRID INTERFACE In this section, we review the reflection properties of the interface between two uniform, finite-difference grids of different

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Fig. 1. One-dimensional, finite-difference grid interface between grids of grid size ratio 1:3.

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is used to denote In the above expressions the notation . Working with the the magnetic field value at position phasor form of (1) and (2) discretized using central differences in the staggered Yee grid, the scheme in [18] is followed to derive a set of discrete equations from which expressions for and can be derived. (8)

grid size in the context of one-dimensional wave propagation. Our development follows the analysis in [12]. In particular, the analysis is carried out for the interface between two grids of grid size ratio of 1:3. This is the ratio used by several investigators for FDTD sub-gridding (e.g., [18]). The finite difference grid interface is depicted in Fig. 1. denoting the variable-length grid size along -axis With in the fine grid, while in the computational domain, in the coarse grid. For the one-dimensional plane wave propagation considered along the axis, the only electric and magnetic field components considered are, respectively, and . Assuming normalized free-space permittivity ( ) and permeability ( ), the source-free Maxwell equations in this case simply become

Elimination of the electric field in the above equations yields two equations that involve only magnetic field values. Using (6) and (7) in these equations yields the linear system for the and . The expressions obtained are given calculation of below

(1)

(13)

(2)

(14)

For time-harmonic solutions with time-dependence of , where and is the angular the form frequency, the phasor of each field component satisfies the Helmholtz equation, shown below for the magnetic field.

(3) where . Using a central difference approximation of (3) on the fine and on the coarse grids, the respective numerical and are readily obtained [13] and are given wave numbers by (4)

(9) (10) (11) (12)

where

and

The magnitude of the reflection coefficient vs. is depicted in Fig. 3. Also shown in the figure is the reflection coefficient obtained from the FDTD simulation of wave propagation across this grid interface. More specifically, the numerical reflection coefficient was obtained through the equation

(5) (15) Next, we examine the reflection and transmission properties of the grid interface. This is done by considering a plane wave propagating from left (fine grid) to right (coarse grid). The total magnetic field phasor in the fine grid is of the form (6) is the reflection coefficient for the grid interface. The where transmitted magnetic field phasor in the coarse grid is given by

(7)

denotes the discrete fourier transIn the above equation form (DFT) operator. The DFT of the incident electric field at the interface node is computed using the recorded field at that node obtained from the FDTD simulation of a plane wave propagating on a uniform fine grid. The DFT of the reflected electric field is calculated as at the interface node , where is the recorded electric field at that node from the simulation of a plane wave propagation on the grid depicted in Fig. 1. As expected, and as clearly evident from the plot in Fig. 3, wavelengths that are well resolved on both grids propagate

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Fig. 2. Staggered electric and magnetic fields, with sub-gridding along one dimension.

Gibb’s phenomenon), one soon concludes that such an approach is not optimized to address the problem of spurious . In reflections for wavelengths which experience light of this, use is made of the design criteria discussed in [19] to specify our spatial filter; these are repeated here for completeness of discussion. We note that these criteria may be changed as necessary to optimize filter performance for different applications. We define the spatial frequency response of the filter (16)

Fig. 3. Magnitude of analytic and computed reflection coefficient, and magnitude of analytic transmission coefficient, vs. k h= .

through the grid interface with minimum reflection. The interval for which the reflection coefficient has magnitude 1 is and is also depicted in Fig. 3. The left end of the interval is at the cutoff wavelength for the coarse grid, , (see (5)), while the right end of the interval is (see (4)). at the cutoff wavelength on the fine grid, Unless suppressed, waves with wavelengths will give rise to spurious reflections that may degrade the accuracy of the numerical wave response. In fact, as the figure suggests, the cutoff wavelength of any spatial filter used to suppress such to ensure wavelengths should be chosen even higher than that the allowed wavelengths in the computational domain propagate through the grid interface with minimum reflection. Equation (13) provides a convenient means for the selection of the desired cutoff wavelength for the spatial filter.

where , is the propagation constant, is the Fourier coefficient, is the integer maximum number of samples (also the filter order), and is an integer. In the above equation, is understood to represent the coarsest grid size relevant to the application of interest. 1) The filter must fully pass the zero frequency component of ). the signal entirely (i.e., 2) The filter must fully eliminate all signal components with (i.e., ). wavelength 3) The filter must have a frequency response as flat as (i.e., possible near ). As such, instead of the traditional FIR digital filter design procedure, we use the aforementioned design criteria to obtain a system of linear algebraic equations, the solution of which are in (16). the Fourier coefficients Once the unknown Fourier coefficients are found, to apply the , filter to the FDTD grid at cell location with field value we multiply the spatial frequency response (16) by the spatial , at that cell. The new frequency domain value of the field filtered value of the cell is and its Fourier series is (17) Given the frequency shifting property of discrete LTI systems [20], and in light of (16) and (17), we obtain the inverse Fourier series of (17) which provides the filtered time domain value at cell location

III. SPATIAL FILTER THEORY Given the results of the reflection analysis in Section II, we conclude that to suppress spurious reflections of from fine-coarse grid interface, we must focus on eliminating their source, namely, discrete waves on the FDTD grid with wavelength prone to such spurious reflections. In this section the application of isotropic spatial filters is proposed as a means for the filtering of such spurious waves. First, we develop the theory for 1D spatial filters, and then transform the 1D to 2D, and then to 3D. A. 1D Spatial Filter If one follows the traditional finite impulse response (FIR) digital filter design procedure [14]–[16] which uses an ideal impulse response together with smoothing functions (such as Hamming or Kaiser windows to suppress the well-known

(18) The spatial frequency response of the 1st order 1D filter is shown in Fig. 4. B. 2D Spatial Filter Using the McClellan transformation technique [17], we expand the 1D filter (16) into a 2D filter. Since is an integer, we as an order Chebyshev polynomial can express with constant coefficients [16] (19) where

is the constant coefficient and

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ZADEHGOL AND CANGELLARIS: ISOTROPIC SPATIAL FILTERS FOR SUPPRESSION OF SPURIOUS NOISE WAVES

Fig. 4. Magnitude of spatial frequency response jF (z )j vs. kh= for a 1D 1st order filter.

Fig.

5. Magnitude

of

spatial

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frequency

k h=; k h=g for a 2D 1st order filter.

response

j

F (z ; z

)j

vs.

f

The filtered discrete value may be obtained through the inverse Fourier series of (24), in a similar fashion as for the 1D case (18)

Using (19), we rewrite (16)

(20) For the 2D spatial filter, we use the McClellan transformation [17] to map a single spatial axis to two orthogonal spatial axes and

(25)

(21)

The spatial frequency response of the 1st order 2D filter is shown in Fig. 5.

where are real constants, and, for a circularly sym. metric contour, it is Using transformation (21) in (20), we obtain

C. 3D Spatial Filter As in the 2D case, the development of the 3D spatial filter starts with the 1D design (16), but instead of applying the McClellan transformation [17] once, we apply it twice to map a single spatial axis to three orthogonal spatial axes , , and

(22) and . where To obtain the 2D FDTD discrete implementation of the above, to a linear combination of terms, we reduce and replace with the new constant coefficient . Thus (22) can be written as

(23) Once the spatial filter is obtained, to apply it to the 2D with field value , FDTD grid at cell location we multiply the spatial frequency response (23) by the spatial , at that cell. The frequency-domain value of the field and its Fourier series new filtered value of the cell is is

(24)

(26) are real constants, and for a spherically symwhere . metric surface it is The 3D frequency response is

(27) where , , , and 3D Fourier coefficient. The 3D frequency-domain field value is

is the

(28)

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the filter should be applied throughout the transient simulation without resulting in attenuation of the temporal frequency content of the response beyond acceptable levels. Clearly, application of the filter should start after wave transmissions through grid interfaces have commenced. Furthermore, given the finite stencil of the filter, its selective use at positions where such interfaces are encountered and at times appropriate for each position seems most appropriate. IV. NUMERICAL STUDIES

Fig. 6. Magnitude of spatial frequency response jF (z ; z ; z )j vs. fk h=; k h=; k h= g for a 3D 1st order filter. The surface contours are for jF (z ; z ; z )j = f0:05; 0:25; 0:5; 0:75; 0:95g.

And finally, the 3D time-stepping formulation (convolution) is

(29) The spatial frequency response of the 1st order 3D filter is shown in Fig. 6 for surface contours plotted at five distinct mag. nitudes of D. Filter Order and Frequency of Its Application By using the expression for the 1D spatial frequency response (16) and given the circular symmetry in 2D and spherical symmetry in 3D, we can easily solve for the spatial knee frequency of a given filter order. The corresponding temporal angular knee frequency is computed as , where is the wave phase velocity in the medium. If a higher is desired, a higher order filter may be used. Alternatively, we note that application of the filter consecutive times during one update of the transient fields is equivalent to raising the magnitude of frequency response of the filter to is desired, we can select the exponent . Thus, if a lower and raise its magnitude to the the filter with the closest power , set the resulting expression equal to the desired magand solve for . For example, for a 3 dB drop in nitude at we solve the following equation for magnitude at (30) Appendix A provides a sample table of vs. for 1st through 5th order spatial filters. Once the order of the filter has been decided, the power obtained using (30) serves as the maximum number of times

To investigate the effectiveness of the proposed filters, we consider their application to the FDTD modeling of a two-dimensional (2D) and a three-dimensional (3D) cavity resonator. For the 2D case, a square box with perfectly conducting electric walls is excited by an infinitely-long electric wire current , source; hence, for the case of a -directed line source, a two-dimensional simulation is performed. For the 3D case, the resonator is a cube with perfectly conducting electric walls excited by a -directed, half-wavelength electric dipole. In both cases the FDTD grid used utilizes a 1:3 fine:coarse sub-gridding along the -axis [18]. Information about the physical size of the resonators and the FDTD grid attributes is provided below. The excitation waveform is taken to be a Gaussian pulse. The Gaussian pulse is chosen to have a frequency bandwidth , and the fine grid is setup for a maximum frequency with spatial discretization , ; we note that these choices for , , where bring the point of the Gaussian bandwidth to and in the FDTD grid. This is to ensure good grid resolution to capture the signal’s power spectral density above the point, while at the same time allowing sufficient energy in the low-resolution wavelength regime to excite the grid where . it is most susceptible to We define as the wavelength at . The side of the 2D . The dissquare resonator and the 3D cube resonator is cretization along utilizes a grid of size over half of the resover the other half. onator domain and a coarse grid of size Uniform grids of size are used along the remaining axes. For the 2D case the source is located in the fine grid at half-distance between PEC on left and fine-coarse transition on right, and the transient electric field is recorded at half-distance between source and fine-coarse transition on -axis. For the 3D case, the half-wavelength dipole is placed along the -axis in the fine grid, at half-distance along the uniform grid axes and half-distance between PEC and fine-coarse transition on the -axis, while the -component of the electric field is recorded at half-distance between source and fine-coarse transition on -axis. The response is computed for the following three cases. • Using a uniform grid of size along . This is the reference solution and is referred to as Uniform in the following. • Using a 1:3 fine:coarse sub-gridding along -axis as described above without any filtering of the response. This is . referred to as • Using a 1:3 fine:coarse sub-gridding along -axis as described above with filtering of the response. This is referred . to as The idea is to let the signal from a source in the fine grid arrive at the fine-coarse interface, then as the spurious reflections

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TABLE I SPATIAL KNEE FREQUENCY k h= VS. NUMBER OF TIMES FILTER IS APPLIED p, FOR 1ST THROUGH 5th ORDER SPATIAL FILTERS

2

propagate back into the fine grid, intercept this noise wave in a pre-defined region of the fine grid and eliminate it by application of spatial filters for a finite time and until noise is attenuated to an acceptable level. Given that in any FDTD simulation, the material properties of the grid are known in advance, we can calculate the propagation velocity in the medium, and from it the flight time of signals. Using the signal flight times, and knowledge of location of the sub-grid interface, it is possible to develop a filter application schedule at specified regions of the fine grid. Alternatively, it is possible to use multi-resolution signal decomposition techniques based on wavelet transforms [21], [22] to quickly identify locations within the grid that require filter application. Below, we implement the former technique and give a demonstration in 2D and 3D simulations. begin to occur near The spurious reflections of . For this example, let’s specify our attenuation at . requirement of the noise waves to be Then, according to Table I, the first-order spatial filter is applied approximately 140 times along the fine-coarse grid interface (plus a 20-cell padding, equivalent to one ) on the fine , at time steps . The side, at increments of filter is applied to both the E-field and H-field components, according to (25) for 2D simulation or (29) for 3D simulation. FDTD simWe run both uniform and sub-gridded 2D ulations, and plot the electric field data at the fine-coarse interface from both simulations, in Fig. 7. It can be seen that the electric field begins to experience spurious reflections starting until , before the first reat approximately ; we flection from left PEC arrives at the interface at would like to eliminate this noise. The 2D simulation results are shown in Fig. 8. In the 3D case, we start filtering at until . The 3D simulation results are shown in Fig. 9. V. CONCLUSION For the standard Yee update algorithm with 1:3 sub-gridding, through a quantitative analysis of the update equations at the fine-coarse interface node, we derived an analytic expression for at the interface node, and correlated the reflection coefficient against computed results in 1D FDTD. We developed the theory of spatial filter operators for 1D, 2D, and 3D by relying on signal/image processing concepts and utilizing spatial transformations to map from 1D to 2D and 3D. Along the way, we demonstrated a simple procedure for a-priori determination of the filter order and application time interval.

Fig. 7. Magnitude of z-component of the electric field E (V =m) vs. time-step  at the fine-coarse interface.

Fig. 8. Magnitude of z-component of the electric field E (V=m) vs. time-step  for a 2D 1st order filter applied at increments of  = 1.

Fig. 9. Magnitude of z-component of the electric field E (V=m) vs. time-step  for a 3D 1st order filter applied at increments of  = 1.

The results showed that the methodology is effective in suppression of noise waves from fine-coarse grid interface for both 2D and 3D sub-gridded FDTD simulations. It is clear that even at high spatial sampling resolution (where ) there exists spurious reflections which will not go away for a broad-band source, without significant increase of the grid resolution; thus, the 2D and 3D spatial filters offer a way to attenuate such noise waves at minimal cost. In this case, the cost associated with filtering is limited to a relatively small

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portion of the grid region (one wide) over a finite portion of the overall simulation time (140 filter operations in 2D and 160 filter operations in 3D, with each filter operation being near the same order as a cell update based on first order central difference), while requiring no additional computer memory. In lieu of filtered sub-gridding, a 3 increase in coarse grid sampling resolution would have increased the memory requirements by 93 in 2D and 273 in 3D, while tripling the number of required time-stepping updates in 2D and 3D for the same amount of simulated time. While the numerical studies used to demonstrate the implementation of the proposed spatial filters involved FDTD grids utilizing a 1:3 sub-gridding along one of the three Cartesian axes, their application to grids with 1:3 sub-gridding along all three axes is straightforward. As far as the impact of the application of these spatial filters on the stability of the FDTD algorithm, the following two points are worth noting. First, the spatial filtering operators are independent of the field updating operations in the FDTD time integration algorithm. Therefore, as long as the stable update scheme described in [12], [18] for the case of 1:3 fine-coarse sub-gridding is being used, the stability of the numerical integration process is unaffected by the spatial filtering operation. The second point concerns the issue of the possible impact of the application of the proposed spatial filters to the late-time instability of the FDTD solution. As it has been pointed out in [23], one can extend the stability of the FDTD algorithm by filtering out high-frequency harmonics, which are not relevant to the bandwidth of the desired response and thus are poorly resolved on the FDTD grid utilized for the numerical solution. Using the terminology in [23], the CFL enhancement factor at for the proposed spatial filters is the cutoff wavelengths of . Thus, even though our proposed implefound to be mentation is for the filter to be selectively applied in the immediate neighborhood of the fine-coarse grid interfaces and, thus, filtering of the spurious wavelengths is not done throughout the entire computational domain, one expects that such filtering contributes to extending the stability of the numerical integration. Our numerical studies to date, limited to the case where the same time step is used for the time integration in both the fine-grid and the coarse-grid regions, support this conjecture. APPENDIX A-Priori FILTER ORDER AND APPLICATION TIME INTERVAL The spatial knee frequency for 1st through 5th order isotropic , where spatial filters are computed for according to (30) and shown in Table I in increments of . As the filter order increases, its spatial pass-band increases while at the same time its roll-off becomes steeper; this occurs while avoiding the well-known Gibbs phenomenon encountered in the traditional design procedure of FIR digital filters where it is resolved by multiplication of the frequency response with a smoothing function such as the Hamming or the Kaiser window. Using this data together with the dispersion relation, and given the PSD of the source excitation, one can determine the appropriate filter order and application time interval to

eliminate undesirable wavelengths, while preserving the signal of interest to within specification; for example, for applications involving digital signals, the specification is usually dictated by and as well the receiver’s input voltage thresholds as sensitivity specifications. REFERENCES [1] K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302–307, May 1966. [2] R. A. Chilton and R. Lee, “Conservative and provably stable FDTD subgridding,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2537–2549, Sep. 2007. [3] P. Chow, T. Kubota, and T. Namiki, “A stable FDTD Subgridding method for both spatial and temporal spaces,” in Proc. Antennas and Propagation Society Int. Symp., Jul. 2008, pp. 1–4. [4] K. M. Krishnaiah and C. J. Railton, “Passive equivalent circuit of FDTD: An application to subgridding,” Electron. Lett., vol. 33, no. 15, pp. 1277–1278, Jul. 1997. [5] Y. Hao, V. Douvalis, and C. G. Parini, “Reduction of late time instabilities of the finite-difference time-domain method in curvilinear coordinates,” Inst. Elect. Eng. Proc. Sci., Meas. Technol., vol. 149, no. 5, pp. 267–271, Sep. 2002. [6] S. R. Cloude, A. M. Milne, and P. D. Smith, “Time domain modeling: integral equations, finite differences and experimental results,” in Proc. Int. Conf. on Computation in Electromagnetics, Nov. 1991, pp. 241–244. [7] A. J. Parfitt and T. S. Bird, “Application of a near-field transform algorithm to antenna coupling using the FDTD method,” in Proc. Antennas and Propagation Society Int. Symp., Jun. 1998, vol. 1, pp. 500–503. [8] M. S. Sarto and A. Scarlatti, “Suppression of late-time instabilities in 3-D FDTD analyses by combining digital filtering techniques and efficient boundary conditions,” IEEE Trans. Magn., vol. 37, no. 5, pp. 3273–3276, Sep. 2001. [9] B. Donderici and F. L. Teixeira, “Improved FDTD subgridding algorithms via digital filtering and domain overriding,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2938–2951, Sep. 2005. [10] B. Donderici and F. L. Teixeira, “Accurate interfacing of heterogeneous structured FDTD grid components,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1826–1835, Jun. 2006. [11] T. Weiland and P. Thoma, “A consistent subgridding scheme for the finite difference time domain method,” Int. J. Numer. Model.: Electron. Netw., Devices Fields, vol. 9, no. 5, pp. 359–374, 1996. [12] P. Monk, “Sub-gridding FDTD schemes,” Appl. Comput. Electromagn. Society J., vol. 11, pp. 37–46, 1996. [13] A. Taflove and S. C. Hagness, Computational Electrodynamics : The Finite-Difference Time-Domain Method, 3rd ed. Boston, MA: Artech House, 2005. [14] A. Antoniou, Digital Filters : Analysis, Design, and Applications, 2nd ed. New York: McGraw-Hill, 1993. [15] R. King and M. Ahmadi, Digital Filtering in One and Two Dimensions : Design and Applications. New York: Plenum Press, 1989. [16] W.-S. Lu and A. Antoniou, Two-Dimensional Digital Filters. New York: M. Dekker, 1992. [17] J. H. McClellan, “On the Design of One-Dimensional and Two-Dimensional FIR Digital Filters,” Ph.D. dissertation, Rice Univ., Houston, TX, 1973. [18] D. M. Sheen, “Numerical Modeling of Microstrip Circuits and Antennas,” Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, 1988. [19] R. Vichnevetsky and J. B. Bowles, Fourier Analysis of Numerical Approximations of Hyperbolic Equations. Philadelphia, PA: SIAM, 1982. [20] A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals & Systems, ser. Prentice-Hall signal processing, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1997. [21] J. Schmeelk, “Wavelet transforms on two-dimensional images,” Math. Comput. Model., vol. 36, no. 7-8, pp. 939–948, 2002. [22] S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Analys. Machine Intell., vol. 11, no. 7, pp. 674–693, Jul. 1989. [23] C. D. Sarris, “Extending the stability limit of the FDTD method with spatial filtering,” IEEE Microw. Wireless Compon. Lett., Dec. 2010, to be published.

ZADEHGOL AND CANGELLARIS: ISOTROPIC SPATIAL FILTERS FOR SUPPRESSION OF SPURIOUS NOISE WAVES

Ata Zadehgol (M’10) is currently working towards the Ph.D. at the University of Illinois, Urbana-Champaign, under the advisorship of Prof. Andreas C. Cangellaris. Since 1997, and prior to joining the University of Illinois, he held various technical positions in industry; most recently, as a senior staff Signal Integrity Engineer at Intel Corporation. His research interests and activities include computational techniques for electromagnetic field analysis, high-speed electronic systems design and optimization, high-speed platform power delivery modeling/simulation, and electromagnetic compatibility.

Andreas C. Cangellaris (F’00) received the Diploma in Electrical Engineering (1981) from the Aristotle University of Thessaloniki, Greece, and the M.S. (1983) and Ph.D. (1985) degrees in electrical engineering from the University of California, Berkeley. He is the M. E. Van Valkenburg Professor in the Department of Electrical and Computer Engineering, University of Illinois, Urbana-Champaign. Prior to joining the University of Illinois, he was on the faculty of the Department of Electrical and Computer

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Engineering, University of Arizona, from 1987–1997. Before joining the faculty at the University of Arizona, he was Senior Research Engineer in the Electronics Engineering Department, General Motors Research Labs. His research interests and activities include computational techniques for electromagnetic field analysis; methodologies and CAD tools for noise-aware analysis and design of high-speed/high-frequency electronics; and numerical methods for modeling and simulation of multi-physics phenomena. On these subjects, he has authored or coauthored more than 200 refereed papers and articles in journals and conference proceedings. Prof. Cangellaris is a Fellow of the IEEE. He is an active member of the IEEE Microwave Theory and Techniques Society and the IEEE Components Packaging and Manufacturing Technology Society, serving as member of technical program committees for conferences and symposia sponsored by these societies. He is the co-founder of the IEEE Topical Meeting on Electrical Performance of Electronic Packaging and Systems. In 2005, he received the Alexander von Humboldt Research Award from the Alexander von Humboldt Foundation, Germany, for his contributions to electromagnetic field modeling. During 2008 to 2010, he served as Distinguished Microwave Lecturer for the IEEE Microwave Theory and Techniques Society. Currently, he serves as Editor of the IEEE Press Series on Electromagnetic Wave Theory.

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High-Order Split-Step Unconditionally-Stable FDTD Methods and Numerical Analysis Yong-Dan Kong and Qing-Xin Chu, Senior Member, IEEE

Abstract—High-order split-step unconditionally-stable finite-difference time-domain (FDTD) methods in three-dimensional (3-D) domains are presented. Symmetric operator and uniform splitting are adopted simultaneously to split the matrix derived from the classical Maxwell’s equations into four sub-matrices. Accordingly, the time step is divided into four sub-steps. In addition, high-order central finite-difference operators based on the Taylor central finite-difference method are used to approximate the spatial differential operators first, and then the uniform formulation of the proposed high-order schemes is generalized. Subsequently, the analysis shows that all the proposed high-order methods are unconditionally stable. The generalized form of the dispersion relations of the proposed high-order methods is carried out. Moreover, the effects of the mesh size, the time step and the order of schemes on the dispersion are illustrated through numerical results. Specifically, the normalized numerical phase velocity error (NNPVE) and the maximum NNPVE of the proposed second-order scheme are lower than that of the alternating direction implicit (ADI) FDTD method. Furthermore, the analysis of the accuracy of the proposed methods is presented. In order to demonstrate the efficiency of the proposed methods, numerical experiments are presented. Index Terms—Finite-difference time-domain (FDTD), highorder, numerical dispersion, split-step scheme, unconditionally stable.

I. INTRODUCTION HE finite-difference time-domain (FDTD) method [1] has been proven to be an established numerical technique that provides accurate predictions of field behaviors for electromagnetic interaction problems. However, the conventional FDTD method is an explicit method, and the time step size is constrained by the Courant-Friedrichs-Lewy (CFL) condition [2], which affects its computational efficiency when fine meshes are required. Recently, to overcome the CFL condition on the time step size of the FDTD method, an unconditionally-stable FDTD

T

Manuscript received June 21, 2010; revised January 19, 2011; accepted January 26, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported by the Science Fund of China (U0635004 and 60801033) and the State Key Laboratory of Millimeter Waves (K201102). Y.-D. Kong is with School of Electronic and Information Engineering, South China University of Technology, Guangzhou, Guangdong 510640, China (e-mail: [email protected]). Q.-X. Chu is with School of Electronic and Information Engineering, South China University of Technology, Guangzhou, Guangdong 510640, China and also with the State Key Laboratory of Millimeter Waves, Nanjing, Jiangsu 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.2011.2161543

method based on the alternating direction implicit (ADI) technique was developed [3]. The ADI-FDTD method has second-order accuracy both in time and space. In addition, the numerical dispersion of two and three dimensions was analyzed in [4], [5]. However, it presents large numerical dispersion error with large time steps. An iterative ADI-FDTD method with reduced splitting error was developed [6]. Similarly, error-reduced ADI-FDTD methods were proposed in [7]. Moreover, the analysis of the accuracy of the ADI-FDTD method was done in [8]. Subsequently, other unconditionally-stable methods such as split-step [9], [10] and locally-one-dimensional (LOD) [11] FDTD methods were developed. The LOD-FDTD method can be considered as the split-step approach (SS1) with first-order accuracy in time, which consumes less CPU time than that of the ADI-FDTD method. Moreover, three-dimensional LOD-FDTD methods with second-order accuracy in time were shown in [12], [13]. Subsequently, to improve the accuracy, unconditionally-stable FDTD methods with high-order accuracy and low dispersion error in 2-D domains were proposed in [14], [15]. Then, the method in [14] was extended to 3-D domains with second-order spatial accuracy in [16]. On the other hand, using high-order schemes is a usual method to reduce the numerical dispersion error. Concretely, a higher-order ADI-FDTD method in 2-D domains was presented in [17]. The numerical dispersion of the 2-D ADI-FDTD method with the higher order scheme was analyzed in [18]. Furthermore, a split-step FDTD method with higher-order spatial accuracy in 2-D domains was developed in [19]. Then, a comprehensive study of the stability and dispersion characteristics for a set of higher order 3-D ADI-FDTD methods was presented in [20]. Subsequently, an arbitrary-order 3-D LOD-FDTD method was proposed in [21]. Novel high-order split-step FDTD methods in 3-D domains are presented in this paper. Firstly, symmetric operator and uniform splitting are adopted simultaneously to split the Maxwell’s matrix into four sub-matrices. Accordingly, the time step is divided into four sub-steps, and the proposed methods are denoted by SS4-FDTD. In addition, high-order central finite-difference operators based on the Taylor central finite-difference method in [22] are used to approximate the spatial differential operators. Moreover, the uniform formulation of the proposed high-order schemes is generalized. Secondly, all the high-order schemes are proven to be unconditionally stable by using the Fourier method. Furthermore, the generalized form of the dispersion relations of the proposed high-order methods is carried out. Thirdly, some numerical results are provided to show the effects of the mesh size, the time step and the order of schemes on the dispersion. Subsequently, the accuracy of the proposed

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KONG AND CHU: HIGH-ORDER SPLIT-STEP UNCONDITIONALLY-STABLE FDTD METHODS AND NUMERICAL ANALYSIS

schemes is analyzed. Finally, to demonstrate the efficiency of the proposed methods, numerical experiments are presented. It can be concluded that the proposed methods achieve better accuracy even with the coarser mesh, and such an improvement actually leads to other advantages such as higher computational efficiency and lower memory requirements.

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TABLE I COEFFICIENTS OF HIGH-ORDER CENTRAL FINITE-DIFFERENCE SCHEMES

II. THE PROPOSED HIGH-ORDER METHODS In linear, isotropic, non-dispersive and lossless medium, and are the electric permittivity and magnetic permeability, respectively. Then, the 3-D Maxwell’s equations can be written in a matrix form as

TABLE II NUMBER OF ARITHMETIC OPERATIONS AND TRI-DIAGONAL MATRICES

(1) where , is the Maxwell’s matrix. Symmetric operator and uniform splitting are simultaneously into four parts. Then, exploited to decompose the matrix (1) can be written as (2) Due to the limitation of space, , , and are not shown here, they can be found in [16]. By using the split-step scheme [9], (2) is divided into four , one time step is divided into sub-equations, from to four sub-steps accordingly, , , and , by successively solving (3a) (3b) (3c) (3d) Furthermore, the right side of the above equations can be approximated by using the Crank-Nicolson scheme [14]. Subsequently, four sub-procedures are generated as follows (4a) (4b) (4c) (4d) where is a 6 6 identity matrix. By using the Taylor central finite-difference method in [22], the first-order spatial difference is defined as operator

(5) and

is equals to different integers . where Based on the Taylor-series analysis [2], the first-order spatial difference operators in the proposed methods are approximated by using the conventional Yee’s elements plus “one-cell-away” or “multi-cell-away” elements. All the elements use a second-order finite-difference formula to eliminate the third-order and high odd-order terms. For instance, the fourth-order accuracy is obtained by the cancellation of the two second-order error terms from the Yee’s elements and the “one-cell-away” elements. Similarly, other even-order central for difference schemes can be obtained. The coefficients , 4, 6, 8, and 10 are given in Table I. Then, the second-, fourth-, sixth-, eighth-, and tenth-order central difference schemes can be developed. Then, the high-order methods are developed. (4a), (4b) are suitable for all the proposed high-order schemes. In order to investigate the computational requirements of the second-order FDTD methods, the number of arithmetic operations and tri-diagonal matrices is shown in Table II. From Table II, at each time step, more arithmetic operations and tri-diagonal matrices are involved in the computations of the proposed method. Therefore, the computational requirement of the proposed method is then larger than those of other FDTD methods at each time step. However, the proposed method has the high-order accuracy, a larger time step and a coarser mesh can be used. So that the total number of iterations required by the proposed method can be reduced. Consequently, the computational requirement of the proposed method is lesser. When the higher spatial-order methods are formulated, the proposed methods with the mixed-order technique [23] can be employed for the treatment of the material interface. On the fine mesh, the proposed low-order method is used near interfaces and boundaries, whereas on the coarse mesh, the proposed high-order method is used. Therefore, existing successfully-applied techniques in the proposed low-order method for the incorporation of discontinuities, boundary conditions, and thin features are available for use on the fine mesh. On the coarse mesh,

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the proposed high-order method is used to mainly simulate the wave propagation in homogeneous media. Therefore, following this approach, high accuracy is obtained around fine geometric features. III. NUMERICAL STABILITY ANALYSIS By using the Fourier method, assuming , , and to be the spatial frequencies along the , , and directions, the field components in spectral domain at the th time step can be denoted as (6)

(10b)

By substituting (6) into (4a)–(4d), the following equations can be generated

It is obvious that the values of corresponding to the second-, fourth-, sixth-, eighth-, and tenth-order central finite-difference schemes are all real numbers, thus the eigenvalues associated with these schemes can be represented as (9). , we can Since conclude that all the high-order schemes are unconditionally stable.

(7) where is the growth matrix, and (see (8a) and (8b) at the bottom of the page) where , , , , , . can be found, as By using Maple 9.0, the eigenvalues of (9) where

IV. NUMERICAL DISPERSION ANALYSIS Assume the field to be a monochromatic wave with angular frequency

, and

(11) Then, (7) can be expressed as (12) where

is related to the initial field vector

and defined by (13)

For a nontrivial solution of (12), the determinant of the coefficient matrix should be zero as follows (10a)

(14)

(8a)

(8b)

KONG AND CHU: HIGH-ORDER SPLIT-STEP UNCONDITIONALLY-STABLE FDTD METHODS AND NUMERICAL ANALYSIS

Fig. 1. Normalized numerical phase velocity error (NNPVE) versus CFLN = 4 and CPW = 20 for the second-order schemes.

Fig. 2. NNPVE versus  with CFLN the second-order schemes.



with

= 2, 5, CPW = 20 and  = 45

Fig. 3. NNPVE versus second-order schemes.

 and  with CFLN

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= 2 and CPW = 20 for the

Fig. 4. Maximum NNPVE versus CFLN with CPW = 50, 100 for the secondorder schemes. for

With reference to the eigenvalues of above, the dispersion relationship of the high-order schemes can be deduced in (15). (15) Based on the previous arguments of the generalized eigenvalues, the dispersion relation in (15) can also be generalized to all the high-order schemes. This is achieved simply by rewith those corresponding to the high-order differplacing ence schemes. V. PERFORMANCES OF THE PROPOSED METHODS To verify the superiority of the proposed methods, the dispersion characteristics of the proposed methods are investigated based on our generalized results. Assume that a wave propagating at angle and is in the spherical coordinate system. , , . Then, By substituting them into the dispersion relation (15), the nucan be solved numerically, merical phase velocity where is the numerical wave number. Subsequently, the normalized numerical phase velocity error (NNPVE) is defined as , where is the speed of light in the medium. Note that is used as reference to find out the NNPVE. For clarity, CFLN is used: it is defined as the ratio between the time step taken and the maximum CFL limit of the FDTD method.

be a ratio factor and Let wavelength (CPW): .

, the cell per

A. Comparisons of Results With Second-Order FDTD Methods In this subsection, the numerical dispersion characteristics of the second-order FDTD methods with different parameters are studied. Since the accuracy of the second-order LOD-FDTD method is similar to that of the ADI-FDTD method, in order to have a clear view, the results are only compared with the ADI-FDTD method in the graphs, and the results of comparison with the LOD-FDTD method are omitted in this paper. and Fig. 1 shows the NNPVE versus with . From Fig. 1, it is apparent that the NNPVE of the SS4-FDTD method is greatly reduced compared with the ADI, the NNPVE of the FDTD method. For instance, with SS4-FDTD method is reduced by more than 67% in comparison with the ADI-FDTD method. , 5, Fig. 2 presents the NNPVE versus with and . As can be seen from Fig. 2, the NNPVE increases as CFLN increases. However, the increase of the NNPVE of the SS4-FDTD method is much less pronounced than that of the ADI-FDTD method. Fig. 3 shows the NNPVE and . From Fig. 3, versus and with the NNPVE of the SS4-FDTD method is lower than that of the ADI-FDTD method for arbitrary and . Fig. 4 shows the maximum NNPVE versus CFLN with , 100. Here, in the entire range of and , the

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Fig. 5. Maximum NNPVE versus CPW with order schemes.

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CFLN = 2, 4 for the second-

Fig. 6. Maximum NNPVE versus CFLN with CPW fourth-order schemes.

= 50 for the second- and

maximum value of the NNPVE is denoted as the maximum and , the NNPVE. From Fig. 4, with maximum NNPVE of the ADI-FDTD method is 13.5%. However, the maximum NNPVE of the SS4-FDTD method is 4%, which is lower than that of the ADI-FDTD method. On the other hand, the maximum NNPVE increases as CFLN increases. has the same error The SS4-FDTD method with . It is compared with the ADI-FDTD method with concluded that the SS4-FDTD method with the coarsest mesh leads to the same level of accuracy as the ADI-FDTD method with the finest mesh. Such an improvement of the accuracy, which is realized by the SS4-FDTD method with the coarsest mesh, leads to other advantages, such as higher computational efficiency and lower memory requirements. Fig. 5 shows the maximum NNPVE versus CPW with , 4. As can be seen from Fig. 5, the maximum NNPVE decreases as CPW increases. For the same CFLN value, the maximum NNPVE of the SS4-FDTD method is lower than that of the ADI-FDTD method. Moreover, the has the same error comSS4-FDTD method with . pared with the ADI-FDTD method with B. Comparisons of Results With Proposed High-Order Schemes In this subsection, the numerical dispersion characteristics of the proposed second-, fourth-, sixth-, eighth-, and tenth-order schemes were investigated based on our generalized results.

Fig. 7. Maximum NNPVE versus CPW with CFLN fourth-order schemes.

Fig. 8. NNPVE versus  with CPW fourth-order scheme.

= 5 for the second- and

= 50 and CFLN = 5 for the proposed

Fig. 6 shows the maximum NNPVE versus CFLN with . Moreover, Fig. 7 shows the maximum NNPVE versus CPW with . As can be seen from Fig. 6, for the SS4-FDTD method, the maximum NNPVE of the fourth-order scheme is smaller than that of the second-order scheme. In addition, from Fig. 7, the maximum NNPVE decreases as CPW increases. Furthermore, the maximum NNPVE of the fourth-order schemes is lower than those of the second-order schemes. When CFLN is smaller, the difference between the fourth-order and the second-order is becoming remarkable, which will be seen from Fig. 11. and Fig. 8 presents the NNPVE versus with for the fourth-order SS4-FDTD method. It can be seen that the NNPVE reaches a minimum at and a , , maximum at three axial directions, . In addition, even with , the and maximum NNPVE is less than 0.3%. for the Fig. 9 shows the NNPVE versus with proposed high-order schemes. It is apparent that the NNPVE is reduced when the higher-order scheme is used. Moreover, the NNPVE of the high-order schemes increases as CFLN increases, which can be seen from the comparisons between Fig. 9(a) and (b). On the other hand, with the fine mesh, when the order is increased beyond four, the NNPVE decreases little. However, with the coarse mesh, the NNPVE can be decreased with the higher order schemes, which can be seen from

KONG AND CHU: HIGH-ORDER SPLIT-STEP UNCONDITIONALLY-STABLE FDTD METHODS AND NUMERICAL ANALYSIS

Fig. 10. Maximum NNPVE versus CFLN with high-order schemes.

CPW = 50 for the proposed

= 45 for the proposed high-order schemes. CFLN = 8, CPW = 20. (c) CFLN = 1,

Fig. 9. NNPVE versus  with  , . (b) (a) .

CFLN = 1 CPW = 20 CPW = 5

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Fig. 9(c). Therefore, the NNPVE of the high-order schemes increases as CFLN increases. Whereas, the NNPVE of the high-order schemes decreases as CPW increases. Fig. 10 shows the maximum NNPVE versus CFLN with for the proposed high-order schemes. From Fig. 10, the maximum NNPVE of the high-order schemes increases as CFLN increases. Moreover, the maximum NNPVE of the fourth- and sixth-order schemes is lower than that of the second-order scheme. However, the maximum NNPVE of the sixth-order scheme is similar to that of the fourth-order scheme. , the maximum NNPVE On the other hand, when of the sixth-order scheme is lower than that of the fourth-order

Fig. 11. Maximum NNPVE versus CPW for the proposed high-order schemes. (a) . (b) .

CFLN = 1

CFLN = 5

scheme (not shown here due to limitation of space). In other words, for the fine mesh, when the order of the approximation increases beyond four, the maximum NNPVE decreases little. For the coarse mesh, the maximum NNPVE of the higher order schemes does decrease. Nevertheless, the reduction of the maximum NNPVE is not linearly proportional to the orders. The reduction becomes flat beyond a certain order. This is most likely due to the fact that the high-order approximation is only applied in the spatial domain. The increased order of the approximation can only improve the accuracy of the approximation by using (5) to replace the first-order spatial difference

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operators, because the truncation error no longer decreases for a fixed mesh resolution beyond a certain order. Fig. 11 presents the maximum NNPVE versus CPW for the proposed high-order schemes with different CFLN values. , it is apparent that the increase of the order When is more effective than the increase of CPW. However, when CFLN is larger, it presents an extreme case that the increase of the order reduces a little in the maximum NNPVE, whereas the increase of CPW does reduce the maximum NNPVE a lot, up to four times. Nevertheless, increasing CPW leads to such disadvantage as higher memory requirements.

(18) where

(19) and . In (19), where at a given time Suppose a numerical solution is transported to that at the next time from the forward Taylor series development

VI. NUMERICAL ACCURACY ANALYSIS The analysis of the accuracy of the proposed methods is based on the paper of [8]. As can be seen from (4a)–(4d), all the high-order methods have the uniform formulation. Therefore, the analytical formulations of the accuracy of all the proposed methods are the same. After the elimination of the intermediate in (4a)–(4d), the following variables of equivalent single-step procedure can be obtained

. Now,

(20) Therefore, by substituting (1) into (20), the following multidimensional Taylor series is obtained.

(21) Subsequently, the analytical fields in (18) are expanded as multidimensional Taylor series about the point . By substituting (21) into (18), and neglecting all the higher order terms with products of the form , the simplified expression of the truncation error is O

(16) Then, the above formula can be rewritten as

(22) where (17) The truncation error is obtained by substituting the analytical field solution of (1) into the left-hand side of (17), which is shown as follows.

+

+

-

+

+

- +

-

+

-

+ +

+

+ +

+

+

-

+

+

+

+ +

-

+

+

The resulting expression of is manipulated by using Maple 9.0, neglecting all the higher order terms with products of the with , the form of form the truncation error with the remaining fourth-order term is given in (23), shown at the bottom of page, where , , ,

-

+

-

+

+

+

+

+

- +

-

+

+ +

+ +

+

-

+ + + (23)

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, and is inside an ellipsoid of center and semi-axes , and

Here, in order to make it straightforward to identify the terms and cases, the vector field corresponding to the 2-D components in the column vector in (23) are ordered in such a way. In addition, the sub 3 by 3 matrices in (23) show some kinds of symmetry. In the 2-D domains, all of the terms in the upper right and lower left quadrants of the matrix of (23) that contain derivatives with respected to z are reduced to zero. The case and the lower upper left quadrant simplifies to the 2-D case. right quadrant simplifies to the 2-D The boxed quantities in (23) represent those terms which are unique to the proposed methods. Moreover, the boxed quantities depend on the cubic or fourth power of the time increment multiplied by the spatial derivatives of the field. VII. NUMERICAL EXPERIMENTS In order to verify the properties of the proposed methods, the FDTD method, the ADI-FDTD method, the LOD-FDTD method, and the SS4-FDTD method are utilized to simulate a in size, all of which have cavity of the second-order accuracy in space. In addition, the cavity is filled with air and terminated with perfect electric conducting (PEC) boundaries. Therefore, on the PEC outer boundary of the FDTD space lattice, the tangential electric fields remain zero for all time steps. Moreover, a sinusoidal modulated Gaussian is used as the pulse of excitation source, where , , . The mesh size is chosen as with 30 samples per wavelength at the highest frequency of the . The excitation source, leading to a mesh number of total simulation time is selected to be 2.587 ns. The simulations are performed on a computer of Pentium IV with 2 GB RAM, and the computer program is developed with C++. -field at the observation point for Fig. 12 shows the three kinds of unconditionally-stable FDTD methods with . For the purpose of comparisons, the result is also shown in of the FDTD method with Fig. 12. From Fig. 12, the result of the SS4-FDTD method is

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TABLE III COMPARISONS OF RESULTS WITH FOUR FDTD METHODS

by using the (2, 4) stencil approach [26]. Recently, to further reduce the dispersion error of the LOD-FDTD method, the parameter optimized method to the fourth-order LOD-FDTD was proposed based on the (2, 4) stencil [27]. Therefore, the similar approach will be used to the fourth-order SS4-FDTD method. It is meaningful to propose the parameter optimized method for the SS4-FDTD method based on the (2, 4) stencil, and it will be included in our future research plan. VIII. CONCLUSION

Fig. 12.

E

-field at the observation point for four FDTD methods.

in better agreement with the FDTD method than those of the ADI-FDTD method and the LOD-FDTD method. Table III shows the comparisons of results of four methods. As can be seen from Table III, the SS4-FDTD method with 0.3 mm mesh size provides the same accuracy as the ADI-FDTD method and the LOD-FDTD method with 0.15 mm mesh size. In addition, the SS4-FDTD scheme requires the CPU time of 53 s and the memory requirement of 3.9136 MB. However, the ADI-FDTD method and the LOD-FDTD method increase the CPU time to 658 s, 490 s and the memory requirement to 31.2995 MB, 31.2995 MB, respectively. Consequently, with the same level of accuracy, the saving in the CPU time and the memory requirement of the SS4-FDTD method can be more than 91% and 86% in comparisons with the ADI-FDTD method and 89% and 86% in comparisons with the LOD-FDTD method. On the other hand, when the proposed fourth-order scheme works, the material interface and boundary conditions are treated using the fourth-order method in order to maintain the overall accuracy. To improve the computational efficiency, optimized FDTD methods based on the (2, 4) stencil were proposed in [24]. In the optimized (2, 4) FDTD methods, the spatial differential operators are approximated by the cell-centered finite difference scheme with four stencils, which features second-order accuracy in general. In addition, optimized three-dimensional FDTD schemes were introduced for the numerical solution of Maxwell’s equations in [25]. Subsequently, the optimized method in [24] was extended into the ADI-FDTD method and further reduced the dispersion errors

High-order split-step unconditionally-stable FDTD methods in 3-D domains have been presented. Symmetric operator and uniform splitting are adopted simultaneously to split the Maxwell’s matrix into four sub-matrices, and accordingly, four sub-steps are produced. In addition, the high-order central finite-difference schemes based on the Taylor central finite-difference method have been used to approximate the spatial differential operators. To trade off between the computational efficiency and accuracy, the effects of the mesh size, the time step and the order of schemes on the dispersion have been illustrated through the numerical results. Moreover, the accuracy of the high-order schemes has been analyzed. Furthermore, numerical experiments have been presented to demonstrate the efficiency of the proposed methods. The schemes proposed in this paper achieve better accuracy accompanied with higher computation efficiency and lower memory requirement. Therefore, the proposed schemes will be of interest and usefulness in electromagnetics problems and can be applied into waveguide, antenna or EMC problems. Generalizing these extensions will be our future work. REFERENCES [1] 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, May 1966. [2] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. Boston, MA: Artech House, 2000. [3] F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1550–1558, Sep. 2000. [4] G. Sun and C. W. Trueman, “Analysis and numerical experiments on the numerical dispersion of two-dimensional ADI-FDTD,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 78–81, 2003. [5] F. Zheng and Z. Chen, “Numerical dispersion analysis of the unconditionally stable 3-D ADI-FDTD method,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 5, pp. 1006–1009, May 2001.

KONG AND CHU: HIGH-ORDER SPLIT-STEP UNCONDITIONALLY-STABLE FDTD METHODS AND NUMERICAL ANALYSIS

[6] S. Wang, F. L. Teixeira, and J. Chen, “An iterative ADI-FDTD with reduced splitting error,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 2, pp. 92–94, Feb. 2005. [7] I. Ahmed and Z. Chen, “Error reduced ADI-FDTD methods,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 323–325, 2005. [8] S. G. Garcia, T. W. Lee, and S. C. Hagness, “On the accuracy of the ADI-FDTD method,” IEEE Antennas. Wireless. Propag. Lett., vol. 1, no. 1, pp. 31–34, 2002. [9] J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math., no. 158, pp. 485–505, Mar. 2003. [10] J. Lee and B. Fornberg, “Some unconditionally stable time stepping methods for the 3D Maxwell’s equations,” J. Comput. Appl. Math., no. 166, pp. 497–523, Mar. 2004. [11] J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Effcient implicit FDTD algorithm based on locally one-dimensional scheme,” Electron. Lett., vol. 41, no. 19, pp. 1046–1047, Sep. 2005. [12] E. L. Tan, “Unconditionally stable LOD-FDTD method for 3-D Maxwell’s equations,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 2, pp. 85–87, Feb. 2007. [13] I. Ahmed, E. Chua, E. Li, and Z. Chen, “Development of three-dimensional unconditionally stable LOD-FDTD method,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3596–3600, Nov. 2008. [14] Q. X. Chu and Y. D. Kong, “High-order accurate FDTD method based on split-step scheme for solving Maxwell’s equations,” Microwave. Optical. Technol. Lett., vol. 51, no. 2, pp. 562–565, Feb. 2009. [15] Q. X. Chu and Y. D. Kong, “Three new unconditionally-stable FDTD methods with high-order accuracy,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2675–2682, Sep. 2009. [16] Y. D. Kong and Q. X. Chu, “A novel three-dimensional unconditionally-stable FDTD method,” in Proc. IEEE MTT-S Int Microw. Symp. Dig., Boston, MA, Jun. 2009, pp. 317–320. [17] Z. Wang, J. Chen, and Y. Chen, “Development of a higher-order ADIFDTD method,” Microwave. Optical. Technol. Lett., vol. 37, no. 1, pp. 8–12, Apr. 2003. [18] M. K. Sun and W. Y. Tam, “Analysis of the numerical dispersion of the 2-D ADI-FDTD method with higher order scheme,” in Proc. Antennas and Propagation Soc. Int. Symp., 2003, vol. 4, pp. 348–351. [19] W. Fu and E. L. Tan, “Development of split-step FDTD method with higher-order spatial accuracy,” Electron. Lett., vol. 40, no. 20, pp. 1252–1253, Sep. 2004. [20] W. Fu and E. L. Tan, “Stability and dispersion analysis for higher order 3-D ADI-FDTD method,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3691–3696, Nov. 2005. [21] Q. F. Liu, Z. Chen, and W. Y. Yin, “An arbitrary order LOD-FDTD method and its stability and numerical dispersion,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2409–2417, Aug. 2009. [22] F. Xiao, X. H. Tang, and H. Ma, “High-order US-FDTD based on the weighted finite-difference method,” Microwave. Optical. Technol. Lett., vol. 45, no. 2, pp. 142–144, Apr. 2005. [23] S. V. Georgakopoulos, R. A. Renaut, C. A. Balanis, and C. R. Birtcher, “A hybrid fourth-order FDTD utilizing a second-order FDTD subgrid,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 11, pp. 462–464, Nov. 2001. [24] G. Sun and C. W. Trueman, “Optimized finite-difference time-domain methods on the (2, 4) stencil,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 832–842, Mar. 2005. [25] T. T. Zygiridis and T. D. Tsiboukis, “Optimized three-dimensional FDTD discretization of Maxwell’s equations on Cartesian grids,” J. Comput. Phys., vol. 226, no. 2, pp. 2372–2388, Oct. 2007.

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[26] W. Fu and E. L. Tan, “A parameter optimized ADI-FDTD method based on the (2, 4) stencil,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1836–1842, Jun. 2006. [27] Q. F. Liu, W. Y. Yin, Z. Chen, and P. G. Liu, “An efficient method to reduce the numerical dispersion in the LOD-FDTD method based on the (2, 4) stencil,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2384–2393, Jul. 2010.

Yong-Dan Kong was born in Heze, Shandong, China, on June 04, 1981. She received the B.S. degree in electronic engineering from Qufu Normal University, Qufu, Shandong, China, in 2006 and the Ph.D. degree in electronic and information engineering from South China University of Technology, Guangzhou, Guangdong, China, in 2011. She is currently working as the Research Assistant at South China University of Technology. Her current research interest is computational electromagnetics.

Qing-Xin Chu (M’99–SM’11) received the B.S., M.E., and Ph.D. degrees in electronic engineering from Xidian University, Xi’an, Shaanxi, China, in 1982, 1987, and 1994, respectively. He is currently a Professor with the School of Electronic and Information Engineering, South China University of Technology, Guangzhou, Guangdong, China. He is also the Director of the Research Institute of Antennas and RF Techniques of the university. From Jan. 1982 to Jan. 2004, he was with the School of Electronic Engineering, Xidian University, and since 1997, he was a Professor and the Vice-Dean of the school of Electronic Engineering, Xidian University. From July 1995 to September 1998 and from July to October 2002, he was a research associate and visiting professor in the Department of Electronic Engineering, Chinese University of Hong Kong, respectively. From February to May 2001 and from December 2002 to March 2003, he was a Research Fellow and Visiting Professor in the Department of Electronic Engineering, City University of Hong Kong, respectively. He visited the school of Electrical and Electronic Engineering, Nanyang Technological University, Singapore from July to October 2004, with the Tan Chin Tuan Exchange Fellowship Award, and visited the Department of Electrical and Electronic Engineering, Okayama University, Japan from January to March 2005, with the Fellowship awarded by Japan Society for Promotion of Science (JSPS). He was also a Visiting Professor of Ecole Polytechnique de I’Universite de Nantes, France from June to July 2008. He has published over 200 papers in journals and conferences. His current research interests include analytical and numerical techniques in electromagnetics, microwave filters, spatial power combining array, and antennas in mobile communication. Prof. Chu is a Senior Member of the China Electronic Institute (CEI). He was the recipient of the top-class Science Award by the Education Ministry of China in 2008 and 2002, and the first-class Educational Award of Shaanxi Province in 2003.

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Modeling of Sloped Interfaces on a Yee Grid Dzmitry M. Shyroki

Abstract—To represent material boundaries in the finite-difference time-domain or frequency-domain method, effective cell permittivity e can be introduced for each grid cell crossed by material interface. In this paper we revisit the derivation of tensorial e for a sloped interface, and describe possible interpolation schemes for coupling of different effective electric field and induction components near the interface. We put the resulting non-symmetric and symmetrized effective permittivity matrices to numerical tests in the frequency domain. For very-high-contrast interfaces the symmetrized schemes perform worse than simple staircasing while non-symmetrized interpolation retains the second-order convergence. Index Terms—Anisotropy, effective cell permittivity, finite-difference frequency-domain (FDFD) and finite-difference time-domain (FDTD) methods, material interface.

I. INTRODUCTION

I

N the finite-difference time-domain [1]–[3] and frequencydomain [4], [5] modeling, major inaccuracies of “physical” origin are due to (i) open boundaries and (ii) material interfaces being represented on a staggered Yee grid. As regards modeling open boundaries, there seems to be a consensus in the community that for mapping of infinitely extended outer space to the bounded computational domain the perfectly matched layer method is the best choice; in its matured, unsplit-field formulation [6], [7] it is indeed an elegant and highly efficient tool, especially when combined with physical coordinate squeezing [8], [9]. As for representing material interfaces on a finite-difference grid, the existing approaches can be classified into two groups: within one group the grid in vicinity of material interfaces is modified; alternatively—or additionally to that—the effective cell permittivity method can be used. The latter requires contensor connecting the averaged electric field struction of and induction vectors in a cell crossed by material interface. For an interface parallel to the grid surface this tensor is represented by a diagonal matrix with its component calculated - (and -) centered cell; this fully preserves the at the structure of conventional finite-difference algorithms for homogeneous medium and was shown to maintain their second-order accuracy in the cell size [10], [11]. which has not been There is a subtlety in definition of discussed in the literature: one has to define the averaging doand and this can be done in different ways. mains for

Manuscript received November 04, 2010; revised January 09, 2011; accepted March 07, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The author is with the Institute of Optics, Information and Photonics, University Erlangen-Nürnberg, 91054 Erlangen, Germany and also with the Max Planck Institute for the Science of Light, 91054 Erlangen, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2161559

One alternative implies simple volume averaging of and over the given cell [12], [13] and seems very intuitive in the context of the finite-difference formalism for Maxwell’s equations; yet we may attempt instead, in a way which stems from a more elegant and physically sound finite-integral interpretation of Maxwell’s grid equations [2], to take into account different geometric nature of electric field and induction [14]–[16] and surface averaging for and do line averaging for in the derivation of for the given grid cell; this would be close to the work of Sullivan et al. [17] where an explicit exhas not however been derived. Following the pression for in two alternatives, we construct two different formulas for Section II of this paper—the resulting effective permittivities differ one from another when material interface is sloped—and test them numerically. Our rather surprising conclusion is that, on a staggered grid, the simpler, volume-average based definition is preferable. Another problem we address in this paper is the rise of when the material nonzero off-diagonal components in boundary is sloped (whatever the subtleties of the definition of are). Physically this amounts to coupling between spatially and , and calls staggered , , and components of for some interpolation of those components which are defined at grid points other than the given one. Different possibilities are described in Section III, some taken from the literature and others newly constructed. In Section IV, through the finite-difference frequency-domain (FDFD) simulations of the mode of a dielectric sphere of permittivity or 36 in the air, we compare the numeric performance of different interpolation schemes to construct non-symmetric and symmetrized effective permittivity matrices. II. EFFECTIVE PERMITTIVITY TENSOR For the finite-difference modeling we need a grid represen(or rather tation for the constitutive relation since one has to determine from in the time-stepping procedure of the finite-difference time-domain method, and it is that enters a closed system of Maxwell’s frequency-domain equations directly) which is assumed to be exact for the local elecand induction flux ; this is not obvitric field vector ously translated to the grid equation when material discontinuity is present. Consider a grid cell crossed by a boundary between two isotropic media. In both media the local and vectors at each are mutually parallel and related via point, but in the finite-difference method we are interested in the and averaged over some region of space. If quantities we assume that the actors in Maxwell’s grid equations are fields averaged over the given grid cell volume , then we have to . define the effective through the relation From the continuity of local vectors and [ stands for the outer product,

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SHYROKI: MODELING OF SLOPED INTERFACES ON A YEE GRID

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] and assuming that they are constant in the cell, i.e., that the cell size is small compared to the light wavelength in both media, we obtain

(1) a uniaxial tensor with its axis pointing along the normal to material interface within a given cell, and its eigenvalues equal to the Wiener bounds which occur in the basic effective medium theory [18]. This tensor was used with success in the finite-difference time-domain [12] and frequency-domain modeling [13]; it is interesting that the same tensor can be derived from perturbation theory for high-index-contrast material interfaces [19]. There yet remains one issue with formula (1) to be discussed: in a topological, metric-free interpretation of Maxwell’s grid equations and [2], [15] the unknowns are not the local fields and fluxes and sampled at appropriate grid nodes, nor the volume-averaged quantities etc., but the field components averaged over the respective cell edges—the etc.; and the flux line-averaged components averaged over the corresponding cell faces—the surface-averaged etc., see Fig. 1. We are therefore interested in relating the quanand tities , assuming for a moment that all these components are known for a particular grid cell (this is not true for a Yee grid and we discuss the consequences of this later). Consider a two-dimensional case for brevity; we may write the following two exact equations:

Fig. 1. In the topological interpretation of Maxwell’s grid equations E is assumed to be averaged over the L line while D over the L L surface. ToLLL. gether they form the E -centered cell of volume V

=

which ocThis is reduced to (1) if curs when points along one of the grid lines or for an interface crossing the center of a given cell, with oriented arbitrarily. Otherwise, (1) and (5) are obviously different. III. FIELD INTERPOLATION SCHEMES There is a problem with (1) or (5) to be settled before using any of these formulas in actual finite-difference computations which, in general, couples the on a Yee grid. In deriving , , and components of and it was assumed that these components are all collocated, that is, are defined for one and the same grid cell. Below we describe different schemes how (1) or (5) can still be used to compute the staggered components from the staggered components. A. D and E Interpolations

(2) (3) (for shorthand both and are used). If we, again, deand be constant through the mand that the local vectors cell we can take the appropriate components out of the aver, aging brackets and then, furthermore, put in order to arrive at a system of equations

- ( -) centered cell. We can get and Consider the at the node in the staggered Yee scheme by simple linear four-point interpolation of the neighboring ’s and ’s to components of that node [17], [20]–[22], and then use the cell (we underline the effective tensor calculated for the corresponding index of and omit the eff subscript here and below) to obtain , and similarly for and —with and components. We call it the D-interpolation scheme and put it in the form with the modified effective matrix

(6) (4) which can be solved, for example, for obtain the effective inverse permittivity:

components to

(5)

where are “forward” and “backward” interpolation operators in the th direction [23], such that for and , and the first-E-then-H staggering in each direction (see Fig. 2) is assumed. A closely related alternative is the E-interpolation scheme components are calculated at the locawhen the tion and are used to compute all three ’s at that location: , and similarly at the and nodes; and after that all the “alien” components of thus obtained all over the grid

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with ’s while others are first converted into ’s and then inschemes similar terpolated. Actually there is a family of can be calto (8) since each of the three pairs of culated at either or location. C.

Interpolation

This is a simple half-sum of (6) and (7) whose off-diagonal blocks

(9)

Fig. 2. Positions of electric and magnetic field vector components hE i and hH i averaged over the corresponding cell edges in the “first-E-then-H” Yee scheme. The averaging brackets are omitted.

are, again, related by simple transpose: , given that is symmetric. This scheme is interpreted as splitting the field into two halves by (9), one of which is D-interpolated and another one is E-interpolated. D. Werner-Cary Interpolation Proposed by Werner and Cary [23], this amounts to construction of with the off-diagonal components

are interpolated to their standard locations, again with a linear four-point procedure:

(7) Although (6) and (7) are different, they lead to identical numeric values for observable physical quantities such as mode eigenfrequency. To prove that actually this must be expected, given the symmetric (1), one can note that and write Maxwell’s grid equations in the frequency domain in with the matrix —this form the form is possible in particular for nonmagnetic media. Then the two and , are transpose matrices, one of another, hence their eigenvalues coincide. To emphasize and interpolation schemes are physically indisthat the tinguishable one may use a single term like the “non-split field interpolation” for any of the two.

(10) ’s (for only) are calculated at one and the where all , , and edges same location—at the vertex where the of the Yee cell meet. In another version of Werner-Cary interpolation, the off-diagonal ’s are attributed to the intersection , , and integration lines, so that the succession of the . It is this inof interpolators in (10) is reversed: terpolation that we use to plot the “Werner-Cary” curves in the plots in Section IV. E. Auxiliary Field Interpolation Another way to symmetrize the D and E interpolations is by , and interpointroducing an auxiliary field lating all its components to the , , and grid nodes in order at those nodes. This gives with to calculate the off-diagonal components (11)

B. Mixed E-D Interpolations The matrices (6) and (7) are not symmetric and were reported to cause late-time instabilities in the time-domain simulations [23], [24]. With symmetric (1) it is easy to symmetrize in many ways, one of which is to define

(8)

In this scheme and are both sampled at the lois symmetric; in that case cation and are therefore equal if the matrix block and similarly for other blocks, which means that is symmetric. We call such interpolation “mixed E-D” because some components are first interpolated and then converted of the

This method was advocated for the non-orthogonal finite-difference time-domain simulations [24] and it was also used in modeling of anisotropic media without discontinuities [25], [26]. It was never claimed to perform well with discontinuous (and shortly we will see why) but for completeness we include it in our survey. IV. NUMERIC RESULTS AND DISCUSSION As a benchmark problem the mode of a dielectric or 36 in the air was chosen. sphere of permittivity The mode has both radial and tangential electric field components, so (unlike the TE modes) it allows to fully probe the ten. The finite-difference frequency-domain sorial structure of (FDFD) codes—a quasi-two-dimensional full-vector body-ofrevolution one [13], and a fully three-dimensional—were used

SHYROKI: MODELING OF SLOPED INTERFACES ON A YEE GRID

Fig. 3. Relative error in the TM mode eigenfrequency jf 0 f j=f computed with the body-of-revolution FDFD code. The permittivity of the sphere is  = 16,  is calculated via formula (1) or (5) and D-interpolated (6). Results with staircased boundary are also included.

with the staggered, equidistant, rectangular grids, so that different grid cells were crossed by the surface of the sphere at mode eigenfredifferent angles. Analytic values for the , where is the vacuum speed quency (in the units of of light, the radius of the sphere) and for the quality factor are: , for the permittivity , , for . and First we compare the numeric performance of two alternative —one according to formula (1) and another definitions of (Fig. 3) both definitions given by (5). For the sphere of lead to similar accuracy improvements as compared to calculations with stair-casing the boundary of the sphere. This picture changes dramatically for the very-high-contrast sphere of : as shown in Fig. 4, the “volume-based” still shows while the “line-surquadratic accuracy with the cell size becomes nearly as poor as simple stair-casing face-based” of the boundary. This can be explained through the following observations: in deriving (5) we assumed that the field components are collocated at a given node, but in any Yee-type grid some are not and we have to interpolate them to that node. Such and surface-averaged interpolation smears line-averaged over some volume which is naturally identified with the given cell volume; thus, (1) can be seen as a more reasonable formula than (5) for Yee-type grids. An immediate question is whether the averaging volume (where and are grid steps in and ) . Fig. 5 allows to comshould coincide with the cell volume -sized against pare the results of standard built on the or edges (in both dimensions) for a moderate-contrast sphere. We see that expanding the averaging region gives no improvement, hence standard averaging over in (1) is recommended. This conclusion the cell volume holds also for very-high-contrast interfaces, as demonstrated in . Fig. 6 for the sphere of is (1) where Having settled that the best formula for surrounding the averaging is performed over the volume the given grid node, we will evaluate the various interpolation schemes described in Section III. Already from Fig. 7 plotted

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Fig. 4. Same as Fig. 3 but for the sphere of permittivity 

= 36.

Fig. 5. Relative error in the TM mode eigenfrequency jf 0 f j=f computed with the body-of-revolution FDFD code. The permittivity  = 16,  is calculated via formula (1) with D-interpolation, with averaging performed over the 1x-, 1:31x-, and 1:61x-wide pixels.

Fig. 6. Same as Fig. 5 but for the sphere of permittivity 

= 36.

for the sphere we see that the auxiliary field interpolation (called Weiland’s here for brevity, though Weiland et al. did not advocate its use for discontinuous media) is inadequate. An explanation, in short, is because the operations of averaging

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Fig. 7. Relative error in the TM mode eigenfrequency jf 0 f j=f computed with the body-of-revolution FDFD code. The permittivity of the sphere is  = 16,  is calculated via formula (1) and interpolated according to one of the schemes from Section III.

Fig. 8. Same as Fig. 7 but for the sphere of permittivity 

= 36.

and taking square root do not commute. Indeed, for the auxil, such that , iary field and gives the continuity of

(12) (13) and if one wishes to relate the averaged , , and , , and with given by formula (1), one has to define

vectors via

(14) (15) none of which equals . Thus, splitting in its square roots as suggested in (11) is not justified for discontinuous . In Fig. 8 the same interpolation schemes are compared for . We the case of the sphere of very high permittivity see that the only method which retains good accuracy and preis the non-split serves second-order convergence rate with or field interpolation when the electric field and induction are treated in their entirety; this is a reasonable tactic since and in Maxwell’s grid equations are not “a sequence of numbers but vectors with exact algebraic properties” [2]. In this context the interpolation can be seen as a mixture of two different representations, (6) and (7), of a correct (second-order accurate) finite-difference operator, and such mixture makes little sense of course. The same could be said about other symmetrized effective matrices. All the numeric illustrations presented by now were done with the body-of-revolution code which is quasi-two-dimensional, i.e., it requires discretization of an object in the axial plane only. Fig. 9 which actually reproduces the results of Fig. 8 but obtained with the three-dimensional, Cartesian-grid-based code

Fig. 9. Relative error in the TM mode eigenfrequency jf 0 f j=f calculated with the three-dimensional FDFD method. The permittivity of the sphere is  = 36,  is calculated according to formula (1) and interpolated according to one of the schemes from Section III.

suggests that our conclusions also hold for the full three-dimensional calculations where the difference between E or D interpolation and its symmetrized alternatives is even more dramatic. V. CONCLUSION Material interfaces should better be modeled on a grid with grid surfaces following those interfaces; in that case, effective inverse permittivity (1) or (5) of the cell is represented by a diagonal matrix and the second order accuracy of standard Yee scheme is maintained at a tiny price of calculating, on a prefor the cells crossed by material interprocessing stage, faces. When building adaptive grid is not possible or not desirable however, the second-order accuracy can still be preserved by use of the effective inverse permittivity (1) with non-split or field interpolation, (6) or (7). We examined a number of alternative, symmetrized interpolation schemes—a total of twelve, of which four families are distinctively different in their numeric performance—that were or could be suggested for use with (1), and we conclude that they should be avoided in highindex-contrast simulations whenever possible as they lead to deteriorated accuracy and convergence rate of the finite-difference algorithm. This is a clear recipe for frequency-domain modeling

SHYROKI: MODELING OF SLOPED INTERFACES ON A YEE GRID

but poses a question of accuracy-versus-stability compromise in the time domain, since late-time instabilities associated with schemes (6), (7) were reported [23], [24]. REFERENCES [1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, pp. 302–307, May 1966. [2] T. Weiland, “Time domain electromagnetic field computation with finite difference methods,” Int. J. Num. Model., vol. 9, pp. 295–319, 1996. [3] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston: Artech House, 2005. [4] T. Weiland, “On the numerical solution of Maxwell’s equations and applications in the field of accelerator physics,” Part. Accel., pp. 245–292, 1984. [5] T. Weiland, “On the unique numerical solution of Maxwellian eigenvalue problems in three dimensions,” Part. Accel., vol. 17, pp. 227–242, 1985. [6] J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: A generalized formalization,” J. Opt. Soc. Am. A, vol. 22, pp. 1844–1849, 2005. [7] D. M. Shyroki and A. V. Lavrinenko, “Perfectly matched layer method in the finite-difference time-domain and frequency-domain calculations,” Phys. Stat. Sol. B, vol. 244, no. 10, pp. 3506–3514, 2007. [8] D. M. Shyroki, “Squeezing of open boundaries by Maxwell-consistent real coordinate transformation,” IEEE Microwave Wireless Comp. Lett., vol. 16, no. 11, pp. 576–578, Nov. 2006. [9] D. M. Shyroki, A. M. Ivinskaya, and A. V. Lavrinenko, “Free-space squeezing assists perfectly matched layers in simulations on a tight domain,” IEEE Antennas Wirel. Propag. Lett., vol. 9, pp. 389–392, 2010. [10] T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, “The second-order condition for the dielectric interface orthogonal to the Yee-lattice axis in the FDTD scheme,” IEEE Microwave Guided Wave Lett., vol. 10, pp. 359–361, Sept. 2000. [11] K.-P. Hwang and A. C. Cangellaris, “Effective permittivities for second-order accurate FDTD equations at dielectric interfaces,” IEEE Microwave Wirel. Comp. Lett., vol. 11, no. 4, pp. 158–160, Apr. 2001. [12] A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett., vol. 31, no. 20, pp. 2972–2974, Oct. 2006. [13] D. M. Shyroki, “Efficient Cartesian-grid-based modeling of rotationally symmetric bodies,” IEEE Trans. Microwave Theory Tech., vol. 55, pp. 1132–1138, Jun. 2007. [14] E. J. Post, Formal Structure of Electromagnetics. Amsterdam: NorthHolland, 1962.

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[15] E. Tonti, “Finite formulation of the electromagnetic field,” Prog. Electromagn. Res., vol. PIER 32, pp. 1–44, 2001. [16] F. W. Hehl and Y. N. Obukhov, Foundations of Classical Electrodynamics: Charge, Flux, and Metric. Boston, MA: Birkhäuser, 2003. [17] J. Nadobny, D. Sullivan, W. Wlodarczyk, P. Deuflhard, and P. Wust, “A 3-D tensor FDTD-formulation for treatment of sloped interfaces in electrically inhomogeneous media,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1760–1770, Aug. 2003. [18] M. Born and E. Wolf, Principles of Optics. Oxford: Pergamon, 1968. [19] C. Kottke, A. Farjadpour, and S. G. Johnson, “Perturbation theory for anisotropic dielectric interfaces, and application to subpixel smoothing of discretized numerical methods,” Phys. Rev. E, vol. 77, pp. 0366111–036611-10, 2008. [20] R. Holland, “Finite difference solutions of Maxwell’s equations in generalized nonorthogonal coordinates,” IEEE Trans. Nucl. Sci., vol. NS-30, pp. 4589–4591, Dec. 1983. [21] M. Fusco, M. V. Smith, and L. W. Gordon, “A three-dimensional FDTD algorithm in curvilinear coordinates,” IEEE Trans. Antennas Propag., vol. 39, no. 10, pp. 1463–1471, Oct. 1991. [22] J.-F. Lee, R. Palandech, and R. Mittra, “Modeling three-dimensional discontinuities in waveguides using nonorthogonal FDTD algorithm,” IEEE Trans. Microwave Theory Tech., vol. 40, no. 2, pp. 346–352, Feb. 1992. [23] G. R. Werner and J. R. Cary, “A stable FDTD algorithm for nondiagonal, anisotropic dielectrics,” J. Comput. Physics, vol. 226, pp. 1085–1101, 2007. [24] S. D. Gedney and J. A. Roden, “Numerical stability of nonorthogonal FDTD methods,” IEEE Trans. Antennas Propag., vol. 48, no. 2, pp. 231–239, Feb. 2000. [25] R. Schuhmann and T. Weiland, “A stable interpolation technique for FDTD on non-orthogonal grids,” Int. J. Numer. Model., vol. 11, pp. 299–306, 1998. [26] S. Feigh, M. Clemens, R. Schuhmann, and T. Weiland, “Eigenmode simulation of electromagnetic resonator cavities with gyrotropic materials,” IEEE Trans. Magn., vol. 40, no. 2, pp. 647–650, Mar. 2004.

Dzmitry M. Shyroki was born in Minsk, Byelorussia, USSR, in 1981. He received the Specialist degree from Byelorussian State University, Minsk, Byelorussia, in 2004 and the Ph.D. degree from the Technical University of Denmark, Lyngby, Denmark, in 2008. Currently he is with the University Erlangen-Nürnberg and the Max Planck Institute for the Science of Light, Erlangen, Germany, where he is involved in the modeling of plasmonic devices.

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Incorporating the G-TFSF Concept into the Analytic Field Propagation TFSF Method John B. Schneider, Senior Member, IEEE, and Zhen Chen, Student Member, IEEE

Abstract—Previously, Anantha and Taflove reported a generalized total-field/scattered-field (G-TFSF) formulation that was developed to facilitate the study of infinite scatterers, such as wedges [IEEE Trans. Antennas and Propag., vol. 50, no. 10, 1337–1349, Oct. 2002]. The G-TFSF formulation relied upon having the TFSF boundary embedded within a perfectly matched layer (PML). To account for the PML, the incident-field terms that appear in the update equations for nodes adjacent to the TFSF boundary were scaled by a constant in accordance with the amount of attenuation produced by the PML. In this work we describe how the analytic field propagation TFSF (AFP TFSF) formulation can be used in a G-TFSF-like way. This new approach possesses various advantages over the previously presented work. For example, owing to the dispersion inherent in PML’s, the spectral components of pulsed excitation propagate at the different speeds within the PML. This dispersive behavior can be accommodated in the AFP-based formulation but not in the original G-TFSF implementation. Additionally, the AFP-based technique can directly model the infinite nature of objects, such as wedges, so that corners need not be embedded within the PML. Index Terms—FDTD methods.

I. INTRODUCTION N finite-difference time-domain (FDTD) simulations the total-field/scattered-field (TFSF) boundary separates the computational grid into two regions: a total-field region (that contains the incident and scattered fields) and a scattered-field region (that contains only the scattered field). In addition to confining the incident field within the total-field region, the TFSF boundary can, in principle, be used to introduce any type of incident field into the FDTD grid. In practice, however, the TFSF boundary is used almost exclusively to introduce plane wave excitation. The original formulations of the TFSF boundary date back to the early 1980’s [1], [2]. The TFSF boundary is implemented by subtracting the incident field from, and adding the incident field to, the update equation for nodes that are both adjacent to and tangential to the boundary (addition is done to nodes on one side of the boundary while subtraction is done to those on the other side). These incident-field terms act like the currents in an equivalence principle formulation or the sources surrounding

I

Manuscript received December 04, 2010; accepted January 27, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161452

a Huygens surface. For non-grid-aligned incidence, the major source of error in the implementation of TFSF boundaries is the mismatch between how fields propagate in the FDTD grid and how the incident field propagates (in whatever form the incident field is specified). If one uses the expression for the incident field that pertains to the continuous world, this mismatch can cause significant spurious fields to leak across the TFSF boundary. Alternatively, the incident field can be obtained from a one-dimensional (1D) “auxiliary-grid” FDTD simulation where the sole purpose of the auxiliary grid is to model the propagation of the incident field. For grid-aligned incidence in 2D and 3D, this auxiliary-grid approach works perfectly: no fields leak from the boundary since the dispersion in the auxiliary grid and the dispersion experienced by the incident field in the higher-dimensional grid are exactly the same. Unfortunately, for non-grid-aligned (oblique) incidence, although the traditional auxiliary-grid approach is very computationally efficient, fields will leak from the boundary and in some applications these leaked fields may be excessively large. An excellent description of both the general implementation of a TFSF boundary and the implementation of a 1D auxiliary grid can be found in [3]. Various authors have sought ways to improve the “traditional” auxiliary-grid approach mentioned above. Guiffaut and Mahdjoubi modified the auxiliary 1D grid so that the dispersion relationship governing the 1D grid was nearly the same as that for the obliquely propagating wave in the higher-dimensional grid [4]. This approach was limited in that, like the traditional approach, it still relied upon interpolation of the fields at nodes in the 1D grid to obtain fields at points corresponding to the projected location of nodes in the higher-dimensional grid. Other authors considered ways in which the errors caused by interpolation could be reduced [5], [6]. Nevertheless, there are some sources of error (leakage) that simply cannot be addressed by the traditional auxiliary-grid approach such as the fact that, unlike in the continuous world, for oblique incidence the fields in the FDTD grid are not completely orthogonal to the propagation vector. An alternative 1D approach was recently put forward by Tan and Potter [7]. The 1D grid Tan and Potter developed had the same dispersion relationship as the higher-dimensional grid and, importantly, the nodes in the 1D grid are located precisely so that interpolation is not needed to obtain the fields in the higherdimensional grid. This approach is attractive in many ways but currently it cannot be used for half-space problems (which is relevant to the work considered here) and it also requires that the incident angle be represented by a rational number (and the computational efficiency is related to this rational number).

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SCHNEIDER AND CHEN: INCORPORATING THE G-TFSF CONCEPT INTO THE ANALYTIC FIELD PROPAGATION TFSF METHOD

Instead of using an auxiliary simulation to obtain the incident field, various authors derived an analytic description for the propagation of a (pulsed) plane wave in the FDTD grid [8]–[12]. This approach, which we identify as the analytic field propagation (AFP) TFSF method, allows fields to be incident at any angle and yet the resulting leakage is negligible. Furthermore, half-space problems can be modeled in which the transmitted and reflected fields associated with that half-space problem are also described analytically and confined to the total-field region. (In fact, the AFP TFSF method can even be used when the incident angle is beyond the critical angle and hence the transmitted field is evanescent [12].) The AFP TFSF method is implemented by first obtaining, via a Fourier transform, the spectral description of the incident field at a single point in the grid. Then, given this “input” and using the FDTD dispersion relationship as the transfer function [13], [14], the spectral description of the incident field can be found at any other point in the grid. One then merely has to take the inverse Fourier transform of the product of the input and the transfer function to obtain the time-domain description of the incident field. This calculation is performed for every node that is tangential to, and adjacent to, the TFSF boundary. Implementation details can be found in [8]–[12]. For half-space problems, in addition to the usual incident or “incoming” wave, one must also calculate the reflected and transmitted fields. They are obtained in much the same way as the incoming field: one simply needs to include in the spectral description of the field the reflection coefficient or the transmission coefficient (which are themselves frequency dependent in the FDTD world). For the reflected field there is a slight change in the transfer function from that which pertains to the incoming field since the reflected field propagates away from the interface rather than toward it (hence there is sign change in the normal component of propagation). For the transmitted field the dispersion relation that gives the transfer function is that which governs propagation in the medium on the other side of the interface from the incident field. In 2002 Anantha and Taflove reported what they called a generalized TFSF (G-TFSF) formulation [15] in which the total-field/scattered-field boundary was partially embedded in the perfectly matched layer (PML) [16] that was used to terminate the grid. If the incident field was propagating from the non-PML region toward the embedded TFSF boundary, the incident field terms in the update equations were scaled by an attenuation constant that was based on the amount of attenuation introduced by the PML. On the other hand, if the incident field originated on the embedded TFSF boundary and from there propagated toward the interior of the grid, the incident field terms were scaled by an amplification factor that compensated for the amount of attenuation the fields experienced when passing through the PML on their way to the interior of the grid. Anantha and Taflove recognized that one could not simply use the continuous-world expression to determine the amount of attenuation a field would experience in an FDTD simulation. Instead the attenuation constants were obtained via calibration simulations in which a pulse was incident, at the particular angle of interest, on a PML and then the amount of attenuation measured at the appropriate locations. The amplification constant

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was the inverse of the attenuation constant. Anantha and Taflove stated that “the wave in the PML region propagates normally to the electric field with the speed of light in vacuum and undergoes an exponential decay with PML depth” [15]. Although this is true in the continuous world, it is not true in the discretized world of the FDTD grid. Instead, the PML region, like the rest of the grid, is dispersive. If the effect of the PML on the incident field is to be represented by a single attenuation constant, one has no choice but to ignore this dispersive behavior. However, using the AFP implementation of a TFSF boundary the incident field is, at an intermediate step, broken into its spectral representation. At that point the dispersive nature of the PML can be included. In order to correct for both the amplitude and phase change caused by the presence of the PML, it is necessary that the calibration consist of two auxiliary simulations that record the full time-domain fields at the appropriate points. In the work by Anantha and Taflove only one auxiliary simulation was used for a particular incident angle. In the AFP-based implementation, one auxiliary simulation is done with the PML present at the nodes of interest and another simulation is done with the PML removed from these nodes. The two sets of time-domain fields are Fourier transformed to the frequency domain where the spectrum of the field with the PML is normalized by the spectrum of the field obtained without the PML. This gives the transfer function representing the effect of the PML. It is then a simple matter to incorporate this transfer function into the AFP TFSF method. The original G-TFSF formulation modeled infinite scatterers, such as wedges, by partially embedding a finite scatterer in the PML and then having the G-TFSF boundary surround that. In this way the scattering from the PML-embedded edges and corners of the finite scatterer (i.e., the edges and corners that do not exist in the corresponding infinite scatterer) could be negligibly small since these scattered fields would have to propagate through the PML before entering the interior of the grid. As will be shown, there is no need for the construction of such finite scatterers when using the AFP TFSF boundary. Since the AFP TFSF method has already been formulated to directly handle half-space problems, this capability can be used to model scattering from wedges even though wedges are not truly half-space problems. Thus, it is not necessary for the TFSF boundary to be four-sided. In fact, it can be two-sided with two edges extending into the PML or, in the case of some perfect electric conductor (PEC) scatterers, only one edge extending into the PML. The remainder of the paper is organized as follows. In Section II we demonstrate how calibration is performed and show that the dispersive nature of the PML is an important consideration when embedding a TFSF boundary within the PML. (We should note that in Anantha and Taflove’s original work [15] they used a split-field PML in accordance with the one first presented by Bérenger [16]. In this work we use an unsplit complex frequency-shifted perfectly matched layer (CPML) formulation employing recursive convolution [17].) That is followed by a section that describes more general geometries and discusses the how calibration is done when the TFSF boundary is embedded in a PML that is itself in a dielectric associated with a half-space problem. Results are then provided to show

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Fig. 1. Depiction of the TM illumination of a PEC half-plane using a twosided AFP TFSF boundary. The portion of the TFSF boundary that extends into the PML on the right side of the grid is enclosed in an oval. The expanded view shows the E , H , and H nodes along this boundary. Only the fields tangential to the boundary are involved in the TFSF formulation. Thus, for this horizontal edge, those are the E and H nodes. The TFSF nodes within the PML require special consideration and are shown as encircled pairs. The incident angle  is defined with respect to the PEC surface normal.

the improvements offered by the approach being advocated here. Additionally, to demonstrate the type of problems that can be addressed with this technique, some snapshots are shown from a simulation involving the termination of a dielectric halfspace in a 90 degree corner. We then conclude. II. CALIBRATING BOTH MAGNITUDE AND PHASE Let us start by considering oblique illumination of a PEC knife-edge, or half-plane, as shown in Fig. 1. As mentioned previously, in the AFP TFSF formulation the “incident field” consists of both the “incoming field” (i.e., what is typically called the incident field) and the reflected field. Were the PEC to span the entire domain, there would be no scattered field. The scattered field in this geometry is produced solely by the termination of the PEC. Thus, the incident field corresponds to that of illumination of an infinite plane wave. The incident angle is defined with respect to the normal to the PEC surface. Since the “incoming angle” and reflected angle are equal in magnitude, the behavior of both the incoming field and the reflected field at the PML can be characterized by this single angle. As shown in Fig. 1, the TFSF boundary is two-sided. One side, the vertical one to the left of the figure, extends from the PEC to a point within the interior of the grid (by interior we mean points not within the PML). The other side is horizontal and extends from the top of the vertical boundary to the right edge of the grid. Thus this horizontal boundary passes through the PML region. The expanded view presented at the top of the and nodes figure shows the nodes along this edge. The adjacent to the TFSF boundary and within the PML are paired together with an enclosing oval. Note that the field is incident . on the PML on the right side of the grid at an angle of

Fig. 2. Auxiliary simulations used for calibrating the AFP-based G-TFSF implementation. (a) The measured nodes are embedded inside the PML. (b) The measured nodes are outside the PML.

Naturally, special consideration is needed to calculate the incident field at the nodes within the PML. This involves two separate calibration runs. The calibration data determine the PML transfer function based on the “depth” of a node in the PML. The transfer function is the same for both electric- and magnetic-field nodes provided they are at the same depth. Thus, the and pairs shown in Fig. 1 have the same transfer function. One could potentially derive a completely analytic description of the effect of the PML. Some authors have developed expressions for the reflection and transmission coefficients associated with a discretized PML (see, for example, [18]–[21]). However, owing to the complexity of performing a completely analytic description of a multilayer CPML, that was not attempted here. Instead, as was done in [15], we simply do auxiliary computations (i.e., calibration runs) where the sole purpose is to measure the effect of the PML. For a given PML geometry, a calibration needs to be performed only once but it is a function of the incident angle. Hence, if the incident angle changes, another calibration needs to be performed. In [15], the goal was to obtain a scalar constant that was subsequently used to scale the incident-field terms in the update equations for nodes tangentially adjacent to the TFSF boundary and in the PML. For a given incident angle, the constant was a function of only the depth of a node within the PML. In the AFP-based implementation, two auxiliary FDTD simulations are done: one with the nodes of interest embedded in the PML and one where the PML is removed from these nodes. This is illustrated in Fig. 2. Fig. 2(a) depicts the simulation where the nodes of interest (i.e., the measured nodes) are embedded in the PML. The measured nodes form a line that spans the PML and is normal to the edge of the grid. One can measure either electric or magnetic fields (in our work we used electric fields). In this simulation there is only an “incoming” field which originates from the upper-left corner of the total-field region. The distance between the vertical TFSF boundary and the PML on the right side

SCHNEIDER AND CHEN: INCORPORATING THE G-TFSF CONCEPT INTO THE ANALYTIC FIELD PROPAGATION TFSF METHOD

of the domain is unimportant—it can be as small as a single cell. For this simulation, the horizontal TFSF boundary, shown near the top of the figure, does not extend into the PML. Thus, there is some spurious radiation from the termination of this boundary. Owing to the oblique incidence (which produces a vertical phase velocity of the incoming field greater than the speed of light), the incoming field arrives at the measured nodes prior to this spurious radiation. Therefore the spurious radiation can be (and is) time-gated out of the measured data. (The computational domain must be of sufficient size to allow this temporal separation to occur. This time-gating approach is not absolutely necessary as one could extend the horizontal TFSF boundary into the PML, but it would take multiple calibrations runs to do this. As will be shown, the calibration data obtained from the two auxiliary simulations depicted in Fig. 2 were sufficiently accurate that no further refinement was deemed necessary.) Fig. 2(b) depicts the simulation where the measured nodes are not embedded in the PML. The only difference between this simulation and the one depicted in Fig. 2(a) is that the computational domain is expanded to the right. The distance from the measured nodes to the TFSF boundary remains unchanged and all the parameters pertaining to the incident field are unchanged. (In theory, to obtain a transfer function that describes only the effect of the PML, in the simulation shown in Fig. 2(b), the right side of the computational domain should be made distant enough that any reflection from the termination of the right side of the grid does not return to measurement points over the duration of the observation. However, in practice, reflections from the PML on the right were found to be small enough that isolating them from the transfer function did not have an appreciable effect on the quality of the final results. Thus, in the results to be shown later, the geometry we used for the second calibration run was similar to the depiction in Fig. 2(b)—the right-side PML was just beyond the measurement points.) We are interested in the effect the PML has on both the magnitude and the phase of the incident field. Thus the entire temporal history at each measured electric-field node in the PML is recorded. The time-domain data are transformed to the frequency domain. The spectral representation of the field for nodes embedded in the PML is normalized by the spectrum of the field at the same node but without the PML. Said another way, the data obtained from Fig. 2(a) are normalized by the data obtained from Fig. 2(b). These normalized spectral data, one set per electric-field node, are the transfer functions that can be used directly in the AFP-based G-TFSF formulation to model the effect of the PML. while If the time-domain field with the PML is the field without the PML is , the transfer function is given by (1)

where

is the Fourier transform. The transfer function would be used, for instance, in Eqs. (45)–(47) in [11] to account for the presence of the PML. However, in our implementation we used a slightly simplified approximation of the transfer function as will be described.

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Fig. 3. Magnitude of the transfer function as a function of discretization at depths of 2, 4, and 6 cells into the PML.

Fig. 3 shows the magnitude of the PML transfer function vs. discretization when measured at a cell that is at a depth of either 2, 4, or 6 cells (the overall thickness of the PML is 8 cells). There were 2048 time-steps in the simulation (zero-padding was used after the time-gating to obtain the desired number of points). . The simulations were run at the 2-D Courant limit of This plot correspond to the first 249 non-dc frequency bins after Fourier transforming the time-domain data. Thus, the lowest non-dc frequency corresponds to approximately 1448 points per wavelength while the highest frequency corresponds to approximately 5.8 points per wavelength. (The discretization is given where is the bin number. An of zero by corresponds to dc.) From Fig. 3 one see that the transfer function magnitude is nearly (but not perfectly) constant with respect to frequency (or discretization). Thus, the attenuation experienced by fields as they propagate into the PML is nearly independent of frequency. This is the basis of the implementation that Anantha and Taflove [15] where a single constant was used to describe the effect of the PML. However, phase is not independent of frequency. Fig. 4 shows the phase of the transfer function as a function of frequency or, more precisely, frequency bin. The data shown are for the same depths and span the same discretizations as shown in Fig. 3. The phase of the transfer function represents the difference in phase between when the PML is and is not present. If one is to use a real constant to model the transfer function, ideally the phase of the transfer function would be zero for all frequencies. At a depth of 2 cells, the phase of the transfer function remains very small, varying from approximately 0.0017 degrees to just under 0.357 deat 1448 points per wavelength grees at 5.8 points per wavelength . This amount of phase difference would not be a practical concern in most applications. However, as seen in Fig. 4, at greater depths, the phase change imparted by the PML can be much more substantial. The phase shift is nearly 35 degrees at a depth of 6 cells and a discretization of 5.8 points per wavelength. For a PML of typical size (i.e., in the neighborhood of eight cells), to obtain a high-fidelity simulation where the penetration of the TFSF boundary

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Fig. 4. Phase of the transfer function as a function of frequency at depths of 2, 4, and 6 cells into the PML.

into PML does not cause significant spurious radiation, it is necessary to account for this phase shift. In principle, it is possible to measure the magnitude and phase of the transfer function at each spectral bin and use that in a given simulation. However, we use a simplified approximation to the transfer function that still provides excellent fidelity. Like Anantha and Taflove, we use a constant for the magnitude. For a given depth, this constant is obtained by averaging a portion of the data shown in Fig. 3. (The average is taken over frequencies that fall within the full-width half-maximum spectrum of the Ricker wavelet pulse that was used for illumination. This effectively discards the highest and lowest frequencies from the average.) For the phase, one notes from Fig. 4 that the phase varies nearly (but not perfectly) linearly with frequency. Thus, we fit a straight line to the phase data shown in Fig. 4. These simplifications allows the calibration data to be decoupled from the duration of the actual simulation of interest. Thus, knowing that the magnitude is constant and the phase varies linearly, one can easily calculate the transfer-function coefficient for any frequency of interest (and it will not matter if the calibration data were calculated with 2048 time-steps while the simulation of interest may use, for example, 5000 time-steps or any other value). (In the CPML implementation used here, the maximum conductivity was obtained from (7.66) of [3] with a polynomial . The and (or ) parameters grading exponent of that can be used to tune a CPML were set to 1.0 and 0.0, respectively. PML parameters other than the ones we employed may make it so that the magnitude is not approximated well by a constant or the phase variation is not approximated well by simple linear variation. That does not change the basic underlying approach being advocated here. For such cases one can simply approximate the frequency-dependent variations in magnitude and phase with higher-order functions, i.e., functions other than a constant or straight line.) In the standard AFP TFSF method the entire time-domain description of the incident field at every node adjacent to the TFSF boundary is pre-computed, stored, and then recalled as needed during the time-stepping of an FDTD simulation. The fact that the incident field in an AFP G-TFSF simulation contains both the incoming and the reflected fields does not change

Fig. 5. A corner illuminated by a field that first encounters the edge of the corner, i.e., the incident field originated in the homogeneous space to the left of the corner. The two places where the TFSF boundary passes into the PML are enclosed in an oval.

the way calibration is done. As indicated previously, this is a consequence of the angle of reflection being equal to the angle of incidence: the way in which the PML affects the magnitude and phase of these two fields is the same. III. OTHER GEOMETRIES For the knife-edge problem shown in Fig. 1, the geometry of the TFSF boundary would not change if, say, one wanted to consider the diffraction from a 90-degree PEC corner. In fact, the corner could have any angle from 0 degrees (the knife-edge problem) to 180 degrees (a perfectly flat plane). However, the geometry does change if the incident angle is such that the incoming field encounters the edge before the rest of the half-plane. This scenario is depicted in Fig. 5 where the corner is illuminated by a plane wave that originates in the homogeneous region to the left of the corner. (We will identify this as the free space region even though it could be any homogeneous dielectric material.) In Fig. 5 the corner is drawn as a 90-degree corner, but any angle is permissible. Furthermore, the corner material is arbitrary. It could either be a PEC or dielectric. In this scenario there is no reflected plane wave: the incident field is comprised solely of the “incoming field.” The TFSF boundary is again two-sided but now each side passes through the PML and is terminated at the edge of the grid. Both these edges are in the free-space region. Two sets of calibration need to be performed to account for the fact that the edges are orthogonal to each other. Nevertheless, the way in which calibration is done is essentially unchanged from the description in the previous section. The only change is the angle of the incoming field. Using a two-sided PML, it is also possible to model the illumination of a dielectric corner when the incident field originates in the half-space region. This is depicted in Fig. 6 where the incident field is introduced from the left and consists of the incoming field, the reflected field, and the transmitted field. In this case one side of the TFSF boundary is terminated in the PML in the dielectric (at the bottom of the figure) and the other side is terminated in the PML in free space (along the right side of the figure). For the case of a penetrable dielectric as shown in Fig. 6, the calibration must be performed in a slightly different way than depicted in Fig. 2. Instead, the calibration set-up is as shown

SCHNEIDER AND CHEN: INCORPORATING THE G-TFSF CONCEPT INTO THE ANALYTIC FIELD PROPAGATION TFSF METHOD

Fig. 6. A dielectric corner where the incident field originated in the halfspace region to the left. The incident field consists of the incoming field, the reflected field, and the transmitted field. There are two places where the TFSF boundary passes into the PML. These are indicated with an oval: one in the dielectric and one in free space.

Fig. 7. Calibration set-up to determined the transfer function for the PML when the TFSF boundary is in the dielectric region. (a) The measured nodes are embedded inside the PML. (b) The measured nodes are outside the PML.

in Fig. 7. In this case the AFP method is used to introduce the incident field in a half-space problem (with no discontinuity in the dielectric). As drawn, the transfer functions obtained for the “measured” nodes in Fig. 7 would be used for the nodes adjacent to the TFSF boundary in the PML in the dielectric region shown at the bottom of Fig. 6. It is important to note that although the incoming and reflected field can be characterized by a single angle of incidence (or reflection), this is not true of the transmitted field. As described in [11], owing to the dispersion inherent in the FDTD grid, the angle of transmission is frequency dependent. Because of this, and the anisotropy of the FDTD grid, the calibration measurement is done in such a way as to reproduce this phenomenon. The distance between the PML and the interface is unimportant. As was done in Fig. 2, the field is measured both with and without the PML present at the measured nodes. Transforming from the time domain to the

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Fig. 8. Field at the observation point shown in Fig. 1 versus time. The simulation is constructed so that over the duration of the observation only the incoming field has passed into the PML. Since the observation point is in the scattered-field region, ideally the field would be zero.

Fig. 9. Field at the observation point shown in Fig. 6 versus time. Similar to Fig. 8, the simulation is constructed so that over the duration of the observation only the transmitted field has passed into the PML. Since the observation point is in the scattered-field region, ideally the field would be zero.

frequency domain and then normalizing the PML-present measurement by the PML-absent measurement gives the transfer function. For the dielectric case we again find it is a sufficiently good approximation to treat the magnitude as a constant and model the phase variation as a linear function of frequency. IV. RESULTS There is an observation point indicated in the upper right portion of the scattered-field region in Fig. 1. This point is two points above the TFSF boundary and two points to the left of the start of the PML. Hence, ideally, the field at this point should be zero until scattered fields propagate to this point from a discontinuity in the total-field region. Fig. 8 shows the fields at this observation point when one does and does not account for the effect the PML has on the phase of the incident field. (Thus one plot corresponds to the field that is present using the original G-TFSF formulation and the other plot corresponds to the AFP-

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Fig. 10. Snapshots of a dielectric corner when the illumination originates in the half-space region to the left of the corner. These gray-scale images use four decades of logarithmic compression such that fields larger than one ten-thousandths of the peak value appear as non-white. Snapshots were taken at time step (a) 180, (b) 230, (c) 280, (d) 330, (e) 380, and (f) 430. See the text for further details.

based G-TFSF formulation being described here.) In the simulation the incoming field was a unit-amplitude Ricker wavelet discretized such that there were 20 points per wavelength at the most energetic frequency of the pulse. The simulation was run at the 2D Courant limit of and the incident angle corresponds 60 degrees as drawn in Fig. 1. Over the duration shown here, no scattered fields from within the total-field region had arrived at that observation point. The fact that the fields at the observation point are non-zero is a consequence of the spurious leaking of the incoming field as it propagates along the portion of the TFSF boundary embedded in the PML.

If one corrects for only the magnitude change caused by the PML, the maximum of the spurious field is found to be . On the other hand, by correcting for both 1.732 the magnitude and phase change caused by the PML, the maximum of the spurious field drops to 3.037 , i.e., a reduction in the spurious field of slightly more than 35 dB. Fig. 9 is similar to Fig. 8 except now the observation point is in the dielectric scattered-field region as indicated in Fig. 6. (The observation point appears in the lower left portion of the scattered-field region in Fig. 6.) This point is two cells above the PML and two cells to the left of the TFSF boundary. The

SCHNEIDER AND CHEN: INCORPORATING THE G-TFSF CONCEPT INTO THE ANALYTIC FIELD PROPAGATION TFSF METHOD

incident angle is 60 degrees and the relative permittivity of the dielectric is 4. The simulation is run at the 2D Courant limit and the incident field is a unit amplitude Ricker wavelet discretized such that there are 40 points per wavelength at the most energetic frequency in free space (thus there are 20 points per wavelength in the dielectric). Fig. 9 shows the electric field at the observation point when there is a correction of only the magnitude and when there is a correction of magnitude and phase. Again, the ideal solution is the absence of any field. The improvement observed in Fig. 9 by incorporating the phase information is not as dramatic as in Fig. 8. In this case, when only the magnitude is corrected, the maximum of the spu. When the phase is also corrious field is 1.630 . This reprerected, the maximum drops to 2.308 sents an improvement of slightly less than 17 dB. Fig. 10 shows snapshots of a simulation in which a pulsed, polarized Ricker wavelet plane wave unit-amplitude, illuminates a dielectric corner. This illumination, discretization, and dielectric constant are the same as described for the calibration measurement. The illumination starts in the half-space region. The TFSF boundary is two-sided and the incident field, consisting of the incoming, reflected, and transmitted waves, emerges from the TFSF “fully formed” for the half-space problem [8], [9], [11], [12]. The computational domain is 201 161 cells. The figures are drawn to scale and the locations of the TFSF boundary, the dielectric, and the PML are indicated. The PML is eight cells thick and spans the edges of the grid (even though, for clarity, the PML is only explicitly draw in the corners). These snapshots use four decades of logarithmic compression such that any field greater than one ten-thousandth of the peak field will show up as non-white. Fig. 10(a) shows the field at time-step 180 when the leading edge of the incoming field has arrived at the discontinuity at the corner. This figure indicates the direction of travel of the incoming, reflected, and transmitted waves by drawing a line along planes of constant phase. For the sake of illustration, these lines are extended into the scattered-field region even though there is no incident field in this region. Fig. 10(c) shows the field at time-step 280 when the incoming field has propagated beyond the right side of the grid. The important thing to notice is that there is no leaked field visible in the scattered-field region—the only fields present in the scattered-field region are those associated with the scattering from the corner. Fig. 10(d) and (e), taken at time-steps 330 and 380, respectively, show the fields after the transmitted wave has encountered the bottom PML. Although difficult to see, close inspection of these figures shows faint non-white patches adjacent to the TFSF boundary. However, these patches are confined to the PML region—there is no visible leakage into the interior of the grid. One additional note is that the AFP technique is able to provide the incident field at any point in the grid (i.e., the incident field that would exist without the discontinuity). Thus, even for points within the total-field region, one can obtain the scattered field simply by subtracting the incident field provided by the AFP method. V. CONCLUSION We have shown that the G-TFSF concept can be folded into an AFP TFSF framework. By incorporating both magnitude and

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phase information about the effect the PML has on the incident field, one can simply terminate the TFSF boundary at the edge of the computational grid—there is no need for a four-sided TFSF boundary. This new approach also ensures less leakage while using smaller PML regions than used in the original G-TFSF work. REFERENCES [1] D. E. Merewether, R. Fisher, and F. W. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci., vol. 27, no. 6, pp. 1829–1833, Dec. 1980. [2] K. R. Umashankar and A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat., vol. EMC-24, no. 4, pp. 397–405, Nov. 1982. [3] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3 ed. Boston, MA: Artech House, 2005. [4] C. Guiffaut and K. Mahdjoubi, “Perfect wideband plane wave injector for FDTD method,” in Proc. IEEE Antennas and Propagat. Soc. Int. Symp., Salt Lake City, UT, Jul. 2000, vol. 1, pp. 236–239. [5] U. Oguz and L. Gürel, “Interpolation techniques to improve the accuracy of the plane wave excitations in the finite difference time domain method,” Radio Sci., vol. 32, no. 6, pp. 2189–2199, Nov.–Dec. 1997. [6] U. Oguz and L. Gürel, “An efficient and accurate technique for the incident-wave excitations in the FDTD method,” IEEE Trans. Microwave Theory Tech., vol. 46, no. 6, pp. 869–882, Jun. 1998. [7] T. Tan and M. Potter, “1-d multipoint auxiliary source propagator for the total-field/scattered-field FDTD formulation,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 144–148, 2007. [8] T. Martin and L. Pettersson, “Modified fresnel coefficients for Huygen’s sources in FDTD,” Appl. Comput. Electromagn. Society J., vol. 17, no. 1, pp. 30–41, Mar. 2002. [9] C. D. Moss, F. L. Teixeira, and J. A. Kong, “Analysis and compensation of numerical dispersion in the FDTD method for layered, anisotropic media,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1174–1184, Sep. 2002. [10] J. B. Schneider, “Plane waves in FDTD simulations and a nearly perfect total-field/scattered-field boundary,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3280–3287, Dec. 2004. [11] J. B. Schneider and K. Abdijalilov, “Analytic field propagation TFSF boundary for FDTD problems involving planar interfaces: PECs, TE, and TM,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2531–2542, Sep. 2006. [12] K. Abdijalilov and J. B. Schneider, “Analytic field propagation TFSF boundary for FDTD problems involving planar interfaces: Lossy material and evanescent fields,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 454–458, 2006. [13] A. Taflove, “Review of the formulation and applications of the finitedifference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion, vol. 10, no. 6, pp. 547–582, 1988. [14] J. B. Schneider and C. L. Wagner, “FDTD dispersion revisited: Fasterthan-light propagation,” IEEE Microw. Guided Wave Lett., vol. 9, no. 2, pp. 54–56, Feb. 1999. [15] V. Anantha and A. Taflove, “Efficient modeling of infinite scatterers using a generalized total-field/scattered-field FDTD boundary partially embedded within PML,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1337–1349, Oct. 2002. [16] J.-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys., vol. 114, no. 2, pp. 185–200, 1994. [17] J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett., vol. 27, no. 5, pp. 334–339, Dec. 2000. [18] Z. Wu and J. Fang, “High-performance PML algorithms,” IEEE Microw. Guided Wave Lett., vol. 6, no. 9, pp. 335–337, Sep. 1996. [19] W. C. Chew and J. M. Jin, “Perfectly matched layers in the discretized space: An analysis and optimization,” Electromagnetics, vol. 16, no. 4, pp. 325–340, 1996. [20] J. Fang and Z. Wu, “Closed-form expression of numerical reflection coefficient at PML interfaces and optimization of PML performance,” IEEE Microw. Guided Wave Lett., vol. 6, no. 9, pp. 332–334, Sep. 1996. [21] J.-P. Bérenger, “Evanescent waves in PML’s: Origin of the numerical reflection in wave-structure interaction problems,” IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 1497–1503, Oct. 1999.

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John B. Schneider (M’91–SM’10) received the B.S. degree in electrical engineering from Tulane University, New Orleans, LA, and the M.S. and Ph.D. degrees in electrical engineering from the University of Washington, Seattle. He is presently an Associate Professor in the School of Electrical Engineering and Computer Science, Washington State University, Pullman. His research interests include the use of computational methods to analyze acoustic, elastic, and electromagnetic wave propagation. Prof. Schneider is a former recipient of the Office of Naval Research Young Investigator Award. He has served as an Associate Editor of the Journal of the Acoustical Society of America and is the Co-Chair of the Technical Program Committee for the 2011 IEEE International Symposium on Antennas and Propagation in Spokane, Washington.

Zhen Chen (S’09) was born in Changsha, China, in 1987. He received the B.S. degree (with high distinction) from Huazhong University of Science and Technology Wuhan, China, in 2009. He is currently a Research Assistant at Washington State University, Pullman, where he is pursuing a doctoral degree involving the application and enhancement of the finite-difference time-domain method.

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A New Boundary Closure Scheme for the Multiresolution Time-Domain (MRTD) Method Pengfei Yao and Shan Zhao

Abstract—This paper introduces a novel boundary closure treatment for the wavelet based multiresolution time-domain (MRTD) solution of Maxwell’s equations. Accommodating non-trivial boundary conditions, such as the Robin condition or time dependent condition, has been a challenging issue in the MRTD analysis of wave scattering, radiation, and propagation. A matched interface and boundary (MIB) method is introduced to overcome this difficulty. Several numerical benchmark tests are carried out to validate the MIB boundary scheme. The proposed boundary treatment can also be applied to other high order finite-difference time-domain (FDTD) approaches, such as the dispersion-relation-preserving (DRP) method. The MIB boundary scheme greatly enhances the feasibility for applying the MRTD methods to more complicated electromagnetic structures. Index Terms—Convergence of numerical methods, finite difference time domain methods.

I. INTRODUCTION

I

T is well known that numerical dispersion is a major limiting factor for the applicability of the finite-difference timedomain (FDTD) scheme to electromagnetic problems involving electrically large structures. Typically, the dimensions of the scatterer in such problems greatly exceed the wavelength of the incident wave so that the grid size required by using the FDTD method could become prohibitively expensive. In order to circumvent this difficulty, numerical approaches that are able to accurately represent the wave solution by using only a few points per wavelength must be employed to relieve the computational cost. This motivates the development of many low dispersion or high order FDTD methods in the past two decades, including FDTD(2,4) scheme [1], wavelet based multiresolution time-domain (MRTD) methods [2], [3], Fourier pseudospectral time-domain (PSTD) methods [4], [5], dispersion-relation-preserving (DRP) FDTD methods [6], [7], and local spectral time-domain (LSTD) methods [8], [9], etc. What are more related to the present paper are the MRTD methods, even though the proposed boundary closure scheme can be applied to other long stencil FDTD approaches, such as the DRP-FDTD or LSTD. The original MRTD schemes Manuscript received August 10, 2010; revised November 22, 2010; accepted February 15, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by NSF grants DMS0616704, DMS-0731503, and DMS-1016579, and by a UA Research Grants Committee Award. The authors are with the Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161441

[2], [3] are derived using cubic spline Battle-Lemarie scaling and wavelet functions. When electromagnetic fields are expanded solely in terms of scaling functions, the corresponding MRTD scheme is usually called as an S-MRTD scheme, while W-MRTD scheme refers to a scheme in which both scaling and wavelet functions are used as basis functions. Since the orthogonal Battle-Lemarie wavelet family is not compactly supported, various MRTD methods using compactly supported wavelet expansions have been developed in the literature. For example, novel MRTD methods based on Coifman and Daubechies scaling functions have been introduced, respectively, in [10] and [11]. A general framework for constructing MRTD algorithms based on biorthogonal scaling and wavelet functions has been established in [12], with a particular realization given to the Cohen-Daubechies-Feauveau (CDF) biorthogonal wavelets. The stencil length of the Daubechies type MRTD methods can be adjusted by the number of vanishing moments, while the latter also determines the order of accuracy of the resulting MRTD spatial discretization [13]. In [14], a systematic procedure is proposed to update the time in the MRTD calculations by using a novel Runge-Kutta scheme so that an arbitrarily high order of convergence in both space and time could be realized. The Fourier dispersive error analysis of some MRTD schemes and a comparison with the standard high order FDTD schemes have been conducted in [15]. The MRTD methods all use wide stencils. Thus, special boundary treatments are required near boundaries where the MRTD approximation may refer to nodes outside the computational domain [16], [17]. Like other time-domain approaches, the perfectly matched layer (PML) absorbing boundary conditions can be naturally incorporated into the collocation procedure of the MRTD schemes [18], while the simple image principle is commonly used in the MRTD calculations to deal with perfect electric conducting (PEC) or perfect magnetic conducting (PMC) boundary conditions [2]. In order to handle the PEC conditions in different scenarios, several advanced boundary closure schemes [19]–[21] have been introduced to the MRTD analysis. The generalization of the image principle at the PEC walls of the multi-layer dielectric structures has been formulated in [19]. An elegant extension of the CDF-MRTD schemes to treat thin metallic irises or infinitely thin perfect electric walls has been presented in [20]. Modification of basis expansion has been suggested in [21] for the purpose of implementing the image principle for the MRTD methods with basis functions being non-symmetric and/or without interpolation property. In summary, the existing MRTD boundary treatments can only handle some regular boundary conditions, such as the PML, PEC, and PMC conditions. No general boundary

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closure procedure is available for the MRTD methods to accommodate more complicated boundary conditions, such as the Robin/mixed condition. This greatly limits the possible application of the MRTD methods to more general electromagnetic calculations. The objective of the present work is to construct a general procedure to implement nontrivial boundary conditions in the MRTD discretization. This is accomplished by introducing a fictitious domain boundary closure via the matched interface and boundary (MIB) method. For regular domain with straight boundaries, the MIB boundary scheme has been constructed for supporting arbitrarily high order central finite difference methods [16], [17]. Successive implementations of the MIB scheme for treating curved dielectric interfaces [22], [23] and curved PEC walls [24] have also been carried out. In the present study, to illustrate the proposed boundary closure scheme, the CDF S-MRTD method [12] will be employed. We note that the proposed procedure can be extended to other S-MRTD and W-MRTD methods. The rest of this paper is organized as the follows. Section II is devoted to the theory and algorithm of the MIB boundary closure scheme. Numerical tests involving Robin and time-dependent boundary conditions are carried out to validate the proposed method in Section III. Finally, a conclusion ends this paper.

TABLE I COEFFICIENTS FOR THE CDF-MRTD SCHEME [12]

where is an appropriate scaling function for the particular scheme being used, and the unknown field expansion coeffi, , and are time depencients dent. The staggered Yee grid can be naturally employed in such a MRTD expansion. In a homogeneous medium, the MRTD semi-discretization of Maxwell’s equations (1) and (2) can be given as

(6) (7)

II. THEORY AND ALGORITHM (8)

A. Multiresolution Time-Domain (MRTD) Analysis Assuming the absence of charge density and current source, and linear isotropic constitutive relations, we consider the transverse magnetic (TM) modes that are governed by the time-dependent two-dimensional (2D) Maxwell’s equations (1) (2) where and are, respectively, the normalized electric and magnetic field intensities and and are the relative electric permittitivy and magnetic permeability of material, respectively. Here, a nondimensional form of the equations is considered, i.e., in free space. Throughout, the medium is assumed . to be nonmagnetic with In the multiresolution time-domain (MRTD) analysis, basis functions can be chosen as scaling functions only or both scaling and wavelet functions. To illustrate the proposed boundary procedure, we concern ourselves with the S-MRTD schemes, i.e., using scaling functions only (3) (4) (5)

and are the spacing in and directions. In the where present MRTD calculations, the Cohen-Daubechies-Feauveau and (CDF) MRTD coefficients [12] will be employed for are given in the Table I. For such MRTD methods, the spatial order of accuracy of the CDF(2, ) biorthogonal family is known [13]. Thus, the order of accuracy in space for the to be CDF(2,2), CDF(2,4), and CDF(2,6) MRTD schemes is, respectively, four, six, and eight. The temporal discretization of (6)–(8) can be simply formulated by using various standard time stepping methods. The classical fourth order Runge-Kutta method will be utilized such that both spatial and temporal orders of accuracy are at least four for the present MRTD analysis. However, the overall high order of accuracy of the present MRTD approach can still be impaired by deficient treatments of boundary conditions, particularly when complicated boundary conditions are encountered. B. Image Principle In the MRTD methods, the image principle is commonly used to deal with the perfect electric conducting (PEC) and perfect magnetic conducting (PMC) boundary conditions. This is a treatment to implement simple boundary conditions by assuming that there is a one-to-one correspondence between the inner grid node and the imaging fictitious node outside the domain. For example, for a problem interval being , consider a uniform grid with . Here the actual mesh size of the partition is , while

YAO AND ZHAO: A NEW BOUNDARY CLOSURE SCHEME FOR THE MRTD METHOD

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represents the maximal number of fictitious points needed in a particular MRTD scheme, in order to ensure the MRTD spatial . Consider approximations (6)–(8) throughout the domain the right boundary as an example and denote a function to be either or . One could assume the following one-to-one image principle at the boundary (9) . In the PEC condition, we have for at the right end . By using Maxwell’s equations, one can at [24]. Thus, the further derive that PEC condition for can be satisfied by choosing in (9). This is also called an anti-symmetric boundary exten, i.e., , can sion [17]. The PEC condition for in (9). This is also called a symbe imposed by taking metric boundary extension [17]. The image principle (9) can actually handle some boundary conditions that are more complicated than the PEC ones. Please see [17] for more details.

this stencil. Based on such a partition, the boundary condition (10) is discretized to be (12)

C. Matched Interface and Boundary (MIB) Treatment However, for more general boundary conditions, such as the Robin condition, the image principle (9) can not be rigorously valid or can only be satisfied up to second order accuracy. Obviously, a more reasonable assumption is that a fictitious value should not depend on one inner value only, it should depend on a set of function values inside the boundary. This motivates the development of the matched interface and boundary (MIB) boundary scheme [16], [17]. We illustrate the idea by considering a Robin type boundary condition at (10) The MIB boundary treatment assumes a function relation which is generalized from the image principle: (11) are the MIB representation coefficients and . Thus, in the MIB discretization, each fictitious value outside the domain will depend on inner values and one inhomogeneous boundary value . An iterative procedure is commonly used in the MIB method to determine these fictitious values one by one. At the first step, since only one boundary condition is avail, able, one can only determine one fictitious point see the Fig. 1. In the MIB scheme, the first derivative in the Robin condition (10) will be approximated by using one-sided grid points inside and one ficfinite difference with titious node outside. In particular, we choose as the differentiation point of the finite difference and denote it as . the origin of the local grid stencil: We denote the corresponding finite difference weights to be . Here the subscript of the weight is for the local grid index, while the superscript represents the fact that there are totally nodes in where

Fig. 1. Illustration of the MIB grid partition and the iterative procedure. Filled circles: regular nodes; Partially filled circles: solved fictitious nodes; Open circles: unsolved fictitious nodes.

in (12) can be solved in terms of The only unknown other values, giving rise to the following representation coeffi, , and cients: for . At the second step, we are about to determine the second , see Fig. 1. One possible way is to fictitious value in a process similar to that in the first step, by solve also considering only one fictitious node outside the domain. A more accurate treatment is usually employed in the MIB scheme by considering two fictitious nodes simultaneously. The same boundary condition (10) is now discretized as (13) are the finite difference weights to approxiwhere mate first derivative at based on a local grid stencil: . Here the superscript innodes involved in this dicates that there are totally in terms approximation. From (13), one can solve of the others. Then, the known representation coefficients for can be substituted in so that will also deregular function values and . Consequently, pend on for can the representation coefficients be attained. Through such an iterative procedure, the total fictitious points can be efficiently determined in steps, see Fig. 1. The MIB treatment of other boundary conditions can be similarly carried out. We note that in the MIB method, boundary conditions are enforced systematically so that it can achieve arbitrarily high orders in principle. In practice, the order of accuracy of the MIB scheme is dominated by the total number of interior support nodes . One has certain flexibility in choosing in the finite different approximation. A large value is usually selected to ensure that the order of accuracy in the MIB boundary treatment is not less than that of the MRTD scheme. Nevertheless, for unsteady problems, a very large may render the MIB method

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unstable. The stability issue of the MIB method has been discussed in [17]. To guarantee the stability, one should choose according to the upbounds established in [17]. An advantage of the generalized image principle (11) is are independent of the that the representation coefficients . Consequently, they are time invariant. boundary data Thus, the MIB scheme actually needs to be carried out only takes different value once at the beginning, even though at different time. Therefore, the computational overhead introduced by the MIB boundary treatment is negligibly small in real MRTD computations. In summary, the MIB method provides a fictitious domain support so that the MRTD methods can be applied in a translation invariant manner throughout the domain. Furthermore, since the proposed MIB boundary treatment does not depend on the MRTD discretizations, this boundary closure scheme can be applied to other high order finite-difference time-domain (FDTD) methods.

TABLE II THE MRTD RESULTS OF EXAMPLE 1 BY USING THE IMAGE PRINCIPLE

0

Here 1.482( 3) denotes 1:482 Case 2, m = n = 10.

2 10

. In Case 1, m = n = 5 and in

TABLE III THE MRTD RESULTS OF EXAMPLE 1 BY USING THE MIB SCHEME

III. NUMERICAL EXPERIMENTS In this section, we examine the usefulness of the MIB boundary treatment by testing its robustness, accuracy, and convergence. Three MRTD schemes, i.e., the CDF(2,2), CDF(2,4), and CDF(2,6), are employed for the spatial discretization and the classical fourth order Runge-Kutta method is used for the temporal integration. Based on the given initial values at time , Maxwell’s equations (1) and (2) will be solved until a . Here we choose for all tests in our stopping time non-dimensionalized unit system. A uniform grid is employed being the mesh size along each in all examples, with direction. Unless otherwise specified, a small time increment is used to ensure that the temporal discretization error is negligible in our present tests. The absolute errors will be reported in all cases. A. Example 1: Hollow Rectangular Waveguide We first validate the MIB boundary closure scheme by considering a air filled rectangular waveguide with perfect conducting walls [9]. Designed to solve complicated boundary conditions, the MIB boundary closure scheme can also handle the simple PEC boundary conditions. Moreover, such a study actually enables us to compare the MIB boundary method with the image principle. The cross section of the hollow waveguide is chosen as . Such a simple structure permits analytical solutions:

(14) where , and and are the wavenumbers. For the present example, two PEC conditions at for magnetic components are involved: and , and at and . For , at four PEC walls. According to we originally have

In Case 1, m =

n

= 5 and in Case 2, m = n = 10.

Maxwell’s equations, it is easy to derive two electric PEC conditions: at and , and at and . With little modification, the MIB treatment discussed in the previous section can be applied to solve these four PEC conditions. In the present computations, the initial values are taken according to the analytical solutions. The physical parameters are and . Two test cases are examchosen as in Case 1 and in Case 2. ined with The high frequency solutions in Case 2 are particularly important to investigate the performance of high order methods. The numerical results of the image principle and the MIB boundary scheme are listed, respectively, in Table II and Table III. The image principle is obviously satisfied for the present analytical solutions (14). Thus, the numerical errors reported in Table II are primarily due to the MRTD spatial discretizations. As mentioned above, the order of accuracy for the CDF(2,2), CDF(2,4), and CDF(2,6) MRTD schemes is, respectively, four, six, and eight. Such orders have been numerically verified in the low frequency case and the high frequency case with a large . With the confidence of the present MRTD spatial discretization, we now turn to the MIB boundary closure. A large enough is used in the MIB scheme. In particular, we choose , 5, and 7, respectively, for the CDF(2,2), CDF(2,4), and CDF(2,6) MRTD schemes. It can seen from the Table III that the MIB results are in good agreement with those of the image principle,

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Fig. 2. Numerical convergence rates of the MIB-MRTD method. (a) Example 1, Case 1; (b) Example 1, Case 2; (c) Example 2, Case 1; (d) Example 2, Case 2; (e) Example 3, Case 1; (f) Example 3, Case 2; (g) Example 4, Case 1; (h) Example 4, Case 2. In all cases, the solid line represents the least squares fitted linear trend. The slope of this line reveals the overall numerical order of the spatial discretization, and is labeled on the graph.

with very minor differences. This validates the MIB boundary closure scheme. We further examine the order of convergence of the MIB-MRTD spatial discretization by plotting the numerical errors in chart (a) and (b) of Fig. 2. These errors are shown as dashed lines. A linear fitting by means of the least squares is then conducted for each case in the log-log scale. The corresponding fitted convergence lines are depicted as solid lines in Fig. 2. Moreover, the fitted slope essentially represents the numerical convergence rate of the MIB-MRTD method. These rates are shown in Fig. 2 too. It is interesting to note that there are some differences between the convergence patterns of the low frequency test (Case 1) and the high frequency test (Case 2). It is known that the high frequency test is challenging to low order methods or coarse numerical meshes. Thus, the , overall rate of CDF(2,2) for the Case 2 is only in Table III. although it attains a value of 3.67 when , all three MRTD schemes On the other hand, with perform poorly. We even have that the higher order method

yields a larger error. This is because the grid resolution is too low in this case, i.e., only 4 grid points per wavelength (PPW). With such a low PPW, only spectral type methods, such as pseudospectral time-domain (PSTD) methods [4], [5] or local spectral time-domain (LSTD) methods [8], [9], can deliver reasonable accuracy. For the present problem, the MRTD methods is larger. Eventually, the numerical perform better when orders of both the CDF(2,4) the CDF(2,6) are quite close to the theoretical ones. are The currently chosen MIB parameter values of within the upbounds given in [17] so that the MIB-MRTD computations are guaranteed to be conditionally stable. An interesting question next is whether the MIB scheme will affect the Courant-Friedrichs-Levy (CFL) factor of the underlying MRTD method or not. To this end, we detect the numerical CFL numbers of the MRTD discretization with or without MIB boundary closure. In particular, we consider the Case I with and a stopping time . Denote the total number . We numerically of time steps to be . We have

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TABLE IV THE NUMERICAL CFL NUMBERS FOR EXAMPLE 1

TABLE V THE RESULTS OF THE MIB-MRTD METHOD FOR EXAMPLE 2

In Case 1,

Fig. 3. Computational domain of Example 2 (left) and Example 3 (right).

search for the critical values such that the computation is still stable. Then, the numerical CFL number is reported to be . See Table IV. It is clear that the CFL numbers with and without MIB scheme are essentially the same. This is in consistent with our previous findings that for hyperbolic equations, the MIB method maintains the same CFL number as the underlying spatial method whenever it is stable [17].

m = n = 5 and in Case 2, m = n = 10.

mating the second derivative. In this manner, the MIB scheme can be proceeded as in the previous studies. The numerical results of the MIB-MRTD method are reported in Table V. These results are also depicted in chart (c) and (d) of Fig. 2. It can be seen that the MIB-MRTD schemes attain the correct numerical orders in both low and high frequency tests. Furthermore, the convergence patterns of the high frequency case become better, in comparing with those of the Example 1. This is because a smaller computation domain in the Example 2 actually implies a larger PPW. The present result shows that the overall performance of the MIB scheme in handling complicated boundaries is satisfactory.

B. Example 2: Shifted Computational Domain We next study a synthetic example based on the hollow rectangular waveguide. Consider the same physical setting as in the Example 1, but with a shifted computational domain . Denote four boundaries of this new domain to be , , , and , see Fig. 3(a). We then have and , although that the PEC conditions are not valid on the same analytical solutions are assumed. Instead, the correct boundary conditions are constructed as the follows: (15) (16) (17) (18) The image principle obviously cannot handle these complicated boundary conditions. conditions (15) and (16) can The MIB treatment of two be carried out similarly. Nevertheless, a subtle point needs to be taken care of in solving magnetic boundary conditions (17) and (18). We illustrate this by considering (17) as an example. By using a staggered grid, there is actually no grid node for located exactly on the boundary point . Thus, the second term in (17), i.e., , cannot be directly evalvalue shall be uated in the MIB discretization. Instead, the interpolated based on the same grid stencil used for approxi-

C. Example 3: Rectangular Open Cavity We then consider a 2D rectangular open cavity embedded in an infinite ground plane, see Fig. 3(b). The study of electromagnetic scattering by such a cavity is of great industrial and military interests, because the open cavity can be regarded as a prototype structure of a more realistic one, such as a jet engine inlet duct or exhaust nozzle [25]. Denote the width of the cavity as and the depth as . The cavity walls and ground plane are usually assumed to be PEC boundaries. The original cavity problem is defined on cavity and the half space above the ground plane with Sommerfeld’s radiation conditions imposed at infinity. A modern approach to solve cavity problem is to introduce a transparent boundary condition above the cavity [25], in Fig. 3(b). This induces a computational domain i.e., along within the rectangular cavity. In the present study, we numerically solve 2D Maxwell’s equations (1) and (2) within the rectangular cavity. To benchmark our numerical results, a set of analytical solutions is constructed as the follows:

(19) where , and and are the wavenumbers. Note that the time harmonic part of the analytical solution (19) actually represents a single mode of the mode-

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TABLE VI THE RESULTS OF THE MIB-MRTD METHOD FOR EXAMPLE 3

In Case 1, n

= 5 and k = 1 and in Case 2, n = 10 and k = 5.

matching solution which converges to the exact solution of the scattering problem [25]. With the PEC conditions on , , and , the Robin type boundary conditions are assumed on : (20) (21) The MIB scheme for conditions (20) and (21) can be carried out exactly as what was described in Section II. We note that even though the boundary data is a spatial function along the , the MIB representation boundary and time variant, i.e., coefficients solved from boundary conditions (20) and (21) are still time independent and independent. Thus, it is sufficient to conduct the MIB treatment only once. The solved representation coefficients can then be applied at any time and at any node along . Therefore, the MIB boundary treatment is computationally very efficient. In the present study, the physical parameters are chosen as and . Again, two test cases are studied with and in Case 1 and and in Case 2. The numerical results are shown in Table VI and chart (e) and (f) of Fig. 2. It can be seen that the correct spatial order of accuracy is achieved in all tests. We note that the MRTD errors are greater for the high frequency case. This is bethan 0.2 when cause the present solution is highly oscillatory along direction, while is subject to a very rapid exponential growth along direction. See Fig. 4. Such a solution is very difficult to be resolved on a coarse grid. Nevertheless, when we refine the mesh, a satisfactory accuracy is achieved by the MIB-MRTD scheme. D. Example 4: A Test Without Analytical Solution We finally consider a test without analytical solution. The hollow rectangular waveguide with four PEC boundaries in the Example 1 is studied again. However, we consider the structure being excited by Gaussian type pulses initially. Such a study can be used to numerically predict the cutoff frequencies of the are considered structure [9]. Two initial values of (22) (23)

=1

Fig. 4. Plot of numerical solution E at time t by using the CDF(2,6) and N . (a): Example 3, Case 2; (b): Example 4, Case 2.

= 80

respectively, for Case 1 and Case 2, while the initial values of and are chosen as zero. Here , , and . The PEC conditions hold in Case 2, while the initial solution of Case 1 satisfies the PEC conditions approximately, since the Gaussian decays to a negligibly small values at the . boundaries. We then integrate Maxwell’s equations to No analytical solution is available to benchmark our MIB results. Thus, an “exact solution” obtained by the MIB-MRTD is used as the referwith CDF(2,6) and a dense mesh ence. In comparing with the reference solution, the MIB-MRTD errors are shown in Table VII and chart (g) and (h) of Fig. 2. It can be seen that the correct orders are numerically achieved, even though the solution is highly irregular, see Fig. 4(b). IV. CONCLUSION In conclusion, we have introduced a novel boundary closure scheme, the matched interface and boundary (MIB) method, for the treatment of general boundary conditions in the multiresolution time-domain (MRTD) calculations of Maxwell’s equations. In the MIB method, boundary conditions are repeatedly utilized to systematically determine a set of fictitious values outside the domain. Consequently, the MRTD approximation can be applied in a translation invariant manner near the boundary.

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TABLE VII THE RESULTS OF THE MIB-MRTD METHOD FOR EXAMPLE 4

Several numerical experiments have been carried out to demonstrate the robustness of the MIB scheme in handling complicated boundary conditions, such as Robin and/or time-dependent ones. The MIB boundary treatment can achieve arbitrarily high order accuracy in principle. In the present study, the MIB orders are guaranteed to be not less than that of the underlying MRTD spatial discretization, so that the MIB-MRTD methods achieve the theoretical orders in all numerical tests. The MIB coefficient generation can be carried out only once to deal with boundary conditions with spatial and temporal dependent inhomogeneous terms. Thus, the MIB boundary treatment is computationally cheap. The MIB fictitious domain treatment does not assume any a priori knowledge of wave solutions, so that it has no limitation to be applied to real world electromagnetic problems. The MIB treatment of nontrivial boundary conditions on irregular domains is currently under our consideration. REFERENCES [1] J. Fang, “Time domain finite difference computation for Maxwell’s equations,” Ph.D. dissertation, Department of Electrical Engineering, Univ. California, Berkeley, CA, 1989. [2] M. Krumpholz and L. P. B. Katehi, “New time-domain schemes based on multiresolution analysis,” IEEE Trans. Microw. Theory Tech., vol. 44, pp. 555–571, 1996. [3] L. P. B. Katehi, J. F. Harvey, and E. Tentzeris, “Time-domain analysis using multiresolution expansions,” in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, Ed. Boston, MA: Artech House, 1998. [4] Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Techn. Lett., vol. 15, pp. 158–165, 1997. [5] Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Remote Sens., vol. 37, pp. 917–926, 1999. [6] S. Wang, F. L. Teixeira, R. Lee, and J.-F. Lee, “Dispersion-relation preserving (DRP) 2D finite-difference time-domain schemes,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., San Antonio, Tx, 2002, vol. 3, pp. 264–267. [7] S. Wang and F. L. Teixeira, “Dispersion-relation-preserving (DRP) FDTD algorithms for large-scale three dimensional problems,” IEEE Trans. Antennas Propag., vol. 51, pp. 1818–1828, 2003. [8] G. Bao, G. W. Wei, and S. Zhao, “Local spectral time-domain method for electromagnetic wave propagation,” Opt. Lett., vol. 28, pp. 513–515, 2003. [9] Z. H. Shao, G. W. Wei, and S. Zhao, “DSC time-domain solution of Maxwell’s equations,” J. Comput. Phys., vol. 189, pp. 427–453, 2003. [10] X. Wei, E. Li, and C. Liang, “A new MRTD scheme based on Coifman scaling functions for the solution of scattering problems,” IEEE Microw. Wireless Compon. Lett., vol. 12, pp. 392–394, 2002.

[11] M. Fujii and W. J. R. Hoefer, “Time-domain wavelet Galerkin modeling of two-dimensional electrically large dielectric waveguides,” IEEE Trans. Microw. Theory Tech., vol. 49, pp. 886–892, 2001. [12] T. Dogaru and L. Carin, “Multiresolution time-domain using CDF biorthogonal wavelets,” IEEE Trans. Microw. Theory Tech., vol. 49, pp. 902–912, 2001. [13] N. Kovvali, W. Lin, and L. Carin, “Order of accuracy analysis for multiresolution time-domain using Daubechies bases,” Microw. Opt. Tech. Lett., vol. 45, pp. 290–293, 2005. [14] Q. S. Cao, R. Kanapady, and F. Reitich, “High-order Runge-Kutta multiresolution time-domain methods for computational electromagnetics,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 3316–3326, 2006. [15] K. L. Shlager and J. B. Schneider, “Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes,” IEEE Trans. Antennas Propag., vol. 52, pp. 1095–1104, 2004. [16] S. Zhao, “On the spurious solutions in the high-order finite difference methods,” Comput. Method Appl. Mech. Engrg., vol. 196, pp. 5031–5046, 2007. [17] S. Zhao and G. W. Wei, “Matched interface and boundary (MIB) method for the implementation of boundary conditions in high-order central finite differences,” Int. J. Numer. Method Engrg., vol. 77, pp. 1690–1730, 2009. [18] E. M. Tentzeris, R. L. Robertson, J. F. Harvey, and L. P. B. Katehi, “PML absorbing boundary conditions for the characterization of open microwave circuit components using multiresolution time-domain techniques (MRTD),” IEEE Trans. Antennas Propag., vol. 47, pp. 1709–1715, 1999. [19] Q. S. Cao, Y. C. Chen, and R. Mittra, “Multiple image technique (MIT) and anisotropic perfectly matched layer (APML) in implementation of MRTD scheme for boundary truncations of microwave structures,” IEEE Trans. Microw. Theory Tech., vol. 50, pp. 1578–1589, 2002. [20] M. Peschke and W. Menzel, “Investigation of boundary algorithm for multiresolution analysis,” IEEE Trans. Microw. Theory Tech., vol. 51, pp. 1262–1268, 2003. [21] N. Kovvali, W. Lin, and L. Carin, “Image technique for multiresolution time-domain using nonsymmetric basis functions,” Microw. Opt. Tech. Lett., vol. 47, pp. 44–47, 2005. [22] S. Zhao, “High order vectorial analysis of waveguides with curved dielectric interfaces,” IEEE Microw. Wireless Compon. Lett., vol. 19, pp. 266–268, 2009. [23] S. Zhao, “High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces,” J. Comput. Phys., vol. 229, pp. 3155–3170, 2010. [24] S. Zhao, “A fourth order finite difference method for waveguides with curved perfectly conducting boundaries,” Comput. Method Appl. Mech. Engrg., vol. 199, pp. 2655–2662, 2010. [25] G. Bao and W. Zhang, “An improved mode-matching method for large cavities,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 393–396, 2005.

Pengfei Yao was born in Hangzhou, China, in 1983. He received the B.Sc degree in mathematics from Shantou University, China, in 2006. Currently, he is a graduate student in the Department of Mathematics, The University of Alabama, Tuscaloosa. His research interests include high order numerical methods for partial differential equations, and computational electromagnetics.

Shan Zhao was born in Guiyang, China, in 1974. He received the B.Sc. degree in mathematics from Lanzhou University, China, in 1997 and the Ph.D. degree in scientific computing from the National University of Singapore, in 2003. From 2003 to 2006, he was a Postdoctoral Fellow with Michigan State University. In 2006, he joined the faculty of the Mathematics Department, University of Alabama, Tuscaloosa, as an Assistant Professor. His current research interests include high order methods for partial differential equations, fast electromagnetic simulation in inhomogeneous media, and mathematical modeling of biomolecular surfaces.

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The Magnetostatic Frill Source Ted Simpson, Life Senior Member, IEEE

Abstract—The magnetostatic frill source (MsFS) is introduced as a convenient way to obtain the electric field from a coaxial aperture in a ground plane. This approximation simplifies the analysis of arbitrarily shaped monopole antennas since both radial and axial components of the electric field may be expressed in closed form. Antenna characteristics computed using the MsFS are shown to agree very well with theoretical and measured results for cylindrical, conical and spheroidal monopoles. To demonstrate the utility of this model, it is shown that a single filamentary loop of magnetic current equal to the driving voltage and with radius equal to the geometric mean of inner and outer radii of the coaxial aperture is quite adequate for most purposes. Limitations of this approximation are discussed. Index Terms—Antenna feeds, antenna input admittance, electric field integral equations (EFIE), Galerkin’s method, method of moments (MoM).

I. INTRODUCTION

T

HE time-harmonic magnetic frill [1]–[7] source model has proven adequate for the analysis of cylindrical antennas fed by a coaxial line through an aperture in a ground plane. For cylindrical monopoles where only the axial component of the incident field is required, the simplification developed in [2] is both convenient and accurate. Unfortunately, for antennas of arbitrary shape that require the radial component, one must resort to both numerical integration and differentiation [1], which is not simple at all. Some improvement in this approach [3] avoided numerical differentiation but only at the price of evaluating numerical integrals of two truncated power series. A simple impulsive source [3] retained the simplicity of the delta-function source and yielded stable convergence under refinement but was neither highly accurate nor capable of yielding the radial field component. Recently a simple source was proposed [4] that offered both axial and radial fields but was not very accurate, as we shall see. Thus an alternative model is still to be desired. A single loop of magnetic current, mentioned briefly by Collin [7], may fill this need. In this paper we set out to determine how well such a simple device performs when applied to the analysis of a variety of antennas. II. FORMULATION In this study we assume that the surface currents on a coaxially driven monopole are to be determined using the electric field integral equation (EFIE) with the method of Manuscript received January 18, 2011; revised February 01, 2011; accepted February 28, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The author is with the Department of Electrical Engineering, University of South Carolina, Columbia, SC 29210 USA (e-mail: [email protected], [email protected]). Digital Object Identifier 10.1109/TAP.2011.2161565

moments (MoM). With this method one obtains the solution where consists of [9] in terms of a matrix equation segment-to-segment coupling terms, the basis coefficients for the current, and the incident voltages (inner products of the incident tangential electric field and the basis functions) across each segment. Here we consider a simpler way to derive the incident field from the electric field in the coaxial aperture that preserves the physical nature of the magnetic frill. As an example of an antenna fed by a coaxial line, consider the prolate spheroidal monopole depicted in Fig. 1(a). The electric field in the coaxial aperture is assumed to be that of the transverse electromagnetic (TEM) mode (1) where is the radial coordinate, and and are the inner and outer radii of the coaxial line, respectively; it is assumed that the radius of the outer conductor is chosen small enough that only the TEM mode propagates in the coaxial line. The equivalence principle [6] enables the physical problem shown in Fig. 1(a) to be replaced by its equivalent in Fig. 1(b). Further simplification results from the use of image theory, yielding the dipole antenna in Fig. 1(c) driven by a frill of magnetic surface current with density (2) Note that in calculations, . For the magnetic current in (2) the electric vector potential [7] is given by (3) , is where is the aperture surface, the distance from the source point to the observation point, and is the wavenumber in free space, . The corresponding electric field is given by (4) Tsai [1] evaluated (4) numerically using different expressions appropriate to different distances from the source. Butler and Tsai [2] improved upon this, at least for the component of the field, enabling the component of the electric field from the frill to be expressed as (5) If the coaxial line is chosen so that it will support only the TEM mode, the higher order modes (generated in the transition

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Note that (6) is an exact expression for the -component of the electrostatic field; unfortunately, a similar expression for the radial component is unavailable. However, for the magnetostatic case, there is an alternative way to compute the radial field as well. This will be addressed next. Consider a single filamentary loop of magnetic current with radius concentric with the -axis and lying in the . From the solution [9] to the dual problem of obtaining the magnetic field of a loop of electric current, it is easy to show that the electric vector potential for a -directed loop of magnetic current is

(7) with and . Substituting (7) into (4) yields [10] both components of the electric field as (8a) (8b)

Fig. 1. Steps in developing a model for analyzing (a) a coaxial line feeding a spheroidal monopole antenna over a ground plane; (b) magnetic current frill replaces coax aperture; (c) image replaces image plane.

region) contribute only weakly to the evanescent electric field that excites the antenna. These fields decay rapidly with distance and, as Butler and Tsai [2] show, have imaginary parts that are several orders of magnitude less than their real parts and may be considered negligible except in special cases, for example, cutoff. Here we when the coaxial line is used near the will assume that the electric field incident on the antenna is significant only within an electrically small region where and , permitting (5) to be written as (6) where

is the complete elliptic integral of the first kind, , , and .

and where is the complete elliptic integral of the second kind. For a one-loop approximation (1-loop source), the total , obtained by integrating (2) over magnetic current the frill and considered to be flowing in a loop of effective radius , may be substituted for (8a) and (8b). in (8a) and (8b) may be replaced For greater accuracy, , by by the differential current , and these expressions integrated numerically over the aper. A singularity encountered at may ture be removed analytically [1], if necessary. However, by using Gaussian integration [11], thereby avoiding the evaluation of the integrand at the limits, it has been found that fairly accurate results may be obtained. In particular, a 3-point Gaussian integral was found adequate; for brevity in what follows this is referred to as the 3-loop model. III. AXIAL FIELDS vanishes as a result of symmetry, and Along the -axis the -component of the electric fields in (6), (8a), and (8b) take on simple forms that are convenient for validating results from the complicated but more general expression. Both (6) and (8) and involve a parameter that vanishes as ; it follows then that along the -axis (6) becomes (9) Similarly, on the -axis (8b) becomes (10)

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TABLE I COMPARISON OF Re(E ) FROM [2] WITH QUASI-STATIC E FROM 1-LOOP AND 3-LOOP SOURCES FOR a = 0:003  AND b = 0:005  ALONG A 45 LINE

In the next section we will investigate the accuracy of results obtained by approximating the electric field incident upon the antenna by one or more loops of magnetostatic current to see how well this type of source model performs when used in the analysis of typical coax-driven monopole antennas. IV. COMPARING 1- AND 3-LOOP MAGNETOSTATIC SOURCES WITH THE OTHER SOURCE MODELS of a single loop could be chosen so that (10) If the radius were a good approximation for (9), then this simple model could also be used to compute the electric field over the region elsewhere, subject to later verification. To this end, RMS values were calculated along the axis for typof ical coaxial lines having and over the range . The result was that this discrepancy was mini. Thus, for convenience, mized for as the effective radius for a 1-loop we will regard source. As pointed out in [2], (5) is exact and applies to all observacalculated tion points not on the frill proper. A comparison of from both (5) (from [2, Table I]), and (8b) is therefore of interest here and is presented in Table I. Since the imaginary part of from (5) is negligible, only the real part was shown. Values of ) the fields are shown in Table I at points along a 45 line ( with increasing . While there is only modest agreement with 1-loop results, there is very good agreement with 3-loop results. Nonetheless, the 1-loop model is still useful for its simplicity and, as may be seen, it yields antenna properties that are often indistinguishable from those obtained by more accurate methods. The appealing simplicity of the 1-loop MsFS is tempered somewhat by its limitations. However, by comparing the admittance vs. frequency curves plotted in Fig. 2, it may be seen that it obviously outperforms other simple source models [4], [5] even for a cylindrical monopole where only the axial component of the incident field is involved. And, although it strictly applies only in the low-frequency limit, it has been shown to be reasonably accurate for full-wave analysis for conventional . Except for the most demanding coaxial lines with situations, this limitation may be removed by the use of loworder Gaussian integrals. In the following examples, the use of a 3-point Gaussian integral is referred to as a “3-loop” source.

Fig. 2. The admittance G + jB of a cylindrical monopole [11] vs. frequency calculated using three source models, the 1-loop MsFS, Trintinalia’s [5] source, and Glisson’s source [4].

V. CURRENT DISTRIBUTION ON A THIN HALF-WAVE MONOPOLE ANTENNA It is well-known that the input current for a thin cylindrical antenna almost vanishes at second resonance; this occurs when the (monopole) antenna length is approximately one-half wavelength long. Therefore, to put an approximate source to a rigorous test, the half-wave case is a good place to start. Since all theoretical models involve approximations of one sort or another, both measured as well as theoretical current distributions will be shown for comparison. In a careful study of the feed region, Butler, Michalski, and Filopovic [11] modeled both the coaxial line (back to a point sufficiently remote from the aperture to avoid higher modes near the ultimate source) as well as the antenna. In Fig. 3, the measured and computed current distribution in [11] agrees very well with that obtained using a 1-loop source. In this case, with , the 1-loop model is obviously adequate, at least for graphical display, for calculating the current on this thin cylin. For large ratios of and for drical antenna with thicker antennas, 3-point integration will be shown to provide a more accurate model of the frill. VI. ADMITTANCE AND IMPEDANCE FOR THREE MONOPOLE ANTENNAS ) Using a 1-loop source, the admittance of a thin ( half-wave cylindrical antenna was calculated and plotted along with both measured and theoretical results [12] in Fig. 4 with excellent agreement. Clear agreement with the theory used by Martin [13] and the 1-loop source is also apparent, although both depart slightly from his measurements. The coaxial aperture in and one might expect the 1-loop results this case has to differ somewhat from the measurements; however, although this is not shown, the 3-loop source gave almost identical results and so were not plotted on this rather cluttered graph. We can only infer from Fig. 3 that the 1-loop source model gave excellent results in comparison to the full-wave theory that was used in [13].

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Fig. 3. Comparing current distributions calculated using a 1-Loop source with theoretical and measured current from [12]. Fig. 5. Comparison of theoretical and measured [14] admittance of a conical antenna.

Fig. 4. Comparison of theoretical and measured input admittance of a cylindrical monopole antenna [13].

Fig. 6. Comparison of theoretical and measured [15] admittance of a spheroidal antenna.

Next, a conical monopole used as a probe in the study of the dielectric properties of fluids [14] was analyzed with both 1- and 3-loop sources and the calculated impedance plotted in Fig. 5 along with measurements from Smith and Nordgard [14]. As in the previous example, the 1-loop results were indistinguishable from the 3-loop results, so only one was shown. Unlike the previous case, however, this antenna is quite thick with and a coaxial aperture with . The measured results disagree only slightly with the 3-loop results transformed through the indicated 2 mm to the surface of the Teflon bead to which the measured data was referenced. Using a 10-point Gaussian integral did not change the calculated impedance. Since no theoretical results were given here (other than the rational function

computed from the measurements themselves), we can only remark that the discrepancy is encouragingly small. As an example of a case requiring the radial incident field, the admittance of the rather thick spheroidal monopole antenna shown in Fig. 1 was calculated using a 3-loop with source and plotted in Fig. 6 along with theoretical and measured results obtained by Smith and Capoglu [15]. The agreement here is excellent even for the full-wave case. Since these theoretical results were obtained from a rigorous solution to the boundary-value problem in which the spheroid is illuminated by an incident field from the same coaxial aperture field (in time-harmonic form) given in (1) and used in this paper, the agreement with 3-loop results is the strongest evidence yet

SIMPSON: THE MAGNETOSTATIC FRILL SOURCE

TABLE II COMPARISON BETWEEN INPUT ADMITTANCES (mS) COMPUTED WITH 30 SEGMENTS FOR 1- AND 3-LOOP SOURCES FOR SEVERAL ANTENNAS

shown for the validity of this approximate source. And, as before, 1-loop results were found to be indistinguishable for those with a 3-loop source. To clarify the difference in the accuracy of the results discussed above, the input admittances of all four antennas studied were computed using both these approximate source models and at critical electrical lengths where the susceptance had a steep slope and listed in Table II. The differences between 1-loop and 3-loop results are obvious, since they agree only to about three significant figures. However, this difference is too small to observe in a plot; this is why both sets of results were not shown. VII. CONCLUSIONS The MsFS was shown to provide a very accurate and useful approximation for the numerical solution of the electric field integral equation (EFIE) applied to monopole antennas of arbitrary shape and driven by a coaxial line over an indefinitely large ground plane. Its advantage over the full-wave treatment [1], [2], except where extreme accuracy is required, is its simplicity. Unlike other alternative source models [4] and [5], it is physically meaningful and does not require ad hoc adjustment. An additional advantage of the use of the MsFS for frequency sweeps is that since it is independent of frequency it need not be re-computed for each frequency. REFERENCES [1] L. L. Tsai, “A numerical solution for the near and far fields of an annular ring of magnetic current,” IEEE Trans. Antennas Propag., vol. 20, no. 5, pp. 569–576, 1972. [2] C. M. Butler and L. L. Tsai, “An alternative frill field formulation,” IEEE Trans. Antennas Propag., vol. 21, no. 1, pp. 115–116, 1973. [3] A. Sakitani and S. Egashira, “Simplified expressions for the near fields of a magnetic frill current,” IEEE Trans. Antennas Propag., vol. 34, no. 8, pp. 1059–1062, 1986.

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[4] G. P. Junker, A. A. Kishk, and A. W. Glisson, “A novel delta-gap source model for center-fed cylindrical dipoles,” IEEE Trans. Antennas Propag., vol. 43, no. 5, pp. 537–540, 1995. [5] L. C. Trintinalia, “Simple excitation model for coaxial driven monopole antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1907–1912, 2010. [6] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961, pp. 110–111. [7] R. E. Collin, Antennas and Radiowave Propagation. New York: McGraw-Hill, 1985, pp. 467–472. [8] T. L. Simpson, “The disk loaded monopole antenna,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 542–550, 2004. [9] A. W. Glisson and D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag., vol. 28, no. 5, pp. 593–603, 1980. [10] J. Van Bladel, Electromagnetic Fields. New York: Hemisphere Publishing, 1985, pp. 155–156. [11] S. C. Chapra and R. P. Canale, Numerical Methods for Engineers. New York: McGraw-Hill, 1985, sec. 14.2. [12] C. M. Butler, K. A. Michalski, and S. Filipovic, “An analysis of the coax-fed monopole in a general medium,” Naval Ocean Systems Center, San Diego, CA, NOSC CR 220. [13] A. Q. Martin, “An analytical investigation of an axially directed antenna in the presence of an infinite conducting cylindrical tube,” Ph.D. dissertation, Clemson Univ., Clemson, SC, 1989. [14] G. S. Smith and J. D. Nordgard, “Measurement of the electrical parameters of materials using antennas,” IEEE Trans. Antennas Propag., vol. 33, no. 7, pp. 783–792, 1985. [15] I. R. Capoglu and G. S. Smith, “The input admittance of a prolate-spheroidal monopole antenna fed by a magnetic frill,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 572–585, 2006.

Ted Simpson (M’56–SM’72–LSM’83) was born in 1935 in Knoxville, TN. He received the BSEE and MSEE degrees from the University of Tennessee in Knoxville, and the Ph.D. degree in engineering and applied physics from Harvard University, Cambridge, MS, in 1970. He was employed by Deco Electronics in Leesburg, VA, from 1956 to 1968, where his principle work was in the design of military antennas with particular attention to high-power VLF transmitting antennas. He joined the faculty of the Department of Electrical Engineering, University of South Carolina, in 1968 where he remained until 1999 when he retired with the faculty rank of Professor Emeritus; during his tenure there he was appointed as a Visiting Professor to the Naval Ocean Systems Center (now SPAWAR) in San Diego, CA, where he worked with the shipboard antenna group. In addition, he spent the 1975–76 year with Electrospace Systems in Richardson, TX, working with the military antenna group. Prof. Simpson was an undergraduate member of the Society of the Sigma Xi, Tau Beta Pi and Eta Kappa Nu. He has served as a Reviewer for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and IEEE Antennas and Propagation Magazine and as Local Arrangements Chairman for the 2009 IEEE APS/URSI symposium in Charleston, SC.

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Generation of Excitation-Independent Characteristic Basis Functions for Three-Dimensional Homogeneous Dielectric Bodies Jaime Laviada, Member, IEEE, Marcos R. Pino, and Fernando Las-Heras, Senior Member, IEEE

Abstract—The aim of this paper is the development of characteristic basis functions (CBFs) for homogeneous dielectric bodies that are modeled with the surface equivalence principle. This approach contributes to generalize the characteristic basis function method (CBFM) for any kind of material. The foundations for the generation of these new basis functions are widely revised and the CBFM is adapted for this purpose. The validity of the method is illustrated with the systematic analysis of canonical geometries as well as with the computation of the specific absorption rate inside a head-like geometry. In all the cases, the error in the current distribution as well as in the near- and far-field is considered. In addition, a detailed comparison of the generation of the CBFs using plane waves and spherical waves is also carried out. Index Terms—Characteristic basis function method, dielectrics, method of moments.

I. INTRODUCTION HE method of moments (MoM) [1] is one of the most versatile numerical strategies for dealing with electromagnetic problems. One of the keystones of the MoM is the expansion of the unknown function into a set of known basis, namely basis functions. In Harrington’s original work [1], he pointed that these functions should be chosen according to: i) the accuracy of the solution; ii) the ease of evaluation of the matrix elements; iii) the realization of a well-conditioned matrix. For instance, one of the basis functions that satisfy the previous criteria and that have been widely and successfully used in the literature are the Rao-Wilton-Glisson (RWG) basis functions [2]. Many researchers have focused on creating aggregate functions of low-level basis functions (e.g., the aforementioned RWG basis functions) that result in a smaller number of unknowns without a significant loss of accuracy. The main advantages of employing these basis functions are: i) reduction

T

Manuscript received July 11, 2010; revised February 23, 2011; accepted March 04, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported by the Ministerio de Ciencia e Innovación of Spain/FEDER under projects TEC2008-01638/TEC (INVEMTA) and CONSOLIDER-INGENIO CSD2008-00068 (TERASENSE); by Gobierno del Principado de Asturias (PCTI)/FEDER-FSE under projects EQUIP08-06, FC-09-COF09-12, EQUIP10-31 and PC10-06, and by Cátedra Telefónica-Universidad de Oviedo. The authors are with the Departamento de Ingeniería eléctrica, University of Oviedo, Spain (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161545

of the memory consume because of the smaller size of the system of equations matrix; ii) better condition number due to the elimination of redundancy; iii) possibility of direct solution (e.g., an LU factorization) of larger problems avoiding the potential converge problems of the iterative schemes. Among the pioneering works on this field, we find the characteristic modes [3], [4] of Harrington that are computed as the eigenvectors of the system of equations matrix. After the Harrington’s research, the work on this field has been dispersed. For instance, it is worthwhile to mention the multilevel method of moments [5] that is based on an iterative scheme that combines a hierarchical aggregation of the basis functions for solving two dimensional scattering problems. This multilevel strategy would be later modified for the analysis of printed circuits [6]. This method is based on breaking the geometry into domains and exciting them with artificial ports to generate the basis functions for each domain. A similar approach would be later followed in [7]. The previous approaches were limited to small geometries or certain shapes (e.g., planar structures). Nevertheless, this problem has been addressed by several authors in the last years. Among other methods, we can cite the synthetic-functions expansion (SFX) [8] or the characteristic basis functions method (CBFM) [9]. Both methods share foundations in order to tackle the analysis of electrically large structures with accurate results. Thus, we will only focus on the CBFM to generate the basis functions, namely characteristic basis functions (CBFs). In particular, we will be based on the methodology as described in [10], [11] since it yields CBFs that are independent of the excitation as opposite to the original “primary and secondary CBFs” approach [9]. This fact together with the possibility of the direct solution of the system of equations enables the efficient analysis of partial modifications [12] or the solution for multiple excitations in a single analysis. This last point is key for the efficient solution of certain problems as, for instance, the computation of monostatic radar cross sections or the synthesis of large phased arrays [13]. It is also worthwhile to mention that the CBFM has been greatly extended in the recent years. Among other methods, it is remarkable the hybridization of the CBFM with the adaptive cross approximation [14] or with the fast multipole method [11] and the extension of the CBFM to a multilevel strategy [15]. The CBFM has been mainly focused in the analysis of PEC bodies (e.g., [10], [11]) thus far. These works are based on the usual formulation for the MoM, where one equivalent problem, with a set of currents in an unbounded and homogeneous

0018-926X/$26.00 © 2011 IEEE

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Fig. 1. Equivalent problems for the analysis of a homogeneous dielectric body. (a) Original Problem, (b) Equivalent outer problem. (c) Equivalent inner problem.

medium, is solved. For these problems, the solution of open isolated pieces of geometry with the electric field integral equation (EFIE) has been proven to yield a complete set of CBFs (e.g., see our discussion in [15]). In the case of dielectric bodies, the solution with the MoM and the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) [16]–[18] formulation involves two equivalent problems instead of only one; each of these problems separately does not correspond to the original physical problem and, furthermore, they need to be combined to take into account both the solution in the inner medium and in the outer medium. Thus, dielectric and PEC bodies analyses are based on different formulations. To our best knowledge, there are not techniques that have been proven to yield a complete set of basis functions for the case of dielectric bodies. In this paper, a procedure involving both equivalent problems is derived to obtain a a set of CBFs that are valid for the overall dielectric problem. Although some approaches are available in the literature to handle metallic elements enclosed by multilayered dielectric media [19], these strategies cannot be directly applied to the case of bodies involving arbitrary-shaped dielectrics since they based on the used of the corresponding Green’s functions which are usually not available. It is also important to note that the analysis of three-dimensional dielectric media has also been recently considered in [20] for the case of arrays analysis. In that work, the generation of the CBFs is based on the “primary and secondary” approach [9] which yields to basis functions that depend on the rest of the geometry. Although this approach is very appropriate for arrays since it typically results in fewer unknowns, it is usually not employed in the case of scattering problems because it generates CBFs that depend on the rest of the geometry. In addition, the CBFs in [20] are generated for relatively small elements that are not connected to each other and, thus, it is possible to generate the CBFs by considering the entire element without the need of truncating it. The remainder of this paper is arranged as follows. Firstly, the PMCHWT [16]–[18] formulation to solve dielectric bodies, as well as the involved operators and integral equations are described. In the third section, the CBFs generation on dielectric bodies is introduced and discussed together with several nu-

merical aspects that are relevant for the implementation of the method. Next, the validity of this new kind of basis functions is illustrated with several examples involving canonical geometries as well as a head-like geometry. Finally, the advantages and the drawbacks are discussed. II. SOLUTION OF HOMOGENEOUS BODIES WITH MOM A. PMCHWT Formulation Let us consider the analysis of a homogeneous dielectric body composed of a medium 1 that is surrounded by a medium 0 (see Fig. 1(a)). The analysis of this kind of problems using the surface equivalence principle (SEP) is based on the solution of an inner (see Fig. 1(b)) and an outer (see Fig. 1(c)) equivalent problems involving equivalent currents radiating in an homogeneous medium [21]. In the case of the outer problem, the boundary conditions on the surface of the dielectric body, , imply that the equivalent and , respectively, must electric and magnetic currents, and , be related to the total electric and magnetic fields, respectively, as (1) (2) where the total fields are the superposition of the impressed incident fields plus the scattered fields, i.e., and . The fields scattered by the equivalent currents can be expressed as (3) (4) or 1) is the complex impedance of the medium where ( computed as1

1In this paper, we will assume that the media have no magnetic losses and, therefore, the complex permeability  ^ is equal to the conventional permeability  .

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 9, SEPTEMBER 2011

is the complex permittivity of the medium (12)

where and are the relative permittivity and the conductivity of the medium , respectively; is the vacuum permittivity and is the angular frequency. The operators and are defined as

The solution of (11) and (12), which constitute the PMCHWT formulation, enables us the computation of the equivalent currents on the surface of the dielectric and, therefore, the evaluation of the electromagnetic fields everywhere inside or outside the dielectric body. B. Discretization of the Integrals

(5)

In order to solve the previous equations with the method of moments [1], the electric and magnetic currents are expanded in terms of a known set of basis functions, , as

(6) (13) Thus, the operator represents the electric field radiated by a certain electric current placed on the medium . This operator and) radiated also represents the magnetic field (scaled by by a magnetic current on the medium . On the other hand, the represents the electric field (scaled by and) operator radiated by a magnetic current or also the magnetic field radiated by an electric current. Expanding the total field as the impressed field plus the scattered field and inserting (3) into (2), and (4) into (1), we can formulate the electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) for the outer problems, which are only valid on , as

(14) and are the unknown weights of the expansions where for the electric and magnetic currents, respectively. In this case, the same basis functions have been used to expand the electric and magnetic currents but other choices are also possible (e.g., [23]). Next, the test functions are chosen equal to the basis functions (Galerkin’s method) yielding the following matrix equation: (15)

(7)

where the system of equations matrix is

(8) where the 1/2 factor is due to the fact of considering the limit of the tangential component of (6) when the observation point approaches [22]. In this case, both operators can be interpreted as Cauchy principal values. A similar development can be carried out for the inner problem yielding the following equations on the surface (9)

(16) and the entries of each submatrix are given by the following reaction terms: (17) (18) where tions:

denotes the symmetric product between two func-

(10) are the electric and magnetic equivalent curwhere and rents for the inner problem. , then the boundary conditions for the conSince and tinuity of the tangential fields forces that . Henceforth, we unify the notation as and . The addition of (7) and (9) together with the addition of (8) and (10) yields the following equations for the surface

The unknowns and right hand side vectors are given by (19) where the

th entry of these subvectors is given by (20) (21) (22)

(11)

(23)

LAVIADA et al.: GENERATION OF EXCITATION-INDEPENDENT CBFs FOR 3-D HOMOGENEOUS DIELECTRIC BODIES

III. SOLUTION OF HOMOGENEOUS DIELECTRICS WITH THE CBFM A. Generation of the CBFs If the geometry is decomposed into non-overlapped pieces and the electric and (blocks) [10] and denoting with magnetic current on the th block, then the total current distribution can be expressed as:

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and Similarly, are the magnetic fields radiated by the electric and magnetic currents of the rest of the blocks. The magnetic field incoming to the th block can also be approximated using a set of modes as previously

(29) (24) (25)

These modes that expand the magnetic field must be related with those that expand the electric field through the Maxwell’s equations and, therefore, must satisfy the following relationship:

If (24) is inserted into (11) and (25) into (12), then the two following integral equations can be formulated for the field radiated by the electric and magnetic currents on the th block:

Since the incident field can also be incorporated to the previous expansions, then (26) and (27) can be expressed as

(30)

(26) (31)

(27)

The terms represent the electric field on the th block due to the electric currents outside the th block considering that the currents are on an homogeneous medium with same constitutive parameters than the ones of the medium . On a similar way, represents the electric field radiated by the magnetic currents of the rest of blocks. Since the field incoming to a certain block can be expressed with arbitrary , then we have accuracy in terms of a known set of modes, that (28)

where the coefficients are chosen to model the external field due to the radiation of the currents on the other blocks together with the impressed incident field. Discretizing the previous equations, the following matrix equations are obtained:

(32)

(33) where and are the submatrices containing the reactions terms between the low-level basis function of the th block. The th entry of the vectors and are defined as follows: (34)

are typically spherical waves or plane where the modes waves on the medium , and are the weights of the expansion. The previous expansion would be exact if an infinite number of modes is considered. In practice, it is truncated to a finite number, . Some rules for choosing this parameter are given in [24] for the case of spherical wave expansion. In the case of the plane wave expansions applied to the CBFM, some empirical rules can be found in [15].

(35) where denotes the th basis function belonging to th contains the incident electric field block. In other words, described by the th mode, i.e., , tested by the basis functions of the th block. Similarly, contains the incident magnetic field, which is described by the th mode, i.e., .

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Equations (32) and (33) reveals that the current distribution on the th block must be a linear combination of the currents generated as a consequence of illuminating . Hence, a complete set the block with each mode of basis functions can be constructed with the induced currents due to each mode. Hence, the induced electric and magnetic currents due to each kind of excitation can be simultaneously computed solving the following system of equations:

(36)

where the th column of the matrices and contains the coefficients of the electric and magnetic currents induced in the th block as a consequence of the th mode. Finally, the coefficients of the CBFs for the electric and magnetic currents can be obtained by merging the currents computed and in (36) for

At this point, one could wonder if the two equations in (36) could be solved independently so that the size of the system of equations is halved. One possible way to carry it out would be radiation due to expand into modes the currents in the to the equation so that this expansion could be also incorporated to the right hand side. In this case, the domain supporting the sources of the field overlap with the observation points and, therefore, they are inside the minimum sphere enclosing the block. Since plane waves in the visible spectrum can only model the far-field radiated by a set of sources (see our discussion in [15]), they are clearly not good candidates for this expansion. In the case of using spherical waves, singular modes at the origin (type in [24]) should be used but the use of these waves is discouraged due to numerical instabilities. Hence, it is not numerically recommended to expand the radiation due to the currents into modes in order to decouple the electric and magnetic unknowns in (36). A final remark should be made regarding the generation of the CBFs with the previous equations, and this is that they do not depend on the impressed field of the final problem—i.e., the right hand side in (15)—as opposite to the very extended approach of generating the CBFs with the “primary and secondary” strategy for conducting bodies [9]. Hence, the herein proposed CBFs are very appropriate for problems involving several excitations as, for instance, the computation of monostatic radar cross sections or the synthesis of large arrays [13].

foundations. Firstly, it is important to realize that the linear independency of the CBFs generated by illuminating with the modes is not guaranteed. Thus, a singular value decomposition (SVD) filtering with threshold [10], [11], [15] is applied to and ) in order to each set of currents (i.e., avoid linear dependency. As a consequence, the number of unknowns is considerably reduced and a better compression of the system of equations matrix is achieved. The previous idea is also applied when joining the two different sets of filtered basis functions, i.e., the SVD and is applied to the matrices , where the matrices and result from the application of the first SVD filtering. The threshold for this second SVD is denoted by . The coefficients surviving to these two SVDs will be the final set of CBFs. Due to splitting the geometry into blocks, some edge effects can appear in the boundary of them and, as pointed in [10], it can slow down the convergence of the CBFs. This convergence can be improved by extending the blocks when solving (36) and, after that, discarding the coefficients belonging to the extension [10]. Although no discontinuities of the currents between blocks have been found in this work, several authors have pointed multiple approaches to the connection scheme for blocks that can improve this aspect. Among these approaches, we can cite the use of overlapped and windowed CBFs [14], [25] or the use of conventional RWGs to connect blocks [8]. Both schemes are fully compatible with the method presented in this paper. Once the coefficients of the CBFs have been computed, the new matrix of the system of equations with the reaction terms between CBFs can be computed from the submatrices of (16) with a basis change as described in [10], [11], [15]. Finally, it should be remarked that the presented approach only alters the way for generating the CBFs and, therefore, the computational complexity of the CBFM remains the same [15]. IV. NUMERICAL RESULTS In order to illustrate the previous formulation, a systematic analysis of two canonical geometries—a sphere and a cube—for multiple constitutive parameters is firstly considered. After that, a head-like geometry made of homogeneous dielectric is analyzed for the case of an impinging plane wave. The low-level basis functions that will be used in the remainder sections are the conventional Rao-Wilton-Glisson (RWGs) basis functions [2]. The error in the computation of the weights for these basis functions will be used to evaluate the accuracy of the CBFM strategy. This error is computed separately for the electric and magnetic currents as follows: %

(37)

%

(38)

B. Numerical Aspects It is worthwhile to remark some numerical aspects that were not previously introduced to avoid distracting from the main

and are the coefficients of the low-level where basis functions expansion of the electric and magnetic currents

LAVIADA et al.: GENERATION OF EXCITATION-INDEPENDENT CBFs FOR 3-D HOMOGENEOUS DIELECTRIC BODIES

TABLE I ERROR OF THE CURRENTS COMPUTATION. SPHERE WITH RADIUS a AND  S/M

=0

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= 0:5 M

computed with the CBFM, and and are the reference coefficients computed with the MoM. The number of waves to generate the CBFs is set to 2008 in the case of plane wave (PW) illumination and to 1920 in the case of spherical wave (SW) illumination. Unless otherwise speciand . fied, the SVD thresholds are set to Finally, it is important to remark that the number of unknowns that is used in the remainder of the paper refers to the number of unknowns for each kind of current (electric or magnetic). Therefore, the total number of unknowns for the MoM analyses . is

Fig. 2. Bistatic RCS of a sphere with radius 0.5 m and constitutive parameters " and  S/m at 300 MHz in xz-plane. The incident plane wave is polarized in the x-direction.

=5

=0

A. Canonical Geometries In this first analysis, we consider the study of a dielectric sphere with radius equal to 0.5 m at 300 MHz. The size of the sphere is chosen to be a reasonable electrically small body so the reference results can be computed with the MoM even for large permittivity values. A cube with edges equal to 0.7 m is also considered at the same frequency to observe the possible discrepancies between both geometries. The edges of the cube are placed parallel to the cartesian axes. The geometries are discretized with a mesh of triangles with , where is the wavelength average edge length equal to inside the dielectric medium. In all the cases, the bodies are divided into 8 symmetric blocks that are extended a 30% during the CBFs generation stage to mitigate edge effects. The equivalent currents on the surface of these bodies are computed considering an incident plane wave propagating toand with linear polarization for the electric field in the ward -direction. Firstly, we consider the analysis of the sphere without losses S/m) for multiple permittivity, , values. The error in ( the currents computations is gathered in the Table I. In this example, the error is under 1% except for a few specific cases. Although the results for SWs and PWs, in terms of error in the currents, are similar, the generation of the CBFs with PWs yields some specific cases where the error is much higher than the average. On the other hand, the results with the SWs are more stable. In order to check the effect of the error of the currents on the far-field, we consider the example with the highest error, i.e., the where the error of the magnetic currents is over case with an 8%. The bistatic radar cross section (RCS) on the xz-plane is shown in Fig. 2 with good agreement with respect to the Mie’s

Fig. 3. CBFs/RWG ratio (left axis) and number of CBFs (right axis) for several S/m at permittivity values of a sphere of radius 0.5 m and conductivity  300 MHz.

=0

series. As it will be shown later, this kind of error in the current computation with the CBFM still keeps a very good accuracy even for the near-field calculation. The number of resulting electric and magnetic CBFs for each is compiled in Fig. 3. As it can be seen, this number is increased if the permittivity of the medium is also increased. This is a coherent result as the wavelength for high permittivities is smaller than for low permittivities and, therefore, a higher number of degrees of freedom is expected to be required to model all the possible currents. In addition, Fig. 3 reveals that the ratio of CBFs to RWG2 decreases if the permittivity is increased. This is associated to the fact of keeping the same block decomposition for all the different permittivities. Thus, the blocks are electrically larger for high permittivity values and it results in higher compression rates as observed in [15]. 2This ratio provides a measurement of the compression achieved with the CBFM.

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Fig. 4. CBFs/RWG ratio (left axis) and number of CBFs (right axis) for several at 300 conductivity values of a sphere of radius 0.5 m and permittivity " MHz.

=5

TABLE II ERROR OF THE CURRENTS COMPUTATION. SPHERE WITH RADIUS a AND "

=5

= 0:5 M

As it can be seen in Fig. 3, the number of resulting CBFs is higher in the case of using the SW illumination. It is attributed to the fact of truncating the PWs to the visible spectrum [15] what introduces a limitation that is not present in the case of the SW illumination. Hence, the resulting set of CBFs due to illuminating with SWs is expected to be more complete and, therefore, larger but, also, more accurate. After that, we consider the same sphere with the permittivity while the conductivity ranges from 0 value fixed to S/m to 0.3 S/m. The error in the current computation is shown in Table II. As in the permittivity sweep, the error with SWs is lower than with PWs and also more stable. On the other hand, the number of CBFs and the ratio CBFs to RWG does not change so significantly as the conductivity increases (see Fig. 4) because the changes in the wavelength inside the dielectric medium are also not so significant. Next, the same analyses as previously described are repeated for the cube with edge 0.7 m. The permittivity sweep is firstly accomplished yielding the results shown in Table III. The obtained error values for this example with PW illumination are slightly larger than in the sphere example. For instance, the case involves an error close to the 30% in the magnetic with currents estimation. As in the case of the sphere, Fig. 5 shows without the bistatic RCS of the cube for the worst case

Fig. 5. Bistatic RCS of a cube with edge 0.7 m and constitutive parameters and  S/m at 300 MHz in the xz-plane. The incident plane wave is " polarized in the x-direction.

=8

=0

TABLE III ERROR OF THE CURRENTS COMPUTATION. CUBE WITH EDGE LENGTH l M AND  S/M

=0

= 0:7

significant loss of accuracy in the far-field. On the other hand, the values for SW illumination are also slightly higher than in the case of the sphere but they do not suffer from such high error peaks. The number of retained CBFs as well as the CBF/RWG ratio is shown in Fig. 6 with an identical behavior to the aforementioned one for the sphere geometry. for several conductivity The analysis of the cube with values is compiled in Table IV and Fig. 7. Again, this example shows higher error values than those of the equivalent conductivity sweep for the sphere. In addition, this error is much more significant in the case of PW illumination. The results for the SW illumination are again more accurate than those of PW illumination. B. Dielectric Head In this example, a head-like geometry (see Fig. 8) is conand sidered. The constitutive parameters are set to S/m, that are on the range of the real permittivity and conductivity of many biological tissues [26]. The geometry is discretized in 7336 triangles involving a total number of 22 008 RWG unknowns. The incident field is a plane wave with ampliand electric field polartude of 1 V/m, propagating toward ized in the -direction.

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Fig. 8. Block decomposition of the phantom head geometry. The plane, where the SAR is to be computed, is also shown. Fig. 6. CBFs/RWG ratio (left axis) and number of CBFs (right axis) for several S/m permittivity values of a cube of edge length 0.7 m and conductivity  at 300 MHz.

=0

xy-plane shown in Fig. 8. This rate is defined as: (39)

Fig. 7. CBFs/RWG ratio (left axis) and number of CBFs (right axis) for several at conductivity values of a cube of edge length 0.7 m and permittivity " 300 MHz.

=5

TABLE IV ERROR OF THE CURRENTS COMPUTATION. CUBE WITH EDGE LENGTH l M AND

"

=5

= 0:7

The generation of the CBFs is carried out only with the SW illumination due to its better accuracy with regards to the PW illumination, as observed in the previous examples. In order to evaluate this example, the computation of the Specific Absorption Rate (SAR) at 900 MHz is considered in the

where is the mass density that is set to kg/m . The problem is solved for multiple values while the other in all the cases. The results for threshold is set to the SAR are depicted in Fig. 9 and the corresponding number of CBFs and the error are compiled in Table V. As expected, the decrease of the SVD threshold results in a smaller number of CBFs but also in a higher error. Nevertheless, the error in the SAR computation only seems to affect the points very near to the currents domain and, therefore, there is not an appreciable distortion once the field is computed slightly further from the head surface. As a consequence, a threshold equal to provides very good results for the SAR computation and also good results for the current computation. However, if only the SAR far away from the head limits is of interest, then , improving the compresthe threshold can be relaxed until sion of the CBFM without a significant loss of accuracy in the SAR computation. V. CONCLUSIONS AND DISCUSSION The CBFM has been extended to handle homogeneous dielectric bodies. The methodology continues the previous works on this method that were developed for conductor bodies and, therefore, it yields CBFs that are independent of the impressed fields and of the rest of the geometry. Furthermore, the application of this methodology results in a set of CBFs that is error-controllable by means of the SVD thresholds. This set of basis functions has been rigorously formulated and it tends to be a complete set, i.e., it can model any possible induced current, as long as the SVD thresholds are sufficiently decreased. Although the presented formulation has been focused on the case of the interface between two different dielectric media, it should be easily extended to other formulations (e.g., Mller formulation [27]) or to the case of the joint of three (or more) dielectrics by following an equivalent development. In addition, the CBFs have been generated with plane waves as well as with spherical waves. The latter kind has provided a better behavior in terms of stability and accuracy so they are

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= 50 and  = 1 S/m at 900 MHz. (a) Method of moments, (b) CBFM: t = 10 = 10 .

Fig. 9. SAR results for the head of Fig. 8 with " t ;t , (d) CBFM: t ;t

= 10

= 10

= 10

TABLE V RESULTING CBFS AND THE ASSOCIATED ERROR FOR MULTIPLE SVD THRESHOLDS

preferred. It has also been empirically illustrated that the so-generated CBFs are accurate not only for the far-field computation but also for the near-field computation even on the presence of significant errors in the current estimation. REFERENCES [1] R. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [2] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, pp. 409–418, May 1982.

;t

= 10

, (c) CBFM:

[3] R. F. Harrington and J. R. Mautz, “Theory of characteristic modes for conducting bodies,” IEEE Trans. Antennas Propag., vol. AP-19, pp. 622–628, Sept. 1971. [4] R. F. Harrington and J. R. Mautz, “Computation of characteristic modes for conducting bodies,” IEEE Trans. Antennas Propag., vol. AP-19, pp. 629–639, Sept. 1971. [5] K. Kalbasi and K. R. Demarest, “A multilevel formulation of the method of moments,” IEEE Trans. Antennas Propag., vol. 41, no. 5, pp. 589–599, May 1993. [6] S. Ooms and D. DeZutter, “A new iterative diakoptics-based multilevel moments method for planar circuits,” IEEE Trans. Microw. Theory Tech., vol. MTT-46, no. 3, pp. 280–291, Mar. 1998. [7] E. Suter and J. R. Mosig, “A subdomain multilevel approach for the efficient MoM analysis of large planar antennas,” Microwave Opt. Technol. Lett., vol. 26, no. 4, pp. 270–277, Aug. 2000. [8] L. Matekovits, V. A. Laza, and G. Vechhi, “Analysis of large complex structures with the synthetic-functions approach,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2509–2521, Sept. 2007. [9] V. Prakash and R. Mittra, “Characteristic basis function method: A new technique for efficient solution of method of moments matrix equation,” Microwave Opt. Technol. Lett., vol. 36, no. 2, pp. 95–100, Jan. 2003. [10] E. Lucente, A. Monorchio, and R. Mittra, “An iteration-free MoM approach based on excitation independent characteristic basis functions for solving large multiscale electromagnetic scattering problems,” IEEE Antennas Propag., vol. 56, no. 4, pp. 999–1007, Apr. 2008.

LAVIADA et al.: GENERATION OF EXCITATION-INDEPENDENT CBFs FOR 3-D HOMOGENEOUS DIELECTRIC BODIES

[11] C. Delgado, M. F. Cátedra, and R. Mittra, “Application of the characteristic basis function method utilizing a class of basis and testing functions defined on NURBS patches,” IEEE Antennas Propag., vol. 56, no. 3, pp. 784–791, Mar. 2008. [12] J. Laviada, J. Gutierrez-Meana, M. Pino, and F. Las-Heras, “Analysis of partial modifications on electrically large bodies via characteristic basis functions,” IEEE Trans. Antennas Propag., vol. 9, pp. 834–837, 2010. [13] J. Laviada, R. G. Ayestarán, M. R. Pino, F. Las-Heras, and R. Mittra, “Synthesis of phased arrays in complex environments with the multilevel characteristic basis function method,” Progr. Electromagn. Res., vol. 92, pp. 347–360, 2009. [14] R. Maaskant, R. Mittra, and A. Tijhuis, “Fast analysis of large antenna arrays using the characteristic basis function method and the adaptive cross approximation algorithm,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3440–3451, Nov. 2008. [15] J. Laviada, M. R. Pino, F. Las-Heras, and R. Mittra, “Solution of electrically large problems with multilevel characteristic basis functions,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3189–3198, Oct. 2009. [16] A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in Computer Techniques for Electromagnetics. Oxford, U.K.: Permagon, 1973. [17] Y. Chang and R. F. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag., vol. AP-25, no. 6, pp. 789–795, Nov. 1977. [18] T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci., vol. 12, pp. 709–718, 1977. [19] L. Matekovits, G. Vecchi, M. Bercigli, and M. Bandinelli, “Syntheticfunctions analysis of large aperture-coupled antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1936–1943, July 2009. [20] C. Craeye, T. Gilles, and X. Dardenne, “Efficient full-wave characterization of arrays of antennas embedded in finite dielectric volumes,” Radio Sci., vol. 44, no. 1, Jan.–Feb. 2009. [21] K. Umashankar, A. Taflove, and S. M. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag., vol. AP-34, no. 6, pp. 758–766, June 1986. [22] J. V. Bladel, Waves and Fields in Inhomogeneous Media, 2nd ed. New York: Wiley, 2007. [23] B. H. Jung, T. K. Sarkar, and Y. S. Chung, “A survey of various frequency domain integral equations for the analysis of scattering from three dimensional dielectric objects,” in Proc. IEEE Int. Workshop on Microelectromechanical Systems (MEMS’97), Nagoya, Japan, Jan. 2002, pp. 193–246. [24] J. E. Hansen, Spherical Near-Field Antenna Measurements. London, U.K.: Peter Peregrinus Ltd., 1988. [25] R. Maaskant, R. Mittra, and A. G. Tijhuis, “Application of trapezoidalshaped characteristic basis functions to arrays of electrically interconnected antenna elements,” in Proc. Int. Conf. on Electromagnetics in Advanced Applications, ICEAA, Sep. 2007, pp. 567–571. [26] P. Bernardi, M. Cavagnaro, S. Pisa, and E. Piuzzi, “Specific absorption rate and temperature increases in the head of a cellular-phone user,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 7, pp. 1118–1126, July 2000. [27] C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin, Germany: Springer-Verlag, 1969.

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Jaime Laviada (S’08–M’10) was born in Gijón, Spain, in 1982. He received the M.S. degree in telecommunication engineering and the Ph.D. degree from the University of Oviedo, in 2005 and 2010, respectively. In 2006, he joined the research group Signal Theory and Communications, Department of Electrical Engineering, University of Oviedo. He was a Visiting Scholar in the Electromagnetics and Communications Lab, Pennsylvania State University, during 2007 and 2008. His main research interests are in numerical efficient techniques and antenna pattern synthesis.

Marcos R. Pino was born in Vigo, Spain, in 1972. He received the M.Sc. and Ph.D. degrees in electrical engineering from the University of Vigo, Vigo, Spain, in 1997 and 2000, respectively. During 1998, he was a Visiting Scholar at the ElectroScience Laboratory, The Ohio State University, Columbus (OH). Since 2001, he has been with the Department of Electrical Engineering, University of Oviedo, where he currently has the title of Associate Professor. He is currently teaching courses on electromagnetic fields. His research areas are radar cross section, rough surface scattering, and applied mathematics for computational electromagnetics.

Fernando Las-Heras (M’86–SM’08) was born in Zaragoza, Spain. He received the M.S. (1987) and Ph.D. degrees (1990) in telecommunication engineering from the Technical University of Madrid, Madrid, Spain. He was National Graduate Research Fellow (1988–1990) and Associate Professor (1991–2000) in the Department of Signal, Systems and Radiocommunications, Technical University of Madrid. Since 2001, he has been with the University of Oviedo, heading the research group Signal Theory and Communications (TSC-UNIOVI) in the Department of Electrical Engineering. Since December 2003, he holds a position of Full-Professor at the University of Oviedo where, from 2004 to 2008, he was Vice-Dean of Telecommunication Engineering at the Technical School of Engineering, Gijon. He has been a Visiting Researcher at Syracuse University, New York, and a Visiting Lecturer at the National University of Engineering, Peru, and ESIGELEC, France. He has authored over 280 technical journal and conference papers in the areas of antenna design, the inverse electromagnetic problem with application to diagnostic, measurement and synthesis of antennas, propagation and computational electromagnetics and engineering education.

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Integral Equation Based Domain Decomposition Method for Solving Electromagnetic Wave Scattering From Non-Penetrable Objects Zhen Peng, Member, IEEE, Xiao-Chuan Wang, Student Member, IEEE, and Jin-Fa Lee, Fellow, IEEE

Abstract—The integral equation (IE) method is commonly utilized to model time-harmonic electromagnetic (EM) problems. One of the greatest challenges in its applications arises in the solution of the resulting ill-conditioned matrix equation. We introduce a new domain decomposition method (DDM) for the IE solution of EM wave scattering from non-penetrable objects. The proposed method is a non-overlapping/non-conformal DDM and it provides a computationally efficient and effective preconditioner for the IE matrix equations. Moreover, the proposed approach is very suitable for dealing with multi-scale electromagnetic problems since each sub-domain has its own characteristics length and will be meshed independently. Furthermore, for each sub-domain, we are free to choose the most effective IE sub-domain solver based on its local geometrical features and electromagnetic characteristics. Additionally, the multilevel fast multi-pole algorithm (MLFMA) is utilized to accelerate the computations of couplings between sub-domains. Numerical results demonstrate that the proposed method yields rapid convergence in the outer Krylov iterative solution process. Finally, simulations of several large-scale examples testify to the effectiveness and robustness of the proposed IE based DDM. Index Terms—Domain decomposition, integral equation (IE) method, method of moments, multilevel fast multipole algorithm (MLFMA), scattering.

I. INTRODUCTION NTEGRAL equation (IE) methods have enjoyed considerable success in solving electromagnetic wave scattering and radiation problems involving large complex bodies [1]. It is very attractive for homogeneous or layered homogeneous objects, as only the discretizations of the surface of the object and the discontinuous interfaces between different materials are needed. However, application of the method usually leads to a dense and ill-conditioned matrix equation. The efficient and robust solution of the IE matrix equation poses an immense challenge. To circumvent this difficulty, in recent literature, there are quite a few proposed approaches. Roughly, we can divide them into two categories: the development of fast algorithms with reduced computational complexity and memory requirements, and the

I

Manuscript received May 08, 2010; revised February 01, 2011; accepted February 23, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by Northrop Grumman Corporation. The authors are with the ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212 USA (e-mail: [email protected]; lee.1863@osu. 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.2011.2161542

construction of an effective preconditioner. Among the fast algorithms, we mention the adaptive integral method (AIM) [2], IE-QR [3], IE-FFT [4] and multilevel fast multi-pole algorithm (MLFMA) [5]. Among the preconditioning strategies, we list the papers by Lee et al. [6], Adams [7], Christiansen and Nedelec [8], Andriulli et al. [9], and Stephanson and Lee [10]. These studies have made a rigorous numerical solution of electromagnetic wave scattering from electrically large objects feasible. However, even with these state of art approaches, slow convergence for solving IE matrix equations are still often observed in many real-life applications. To understand the nature of the difficulties that we are facing, we may take a real-life aircraft as an example. The aircraft frame (metallic, composite, and thin coatings) is electrically large; the cables/wirings are of very small diameters; of particular concern is this large platform usually comprises a variety of antennas and weapons. Trying to solve/model such a problem in its entirety is very difficult, if not impossible. First of all, it will be extremely demanding to generate a good quality mesh of the entire problem, including all the needed tiny fine features, let alone solving them efficiently. Secondly, even if such a mesh can be constructed, the drastically different mesh sizes existing in the same discretization will result in an extremely ill-conditioned matrix equation. Additionally, we should also mention that for the inhomogeneous discretizations, the performance of the fast algorithms that we mentioned earlier would deteriorate dramatically. Thirdly, another particular concern in this work is the existence of some very thin structures, such as aircraft wings or thin plates, where two original surfaces on two sides almost coincide in space. In this situation, the “differential mode” current radiates very weakly and contributes to a few very small eigenvalues. This again will significantly slow down convergence in the Krylov iterative solvers. In this regard, it will be greatly beneficial if isolation of those problematic regions is feasible. Clearly, a breakaway from conventional IE approaches needs to be employed in order to address such a strenuous challenge with a good degree of success. In this paper, we propose a new integral equation domain decomposition method (IE-DDM) for solving electromagnetic wave scattering from large non-penetrable objects. As is well known, DDM is quite successful within the finite element methods (FEMs) and finite difference methods (FDMs). Many references are available in the literature ([11]–[15]). There are also some studies on the marriage of the DDM and IE methods [16]–[23]. Among them, it is worthwhile to mention the work presented by Li and Chew [24], [25], which

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PENG et al.: IE BASED DDM FOR SOLVING EM WAVE SCATTERING FROM NON-PENETRABLE OBJECTS

is able to dissociate and isolate the solution of one region from another by using the virtual non-connected equivalence principle algorithm (EPA). The original unknowns are then transferred to the unknowns on the equivalence surface that encloses the elements. The proposed IE-DDM directly addresses the multi-scale issue faced by the surface integral equation (SIE) methods for real-life applications. It first decomposes the original problem domain into smaller disjoint sub-domains, in which local sub-problems are to be solved. Sub-domains are coupled to one another via the MLFMA using the Stratton-Chu representation formulas. There are three major benefits of the proposed IE-DDM: (a) the method belongs to the class of non-overlapping DDMs and it provides a computationally efficient and effective preconditioner for the corresponding SIE. Each sub-domain is described by a closed surface and the continuity of tangential fields on the touching interfaces between sub-domains are enforced by Robin type transmission conditions [12], [26]; (b) The presented DDM is non-conformal and very suitable for dealing with multi-scale EM problems since each sub-domain has its own characteristic length and will be meshed completely independent. Moreover, each sub-domain is also allowed to choose its own appropriate sub-domain SIE solver, such as AIM, MLFMA, or adaptive cross approximation (ACA) [33], based on its own local EM characteristics and geometrical features; and, (c) This method provides unprecedented flexibility and convenience for design and parameter studies, since it is possible to only re-mesh the portion of the geometry that has changed during the design process. The rest of paper is organized as following: In Section II, we present the notations and formulation of the IE-DDM. We then discuss the acceleration of IE-DDM using MLFMA and an efficient Krylov subspace recycling iterative solver for solving the sub-domain matrix equations. In Section III we illustrate the performance of IE-DDM on solving EM scattering from a model metallic cylinder, a helicopter, a mock-up tank and finally a mock-up fighter jet. A summary and conclusion are included in Section IV.

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Fig. 1. Electromagnetic wave scattering by a PEC cylinder.

Fig. 2. Notations for decomposition of the domain.

is the intrinsic impedance and is the Here wavenumber in free space, and are the coordinates of the obgives servation and source points, respectively; and, the radial frequency of operation with frequency in Hz. and are the permittivity and permeability in free space, respec, where tively. is the distance between the source and observation points. The surface integral equations that govern the surface electric and magnetic fields on the PEC surface can then be obtained by (2) and (3) where denotes the tangential electric field, and and are the tangential electric and magnetic field of plane wave illumination on , respectively. , the twisted tangential Moreover it is worth noting that for , components of the exterior scattered magnetic field, can be expressed as

II. INTEGRAL EQUATION DOMAIN DECOMPOSITION METHOD (4)

A. Notations To introduce the notations in this paper, we first consider the problem of electromagnetic wave scattering by a PEC domain , as illustrated in Fig. 1. is the exterior surface of , and is the outward unit normal on pointing away from . is the electric current on the surface. Through the Stratton-Chu representation formulas, the scattered fields can be written as

where p.v. denotes the principle value integral. Equations (2) and (3) are the well-known electric field integral equation (EFIE) and magnetic field integral equation (MFIE), respectively. To eliminate the interior resonance problem, one usually combines (2) and (3) together to form the combined field integral equation (CFIE) formulation. B. Formulation

(1)

As shown in Fig. 2, we partition the original problem domain into two non-overlapping sub-domains . Acand are defined as the exterior surfaces of cordingly, and , and and are the electric currents on and , respectively. It is worth noting that through this decomposition, and are closed conducting surfaces. Subsequently, both

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The incident electric and magnetic fields for sub-domain can then be written as

(8)

Fig. 3. A decomposition of the domain with touching surfaces.

the incident electric and magnetic fields, can be expressed as for sub-domain

and

where and are somewhat of concern as we need to consider the contributions from current residing on , which are of the forms ,

(5)

and are the tangential fields of Note that the incident plane wave (or more precisely, the tangential and the twisted tangential components of the electric and magnetic and fields of the incident wave on , respectively), and are the scattered tangential fields contributed from . Similar expressions of the incident fields for sub-domain are as follows: sub-domain

(9) and . Note that the observation point where and the source points are located on two touching interfaces, and . In a discrete regime, the meshes and basis functions and are non-conformal, so it is taxing to evaluate on these integrations with desired high accuracy. To remedy this as difficulty, we may rewrite the problematic

(6) Simply put it, (5) and (6) are the direct consequences of an obbehaves as a secondary servation that the scattered field from incident wave for sub-domain , and vice verse. Subsequently, the desired EFIEs and MFIEs for sub-domains are

(10) via the tangential continuity of the electric fields and (7). Similarly, we have

(11) (7)

Straightforwardly combining the EFIEs and MFIEs in (7) results in the general CFIEs to solve for the decomposed problem, which is equivalent to that of the original problem without decomposition. However, the integrations (1) on the touching surfaces of two adjacent sub-domains are somewhat troublesome . To address this issue, we because of the singularity of and into and further divide the surfaces , as depicted in Fig. 3. and are the exterior boundaries defined as and , whereas and are the touching interfaces between sub-domains and , and we write , whose outward unit , and denotes the side normal pointing to whose outward unit normal pointing to sub-domain . We do and so to emphasize the fact that the interface meshes on are allowed to be non-conformal (non-matching).

Thus, by substituting (10) and (11) into (8), the incident electric and magnetic fields for sub-domain can then be rewritten as

(12) with the aid of the following identity (13) Next we can also obtain the incident electric and magnetic as field for sub-domain

(14)

PENG et al.: IE BASED DDM FOR SOLVING EM WAVE SCATTERING FROM NON-PENETRABLE OBJECTS

C. Matrix Equation Denote the surface integral of two complex-valued vector functions as

The matrix blocks usual CFIE matrix

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, and for sub-domain

(15)

form the

(20) The remaining blocks are given as

Moreover, on each surface , we introduce a quasi-uni. Additionally, form independent triangular mesh, for each sub-domain, we define discrete trial functions , with the [27], and usual div-conformal vector function space on . Here, is test functions taken to be the space of the surface div-conforming vector Rao-Wilton-Glisson (RWG) basis functions described in [28]. Subsequently, the discrete formulation can be formally stated as

(21) (16) is the combination parameter of CFIE and we choose throughout this study. Furthermore, and denote the Galerkin testing of EFIE and MFIE for closed objects, respectively. Explicitly, they are written as Here,

(17)

Here, we omit the explicit forms of the matrix blocks and as they can and be obtained in a similar way. As the meshes on are non-conformal, the integrations of and are implemented on a union of the two interface meshes to accurately perform the numerical integrations. We refer the interested readers to [12] and [26] for details. Note that the approach presented herein offers great flexibility in terms of mesh non-conformity. By writing the system matrix as

. The right-hand-side (RHS) of sub-domain , see (16), can be computed through the following substitutions:

(22) where

(23)

(24)

(18) can be evaluated in much the same The RHS of sub-domain way as in (18). As a consequence, the following matrix equation is obtained:

We solve (22) via a preconditioned Krylov subspace method, and examine the behavior of the iterative solution of the preconditioned system (25) which can also be written as (26)

(19)

Note that the preconditioner that we espoused herein, , does involve computing the sub-domain solution for each of the sub-domains.

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D. Acceleration of IE-DDM Using the Multilevel Fast Multipole Algorithm In this section, we address the efficient computation of the matrix-vector multiplications. In the proposed IE-DDM herein, we use the MLFMA to accelerate the matrix-vector multiplications. Conventionally, to implement the MLFMA, the entire object is first enclosed in a large box, which is divided into eight smaller boxes. Each sub-box is then recursively subdivided into smaller cubes until the edge length of the finest cube is about half of a wavelength. The interactions between these boxes can be divided into two categories: the near-region couplings, where the interactions are computed much the same way as in the direct method of moment; and, the far-region couplings, and the interactions between boxes are calculated using the fast multi-pole method (FMM). We shall recall the basic formula to calculate the matrix elements in the far-region couplings, which is (27) where and are the radiation and receiving patterns, respectively. Specifically, we have

Fig. 4. Implementation of MLFMA in IE-DDM.

different from the one corresponds to , but we consider its implementation straightforward since only sparse MV multiplications are involved. , However, the acceleration of the computations, is different in several aspects. The details of the MV operation are given as in

(28) and

is the translator defined as

(29) denotes the spherical Hankel function of the second kind, the Legendre polynomial of degree , and the number of multipole expansion terms. Detailed description of the MLFMA can be found in [1] and [5], and it will not be repeated here. In what follows, we shall assort the matrix-vector (MV) multipliand , cations, in (26), into three categories, and discuss the practical issues involved in accelerations. As described earlier, forms the CFIE matrix for sub-domain , therefore the MV operation (30) can be accelerated using the usual MLFMA. Moreover, once the sub-domain matrices, structed, it is considerably easy to compute by

, are con, given

The reason is that the operations make use of the same radiating and receiving patterns of in the MLFMA, except that the do not radiate into the exterior surface electric currents on . Note also that the near field calculation in is slightly

Evident from the expression, we observe that all electric curof sub-domain radiate into only the rents on the surface of sub-domain . Subsequently, we bring exterior surface about the acceleration in two separate procedures: for the nearregion calculations, we need to compute only the couplings in the buffer region, depicted as red boxes in Fig. 4; as for the far-region computations, we may use the same radiating and receiving patterns, which are readily available during the conand . structions of E. Recycling Krylov Subspace Method In general, the number of degrees of freedom (DOFs) needed for solving real-life multi-scale problems is large. Consequently, it is natural to embrace a Krylov subspace method to solve the matrix equation (26). Furthermore, as suggested by the preconditioner that we employed in (26), every Krylov-iteration requires solving each of the sub-domain matrix equations once. Therefore, the solution of sub-domain matrix equations needs to be effectively handled. In essence, this is similar to solving the matrix equation with multiple right-hand-sides (RHSs). However, these RHSs are computed sequentially. For example, the 2nd RHS is obtained only after the 1st outer Krylov iteration is completed. Moreover, the number of RHSs, , is the number of iterations required for the outer Krylov method to converge for solving (26). To speed up the computations of sub-domain solutions, we exploit a Krylov recycling algorithm, GCRO-DR [31], [32]. The specific GCRO-DR algorithm employed herein is outlined in [30], and we refer the interested readers to it for details.

PENG et al.: IE BASED DDM FOR SOLVING EM WAVE SCATTERING FROM NON-PENETRABLE OBJECTS

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Fig. 5. A PEC cylinder.

Fig. 7. Domain partitions of EM scattering from a helicopter, with moving parts, by using the proposed IE-DDM.

Fig. 8. The study of GCRO-DR(30,20) for fuselage sub-domain.

Fig. 6. Using IE-DDM to solve a PEC cylinder scattering problem, (a) current distribution using IE-DDM and a single domain CFIE; (b) far field patterns.

III. NUMERICAL RESULTS Four numerical examples are included in this section to validate the accuracy and to illustrate the efficiency of the proposed approach. They are: electromagnetic wave scatterings from a metallic cylinder, a helicopter, a mock-up tank and finally a mock-up fighter jet operated at X band. The relative residual of solving the matrix equation in (26) is denoted as . All computational statistics are reported using a workstation with two quad-core 64-bit Intel Xeon E5520 CPUs and 48 GB of RAM. A. Validation by EM Scattering From a Metallic Cylinder To begin, we study the EM wave scattering from a PEC cylinder. The dimensions of the PEC cylinder are depicted direction, in Fig. 5. The incident wave, polarized in the

Fig. 9. Convergence study in using IE-DDM to compute EM scattering from a helicopter with moving parts.

illuminates the cylinder from and at 300 MHz. A transverse plane partitions the cylinder into two equal sized sub-domains. Each of the two sub-domains is meshed completely independently from each other, such that the interface meshes between them do not match. To examine the accuracy

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Fig. 11. Domain partition by using IE-DDM to solve tank scattering problem.

TABLE I COMPUTATIONAL STATISTICS OF HELICOPTER 

= 10

(SD-CFIE), with near-field diagonal block preconditioner. As can be seen from the figure, the solution from IE-DDM is in good agreement with the SD-CFIE. Additionally, it is worth noting that the electric currents across the interfaces of two sub-domains are normal continuous, satisfied through the Galerkin testing procedures. Through this example, we have validated the accuracy of the proposed IE-DDM method. B. Plane Wave Scattering From a Helicopter With Moving Parts

Fig. 10. Using IE-DDM to solve helicopter scattering problem (a) current distribution using IE-DDM and SD-CFIE; (b) far field pattern, co-polarization; (c) far field pattern, cross-polarization.

of IE-DDM, the electric currents on the surface and scattered field at and are shown in Fig. 6, compared with the results using conventional single domain CFIE

In the second numerical example, we study the scattering from a mock-up helicopter. The EM scattering from the helicopter is further modulated by the rotations of the top and tail blades. The overall geometry is divided into 3 closed objects, top rotor, tail rotor and fuselage, as shown in Fig. 7. They are meshed independently and touch each other through common interfaces. Furthermore, due to the existence of thin blades and very fine geometric features of the rotors, a multi-frontal direct method [29] is adopted as the sub-domain solver for the rotors. In contrast, the Krylov recycling subspace solver GCRO-DR is used for the fuselage sub-domain. We first examine the method for the fuselage and compare its perGCRO-DR formance with the truncated GCR(30) method. Here we choose and the size the maximum size of the subspace to be of the selected recycling subspace to be in GCRO-DR [30]. The numbers of matrix-vector (MV) multiplications for solving the sub-domain matrix equation with 6 RHSs are plotted in Fig. 8. We note that GCRO-DR shows significant improvement over the truncated GCR. The overall number

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Fig. 12. The convergence study of IE-DDM to solve tank scattering problem.

Fig. 14. Using IE-DDM to compute EM scattering from a mock-up tank: (a) far field pattern, co-polarization; (b) far field pattern, cross-polarization.

Fig. 13. Current distributions of tank scattering at 2 GHz using IE-DDM and SD-CFIE (a) front view; (b) side view.

of MV multiplications required for GCRO-DR(30,20) is 300, whereas 720 MV multiplications are needed for GCR(30). Moreover, the performance of the proposed IE-DDM is compared against the SD-CFIE for solving this example. The incident wave, polarized in the direction, illuminates the heliand at 500 MHz. Two benefits copter from can be clearly identified from the comparison. One is the superior convergence of the IE-DDM, as shown in Fig. 9, compared

to the SD-CFIE. Only 15 outer Krylov iterations are needed for the proposed IE-DDM approach, whereas the SD-CFIE fails to within 500 iterations. This is mainly due converge to to the presence of the thin blades and very fine geometric features of the rotors. Table I summarizes the computational statistics. The other benefit is the flexible domain partitioning in the IE-DDM approach. Unlike the SD-CFIE formulation, the proposed IE-DDM does not require re-meshing the helicopter every time when the rotation angles of the rotors changed. Consequently, it provides great benefit in performing parameter study of the EM scattering due to different rotation angles of the rotors. In Fig. 10(a), we plot the electric currents on the surface; and , are included and the echo area, for in Fig. 10(b) for co-polarization and Fig. 10(c) for cross-polarization. C. Plane Wave Scattering From a Mock-Up Tank The third example that we considered herein is the electromagnetic plane wave scattering from a mock-up tank. As depicted in Fig. 11, we divide the entire tank into 4 closed objects.

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TABLE II COMPUTATIONAL STATISTICS OF TANK AT 2 GHZ 

= 10

Fig. 15. Current distribution on the surface of a mock-up tank, with an incident EM plane wave at 10 GHz.

TABLE III

Fig. 16. Domain partition by using IE-DDM to solve fighter jet scattering problem.

COMPUTATIONAL STATISTICS OF TANK AT 10 GHz

Each object has its own characteristics length and is meshed independently from the others: the turret (containing small features), two trucks, and the main body. Also note that they are touching each other through common interfaces. Instead of applying the CFIE to one single large domain, as is commonly done, we have espoused IE-DDM with GCRO-DR (30,20) for each of the four regions separately, and iterated the numerical solutions until they converge. The incident wave, polarized in the direction, illuminates and at 2 GHz. As seen from the tank from Fig. 12, we observed fast convergence (16 outer Krylov iterations) of the proposed IE-DDM approach compared to many iterations needed by the SD-CFIE formulation (which fails to within 500 iterations). The electric curconverge to and rent on the surface and scattered field at are shown in Figs. 13 and 14, respectively, compared with the results using the SD-CFIE. The computational statistics using IE-DDM and SD-CFIE are given in Table II. Furthermore, we continue our simulations of EM scattering from a tank at a higher frequency, 10 GHz, using the IE-DDM.

Fig. 17. The convergence study of IE-DDM to solve EM scattering from a mock-up PEC fighter jet scattering problem.

The total number of unknowns is 16 224 487, peak memory usage is 25 GB, and overall computational time is 24 hours and 34 minutes. We also notice that IE-DDM again exhibits fast convergence (only 18 outer iterations are required to converge to ). Additionally, the computational statistics of the four sub-domains at 10 GHz are summarized in Table III. In Fig. 15, we plot the electric currents on the surface. Numerical results corroborate the superior performance and robustness of the proposed IE-DDM.

PENG et al.: IE BASED DDM FOR SOLVING EM WAVE SCATTERING FROM NON-PENETRABLE OBJECTS

TABLE IV COMPUTATIONAL STATISTICS OF FIGHTER JET ( = 10

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)

Fig. 19. Current distribution on the mock-up fighter jet at 10 GHz.

from 2 GHz to 10 GHz, which speaks directly about the robustness of the method. Table IV encompasses the computational statistics for solving the EM scattering from the mock-up fighter jet using the IE-DDM. For comparison, we also include the results of using the usual SD-CFIE. Aside from the reduction of memory requirements by using IE-DDM, the CPU times of using IE-DDM also reduces significantly versus the SD-CFIE. The electric currents on the surface at 2 GHz and 10 GHz are plotted in Figs. 18 and 19, respectively. IV. CONCLUSION

Fig. 18. Current distributions of fighter jet scattering at 2 GHz using IE-DDM and SD-CFIE (a) top view; (b) bottom view.

A new integral equation based domain decomposition method has been introduced in this paper to solve real-life multi-scale problems. The proposed approach first decomposes the entire structure according to the local complexities of the problem domain. Then a domain decomposition strategy is employed to obtain the final solution. Superior performance and robustness are witnessed on several numerical examples. The proposed method greatly relieves the burden of mesh generation of complicated real-life systems. Also, the condition number of the preconditioned matrix equation is significantly better than the usual SD-CFIE. Thus, the new IE-DDM solver can also be viewed as an efficient and robust preconditioner to improve the ill-conditioned nature of the IE matrix equations. ACKNOWLEDGMENT

D. EM Scattering From a Mock-Up Fighter Jet As the last example, we examine the numerical scalability of the proposed IE-DDM versus frequency. The discretizations remain almost the same density for solving an EM plane wave scattering from a mock-up fighter jet. The operating frequency varies from 2 GHz to 10 GHz. Fig. 16 shows the geometry and partitioning of this fighter jet. As indicated in Fig. 16(a) and (b), the entire platform is divided into 6 closed objects: two aircraft wings (containing small features), tail wings (very thin structures), and the fuselage parts, according to the local complexities of the problem domains. The GCRO-DR(30,20) is used as the sub-domain solver. The numbers of outer iterations required for IE-DDM versus different operating frequencies are graphed in Fig. 17. The new IE-DDM provides a good numerical scalability with respect to the operating frequency. The number of iterations only grows from 9 to 11 when frequency increases

The authors wish to thank Y.-h. Zhao in the ElectroScience Lab, The Ohio State University, for preparing some of the geometries in Section III. REFERENCES [1] W. C. Chew, J.-M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagneitcs. Boston, MA: Artech House, 2001. [2] F. Ling, C.-F. Wang, and J.-M. Jin, “Efficient algorithm for analyzing large-scale microstrip structures using adaptive integral method combined with discrete complex-image method,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 5, May 2000. [3] K. Zhao and J.-F. Lee, “A single-level dual rand IE-QR algorithm to model large microstrip antenna arrays,” IEEE Trans. Antennas Propag., vol. 52, no. 10, pp. 2580–2585, Oct. 2004. [4] S. M. Seo and J.-F. Lee, “A fast IE-FFT algorithm for solving PEC scattering problems,” IEEE Trans. Magn., vol. 41, no. 5, pp. 1476–1479, May 2004. [5] J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equation of electromagnetic scattering,” Mico. Opt. Tech. Lett., vol. 10, no. 1, pp. 14–19, Sept. 1995.

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[6] J. Lee, J. Zhang, and C.-C. Lu, “Sparse inverse preconditioning of multilevel fast multipole algorithm for hybrid integral equations in electromagnetic,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2277–2287, Sept. 2004. [7] R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equations,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 362–370, Feb. 2004. [8] S. H. Christiansen and J.-C. Nedelec, “A preconditioner for the electric field integral equation based on Calderon formulas,” SIAM J. Numer. Anal., vol. 40, no. 3, pp. 1100–1135, 2002. [9] F. P. Andriulli, K. Cools, F. Olyslager, A. Buffa, S. H. Christiansen, and E. Michielssen, “A multiplicative Calderon preconditioner for the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2398–2412, Aug. 2008. [10] M. B. Stephanson and J.-F. Lee, “Preconditioned electric field integral equation using Calderon identities and dual loop/star basis functions,” IEEE Trans. Antennas Propag., vol. 57, no. 3, pp. 1274–1279, Apr. 2009. [11] B. Stupfel and M. Mognot, “A domain decomposition method for the vector wave equation,” IEEE Trans. Antennas Propag., vol. 48, no. 5, pp. 653–660, May 2000. [12] K. Zhao, V. Rawat, S.-C. Lee, and J.-F. Lee, “A domain decomposition method with non-conformal meshes for finite periodic and semi-periodic structures,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2559–2570, Sept. 2007. [13] Y.-J. Li and J.-M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of 3-D large-scale electromagnetic problems,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2803–2810, Oct. 2007. [14] Z. Peng and J.-F. Lee, “Non-conformal domain decomposition method with second order transmission conditions for time-harmonic electromagnetic,” J. Comput. Phys., DOI: 10.1016/j.jcp.2010.03.049. [15] Y. Lu and C. Y. Shen, “A domain decomposition finite-difference method for parallel numerical implementation of time-dependent Maxwell’s equations,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 556–562, Mar. 2007. [16] C. C. Lu and W. C. Chew, “The use of Huygens’ equivalence principle for solving 3-D volume integral equation of scattering,” IEEE Trans. Antennas Propag., vol. 43, no. 5, pp. 500–507, May 1995. [17] M. A. Jensen and J. D. Freeze, “A recursive Green’s function method for boundary integral analysis of inhomogeneous domains,” IEEE Trans. Antennas Propag., vol. 46, no. 12, pp. 1810–1816, Dec. 1998. [18] G. C. Hsiao, O. Steinbach, and W. L. Wendland, “Domain decomposition methods via boundary integral equations,” J. Comput. Appl. Math., vol. 125, no. 1–2, pp. 521–537, 2000. [19] Y.-H. Chu and W. C. Chew, “A multilevel fast multipole algorithm for electrically small composite structures,” Microw. Opt. Tech. Lett., vol. 43, no. 3, pp. 202–207, Nov. 2004. [20] G. Of, O. Steinbach, and W. L. Wendland, “Boundary element tearing and interconnecting domain decomposition methods,” in Lecture Notes in Applied and Computational Mechanics. Berlin, Germany: Springer-Verlag, 2006, vol. 28/2006, pp. 461–490. [21] A. Bendali, Y. Boubendir, and M. Fares, “A FETI-like domain decomposition method for coupling finite elements and boundary elements in large-size problems of acoustic scattering,” Comput. Struct., vol. 85, no. 9, pp. 526–535, 2007. [22] U. Langer, G. Of, O. Steinbach, and W. Zulehner, “Inexact fast multipole boundary element tearing and interconnecting methods,” in Lecture Notes in Computational Science and Engineering. Berlin, Germany: Springer-Verlag, 2007, vol. 55, pp. 405–412. [23] M. Maischak, “A multilevel additive Schwarz method for a hypersingular integral equation on an open curve with graded meshes,” Appl. Numer. Math., vol. 59, no. 9, pp. 2195–2202, 2009. [24] M.-K. Li and W. C. Chew, “Wave-field interaction with complex structures using equivalence principle algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 130–138, Jan. 2007. [25] M.-K. Li and W. C. Chew, “Multiscale simulation of complex structures using equivalence principle algorithm with high-order field point sampling scheme,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2389–2397, Aug. 2007. [26] Z. Peng, V. Rawat, and J.-F. Lee, “One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems,” J. Comput. Phys., vol. 229, pp. 1181–119, Feb. 2010. [27] R. A. Adams, Sobolev Space. Pure and Applied Mathematics. New York-London: Academic Press, 1975, vol. 65. [28] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, pp. 409–418, May 1982.

[29] P. Amestoy, I. S. Duff, and C. Voemel, “Task scheduling in an asynchronous distributed multifrontal solver,” SIAM J. Matrix Anal Appl., vol. 26, no. 2, pp. 544–565, 2005. [30] Z. Peng, M. B. Stephanson, and J.-F. Lee, “Fast computation of angular responses of large-scale three-dimensional electromagnetic wave scattering,” IEEE Trans. Antennas Propag., vol. 58, no. 9, pp. 3004–3012, Sept. 2010. [31] E. De Sturler, “Truncation strategies for optimal Krylov subspace methods,” SIAM J. Numer. Anal., vol. 36, pp. 864–889, 1999. [32] M. L. Parks, E. De Sturler, G. Mackey, D. D. Johnson, and S. Maiti, “Recycling Krylov subspaces for sequences of linear systems,” SIAM J. Sci. Comput., vol. 28, no. 5, pp. 1651–1674, 2006. [33] K. Zhao, M. N. Vouvakis, and J.-F. Lee, “The adaptive cross-approximation algorithm for accelerated method of moments computations of EMC problems,” IEEE Trans. Electromagn. Compat., vol. 47, no. 4, pp. 763–773, Nov. 2005.

Zhen Peng (M’09) received the B.S. in electrical engineering and information science from the University of Science and Technology of China, in 2003 and the Ph.D. degree from the Chinese Academy of Science, in 2008. From 2008 to 2009, he was a Postdoctoral Fellow at the ElectroScience Laboratory, Ohio State University. He has been working as a Senior Research Associate at the ElectroScience Laboratory, Ohio State University since 2009. His research interests are in scientific computing, specifically in the area of fullwave numerical methods in computational electromagnetic. Recently research directions include the domain decomposition methods for both finite element method and boundary integral method, the hybrid finite element-boundary integral method, and the multilevel fast multipole method. Applications of his research include: novel antennas for wireless communication systems, electromagnetic compatibility and interference analysis of multiple antenna systems on military and commercial aircrafts, signal integrity and package analyses for modern ultra-large integrated circuits, and the design tolls for energy efficient LCD back-light unit.

Xiaochuan Wang (S’09) was born in Shandong Province, China. He received the B.Eng. and M.Sc. degrees in electrical engineering from Tsinghua University, Beijing, China, in 2005 and 2007, respectively. He is currently a Graduate Research Associate at The Ohio State University, where he is working towards the Ph.D. degree. His research interests include domain decomposition method in computational electromagnetics.

Jin-Fa Lee (F’05) received the B.S. degree from National Taiwan University, in 1982, and the M.S. and Ph.D. degrees from Carnegie-Mellon University, in 1986 and 1989, respectively, all in electrical engineering. From 1988 to 1990, he was with Ansoft Corp., where he developed several CAD/CAE finite element programs for modeling three-dimensional microwave and millimeter-wave circuits. From 1990 to 1991, he was a Postdoctoral Fellow at the University of Illinois at Urbana-Champaign. From 1991 to 2000, he was with Department of Electrical and Computer Engineering, Worcester Polytechnic Institute. Currently, he is a Professor at ElectroScience Lab., Department of Electrical Engineering, Ohio State University. His research interests mainly focus on numerical methods and their applications to computational electromagnetics. Current research projects include: analyses of numerical methods, fast finite element methods, fast integral equation methods, three-dimensional mesh generation, domain decomposition methods, hybrid numerical methods and high frequency techniques based on domain decompositions approach, LCD modeling, large antenna arrays and co-design for signal integrity and packaging. Prof. Lee was elected an IEEE Fellow in 2005.

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Scattering and Absorption of Waves by Flat Material Strips Analyzed Using Generalized Boundary Conditions and Nystrom-Type Algorithm Olga V. Shapoval, Student Member, IEEE, Ronan Sauleau, Senior Member, IEEE, and Alexander I. Nosich, Fellow, IEEE

H

E

Abstract—The scattering of the - and -polarized plane waves by a thin flat homogeneous magneto-dielectric strip is considered. Assuming the strip to be thinner than the wavelength, we shrink its cross-section to the median line where the generalized boundary conditions are imposed. The numerical solution is built on two singular integral equations discretized using Nystrom-type numerical algorithm. The obtained results demonstrate fast convergence and good agreement with data known for the limiting values of the strip parameters. This opens a way to the accurate numerical analysis of various striplike configurations simulating natural objects and electromagnetic circuit components, both in traditional microwave applications and nanophotonics. Index Terms—Discrete mathematical model, generalized two-side boundary conditions, scattering cross-sections, singular and hyper-singular integral equations, strip scatterer.

I. INTRODUCTION TRIPS and striplike scatterers are frequently met in microwave and photonic devices because of their simple manufacturing with the existing etching and printing technologies. Their thickness usually makes a fraction of the free-space electromagnetic wave length while the width can be smaller, comparable to and larger than the wavelength. This combination of parameters makes both quasistatic and high-frequency methods of analysis inapplicable and thus the full-wave methods are mandatory. Still the small thickness suggests that the analysis can be simplified by neglecting the internal field of the strip and considering only the limiting values of the field components. Materials of such scatterers vary greatly, and so vary their theoretical models, from perfect electrically conducting (PEC) to resistive

S

Manuscript received July 22, 2010; revised December 24, 2010; accepted February 23, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the National Academy of Sciences of Ukraine in the framework of Target Program “Nanotechnologies and Nanomaterials,” the State Committee for Science, Ukraine via the project M/146-2009, and in part by the European Science Foundation via research network project “Newfocus.” The work of O. V. Shapoval was supported by a Doctoral Research Award from the IEEE Antennas and Propagation Society. O. V. Shapoval is with the Institute of Radio-Physics and Electronics, National Academy of Sciences of Ukraine, Kharkiv 61085, Ukraine (e-mail: [email protected]). R. Sauleau is with the Institute of Electronics and Telecommunications of Rennes, Universite de Rennes 1, 35042 Rennes, France A. I. Nosich is with the Institute of Radio-Physics and Electronics, National Academy of Sciences of Ukraine, Kharkiv 61085, Ukraine and also with the Université Européenne de Bretagne, c/o Université de Rennes 1, Rennes Cedex 35042, France. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161547

and impedance strips. In the optical range, the model of PEC scatterer is not applicable at all, even for the scatterers made of the noble metals like gold and silver. All mentioned dictates a necessity of the development of new accurate mathematical models and convergent numerical methods for the scattering by the thin magneto-dielectric strips. In this paper, the two-dimensional (2-D) electromagnetic wave scattering and absorption by a flat material strip is characterized using the two-side generalized boundary conditions (GBC) and singular integral equations (IEs) [1]. The novelty of the paper is, first, in the use of the special type of GBC proposed in [2]–[4], valid for a thin and high-contrast material layer. This is unlike the important earlier studies [5]–[8], where a small material contrast was implied. Second, to solve the IEs we develop a fast and convergent Nystrom-type numerical algorithm having controlled accuracy. There are many techniques for building the numerical solutions to the IEs in the 2-D scattering by the striplike structures including PEC and imperfect strips, e.g., dielectric and impedance, both stand-alone and periodically structured: the Galerkin moment method [9], the inverse Fourier transform method [10], and, apparently the most advantageous, the method of analytical regularization [11]–[17]. The last of the mentioned guarantees convergence of numerical solutions and provides 3–4 digit accuracy with a relatively small number of unknowns. Besides, recently the Nystrom-type numerical techniques using the quadratures and interpolation-based discretization have attracted attention of researchers [18]–[23]. They have been already demonstrated as convergent, economic and simple in implementation for the modeling of the scattering by PEC zero-thickness flat and curved strips [20]–[27]. Therefore here we are going to extend them to the analysis of stand-alone imperfect flat strips. The paper is organized as follows. In Section II, we formulate the plane-wave scattering problem for a thin material (magneto-dielectric) flat strip using GBS and obtain the basic IEs. Section III describes details of the proposed Nystrom-type numerical algorithm. In Section IV, we demonstrate and discuss the numerical results for various strips. Conclusions are summarized in Section V. II. FORMULATION AND BASIC EQUATIONS A. Formulation Consider the 2-D scattering problem for a magneto-dielectric strip of the width and thickness , characterized with the

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C. Singular Integral Equations We assume that in the case of arbitrary -polarized (or -polarized) incident wave the solution of the problem can be found as a sum of the single-layer and double-layer potentials

Fig. 1. Geometry of the problem.

(4) relative material constants and . Suppose that the strip is illuminated by an -polarized electromagnetic plane wave incident at the angle measured from the -axis (Fig. 1). The time factor is assumed as and omitted. The total magnetic field has only -component, which can be , where represented as a sum, is the incident plane wave ( being the freespace wavenumber) and is the scattered field. Outside of the strip surface the total field is required to satisfy the Helmholtz equation. Assuming that the strip is thin, we follow [2]–[4], replace the strip cross-section with its central , and impose, on , the line, following two-side generalized boundary conditions,

where is the Hankel function, and and are the unknown surface currents of magnetic and electric type induced on the strip. It is obvious from the characteristics of the potentials that thus defined function satisfies Helmholtz equation off and the referred above radiation condition. Using GBC and the properties of the limit values of potentials when crossing the integration contour, we reduce our problem to two decoupled IEs (see also [4], [7]) with logarithmic-type and hyper-type singularities, respectively

(5)

(1) (6)

(2) where is the one-side unit normal vector on . Note that such GBC appear provided that two sides of the strip have identical properties. They allow us to eliminate from consideration the field inside the strip. This is done at the expense of introducing the so-called relative electric and magnetic resistivities, and . Note also that PEC boundary conditions and . In the case of follow from (1) and (2) if -polarization, the function should be understood as the electric field -component and the values and in (1) and must satisfy (2) exchange their places. Additionally, Sommerfeld radiation condition at infinity and condition of the local energy finiteness in any finite domain enclosing an edge point. This scattering problem is uniquely solvable (see [1]).

where . Note that in the case of the -polarization the scattering problem is reduced to two IEs like (5) and (6) where and exchange their places. III. DISCRETE MODEL OF THE PROBLEM A. Discretization Using the Quadratures Introduce dimensionless value tion , the variable to

and unknown func, and change . Then the IEs take form as (7)

B. Resistivities According to three independent derivations [2]–[4] that are in agreement with each other, relative electric and magnetic resistivities of a thin homogeneous material layer are found as

(3)

(8) The principal terms of the asymptotic expansions of the Hankel functions at are given by and , respectively. Hence, the kernel functions of (7) and (8) can be presented as (9)

where is the relative material impedance. The formulas (3) are valid under the conditions that and , i.e., a high contrast is implied [3]. This is important difference from the model used in [5]–[8] that required a small . Note that and can, in principle, contrast, be the functions of .

(10) where and are smooth functions. Therefore, the hyper-singular integral in (8) is understood in the sense of finite part by Hadamard.

SHAPOVAL et al.: SCATTERING AND ABSORPTION OF WAVES BY FLAT MATERIAL STRIPS ANALYZED USING GBC

For the discretization of (7) we use the Simpson quadrature , formulas (see [28]) with the equidistant grid after performing the transformation,

(11) . For IE (8), where we use quadrature formulas of interpolation type (see [19]) with , , which are the nodes at nulls of the Chebyshev polynomials of the second kind. After with the Lagrange insubstituting the unknown function of order , we have terpolation polynomial

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we obtain approximate solutions to our SIEs in the form of interpolation polynomials for the unknown surface currents. The quadrature formulas ensure rapid convergence of the approximate solutions to the accurate ones (for the proof, see [18]–[23]) and , respectively. if B. Scattering Characteristics Using the large-argument asymptotics for the Hankel functions in the kernels of (4), the scattered field in the far zone can be represented as , where is the scattering pattern. It is found as

(18) The total and backward scattering cross sections (TSCS and BSCS) (the latter quantity is also known as monostatic radar cross-section) characterize the total scattered power and that reflected back to the source, respectively (19)

(12)

If the strip material is lossy, the heating losses are characterized by the absorption cross-section (ACS) (20) that is linked to TSCS by the optical theorem (13)

(21) IV. NUMERICAL RESULTS AND DISCUSSION

(14) As a result, we obtain two decoupled matrix equations, (15)

A. Convergence To see the actual rate of convergence, we computed the rootmean-square deviations of the norms of the surface current functions, and , versus the matrix orders , (22)

(16) Here, and are the unknown values of the surface current functions in the Simpson and Chebyshev nodes, , are smooth functions are the respectively, and right-hand parts of IEs (7) and (8) in the same nodes. as they are easily We do not write here the coefficients derived using the classical Simpson quadratures [28] however

(17) Note that , are the Chebyshev polynomials of the 1st and 2nd kind, respectively. On solving the matrix equations

The plots in Fig. 2 demonstrate fast decrement of error in the H-wave case if the size of the quadrature formulas increases (the errors in the scattering cross-sections decay even faster). B. H-Case: Validation To validate our algorithm, in Fig. 3, we present the normalized-frequency dependences of TSCS and BSCS for three strips illuminated by the normally incident -wave. Note that the cross sections are normalized by their high-frequency limits for a PEC strip under broadside illumination, that and , respectively. The PEC-strip results are preis by sented with solid curves and show perfect agreement with the data known from the published papers and reference books (see [29]–[31]). Two other sets of curves are for the heavy-lossy dielectric strips. Note that the curves for the most lossy strip are very close to the PEC-strip curves. This is expected as, if , then and .

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Fig. 4. Normalized TSCS (a) and BSCS (b) as a function of the H-wave inci; other parameters are as in Fig. 3. dence angle for the strip with

 = 10

m and

Fig. 2. The computation errors as a function of the quadrature orders, , for the strips with and different values of ( ).

n

" = 10 + i

 = =2



Fig. 3. Normalized TSCS and BSCS as a function of for the scattering of a normally incident ( ) -wave by the PEC strip (solid) and by the lossy dielectric strips with (dashed) and (dotted); , .

= =2 H " = 1 + 30i h = d=400 m = n = 100

" = 1 + 3000i

In Fig. 4, TSCS (a) and BSCS (b) are presented as a function of the incidence angle for the three dielectric strips and a PEC . strip of the normalized width As one can see, the curves corresponding to the densest strip nicely coincide with the PEC-strip curves for all angles of inciwhere the latter curves go to zero. dence except of Additional validation comes from the fact that in all cases the optical theorem (21) was satisfied with minimum 7 digits. We do not make comparison here with MoM computations because this has been already highlighted in [22], [23].

C. H-Case: Transversal Resonance As mentioned above, the material strip model that is based on the GBC (1), (2) and resistivity formulas (3) takes into account the thickness of the strip even if it is larger than the wavelength in the strip material. Therefore it is interesting to analyze the scattering by the thin strips of varying width made of the dielectrics with relatively large real parts. In Figs. 5 and 6, we present the frequency scans of the nor, under the normal and malized cross-sections for inclined incidences, respectively. Note that the permittivity of approximately corresponds to dry wood or paper at microwave frequencies. The “paper” strip scattering characteristics remain small and vary monotonically in the whole studied , is less usual range. Another value taken here, however can be associated with one of the novel colossal-permittivity materials, see [32]. . For such a strip, a single resonance appears around This is the frequency at which the infinite material slab is almost transparent for a normally incident plane wave as its thickness ). is a half-wavelength in the material ( Therefore this is a transversal resonance of infinite slab. For a finite strip, satisfaction of the same condition leads to the drop in BSCS at the normal incidence (Fig. 5(a)) and to the disappearance of the specular-scattering lobes in the in-resonance far-field patterns. Visualization of the in-resonance total near-field patterns shows that the central part of the strip remains transparent and the shadows are produced only by the strip edges—see Figs. 5(b) and 6(b). In this resonance, ACS displays maximum for any incidence angle.

SHAPOVAL et al.: SCATTERING AND ABSORPTION OF WAVES BY FLAT MATERIAL STRIPS ANALYZED USING GBC

Fig. 5. Normalized TSCS, BSCS and ACS (a) as a function of  for the normally incident H-wave ( = ) scattering by thin dielectric strips with " i (dotted curves) and " i (solid curves); the scattered far-field and the total near-field patterns at  . : (b); m n

= 10 +

=

2

= 1000 + = 19 85

= = 100

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Fig. 7. Normalized TSCS and BSCS and relative resistivities jRj and jQj as a function of  (a), and scattered far-field and total near-field patterns for  i and h d= ; (b) for the scattering by a dielectric strip with " m n under the edge-on incidence of H-wave.

= = 100

= 10 +

= 10 = 100

D. H-Case: Edge-on Incidence

= =4) incidence of H-wave = 1000 + i (b).

Fig. 6. The same as in Fig. 6 for the inclined ( (a); the field patterns at  : and "

= 19 85

If the incidence is inclined then the BSCS plot displays a sequence of periodic low-intensity oscillations (Fig. 6(a)). Their nature will be discussed in the next sub-section.

Flat zero-thickness PEC strips are invisible at the edge-on incidence of the -polarized plane wave. In contrast, material strips remain visible even at the edge-on incidence as they still and scatter and absorb the -wave. Note that in this case as follows from (6). hence In Figs. 7(a) and 8(a), we present the dependences of the normalized TSCS, ACS and BSCS on the normalized strip width, and for two dielectric strips with under the edge-on illumination, respectively. Note that BSCS remains small however displays a sequence that correof low-level periodic oscillations with period of . These maxima sponds to the increment of the strip width by and minima are explained by the in-phase and anti-phase contributions of the leading and trailing edges of the strip to the backscattered field. They are not connected to the natural modes of the strip as open resonator. Such explanation is proven be the fact that the locations of the minima and maxima do not depend on the value of . The “paper” strip TSCS and ACS also remain small and vary monotonically in the whole studied frequency range. However the denser strip characteristics (Fig. 8(a)) demonstrate the that has been discussed transversal resonance around in the previous section. The far-field scattering patterns and the total near-field patterns in Figs. 7(b) and 8(b) reveal that, in either case, the field is dominated by the forward scattering and the shadow, respectively.

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Fig. 8. The same as in Fig. 7 for "

E.

= 1000 + i and  = 18:85 in (b).

-Polarization Case: Longitudinal Resonances

The alternative case of the -polarized plane wave scattering is considered similarly to the -case. It reduces to two IEs like (7) and (8) however with and exchanging their places. Formally small difference, it entails major modifications in the behavior of the cross sections. In Fig. 9, presented are the normalized TSCS, BSCS and ACS values as a function of the frequency parameter for -polarized plane wave incident normally and in the edge-on manner. and thickness The strip has the permittivity . Unlike the -wave case, here one can see many sharp resonances in the scattering and absorption. These resoof nances correspond to the natural Fabry-Perot-like modes and hence are lona thin dielectric strip where gitudinal resonances. As it is suggested by the effective refractive index model of a thin dielectric layer [33], the approximate characteristic , where equation for such resonances is as follows from (1) and (2). This quantity is a function of the frequency and layer material; it appears as the normalized propagation coefficient of the -polar) of the infinite layer. ized principal guided wave ( In Fig. 10, we show the dependences of the real and imagion for the corresponding dielectric layer. nary parts of One can verify that the roots of the mentioned equation, marked in Fig. 9 as stars above the axis, are indeed close to the resonances. Note that in the case of the -polarization the corresponding effective refractive index of a thin strip is found to be . This value is much closer to 1 than (even if ) and therefore no associated sharp longitudinal resonances are visible in Figs. 6 and 8.

Fig. 9. Normalized TSCS, BSCS and ACS as a function of  for the scattering : i for the normal (a) and edge-on (b) incidence by the strip with " , and m n . of -wave; other parameters are h d=

E

= 100+ 0 1

= 100

Fig. 10. The real and imaginary parts of  function of  for the strip with "

= = 150

and relative resistivity jRj as a

= 100 + 0:1i.

Finally, Fig. 11 presents the field patterns in four resonances for the normally-incident plane -wave scattering (because of the symmetry of excitation only the longitudinal strip modes corresponding to odd values of are excited). As one can see, the in-resonance far fields display sizable scattering in the strip plane, in addition to the intensive shadow and specular scattering. Note that these results can be viewed as partial justification of the empiric effective refractive index model of a thin dielectric strip that enables one to reduce the dimensionality when searching for the frequencies of longitudinal resonances.

SHAPOVAL et al.: SCATTERING AND ABSORPTION OF WAVES BY FLAT MATERIAL STRIPS ANALYZED USING GBC

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properties of the lossy dielectric strips, including the field behavior in the far and near zones. In the case of the -polarization, a transversal resonance has been studied. This resonance results in good transparency of the strip at the normal incidence of plane wave, so that only the edges contribute to the scattering. We have also given special consideration to the edge-on incidence case because thin magneto-dielectric strips remain visible in this case even under the -wave illumination, unlike the PEC strips. In the case of the -polarization, we have demonstrated the sequence of resonances in the scattering and absorption. They are explained as the longitudinal Fabry-Perot resonances of the natural wave of the corresponding dielectric layer traveling from one end of the thin strip to another. Such an effect is absent in the -polarization case because of much smaller propagation constant of the corresponding natural wave of the same dielectric layer. Thus we have built a relatively universal mathematical model that enables one to investigate a wide class of the flat material striplike scatterers in the resonance range. ACKNOWLEDGMENT The authors are grateful to Dr. E.I. Smotrova and Dr. V.S. Bulygin for many helpful discussions and to the anonymous reviewer for valuable comments. REFERENCES

= 1,3,5,7) for the = 100 + 0:1i and h = = 4 45  = 5:63 (d);

Fig. 11. The field patterns in the E mode resonances (p normal incidence of plane wave on the strip with " d= at  : (a),  : (b),  : (c) and m n .

400 = 0 77 = = 150

= 3 08

V. CONCLUSIONS We have presented an efficient and accurate method to analyze the scattering by a thin flat magneto-dielectric strip in free space illuminated by a plane electromagnetic wave. This method of numerical modeling is based on two decoupled logsingular and hyper-singular IEs for the electric and magnetic surface currents on the strip and uses their direct discretization with the aid of the special quadrature formulas of interpolation type. In contrast to the conventional moment method with local basis functions it has guaranteed convergence and controlled accuracy of computations. Besides, it is simpler in implementation than the analytical regularization methods. We have presented the numerical results for the - and -polarization cases and investigated the scattering and absorption

[1] D. Colton and R. Kress, Integral Equations Methods in Scattering Theory. New York: Wiley, 1983. [2] G. A. Grinberg, “Boundary conditions for the electromagnetic field in the presence of thin metallic shells,” (in Russian) Radio Engrg. Electron., vol. 26, no. 12, pp. 2493–2499, 1981. [3] G. Bouchitté, “Analyse limite de la diffraction d’ondes électromagnétiques par une structure mince,” C.R. Acad. Paris, ser. II, vol. 311, pp. 51–56, 1990. [4] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “Surface-integral equations for electromagnetic scattering from impenetrable and penetrable sheets,” IEEE Antennas Propag. Mag., vol. 36, no. 6, pp. 14–25, 1993. [5] T. B. A. Senior, “Backscattering from resistive strips,” IEEE Trans. Antennas Propag., vol. 27, no. 6, pp. 808–813, 1979. [6] S. Dowerah and A. Chacrabarty, “Extinction cross section of a dielectric strip,” IEEE Trans. Antennas Propag., vol. 38, no. 5, pp. 696–706, 1988. [7] T. B. A. Senior and J. L. Volakis, “Sheet simulation of a thin dielectric layer,” Radio Sci., vol. 22, no. 7, pp. 1261–1272, 1987. [8] T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics. London: The IEE Press, 1995. [9] R. C. Hall and R. Mittra, “Scattering from a periodic array of resistive strips,” IEEE Trans. Antennas Propag., vol. 33, no. 9, pp. 1009–1011, 1985. [10] R. Petit and G. Tayeb, “Theoretical and numerical study of gratings consisting of periodic arrays of thin and lossy strips,” J. Opt. Soc. Amer., vol. 7, no. 9, pt. A, pp. 1686–1692, 1990. [11] Z. S. Agranovich, V. A. Marchenko, and V. P. Shestopalov, “Diffraction of a plane electromagnetic wave from plane metallic lattices,” Soviet Phys. Tech. Phys., vol. 7, pp. 277–286, 1962. [12] A. Matsushima and T. Itakura, “Singular integral equation approach to plane wave diffraction by an infinite strip grating at oblique incidence,” J. Electromagn. Waves Applicat., vol. 4, no. 6, pp. 505–519, 1990. [13] A. I. Nosich, “Green’s function – Dual series approach in wave scattering from combined resonant scatterers,” in Analytical and Numerical Methods in Electromagnetic Wave Theory, M. Hashimoto, M. Idemen, and O. A. Tretyakov, Eds. Tokyo: Science House, 1993, ch. 9, pp. 419–469.

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[14] A. I. Nosich, “Method of analytical regularization in wave-scattering and eigenvalue problems: Foundations and review of solutions,” IEEE Antennas Propag. Mag., vol. 42, no. 3, pp. 34–49, 1999. [15] A. I. Nosich, Y. Okuno, and T. Shiraishi, “Scattering and absorption of E- and H-polarized plane waves by a circularly curved resistive strip,” Radio Sci., vol. 31, no. 6, pp. 1733–1742, 1996. [16] T. L. Zinenko, A. I. Nosich, and Y. Okuno, “Plane wave scattering and absorption by resistive-strip and dielectric-strip periodic gratings,” IEEE Trans. Antennas Propag., vol. 46, no. 10, pp. 1498–1505, 1998. [17] T. L. Zinenko and A. I. Nosich, “Plane wave scattering and absorption by flat gratings of impedance strips,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2088–2095, 2006. [18] Z. T. Nazarchuk, Numerical Investigation of Wave Diffraction by Cylindrical Structures (in Russian). Kyiv: Naukova Dumka, 1989. [19] Z. Nazarchuk and K. Kobayashi, “Mathematical modelling of electromagnetic scattering from a thin penetrable target,” Progr. Electromagn. Res., vol. 55, pp. 95–116, 2005. [20] Y. V. Gandel, Introduction to the Methods of Computations of Singular and Hyper-Singular Integrals. Kharkiv: KhNU Press, 2001. [21] J. Tsalamengas, “Exponentially converging Nystrom’s methods for systems of SIEs with applications to open/closed strip or slot-loaded 2-D structures,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1549–1558, 2006. [22] J. Tsalamengas, “Exponentially converging Nystrom methods in scattering from infinite curved smooth strips–Pt. 1: TM case,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3265–3274, 2010. [23] J. Tsalamengas, “Exponentially converging Nystrom methods in scattering from infinite curved smooth strips–Pt. 2: TE case,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3275–3281, 2010. [24] A. A. Nosich and Y. V. Gandel, “Numerical analysis of quasi-optical multireflector antennas in 2-D with the method of discrete singularities,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 399–406, 2007. [25] A. A. Nosich and Y. V. Gandel, “Role of edge illumination in the mm-range elliptic reflector beam waveguide performance,” in Proc. Eur. Microwave Conf. (EuMC-07), Munich, 2007, pp. 376–379. [26] A. A. Nosich, Y. V. Gandel, A. Matsushima, and R. Sauleau, “Collimation and focusing of wave beams with metal-plate lens antennas analyzed using Nystrom-type MDS algorithm,” presented at the Proc. IEEEAP-S Int. Symp. (APS-08), San Diego, 2008, session 237.10. [27] A. A. Nosich, R. Sauleau, and Y. V. Gandel, “Classical ADE and PACO 2-D omnidirectional dual-reflector antennas simulated in 2-D using a Nystrom-type MDS algorithm,” in Proc. Eur. Microwave Conf. (EuMC-09), Rome, 2009, pp. 858–861. [28] M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions New York, Dover, 1979. [29] J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes. Amsterdam: North-Holland, 1969. [30] G. T. Ruck, Ed., Radar Cross-Section Handbook New York, PlenumPress, 1970. [31] H. C. van de Hulst, Light Scattering by Small Particles. New York: Dover Publications, 1981. [32] S. Krohns et al., “Colossal dielectric constant up to gigahertz at room temperature,” Appl. Phys. Lett., vol. 94, no. 12, pp. 2903–2905, 2009. [33] E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: Quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Topics Quant. Electron., vol. 11, no. 5, pp. 1135–1142, 2005.

Olga V. Shapoval (S’10) was born in Nikopol, Ukraine, in 1987. She received the M.S. degree in applied mathematics (with honors) from Kharkiv National University, in 2009. She is currently working toward the Ph.D. degree at the National Academy of Science of Ukraine, Kharkiv. Her current research interests are in the scattering problems by striplike structures, efficient mathematical and numerical solution techniques, and singular and hyper-singular integral equations. Ms. Shapoval was a recipient of the 2010 URSI Young Scientist Award for attending the Asia-Pacific Radio Science Conference in Toyama and the Doctoral Research Award from the IEEE Antennas and Propagation Society.

Ronan Sauleau (M’04–SM’06) was born in Rennes, France, in 1972. He received the Electronic Engineering and Radiocommunications degree and the French DEA degree in electronics from the Institut National des Sciences Appliquees (INSA), Rennes, France, in 1995, the Aggregation degree from Ecole Normale Superieure de Cachan, France, in 1996, and the Doctoral degree in signal processing and telecommunications from the Institut d’Electronique et Telecommunications de Rennes (IETR), University of Rennes 1, in 1999. Since 1999, he has been on the staff of IETR and was elected Professor in 2009. His main fields of interest are millimeter wave printed antennas, focusing devices, and periodic structures including electromagnetic bandgap materials and metamaterials. Dr. Sauleau received the 2004 ISAP Conference Young Scientist Travel Grant and the first Young Researcher Prize in Brittany, France, in 2001 for his work on gain-enhanced Fabry-Perot antennas. In 2007, he was elected a Junior Member of the Institute Universitaire de France.

Alexander I. Nosich (M’94–SM’95–F’04) was born in 1953 in Kharkiv, Ukraine. He received the M.S., Ph.D., and D.Sc. degrees in radio physics from the Kharkiv National University, Ukraine, in 1975, 1979, and 1990, respectively. Since 1979, he has been with the Institute of Radio Physics and Electronics, National Academy of Science of Ukraine, Kharkiv, where he is currently Professor and Principal Scientist heading the Laboratory of Micro and Nano-Optics. Since 1992, he has held a number of guest fellowship and professorship in the EU, Japan, Singapore, and Turkey. His research interests include the method of analytical regularization, propagation and scattering of waves in open waveguides, simulation of the semiconductor lasers and antennas, and the history of microwaves. Prof. Nosich was one of the initiators and technical committee chairman of the international conference series on Mathematical Methods in Electromagnetic Theory (MMET). In 1995, he organized the IEEE AP-S East Ukraine Chapter, the first one in the former USSR. Currently, he represents Ukraine, Georgia, and Moldova in the European Association on Antennas and Propagation.

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Adaptive Aperture Partition in Shooting and Bouncing Ray Method Yu Bo Tao, Hai Lin, and Hu Jun Bao

Abstract—The shooting and bouncing ray (SBR) method may give rise to a divergence problem when ray tubes intersect discontinuous parts of the target, such as the boundary, and this affects the accuracy to some extent. This paper proposes an adaptive aperture partition algorithm to solve this problem. The proposed algorithm adaptively splits the virtual aperture into continuous irregular beams instead of discrete uniform ray tubes according to the geometry of the target during the recursive beam tracing. These beams form a beam tree, the level of which represents the number of reflections. Geometric optics is applied to the representative propagation path of each leaf beam to obtain the exit field, and then physical optics is used to evaluate each leaf beam’s scattered field. The proposed algorithm could generate the convergent solution of the SBR method when the ray-tube size tends toward infinitesimal. Additionally, this paper describes the beam-triangle intersection in detail and utilizes the kd-tree to accelerate the beam-target intersection. Numerical experiments demonstrate that adaptive aperture partition can greatly improve the accuracy of the SBR method, and the computational efficiency can be also significantly enhanced in several applications, such as the RCS prediction in the THz band. Index Terms—Adaptive aperture partition, beam tracing, beamtriangle intersection, kd-tree, radar cross section (RCS), shooting and bouncing ray (SBR).

I. INTRODUCTION HE prediction of the high-frequency scattering from arbitrarily shaped targets is of growing importance for the simulation of radar systems, such as the radar cross section (RCS) and inverse synthetic aperture radar (ISAR) applications. The shooting and bouncing ray (SBR) [1], [2] method is one of the principal ways to predict the scattered field of electrically large and complex targets with great accuracy and efficiency. In the SBR method, the incident plane wave is described by means of a uniform grid of ray tubes, and the density of ray tubes should be greater than about ten rays per wavelength in view of the convergence of results. Four corner rays and the central ray of each ray tube are recursively traced to obtain the exit positions. The exit field of each ray tube is also traced and calculated during the central ray tracing according to the law of geometrical optics (GO) [3]. The field scattered from each

T

Manuscript received April 30, 2010; revised December 20, 2010; accepted January 15, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. 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]; lin@cad. zju.edu.cn; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161435

Fig. 1. The illustration of the ray-tube divergence problem. The target is three orthogonal rectangles with an electrically small structure on the middle rectangle, and its projection on the virtual aperture is the gray polygon. Two corner rays of the ray tube A do not intersect the target, the standard approach is to discard A. Actually, the intersected area of A is shown in the enlarged ray tube A on the right. For the ray tube B , the standard approach uses the four intersection positions to construct an equivalent ray tube, as shown in the enlarged ray tube B . However, the intersected area of the ray tube B is shown in the enlarged ray tube B . It is more complex to process the divergent ray tubes C and D .

ray tube can be evaluated through physical optics (PO), and the scattered field of the target is the sum of all scattered fields of ray tubes. The independence of corner/central ray tracing makes the SBR method easily implementable and highly effective. However, it may result in a divergence problem, i.e., the ray tube diverges in the recursive ray tracing, as illustrated in Fig. 1. This is different from the divergence factor in the original SBR method [1], which describes the change in the field magnitude with the size of the ray tube when it intersects the target defined by the parametric surface. In this paper, the target is described in terms of triangles, and the divergent ray tube may partially intersect the target or may be reflected without one dominant propagation direction. Concretely, when any one of corner rays does not intersect the target, such as the ray tube in Fig. 1, this ray tube is divergent. The current approach to processing divergent ray tubes is to discard these invalid ray tubes directly, not including them in the calculation of the scattered field. However, the simple approach would affect the accuracy of the SBR method, especially when the frequency is down to 500 MHz [4]. Although the high-frequency approximation would not be accurate enough in the low frequency, the divergence problem may be another factor for inaccurate results. This is because the size of ray tubes is so large enough that the scattered fields of these discarded ray tubes can not be simply ignored. More complexly, ray tubes would diverge in the recursive ray tracing, and it is very difficult to identify this divergence accurately due to the discrete , and in Fig. 1. Even if sampling, such as the ray tube

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the equivalent cross section of the ray tube is constructed from the four intersection positions to calculate the scattered field, it is still different from the actual scattered field, especially for the ray tube and . Therefore, diverged ray tubes, whether it is identified or not, would affect the accuracy of the SBR method. The ray-tube divergence problem arises from the procedure of the SBR method: first divide the grid into dense uniform ray tubes, and then trace each ray tube individually. This pre-partition procedure does not consider the geometry of the target. Therefore, ray tubes may intersect the discontinuous area of the target in the recursive ray tracing, such as the boundary and the electrically small and complex part, and this leads to the ray-tube divergence problem. In this paper, we propose an adaptive aperture partition algorithm based on beam tracing to solve the divergence problem. The basic idea is to delay the virtual aperture partition until the recursive beam tracing and divide the aperture into beams only if necessary. A beam is a continuous volume of rays, while a ray tube is simplified to five discrete rays (four corner rays and one central ray) without taking into account the space in the ray tube. Beam tracing was first introduced by Heckbert and Hanrahan [5] in 1984. In computer graphics, beam tracing utilizes the geometric coherence of rays, i.e., neighbor rays usually intersect the same triangle of the target and have the same propagation path, to improve the efficiency of ray tracing. Beam tracing has been employed in antialiasing [6] and the calculation of exact soft shadows [7]. In acoustic modeling, beam tracing has been widely used to calculate exact sound propagation paths from the source to the receiver in virtual environments [8], [9]. Beam tracing has also been applied in the radio propagation prediction to find all exact propagation paths from a transmitter to a receiver. Son and Myung [10] presented a ray tube tree based on uniform geometrical theory of diffraction (UTD) for quasi 3D environments. An enhanced three-dimensional beamtracing algorithm including diffraction phenomena has been developed by Bernardi et al. [11] to evaluate the field distribution in complex indoor environments. Recently, Kim and Lee [12] presented the concept of ray frustums, which is similar to the beam. They also introduced several acceleration techniques, such as quad tree, for fast ray frustums traversal in the environment. Compared to ray-tracing methods, these beam-tracing algorithms eliminate reception tests and existence tests of propagation paths, and improve the numerical efficiency and accuracy. There have been several publications on how to partition the virtual aperture more effectively. Suk et al. [13] presented a multiresolution grid algorithm, which recursively subdivide the divergent ray tube into four uniform children ray tubes until the size of the divergent ray tube is less than the criterion. Although the multiresolution grid algorithm greatly reduces the initial number of ray tubes and accelerates the ray tracing of the SBR method, it still suffers from the ray-tube divergence problem, as the partition of the grid does not take into account the geometry of the target. Weinmann [14] introduced a random sampling strategy on the virtual aperture plane. For each sample, a ray is constructed and recursively traced, and a equivalent ray tube is generated at the intersection position to assess the scattered field. Although it is obvious that using a ray to represent

a ray tube can avoid the divergence problem to a certain extent, more additional samples are required to reduce the prediction error. Recently, Xu and Jin [15] developed analytic tracing of polygonal ray tubes for precisely calculating the illumination and shadowing of triangles. However, they did not describe the 3D implementation in detail. For arbitrarily shaped targets, the effective implementation of adaptive aperture partition requires the exact understanding of the beam-triangle intersection. As a result, we systematically discuss the procedure of the beam-triangle intersection in this paper. In addition, various acceleration techniques have been proposed to reduce the computational time of the SBR method, especially the ray tracing. Jin et al. [16] introduced the octree to decrease the number of ray-triangle intersection tests. The kd-tree has been proved as the best general-purpose acceleration structure for ray tracing of static scenes in computer graphics [17]. Therefore, Tao et al. [18] utilized the kd-tree to accelerate the ray tracing of the SBR method, and they extended this work to a GPU-based SBR method fully executed on the graphics processing unit (GPU) [19] for fast RCS prediction. In this paper, we also use the kd-tree to accelerate the beam-target intersection based on the work of a beam tracer [7]. The key difference is that the beam in the beam tracer is a polygonal pyramid emitting from a point source, while the beam here is a polygonal prism launched from the virtual aperture with the same direction. It is necessary, therefore, to adapt the existing beam tracing in computer graphics to simulate the plane wave. This paper is organized as follows. We first introduce an overview of the proposed adaptive aperture partition in Section II. The beam-triangle intersection and kd-tree beam traversal are described in Section III, and this procedure generates a beam tree representing all possible propagation paths. Section IV is dedicated to the use of the beam tree to predict the scattered field of the target. The experimental results and discussions are covered in Section V. Finally, the conclusions are drawn in Section VI. II. METHOD OVERVIEW In the SBR method, the grid on the virtual aperture is divided into dense ray tubes uniformly before the ray tube tracing, and the ray tube is a square prism started from the virtual aperture. In this paper, a beam is a quadrangular prism or a triangular prism with an irregular cross-section, and a beam is marked as hit or miss depending on whether it intersects the target. The initial beam is launched from the virtual aperture, and the reflected beam is started from the intersected area on the hit triangle. Adaptive aperture partition is performed during beam tracing and is described as follows. Once the incident direction of the electromagnetic wave has been specified, we first construct the polygonal virtual aperture perpendicular to the incident direction, which should be large enough to cover at least the projected area of the target. The whole aperture as an initial beam is traced in the space of the target. When the beam encounters the geometry of the target, it is dynamically split according to the projection of the triangle and generates several irregular hit beams and miss beams. The generated beam continues to be traced until it hits the nearest triangle or exits the target. When the intersection between the initial beam and the target is finished, the virtual

TAO et al.: ADAPTIVE APERTURE PARTITION IN SHOOTING AND BOUNCING RAY METHOD

aperture has been adaptively partitioned and a group of primary beams are generated. For each hit beam, the intersected area on the hit triangle acts as the virtual aperture of the reflected beam. The reflected beam is recursively traced and its virtual aperture is also adaptively split until it exits the target or the number of intersections is larger than the maximum order of the reflection. The kd-tree can be used to accelerate the beam-target intersection. Finally, all hit beams form a beam tree. With the beam tree, we can construct a representative propagation path for each leaf beam. The exit field of each leaf beam can be calculated by GO based on the representative propagation path, and the PO integral is applied to each leaf beam to evaluate the scattered field of the target. As the splitting lines are these projected edges of triangles visible from the incident direction, each hit beam intersects only one triangle. Thus, the virtual aperture is divided according to the geometry of the target and adaptive aperture partition overcomes the divergence problem. The fast generation of the beam tree and electromagnetic computing based on the beam tree are detailed in the following sections. III. ADAPTIVE APERTURE PARTITION Adaptive aperture partition is actually the process of the beam-target intersection: the primary beams are the intersection result between the root beam and the target, while the secondary beams are generated through the intersection of the hit beam and the target. In order to describe the adaptive aperture partition algorithm clearly, we first introduce the basic beam-target intersection procedure, and then describe how to accelerate this intersection procedure using the kd-tree. A. Beam-Target Intersection Since the target in this paper is modeled by triangles, the fundamental operation of the beam-target intersection is the beamtriangle intersection. The beam-triangle intersection would split the beam into the hit part and the miss part, which are equal to the blocked part and the passed part of the beam. This intersection is similar to standard geometry set operations [5], [7], such as the intersection and difference of two polygons. In the beam-triangle intersection, the part of the triangle behind the virtual aperture plane of the beam is first clipped and the remaining part is projected onto the virtual aperture plane. There are three simple cases that can be determined by checking the relative position between the beam and the projected triangle on the virtual aperture plane. When the beam is on the outside of one edge of the projected triangle or the projected triangle is on the outside of one edge of the beam, as shown in Fig. 2(a) and (b), the beam does not intersect the triangle and the intersection is terminated. When the beam is on the inside of all edges of the projected triangle, as shown in Fig. 2(c), the beam is fully inside the projected triangle. In this case, if the beam is the miss beam, we simply mark the beam as the hit beam and record the intersection information, i.e., the intersected triangle and the distance from each corner of the beam to the intersected triangle. Otherwise, if the beam is a hit beam, we need to judge the order of the previous intersected triangle and the current intersected triangle from the direction of the beam, and the nearest triangle is recorded in the beam. However, when two triangles pierce

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Fig. 2. The illustration of three simple cases in the beam-triangle intersection. The gray polygon is the projected triangle. (a) The beam is on one side of the projected triangle. (b) The projected triangle is on one side of the beam, which is a triangular prism. (c) The beam is fully inside the projected triangle.

Fig. 3. The illustration of the beam-triangle intersection. The dark gray triangle is the projected triangle. The beam is iteratively split by each edge. The light gray polygon in each step is the miss beam, and the final miss beam requires an additional splitting indicated by the dashed line. The initial beam is split into one hit beam and four miss beams.

each other, that is neither of the triangles is wholly closer to the virtual aperture plane, and the beam should be split based on the intersection line of the two triangles. Furthermore, there is another simple case for the hit beam. If the current triangle is wholly behind the intersected triangle of the hit beam from the beam’s direction, i.e., the distances from corners of the beam to the triangle are all further than its corresponding distances to the intersected triangle, the beam is blocked by the intersected triangle and does not intersect the triangle. In this case, there is no need to perform the intersection between the hit beam and the triangle. When the relationship between the beam and the triangle does not belong to the above simple cases, the beam and the projected triangle overlap on the virtual aperture plane. The classical Sutherlan-Hodgman polygon clipping algorithm [20] can be used to compute the intersection of the beam and the triangle. This algorithm deals with one triangle edge at one time. The current edge splits the beam into two parts: the miss beam and the hybrid beam, and the remaining edges only need to deal with the hybrid beam. In some situations, the miss beam may not be generated during the triangle edge’s splitting. Finally, the original beam is split into several beams, at least one hit beam and one miss beam. It is worthwhile to point out that hit/miss here is relative to the current triangle. The generated miss beams have the same intersection information with the original beam, while the generated hit beams should update the intersection information. The update procedure is the same as the case that the beam is fully inside the projected triangle. Fig. 3 illustrates the splitting procedure of the beam-triangle intersection. When the corner number of the generated beam is larger than four, an additional splitting is required to ensure that the corner number is less than or equal to four for the consistency of beams. The beam-target intersection performs the beam-triangle intersection for all triangles iteratively. In this process, old beams

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Fig. 4. The illustration of the initial beam intersection with the right-angle dihedral corner modeled by four triangles. The initial beam is iteratively split by each triangle from left to right. The dashed lines in each step are splitting lines based on the projected edges of the current processed triangle (the dark gray color), and the previous generated hit beams are indicated by the light gray color. The intersection process of the initial beam and the first triangle is illustrated in Fig. 3, and the hit beam h is generated. The second triangle splits two miss beams and generates two hit beams h and h and three miss beams, while the third triangle splits one miss beam and generates one hit beam h and three miss beams. The final triangle splits two miss beams and generates two hit beams h and h and four miss beams.

Fig. 5. The primary hit beams on the virtual aperture of the airplane from the incident direction.

are split and new beams are generated gradually, and all current beams including hit beams and miss beams continue the beam-triangle intersection for the remaining triangles. Finally, the virtual aperture is adaptively divided into irregular areas, and a group of primary beams with these irregular cross-sections are generated. Among these beams, miss beams have no intersection with the target, and each hit beam intersects only one triangle of the target. As the triangles’ edges of the target are all located at the splitting lines of the virtual aperture, adaptive aperture partition generates convergent beams and solves the ray-tube divergence problem. The initial beam is the root beam, and all hit beams form the first level of the beam tree. Fig. 4 illustrates the intersection process of the initial beam with the right-angle dihedral corner, and the initial beam is divided into six primary hit beams. Fig. 5 shows primary hit beams on the virtual aperture divided based on the visible triangles of the airplane from the incident direction, and each hit beam intersects only one visible triangle of the airplane. For multiple reflections, we take the intersected area on the hit triangle and the reflected direction of the hit beam as the virtual aperture and the propagation direction of the reflected beam. The reflected beams continue to perform the beam-target intersection, and the only difference is that all generated miss beams are hit-exit beams and should become sibling nodes of the current reflected beams. As the virtual aperture of the reflected beam is adaptively divided according to the geometry of the target, it also eliminates the divergence of the reflected beam.

Fig. 6. The illustration of the primary reflected beam intersection with the right-angle dihedral corner. The left (right) three reflected beams and the partitions on their virtual apertures are shown in light (dark) gray color on the left (right), and the dashed lines on the virtual aperture are splitting lines. The primary hit beam h intersects triangles T and T , and is split into two secondary hit beams h and h , while the primary hit beams h (h ) only intersects the triangle T , and generates one secondary hit beam h (h ). The primary hit beam h intersects the triangle T and generates one secondary hit beam h , and it also intersects the triangle T and generates one secondary hit beam h and one miss beam h . The primary hit beam h (h ) only intersects the triangle T , and generates one secondary hit beam h (h ) and one miss beam h (h ).

The reflected beam is recursively traced until the beam exits the target or the number of intersections is larger than the maximum order of the reflection. Fig. 6 illustrates the reflected beams and the partition on their virtual apertures of the right-angle dihedral corner, and the reflected beams are constructed based on the primary hit beams in Fig. 4. Finally, all hit beams constitute a beam tree. The root (zero level) of the beam tree is the initial beam without any splitting, and the nodes in the level of the beam tree are the th-reflection hit beams. As the virtual aperture of each level is adaptively divided based on the geometry of the target, new beams are dynamically generated to avoid the divergence problem. Fig. 7 depicts a beam tree generated from the right-angle dihedral corner. The leaf nodes are the exit beams, and electromagnetic computing is only performed on these beams. B. Kd-Tree Beam Traversal The beam-target intersection discussed above adaptively splits the virtual aperture based on the geometry of the target during the recursive beam tracing, and it eliminates the divergence problem in a unified manner. However, all triangles are required to be projected on the virtual aperture of each beam to perform the beam-triangle intersection. It is, therefore, very time-consuming when the geometry of the target is complex. Actually, this efficiency problem of the beam-target intersection is analogous to that of the ray-target intersection in the SBR method. Various acceleration techniques have been proposed to improve the efficiency of the ray-target intersection, such as the octree [16] and kd-tree [18]. In this section, we introduce the kd-tree to accelerate the beam-target intersection. This basic procedure is based on the traversal algorithm [7], which is used to calculate the exact soft shadows for point lighting. The essential difference is that beams in the SBR method are directional beams with the same ray directions instead of beams emitting from a point source. The kd-tree is a simplified version of the binary space partitioning tree and has been used in the SBR method [18], [19]. The octree, which is common in computational electromagnetic [21], recursively uses the middle point of the extend in each direction as the splitting position to subdivide the target space into

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Fig. 7. The illustration of the beam tree generated from the right-angle dihedral corner. The beams in the first and second level correspond to the primary hit beams and the secondary hit beams as shown in Figs. 4 and 6, respectively. The miss beams generated in Fig. 6 are sibling nodes of the primary hit beams in Fig. 4. A representative propagation path from the leaf beam h to the initial beam s (P P P ) is indicated in a bold line.

! !

eight equal sub-spaces. The kd-tree takes into account the triangle distribution in the target space to search for the optimal splitting axis and position based on the ray-tracing cost estimation model, and recursively employs the optimal axis-perpendicular plane to divide the target space into two uneven axis-aligned sub-spaces. The ray traversal algorithm starts at the root node of the kd-tree and searches for the nearest intersected triangle of the ray in the target, and most rays could find the intersection in the first leaf nodes visited [17]. The construction procedure of the kd-tree has been introduced in [18], and Pharr and Humphreys’ book [22] gives the detailed description on the ray traversal algorithm in the kd-tree. The beam traversal procedure is based on the ray traversal procedure, as a beam can be taken as three/four corner rays in the kd-tree traversal. However, we also need to take into account other rays in the beam, and there are several differences between the two procedures: (1) the choice of the child node to traverse, (2) the maintenance of the stack, (3) the beam-triangle intersection in the leaf node, (4) the termination condition of the traversal. The beam-triangle intersection in the leaf node has been introduced in the Section III.A, and the Appendix provides detailed descriptions about other three differences and pseudocode for the interested reader. Each beam is recursively traced based on the proposed kd-tree traversal algorithm. The use of the kd-tree can significantly accelerate the beam-target intersection, as it eliminates a large number of unnecessary beam-triangle intersections. In addition, the virtual aperture of the beam is split only by visible triangles, not all triangles, and this reduces the final number of hit beams. IV. BEAM TREE BASED ELECTROMAGNETIC COMPUTING The beam-target intersection adaptively splits the virtual aperture of each beam, and finally it generates a beam tree. A leaf beam is a group of rays with the same propagation path, and a beam tree contains all possible propagation paths. The propagation path here refers to all rays of the leaf beam are reflected by the same sequence of triangles of the target. The scattered field of the target can be obtained by evaluating the scattered fields of leaf beams only. We calculate the cen-

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Fig. 8. The illustration of backward ray tracing. The beam tree of the rightangle dihedral corner is displayed in Fig. 7. The representative propagation path P P . of the leaf beam h is P

! !

tral position of the leaf beam’s virtual aperture, and employ backward ray tracing to produce a representative ray path for each leaf beam. More specifically, the origin and the direction and the inverse of the backward ray are the central position propagation direction of its parent beam, respectively. The backward ray is tested for intersection with the virtual aperture of its parent beam, and it obtains the intersection position . The origin and the direction of the backward ray are replaced with and the inverse propagation directhe intersection position tion of the corresponding parent beam, respectively. The backward ray is recursively traced until it hits the virtual aperture of . the root beam and produces the final intersection position Thus, the representative ray path of the leaf beam is from to : . Fig. 8 displays an example of backward ray tracing in the right-angle dihedral corner, and its beam tree is illustrated in Fig. 7. Geometric optics is applied to each intersection of the representative ray path to evaluate the exit field of the leaf beam. The reflected field is calculated based on the field before the intersection and the geometric information of the intersected triangle as follows:

(1) where , and . The vector is the propagation direction before the intersection, is the propagation direction after the intersection, and is and the the normal of the intersection. The incident field is reflected field is . The detailed formulas about the reflection coefficients are explained in [1], [3]. Since the leaf beam is reflected by a group of triangles, the exit field on the leaf beam has the same amplitude and a linear phase variation with the exit field of the representative ray. The scattered field of the leaf beam can be approximated by the PO integral as follows:

(2)

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Fig. 9. The VV-polarization comparison of uniform aperture partition, adaptive aperture partition and MLFMM results for a simple 1 m 1 m square patch at 500 MHz.

2

where is the observation direction. The of the polygon expressed as the exit field

can be

(3) Generally, the coefficients and in the EH formulation (0.5) provide a better result [2]. The PO integral on the planar aperture can be approximately in a more computable form [2], and the shape function can be solved through the 2D Fourier transform [23]. The scattered field of the target is generated by summing all leaf beams’ scattered fields. V. RESULTS AND DISCUSSION Several numerical experiments were performed to verify the accuracy and efficiency of the proposed adaptive aperture partition. The original partition of the SBR method is referred to as uniform aperture partition in this section. In our implementation, the identified divergent ray tubes are simply discarded in uniform aperture partition. These experiments were tested on an Intel Core 2 Quad Q9550 (2.83 GHz) processor with an NVIDIA GeForce 285 GTX (CUDA Toolkit 3.0), and at most fifth-order reflection is considered. The target is described by triangles and meshed according to the geometric error instead of the wavelength. Thus, it is desirable that the flat part is modeled by large triangles, as this reduces the beam-triangle intersections and avoids many unnecessary beam splittings. A simple 1 m 1 m square patch is used to analyze the influence of the ray-tube divergence problem on the accuracy of the SBR method. Fig. 9 shows the VV-polarization RCS result, which are predicted using an angular resolution of 1 at 500 . Adaptive aperture partition acMHz frequency with tually generates two large hit beams covering the whole patch, and the result is equal to the PO integral of the total patch. The

Fig. 10. The VV-polarization comparison of uniform aperture partition, adaptive aperture partition and MLFMM results for a simple 1 m 1 m square patch at 3 GHz.

2

result of uniform aperture partition at is largely different from the one of adaptive aperture partition. The ray-tube divergence problem severely affects the accuracy in this case, since ray tubes that intersect the boundary of the patch are not included in the calculation of the scattered field and the ray-tube is considerable large compared to the target. The size result of uniform aperture partition at is also shown in Fig. 9. In this case, the relative error due to the ray-tube divergence problem is greatly reduced, and the result tends toward the one of adaptive aperture partition. The result of adaptive aperture partition which eliminates the ray-tube divergence problem is more accurate than the one of uniform aperture partition in the SBR method. In other words, adaptive aperture partition can generate the convergent solution of the SBR method when the ray-tube size tends toward infinitesimal. The result of the multilevel fast multipole method (MLFMM) is used as a comparison to verify the result. As PO is mainly valid in the nearby direction of specular reflection and it is also not accurate enough in the low frequency, there are some deviations between the results of the SBR method and MLFMM. However, the result of adaptive aperture partition matches the MLFMM result and is more accurate than the one of uniform aperture partition. With the increasing frequency, the SBR method is highly effective in predicting the scattered field of arbitrarily shaped targets. However, the ray-tube divergence problem still affects the accuracy of the SBR method. Fig. 10 shows the VV-polarization RCS result for the same square patch, which are predicted using . an angular resolution of 1 at 3 GHz frequency with As can be observed clearly from Fig. 10, there are obvious differences between the results of uniform aperture partition at and adaptive aperture partition, and the result of uniform tends to the one of adaptive aperaperture partition at ture partition. Adaptive aperture partition generates the optimal result of the SBR method and its result is more similar to the MLFMM result. In addition, the result of uniform aperture partition depends on the boundary of the virtual aperture and the partition criterion, and there are differences among results under different configurations of the virtual aperture. In contrast to

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Fig. 11. The VV-polarization comparison of adaptive aperture partition, adapTW-ILDC and MLFMM results in the bistatic RCS tive aperture partition calculation for the pencil at 3 GHz.

+

uniform aperture partition, the result of adaptive aperture partition does not rely on the boundary of the virtual aperture and the order of partition. Thus, adaptive aperture partition is more stable than uniform aperture partition. The agreement between adaptive aperture partition and MLFMM is less good for angles higher than 35 , as the observation direction in these angles deviates from the specular reflection direction, and PO could not evaluate the scattered field accurately in this case. The pencil illustrated in Fig. 11 is used to verify the accuracy of adaptive aperture partition in the SBR method. It was first employed by Hastriter [24] to verify the Fast Illinois Solver Code (FISC), and then it was also used to verify the effectiveness of truncated-wedge incremental-length diffraction coefficients and (TW-ILDC) [25], [26]. The incident direction is at , and the observation directions are from to on the plane using an angular resolution of 1 . These calculation parameters are the same as [26]. Fig. 11 illustrates the bistatic VV-polarization RCS comparison of adaptive aperture partition and MLFMM results at 3 GHz. When specular scattering is not the dominant mechanism, the edge-diffraction effect could not be ignored. The visible edges can be identified by checking the boundaries of primary hit beams, as the projected triangles’ edges are the splitting lines of beams. Then, TW-ILDC is applied to these visible edges to evaluate the edge diffracted field, and the detailed formulas of TW-ILDC are described in [25]. It is clear that the SBR TW-ILDC result matches well the MLFMM result, and the SBR TW-ILDC result is nearly the same as the one in [26]. The deviation in observation angles higher than 300 may be due to the pencil-tip diffraction, which is not included in our current implementation. The trihedral corner reflector, which is a typical benchmark target, is used to verify the high frequency multiple-bounce scattering [2]. The geometry of trihedral corner reflector is depicted in Fig. 12, three right-angled triangles with the side length 1 m. The 91 equal-spaced incident directions are from to on the plane. The monostatic HH-polarization RCS comparison of uniform aperture partition, adaptive

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Fig. 12. The HH-polarization comparison of uniform aperture partition, adaptive aperture partition and MLFMM results in the monostatic RCS calculation for the trihedral corner reflector at 3 GHz.

aperture partition, and MLFMM results at 3 GHz is illustrated is in Fig. 12. The result of uniform aperture partition at smaller than the one of adaptive aperture partition due to the ray-tube divergence problem. A good agreement between the adaptive aperture partition and MLFMM results further verifies the accuracy of adaptive aperture partition in the SBR method. The impact of the ray-tube divergence problem is more obvious for electrically large and complex targets. We calculate the scattered field of the airplane as shown in Fig. 5 to show such influence on accuracy. The size of the airplane is approximately 14 m 17 m 4.5 m, and the structure of the airplane is much more complex. The 181 equal-spaced incident directions to on the plane. Fig. 13 are from shows the RCS comparison of adaptive aperture partition and uniform aperture partition results at 3 GHz. It can be seen obviously that the result of uniform aperture partition is slightly smaller than the one of adaptive aperture partition due to the discarding of divergent ray tubes. There is no comparison with the MLFMM result, as the MLFMM becomes unusable for the airplane due to the limited computational resources. Table I shows the total computational times of all incident angles using CPU uniform aperture partition, GPU uniform aperture partition, and the proposed adaptive aperture partition for experimented targets. As demonstrated in [19], GPU uniform aperture partition is at least 30 times faster than CPU uniform aperture partition for most cases. Especially, it fully exploits the , achieving an acceleration ratio about potential of GPU at 100 for the trihedral corner reflector and airplane. As the trihedral corner reflector has a very simple geometric shape, adaptive aperture partition only needs 0.078 seconds for all incident angles, while the computational times of CPU uniform aperture and are 18.03 and 1974.76 seconds, repartition at spectively. Adaptive aperture partition is even faster than GPU uniform aperture partition for this target, about 2 and 60 times and , respectively. Thus, adaptive aperture faster at partition is very well suited to the target with large flat regions, such as the patch and trihedral corner reflector.

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Fig. 13. The comparison of uniform aperture partition and adaptive aperture partition results in the monostatic RCS calculation for the airplane as shown in Fig. 5 at 3 GHz. (a) VV-polarization result, (b) HH-polarization result.

TABLE I THE COMPUTATIONAL TIMES OF THE CPU UNIFORM APERTURE PARTITION, GPU UNIFORM APERTURE PARTITION [19], AND ADAPTIVE APERTURE PARTITION FOR THE RESULTS IN FIGS. 9–14 (SEC)

In computer graphics, beam tracing is much slower than ray tracing for models of complex structures, especially when the pixel size on the sampling plane is larger than the average visible triangle size. In the SBR method, adaptive aperture partition also has such a problem. The beam number is proportional to the number of visible triangles. In our experiments, the average primary hit beam number/visible triangle number is 292609/33874, 5/3, and 16251/1128 for the targets in Figs. 11, 12, and 13, respectively. This ratio reaches up to 14.4 for the airplane. The reason is that as visible triangles split the beam iteratively, the area on the virtual aperture corresponding to one visible triangle may be already split into several beams before encountering this visible triangle. This also explains why adaptive aperture partition may be slower than uniform aperture partition in some situations, such as the pencil and the airplane. As shown in Table I, adaptive aperture partition is much slower , as it spends addithan CPU uniform aperture partition at tional computational burden on the generation of a high number of beams and the size of many beams would be much smaller than ray tubes in uniform aperture partition. However, the accuracy of adaptive aperture partition is greatly improved com. Although the result pared to uniform aperture partition at is very similar to the one of uniform aperture partition at of adaptive aperture partition, the computational time is terribly long due to a large number of ray tubes. For example, the total

computational time of CPU uniform aperture partition at is about 100 times slower than uniform aperture partition at for the airplane, and adaptive aperture partition is faster than for the pencil and airCPU uniform aperture partition at plane. Although the performance of adaptive aperture partition is much worse than GPU uniform aperture partition, adaptive aperture partition can also explore the GPU power for acceleration, which would be our future work. Another feature of adaptive aperture partition is that the partition of the virtual aperture is insensitive to the incident frequency, and it is only related to the geometry of the target. This is particularly useful for the high-resolution range and inverse synthetic aperture radar (ISAR) applications, since we only need to generate one beam tree for all frequencies in the bandwidth. In the terahertz (THz) band, the electrical size of the target is extremely large, and the number of ray tubes in uniform aperture partition is also significantly increased. Thus, the computational time is seriously affected by such a large number of ray tubes [27]. Adaptive aperture partition is very suited to the THz band, as it is no concern of the frequency. For instance, Fig. 14 shows the RCS result of the pencil at 1 THz, and other computational parameters are the same to the one in Fig. 11. As can be seen from Table I, the computational time of adaptive aperture partition is almost the same for all frequencies, 822.40 seconds, while the computational time of GPU uniform aperture partition at is 3384.81 seconds, about 4 times slower, and its computational time at is extremely time-consuming, more than one day. Compared to the result in the GHz as shown in Fig. 11, the maximum/minimum RCS value is increased/decreased in the THz. VI. CONCLUSION This paper presents an adaptive aperture partition algorithm to solve the ray-tube divergence problem, and also proposes the kd-tree to accelerate the beam-target intersection. Adaptive aperture partition is more stable than uniform aperture partition in the original SBR method and yields the optimal result of the

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Fig. 14. The bistatic RCS result of the pencil at 1 THz.

SBR method. This is because adaptive aperture partition takes into account the geometry of the target and the generated hit beams are convergent. In contrast, the result of uniform aperture partition is sensitive to the boundary of the virtual aperture and the partition criterion to some extent, and it is also influenced by the divergent ray tubes for complex targets. Another feature of adaptive aperture partition is that the aperture partition depends only on the geometry of the target, and it is insensitive to the incident frequency. Therefore, adaptive aperture partition can significantly accelerate the high-resolution range and inverse synthetic aperture radar (ISAR) applications as well as the RCS prediction in the THz band. Numerical results demonstrate the accuracy and effectiveness of the proposed algorithm. Furthermore, the beam-target intersection accelerated by the kd-tree can be adapted to other electromagnetic applications, such as the radio propagation prediction [10]–[12].

APPENDIX DIFFERENCES BETWEEN RAY TRAVERSAL AND BEAM TRAVERSAL IN THE KD-TREE The other three differences between ray traversal and beam traversal in the kd-tree are described in detail as follows: a) The Choice of the Child Node to Traverse: When a beam encounters an interior node, we first need to choose which child node to traverse. The near and far child nodes are decided by the beam’s corners with respect to the splitting plane. If the corners straddle the splitting plane, the beam’ propagation direction is taken into account to determine the near and far nodes. It is not accurate enough to simply use the beam’s corner rays to determine the traversal order of all rays in the beam [7], [28]. As illustrated in Fig. 15, all four corner rays of the beam need to traverse the far node only. However, the rays in the beam actually hit the near node, and the near node should be traversed first. This problem can be solved by maintaining three ranges, i.e., one range for each axis, and each range defines the part of the ray within two bounding planes perpendicular to its corresponding axis. If distances from the corners to the splitting

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Fig. 15. The illustration of the beam traversal. All four corner rays of the beam need to visit the far node (the back node) only. However, the beam hits the near node and should traverse the near node firstly.

plane are all further than either of the other axes’ maximum dis, or all corner rays face away from tances only the near node needs to be traversed. the far node If distances are all less than either of the other axes’ minimum , only the far node needs to be distances traversed. Otherwise, both nodes need to be traversed. b) The Maintenance of the Stack: The ray traversal employs a node stack as a priority queue of nodes left to visit according to how close to the origin of the ray. The ray is not changed during the ray traversal, but the beam would be split and gradually generate new beams during the beam traversal. As a result, besides the node stack, the beam traversal requires an additional beam stack. The node stack preserves the to-be-visited tree node and its corresponding beam, and the beam stack maintains the new generated to-be-processed beam. c) The Termination Condition of the Traversal: In the ray traversal, when the ray’s distance to a triangle is less than the ray’s maximum distance to the node, the intersection is considered as the nearest intersection along the ray and the traversal terminates. Likewise, when the corner rays’ distances to the triangle are all less than the corner rays’ maximum distances to the node in any dimension, the triangle can be guaranteed to the nearest triangle along the beam and the hit beam can stop its traversal. Otherwise, the hit beam still needs to traverse the kd-tree until the entry distance to the next to-be-visited node in any one axis is larger than the distances to the hit triangle. The miss beam terminates its traversal after passing through the target space, i.e., there is no element left in the node stack. After we discussed the differences between the ray traversal procedure and the beam traversal procedure, a kd-tree traversal algorithm for beam tracing is shown in Algorithm 1. It should be and as well as noted that the comparison between corresponds to the array comparison of all corner rays. Each node in the node stack should be traversed by all new generated beams after the node is pushed into nodeStack. The function beamTriIntersect deals with the intersection of the beam with one triangle in the leaf node, and keeps the original beam or new generated beams if split into stack. All beams in the stack

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should be tested for intersection with the remaining triangles to find the nearest intersected triangle. Finally, the input beam is split into a group of beams, and these beams are children beams of the input beam. Algorithm 1 Kd-Tree Traversal Algorithm for Beam Tracing

while

and

do

// process interior nodes while do

if

or

or then

else if

or then

else

end if end while // intersection test with the triangles in the leaf node stack.push(beam) for triangle in node.triangles do for beam in stack do beamTriIntersect(beam, triangle, stack) end for end for beamStack.push(stack) // get the next beam and node from the stack (beam, node) getBeamNode(nodeStack, beamStack) end while return beam ACKNOWLEDGMENT The authors would like to thank Prof. T. J. 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] M. L. Hastriter and W. C. Chew, “Comparing Xpatch, FISC, and ScaleME using a cone-cylinder,” in Proc. IEEE Antennas and Propag. Society Int. Symp., Monterrey, CA, Jun. 2004, vol. 2, pp. 2007–2010. [5] P. S. Heckbert and P. Hanrahan, “Beam tracing polygonal objects,” in Proc. SIGGRAPH’84, New York, 1984, pp. 119–127. [6] D. Ghazonfarpour and J.-M. Hasenfratz, “A beam tracing method with precise antialiasing for polyhedral scenes,” Comput. Graph., vol. 22, no. 1, pp. 103–115, 1998. [7] R. Overbeck, R. Ramamoorthi, and W. R. Mark, “A real-time beam tracer with application to exact soft shadows,” in Proc. EuroGraphics Symp. on Rendering, Jun. 2007, pp. 85–98. [8] T. Funkhouser, I. Carlbom, G. Elko, G. Pingali, M. Sondhi, and J. West, “A beam tracing approach to acoustic modeling for interactive virtual environments,” in Proc. SIGGRAPH’98, New York, 1998, pp. 21–32. [9] N. Tsingos, T. Funkhouser, A. Ngan, and I. Carlbom, “Modeling acoustics in virtual environments using the uniform theory of diffraction,” in Proc. SIGGRAPH’01, New York, 2001, pp. 545–552. [10] H.-W. Son and N.-H. Myung, “A deterministic ray tube method for microcellular wave propagation prediction model,” IEEE Trans. Antennas Propag., vol. 47, no. 8, pp. 1344–1350, 1999. [11] P. Bernardi, R. Cicchetti, and O. Testa, “An accurate UTD model for the analysis of complex indoor radio environments in microwave WLAN systems,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1509–1520, 2004. [12] H. Kim and H. Lee, “Accelerated three dimensional ray tracing techniques using ray frustums for wireless propagation models,” Progress Electromagn. Res. (PIER), vol. 96, pp. 21–36, 2009. [13] 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. [14] 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. [15] F. Xu and Y.-Q. Jin, “Bidirectional analytic ray tracing for fast computation of composite scattering from electric-large target over a randomly rough surface,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1495–1505, 2009. [16] 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. [17] V. Havran, “Heuristic ray shooting algorithms” Ph.D. dissertation, Univ. of Czech Technical, Prague, Nov. 2000 [Online]. Available: http://www.cgg.cvut.cz/~havran/phdthesis.html [18] 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. [19] Y.-B. Tao, H. Lin, and H.-J. Bao, “GPU-based shooting and bouncing ray method for fast RCS prediction,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 494–502, 2010. [20] D. D. Hearn and M. P. Baker, Computer Graphics With Open GL, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 2003. [21] W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Boston, MA: Artech House, 2001. [22] M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation. San Fransisco, CA: Morgan Kaufmann, 2004. [23] S.-W. Lee and R. Mittra, “Fourier transform of a polygonal shape function and its application in electromagnetics,” IEEE Trans. Antennas Propag., vol. 31, no. 1, pp. 99–103, 1983. [24] M. L. Hastriter, “A study of MLFMA for large-scale scattering problems,” Ph.D. dissertation, Univ. of Illinois at Urbana-Champaign, Champaign, IL, 2003. [25] 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. [26] J. T. Moore, A. D. Yaghjian, and R. A. Shore, “Shadow boundary and truncated wedge ILDCs in Xpatch,” in Proc. IEEE Antennas and Propag. Society Int. Symp., 2005, vol. 1, pp. 10–13.

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[27] Z. Li, T.-J. Cui, X.-J. Zhong, Y.-B. Tao, and H. Lin, “Electromagnetic scattering characteristics of PRC targets in the terahertz regime,” IEEE Antennas Propag. Mag., vol. 51, no. 1, pp. 39–50, 2009. [28] A. Reshetov, A. Soupikov, and J. Hurley, “Multi-level ray tracing algorithm,” ACM Trans. Graph., vol. 24, no. 3, pp. 1176–1185, 2005.

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Hai Lin received the B.Sc. and M.Sc. degrees in electrical engineering from Xidian University, Xi’an, China, in 1987 and 1990, respectively, and the Ph.D. degree in computer science from Zhejiang University, Hangzhou, China, in 2000. Currently, he is a Professor of visual computing in the State Key Lab. of CAD&CG, Zhejiang University. He is also a Visiting Professor at the Department of Computing and Information Systems, University of Bedfordshire, U.K. His research interests include computational electromagnetic, computer graphics, scientific visualization.

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, Zhejiang University. His research interests include computational electromagnetics and data visualization.

Hujun Bao received the B.S. and Ph.D. degrees in applied mathematics from Zhejiang University, Hangzhou, China, in 1987 and 1993, respectively. Currently, he is a Professor and the Director of State Key Laboratory of CAD&CG, Zhejiang University. His main research interest is computer graphics and computer vision, including realtime rendering technique, geometry computing, virtual reality and 3D reconstruction.

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On the Relation Between Optimal Wideband Matching and Scattering of Spherical Waves Sven Nordebo, Senior Member, IEEE, Anders Bernland, Mats Gustafsson, Member, IEEE, Christian Sohl, and Gerhard Kristensson, Senior Member, IEEE

Abstract—Using an exact circuit analogy for the scattering of vector spherical waves, it is shown how the problem of determining the optimal scattering bounds for a homogeneous sphere in its high-contrast limit is identical to the closely related, and yet very different problem of finding the broadband tuning limits of the spherical waves. Using integral relations similar to Fano’s broadband matching bounds, the optimal scattering limitations are determined by the static response as well as the high-frequency asymptotics of the reflection coefficient. The scattering view of the matching problem yields explicitly the necessary low-frequency asymptotics of the reflection coefficient that is used with Fano’s broadband matching bounds for spherical waves, something that appears to be non-trivial to derive from the classical network point of view. Index Terms—Fano matching bounds, Herglotz functions, positive real functions, spherical waves, sum rules, wideband matching.

I. INTRODUCTION

I

NTEGRAL identities based on the properties of Herglotz functions [1], or positive real (PR) functions [2], constitute the basis for deriving Fano’s broadband matching bounds [3] and have been used recently to describe a series of new sum rules for the scattering of electromagnetic waves [1], [4]–[6]. Hence, under the assumptions of linearity, continuity, time-translational invariance and passivity, sum rules can be derived from the analytic properties of the forward scattering dyadic, see, e.g., [5], [6], and have also applications in antenna theory, see, e.g., [7]–[11]. In [12], similar relations are used to determine the ultimate thickness to bandwidth ratio of radar absorbers. Limitations on the scattering of vector spherical waves have been considered in [4]. The sum rules rely on the well-known connection between the transfer functions of causal and passive systems and Herglotz functions, or positive real (PR) functions, as well as the analytic properties of these functions, see, e.g., [2], [13], [14].

Manuscript received November 12, 2010; revised January 20, 2011; accepted February 09, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the High Speed Wireless Communications Center of the Swedish Foundation for Strategic Research (SSF). S. Nordebo is with the School of Computer Science, Physics and Mathematics, Linnaeus University, 351 95 Växjö, Sweden (e-mail: [email protected]). A. Bernland, M. Gustafsson, C. Sohl, and G. Kristensson are with the Department of Electrical and Information Technology, Lund University, Box 118, 221 00 Lund, Sweden (e-mail: [email protected]; mats.gustafsson@eit. lth.se; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161451

Consequently, sum rules and limitations on arbitrary reflection coefficients stemming from passive systems can be derived, as described in [1]. The procedure is reviewed briefly in this paper. By using Fano’s approach, optimum broadband tuning limits of the higher-order spherical waves are considered in [7], [8], giving important physical insight into the matching limitations for ultrawideband (UWB) antennas, see also [15]–[21]. Previously, the Fano broadband matching bounds have been applied mainly to the lowest order spherical waves. However, methods of finding solutions to the matching limitations for the spherical waves of higher orders have recently received further development, see, e.g., [8], [22], [23] with references. Hence, there is a need to further develop analytical results as an aid in the related numerical analysis. In [7], [8], it is conjectured that the low-frequency asympis of the form totics of the positive real function (1) where

is the reflection coefficient corresponding to a TE or TM spherical wave of order , the Laplace the speed variable, the radius of a circumscribing sphere, of light in free space, and constants to be determined from network analysis and the circuit analogy of the spherical wave impedance. The aim of this paper is to give an analytical solution to the conjecture (1) made in [7], [8], by using the recent developments in the application of sum rules for passive scatterers. The rest of the paper is organized as follows: In Section II is given a brief outline on the general approach to obtain sum rules and physical limitations for reflection coefficients stemming from passive systems, and the Fano broadband matching bounds for spherical waves is put in this context. The related conjecture (1) is also given a precise formulation. In Section III is treated the problem of finding the optimal limitations for scattering of vector spherical waves. Here the geometry of the spherical object is known but the dispersion is unknown. A detailed study of the high-frequency asymptotics of the reflection coefficient is performed including, e.g., the Debye and the Lorentz dispersion models, and is given in the Appendix A. Using the integral relations derived in [1], which are similar to the relations in the derivation of Fano’s broadband matching bounds [3], the optimal scattering limitations are determined by the static response as well as the high-frequency asymptotics of the reflection coefficient. As with the Fano approach, the integral relations yield a non-convex global optimization problem which in general is difficult to handle.

0018-926X/$26.00 © 2011 IEEE

NORDEBO et al.: ON THE RELATION BETWEEN OPTIMAL WIDEBAND MATCHING AND SCATTERING OF SPHERICAL WAVES

Fig. 1. The cone fz : # 

arg

z   0 #g for some # 2

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denotes the complex conjugate [1]. Herglotz funcwhere tions stemming from reflection coefficients in real physical systems exhibit this symmetry property [2], [13], [14]. For all Herand glotz functions it holds that . Throughout this paper, means in the cone for any , , see Fig. 1. and likewise for It is assumed that the low- and high-frequency asymptotic expansions are given by

; =2].

(0

as In Section IV, the two previous sections are tied together. An exact circuit analogy for the scattering of spherical waves is used similar to [24], [25], to show how the problem of determining the scattering limitations for a homogeneous sphere in its highcontrast limit becomes identical to the closely related, and yet very different problem of finding the broadband tuning limits of the spherical waves [7], [8]. Furthermore, the scattering view of the matching problem yields explicitly the necessary lowfrequency asymptotics of the reflection coefficient (1), i.e., the that are used with Fano’s broadband matching coefficients are given by bounds for spherical waves. The coefficients the (50) in this paper. This is something that appears to be nontrivial to derive from the classical network point of view. Finally, in Section V is given a numerical example where a relaxation of the Fano equations is considered which is easily solved, and which is especially useful in the regime of Rayleigh scattering. II. LIMITATIONS ON PASSIVE REFLECTION COEFFICIENTS This section reviews the general approach presented in [1] to find sum rules and physical limitations for reflection coefficients stemming from linear, continuous, time-translational invariant, and passive physical systems. The approach, which is used for the matching and scattering problems in the following sections, relies on the well-known connection between the transfer functions of causal and passive systems and Herglotz (or positive real) functions, as described in, e.g., [1], [2], [13], [14]. A set of integral identities for Herglotz functions was proved in [1]. Applied to a reflection coefficient , they give a set of sum rules. The sum rules relate integrals of over infinite frequency intervals to the static and high-frequency properties of the system, and so are much like Fano’s matching equations. Physical limitations for the reflection coefficient are derived by considering finite frequency intervals. The general approach is presented in more detail in [1], where also all the necessary proofs are given.

(3) as and where the little ordo notation is defined as in [27], and are non-negative integers (or possibly infinity), chosen so that all the coefficients and are real valued, and hence that all the even indexed coefficients are zero [1]. The asymptotic exfor any argument pansions are clearly valid as in the case is analytic in a neighbourhood of the origin (infinity). The following integral identities have been derived in [1], and they are the starting point to derive limitations on reflection coefficients: (4)

. It should be noted that the integral for identities in (4) do not apply in the case when the largest possible in (3). In case the imaginary integers and are is not regular on the real axis, the integral should part be interpreted as

(5) , and where i denotes the imagifor . This is equivalent to interpreting (4) in the nary unit, distributional sense [1]. Equation (4) is assumed to be replaced by (5) whenever necessary throughout this paper. The identities (4) can be used to derive Fano’s matching equations [3]. They have also been used recently to derive a series of new sum rules for the scattering of electromagnetic waves [4]–[6], with applications in antenna theory [7]–[9]. There are other applications for the identities (4) as well, see, e.g., [12], [28]–[31].

A. Herglotz Functions and Integral Identities

B. Limitations on Passive Reflection Coefficients

Here the class of Herglotz functions is reviewed briefly, and the integral identities used to obtain sum rules and limitations for reflection coefficients are presented. A Herglotz function is defined as an analytic function for with the property that , cf., [1], [26]. It is assumed that obeys the symmetry

denote a reflection coefficient of a system where the Let reflected signal is related to the incoming signal as

(2)

where is the angular frequency. It is assumed that the reflection coefficient is the Fourier transform of a real valued convolution kernel . The Fourier transform is defined as

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when is sufficiently regular, and it is otherwise defined in the appropriate distributional sense [1], [14]. corresponds to a passive system, it is bounded with If . The system is causal if the reflection coefficient which vanishes corresponds to a causal convolution kernel . It is a well-known result that the reflection coeffifor of a passive and causal system is an analytic function cient bounded in magnitude by one in the open upper half plane, i.e., is analytic and for [2], [13]. The scattering of electromagnetic waves is always causal. However, sometimes the scattering of electromagnetic waves, such as the scattering of incoming and outgoing spherical waves, may be perceived as non-causal depending on the definition of the scattering coefficient [4], [26]. Hence, for a non-causal system corresponds a time delay can be introduced so that to a causal convolution kernel . The reflection coeffiis thus an analytic function for , and it is cient . bounded according to A Herglotz function can be constructed by taking the complex logarithm of [1]. It requires that the zeros of are removed, which is done with a Blaschke-product [32]. The Herglotz func): tion is therefore (with (6) of in are repeated according to their where the zeros is real-valued, and so multiplicity. The convolution kernel is real valued on the imaginary axis with the symmetry for . Without loss of generality it may , in which case obeys the symbe assumed that , consider the function instead. metry (2). If Suppose that the low-frequency asymptotics of is given by as . The low-frequency asymptotics of is then

(7)

. The integral identities (8) then yield bandwidth the following inequalities:

(9) where it has been used that by

. The factor

is defined

(10) Note that for all the narrowband approximation when

, and .

in

C. Fano Broadband Matching Bounds for Spherical Waves The classical broadband matching bounds for lossless networks by Fano [3] are revisited using the Herglotz function formulation and integral identities (4) and (8). The Fano matching bounds are then used to formulate the problem of finding the broadband tuning limits of the wave impedance of the spherical waves as in [7], [8]. In circuit theory it is convenient to employ the Laplace vari, with . The Herglotz function able then corresponds to a positive real (PR) function with the property that is analytic with for , cf., [14], [33]. The symmetry (2) takes the form . The low- and high-frequency asymptotics are given by as (11)

Note that there are only odd indices in the last summation above since the complex zeros appear in symmetric pairs . With , the following relationships are now obtained from (4):

as where all coefficients are real valued and the even indexed coefficients are zero. Furthermore, the PR function property implies and . Note also that the mapping that implies the relations and for the coefficients in (3) and (11). The following integral identity now follows directly from (4):

(8)

(12)

denotes the Kronecker delta. Note that the term Here originates from the high-frequency asymptotics. where the maximum is taken over Denote the angular frequency interval , is the center angular frequency and the relative

for . denote the reflection coefficient corresponding to an Let netarbitrary impedance function defined by a passive work. Such an impedance function can always be represented by a lossless two-port which is terminated in a pure resistance

NORDEBO et al.: ON THE RELATION BETWEEN OPTIMAL WIDEBAND MATCHING AND SCATTERING OF SPHERICAL WAVES

N N N

 

%

Fig. 2. Cascade of two reciprocal two-ports and . Here, , and denote the overall scattering parameters, and the corresponding primed scattering parameters refer to the two networks and .

N

[3]. The appropriate PR function corresponding to (6) is given by (13) where are the zeros of with . Note that the is not needed here, since the reflection cocausality factor efficient corresponds to a causal convolution kernel. is Suppose that the low-frequency asymptotics of given by , as . The low-frequency asymptotics of is then given by

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Fig. 3. Matching network and equivalent circuit for the impedance of a TM wave at a spherical boundary of radius . The circuit is drawn for a TM wave of odd order .

l

a

where the primed scattering parameters corresponding to the and have been defined as indicated in two networks is a high-pass Fig. 2. It is assumed that the network ladder with (where ). Without loss of gener. Furthermore, it is ality, it may then be assumed that also assumed that so that there is no cancellain (16). This condition is easily achieved tion of zeros at ladder structure for if by choosing the appropriate is known, cf., also [3]. Suppose that the transmission coefficient has a zero of at . This implies that order , where the big ordo notation is defined as in [27]. Suppose further that the low-frequency is given by asymptotics of as . Hence,

(14) , the following relationships are now obWith tained from (12) (cf., [3]):

(15) if the circuit consists of only lumped elements, Note that is a rational function in this case. Furthermore, for since the asymptotic expansions (11) are valid rational functions as and , respectively. Consider now the broadband matching problem as described in [3]. In Fig. 2 is shown the cascade of two lossless and reand with a source at one side and a ciprocal two-ports resistive termination at the other side. Let be the fixed netthe matching network. The reflection and transwork and mission coefficients for the overall two-port are denoted by , and . Since the overall two-port is lossless with , the optimal matching limitations for the input port of interest with coefficient may be conveniently analyzed by considering the opposite port with coefficient , as depicted in Fig. 2. The reflection coefficient for the overall two-port is given by (16)

(17) implying that for . Furthermore, from (16) follows that for , and hence the invariance of the Taylor coefficients for . Thus, (15) can now be applied with for . These are the original Fano matching equations formulated in [3]. Consider now the problem of finding the optimum broadband tuning limits of the wave impedance of the spherical waves, as described in, e.g., [7], [8]. Hence, consider the matching or spherical problem of an outgoing wave of order . As was shown by Chu [24], the wave impedance of the spherical waves as seen at a spherical boundary can be high-pass ladder network terminated represented by a finite in a fixed resistance, cf., Fig. 3. The impedance is the normalized wave impedance as seen at a spherical boundary of radius , i.e., at the left (antenna) side of the equivalent circuit in Fig. 3. The input impedance used in the Fano analysis is the as seen from the opposite, right-hand side of impedance the equivalent circuit when it is correctly terminated in a pure resistance. The corresponding reflection coefficient is given by . It has been conjectured [7], [8] that the low-frequency asympis of the form totics of

(18)

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where

(19) and where is a constant determined from network analysis. The conjecture (19) may be verified by using the equivalent . However, from a network circuits for a fixed order analysis point of view, it seem to be non-trivial to prove it for general order . In the next two sections, it is shown that the conjecture (19) is true and an explicit expression for is given by showing that the matching problem is equivalent to the problem of finding the optimal scattering limitations for a homogeneous sphere in its high-contrast limit, i.e., in the limit as the permittivity or the permeability tends to infinity. III. OPTIMAL LIMITATIONS FOR SCATTERING OF VECTOR SPHERICAL WAVES Consider the scattering of vector spherical waves which is associated with an isotropic and homogeneous sphere of radius , and with relative permeability and permittivity and , reand the relative spectively. The refractive index is . The exterior of the sphere is wave impedance free space, and and are the speed of light and the wave impedance of free space, respectively. For convenience, intro. Allow to take duce the angular wave number in values in the upper-half plane, so that corresponds to Section II-B. Let denote the spherical coordinates and the corresponding radius vector. A. Exterior of the Sphere

are the vector spherical harmonics and where the spherical Hankel functions of the th kind, , 2, and order , see, e.g., [34]–[36]. Here, denotes differentiation with respect to the argument . The vector spherical harmonics are given by (24)

where

are the scalar spherical harmonics given by

(25) and where are the Associated Legendre functions, see, are ore.g., [34]. The vector spherical harmonics thonormal on the unit sphere and have the directional properties for , 2 and . B. Interior of the Sphere and Scattering Coefficients The electric and magnetic fields inside the sphere for are given by (26)

(27) where are regular vector spherical waves, and the corresponding multipole coefficients. The regular vector spherical waves are defined by

The electric and magnetic fields outside the sphere, i.e., for , are given by

(28)

(20)

(29)

and

(21) and are outgoing and incoming where vector spherical waves, respectively, see, e.g., [34]–[36], and the corresponding multipole coefficients. Here corcorresponds responds to transverse electric (TE) waves, denotes to transverse magnetic (TM) waves, and the complimentary index. The other indices are , , where denotes the order of the and spherical wave. The vector spherical waves are given by (22)

(23)

are the spherical Bessel functions of order , see, where denotes differentiation with respect to e.g., [34]–[36]. Here, . the argument and in (20), (21), Continuity of the tangential fields (26) and (27) for yields the following solution for the reflection coefficient defined by :

(30) and , cf., [37]. It is assumed that where can be represented by an asymptotic series at . It has been , where is the transform of a shown that causal kernel, see [4]. It can be expected that as , which means that for the Herglotz function corresponding to (6). A detailed study of the high-frequency

NORDEBO et al.: ON THE RELATION BETWEEN OPTIMAL WIDEBAND MATCHING AND SCATTERING OF SPHERICAL WAVES

asymptotics of the reflection coefficient has been performed in the Appendix A, including, e.g., the Debye and Lorentz dispersion models, and it asserts this expectation for these material models. The low-frequency asymptotics is obtained from a , Taylor series expansion yielding or (31) where (32) and is the static response. The symbol denotes asymptotic equivalence and is defined in, e.g., [27]. Note that the lowfrequency asymptotics of the TM (TE) wave reflection is inde. pendent of

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use one single zero, and the solution can be uniquely obtained from a 2 2 non-linear system of equations, see [3]. However, the numerical solution to the non-convex optimizawhen tion problem (34) will in general require a global optimization the routine and an exhaustive search. Furthermore, for optimal number of zeros is not known. A straightforward relaxation of the narrowband Fano equations (34) is considered in Section V below. IV. EXACT CIRCUIT ANALOGY FOR THE SCATTERING OF A HOMOGENEOUS SPHERE Recursive relationships for the spherical Hankel functions can be used to obtain an exact circuit analogy for the scattering of spherical waves as described below, cf., also [24], [25]. satisfy the following The spherical Hankel functions initial relations:

C. Optimization Formulation The following inequalities are obtained from (9) when applied to the reflection coefficient (30) using (31), where , and :

(35)

(33) and the recursive relations

where is defined as in (9), is defined by (10), and where are the zeros of the reflection coefficient , defined as in (6) and (9). Note that the same relations are obtained by using (15) and (19) and the substitution . in The narrowband model is now assumed, i.e., let . Hence, the assumption (33). Note also that in general, will simplify the analysis below without loss of generality. Let , where , and , and let . The optimum solution to the inequalities in (33) can then be formulated as the solution to the following constrained optimization problem:

(36)

for , 2 and , see, e.g., [34]–[36]. There are two possible dual circuits associated with the recursions in (35) and (36), cf., Fig. 4. For circuit a), define (37) and (38)

(34) . The second constraint where the variables are . Note that is equivalent to above is ignored when removing the corresponding zeros from the summations above. The solution to the optimization problem (34) defines the Fano limit1 for the reflection coefficient of the spherical waves, . When , it is sufficient to i.e., 1The

term Fano limit is used here even though the scattering problem is different from the matching problem. This is motivated by the equivalence of these problems as discussed in this paper.

where and represent voltages and currents, respectively. , where and . By introLet and ducing the normalized Laplace variable employing the definitions in (37) and (38), the following initial and corresponding to (35) are obtained: relations for

(39)

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Fig. 4. The two dual electric circuits with termination representing spherical Hankel functions of the first kind h (z ), i.e., outgoing waves outside a sphere of radius a.

and the recursive relations corresponding to (36) are given by

where

(40) , 2. The dual circuit b) is obtained by interchanging , or equivalently, by simultaneously interchanging and .

Fig. 5. Electric circuit analogy for TM and TE waves of odd and even order, corresponding to spherical Hankel functions of the first kind h (z ), i.e., outgoing waves outside a sphere of radius a.

A. Exterior of the Sphere Consider now the free space exterior of the sphere where and . In Fig. 4 is shown the two dual electric circuits with termination representing spherical Hankel functions of the first kind , corresponding to outgoing vector spherical waves. There are four different circuits representing the TM and TE waves of odd and even order, as depicted in Fig. 5. In Fig. 6 is shown the excitation with a Hankel function generator for the two dual electric circuits representing spherical Hankel functions of the second kind , corresponding to incoming vector spherical waves. From the field definition (20) and (21) and the circuit (and and its dual) definition (37) and (38), the tangential fields (spherical wave indices for , 2) outside the sphere are given by

(41)

have been suppressed for where the arguments , and simplicity, and the upper and lower signs refer to even and odd orders, respectively. The normalized TE and TM wave impedances are given by

(42)

Fig. 6. Excitation with a Hankel function generator for the two dual electric circuits representing spherical Hankel functions of the second kind h (z ), i.e., incoming waves outside a sphere of radius a.

where , 2 correspond to the outgoing and incoming waves, respectively. B. Interior of the Sphere Next, consider the interior of the sphere where , and . In Fig. 7 is shown the two dual electric circuits with termination representing spherical Hankel functions of the second kind , corresponding to incoming vector spherical waves. The circuit definitions (37) and (38) and recursions (39) and (40) are the same, but the circuit interpretaand a sign tion is different with an opposite direction for change of and . These changes correspond precisely to the symmetry of the incoming and outgoing wave impedances (43) defined in (42). The four different circuits representing odd and even TM and TE waves in Fig. 5 are changed accordingly. In Fig. 8 is shown the excitation with a Hankel function generator for the two dual electric circuits representing spherical Hankel functions of the first kind , corresponding to outgoing vector spherical waves. The circuit elements with impedances

NORDEBO et al.: ON THE RELATION BETWEEN OPTIMAL WIDEBAND MATCHING AND SCATTERING OF SPHERICAL WAVES

Fig. 7. The two dual electric circuits with termination representing spherical Hankel functions of the second kind h (z ), i.e., incoming waves inside a sphere of radius a.

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Fig. 9. Scattering model with Hankel function generators and matching. The interior generator is dependent, creating Bessel functions corresponding to standing waves within the sphere. The circuits are drawn for TM waves. The TE waves are similar.

dependent interior generator and its internal resistance correspond to a reflection coefficient (45)

Fig. 8. Excitation with a Hankel function generator for the two dual electric circuits representing spherical Hankel functions of the first kind h (z ), i.e., outgoing waves inside a sphere of radius a.

and are regarded as “generalized” inductors and capacitors in case the material is dispersive. However, these circuit elements behave asymptotically as “true” inductors and caand pacitors in the low-frequency limit. Hence, when . From the field definition (26) and (27) and the circuit (and its and dual) definition (37) and (38), the tangential fields (spherical wave indices for , 2) inside the sphere are given by

Note that in the equivalent circuit analogy depicted in Fig. 9, and with , the voltage and current constituents 2, correspond to a wave splitting with respect to the generator or termination impedance , cf., also Figs. 6 and 8. The circuit problem, and hence the scattering problem, has a unique solution through the scattering (S-matrix) relations

(46) are the scattering parameters of where the equivalent circuit representing the exterior as well as the interior of the sphere. Here, is the amplitude of the incoming wave and (46) can be solved for the amplitudes of the outgoing and the Bessel function (standing wave) amplitude wave . The overall reflection coefficient for the equivalent circuit is given by (47)

(44)

where the arguments , and have been suppressed for simplicity, and the upper and lower signs refer to even and odd orders, respectively. The normalized TE and TM wave impedare given by (42) with . ances C. Exact Circuit Analogy for the Scattering The scattering problem in Section III can now be interpreted by using an exact (equivalent) circuit analogy where the exterior and the interior tangential fields (41) and (44) are perfectly matched as depicted in Fig. 9. An independent exterior generator is used to generate the incoming waves, and a dependent interior generator is used to create the outgoing waves and hence the Bessel functions (obtained as the superposition of the two kinds of Hankel functions) within the sphere, see also [25]. The

is the reflection coefficent given by (30). where Note that the presence of the negative circuit elements in Fig. 9 is consistent with the fact that the wave impedance for incoming waves at is anticausal, cf., (42) and (43). However, note also that the overall equivalent circuit is causal due to the delay factor in (47) above. The low-frequency asymptotics of the function corresponding to (47) is given by (48) where as

is given by (32). The high-frequency asymptotics is

, where . Furthermore, it is expected that for many material models as discussed in Section III-B. Note that the circuit elements corresponding to the interior and when in Fig. 9 behave as . Note also that the low-frequency asymptotics , i.e., the coefficient of the TM (TE) reflection coefficient , is independent of . Hence, when considering the

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In order to relax the consider the minimization

constraints of the

in (34), expression

when is fixed. This implies the stationarity condition yielding the soluwhere tions, . Hence, by choosing

!1

Fig. 10. Interpretation of the scattering model in the high-contrast limit, (0) = . The exterior circuit is drawn for TM waves. The TE waves are similar.

(0)

(51) where

high-contrast limit of the low-frequency asymptotics (48) in the may be carried TM (TE) case, the limit . In this limit, the circuit elements out using and behave as open and short with impedances circuits, respectively. Further, the low-frequency asymptotics of as . Hence, the high-contrast (45) is limit of the low-frequency asymptotics in (48) may be obtained equivalently by using the exterior circuit with open or short termination as depicted in Fig. 10. This means that the low-freaccording to the conjecture (18) and quency asymptotics of , and hence (19) is identical to (48) with

(49) where (50) is the high-contrast limit of (32) when . Note that at the exterior circuit has a transmission zero of order and the term of the reflection coefficient is therefore invariant to whether the circuit is terminated with a short, open or match, cf., (16). Note also the interesting distinguishing feature that the integral identity (15) contains no causality term as in (8), instead this term appears in the low-frequency asymptotics of as in (18) and (49). In conclusion, the optimal Fano matching problem for the exterior circuit as described in Section II-C is equivalent to the problem of determining the optimal limitations for scattering of spherical waves in the high-contrast limit as described in Section III-C. An exact expression for the low-frequency is given by (49) and (50). The asymptotics of exact expression agree perfectly with the numerical results given in [8]. V. NUMERICAL EXAMPLE: RELAXATION OF THE FANO EQUATIONS As a numerical example, a relaxation of the narrowband Fano , one equations (34) is considered below. To solve (34) for has to resort to global optimization and computationally expensive numerical experiments. Hence, a straightforward relaxation yielding an upper bound on the objective function may be useful.

, and by employing , a relaxation of (34) valid for all

is given

by

(52)

where there are two variables . The solution to (52) yields an upper bound for the corresponding Fano limit in the variable . Hence, . When , the relaxation becomes tight and the solution to (52) is identical to the Fano . Furthermore, for there is limit a transition point where the second constraint becomes inactive for . To solve (52) for and hence , it is noted that the first (linear) constraint is always active. Since the polynomial constraints are monotonic in for , the optimum solution is found as the minimum of over the constraint subsets corresponding to a 2 2 non-linear system . of equations containing the first linear constraint Note that each such constraint subset has a unique solution for . The asymptotic solution to (52) when is given by (53) For , the asymptotic solution to (52) when is and is hence given by the governed by the lowest index solution to the first two constraints, i.e., the real valued root of . In Fig. 11 is shown the upper Fano limit as a function of for and , respectively. VI. SUMMARY Optimal limitations for the scattering of vector spherical waves is considered where the geometry of the object is known but the temporal dispersion is unknown. Using integral relations similar to the derivation of Fano’s broadband matching bounds, the optimal scattering limitations are determined by the static response as well as the high-frequency asymptotics of the reflection coefficient. Using an exact circuit analogy for the scattering of spherical waves, it is shown how the problem of determining the optimal scattering bounds for a homogeneous sphere in its high-contrast limit becomes identical to the closely related, and yet very different problem of finding the broadband

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parameters as a power series at infinity, i.e.,

(56)

where

, and where and . The last power series includes the Debye and the Lorentz dispersion models [35]. In particular, the Debye dispersion model (with real valued and positive parameters , and ) is given by

= 1 2 ... 5

Fig. 11. Upper Fano limit f as a function of k a for l ; ; ; . Graphs f for  a and b show , respectively. The dashed and  lines show the asymptotic upper bounds d c k a in the narrowband B. approximation where G

log

(0) = 1 =

(0) = 100 ( + )( )

,

tuning limits of the spherical waves. Furthermore, the scattering view of the matching problem yields explicitly the necessary low frequency asymptotics of the reflection coefficient that is used with Fano’s broadband matching bounds for spherical waves, something that appears to be non-trivial to derive from the classical network point of view. As with the Fano approach, the integral relations yield a nonconvex global optimization problem which in general is quite difficult to handle. As a numerical example, a relaxation of the Fano equations is considered which is easily solved and which is especially useful in the regime of Rayleigh scattering.

(57) and the Lorentz dispersion model (with real valued and positive , , and ) parameters

(58) as , which also motivates the assumption of real-valued or is non-zero, then coefficients in the expansion. If corresponds effectively to a Debye model or a conductivity model. If both are zero, the model is of Lorentz’ type. These expansions imply

APPENDIX A HIGH-FREQUENCY ASYMPTOTICS OF SCATTERING COEFFICIENTS To find the dominant behavior of the reflection coefficients in (30) for high frequencies, the asymptotic behavior of the spherical Bessel and Hankel functions are needed. For large arguments the spherical Hankel functions behave as [38]

(59) where

(60) are now studied. Introduce the appropriate The quantities numerator and denominator such that (54) is complex valued, and , and the big ordo notation is defied as in [27]. Moreover, as the spherical Bessel functions behave as [38] as

, where

(61) where the numerator

is

(62) (55)

To find the high-frequency behavior of (30), special care must be taken to separate the exponential behavior of and the algebraic behavior of . To this end, expand the material

and the denominator is

(63)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 9, SEPTEMBER 2011

Moreover, as the power series expansions defined above yield after some algebra

(64) For simplicity, assume that there is no optical response i.e., . Then (64) implies as

(65)

Along the real axis all the exponential terms contribute, and the quotient is

(66) , the term is exponenIn the upper half-plane as tially small and the main contribution comes from terms of the . Therefore, the dominant contribution is given by form

(67) where

has been inserted. REFERENCES

[1] A. Bernland, A. Luger, and M. Gustafsson, “Sum rules and constraints on passive systems,” J. Phys. A: Math. Theor., vol. 44, no. 14, p. 145205, 2011. [2] D. Youla, L. Castriota, and H. Carlin, “Bounded real scattering matrices and the foundations of linear passive network theory,” IRE Trans. Circuit Theory, vol. 6, no. 1, pp. 102–124, 1959. [3] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances,” J. Franklin Inst., vol. 249, no. 1, 2, pp. 57–83, 1950, and 139–154. [4] A. Bernland, M. Gustafsson, and S. Nordebo, “Physical limitations on the scattering of electromagnetic vector spherical waves,” J. Phys. A: Math. Theor., vol. 44, no. 14, p. 145401, 2011. [5] C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on broadband scattering by heterogeneous obstacles,” J. Phys. A: Math. Theor., vol. 40, pp. 11 165–11 182, 2007.

[6] C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on metamaterials: Restrictions on scattering and absorption over a frequency interval,” J. Phys. D: Applied Phys., vol. 40, pp. 7146–7151, 2007. [7] M. C. Villalobos, H. D. Foltz, J. S. McLean, and I. S. Gupta, “Broadband tuning limits on UWB antennas based on Fano’s formulation,” in Proc. Antennas and Propagation Society Int. Symp., 2006, pp. 171–174. [8] M. C. Villalobos, H. D. Foltz, and J. S. McLean, “Broadband matching limitations for higher order spherical modes,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1018–1026, 2009. [9] M. Gustafsson, C. Sohl, and G. Kristensson, “Physical limitations on antennas of arbitrary shape,” Proc. R. Soc. A, vol. 463, pp. 2589–2607, 2007. [10] C. Sohl and M. Gustafsson, “A priori estimates on the partial realized gain of ultra-wideband (UWB) antennas,” Quart. J. Mech. Appl. Math., vol. 61, no. 3, pp. 415–430, 2008. [11] M. Gustafsson, C. Sohl, and G. Kristensson, “Illustrations of new physical bounds on linearly polarized antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1319–1327, May 2009. [12] K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag., vol. 48, no. 8, pp. 1230–1234, Aug. 2000. [13] M. Wohlers and E. Beltrami, “Distribution theory as the basis of generalized passive-network analysis,” IEEE Trans. Circuit Theory, vol. 12, no. 2, pp. 164–170, 1965. [14] A. H. Zemanian, Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, With Applications. New York: McGraw-Hill, 1965. [15] H. A. Wheeler, “The wide-band matching area for a small antenna,” IEEE Trans. Antennas Propag., vol. 31, pp. 364–367, 1983. [16] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and of antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, 2005. [17] A. Hujanen, J. Holmberg, and J. C.-E. Sten, “Bandwidth limitations of impedance matched ideal dipoles,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3236–3239, 2005. [18] M. Gustafsson and S. Nordebo, “Bandwidth, Q factor, and resonance models of antennas,” Progr. Electromagn. Res., vol. 62, pp. 1–20, 2006. [19] M. Gustafsson and S. Nordebo, “Characterization of MIMO antennas using spherical vector waves,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2679–2682, 2006. [20] W. Geyi, “Physical limitations of antenna,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 2116–2123, Aug. 2003. [21] J. L. Volakis, C. C. Chen, and K. Fujimoto, Small Antennas: Miniaturization Techniques & Applications. New York: McGraw-Hill, 2010. [22] B. Kogan, “Comments on “broadband matching limitations for higher order spherical modes”,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1826–1826, 2010. [23] M. Villalobos, H. Foltz, and J. McLean, “Reply to comments on “broadband matching limitations for higher order spherical modes”,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1827–1827, 2010. [24] L. J. Chu, “Physical limitations of omni-directional antennas,” Appl. Phys., vol. 19, pp. 1163–1175, 1948. [25] H. L. Thal, “Exact circuit analysis of spherical waves,” IEEE Trans. Antennas Propag., vol. 26, no. 2, pp. 282–287, Mar. 1978. [26] H. M. Nussenzveig, Causality and Dispersion Relations. London: Academic Press, 1972. [27] F. W. J. Olver, Asymptotics and Special Functions. Natick, Massachusetts: A.. K Peters, Ltd., 1997. [28] M. Gustafsson and D. Sjöberg, “Sum rules and physical bounds on passive metamaterials,” New J. Phys., vol. 12, p. 043046, 2010. [29] M. Gustafsson, “Sum rule for the transmission cross section of apertures in thin opaque screens,” Opt. Lett., vol. 34, no. 13, pp. 2003–2005, 2009. [30] C. R. Brewitt-Taylor, “Limitation on the bandwidth of artificial perfect magnetic conductor surfaces,” IET Microw. Antennas Propag., vol. 1, no. 1, pp. 255–260, 2007. [31] M. Gustafsson, C. Sohl, C. Larsson, and D. Sjöberg, “Physical bounds on the all-spectrum transmission through periodic arrays,” EPL Europhys. Lett., vol. 87, no. 3, p. 34002, 2009, (6 pp). Spaces. New York: Dover Publications, [32] P. L. Duren, Theory of 2000. [33] A. Papoulis, The Fourier Integral and Its Applications. New York: McGraw-Hill, 1962. [34] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. New York: Academic Press, 2001.

Q

H

NORDEBO et al.: ON THE RELATION BETWEEN OPTIMAL WIDEBAND MATCHING AND SCATTERING OF SPHERICAL WAVES

[35] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1999. [36] Spherical Near-Field Antenna Measurements, ser. IEE electromagnetic waves series, J. E. Hansen , Ed. Stevenage, U.K.: Peter Peregrinus, 1988, 0-86341-110-X, no. 26. [37] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [38] Handbook of Mathematical Functions, ser. Applied Mathematics Series No. 55, M. Abramowitz and I. A. Stegun, Eds. Washington D.C.: National Bureau of Standards, 1970. Sven Nordebo received the M.S. degree in electrical engineering from the Royal Institute of Technology, Stockholm, Sweden, in 1989, and the Ph.D. degree in signal processing from Luleå University of Technology, Luleå, Sweden, in 1995. He was appointed Docent in signal processing in 1999. Since 2002, he is a Professor of Signal Processing at the School of Computer Science, Physics and Mathematics, Linnaeus University. Since 2009, he has been a Guest Professor of Signal Processing at the Department of Electrical and Information Technology, Lund University. His research interests are in statistical signal processing, sensor array and multichannel processing, wireless communications, antennas and propagation, electromagnetics, inverse problems and imaging, microwave tomography, electrical impedance tomography.

Anders Bernland (M’10) received the M.Sc. degree in engineering mathematics from Lund University, Lund, Sweden, in 2007, where he is currently working toward the doctoral degree. In May 2010 he obtained the degree of Licentiate in Engineering in Electromagnetic Theory. His research interests include electromagnetic scattering, antennas and propagation, wireless communication, and sum rules and physical limitations.

Mats Gustafsson (M’05) received the M.Sc. degree in engineering physics in 1994, the Ph.D. degree in electromagnetic theory in 2000, and was appointed Docent in Electromagnetic Theory in 2005, all from Lund University, Sweden. In 2000, he joined the Electromagnetic Theory Group, Lund University, where he is presently an Associate Professor. He co-founded the company Phase Holographic Imaging AB in 2004. His research interests are in scattering and antenna theory and inverse scattering and imaging with applications

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in microwave tomography and digital holography. He has written over 50 peer reviewed journal papers and over 60 conference papers. Dr. Gustafsson received the Best Antenna Poster Prize at EuCAP 2007 and the IEEE Schelkunoff Transactions Prize Paper Award 2010.

Christian Sohl was born in Hässleholm, Sweden, on October 1, 1981. He received the M.Sc. degree in engineering physics and the Ph.D. degree in electromagnetic theory from Lund University, Lund, Sweden, in 2004 and 2008, respectively. Since 2010, he is a Postdoctoral Fellow in electromagnetic theory at Lund University. His major research interests are in the fields of scattering theory and fundamental limitations on electrically small antennas.

Gerhard Kristensson (SM’90) was born in 1949. He received the B.S. degree in mathematics and physics in 1973 and the Ph.D. degree in theoretical physics in 1979, both from the University of Göteborg, Sweden. In 1983 he was appointed Docent in theoretical physics at the University of Göteborg. During 1977–1984, he held a research position sponsored by the National Swedish Board for Technical Development (STU) and he was Lecturer at the Institute of Theoretical Physics, Göteborg from 1980–1984. In 1984–1986, he was a Visiting Scientist at the Applied Mathematical Sciences group, Ames Laboratory, Iowa State University. He held a Docent position at the Department of Electromagnetic Theory, Royal Institute of Technology, Stockholm during 1986–1989, and in 1989 he was appointed the Chair of Electromagnetic Theory at Lund Institute of Technology, Sweden. In 1992, 1997, and 2007, he was a Visiting Erskine Fellow at the Department of Mathematics, University of Canterbury, Christchurch, New Zealand. He is the author of four textbooks and the editor of three scientific books. He has written 12 chapters in scientific books and is the author of over 70 peer reviewed journal papers and over 70 reviewed contributions in conference proceedings. His major research interests are focused on wave propagation in inhomogeneous media, especially inverse scattering problems. High frequency scattering methods, asymptotic expansions, optical fibers, antenna problems, and mixture formulas are also of interest, as well as radome design problems and homogenization of complex materials. Dr. Kristensson is (or has served as) a member of the Editorial Board and the Advisory Board of Inverse Problems, the Board of Editors of Wave Motion, and the Editorial and Review Board of Journal of Electromagnetic Waves and Applications and Progress in Electromagnetic Research. Currently, he is the Chairman of the Swedish National Committee of Radio Science, SNRV, and he has organized or has been a member of the scientific committee of several international and national conferences. He is a Fellow of the Institute of Physics, UK, Board member of the IEEE MTT/AP Chapter of Sweden, and he is the official member of the International Union of Radio Science, URSI, for Sweden.

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A New Solution for Characterizing Electromagnetic Scattering by a Gyroelectric Sphere Joshua Le-Wei Li, Fellow, IEEE, and Wee-Ling Ong, Student Member, IEEE

Abstract—A new solution to electromagnetic scattering by a gyroelectric sphere is obtained. Gyroelectric characteristics are considered, where both internal transmitted fields and external scattered fields are derived theoretically. The derived solutions are capable of dealing with incident electromagnetic waves at an arbitrary incident angle and arbitrary polarization. After the theoretical formulas are obtained, numerical validations are made by comparing our present results with those obtained using the Fourier transform method. Good agreements are observed between the present results obtained in this paper and those obtained using the other method. Some new numerical results are presented to investigate effects of electric anisotropy ratio and gyroelectric ratio on the radar cross section for a gyroelectric sphere and a left-handed metamaterial gyroelectric sphere. The new formulation of the problem is expected to have wide practical applications. In addition, some critical mistakes in literature were corrected. Index Terms—Anisotropic media, eigenvalues and eigenfunctions, electromagnetic scattering, electromagnetic scattering by anisotropic media, electromagnetic theory, radar cross sections, vector wave equation.

I. INTRODUCTION

R

ECENTLY, considerable attention has been paid to characterizing the interactions between electromagnetic fields and anisotropic media owing to its vast and promising technological and biological applications [1]–[3]. In fact, it has been a classic topic over the past many years and it has attracted a lot of interest in the electromagnetic community [4]–[7]. By adjusting the radial ratio of radially anisotropic coated spheres, classic spherical cloaking can be realized [8]. Various numerical and analytical methods have been reported to study this problem, such as the method of moments (MoM) [9], transmission line modeling [10], the coupled dipole approximation method [11], integral equation method [12], spectral domain Fourier transforManuscript received April 25, 2010; revised December 07, 2010; accepted February 28, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported by the Defence Science and Technology Agency of Singapore and the National University of Singapore through the Defence Innovative Research Program with Project No: DSTA-NUS-DIRP/ 2007/02. The work of J. Li was supported by a QRJH Chair Professorship at the University of Electronic Science and Technology of China. The work of W. L. Ong was supported by an Undergraduate/Pre-Graduate Scholarship awarded in March 2010 by the IEEE Microwave Theory and Techniques Society (MTT-S), Piscataway, NJ. J. L.-W. Li is with the Institute of Electromagnetics and School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: [email protected]). W.-L. Ong is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singpaore. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161541

mation [13], [14], and mode expansion method [8], [15]. Among them, a few are numerical approaches which employed computational techniques to deal with the meshed problems for numerical solutions while the rest are analytical approaches aiming at obtaining analytical solutions and more physical insights into electromagnetic scattering by, and also antenna radiation in the presence of, various anisotropic spheres. Along the line of the analytical solutions to electromagnetic scattering by anisotropic spheres, we could summarize them into the following three techniques: (a) the expansion technique in terms of vector spherical wave functions (VSWFs) together with the Fourier transform (FFT) method, (b) the dyadic Green’s function (DGF) technique based on modified VSWFs, and (c) the multiple scattering method along with T-matrix method. The expansion technique in terms of VSWFs together with the FFT method was employed to derive analytical solutions for the plane wave scattering by a plasma anisotropic sphere [16], a uniaxial anisotropic sphere [1], [17], a multilayered plasma anisotropic sphere [18], and an impedance sphere coated with an uniaxial anisotropic layer [3]. The dyadic Green’s function (DGF) technique based on modified VSWFs was proposed and DGFs were constructed to study multilayered radial anisotropic spheres [4], [19]. The multiple scattering method along with T-matrix method was used to study the scattering problem by a single gyromagnetic sphere [20]–[23]. Different approaches have their own advantages. In general, the expansion technique using vector spherical wave functions and Fourier transform technique are straightforward in theoretical formulation (which is derived in a compact form while its numerical solutions are relatively tedious to be obtained primarily due to repeated timeconsuming numerical integrations). The dyadic Green’s function technique developed based on modified VSWFs is very general, which can be applied to a large class of radiation and scattering problems, however it is difficult and lengthy to derive the scattering coefficients of these dyadic Green’s functions in analytical and compact form. The multiple scattering method along with T-matrix method is relatively easier in handling the numerical solutions in terms of the eigenvalues and eigenvectors, but its formulation looks complicated and lengthy in the solution derivation procedure. The common features of the first two approaches, i.e., techniques (a) and (b), are that they both use the physical wave numbers as the propagation constants derived from the resulting determinant of the vector wave equations, while the latter approach, i.e., technique (c), uses the mathematical eigenvalues for the wave numbers and they are derived purely from mathematics instead of physics. Here in this paper, we consider a sphere whose material is characterized by a gyrotropic permittivity tensor. Both electromagnetic wave propagation in, and scattering by, an anisotropic

0018-926X/$26.00 © 2011 IEEE

LI AND ONG: A NEW SOLUTION FOR CHARACTERIZING ELECTROMAGNETIC SCATTERING BY A GYROELECTRIC SPHERE

spherical scatterer are also considered, where the scattered and transmitted fields are formulated and their scattering coefficients derived theoretically (based on the method of multiple scattering) and computed numerically (using the T-matrix method). Various effects of gyrotropic and anisotropic parameters of the gyroelectric and left-handed gyroelectric sphere materials on radar cross sections (RCS’s) are studied extensively. Validation of the present formulation and also the developed codes based on the Mathematica™ software package is made at first by comparing our present results with those results obtained in literature using the Fourier transform method. An excellent agreement is obtained between the two sets of results for a number of special cases. Of course, some new results are also obtained here in this paper for the first time. In the subsequent formulation and analysis, a time dependence of the form is assumed but will be suppressed throughout the treatment for simplicity. II. BASIC FORMULATIONS A. Expansion of Electromagnetic Field Inside Sphere Consider an anisotropic sphere shown in Fig. 1, where the spherical radius is and the center is located in the free space in spherical coordinates. The sphere is characterized by a scalar permeability, , and a permittivity tensor of the following form (1)

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(6b) Due to the relation in (3c), we can express

as (7)

where is yet to be determined, (for TE-modes) and (for TM-modes) denote the vector spherical wave functions (VSWFs) of the first kind, and with the coefficient given by [25], [26] (8) and being the amplitude of the incident electric field. Unless implies explicitly specified, hereinafter the summation , while sums up from that the index runs from 1 to to for each . In practical calculations, the expansion is supposed to be uniformly convergent and can be truncated at [24], [27] for the suggested values of where Mie scattering by an isotropic sphere where is the wave number inside the isotropic sphere. In the present case, we have used the suggested truncation number as a reference, but checked the convergence for all the summations and . In addition, it is truncated them at a relative error of 1.0 indicated that the convergence number depends very much on of the spherical scatterer. the electric size With the use of the properties of VSWFs, it can be worked out that

Therefore, the following constitutive relations are defined in the sphere of composite material

(9a)

(2a) (2b)

(9b)

Maxwell’s equations for a time-harmonic field inside the source-free and homogeneous sphere can be expressed as

where the coefficients , , , given in Appendix. Therefore, we have

,

, and

(3a) (3b) (3c) (3d)

are

(10a)

From (3), we obtain where the expansion coefficients are defined1 as follows: (4) (11a) where

and (11b) (5)

with (6a)

1The expression of in literature was incorrect in the paper by Lin et al. [20]. The second column and row element of the inverse permeability tensor in (5) of that paper should read  instead of  and the value of w should be multiplied by a factor of E . In the H expression, instead of an addition of the last term, it should be a substraction and the value of n should be n [24]. jz j; n

max(

) + 15

=

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and denoting the row and column indices respecwith tively. Equation (15) implies an established eigen-system, with where deeigenvalues and eigenvectors notes the index of eigenvalues and corresponding eigenvectors. based on the eigenvecWe can then construct a new function tors

Fig. 1. Geometry for electromagnetic scattering of plane wave by an anisotropic sphere.

(17) with

. It is easy to show that

satisfies (18a) (18b)

(11c) Thus, we can express

as

(11d)

(19)

(12a)

with [28], [29] and the expansion coefficients, , are to be determined by matching the boundary conditions at the surface of the sphere. With given by (19), we can and as write

Since the VSFWs satisfy

(12b) (12c) substituting (10a) and (7) into (4), we obtain, after some simple manipulations, the following: (13) with

(20a) (14a) (14b) (20b)

Equation (13) represents the characteristic equation and it suggests that and must be equal to zero. Thus, we can express (14) in the following matrix form

Since , its expansions include the term is absent in the isotropic case.

(15)

B. Expansion of the Scattered and Incident Fields

where are defined by

, and the sub-element matrices , , , and

which

and the incident fields The scattered fields in the isotropic medium have the same form as those in the Mie solution [24], [30]. The scattered fields are given explicitly as

(16a) (21a) (16b) (21b) (16c) (16d)

where . The coefficients and determined by matching boundary conditions.

are to be

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In spherical coordinates system, the incident fields are expressed as

and

where the electric size parameter is defined as other inter-parameters are defined as

(25)

(22a) (22b) where denotes the normalized complex , and the unit vectors and polarization vector, with are defined in the direction of increasing and to constitute , as shown a right-hand base system together with in Fig. 1. In terms of VSWFs, the incidents fields read

The Riccati-Bessel functions are given by [24]

and

utilized as above

(26) with and denote the spherical Bessel functions of the first and third kinds, respectively. Details on the numerical solution of (24) can be found from [20]. From the coefficients of scattered fields, the scattering effidefined as ciency factor (27)

(23a) the radar cross section (RCS)

defined [29] as (28)

(23b) and of the incident wave and the where the coefficients details on their deduction can be found in [20]. So, more details will not be provided herein.

the differential scattering cross section in [20], [31]

represented

(29) and the asymptotic far field

expressed as

C. Matching Boundary Conditions

(30)

The continuity at of the tangential electric field component for (24a)–(b) and magnetic field component for (24c)–(d) yields

can be computed, where denotes the scattering amplican be tude. The asymptotic forms of VSWFs when readily found from [32] and thus are omitted herein. III. NUMERICAL RESULTS AND DISCUSSIONS

(24a)

In the previous section, we have presented the necessary theoretical formulations of the electromagnetic fields due to a plane wave incidence in the presence of an gyroelectric sphere. Validation is made first here in this section; and thereafter some new results depicted to demonstrate roles of anisotropy ratio and gyroelectric contributions are to be examined and presented for the first time, in the case of a gyroelectric and left-handed material (LHM) sphere.

(24b) A. Theory and Code Validations

(24c)

(24d)

In order to check the accuracy of the new numerical results obtained, we compared our results with the results obtained using the Fourier transform technique. We verify our results for three special cases, namely, gyroelectric sphere in [16], lefthanded material uniaxial gyroelectric sphere in [33] and uniaxial nonmagnetic sphere in [17]. These results and comparisons are depicted in Figs. 2 and 3, respectively. An excellent and the agreement of the RCS values in the -plane -plane is achieved between our proposed solutions and

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Fig. 2. RCS versus scattering angle,  , in both the E -plane and the H -plane.  , : i  ,  (a) A lossy gyroelectric sphere:  : i  and    of x k a  . (b) A lossy left-handed material  , : i  ,  : i  and  sphere:  of size parameter x k a : .

05)

= = (1 + 0 5 ) =2 = =3 =0 = (04+0 02 ) = (02+0 01 ) = = 0 75

= (4 + =0

Fig. 4. (a) E -plane RCS, (b) H -plane RCS. Effects on RCS values due to elec. Specifically, tric anisotropy ratios of a lossless uniaxial sphere with  three electric anisotropy ratios are considered where the transversal permittivity is fixed at    : (1) negative uniaxial medium, A (solid line); (2) isotropic medium, A (dashed line); (3) positive uniaxial medium, A : (dash dotted line).

=3

= 06

=1

=0 =3

solutions by the Fourier transform method adopted by Geng et al. Thus, the correctness and applicability of our theoretical formulas as well as our programming codes are partially verified. is always Herein and subsequently, a relative error of 1.0 kept in the calculations and all convergence and proper truncations are well checked. B. Gyroelectric Spherical Scatterer

Fig. 3. RCS versus scattering angle,  , in both the E -plane and the H -plane,  . (a) A lossless uniaxial sphere: the permittivity and permewhere we  :  ,  :  , ability tensor elements are taken to be  , and p ; p ; . (b) An absorbing uniaxial sphere: x k a ,  : i  ,  : i  , , and p ; p ; .

= = 4 9284 ) = (1 0) 0 ( = (4 + 0 2 ) = (2 + 0 1 ) =0

= 5 3495 = = =4 ( ) = (1 0)

In this section, the properties of a nonmagnetic (i.e., which are assumed in some literature) spherical scatterer of radius with a plane wave (whose incident field of electric field amplitude equal to unity, which is polarized parallel to the -direction, and that propagates in the direction) are investigated numerically. From Fig. 4, it is observed that the effect of the electric upon bistatic RCS is anisotropy ratio quite noticeable in the far-field compared with the isotropic . Interestingly in the -plane, results in case, a dramatic decrease of RCS level at , as compared to the other two cases. On the contrary in the -plane, enhances RCS significantly in the vicinity of at 55 and 110 with respect to the other two cases. Also in the -plane, as decreases, resonances become sharper and start to shift toward right side. Fig. 5 depicts another interesting phenomenon when the gyroelectric cross terms are included to a nonmagnetic uniaxial

LI AND ONG: A NEW SOLUTION FOR CHARACTERIZING ELECTROMAGNETIC SCATTERING BY A GYROELECTRIC SPHERE

Fig. 5. (a) E -plane RCS, (b) H -plane RCS. Effects on RCS values due to gyroelectric parameters of a lossless gyroelectric sphere. The permittivity elements are assumed to be   :  and  :  with an electric size k a . of x

=

=2

= 5 3495

= 4 9284

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Fig. 7. (a) E -plane RCS, (b) H -plane RCS. Effect of gyroelectric parameters in RCSs of a lossy nonmagnetic uniaxial sphere. The permittivity elements are : i  and  : i  with an electric assumed to be   size of x k a .

=

= (2 + 0 1 ) =4

= (4 + 0 2 )

sphere. The RCS level at the resonances increases in general. The RCS level nearby the resonances starts to, however, decrease slowly for both the -plane and the -plane as the gyroelectric effect increases. This phenomenon becomes particularly apparent in the -plane. For absorbing sphere, it is expected that the higher imaginary part of the complex permeability leads to lower RCS values and this is shown in Fig. 6. However, this does not hold in the vicinity and 155 . Furthermore, there is little change in RCS of to 20 . from The lossy gyroelectric sphere shows weaker oscillations, as from compared to the lossy uniaxial sphere, where to 120 in Fig. 7. There is no regular fluctuation as it is usually observed. It is also noticed that for a very large , the oscillations of sized sphere, for instance the uniaxial case ( in Fig. 7) are very vigorous. C. Left-Handed Material Spherical Scatterer

Fig. 6. (a) E -plane RCS, (b) H -plane RCS. Effects on RCS values due to imaginary part of complex permittivity of a lossy gyroelectric sphere.

In this subsection, the properties of nonmagnetic (i.e., ) LHM spherical scatterer of radius is considered with a plane wave incident field of electric field amplitude equal to -direction, and that propagates unity, polarized parallel to the in the - direction. In Fig. 8, RCS values for three anisotropy ratios are shown. It exhibits characteristics similar to those in Fig. 4 that is either to enhance or reduce the RCS level significantly. To illustrate,

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Fig. 8. (a) E -plane RCS, (b) H -plane RCS. Effects on RCS value due to electric anisotropy ratios of a uniaxial lossless LHM sphere. The permittivity el and  for an electric size of ements are assumed to be   x k a :  . Specifically, three electric anisotropy ratios are considered  : (1) negative uniwhere the transversal permittivity is fixed at   (solid line); (2) isotropic, A (dashed line); (3) negative axial, A uniaxial, A : (dash dotted line).

=

= 0 75 =2 = 0 667

= 02

=0 = 02 =1

when , the RCS level at the vicinity of enhances in both the - and -planes (increase by about 30 dB and 20 dB in the -plane and the -plane, respectively). Fig. 9 shows effects on radar cross sections due to various gyroelectric properties a lossless nonmagnetic LHM sphere. Compared to Figs. 5 and 9 also exhibits a similar characteristic, but it has less oscillations for LHM sphere which may due to the smaller size parameter chosen. Fig. 10 shows the radar cross section as a function of gyroelectric cross term of a lossy nonmagnetic LHM sphere. Ap, the resonance of the radar cross section parently at in the -plane has been significantly reduced. In addition, at , the RCS level in the -plane decreases and side lobes start to disappear as gyroelectric cross term increases from 0.4 to 0.8. In a nutshell, we have demonstrated the correctness of our formulas derived, the accuracy of our codes developed, and the applicability of the results obtained, to study the far field characteristics for both gyroelectric and left-handed material sphere even of a very large electric size. Of course, when the electric size becomes larger, the convergence number increases at a rate of polynomial function relations. With this increased convergence number, more eigenvalues of are needed to achieve an accurate solution. Corresponding to such an increase, the sparse matrix of large dimension may be ill-conditioned, so a proper numerical iterative procedure should be seriously considered in the numerical solution procedure. It is observed that as we include the gyroelectric cross term into nonmagnetic uniaxial gy-

Fig. 9. (a) E -plane RCS, (b) H -plane RCS. Effects on RCS values due to gyroelectric terms of a lossless LHM sphere. The permittivity elements are assumed  and   with an electric size of x k a :  . to be  

= 01

= 02

=

=05

Fig. 10. (a) E -plane RCS, (b) H -plane RCS. Effects on RCS values due to gyroelectric properties of a lossy LHM sphere. The permittivity elements are assumed to be   : i  and  : i  with an electric size of x k a .

=

= (02 + 0 01 ) =

= (04 + 0 02 )

roelectric or LHM expression for the sphere, the resonances become weaker and the nearby RCS level gets stronger. Also, the

LI AND ONG: A NEW SOLUTION FOR CHARACTERIZING ELECTROMAGNETIC SCATTERING BY A GYROELECTRIC SPHERE

electric anisotropy ratio can be adjusted so as to enhance or reduce RCS levels at resonances. IV. CONCLUSIONS New solutions to the scattering of plane waves by a gyroelectric sphere are derived by employing the multiple scattering spheres method along with T-matrix method. Numerical results are then yielded from the new formulas and found to agree very well with those of Fourier transform method. Some general numerical results not yet published elsewhere are also presented. The present formulations can be extended to more general constitutive relations where both permittivity and permeability are both tensors in form, or to more complicated geometries, for instance, a composite material coated conducting sphere, a layered sphere and multiple spheres of composite materials. APPENDIX With

, the coefficients are obtained as follows:

(31a)

(31b)

(31c)

(31d)

(31e)

(31f) Detailed procedures for derivations of these matrix elements can be found in [20]. For the definition and the orthogonal relations of VSWFs, readers can refer to [23] for more details.

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ACKNOWLEDGMENT The authors are also grateful for the useful discussions with Dr. C. Wan and K. H.R. Zheng in the Department of Electrical and Computer Engineering, National University of Singapore. REFERENCES [1] S. Liu, L.-W. Li, M.-S. Leong, and T.-S. Yeo, “Field representations in general rotationally uniaxial anisotropic media using spherical vector wave functions,” Microw. Opt. Technol. Lett., vol. 25, no. 3, pp. 159–162, May 2000. [2] G. C. Kokkorakis, “Scalar equations for scattering by rotationally symmetric radially inhomogeneous anisotropic sphere,” Progr. Electromagn. Res. Lett., vol. 3, pp. 179–186, 2008. [3] Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution of electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag., vol. 57, pp. 572–576, 2009. [4] S. Liu, L.-W. Li, M.-S. Leong, and T.-S. Yeo, “On the constitutive relations of chiral media and Green’s dyadics for a unbounded chiral medium,” Microw. Opt. Technol. Lett., vol. 23, no. 6, pp. 357–361, Dec. 1999. [5] S. Liu, L.-W. Li, M.-S. Leong, and T.-S. Yeo, “Scattering by an arbitrarily shaped rotationally uniaxial anisotropic object: Electromagnetic fields and dyadic Green’s functions (Abstract),” J. Electromagn. Waves Applicat., vol. 14, no. 7, pp. 903–904, Jul. 2000. [6] L.-W. Li, N.-H. Lim, and J. A. Kong, “Cylindrical vector wave function representation of Green’s dyadic in gyrotropic bianisotropic media,” J. Electromagn. Waves Applicat., vol. 17, no. 11, pp. 1589–1590, Nov. 2003. [7] L.-W. Li, N.-H. Lim, W.-Y. Yin, and J. A. Kong, “Eigenfunctional expansion of dyadic Green’s functions in gyrotropic media using cylindrical vector wave functions,” J. Electromagn. Waves Applicat., vol. 17, no. 12, pp. 1731–1733, Dec. 2003. [8] L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E, vol. 78, pp. 046609-1–046609-11, 2008. [9] R. D. Graglia, P. L. E. Uslenghi, and R. E. Zich, “Moment method with isoparametric elements for three-dimensional anisotropic scatterers,” Proc. IEEE, vol. 77, pp. 750–750, 1989. [10] A. F. Yagli, “Electromagnetic scattering from three dimensional gyrotropic objects using the transmission line modeling (TLM) method,” Ph.D. dissertation, Graduate School of Syracuse University, New York, 2006. [11] V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by 3-D anisotropic scatterers,” IEEE Trans. Antennas Propag., vol. AP-37, no. 6, pp. 800–802, Jun. 1989. [12] S. N. Papadakis, N. K. Uzunoglu, and C. N. Capsalis, “Scattering of a plane wave by a general anisotropic dielectric ellipsoid,” J. Opt. Soc. Am. A, vol. 7, pp. 1708–1712, Jun. 1990. [13] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: Van Nostrand Reinhold, 1990. [14] W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E, vol. 47, no. 1, pp. 664–673, Jan. 1993. [15] K. L. Wong and H. T. Chen, “Electromagnetic scattering by a uniaxially anisotropic sphere,” IEE Proc.-H, vol. 139, pp. 314–318, Aug. 1992. [16] Y. L. Geng, X. B. Wu, and L. W. Li, “Analysis of electromagnetic scattering by a plasma anisotropic sphere,” Radio Sci., vol. 38, pp. 12-1–12-12, 2003. [17] Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E, vol. 70, pp. 0566091–056609-8, 2004. [18] Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Electromagnetic scattering by an inhomogeneous plasma anisotropic sphere of multilayers,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 3982–3989, Dec. 2005. [19] C. W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropic ratio: Application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3515–3523, Dec. 2007. [20] Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E, vol. 69, pp. 056614-1–056614-14, 2004. [21] M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E, vol. 72, pp. 056610-1–056610-13, Nov. 2005.

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[22] N. Ji, M. K. Lin, J. Zhou, and Z. F. Lin, “Radiation torque on a spherical birefringent particle in the long wave length limit: Analytical calculation,” Appl. Opt., vol. 13, no. 14, pp. 5192–5204, Jul. 2005. [23] S. Y. Liu and Z. F. Lin, “Opening up complete photonic bandgaps in three-dimensional photonic crystals consisting of biaxial dielectric spheres,” Phys. Rev. E, vol. 73, pp. 066609-1–066609-11, Jun. 2006. [24] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983. [25] Y.-L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A, vol. 20, no. 11, pp. 2093–2105, Nov. 2003. [26] Y.-L. Xu, “Radiative scattering properties of an ensemble of variously shaped small particles,” Phys. Rev. E, vol. 67, pp. 046620–046620, 2003. [27] W. J. Wiscombe, “Improved MIE scattering algorithms,” Appl. Opt., vol. 19, no. 9, pp. 1505–1509, May 1980. [28] L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing. New York: Wiley-Interscience, 1985. [29] B. Stout, M. Nevière, and E. Popov, “T matrix of the homogeneous anisotropic sphere: Applications to orientation-averaged resonant scattering,” J. Opt. Soc. Am. A, vol. 24, pp. 1120–1130, Apr. 2006. [30] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [31] L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Theories and Applications. New York: Wiley-Interscience, 2000. [32] Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: Far field,” Appl. Opt., vol. 36, no. 36, pp. 9496–9508, Dec. 1997. [33] Y. L. Geng and S. He, “Analytical solution for electromagnetic scattering from a sphere of uniaxial left-handed material,” J. Zhejiang Univ. Sci. A, vol. 7, pp. 99–104, 2006.

Joshua Le-Wei Li (S’91–M’92–SM’96–F’05) received the Ph.D. degree in electrical engineering from Monash University, Melbourne, Australia, in 1992. In 1992, he was with Department of Electrical and Computer Systems Engineering, Monash University, sponsored by the Department of Physics, La Trobe University, Melbourne, Australia, as a Research Fellow. Between 1992 to 2010, he was with the Department of Electrical and Computer Engineering, National University of Singapore, where he was a Full Professor and Director of NUS Centre for Microwave and Radio Frequency. In 1999 to 2004, he was seconded to the High Performance Computations on Engineered Systems (HPCES) Programme of Singapore-MIT Alliance (SMA) as a Course Coordinator and SMA Faculty Fellow. In May–July 2002, he was a Visiting Scientist with Research Laboratory of Electronics , Massachusetts

Institute of Technology; and in October 2006, he was an Invited Professor with University of Paris VI, France. He was an Invited Visiting Professor at the Swiss Federal Institute of Technology, Lausanne (EPFL) between January and June 2008; and a Visiting Guest Professor at Swiss Federal Institute of Technology, Zurich (ETHZ), between July and November, 2008; both in Switzerland. Currently, he has been with School of Electronic Engineering, University of Electronic Science and Technology of China, where he is a QRJH Chair (or National) Professor and the Founding Director of Institute of Electromagnetics. His current research interests include electromagnetic theory, computational electromagnetics, radio wave propagation and scattering in various media, microwave propagation and scattering in tropical environment, and analysis and design of various antennas. In these areas, he has (co-)authored a book, Spheroidal Wave Functions in Electromagnetic Theory (New York: Wiley, 2001), 48 book chapters, over 310 international refereed journal papers, 48 regional refereed journal papers, and over 350 international conference papers. Dr. Li was the recipient of a few awards including two best paper awards, the 1996 National Award of Science and Technology of China, the 2003 IEEE AP-S Best Chapter Award when he was the IEEE Singapore MTT/AP Joint Chapter Chairman, and the 2004 University Excellent Teacher Award of National University of Singapore. He has been a Fellow of The Electromagnetics Academy since 2007 and was IEICE Singapore Section Chairman between 2002–2007. As a regular reviewer of many archival journals, he is an Associate Editor of Radio Science and International Journal of Antennas and Propagation; an (Overseas) Editorial Board Member of five international and regional archival journals and one book series by EMW Publishing. He is honored as an Advisory, Guest, or Adjunct Professor at one State Key Laboratory and other four leading universities in related areas of electromagnetics in China. He also serves as a General Chairman of ISAP2006 and TPC Chairman of PIERS2003 and iWAT2006.

Wee-Ling Ong (S’08) received the B.Eng. degree in electrical engineering (with first class honors) from the National University of Singapore (NUS), in July 2010. From August 2009 to July 2010, she conducted research at the Department of Electrical and Computer Engineering, National University of Singapore, under the supervision of Professor Joshua Le-Wei Li in the Centre for Microwave and RF at NUS. Her current research interests include electromagnetic wave propagation in, and scattering by, composite materials and metamaterials. Miss Ong was a recipient of the IEEE Microwave Theory and Techniques Society (MTT-S) Fall Undergraduate/Pre-Graduate Scholarship 2009/2010, NJ, and also the 2010 First Place Prize-Undergraduate Category in the IEEE Regional 10 Student Paper Contest by IEEE Regional 10 (Asia-Pacific) in October 2010.

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Exact Complex Source Point Beam Expansions for Electromagnetic Fields Koray Tap, Member, IEEE, Prabhakar H. Pathak, Fellow, IEEE, and Robert J. Burkholder, Fellow, IEEE

Abstract—Complex source point (CSP) beams (or CSPBs) are known to be exact solutions of Maxwell’s equations. Hence they can be used as basis functions to represent the electromagnetic fields generated by arbitrary sources. In this work, it is shown that any fully vector electromagnetic field can be represented exactly as an expansion of CSPBs originating from a relatively arbitrary surface enclosing the sources. Three different variants of the CSPB representation are first considered for a spherical expansion surface and their properties are discussed. It is shown that the CSPB expansion set can be conveniently truncated by selecting only the significantly contributing beams for efficient field calculations. The CSPB representations are next extended to the case of relatively arbitrary expansion surfaces. Numerical results are presented to demonstrate and compare the convergence and efficiency of the three approaches. Index Terms—Complex source point beams, Gaussian beams.

I. INTRODUCTION HE complex source point (CSP) concept was first introduced in [1]–[3] as a new way of generating Gaussian beams. An electromagnetic (EM) CSP can be obtained with this method by analytically continuing the spatial coordinates of a real EM point source into the complex domain. The field of a CSP is a beam that is an exact solution of Maxwell’s equations, since it can be represented in terms of dyadic Green’s function, for an unbounded homogeneous isotropic medium, with a complex source location. It is assumed that the medium external to the source is free space. An important property of the CSPB is that it reduces to the fields of an EM Gaussian beam (GB) within the paraxial region [1]–[3]. Since CSPBs are exact solutions of Maxwell’s equations, it is reasonable to expect that arbitrary EM fields can be expanded in terms of a set of CSPBs. In [4], an approximate CSPB representation for a planar aperture type source distribution is arrived at indirectly by using a Gabor expansion along with an asymptotic procedure. In a related approach, a GB summation method (GBSM) is used to expand the field of a scalar source into a set of GBs that emanate from the source domain and propagate radially outward in all directions. The GB coefficients are computed in [5], [6] by matching the GB fields to the radiated field

T

Manuscript received June 29, 2010; revised December 06, 2010; accepted January 15, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. K. Tap is with ASELSAN Inc., Ankara 06370, Turkey (e-mail: [email protected]). P. H. Pathak and R. J. Burkholder are with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43212 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161438

in the far zone. It is noted that unlike CSPBs, GBs constitute only paraxial solutions of the wave equation. Thus, the GBSM is strictly valid only within the domain of paraxial asymptotics. The idea of a CSPB expansion based on a complex Huygens’ principle appears to have been first introduced in [7], [8]. An exact CSP representation for the field of a conventional, simple, scalar point source at a real location in free space was shown in [7] by using an exact integral representation of the scalar free space Green’s function over a spherical domain. In [8], the field of a scalar line source of infinite length was expanded in terms of CSPs located on a circle enclosing the line source. The complex Huygens’ principle was extended in [9] for more general scalar sources enclosed by the Huygens’ surface. The beam weights were computed through a series summation involving spherical harmonic functions only for the special case of a spherical Huygens’ surface on which the source distribution was assumed to be known. An exact complex source representation for arbitrary scalar fields was presented in [10], where the CSPBs are launched from a single point in space. The unknown beam weights are computed through an analytical solution to an associated integral equation. The latter procedure requires a knowledge of the complete spherical harmonic expansion of the sources inside the Huygens’ surface; however while it is a rigorous procedure, it can become cumbersome for sources with a large spatial support. Complex source representations for scalar transient radiation are also available [11]–[13]. Some important applications of complex point sources in acoustics can be found in [14] to represent the acoustic fields in cylindrical acoustic near field measurements and in [15] to model the receiving electroacoustic transducers. The idea in [14] is next extended to the EM case in [16]. Some related work in the scalar domain has recently been presented in [17]. A CSPB expansion for the vector case has recently been treated in [18], where only electric CSPs are used (as in method 2 of this paper). The unknown coefficients are however computed through a spherical wave expansion procedure, which requires the knowledge of the spherical wave coefficients. The idea of CSPB representation was also previously introduced in [19]–[22] and utilized to accelerate the iterative EM solutions of large method of moments problems without the need for spherical vector wave functions. In this paper, it is shown that any arbitrary and fully vector EM field can be represented by CSPBs launched from a closed surface, , enclosing the sources. The surface equivalence theorem is first used to represent the field outside with equivalent sources located on [23]. The surface parameters are next assigned complex values such that the corresponding new equivalent currents are now located on the complex surface . Three different variations of the surface equivalence theorems

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are considered in this work, which give three different CSPB representations. In two of these three cases, the free space Green’s function is employed corresponding to two different free space equivalences, whereas in the third form of the equivalence theorem it becomes necessary to use a special Green’s function for a sphere that is a perfect magnetic conductor (PMC) in free space; the latter Green’s function does not have a closed form solution. However, it is demonstrated here that this special Green’s function can be approximated very accurately in closed form by employing the image theorem, thus providing an accurate and fast CSPB representation especially for electrically large sources. These three cases are first investigated with a spherical expansion surface. The idea is next extended to relatively arbitrary expansion surfaces. The analytical formulations of the CSPB expansions on a sphere are described in Section II, and the properties of the CSPB expansions are presented in part D of Section II. It is shown that the beam representations can be truncated to include only beams which significantly contribute to the field at any point due to the natural windowing property of the CSPBs. The discretization of the CSPB representation integrals and their regions of validity are explained. The extension to relatively arbitrary expansion surfaces is presented in Section III. Utilizing an expansion surface other than a sphere can be useful for applications in some cases, for instance when the source domain is large in one dimension (e.g., linear arrays). Numerical results are presented in Section IV to demonstrate and compare the contime vergence and efficiency of the three approaches. An dependence for the fields is assumed and suppressed in the following; here is the angular frequency of the wave.

Consider a set of general electric and magnetic volume cur, respectively, which radiate the elecrent densities tric field and the magnetic field in free space as shown in Fig. 1(a). These primary source currents may represent, for example, an aperture antenna opening (e.g., a reflector feed horn) or a scattering structure (e.g., a subreflector in a multireflector antenna system). Let a spherical surface of radius encapsulate these sources. The aim is to represent the fields in terms of CSPBs radiated from a complex extension of that sphere. To this end, three different types of the surface equivalence theorems are considered. and

where is the radial vector on the sphere. and radiate in free space as shown in Fig. 1(b), with the primary sources absent. In (1), and denote the electric type dyadic Green’s functions pertaining to electric and magnetic and current sources, respectively, in free space. These are available in closed form [23]. Noting that defines the sphere in (1), one can thus analytically extend it to a new complex value analogous to that in [8]–[13] by replacing with a complex value, (2) The points on this new complex sphere relation

II. 3-D CSPB REPRESENTATION ON A SPHERE

A. Method 1: CSPB Expansion With Both Equivalent Sources

Fig. 1. Different variations of the surface equivalence theorem. (a) Original problem. (b) Equivalent problem 1. (c) Equivalent problem 2. (d) Equivalent problem 3.

Type

satisfy the following (3)

where the coordinates are now allowed to be complex. The new complex sphere has the following parametric representation: (4) where

is the position vector on , with (5) (6)

and with

being real. Substituting

and

in (1) yields

The fields outside are represented in terms of the equivand , realent electric and magnetic surface currents spectively, located on the sphere [23] (7)

(1)

. It is known where the unit normal stays real since with that the dyadic Green’s functions a complex source location give rise to a vector CSPB [3]. The designates the real part of the complex location vector

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component of or on the sphere is required to be continuous. Referring to Fig. 1(c), we thus choose

(8) Since the tangential electric field is continuous across , no equivalent magnetic currents exist on . However, an equivacan be defined which generates the field lent electric current inside and outside of , respectively. Note that this is different from that in (1) for the previous case. The equivalent representation in this case takes the following form:

(9)

Fig. 2. Different variations of the CSP representations. (a) Equivalent problem 1. (b) Equivalent problem 2. (c) Equivalent problem 3.

beam centers (centers of beam waist) from which the CSPBs are launched. In this case, it is the real sphere . The imaginary part determines the beam axis direction and size [24]. Therefore, the repreof the beam waist sentation in (7) is in the form of a continuous CSPB expansion with the beam centers located on the sphere and the beam ) as seen in Fig. 2(a). axes directed radially outward (for The beam coefficients or weights, namely the equivalent surface on , are readily computed through this sources of assumed known primary formulation from the fields inside and evaluated at the complex locasources tions as designated above the brackets of (7). The integration in (7) may be evaluated numerically via discretization based on a quadrature rule; hence (7) gives rise to a discrete set of beams on the order of the numerical sampling or discretization density. This numerically rigorous CSPB expansion may now be used to efficiently track the radiated fields as spectrally (or angularly) compact beams. It is noted that the complex distance in . the Green’s functions is evaluated as is chosen here to satisfy the radiation The branch condition.

As in the case of method 1 (see (1) and (7)), the radius of the sphere is assigned a complex value denoted by and the are replaced by their complex counterparts variables to obtain the desired CSPB representation. The coefficients of the beams, namely for the unknown , can be obtained by solving this integral equation on a test surface. A simple numerical solution approach is described below. The above integral equation in (9) with complex equivalent sources is first discretized by using point sampling on the complex sphere

(10) where is the total number of beams in this discrete expansion, (for ) represent the unknown weights while denotes the complex locations of the to be solved and . These vector weights can be split into two tangensources tial components yielding

(11) The above equation is tested pointwise on a spherical surface of radius , where .

(12)

(13) B. Method 2: CSPB Expansion With Only Sources

Type Equivalent

An alternative CSPB expansion can be achieved by starting with a different equivalent problem formulation as shown in Fig. 1(c). In this equivalent problem, the constraint on fields to be zero inside the sphere is removed. Instead, the tangential

where denotes the test points which are located . These linear equations can be converted into a at matrix equation as follows: (14)

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where (15) (16) (17) (18) (19)

Fig. 3. Discretization of the unit sphere. The points on the surface denote the beam centers.

(20) (21) (22) The solution to the matrix equation yields the desired unknown and of (21) and (22), respectively beam weights, since are known via . type CSPBs are The advantage of method 2 is that only generated, in contrast to method 1, which additionally requires type CSPBs. The trade-off is the overhead computation cost required to find the CSPB coefficients numerically. However, method 2 is the only one of the three methods described here that allows an arbitrary set of electric fields to be defined on a real spherical surface. This allows experimental or numerical data to be used directly as input for the CSPB expansion. C. Method 3: CSPB Expansion With Complex PMC Sphere

Type Sources on a

A third type of equivalent problem can be obtained by filling the enclosed volume of 1(b) by a PMC (or PEC) as shown in (or ) are Fig. 1(d). In this case, the equivalent currents shorted out and the remaining currents radiate in the presence of the PMC (or PEC) closed spherical surface. For the PMC case, the electric field is expressed as follows:

(23) where is the dyadic Green’s function for radiation from sources in the presence of the PMC spherical surface. As before, the radius of the sphere is assigned a complex value are replaced by their denoted by and the variables counterparts in the complex domain ( to obtain the desired CSPB representation. One can use the well known exact [25] and analytically coneigenfunction series solution for tinue it for the complex radius. However, the number of terms in the series of the eigenfunction solution grows rapidly with the increase in . Alternatively, it is demonstrated here that this Green’s function can be approximated very accurately by using the image theorem. It is known that the CSP radiates strongly in the forward direction and negligibly in the backward direction. In addition, the beam axis, along which the beam field is the strongest, lies in the unit normal direction to the complex sphere. Therefore, the complex creeping waves are not sphere, and most of strongly excited on a moderate to large the beam power is thus directed towards the “deep” lit region of

on a PMC sphere if and are each elemental source not too small in terms of wavelength. Therefore, the complex sphere can be replaced by a PMC plane tangent to the complex sphere at the CSP location and the image theorem can be subsequently employed to approximately evaluate the CSPB field. Hence the CSPB representation in this case is obtained by rein (23) by , respecplacing tively. The accuracy of this approximation increases for larger . The numerical results indicating the accuracy of this approximation are presented in Section IV. type CSPBs are The advantage of method 3 is that only generated, as in method 2, but a numerical solution is not required to find the CSP coefficients. D. The Properties of the CSPB Representation 1) Discretization of the Integral Representations and Parameter Selection Rules: For the CSPB expansion method 2, the integral representation is descretized with point sampling as previously given in (10). The grid shown in Fig. 3 is selected for point sampling. The dots located on the spherical surface represent the beam centers that are distributed over the sphere in a nearly uniform fashion. In this grid, the discretization intervals or distances are nearly same for all rings and approximately equal , to . This is achieved by selecting and are the samples along and directions, rewhere spectively. The function takes the integer value nearest to its argument . A numerical study is performed to estimate the minimum number of grid points required for a given sized source distribution. The details of this study are explained in Appendix B. Based on this study, the number of samples (or the number of rings on the figure) is related to the source size via (24) where is the diameter of the minimum sphere enand are parameters related to closing the sources, and the given error tolerance and beam parameter . Once is known, one can form the full grid as described in the previous paragraph. It is noted that the empirical formula given in (24) should be taken as an initial guideline for selecting the number of beams for a given error tolerance. The actual number may depend on the distribution of sources within the expansion sphere for the specific problem of interest. The CSPB expansion representations in (7) and (23) (with complex domain substitutions) are in the form of integrals over the complex sphere and have to be discretized for numerical

TAP et al.: EXACT COMPLEX SOURCE POINT BEAM EXPANSIONS FOR ELECTROMAGNETIC FIELDS

evaluation. Well known numerical integration schemes can be used to evaluate these integrals. It is verified by numerical studies that a Legendre type Gauss quadrature scheme along and mid-point integration rule in can provide a compact beam set. The relation in (24) can be taken as a starting point to estirequired for the numerical mate the number of samples integration. One may however need to perform a few numerical required for the desired accuracy. In trials to arrive at the order to avoid the unnecessary accumulation of beam samples around poles, one can reduce the number of samples on the rings closer to the poles. This can be achieved by selecting for the mid point integration rule, . The selection yields an efficient where evaluation of the integral. A set of numerical studies have been performed to understand the effects of the parameters and on the efficiency of the expansion [22]. It is deduced from these studies that for an efficient fast converging beam representation should be chosen such that becomes the smallest sphere tightly enclosing all of the primary sources . This is reasonable since as the surface gets larger, one needs to place more beams on it to satisfy (24). The number of beams required for a given accuracy has a relatively weaker dependence on the beam parameter . As becomes very large, however, the integrand of the CSPB integral representation starts to vary more rapidly, which necessitates a parameter are denser grid on the sphere. CSPBs with large narrow in angle in the far field pattern. Hence, one may need a greater number of large waisted beams to synthesize a given far field pattern. On the other hand, when becomes too small, the locations of the equivalent currents approach the real sphere with radius . In this case, should be chosen to be somewhat greater (about one wavelength) than the minimum sphere enclosing the original sources, in order to avoid fast variations of the source fields in the extreme near zone or reactive region. 2) Computational Effort: As explained previously, the number of samples along direction is proportional , which is subto number of samples along direction by (24). sequently related to maximum source dimension is related to by Therefore the number of total beams . For methods 1 and 3, the computational effort , to calculate the beam coefficients is in the order is the number of the original source points inside . where may also represent the number of operations Alternatively, required for the calculation of the fields due to an extended source distribution in a given volume inside . To compute the observation points the required effort for all field value at where is the number of truncated three methods is beams (discussed below). To find the beam coefficients via or operations for a method 2, one needs direct solution or an iterative solution of the associated matrix denotes the number of iteraequation, respectively, where tions. Clearly method 2 is computationally disadvantageous for large expansion problems as compared to methods 1 and 3 since it requires the solution of a matrix equation to compute the beam coefficients. However method 2 is directly applicable when the source is unknown and only the knowledge of the electric field at a proper set of locations on a test surface is present, for instance when the electric field values are given as a result

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of measurement or simulation. The application of methods 1 and 3, on the other hand, assumes that the expressions for the fields of the sources are known analytically on the complex locations. For example, if the sources are not known, but the measured near field or far field is known, one can match this data to the spherical vector wave function (SVWF) expansion to obtain the analytical expressions for the fields for methods 1 and 3. Efficient solution of the matrix equation for method 2 is considered as a part of future work. The drawback of method 1 is that it utilizes both electric and magnetic equivalent currents on . On the other hand, method 3 alleviates the numerical drawbacks of the other two methods; it only requires electric type CSPBs and the coefficients of these CSPBs can directly be computed from the assumed source fields as in method 1. The disadvantage of the method 3 is that it is not exact since it employs an approximate form of the Green’s function for the radiation in the presence of PMC sphere as discussed in Subsection II.C. 3) Truncation of the Beam Set: The CSPBs are directional field elements with a Gaussian-like natural window within the paraxial region. Therefore only a small number of the total beams in the expansion significantly contribute to the field at a given point, and hence the remaining beams can be discarded without introducing significant error. This property was previously investigated in the literature in a similar context (for example [8], [12], [24], [26]). In order to illustrate the truncation idea, the integrals of (7), (9) and (23) in their complex forms can be first written in the following generic form: (25)

represents the fully 3-dimensional angular space, where is the position vector to point P shown in Fig. 4 and the term includes the parts of the integrand other than the exponential term . This exponential term provides the exponential decay as the beam axis moves away from the observation point. It is the dominant factor in magnitude as compared to and serves as a natural window function for truncation. Splitting the complex distance into real and imaginary parts as , the window function can be defined as the magnitude of the exponential term (26) is added to normalize the function to where the constant one. Incorporating (26) into (25) yields (27)

As shown in Appendix A, the exponential term approximated in the paraxial region as follows:

can be

(28)

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Fig. 5. The branch cut and branch points of a z-directed CSPB.

Fig. 4. The beam set is truncated to include only significantly contributing beams. (a) The observation point P lies in the paraxial region of the beam. (b) r ) = . The truncation cone for the threshold  is defined as W (r; ~

Using this approximation along with (26) yields the following parametric form for the window function: (29) denotes the real distance between the beam The parameter center (i.e., the center of beam waist) and the observation point, , as shown in Fig. 4(a). It is clear that when , one obtains and hence in this case. In order to define a truncation cone, a threshold number is first chosen. Truncation cone is chosen such that the value of the window function on this , where denotes the complex locations cone is of the beams whose axes lie on the cone. It is evident that the are inside the cone of solid angle , beams with are outside, where is a whereas the ones with . Hence the expansion integrals can be truncated subset of as below to include only the significantly contributing beams within this cone (30)

It is also worth emphasizing that there is no need to make a search to find the significantly contributing beams. One can start adding the beams whose axes pass close to the observation point. In the process of summation, as the beam axis moves away from the observation point, the window function is computed and compared with the threshold. The summation is halted is reached. The window funcwhen the condition tion is therefore not computed for the neglected beams in this way and hence no extra computation is necessary. 4) Validity Region of CSP Representations: The CSPBs possess branch point and branch cut singularities [24], which are . pictorially described in Fig. 5 for a CSP located at As seen from the figure, the branch points are located on a circle of radius and the branch cut is defined as the disk enclosed by this circle. When considering the CSPB expansion from a spher, the locus of branch ical surface with complex radius point locations form a sphere of radius as shown in the cross sectional view of Fig. 6(a). The branch cuts are located . in the volume between the spheres with radii and In order for the beam expansion to be valid (or exact), the observation point should not be on the branch point singularities and

Fig. 6. Cross sectional view showing the branch points and branch cuts of the CSPBs in the expansion. (a) The expansion is exact outside the sphere with a + b , the field can be radius a + b . (b) In the region a < r < computed only with a limited accuracy.

branch cuts. Therefore, the validity region is defined as the infi. If the nite volume excluding the sphere with radius observation point lies inside the region as in Fig. 6(b), the expansion does not abruptly become inaccurate and it is still possible in this case to compute the field with a limited accuracy [22]. Noting that the CSPBs radiate significantly only in the forward direction, the dark colored beams shown with thicker lines in Fig. 6(b) contribute strongly to the field, whereas the contribution of the light colored beams is negligible and they can be excluded from the expansion. If the darkened portion of the sphere in the figure contains sufficient information for the field at P, the beams launched from this portion can still provide sufficient accuracy. This accuracy increases as P moves closer to the outer sphere, since the number of dark colored beams increases. III. EXTENSION TO RELATIVELY ARBITRARY SURFACES The CSPB expansion methods described in Section II for a sphere can be extended to the case where the surface enclosing the sources is allowed to be a relatively arbitrary closed surface . Consider the following parametric representation of : (31) and are the polar and azimuthal where describing can be angles, respectively. The function expanded exactly, for example, in terms of a two dimensional Fourier-Legendre series as follows [23]:

(32)

TAP et al.: EXACT COMPLEX SOURCE POINT BEAM EXPANSIONS FOR ELECTROMAGNETIC FIELDS

where are the coefficients of the expansion and denotes the associated Legendre function of the first kind. The first type of the surface equivalence theorem is next employed with the equivalent sources on

(33) where the surface Jacobian is defined as (34) is the outward unit normal vector to . It is noted that and and all depend on and , the parameters respectively. Noticing that the surface coefficients in the integral of (33) appear as free parameters, one can extend them analytically to complex values as shown below. This is similar to extending the sphere radius to a complex value, namely (35) (36) where and are real quantities and are associated with the imaginary part of the resulting complex source location vector , which is defined as (37) where

(38) The CSPB representation in this case is obtained by replacing in (33) by their counterparts in the complex do, respectively. The unit normal vector is main in general complex and can be computed from the complex tangent vectors as follows:

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Recalling that the dyadic Green’s functions in (33) with sources in the complex domain generate the vector CSPBs, this representation is therefore a CSPB expansion with the beam centers located on the surface . The size of the beam waist and the direction of beam axis are again determined by the imaginary part of the complex position vector . As mentioned previously, the CSPBs possess branch cut and branch points over which the beam function becomes undefined (see Fig. 5). In order for the beam representation in (33) to be exact, it is again required that both the source and observation points are not located on the branch cuts and branch points of any of the beams radiated from and the surface . This condition restricts the values that can take in the complex substitution. Clearly any arbitrary closed convex surface (of which a sphere or ellipsoid, etc. are special cases) satisfies the above requirement when the beam axes are normal to the surface. This case is of primary interest in practice since the branch cut disks of all beams are tangential to and the volume enclosed by is free of singularities. The beam axes however may not be normal to the convex surface in and . In the the general case depending on the values of latter case, or when is chosen to be a general non-convex surface, care should be taken not to violate the above conditions. Hence the expansion surface can be relatively arbitrary in the sense that i) It can be any closed convex surface, for which the beam axes can be selected to be normal to the surface, ii) It can be a general non-convex closed surface, for which the beam parameters can be properly selected not to violate the above stated in (37) requirements. In addition, the complex function should be differentiable with respect to and such that the complex Jacobian can be defined as in (39)–(43). Utilizing an expansion surface other than sphere can be more useful in some cases. For example when the source is a long linear or rectangular planar array of length , the minimum rato enclose the array. dius of the expansion sphere should be As explained previously in II.D.4, the validity region of the expansion in this case excludes the volume defined by a sphere of . Increasing the size of the validity region radius is possible by choosing an ellipsoidal expansion surface which can encapsulate the linear array in a tighter manner as shown in Fig. 7. The parametric representation of an ellipsoid is given below

(44)

(39) (40) (41)

where are the lengths of the semi-axes. By comparing (44) with (32) and using the properties of the associated Legendre functions, it is deduced that the only non-zero parameters in this case are the following:

(42) The complex Jacobian is defined as below (43)

(45) (46) (47)

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Fig. 8. The x-directed electric point sources are distributed over a circular disk ;  :  and  . with radius r

= 2 1 = 0 25

1 = 20

Fig. 7. Ellipsoidal expansion surface.

Performing the analytical extension as in (35), (36), one can write the position vector to any point on the new complex surface as follows:

(48) are real parameters. The imaginary part of the where surface in this case becomes the ellipsoid defined by

(49) The CSPB representation is obtained by substituting of (48) into the radiation integral of (33). Since the real part of the CSP location is on the given ellipsoid in (44), the beams are launched from this surface. The axes of the resulting beams with the analytical extension of (48) however are not normal to the ellipsoid in general. Therefore one needs to select the values of such that the singularities of the beams do not coincide with the linear array. In order to make the beam axes normal to the ellipsoid, the complex surface can be defined as follows: (50) is the unit normal vector to the ellipsoid where and is a real constant that determines the beam size. The numerical convergence of the representations in (48) and (50) is presented in Section IV.B.

Fig. 9. The convergence of the expansion method 1 with a spherical surface.

in Section II.D.1, the Gauss quadrature integration scheme is used along and mid-point integration rule is used along with for methods 1 and 3. The special grid described in Section II.D.1, and shown in Fig. 3, is used for method 2. 1) Convergence of the Expansion Approaches: In order to demonstrate the convergence of CSPB expansions, the field rais expanded in terms diated by the disk source of radius . of CSPBs originating from a spherical surface of radius The observation is made on an angular grid of points distributed with increments. over a sphere of radius The normalized average error, defined below is evaluated and plotted in Figs. 9–11 with respect to the number of beams N for method 1, method 2, and method 3 respectively.

IV. NUMERICAL RESULTS A. Spherical Expansion Surface In this section, numerical results are presented to demonstrate the performance of the CSPB expansion approaches with a spherical expansion surface. The source distribution used in the examples is a set of x-directed electric point sources that are distributed discretely over an area defined by a circular disk uniformly excited to radiate a maximum in the of radius broadside direction as shown in Fig. 8. The point current elin the radial direction ements are separated by and by in the azimuthal direction. As mentioned

(51)

where and denote the exact field values and the field values obtained by CSPB expansion, respectively. As seen from the Figs. 9 and 10, the error decays down to the precision level (double precision is used here) for both methods 1 and 2 of the CSP expansion. The error converges to higher values for method 3 due to the approximations in the Green’s function as explained

TAP et al.: EXACT COMPLEX SOURCE POINT BEAM EXPANSIONS FOR ELECTROMAGNETIC FIELDS

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Fig. 10. The convergence of the expansion method 2 with a spherical surface.

Fig. 11. The convergence of the expansion method 3 with a spherical surface.

in Section II.C. As seen from the plots in Fig. 11, the error increases, which indicates that the applicagets smaller as tion of the image theorem becomes a better approximation for the Green’s function. The accuracy is also observed to improve for larger sources (or for spheres with larger ). 2) Truncation of the Beam Expansions: In this section, the beam expansions are truncated based on the selection scheme described in Subsection II.D.3. The radius of the circular disk and the complex radius of in Fig. 8 is selected to be . The observation the sphere is taken to be in the plane defined by is made over a circular arc of radius . As seen in Fig. 12, all three expansion approaches give accurate field representations as compared with the exact field. For the first and third method, a total of 6510 beams are used in the expansion. But after the truncation, on average of 266 beams remained significant at an observation point. This is an order of magnitude reduction in the number of CSPBs contributing to the total field at a given point outside the surface. With the special grid used in Fig. 3 for method 2, a total of 3548 beams were necessary and 144 of them are retained after truncation. It is noted that in method 2, the grid points beyond are not included in the expansion in order to keep total number of beams sufficiently small for numerical evaluation of beam coefficients. The computational effort required for the different expansion methods was explained in Section II.D.2.

Fig. 12. Normalized field with three methods of CSPB expansion. (a) Method 1. (b) Method 2. (c) Method 3.

B. Ellipsoidal Expansion Surface The field radiated by a linear array of x-directed electric point sources is expanded by CSP expansion method 1 with beams originating from an ellipsoidal surface of semi-axis dimensions . The linear array is placed on the z-axis and centered at the origin as shown in Fig. 7. The length and the elements are apart. of the array is denoted by Three different examples are considered for the imaginary part of the CSP locations. In the first example, the function is taken to be an ellipsoid as in (49) with

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Fig. 14. The convergence of the expansion method 1 with an ellipsoidal expansion surface.

The length of the linear array is chosen such that the branch cut singularities do not overlap with the sources for the first two examples. The observation points are located on the ellipse, defined by (52) which is away from the branch cut disks of the CSPBs for all three examples. The convergence of the normalized average errors are plotted in Fig. 14. As seen from the figure, the error decays as the number of beams is increased for all three examples. It is observed that for the same error level the beam expansion in example 3 requires less beams than the expansion in examples 1 and 2. The reason may be the fast variation of the integrand due to the proximity of the branch point singularities of the beams to the sources for examples 1 and 2. V. CONCLUSION

= 0 . (a) Example 1: = 18.

Fig. 13. The branch point circles of the beams for  L , (b) Example 2: L , (c) Example 3: L

= 10

= 16

. In the second example, the parameters are sesuch that defines a lected as is sphere of radius . Finally, in the third example, selected to be the normal vector to the ellipsoidal expansion sur. face as in (50) with The branch point circles of the beams located at are shown in Fig. 13(a)–(c) for the three examples, respectively. The dots on the ellipse in the figure represent the CSPB centers and the circles around each dot show the branch points of the corresponding CSPB. As seen from the figures, for the first two examples, the branch cut disks of the beams are not tangential to the ellipsoidal surface and they enter the region enclosed by the expansion ellipsoid. In addition, in the first example, the sizes of the beam waists differ for beams located on different positions of the ellipse. In the third example the beam axes are normal to the surface and branch cut disks are tangential to the surface.

A numerically rigorous expansion of an arbitrary source region in terms of CSPBs has been presented. Convergence is demonstrated in terms of the number of CSPBs and the single parameter that determines the beam nature of the CSPB field. It is this beam nature that reduces the number of CSPBs that contribute to a given field point by an order of magnitude or more. To this end, three different variants of the surface equivalence theorem are considered, which give three different CSPB representations. CSPB expansion idea is also extended to the case of relatively arbitrary expansion surfaces. This expansion may be applied to the problem of antenna radiation in complex environment using beam tracking [3], for speeding up fast integral equation methods [21], and for fast near field to far field transformations, to name just a few examples. APPENDIX A. The Paraxial Behavior of the CSPBs The dyadic Green’s functions for the electric field due to electric and magnetic CSPs are obtained by replacing the source

TAP et al.: EXACT COMPLEX SOURCE POINT BEAM EXPANSIONS FOR ELECTROMAGNETIC FIELDS

argument with the complex one as follows:

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 ;

TABLE I VALUES FOR DIFFERENT c AND 

(53) (54) where is the identity dyad, . The CSPs are assumed to be located over a complex . The exponential sphere with position vectors terms in (53), (54) dominate the magnitude behavior of the beam is field. The complex displacement next paraxially approximated to obtain an interpretable analytical form for this dominant behavior. Defining as the real vector between the observation point and the beam center as shown in Fig. 4(a), can be rewritten in the following form [22]: (55) For (56) which describes the paraxial region,

can be approximated as (57)

The above approximation of ponential term yielding

is next substituted into the ex-

(58) Defining for small

as

and approximating the cosine function yields (59)

The angle Fig. 4(a).

and the direction of the beam axis are labeled in

B. Numerical Study to Estimate the Number of Grid Points In order to estimate the number of grid points required for a CSPB expansion, eight x-directed electric point sources are located on the eight vertices of a cube. The side length of the so that the maximum dimension of cube is taken to be the cube is L. The field radiated by these sources is expanded in terms of CSPBs with method 2 by using a sphere having com, where , and is a plex radius real constant. The field values are evaluated over an observation sphere in far zone and compared with the exact solution. The normalized average error is computed as in (51). The number of

samples used in the expansion is increased in an iterative manner until the calculated error is smaller than the desired . This procedure is repeated for a level of error tolerance is calculated for each . A line is fit number of L values and for a given to obtain the to the resulting data to estimate relation previously given in (24). The function in (24) takes the upper nearest integer to its argument. The same procedure is repeated for different values and error tolerances . coefficients are tabulated in Table I. The computed REFERENCES [1] J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beam,” J. Opt. Soc. Amer., vol. 61, pp. 40–43, 1971. [2] G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett., vol. 7, pp. 684–685, 1971. [3] L. B. Felsen, “Complex source point solution of the field equations and their relation to the propagation and scattering of Gaussian beams,” in Proc. Symp. Math., 1976, vol. 18, pp. 39–56. [4] J. J. Maciel and L. B. Felsen, “Gaussian beam analysis of propagation from an extended plane aperture distribution through dielectric layers—Part 1: Plane layer,” IEEE Trans. Antennas Propag., vol. 38, no. 10, pp. 1607–1617, Oct. 1990. [5] M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion, vol. 4, pp. 85–97, 1982. ˇ [6] V. Cervený, M. M. Popov, and I. Pˇsenˇcik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astr. Soc., vol. 70, pp. 109–128, 1982. [7] A. N. Norris, “Complex point-source representation of real point sources and the Gaussian beam summation method,” J. Opt. Soc. Am., vol. 3, no. 12, pp. 2005–2010, Dec. 1986. [8] I. T. Lu, L. B. Felsen, and Y. Z. Ruan, “Spectral aspects of the Gaussian beam method: Reflection from homogenous half space,” Geophys. J. R. Astron. Soc., vol. 89, pp. 915–922, 1987. [9] Y. Dezhong, “Study of complex Huygens principle’,” Internat. J. Infrared Milli. Waves, vol. 16, pp. 831–838, 1995. [10] A. N. Norris and T. B. Hansen, “Exact complex source representations of time harmonic radiation,” Wave Motion, vol. 25, pp. 127–141, 1997. [11] E. Heyman and L. B. Felsen, “Complex source pulsed beam fields,” J. Opt. Soc. Am. A, vol. 6, pp. 806–817, 1989. [12] E. Heyman, “Complex source pulsed beam representation of transient radiation,” Wave Motion, vol. 11, pp. 337–349, 1989. [13] T. B. Hansen and A. N. Norris, “Exact complex source representations of transient radiation,” Wave Motion, vol. 26, pp. 101–115, 1997. [14] T. B. Hansen, “Complex point sources in probe-corrected cylindrical near-field scanning,” Wave Motion, vol. 43, pp. 700–712, 2006. [15] S. Zeroug, F. Stanke, and R. Burridge, “A complex-transducer-point model for finite emitting and receiving ultrasonic transducers,” Wave Motion, vol. 24, pp. 21–40, 1996. [16] T. B. Hansen, “Complex-point dipole formulation of probe-corrected cylindrical and spherical near-field scanning of electromagnetic fields,” IEEE Trans. Antennas Propag., vol. 57, pp. 728–741, Mar. 2009. [17] T. B. Hansen and G. Kaiser, “Generalized Huygens principle with pulsed-beam wavelets,” J. Phys. A: Math. Theor., vol. 42, 2009.

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[18] E. Martini, G. Carli, and S. Maci, “A direct non redundant complex source point expansion of the electromagnetic field radiated by an arbitrary source,” presented at the EUCAP, Barcelona, Spain, Apr. 2010. [19] K. Tap, P. H. Pathak, and R. J. Burkholder, “Exact complex source point beam expansion of electromagnetic fields from arbitrary closed surfaces,” presented at the IEEE AP-S Int. Symp., Honolulu, Hawaii, Jul. 2007. [20] K. Tap, P. H. Pathak, and R. J. Burkholder, “An exact CSP beam representation for EM wave radiation,” presented at the ICEAA, Torino, Italy, Sep. 2007. [21] K. Tap, P. H. Pathak, and R. J. Burkholder, “Fast complex source point expansion for accelerating the method of moments,” presented at the ICEAA, Torino, Italy, Sep. 2007. [22] K. Tap, “Complex source point beam expansions for some electromagnetic radiation and scattering problems,” Ph.D. dissertation, The Ohio State Univ., Columbus, 2007. [23] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: Wiley, 2001. [24] E. Heyman and L. B. Felsen, “Gaussian beam and pulsed-beam dynamics: Complex-source and complex spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A, vol. 18, no. 7, pp. 1588–1611, 2001. [25] J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes. New York: Hemisphere, 1987. [26] T. Heilpern, E. Heyman, and V. Timchenko, “A beam summation algorithm for wave radiation and guidance in stratified media,” J. Acoust. Soc. Am., vol. 121, no. 4, pp. 1856–1864, Apr. 2007.

Koray Tap (S’00–M’07) received the B.S. degree in electrical engineering from Bilkent University, Ankara, Turkey, in 2001, and the M.S. and Ph.D. degrees in electrical engineering from the Ohio State University (OSU), Columbus, in 2004 and 2007, respectively. From 2001 to 2007, he was a Graduate Research Associate with the Electro-Science Laboratory, OSU. Since 2007, he has been with ASELSAN Inc., Ankara, as an Antenna Engineer. His research interests include the beam methods and hybrid methods combining high-frequency asymptotic techniques with numerical techniques for solving large scale electromagnetic radiation and scattering problems.

Prabhakar Pathak (F’86) received the Ph.D. degree from the Ohio State University (OSU), Columbus, in 1973. Currently he is a Professor Emeritus at OSU. He is regarded as a co-contributor to the development of the uniform geometrical theory of diffraction (UTD). Presently, he is developing new UTD ray solutions, for predicting the performance of antennas near, on, or embedded in, thin material/metamaterial coated metallic surfaces. Recently his work has also been involved with the development of new and fast hybrid asymptotic/numerical methods for the analysis/design of very large conformal phased array antennas for airborne/spaceborne and other applications. In addition, he is working on the investigation and development of Gaussian beam summation methods for a novel and efficient analysis of a class of large modern radiation and scattering problems including the analysis/synthesis of very large spaceborne reflector antenna systems. He has published over 100 journal and conference papers, as well as authored/coauthored chapters for seven books. Prof. Pathak was elected an IEEE Fellow in 1986, and is an elected member of US Commission B of the International Union of Radio Science (URSI). He received the 1996 Schelkunoff best paper award from the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and the ISAP 2009 conference best paper award. He received the George Sinclair award in 1996 for his research contributions to the O.S.U. ElectroScience Laboratory, and the Lumley Research Award in 1990, 1994 and 1998 from the O.S.U. College of Engineering. In July 2000, he received the IEEE Third Millennium Medal from the Antennas and Propagation Society. In 2010, he served as an elected member of the IEEE Administrative Committee (AdCom) for the Antennas and Propagation Society. He has presented several short courses and invited lectures both in the U.S. and abroad. He has chaired and organized several technical sessions at national and international conferences. He was invited to serve as an IEEE Distinguished Lecturer from 1991 through 1993. He also served as the chair of the IEEE Antennas and Propagation Distinguished Lecturer Program during 1999–2005. Prior to 1993, he served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION for two consecutive terms.

Robert J. Burkholder (S’85–M’89–SM’97–F’05) received the B.S., M.S., and Ph.D. degrees in electrical engineering from The Ohio State University, Columbus, in 1984, 1985, and 1989, respectively. Since 1989, he has been with the Ohio State University ElectroScience Laboratory, Department of Electrical and Computer Engineering, where he is a Research Professor. He has contributed extensively to the EM scattering and imaging analysis of large and complex geometries, and targets in the presence of rough surfaces and inside urban structures. His research specialties are high-frequency asymptotic techniques and their hybrid combination with numerical techniques for modeling large-scale electromagnetic radiation, propagation, and scattering problems. Dr. Burkholder is an elected Full Member of URSI, Commission B, a member of the American Geophysical Union, and a member of the Applied Computational Electromagnetics Society. He is currently serving as an Associate Editor for IEEE Antennas and Wireless Propagation Letters.

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Scattering by an Eccentrically Loaded Cylindrical Cavity With Multiple Slits Santosh Seran, J. Patrick Donohoe, Senior Member, IEEE, and Erdem Topsakal, Senior Member, IEEE

Abstract—Rigorous solutions for the scattered fields of an eccenor trically loaded cylindrical cavity with multiple slits under plane wave illumination are formulated using a Neumann series expansion. The series solution exhibits favorable convergence properties resulting in a computationally efficient scheme. The solution scheme is applied to several geometries, showing that the incidence can back scattered RCS due to the slot mode under be suppressed by reducing the continuous circumferential metallic length of the cylinder. Also, the low frequency RCS for excitation can be reduced by blocking large slit apertures with thin strips.

TE

TM

TE

TM

Index Terms—Apertures, cavities, electromagnetic diffraction, electromagnetic scattering.

I. INTRODUCTION LECTROMAGNETIC field penetration through longitudinal slots in cylindrical geometries has received considerable attention in the literature given the importance of this geometry with regard to electromagnetic compatibility and RCS reduction. The problem of a single slit on a coaxial cable and illumination by Ziolkowski was solved for both [1] using the dual series approach. Felsen solved the same case using ray-mode parameterization [2] problem for the illumination using the while Arvas solved the problem for illumimethod of moments (MoM) [3]. Yu [4] considered nation of the coaxial geometry with a thick outer boundary by matching the field at the boundary and using the orthogonality of the Fourier series. The scattered fields of a cylinder with a slit were determined in [5] and [6] by incorporating a modal expansion of the aperture fields in MoM solution. Also, Shumpert and Bulter proposed three methods to study penetration of a slotted conducting cylinder in [7] and [8]. The slotted cylinder with inner and outer lossy coating has been analyzed by Colak in [9] and [10]. The multiple slit on a concentric multilayer cylinder was solved by Yin [11] using a Chebyshev polynomial current expansion. More recently, the problem of an unclosed cylinder with an eccentric dielectric inner coating was solved by Ioannidou [12] using a Riemann Hilbert solution. In this paper, we consider scattering by a loaded cylindrical cavity with multiple longitudinal slits in the outer conductor and an eccentrically located inner conductor. The details of the scatterer geometry are defined in Section II. The motivation behind

E

Manuscript received December 02, 2009; revised October 04, 2010; accepted February 23, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Department of Electrical and Computer Engineering, Mississippi State University, MS 39762 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2161536

Fig. 1. TE/TM plane wave incident on an eccentrically loaded cylindrical cavity with multiple slits.

the analysis is to characterize the backscattered radar cross section (RCS) relative to the number and position of the slits in the outer conductor and the position of the inner conductor. The solution is based on the Neumann series approach described by Delfino [13]. The Neumann series approach is similar to the Chebyshev series expansion described by Yin, but the Neumann series expansion provides a more straightforward formulation. In addition, the Neumann series solution incorporates the required edge condition in a flexible way and also provides a linear system which is invertible [13]. The Neumann series solution approach is described in Section III. The numerical procedure and results are described in Section V. II. GEOMETRY OF THE PROBLEM The geometry of the eccentrically loaded cylindrical cavity with multiple slits illuminated by a plane wave is shown in Fig. 1. The radii of the inner and outer perfectly conducting and , respectively. The outer cylinders are defined by cylinder, with multiple slits, is assumed to be infinitesimally thin, with its axis lying along the axis of the cylindrical . The regions exterior and interior coordinate system to the outer cylinder are assumed to be ideal dielectrics of and , respectively. The axis of the inner permittivities . A shifted cylindrical coordinate conductor is located at is used to define the fields internal to the system outer cylinder where the axis of the inner conductor lies along axis of the shifted coordinate system. The structure is the to (electric field parallel to -axis) or illuminated by a

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to (magnetic field parallel to -axis) plane wave incident at an angle of . The time dependence assumed here is . The total number of metallic cylindrical segments on the unclosed cylinder is denoted by . The angular width metallic segment is denoted by and the angular of the position of the midpoint of each cylindrical segment is denoted . The region defined by is denoted as region 1 by while the region defined by is denoted as region 2. The wave number of the external region is denoted by and that of the internal region is denoted by .

can be obtained using (5). The fields while the coefficient and can be written as (8)

(9) Using the addition theorem for Bessel functions, (8) can be exas pressed in terms of the shifted coordinates

III. INTEGRAL EQUATION FORMULATION Given the structure of the scatterer and the incident field, the case) or total magnetic field ( case) total electric field ( can be solved via a scalar boundary value problem. Let represent the total electric/magnetic field in region assuming / excitation, respectively. The total field can be written as

(10) where (11)

(1) (12) where

is the incident plane wave given by The fields on the outer cylinder, the inner cylinder, and throughout region 2 can be written as

(2) is the field scattered by the dielectric structure interior to the is the field scattered by the currents on slotted cylinder and the conducting cylinders.

(13) (14)

A. Formulation for Scattered Fields (15)

Consider the structure in Fig. 1 with the metallic cylinders removed. The exterior field (region 2) is written as (3)

respectively, where Table I.

,

, and

are defined in

B. Formulation for Scattered Fields while the interior field (region 1) is written as (4) and are determined by enforcing the The coefficients continuity of the tangential electric and magnetic fields across and utilizing the orthogonality of the the boundary at Fourier series, which yields the following equations (5) (6) Solving for

gives

The fields scattered by the conducting cylinder currents can be determined using the specific current and appropriate Green’s function for the structure. The approach applied here is similar to the one described in [17] and [18] for an unloaded can be unclosed cylinder configuration. The scattered field written as (16) where the primed and unprimed vector locates source and observation point. The vector locates points on the contour ( and define the inner and outer conducting cylinders, respectively). Enforcing the PEC boundary condition of zero tangential electric field on the cylinders yields (17)

(7)

where the field point in (17) is located on the contour

.

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TABLE I FUNCTION VALUES FOR TM/TE CASES (n AND n ARE NORMALS TO THE SURFACE AT THE POSITION r AND r RESPECTIVELY)

(23)

(24) The Green’s function for currents on the inner cylinder ( ) and observation points exterior to the outer cylinder is written in terms of the shifted coordinate system as (25) The Green’s function for currents on the inner cylinder and observation points interior to the outer cylinder is written as

(26) where (27) Using the addition theorem for Bessel functions, the field in region 1 can be expressed in terms of the original coordinates as The Green’s function for source currents on the outer cylinder ) and observation points exterior to the outer cylinder is ( written as

(28) where

(18) (29) The Green’s function for source currents on the outer cylinder and observation points interior to the outer cylinder is written as

and (30)

(19)

Enforcing continuity of the tangential electric and magnetic fields at the interface between regions 1 and 2 along with the orthogonality of the Fourier series yields

where

(31) (32)

(20) The Green’s function coefficients are determined by enforcing the continuity of tangential electric and magnetic fields at the interface between regions 1 and 2 along with the orthogonality of the Fourier series. The resulting coefficients are (21) (22)

C. Integral Equation Solution The current densities and on the cylinder conand are expanded in terms of Fourier series as tours (33)

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(34)

where is a parameter chosen based on the required edge behavior, and the inverse Fourier transform for each term in (40) [14] is given by

The Green’s functions in (19) and (26) can be expressed in terms by again applying the Bessel of the shifted coordinates function addition theorem, yielding

otherwise (42)

(35)

(36)

where is function of and only, and is the Gegenbauer polynomial. The proper behavior of the current for the density at the edges of the cylinder is given by case and for the case. For the metallic unclosed slits, expressing the Fourier coefficient of cylinder with the original integral equation formulation in terms of the Neumann series yields

(37)

(43)

where

Substituting the appropriate Green’s functions into the integral equation of (17) for points on the inner cylinder, and using the orthogonality of Fourier series yields

The current density becomes

(44) (38) is defined in Table I. Substituting (19) and (28) into where the integral equation of (17) for points on the outer cylinder yields

The above expression for the current density satisfies the edge condition for each segment for the metallic cylinder. Substituting (43) into (38) yields

(45) where (39)

(46) Also, substituting (43) into (39) yields

where (40) , are given in Table I. and Consider first a cylinder with a single slit, where the angle . As shown in [13], each range of the conductor is Fourier coefficient of the respective current expansion can be expressed in terms of the following Neumann series expansion: (41)

(47)

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The solution of the above equation, as described in [13], involves testing the equation with

which yields the following linear equation

Fig. 2. Geometry of the single slit cylinder (concentric loading) used in (a.) , , . (b) Fig. 10. with , , Fig. 7. with .

N =2

N =4 N =1 N =0

N =3 N =1

IV. NUMERICAL PROCEDURE AND RESULTS (48) Solving (48) for the unknowns is given by where. The field

yields the solution every-

(49) D. Determination of Backscattered RCS The normalized backscattered RCS is given by (50) and from (15) and (49) into (50) and using Substituting the large argument form of the Hankel function [19] yields

As shown in Fig. 1, the slits in the loaded conducting cylinder are assumed to be symmetrically spaced with the first slit cen. The angular width of each slit is defined by tered about . The loaded conducting cylinder geometry is modified by symmetrically adding small gaps in each conducting segment case or small conducting strips in the slit openings for the case. The angular width of each added gap is defor the fined as and the angular width of each added blocking strip as . The total number of individual conductors ( ) is equal to the number of conducting segments ( ) plus the number of blocking strips ( ). For the single slit problem, the angular width and angular position of the center of each metallic strip is given by (52)-(53), shown at the bottom of the page. Fig. 2 illustrates two single slit cylinder examples with (a.) three added , , ) and (b.) two blocking strips gaps ( ( , , ). The solution of the linear system can be obtained by trunseries from to and the series from cating the to . The resulting equation pair in (45) and (48) can be written in matrix form as (54) The each element of the A matrix consists of summation

(51)

(55)

(52)

(53)

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N = 1, N = 1, N = =5 =05 =1 =0 =0 = ( 0 =36),  = 0, = 2 =1 =1 = 3 45 + j0:25,  = 1 (3) = 7 3 = 0 9 + j0 32

Fig. 3. Truncation number versus truncation error for , kb ,a : ,b , , ,r = , (1) " , :  (2) " : , : : . "

0

The above series converges slowly as such that a large number of terms is required for accurate commutation of the series. An equivalent series yielding faster convergence for , certain configurations is defined in [15] and [16] for , and , , but for general values of and , such methods are not applicable and direct commutation of the series is required. The series in (55) is comto , using the actual value of puted from for to . For faster computation, the asympfor large (given in Appendix A) is used totic value of for remaining terms in the series. The value of used here as described by Elsherbeni [20]. In order to determine is the number of terms , required to obtain an accurate solution for the backscattered RCS ( ), we define the truncation error as

Fig. 4. Truncation number N versus 2kb for TE case. (b = 1, w = = =2, =18, w = =360). 0 " = 1,  = 1, r = 0,  = 0,  a = 0 :5 , N = 1 , N = 1 , N = 0 . + 0 " = 1 ,  = 1 , r = 0 , =  , a = 0 :5 , N = 1 , N = 1 , N = 0 . x 0 " = 1 ,  = 0,  =  , a = 0:8, N = 1, N = 1, N = 0.  = 1, r = 0,  = 0,  o 0 " = 1 ,  = 1 , r = 0 :2 ,  = 0 ,  =  , a = 0 :5 , N = 1 , = , N = 1, N = 0. 0 " = 1,  = 1, r = 0:2,  = =2,  0 " = 1,  = 1, r = 0, a = 0 :5 , N = 1 , N = 1 , N = 0 . = 0 , a = 0 :5 , N = 2 , N = 1 , N = 0 . 1 0 " = 7 :3 ,  = 0,  = 0, a = 0:5, N = 2, N = 1,  = 0:9 + j0:32, r = 0,  = 0,  N = 0. TABLE II TRUNCATION NUMBER N TO ACHIEVE A TRUNCATION ERROR LESS THAN

10

(56) is the backscattered RCS computed using terms of . The truncation error versus plotted in Fig. 3 shows that the Neumann series solution has excellent (the number of terms convergence properties. A plot of ) verses required to obtain a truncation error less than ( times the segment arc length in wavelengths) is solution by varying the incident shown in Fig. 4 for the ), the inner cylinder radius ( ), angle ( , and the eccentricity of the inner cylinder ( , ) and the inner cavity material ( , and , ). The results in Fig. 4 show that the truncation number depends primarily on the segment length and is practically independent of all other parameters. Simple approximate equations for calculating the required to obtain a truncation error less than value of for the and solutions are given in Table II. The solution scheme presented here is validated by combackscattered RCS found in [3] for paring results for the the single slit as shown in Fig. 5 with excellent agreement. Similar agreement is obtained for concentric loading ( case) as given in [2] but is not included here for brevity. The material used for filling region 1 is shellac natural XL or 2.5 where

Fig. 5. Normalized backscattered RCS (TM) for (b = 1, a = 0:3, r =  = 0,  =  , N = 1, N = 1, N = 0, w = =18, " = 1,  = 1).

dichlorostyrene as used in [9] and [10]. The effect on the RCS case) is shown of loading region 1 with the lossy material ( in Fig. 6, where significant RCS reduction is shown, which is consistent with the results shown in [10]. The RCS of the 2.5 dichlorostyrene-loaded cylinder approaches that of the equivalent closed cylinder. The effect of increasing the number of gaps in the outer case) with a single slit is presented in Fig. 7. It cylinder ( should be noted that the total gap width on the outer cylinder is kept constant as the number of gaps is increased. Thus, the gap width decreases as the number of gaps increase. As shown if Fig. 7, the slot mode can be reduced by increasing the number

SERAN et al.: SCATTERING BY AN ECCENTRICALLY LOADED CYLINDRICAL CAVITY WITH MULTIPLE SLITS

Fig. 6. Normalized backscattered RCS (TE) for different material loading for b = 1, a = 0:3, r =  = 0,  =  , N = 1, N = 1, N = 0, w = =18. (1) " = 1,  = 1, (2) " = 3:45 + j0:25,  = 1, (3) " = 7:3,  = 0:9 + j0:32.

Fig. 7. Normalized backscattered RCS (TE) for single slit cylinder with different numbers of gaps (b = 1, a = 0:3, r =  = 0,  =  , N = 1, N = 0, w = =18, w = =360, " = 1,  = 1). (1) N = 1 (no gaps), (2) N = 4 (3 gaps), (3) N = 10 (9 gaps).

of segments forming the outer cylinder, which decreases the continuous current path around the cylinder. The impact of the inner cylinder eccentricity on the RCS is presented in Fig. 8. At lower frequencies, the RCS is not affected by changing the eccentricity of the inner cylinder. However, at higher frequencies, the RCS becomes sensitive to the eccentricity of the inner cylinder, as the internal resonances are affected by the position on the inner conductor. The effect of loading the inner region with a lossy material for case is shown in Fig. 9. Similar levels of RCS reducthe case. The introduction tion are obtained here as seen for the of small gaps in the outer cylinder has minimal impact on the case, given the axial current for this polarizaRCS for the tion. The effect of employing small blocking strips in a large case is presented in Fig. 10. The results show slit for the that the low frequency RCS can be made to approach that of a closed cylinder by using a large number of blocking strips. case The blocking strips have no effect on the RCS for the since the induced circumferential current is negligible and hence would be essentially transparent at low frequencies. Reducing case can the continuous length of the outer cylinder for the reduce the average value of the low frequency RCS. The effect case is shown in of the inner cylinder eccentricity for the

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Fig. 8. Normalized backscattered RCS (TE) for variation in the eccentricity of the inner cylinder (b = 1, a = 0:3,  =  , N = 4, N = 1, N = 0, w = =3, w = =360), " = 1,  = 1. (1) r = 0:6,  = 0, (2) r = 0:3,  = 0, (3) r =  = 0, (4) r = 0:3,  =  .

Fig. 9. Normalized backscattered RCS (TM) for different material loading with (b = 1, a = 0:3, r =  = 0,  =  , N = 1, N = 1, N = 0, w = =3, w = =360). (1) " = 1,  = 1, (2) " = 3:45 + j0:25,  = 1, (3) " = 7:3,  = 0:9 + j0:32.

Fig. 10. Normalized backscattered RCS (TM) for variation in the number of blocking strips in a single slit cylinder with no gaps (b = 1, a = 0:3, r =  = 0,  =  , N = 1, w = =3, w = =90, " = 1,  = 1).

Fig. 11. Similar to the case, the RCS is essentially independent of eccentricity at low frequency but sensitive to eccentricity at high frequency. The method presented here is computationally efficient. For example, the computation time required for the single slit , , , , problem defined by ( , , , , using MATLAB

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Similarly, for the

case, we find

(60)

REFERENCES

Fig. 11. Normalized backscattered RCS (TM) for variation in the eccentricity of the inner cylinder (b = 1, a = 0:3,  =  , N = 1, N = 1, N = 0, w = =3, w = =90, " = 1,  = 1).

on a personal computer with a 2.53 GHz processor and 3 GB of RAM is less than 11 seconds. V. CONCLUSION A method for computing the field scattered from an eccentrically loaded cylinder with multiple slits is presented. The proposed technique is found to be accurate and efficient, and is also shown to exhibit good convergence properties. Various geometries have been considered to determine the impact on the illumination, the backscattered backscattered RCS. Under RCS due to the slot mode can be reduced by introducing more slits in the outer cylinder, thus reducing the continuous path for illutransverse current flow. The low frequency RCS under mination for large slits can be reduced by blocking the aperture a sufficient number of thin strips. APPENDIX The asymptotic value of for large is determined by first computing the asymptotic value of the following terms:

(57) Using the expression for Bessel and Hankel functions of large order from [19] and neglecting higher order terms, the asympare totic value of the above terms as (58) respectively. Substituting the above asymptotic values in (47), for the case is given the asymptotic value of (59)

[1] R. W. Ziolkowski and J. Grant, “Scattering from cavity-backed apertures: The generalized dual series solution of the concentrically loaded E-pol slit cylinder problem,” IEEE Trans. Antennas Propag., vol. AP-35, pp. 504–528, May 1987. [2] L. B. Felsen and G. Vecchi, “Wave scattering from slit coupled cylindrical cavities with interior loading: Part II – Resonant mode expansion,” IEEE Trans. Antennas Propag., vol. 39, no. 8, Aug. 1991. [3] E. Arvas, “Electromagnetic diffraction from a dielectric filled slit-cylinder enclosing a cylinder of arbitrary cross section: TM case,” IEEE Trans. Electromagn. Compat., vol. 31, pp. 91–102, Feb. 1989. [4] J.-W. Yu and N.-H. Myung, “Oblique scattering and coupling to a slit coaxial cable: TM case,” JEWA, vol. 14, pp. 931–942, 2000. [5] J. R. Mautz and R. F. Harrington, “Electromganetic penetration into a conducting circular cylinder through a narrow slot, TE case,” JEWA, vol. 3, no. 4, pp. 307–336, 1989. [6] J. R. Mautz and R. F. Harrington, “Electromganetic penetration into a conducting circular cylinder through a narrow slot, TM case,” JEWA, vol. 2, no. 3/4, pp. 269–293, 1988. [7] J. D. Shumpert and C. M. Butler, “Penetration through slots in conducting cylinders – Part 1: TE case,” IEEE Trans. Antennas Propag., vol. 46, no. 11, pp. 1612–1621, Nov. 1998. [8] J. D. Shumpert and C. M. Butler, “Penetration through slots in conducting cylinders – Part 1: TM case,” IEEE Trans. Antennas Propag., vol. 46, no. 11, pp. 1621–1628, Nov. 1998. [9] D. Colak, A. I. Nosich, and A. Altintas, “Radar cross-section study of cylindrical cavity-backed apertures with outer or inner material coating: The case of E-polarization,” IEEE Trans. Antennas Propag., vol. AP-41, no. 11, pp. 1551–1559, Nov. 1993. [10] D. Colak, A. I. Nosich, and A. Altintas, “Radar cross-section study of cylindrical cavity-backed apertures with outer or inner material coating: The case of H-polarization,” IEEE Trans. Antennas Propag., vol. AP-43, no. 5, pp. 440–447, May 1995. [11] W.-Y. Yin, L.-W. Li, T.-S. Yeo, and M.-S. Leong, “Multiple penetration of a TEz-polarized plane wave into multi-layered cylindrical cavity-backed apertures,” IEEE Trans. Electromagn. Compat., vol. 42, no. 4, pp. 330–338, 2000. [12] M. P. Ioannidou, “EM wave scattering by an axial slot on a circular PEC cylinder with an eccentrically layered inner coating: A dual-series solution for TE polarization,” IEEE Trans. Antennas Propag., vol. 57, no. 11, pp. 3512–3519, 2009. [13] F. Delfino, R. Procopio, and M. Rossi, “A new method for the solution of convolution-type dual integral-equation systems occurring in engineering electromagnetic,” J Eng Math, vol. 63, pp. 51–59, 2009. [14] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey and D. Zwillinger, Eds., 7th ed. New York: Academic Press, 2007. [15] R. W. Scharstein, M. L. Waller, and T. H. Shumpert, “Near-field and plane-wave electromagnetic coupling into a slotted circular cylinder: Hard or TE polarization,” IEEE Trans. Electromagn. Compat., vol. 48, no. 4, pp. 714–724, Nov. 2006. [16] L. Glasser, “A class of Bessel summations,” Math. Comput., vol. 37, no. 156, p. 499, Oct. 1981. [17] V. V. Veremey and R. Mittra, “Scattering from structures formed by resonant elements,” IEEE Trans. Antennas Propag., vol. AP-46, no. 4, pp. 494–501, Apr. 1998. [18] Y. A. Tucklin, “Wave Scattering by an open cylindrical screens of arbitrary profile with Dirichlet boundary value,” Soviet Phys. Doklady, vol. 30, pp. 1027–1030, Apr. 1985. [19] J. A. Stratton, Electromagnetic Theory. New York: McGraw Hill, 1941. [20] A. Z. Elsherbeni, “A Comparative study of two dimensional multiple scattering technique,” Radio Sci., vol. 29, pp. 1023–1033, 1994.

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Santosh Seran received the B.E. degree in electronics and telecommunication from Mumbai University, Mumbai, India, in 2001, the M.E. degree in communication systems from Anna University, Chennai, India, in 2005, and the Ph.D. degree from Mississippi State University, Mississippi State, in 2010. He is currently working as Postdoctoral Associate at Mississippi State University. His research interest includes electromagnetic theory, Wiener-Hopf technique, and scattering theory.

J. Patrick Donohoe (SM’99) was born in Jackson, MS, on December 17, 1958. He received the B.S. and M.S. degrees in electrical engineering from Mississippi State University, Mississippi State, in 1980 and 1982, respectively, and the Ph.D. degree in electrical engineering from the University of Mississippi, University, in 1987. He joined the Department of Electrical and Computer Engineering, Mississippi State University, in 1986 where he currently holds the title of Professor. His primary research interests include computational electromagnetics, electromagnetic compatibility, electromagnetic properties of composite materials, and lightning protection. Dr. Donohoe is a registered professional engineer in the state of Mississippi, and a member of Eta Kappa Nu.

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Erdem Topsakal (SM’03) was born in Istanbul, Turkey, in 1971. He received the B.Sc., M.Sc., and Ph.D. degrees in electronics and communication engineering from Istanbul Technical University, in 1991, 1993, and 1996, respectively. He was a Postdoctoral Fellow from 1998 to 2001 and an Assistant Research Scientist from 2001 to July 2003 in the Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor. In August 2003, he joined the Electrical and Computer Engineering Department, James Worth Bagley College of Engineering, Mississippi State University, Starkville, as an Assistant Professor. In 2008, he was promoted to Associate Professor. His research areas include implantable antennas, numerical methods, fast methods, antenna analysis and design, frequency selective surfaces/volumes, electromagnetic coupling and interference, direct and inverse scattering. He has published over 100 refereed journal and conference papers in these areas. Dr. Topsakal is a member of electrical engineering honor society Eta Kappa Nu. He is an elected member of the URSI commissions B and K. He received the URSI Young Scientist Award in 1996 and a NATO Fellowship in 1997. He is the recipient of 2004–2005 MSU Electrical and Computer Engineering Department’s Outstanding Educator Award, and the 2009 Bagley College of Engineering Outstanding Research Paper Award. He currently serves as an Associate Editor for the Applied Computational Electromagnetics Society Journal, and the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS (AWPL). He is the Vice Chair for URSI-USNC Commission K, Electromagnetics in Biology and Medicine and also serves on IEEE USA Committee on Communications and Information Policy as a representative of IEEE Engineering in Medicine and Biology Society (EMBS).

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Controllable Metamaterial-Loaded Waveguides Supporting Backward and Forward Waves Fan-Yi Meng, Member, IEEE, Qun Wu, Senior Member, IEEE, Daniel Erni, Member, IEEE, and Le-Wei Li, Fellow, IEEE

Abstract—Rectangular waveguides loaded by anisotropic metamaterials are analyzed to assess the controllability of transmission characteristics of the involved electromagnetic waves. Dispersion relations of m0 modes in the metamaterial-loaded waveguide (MLW) are theoretically investigated. It is shown that all propagating modes (the forward wave, the backward wave and the evanescent wave) in the MLW can be realized below the cut-off frequency by changing transverse and longitudinal components of permeability tensors of the loading metamaterials. Numerical simulations are carried out to verify the proposed theory and the controllability. Transmission characteristics and effective constitutive parameters of three MLWs with different cells, which should theoretically support forward waves, backward waves and evanescent waves, respectively, are numerically calculated. Dispersion curves and magnetic field distribution for the backward wave MLW and the forward wave MLW are simulated. It is shown that the simulated results are in a good agreement with theoretical predictions. Implementation of the controllable MLW was achieved by using axially rotating control rods. Rotating the control rods can reconfigure the metamaterial and make propagating modes in the MLW switch from backward waves to forward waves or evanescent waves.

TE

Index Terms—Backward wave, controllability, evanescent wave, forward wave, metamaterial-loaded waveguide.

I. INTRODUCTION ECTANGULAR waveguides are commonly used as basic guiding structures in microwave, radar and antenna technology. Moreover, there are many applications of the wave-

R

Manuscript received April 01, 2010; revised December 07, 2010; accepted February 09, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the National Natural Science Foundation of China under Grants 60801015 and 60971064, the Open Project Program of the State Key Laboratory of Millimeter Wave under Grants K201007 and K201006, the Development Program for Outstanding Young Teachers in Harbin Institute of Technology under Grant HITQNJS.2008.07, the Fundamental Research Funds for the Central Universities under Grant HIT.IBRSEM.2009, and in part by the Aviation Science Funds under Grant 20080177013. F.-Y. Meng and Q. Wu are with the Department of Microwave Engineering, Harbin Institute of Technology, Harbin 150001, China, and also with the State Key Laboratory of Millimeter Waves, Nanjing 210096, China (e-mail: [email protected]; [email protected]). D. Erni is with the Laboratory for General and Theoretical Electrical Engineering (ATE), Faculty of Engineering, University of Duisburg-Essen, and CeNIDE-Center for Nanointegration Duisburg-Essen, D-47048 Duisburg, Germany (e-mail: [email protected]). L.-W. Li is with Department of Electrical and Computer Engineering, National University of Singapore, Kent Ridge 119260, Singapore (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161540

guide as radiating elements. The most familiar example is a slotted waveguide antenna, which is popular in navigation, radar and other high-frequency systems. It has a high efficiency and supports linear polarization with low cross-polarization. Such an antenna is often used in aircraft applications because it can be made to conform to the surface on which it is mounted. A leaky-wave antenna is basically also a waveguide structure that possesses a controllable mechanism for power leakage all along its length. The radiated beam of the leaky wave antenna may be frequency scannable, producing a fan beam that is narrow in the scan plane. The frequency scannable leaky wave antennas are cost effective in systems like low cost radars, side looking sensors in vehicles and imaging because phase shifters and associated circuit elements are not required to provide the beam steering [1]. Recently, a very unusual rectangular waveguide was proposed by Marques et al. in [2], [3] and then extensively studied by Hrabar et al. in [4]–[6]. The rectangular metallic waveguide is loaded with split ring resonators (SRRs) [7]–[11], which have the property of negative permeability and are also used to realize the left-handed metamaterial (LHM) along a composite with both, negative permittivity and permeability [12]–[17]. The metamaterial loaded waveguide (MLW) supports the propagation of backward waves below the cut-off frequency of the hollow waveguide [18]. The MLW has attracted extensive attention [19]–[32] since it provides an alternative approach for the realization of negative index metamaterials as well as a unique method for the miniaturization of waveguide-based devices. Hrabar et al. [4] showed that backward propagation occurs when the longitudinal permeability is positive and the transversal permeability is negative and that a MLW with broad bandwidth can be realized by proper feedings. Eshrah et al. [28], [29] designed and fabricated a sub-wavelength rectangular waveguide with dielectric-filled corrugations supporting backward waves. Belov et al. [33] theoretically studied rectangular metallic waveguides that are periodically loaded with uniaxial resonant scatterers based on the local field approach (i.e. the dipole approximation). Xu et al. [23] performed experiments and simulations to study the transmission of electromagnetic waves through waveguides loaded by resonance structures of electric and magnetic type. It is worth mentioning that in case of a metamaterial with anisotropic unit cells the electromagnetic characteristics of the metamaterial can be effectively tuned simply by changing the arrangement of unit cells. Hence, when integrated in a rectangular waveguide the tunability of the metamaterial enables the waveguide to provide the further advantage of controllable transmis-

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MENG et al.: CONTROLLABLE METAMATERIAL-LOADED WAVEGUIDES SUPPORTING BACKWARD AND FORWARD WAVES

sion characteristics, in addition to the smaller waveguide cross section. Making rectangular waveguides controllable so that their behavior may adapt to changing system requirements or environmental conditions can ameliorate or eliminate restrictions and provide additional functionality on the system level. For example, the controllability of the propagating rectangular waveguide mode can give rise to leaky wave antennas with fixed frequency scannability. In fact, the fixed frequency scannability is often preferable to frequency scannability in certain applications where a narrow frequency band is available and wide angle of coverage is required. Such controllable waveguides can also help to broaden the scanning range of leaky wave antennas, which increases link capacity of e.g. multiple input multiple output (MIMO) communication systems [34]–[37]. In this paper, we focus on the effects emerging from corresponding changes in the position and direction of the loading metamaterial cells in MLWs. Transmission characteristics of electromagnetic waves in the MLWs are theoretically analyzed. Moreover, numerical simulations are performed to demonstrate the controllability of the MLWs. The work done in this paper is of practical relevance in designing controllable MLWs and corresponding tunable microwave and RF devices.

Fig. 1. Outline of the rectangular waveguide with cross-section a waveguide is loaded with metamaterial along the z -direction.

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2 b. The

From (5) the quantity equation for the z-component is obtained as (6) Inserting (4) into (6) leads to

II. THEORY DESCRIPTION Consider a -directional rectangular waveguide whose width and height are denoted by and , respectively, as shown in Fig. 1. The walls of the waveguide are perfect electric conductors (PEC). The waveguide is loaded with lossless anisotropic metamaterial having the following permeability tensor and permittivity

(7) Solving this differential equation yields the dispersion equation for the rectangular waveguide loaded with anisotropic metamaterial (8) From (8) one finds the expression for the propagation factor

(1) The field distribution inside the waveguide satisfies source-free curl Maxwell equations (9)

(2) (3) For and , following relation can be derived from (2) and (3) for TE modes

modes, the dispersion relation (9) is simplified to

Assuming

(10) On the other hand, corresponding calculations show that the is given by wave impedance of

(4) On the other hand, inserting (3) into (2) leads to the vector wave equation for the electric field (5)

(11)

Let us consider a homogeneous medium with effective permittivity and permeability . For a plane EM wave propagating along the direction inside the medium, the dispersion

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relation and the wave impedance are

(12) The comparison between (10), (11) and (12) suggests that the MLW can be viewed as an effective medium with [2], [3]

(13) Moreover, there is

(14) Transmission characteristics of MLWs in the case of have been analyzed and discussed in detail in [4]. It has been and support shown that MLWs with backward wave propagation below the cut-off frequency of the corresponding empty waveguide. Here, we focus on the case , where from (13) one can setup the following of relation

(15) This means that is positive at any frequency. There, it can be inferred that MLWs with fore, considering and support propagation of forward waves, whereas MLWs with and yields exponentially attenuated waves, irrespective of whether the operating frequency is above or below the cut-off frequency of the empty waveguide. III. CONTROLLABLE MLWS According to the analysis in the above section, it is now known that propagating modes of MLWs can be controlled by changing the permeability tensors of the underlying metamaterials. Thus, controllable MLWs can be realized. A possible scheme for the implementation of controllable MLWs is shown in Fig. 2. The controllable MLW is set up as a conventional waveguide loaded with a metamaterial which consists of an array of disconnected modified split ring resonators (MSRRs) [38], [cf. Fig. 3(a) and 3(b) (inset)]. The MSRR has much higher mutual capacitance between the rings compared to the ordinary SRR. Therefore, it can lower the resonance frequency and allow the fabrication of inclusions of smaller “electrical dimensions”, which are defined as the ratio of physical dimensions to the operating wavelength. In addition, the MSRR can suppress the unwanted bianisotropic effects [4], [38]. Each metamaterial cell (MSRR) is attached to an axially rotating dielectric control rod that is vertically fixed through holes in the upper and lower wall of the waveguide. Control rods are periodically arranged along the -directional line of symmetry

Fig. 2. Geometry of the controllable MLW in the case of backward wave. (a) The perspective view. (b) The top view. (c) The front view. (d) The side view.

of the waveguide. Thus, as these control rods are rotated, the metamaterial cells are rotated too, which changes the permeability tensor of the loading metamaterial in the MLW because of the varying anisotropy of the MSRR. Consequently, the controllability of the MLW is carried out this way. For example, if the odd-numbered rods (#1, 3, 5, 7, and 9) in Fig. 2(a) are turned anti-clockwise by 90 and the even-numbered rods (#2, 4, 6, 8, and 10) are turned clockwise by 90 , the MLW as shown in Fig. 14 is realized. If only the odd-numbered rods in Fig. 2(a) undergo a clockwise rotation by 90 and the even-numbered rods are kept unchanged, the MLW yields a structure as shown in Fig. 20. In this study, physical parameters of the MSRR have been extracted from an experimental analysis in [4]. The MSRR comprises two copper rings placed back to back on CuClad sub) with slots oriented strate (substrate thickness 0.7 mm, in opposite directions. The rings have an outer diameter of 4 mm with track width of 1 mm and a slit width of 0.5 mm. In addition, the waveguide has a cross-section of 12 mm 12 mm and a cut-off frequency of 12.5 GHz. In the following section, the electromagnetic properties of the MSRR are investigated.

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Note that the expression for is a complex function with multiple branches. The ambiguity can be resolved using the condi. The combination of (18) and (19) yields tion (20) In the same sense one gets the following exact expression [39] (21) Combining (20) and (21) yields the closed-form relation for the and the effective permittivity effective permeability (22) (23)

Fig. 3. (a) Dispersion curve for the parallel polarization . curve for the perpendicular polarization

H

H

. (b) Dispersion

Using HFSS, a commercial electromagnetic software package based on the finite-element method (FEM), dispersion curves are computed for an infinite periodic array of the MSRRs with the thickness of one unit cell. There are two incident polarizations of interest: magnetic field polarized along the split ring axes [ , cf. inset of Fig. 3(a)], and perpendicular to the , cf. inset of Fig. 3(b)]. In both cases, the split ring axes [ electric field is in the plane of the rings. As shown by the curves in Fig. 3(a) and 3(b), a band gap is found in either case. We will gap is due to negative effective permeability show that the while the gap is due to a small resonance, which nor negative causes neither negative effective permeability effective permittivity . and of the MSRR Effective constitutive parameters metamaterial are extracted from transmission and reflection data. Similar to [39], our extraction approach also begins by and with -parameters introducing the composite terms

(16) and deriving following quantities [39] (17) where the transmission term (18) Consequently, from (17) one obtains (19)

The expressions for and are equivalent to those in [40], [41], but their calculation procedures are different. We use this in such extraction approach because the condition a calculation can help to effectively select the correct branch in and for all reported cases in this the expressions for paper. polarization is necessary An incident plane wave with to calculate the -parameters for the extraction of the effective permeability tensor component along the split ring axes while an incident plane wave with is required for the extraction of the effective permeability tensor component parallel to . There are also other issues worth the split ring plane noting in the calculation of the -parameters for extraction, which are elaborately discussed in [39]–[41]. Fig. 4(a) and (b) and for the incident wave acdisplay the retrieval of cording to the inset in Fig. 3(a), respectively. It can be observed is negative near 8.35 GHz while the that the real part of is positive. The oscillation in the curves real part of indicates that the MSRR provides not only a magnetic response, but also an electric one. It is worth mentioning that the imagiis positive although the imaginary part of nary part of is negative. This seems nonphysical because imaginary parts of both the complex permeability and permittivity are always negative for conventional passive materials. However, such positive imaginary part is a common phenomenon for metamaterials. For example, the single-SRR-metamaterial in [42] and the and negative SRR-metamaterial in [40] have positive . Because [40] and [42] used the convention to extract and , their positive and negative have the same physical meaning with the negative and compared to our notation. We think that the positive shown in Fig. 4(b), which represents active the positive materials, is closely related to radiation losses [43]. As stated of SRR in [43], the total loss leading to the negative includes two principal components: radiation loss and dissipation loss. Moreover, radiation loss dominates and far exceeds dissipation loss, even in the optical regime, where ohmic losses are usually high. Radiation loss results from the electromagnetic energy that is scattered by the metamaterial elements away from the incident wave. Therefore, different from dissipation

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Fig. 4. Extracted real part and imaginary parts of (a)  and (b) " of the MSRR metamaterial. The solid line represents the real part while the dashed line represents the imaginary part.

loss, radiation loss is not a real irreversible loss that is associated to the conversion of electromagnetic energy into other energy forms such as e.g. heat energy. Because of the conservation of electromagnetic energy, it can be inferred that the positive [Fig. 4(b)] should arise from the scattered electromagnetic wave, which causes the radiation loss and contributes to [Fig. 4(a)]. the negative Fig. 5 shows the extracted together with the corfor an incident wave that is polarized as responding displayed in the inset of Fig. 3(b). It can be seen that the real and are always positive, although there is parts of a small resonance near 8.4 GHz. In addition, while comparing Fig. 4 with Fig. 5, it can be observed that for different incident waves, the resonance frequencies of the MSRR are slightly differing. The reason is that the dimension of the MSRR is not very small [about 0.18 of the wavelength], and thus the quasi-static equivalent circuit elements of the MSRR slightly change with respect to the different directions and forms of the incident wave. Fig. 6 depicts the dispersion curves for the MSRR when the split ring axis of the MSRR encompasses angles of 15 , 30 , 45 , 60 , and 75 with respect to the magnetic field polarization direction. One finds that, as the angle increases, the width of the band gap reduces and the resonance of the MSRR weakens. A comparison between the dispersion curves in Fig. 6 and Fig. 3 gap for the angle of 0 reveals that the bandwidth of the is wider than those in the other cases, where the angle deviates from 0 . This means that the resonance of the MSRR is strongest when the magnetic field is polarized along the split ring axis. The influence of the rotation angle on the negative permeability effect of the MSRR is also worth investigating on its own. The

Fig. 5. Extracted real part and imaginary parts of (a)  and (b) " of the MSRR metamaterial. The solid line represents the real part while the dashed line represents the imaginary part.

Fig. 6. Dispersion curves for the MSRR when split ring axis of the MSRR encompasses angles of 15 , 30 , 45 , 60 , and 75 to the polarization direction of the magnetic field. The line with circles corresponds to the angle of 15 . The line with squares corresponds to the angle of 30 . The line with up-triangles corresponds to the angle of 45 . The line with down-triangles corresponds to the angle of 60 . The thick line corresponds to the angle of 75 .

effective permeability of metamaterials consisting of MSRRs case is that are rotated by 30 and 50 with respect to the extracted and their real parts are shown in Fig. 7. The spectral responses feature a negative permeability (cf. negative peak) that still exists when the angle is 30 but tends to disappear when approaching 50 . The effective permeability of MSRR metamaterials, where the split ring axes encompass other polarization angles to the magnetic field, is also extracted although not shown here. It is observed that such metamaterials provide negative effective permeabilities when the angles are smaller than 50 . The dispersion curve for a two-dimensional (2-D) metamaterial composed of crossed MSRRs [cf. Fig. 8 (inset)] is simulated and depicted in Fig. 8. It can be seen that the dispersion curve is very similar to the one shown in Fig. 3(a) because the effect

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Fig. 9. Simulation system for the transmission characteristics of the backward wave MLW.

Fig. 7. Extracted real parts of the effective permeability of the metamaterials case. The solid consisting of MSRRs turned 30 and 50 with respect to the line corresponds to the MSRRs turned 30 . The dashed line corresponds to the MSRRs turned 50 .

H

Fig. 10. Magnitude of the transmission coefficient (s ) of the backward wave MLW.

Fig. 8. Dispersion curve for two-dimensional (2-D) metamaterial composed of crossed MSRRs.

of the MSRR, whose split ring axis is perpendicular to the magnetic field polarization, on the electromagnetic characteristics of the 2-D metamaterial is weak. In addition, for the crossed MSRRs, there is also a quite narrow dispersion curve caused by the MSRR, whose split ring axis is perpendicular to the magnetic field polarization, although it is not shown here. IV. NUMERICAL SIMULATION According to the analyses in Sections II and III, we have shown that as the control rods turn, the propagating mode field in the controllable MLW can be either a forward, backward or evanescent wave. Numerical simulations are performed to validate the controllability of the MLW in following three cases. Note that the effect of the control rods is not considered in this section but in the subsequent Sections V. Moreover, the material losses of copper associated to the MSRR structures are considered in all simulations in this paper. Case A. Backward Wave MLW: In the case as shown in Fig. 2, the split rings axes are parallel to the -direction, and the MSRR and a positive near 8.35 metamaterial provides a negative and . Therefore, according GHz because to (13) and (14), the MLW will have a negative effective permittivity and a negative effective permeability to support the propagation of a backward wave near 8.35 GHz. Transmission characteristics of the MLW are simulated using CST MW STUDIO. Fig. 9 shows the corresponding simulation system. The MLW is

introduced between two feeding waveguides (with a cross-section of 45 mm 15 mm, and a cut-off frequency of 3.3 GHz), and directly connected with the feeding waveguides. The walls of both the feeding waveguides and the MLW are made of perfectly electrical conductor (PEC). The input and output ports are set up as waveguide ports, which represent a special kind of boundary condition of the computation window, enabling a considerably adapted wave excitation as well as the absorption of energy. In principle this kind of port mimicks an infinitely long waveguide that is connected to the structure. In this case, the waveguide modes leave the structure (and hence the computational domain) toward the boundary planes with very low levels of reflections. In the simulation, the MLW has a length of 60 mm and is loaded with an array of ten MSRRs. of the MLW is The simulated transmission coefficient depicted in Fig. 10 [solid line], where the spectral response displays a narrow passband near 8.35 GHz. The cut-off frequency of the waveguide is lowered from 12.5 GHz to about 10.2 GHz because of the CuClad substrate with a relative permittivity of 2.6. It is worth mentioning that the MLW shown in Fig. 2 was experimentally investigated in [4] and the obtained results were further discussed in [24], [44]. From the experimental results one can identify a propagation band around 7.9 GHz, which is close to our simulated result of 8.35 GHz. However, for this below-cut-off (sub-wavelength) passband, there is a large difference between the experimental insertion loss of 30 dB in [4] and our simulated value of 10 dB as shown in Fig. 10. Two reasons mainly contribute to the difference. On one hand, the loss of the CuClad substrate is not considered in the simulation. At resonance, a significant amount of electromagnetic field extends outside the metallic strips and dissipates in the substrate. Therefore, the dielectric loss of the substrate has a significant influence on the insertion loss within this below-cut-off propagation

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Fig. 11. Extracted real and imaginary parts of the effective permittivity of the backward wave MLW. The solid line represents the real part while the dashed line represents the imaginary part.

band. The dashed line in Fig. 10 shows the simulated magnifor the backward wave MLW with a lossy CuClad tude of substrate, which has dielectric loss tangent of 0.002. It is now confirmed that in this case the simulated insertion loss within the below-cut-off propagation band increases to a value around 23 dB, which is closer to the experimental result. On the other hand, it is hard to accurately simulate the very thin copper cladding of the MSSR, due to the need of a high resolution in the computational mesh that should be fine enough to accommodate the very short penetration length. In this case, down scaling of the copper conductivity in order to cope with the artificial cladding thickness that is dictated by the minimum mesh size should lead to more accurate simulation results. When both, the lossy CuClad substrate and the metallic cladding with smaller conductivity [i.e. a third of the Copper conductivity] are considered, transmission characteristics of the backward wave is depicted by the MLW are simulated and the magnitude of dash-dotted line in Fig. 10. In this case, the insertion loss within the below-cut-off propagation band is around 28 dB and very close to the measured 30 dB. Effective constitutive parameters and of the MLW are extracted and the corresponding spectral responses are depicted in Figs. 11 and 12, respectively. The real parts of both the effective permittivity and the effective permeability are simultaneously negative near 8.35 GHz, which means that electromagnetic wave propagation in the MLW appears in the form of backward waves. Worth noticing are the curves. They arise from the electric reoscillations in the sponse of the MSRR and can be predicted by (13) considering shown in Fig. 4(b). A siman in (13) to be equal to the ilar phenomenon is also present in metamaterials composed of SRRs and thin wires [41]. The dispersion curve for the backward wave MLW [the solid line] is displayed Fig. 13, where the dispersion curve for the MSRRs from Fig. 3(a) is also included [dashed line] for comparison. It can be seen that a passband occurs within the bandgap of the MSRR, and the phase delays as the frequency increases within this passband. From this phenomenon, one can easily underpin the presence of backward waves. Case B. Forward Wave MLW: When the odd-numbered rods (#1, 3, 5, 7, and 9) in Fig. 2(a) are turned anti-clockwise by 90 and even-numbered rods (#2, 4, 6, 8, and 10) are turned clockwise by 90 , the MLW as shown in Fig. 14 can be realized. In

Fig. 12. Extracted real and imaginary parts of the effective permeability of the backward wave MLW. The solid line represents the real part while the dashed line represents the imaginary part.

Fig. 13. Dispersion curves for the backward wave MLW and MSRRs only. The solid line corresponds to the backward wave MLW. The dashed line corresponds to the dispersion curve of the MSRRs only [cf. Fig. 3(a)].

this case, the axes of the split rings are parallel to the -direction, and positive thus the MSRR metamaterial provides negative near 8.35 GHz. According to (13) and (14), the MLW will have positive effective permittivity and positive effective permeability to support the propagation of a forward wave near 8.35 GHz. The transmission characteristics of the MLW are simulated using the same system as given in Fig. 9. The length of the simulated forward wave MLW amounts to 60 mm and the underlying waveguide is loaded with an array of ten unit cells to form the MSRR metamaterial. Fig. 15 depicts the spectral response of the transmission coeffor the aforementioned MLW structure. A passband ficient can be identified near 8.35 GHz while the cut-off frequency of the MLW is about 11.5 GHz, which is lower than the cutoff of the corresponding empty waveguide due to the presence of the CuClad substrate. A close inspection of Fig. 10 and Fig. 15 reveals that the cut-off frequencies of backward wave MLWs are smaller compared to the cutoff frequencies of forward wave MLWs. This is due to the specific transversal distribution of the electric field in the waveguide, where e.g. the unperturbed mode yields a single-lobed transversal field profile centered around the waveguide axis so that the influence of the CuClad substrate on the cut-off frequency—namely the detuning towards smaller frequencies—is strongest when the substrate is placed along the center of the waveguide, and is weakest when the substrate is put adjacent to the waveguide walls. For the backward wave MLW, the CuClad substrate is placed along the

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Fig. 16. Extracted real and imaginary parts of the effective permittivity of the forward wave MLW. The solid line represents the real part while the dashed line represents the imaginary part.

Fig. 14. Geometry of the forward wave MLW.

Fig. 17. Extracted real and imaginary parts of the effective permeability of the forward wave MLW. The solid line represents the real part while the dashed line represents the absolute value of the imaginary part.

Fig. 15. Magnitude of the transmission coefficient (s ) of the forward wave MLW.

center of the waveguide, whereas for the forward wave MLW, the CuClad substrate extends between the center and the walls of the waveguide, yielding the observed cut-off behavior. In order to determine the propagating mode for the passband, and of the forward effective constitutive parameters wave MLW are extracted and displayed in Figs. 16 and 17, reand spectively. Both spectral responses confirm that are simultaneously positive near 8.35 GHz, underpinning the presence of forward waves in the MLW. Note that outshown in Fig. 17 side the resonance region, the value of approaches a value of 0.78 rather than one. According to [45], the physical reason for this inaccuracy is related to the presence of (evanescent) fringe fields at the interface between the MLW (formed by the stack of waveguide sections each containing a MSRR) and the free space, where higher-order modal contributions are expected. In addition, Figs. 16 and 17 reveal that the spectral response and alternates between positive and of both negative values. Actually, such imaginary parts with both positive and negative signs have also been observed in [43] and and of fishnet-metamaterial in [46]. In particular, the

Fig. 18. Dispersion curves for the forward wave MLW and MSRRs only. The solid line corresponds to the forward wave MLW. The dashed line corresponds to the MSRRs only.

[46] and the of EIT-metamaterial in [43] provide imaginary parts with both signs. Referring to the positive definiteness of the imaginary parts shown in Figs. 4(b) and 5(b), imaginary parts with both signs have a more complicated physical origin. The bipolar range of values is not only caused by the aforementioned radiation loss [refscattered electromagnetic wave], but also related to the exchange of induced electromagnetic energies between the MSRR and the rectangular waveguide [43]. The dispersion curve for the forward wave MLW is calculated and depicted in Fig. 18 [solid line]. For comparison the dispersion curve of the MSRRs from Fig. 3(a) is included as well [dashed line]. The passband occurs within the forbidden band of the MSRR supporting forward waves due to the positive

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Fig. 19. Geometry of the connected forward wave and backward wave MLWs. Fig. 21. Geometry of the evanescent wave MLW hosting a 2-D metamaterial.

Fig. 20. Snapshots of simulated distributions of the x-component of the magnetic field in the connected MLWs at different time instants at 8.35 GHz.

the slope of the dispersion curve. As a further step the backward wave MLW and the forward wave MLW are connected as shown in Fig. 19, and field distributions in the connected MLWs are computed in order to analyze the propagating modes in the backward wave MLW and the forward wave MLW. The simulation system is similar to that shown in Fig. 9, but the backward MLW is replaced with the connected MLWs. Moreover, the system is excited to get electromagnetic waves propagating from the forward wave MLW to the backward wave MLW. Fig. 20 shows the computed distributions of the -component of magnetic field at three instants of time corresponding to phase angles of 0 , 180 and 360 at 8.35 GHz. It is clearly observable that the field pattern in the forward wave MLW moves away from the source while the pattern in the backward wave MLW moves toward the source. Hence, forward waves propagate in the forward wave MLW while backward waves propagate in the backward wave MLW. Woth noting is that the electromagnetic energy always propagates forward from the source, irrespective of whether it propagates in the forward wave MLW or in the backward wave

MLW, because for the forward wave MLW the electromagnetic energy has the same propagation direction with the phase pattern, but for the backward wave MLW the electromagnetic energy and the phase pattern have opposite propagation direction. Case C. Evanescent Wave MLW: When the odd-numbered rods (#1, 3, 5, 7, and 9) in Fig. 2(a) are turned clockwise by 90 and the even-numbered rods (#2, 4, 6, 8, and 10) are kept unchanged, the evanescent wave MLW as shown in Fig. 21 can be realized. In this case, the loading metamaterial is two-dimensional (2-D) and isotropic, and has negative near 8.35 GHz. According to (13), the MLW will have positive effective permittivity and negative effective permeability which make the passbands near 8.35 GHz in Fig. 10 and Fig. 15 disappear. of the evanescent MLW The transmission coefficient is simulated and depicted in Fig. 22. Comparing the displayed spectral response with the ones in Figs. 10 and 15 indicates an increase of the insertion loss from 10 dB to 30 dB within the propagation band below the cut-off. In addition, the cut-off frequency of the evanescent wave MLW is about 11 GHz, which is higher than the cut-off frequency 10.2 GHz of the backward wave MLW but still lower than the cut-off frequency 11.5 GHz of the forward wave MLW. This also conforms to the layout of the evanescent wave MLW from Fig. 21, where half of the CuClad substrate is placed along the center of the waveguide and the other half is located between the center and the waveguide walls. Hence, the impact of the CuClad substrate on the cut-off frequency for the evanescent wave MLW lies between those for the backward wave MLW and the forward wave MLW. and effective permeability Both effective permittivity of the evanescent MLW are extracted and illustrated in Figs. 23 and 24, respectively. Near 8.4 GHz the spectral reincreases from negative to positive values, sponse of decreases from positive whereas the magnitude of

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Fig. 22. Magnitude of the transmission coefficient (s ) of the evanescent wave MLW.

Fig. 25. Effects of control rods with different " on the magnitude of the transmission coefficient of the backward wave MLW when the rods diameter is 1.0 mm.

Fig. 23. Extracted real and imaginary parts of the effective permittivity of the evanescent wave MLW. The solid line represents the real part while the dashed line represents the imaginary part.

Fig. 26. Effects of control rods with different diameters on the magnitude of the transmission coefficient of the backward wave MLW when " = 2:6.

Fig. 24. Extracted real and imaginary parts of the effective permeability of the evanescent wave MLW. The solid line represents the real part while the dashed line represents the imaginary part.

to negative values because of a sharp resonance. The results correspond to the suppressed propagation shown in Fig. 22. The above results infer that the simulated electromagnetic characteristics correspond to those predicted theoretically, confirming both the theory proposed in Section II and the controllability of MLWs. V. IMPACT OF CONTROL RODS In this section, the effect of the control rods on electromagnetic characteristics of the MLW is investigated. In order to reduce a possible impact of the control rods on the overall performance, they should be better made of a non-metallic material that is then characterized with the relative permittivity .

Fig. 27. Intensity distribution of electric field strength within the backward wave MLWat 8.35 GHz.

Therefore, one just has to determine the effects of the relative permittivity and the diameter of control rods. Figs. 25 and 26 provide an overview of these effects where Fig. 25 shows the influence of the material on the performance of the backward MLW for a rod diameter of 1 mm. The introduction of dielectric control rods into the simulation model slightly decreases the center frequency of the passband in the MLW. This redshift of the passband is further pronounced with increasing relative permittivity of the rods. Fig. 26 shows the impact of diameter changes of the control rods in the backward . As the diameter increases, the MLW while keeping center frequency of the passband slightly decreases. This effect is also very small. Fig. 27 depicts the intensity distribution of the electric field strength within the fully equipped backward wave MLW. The figure also shows that the electric field aggregates

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around the MSRRs with virtually now radiation leakage from the MLW through the holes on the walls. The impact of the control rods on the performance of the forward wave MLW are studied too, and turned out to be comparable to the previous case of the backward wave MLW. VI. CONCLUSION In this paper, a theoretical model for rectangular waveguides loaded by metamaterials was proposed in order to analyze the controllability of the underlying wave propagation, and hence the transmission characteristics of the waveguide system. Numerical simulations were carried out to verify both the proposed theory and the controllability, and the results have proven to be quite accurate. The controllability of the MLWs was enabled using axially rotating control rods that allow the waves in the MLWs to switch from backward to forward propagation or to an evanescent decay. The theory about controllable waveguides loaded by metamaterials and the implementation method of the controllability proposed in this paper are of great practical interest especially in the context of functional MLW devices. ACKNOWLEDGMENT The authors would like to thank Prof. W. Hong for very fruitful discussions. REFERENCES [1] G. W. Slade, L. Carin, Q. Xu, S. E. Borchardt, and K. J. Webb, “A study of slotline leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 38, pp. 411–414, 1990. [2] 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(4), 2002. [3] R. Marques, J. Martel, F. Mesa, and F. Medina, “A new 2D isotropic left-handed metamaterial design: Theory and experiment,” Microw. Opt. Technol. Lett., vol. 35, pp. 405–408, 2002. [4] S. Hrabar, J. Bartolic, and Z. Sipus, “Waveguide miniaturization using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag., vol. 53, pp. 110–119, 2005. [5] S. Hrabar and D. Zaluski, “Subwavelength guiding of electromagnetic energy in waveguide filled with anisotropic mu-negative metamaterial,” Electromagnetics, vol. 28, pp. 494–512, 2008. [6] S. Hrabar and G. Jankovic, “Basic radiation properties of waveguides filled with uniaxial single-negative metamaterials,” Microw. Opt. Technol. Lett., vol. 48, pp. 2587–2591, 2006. [7] M. W. Feise, J. B. Schneider, and P. J. Bevelacqua, “Finite-difference and pseudospectral time-domain methods applied to backward-wave metamaterials,” IEEE Trans. Antennas Propag., vol. 52, pp. 2955–2962, 2004. [8] R. Marques, F. Mesa, J. Martel, and F. Medina, “Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design—Theory and experiments,” IEEE Trans. Antennas Propag., vol. 51, pp. 2572–2581, 2003. [9] P. Markos and C. M. Soukoulis, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. E, vol. 65, p. 36622(8), 2002. [10] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2075–2084, Nov. 1999. [11] J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 1451–1460, 2005.

[12] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, May 2000. [13] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [14] R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett., vol. 78, pp. 489–491, 2001. [15] B. Ivsic, Z. Sipus, and S. Hrabar, “Analysis of uniaxial multilayer cylinders used for invisible cloak realization,” IEEE Trans. Antennas Propag., vol. 57, pp. 1521–1527, 2009. [16] S. Hrabar, D. Bonefacic, and D. Muha, “Numerical and experimental investigation of basic properties of wire medium-based shortened horn antennas,” Microw. Opt. Technol. Lett., vol. 51, pp. 2748–2753, 2009. [17] D. Bonefacic, S. Hrabar, and D. Kvakan, “Experimental investigation of radiation properties of an antenna embedded in low permittivity thin-wire-based metamaterial,” Microw. Opt. Technol. Lett., vol. 48, pp. 2581–2586, 2006. [18] G. Lubkowski, C. Damm, B. Bandlow, R. Schuhmann, M. S. ßler, and T. Weiland, “Broadband transmission below the cutoff frequency of a waveguide loaded with resonant scatterer arrays,” IET Microw., Antennas Propag., vol. 1, pp. 165–169, 2007. [19] A. Alu and N. Engheta, “Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or doublepositive (DPS) layers,” IEEE Trans. Microw. Theory Tech., vol. 52, pp. 199–210, 2004. [20] S. Antipov, L. Spentzouris, W. Gai, M. Conde, F. Franchini, R. Konecny, W. Liu, J. G. Power, Z. Yusof, and C. Jing, “Observation of Wakefield generation in left-handed band of metamaterial-loaded waveguide,” J. Appl. Phys., vol. 104, p. 014901(6), 2008. [21] J. D. Baena, L. Jelinek, and R. Marques, “Reducing losses and dispersion effects in multilayer metamaterial tunnelling devices,” New J. Phys., vol. 7, p. 166(13), 2005. [22] H. Bahrami, M. Hakkak, and A. Pirhadi, “Analysis and design of highly compact bandpass waveguide filter utilizing Complementary Split Ring Resonators (CSRR),” Progr. Electromagn. Res., vol. 80, pp. 107–122, 2008. [23] H. Xu, Z. Wang, J. Hao, J. Dai, L. Ran, J. A. Kong, and L. Zhou, “Effective-medium models and experiments for extraordinary transmission in metamaterial-loaded waveguides,” Appl. Phys, Lett., vol. 92, p. 041122(3), 2008. [24] S. Hrabar, J. Bartolic, and Z. Sipus, “Reply to “Comments on waveguide miniaturization using uniaxial negative permeability metamaterial”,” IEEE Trans. Antennas Propag., vol. 55, pp. 1017–1018, 2007. [25] J.-C. Liu, C.-Y. Liu, Y.-S. Hong, C.-Y. Wu, and D.-C. Lou, “Waveguide miniaturization with SR(ZNTI)FE10O19 hexaferrite metamaterial,” Microw. Opt. Technol. Lett., vol. 49, pp. 201–203, 2007. [26] I. A. Eshrah and A. A. Kishk, “Electric-type dyadic Green’s functions for a corrugated rectangular metaguide based on asymptotic boundary conditions,” IEEE Trans. Antennas Propag., vol. 55, pp. 355–363, 2007. [27] I. A. Eshrah and A. A. Kishk, “Magnetic-type dyadic Green’s functions for a corrugated rectangular metaguide based on asymptotic boundary conditions,” IEEE Trans. Microw. Theory Tech., vol. 55, pp. 1124–1131, 2007. [28] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “Spectral analysis of left-handed rectangular waveguides with dielectric-filled corrugations,” IEEE Trans. Antennas Propag., vol. 53, pp. 3673–3683, 2005. [29] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “Rectangular waveguide with dielectric-filled corrugations supporting backward waves,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 3298–3304, 2005. [30] F. Y. Meng, Q. Wu, J. H. Fu, X. M. Gu, and L. W. Li, “An anisotropic metamaterial-based rectangular resonant cavity,” Appl. Phys, A, vol. 91, pp. 573–578, 2008. [31] J. Carbonell, L. J. Rogla, V. E. Boria, and D. Lippens, “Design and experimental verification of backward-wave propagation in periodic waveguide structures,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 1527–1532, 2006. [32] A. M. Attiya, A. A. Kishk, and A. W. Glisson, “Analysis of two-dimensional magneto-dielectric grating slab,” Progr. Electromagn. Res., vol. 74, pp. 195–216, 2007.

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[33] P. A. Belov and C. R. Simovski, “Subwavelength metallic waveguides loaded by uniaxial resonant scatterers,” Phys. Rev. E, vol. 72, p. 036618(11), 2005. [34] A. Alu, F. Bilotti, N. Engheta, and L. Vegni, “Theory and simulations of a conformal omni-directional subwavelength metamaterial leakywave antenna,” IEEE Trans. Antennas Propag., vol. 55, pp. 1698–1708, 2007. [35] F. P. Casares-Miranda, C. Camacho-Penalosa, and C. Caloz, “High-gain active composite right/left-handed leaky-wave antenna,” IEEE Trans. Antennas Propag., vol. 54, pp. 2292–2300, 2006. [36] T. Ueda, K. Horikawa, M. Akiyama, and M. Tsutsumi, “Nonreciprocal phase-shift composite right/left handed transmission lines and their application to leaky wave antennas,” IEEE Trans. Antennas Propag., vol. 57, pp. 1995–2005, 2009. [37] T. Ueda, N. Michishita, M. Akiyama, and T. Itoh, “Dielectric-resonator-based composite right/left-handed transmission lines and their application to leaky wave antenna,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 2259–2269, 2008. [38] R. Marques, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B, vol. 65, p. 144440(6), 2002. [39] R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag., vol. 51, pp. 1516–1529, 2003. [40] D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E, vol. 71, p. 036617(11), 2005. [41] D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B, vol. 65, p. 195104(5), 2002. [42] A. F. Starr, P. M. Rye, D. R. Smith, and S. Nemat-Nasser, “Fabrication and characterization of a negative-refractive-index composite metamaterial,” Phys. Review B, vol. 70, p. 113102(4), 2004. [43] K. L. Tsakmakidis, M. S. Wartak, J. J. H. Cook, J. M. Hamm, and O. Hess, “Negative-permeability electromagnetically induced transparent and magnetically active metamaterials,” Phys. Rev. A, vol. 81, p. 195128(11), 2010. [44] F.-Y. Meng, Q. Wu, B.-S. Jin, H.-L. Wang, and J. Wu, “Comment on ‘Waveguide miniaturization using uniaxial negative permeability metamaterial’,” IEEE Trans. Antennas Propag., vol. 55, pp. 1016–1017, 2007. [45] M. G. Silveirinha, A. Alu, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E, vol. 75, p. 036603(16), 2007. [46] N. Liu, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Plasmonic building blocks for magnetic molecules in three-dimensional optical metamaterials,” Adv. Mater., vol. 20, pp. 3859–3865, 2008.

Qun Wu (SM’05) received the B.Sc. degree in radio engineering, thr M.Eng. degree in electromagnetic fields and microwave technology, and the Ph.D. degree in communication and information systems engineering, all at Harbin Institute of Technology (HIT), Harbin, China, in 1977, 1988, and 1999, respectively. He worked as a Visiting Professor at Seoul National University (SNU), Korea, from 1998 to 1999, and Pohang University of Science and Technology, from 1999 to 2000. Since 1990, he has been with Department of Electronic and communication Engineering at HIT, China, where he is currently a Professor. He has published over 30 international and regional refereed journal papers. His current research interests are in microwave active circuits, electromagnetic compatibility, MMIC, and millimeterwave MEMS devices. Dr. Wu received two Third-Class Prizes and one Second-Class Prize of Scientific Progress Awards from the Ministry of Aerospace of China in 1989 and 1992, respectively.

Fan-Yi Meng (M’07) received the B.S., M.S., and Ph.D. degrees in electromagnetics from Harbin Institute of Technology, Harbin, China, in 2002, 2004 and 2007, respectively. Since Aug. 2007, he has been with the Department of Microwave Engineering at the Harbin Institute of Technology where he is currently an Assistant Professor. He has (co-)authored 3 books, 30 international refereed journal papers, over 20 regional refereed journal papers, and 20 international conference papers. His current research interests include electromagnetic and optical metamaterials, plasmonics, and EMC. Dr. Meng was the recipient of a number of awards including the 2010 Award of Science and Technology from the Heilongjiang Province Government of China, the 2010 “Microsoft Cup” IEEE China Student Paper Contest Award, two best paper awards from the National Conference on Microwave and Millimeter Wave in China, in 2009 and 2007, respectively, the Young Scientist Travel Grant (YSTG) from the International Symposium on Antennas and Propagation in Japan, in 2007, the 2007 Excellent Graduate Award of Heilongjiang Province of China, and the Outstanding Doctor Degree Dissertation Award of Harbin Institute of Technology.

Le-Wei Li (F’10) received the B.Sc. degree in physics from Xuzhou Normal University (XNU), Xuzhou, China, in 1984, the M.Eng.Sc. degree in electrical engineering from China Research Institute of Radiowave Propagation (CRIRP), Xinxiang, China, in 1987, and the Ph.D. degree in electrical engineering from Monash University, Melbourne, Australia, in 1992. Since 1992, he has been with the Department of Electrical & Computer Engineering, National University of Singapore, where he is currently a Professor and the Director of NUS Centre for Microwave and Radio Frequency. From 1999 to 2004, he was seconded to the High Performance Computations on Engineered Systems (HPCES) Programme of Singapore-MIT Alliance (SMA) as an SMA Faculty Fellow. His current research interests include electromagnetic theory, computational electromagnetics, radio wave propagation and scattering in various media, microwave propagation and scattering in tropical environment, and analysis and design of various antennas. In these areas, he has (co-)authored 3 books, 45 book chapters, over 280 international refereed journal papers, 31 regional refereed journal papers, and over 320 international conference papers.

Daniel Erni (S’88–M’93) received the diploma degree in electrical engineering from the University of Applied Sciences, Rapperswil (HSR), Switzerland, in 1986 and the diploma degree in electrical engineering and the Ph.D. degree from ETH Zürich, Zürich Switzerland, in 1990 and 1996, respectively. Since 1990, he has been working at the Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zürich. From 1995 to 2006, he was the Founder and head of the Communication Photonics Group, ETH Zürich. Since Oct. 2006, he is a Full Professor for General and Theoretical Electrical Engineering at the University of Duisburg-Essen, Germany (http://www.ate.uni-due.de/). His current research includes advanced data transmission schemes (i.e. O-MIMO) in board-level optical interconnects, optical on-chip interconnects, ultra-dense integrated optics, nanophotonics, plasmonics, electromagnetic and optical metamaterials, and quantum optics. On the system level he has pioneered the introduction of numerical structural optimization into dense integrated optics device design. He is a member of the editorial board of the Journal of Computational and Theoretical Nanoscience and edited the 2009 Special Issue on Functional Nanophotonics and Nanoelectromagnetics. Dr. Erni is a Fellow of the Electromagnetics Academy, a member of the Center for Nanointegration Duisburg-Essen (CeNIDE), an associate member of the Swiss Electromagnetics Research Centre (serec), as well as a member of the Swiss Physical Society (SPS), the German Physical Society (DPG), and of the Optical Society of America (OSA).

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Anomalous Negative Group Velocity in Coupled Positive-Index/Negative-Index Guides Supporting Complex Modes Hassan Mirzaei, Student Member, IEEE, Rubaiyat Islam, Member, IEEE, and George V. Eleftheriades, Fellow, IEEE

Abstract—Anomalous negative group velocity (NGV; group velocity antiparallel to the power flow) is reported in a guided-wave structure supporting complex modes. This structure consists of a coupled-line system comprising a positive-index microstrip line edge coupled to a negative-index line. The NGV can be observed on the positive-index line under suitable excitation and termination conditions. What is remarkable about this structure is that the anomalous NGV neither requires any material losses nor any strong reflections on the observed line. This work verifies that absorption or reflection are not necessary conditions for observing NGV, rather NGV can be observed in specific lossless coupled-line structures as well. The general conditions for obtaining NGV in coupled positive-index/negative-index guides are analytically derived and corresponding NGV observations are experimentally reported at microwave frequencies. Specifically, we report the propagation of modulated pulses exhibiting a negative group delay as well as phase shifters maintaining a constant phase over a broad bandwidth. This opens up the possibility of utilizing such coupled-line guides for applications in pulse shaping, delay control, constant phase shifters (vs. frequency) and reducing beam squinting in series-fed antenna arrays. Index Terms—Anomalous dispersion, backward waves, complex modes, coupled-mode theory, metamaterials, negative group delay (NGD), negative group velocity (NGV).

I. INTRODUCTION

P

ROPAGATION of electromagnetic waves in dispersive media can lead to counterintuitive phenomena including abnormal phase and group velocities [1]. Abnormal group velocities include superluminal ones (velocities faster than the speed of light in vacuum) and even negative group velocities (NGVs). A negative group velocity, , and hence a negative ) implies that the peak of a group delay (NGD, pulse envelope appears to exit the medium before the peak even enters that medium. However, turn on and off points of the wave packet propagate with a positive delay at the output in agreement with causality requirements. Over the years, negative

Manuscript received October 25, 2010; revised January 25, 2011; accepted February 23, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 2E4, Canada (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161558

group velocities have been theoretically and experimentally studied at different bands ranging from microwave to optical frequencies (for example see [2]–[9]). Starting with the original experimental demonstration of NGV [3], the corresponding structures where NGV propagation takes place, either rely on material losses (e.g., absorption lines, resistors, metal losses) or operate in conventional bandgaps thus leading to strong reflection of the incident power in the observed medium. These observations have led to a widely accepted conclusion over the years that passive structures exhibit abnormal group velocities only in association with material losses (e.g., operation close to absorption lines) or reflection. Although, based on the Kramers-Kronig relations, absorptive and reflective systems are required to have spectral regions of abnormal group velocities [4], being absorptive or reflective is not a necessary condition for a structure to exhibit those properties [10]. Indeed, it has been shown in [10] that exchange or transfer of energy between two interacting modes in a passive structure is another way for observing abnormal group velocities. In [10], a birefringent photonic crystal with bandgaps at microwave frequencies has been used. A linearly-polarized incident wave with components along both the fast and slow axes of a photonic crystal (and in the frequencies far from the bandgaps) passed through the photonic crystal. It has been demonstrated that, as a result of interference between these wave components (rather than absorption or reflection), abnormal group velocities could be observed. In this work, we investigate a guided-wave structure, composed of coupled positive-index/negative-index guides, which is essentially lossless1 and does not reflect the incident wave in the excited and observed guide, yet the wave propagating on the positive-index guide exhibits a NGV [11]. The structure is the microstrip form of coupled positive-index/negative-index guides and we call it the microstrip/negative-refractive-indextransmission line (MS/NRI-TL) coupler [12]. In addition to the frequency-domain experimental results for a fabricated NGV circuit based on MS/NRI-TL couplers presented in [11], an experiment verifying directly the presence of NGD in the time domain is presented. Moreover, the theory in [11] is considerably expanded to analytically show that, in a general system of two coupled positive-index/negative-index guides, propagation with NGV characteristics can exist on the positive-index guide. Furthermore, the condition to observe such a phenomenon is analytically derived. Unlike [10], the observation of NGV in this 1Although in practice every microstrip structure has some loss, the NGV phenomenon discussed in this paper neither requires nor depends on loss.

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MIRZAEI et al.: ANOMALOUS NGV IN COUPLED POSITIVE-INDEX/NEGATIVE-INDEX GUIDES SUPPORTING COMPLEX MODES

work takes place in association with the propagation of peculiar eigenomodes called “complex modes” which are excited in MS/NRI-TL couplers, as was shown for the first time in [12]. Complex modes are remarkable eigenmodes of guided-wave structures and their existence has been reported in dielectric loaded waveguides [13], isotropic planar plasma layers [14], isotropic dielectric waveguides [15], shielded printed lines [16] and recently MS/NRI-TL couplers [12], as well as plasmonic optical couplers [17]. Complex modes exhibit both exponential decay and phase progression in the axial direction of the guide, although the corresponding guides are in principle lossless. The net power carried by each complex mode is zero [18]. However, there can be local power flow within the structure. In the structure studied here, composed of positive-index/negative-index guides, there is power flow on each guide separately, although the net power flow across every axial cross section is zero. The paper is organized as follows. Section II summarizes the main characteristics of waves propagating with a NGV and the corresponding dispersion of the media in which such waves propagate. Furthermore, the presence of NGV in the complex-mode band of positive-index/negative-index coupled guides is analytically investigated. In Section III, we present guidelines for designing MS/NRI-TL couplers. In addition, a typical dispersion diagram which illustrates the excitation of complex modes and the corresponding negative group velocity regime is discussed. After the key discussion on the possibility of the dominant excitation of a single complex mode exhibiting a NGV in a semi-infinite MS/NRI-TL coupler on the MS line, the design of a matched boundary condition is described in Section IV by which a finite-length coupler can be treated as a semi-infinite one. Section V demonstrates the experimental observation of NGV in a fabricated MS/NRI-TL coupler in both the frequency and time domains. Concluding remarks are presented in Section VI.

II. NGV IN COUPLED POSITIVE-INDEX/NEGATIVE-INDEX GUIDED-WAVE STRUCTURES To set the stage for the discussion about the NGV in coupled positive-index/negative-index guided-wave structures, consider a dispersion diagram for the eigenmodes of a structure at some frequency band having the form shown in Fig. 1(a). The structure is assumed to have axial symmetry along the z-axis so that the axial dependence of the propagating fields can be expressed . Since the propagation constant and the group veby have opposite signs, such a dispersion dialocity gram can be related to a backward wave (as in [19]) for which the direction of energy flow (i.e., Poynting vector ) is identical as shown in Fig. 1(b). to the direction of the group velocity On the other hand, such a dispersion diagram could also correspond to an eigenmode having abnormal NGV. If a guide within the structure having such a dispersion relation only supports forward waves, then the direction of and should be the same while would be antiparallel with them as depicted in Fig. 1(c). When the Poynting vector is contra-directional to the group velocity, the propagating wave will exhibit an abnormal NGV.

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Fig. 1. (a) A typical dispersion diagram which can be representing a backward wave or a wave with NGV. (b) In propagation of a backward wave, the propagation constant is antiparallel to both the Poynting vector and the group velocity. (c) In propagation of a wave with NGV, the group velocity and Poynting vector are antiparallel.

Fig. 2. (a) A MS/NRI-TL coupler unit cell. (b) Dimensions of the designed coupler at 900 MHz. (c) The dispersion diagram of the isolated MS line ( ) and NRI transmission line ( ) intersect at a frequency ! ; the NRI transmission line is designed to have a closed stopband at ! .

In this work, using coupled-mode theory, we show that in an axial guided-wave structure composed of lossless coupled positive-index/negative-index guides there can exist complex modes at part of the frequency spectrum. One of these complex modes has a dispersion characteristic similar to the one in Fig. 1(a). Related to that mode, it is possible to observe propagation with a NGV on the positive-index line, which only supports forward waves, under suitable excitation conditions. This is a peculiar and unique phenomenon since the structure is in principle lossless and there are no strong reflections on the line where the NGV is observed. A specific structure of this kind is a MS/NRI-TL coupler which consists of a microstrip line edge coupled to a negative-refractive-index line (i.e., a microstrip line periodically loaded with series capacitors and shunt inductors) a unit cell of which is shown in Fig. 2(a) [12]. An optical implementation of this coupler using plasmonic guides has been reported in [17]. Let unit vector be the axial direction of the coupler. Also, let the axial and temporal dependence of the fields be denoted where is the complex frequency and by is the complex propagation constant. In general, the linear coupling between transmission lines with voltage

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and current vector

can be described by: (1)

In the case of lossless and reciprocal lines, the and matrices are symmetric, purely imaginary on the imaginary ) and analytic in the right-half plane. For , axis ( the eigenvalues of (1) are obtained from the following characteristic equation:

arc in the right-half plane which circumvents the branch-point. When is positive, it is likely that the angle of the right-hand . To see this, one can make side of (5) will be less than so small such that setting in the denominator of (5) will result in negligible error. . If the angle of We can write is less than , then is negative implying that the group velocity is negative as well. Letting and in (3), the frequency derivative of in the limit of small coupling can be expressed as (6)

(2) and in (2) are real along the The rational functions imaginary axis. We notice that the eigenmodes of a lossless system can be complex if the radicand in (2) is negative. The limiting form of (2) in the absence of coupling is

Considering the coupling between a ( ) and a backward wave ( is positive provided that

forward-wave ),

(3)

(7)

are the propagation constants of the two In this equation, isolated lines. Comparing (2) and (3), it is possible to anticipate the existence of the complex modes when the propagation constants of the isolated modes are of equal magnitude. From (2) it can be verified that when is complex on the imaginary axis, the rest of the eigenvalues are and . Of these four solucorresponding to the tions, only two are bounded at proper modes of the semi-infinite problem in which the lines to . We will refer to the solutions stretch from in this proper pair as the and modes. ( ) denote the critical frequency between Let the excitation of regular propagating modes and complex modes in the system — i.e., is purely imaginary for lower than and complex above it. Hence, the radicand in (2) changes sign about this frequency point implying that . Let ( ) be the eigenvalue of one of the proper . On this particular Riemann modes at the branch-point sheet and in the vicinity of the branch-point, shown in Fig. 10, we can expand (2) as

This is always possible in practice and in the design of a MS/NRI-TL coupler this condition is automatically satisfied at the lower edge of the complex-mode band where this equation is evaluated (for details, see the Appendix B).

(4) branch point and a descripA discussion about the tion of the corresponding Riemann surfaces can be found in Appendix A. In (4), is a real quantity equal to the negative of the slope of along the imaginary axis. The constant is real and poschanges from itive so that when the angle of to , the eigenmode transforms from being purely imaginary to one which is complex with a positive real part. We can ( ) now differentiate (4) at a point slightly above the branch-point frequency to obtain (5)

At , must have positive real and imaginary parts when it is analytically continued on the imaginary axis through an

III. DESIGN GUIDELINES AND DISPERSION PROPERTIES OF MS/NRI-TL COUPLERS In this section, a particular guided-wave system of the form studied in the previous section is investigated which consists of a microstrip (MS) transmission line coupled to a negative-refractive-index (NRI) transmission line. In Particular, a MS/NRI-TL coupler is designed around 900 MHz using the TMM4 laminate from Rogers Corp. as a substrate with a dielectric constant equal to 4.5 and height of 1.27 mm. A. Design of the MS/NRI-TL Coupler First, guidelines for the design of unit cell of a MS/NRI-TL coupled structure are presented [20]. Such a unit cell is depicted in Fig. 2(a) where the width of the transmission lines, their physical length, the spacing between them and the value of the loading elements have to be determined. The width of the transmission lines, and , are simply selected to result in a characteristic impedance equal to , the characteristic impedance of the system (here 50 ). The dispersion diagrams of the isolated MS line ( ) and the NRI transmission line ( ) are shown in Fig. 2(c). In this figure, is a straight line and is obtained from the following equation: (8) where and are the per-unit-length inductance and capacitance of the MS line respectively. The value of at a given frequency can be obtained by a commercial software or using the approximate closed-form formulas for the microstrip effective dielectric constant [21]. In Fig. 2(c), the NRI transmission line dispersion diagram . This happens when the loading shows a closed stopband at

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components of the NRI transmission line, and , are selected such that the matching condition for the NRI-TL is satisfied, that is [22] (9) For the NRI transmission line, when the stopband is closed and the “effective medium” approximation is valid, its dispersion relation takes a simple form [23] (10) From this equation the frequency of the closed stopband, in Fig. 2(c), can be calculated by letting (11) The dispersion diagrams of the isolated MS line and the NRI (i.e., at transmission line intersect at a frequency ) as shown in Fig. 2(c). This is the frequency around which the ). The “efcomplex-mode band is formed ( here fective medium” approximation for the NRI transmission line , where is the guided wavelength, is valid when or equivalently when . To make this approximation valid, the unit cells of the MS/NRI-TL coupler have an at and from here the physical electrical length of about , is calculated. length of the unit cell, Evaluating (10) at using the fact that at , , as well as employing (9), the value of the NRI and , can be caltransmission-line loading components, culated. The spacing between the MS line and the NRI transmission line, , determines the bandwidth of the complex-mode band. The closer the lines, the larger the coupling and the broader the complex-mode band. The parameters of the designed unit cell are shown in Fig. 2(b). B. Dispersion Diagram of the MS/NRI-TL Coupler According to the Appendix B, for a MS/NRI-TL coupler, (7) is automatically satisfied and one of the modes in the complexmode band will have a desired dispersion characteristic of the form shown in Fig. 1(a). The dispersion diagram of the designed coupler is shown in Fig. 3. The corresponding complex-mode band is created around the target frequency. A coupler of arbitrary length can be created by cascading those unit cells. As shown in Fig. 3, the complex-mode band extends from 750 to 1010 MHz for a gap of 0.23 mm between the coupled lines. By assuming a small amount of material loss, the proper . eigenmodes are those which remain bounded as There are two eigenmodes at each frequency which are labeled as the and mode in Fig. 3. These are the modes that will axis. The slope of the reach an observer located along the -mode changes sign as it enters the complex-mode band. In this band, the -mode has a dispersion diagram similar to the one previously discussed in Fig. 1(a) and can correspond to either a backward wave or a wave with a NGV when observed on a transmission line supporting forward waves. Since the MS

Fig. 3. Dispersion diagram of the MS/NRI-TL coupler showing all eigenmodes. The proper c and  -modes are highlighted.

line only supports forward waves, excitation of the -mode in the structure and its propagation on this line2 results in a NGV in agreement with the discussion made for the generic case in Fig. 1(a). Nevertheless, for observing NGV there should exist a mechanism to dominantly excite the appropriate -mode. In general, the superposition of the and -modes results in properties which are different from those of the individual ones. Despite the common belief over the years that complex modes cannot be excited independently of each other, it has been recently shown in [24] that the independent excitation of one mode in a complex pair is possible over a discrete set of frequencies (or dominantly excited over a finite bandwidth). For the case of a semi-infinite MS/NRI-TL coupler the method to excite the -mode ( -mode) amounts simply to connect the source to the MS (NRI) line. IV. MATCHED BOUNDARY CONDITION FOR FINITE-LENGTH MS/NRI-TL COUPLERS The possibility of exciting individual modes in a MS/NRI-TL coupler paves the way for the experimental observation of NGV on the MS line in these structures, but, to that end one should tackle the problem of the semi-infinite coupler requirement as stated above. This can be resolved by providing a matched boundary condition for the guided-wave structure at one end [25]. For the MS/NRI-TL coupler, this can be realized by terminating the coupler at one end with a suitable two-port network as can be seen in Fig. 4(a). Using such a termination, the structure effectively acts as a semi-infinite one. On line 1 (MS line), it is assumed that there is a traveling-wave voltage (and current) which is a superposition of the corresponding and -modes. Therefore, the voltages and currents can be related by the following equation:

(12)

In this equation, are the characteristic impedances are the ratios of the of the and -modes on each line and 2Note that, when the c-mode is excited in a system, it carries no net power across every cross section of the system. However, the excitation of this mode in the structure excites currents and voltages on both lines and there is nonzero power flow on each line.

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Fig. 6. Group delay at port 6 (the sampling port at the end of the MS line in the fabricated MS/NRI-TL coupler) compared to that of an isolated microstrip line of the same length.

Fig. 4. (a) MS/NRI-TL coupler with a two-port termination providing the matched boundary condition for the eigenmodes. (b) Fabricated coupler in which the sampling ports and two-port termination can be observed.

Fig. 5. Transmission phase at output ports. The positive slope of the phase response indicates the existence of NGV.

modal voltages. It can be shown that [26], [27] these parameters are not independent and for a general system of two coupled . lines are related by Assuming that the -mode is excited on the coupler, we select the two-port termination parameters such that no reflection takes place and no conversion to the other mode happens. This implies that the same relation as the modal one should hold between the terminal voltages and currents of the two-port termination. Using the parameters for the two-port network representation we can write (13) The two-port network is lossless and reciprocal and therefore the parameters are purely imaginary and . Also we define the real and imaginary parts of the modal parameters , and by . Equating the real and imaginary parts of (13) gives rise to four equations and because of reciprocity, there , and ). If one are only three unknowns (i.e., selects three equations and solves for the -parameters of the termination, it can be easily shown, using the modal relations,

Fig. 7. Time-domain test setup for measuring the group delay of a device under test (DUT). (a) Block diagram. (b) Image of the test setup.

that the solution also satisfies the unused equation

(14) The two-port termination for the -mode can be determined using the same method, however, using the fact that in the comand [12], [24], it is clear plex mode region from (14) that the required termination for both modes is exactly the same. V. EXPERIMENTAL OBSERVATION OF NGV ON THE MS LINE IN A MS/NRI-TL COUPLER By fabricating the designed MS/NRI-TL coupler, the propagation with NGV on the MS line, when the -mode is independently excited in the system, can be experimentally demonstrated. The wave propagating in the MS/NRI-TL coupler is

MIRZAEI et al.: ANOMALOUS NGV IN COUPLED POSITIVE-INDEX/NEGATIVE-INDEX GUIDES SUPPORTING COMPLEX MODES

Fig. 8. Time-domain experimental results showing NGD for the MS/NRI-TL coupler (sampled out on the MS line) and PGD for an isolated microstrip line of the same length shown both in an expanded (left) and a reduced (right) time interval.

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Fig. 9. Constant phase-shift circuit obtained by adding transmission lines to the negative group velocity circuit. (a) Modified circuit. (b) Phase at P4A. (c) Phase at P5A (d) Phase at P6A.

sampled out along the microstrip line. The number of the sampling points can be equal to the number of the unit cells (here 16), however, for the proof of concept, only four equispaced sampling points have been selected on the MS line. Sampling is done with a transistor having a very high input impedance and a very low input capacitance. This transistor acts as a buffer and therefore, has minimum perturbation on the propagating wave. The fabricated coupler can be seen in Fig. 4(b). A. Frequency-Domain Results for NGV in MS/NRI-TL Couplers The phase response of the four outputs with respect to the input can be seen in Fig. 5 in both simulation and measurement. The simulations are performed using Agilent’s Advanced Design System (ADS). The measurements are carried out by an Agilent E8364B Network Analyzer. A good agreement between the measurement and simulation results can be observed in this figure. The positive slope of the phase response confirms the presence of NGD (and equivalently NGV) based on the rela. The NGD results can be seen in Fig. 6 for tion port 6 in Fig. 4(b). For the sake of comparison, the group delay of a microstrip line with the same length is also measured and shown in this figure. As can be seen, a group delay of 1.2 ns is measured for the microstrip line, while the group delay for the ( ). This reMS/NRI-TL circuit is about sult shows that a MS line coupled to a NRI-TL line exhibits a compared to an isolated microstrip line delay equal to due to the propagation of complex modes in the former case. B. Time-Domain Observation of NGD in MS/NRI-TL Couplers For further verification, a time-domain test is also conducted. The time-domain test setup can be seen in Fig. 7 showing a block diagram, as well as, a picture of the test setup and instruments used. A 770 MHz sinusoidal wave is modulated by a Gaussian pulse of full width half maximum (FWHM) equal to 10 nS and then divided by two by a power splitter. Half of the wave is connected by a coaxial cable to one channel of a high frequency oscilloscope as a reference and the other half is passed through the device under test (DUT) and applied to the other channel

Fig. 10. Branch-points (indicated by circles) that connect the four complex frequency Riemann sheets on which the two modes of the MS/NRI-TL coupler (and their reflected counterparts) are described. The crosses indicate poles, and the “ ” and “ ” superscripts denote proper and improper modes respectively. The dotted lines refer to mode conversion when analytically continuing around the first order branch-points they connect.

+

0

of the oscilloscope by a cable of the same type and length. The time-domain experiment results can be seen in Fig. 8. In this figure the reference signal, the output of the MS/NRI-TL coupler sampled out at the end of the MS line (port 6 in Fig. 4(b)) and the output of an isolated microstrip line of the same length are shown. Fig. 8 demonstrates that the peak of the Gaussian pulse is delayed in the microstrip line, whereas it is advanced for the MS/NRI-TL coupler with respect to the reference signal. C. Modified Circuit With Constant Phase-Shift Response In order to obtain a circuit with constant phase-shift response (i.e., zero group delay; infinite group velocity), the NGD circuit in Fig. 4(b) can be cascaded with a transmission line having a positive group delay (PGD). The modified circuit in which transmission lines are added to the P4, P5 and P6 ports can be seen in Fig. 9(a). The results depicted in Fig. 9(b) to (d) exhibit a good agreement between experiment and simulation. As can be seen in Fig. 9(a), the length of the transmission lines

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with PGD, added to the output ports with NGD, increase as one moves toward the end of the coupler; meaning that a larger PGD is required for compensating the NGD as the distance from the beginning of the coupler increases. This is because the excitation of the -mode in the structure and its propagation along the , where is MS line results in a phase shift equal to the distance from the beginning of the coupler. From this, the . group delay can be calculated as This equation implies that the group delay is negative because is negative when referring to the dispersion diagram in Fig. 3. In addition, the NGD is proportional to , the distance from the beginning of the coupler. Therefore, as one moves away from the input on the MS line, a longer NGD is incurred and correspondingly a longer PGD is required to compensate that. By properly choosing the electrical length of the coupler unit cells, any value of the phase shift with any step can be achieved all within the same circuit. VI. CONCLUSION In summary, we have analytically and experimentally studied the propagation of eigenmodes in a MS/NRI-TL coupler as an example of a coupled positive-index/negative-index axial guide which is capable of supporting abnormal NGV on the positiveindex guide. Despite the fact that NGV is usually observed in association with material losses or strong reflections in guidedwave structures, the structures studied in this paper are essentially lossless and there is no strong reflection in the guide where the NGV is observed. The key attribute of these guided-wave structures, which allows this kind of NGV phenomenon, is the dominant excitation of a single mode in a complex-mode pair (which was thought not to be possible in the past [24]). A particular MS/NRI-TL coupler supporting complex modes has been fabricated and the microwave propagation corresponding to NGV on the MS line has been experimentally demonstrated both in the frequency and time domains. A modified circuit exhibiting a constant phase-shift response over a broad bandwidth has been presented as well. This study opens up the possibility of utilizing such coupled positive-index/negative-index guides for applications in pulse shaping, delay control and in phase shifters with a constant phase-shift response vs. frequency. In addition, the NGD in these coupled structures can be utilized in the feed network of series-fed antenna arrays to reduce beam squinting (by using zero-phase broadband phase shifters) [28]–[30].

ical points). Such a multi-valued function is conveniently described through the construction of a four-sheeted Riemann surface as depicted in Fig. 10. This frequency domain representation allows one to map each of the four imaginary axes in Fig. 10 to one of the traces in Fig. 3. A careful study (which is beyond the scope of this paper) of (2) is required to arrive at the exact arrangement of the various sheets. The Riemann surface depicted is an oversimplification of the actual topography where the sheets smoothly merge into each other for every circumnavigation around the common branch-points. The modes are defined for complex frequencies on this smooth surface and all traces in Fig. 3 are connected together analytically as different segments of a single-valued function whose domain is the multi-sheeted Riemann surface. When we approximate (2) with (4), we ensure that a continuous assignment of values that circle the branch-point at in Fig. 10, results in the conmode to the mode and vice-versa. The version of the branch-point that appears on the right-hand side of the imaginary axis does not result in system instability or non-causality [24]. The Riemann surface serves as an essential visual tool that aids in the computation of the time domain response through an inverse Laplace transform integral. APPENDIX B EVALUATION OF THE NGV CONDITION IN MS/NRI-TL COUPLERS In Section II, it was shown that in a system of coupled positive-index/negative-index guides, a complex-mode band is formed and the dispersion diagram for one of the proper modes of this system, namely the -mode, exhibits a negative slope, provided that (7) is satisfied at the lower edge of the complex-mode band. In this appendix, the validity of a (7) in a MS/NRI-TL is evaluated. The dispersion relations for the isolated MS line and the NRI transmission line (with the enforced “matching condition” and under the “effective medium approximation”) can be seen in (8) and (10). Using these two equations, the constituent sides of (7) can be formed (A.15) (A.16) In order to satisfy (7), one should have (A.17)

APPENDIX A RIEMANN SURFACES AND CORRESPONDING BRANCH-POINTS OF THE EIGENMODES IN A MS/NRI-TL COUPLER As stated in Section II, from the four solutions of (2), only two of them correspond to the propagation constant of the proper and modes in a semi-infinite MS/NRI-TL coupler which extends from to . This can be visualized in the dispersion diagram of the MS/NRI-TL coupler depicted in Fig. 3. The four traces that appear in Fig. 3 assign four values of to each frequency point (except for a denumerable set of crit-

For the dimensions in Fig. 2(b) and for the TMM4 substrate, and for the host one can find and from the same MS line. Calling the values of , and at frequencies figure, the condition of (7) is satisfied. As can be seen in the dispersion diagram in Fig. 3 and also in the simulation and the experimental results in Fig. 5, the entire complex-mode band including its lower edge lie below this frequency. Nevertheless, in the design of a MS/NRI-TL coupler this condition is automatically satisfied. To show this, recall that at ,

MIRZAEI et al.: ANOMALOUS NGV IN COUPLED POSITIVE-INDEX/NEGATIVE-INDEX GUIDES SUPPORTING COMPLEX MODES

the center frequency of the complex-mode band, using (8) and (10) one can find

and

(A.18) This means that the center of the complex-mode band, hence, also its lower edge, automatically satisfy (A.17) and (7).

REFERENCES [1] L. Brillouin, Wave Propagation and Group Velocity. New York: Academic Press, 1960. [2] C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A, vol. 1, no. 2, pp. 305–313, Feb. 1970. [3] S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett., vol. 48, no. 11, pp. 738–741, Mar. 1982. [4] E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A, vol. 48, no. 5, pp. 3890–3894, Nov. 1993. [5] R. W. Ziolkowski, “Superluminal transmission of information through an electromagnetic metamaterial,” Phys. Rev. E, vol. 63, no. 4, p. 046604, Mar. 2001. [6] G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science, vol. 312, no. 5775, pp. 892–894, 2006. [7] O. Siddiqui, M. Mojahedi, and G. V. Eleftheriades, “Periodically loaded transmission line with effective negative refractive index and negative group velocity,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2619–2625, Oct. 2003. [8] S. Lucyszyn and I. D. Robertson, “Analog reflection topology building blocks for adaptive microwave signal processing applications,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 3, pp. 601–611, Mar. 1995. [9] M. Kitano, T. Nakanishi, and K. Sugiyama, “Negative group delay and superluminal propagation: An electronic circuit approach,” IEEE J. Sel. Topics Quantum Electron., vol. 9, no. 1, pp. 43–51, Jan.-Feb. 2003. [10] D. R. Solli, C. F. McCormick, C. Ropers, J. J. Morehead, R. Y. Chiao, and J. M. Hickmann, “Demonstration of superluminal effects in an absorptionless, nonreflective system,” Phys. Rev. Lett., vol. 91, no. 14, p. 143906, Oct. 2003. [11] H. Mirzaei and G. V. Eleftheriades, “Negative and zero group velocity in microstrip/negative-refractive-index transmission-line couplers,” in Proc. IEEE MTT-S Int. Microwave Symp. Digest, MTT’10, 2010, pp. 37–40. [12] R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow,” Electron. Lett., vol. 40, no. 5, pp. 315–317, Mar. 2004. [13] P. J. B. Clarricoats and K. R. Slinn, “Complex modes of propagation in dielectric-loaded circular waveguide,” Electron. Lett., vol. 1, no. 5, pp. 145–146, Jul. 1965. [14] T. Tamir and A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE, vol. 51, no. 2, pp. 317–332, Feb. 1963. [15] T. F. Jablonski, “Complex modes in open lossless dielectric waveguides,” J. Opt. Soc. Am. A, vol. 11, no. 4, pp. 1272–1282, 1994. [16] M. J. Freire, F. Mesa, and M. Horno, “Excitation of complex and backward mode on shielded lossless printed lines,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 7, pp. 1098–1105, Jul. 1999. [17] Y. Wang, R. Islam, and G. V. Eleftheriades, “An ultra-short contra-directional coupler utilizing surface plasmon-polaritons at optical frequencies,” Opt. Express, vol. 14, no. 16, pp. 7279–7290, 2006. [18] S. Laxpati and R. Mittra, “Energy considerations in open and closed waveguides,” IEEE Trans. Antennas Propag., vol. 13, no. 6, pp. 883–890, Nov. 1965. [19] M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett., vol. 92, no. 6, p. 063903, Feb. 2004.

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[20] R. Islam and G. V. Eleftheriades, “Printed high-directivity metamaterial MS/NRI coupled-line coupler for signal monitoring applications,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 4, pp. 164–166, Apr. 2006. [21] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2004. [22] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [23] M. A. Antoniades and G. V. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 103–106, 2003. [24] R. Islam and G. V. Eleftheriades, “On the independence of the excitation of complex modes in isotropic structures,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1567–1578, May 2010. [25] A. M. Belyantsev and A. V. Gaponov, “Waves with complex propagation constants in coupled transmission lines without energy dissipation,” Radio Eng. Electron. Phys. (USSR), vol. 9, pp. 980–988, 1964. [26] K. C. Gupta, R. Garg, I. J. Bahl, and P. Bhartia, Microstrip Lines and Slotlines. Norwood, MA: Artech House, 1996. [27] P. B. R. K. Mongia, I. J. Bahl, and J. Hong, RF and Microwave Coupled-Line Circuits, 2nd ed. Norwood, MA: Artech House, 2007. [28] G. V. Eleftheriades, M. A. Antoniades, and F. Qureshi, “Antenna applications of negative-refractive-index transmission-line structures,” IET Microw. Antennas Propag., vol. 1, no. 1, pp. 12–22, Feb. 2007. [29] S. S. Oh and L. Shafai, “Compensated circuit with characteristics of lossless double negative materials and its application to array antennas,” IET Microw. Antennas Propag., vol. 1, no. 1, pp. 29–38, Feb. 2007. [30] S. Keser and M. Mojahedi, “Removal of beam squint in series fed array antennas using abnormal group delay phase shifters,” presented at the IEEE Int. Symp. on Antennas and Propagation, Jul. 2010.

Hassan Mirzaei (S’07) received the B.Sc. degree (honors) in electrical and computer engineering from Isfahan University of Technology, Iran, in 1998 and the M.Sc. degree in electrical engineering from Sharif University of Technology, Iran, in 2000. He is currently working toward the Ph.D. degree at the University of Toronto, Toronto, ON, Canada. From 2001 to 2007, he worked as an RF and microwave Engineer at several Iranian telecommunication companies where he was involved in the design of microwave circuits and systems. From 2007 to 2008, he was a Research Assistant at the University of Waterloo, Waterloo, ON, Canada. His research interests include dispersion engineering, application of wave propagation and interaction theory to circuit design and circuit-enabled electromagnetic structures. Mr. Mirzaei is the recipient of a Natural Sciences and Engineering Research Council of Canada Postgraduate Scholarship (NSERC PGS D) from 2008 to 2011.

Rubaiyat Islam (S’08–M’11) received the B.A.Sc. degree in engineering science (electrical option) and the Ph.D. degree in electrical engineering (electromagnetics) from the University of Toronto, Toronto, ON, Canada, in 2002 and in 2011, respectively. He is currently a Sr. Analog Designer in the signal integrity group at Advanced Micro Devices, Inc. (AMD). His research interests include the network theory of guided modes, metamaterials, RF/microwave passive devices such as couplers, filters, phase-shifters and power distribution networks, microwave circuit miniaturization, coupled-mode theory and the theory of complex/leaky modes. Dr. Islam held the NSERC PGS D doctoral award from 2005 to 2008.

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George V. Eleftheriades (S’86–M’88–SM’02– F’06) received the Diploma degree in electrical engineering from the National Technical University of Athens, Athens, Greece, in 1988, and the Ph.D. and M.S.E.E. degrees in electrical engineering from The University of Michigan at Ann Arbor, in 1993 and 1989, respectively. From 1994 to 1997, he was with the Swiss Federal Institute of Technology, Lausanne, Switzerland, where he was engaged in the design of millimeter wave and sub-millimeter-wave receivers and the creation of fast computer-aided design (CAD) tools for planar packaged microwave circuits. He is currently a Professor with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada, where he holds the Canada Research/Velma M. Rogers Graham Chair in Engineering. He currently leads a group of talented graduate students in the areas of electromagnetic negative-refraction microwave and optical metamaterials, integrated circuit (IC) antennas and components for broadband wireless communications, novel an-

tenna beam-steering techniques, and electromagnetic design for high-speed digital circuits. Prof. Eleftheriades has served as an IEEE Antennas and Propagation Society (AP-S) Distinguished Lecturer (2004–2009) and as a member of the IEEE AP-S Administrative Committee (AdCom, 2008–2010). He is an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and a member of Technical Coordination Committee MTT-15 (Microwave Field Theory). He has been the General Chair of the 2010 IEEE Int. Symposium on Antennas and Propagation and CNC/USNC/URSI Radio Science Meeting held in Toronto. He was the recipient of the 2008 IEEE Kiyo Tomiyasu Technical Field Award, the 2001 Ontario Premiers’ Research Excellence Award, the 2001 Gordon Slemon Award, presented by the University of Toronto, and the 2004 E. W. R. Steacie Fellowship presented by the Natural Sciences and Engineering Research Council of Canada. He is the co-recipient of the inaugural 2009 IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS Best Paper Award. One of the papers he co-authored received the R.W.P. King Award in 2008. In 2009, he was elected a Fellow of the Royal Society of Canada.

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Communications The Half-Width Microstrip Leaky Wave Antenna With the Periodic Short Circuits Yuanxin Li, Quan Xue, Hong-Zhou Tan, and Yunliang Long

Abstract—A half-width microstrip leaky wave antenna (LWA) with periodic short circuits is presented. The backward-to-forward beam-scanning capability is achieved by periodic construction. The proposed antenna consists of a long rectangular patch with the short circuits that are placed with a series of shorting pins between the antenna patch and the ground plane periodically. The measurement results show that the main lobe scans elecditronically from 144 to 41 in H-plane ( - plane) toward endfire ( rection) with an open stopband when the operating frequency is increased from 4.4 GHz to 8.8 GHz.

+

Index Terms—Backward to forward, half-width microstrip leaky wave antenna, periodic.

I. INTRODUCTION Since 1979, the microstrip leaky wave antenna (LWA) has been studied and used in many applications because of its attractive properties, such as the beam-scanning capability, low profile, and ease of analysis by the multimode cavity model [1], [2]. The half-width microstrip LWA was studied by G.M. Zelinski in 2007 [3]. Compared with the conventional microstrip LWA, the half-width microstrip LWA has the advantages of reduced size with a similar beam-scanning capability. Both the conventional whole-width antenna and the half-width antenna can work in the first higher-order mode with positive phase constants. The frequency-scanning pencil beam in the H-plane scans from the broadside direction to the endfire direction with the increase of the operating frequency in the H-plane when 0 < z  z < k0 , where z is the attenuation constant, z is the phase constant, and kz = z 0 jz is the propagation constant [4], [5]. Several LWA designs with backward-to-forward beam-scanning capability have been reported in the past few years, owing to the positive or the negative phase constant [6], [7]. The backward-to-forward capability is easy to achieve by applying the composite right/left-handed (CRLH) metamaterial as the LWA with dominant mode [8], [9]. The fix-frequency beam-scanning antenna using the CRLH structure was reported in 2004 [10]. On the other hand, the power could leak as the space wave from the periodic waveguide structure, which has been applied as the antenna and called the periodic LWA. The periodic construction creates a guide wave that consists of an infinite number of the space harmonics (Floquet waves). With the appropriate structure of the radiating period, the period p and the space harmonics n, the periodic construction makes the slow wave radiate out along the radiating edge Manuscript received July 26, 2010; revised February 03, 2011; accepted February 09, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the Natural Science Foundation of China under Grant 60901028, the Fundamental Research Funds for the Central Universities, and in part by NSFC-Guangdong under Grants U0635003 and U0935002. Y. Li, H.-Z. Tan, and Y. Long are with the Department of Electronics and Communication Engineering, Sun Yat-sen University, Guangzhou, China (e-mail: [email protected]). Q. Xue is with the State Key Laboratory of Millimeter Waves, City University of Hong Kong, Kowloon, Hong Kong SAR, China. Digital Object Identifier 10.1109/TAP.2011.2161439

when j zn =k0 j < 1. The periodic construction makes the phase constant increase from negative to positive values with the increase of the frequency. The periodic LWA, in comparison to the uniform LWA, has the advantage of its main lobe scanning in backward direction. It consists of a rectangular waveguide that is loaded with a periodic array of holes in the narrow wall of the waveguide, and the constructions are complex [11], [12]. In their previous works, the authors presented a periodic half-width microstrip LWA whose main lobe is scanned from 149 to 28 in the H-plane (y -z plane) [13]. The antenna consists of a series of half-width microstrip LWA, which are uniform structures and placed alternatively on the different sides of a transmission line. The complex propagation constant kzn of the wave in this periodic half-width LWA is given by Floquet’s theorem [6]

kzn = zn 0 jz = kz + 2n d ; n = 61; 62; 63 . . .

(1)

where d is the period, n is the order of the space harmonics. j zn =k0 j < 1 when an appropriate period p is designed and the space harmonics n = 01. With an increase in the operating frequency, zn of the periodic half-width microstrip LWA varies from negative to positive values, and the main lobe of the periodic antenna scans from the backfire direction to the endfire direction. The attenuation constant z is the same for all space harmonics, and its value is the same as the attenuation constant of the fundamental Floquet harmonic. In (1), kz = z 0 jz is the approximate wave number of the fundamental Floquet wave in the uniform structure, which is the half-width microstrip LWA in [13]. The methods to obtain the complex propagation constant of the half-width microstrip LWA express are as follows [5], [14], [15]:

kz = !2 "r 0 kx2 kx 0 !y! exp(jkx 2W ) = 0 kx + !y! "r 1W y! = 120h0 + j k0120 

(2) (3) (4)

where 1W is the equivalent extension of the half-width microstrip LWA [16]

"e + 0:3 l=h + 0:262 "e 0 0:258 l=h + 0:813 a ln 1 0 42 r2 + 0:601 a2"r (5) + 2 r a2 20 0 5h "e = "r 2+ 1 + "r 20 1 1 + W (6) h is the thickness of substrate, l is the length of half-width microstrip LWA, a is the space between each of pins in the short circuits, r is the radius of shorting pins, and W is the width of the half-width microstrip 1W = 0:412h

LWA. Compared with the conventional periodic waveguide LWA, this design has the advantages of being a simple and compact structure. An improved half-width microstrip LWA with the periodic short circuits is now presented. The proposed antenna shown in Fig. 1 consists of a long rectangular patch with a series of short circuits, which make it

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Fig. 1. (a). The 3D view of the half-width microstrip LWA with the periodic short circuits. (b). The layout of the proposed half-width microstrip LWA. (L : mm, s : mm, T mm, W mm, l mm, l : mm, h : mm, " : ). mm, a mm, r

210

= 11 6 = 22 =3 =05 =08

= 15 = 21 = 2 65

= = 18

the radiating periodic structures. The far-field H-plane (y -z plane) radiation patterns of the proposed antenna have been shown. The measured and the calculated normalized complex propagation constant have been compared. The reflection coefficient of the antenna has been studied, too. II. DESIGN As shown in Fig. 1, a long rectangular patch was printed on the substrate board with relative dielectric constant, i.e., "r = 2:65, dielectric loss tangent, i.e., tan  = 0:0015 and thickness h = 0:8 mm. The patch was 210-mm long (L) and 11.6-mm wide (W ). In the feeding circuit, the width of the transmission line is 50 , i.e., s = 2:2 mm, and the length of the feeding line, i.e., T = 15 mm. The width of the proposed antenna is just a half of the one presented in [13]. The periodic construction consists of a series of the short circuits. The short circuits appear to be on two alternate edges of the long rectangular patch. Along the long edge of the patch, the antenna patch and the ground plane are connected by the shorting pins periodically. The dimensions of the radiating period with the shorting pins are as follows: l = a + l0 = 21 mm. Due to the power leakage, there should not be two shorting pins on two long radiating edges in the same location. So, there are only 7 shorting pins in each short circuit, except for the short circuit on the feeding terminal of the patch, which are with 8 shorting pins. The space between shorting pins is a = 3 mm, the radius of these pins is r = 0:5 mm. The design process of the proposed antenna is simple. The reduced size of this new design makes it cheaper. III. MEASURED RESULTS The radiation patterns of the half-width microstrip LWA with periodic short circuits are measured in the far-field condition. Fig. 2 shows the plots of the beam-scanning radiation patterns measured in y -z plane for the proposed half-width antenna at different frequencies. With the increase in the frequency of operation, the main lobe elevation steers from the backward direction to the forward direction in y -z plane. In Fig. 2(a), when the antenna is at 4.6 GHz, the main lobe directs at

Fig. 2. The measured radiation patterns of the proposed half-width microstrip LWA with the periodic short circuits at different frequency of operation, (a) : GHz, F GHz, and F : GHz, (b) copolarization patterns, F copolarization patterns, F : GHz, F : GHz, and F : GHz, (c) cross polarization patterns, F : GHz, F GHz, and F : GHz, (d) cross polarization patterns, F : GHz, F : GHz, and F : GHz.

= 46 =65 =46 =65

=5 =75 =5 =75

= 52 =85 =52 =85

 = 133 . At 5 GHz and 5.2 GHz, the measured scanning angles for the upper beams are  = 111 and  = 103 , respectively. When the operating frequency changes from 6.5 GHz to 8.5 GHz, the main lobe scans from  = 79 to  = 47 in the forward direction, as illustrated in Fig. 2(b). Due to the reflection from the open end of the patch, there are huge side lobes in the radiation patterns. The cross polarization radiation patterns at different frequency are shown in Fig. 2(c) and (d). The normalized phase constants zn =k0 and the attenuation constants z =k0 are illustrated in Fig. 3. The phase constant zn determines the direction of the main lobe. The attenuation constant z =k0 is a measurement of the power that leaks per unit length along the antenna. The measured phase constant zn =k0 is obtained by [4] zn k0

= sin 2 0 

(7)

where  is the measured angle of the maximum of the main lobe in y -z plane, elevated from the endfiring direction (+z direction). As shown in Fig. 2, the main lobe steers from the backfire direction to the forward direction with an increase of the operating frequency in the y -z plane. There is an open stopband from 5.3 GHz to 6.2 GHz, in which the radiation is hard to achieve. The measurement results agree closely with the simulating results obtained by (1), considering that the period d = 22l = 42 mm and the space harmonics n = 01. The measured z =k0 of the wave in this half-width antenna are given by [2]

z k0

= 0:18 1 

HPBW

1 cos 2 0 

(8)

where HPBW is the measured half-power beamwidth of the main lobe. Fig. 2 shows that the beamwidth of the main lobe linearly related to z =k0 . The attenuation constant drops near to zero when the phase

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Fig. 3. The attenuation constant  =k and the phase constant =k of the half-width microstrip LWA with the periodic short circuits. (d mm, n ), – – Calculated normalized phase constant  =k by (1), * Measured normalized attenuation constant  =k by (8).—Calculated normalized phase constant =k by (1), Measured normalized phase constant =k by (7), Measured normalized phase constant =k in [13].

= 42

01



=

1

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Fig. 5. The gain and the direction of main lobe against frequency, * The gain, Measured direction of the main lobe.



(+z direction) when the operating frequency increases from 4.4 GHz to 8.8 GHz. This new antenna design is simpler and cheaper than the conventional periodic antenna and the right/left-handed leaky wave antenna. This antenna design will be useful in manufacture of the automobile radar system or other consumer products.

REFERENCES

(S )

S

Fig. 4. The reflection coefficient against frequency,— of the halfwidth microstrip LWA with the periodic short circuits, – – of the periodic half-width microstrip LWA in [13].

S

constant is near to 0, which is called as the open stop-band point. In this region, the main lobe is difficult to obtain in the broadside direction. The reflection coefficient (S11 ) of this half-width microstrip LWA with a periodic short circuit is measured and shown in Fig. 4. The solid line is the S11 with respect to the operating frequency of the proposed antenna. The wide impedance bandwidth has been achieved. Due to the similar radiation characteristics, the S-parameter agrees closely with the one from the periodic half-width microstrip LWA present in [13]. The gain and the directive of the proposed antenna are plotted in Fig. 5. The open stopband is evident in the reflection coefficient (S11 ) and the gain from 5.3 GHz to 6.2 GHz. The directive of the antenna shows that the main lobe steers from the backward direction to the forward direction. IV. CONCLUSION This communication presents an improved half-width microstrip LWA with the periodic short circuits. The proposed antenna consists of a long rectangular patch, where the ground plane is connected by the short circuits periodically. The backward-to-forward beam-scanning radiation patterns are achieved by this simple periodic construction. The experimental results show the main lobe scans electronically and continuously from 144 to 41 in H-plane (y -z plane) toward endfire

[1] W. Menzel, “A new travelling-wave antenna in microstrip,” AEU, vol. 33, no. 4, pp. 137–140, 1979. [2] A. A. Oliner, “Leaky waves: Basic properties and applications,” APMC’97, vol. 1, pp. 397–400, Dec. 2–5, 1997. [3] G. M. Zelinski, G. A. Thiele, M. L. Hastriter, M. J. Havrilla, and A. J. Terzuoli, “Half width leaky wave antennas,” IET Proc.-Microw. Antennas Propag., vol. 1, no. 2, pp. 341–348, Apr. 2007. [4] Y. D. Lin and J. W. Sheen, “Mode distinction and radiation-efficiency analysis of planar leaky-wave line source,” IEEE Trans. Microwave Theory Tech., vol. 45, no. 10, pp. 1672–1680, Oct. 1997. [5] C. Luxey and J. M. Laheurte, “Simple design of dual-beam leaky-wave antennas in microstrips,” IEE Proc.-Microw. Antennas Propag., vol. 144, no. 6, pp. 397–402, Dec. 1997. [6] D. R. Jackson and A. A. Oliner, , C. A. Balanis, Ed., “Modern Antenna Handbook,” in Leaky-Wave Antennas. Hoboken, NJ: Wiley, 2008. [7] A. Grbic and G. V. Eleftheriades, “Periodic analysis of a 2-D negative refractive index transmission line structure,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pt. 1, pp. 2604–2611, Oct. 2003. [8] L. Liu, C. Caloz, and T. Itoh, “Dominant mode leaky-wave antenna with backfire-to-endfire scanning capability,” IEE Electron. Lett., vol. 38, no. 23, pp. 1414–1416, Nov. 2002. [9] C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antenna Propag. Mag., vol. 50, no. 5, pp. 25–39, Oct. 2008. [10] S. J. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 12, pp. 2678–2690, Dec. 2004. [11] J. D. Kraus, “A backward angle-fire array antenna,” IEEE Trans. Antennas Propag., vol. 12, no. 1, pp. 48–50, Jan. 1964. [12] P. Burghignoli, G. Lovat, and D. R. Jackson, “Analysis and optimization of leaky-wave radiation at broadside from a class of 1D periodic structures,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2593–2604, Sep. 2006. [13] Y. X. Li, Q. Xue, E. K.-N. Yung, and Y. L. Long, “The periodic halfwidth microstrip leaky-wave antenna with a backward to forward scanning capability,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 963–966, Mar. 2010. [14] A. K. Bhattacharyya, “Long rectangular patch antenna with a single feed,” IEEE Trans. Antennas Propag., vol. 38, no. 7, pp. 987–993, Jul. 1990. [15] Y. X. Li, Q. Xue, E. K.-N. Yung, and Y. L. Long, “Quasi microstrip leaky-wave antenna with a 2-dimensional beam-scanning capability,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 347–354, Feb. 2009. [16] J. R. James, P. S. Hall, and C. Wood, Microstrip Antenna Theory and Design. London, U.K.: Peter Peregrinus, 1981.

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An Electronically Reconfigurable Microstrip Antenna With Switchable Slots for Polarization Diversity M. S. Nishamol, V. P. Sarin, D. Tony, C. K. Aanandan, P. Mohanan, and K. Vasudevan

Abstract—An electronically reconfigurable microstrip antenna with circular and linear polarization switching is presented. The prototype fabricated on a substrate of dielectric constant ( ) 4.4 and height (h) 1.6 mm is fed by a proximity feed fabricated using the same substrate. By controlling the bias voltage of two PIN diodes, the polarization of the antenna can be switched between three states; two states for linear polarization (horizontal and vertical) and one state for circular polarization (RHCP). Simulation and experimental results show that the proposed antenna has a cross polar level better than 10 dB in the linear polarization state and 18 MHz axial ratio bandwidth in the circular polarization state. The frequency and polarization diversities of this design could potentially improve the reliability of wireless communication systems. Index Terms—Patch antenna, PIN diodes, polarization diversity, reconfigurable, switchable slots.

Fig. 1. Geometry of the proposed antenna (L = 30 9 mm, W = = 17 6 mm, w = 2 3 mm, 43 5 mm, l = 5 1 mm, l C = C = 33 pF, h = h = 1 6 mm and = 4 4, X = 8 49 mm and Y = 10 9 mm) (a) front view and (b) side view.

I. INTRODUCTION Frequency hopping spread spectrum systems switch between many different frequencies at a high rate to provide immunity to jamming, narrow-band interferences and multi-path fading. Software defined radios can be reconfigured to communicate many different protocols at different frequencies and polarizations. In cognitive radios the frequency and data rate are automatically determined depending on the available spectrum at runtime [1]. Reconfigurable antennas with stable radiation characteristics at different frequencies and polarizations offer several degrees of freedom to antenna designers. In microwave tagging systems polarization diversity antenna provides a powerful modulation scheme such as the circular polarization modulation [2]. In addition, many reconfigurable antennas with polarization diversity between two circular polarizations [3]–[5] have been discussed in past literature. A slot ring antenna with perturbations [6] allowing switching either between linear and circular polarization (CP) or between two CP was introduced. However, this requires a relatively complex biasing network using a few passive elements and in addition, needs several rectangular discontinuities which increase the design complexity. In [7], the polarization can be switched between circular and linear polarization (LP) by changing the shape of the slit according to the ON or OFF state of the PIN diode. However, this antenna requires a larger area to occupy the patch and dc-bias circuit. In addition complex biasing circuit causes an inconvenience for fabrication. The primary advantage of the reconfigurable antenna lies in its ability to support two separate applications at two different frequency bands with different radiation patterns and polarization characteristics using a single radiating aperture.

Manuscript received September 15, 2010; revised February 03, 2011; accepted February 15, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the University Grants Commission (UGC) and in part by the Department of Science and Technology (DST), Government of India. The authors are with Centre for Research in Electromagnetics and Antennas (CREMA), Department of Electronics, Cochin University of Science and Technology, Cochin-22, Kerala, India (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161446

Design of matching networks and the choice of switches are crucial in these types of antennas [8]–[10]. In this communication, the design, fabrication and measurement of a novel single feed rectangular microstrip antenna with reconfigurable polarization capability is presented. The basic structure utilizes an X shaped slot at the center of a rectangular cross-shaped patch. The X shape is chosen to induce symmetric current distributions for TM10 and TM01 modes as well as to obtain greater area reduction [11]. The polarization can be switched between CP and LP by changing the shape of the slot according to the ON or OFF state of the PIN diode without changing the geometrical parameters and is devoid of any impedance matching circuitry. Furthermore the antenna has an added advantage of reduced size with low levels of cross-polarized radiation in LP state and an 18 MHz axial ratio bandwidth in CP state. In addition, the antenna is simple because it requires only two PIN diodes as well as less area to occupy the patch and dc-bias circuit compared to conventional polarization diversity antennas available in literature. II. ANTENNA GEOMETRY The geometry of the proposed antenna is shown in Fig. 1. The antenna is fabricated on a substrate of thickness h (1.6 mm) and relative permittivity "r (4.4). The initial cross patch is obtained by removing the four square regions of side ls mm from the corners of a rectangular patch of size L 2 W mm2 . An X-slot of arm length lx mm and width wx mm is then carved at the center of the cross patch. The antenna is electromagnetically coupled using a 50 microstrip line fabricated using the same substrate material. The dimension of the ground plane is 100 2 100 mm2 . Two PIN diodes are inserted into the center of the slot in which D1 is oriented parallel to the feed line and D2 is oriented normal to the feed line. There is a printed crossed section in the center of the X-slot that connects both the diodes to the patch. For the proper biasing of the diodes, three narrow slot lines are carved in the patch. Three small smd capacitors C1 , C2 and C3 are soldered at these slot lines which block the dc bias current as well as provide good RF continuity. The PIN diode (BAR 64-04) requires a bias voltage of 1.1 V which is supplied from a battery through chip inductors. The dc bias circuit, used to control the ON/OFF state of diodes, is located on the right edge of the patch.

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Fig. 2. Simulated surface current distribution of the Cross patch antenna with X-slot at (a) 1.12 GHz and (b) 1.44 GHz.

TABLE I STATES OF THE EXTERNAL DC BIAS VOLTAGE AND PIN DIODES FOR DIFFERENT ANTENNA PROTOTYPES

Fig. 3. Simulated surface current distribution of (a) antenna 1 at 1.48 GHz, (b) antenna 2 at 1.53 GHz, and (c) antenna 3 at 1.53 GHz.

III. RESULTS AND DISCUSSION The antennas are tested using HP8510C vector network analyzer. The fundamental resonant modes (TM10 and TM01 ) of the unslotted cross shaped patch are at 1.8 GHz and 2.4 GHz with orthogonal polarizations. The proper selection of the slot size modifies the horizontal and vertical electrical lengths of the patch equally so that the two resonant frequencies are lowered to 1.12 GHz and 1.44 GHz. The simulated surface current distribution of the cross patch antenna with X-slot is depicted in Fig. 2. It is well evident from the figure that the insertion of the slot increases the current path thereby lowering the resonant frequency. The X-slot length is optimized to achieve maximum area reduction using Ansoft HFSS. The antenna exhibits good radiation characteristics for both resonant frequencies with an area reduction of 77% and 64% for the first and second frequency respectively when compared to a standard rectangular patch operating at the same frequencies. Two PIN diodes are inserted at the center of the X-slot to achieve the reconfigurable polarization capability. The orthogonally polarized dual frequency cross patch antenna can be reconfigured for different polarization with respect to the bias voltage applied to the diodes. The bias circuit consists of three dc block capacitors, RF chokes, two switches and input voltage. The dc bias lines are connected to the patch through RF chokes. The ON state of the diode is represented by a series resistor, R = 1:35 , while the OFF state is represented by a capacitor of C = 0:35 pF. Three dc block capacitors of C = 33 pF are chosen to isolate the RF components from the dc signal and RF choke inductor isolate the RF signal from flowing into the dc signal. The ON/OFF state of the diodes are controlled with respect to the potential applied to the terminals V1 , V2 and V3 , which is described in Table I. To radiate linearly polarized waves, both PIN diodes on the cross patch should be biased either in the “ON state” (Antenna 1) or in the “OFF state” (Antenna 2). When the two diodes are in “ON state”, they act as electrically short circuits (1.35 ). Hence, the shape of the X slot is modified with a cross shape at the center. From Fig. 3(a), it is clear that the new slot shape forces the currents on the patch to flow directly through it so that the effective current path is shortened. Therefore, the antenna excites TM10 mode at 1.48 GHz and TM01 mode at 1.95 GHZ. Thus antenna 1 is linearly polarized along X-direction with 2:1 VSWR bandwidth of 25 MHz at 1.48 GHz. When both the PIN diodes are in the “OFF state”, they act as electrically open circuits. Hence, the shape of the slot is modified with a

printed short circuit from left to bottom and open circuit from right to top at the X-slot center which makes the structure asymmetric with respect to the horizontal central line. Therefore, the currents flow through the center of the X-slot and along the edges of the patch is asymmetric as shown in Fig. 3(b). This increased current path shifts the TM01 mode towards the lower frequency region. Hence, an “X” slot loaded cross patch antenna excites linearly polarized radiation along Y-direction at 1.53 GHz with 2:1 VSWR bandwidth of 33 MHz. To radiate circularly polarized waves, one of the diodes on the patch (D1 ) should be in the “ON state” while the other should be in the “OFF state” (antenna 3). In this case, the X-slot shape is changed to two V slot connected back-to-back. As shown in Fig. 3(c), this new slot shape forces the currents to flow through the center of the X-slot as well as along the edges of one of the slot arm. This behavior results in the splitting of the current into two near orthogonal resonant modes at 1.5 GHz and 1.535 GHz respectively. The surface current distribution is along Y-direction at 1.5 GHz and is along X-direction at 1.535 GHz. Also, the input reactance of lower mode is inductive (39 + j4:3 ) while that of the other mode is capacitive (300 j3:9 ) in nature for the two resonant modes. The 2:1 VSWR bandwidth of antenna 3 is measured to be 65 MHz (4.3%) with respect to the center frequency of 1.495 GHz with 1.18% CP (3-dB axial ratio) bandwidth. Its narrow axial-ratio bandwidth is the consequence of imperfect excitation due to a single feed. However, the attractiveness of single-feed circularly polarized (SFCP) antenna is that it requires no polarizer for CP generation and makes the overall system compact. When D1 is in the “OFF state” and D2 in the “ON state” (antenna 4), the antenna excites TM10 mode at 1.27 GHz and TM01 mode at 1.95 GHz. This case is not considered on later discussion since this condition is not in the range where frequency and polarization switching is obtained. Extensive parametric analysis is conducted to optimize the functions of four corner notches of the rectangular patch. In antenna 1 or antenna 2, the corner notches cause a small shift in resonances. The effects of corner notches are significant in antenna 3 than that of antenna 1 or antenna 2. Fig. 4 shows that ls = 5:1 mm is a good selection in antenna 3 to achieve two near orthogonal resonant modes. From all these analyses, the optimum parameters of the antenna are found to be L = 0:28g , W = 0:39g , ls = 0:05g , lx = 0:15g , and Wx = 0:02g where g is the guided wavelength of the required frequency. The design parameters are validated for different frequencies.

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Fig. 4. Effect of the chamfers in antenna 3 (L = 43 5 mm, l = 17 6 mm, and W = 2 3 mm).

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30 9 mm, W = Fig. 7. Measured axial ratio of the antenna.

Fig. 5. Simulated and measured reflection coefficient of the antennas.

Fig. 8. Radiation pattern (a) antenna 1 at 1.48 GHz (b) antenna 2 at 1.53 GHz and (c) antenna 3 in two orthogonal planes at 1.53 GHz.

Fig. 6. A prototype of the fabricated antenna.

A prototype of the proposed antenna is fabricated with L = 30:9 mm, W = 43:5 mm, ls = 5:1 mm, lx = 17:6 mm and Wx = 2:3 mm. The simulated and measured reflection coefficients (S11 ) of the antenna are given in Fig. 5. The agreement between simulation and measurement are good as the non-ideal characteristics of the diodes were taken into account. The prototype of the fabricated antenna is shown in Fig. 6.

The axial ratio graph of antenna 3 is plotted in Fig. 7. The best CP performance in the broadside direction is achieved at 1.53 GHz with 1.18% CP bandwidth. Fig. 8(a) and (b) shows the measured radiation patterns at 1.48 GHz and 1.53 GHz respectively for the LP states. The radiation pattern at 1.53 GHz in ' = 00 and ' = 900 planes for the CP state is given in Fig. 8(c). The level of cross-polarization (i.e., the left-hand circular polarization, LHCP) is lower than 015 dB over the main beam direction. All the radiation patterns are broadside in nature with good LP and CP characteristics at the respective resonant frequency. The gain of the proposed antenna is measured using a double-ridged horn as reference. Antenna 1 follows a direct path through the center of the X-slot, which provides peak gain of 2.55 dBi for TM10 mode. Since both the diodes are OFF at 1.54 GHz the currents have to flow around the edges of the slot. The opposing currents on either side of the slot arm produce an equivalent magnetic field. However, the radiation due to this magnetic field on one-arm cancels with that of the opposite arm in the far-field. Thus the peak gain of antenna 2 is reduced to1.97

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dBi at the resonance frequency. Antenna 3 provides a maximum gain of 2.2 dBi at 1.52 GHz for the CP state.

A Novel Hexa-Band Antenna for Mobile Handsets Application

IV. CONCLUSION

Chia-Mei Peng, I-Fong Chen, and Chia-Te Chien

A single feed electronically reconfigurable microstrip antenna with switchable slots for frequency and polarization diversities has been presented in this communication. The antenna can produce linear and circular polarization by controlling the bias conditions of two PIN diodes. A good impedance matching performance for all polarization states is observed without any matching networks. The proposed design achieves a cross polar level better than 010 dB in linear polarization and 1.18% CP bandwidth in circular polarization state with broadside radiation characteristics and moderate gain. In addition, the antenna is simple and compact because it uses only a few active and passive components and requires less area to occupy the patch and dc-bias circuit compared to conventional polarization diversity antennas. The frequency and polarization diversities of this design provide some potential applications for wireless communications.

REFERENCES [1] J. Mitola and G. Q. Maguire, “Cognitive radio: Making software radios more personal,” IEEE Personal Commun. Mag., vol. 6, no. 4, pp. 13–18, Aug. 1999. [2] M. A. Kossel, R. Kung, H. Benedickter, and W. Bachtold, “An active tagging system using circular polarization modulation,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2242–2248, Dec. 1999. [3] M. Boti, L. Dussopt, and J.-M. Laheurte, “Circularly polarized antenna with switchable polarization sense,” Electron. Lett., vol. 36, no. 18, pp. 1518–1519, Aug. 2000. [4] F. Yang and Y. Rahmat-Samii, “A reconfigurable patch antenna using switchable slots for circular polarization diversity,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 3, pp. 96–98, Mar. 2002. [5] M.-H. Ho, M.-T. Wu, C.-I. G. Hsu, and J.-Y. Sze, “An RHCP/LHCP switchable slotline-fed slot ring antenna,” Microw. Opt. Technol. Lett., vol. 46, no. 1, pp. 30–33, Jul. 2005. [6] M. K. Fries, M. Grani, and R. Vahldieck, “A reconfigurable slot antenna with switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 11, pp. 490–492, Nov. 2003. [7] Y. J. Sung, “Reconfigurable patch antenna for polarization diversity,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 3053–3054, Sep. 2008. [8] B. Kim, B. Pan, S. Nikolaou, Y.-S. Kim, J. Papapolymerou, and M. M. Tentzeris, “A novel single-feed circular microstrip antenna with reconfigurable polarization capability,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 630–638, Mar. 2008. [9] M. Ali, A. T. M. Sayem, and V. K. Kunda, “A reconfigurable stacked microstrip patch antenna for satellite and terrestrial links,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 426–435, Mar. 2007. [10] S. V. Shynu, G. Augustin, C. K. Aanandan, P. Mohanan, and K. Vasudevan, “A compact electronically reconfigurable dual frequency microstrip antenna for L-band applications,” Int. J. Wireless Opt. Comm., vol. 2, no. 2, pp. 181–187, 2004. [11] M. S. Nishamol, V. P. Sarin, A. Gijo, V. Deepu, C. K. Aanandan, P. Mohanan, and K. Vasudevan, “Compact dual frequency dual polarized cross patch antenna with an x-slot,” Microw. Opt. Technol. Lett., to be published.

Abstract—A novel hexa-band antenna for mobile handsets application is proposed and analyzed in this communication. An asymmetric T-type monopole antenna with a shorted-line is designed to be operated in codedivision multiple access (CDMA, 824–894 MHz), global system for mobile communications (GSM, 880–960 MHz), digital communication system (DCS, 1710–1880 MHz), personal communication system (PCS, 1850–1990 MHz), wideband code division multiple access (WCDMA, 1920–2170 MHz) and Bluetooth (2400–2484 MHz) bands. A prototype of the proposed antenna with 50 mm in length, 3 mm in height and 15 mm in width is fabricated and experimentally investigated. The experimental results indicate that the VSWR 2.5:1 bandwidths achieved were 17.8% and 37.1% at 900 MHz and 2100 MHz, respectively. The specific absorption rate (SAR) for an input power of 24 dBm in CDMA, GSM and WCDMA bands, and an input power of 21 dBm in DCS and PCS bands all meet the SAR limit of 1.6 mW/g. Experimental results are shown to verify the validity of theoretical work. Index Terms—Hexa-band antenna, mobile handsets, shorted-line, specific absorption rate (SAR), T-type monopole.

I. INTRODUCTION Wireless communications continue to enjoy exponential growth in the cellular telephony, wireless Internet, and wireless home networking arenas. In order to roam worldwide, the operation bands of major wireless services, such as code-division multiple access (CDMA), global system for mobile communications (GSM), digital communication system (DCS), personal communication system (PCS), wideband code division multiple access (WCDMA) and Bluetooth should be simultaneously considered [1]. Downsizing the handset unit, which has seen remarkable progress in recent years, requires the size reduction of the antenna element also. However, as a small antenna element is used, the utilization of the handset body is beneficial to enhance antenna performance of the handset because the handset body is usually larger than the antenna element. Therefore, the overall effective antenna dimensions augment dramatically. As a consequence, the corresponding gain and the bandwidth of the antenna system are increased [2]–[10]. While the use of the handset body as a part of the radiator is advantageous, it also caused disadvantage at the same time in practical operation. The antenna performance in terms of gain and input impedance varies due to the influence of the human head and hand. In this communication, an asymmetric T-type monopole antenna is designed jointly with a solid shorting-line to achieve hexa-band

Manuscript received September 16, 2010; revised November 13, 2010; accepted January 26, 2011. Date of publication July 07, 2011; date of current version September 02, 2011. This work was supported by the National Science Council, R.O.C., under Contract 97-2221-E-228-004. C.-M. Peng and I.-F. Chen are with the Department of Electronic Engineering and Institute of Computer and Communication Engineering, Jinwen University of Science and Technology, Taipei, Taiwan, R.O.C. (e-mail: [email protected]. tw). . C.-T. Chien us with the Department of Electronic Engineering and Institute of Computer and Communication Engineering, Jinwen University of Science and Technology, Taipei, Taiwan, R.O.C. and also with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161447

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WCDMA and Bluetooth bands, the shape of part A is designed for wideband operation, the tuning of broad bandwidth is obtained by increasing strip-area and inserting some slits. These slits cause the discontinuities of the current distribution on the surface of radiating-strip which improves the impedance bandwidth [11], [12]. In part B, the resonant frequency is designed to occur at 900 MHz, the electrical-length of the planar-strip is equal to 80 mm (which is 15 mm + 25 mm + 6 mm + 22 mm + 5 mm + 7 mm). For covering CDMA and GSM bands, the solid-open stub is used as a top-loading of part B and it increases the electrical-length and impedance bandwidth in the antenna’s lower-operating band. The impedance matching at lower- and upper-operating bands can be also tuned by solid shortingline of part A and extended strip of part B. The solid shorting-line is found to be effective in obtaining a wider impedance bandwidth in the antenna’s upper-operating band. Note that the widths of these strips, slits, solid shorting-line and solid open-stub, etc., are not identical. By selecting appropriate dimensions (part A, part B) of the antenna structure, good impedance matching of the asymmetric T-type monopole can be obtained, and thus the bandwidth is also extended. Besides, [7] indicated that the ground-plane mode is responsible for SAR. Hence, in order to demonstrate the low current distribution on the handset body, the effect of varying the ground-plane length of the proposed antenna structure is investigated by simulations. Detail results will be presented and discussed in the next section. III. EXPERIMENTAL RESULTS AND DISCUSSION Fig. 1. (a) Geometry of the proposed antenna for hexa-band operation in the mobile handset. (b) Dimensions of the proposed antenna.

(CDMA, GSM, DCS, PCS, WCDMA and Bluetooth) performance. A hexa-band antenna is constructed to operate in the range of a dual operating-band: lower-operating band (CDMA and GSM) and upper-operating band (DCS, PCS, WCDMA and Bluetooth). The proposed antenna has a dual asymmetric radiated-strip structure that is developed by modifying the structure of a printed T-type monopole. The feasibility of wide bandwidth operation has been proven by the design of a solid shorted-line and a solid open-stub radiating structure to operate in the dual operating bands. Smaller power loss (dB absorption) due to the influence of phantom-head model is shown. It is also demonstrated that the proposed antenna structure produces a low specific absorption rate (SAR) value. Details of the design considerations and the experimental results of the constructed prototype are presented and discussed in the following sections. II. ANTENNA STRUCTURE AND DESIGN Fig. 1(a) shows the geometry of the proposed antenna for hexa-band operation in the mobile handset. The presented antenna structure is composed of an asymmetric T-type monopole which is printed on a FR4 glass epoxy substrate with the thickness of 1.6 mm, relative permittivity of 4.3 and loss tangent of 0.023. The proposed antenna is placed on a portion without metal ground on the backside. All sections are at the same layer. One of the asymmetric T-type strips combined with solid shorting-line to form a loop structure is denoted as part A. The other strip combined with solid open-stub is denoted as part B. The electrical-length of the radiating elements can be determined from the quarter-wave length at the resonant frequencies. Detailed dimensions of the proposed antenna are given in Fig. 1(b). In part A, the resonant frequency is designed to occur at 1800 MHz, the electrical-length of the planar-strip is equal to 40 mm (which is 15 mm + 25 mm). For covering DSC, PCS,

In the experiment, the feeding-point and ground-plane are connected to a 50 SMA connector. By using the described design procedure, a hexa-band antenna is constructed to operate in the range of a dual operating-band: lower-operating band (CDMA and GSM) and upper-operating band (DCS, PCS, WCDMA and Bluetooth). Fig. 2(a) shows the measured and simulated V.S.W.R plot of the dual band antenna and the V.S.W. R  2 bandwidths are 135 MHz (15%) and 790 MHz (37.6%) at 900 MHz and 2100 MHz, respectively. The simulated results are obtained by using the Ansoft HFSS. We can also find that a good agreement between the simulation and measurement is obtained. Fig. 2(b) shows the measured V.S.W.R of the proposed antenna in terms of part A and part B. For part A only, the radiated-strip and the shorting-line is matched at the DCS, PCS and WCDMA bands, the 560 MHz (28% at 2000 MHz) operating bandwidth is shown. This is due to the fact that the surface current distribution of the asymmetric radiated-strip is discontinuous. For part B only, the modified bended monopole antenna is matched at the GSM and PCS bands. As expected, the measured results indicate that part A and part B introduce an upper- and lower-operating band, respectively. The measured Smith Chart as shown in Fig. 2(c), the full characteristics of the proposed antenna are shown. Fig. 3 presents the measured 3-D and 2-D radiation patterns in the free space at 850 MHz and 902 MHz in the xy-plane and yz-plane, respectively. It is obvious that the dipole-like radiation patterns are observed. In other words, at the lower-operating bands, the ground-plane becomes a part of the antenna, and is responsible for the radiation [7]. The measured radiation patterns at 1720, 1920, 2045 and 2450 MHz are shown in Fig. 4. From Fig. 4, more variations in the radiation pattern-shapes are obtained, as compared to those in Fig. 3. This is probably because the ground-plane still acts as a part of the antenna at the upper-operating band. The overall ground-plane length is about one wavelength long and there are normally four main lobes at the upper-operating band. Table I. presents the measured antenna total efficiency of the proposed antenna in the free space (without phantom-head and phantomhand), and with phantom-head and phantom-hand.

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Fig. 3. Measured 3-D and 2-D radiation patterns at (a) 850 MHz and (b) 902 MHz for the proposed antenna.

A. Analysis of the Proposed Antenna Structure

Fig. 2. (a) The measured and simulated V.S.W.R against frequency. (b) The measured V.S.W.R in terms of the part A and part B. (c) The measured Smith Chart of the proposed antenna.

Acceptable radiation characteristic for the practical applications is obtained for the proposed antenna. The omni-directional feature of the proposed antenna can be observed from the Horizontal-plane. The effect of the proposed antenna structure on the antenna performance is also studied and the results are described below. In addition, the SAR results of the proposed antenna are also analyzed.

The design parameters and the corresponding characteristics of the resonant frequency, input impedance and bandwidth are a function of the geometrical parameters of the proposed antenna. The simulated current distribution of the proposed antenna structure on the handset body is shown in Fig. 5. In the upper-operating band, only a few current is distributed on the handset body. Note that a small loop antenna can be regarded as a magnetic dipole normal to the loop plane and it reduces the current flow on the handset body [6], [7]. However, in the lower-operating band, more current are distributed on the handset body as compared to those in the upper-operating band. That is because in the lower-operating band, the electrical-length of the modified bended monopole is over one quarter-wavelength, as a consequently, the input impedance of the modified bended monopole is matched to the handset body [3]–[7]. When a mobile handset is used in close proximity to a human head, dielectric-loading effect can be expected, there may also be a detuning issue. In order to demonstrate the distinctive performance of the proposed antenna in the presence of a human head, the measurement efficiency set-up with the phantom-head is shown in Fig. 6. The liquid parameters used in the measurements are listed in Table II. The measured V.S.W.R against frequency of antenna with phantom-head is shown in Fig. 7.

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TABLE I THE MEASURED ANTENNA GAINS AND THE TOTAL EFFICIENCY WITHIN THE OPERATING BANDWIDTH OF THE PROPOSED ANTENNA

Fig. 4. Measured 3-D and 2-D radiation patterns at (a) 1720 MHz, (b) 1920 MHz, (c) 2045 MHz, (d) 2450 MHz for the proposed antenna.

The degradation of total efficiency of antenna with phantom-head is shown in Table I.

Fig. 5. The simulated current distribution of the proposed antenna structure on the handset body (the ground-length is 100 mm) at (a) 850 MHz, (b) 902 MHz, (c) 1720 MHz, (d) 1920 MHz, (e) 2045 MHz, (f) 2450 MHz.

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TABLE II THE LIQUID PROPERTY OF PHANTOM-HEAD/HAND

Fig. 6. Photo of experimental arrangement for efficiency measurement with (a) Phantom-head. (b) Phantom-hand.

Fig. 8. Photo of experimental arrangement for SAR measurement.

Fig. 9. The physical model for measuring SAR with the proposed antenna at the top and bottom position of the handset body.

Fig. 7. Measured V.S.W.R against frequency of antenna with phantom-head and hand.

B. Analysis of the SAR The SAR in passive mode has been measured using Dasy-4 system [13], as shown in Fig. 8. The antenna is placed at the cheek position of the right-hand side of the phantom, and the spacing between the ground-plane and the cheek is 3 mm. Two cases for the proposed antenna test are shown in Fig. 9. The input power of the proposed antenna at GSM, CDMA and WCDMA bands is 24 dBm. However, the

input power at DCS and PCS bands is 21 dBm (both considering a user channel being 1/8 of a time slot) [2]. The liquid parameters used in the measurements are listed in Table III. The measured SAR results in 1 g mass of simulated tissue from exposure to the antenna radiation are listed in Table IV. When the proposed antenna is to be located at the top position (normal using mode), it is seen that the 1 g mass SAR results at all frequencies meet the SAR limit of 1.6 mW/g. We can also observe that the difference between the measured SAR at the top and bottom positions is large. Obviously, this is due to the high current density concentration around the antenna. In general, the SAR passive test is only a preliminary measurement and the test results are used to analyze the antenna. In practical application, SAR is finally tested with an active device which may result in a different SAR value due to extra device elements.

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TABLE III THE LIQUID PROPERTY OF PHANTOM

TABLE IV THE MEASURED SAR RESULTS IN 1-g OF THE SIMULATED TISSUE FROM EXPOSURE TO THE ANTENNA RADIATION WITH TWO CASES OF THE PROPOSED ANTENNA TO LOCATE AT THE TOP AND BOTTOM POSITIONS OF THE HANDSET BODY

IV. CONCLUSION In this communication, the proposed hexa-band antenna is practically capable to operate at the CDMA, GSM, DCS, PCS, WCDMA and Bluetooth bands. We demonstrated that a printed asymmetric T-type monopole with a solid shorting-line and a solid open-stub structure provides the hexa-band operation. By correctly choosing the shorting-line parameters and by modifying the shape of the T-type monopole arms, two bandwidths, 17.8% and 37.1%, can be obtained. The contribution of this communication is to implement a simple and low profile antenna for the practical mobile handset application. Measurement results show that a broad bandwidth is obtained. Although this antenna is designed for mobile handset applications, this design concept can be extended to the antenna design for laptop computers. ACKNOWLEDGMENT The authors would like to thank Prof. C.-Y. Wu (IEEE Life Fellow, Dept. of E.E, Jinwen University of Science and Technology), Prof. C.-W. Hsue (IEEE Fellow, Dept. of E.E, National Taiwan University of Science and Technology) for their help. The authors also appreciate the reviewer’s comments to improve the quality of this communication.

REFERENCES [1] Ramiro and Chaouki, “Wireless communications and networking: An overview,” IEEE Antennas Propag. Mag., vol. 44, pp. 185–193, Feb. 2002. [2] C.-H. Chang and K.-L. Wong, “Printed 8-PIFA for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, pp. 1373–1381, May 2009. [3] J. D. Kraus and R. J. Marchefka, Antennas, 3rd ed. New York: Mc Graw-Hill, 2002, pp. 804–805. [4] K.-L. Wong, G. Y. Lee, and T.-W. Chiou, “A low-profile planar monopole antenna for multiband operation of mobile handsets,” IEEE Trans. Antennas Propag., vol. 51, no. 1, pp. 121–125, Jan. 2003.

[5] Z. Li and Y. Rahmat-Samii, “Optimization of PIFA-IFA combination in handset antenna design,” IEEE Trans. Antennas Propag., vol. 53, pp. 1770–1777, May 2005. [6] 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. [7] A. Cabedo, J. Anguera, C. Picher, M. Ribó, and C. Puente, “Multi-band handset antenna combining a PIFA, slots, and ground plane modes,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2526–2533, Sep. 2009. [8] 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, Jun. 2004. [9] 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 2006), New York, Mar. 2006. [10] 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. [11] C.-M. Peng, I.-F. Chen, and C.-W. Hsue, “Modified printed folded 8 dipole antenna for DVB applications,” IEICE Trans. Commun.., vol. E90-B, pp. 2991–2994, Oct. 2007. [12] I.-F. Chen, C.-M. Peng, and S.-C. Liang, “Single layer printed monopole antenna for dual ISM-band operation,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1270–1273, Apr. 2005. [13] Schmid and Partner Engineering, AG (SPEAG) [Online]. Available: http://www.speag.com/speag/products.php

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Design of Single-Feed Dual-Frequency Patch Antenna for GPS and WLAN Applications Shun-Lai Ma and Jeen-Sheen Row

Abstract—A design of single-feed dual-frequency patch antennas with different polarizations and radiation patterns is proposed. The antenna structure is composed of two stacked patches, in which the top is a square patch and the bottom is a corner-truncated square-ring patch, and the two patches are connected together with four conducting strips. Two operating frequencies can be found in the antenna structure. The radiations at the lower and higher frequencies are a broadside pattern with circular polarization and a conical pattern with linear polarization, respectively. A prototype operating at 1575 and 2400 MHz bands is constructed. Both experimental and simulated results show that the prototype has good performances and is suitable for GPS and WLAN applications. Index Terms—Circular polarization, conical radiation, dual frequency, patch antenna.

I. INTRODUCTION Dual-frequency antennas are often used in multi-functional wireless products to reduce the number and volume of the required antennas. Several designs related to the dual-frequency antenna have been reported [1]–[7], and these designs can provide the antenna to be with specific radiation characteristic at each operating frequency to comply with system requirements, especially in far-field pattern and polarization. For example, the design in [1] uses two stacked corner-truncated patches to achieve a dual-frequency circularly-polarized (CP) antenna with broadside radiation, and it can be applied to the L1/L2 bands of the global positioning system. The similar antenna performances can also be carried out by a single-layer circular patch antenna surrounded by two concentric annular rings [2]. To obtain different radiation characteristics at the two operating frequencies, [3] proposes a two-element coplanar structure for GPS/DCS applications, including an annular-ring patch antenna and a corner-truncated square patch antenna. The annular-ring patch antenna operates at the TM21 mode, and consequently it can radiate a conical pattern with linear polarization (LP); however, the corner-truncated square patch antenna operating at its fundamental mode can generate CP broadside radiation. Because the square patch is fabricated on a ceramic substrate with higher permittivity, it can be placed inside the annular-ring patch to form a compact structure. A reverse arrangement for the two radiating patches is also presented for the applications of GPS/UMTS [4]. In this design, the GPS antenna is a corner-truncated square-ring microstrip antenna fabricated in air substrate, and the UMTS antenna uses a monopole antenna top loaded a shorted patch to produce uniform conical radiation. The inner slot of the square-ring patch of the GPS antenna is big enough to accommodate the shorted patch of the UMTS antenna, and therefore the two radiating elements can form a coplanar structure. Such dual resonant elements with different radiation characteristics are also arranged in a stacked structure to realize the dual-frequency design for the applications of GPS/DCS [5] or GPS/DSRC [6]. The advantage Manuscript received December 08, 2010; revised January 20, 2011; accepted February 05, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Department of Electrical Engineering, National Changhua University of Education, Chang-Hua, Taiwan 500, R.O.C. (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161453

Fig. 1. Geometry of the proposed dual-frequency patch antenna.

of employing two elements is that their operating frequencies can be tuned separately. From these past designs, it is found that the dual-frequency antenna can be excited with a single feed if the resultant two operating frequencies have similar radiation characteristics; however, it is relatively difficult to use a single feed to generate different radiation patterns and polarizations at the two frequencies. Although the design in [7] presents a single-feed dual-frequency antenna with different patterns, its structure is not suitable for generating CP radiation. In addition, [7] only has one radiating element, and consequently it is relatively difficult to simultaneously tune the dual operating frequencies to given frequencies. In this communication, a new design for dual-frequency antennas is proposed. The antenna consists of two radiating elements which are arranged in a stacked structure. One element is designed for GPS operation and it has CP broadside radiation. The other element operates at the WLAN band and its radiation is LP conical pattern. Moreover, the two elements can be simultaneously excited by a single feed. Details of the antenna designs are described, and the key parameters to tune the two operating frequencies are also analyzed and discussed. II. ANTENNA STRUCTURE Fig. 1 shows the geometry of the proposed dual-frequency patch antenna. A corner-truncated square-ring patch, having an outer side length of a1 and an inner side length of b1 , is printed on an FR4 substrate (thickness 1.6 mm and permittivity 4.4) that is placed at a height of h1 above the ground plane. The side length of the truncated corners is l. A square loading patch, which has the same dimensions as the inner slot of the square-ring patch, is fabricated on another FR4 substrate (thickness 0.8 mm) which is supported by four conducting strips at a height of h2 above the square-ring patch. All the conducting strips have the same width w and they are respectively attached to four edges of the square patch. In addition, a square-ring slot with outer side length a2 and inner side length b2 is embedded into the square patch. The center of the square-ring slot is located on the y -axis and it is at a distance of s away from the center of the square patch. A coaxial probe with radius 0.7 mm is used to excite the antenna, and the distance between the feed point and the center of the square-ring slot is d. The structure in Fig. 1 involves two stacked radiating elements. One is the corner-truncated square-ring patch antenna. The other is a top-loaded monopole antenna formed by the coaxial probe and the loading patch. The two elements are connected together with the four conducting strips, and they share the same ground plane. To excite the two elements well, gap-coupled feed through the square-ring slot is adopted. Such a feed mechanism can be regarded as an L-type impedance matching network, in which the inductance is introduced by the thin probe and the capacitance is contributed by the gap and

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Fig. 2. Measured and simulated return loss of the example antenna.

Fig. 3. Measured and simulated axial ratios of the example antenna.

the parallel plate between the ground plane and the inner patch of the square-ring slot. Therefore, impedance matching can be achieved by tuning the parameters a2 , b2 , s, and d. The related parametric analyses can be referred to [8].

III. MEASURED AND SIMULATED RESULTS An example design with the dimensions of a1 = 61 mm, b1 = 41 mm, a2 = 13 mm, b2 = 11 mm, l = 8:5 mm, w = 1 mm, h1 = 4 mm, h2 = 2 mm, s = 11 mm, d = 2 mm, and L = 100 mm

is selected to demonstrate the performance of the proposed dual-frequency antenna. The example antenna has also been implemented, and the measured results are exhibited in Fig. 2 along with the numerical analysis results obtained by HFSS. Good agreements between them are observed except the difference of the impedance bandwidth at the higher operating frequency, and it could be due to the dimension error of h2 during the manufacture. From the obtained results in Fig. 2, it is clearly seen that the antenna has two operating frequencies, and each operating frequency is formed by two coupled resonant modes. The lower operating frequency is due to the square-ring patch antenna operating at the TM11 mode, and the perturbations of the truncated corners allow the fundamental mode to split into two orthogonal degenerated modes. The measured impedance bandwidth, referred to 10 dB return loss, is from 1515 to 1630 MHz. Within the impedance bandwidth, a CP operation is also found, and the variation of the axial ratio against frequency is presented in Fig. 3. The experimental results in Fig. 3 clearly indicate that the antenna has good CP performance and its CP bandwidth, determined by 3 dB axial ratio, is about 1.4% with respect to the center frequency of 1568 MHz. Fig. 4 also plots the measured radiation patterns at 1575 MHz together with the simulated results, and it can be observed that the antenna produces broadside radiation with

0

Fig. 4. Measured and simulated radiation patterns at 1575 MHz. (a) x z plane (b) y z plane.

0

the cross polarization level of less than 016 dB at the lower operating frequency. The peak gain is around 7.5 dBic. As for the higher operating frequency, the two coupled resonant modes are due to the top-loaded monopole antenna and the square-ring patch antenna operating at the TM21 mode. Both the resonant modes can generate conical radiation patterns. From the measured results in Fig. 2, the resultant impedance bandwidth of the two coupled modes is from 2360 to 2560 MHz. It has to be mentioned that the polarization of the top-loaded monopole antenna is mainly E but the polarization of the TM21 mode square-ring patch antenna involves E and E . Therefore, the example antenna operating at the higher frequency can radiate LP patterns with rich cross polarization, which is suitable for the application of WLAN. The far-field patterns at 2400 MHz are exhibited in Fig. 5. Both the measured and simulated results indicate that the antenna has conical radiation and the measured peak gain is about 2.4 dBi. In Figs. 4 and 5, a significant difference between the simulation and experimental radiation pattern results in the back side of the antenna is observed. This could be due to the scattering of the feeding cable connected to the antenna under test.

IV. PARAMETER ANALYSES AND DISCUSSION For the proposed dual-frequency antenna design, h2 , l, and w are the key parameters to achieve the requirements of GPS and WLAN. To investigate their effects on antenna performance, parameter analyses are performed and only one parameter is varied at a time. When other dimensions are fixed and the same as those of the example antenna, the effects of various h2 on the input impedance are exhibited in Fig. 6. As h2 is increased, the CP operating frequency is slightly moved to a

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Fig. 7. Simulated return loss of various l for the example antenna.

0

Fig. 5. Measured and simulated radiation patterns at 2400 MHz. (a) x z plane (b) y z plane.

0

Fig. 8. Simulated return loss of various w for the example antenna.

Fig. 6. Simulated return loss of various h for the example antenna. Fig. 9. Simulated axial ratio of various w for the example antenna.

lower frequency; however, the impedance bandwidth of the LP operating frequency is enhanced. Therefore, a proper h2 is required to cover the WLAN band. Fig. 7 shows the simulated return loss results for the cases of l = 0, 8.5, 10, and 13 mm. From the obtained results, it can be obviously seen that the antenna has two operating frequencies for each case, and the two coupled modes at each operating frequency are gradually separated with increasing l. The results imply that the parameter l needs to be properly selected not only for generating CP radiation at the lower operating frequency but also for obtaining more impedance bandwidth at the LP operating frequency. As for the effects of w, the simulated results are given in Fig. 8. Observing the case of w = 0 mm, the lower operating frequency obviously disappears, which represents that the square-ring patch antenna cannot be excited if the conducting strips are absent. By inspecting the current distribution and radiation patterns, it is found that the sole operating frequency, located around 2435 MHz, is due to the TM01 mode

of the square patch antenna excited by the gap-coupled feed rather than the monopole mode. It suggests that the conducting strip is essential to excite the fundamental mode of the square-ring patch antenna and the top-loaded monopole mode. Moreover, Fig. 8 also indicates that when w is increased, the resonant frequency of the top-loaded monopole mode, defined as the frequency with minimum return loss, is moved to a higher frequency but the resonant frequency of the TM21 mode is almost constant. The variations of w somewhat affect the CP performance at the lower operating frequency as well. Fig. 9 shows that the CP operating frequency only has an increase of 1.2% as the width of the conducting strip is widened from 0.5 mm to 2 mm. Based on the above analyses, it may be concluded that the proposed antenna structure can simultaneously excite two operating modes with different radiation characteristics, and their respective frequencies and performances can be tuned separately.

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V. CONCLUSIONS A new design for dual-frequency patch antennas has been presented. The antenna is single-feed and it possesses distinct radiation characteristics at the two operating frequencies. At the lower operating frequency, the antenna can radiate broadside patterns with good circular polarization, but it generates linearly-polarized conical radiation patterns with rich cross polarization at the higher operating frequency. These characteristics have been proved with experiments and the key parameters of the antenna have also been investigated. The antenna can be applied to the wireless products which integrate satellite and terrestrial communications, such as GPS and WLAN.

REFERENCES [1] C. M. Su and K. L. Wong, “A dual-band GPS microstrip antenna,” Microwave Opt. Technol. Lett., vol. 33, pp. 238–240, May 20, 2002. [2] X. L. Bao and M. J. Ammann, “Dual-frequency circularly-polarized patch antenna with compact size and small frequency ratio,” IEEE Trans. Antennas Propag., vol. AP-55, pp. 2104–2107, Jul. 2007. [3] S. Y. Lin and K. C. Huang, “A compact microstrip antenna for GPS and DCS application,” IEEE Trans. Antennas Propag., vol. AP-53, pp. 1227–1229, Mar. 2005. [4] C. Y. D. Sim, J. S. Row, and S. H. Chen, “A dual-band antenna design for GPS and UMTS applications,” Microwave Opt. Technol. Lett., vol. 49, pp. 1935–1939, Aug. 2007. [5] G. Z. Rafi, M. Mohajer, A. Malarky, P. Mousavi, and S. Safavi-Naeini, “Low-profile integrated microstrip antenna for GPS-DSRC application,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 44–48, 2009. [6] J. Y. Wu, J. S. Row, and K. L. Wong, “A compact dual-band microstrip patch antenna suitable for DCS/GPS operations,” Microwave Opt. Technol. Lett., vol. 29, pp. 410–412, Jun. 20, 2001. [7] S. H. Chen, J. S. Row, and C. Y. D. Sim, “Single-feed square-ring patch antenna with dual-frequency operation,” Microwave Opt. Technol. Lett., vol. 49, pp. 991–994, Apr. 2007. [8] J. S. Row and S. H. Chen, “Wideband monopolar square-ring patch antenna,” IEEE Trans. Antennas Propag., vol. AP-54, pp. 1335–1339, Apr. 2006.

A Methodology for the Design of Frequency and Environment Robust UHF RFID Tags H. Chaabane, E. Perret, and S. Tedjini

Abstract—A methodology for the design of frequency and environment “robust” UHF RFID Tags is proposed and applied for the automatic generation of tag antennas. This is an automated process of antenna generation mainly associated with a Genetic Algorithm optimizer. Original antenna topologies are generated and selected automatically according to an evolutionary algorithm. The efficiency property is implemented as a set of constraints and integrated in the cost functions of the design process. Moreover, the proposed methodology is fully automatic and avoids classical tedious simulation steps. The methodology is applied to design “robust” RFID tags, under constraints of size, UHF frequency bands and environment characteristics. The design of efficient tags showing good performance for the worldwide UHF band is demonstrated. Some designs have been realized and measured in an anechoic chamber owing to an experimental set up dedicated to RFID communication characterization. Good agreement between simulation and measurement is observed. Index Terms—Antenna design, efficiency, genetic algorithm, robustness, UHF RFID tags.

I. INTRODUCTION UHF RFID tags have become very useful in numerous applications due to their two main practical advantages of being all passive and wireless. However their deployment in real situations and scenarios is difficult as their behavior can be very sensitive to the characteristics of the tagged items and more generally to the physical environment of the application. The UHF tags are composed of two main elements: the RFID chip and the tag antenna. In recent years, many advances have been made in microelectronics and embedded systems leading to high performance chips as the minimum operating power tends to -20 dBm [1], and multiport chip configurations are commercially available. On the other hand, the design and optimization of tag antennas have received strong attention and numerous results are available in the literature [2], [3]. One of the challenges encountered in tag antenna design is to ensure the interoperability of tags and readers regardless of the region. Indeed, there are three different frequency bands 868–869 MHz, 902–928 MHz and 950–956 MHz, depending on the regulation in each country [4]. However, the problem of tag sensitivity to the tagged item and the environment is generally neglected during the design process except for some special cases like metallic items [5], [6]. In most cases, the antenna design process is based on standard electromagnetic (EM) simulators [3]. But traditional antenna design approaches are mainly empirical and based on the knowledge and experience of the designer [7]–[9]. The designer starts with a priori antenna topology based on his EM experience and expertise [7], [8]. In order to enable large spatial coverage, many of the tags are based on the dipole antenna. For sake of compactness, folded dipoles are preferred and sometimes, distributed elements are added to improve the tuning performance [2]. Antennas with loop and radiator elements can also be used. In this case, coupling methods, between the antenna and the tag, are carefully investigated Manuscript received September 28, 2010; revised February 04, 2011; accepted March 07, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with Grenoble-INP LCIS, 26902 Valence Cedex 09, France (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161556

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[7], [8]. A parametric study on different antenna geometrical parameters allows a good grasp of the antenna potential. It is challenging to identify the key parameters, i.e. the minimal number of independent geometrical parameters, which can be used to achieve the desired frequency response and good impedance matching [2], [3]. An optimization step is generally needed in order to conclude the design [3]. The other remarkable factor in antenna design is the context in which UHF tags are used. The physical environment affects their characteristics considerably, and can decrease their performance [2]. In practice, the disturbing element is no more than the object above which the tag is placed [2], [6]. It is quite evident that the tag must be “usable” with the greatest number of items. This fact is essential in practice as the items to be tagged are quite different in terms of size, shape and packaging composition. In other words, the tag must be “robust” in terms of frequency in order to be used worldwide, but it must be “robust” regarding the environment as it will be placed on different items having different EM characteristics in a variety of physical environments. So, the challenge is to design what we call in this article “robust” tag, i.e., tag having enough efficiency both in terms of read range stability on frequency and insensitivity to physical environment. These tags will be thereafter named frequency and environment robust tag (FER-Tag). In this communication, for the first time, a methodology for the design of FER-Tags is introduced. One of the main advantages of this approach is its ability to generate automatically the shape of the antenna according to the input constraints defined by the user. The developed approach has two main steps: shape generation and antenna selection. Both steps are governed by the well known genetic algorithm (GA) technique. A tutorial introduction to GA optimization and its application to various electromagnetic devices and problems can be found in [10]. So, we demonstrate a design approach that automatically generates the shape of the antenna without the need of a priori antenna shape. Indeed the antenna is built from a set of basic elements chosen by the designer. The principle of the methodology is presented in Section II and the developed tool is described in Section III. Selected examples of application for UHF tags and discussion on FER-Tag potential are given in Section IV. The experimental validation of the method is also presented in Section IV. Then concluding remarks are addressed at the end of the communication. II. PRINCIPLE OF THE METHODOLOGY The robustness property that a given specific application would require introduces a large number of constraints in the antenna design process. In many designs of RFID antennas, the shape of the antenna is preset and the optimization process runs on the sizes of some parts of the antenna. In order to improve the performance or to add new constraints, further degrees of freedom must be added [3]. Thus, to satisfy the constraints imposed by the insensitivity to the environment several freedom degrees are necessary. One way to maximize the number of freedom degrees is to avoid any preset shape of the antenna. So, contrary to the traditional design approaches, we will not impose any specific or generic antenna shape. The effectiveness and flexibility of this methodology are due to its ability to generate unspecified and versatile antenna shapes. The technique of generating antennas is the crucial aspect of our methodology. The generation method will directly impact the performance of the design process. The developed methodology is depicted in Fig. 1. The input parameters are chip characteristics, antenna size and all the constraints modeling the physical environment. Two software have been combined. In our case, MATLAB is used for shape generation and performance evaluation, and Ansoft Designer for electromagnetic simulation. It is very interesting to automatically generate versatile antenna forms, but this will not be effective if the design process is not able

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Fig. 1. Automated antenna design tool process. On the top, constraints are imposed. The optimization algorithm is implemented on MATLAB code, while Ansoft Designer, called by MATLAB, is used to perform EM simulations. The final generated antenna fits the input constraints.

to generate mandatory parts of the antenna such as loops or patches. In case of RFID tags, the generation of loops is necessary as the chip impedance is usually capacitive and must be compensated with inductive elements for matching purposes. On the other hand, loops are very useful for short range or near field communication. In this methodology the antenna is generated from a set of basic predefined radiating elements by interconnecting and placing these elements in a constrained space (surface of the tag) with respect each to other. In order to obtain the more general antenna shape, one can choose very simple basic elements. In this work, we used only strips to form a set of basic radiating elements (see Fig. 2). Thus, the design is based on end-to-end connection of different strips to obtain antennas like the one shown at the bottom of Fig. 1. This approach allows obtaining meanders and offers the possibility of getting loops. The chip location in the tag structure can be in numerous positions and allows us to obtain a large panel of antenna topologies, symmetric and asymmetric tags. Two radiating elements (patches) are added to the ends of the obtained structure. These elements are inserted into the design in order to add shapes other than the folded dipole. The patches also offer the possibility of adding more metal surface to increase the antenna radiating efficiency. The larger of the patches could also be exploited for labeling purposes. Several parameters are then introduced in Fig. 2: 1) the total strip number “N;” 2) the strip width “w;” 3) the strip length “Li ” (i = [1 : N]); 4) “i ” the angle formed by the strips “i” and “i + 1.” We impose i = [0=2; 0; =2;  ]; 5) “Lp ” and “Wp ” the patch dimensions and; 6) the tag size. All these parameters are introduced into the GA process to get the antenna shape. So, 2N + 3 parameters are optimized in the antenna design process to get original topologies. III. THE AUTOMATED ANTENNA DESIGN TOOL Genetic Algorithm method was used during the design process. This method has been considered successfully for antenna optimization in several works [10]–[12]. It starts with a large number of antennas as the

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is used for electromagnetic simulation which requires three minutes CPU time for each individual. The number of individuals per generation is fixed at 25. In order to evaluate the antenna topologies generated in the GA process, we define a cost function that is based on the antenna Read Range calculated from EM simulation on Ansoft Designer. The Read Range is obtained by using the formula (1)

R=

2

Fig. 2. Obtained antenna topologies for three different sizes: Tag 1: 80 80 mm (N = 132); Tag 2: 40 40 mm (N = 152); Tag 3: 95 8 mm (N = 92). The antenna designs are not to scale.

2

2

initial population. An electromagnetic simulator is used for the study of each member of the initial population. Then, it evaluates the performance of each member with the help of cost functions that compare individual performances to the target performance. The cost functions are defined by the user; usually, they comprise many constraints combined with the weighting factors. The cost functions return a measure of the fitness of each individual. The GA will select the best or fittest individuals and generate a new population by mutation and crossover processes. In this evolutionary process of “survival of the fittest,” the better quality individuals mate and produce offsprings, while poor quality individuals disappear [10]. So, the antenna shape changes, from one iteration to another, based on the evolutionary selection principle. This process is repeated until an antenna design, which satisfies the project specification is obtained (Fig. 1). In this work, the standard fabrication process constraints of the RFID tags are considered. The antenna is made up of aluminum with thickness 9  and printed on PET substrate with a relative permittivity of 3.2 and thickness 50  . The RFID chip considered in this work is the NXP G2XL/XM chip with an equivalent input impedance of 0 3 at 915 MHz and a minimum operating power of 015 dBm [13]. Some examples of the obtained designs are given in Fig. 2. They correspond to three standard UHF RFID tag sizes: 95 2 8 mm, 80 2 80 mm and 40 2 40 mm. In order to get a larger panel of antenna topologies, the antenna shape generated could be either symmetric or asymmetric. It can be noticed that the obtained tags are asymmetric and the chip is located at the center of the antenna. This particularity was not imposed but obtained according to the best fit selection method used in Genetic Algorithm. We may note the existence of a loop connected to the RFID chip in the three tags. This loop was imposed neither in the design process nor in the initial population, but the GA process “naturally” converged to the generation of this inductive element. To study the sensitivity of the tag with respect to the environment, we consider that the antenna is either tagged on different dielectric materials having thickness 10 mm and relative permittivity varying from 2.2 to 5.7 or in free space [14]. This dielectric material models a large panel of classical RFID tagged items such as Nylon, Plexiglas, Porcelain, Polyester, PVC, etc. The optimization process returns the less sensitive tag on the entire UHF frequency band with a Read Range between 5 and 8 m. The CPU time of this automated approach is in nearly thirteen hours (Dual Core Processor: 2 GHz, Memory: 4 Go). This simulation time is quite important, but it does not require any human interaction. Everything was done to automate all the design processes. This time depends on the speed of the electromagnetic simulator and the number of individuals considered by the GA algorithm. In this study Ansoft Designer

m

(22 j 195 )

m



4

P3G30 P

(1)

ic

where G is the antenna gain, 0 the mismatch impedance between antenna and chip, Pic the chip minimum operating power and P the maximum permitted power emitted by the reader. P and ic are set by the user (in our study, P is 2 W ERP (European standards [4]) and (NXP G2XL/XM chip). ic 0 In our case, the GA process used is multi-objective in order to take into account several constraints on frequency and environment. Each cost function corresponds to an environment constraint and is calculated separately on ANSOFT Designer. The Read Range is evaluated for each environment configuration and each frequency band (868 MHz, 915 MHz and 955 MHz), but the minimum Read Range value is used to evaluate the fitness function F as shown in (2).

P

P

15 dBm

min ( R(f ))when tag on dielectric (" =2:2) min ( R(f ))when tag on dielectric (" =3:2) (2) F= min ( R(f ))when tag on dielectric (" =4:1) min ( R(f ))when tag on dielectric (" =5:7) where R(f ) is the simulated tag Read Range at the frequency f . (f 1 = 868 MHz; f 2 = 915 MHz; f 3 = 955 MHz) and  is the i=1:3

i

i

r

i=1:3

i

i

r

i=1:3

i

i

r

i=1:3

i

i

r

i

i

i

corresponding weighting factors. The cost function can be defined differently by using the mean Read Range value on the whole frequency band [10]. The cost function defined in our work imposes many more constraints than the mean value, and guarantees optimization on the whole frequency band. We can also use weighting factors to get more constraints on a specific frequency band or for a specific environment configuration. So, the antenna design tool developed offers flexibility in the choice of frequency and environment constraints. For the efficient implementation of this methodology, the convergence problem must be carefully addressed. Typical convergence behavior is pointed out in Fig. 3, where the best individual regarding fitness functions based on the Read Range versus the generation number is presented. In this configuration, 25 is typically the minimum number of generations required for convergence. IV. RESULTS

Tags, with different sizes, presented in Fig. 2 are simulated on the entire UHF frequency band. Read Range obtained when tags are on dielectric objects is shown in Fig. 4. According to these results, a very high insensitivity to the dielectric permittivity variation is observed, particularly in the European frequency band. This behavior is due to the fact that more constraints have been imposed for the European frequency band during automatic antenna design. So, for different imposed tag shapes and sizes, as expected, the Read Range is stable, between 5 and 8 m in nearly the entire UHF frequency band, whatever may be the dielectric tagged object used. For the sake of comparison, the NXP commercial tag FFL 95-8 which has roughly the same size as the FER-Tag3, is considered. Fig. 5 shows the pictures of two tags built by using standard UHF RFID fabrication process.

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Fig. 5. Prototype pictures of the NXP FFL-95-8 tag (up) and our FER-Tag 3.

Fig. 3. Convergence study of the automated antenna design tool versus the generation number for FER-Tag 3. Only the best individual regarding the Read Range cost function is represented for each generation. The four Read Range cost functions for environment constraints, with permittivities of 2.2 (solid line), 3.2 (1), 4.1 ( ) and 5.7 (dashed line) are compared.

2

Fig. 6. Read Range versus frequency for FER-Tag 3 with size 95 8 mm (no marker) and NXP FFL95-8 ( ). Measurements are done on a dielectric with permittivities of 2.2 (dashed line) and 5.7 (solid line).

Fig. 4. Read Range versus frequency for the three designed FER-tags, depicted in Fig. 2, of different sizes: 95 8 mm (+), 80 80 mm (o) and 40 40 mm ( ). Simulations are done on dielectric with permittivities of 2.2 (dashed line) and 5.7 (solid line).

2

2

2

The Read Range of the two tags is given in Fig. 6, where, two environments are considered. As can be observed, the FER-Tag has roughly the same Read Range for all the frequency bands in both environments. On the contrary, the FFL-95-8 has more sensitivity with respect to the frequency and environment. Significant degradation of Read Range, especially for high frequency, is quite evident. The experimental validation was done in an anechoic chamber using an RFID characterization monostatic method [15]. The measurement setup is composed of a Vector Signal Generator (VSG; Agilent MXG-N5182A) used as transmitter, a Spectrum Analyzer (Tektronix RSA3408A) used as receiver, a Double Ridge Guide Horn with 7 dBi of gain and a circulator in order to isolate the transmission and reception channels. The VSG allows the emulation of any RFID commands. For the Read Range measurement, we consider forward RFID link from the reader to the tag. Thus, the Read Range is limited by the tag sensitivity (the antenna gain, the chip sensitivity, the mismatch impedance between antenna and chip. . .). Hence, we generate a query command with modulation and coding format corresponding to the EPC class1 Gen2 protocol. The command is sent with a fixed carrier

Fig. 7. Read Range versus frequency for FER-Tag 3 (no marker) and NXP FFL95-8 ( ) with measurement error bars: Simulations (dashed line) and measurements (solid line) are done on dielectric with permittivity of 2.2 or 5.7.

frequency and the output power is increased till the tag response is detected. The Read Range is then calculated from the minimum power emitted by the VSG in order to get a full response from the tag to the QUERY sent. The tag response is received on the real-time spectrum analyzer. This is followed by the post processing in order to get the main characteristics of the tag like radar cross section and Read Range. We also estimate the measurement uncertainty to 618% of the nominal value [15]. This uncertainty integrates all the error sources like signal levels, cables, antennae, circulators, etc. A comparison between simulation and measurement is given in Fig. 7. A very good agreement can be noticed, as the maximum difference between simulation and measurement is less than 15%, which is within the estimated uncertainty of the experimental setup.

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Fig. 8. Measurement setup for Read Range evaluation. Top view of the anechoic chamber, the tag is in the vertical position.

2

Fig. 10. Read Range [m] versus  for FER-Tag 3 with size of 95 8 mm (no marker) and NXP FFL95-8 ( ) at 915 MHz (US Band). Measurements are done on dielectric with permittivities of 2.2 (solid line) and 5.7 (dashed line).

2

Fig. 9. Read Range [m] versus  for FER-Tag 3 with size of 95 8 mm (no marker) and NXP FFL95-8 ( ) at 868 MHz (EU Band). Measurements are done on dielectric with permittivity of 2.2 (solid line) and 5.7 (dashed line).

We also studied the directivity of the tag and its space coverage. Hence, we measured the Read Range as a function of the relative orientation between the reader major axis and the tag. The measurement setup is described in Fig. 8. Figs. 9, 10 and 11 shows the Read Range as function of the orientation angle ( = 0 corresponds to a perfect alignment between the reader and the tag, see Fig. 8) with two different environments (permittivity of 2.2 and 5.7) and for three different frequencies: 868, 915 and 955 MHz. As expected, we observe a common RFID tag Read Range pattern. Maximum Read Range is observed in case of perfect alignment between reader and tag. The performance decreases considerably when the reader is perpendicular to the tag. Even if better Read Range is obtained with the NXP tag under perfect alignment condition with low dielectric perturbation (permittivity of 2.2), the best performance is observed for the FER-Tag under strong dielectric perturbation (permittivity of 5.7) irrespective of the orientation between the reader and the tag. So, the FER-Tag is less sensitive to dielectric perturbation and presents good performance stability when tagged on different dielectric materials. This is confirmed by results shown in Fig. 11: the NXP tag Read Range is less 3 m while the FER-Tag exhibits good performance (Read Range higher than 5 m) for both dielectric perturbations, irrespective of the orientation between the reader and the tag. For the three frequencies used in our measurement, the aperture angle (defined with a Read Range superior to 5 m) is more than 50 degrees for all the environment configurations of the FER-Tag, and between 0 and 60 degrees for the NXP Tag. This peculiarity of the FER-Tag makes it much more “robust” in the actual use when attached to an entire pallet of objects, as in an industrial supply chain. V. CONCLUSION A new antenna design methodology for Frequency and Environment Robust Tag has been developed and used for the development

2

Fig. 11. Read Range [m] versus  for FER-Tag 3 with size of 95 8 mm (no marker) and NXP FFL95-8 ( ) at 955 MHz (ASIA Band). Measurements are done on dielectric with permittivities of 2.2 (solid line) and 5.7 (dashed line).

of UHF tags. The methodology consists of coupling two software: an Electromagnetic Simulator (ANSOFT Designer) and a General Computing Tool (MATLAB). In order to satisfy the large number of constraints associated with the robustness property, our methodology does not impose any specific or a priori antenna shape. The effectiveness of this methodology is due to its ability to generate unspecified and versatile antenna shapes. Indeed, antenna generation is an automatic process controlled by a genetic algorithm. The GA used in our case is a multi-objective process. Consequently, several constraints can be imposed by using multiple cost functions. This is very useful because these cost functions allow us to integrate the complexity of real RFID environment into our design methodology, which is mandatory for robustness property. In order to obtain the more general antenna shape, we use only strips to form a set of basic radiating elements. This choice allows the generation of loops as a part of the antenna, which is necessary for matching purposes and near field communication. Several designs based on the NXP G2XL/XM chip have been obtained. A high robustness level with respect to frequency and dielectric environment is demonstrated. The experimental results are obtained owing to a universal and automatic experimental setup. The measurements of Read Range and the tag spatial coverage validate the proposed methodology. The good performance of the obtained antennas shows the efficiency of the approach whatever the imposed constraints. On the other hand, due to the flexibility of our approach it is evident that its application for other environments like metallic situations, high permittivity and dense cases, is straightforward. In particular, we have applied it to metallic environment and the results will soon be published elsewhere.

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REFERENCES

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A Planar UWB Patch-Dipole Antenna

[1] Impinj, Monza 4 Tag Chip Datasheet Feb. 2010 [Online]. Available: www.impinj.com/Documents/Tag_Chips/ [2] 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. 42, pp. 3870–3876, 2005. [3] G. Marrocco, “The art of UHF RFID antenna design: Impedancematching and size-reduction techniques,” IEEE Antennas Propag. Mag., vol. 50, no. 1, pp. 66–79, 2008. [4] D. M. Dobkin, The RF in RFID: Passive UHF RFID in Practice. London, U.K.: Newnes, 2007, pp. 24–45. [5] L. Ukkonen, M. Schaffrath, J. Kataja, L. Sydanheimo, and M. Kivikoski, “Evolutionary RFID tag antenna design for paper industry applications,” Int. J. Radio Freq. Identification Technol. Applicat., vol. 1, no. 1, pp. 107–122, 2006. [6] S.-L. Chen, “A miniature RFID tag antenna design for metallic objects application,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1043–1045, 2009. [7] S. L. Chen and R. Mittra, “Indirect coupling method for RFID tag antenna design,” Electron. Lett., vol. 46, no. 1, pp. 8–10, 2010. [8] H. W. Son and C. S. Pyo, “Design of RFID tag antennas using an inductively coupled feed,” Electron. Lett., vol. 4, no. 18, pp. 995–996, 2005. [9] G. Marrocco, “Gain-optimized self-resonant meander line antennas for RFID applications,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 302–305, 2003. [10] J. M. Johnson and Y. Rahmat-Samii, “Genetic algorithms in engineering electromagnetics,” IEEE Antennas Propag. Mag., vol. 39, no. 4, pp. 7–21, 1997. [11] E. E. Altshuler and D. S. Linden, “Wire-antenna designs using genetic algorithms,” IEEE Antennas Propag. Mag., vol. 39, pp. 33–43, 1997. [12] K. Lee, Y. Kim, and Y. C. Chung, “Design automation of UHF RFID tag antenna using a genetic algorithm linked to with MWS CST,” presented at the 4th IEEE Int. Symp. on Electronic Design, Test & Applications, Delta, 2008. [13] NXP Application Note, An 1629 UHF RFID label antenna design, UHF antenna design,” Sep. 2008 [Online]. Available: http://www.nxp.com [14] H. Chaabane, E. Perret, and S. Tedjini, “Towards UHF RFID robust design tag,” presented at the 4th IEEE Int. Conf. on RFID, FL, 2010. [15] A. Pouzin, T. P. Vuong, S. Tedjini, M. Pouyet, J. Perdereau, and L. Dreux, “Determination of measurement uncertainties applied to the RCS and the differential RCS of UHF passive RFID tags,” presented at the Proc. IEEE A&P-S, Charleston, 2009.

Wee Kian Toh, Xianming Qing, and Zhi Ning Chen Abstract—A planar dipole antenna is presented for ultrawideband applications. The proposed antenna consists of two patches forming a half-wavelength center-fed dipole. The differentially-fed dipole couples with an integrated balun-transformer and ground plane to achieve a stable broadside radiation pattern and consistent gain of 7.5–8.8 dBi. An impedance bandwidth from 3.1–4.8 GHz (43%) for a reflection coefficient of 10 dB has been achieved. Index Terms—Broadband antennas, dipole antennas, microstrip antennas, patch antennas, reflector antennas, UWB antennas.

I. INTRODUCTION The ultrawideband (UWB) technology employs broadband antennas for communication, imaging, and radar applications. The lower and upper spectra authorised by the Federal Communication Commission (FCC) [1] are 3.1–4.8 GHz (43%) and 6.0–10.6 GHz (55%), respectively. A low profile planar antenna with a stable broadside radiation pattern, consistent gain, and low back radiation are required for providing a wide sectorial coverage. Omni- and bi-directional antennas, such as planar monopole [2], [3], loop [4], Kandoian discone, and slot antennas are unsuitable candidates, due to their low gain profile and radiation pattern. Also, directional antennas such as the Vivaldi [5], horn, log-periodic, conical spiral, waveguide and dish antennas are unsuitable, as they are electrically large and/or have a high profile along the direction of wave propagations. Microstrip patch antennas have a low profile and wide beamwidth. To increase the impedance bandwidth, the substrate material is replaced by air dielectric, and the patch is elevated to lower the Q-value. The high reactance of the antenna feed limits the impedance bandwidth. This is overcame by introducing reactance loading to nullify the reactance, examples include the E-patch [6], U-slot [7], L-probe [8], electric dipole with shorted patch [9], aperture coupled feed [10], and concave center feed [11]. Several patch designs addressed the cross-polarization level problems due to high-order modes by applying the suspended probe feed [12], center slot feed [13], crossed exponentially tapered slot [14], and folded plate pair [15]. Other patch designs addressed the squinting and unstable radiation pattern issues, due to the frequency dependent radiation aperture variation, by using the coupling between the feed and radiator; examples includes the folded feed [16] and n-shaped feed [17]. A reflector antenna consisting of a dipole placed =4 above a sheet reflector has a directional radiation pattern, low cross polarization level and low back radiation. However it has a high profile, narrow beamwidth, and requires a broadband dipole with a balun. In this communication, a low profile broadband dipole antenna with an integrated balun-transformer is presented. II. ANTENNA DESIGN Fig. 1 shows the geometry of the dipole antenna. It consists of a 15 2 50 mm (0.15 2 0.50 0 ) half wavelength dipole, where 0 is the Manuscript received September 20, 2010; revised January 16, 2011; accepted February 17, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Institute for Infocomm Research (IIR), RF & Optical Department, Singapore 138632, Singapore (e-mail: weekiantoh@gmail. com; [email protected]; [email protected]; [email protected]. edu.sg). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161553

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Fig. 1. Antenna geometry.

Fig. 3. Measured co- and cross-polarization radiation patterns at (a) 3.0 GHz, (b) 4.0 GHz, and (c) 4.8 GHz.

Fig. 2. Measured and simulated S -parameters and gain profiles.

free space wavelength at 3 GHz. The center-fed dipole is suspended at a height of h2 = 13 mm (0.13 0 ) by a pair of 15 2 10.5 mm vertical strips extended from a 5 mm wide microstrip line. The microstrip line is supported by Styrofoam material at h1 = 2:5 mm, and soldered to a 50 SMA connector. A gradual 50–70–89 broadband impedance transformation is achieved by varying the width of the microstrip line. The square ground plane is 100 2 100 mm (1 2 1 0 ). To achieve symmetrical broadside radiation patterns in the E- and H-planes, a differential feed was employed to excite the dipole for a TM01 mode resonance. The vertical strips were connected to the

suspended microstrip line spaced at half guided wavelength g =2 ( 40 mm) apart at 4.0 GHz. A g =2 bypass-balun [18], [19] performs both single-unbalanced to dual-balanced terminal and impedance transformations. With these transformations, no matching network or hybrid ring [20] coupler is required prior to the antenna. No shorting post(s) [21] are required to achieve impedance matching, suppression of high-order modes, or symmetrical radiation patterns. III. RESULTS AND DISCUSSIONS A. Measurements All simulations were conducted using IE3D, a full wave EM simulator. A finite ground plane size is used in all simulations. The measurements were taken using a HP 8510C Vector Network Analyzer (VNA) and the MIDAS far-field measurement system in an anechoic chamber. Fig. 2 compares the measured and simulated S -parameters and gain profile. The impedance bandwidth for reflection coefficient

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Fig. 4. Simulation co- and cross-polarization radiation patterns at 4.0 GHz.

S11 j < 010 dB is 3.1–4.8 GHz (43%). There is a 0.1 GHz down shifting for the measured results. The discrepancies between the measured and simulated reflection coefficients from 4.5 GHz onwards are due to fabrication tolerances. Fig. 3(a)–(c) shows the co- and cross-polarization radiation patterns in the principle planes. The E- and H-planes radiation patterns are symmetrical across the lower UWB band as the dipole is fed differentially at the center. Their 3-dB beamwidth are 48 –58 and 70 –82 respectively. The unwanted radiation null at  = 45 along the E-plane, which results in a squinted E-plane radiation pattern for non-symmetrically fed broadband patch antenna has been circumvented. The crosspolarization levels of this antenna remained low, especially at low elevation angle. Due to the prototyping tolerance and the dynamic range of the chamber, the measured cross-polarization levels are higher than that of the simulation—compare Figs. 3(b) and 4. The relatively wider width of the dipole and its coupling with the integrated balun increase the cross-polarization slightly. The low back radiation minimizes the interference to the adjacent sector antennas. This antenna has a simulation radiation and antenna efficiencies of > 99% and > 90% across the operating frequency respectively. j

B. Parametric Studies Fig. 5(a)–(e) show the simulated parametric studies on the reflection coefficient jS11 j and gain profile, when the half dipole length l, feed gap g , microstrip height h1 , overall antenna height h2 , and ground plane size are varied. Fig. 5(a) shows that increasing the length l by 2 mm (8.5%) increases the lower 3.4 GHz resonance to 3.25 GHz (4.4%) while the higher 4.75 GHz resonance remains unchanged. Fig. 5(b) shows that increasing the gap g from 2 mm to 3 mm (50%), results in a 0.1 GHz (5.8%) reduction of impedance bandwidth. Fig. 5(c) shows that the dipole antenna is not sensitive to the +1 mm (40%) height h1 variation of the 5 mm wide microstrip line. Fig. 5(d) shows that this antenna is not sensitive to the +2 mm (15%) variation in the overall height h2 . Finally, Fig. 5(e) shows that the performance of this dipole antenna is not ground dependent from 80 2 80 mm (0.8 2 0.8 0 ) onwards. Throughout these parametric studies, the gain fluctuate less than 1 dBi. Therefore this dipole antenna has a high manufacturing tolerance. IV. ANALYSIS A. Resonance The lower fLow and higher fHigh resonances of this antenna form a broadband impedance bandwidth. The half wavelength resonances of this antenna are determined by (a) the approximated total length of the dipole 2l and (b) the sum of one arm of the dipole and vertical feed l + h2 0 h1 . The coupling between the feed gap, balun, and dipole affect the actual resonances slightly. The resonances are calculated to be fLow = 3:2 GHz (5.8%) and fHigh = 4:4 GHz (4.3%).

Fig. 5. Parametric studies by varying the (a) length l , (b) gap g , (c) height h , (d) height h , and (e) ground plane size.

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to the square of the current on the dipole, and inversely proportional to the bandwidth. This results in conductor and mismatch losses. This antenna achieved a relatively constant gain profile across a broadband of frequencies, compared to a thin dipole antenna with a reflector. V. CONCLUSION A low profile dipole antenna with an integrated balun-transformer is presented. The balun does not affect the operation of the antenna. This broadband UWB antenna is fed differentially at the center, and it featured a symmetrical broad beamwidth radiation patterns in the -plane with low cross-polarization and consistent gain. The parametric studies showed that this antenna has a high manufacturing tolerance. Also, the analytical and simulated gain agrees well with that of measurement.

H

Fig. 6.

S -parameters for two 10 M probes applied at loci A and B of Fig. 1.

B. Balun In order to illustrate the phase difference between the two vertical probes were placed at loci A and strips using simulation, two 10 B as shown in Fig. 1. They are assigned as ports 2 and 3 respectively. The simulation results are presented in Fig. 6. There are no significant changes in the reflection coefficients jS11 j, as compared to that of . Fig. 2. Also, both the magnitudes of jS21 j and jS31 j are < 0 Therefore, the probes do not overload the operation of the antenna. The amplitude difference between jS21 j and jS31 j is less than 1 dB from 3.1–4.5 GHz. The phase difference between 6 S21 and 6 S31 is 180 from 3.0–4.8 GHz. Therefore, a broadband differential feed for the dipole antenna is achieved.

M

50 dB

C. Gain and Radiation Pattern

24

The gain and radiation pattern of a horizontal center-fed = dipole above a perfectly conducting ground (xy plane) are expressed using (1) and (2) respectively [22]

R11 + R1L R11 + R1L Rm

G ()=10log

0

2sin(hr cos ) 1:62 (dBi)

j

j2

 E (; )= cos 2 cos2  cos2  sin(hr sin ) 1 cos cos 

(1) (2)

0

= (2 )

2

where hr = h, h is the height of the = antenna above a perfectly conducting ground plane, R11 is the self-resistance of = antenna, R1L is the loss resistance of = antenna, Rm is the mutual resistance of = antenna and its image at a distance of h, R1 R11 0 Rm is the driving-point radiation resistance, and  is the angle of elevation. The feed-point resistance R1 is plotted as a function of the height of a horizontal = dipole over a perfectly conducting ground in [22]. It varies towards the natural 73 as the effect of mutual resistance , Rm reduces with greater separation. Using (1) with R1L , and adding 3 dBi for the enlarged radiation aperture, the gain for the dipole antenna is calculated to be 8–8.7 dBi. This agrees well with that of simulation and measurement in Fig. 2. The reflected wave from the ground plane has an equal magnitude and 180 out of phase with respect to that of the dipole. The maximum radiation pattern is always at boresight for a spacing less than = . The is 13% and 20.8% of a 3.0 GHz and 4.8 GHz waveheight h2 lengths, respectively. Hence this antenna operates as a patch antenna at the lower frequency, and a reflector antenna at the higher frequency. The maximum gain increases slightly as the dipole is brought closer to the ground plane, due to the increase of the coupling factor. However it reduces drastically as the spacing is further reduced, due to the sharp decrease in pattern factor. The increase in the -value is proportional

2

2

2

13



2

2 =

= 1

=

4

= 13mm

Q

REFERENCES [1] First Report and Order Federal Communication Commission (FCC), 2002. [2] A. F. Gangi, S. Sensiper, and G. R. Dunn, “The characteristics of electrically short umbrella top-loaded antenna,” IEEE Trans. Antennas Propag., vol. 13, no. 6, pp. 864–871, Nov. 1965. [3] K.-L. Lau, P. Li, and K.-M. Luk, “A monopolar patch antenna with very wide impedance bandwidth,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 655–661, Feb. 2005. [4] N. Behdad and K. Sarabandi, “A compact antenna for ultrawide-band applications,” IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2185–2192, Jul. 2005. [5] P. J. Gibson, “The Vivaldi aerial,” in Proc. Eur. Microw. Conf., Oct. 1979, pp. 101–105. [6] F. Yang, X. Zhang, X. Ye, and Y. Rahmat-Samii, “Wideband E-shaped patch antennas for wireless communications,” IEEE Trans. Antennas Propag., vol. 49, no. 7, pp. 1094–1100, Jul. 2001. [7] T. Huynh and K. F. Lee, “Single-layer single patch wideband microwave antenna,” IEE Electron. Lett., vol. 31, no. 16, pp. 1310–1312, Aug. 1995. [8] K. M. Luk, C. L. Mak, Y. L. Chow, and K. F. Lee, “Broadband microstrip patch antenna,” IEE Electron. Lett., vol. 34, no. 15, pp. 1442–1443, Jul. 1998. [9] H. Wong, K.-M. Mak, and K.-M. Luk, “Wideband bowtie patch antenna with electric dipole,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 2098–2101, Jul. 2008. [10] D. M. Pozar, “A microstrip antenna aperture coupled to a microstip line,” Electron. Lett., vol. 21, pp. 49–50, Jan. 1985. [11] Z. N. Chen, “Broadband suspended plate antenna with concaved center portion,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1550–1551, Apr. 2005. [12] Z. N. Chen and M. Y. W. Chia, “Broad-band suspended probe-fed plate antenna with low cross-polarization antenna,” IEEE Trans. Antennas Propag., vol. 51, no. 2, pp. 345–347, Feb. 2003. [13] Z. N. Chen and M. Y. W. Chia, “A novel center-slot-fed suspended plate antenna,” IEEE Trans. Antennas Propag., vol. 51, no. 6, pp. 1407–1410, Jun. 2003. [14] C. R. Medeiros et al., “Wideband slot antenna for WLAN access points,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 79–82, 2010. [15] C. H. K. Chin, Q. Xue, and H. Wong, “Broadband patch antenna with a folded plate pair as a differential feeding scheme,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2461–2467, Sep. 2007. [16] W. K. Toh and Z. N. Chen, “On a broadband elevated suspended-plate antenna with consistent gain,” IEEE Antennas Propag. Mag., vol. 50, no. 2, pp. 95–105, Apr. 2008. [17] W. K. Toh, Z. N. Chen, and X. Qing, “A planar UWB antenna with a broadband feeding structure,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 2172–2175, Jul. 2009. [18] L. Walter, A. Gothe, and H. D. Rosenstein, DRP, vol. 568, p. 559, Nov. 1931. [19] J. D. Kraus and R. J. Marhefka, Antennas, 3rd ed. New York: McGrall-Hill, 1988, p. 813. [20] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998, ch. 7, pp. 401–407. [21] D. Schaubert, F. Farar, A. Sindoris, and S. Ayes, “Microstrip antennas with frequency agility and polarization diversity,” IEEE Trans. Antennas Propag., vol. 29, pp. 118–123, Jan. 1981. [22] J. D. Kraus and R. J. Marhefka, Antennas, 3rd ed. New York: McGraw-Hill, 1988, p. 557.

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Compact Dielectric Resonator Antennas With Ultrawide 60%–110% Bandwidth Yuehe Ge, Karu P. Esselle, and Trevor S. Bird

Abstract—The design of rectangular dielectric resonator antennas (DRA) with ultrawide bandwidths, in the range of 60–110%, is described. The DRA exploits multiple low-Q modes with overlapping bandwidths to achieve a wide contiguous bandwidth. This is achieved using a full-length, low-permittivity inset between a higher permittivity dielectric volume and a ground plane. With the proposed dielectric arrangement and a feed inside DR, it is possible to efficiently couple a sufficient number of such feedline. The volume of such DRAs is also overlapping modes to a 50 reduced by means of a finite planar conducting wall. These advantages led to an example design with a bandwidth that is significantly wide and at the same time has a smaller DR volume than conventional DRAs. A prototype antenna designed to operate in the FCC UWB band from 3.1 to 10.6 GHz at 3.1 has a dielectric volume of 12 8 15.2 mm (or GHz), and an average measured gain of 5 dB over the band.



1 7 10

Index Terms—Dielectric resonator antenna (DRA), FCC, stacked antenna, ultrawideband (UWB).

I. INTRODUCTION Dielectric resonator antennas (DRAs) have attracted considerable attention since 1980s due to many advantages such as high efficiency, small size and simplicity. Various investigations offered significant enhancements to the DRAs [1], improving performance figures such as bandwidth and polarization. Methods proposed to broaden the bandwidth of DRAs [2]–[12] include the use of multiple stacked dielectric resonator (DR) elements [2]–[5], a thinner dielectric segment with a higher permittivity between the DR and the ground [1], [6], specially shaped DRs and a notch in the base of the DR [1], [8]. The hybridization of the DR with a metallic patch [9], [10] is another approach to achieve a wider bandwidth. A combination of a DR and a patch antenna, each resonating at different frequencies, provides a bandwidth that is wider than the individual bandwidth of the corresponding DRA and patch antennas. In addition, some special feeding structures can be applied to obtain wideband DRAs [11], [12]. Although the high permittivity of the DR itself leads to a small volume, sometimes the resulting DRAs are still too large and hence additional techniques are required to reduce their size. In one approach [13], a shorting plate was used to design half-volume DRAs and image theory has been applied in [14] for the same purpose. In addition, a multi-segment DRA with a shorting wall (e.g., edge ground) and a probe feed on DR surface has been investigated [15]. All result in about 50% volume reduction of the DR. In some special cases, a DRA can be designed to have an ultrawide bandwidth (UWB). In [16], [17], UWB DRAs have been realized through a hybrid approach, by loading a metal monopole with an annular DR, and in [18], an airgap between the DR and ground plane is used to obtain an ultrawideband impedance bandwidth. Manuscript received March 15, 2010; revised February 07, 2011; accepted February 19, 2011. Date of publication July 14, 2011; date of current version September 02, 2011. This work was supported by the Australian Research Council. Y. Ge is with the College of Information Science & Engineering, Huaqiao University, Xiamen, Fujian Province 361021, China (e-mail: [email protected]). K. P. Esselle is with the Department of Electronic Engineering, Faculty of Science, Macquarie University, Sydney, NSW 2109, Australia. T. S. Bird is with Macquarie University, Sydney, NSW 2109, Australia and also with the CSIRO ICT Centre, Epping, NSW 1710, Australia. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161538

Fig. 1. A dielectric resonator antenna with a low-permittivity insert.

Recently we have investigated a class of DRAs, in which a lower-permittivity dielectric insert of full length is employed between the DR and thegroundplane,andasignificant bandwidth enhancement was achieved [7],[19]–[21]. Ourpreliminaryresultsindicated the potentialofthis technique to provide impedance bandwidths as large as 100% that are required for FCC UWB and other ultrawideband applications. The aim here is to fully describe and exploit this technique to design UWB antennas with close to 100% bandwidths for such communication systems. II. CHARACTERIZATION OF HALF-SIZE DRAS An example configuration is shown in Fig. 1. The DR, which has a dielectric constant "1 , is placed above a thin low-permittivity insert (LPI) of a dielectric constant "2 ("2 < "1 ). The basic concept behind this approach is that when a dielectric insert of lower permittivity is introduced between the DR and the ground, the bandwidth of each resonance mode will increase, due to the decrease of the Q-factor. This is the same concept employed in DRA designs with air-filled notches but, unlike in notched DRAs, the low-permittivity insert in our designs are of full-length and it completely separates the high-permittivity DR from the ground plane, forcing all electric field lines of force to pass through the low-permittivity region. This feature, and the use of a probe feed inside the DR, enabled us to couple a sufficient number of overlapping modes to a simple 50 feed and achieve a reasonable VSWR over an ultrawide (60%–100%) contiguous band. Furthermore, following previous investigations [13]–[15], a conducting wall is placed at one side of the rectangular DRA, in order to halve the volume of the DRA. In our rectangular DRA example, the DR and the LPI have dimensions of a 2 b 2 c and a 2 b 2 d, respectively. The DRA is fed by a coaxial probe. A. Effect of a Conducting Wall in a Half-Size DRA The predicted and measured results of a full-size wideband rectangular DRA, which is shown in Fig. 2(a), was reported in [7]. The dimensions of this DRA are a1 2 b 2 c (2a 2 b 2 c). A wide bandwidth of 66% was achieved by using the LPI (e.g., hard foam) inserted between a rectangular DR and the ground plane. The application of a shorting conducting wall to a standard rectangular DRA halves the dielectric volume and footprint of the DRA while maintaining a wide bandwidth. This is illustrated in Figs. 1 and 2. The full-size DRA in Fig. 2(a) is cut in half and an infinite PEC wall is introduced, as shown in Fig. 2(b), to form the image. Then the infinite PEC wall is truncated to the size of the DRA face, shown in Fig. 2(c), for convenience in implementation. The result is the half-size DRA with a low-permittivity insert, shown in Fig. 1. To investigate the effect of the size-reduction technique mentioned above, let us first consider a conventional DRA that has no LPI, i.e.,

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Fig. 2. Application of a shorting conductor to reduce size: (a) a full-size DRA; (b) a half-size DRA with infinite PEC wall; (c) a half-size DRA with a finite conductor on one face.

Fig. 4. VSWR of half-size DRAs with and without low-permittivity inserts. Insert thickness is d; probe length is h. TABLE I BANDWIDTH VERSUS INSERT PERMITTIVITY

B. Bandwidth Enhancement of Half-Size DRAs Using Lower-Permittivity Inserts

Fig. 3. VSWR of three rectangular DRAs: without a PEC wall (i.e., full-size), with an infinite PEC wall, and with a finite conductor.

= 0. Let a full-size rectangular DRA has the following parameters: length = 20 mm, b = 12 mm, c = 12 mm and "1 = 9:2. After cutting this DR in half along its length, the length a of the corresponding half-size DRA is 10 mm. Particular attention needs to be paid to the first resonance frequency because achieving a reasonable VSWR at lower frequencies is a major challenge in UWB antenna design. The VSWR values predicted by CST Microwave Studio for the three DRAs are shown in Fig. 3. It can be seen that the resonance frequency of the lowest mode of the half-size DRA with an infinite PEC wall is very close to that of the full-size DRA, which can be approximately estimated using the well-known dielectric waveguide model [22], whereas the first resonance frequency of the half-size DRA with a finite conducting wall has decreased slightly when the conducting wall is truncated only to cover the face of the DR. The lowest resonance frequency of the antenna is not significantly affected when the DR volume is halved with a large conducting wall. When the conducting plane is truncated, the lowest resonance frequency decreases, giving an additional advantage in size reduction, although the bandwidth slightly decreases. For example, when the ground plane is 2 2 2 at 3 GHz, the first resonance frequencies of the two half-size DRAs with infinite and finite walls are 3.90 GHz and 3.26 GHz, respectively. The bandwidths of the same two DRAs are 7.2% and 6.8%, respectively. Note that these bandwidths are less than the 9.6% bandwidth of the corresponding full-size DRA, which has a resonance frequency of 3.86 GHz. In brief, with reasonably large ground planes, a DRA with a finite conductor enables approximately 50% volume and footprint reduction with a small decrease in resonance frequency.

d

The technique based on full-length lower-permittivity inserts is now applied to make the radiating modes to overlap and hence to extend the bandwidth of half-size DRAs to ultrawideband scale. To illustrate, a RT/Duroid 5880 insert that has a dielectric constant of 2.2 was introduced and the probe length was readjusted to obtain a wide VSWR bandwidth. The side-wall conductor covers both the DR and the dielectric insert faces, as shown in Fig. 1. The results obtained from CST Microwave Studio for three combinations of inserts and probes, shown in Fig. 4, indicate that a VSWR bandwidth up to 74% (3.13–6.79 GHz) is achievable when d = 3 mm and h = 4 mm. This compares with only 6.8% for the DRA without the insert, i.e., d = 0. Note that the optimization of the position and the length (h) of the probe is crucial to obtain the best bandwidth from this type of antennas. The inserts with other dielectric constants, from 1 to 9.2, have also been considered and their effect on VSWR bandwidth is listed in Table I. In these simulations, the dimensions of the DRA are the same as those in Fig. 4 and the length (h) of the probe was adjusted to get the best bandwidth for each case. It can be seen from Table I that better VSWR bandwidths can be obtained when "2 is in the range of 2.2–7.7, due to the overlapping of different operating modes. The best bandwidth is found when "2 = 6:6, where different operating modes overlap to create a contiguous bandwidth. Multiple layers of full-length inserts with different low permittivities were also considered and no clear advantage was noted over the use of a single-layer LPI. The radiation patterns of the half-size stacked DRAs were also studied, using CST Microwave Studio. It is found that the levels of E and E' will not change much in E and H planes, but the pattern shapes changed due to the presence of the conducting wall, as shown in Fig. 5. In summary, a DRA with a side-wall conductor and a LPI can be designed to exhibit an extremely wide VSWR bandwidth and a reduced size. When designing reduced-size wideband DRAs, the dielectric waveguide model [22] can be used to estimate the initial values of the DR dimensions—a; b and c. The resonance frequency of the lowest mode of the DR may be set close to the lower end of the desired operating band as the starting point. Assuming that dielectric permittivity

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Fig. 5. E patterns on the E-plane at the first resonant modes of the full-size and half-size DRAs, respectively.

Fig. 7. Measured radiation patterns (E and planes (XOZ and YOZ).

E

) in the principal orthogonal

Fig. 6. Measured and predicted VSWR of the half-size DRA with a low permittivity insert.

values are fixed, the remaining design parameters of the antenna are a; b; c; d; s1 ; s2 , and h. Parameters a; b; c and d can be selected to determine the overall operating band of the DRA and the parameters s1 ; s2 , and h can be used to fine-tune the operating band and/or to achieve good impedance matching within the band. III. UWB DRA DESIGN The design techniques described above were applied to obtain a compact DRA design that has an acceptable VSWR over the FCC UWB bandwidth of 3.1–10.6 GHz (i.e., 109.5%). The material selected for the DR and dielectric insert are TMM10 and RT/Duroid 5880 where the dielectric constants are 9.2 and 2.2, respectively. In the design process, the dielectric waveguide model [22] was used to estimate the initial dimensions (a; b and c) of the DRA. Then HFSS and CST Microwave Studio software packages were used to fine-tune and optimize the design, while adjusting the probe position (s1 and s2 ), probe length (h) as well as the thickness (d) of the dielectric insert. The parameters of the final design are: a = 12 mm, b = 8 mm, c = 12 mm, d = 3:2 mm, h = 5:2 mm, s1 = 3 mm and s2 = 4 mm. The size of the ground plane in simulation model is 40 2 40 mm2 . The total dielectric volume is 12 2 8 2 15.2 mm3 or 0:1240 2 0:0830 2 0:1570 at the lowest operating frequency of 3.1 GHz. The design was fabricated and the planar conductor was initially implemented using copper tape. The geometry of this antenna is identical to that in the simulations, except the ground plane was increased in

Fig. 8. Measured and computed gain of the wideband antenna design.

size to 100 2 100 mm2 . The measured VSWR of the antenna prototype is shown in Fig. 6, together with the VSWR predicted by HFSS. The theoretical VSWR is less than 2 over a 109% bandwidth, from 3.1 GHz to 10.7 GHz. The measured VSWR is close to the predicted values although around 6.3 GHz and 8.3 GHz it slightly exceeded 2 (VSWR = 2:2). Fig. 7 illustrates the measured radiation patterns of the antenna at frequencies of 3.2 GHz, 6 GHz and 10 GHz. It can be seen that, in the YOZ plane, radiation patterns at the three frequencies have similarities (such as cross-polar minima in the upward +z direction indicated by the E curves on YOZ plane). The patterns on the orthogonal XOZ plane change somewhat with frequency because different dielectric resonance modes become dominant as the frequency is swept. Nevertheless, there are no significant nulls in the upper hemisphere at any frequency. The total power radiated into the lower hemisphere is significantly less than that into the upper hemisphere. Obviously these patterns will be distorted when the antenna is integrated into a small space in a compact wireless communication device but the antenna is still expected to radiate well into the upper hemisphere. The gain of the DRA was also measured over the FCC UWB band and the results are

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plotted in Fig. 8, together with the computed gain, obtained from CST Microwave Studio, for comparison. IV. CONCLUSION We have demonstrated that the VSWR bandwidth of rectangular DR antennas can be increased substantially by introducing a full-length lower-permittivity insert between the DR and the ground plane. The technique was verified and its advantages have been demonstrated by applying it to half-size rectangular DRA designs. For example, an experimental half-size DRA antenna with a lower-permittivity insert was able to cover the FCC UWB band from 3.1 to 10.6 GHz with a maximum VSWR of 2.2. The theoretical 2:1 VSWR bandwidth of this design is over 109.5%. It has a small footprint of 12 2 8 mm2 , or 0:1240 2 0:08320 at 3.1 GHz. Its dielectric volume is 1459 mm3 , or 1:7 2 1003 30 at 3.1 GHz, and overall height is 15.2 mm or 0:1570 .

REFERENCES [1] A. Petosa, A. Ittipiboon, Y. M. M. Antar, D. Roscoe, and M. Cuhaci, “Recent advances in dielectric-resonator antenna technology,” IEEE Antennas Propag. Mag., vol. 40, no. 3, pp. 35–47, 1998. [2] A. A. Kishk, B. Ahn, and D. Kajfez, “Broadband stacked dielectric resonator antennas,,” Electron. Lett., vol. 25, no. 18, pp. 1232–1233, Aug. 1989. [3] S. M. Shum and K. M. Luk, “Stacked annular-ring dielectric resonator antenna excited by axis-symmetric coaxial probe,” IEEE Trans. Antennas Propag., vol. 43, pp. 889–892, Aug. 1995. [4] A. A. Kishk, X. Zhang, A. W. Glisson, and D. Kajfez, “Numerical analysis of stacked dielectric resonator antennas excited by a coaxial probe for wideband applications,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1996–2006, Aug. 2003. [5] A. G. Walsh, S. D. Young, and S. A. Long, “An investigation of stacked and embedded cylindrical dielectric resonator antennas,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 130–133, 2006. [6] A. Petosa, N. Simons, R. Siushansiana, A. Ittipiboon, and M. Cuhaci, “Design and analysis of multisegment dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 48, pp. 738–742, May 2000. [7] Y. Ge, K. P. Esselle, and T. S. Bird, “A wideband probe-fed stacked dielectric resonator antenna,” Microw. Opt. Tech. Lett., vol. 48, no. 8, pp. 1630–1633, Aug. 8th, 2006. [8] A. Kishk, Y. Yin, and A. W. Glisson, “Conical dielectric resonator antennas for wideband applications,” IEEE Trans. Antennas Propag., vol. 50, pp. 469–474, 2002. [9] K. P. Esselle and T. S. Bird, “A hybrid-resonator antenna: Experimental results,” IEEE Trans. Antennas Propag., vol. 53, pp. 870–871, 2005. [10] J. Janapsatya, K. P. Esselle, and T. S. Bird, “Compact wideband dielectric-resonator-on-patch antenna,” Electron. Lett., vol. 42, no. 19, pp. 1071–1072, 2006. [11] S. K. Menon, B. Lethakumary, P. Mohanan, P. V. Bijumon, and M. T. Sebastian, “Wideband cylindrical dielectric resonator antenna excited using an L-strip feed,” Microw. Opt. Tech. Lett., vol. 42, no. 4, pp. 293–294, Aug. 2004. [12] M. Lapierre, Y. M. M. Antar, A. Ittipiboon, and A. Petosa, “Ultrawideband monopole/dielectric resonator antenna,” IEEE Microw. Wireless Compon Lett., vol. 15, no. 1, pp. 7–9, Jan. 2005. [13] M. T. K. Tam and R. D. Murch, “Half volume dielectric resonator antenna designs,” Electron. Lett., vol. 33, no. 23, pp. 1914–1916, 1997. [14] S. G. O’Keefe, S. P. Kingsley, and S. Saario, “FDTD simulation of radiation characteristics of half-volume HEM and TE-mode dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 50, no. 2, pp. 175–179, Feb. 2002. [15] A. Petosa, Dielectric Resonator Antenna Handbook. Norwood, MA: Artech House, 2007, ch. 6. [16] Y.-W. Chan and K.-M. Luk, “The small UWB hybrid antenna,” Microw. Opt. Tech. Lett., vol. 49, no. 9, pp. 2157–2159, 2007. [17] K. S. Ryu and A. A. Kishk, “UWB dielectric resonator antenna mounted on a vertical ground plane edge,” presented at the IEEE Int. Symp.on Antennas and Propagation, Charleston, SC, Jun. 2009. [18] T. A. Denidni and Z. Weng, “Rectangular dielectric resonator antenna for ultrawideband applications,” Electron. Lett., vol. 45, no. 24, pp. 1210–1212, 2009.

[19] Y. Ge and K. P. Esselle, “A UWB probe-fed dielectric resonator antenna,” presented at the IEEE 69th Vehicular Technology Conf. (VTC2009), Barcelona, Spain, Apr. 2009. [20] Y. Ge and K. P. Esselle, “An extremely compact DRA for space-limited UWB communications,” presented at the Eur. Wireless Technology Conf. (EuWIT), Rome, Italy, Sep. 28–Oct. 2 2009. [21] Y. Ge and K. P. Esselle, “A dielectric resonator antenna for UWB applications,” presented at the IEEE Int. Antennas Propag. Symp. Dig., , North Charleston, SC, Jun. 1–5, 2009. [22] R. K. Mongia, “Theoretical and experimental resonant frequencies of rectangular dielectric resonators,” in Proc. Inst. Elect. Eng., Feb. 1992, vol. 139, pp. 98–104.

Zeroth-Order Resonator Antennas Using Inductor-Loaded and Capacitor-Loaded CPWs Chien-Pai Lai, Shih-Chia Chiu, Hsueh-Jyh Li, and Shih-Yuan Chen Abstract—Two compact zeroth-order resonator (ZOR) antennas designed with inductor-loaded (IL) and capacitor-loaded (CL) coplanar waveguides (CPWs) are proposed. A generalized definition of the zeroth-order resonance is also presented, taking into account the losses of the host transmission lines. Analytic formulas are also derived to predict the reactance values needed to sustain a specified zeroth-order resonant frequency in a lossy host line. With these formulas, one may predict the input reflection coefficients for the proposed ZOR antennas. These antennas, using only two unit cells, are designed, fabricated, and tested, and they have a very compact size and a via-free uniplanar structure. The measured results agree very well with those acquired from the proposed method and full-wave simulations, thus verifying the accuracy of the proposed formulation. It is also demonstrated that the ZOR antenna using the CL CPW exhibits higher radiation efficiency. Index Terms—Coplanar waveguides, periodic structures, zeroth-order resonator antennas.

I. INTRODUCTION Recently, composite right/left-handed (CRLH) transmission lines (TLs), composed of shunt inductors and series capacitors periodically loaded along host TLs, have drawn increasing attention because of several unusual properties that they possess. One especially unusual property is zeroth-order resonance [1], [2]. A CRLH TL operating at its zeroth-order resonance has a zero phase constant at a nonzero frequency, also called the infinite-wavelength property. This property can be exploited to design miniature antennas since the resonance condition is independent from the physical length of the antenna [3]–[6]. It is also known that unbalanced CRLH TLs can provide two distinct zeroth-order resonances, namely the shunt and series resonances [1], [2]. However, only one of them would be practical in input impedance Manuscript received November 24, 2010; revised February 17, 2011; accepted February 19, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the National Science Council, Taiwan under Contract NSC 99-2221-E-002-059 and in part by the National Taiwan University under Excellent Research Project NTU-ERP98R0062-01. C.-P. Lai and S.-C. Chiu are with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: [email protected]; [email protected]). H.-J. Li and S.-Y. Chen are with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161561

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Fig. 2. Geometries of the unit cells of (a) the IL CPW realized by a pair of folded shorting stubs, and (b) the CL CPW realized by an IDC. Fig. 1. Transmission line models of (a) an inductor-loaded transmission line (IL TL) and (b) a capacitor-loaded transmission line (CL TL).

matching since the CRLH TL is either open- or short-circuited [4]. Consequently, the resonance that is not used may be removed to decrease the design complexity of CRLH TLs, leading to the so-called inductor-loaded (IL) TLs [7], [8] and the proposed capacitor-loaded (CL) TLs. The former sustains only the shunt resonance, while the latter sustains the series resonance. The zeroth-order shunt resonance of IL TLs has been investigated in [7] and [8]; however, the host TLs were assumed to be lossless to facilitate the dispersion analysis. To the authors’ best knowledge, no research has focused on the CL TLs. In this communication, a methodology capable of analytically solving the dispersion relations of lossy IL and CL TLs is proposed and employed to analyze the IL and CL CPWs. A CPW is adopted as the host TL due to its uniplanar structure and easy realization of series and shunt connections of inductors and capacitors. For both lossy and lossless structures, the resonance conditions are discussed, and analytic formulas are derived to predict the reactance values needed to sustain a specified zeroth-order resonant frequency. Moreover, two ZOR antennas formed by the IL and CL CPWs are proposed, and the design process, which is greatly simplified by the above methodology, is presented. Finally, we compare the performances of our proposed antennas to those of other ZOR antennas [7]–[10]. II. INDUCTOR-LOADED AND CAPACITOR-LOADED CPWS Both the IL and CL TLs are periodic wave-guiding structures. The unit cell of an IL TL is in general composed of two identical sections of the host TL and a shunt inductor in between, while a CL TL has a series capacitor instead. Figs. 1(a) and (b) depict the transmission line models of the unit cells of IL and CL TLs, respectively. Each host TL section may be further replaced by a pair of a series inductor and a shunt capacitor since its length is always much shorter than a guided wavelength [11], [12]. Therefore, in the IL TL, the shunt pair of the loading inductor and the inherent capacitor of the host TL would produce a zeroth-order resonance, which is also called the shunt resonance. Similarly, in the CL TL, the series pair of the loading capacitor and the inherent inductor of the host TL would also produce a zeroth-order resonance, the so-called series resonance. Although other wave guiding structures can also be used as the host TL for forming an IL or CL structure, the CPW is utilized in this work due to its conformity, uniplanar structure, and easy realization of series capacitors and shunt inductors without using via holes. The unit cells of the IL and CL CPWs, which are the constituent parts of the proposed ZOR antennas, are described as follows. A. Inductor-Loaded Coplanar Waveguides The geometry of the unit cell of the IL CPW is shown in Fig. 2(a). The shunt inductance is realized using a pair of shorting stub inductors (SSIs) protruding from the central strip into the bilateral ground planes. The

value of the shunt inductance LL can be estimated based on the transmission line theory with LL = Zc;SSI 2 tan( SSI lSSI )=2! , where Zc;SSI , SSI , and lSSI are the characteristic impedance, propagation constant, and total length (lSSI = a + b + c) of the SSI structure, respectively. The factor of two in the denominator is added since a pair of SSIs are used to keep a symmetric structure. The stubs are folded to keep a compact unit cell. The only drawback from folding the shorting stubs is the presence of unwanted resonances, which may affect the designed zeroth-order resonance. However, this can be mitigated by properly choosing the dimensions of the stubs, especially the arm lengths a, b, and c. B. Capacitor-Loaded Coplanar Waveguides The geometry of the unit cell of the CL CPW is shown in Fig. 2(b). An interdigital capacitor (IDC) is used as the series capacitor. The commonly used  -model for an IDC is simplified into a series capacitor, since the capacitance between the IDC and the bilateral ground plane is much smaller than the shunt capacitance of the host CPW. The series capacitance of the IDC can be calculated with the empirical formulas presented in [13]. Both the unit cells can provide excellent control over the zeroth-order resonant frequency. III. DISPERSION ANALYSIS A. Dispersion Relations Since the IL and CL CPWs are periodic structures, the propagation constant and characteristic impedance ZB of the Bloch wave propagating along the structure can be expressed in terms of the parameters of the ABCD matrix of the unit cell [14]:

cosh L = A 0 ZB = p BZ : A2 0 1

(1a) (1b)

where Z0 is the characteristic impedance of the host CPW and L is the length of the unit cell. The closed-form expressions for Z0 and the propagation constant of the host CPW CPW = CPW + j CPW are given in [15] and [16]. Note that (1a) and (1b) are derived based on the assumption that the unit cells are symmetric, and hence A = D . First, consider an IL CPW composed of unit cells as depicted in Fig. 2(a). The ABCD matrix for the unit cell can be derived as:

cosh 2 L Z0 sinh 2 L

L cosh 2 L Z sinh 2 2 11 10 j!L

L Z0 sinh 2 L : 2 2 1cosh

L cosh 2 L Z sinh 2

A B C D =

1

(2)

With the expression for the matrix parameter A, (1a) can be rewritten as:

cosh L cos L + j sinh L sin L = cosh  cos  Z0 (sinh  cos  + j cosh  sin ): +j sinh  sin  + 2j!L L

(3)

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where =  + j and CPW L = CPW L + j CPW L Taking the real and imaginary parts of (3) yields:

=  + j.

 sin  cosh L cos L = cosh  cos  + Z0 cosh 2!LL  cos  sinh L sin L = sinh  sin  0 Z0 sinh 2!LL :

(4a) (4b)

One may solve these two equations for the attenuation and phase constants of the Bloch wave, namely  and , for the IL unit cell. Since the solution process is lengthy, it is omitted. The closed-form expressions for  and thus obtained are:

N 2 +(M +1)2 +

1 cosh01

=

L

=

1 cos01

2 2 2 N +(M +1) 0 2

L

N 2 +(M 01)2 N2

+(M 01)

2

(5)

:

S = 0:2 mm L = 7 mm H = 1:6 mm

W = 3:9 mm, L = 4:65 nH

Fig. 3. Dispersion diagrams of lossless and lossy IL CPWs. ( .) , , , and

(6)

where M and N represent the right-hand sides of (4a) and (4b). Likewise, the associated formulas for  and for the CL unit cell shown in Fig. 2(b) can also be derived, and they are of the same mathematical form as (5) and (6), except for the expressions for M and N , which should be modified like so:

 sin  = cosh  cos  + cosh 2!Z0 CL sinh  cos  : N = sinh  sin  0 2!Z0 CL

M

(7a) (7b)

Consequently, (5) and (6) can be utilized to acquire the dispersion relations of both the IL and CL CPWs. Most important of all, they are generally applicable to IL and CL structures formed by any host transmission lines, such as the microstrip line, coplanar strip, or stripline, as long as the closed-form or empirical expressions for the characteristic impedance and propagation constant of the host line are given. B. Zeroth-Order Resonance In this subsection, the dispersion characteristics of the IL and CL CPWs will be investigated by using the above formulas, and the condition of the zeroth-order resonance will be discussed. First, consider the IL CPW, of which the unit cell is depicted in Fig. 2(a), and assume the host CPW to be lossless. In this case, the IL CPW is designed on a lossless substrate, say the FR4 substrate ("r = 4:4) but with a vanishing loss tangent (tan  = 0), with its metallic cladding made of a perfect electrical conductor. For demonstration, the design parameters of the IL unit cell are chosen as W = 3:9 mm, S = 0:2 mm, L = 7 mm, H = 1:6 mm, and LL = 4:65 nH. The dispersion relation of such a lossless IL CPW can be calculated with (5) and (6), and the results obtained are plotted in Fig. 3. One can observe that as the frequency is increased up to 2.5 GHz, the attenuation constant  decreases monotonically to zero while the phase constant remains zero. Beyond 2.5 GHz,  is constantly zero while increases monotonically. For a lossless CL CPW, the design parameters of the unit cell shown in Fig. 2(b) are chosen as W = 4:8 mm, S = 4 mm, L = 7 mm, H = 1:6 mm, and CL = 1:2 pF. The calculated dispersion curves are plotted in Fig. 4, and they closely resemble their IL counterparts. For the lossless cases, the threshold frequency, at which  = = 0, is known as the zeroth-order resonant frequency. The most interesting feature of this resonance is the so-called infinite wavelength property, which in theory could produce a uniform field distribution along the entire structure. As a result, the zeroth-order resonant frequency is independent of its physical dimensions. As mentioned in Section II, the zeroth-order resonances of the IL and CL CPWs are in the form of shunt and series resonances, respectively. The associated resonant frequencies are hereafter denoted as !sh and !se .

S = 4 mm L = 7 mm H = 1:6 mm

C = 1:2 pF

Fig. 4. Dispersion diagrams of lossless and lossy CL CPWs. ( .) , , , and

W = 4:8 mm,

In practice, the losses within the host CPW must be considered. By setting tan  = 0:02 for the FR4 substrate and  = 5:82107 S=m for the copper cladding in the above designs, the dispersion relations for the lossy IL and CL CPWs can also be obtained through (5) and (6), and they are also plotted in Figs. 3 and 4, respectively, for comparison. One can see that the responses resemble those of the lossless cases except for the nonzero phase constant at frequencies below 2.5 GHz and nonzero attenuation constant beyond that. Therefore, the condition of the zeroth-order resonant frequency is amended as the frequency at which  = for general applicability. With (5), (6), and this modified resonant condition, explicit formulas for calculating LL and CL needed to sustain a specified resonant frequency can also be derived. For the lossy IL CPW, the relation between the shunt inductance LL and zeroth-order resonant frequency !sh is obtained with:

LL

 sin  = 2!shZ(10 cosh 0 cosh  cos ) :

(8)

Similarly, for the lossy CL CPW, the relation between the series capacitance CL and the resonant frequency !se is derived as:

CL

 sin  = 2!se Z0cosh (1 0 cosh  cos ) :

(9)

For a given zeroth-order resonant frequency, one may use (8) and (9) to predict the values of the shunt inductance LL and series capacitance CL needed in the IL and CL unit cells, respectively. Note that by setting  = CPW L = 0, (8) and (9) reduce to their lossless counterparts. Moreover, for LL and CL to be positive-valued, we have:

cosh  sin (1 0 cosh  cos ) > 0: (10) where  = CPW L and  = CPW L. This indicates that the length

of a unit cell L should be shorter than about half a guided wavelength. This is always satisfied because the length of a unit cell is recommended to be shorter than a quarter of a guided wavelength [11], [12].

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TABLE I DESIGN PARAMETERS OF THE PROPOSED ZOR ANTENNAS

Fig. 5. Layouts of the proposed ZOR antennas using (a) IL CPW and (b) CL CPW.

C. Verification by Full-Wave Simulations In this subsection, the dispersion relations of the lossy IL and CL CPWs calculated by (5) and (6) are compared to those extracted from full-wave simulations. The IL or CL unit cell is modeled as a two-port device and simulated by the Ansoft HFSS v.11. Then, the attenuation and phase constants of the Bloch wave  and are extracted from the simulated scattering matrix [17]. The resulting responses of  and for the aforementioned lossy IL and CL CPWs are plotted in Figs. 3 and 4, respectively. Obviously, they are all in excellent agreement with those calculated from the proposed formulas. IV. ZEROTH-ORDER RESONATOR ANTENNAS A. Antenna Structure and Design The geometries of the proposed ZOR antennas using IL and CL CPWs are depicted in Figs. 5(a) and (b). In both designs, they are constructed by cascading only two unit cells to ensure a compact size, and are fed through a chip capacitor of capacitance Cfeed for better input-impedance matching. Note that more unit cells can also be used to enhance the directivity of the proposed antennas [18]. To obtain the shunt and series modes of the zeroth-order resonances, the IL CPW is terminated by a truncated open end, while the CL CPW is short-circuited through a short section of CPW line of relatively higher characteristic impedance. This CPW section is needed only in the latter case to compensate for the degradation of the input matching condition at the series resonance when fewer unit cells are used. The design process of the proposed ZOR antennas is described as follows. First of all, for a specified zeroth-order resonant frequency, one can determine with (8) and (9) the shunt inductance LL and series capacitance CL needed in the unit cells of the IL and CL CPWs, respectively. The detailed dimensions of the unit cell can then be determined with the aid of the equivalent circuit model for the folded SSIs or the IDC. Based on the ABCD matrix of the unit cell, one can cascade multiple unit cells, two in the proposed antennas, and further include the associated termination, namely open or short circuit, as well as the feed-in capacitor (Cfeed ), to acquire a closed-form expression for the input reflection coefficient or S11 of the proposed ZOR antennas. Note that the additional CPW section in the CL design must also be taken into account in the formulation. Such a closed-form expression drastically facilitates the fine-tuning of the feed-in capacitance Cfeed and the parameters of the additional CPW section. Finally, for verification, a full-wave simulation of the entire antenna is done by the Ansoft HFSS. B. Prototype Antennas and Results The proposed ZOR antennas using the IL and CL CPWs are both designedat2.5GHzfollowingtheaboveproceduresandfabricatedonaFR4 substrate ("r = 4:4 and tan  = 0:02) of thickness H = 1:6 mm. All design parameters of the prototype antennas are tabulated in Table I. They are both fed by a section of CPW with a characteristic impedance equal to

Fig. 6. Photograph of the two prototype antennas. Left: IL case. Right: CL case.

approximately 50 . In the IL case shown in Fig. 5(a), the gap width betweenthecentralstripandthesidegroundplanes S ismadeassmallaspossibletoincreasetheshuntcapacitanceofthehostCPWsothattherequired shunt inductance LL can be realized via the folded shorting stubs. Here, S = 0:2 mm is chosen, which is the minimal achievable gap width using our fabrication process. In order to minimize the parasitic effects caused by structural discontinuities, the total width of the host CPW W + 2S is kept equal to that of the feeding CPW Wf + 2Sf = 4:3 mm, and therefore, thecentral strip ofthe host CPW is of width W = 3:9 mm. As forthe CLcaseshowninFig.5(b),however,awidercentralstrip (W = 4:8 mm) is chosen such that the IDC can provide enough series capacitance CL to sustain the desired zeroth-order resonance. A larger gap width (S = 4 mm) is adopted simply because the parasitic capacitance between the IDC and the side ground planes thus obtained can be ignored, simplifying theclosed-formexpressionsfor theABCDmatrixof the unitcell.Itisalso found that increasing S may enhance the radiation performance to some extent. Photographs of the fabricated ZOR antennas are shown in Fig. 6. The measured input reflection coefficients are plotted in Fig. 7 along with those calculated via the closed-form expressions and full-wave simulations by the HFSS. Obviously, they are all in excellent agreement. The frequency shifts between the measured and calculated results is only 1.7% and 4% in the IL and CL cases, respectively, mainly due to the fabrication error and the approximate circuit models of the folded SSIs and the IDC adopted in our formulation. The measured impedance bandwidths (jS11 j < 010 dB) for the IL and CL designs are 2.6% (65 MHz) and 2.4% (60 MHz), respectively. Since the zeroth-order resonance occurs within the fast-wave region, the proposed antennas are inherently leaky-wave structures. Their radiation patterns, including the measured and simulated results, are plotted in Figs. 8 and 9. The proposed antennas, when operating at the zeroth-order resonance, both radiate in slot-like patterns. The associated peak gains and simulated radiation efficiencies are summarized in Table II. It has been found that the radiation efficiencies are mainly affected by the dielectric loss rather than conductor loss. However, one can see that the proposed ZOR antenna using a CL CPW exhibits a higher gain and efficiency as

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Fig. 9. Radiation patterns of the proposed ZOR antenna using a CL CPW in (a) XZ and (b) YZ planes. (Solid line: simulated E ; dashed line: simulated E ; dotted dashed line: measured E ; dotted line: measured E ). TABLE II RADIATION PERFORMANCES OF ZOR ANTENNAS USING IL CPW, CL CPW, AND IL MSL

TABLE III COMPARISON BETWEEN VARIOUS ZOR ANTENNAS Fig. 7. Input reflection coefficients of the proposed ZOR antennas using (a) IL CPW and (b) CL CPW.

Fig. 8. Radiation patterns of the proposed ZOR antenna using an IL CPW in (a) XZ and (b) YZ planes. (Solid line: simulated E ; dashed line: simulated E ; dotted dashed line: measured E ; dotted line: measured E ).

compared to its IL counterpart. The radiation efficiency of over 50% is achieved by the CL prototype on the FR4 substrate. The reason may be that the field is loosely confined within the CL CPW due to the larger gap width (S ) used. C. Discussion The classical ZOR antenna formed by IL microstrip line (MSL) [7], [8] is also designed at 2.5 GHz and fabricated on the same FR4 substrate for further comparison. Two unit cells are used in this design, in which the shunt inductance is implemented by a thin via hole down to the ground plane. The associated radiation performances are listed in Table II. It is clear thattheproposed ZOR antennas usingILand CL CPWs both exhibit higher gain and efficiency than the design using an IL MSL. In fact, the proposed designs are less susceptible to dielectric loss because the fields are loosely confined within the dielectric substrate. In addition, the characteristics of other published ZOR antennas [7]–[10] are tabulated in Table III and compared to the proposed CL

design. Among these, the proposed antenna possesses not only a compact size but also high radiation efficiency, which is comparable to those designs employing relatively low-loss substrates. Although the electrical size of the proposed ZOR antenna using CL CPW is slightly larger than that of the antenna presented in [10] according to Table III, further size reduction of the proposed CL ZOR antenna can be achieved as long as narrower gaps between the fingers of the IDCs are used. For a fixed capacitance value or design frequency, the area occupied by the IDCs and thus the proposed antenna shrinks as the gaps between the IDC fingers become narrower. In the prototype antenna, the gap width of 0.2 mm was chosen since it is the minimum gap width that can be realized

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using our in-house etching process. In addition, the radiation efficiency of the proposed design is lower than that of the antenna of [10] simply because the FR4 substrate adopted in the proposed design is relatively lossy compared to the Rogers Duroid 5880 substrate used in [10]. For fair comparison, the proposed CL ZOR antenna is also designed on the same Duroid 5880 substrate, and the resonant frequency is set equal to that of the antenna of [10], namely 2.0 GHz. Moreover, a slightly smaller gap width of 0.15 mm, which is feasible for commercial PCB fabrication processes, is used in this extra design. All the design parameters and simulated performance of the extra design are compared to those of the antenna of [10] and also summarized in Table III. One can see that this extra design exhibits a higher radiation efficiency than the antenna of [10] and that they have equivalent configurations in both physical and electrical sizes. Also, please notice that the proposed design has a uniplanar structure with only one layer of metallic cladding, while the antenna of [10] needs two metallic layers, which require precise alignment during the fabrication process. V. CONCLUSION In this communication, a simple methodology capable of analytically solving the dispersion relations of IL and CL transmission lines has been presented and employed to analyze the proposed IL and CL CPWs. From the obtained dispersion diagrams, the condition for zeroth-order resonance is revised as  = in practical designs with conductor and dielectric losses. The formulas used to calculate the required shunt inductance and series capacitance to sustain a given zeroth-order resonant frequency are also derived since they are useful in designing IL and CL CPWs. By adding a feed-in capacitor and properly terminating the IL and CL CPWs, two novel ZOR antennas have been proposed; one formed by an IL CPW and the other by a CL CPW. To facilitate the design process, closed-form expressions for the input reflection coefficients of the antennas are introduced. The results from the expressions agree very well with those measured and simulated, verifying the antenna model as well as the presented formulation. More importantly, the proposed ZOR antenna using the CL CPW, though implemented on a FR4 substrate, still exhibits a peak gain of 1.1 dBi and a radiation efficiency of 55%, comparable to those of other published ZOR antennas using low-loss substrates. The proposed antennas are also very compact in size and have a via-free uniplanar structure, making them suitable for use in wireless hand-held devices.

[9] S. Pyo, S.-M. Han, J.-W. Baik, and Y.-S. Kim, “A slot-loaded composite right/left-handed transmission line for a zeroth-order resonator antenna with improved efficiency,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 11, pp. 2775–2782, Nov. 2009. [10] T. Jang and S. Lim, “A compact zeroth-order resonant antenna on vialess CPW single layer,” ETRI J., vol. 32, no. 3, pp. 472–474, Jun. 2010. [11] A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag., vol. 5, no. 3, pp. 34–50, Sep. 2004. [12] C. Caloz and T. Itoh, “Metamaterials for high-frequency electronics,” Proc. IEEE, vol. 93, no. 10, pp. 1744–1752, Oct. 2005. [13] N. Dib, J. Ababneh, and A. Omar, “CAD modeling of coplanar waveguide interdigital capacitor,” Inte. Jo. RF Microw. Comput.-Aided Engrg., vol. 15, no. 6, pp. 551–559, Nov. 2005. [14] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [15] R. Collin, Foundations for Microwave Engineering. Piscataway, NJ: Wiley-IEEE Press, 2001. [16] R. N. Simons, Coplanar Waveguide Circuits, Components and Systems. New York: Wiley-Interscience, 2001. [17] G.-S. Mao, S.-L. Chen, and C.-W. Huang, “Effective electromagnetic parameters of novel distributed left-handed microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1515–1521, Apr. 2005. [18] A. Rennings, T. Liebig, S. Otoo, C. Caloz, and I. Wolff, “Highly directive resonator antennas based on composite right/left-handed (CRLH) transmission line,” in Proc. 2nd Int. ITG Conf. Antennas (INICA), Munich, Mar. 2007, pp. 190–194.

Corrugated-Conical Horn Analysis Using Aperture Field With Quadratic Phase Arthur Densmore, Yahya Rahmat-Samii, and Gerry Seck Abstract—An approximation of the far-field radiation pattern of the corrugated-conical horn, by integration of the aperture field, augmented with perimeter-matched quadratic radial phase, is sufficient to establish a baseline design for the feedhorn of a satcom antenna system. The approximated far-field patterns compare well to more formal analysis methods, even reasonably so in the main beam with a semi-flare angle as wide as 45 deg for the first two angular modes. The field equations are presented in a compact complex exponential form, from which the components corresponding to excitation with any polarization sense may be obtained (either sense of circular/elliptical polarization or linear polarization with arbitrary orientation). Index Terms—Antenna theory, corrugated antennas, corrugated horn antennas, corrugated waveguides.

REFERENCES [1] A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission line,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 2, pp. 68–70, Feb. 2004. [2] C. Caloz and T. Itoh, Electromagnetic Metamaterials. PiscatawayHoboken, NJ: Wiley-IEEE Press, 2005. [3] A. Sanada, M. Kimura, I. Awai, C. Caloz, and T. Itoh, “A planar zerothorder resonator antenna using a left-handed transmission line,” in Proc. 34th Eur. Microw. Conf., Amsterdam, Oct. 2004, pp. 1341–1344. [4] A. Rennings, T. Liebig, S. Abielmona, C. Caloz, and P. Waldow, “Triband and dual-polarized antenna based on composite right/left-handed transmission line,” in Proc. 37th Eur. Microw. Conf., Munich, Oct. 2007, pp. 720–723. [5] C.-J. Lee, K. M. K. H Leong, and T. Itoh, “Compact dual-band antenna using an anisotropic metamaterial,” in Proc. 36th Eur. Microw. Conf., Manchester, Sep. 2006, pp. 1044–1047. [6] J.-G. Lee and J.-H. Lee, “Zeroth order resonance loop antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 994–997, Mar. 2007. [7] A. Lai, K. M. K. H. Leong, and T. Itoh, “Infinite wavelength resonant antennas with monopolar radiation pattern based on periodic structures,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 868–876, Mar. 2007. [8] J.-H. Park, Y.-H. Ryu, J.-G. Lee, and J.-H. Lee, “Epsilon negative zeroth-order resonator antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3710–3712, Dec. 2007.

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I. INTRODUCTION Using aperture field with quadratic phase is a popular method for smooth-walled conical and rectangular horns, for far-field pattern estimation, and here we investigate the utility of this method for corrugated horns. A simple means for quickly obtaining a reasonable Manuscript received September 08, 2010; revised February 15, 2011; accepted February 23, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. A. Densmore is with the Department of Electrical Engineering, University of California, Los Angeles, Los Angeles, CA 90095-1594 USA and also with L-3 Communications Datron Advanced Technologies Division, Simi Valley, CA 93065-1650 USA (e-mail: [email protected]; Art.Densmore@L-3Com. com). Y. Rahmat-Samii is with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095-1594 USA (e-mail: [email protected]. edu; www.antlab.ee.ucla.edu; www.ee.ucla.edu). G. Seck is with L-3 Communications Datron Advanced Technologies Division, Simi Valley, CA 93065-1650 USA (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161552

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TABLE I FAR-FIELD ESTIMATION METHODS COMPARED IN THIS COMMUNICATION, IN ORDER OF ACCURACY AND COMPUTATIONAL COMPLEXITY

approximation of the far-field patterns can be quite valuable for a proposal or project planning. Corrugated-conical horns are commonly used in reflector antenna systems for microwave communication systems, and the design of these systems requires that the performance of the horns be understood reasonably well. Having a low-cost means, with negligible learning curve, of approximating feedhorn performance is useful, at least for planning purposes. Several methods have been investigated for such approximations and are compared in this communication. Olver, Clarricoats, Kishk and Shafai [18] suggest using aperture field integration with first-order (Taylor) quadratic radial phase—a first-order approximation of the spherical phase implied by the conical flare. We refer to that method as first-order quadratic radial phase (FOQRP). This communication investigates an alternative quadratic phase function, perimeter-matched quadratic radial phase (PMQRP), which provides an improved approximation of the radiation pattern, especially so for wider flare angles. The prior work apparently presumed small flare angle and did not distinctly consider the PMQRP variation of the quadratic radial phase approximation. In this communication, the basic method of far-field determination by far-field vector potential is summarized, and we present the results of the PMQRP approximation method for 12 and 45 deg half-flare, conical-corrugated horns for both first and second angular modes. We compare the approximation with more formal methods of analysis, summarized in Table I. Analysis of corrugated horns has a rich background. The literature [1]–[28] presents a vast collection of articles on corrugated horns from a variety of perspectives. Minnett and Thomas [1], [2] and [7], Jansen, Jeuken and Lambrechtse [6], James [9], and Clarricoats, Saha, Olver, Kishk and Shafai [3], [11] and [18] address fundamentals. Love [5] offers a full collection of papers on the subject. Sakr [24] presents closed form radiation patterns for uniform-phase aperture, Sinton, Robinson and Ramat-Samii [25] addresses optimized horn profile design, Granet and James [26] overview the general design and Lee [27] addresses field modes and dispersion. In [18, Fig. 6.1, Eqs. 6.9 and 9.6], Olver, Clarricoats, Kishk and Shafai suggest the use of aperture integration with quadratic phase. They specify use of a first-order quadratic radial phase (FOQRP) function—a 1st term Taylor approximation of the spherical phase function that’s implied by the cone that fits inside the horn—to represent the affect of the flare. We consider an alternative PMQRP function instead and find that it yields better main bean pattern estimation, especially for wider flares and for both of the first two angular modes. This article focuses on the approximation of the circularly polarized (CP) far-fields by aperture field integration augmented by quadratic phase. Typically with smooth-walled horns the aperture mode is assumed to be the same as that which excites the horn; although, with corrugated horns the aperture mode is different than the excitation mode.

We presume that the corrugated horn has a well designed mode converter (e.g., half-wave depth corrugations at throat tapering to quarterwave depth in the flare) so that balanced hybrid HE mode dominates in the aperture. Note [14] that the 45 deg half-flare horn can be matched without increasing the depth of the corrugations from quarter wavelength. We present results from four different analysis methods, identified in Table I, and find that adding PMQRP to the balanced HE aperture field yields far-fields for the main beam in good agreement with more formal analysis methods. This aperture integration analysis effectively models the corrugated-conical horn as a stepped cone consisting of short sections cylindrical waveguide, with general radius a, each supporting a single cylindrical mode. The cylindrical field distributions of the balanced hybrid modes are addressed, and the far-field radiation patterns are derived by aperture integration, augmented by radial quadratic phase, and compared with far-field patterns determined by Clarricoats’ spherical wave expansion (SWE) method for conical horns, as well as cylindrical mode matching (CMM) of the detailed internal geometry of the corrugated horn with integral equation (IE) method of moments solution for the surface currents on the horn exterior. Both the hybrid HE11 and HE21 modes are addressed. Corrugated horn HE11 radiation patterns are efficient for dual shaped reflector designs where circular symmetry and precise edge illumination control are critical. HE21 patterns provide similar efficiency for difference patterns for single channel monopulse auto-tracking feeds for large ground antennas. All higher angular modes HEn1 , n > 1, provide an on-axis pattern null; although, HE21 uniquely provides tracking error detection in two orthogonal axes, which are typically aligned with the antenna positioner’s elevation and cross-elevation axes. The field equations use rotational polarization (L or R), since this allows compact, general forms. The fields corresponding to linearly polarized excitation are obtained by linear combination of the two senses of rotational polarization. II. APERTURE FIELD AND RADIATION INTEGRAL A. Amplitude Corrugated horns propagate hybrid field modes, and hybrid field modes involve simultaneous propagation of both TEz and TMz modes. Hybrid modes are designated HEnm , or EHnm , where the first index, n, is the angular mode index and represents the number of field cycles for every physical rotation in ' angle, and m is the radial mode index (mth Bessel root), representing the number of field cycles along the radial extent of the waveguide. The HE mode is the sum of TE + TM modes, and the EH mode is their difference, TE 0 TM. A well-designed corrugated horn establishes the balanced HE11 mode at the horn aperture (and also balanced HE21 mode if used for tracking.) Note the dispersion diagram on [11, p. 27]. The balanced hybrid HE11 mode provides purely linearly-polarized transverse fields across the entire aperture, and involves the vector potential amplitudes being related simply through the free-space wave impedance [1, Eq. (10)]. At the corrugation boundary the balanced HE fields have only an axial component. What uniquely characterizes the balanced hybrid mode condition is that the E and H vector fields have exactly the same form but with one rotated (90=n) deg in ' from the other, where n is the angular mode index (see discussion of [10, Fig. 3(d)]). This balanced hybrid mode condition naturally occurs well above cutoff with quarter-wavelength corrugation, both of which are typical at the radiating aperture of a large, corrugated horn. Equation (1) defines the balanced hybrid HE transverse aperture mode fields. To simplify (1) the phasor exp[6jn'] is excluded (in addition to the time- and

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C. Radiation Integral

Fig. 1. Corrugated-conical horn parameters. ('

=  = 0 on positive x-axis).

axial-propagation phasors). J is the cylindrical Bessel function of the first kind

EBAL = HBAL = E BAL = '

H'BAL =

7 j( nm =")Jn01 ( nm ) R 0 (0 nm =)Jn01( nm ) L

+ ( nm =")Jn01 ( nm )

7 j (0 nm =)Jn01( nm )

(1a)

(1c) L R

:

(1d)

L R

:

(2)

The balanced EH modes have the same magnitudes of field coefficients but different signs and with Bessel order n + 1. The boundary conditions of quarter-wave corrugation on the total fields at the mouth of a large horn aperture, where the balanced hybrid modes are supposed to exist, yield the propagation roots of the characteristic equation as

11 = 2:405=b;

21 = 3:832=b;

from

J0 ( 11 b) = 0

from J1 ( 21 b) = 0

(HE11 mode)

(HE21 mode):

(3a) (3b)

B. Various Phase Approximations In the aperture plane (z = h), the radial dependence of the phase, (), of a spherical wave that originates at the apex of the cone, marked by a circled cross in Fig. 1, is

() = 0 [r() 0 r(0)] r() = h2 + 2

(4a) (4b)

and h is the height of the cone, as indicated in Fig. 1. () is fairly well approximated as a quadratic function,

()  u(=b)2

(4c)

where b is the aperture radius. u represents the quadratic phase at the perimeter of the aperture and is defined, respectively for the two different quadratic functions, as 1

u1 = 0 b tan &; FOQRP 2 u2 = 0 b cot & (sec & 0 1); PMQRP

(4d)

(0 ); spherical phase u(0 =b)2 ; quadratic phase.

 (0 ) = E =

(1b)

For the balanced hybrid HE mode the transverse fields relate as

E = 6j0 H = 7jE' = 0 H' :

The far-field radiation patterns are estimated by far-field vector potentials. The radiated field vector potentials are obtained by integration of the equivalent aperture currents, and we combine the fields due to both the electric and magnetic currents to represent a free-standing aperture with neither an electric nor a magnetic ground plane. In the far-field the vector potentials are proportional to the radiated fields. The balanced HE mode, far-field radiation aperture-integral, including a radial phase term, exp[j (0 )], is given in (5). Averaging the far-fields due to equivalent electric and magnetic currents for the aperture field causes the factor of (1 + cos )=2. The second Bessel function in (5) results from the ' integral, which is done analytically

=

6 j E

0 "

1

b

0

(4f)

L R

0j 0r]

exp[

r

1 + cos  2

Jn01 ( 0 )Jn01 ( 0 0 sin )

1 exp j(0 ) 0 d0 1 exp[6jn] j;(6nj n=);1n > 1: L (5) R L UNIFORM = 6 j E E  R

0 exp[0j 0 r] 1 + cos  = 1 b3 Jn02 ( b) "

r 2 Jn01 [ 0 b sin ] 1 ( 0 b sin )2 0 ( b)2 1 exp[6jn] j;(6nj n=);1;n > 1: L R

(6)

where is the radial propagation root at the aperture for mode (n; m), and j is the square root of minus one. With uniform phase a closed-form (6) is obtained using the Lommel integral formula [30, p. 337] and is useful as a check of the numerical integration of (5) when  is zero. These far-field equations concur with those published by Sakr [24] and reveal that the one arbitrary constant, C , in Sakr’s field equations for both HE11 and HE21 depends on the angular mode index and the root of the characteristic equation. Note that the observation frame, (; ) rotates with both  and , with respect to the horn’s fixed frame, which affects the observed radiated phase (with respect to the horn’s fixed frame) for a rotationally polarized wave. Expressions for linearly (or elliptically) polarized field components are obtained by a linear combination of the field expressions for the two rotational senses and using appropriate coordinate transformation [29] from the (; ) observation frame to the horn’s fixed frame in which, for example, vertical or horizontal linear polarization may be defined. Adding the balanced hybrid mode field expressions for the two rotational sense, L + R, represents linearly polarized excitation of the horn with the E field in the positive y -axis direction (for the HE11 main beam, with the y axis in the  direction with  = 0).

(4e)

where  is the half-flare angle of the horn. With PMQRP, u is the difference in electrical length between the slant and axial lengths of the cone that fits inside the horn, and thereby matches the spherical and quadratic phase functions at the aperture perimeter. FOQRP effectively matches the spherical and quadratic phase functions only for small  values.

III. FAR-FIELD RESULTS AND COMPARISON OF METHODS Figs. 2 and 3 reveal how well augmenting the aperture field integration with true spherical or quadratic radial phase approximates the radiated fields for HE11 mode (used for communications) in comparison with the SWE and CMM+IE results, and Figs. 4 and 5 for HE21

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Fig. 2. 12-deg half-flare angle HE11 CP co-pol far-field by integration of aperture fields augmented by radial phase (22.2 dBi), SWE (22.30 dBi), and cylindrical MM+IE (TICRA CHAMP) numerical modeling of the corrugated horn geometry (22.62 dBi) with 33 corrugations (including 5 in mode converter with : , t : , and = . Excitation linear depth profile), b is CP TE11.

Fig. 4. 12-deg half-flare angle HE21 CP co-pol by integration of aperture fields augmented by radial phase (peak 17.1 dBi), SWE (peak 17.19 dBi), and cylindrical MM+IE (MICIAN Wave Wizard) numerical modeling of the corrugated horn geometry (peak 17.17 dBi) with 33 corrugations (including 5 in mode con: , t : , and verter with linear depth profile), b = . Excitation is CP TE21.

Fig. 3. 45-deg half-flare angle HE11 CP co-pol far-field by integration of aperture fields augmented by radial phase, SWE (peak 13.54 dBi), and cylindrical MM+IE (TICRA CHAMP) numerical modeling of the corrugated horn geometry (13.12 dBi) with 11 corrugations (including 5 in mode converter with linear : , t : , and = . Excitation is CP depth profile), b TE11.

Fig. 5. 45-deg half-flare angle HE21 CP co-pol by integration of aperture fields augmented by radial phase, SWE (peak 10.63 dBi), and cylindrical MM+IE (MICIAN Wave Wizard) numerical modeling of the corrugated horn geometry (peak 11.54 dBi) with 11 corrugations (including 5 in mode converter with linear : , t : , and = . Excitation is CP depth profile), b TE21.

mode (used for tracking). All figures represent CP polarization and also present uniform-phase patterns, which have deep nulls that the quadratic phase patterns have filled in. The uniform-phase patterns do well in the main beam with narrow flare angles. With a 12 deg horn the numerical integration of (5) yields good agreement with SWE especially about the main beam (near-axis), as shown in Fig. 2, which also presents the patterns calculated by numerical modeling of the exact geometry of the corrugated horn (by CMM the fields inside the horn and IE solution of the horn exterior currents). The 12 deg horn has thirty-three corrugations, including five for mode converter. The differences that are noticed, mainly off axis, between the aperture integration results and those of the more formal methods are due to the presence of other (higher order) modes other than HE11 in the aperture, as well as the aperture phase not being modeled exactly. (The aperture integration and SWE methods also do not account for currents on the horn exterior.) Fig. 3 presents similar results for a horn with half-flare angle of 45 deg, eleven total corrugations, and slightly shorter pitch; whereas, all the other horn parameters remain the same as in the previous figure. The figure indicates that among the three aperture integration methods considered, PMQRP provides the best approximation of the SWE or CMM+IE results. Note that an on-axis dip of about one and one quarter

dB occurs in the pattern on-axis for the 45 deg half-flare horn using aperture integration with spherical phase; although, the on-axis dip is only about one quarter dB for all the other methods: Aperture integration quadratic phase, SWE, and full-wave CMM+IE. The agreement of the aperture integration results with the SWE or CMM+IE results is less with the 45 deg half-flare than the 12 deg since the reasons for the differences, mentioned above, are more pronounced for a wider flare. For increasing flare angle, the aperture integration results—based on modeling the horn as short sections of cylindrical waveguide, each supporting a single cylindrical mode—become less accurate; in which case, method 3 in Table I, SWE, is a better model. The SWE results presented in the figures satisfy Maxwell’s equations in spherical coordinates with the boundary condition of a conical, anisotropic surface-impedance, as an ideal approximation of the corrugated flare. Figs. 4 and 5 compare the aperture integration with the more formal analysis methods for the second angular (tracking) mode. The agreement is comparable to that for the first angular mode. PMQRP appears to again yield the best approximate results in the main beam. A typical corrugated horn will have cross-pol patterns with a null on-axis and peaking just outside the main beam shoulder about 10–20 dB below the co-pol pattern level, and follow and tend toward the

= 16 9

= 16 9

= 4 21

= 4 21

pitch  3

pitch  5

= 16 9

= 16 9

= 4 21

= 4 21

pitch  3

pitch  5

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co-pol pattern farther off axis. Numerical optimization of the corrugated horn geometry (mainly of the mode converter geometry) can reduce the cross-pol level substantially. IV. CONCLUSION Simple equations for the far-fields of corrugated-conical horns have been presented and discussed, demonstrating a practical approximation method using aperture field augmented with quadratic phase. It provides good agreement in the main beam with more formal computational analysis techniques, such as SWE and accounting for the complete corrugated horn geometry (CMM + IE). The equations can be used to establish a baseline design of a feedhorn for a satcom antenna. Aperture integration methods using spherical, and two different quadratic, radial phase functions are compared. PMQRP yields significantly better results than FOQRP. Numerical results for 12 and 45 deg half-flare horns have been presented and discussed. The results suggest that the aperture integration method works well for modest flare angles, and PMQRP in particular works well for moderate flare angles. But for wide flared horns the aperture integration method may not represent the true aperture fields completely, especially at far, off-axis angles, for which case SWE is a better—yet more numerically complicated—approximate model. ACKNOWLEDGMENT The authors would like to thank D. Hoppe of the Jet Propulsion Laboratory, California Institute of Technology, for helpful discussions, and to EMSS (FEKO), MICIAN (Wave Wizard) and TICRA (CHAMP) for the evaluation of their numerical analysis software codes in support of this research.

REFERENCES [1] H. Minnett and B. Thomas, “A method of synthesizing radiation patterns with axial symmetry,” IEEE Trans. Antennas Propag., vol. AP-14, pp. 654–656, Sep. 1966. [2] B. M. Thomas, “Theoretical performance of prime-focus paraboloids using cylindrical hybrid-mode feeds,” Proc. Inst. Elect. Eng., vol. 118, no. 11, pp. 1539–1549, Nov. 1971. [3] P. J. B. Clarricoats and P. K. Saha, “Propagation and radiation behaviour of corrugated feeds (Part 1—Corrugated-waveguide feed, and Part 2—Corrugated-conical-horn feed),” Proc. Inst. Elect. Eng., vol. 118, no. 9, Sep. 1971. [4] P. D. Potter, “Efficient antenna systems: A new computer program for the design and analysis of high-performance conical feedhorns,” Pasadena, CA, JPL Tech. Rep. 32-1526, Nov./Dec. 1972, vol. 13, Deep Space Network Progress Rep. [5] Electromagnetic Horn Antennas, A. Love, Ed. New York: IEEE Press, 1976. [6] J. K. M. Jansen, M. E. J. Jeuken, and C. W. Lambrechtse, “The scalar feed,” in Electromagnetic Horn Antennas, A. Love, Ed. New York: IEEE, 1976. [7] B. M. Thomas, “Design of corrugated conical horns,” IEEE Trans. Antennas Propag., vol. AP-26, no. 2, pp. 367–372, Mar. 1978. [8] B. M. Thomas and H. C. Minnett, “Modes of propagation in cylindrical waveguides with anisotropic walls,” Proc. Inst. Elect. Eng., vol. 125, no. 10, pp. 929–932, 1978. [9] G. L. James, “Analysis and design of TE11-toHE11 corrugated cylindrical waveguide mode converters,” IEEE Trans. Microw. Theory Tech., vol. MTT-29, no. 10, Oct. 1981. [10] G. L. James, “Propagation and radiation properties of corrugated cylindrical coaxial waveguides,” IEEE Trans. Antennas Propag., vol. AP-31, no. 3, pp. 477–483, May 1983. [11] P. J. B. Clarricoats, Corrugated Horns for Microwave Antennas, ser. Inst. Elect. Eng. Electromagnetic Wave Series, no. 18. London, U.K.: Inst. Elect. Eng., 1984. [12] D. Hoppe, “Propagation and radiation characteristics of a multimoded corrugated waveguide feedhorn,” Pasadena, CA, JPL TDA Progress Rep. 42-82, Apr.–Jun. 1985, pp. 57–67.

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[13] C. M. Knop, Y. Cheng, and E. L. Ostertag, “On the fields in a conical horn having an arbitrary wall impedance,” IEEE Trans. Antennas Propag., vol. 34, no. 9, pp. 1992–1098, Sep. 1986. [14] A. D. Olver and J. Xiang, “Wide angle corrugated horns analysed using spherical modal-matching,” Inst. Elect. Eng. Proc., vol. 135, no. I, pt. H, Feb. 1988. [15] P.-S. Kildal, “Artificially soft and hard surfaces in electromagnetics,” IEEE Trans. Antennas Propag., vol. 38, no. 10, pp. 1537–1544, Oct. 1990. [16] R. Padman and J. A. Murphy, “Radiation patterns of ‘scalar’ lightpipes,” Infrared Phys., vol. 31, no. 5, pp. 441–446, 1991. [17] J. A. Murphy and R. Padman, “Radiation patterns of few-moded horns and condensing lightpipes,” Infrared Phys., vol. 31, no. 3, pp. 291–299, 1991. [18] A. D. Olver, P. J. B. Clarricoats, A. A. Kishk, and L. Shafai, Microwave Horns and Feeds. London, U.K.: Inst. Elect. Eng. Press, 1994. [19] R. R. Collmann and F. M. Landstorfer, “Calculation of the field radiated by horn-antennas using the mode-matching method,” IEEE Trans. Antennas Propag., vol. 43, no. 8, Aug. 1995. [20] C. Granet, “Profile options for feed horn design,” in Proc. Asia Pacific Microw. Conf., Sydney, Australia, 2000, vol. 1, pp. 1448–1452. [21] R. Maffei et al., “Shaped corrugated horns for cosmic microwave background anisotropy measurements,” Int. J. Infrared Millimeter Waves, vol. 21, no. 12, pp. 2023–2033, 2000. [22] J. A. Murphy, R. Colgan, C. O’Sullivan, B. Maffei, and P. Ade, “Radiation patterns of multi-moded corrugated horns for far-IR space applications,” Infrared Phys. Tech., vol. 42, pp. 515–528, 2001. [23] H. S. Lee, “Radiation from a corrugated circular waveguide with a flange,” J. Electromagn. Waves Applicat., vol. 16, no. 9, pp. 1255–1274, 2002. [24] L. Sakr, “The higher order modes in the feeds of the satellite monopulse tracking antennas,” in Proc. IEEE MELECON, Cairo, Egypt, May 7–9, 2002, pp. 453–457. [25] S. Sinton, J. Robinson, and Y. Rahmat-Samii, “Standard and micro genetic algorithm optimization of profiled corrugated horn antennas,” Microw. Opt. Tech. Lett., vol. 35, no. 6, pp. 449–453, Dec. 2002. [26] C. Granet and G. L. James, “Design of corrugated horns: A primer,” IEEE Antennas Propag. Mag., vol. 47, no. 2, pp. 76–84, Apr. 2005. [27] H. S. Lee, “Dispersion relation of corrugated circular waveguides,” J. Electromag. Waves Applicat., vol. 19, no. 10, pp. 1391–1406, 2005. [28] E. Gleeson et al., “Corrugated waveguide band edge filters for CMB experiments in the far infrared,” Infrared Phys. Tech., vol. 46, pp. 493–505, 2005. [29] Y. Rahmat-Samii, “Useful coordinate transformations for antenna applications,” IEEE Trans. Antennas Propag., vol. 27, no. 4, pp. 571–574, 1979. [30] Microwave Antenna Theory and Design, ser. MIT Radiation Laboratory Series, S. Silver, Ed. New York: McGraw-Hill, 1949, vol. 12.

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High-Strength, Metalized Fibers for Conformal Load Bearing Antenna Applications

TABLE I CONDUCTIVE FILM MATERIAL PROPERTIES

Steven E. Morris, Yakup Bayram, Lanlin Zhang, Zheyu Wang, Max Shtein, and John L. Volakis Abstract—we propose the use of high strength, metal-coated Kevlar yarns to weave flexible, conformal, and load-bearing antennas for an emerging class of applications emphasizing multiple functionality. In particular, here we present a unified, quantitative analysis of multiple properties of conductors as load-bearing materials in stress-, weight-, and shape-critical applications (e.g., in aerial vehicles), suggesting advantageous electrical conductor configurations to be metal-coated, multi-filament, high strength fibers. We then describe the fabrication of highly conductive metal coated Kevlar yarns, their mechanical and electrical properties, and the weaving of a flexible, stretchable, volumetric spiral antenna. The high frequency response of the antenna is found to match that of a traditionally made antenna comprised of electroplated copper on a rigid ceramic (Rogers TMM4) substrate. At low frequencies, the relatively lower conductivity of the metal-coated kevlar yarn leads to higher resistive losses compared to the traditional electroplated copper. We discuss strategies for mitigating such losses, and other means of improvement. More broadly, the results described here suggest a novel direction for multi-functional antenna design and applications, enabled by the superior mechanical characteristics of the composite conducting fibers, and the flexible, conformable, woven antenna architectures they help achieve. Index Terms—Antennas, conformal, conformal antennas, e-fiber antennas, fiber devices, flexible antennas, polymer antennas.

I. INTRODUCTION Conformal, light-weight, damage-resistant antennas are desirable for many applications, including wearable devices and unmanned aerial vehicles (UAVs). For small UAVs in particular, high data rate communication occurs at wavelengths that can exceed significantly the size of the craft, in which case the entire UAV airframe, including non-planar surfaces, must serve as an antenna. Equally important design constraints arise from the requirements for low weight and high stress tolerance in UAV construction. Similar considerations apply for wearable communication applications, and motivate research into novel antenna configurations and materials combinations that facilitate structural integration and improve system-wide, multi-functional performance. Consider the candidates for the dielectric and the conductive components of the novel multi-functional antenna design. Conventional dielectric substrates used in sub-wavelength size antennas are not well suited for conformal and light-weight applications, due to being too rigid, brittle, and heavy. Light-weight, low loss, stretchable substrate materials such as polymers are key to developing conformal, flexible, compact and light-weight designs. Polymer-ceramic composites exhibit low loss dielectric properties and are well suited for RF applications with multiple additional functional requirements. In particular, Manuscript received August 18, 2010; revised December 08, 2010; accepted January 15, 2011. Date of publication July 07, 2011; date of current version September 02, 2011. This work was supported in part by the U.S. Air Force Office of Scientific Research (AFOSR) under Grant # FA9550-07-1-0462. S. E. Morris and M. Shtein are with Materials Science and Engineering department University of Michigan, Ann Arbor, MI 48109 USA. Y. Bayram is with the ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212 USA and also with PaneraTech, Inc., Falls Church, VA 22043 USA (e-mail: [email protected]; [email protected]). L. Zhang, Z. Wang, and J. L. Volakis are with ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212 USA. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161442

polydimethylsiloxane (PDMS) has the flexibility necessary to be applied as the substrate for millimeter-wave patch array antennas [1]. A PDMS-based polymer-ceramic composite has been shown suitable for RF applications, with its tunable dielectric constant up to 20 and a low loss tan( ) < 0:01 [2]. The application of traditional metals (e.g., plated copper) as the conductive medium in the antenna is not trivial when PDMS-based dielectrics are considered. The antennas used in UAV and many other applications experience considerable, high and cyclic mechanical stresses, resulting in fatigue- and strain induced cracking, elongation, delamination and response hysteresis. Therefore, a need arises for alternative conducting media to achieve the desired structural and RF performance for conformal, flexible, light-weight RF electronics and antennas. Several E-textile technologies have been considered to improve flexibility and fatigue resistance of antennas [3]–[5]. These technologies are based on metal coating of textile yarns for improved conductivity. However, such yarns often lack high tensile strength required for embedding or weaving into dense, ceramic-loaded dielectrics, and to eventually function in deformable and/or load bearing UAV and wearable antennas. The selection of conductive materials for the multi-functional antenna applications considered above requires a combination of high tensile strength, flexibility, and electrical conductivity. A survey of conventional materials’ properties shows that metals such as copper and silver offer the highest electrical conductivity (Table I), but have tensile strength an order of magnitude lower than that of carbon-based fibers (e.g., Zylon® and Kevlar) [6]–[8] (Table II). An opposite relationship holds for electrical conductivity. Combining a high strength composite fiber core with a conductive metallic coating can potentially result in the desired overall performance characteristics. Examples of multi-material fibers incorporating conductive coatings have been realized for power or data transmission applications, as well as EMI shielding [9]–[11], achieving 390 S-cm conductance and 1700 MPa tensile strength. For application in shape-conforming, space-filling antennas in small aerial vehicles, weight limits potentially place greater constraints on the fiber design. The strength-to-weight ratio of the conductive fiber may need to be optimized subject to achieving some minimum electrical conductivity needed for low-loss antenna performance, which has been the point of competition on commercially available conductive yarns. Here we demonstrate a twisted and protected yarn that is metal-coated at the level of the constituent filaments, and quantify the effects of fiber and coating thickness on the mechanical and electrical characteristics of the composite fiber, as well as the benefits of multi-strand architectures.

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TABLE II STRUCTURAL FIBER MATERIAL PROPERTIES

Fig. 2. Kevlar yarn (a) plated with copper using electroless process, 34 = (b) suspended and torque to create a twist, 21 = (c) encapsulated with fabric glue jacket.

m cm

m cm

as it has been shown to be the strongest and need no cleaning prior to chemical treatment [11]. Uncoated yarns were etched in “polymer film etchant,” sodium hydroxide, for 10 minutes, followed by 4 minutes in “RTM solution C,” sodium hypophosphite, 4 minute in “RTM solution B,” palladium chloride, and 15 minutes in proprietary electroless copper bath, a 1-to-1 ratio of “PC electroless copper solution A,” and “PC electroless copper solution B,” with main constituents being copper sulfate and formaldehyde. Conductive yarns were subsequently fabricated from 1140 denier Kevlar, consisting of 768 filaments of 12  diameter each [8], surrounded by a 1  copper shell, Fig. 2(a), to achieve a conductance of 64 S-cm, as averaged over multiple segments of the fiber.

m

Fig. 1. Summary of experimental results points plotted against modeled lines. E-fiber results maintain 91% of ideal yarn strength and 34% of ideal conductance.

To normalize the performance of various composite fiber architectures, we first calculate the theoretical specific strength (i.e., ultimate tensile strength divided by the fiber density) and plot it versus the fiber conductance multiplied by unit length, as shown in Fig. 1. The desired combination of high tensile strength, high conductivity, and low weight is to be found in the upper right region of the graph. For a given combination of substrate and coating materials, the conductance of the composite fiber increases with the metal film thickness, while the specific strength of the fiber decreases, due to the fact that the metal coating is always more dense than the carbon-based substrate fiber. As a result, the theoretical curves exhibit a universal behavior, trading off specific strength for conductance, with a reversed S-shaped curve. As the theory shows (solid lines in Fig. 1), microfilament yarns offer the greatest strength-to-weight ratio for the core. Superimposed on the theoretical predictions are experimentally realized metal-coated fibers, both for mono- and multi-filament architectures. In the next section, we describe the process used to realize some of the composite fibers analyzed in Fig. 1, followed by an analysis of their performance; methods of improving the conductivity and robustness of the yarns are also addressed. In the last section, we describe the performance of 3-dimensional antennas woven using copper-coated, multi-filament Kevlar fibers and discuss its radiation performance. II. E-FIBER FABRICATION AND CHARACTERIZATION A. Fabrication of Conductive Composite Fibers Electroless copper plating solutions were supplied by Transene Company, Inc. Kevlar 49 yarns of type 965 were selected as a substrate,

m

B. Yarn Twist Yarns are often twisted to achieve peak strength. Twisting increases the friction between adjacent fibers, distributing the load over greater lengths. This technique mitigates irregularities and defects that may cause failure of the yarn prior to peak load, and allow for multiple breaks within the same filament [13]. Yarn denier and twist per inch (TPI) are used to calculate the twist multiple (TM), (1), as a measure of the helix formed at the yarn surface upon twisting

p

TM = TPI 273 denier :

(1)

Previous studies have found that the optimal strength for Kevlar occurs with a TM of 1.1 [7], [13], beyond which strength begins to decrease, though this number is denier-dependent. From the quadratic fit in Fig. 3, strength is maximized at a TM of 0.9 for 1140 denier yarns, attaining a value of 3.0 GPa. In addition to improving strength, twisting of the fiber can improve conductivity. Generally, metal films deposited on the filaments will have defects, limiting the conductance and useful lifetime of the metalized filament. Twisting of the metalized yarn can allow for bridging of high resistance defects in filaments through adjacent films. While we found no optimal value of TM with regard to electrical performance, < : was sufficient to observe overall improvements in conductance (Fig. 3). To protect the underlying conductive film, to preserve the twisted structure of the fiber, and to electrically insulate the conducting composite yarn, commercially available fabric glue was diluted with

TM 1 1

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Fig. 3. 1140 denier coated Kevlar yarn shows enhanced strength at a TM of 0.7, as opposed to 0.9 for bare yarns. In contrast, conductance of the yarn shows no maximum with increasing twist.

Fig. 5. (a) Failure of encapsulating jacket and resulting sharp increase in resistance along fiber. Repairs can be made by conventional (b) soldering techniques or by forming (c) a conductive knot.

Fig. 4. Broadband spiral antenna design and built by the Ohio State University on a Rogers TMM4 substrate (after [15]) Antenna has 16.3 cm diameter and 3.3. mm thickness.

de-ionized water at a ratio of 3-to-1 and coated on the yarn. The film was allowed to dry for 12 hours before applying a second coating. Fabrication of these fibers is illustrated in Fig. 2. A summary of experimental results is shown in Fig. 1, where the copper on Kevlar most closely achieved modeled results, as compared to previously tested monofilament fibers, achieving 30 S-cm. Deviations from the model can be attributed to film defects, as well as less than ideal resistivity as a result of impurities that present themselves in wet electrochemical deposition techniques [14]. (That is, the experimental data points are displaced to the left from the theoretical curve, rather than downward from it.) An ultimate strength of 2.8 GPa was measured for the E-fiber, which is lower than that of the bare Kevlar, as a result of the increased cross-sectional area required by the copper film. Prior work that shows little deviation from untreated yarn samples [7]. However, strength of the E-fiber is maximized at a TM of 0.7, which is possibly due to enhanced inter-filament friction at lower shear stress, enhancing load distribution at lower twist. III. BROADBAND ANTENNA FABRICATION AND TESTING A broadband spiral antenna was considered to demonstrate proof-ofconcept for the proposed technology. In particular, we considered a volumetric spiral antenna realized by electro-plating copper lines into a machined Rogers TMM4 substrate, as shown in Fig. 4. This antenna was optimized for a large frequency range from 200 MHz to frequencies above 1000 MHz, and is intended for low frequency broadband applications. (A detailed description of the spiral antenna design and the theory behind was published elsewhere. [15])

Fig. 6. (a) Sample of encapsulated conductive yarn. (b) Patterned PDMS substrate. (c) Example of weaving and gluing (inset). (d) Completed flexible, woven antenna.

A. Conformal Antenna Fabrication To fabricate the conformal spiral antenna replicating the design shown in Fig. 4, we first built a polymer-ceramic substrate by carrying out the process described in [2]. This substrate has the same dielectric properties as the Rogers TMM4 but is much more flexible, stretchable and therefore conformable. Subsequently, the 4 mm thick, polymer-ceramic substrate was drilled using a 50 W laser to create a pattern for weaving, as shown in Fig. 2(b). The jacketed yarn was then woven into the hole pattern using a common sewing needle, as shown in Fig. 2(c). The inner antenna spiral was covered in generic acrylic glue. Each pole of the antenna comprised 185 complete rotations in and out of the substrate. An ohm-meter having one lead as a needle tip was used to pierce the jacket and map the resistance of the pole from the center and continuing outwards. The resulting structure was flexible and robust, with line resistances of 28 and 42 over a length of 5.2 m each, as shown in Fig. 5(a). These resistance values remained unchanged after flexing of the substrate. We note that weaving caused a gradual increase in the resistance of the wire, attributed to the deterioration of the metal film under cyclic flexing and torsion. This is further supported by a large change in resistance, observed within coil #106, a region in which the jacket had

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woven antenna performance was then compared to that of a traditional plated copper antenna that used a rigid Rogers TMM4 substrate. Referring to Fig. 8 which displays the measured realized gain and reflection coefficient, we note that the woven polymer-composite spiral antenna performs as well as the traditionally made spiral antenna at high frequencies. At lower frequencies, E-fiber performance is noticeably lower than that of traditional antenna. Higher input mismatch at the lower frequencies contributes to such degraded performance. However, we think that the primary contribution for the degraded gain performance is due to that spirals at the outer circle resonate at lower frequencies, requiring the current to flow through a greater length of E-fiber, increasing the resistive losses in the less conductive, metal-coated Kevlar yarn. However, at higher frequencies, the shorter inner section of the spiral resonates, effectively carrying current in shorter yarn lengths. Fig. 7. Volumetric efiber-polymer composite antenna test platform.

IV. SUMMARY We presented a new technology wherein high strength metal-coated Kevlar yarns were woven into polymer-ceramic substrates for conformal, load-bearing antenna applications. With the constructed volumetric spiral antenna using metal coated, twisted yarns woven onto a polymer-ceramic composite, we found that the woven spiral antenna exhibited similar performance characteristics to those of a traditional antenna at high frequencies. Woven antenna performance suffered at low frequencies due to relatively higher resistive losses of the coated yarns, suggesting that further research is needed to optimize conductivity of the yarns for low frequency applications. These results further suggest potential broader benefits to energy harvesting or lighting applications in the DC regime using conducting fibers. [16]–[18]

REFERENCES

Fig. 8. Measurement comparison of the realized gain and reflection coefficient of conformal efiber –polymer composite antenna with traditionally made copper spiral on Rogers TMM4 substrate (see Fig. 4).

failed as a result of mechanical stresses, exposing and untwisting the fiber and abrading the metal, as shown in Fig. 6. The failed section was removed and replaced by a freshly made length of fiber. Antenna reparability was tested using conventional soldering of wires, as well as by tying a knot with the yarn. In the latter approach, the jacket in the knot was perforated and soaked in conductive silver paste, resulting in a repair that was transparent to electrical measurements. The completed antenna comprised 3 E-fiber segments, with a single knot on each pole; one to repair a failed jacket, and one to attain the full length required. It should be noted that using a more uniform and elastic polymer substrate, e.g., silicone, should prevent jacket delamination and substantially mitigate the loads on the yarn during weaving. B. Antenna Performance To demonstrate the RF performance of the proposed metallic Kevlar yarns woven on polymer substrates, we carried out realized gain and reflection coefficient measurements in an anechoic chamber. The e-fiberpolymercompositeantennawas fed with a1800 powercouplerto achieve input impedance of 100 Ohms as required by the antenna design. The

[1] Volakis and G. Kiziltas, “Novel materials for RF devices,” presented at the IEEE Antennas and Propagation Society Int. Conf., Jun. 2007. [2] S. Koulouridis, G. Kizitas, Y. Zhou, D. J. Hansford, and J. L. Volakis, “Polymer-ceramic composites for microwave applications: Fabrication and performance assessment,” IEEE Trans. Microw. Theory, vol. 54, no. 12, pp. 4202–4208, 2006. [3] Y. Ouyang, E. Karayianni, and W. J. Chappell, “Effect of fabric patterns on electrotextile patch antennas,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 3–8, 2005, vol. 2B, pp. 246–249. [4] P. Salonen, Y. Rahmat-Samii, H. Hurme, and M. Kivikoski, “Dualband wearable textile antenna,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 20–25, 2004, vol. 1, pp. 463–466. [5] T. F. Kennedy, P. W. Fink, A. W. Chu, and G. F. Studor, “Potential space applications for body-centric wireless and E-textile antennas,” in Proc. IET Seminar Antennas and Propagation for Body-Centric Wireless Communications, Apr. 24–24, 2007, pp. 77–83. [6] PBO Fiber Zylon Technical Information. Osaka, Japan, Toyobo Co., Ltd., 2005. [7] Kevlar Aramid Fiber Technical Guide. Richmond, VA, DuPont. [8] D. R. Lide, CRC Handbook of Chemistry and Physics, 78th ed. Boca Raton, FL: CRC Press, 1997, p. 12-43,12-44, 12-191, 12-192. [9] J. W. Lee et al., “Multifunctional metal/polymer hybrid for space and aerospace applications,” in Proc. Mater. Res. Soc. Symp., Warrendale, PA, 2005, vol. 851, NN5.3.1. [10] S. M. Lawrence, G. S. Dhillon, and K. W. Horch, “Fabrication and characteristics of an implantable, polymer-based, intrafascicular electrode,” J. Neurosci. Methods, vol. 131, no. 1–2, pp. 9–26, Dec. 2003. [11] R. A. Holler, “Metallized Kevlar for undersea electromechanical cables,” in Proc. Oceans, Sep. 1984, vol. 16, pp. 668–673. [12] Y. Shacham-Diamond, V. Dubin, and M. Angyal, “Electroless copper deposition for ULSI,” Thin Solid Films, vol. 262, no. 1–2, pp. 93–103, Jun. 1995. [13] Y. Rao and R. J. Farris, “A modeling and experimental study of the influence of twist on the mechanical properties of high-performance yarns,” J. Appl. Polymer Sci., vol. 77, no. 9, pp. 1938–1949, 2000. [14] S. Lagrangre et al., “Self-annealing characterization of electroplated copper films,” Microelectron. Engrg., vol. 50, pp. 449–457, 2000. [15] B. A. Kramer, “Size reduction of an UWB low-profile spiral antenna,” Ph.D. dissertation, The Ohio State Univ., Columbus, 2007.

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[16] B. O’Connor et al., “Fiber shaped organic light emitting device,” Adv. Mater., vol. 19, no. 22, pp. 3897–3900, Nov. 2007. [17] B. O’Connor, K. P. Pipe, and M. Shtein, “Fiber based organic photovoltaic devices,” Appl. Phys. Lett., vol. 92, no. 19, May 2008. [18] A. Yadav, K. P. Pipe, and M. Shtein, “Fiber-based flexible thermoelectric power generator,” J. Power Sources, vol. 172, no. 2, pp. 909–913, Jan. 2008.

A Compact MIMO Array of Planar End-Fire Antennas for WLAN Applications Antonio-Daniele Capobianco, Filippo Maria Pigozzo, Antonio Assalini, Michele Midrio, Stefano Boscolo, and Francesco Sacchetto Abstract—An approach to the design of multiple-input multiple-output (MIMO) arrays exploiting planar directive antennas is presented. It is well known that pattern orthogonality is a key aspect to reach low correlation, and thus to improve channel capacity in rich multipath environments. However, attention is often focused on reducing mutual coupling rather than optimizing the active element patterns. In this communication a planar MIMO array of printed Yagi-Uda antennas with integrated balun is presented. The end-fire radiation mechanism of the Yagi-Uda is exploited to obtain a triangular array of three sectoral antennas. This allows to achieve nearly orthogonal patterns, while keeping a low mutual coupling among radiating elements. A properly shaped ground at the feeding points allows to increase the isolation between the antennas, even in such a compact layout. A laboratory model has been characterized experimentally, and the effectiveness of the proposed design in terms of theoretical achievable capacity is demonstrated through numerical simulations considering IEEE 802.11n multipath fading channel models. Index Terms—Antenna arrays, directive antennas, information rates, multipath channels, multiple-input multiple-output (MIMO) systems, wireless LAN.

I. INTRODUCTION The design of antenna elements to be employed in multiple-input multiple-output (MIMO) arrays is recognized as a key feature to reach high channel capacity. However, current implementations often employ uniform linear arrays (ULA) of dipoles or monopoles (i.e., nearly omnidirectional radiators) with quite electrically large spacings (i.e., large fractions of the free space wavelength), in order to reduce the mutual coupling and to minimize the correlation between signals at the antenna ports. To this aim, a half-wavelength inter-element spacing is quite common. Such a relatively large distance among radiating elements thus allows to exploit the principle of spatial diversity to improve the channel capacity, reasonably neglecting the effects of pattern distortion due to mutual coupling. Nevertheless, in order to reach very compact MIMO array configurations, where antennas are close to each other, Manuscript received October 15, 2010; revised December 21, 2010; accepted February 17, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the University of Padova, project CPDA 081514/08. A. D. Capobianco, F. M. Pigozzo, and A. Assalini are with the Department of Information Engineering, University of Padova, Padova 35131, Italy (e-mail: [email protected]). M. Midrio, S. Boscolo, and F. Sacchetto are with the Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica, University of Udine, Udine 33100, Italy. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161557

Fig. 1. Layout of the proposed array (top and bottom layers). The whole structure is 55 mm 48 mm large. Each antenna is 22 mm 22 mm large and lays on the side of a equilateral triangle. The separation among the feeding points is 5.77 mm (0:1 in free space at the 5.25 GHz operation frequency).

2

2

the pattern distortion can no longer be neglected, and the displacement between antennas will result not only in space diversity but also in an induced pattern diversity. Therefore, to account for this effect, the antenna active pattern [1] (i.e., the pattern produced by the excitation of one antenna when the others are closed on a matched load) instead of the isolated antenna pattern has to be employed in the calculation of the correlation coefficients [2] of the transmitter/receiver array. Nevertheless, practical MIMO array designs often focus on the isolated element, and the active pattern resulting from the interaction among antennas is optimized in a second step, by tuning, for instance, the relative orientation among radiators [3]. More sophisticated designs aim to achieve pattern orthogonality in different ways, for example by exciting orthogonal modes within the same geometrical structure in co-located patch antennas [4] or in spirals [5]. Also pattern synthesis based on properly tailored current distribution in a theoretical array [6], or reconfigurable planar arrays made of combined Landstorfer and Yagi-Uda antennas have recently been proposed [7]. In this communication, we present a simple and easy-to-implement solution to achieve nearly orthogonal patterns. Starting from a sectoral isolated element pattern, such as the one of an end-fire radiator, angular diversity is exploited. In particular, a very compact, printed Yagi-Uda antenna with integrated balun has been developed [8]. Then, a MIMO array has been obtained in a triangular configuration (Fig. 1). In the following, we present design criteria for optimizing both the single antenna element and the overall array. A laboratory model has been realized and experimentally characterized in terms of return loss and active element pattern. The effectiveness of the proposed layout is assessed against the resulting channel capacity with IEEE 802.11n propagation models [9], in the 5.15 GHz–5.35 GHz operational band. II. ARRAY LAYOUT In this section we describe the structure of the proposed MIMO antenna array. A. Single Antenna Design The use of non isotropic antennas as a MIMO array element has been recently investigated through channel measurements in [10] with patch antennas. These kind of radiators, however, are backed by a ground plane which is orthogonal to the direction of the main beam, thus preventing their use in a low-profile configuration. Aiming to realize a directive pattern in the horizontal plane, we developed a planar Yagi-Uda antenna with an integrated balun that is 22 mm long (less than half a wavelength in free space at the 5.25 GHz operation frequency). A printed Yagi-Uda antenna has been presented in [11] where a broadband microstrip-to-coplanar strips (CPS) transition is employed. The ground

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Fig. 2. Schematic of the proposed planar Yagi-Uda antenna (top and bottom layers are depicted in the right and left panel, respectively). Dimensions in mil;b ;c : ;d ; ls ; ws ;w limeters are: a

= 6 = 3 = 0 5 1 = 3 = 22 = 22 1 = 1; a2 = 3; b2 = 0:5; d2 = 4:25; d3 = 5; d4 = 4; lg = 5:4; l1 = 20:5; l2 = 16:4; l3 = 11:28; s = 0:5; wg = 8; w0 = 1:6; w2 = 1. The ground plane extension (contained within dashed lines) is delimited by a Y-shaped slot whose branches are spaced by 120 degrees and are 1 mm in width, as shown in Fig. 1.

plane below the transition acts also as the Yagi-Uda reflector. Here we propose a layout consisting of a different microstrip-to-CPS transition with a shaped ground that allows for both the reduction of the metallization near the feeding point and the optimization of the reflector element (Fig. 2). A Y-shaped slot on a face of the substrate prevents neighbor antenna ground planes to overlap each other when three elements of this kind are employed in a triangular array configuration (Fig. 1). Indeed, every single radiator has its own ground plane, without electrical continuity with its neighbor. This has been done to improve antenna-ports isolation. Moreover, instead of connecting the driven element of the Yagi-Uda antenna directly to the feeding microstrip, as in [11], we excite the entire structure by proximity coupling, using an open ended microstrip above a rectangular hole in the ground plane. It is worth mentioning that reflector, driven element and director are on the same face of the substrate. We performed the optimization of the antenna and its feeding line through CST Microwave Studio [12]. As a first design step, the effectiveness of the transition was studied with and without the reflector extensions (l1 in Fig. 2). Simulations have shown that the return loss and bandwidth are very sensitive to the ratio a=b and to the magnitude of the cut c, as well as the ground hole dimensions a2 and b2, once the a=b ratio has been fixed. We used a 50 impedance microstrip line on a Rogers RO3010 (r = 10) substrate of height h = 1:28 mm that converts to a balanced CPS line with 100 characteristic impedance. As a second design step, we set an initial length and width of the driven element alone, in order to achieve resonance at the desired center frequency along with an appreciable bandwidth. As a third step, we included the director element and performed a sweep on the distance from the active element. This had been done for different values of the director length. Particular attention was devoted to reach high directivity and reduced back lobe magnitude, since the coupling with the neighbor antennas is expected to decrease with a reduction of backward radiated energy. To this end, the introduction of the reflector further improves the back lobe suppression. The final step consisted in optimizing the entire structure with fixed CPS line length. The impedance matching was realized by acting again on the a=b ratio of the transition and on the driven element-to-director distance. A further adjustment is obtainable by multiplying the lengths of the three dipoles by a common scale factor. B. Array Behavior The MIMO array is obtained by placing three printed Yagi-Uda antennas in a equilateral triangular configuration, see Fig. 1. As expected,

Fig. 3. Measured E active element pattern for Yagi-Uda 1, 2 and 3 (normalized values) in the azimuthal plane. The cross polarized component E , which is not reported in the plot for the sake of clearness, is below 0.1.

Fig. 4. Photo of a laboratory model of the proposed MIMO array.

the sectoral radiating properties of each element, as well as the absence of a common ground plane, results in an almost undistorted active pattern (Fig. 3), that is very low mutual coupling is achieved. This result is important, as typically the active patterns are strongly deformed by mutual coupling in small arrays. Moreover, there is no need for additional matching line as the scattering parameter jsii j of each antenna is well below the 010 dB level, along with low values of the transmission coefficients jsij j in the desired 5.15 GHz–5.35 GHz band. This was confirmed by measurements on a laboratory model, employing a Quickform-type coaxial cable at the desired antenna port, and connecting the others to 50 (matched) surface mounted device (SMD) resistors (Fig. 4). Simulated and measured values of js11 j are shown in Fig. 5. The active element patterns (Fig. 3) exhibit a directivity of 6 dBi and a weak overlap in the azimuthal plane, i.e., they are nearly orthogonal. Therefore, as shown in the following, a high channel capacity is expected, in a rich scattering environment. III. PERFORMANCE IN 2D MIMO SCENARIOS The proposed array is intended to be used in rich multipath scenarios. Its performance is evaluated in terms of the correlation coefficient and the ergodic channel capacity.

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Fig. 5. Simulated (dashed line) and measured (solid line) js j of the Yagi-Uda antenna 1 in array configuration, when the others are terminated by a 50 load.

The correlation coefficient between the generic ith and j th antenna of the array, at either transmitter or receiver side, is given by [2]

ij

=

( ) ( ) ( ) + p ()E ()E () 3

p  Ei  Ej 



i

i2 j2

3

j

2

Ek

j

k

j



j

k

H

(1)

= p () E () 2 + p () E () 2 d; k = i; j (2) () and E () are, respectively, the components  and  of the 

The channel matrix H is calculated as

d

where k

Fig. 6. Ergodic capacity over a 1000 channel realization as a function of SNR, for a uniform PAS p() = 1=2 in the [0; 2] range (single-cluster flat-fading channel model).

j

=

=

(3)

where H iid is a 3 2 3 matrix whose entries are independent and identically distributed complex Gaussian random variables with unitary power. The ergodic channel capacity C for a given average signal-tonoise ratio (SNR) value results

k

electric field radiated by the k th antenna, in the azimuthal plane, intended as active element pattern in the far-field region [1]. The co-polar component is assumed to be the  component of the field, which lays on the plane parallel to the substrate.  is the cross-polar discrimination, p () and p () are the power azimuth spectrum (PAS) of the two polarizations, in the azimuthal plane, and, finally, ( 1 )3 denotes the complex conjugate. In our analysis we assume  = 1: this means that the scenario provides half of the received power with the  polarization and half of the received power with the  polarization. In addition, we consider p () = p () = p(). These assumptions are widely accepted in the literature for transmissions in rich multipath environments [13]. Since a careful evaluation of the correlation between antennas requires the complex-valued active element patterns, simulated values have been used in the computation of (1) and (2). For a first performance evaluation, we consider a uniform PAS p() = 1=2 in the interval [0; 2 ) and a flat Rayleigh fading channel with a single-cluster. The correlation matrices RTx and RRx for the transmitter and the receiver have elements of position (i; j ) given by (1), with parameters depending on the array adopted at either the transmitter or the receiver side, respectively. Therefore, the overall channel correlation matrix is R = R Tx R Rx , where indicates the Kronecker product. We fix the trace of R , i.e., the SNR is fixed at the receiver. In particular, the transmitter is equipped with a ULA of dipoles with a =2 inter-element spacing. Such a setting is representative of the WLAN downlink transmission, where the fixed access point has not severe size constraints. This configuration also assures that the assumptions used to define the IEEE 802.11n channel models are fulfilled [9]. The well-known Bessel-type correlation coefficient ij (d) = J0 (2d=) is used to evaluate the correlation among the three dipoles at the transmitter [2]. At the receiver side, our sectoral array of Yagi-Uda elements is compared, in terms of channel capacity, with an ideal receiver, whose correlation matrix is the identity matrix.

1 2 = R1Rx2H iidRTx

C

=

log2 det

I

+ N Tx H H y

(4)

where [1] denotes expectation with respect to different channel realizations H iid ; I is the 3 2 3 identity matrix, NTx = 3 is the number of transmit antennas, and ( 1 )y denotes Hermitian transposition. Furthermore, to evaluate the performance of the Yagi-Uda array in other practical scenarios, we resorted to numerical simulations involving clustered, frequency selective MIMO channel models [9]. For this class of channels we considered a truncated Laplacian PAS [14]

( )=

2  2 =1 1 )] 1[ (0 1[ (0 + 1 )]

N

p 

k

Qk

k

2 f

p

0

0

0

exp

p

j

0

0

0;k j

2 111

k

;k 0

;k

k

k

(5)

g

where Nc is the number of clusters to be considered for a given tap of the channel impulse response, Qk is a normalization constant for the k th cluster, 0;k is the angle of arrival (AoA) of the k th cluster, which is centered on the [0;k 01k ; 0;k +1k ] interval with 21k being the angular spread (AS), k is the standard deviation of the non-truncated distribution relative to the k th cluster and, finally, 1[1] is the Heaviside step function. Let L be the number of channel taps, then for each lth tap, l = 0; 1; . . . ; L 0 1, we first compute the PAS in (5) and, similarly to the flat Rayleigh fading case, by using (1) we obtain the resulting correlation matrices. Therefore, as for (3), for each lth channel tap, we find the corresponding channel matrix H l . The computation of the capacity of multipath clustered MIMO channels requires the discrete Fourier transform of the channel [15]

~ =

L01

0j 2l(k=N )

H le

Hk

l=0

;

k

= 0; 1; . . . ; N 1 0

(6)

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 9, SEPTEMBER 2011

Fig. 7. Ergodic capacity over a 1000 channel realization as a function of SNR, for a truncated Laplacian PAS in a clustered, frequency selective fading channel model (model D: 18 taps—3 clusters channel impulse response). A rotated version of the proposed array is also considered (60 degrees counterclockwise rotation, with respect to Fig. 3).

where the transform is computed over godic capacity is obtained as

C=

N > L points. Hence, the er-

1 N01 log det I + H~ kH~ y k N k=0 2 NTx

:

L

:;

:



:; :; :

[5] C. Waldschmidt and W. Wiesbeck, “Compact wide-band multimode antennas for MIMO and diversity,” IEEE Trans. Antennas Propag., vol. 52, pp. 1963–1969, Aug. 2004. [6] B. T. Quist and M. A. Jensen, “Optimal antenna radiation characteristics for diversity and MIMO systems,” IEEE Trans. Antennas Propag., vol. 57, pp. 3474–3481, Nov. 2009. [7] A. C. K. Mak, C. R. Rowell, and R. D. Murch, “Low cost reconfigurable Landstorfer planar antenna array,” IEEE Trans. Antennas Propag., vol. 57, pp. 3051–3061, Oct. 2009. [8] A. D. Capobianco, F. M. Pigozzo, S. Boscolo, M. Midrio, F. Sacchetto, A. Assalini, L. Brunetta, N. Zambon, and S. Pupolin, “A novel compact MIMO array based on planar Yagi antennas for multipath fading channels,” in Proc. Eur. Wireless Technology Conference (EuWIT 2010), 2010, pp. 93–96. [9] TGn Channel Models, IEEE 802.11-03/940r4, May 2004. [10] C. Hermosilla, R. Feick, R. Valenzuela, and L. Ahumada, “Improving MIMO capacity with directive antennas for outdoor-indoor scenarios,” IEEE Trans. Wireless Commun., vol. 8, pp. 2177–2181, May 2009. [11] N. Kaneda, W. R. Deal, Y. Qian, R. Waterhouse, and T. Itoh, “A broadband planar quasi-Yagi antenna,” IEEE Trans. Antennas Propag., vol. 50, pp. 1158–1160, Aug. 2002. [12] CST Microwave Studio 2009. Darmstadt, Germany. [13] R. Vaughan and J. B. Andersen, Channels, Propagation and Antennas for Mobile Communications, ser. IEE Electromagnetic Waves. London: IEE Press, 2003, vol. 50. [14] L. Schumacher, K. I. Pedersen, and P. E. Mogensen, “From antenna spacings to theoretical capacities—Guidelines for simulating MIMO systems,” in Proc. IEEE PIMRC, 2002, vol. 2, pp. 587–592. [15] H. Bölcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDMbased spatial multiplexing systems,” IEEE Trans. Commun., vol. 50, pp. 225–234, Feb. 2002.

(7)

Figs. 6 and 7 show that the proposed array offers a theoretical achievable capacity that is remarkably close to the ideal (i.e., uncorrelated) receiver case, both for a uniform PAS scenario and for a clustered channel (model D in [9]), respectively. In the latter case, a 60 rotation of the proposed array is also considered, to verify the robustness of the design. It is worth mentioning that the channel model D corresponds to a channel with a root-mean square time delay spread equal to 50 ns, = 18 taps, 3 clusters with angle of arrival (AoA) 0;k 2 f158 9 320 2 276 2g[ ] and AS = 21 k 2 f27 7 31 4 37 4g[ ].

:;

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IV. CONCLUSION In this communication it was shown that the use of short end-fire antennas as sectoral radiators, in a planar compact MIMO array, results in almost orthogonal patterns. Specifically, we considered printed Yagi-Uda antennas. The proposed design leads to a low signal correlation among the radiating elements. Therefore, channel capacity can be maintained high even in such a small array (55 mm 2 48 mm). This can be particularly attractive for very compact devices to be employed in indoor applications. Numerical evaluation of the array performance in typical 2D MIMO scenarios, by means of IEEE 802.11n channel models, confirmed the validity of the considered approach. The reported design principle can be extended to arrays having a large number of antennas.

REFERENCES [1] D. M. Pozar, “The active element pattern,” IEEE Trans. Antennas Propag., vol. 42, pp. 1176–1178, Aug. 1994. [2] R. G. Vaughan and J. B. Andersen, “Antenna diversity in mobile communications,” IEEE Trans. Veh. Technol., vol. 36, pp. 149–172, Nov. 1987. [3] S. H. Chae, S.-K. Oh, and S.-O. Park, “Analysis of mutual coupling, correlations, and TARC in WiBro MIMO array antenna,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 122–125, 2007. [4] A. Forenza and R. W. Heath, “Benefit of pattern diversity via twoelement array of circular patch antennas in indoor clustered MIMO channels,” IEEE Trans. Commun., vol. 54, pp. 943–954, May 2006.

Axial Ratio Enhancement for Circularly-Polarized Millimeter-Wave Phased-Arrays Using a Sequential Rotation Technique A. Bart Smolders and U. Johannsen Abstract—Circular polarization is indispensable for robust wireless communication between mobile devices that operate at mm-wave frequencies. Additionally, phased-array solutions are required to cope with the associated free space path loss. In view of the size constraints for antennas integrated on (Bi)CMOS chips, an array of linearly polarized dipoles using a sequential rotation scheme is an attractive approach to comply with all mentioned requirements. When steering such an array off broadside, however, the axial ratio will severely degrade. It is the purpose of this communication to demonstrate how the axial ratio can be retained by compensating the amplitudes and phases of the individual antenna elements. Measured results on a 6 GHz test-bed show that the axial ratio with the proposed calibration scheme remains below 3 dB within the 3 dB beamwidth of the scanned beam. Results from a 60 GHz test-bed confirm the effectiveness of the method. Index Terms—Antenna-on-chip, axial ratio, calibration, circular polarization, integrated antennas, mm-wave antennas, phased arrays, printed dipoles.

I. INTRODUCTION At millimeter-waves (mm-waves), the integration of the antenna into low-cost silicon IC technologies with high transit frequencies fT becomes feasible [1], [2]. At these frequencies, the effective wavelength for an antenna-on-chip (AoC) on silicon is close to or smaller than a Manuscript received August 26, 2010; revised November 19, 2010; accepted February 09, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161443

0018-926X/$26.00 © 2011 IEEE

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millimeter, allowing for a cost-effective integration of small arrays of AoCs, which are crucial to cope with the high free space path loss at these frequencies. Even the integration of a small antenna array on a single chip might be a feasible option. Both variants shall be consolidated under the term “antenna-array-on-chip” (AAoC) throughout this communication. One advantage of an AAoC is the possibility to directly match the individual antenna elements to the differential electronic circuits. Moreover, low-cost wire-bonding assembly techniques can be applied since there is no need for any mm-wave interconnect outside the IC. Circular polarization is indispensable for robust wireless communication between portable/mobile devices at mm-waves. Otherwise, due to the line-of-sight nature of the propagation channel at mm-waves [3], linearly-polarized (LP) solutions would need a highly accurate polarization alignment, which is not very practical to implement for portable/mobile applications. Recent work in [4] describes circularlypolarized (CP) antennas at mm-waves. However, this concept cannot be used in the case of an AAoC due to size and technology constraints and the reported results were limited to non-scanning arrays. Since dipoles have proven to be an effective antenna concept for AAoC integration in terms of size and functionality [2], we chose dipoles as a starting point for circular polarization. An elegant way to create circular polarization with linearly polarized elements in an array environment is by using the sequential rotation technique as introduced by John Huang in 1986 [5]–[7]. However, up to now sequential rotation has mostly been used for non-scanning arrays. In this communication, we describe a straightforward method to use this technique also effectively for beam-steering arrays. This is done by using an adapted calibration technique that retains circular polarization over a wide steering range. This communication presents the concept of sequential rotation for mm-wave phased-arrays that can be integrated on-chip. This is done by, first, demonstrating the proposed calibration method on a scaled 12-element antenna array operating at 6 GHz. Next, the polarization characteristics of a small 2 2 2 array operating at 60 GHz are presented. By using a low-cost, low-complexity, scaled test-bed at 6 GHz, we could avoid the issue of measurement inaccuracies that are typically faced at mm-waves, e.g., imperfect antenna alignment. Hence, we can prove the concept first before applying it at higher frequencies. Differential dipole structures are used for compatibility with on-chip integration. All results and derivations will be given for the '0 = 00 plane. However, the concept is general and can also be used in other planes. II. SEQUENTIAL ROTATION TECHNIQUE FOR SCANNING ARRAYS USING A ROBUST CALIBRATION TECHNIQUE CP radiation can be achieved by applying proper phase shifts to sequentially rotated linearly polarized (LP) antenna elements [5], as illustrated in Fig. 1. By integrating a sub-array consisting of 2 2 2 LP dipoles on a chip, using the approach of [2], we can create integrated mm-wave CP phased-arrays. The required phase-shift and amplitude tapering can be made on-chip by using, for example, an electronic vector modulator. However, in this work we use printed-circuit-board (PCB) dipoles as radiating elements for the proof of concept. The basic configuration as shown in Fig. 1 has limitations when applying beam scanning. This is best illustrated by considering an array of small electric dipoles. Fig. 2 shows the predicted axial ratio (AR) versus scan angle 0 ('0 = 0 plane) for an array of ideal electric dipoles using the configuration of Fig. 1. From this figure we can observe that even for ideal electric dipoles the AR deteriorates when scanning the main beam away from broadside. This is due to the basic differences between the E-plane and H-plane patterns of an ideal electric dipole. When we consider more realistic antenna elements we also need to include the cross-polarization components. Furthermore, in a realistic

Fig. 1. Array that generates CP radiation using sequentially rotated LP antenna elements (top-view). (The amplitude and phase of each element can be adjusted for scanning and calibration purposes.)

Fig. 2. Calculated axial ratio in main beam direction of an array of Hertzian plane). dipoles using the configuration of Fig. 1 ('

=0

array we also need to account for variations over time of the antenna and electronics, requiring an on-line calibration scheme. We therefore propose the following robust calibration scheme for an array with sequential rotation. A. Off-Line Calibration The basic idea of the off-line calibration is to pre-measure or simulate the element patterns of each array element for both the co- and cross polarization. From the measured patterns we can then compute the required amplitude and phase correction values for each scan angle. These complex values can be stored in a look-up table. We will illustrate the basic procedure using a 2 2 2 subarray with sequential rotation. In the ' = 0 plane, we can write the far-field pattern of such an array in the following general form:

Etot () = a1 E1;co ()+ a3 E3;co()+ a2 E2;X ()+ a4 E4;X () E'tot ()= a1 E'1;X ()+ a3 E'3;X ()+ a2 E'2;co()+ a4 E'4;co()

(1)

where Etot and E'tot represent the  – and '–component of the total electric field, E1;co is the co-polarized component due to antenna element 1 and E2;X is the cross-polarized component due to element 2. A similar explanation holds for the other field components. Note that we assumed that the required phase related to beam steering is already included in the various field components. Furthermore, ai (with

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i = 1,2,3,4) represents the complex excitation of element i, excluding the phase associated with beam steering. Since we apply sequential rotation, we will choose a3 = 0a1 and a4 = 0a2 . Therefore, we can write expression (1) in the following simplified form:

tot

E

tot E'

() = a1 (E 1 ()0 E 3 ()) + a2 (E 2 ()0 E 4 ()) = a1 E 103 () + a2 E 204 () () = a1 (E 1 ()0 E 3 ()) + a2 (E 2 ()0 E 4 ()) = a1 E 103 () + a2 E 204 () (2) 

;co



;co



;X ' ;X '

;co

;X '



'



;X

;X



;co '

;X

;co '

;co

where we have simplified the notation by introducing the notation 1;co () 0 E3;co() = E103;co(). The correction procedure is based on solving the far-field components (2) for optimal circular polarization at the required scan angle 0 . This is done by solving the following set of linear equations: E

tot

E

tot

E'

(0 ) = a1 E 103 (0 ) + a2 E 204 (0) = 1 (0 ) = a1 E 103 (0) + a2 E 204 (0) = j ;co





;X

'

'

Fig. 3. Photograph of the 6 GHz test-bed, consisting of an array of 6 dipoles placed according to the sequential rotation technique.

2 2 printed

;X

;co

(3)

in which the total field components have been normalized. Solving this set of linear equations results in: 103 204 ( ) = 204 jE103 (0 ) 0 E103 (0 ) 204 E (0)E (0 )0 E (0 )E (0) 204 ( 0 ) 1 0 a2 (0 )E a1 (0 ) = (4) 103 E ( ) : a2 0

'

;co



;co





 ;co

;co

;X

'

'

;X

;X



;X

0

In a large array, the calibration can be done on a 2 2 2 sub-array level, using (4) for each of the sub-arrays. B. On-Line Calibration Besides the off-line calibration of the element patterns, we need to adjust for (time-varying) errors in the array system. This can be done by using the multi-element phase toggling (MEP) technique based on the work presented in [9]–[12]. In the experiments presented in this communication we did not use the on-line calibration scheme since our test-bed only consists of passive structures in a conditioned test-environment. III. RESULTS FROM A SCALED MODEL AT 6 GHZ First, we will test the calibration procedure on a scaled test-bed operating at 6 GHz. The intention of the test array was not to make an optimal design in terms of intrinsic polarization purity. In fact, we wanted to make an array with moderate polarization properties in order to test our calibration strategy. A. Array Design A 6 2 2 array of dipoles with sequential rotation was developed as shown in the photograph of Fig. 3. A microstrip balun was designed to connect the 100 differential transmission line from the dipoles to standard 50 Ohm SMA connectors. The dipoles (length l = 21 mm, width w = 2 mm, slot s = 1 mm) are printed on a RO3003 ("r = 3 and tan  = 0:0013) substrate with a thickness of 0.5 mm. The distance between the elements in x and y-direction is dx = dy = 0 =2 @ 6 GHz. For practical reasons, four out of 12 array elements have been terminated directly at the differential dipole input with a 100 resistor. The predicted and measured S11 of a single element with balun within the array environment are shown in Fig. 4. The simulation was done

Fig. 4. Simulated (MoM) and measured S connector in Fig. 3).

of array element 2 (middle lowest

with a method-of-moments (MoM) code [13] and agrees well with the measurements. B. Test Results Fig. 5 shows the measured E- and H-plane pattern of an individual array element at 6 GHz. The element is measured within its array-environment, while the other elements have been terminated with the characteristics impedance. The polarization characteristics are quite moderate, which is partly due to the balun structure that is used on the PCB. Without any calibration the measured AR when scanning the beam off broadside is very poor as illustrated in Fig. 6, where the main beam of the array is scanned towards 0 = 25 and 0 = 60 , respectively ('0 = 0 plane). The poor AR is mainly due to the large difference between the E-plane and H-plane pattern of the individual antenna elements. A further reason is the non-negligible cross-polarization. Next we will apply the calibration scheme of Section II. The calibration is done on a 2 2 2 subarray level by using (4). Note that we were not able to measure the element patterns of the four array elements of Fig. 3 which are directly terminated with 100 resistors. For these elements, we have used the approximation that E'4;co ()  0E'2;co (0), and E4;X ()  0E2;X (0) (refer to Fig. 1 for the numbering). The optimal phase and amplitude for each of the array elements has been determined by post-processing the recorded element patterns. We have assumed an uniform amplitude tapering along the array. The whole procedure has been implemented in Matlab. When we apply the calibration scheme, the AR significantly improves near the main beam area, as shown in Fig. 6. Now the AR is well below 3 dB within the 3 dB beamwidth of the main beam of the array. The region of an AR below 3 dB is highlighted between the grey areas

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Fig. 5. Measured normalized element pattern of an individual dipole within ( -plane) and ' ( -plane). the 6 2 array at 6 GHz for the ' Element 2 (middle lowest connector in Fig. 3) results are shown.

2

=0 E

= 90 H

Fig. 7. Predicted and measured left-hand and right-hand circularly polarized array patterns of the test array with calibration; main beam scanned to  (' plane).

= 25

=0

TABLE I OFF-LINE CALIBRATION COEFFICIENTS FOR SUBARRAY 1 (ON LEFT SIDE) , NORMALIZED W.R.T. a . THIS OF FIG. 3, ACCORDING TO (4), EXCLUDES THE ADDITION PHASE REQUIRED FOR BEAMSTEERING

f = 6 GHz

Fig. 8. Photograph of the 60 GHz test-bed, consisting of 2 placed according to the sequential rotation technique.

2 2 printed dipoles

IV. RESULTS AT 60 GHZ

After proving the concept at 6 GHz, a small 2 2 2 array of printed dipoles with sequential rotation operating at 60 GHz was developed and tested. Although the array was fabricated in standard PCB technology, this setup already closely mimics the targeted AAoC scenario. Fig. 6. Axial ratio of the test array with and without calibration based on the measured co/cross element patterns at 6 GHz; main beam scanned to a)  , b)  (' plane).

25

= 60

=0

=

in Fig. 6. The calibration coefficients a1 and a2 are given in Table I. The corresponding left-hand (LH) and right-hand (RH) circular polarization beam patterns for the case of 0 = 25 are presented in Fig. 7. The predicted patterns with calibration from a MoM code are also shown. The quality of the patterns is very good, except for some higher far-out sidelobes. The sidelobe near  = 035 corresponds with a grating lobe of an array with element spacing of 2dx .

A. Array Design

Fig. 8 shows a photograph of the 2 2 2 array of dipoles. The dipoles are printed on RO3003 (thickness 0.5 mm), and are compatible with on-chip integration as described in [2]. Unlike the 6 GHz dipoles, the 60 GHz dipoles are placed inside a cavity, which is composed of a ground plane surrounding the dipoles (see Fig. 8), and a reflecting ground plane on the opposite side of the PCB. By this, substrate modes that can severely deteriorate the radiation pattern of the antennas are efficiently suppressed. One dipole is fed via a differential line, which is connected to a GSSG probe. All other dipoles are matched via resistors into 100

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the calibration frequency. This corresponds to a bandwidth of approximately 3.2%. V. CONCLUSION

Fig. 9. Measured E-plane and H-plane element pattern of a printed dipole in the 2 2 array with sequential rotation at 60 GHz.

2

Fig. 10. Measured axial ratio of the 60 GHz test array with and without calibration for a scan angles of  = 25 (' = 0 plane).

(specified up to 20 GHz). Together with the long feed-lines of seven guided wavelengths and their associated loss, these elements can be considered as sufficiently matched. B. Test Results The mm-wave antenna set-up as described in [14] was used to measure the radiation characteristics of each of the elements. The effect of the environment (e.g., probe-station) was reduced by employing time-gating. The measured E- and H-plane element patterns are shown in Fig. 9. Note that the H-plane pattern for angles smaller than 040 is deteriorated due to the probe that blocks the line-of-sight between the antenna-under-test (AUT) and the reference antenna. With future improvement of the measurement set-up the accuracy of the obtained correction factors for larger scan angles can be further improved. The measured axial ratio of the 2 2 2 array with and without calibration is shown in Fig. 10 for the case of scanning towards 0 = 25 . From this, it is apparent that the AR is noticeably improved by applying the off-line calibration scheme and is below 3 dB over a wide range around the scan angle. The required dynamic range of the correction factors is within a += 0 3 dB range in this example. The frequency dependence of the calibration factors was also investigated. We found that the AR is deteriorated with 2 dB for a frequency offset of 1 GHz as compared to

In this communication we have shown that excellent circular polarization purity can be obtained for scanning antenna arrays consisting of elements with moderate linear polarization characteristics. This is done by applying a sequential rotation technique in combination with a straightforward calibration scheme. The proposed differential antenna structures can be easily transferred to AAoC operating at mm-waves, where the amplitude and phase correction can be directly implemented in the active circuits. However, due to limitations of the mm-wave radiation pattern measurement setup, e.g., AUT alignment inaccuracies, the considerations for the mm-wave array were limited to 0 = 25 in this communication. For larger steering angles the high cross-polarization components result in a significantly smaller range for angles with an AR below 3 dB. With future improvements of the setup, an enhanced mm-wave off-line calibration will be possible.

REFERENCES [1] W. D. van Noort et al., “BiCMOS technology improvements for microwave application,” in Proc. IEEE Bipolar/BiCMOS Circuits and Technology Meeting, Oct. 2008, pp. 93–96. [2] U. Johannsen, A. B. Smolders, R. Mahmoudi, and J. A. G. Akkermans, “Substrate loss reduction in antenna-on-chip design,” in Proc. IEEE Antennas and Propagation Symp., 2009, pp. 1–4. [3] P. Smulders, “Exploring the 60 GHz band for local wireless multimedia access: Prospects and future directions,” IEEE Commun. Mag., vol. 40, no. 1, pp. 140–147, Jan. 2002. [4] A. R. Weily and Y. J. Guo, “Circularly polarized ellipse-loaded circular slot array for millimeter-wave WPAN applications,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2862–2870, Oct. 2009. [5] J. Huang, “A technique for an array to generate circular polarization with linearly polarized elements,” IEEE Trans. Antennas Propag., vol. 34, no. 9, pp. 1113–1124, Sep. 1986. [6] P. S. Hall, “Application of sequential feeding to wide bandwidth, circularly polarized microstrip patch arrays,” Proc. Inst. Elect. Eng., vol. 136, pt. H, pp. 390–398, May 1989. [7] A. K. Bhattacharyya, “Comparison between arrays of rotating linearly polarized elements and circularly polarized elements,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2949–2954, Sep. 2008. [8] K. W. Weiler, “The synthesis radio telescope at Westerbork: Methods of polarization measurement,” Astron. Astrophys., vol. 26, pp. 403–407, 1973. [9] K. M. Lee, R. S. Chu, and S. C. Liu, “A built-in performance-monitoring fault isolation and correction (PMIFIC) system for active phased-array antennas,” IEEE Trans. Antennas Propag., vol. AP-41, pp. 1530–1540, 1993. [10] A. B. Smolders and G. A. Hampson, “Deterministic RF nulling in phased arrays for the next generation of radio telescopes,” IEEE Antennas Propag. Mag., vol. 44, no. 4, pp. 13–22, Aug. 2002. [11] G. A. Hampson and A. B. Smolders, “A fast and accurate scheme for calibration of active phased-array antennas,” in Proc. IEEE lnt. Symp. on Antennas and Propagation Digest, Orlando, Jul. 11–16, 1999, pp. 1040–1043. [12] H. Pawlak and A. F. Jacob, “An external calibration scheme for DBF antenna arrays,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 59–67, Jan. 2010. [13] Momentum [Online]. Available: www.agilent.com [14] J. A. G. Akkermans, R. v. Dijk, and M. H. A. Herben, “Millimeter-wave antenna measurement,” in Proc. Eur. Microwave Conf., pp. 83–86.

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Gain-Enhanced 60-GHz LTCC Antenna Array With Open Air Cavities Siew Bee Yeap, Zhi Ning Chen, and Xianming Qing

Abstract—The gain of low temperature co-fired ceramic (LTCC) patch antenna arrays operating at 60 GHz is enhanced by introducing open air cavities around their radiating patches. The open air cavities reduce the losses caused by severe surface waves and dielectric substrate at millimeter-wave (mmW) bands. The arrays are excited through either a microstrip-line or stripline feed network with a grounded coplanar-waveguide (GCPW) transition. The GCPW transition is designed so that the antenna can be measured with the patch array facing free space therefore reducing the effect of the probe station on the measurement. The proposed antenna arrays with the open air cavities achieves gain enhancement of 1–2 dB compared to the conventional antenna array without any open air 10 cavity across the impedance bandwidth of about 7 GHz for dB at 60-GHz band.

Fig. 1. Conventional aperture-coupled patch antenna.

Index Terms—60 GHz, arrays, millimeter wave antennas, low temperature co-fired ceramic (LTCC), surface waves. Fig. 2. Three LTCC aperture-coupled patch antennas with and without open air cavity operating at 60 GHz.

I. INTRODUCTION At millimeter-wave (mmW) frequencies, conductor loss, dielectric loss and surface wave loss become higher and are critical to the gain of antennas [1]. In particular, the larger electrical thickness and higher permittivity of low temperature co-fired ceramic (LTCC) substrate used in antenna array design at mmW result in significant losses such that enhancing the gain of the antennas becomes much more challenging. There have been quite a few reported methods on how to suppress the losses, specifically caused by severe surface waves. The use of high impedance surfaces around patch antennas has been applied in particular the uni-planar electromagnetic band-gap (UC-EBG) on a 60 GHz LTCC array [2] but with the increased size of antenna arrays. The use of embedded cavity to lower the effective dielectric constant has been reported [3], [4]. However, the required extra processing increases the complexity and price, and has a high chance of deformation. An alternative method is to partially remove the substrate surrounding the radiating patches [5]–[8]. The patch antennas at 2.4 GHz on printed circuit board (PCB) could attain up to 2-dB gain enhancement [8]. However, in LTCC process, it is impossible to fully remove the substrate surrounding the four sides of the radiating patch. Therefore, only the substrate around the main radiating edges of the patch is removed to effectively suppress surface wave loss. This communication presents the method to improve gain by introducing the open air cavities around radiating edges of the patches in the arrays conforming to the constraints of LTCC process at millimeter-wave frequencies. The effects of the open air cavities on the performance of the patch antenna arrays operating at 60-GHz mmW bands are numerically and experimentally investigated by comparing with the conventional array design without open air cavities. Manuscript received August 25, 2010; revised December 02, 2010; accepted January 15, 2011. Date of publication July 14, 2011; date of current version September 02, 2011. This work was supported by Terahertz Science & Technology Inter-RI Program under Grant #082 141 0040 by the Agency for Science, Technology and Research (A*STAR), Singapore. The authors are with Institute for Infocomm Research, Singapore (e-mail: [email protected]; [email protected]; [email protected]. sg). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161549

II. 60-GHz PATCH ANTENNA ARRAY WITH AND WITHOUT OPEN AIR CAVITIES A. 60-GHz Single Patch Elements With and Without Open Air Cavity A conventional aperture-coupled patch antenna is shown in Fig. 1. The substrate used is LTCC Ferro A6-M with "r = 5:9 and tan  = 0:001. The antenna has dimensions of l = 4 mm, w = 4 mm, h1 = 0:38608 mm (4 LTCC layers), and h2 = 0:09652 mm (1 LTCC layer) with a 50- microstrip feed of width wm = 0:15 mm. For comparison studies, the patch with substrate removal around its four sides and the patch with substrate removal only around its two radiating sides are considered. Fig. 2 shows the designs, namely the conventional patch (Design A) without any open air cavity, the patch with surrounding open air cavity (Design B), and the patch with open air cavity only at both sides of the radiating edges (Design C). All studies were carried out by simulation using CST Microwave Studio. Design B was included for comparison purposes only even though it is not realizable with the LTCC process. Fig. 3(a) shows the simulated impedance matching of the three designs. The impedance bandwidth of Design A is 17.7% while Design C is 7.1% for jS11 j  010 dB. The abrupt change in the dielectric at the patch edges for Design B and C results in higher concentration of field directly under the patch. This increases the Q factor and hence the bandwidth for Design B and C becomes narrower. However, the gain of Design B and Design C is improved since most of the trapped fields are radiated at the patch edges and not at the edges of the substrate as in Design A. Fig. 3(b) shows that Designs B and C have achieved a gain increment of 2.3–2.8 dB compared to Design A over the frequency band of 57–64 GHz. B. 60-GHz Patch Arrays With and Without Open Air Cavity Designs A and C were chosen as the antenna elements for the 4 2 4 array designs, respectively. The inter-element spacing of the antenna array is set at 0:60 . Fig. 4(a) shows the array where the concept of Design C is implemented. Due to process limitations of LTCC, the substrate in between the rows also could not be removed entirely in the radiating sides of the patches as shown in Fig. 4(b).

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Fig. 3. (a) Simulated jS

j

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and (b) gain of Design A, B and C.

Fig. 4. (a) Long rectangular open air cavities in between rows of the array and (b) shorter open air cavities positioned at the radiating edges of patch array. Fig. 6. (a) Microstrip-to-GCPW transition, (b) stripline-to-GCPW transition, and (c) simulated jS j and jS j of the transitions.

Fig. 5. (a) Cavity with a microstrip feed-line, (b) cavity covered by an extra LTCC layer with a microstrip feed-line, and (c) cavity with a stripline feed-line.

The microstrip feed network is designed on a single LTCC layer which increases the fabrication difficulty since the area with open air cavity is supported by this single LTCC layer as shown in Fig. 5(a). A modified design was proposed to have an extra LTCC layer on the patch side, as depicted in Fig. 5(b) to make it more robust. The stripline feed is more preferable since it incorporates four LTCC layers and has a bottom ground plane to separate the antenna from the rest of the chip module as shown in Fig. 5(c). A grounded co-planar waveguide (GCPW) transition is designed so that the antenna can be measured with the patch facing upwards into free space [9]. Fig. 6(a) shows the microstrip-to-GCPW transition while Fig. 6(b) shows a stripline-to-GCPW transition. Fig. 6(c) shows the simulated jS11 j and jS21 j of a back-to-back transition. Fig. 7 shows the photos of the antenna arrays and their feeding structures on the back, respectively. Design I is the conventional microstrip fed patch array, Design II is the microstrip fed open air cavity patch array, Design III is the microstrip fed open air cavity patch with an extra layer of LTCC substrate, and Design IV is the stripline fed open air cavity patch. For the microstrip fed arrays, there is an open cavity on the microstrip-side that prevents the microstrip feed network to be in direct contact with the plate below when performing measurements. III. MEASUREMENTS AND RESULTS The measurements were conducted using a Cascade Microtech Summit 11000 probe station and the Agilent E8361A vector network analyzer. The S11 , gain and radiation patterns were measured on-wafer, with the patch array facing upwards into free space, away from the antenna holder and probe station in a mini-chamber as shown

Fig. 7. 60-GHz patch arrays with backing feeds.

Fig. 8. 60-GHz on-wafer measurement setup for jS j, radiation patterns, and gain: (a) antenna under test (AUT) i.e., antenna array, with on-wafer probe sitting on a holder and (b) transmit horn rotation.

in Fig. 8. The probe arms are connected to a straight and L-shape holder, respectively for the E- and H-plane pattern measurements. Fig. 9 compares the simulated and measured jS11 j of the antenna arrays. The bandwidths for jS11 j  010 dB are as follow: 17% (Design I), 13% (Design II), 12% (Design III), and 7.8% (Design IV). Table I compares the simulated and measured gain, and HPBW of Designs I–IV at 60 GHz, respectively. The gain of Designs II–IV is 1.4–2.1 dB higher than that of Design I at 60 GHz. All the designs have similar beamwidths in both E- and H-planes although the gain for the four designs is different. It suggests that the arrays achieve different antenna efficiency although the directivities of the designs are almost the same because of the same size of the array apertures. Fig. 10 shows the simulated and measured gain of Designs I–IV. In particular, Design IV realizes the highest gain increment compared to Design I. Designs II and III have slight difference due to the presence

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j

of the 60 GHz antenna arrays.

TABLE I DIMENSION, GAIN AND BEAMWIDTH OF THE 60-GHz ANTENNA ARRAYS

Fig. 11. Measured radiation patterns of the antenna arrays at 60 GHz.

observed, which is mainly caused by fabrication tolerance and possible effect of the measurement setup. Fig. 10. Simulated and measured gain of antenna arrays.

of an extra layer of substrate for mechanical stability consideration. Designs II and IV have difference being attributed by the losses in the microstrip line feed as compared with the stripline and also the difference in patch dimensions. Fig. 11 shows the measured co-polar radiation patterns for the antenna arrays in both the E- and H-planes. The measured cross-polarization levels in all the designs are almost 20 dB below the peak gain. All designs have similar beamwidth but Design IV has the best front-toback ratio as expected due to the backing ground plane in the stripline structure. Slight discrepancy between simulation and measurement is

IV. CONCLUSION The technique of using open air cavity has been presented for 60-GHz LTCC antenna array designs and validated experimentally for gain enhancement. It has been shown that the surface waves have been suppressed in the arrays by partially removing the substrate around the radiating edges of the patch elements. Compared with the conventional patch antenna arrays without any open air cavity, such suppression of the surface waves have increased the gain up to 2.8 dB over the band of 57–64 GHz. By incorporating a stripline feed, the structure is able to provide an extra ground plane to lessen the effects of the circuit underneath the antenna in a chip module. The use of open air cavities to improve gain in the designs has been shown to be feasible in the

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LTCC process. When compared to embedded air cavities, the design is also more mechanically reliable.

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TABLE I PUBLISHED RESULTS FOR THE 25-ELEMENT, 50-WAVELENGTH PROBLEM

ACKNOWLEDGMENT The authors would like to thank J. Khoo from the Institute for Infocomm Research, Singapore, and K. Kautio, M. Lahti and K. Ronka from VTT Technical Research Centre of Finland for their effort in the fabrication of the antennas.

REFERENCES [1] D. M. Pozar, “Considerations for millimeter wave printed antennas,” IEEE Trans. Antennas Propag., vol. 31, no. 5, pp. 740–747, Sept. 1983. [2] A. E. I. Lamminen, A. R. Vimpari, and J. Saily, “UC-EBG on LTCC for 60-GHz frequency band antenna applications,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2904–2912, Oct. 2009. [3] A. Panther, A. Petosa, M. G. Stubbs, and K. Kautio, “A wideband array of stacked patch antennas using embedded air cavities in LTCC,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 12, pp. 916–918, Dec. 2005. [4] A. E. I. Lamminen, J. Saily, and A. R. Vimpari, “60-GHz patch antennas and arrays on LTCC with embedded-cavity substrate,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2865–2874, Sep. 2008. [5] R. A. R. Solis, A. Melina, and N. Lopez, “Microstrip patch encircled by a trench,” in Proc. IEEE Int. Symp. on Antennas Propag. Society, Jul. 2000, vol. 3, pp. 1620–1623. [6] Q. Chen, V. F. Fusco, M. Zheng, and P. S. Hall, “Micromachined silicon antennas,” in Proc. Int. Conf. on Microwave and Millimeter-Wave Tech., Aug. 1998, pp. 289–292. [7] Q. Chen, V. F. Fusco, M. Zhen, and P. S. Hall, “Trenched silicon microstrip antenna arrays with ground plane effects,” in Proc. 29th Eur. Microwave Conf., Oct. 1999, vol. 3, pp. 263–266. [8] S. B. Yeap and Z. N. Chen, “Microstrip patch antennas with enhanced gain by partial substrate removal,” IEEE Trans. Antennas Propag., vol. 58, no. 9, pp. 2811–2816, Sept. 2010. [9] S. B. Yeap, Z. N. Chen, A. C. W. Lu, V. Sunappan, and L. L. Wai, “60-GHz LTCC antenna array with microstrip to CPW transition,” in Proc. Asia Pacific Microwave Conf., Dec. 2009, pp. 1938–1941.

Weighted Thinned Linear Array Design With the Iterative FFT Technique Warren P. du Plessis Abstract—A version of the iterative Fourier technique (IFT) for the design of thinned antenna arrays with weighted elements is presented. The structure of the algorithm means that it is well suited to the design of weighted thinned arrays with low current taper ratios (CTRs). A number of test problems from the literature are considered, and in each case, the IFT produces results with improved sidelobe level (SLL) at lower CTR. Index Terms—Array antennas, linear arrays, thinned arrays.

I. INTRODUCTION Thinned arrays are formed from normal equally-spaced filled arrays by deactivating a number of the elements. The aperture of the filled array is maintained, so the width of the main beam is comparable to that of the filled array and similar angular resolution is thus achieved. Manuscript received October 08, 2010; revised February 06, 2011; accepted February 09, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The author is with the Defence, Peace, Safety and Security (DPSS), Council for Scientific and Industrial Research (CSIR), Pretoria 0001, South Africa (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2161450

However, the reduced number of active elements means that the size, weight, cost and complexity of the antenna array, its feed network and any signal processing are reduced [1]–[5]. Thinned arrays can be designed to have identical weights for all elements leading to benefits including simplified feed networks, and identical drive for power amplifiers when the array is used for transmission [1], [6]. However, the additional degrees of freedom offered by control of the weights of the antenna elements can lead to significant improvements to the array parameters including the sidelobe level (SLL) [7], [8]. One of the key figures of merit of any array that utilizes weighted element excitations is the ratio of the largest excitation magnitude to the smallest excitation magnitude—the current taper ratio (CTR). Larger CTRs are indicative of increased design complexity because of increased challenges associated with issues such as realizing an appropriate feed network and higher dynamic ranges for the transmitter and receiver systems. Designing for a low CTR is thus desirable [1], [3], [4] with equally-excited arrays having the lowest possible CTR of 1. Sparse arrays are similar to thinned arrays except that the positions of the antenna elements are not quantized. While this approach increases design freedom, potentially leading to improved array performance, periodic quantization of the element positions has a number of advantages. Coupling between antenna elements is essentially identical when the element positions are quantized, simplifying the design. Quantization also means that the results are valid for all frequencies below the design frequency because of the polynomial nature of the results. Furthermore, no limitation is placed on the maximum scan angle of the array when the element spacing is quantized to multiples of half a wavelength. These points are clearly demonstrated through the use of the example of 25 elements in a linear aperture 50 wavelengths long. A number of published results for this problem are summarized in Table I. The first solution uses a cyclic difference set and represents the best value that has been obtained without resorting to iterative numerical methods. Solutions L2 and L3 and are equally-excited arrays and represent compromises between sidelobe level (SLL) and beamwidth. Solutions L4 to L8 show that considerable SLL improvements can be achieved when the elements are weighted at the cost of increasing the CTR. Solutions L9 and L10 give results whose element positions are quantized twentieths of a wavelength and achieve significantly better SLL than the

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Fig. 2. Excitation selection and pattern modification in the IFT, (a) excitation selection and modification, (b) pattern modification.

Fig. 1. Flowchart of the IFT.

other equally-excited cases, but at the cost of reducing the scan angles. Lastly, solutions L11 and L12 show that the best SLL values are achieved when the element positions are not quantized, though this improvement comes at the cost of reduced ability to scan the beam. Despite the benefits of weighted thinned arrays, the literature considering the design of such arrays is limited. Proposed design techniques utilize simulated annealing [3]–[5], [10], mixed integer linear programming [7], genetic algorithms [9], and a hybrid approach combining a genetic algorithm and a local optimizer [8]. The iterative Fourier technique (IFT) developed by Keizer [2], [12] is a version of the alternating projection technique [13] for the design of antenna arrays that exploits the fact that the excitations and pattern of an array are related by a Fourier transform pair. The IFT has been successfully applied to the synthesis of equally-excited thinned arrays and was shown to reliably produce results that are better than the best published results [2]. The extension of the IFT to the design of weighted thinned linear arrays is considered below, with the resulting algorithm being wellsuited to obtaining low CTRs. Test problems from the literature are considered, and the results obtained with the IFT considerably exceed those achieved with other algorithms. II. DESCRIPTION OF THE ALGORITHM A flowchart describing the IFT is given in Fig. 1. Each of the steps is considered below and the modifications necessary for the IFT to be used for the design of weighted thinned arrays are highlighted.

Each iteration commences by generating a random excitation where each element in the allowable aperture has a value uniformly distributed between 0 and 1. The selection of the excitations that will be used is achieved by ranking the excitations in the allowable aperture and selecting the strongest excitations. The number of excitations selected is determined by the desired number of active elements, so the correct filling factor is implicitly achieved by the algorithm. This approach means that only the strongest elements are retained, increasing the likelihood of achieving a low CTR. The elements at the edges of the array are always retained when the array aperture length is specified, irrespective of their excitations. The specified maximum CTR is achieved by modifying any excitations that violate the CTR requirement. The values of the selected excitations are normalized to the largest selected excitation to ensure that the largest excitation is 1. Any selected excitations with values less than 1/CTR are set equal to 1/CTR to ensure that the specified maximum CTR is achieved. An example of this process is shown in Fig. 2(a) for ten elements distributed over a 30-wavelength aperture with a specified maximum CTR of 2. The outside elements and the elements with eight strongest excitations within the allowable aperture are selected as the ten active elements. The outside elements amplitudes are too low to achieve the specified CTR, so their amplitudes are increased to 1/2. The antenna pattern of the modified excitation is then obtained using an inverse fast Fourier transform (IFFT) by zero-padding the excitation to obtain the required number of points. This pattern is then modified by setting all pattern values in the sidelobe region whose amplitude exceeds a target SLL to some constant level below the target SLL. The phase of each point in the antenna pattern is retained during this process. An example of the pattern-modification process is shown in Fig. 2(b). The main beam is 10 wide, the target SLL is 012:5 dB and values that exceed this level are set to an SLL of 022:5 dB. The next candidate excitation is then computed from the modified pattern using a fast Fourier transform (FFT), and the process is repeated

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TABLE II ADDITIONAL TEST PROBLEMS

TABLE III SOLUTIONS TO THE 25-ELEMENT, 50-WAVELENGTH TEST PROBLEM

until the SLL does not improve for 50 iterations. This procedure allocates more tests to solutions that are improving while wasting fewer iterations on solutions that are not improving. The algorithm checks whether the current result exceeds the best result each time a new antenna pattern is computed, and the overall best result is returned when the IFT terminates. If the achieved SLL is better than the target SLL, the target SLL is set to the nearest 0.01 dB that is better than the best achieved SLL. This approach means that the initial target SLL is not crucial because the target SLL will rapidly progress to a useful value. After 1000 runs have been completed since the last target SLL change, the target SLL is decreased by 0.01 dB if there has been an improvement during the last five cycles. This step is necessary because the achieved SLL does not improve every time the target SLL decreases. This procedure to automatically update the target SLL represents a significant improvement over previous versions of the IFT where the target SLL had to be specified [2], [12]. An initial target SLL of 0 dB was used to obtain all the results in Section IV, demonstrating the robustness of the algorithm. The IFT is insensitive to the new parameters introduced, and the heuristically-determined values in Fig. 1 were used to obtain all the results presented in Section IV.

Fig. 3. Solution S2 to the 25-element, 50-wavelength test problem, (a) pattern, (b) excitation.

III. TEST PROBLEMS Three groups of test problems were used to test the IFT, and Tables I and II summarize the published results for these problems. The first test problem has already been described in Section I and considers 25 elements in an aperture 50 wavelengths long with the sidelobe region starting at u = 0:04. This problem will be considered in some detail because its extensive coverage in the literature allows comparisons to a large number of other algorithms. The second test problem considers the symmetric placement of 48 elements on a grid of 64 locations and allows comparisons to the mixed integer linear programming algorithm developed in [7]. The start of the sidelobe region was taken to be the same as that achieved in [7] to ensure that the main beam obtained is no broader than the published result, and the maximum CTR was set to 2. The last four problems are similar to the problems considered in [2], except that the element excitations can be weighted. These problems are useful for testing because they have large numbers of elements and positions, and half the problems require symmetric arrays. Unlike the first two test problems, there is no requirement that the full aperture be used. The improvements that are possible by weighting the element excitations can also be demonstrated using these problems. The values for the start of the sidelobe region obtained for the equally-weighted arrays in [2] were used for the weighted arrays to ensure that the main beams of the weighted thinned arrays are no wider than for the equallyweighted case, and the maximum CTR was set to 2. The IFFT and FFT calculations used 2048 points for the first two problems and 4096 points for the remaining four test problems, giving an average of more than 20 points per pattern root and agreeing with

values used in the literature [2], [5]. At the end of each IFT cycle, the properties of the best solutions were calculated using 16 times more points to ensure that the final results are accurate. The beamwidth and SLL start angles were determined using cubic spline interpolation. IV. ANALYSIS OF RESULTS AND DISCUSSION The solutions to the first test problem obtained using the IFT are summarized in Table III. These solutions represent a number of compromises between SLL and CTR, and significantly improve on the published results summarized in Table I. The IFT has never been applied to this problem before, so Solution S1 considers the equally-excited case. While solution L3 in Table I achieves a better SLL, its main beam is wider than that of S1 and it has a much more significant shoulder (similar to the pattern in Fig. 2(b)). It should be noted that this shoulder is an indication that the design requirements are unrealistic, but it is only recently that design algorithms have developed to the point that this limitation of the test problem has become apparent. The remainder of the solutions to the first test problem consider weighted excitations with specified maximum CTRs varying from 2 to 6 (Solutions S2 to S6 in Table III). The patterns and excitations obtained for a maximum CTR of 2 are plotted in Fig. 3. The results obtained considerably improve on the best published results (L4 to L6 Table I) with the improvement being most marked when the CTR is low. For example, S2 in Table III has an SLL almost 0.5 dB better than L4 in Table I while more than halving the CTR. The result for the second test problem is summarized in the first line of Table IV and plotted in Fig. 4. The IFT result substantially improves

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TABLE IV SOLUTIONS TO THE ADDITIONAL TEST PROBLEMS FROM [7] AND [2]

Fig. 4. Solution to the 48-element, 64-position test problem from [7], (a) pattern, (b) excitation.

TABLE V RESULTS OBTAINED USING A TWO-STAGE IFT PROCESS

but the 78/200 problem. This is achieved while marginally reducing the beamwidths and maintaining CTR values of no more than 2. The limited improvement in the 78/200 case is due to the fact that the beamwidth specification is narrower than for the other 200-position problems, yet fewer elements are available. Furthermore, the beamwidth specification is based on an equally-excited array so it is likely that the specifications favour equal or nearly-equal excitations. This hypothesis is supported by the way the excitation in Fig. 5(b) resembles an equally-excited array. It is also possible to design weighted thinned arrays by applying IFT once to determine the positions of the active elements by designing an equally-excited array, and then applying the IFT a second time to determine the active-element weighting. The results obtained using this approach are summarized in Table V where the active-element positions are determined by the difference-set solution L1 from Table I in the first line, S1 from Table III in the second line and the solutions from [2] in Table II for the rest. Only thelastsolution inTableV shows animprovement overtheresults inTablesIIIandIV,andeventhen,bylessthan0.1dB.Giventheadditional complexity required by the two-stage approach and the small number of known difference sets, the active-element positions and weights should be determined simultaneously as described in Section II. V. CONCLUSION A modification of the IFT for the design of thinned arrays with weighted elements is presented. The structure of this algorithm makes it ideal for the design of arrays with low CTR values because the strongest excitations are selected at each iteration. Results for a number of test problems from the literature are presented, and in each case, the IFT produces substantially better SLL levels at lower CTR values. The use of weighted elements with CTR values of only 2 was also shown to produce substantial SLL improvements (more than 1 dB in the majority of cases) over otherwise identically-specified equally-excited thinned arrays. ACKNOWLEDGMENT The author would like to thank W. P. M. N. Keizer for his willingness to explain the subtleties of the IFT, and the anonymous reviewers for their valuable comments and suggestions.

REFERENCES

Fig. 5. Solution to the 78-element, 200-position test problem from [2], (a) pattern, (b) excitation.

on the published result and achieves a lower SLL at a smaller CTR while maintaining essentially the same beamwidth. The results for the 200-position test problems are summarized in the last four lines of Table IV, and the solution to the 78/200 problem is plotted in Fig. 5. The use of weighted excitations leads to SLL improvements of more than 1 dB over the equally-excited cases for all

[1] R. J. Mailloux, Phased Array Antenna Handbook. Boston. MA: Artech House, 1994. [2] W. P. M. N. Keizer, “Linear array thinning using iterative FFT techniques,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2757–2760, Aug. 2008. [3] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing,” IEEE Trans. Signal Processing, vol. 44, no. 1, pp. 119–122, Jan. 1996. [4] A. Trucco and V. Murino, “Stochastic optimization of linear sparse arrays,” IEEE J. Ocean. Engrg., vol. 24, no. 3, pp. 291–299, July 1999. [5] J.-F. Hopperstad and S. Holm, “Optimization of sparse arrays by an improved simulated annealing algorithm,” in Proc. Int. Workshop on Sampling Theory and Applications, Aug. 1999, pp. 91–95.

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[6] D. G. Leeper, “Isophoric arrays—Massively thinned phased arrays with well-controlled sidelobes,” IEEE Trans. Antennas Propag., vol. 47, no. 12, pp. 1825–1835, Dec. 1999. [7] S. Holm, B. Elgetun, and G. Dahl, “Properties of the beampattern of weight- and layout-optimized sparse arrays,” IEEE Trans. Untrason., Ferroelect. Freq. Contr., vol. 44, no. 5, pp. 983–991, Sept. 1997. [8] M. Donelli, S. Caorsi, F. D. Natale, M. Pastorino, and A. Massa, “Linear antenna synthesis with a hybrid genetic algorithm,” Progr. Electromagn. Res., vol. 49, pp. 1–22, 2004. [9] A. Lommi, A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett., vol. 32, no. 3, pp. 194–196, Feb. 5, 2002. [10] A. Austeng and S. Holm, “The impact of “non-half-wavelength” element spacing on sparse array optimization,” presented at the IEEE Nordic Signal Processing Conf., NORSIG-02, Oct. 2002. [11] T. Isernia, F. J. Ares Pena, O. M. Bucci, M. D’Urso, J. F. Gomez, and J. A. Rodriguez, “A hybrid approach for the optimal synthesis of pencil beams through array antennas,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2912–2918, Nov. 2004. [12] W. P. M. N. Keizer, “Low-sidelobe pattern synthesis using iterative Fourier techniques coded in MATLAB,” IEEE Antennas Propag. Mag., vol. 51, no. 2, pp. 137–150, Apr. 2009. [13] O. M. Bucci, G. D’eila, G. Mazzarella, and G. Panariello, “Antenna pattern synthesis: A new general approach,” Proc. IEEE, vol. 82, pp. 358–371, 1994.

Robust Beamforming With Magnitude Response Constraints Using Iterative Second-Order Cone Programming B. Liao, K. M. Tsui, and S. C. Chan

Abstract—The problem of robust beamforming for antenna arrays with arbitrary geometry and magnitude response constraints is one of considerable importance. Due to the presence of the non-convex magnitude response constraints, conventional convex optimization techniques cannot be applied directly. A new approach based on iteratively linearizing the non-convex constraints is then proposed to reformulate the non-convex problem to a series of convex subproblems, each of which can be optimally solved using secondorder cone programming (SOCP). Moreover, in order to obtain a more robust beamformer against array imperfections, the proposed method is further extended by optimizing its worst-case performance using again SOCP. Different from some conventional methods which are restricted to linear arrays, the proposed method is applicable to arbitrary array geometries since the weight vector, rather than its autocorrelation sequence, is used as the variable. Simulation results show that the performance of the proposed method is comparable to the optimal solution previously proposed for uniform linear arrays, and it also gives satisfactory results under different array specifications and geometries tested. Index Terms—Adaptive beamforming, linear and arbitrary arrays, magnitude response, second-order cone programming (SOCP), worst-case optimization.

I. INTRODUCTION Sensor array processing using antenna arrays have been successfully applied to many engineering fields including wireless communications, Manuscript received September 14, 2010; revised December 10, 2010; accepted February 09, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161445

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radar, radio astronomy, etc. In particular, the theoretical and applied aspects of beamforming have received great research interests during the last decades [1]. One of the most popular beamformers is the minimum variance distortionless response (MVDR) beamformer, which is developed based on an ideal antenna array with exactly known array manifold. However, antenna arrays in real systems may suffer from various types of uncertainties or mismatches, such as look direction mismatch, imperfectly known sensor positions and orientations, and the mismatch between the estimated array covariance matrix and theoretical one. It is known that the performance of the MVDR beamformer may considerably degrade due to the existence of these imperfections. Therefore, much effort has been spent on improving the robustness of the MVDR beamformer with appropriate additional linear or quadratic constraints [1]–[9]. However, most of the above mentioned robust methods lack the flexibility in controlling the beamwidth and response ripple in the look direction. As a result, they may not offer sufficient robustness against large look direction errors [10]. Recently, a number of advanced robust beamforming methods have been proposed to address this issue by imposing prescribed magnitude response constraints over a given beamwidth in the look direction [11], [12], where the magnitude constraints are non-convex. In particular, the approach in [12] simplified the problem by expressing the beamformer weight vector in terms of its autocorrelation sequence. Hence, similar to filter design method in [13], the autocorrelation sequence can be solved optimally using linear programming, and the beamformer weight vector can be determined using an additional step of spectral factorization. However, this method is derived based on the assumption that the array covariance matrix is a Hermitian Toeplitz matrix. Hence, its performance may degrade when the assumption is violated due to array imperfections. Also, it is found that this method may further be affected by the numerical error of the spectral factorization employed. More importantly, it is restricted to linear arrays with inter-element spacing being integer multiples of a base distance, because the conventional spectral factorization is only well developed for one-dimensional Laurent polynomials [13]. As a result, it may not be applicable directly to other array geometries. In this communication, we propose a new method for addressing the robust beamforming problem with magnitude response constraints using iterative SOCP. The basic idea of the proposed approach is to linearize the non-convex magnitude squared response constraints in a neighborhood of the complex array weights in each iteration. For this linearization, it is shown that the problem of finding the optimal updates around the previous iterates is a convex SOCP problem that can be efficiently solved. It should be noted that, different from the outer product matrix used in [11] and autocorrelation sequence used in [12], the beamformer weight vector can be directly obtained in the proposed approach, and hence the extra spectral factorization is not required. Thus, the proposed approach is generally applicable to arrays with arbitrary geometries. Motivated by the conventional robust adaptive beamformers [14], we further extend the proposed approach to deal with the optimization of worst-case performance in order to obtain a more robust beamformer against possible array imperfections. This suggests that the proposed approach offers a general framework for the design of beamformers of arbitrary array geometries in satisfying different commonly used robustness requirements. II. ROBUST BEAMFORMER DESIGN A. MVDR Beamformer

M

Consider an arbitrary antenna array with sensors, it is well known that the conventional MVDR beamformer is chosen by minimizing the

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output power subject to a constraint of unity array response at the direction of arrival (DOA) of the signal of interest (SOI). That is

1

min w H R xw w s:t: w H a(0 ; 0 ) = 1

H (z ) = z T Qz = w H Raw , z = [Refw T g; Imfw T g]T , RefRa g 0ImfRa g , Ra = a(; )aH (; ), and C is the Q = Im fRa g RefRa g RefRxg 0ImfRx g T Cholesky factor of R = ImfR g RefR g = C C . where

x

(1)

where w is the M 2 complex array beamformer weight vector and the superscript H denotes the Hermitian transpose. a 0 ; 0 is the M 2 steering vector corresponding to the DOA of the SOI, i.e., 0 ; 0 . R x is the M 2 M array covariance matrix, which can be estimated using N received samples fx ; ; x N g as Rx N 01 Nt=1 x t xH t . H The constraint w a 0 ; 0 prevents the gain in the DOA of SOI from being reduced, and the solution of (1) can be easily determined using Lagrange multiplier method as:

(

(1) . . . ( ) ^ = ( )=1

)

(

)

1

() ()

01 w MV DR = H Rx a(001; 0 ) : a (0 ; 0 )Rx a(0 ; 0 )

(2)

However, it is known that the performance of the MVDR beamformer in (2) is sensitive to the mismatch between the nominal and true steering vectors due to the uncertainty in the DOA of the SOI as well as other array imperfections [1]–[12].

()

H (z k +  )  H (z k ) + g T (z k ) (6) where g (z ) is the gradient of H (z ) with respect to z and  is the linear

update vector to be determined to satisfy (3) under the approximation Qz k . Once  is available, the in (6). As Q Q T , we have g z k new solution can be updated as z k+1 z k  . This process is repeated until the relative change of two successive solutions is sufficiently small or the maximum number of iterations is reached. To determine  , we z k  into (5) and obtain substitute z

=

L(; )  jG(; )j  U (; ); (; ) 2

( )=

( )

(3)

( )

where G ;  w H a ;  denotes the array response, L ;  and U ;  are respectively the prescribed lower and upper bounds of the magnitude response, and denotes the ROI. Consequently, the robust beamforming with the magnitude constraints in (3) can be written as:

( )

min w H Rxw w s:t: L(; )  w H a(; )  U (; ); (; ) 2 :

(4)

It can be seen that (4) is not a convex optimization problem due to the presence of the non-convex constraint L ;   jw H a ;  j. As a result, conventional convex programming techniques are not directly applicable. Recently, a number of studies have been devoted to solving this problem [11], [12]. However, these approaches are only suitable for some specific antenna arrays, such as uniform linear arrays (ULAs). We now propose to solve the non-convex optimization problem (4) with an iterative SOCP technique, which has been successfully applied to power pattern synthesis [15]. An important advantage of the proposed method is that it can be applied to arbitrary array geometries. To start with, we first rewrite the problem in (4) as (e.g. [7])

( )

kCz k min z s:t: H (z )  U 2 (; ); (; ) 2

H (z )  L2 (; ); (; ) 2

( )

(5)

( )=2 = +

= +

kCz k + C k min  s:t: H (z k ) + g T (z k )  U 2 (; ); (; ) 2

H (z k ) + g T (z k )  L2 (; ); (; ) 2

B. Robust Beamforming Using Iterative SOCP (RB-ISOCP) A possible way to improve the robustness of the MVDR beamformer is to impose additional linear equality constraints. However, this approach may lead to a decrease in degrees of freedom in interference rejection. Recently, much effort has been made to overcome this problem [11], [12]. More precisely, instead of equality constraints, inequality constraints are used to control the array response in the region of interest (ROI), where the SOI comes with a high probability. To maintain a fairly stable gain in the ROI, the following inequality constraints on the magnitude response are imposed [12]:

x

In what follows, we shall describe the proposed algorithm for solving the non-convex problem in (5). Suppose that our algorithm starts with a reasonably feasible initial guess z 0 and arrives at a point z k after k iterations. At a sufficiently small neighborhood of z k , the magnitude squared response of the array, H z , which is smooth, can be approximated by the following linear approximation:

k k   max

(7)

where an additional quadratic constraint k k   max is imposed to ensure that the linear approximation in (6) is sufficiently accurate. Obviously, we can see that the optimization problem in (7) is convex, and it can be solved using SOCP by discretizing the ROI as in [12] and [15]. Hence, the new iteration can be updated using the optimal  as z k+1 z k  . For the sake of presentation, the proposed method described above is referred to as robust beamforming using iterative SOCP (RB-ISOCP). With appropriate choice of initial guess to be presented in Section II-D, the proposed algorithm converges quickly to a satisfactory solution as we shall elaborate further by the simulation results in Section III.

=

+

C. RB-ISOCP via Worst-Case Performance Optimization (RB-ISOCP-WC) It is worth noting that the robust beamformer derived in the previous sub-section is based on the assumption that the covariance matrix R x is known exactly or well-estimated. However, certain mismatches between the nominal covariance matrix R x and the actual one R x always exist in practice due to various kinds of array imperfections as mentioned in Section I. To address this uncertainty in the true covariance matrix, the following mismatch model is adopted in this communication:



R x = Rx + 1

1

(8)

where is an unknown Hermitian error matrix of R x , and its Frobenius norm is bounded by a certain known constant > as k k  . Using a similar idea as in the conventional method [14], the worstcase performance of the robust beamformer in (4) can be optimized by solving the following problem

0

min max w H (Rx + 1)w w 1 s:t: k1k  L(; )  w H a(; )  U (; ); (; ) 2

1

(9)

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which is again a non-convex problem. Following the descriptions in [14], we first solve the following problem

H max 1 w (Rx + 1)w s:t: k1k 

1

1=

( + )

H min w w (Rx + II )w s:t: L(; )  w H a(; )  U (; ); (; ) 2 :

(11)

It can be seen that the problems in (4) and (11) are identical except for the covariance matrices in the respective objective function. Therefore, the iterative SOCP algorithm in the previous sub-section can be similarly employed to solve (11). More precisely, at the k th iteration, the problem in (11) can be approximated as the following convex problem

C z C  min  k k+ k s:t: H (z k ) + g T (z k )  U 2 (; ); (; ) 2

H (z k ) + gT (zk )  L2 (; ); (; ) 2

k k   max



(12)

where C is obtained by the Cholesky factorization of the regularized array covariance matrix Rx II as

+

RefRx + II g 0ImfRx + II g = C T C : R = Im fRx + II g RefR x + II g

(13)

Consequently, the problem in (12) can be solved using SOCP by discretizing the ROI.

As described earlier, the proposed robust beamformers are obtained through an iterative procedure. Hence, it is important to choose a reasonably good initial guess w 0 for the problem in (4) T fw 0T g; fw 0T g for the problem in (7) to obtain a or z 0 satisfactory solution. Following the recommendations in [15], the non-convex constraint in (3) is first rewritten as

Im

w H a(; )

0F

H min w w (Rx + II )w s:t: w H a(; ) 0 F (; )  E (; ); (; ) 2

]

(; )  E (; ); (; ) 2

E. Convergence Behavior and Complexity It should be noted that the algorithm presented above converges to a local solution due to the linear approximation. Fortunately, as illustrated subsequently in a representative example for the design of robust beamformers in ULAs (Example 1 in Section III-A), the proposed algorithm is capable of finding solutions that are very close to the optimal ones obtained using the method in [12]. In fact, when applying the proposed algorithm to this problem, we did not fail in finding a nearly optimal solution for every specification we have tried. The good convergence performance of the proposed algorithm is largely attributed to the global convergence of individual subproblem, as suggested in [15]. On the other hand, the proposed algorithm might also be viewed as a SOCP-based trust region method with simplified update steps [16]. Thanks to the efficient interior-point method, the step size and step direction characterized by the norm bound constraint in each convex subproblem can be optimally solved. Through extensive computer simulations, it is also interesting to note that the norm of  tends to converge to zero as the iteration increases regardless of the value of  max . Therefore, the choice of  max becomes less critical. This allows us to set a larger norm bound initially to speed up the convergence and hence significantly reduce the computational complexity. Similar to the conventional methods in [8], [12], the total complexity of the proposed algorithm mainly depends on the complexity of solving J each subproblem using convex optimization, which is O M 3:5 M 2:5 , where J is the number of sampled points in the ROI [8]. With the dramatic increase in computing power and advanced coding techniques, it is suggested in [17] that nowadays convex optimization can almost be carried out in real-time for a modest-size problem. Nevertheless, taking the number of iterations into account, the proposed algorithm should in general have higher computational complexity than the conventional methods in [12]. Fortunately, since the number of iterations is usually small as mentioned earlier, the increase in computational complexity is still affordable by the virtue of the efficient convex optimization solver.

straint:

(; ); (; ) 2 :

(15)

Hence, the initial guess can be obtained by solving the following SOCP problem

H min w w Rxw s:t: w H a(; ) 0 F (; )  E (; ); (; ) 2

+(2 +

(14)

(; ) = (U (; ) 0 L(; ))=2 and F (; ) = (U (; ) + L(; ))=2. Then, it is approximated as the following convex conE

(

)

where E

w H a(; ) 0 F (; )

(17)

for the worst-case optimization problem in (12). It should be noted that there may be other approaches to find such feasible initial guess. Nevertheless, given the initial guess designed above, the proposed algorithm converges quite quickly to a satisfactory result. For instance in Example 1, the initial guess converges to a nearly optimal solution in only five iterations for a ULA with a beamwidth of 8 and a ripple of 0.4 dB.

1)

D. Choice of the Initial Guess z 0

= [Re

for the problem in (7), and

(10)

with respect to . It can be shown that the solution of (10) is given wwH =kw k2 . Substituting this solution back to (10) yields

ww by the maximum value w H Rx II w , where I is an M 2 M identity matrix. Hence, the problem in (9) can be rewritten as

3479

(16)

III. SIMULATION RESULTS In our simulations, robust beamformer designs for ULAs and uniform circular arrays (UCAs) are considered to evaluate the performance of the proposed methods. In all experiments, the CVX Matlab Toolbox [18] is employed to solve the SOCP optimization problems. Also, we 0r =20 and U ; ' r =20 for a given ripple let L ; ' rdb in decibel scale.

( ) = 10

( ) = 10

A. Example-I: ULA

= 10

sensor elements separated by In this example, a ULA with M L ; U so that the half wavelength is considered. Let the ROI be beamwidth of the ROI is U 0L . The ROI is discretized with a step size of 0.1 . Two equal power interferences with an interference-to-noise

=[

]

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Fig. 1. Beampatterns of various beamformers with beamwidth of 8 and ripple of r different numbers of iteration.

= 0:4 dB. (a) Optimal beamformer. (b) RB-CMR. (c) RB-ISOCP with

= 0 2 dB

Fig. 2. (a) Beampatterns of the RB-CMR with different beamwidths and a fixed ripple r : . (b) Beampatterns of the RB-ISOCP with different : . (c) Beampatterns of the RB-ISOCP with different ripples and a fixed beamwidth of 16 . beamwidths and a fixed ripple r

= 0 2 dB

ratio (INR) of 20 dB are assumed to impinge on the array from far-field at angles 1 = 040 and 2 = 60 . The DOA of the SOI is assumed to be 0 = 3 , whereas the nominal direction is 0 . 1) Infinite Sample Case: Firstly, we test the performance of the proposed method with an ideal array covariance matrix R x by assuming that the number of snapshots N is infinite. The signal-to-noise ratio (SNR) of the SOI is 10 dB. Let the ripple be rdb = 0:4 dB and the ROI be = [04 ; 4 ], i.e., the designed beamwidth of the ROI is 8 . The maximum norm of the linear update vector  is chosen to be  max = 0:3. Fig. 1(c) shows beampatterns obtained using the RB-ISOCP with different number of iterations. It can be seen that the proposed method converges in five iterations, so that the beam patterns so obtained after five and ten iterations are nearly identical. As a comparison, we consider the robust beamformer with constraints on magnitude response (RB-CMR) studied in [12] (see also Matlab code therein). The optimal pattern based on the autocorrelation of w is shown in Fig. 1(a), and it will be used as a gold standard to assess the performance of the proposed approach in this particular problem. It can be seen from Fig. 1(b) that the pattern obtained after spectral factorization (RB-CMR [12]) is quite different from the optimal one and its interference rejection level are significantly degraded. This is mainly attributed to the numerical error caused by spectral factorization. It should be noted there may be other spectral factorization methods with better numerical behaviors. However, we did not intend to modify the code used in [12] for a fair comparison. On

the other hand, Figs. 1(c) show the proposed beampatterns obtained within ten iterations. It can be seen that the proposed RB-ISOCP exhibits deeper nulls than RB-CMR, and its beampattern is very close to the optimal one. Next, we show the beampatterns of the conventional RB-CMR and the proposed RB-ISCOP with different beamwidths and a fixed ripple of 0.2 dB. From Figs. 2(a) and (b), it can be seen that all beampatterns obtained from these two methods are comparable and they are very close to the optimal ones in the sense that the main beam specifications are precisely satisfied. However, the proposed RB-ISCOP can generally form deeper nulls at the directions of the interferences. Similar arguments hold for other simulation results as shown in Fig. 2(c), where the beamwidth is fixed and the ripple varies. The above results suggest that the proposed approach is effective in finding nearly optimal solutions. 2) Finite Sample Case: It is known that any kinds of array imperfections would result in uncertainties of the array covariance matrix. In this example, the uncertainties caused by insufficient snapshots will be considered to evaluate the robustness of the proposed method. The simulation settings are summarized as follows: The ripple is 0.2 dB and the beamwidth of the ROI is 16 . Fig. 3(a) shows the resultant beampatterns of the proposed RB-ISOCP with different numbers of snapshots N . It can be seen that the sidelobe level of the proposed RB-ISOCP is severely affected by insufficient snapshots, although the beam gain in the ROI can be well controlled. Also, its performance is improved as the number of snapshots increases.

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Fig. 3. (a) Beampatterns of RB-ISOCP with different numbers of snapshots. (b) Beampatterns of RB-ISOCP-WC with different numbers of snapshots and a fixed : . (c) The output SINR of RB-ISOCP versus SNR with different relative regularization factors. (d) Comparison of the output relative regularization factor : and the beamwidth is 16 . SINR of RB-ISOCP-WC and RB-CMR-WC. Common settings: The ripple is r

=01

= 0 2 dB

=[

]2[

Fig. 4. Beampatterns of RB-ISOCP and RB-ISOCP-WC for a ten-element UCA. The ROI is  ;  ; : . (a) RB-ISOCP with infinite samples. (b) RB-ISOCP with N . (c) RB-ISOCP-WC with N ripple is r

= 0 3 dB

To further improve the limited performance due to insufficient snapshots, the proposed RB-ISOCP-WC is employed. Following the approach in [12], is chosen as = r Rx (1; 1), where r denotes the relative regularization factor, and it is chosen as r = 0:1 for illustrative purpose. It can be seen in Fig. 3(b) that the performance is greatly improved by considering the uncertainty of the array covariance matrix. Next, with a fixed number of snapshots N = 200, a hundred of Monte-Carlo simulations are run to estimate the output signal-tointerference-plus-noise ratio (SINR) of the proposed RB-ISOCP-WC versus the SNR ranging from 010 dB to 10 dB. It can be seen from Fig. 3(c) that the RB-ISOCP-WC gives better performance than the RB-ISOCP (i.e., r = 0). Also, a small regularization factor is sufficient to achieve a satisfactory improvement. For larger regularization factors, the proposed method gives nearly the same performance, as we can see that the curves of r = 5 and r = 6 almost overlap. For a comparison, the conventional robust beamforming with constraints on magnitude response using worst-case optimization (RBCMR-WC) [12] is also considered. As suggested in [12], the regularization factor for the RB-CMR-WC should be selected as " = "r Rx (1; 1), where "r denotes the relative regularization factor. Comparing our definition of with " in (16) of [12], ( r ) should generally be larger than "("r ) for the same uncertainty matrix 1. For a fair comparison, we choose r = 6 for the proposed algorithm and try to find the best performance of the RB-CMR-WC from the set of different relative regularization factors given by "r = [0; 0:5; . . . ; 6]. Fig. 3(d) shows the performance comparison of the proposed RB-ISOCP-WC and the RB-CMR-WC. For clarity, only the two best results of the

= 100

] = [75 ; 85 ] 2 [015 ; 15 ], and the = 100 and = 0:1.

latter are shown. It can be seen that the RB-CMR-WC achieves the best performance when "r = 6, which is outperformed by the proposed RB-ISOCP-WC. As discussed previously, the inferior performance of the RB-CMR-WC is probably due to the invalid assumption of Hermitian Toeplitz array covariance matrix and the numerical error introduced in spectral factorization. Therefore, a larger regularization factor may be required for the RB-CMR-WC to combat such unexpected errors. Nevertheless, the major advantage of the proposed approach over the conventional method is its usefulness in designing robust beamformers with arbitrary geometries, as we shall demonstrate in the next example. B. Example-II: UCA In Example-I, we have shown that the proposed method works well in the case of ULAs. Though the conventional RB-CMR in [12] is able to achieve comparable performance as ours, its application to other array geometries, say two-dimensional arrays, may not be straightforward as described previously. On the other hand, the proposed method does not have such limitation and it is applicable to arbitrary arrays. The general experimental settings are as follows: The number of sensor elements of the UCA is M = 10, the radius is 5=2 . Two equal power interferences with an INR of 20 dB are assumed to impinge on the array from far-field at angles (1 ; 1 ) = (82 ; 50 ) and (2 ; 2 ) = (78 ; 0100 ). The DOA of the SOI is assumed to be of the SOI is (80 , (0 ; 0 ) = (80; 3), whereas the nominal direction 0 ). The ROI is = [L ; U ]2[L ; U ] = [75 ; 85 ]2[015 ; 15 ], and the target ripple is rdb = 0:3 dB.

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1) Infinite Sample: Similar to the first example, we firstly consider the performance of the RB-ISOCP with ideal array covariance matrix Rx . Fig. 4(a) shows the beampatterns of the RB-ISOCP in discretized -planes within the region  2 [75 ; 85]. It can be seen that the ROI is very flat and the array gain in this region can be well controlled according to the prescribed ripple size. Moreover, deep nulls are imposed at the directions of interferences. 2) Finite Sample: In this experiment, the performance of the proposed method is evaluated when there are uncertainties in the array covariance matrix due to insufficient samples. For simplicity, the experimental settings are identical to the previous infinite sample case, except that the array covariance matrix is approximated from one hundred snapshots. Fig. 4(b) shows the beampattern obtained using the RB-ISOCP within the region  2 [75 ; 85 ]. Compared with the results for the case of ideal array covariance matrix in Fig. 4(a), it can be seen that the sidelobe level degrades significantly, because the RB-ISOCP fails to take the uncertainty of the array covariance matrix into account. Fig. 4(c) shows the beampattern obtained using the RB-ISOCP-WC with a relative regularization factor of r = 0:1. As expected, the performance can be greatly improved, and the result is close to that in the case of ideal array covariance matrix.

[11] Z. L. Yu, M. H. Er, and W. Ser, “Novel adaptive beamformer based on semidefinite programming (SDP) with constraints on magnitude response,” IEEE Trans. Antennas Propag., vol. 56, pp. 1297–1307, May 2008. [12] Z. L. Yu, W. Ser, M. H. Er, Z. Gu, and Y. Li, “Robust adaptive beamformers based on worst-case optimization and constraints on magnitude response,” IEEE Trans. Signal Processing, vol. 57, pp. 2615–2628, Jul. 2009. [13] S. P. Wu, S. Boyd, and L. Vandenberghe, “FIR filter design via spectral factorization and convex optimization,” in Applied and Computational Control, Signals and Circuits, B. Datta, Ed. : Birkhauser, 1998, vol. 1, pp. 215–245. [14] S. Shahbazpanahi, A. B. Gershman, Z. Q. Luo, and K. M. Wong, “Robust adaptive beamforming for general-rank signal models,” IEEE Trans. Signal Processing, vol. 51, pp. 2257–2269, Sep. 2003. [15] K. M. Tsui and S. C. Chan, “Pattern synthesis of narrowband conformal arrays using iterative second-order cone programming,” IEEE Trans. Antennas Propag., vol. 58, pp. 1959–1970, Jun. 2010. [16] J. Nocedal and S. J. Wright, Numerical Optimization. Berlin, Germany: Springer Series in Operations Research, 1999. [17] J. Mattingley and S. Boyd, “Real-time convex optimization in signal processing,” IEEE Signal Processing Mag., vol. 27, pp. 50–61, May 2010. [18] M. Grant and S. Boyd, “CVX: Matlab Software for Disciplined Convex Programming,” ver. 1.21 [Online]. Available: http://cvxr.com/cvx, May 2010

IV. CONCLUSIONS An iterative SOCP method for designing robust beamformers with magnitude response constraints has been presented. A locally optimal solution to the original non-convex problem is efficiently obtained by solving a sequence of convex SOCP subproblems, which are obtained via a local linearization of the magnitude squared response of the array. The proposed method is further extended to handle uncertainties of the array covariance matrix due to array imperfections. By incorporating uncertainties in form of bounded variation in the design procedure, the robustness of the beamformers can be significantly improved. Design results show that the proposed method is an attractive alternative to traditional design methods in tackling the robust beamforming problem, especially for arrays with arbitrary geometries.

REFERENCES [1] J. Li and P. Stoica, Robust Adaptive Beamforming. Hoboken, NJ: Wiley, 2006. [2] O. L. Frost, “An algorithm for linearly constrained adaptive array processing,” Proc. IEEE, vol. 60, pp. 926–935, Aug. 1972. [3] K. L. Bell, Y. Ephraim, and H. L. Van Trees, “A Bayesian approach to robust adaptive beamforming,” IEEE Trans. Signal Processing, vol. 48, pp. 386–398, Feb. 2000. [4] M. H. Er and A. Cantoni, “Derivative constraints for broadband element space antenna array processors,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-31, pp. 1378–1393, Dec. 1983. [5] H. Cox, R. M. Zeskind, and M. H. Owen, “Robust adaptive beamforming,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-35, pp. 1365–1376, Oct. 1987. [6] C. Liu and G. Liao, “Robust capon beamformer under norm constraint,” Signal Processing, vol. 90, pp. 1573–1581, May 2010. [7] S. A. Vorobyov, A. B. Gershman, and Z. Q. Luo, “Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem,” IEEE Trans. Signal Processing, vol. 51, pp. 313–324, Feb. 2003. [8] J. Liu, A. B. Gershman, Z. Q. Luo, and K. M. Wong, “Adaptive beamforming with sidelobe control: A second-order cone programming approach,” IEEE Signal Lett., vol. 10, pp. 331–334, Nov. 2003. [9] R. G. Lorenz and S. P. Boyd, “Robust minimum variance beamforming,” IEEE Trans. Signal Processing, vol. 53, pp. 1684–1696, May 2005. [10] S. E. Nai, W. Ser, Z. L. Yu, and S. Rahardja, “A robust adaptive beamforming framework with beampattern shaping constraints,” IEEE Trans. Antennas Propag., vol. 57, pp. 2198–2203, Jul. 2009.

Performance Improvement of a U-Slot Patch Antenna Using a Dual-Band Frequency Selective Surface With Modified Jerusalem Cross Elements Hsing-Yi Chen and Yu Tao Abstract—A dual-band FSS consisting of regular Jerusalem cross elements was first used to study its impact on the bandwidths and resonant frequencies of a U-slot patch antenna. Based on the simulation experience of the first partial study, another FSS with modified Jerusalem cross elements was proposed to improve the bandwidths, antenna gains, and return losses of a smaller U-slot patch antenna at 2.45 and 5.8 GHz for Bluetooth and WLAN applications, respectively. Measured data of the return loss, radiation pattern, and antenna gain of this smaller U-slot patch antenna were also presented. It is proven that the smaller U-slot patch antenna implanted with a FSS consisting of modified Jerusalem cross elements has a good performance with sufficient bandwidth and higher gain and is capable of dual-band operation. Index Terms—Bandwidths, dual-band FSS, Jerusalem cross elements, return loss, U-slot patch antenna.

I. INTRODUCTION For wireless communications, multi-band and wide-band patch antennas will become the requirements for accurately transmitting the voice, data, video, and multimedia information in wireless communication systems, such as ultra wide-band measurement applications, intelligent transportation systems (ITS), wireless local area networks Manuscript received August 11, 2010; revised January 18, 2011; accepted February 05, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. The authors are with the Department of Communications Engineering, Yuan Ze University, Chung-Li, Taiwan, R.O.C. (e-mail: [email protected]. tw). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161440

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(WLANs), Global positioning system (GPS) services, radio-frequency identification (RFID) applications, and biomedical telemetry services. Microstrip patch antenna is the natural favorite due to its inherent advantages of small size, low profile, lightweight, cost-effect, and its ease of integration with other circuits. However, it is well-know that a patch antenna on a dielectric substrate may have a very narrow bandwidth due to surface wave losses. The surface wave existed on the patch antenna will continue to propagate until it meets a discontinuity. When the surface wave meets the discontinuity, it may radiate and couple energy to the discontinuity. The surface wave will reduce antenna efficiency, gain, and bandwidth. To achieve multi-band and wide-band operation in a patch antenna design, the frequency selective surface (FSS) is implemented or imbedded in a patch antenna in recent years. For more than 4 decades, the FSS has a variety of applications in antennas [1]–[7], spatial microwave and optical filters [8]–[13], absorbers [14]–[16], polarizers [17], planar metamaterials [18], and artificial magnetic conductor (AMC) designs [19]–[21]. The FSS is usually constructed with periodic arrays of metallic patches of arbitrary geometries or slots within metallic screens. Typical FSS geometries are designed by dipoles, rings, square loops, fractal shapes,. . .etc. Because the substrate thickness of a patch antenna is usually much smaller than a half-wavelength in the dielectric material, the ground plane of the patch antenna destroys the patch antenna performance. The FSS structure has a phenomenon with high impedance surface that reflects the plane wave in-phase and suppresses surface wave [3]. These characteristics of FSS structures can be used to improve the radiation efficiency, gain, and bandwidth of a patch antenna. The impact of a FSS on patch antenna performance depends on the lattice geometry, element periodicity, and the electrical properties of the substrate materials. U-slot patch antennas have been proposed to overcome the inherent problem of the narrow bandwidth of the microstrip patch antenna [22]–[24]. However, the resonant frequency may be shifted from operating frequency and the bandwidth may be narrowed down when a wide-band U-slot patch antenna changes its geometry and size to fit different environments. In the first part of this communication, we report on a dual-band FSS consisting of regular Jerusalem cross elements which was used to study the impact on the bandwidths and resonant frequencies of a U-slot patch antenna near 2.45 and 5.8 GHz. The frequency bands of 2.4  2.485 and 5.725  5.825 GHz are regulated by IEEE 802.11b/g and 802.11a (upper band) for Bluetooth and WLAN applications, respectively. Based on the simulation experience of the first partial study, another FSS with modified Jerusalem cross elements implanted in a new U-slot patch antenna with a smaller size was used for further studies to improve the antenna bandwidths, antenna gains, and resonant frequencies at 2.45 and 5.8 GHz, respectively. In simulations, the characteristics of U-slot patch antennas were obtained by using the Ansoft high-frequency structure simulator (HFSS). Simulation results of the return loss, radiation pattern, and gain of this new U-slot patch antenna were validated by measurement data.

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Fig. 1. A U-slot patch antenna and its dimensions.

Fig. 2. A FSS with regular Jerusalem cross elements implanted in the U-slot patch antenna.

respectively [25]. This U-slot patch antenna implanted with a FSS consisting of regular Jerusalem cross elements is shown in Fig. 2. The FR4 material is also used for the upper and lower dielectric substrates of the U-slot patch antenna with a FSS. The thickness of the upper and lower dielectric substrate is kept at 4.4 mm, while the thickness of the lower dielectric substrate H, may range from 2.2 to 3.2 mm. The FSS constructed with regular Jerusalem cross elements is used to improve the antenna performance. The thickness of the top metallic patch, the FSS, and the bottom metallic plate is 0.035 mm. Fig. 3 shows a regular Jerusalem cross element. The detailed dimensions of a regular Jerusalem cross element are W1 = 0:5  2 mm (width of the vertical or horizontal end loading), W2 = 0:5  3 mm (width of the cross-dipole), L1 = 7  10 mm (length of the vertical/horizontal end loading), L2 = 12  14 mm (length of a Jerusalem cross element), and P = 16 mm (periodicity). The Jerusalem cross element itself with the current flow produces an inductor (L). The gap between two vertical/horizontal end loadings gives rise to a series capacitor (Cs ). A parallel capacitor (Cp ) between the Jerusalem cross element and the ground plate is also produced. Therefore, the structure of the FSS can be viewed as behaving like a tuned network of equivalent LC circuits. By altering the Jerusalem cross element geometry the values for L and C can be modified, and the resonant frequency is changed accordingly.

II. THE U-SLOT PATCH ANTENNA WITH A FSS Fig. 1 shows a U-slot patch antenna and its dimensions. In our studies, a coaxial line with a characteristic impedance of 50 ohms is used as the feed of the U-slot patch antenna. The inner conductor of the coaxial line is attached on the top patch going through the dielectric substrate, and the outer conductor is shorted to the metallic plate on the other side of the patch antenna. The FR4 material is used for the dielectric substrate with a thickness of 4.4 mm. The relative dielectric constant and electrical loss tangent of the substrate are adopted to be 4.4 and 0.02 at frequencies 2 to 6 GHz,

III. MEASUREMENT AND SIMULATION RESULTS Simulation results of return losses for the U-slot patch antenna implanted with and without a FSS consisting of regular Jerusalem cross elements are obtained from the commercial software package HFSS, which is the industry-standard simulation tool coded by a finite element program for 3D full-wave electromagnetic field simulation. In order to obtain optimum values of geometrical parameters for antenna design, the effects of each geometrical parameter are analyzed. In each analysis, the periodicity P is always kept unchanged, but

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Fig. 3. Details of a regular Jerusalem cross element.

Fig. 5. A smaller U-slot patch antenna with a modified FSS. (a) Illustration of a smaller U-slot patch antenna with a modified FSS. (b) The Prototype of a smaller U-slot patch antenna with a modified FSS.

Fig. 4. The modified Jerusalem cross element and its dimensions.

W W L L

one of the parameters 1; 2 ; 1 ; 2 , (shown in Fig. 3) and H (shown in Fig. 2) is changed and the other four parameters are kept unchanged. Comparisons of simulation results of return losses for the U-slot patch antenna implanted with and without a FSS are studied. Simulation results were investigated by checking the impedance matching with better than 10 dB return loss. It is found that there are 2 to 4 resonant frequencies when the length of end loading 1 increases from 7 to 10 mm. It is also observed that changing Jerusalem cross element’s length 2 , width of the end loading 1 , width of the cross-dipole 2 , and thickness of the lower dielectric substrate (H) can significantly increase the bandwidth at higher resonant frequencies near 5.6 GHz, but the improvement of bandwidth at lower resonant frequencies near 2.5 GHz is not significant. From observations, the resonant frequencies of the U-slot patch antenna implanted with and without a FSS are found to be near 2.5 and 5.6 GHz for the impedance matching with better than 10 dB return loss. Although the bandwidths have been improved near the resonant frequencies of 2.5 and 5.6 GHz for the U-slot patch antenna implanted with a FSS, however, the resonant frequencies of 2.5 and 5.6 GHz are not in the frequency bands of 2.4  2.485 and 5.725  5.825 GHz regulated by IEEE 802.11b/g and 802.11a (upper band). In addition, the improvement of bandwidth at 2.5 GHz is not significant.

L (W )

(L ) (W )

Fig. 6. Comparison of return losses for the smaller U-slot patch antenna with a modified FSS and the original U-slot patch antenna without using a FSS.

Based on above studies, a FSS with modified Jerusalem cross elements implanted in a new U-slot patch antenna with a smaller size was proposed to further improve the performance of the U-slot patch antenna. Fig. 4 shows the modified Jerusalem cross element and its dimensions. In the modified Jerusalem cross element, there are four smaller cross-dipoles and one larger cross-dipole used to improve the bandwidth at 5.8 and 2.45 GHz, respectively. Fig. 5 shows the smaller U-slot patch antenna implanted with modified FSS and its prototype.

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Fig. 7. Comparison of antenna gain for the smaller U-slot patch antenna with a modified FSS and the original U-slot patch antenna without using a FSS.

Fig. 9. Radiation patterns of the smaller U-slot patch antenna with a modified FSS at 5.8 Ghz. (a) 3-D radiation pattern obtained by measurement; (b) The measured co-and cross-polarization of antenna on E-plane.

Fig. 8. Radiation patterns of the smaller U-slot patch antenna with a modified FSS at 2.45 Ghz. (a) 3-D radiation pattern obtained by measurement; (b) The measured co- and cross polarization of antenna on E-plane.

The dimensions of the smaller U-slot patch antenna with a modified FSS are 68 2 64 mm2 . The FR4 material is also used for the upper and lower dielectric substrates with a thickness of 1.6 mm and 3.2 mm, respectively. The dimensions of the radiator patch are the same as those of the original one.

Fig. 6. shows the comparison of return losses obtained from simulations and measurements for the smaller U-slot patch antenna implanted with a modified FSS and the original U-slot patch antenna without using a FSS. Measurement data were obtained by using an Anritsu37369C antenna-measurement system in the Yuan Ze anechoic chamber. From Fig. 6, it is found that simulation results of return loss makes good agreement with measurement data for the smaller U-slot patch antenna implanted with a modified FSS. It is very clear that much broader bandwidths are obtained by using the smaller U-slot patch antennas implanted with a modified FSS at 2.45 and 5.8 GHz, respectively. Besides, the resonant frequencies of 2.45 and 5.8 GHz fall in frequency bands of 2.4  2.485 and 5.725  5.825 GHz regulated by IEEE 802.11b/g and 802.11a, respectively. The measured antenna gains of the smaller U-slot patch antenna implanted with a modified FSS and the original U-slot patch antenna without using a FSS are shown in Fig. 7. This figure shows that the smaller U-slot patch antenna implanted with a modified FSS produces better gains than the original U-slot patch antenna without using a FSS at resonant frequencies 2.45 and 5.8 GHz, respectively. The higher gains obtained at frequencies 2.45 and 5.8 GHz are 4.69 and 6.54 dBi, respectively. It should be noted that the gain at 2.45 GHz is improved by a little bit, but the gain at 5.8 GHz is significantly improved. Radiation patterns of the smaller U-slot patch antenna implanted with a modified FSS measured at 2.45 and 5.8 GHz are shown in Figs. 8 and 9, respectively. The radiation patterns shown in Figs. 8 and 9 have low side-lobes and back-lob levels.

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It looks acceptable and unidirectional for the smaller U-slot patch antenna implanted with a modified FSS. It is obvious that the smaller U-slot patch antenna implanted with a modified FSS has a better performance with sufficient bandwidth and higher gain and is capable of dual-band operation.

IV. CONCLUSIONS In this communication, a dual-band FSS consisting of regular Jerusalem cross elements was first used to improve the bandwidths and resonant frequencies of a U-slot patch antenna near 2.45 and 5.8 GHz. From simulation results, it is found that the bandwidths have been improved near the resonant frequencies of 2.5 and 5.6 GHz for the U-slot patch antenna implanted with a FSS consisting of regular Jerusalem cross elements; however, the resonant frequencies of 2.5 and 5.6 GHz are not in the frequency bands of 2.4  2.485 and 5.725  5.825 GHz regulated by IEEE 802.11b/g and 802.11a (upper band). In addition, the bandwidth improvement at 2.5 GHz is not significant. For further improvement on the performance of the U-slot patch antenna, a FSS consisting of modified Jerusalem cross elements was proposed to improve the performance of the U-slot patch antenna. It is demonstrated that the FSS consisting of modified Jerusalem cross elements can successfully be used to improve the bandwidths and antenna gains at resonant frequencies of 2.45 and 5.8 GHz for the U-slot patch antenna with a smaller size, respectively. The modified U-slot patch antenna implanted with a FSS consisting with modified Jerusalem cross elements has been proven to provide dual-band, higher gain, and wide-band operation for Bluetooth and WLAN applications. With this modified U-slot patch antenna implanted with a FSS consisting of modified Jerusalem cross elements, the Bluetooth and WLAN systems operating in the 2.4  2.485 (IEEE 802.11b/g) and 5.725  5.825 GHz (IEEE 802.11a, upper band) can provide reliable high-speed connectivity between notebook computers, PCs, personal organizers, and other wireless digital appliances. The structure of the modified U-slot patch antennas implanted with a FSS consisting of modified Jerusalem cross elements is easily fabricated using printed circuit board (PCB) technology.

REFERENCES [1] F. Yang and Y. Rahmat-Samii, “Reflection phase characterizations of the EBG ground plane for low profile wire antenna applications,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2691–2703, Oct. 2003. [2] J. Liang and H. Y. David Yang, “Radiation characteristics of a microstrip patch over an electromagnetic bandgap surface,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1691–1697, Jun. 2007. [3] D. Sievenpiper, L. Zhang, R. F. Jimenez Broas, N. G. Alex’opolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [4] X. L. Bao, G. Ruvio, M. J. Ammann, and M. John, “A novel GPS patch antenna on a fractal hi-impedance surface substrate,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 323–326, 2006. [5] H. Mosallaei and K. Sarabandi, “Antenna miniaturization and bandwidth enhancement using a reactive impedance substrate,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2403–2414, Sep. 2004. [6] A. P. Feresidis, G. Goussetis, S. Wang, and J. C. Vardaxoglou, “Artificial magnetic conductor surfaces and their application to low-profile high-gain planar antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 209–215, Jan. 2005. [7] B. A. Munk, R. J. Luebbers, and R. D. Fulton, “ Transmission through a 2-layer array of loaded slots,” IEEE Trans. Antennas Propag., vol. AP22, no. 6, pp. 804–809, Nov. 1974. [8] B. A. Munk, Frequency Selective Surfaces—Theory and Design. New York: Wiley, 2000.

[9] F. R. Yang, K. P. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuits,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 8, pp. 1509–1514, Aug. 1999. [10] C. N. Chiu, C. H. Kuo, and M. S. Lin, “Bandpass shielding enclosure design using multipole-slot arrays for modern portable digital devices,” IEEE Trans. Electromagn. Compat., vol. 50, no. 4, pp. 895–904, Nov. 2008. [11] M. S. Zhang, Y. S. Li, C. Jia, and L. P. Li, “Signal integrity analysis of the traces in electromagnetic-bandgap structure in high-speed printed circuit boards and packages,” IEEE Trans. Microwave Theory Tech., vol. 55, no. 5, pp. 1054–1062, Nov. 2007. [12] T. K. Wu and S. W. Lee, “Multiband frequency selective surface with multiring patch elements,” IEEE Trans. Antennas Propag., vol. 42, no. 11, pp. 1484–1490, Nov. 1994. [13] G. I. Kiani, K. L. Ford, K. P. Esselle, A. R. Weily, C. Panagamuwa, and J. C. Batchelor, “Single-layer bandpass active frequency selective surface,” Microw. Opt. Technol. Lett., vol. 50, no. 8, pp. 2149–2151, Aug. 2008. [14] G. I. Kiani, K. L. Ford, K. P. Esselle, A. R. Weily, and C. J. Panagamuwa, “Oblique incidence performance of a novel frequency selective surface absorber,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2931–2934, Oct. 2007. [15] B. A. Munk, P. Munk, and J. Pryor, “On designing Jaumann and circuit analog absorbers (CA absorbers) for oblique angle of incidence,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 186–193, Jan. 2007. [16] A. K. Zadeh and A. Karlsson, “Capacitive circuit method for fast and efficient design of wideband radar absorbers,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2307–2314, Aug. 2009. [17] R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys., vol. 7, no. 1, pp. 37–50, 1967. [18] N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations. Hoboken/Piscataway, NJ: Wiley-IEEE Press, 2006. [19] D. J. Kern, D. H. Werner, A. Monorchio, L. Lanuzza, and M. J. Wilhelm, “The design synthesis of multiband artificial magnetic conductors using high impedance frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 8–17, Jan. 2005. [20] J. McVay, N. Engheta, and A. Hoorfar, “High impedance metamaterial surfaces using Hilbert-curve inclusions,” IEEE Microw. Wireless Compon. Lett., vol. 14, pp. 130–132, 2004. [21] J. Bell and M. Iskander, “ A low-profile archimedean spiral antenna using an EBG ground plane,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 223–226, 2004. [22] B. L. Ooi, “A double-II stub proximity feed U-slot patch antenna,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2491–2496, Sep. 2004. [23] R. Chair, C. L. Mak, K. F. Lee, K. M. Luk, and A. Kishk, “Minuature wide-band half U-slot and half E-shaped patch antenna,” IEEE Trans. Antennas Propag., vol. 53, no. 8, p. 26452652, Aug. 2005. [24] H. Y. Chen and S. H. Chen, “Analysis of characteristics of a U-slot patch antenna using finite-difference time-domain method,” Microw. Opt. Tech. Lett, vol. 48, no. 9, pp. 1687–1694, Sep. 2006. [25] M. Xu, T. H. Hubing, J. Chen, T. P. Van Doren, J. L. Drewniak, and R. E. DuBroff, “Power-bus decoupling with embedded capacitance in printed circuit board design,” IEEE Trans. Electromagn. Compat., vol. 45, no. 1, pp. 22–30, Feb. 2003.

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Numerical Reflection Coefficients of ID-FDTD Scheme for Planar Dielectric Interface Pingping Deng and Il-Suek Koh

Abstract—We present analytical formulations of the numerical reflection coefficients of the isotropic-dispersion finite-difference time-domain (IDFDTD) scheme for a planar dielectric boundary. The reflection coefficients and the effective dielectric constants are exactly formulated by using the eigenvalue method [1]. The formulations are numerically verified. Index Terms—Effective dielectric constant, ID-FDTD scheme, reflection coefficient.

I. MOTIVATION To remedy the anisotropic dispersion of the standard FDTD scheme (Yee scheme), several low-dispersion FDTD schemes have been proposed [2]–[5], one of which is the ID-FDTD scheme [6], [7]. The behavior of the ID-FDTD scheme has not been clearly characterized on the material (dielectric or conductor) interface since the field on the interface is updated based on the field values inside the two adjacent media. This is a major difference from the Yee scheme. In [8], the numerical reflection coefficient of the ID-FDTD scheme is approximately formulated for a TE case. The analytical procedure in [8] is not mathematically exact. Hence, it is impossible to exactly determine the order of the accuracy of the ID-FDTD scheme for the material interface based on the approximate formulation. In Sections II and III, we formulate the exact numerical reflection coefficients of the ID-FDTD scheme for a planar dielectric interface for TE and TM modes, respectively. Also, the effective dielectric constant on the boundary is rigorously obtained from the exact formulation. In Section IV, the obtained numerical reflection coefficients are numerically verified. II. TE POLARIZATION Fig. 1 shows the geometry for the reflection problem in the Yee grid for TE and TM modes. The numerical wave numbers of the incident, ~i , k~r and k~t , respecreflected, and transmitted waves are given by k tively. The cubic grid is assumed with the grid size, 1. We consider the lossless dielectric media with the dielectric constant, "m , and the relative permeability, m (m = 1, 2). The numerical wave numbers of each medium are calculated from the known dispersion relation of ~m . k~mx , and k~my , the ID-FDTD scheme [6], which are denoted as k and represent the x- and y - components of the numerical wave number for each medium, respectively. Snell’s law is satisfied as i = r in the discretized domain [1]. Here, i and r represent the incident and reflected angles, respectively. The following identities are also satisfied [1]:

k~y = k~1y = k~2y = k~1 sin i ; k~1x = k~1 cos i ; k~2x =

k~22 0 k~12y :

Manuscript received July 21, 2010; revised February 10, 2011; accepted February 17, 2011. Date of publication July 22, 2011; date of current version September 02, 2011. This work was supported by the Defense Acquisition Program Administration and Agency for Defense Development under contract UD100002KD. P. Deng is with the Graduate School of Information Technology & Telecommunications, Inha University, Korea (e-mail: [email protected]). I.-S. Koh is with the Department of Electronics Engineering, Inha University, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161546

Fig. 1. Grid configuration of the reflection problem for a planar dielectric interface: (a) TE mode and (b) TM mode.

In the two media, the electromagnetic fields are denoted as Ez1 , Hx1 , Hy1 and Ez2 , Hx2 , Hy2 , respectively. At the interface, Ez3 and Hx3 are assumed. The numerical waves are written [1] as

0k~ 1y) 2 (e0jk~ 1x + RT E ID 1 ejk~ 1x); x < 0 ~ ~ Ez2 (x; y; t) = B 1 ej (! t0k 1x0k 1y) ; x > 0 ~ ~ Ez3 (x; y; t) = C 1 ej (! t0k 1x0k 1y) ; x = 0 ~ Hx1 (x; y; t) = D 1 ej (! t0k 1y) 2 (e0jk~ 1x + RT E ID 1 ejk~ 1x); x < 0 ~ ~ Ex2 (x; y; t) = E 1 ej (! t0k 1x0k 1y) ; x > 0 ~ ~ Hx3 (x; y; t) = F 1 ej (! t0k 1x0k 1y) ; x = 0 ~ Hy1 (x; y; t) = G 1 ej (! t0k 1y) 2 (e0jk~ 1x 0 RT E ID 1 ejk~ 1x); x  0 1 Ez1 (x; y; t) = A 1 ej (!

Hy2 (x; y; t) = H 1 e

t

0k~ 1x0k~ 1y) ; x  1

j (! t

2

2

(1)

where A; B; . . . ; H represent arbitrary constants, RT E ID is the numerical reflection coefficient of the TE wave, and !0 is the angular frequency. To obtain the closed-form expression of RT E ID , we use

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for a fine grid. For this case, the reflection coefficient can be simplified to

~ ~ RT E ID = sin k~1x 1 1 B1 0 sin k~2x 1 1 B2 = Rexact + O(12 ) (5) sin k1x 1 1 B1 + sin k2x 1 1 B2 2 where Bm = fmy =m 0 Sy2 m Am . For the Yee scheme (m = 0),

(4) and (5) can exactly reduce to those for the Yee scheme [1]. It can also be observed that (5) approaches the exact reflection coefficient, Rexact [9], as 1 goes to zero. III. TM POLARIZATION

The same procedure for the TE case can be used to formulate a closed-form expression of the numerical reflection coefficient for a TM wave. Fig. 1(b) and Fig. 2(b) show the grid configuration and the nine field sampling points, respectively. For simplicity, only non-magnetic material is considered for the TM mode. The eigenvalue problem can be established. The characteristic equation is the determinant of 9 2 9 matrix, whose non-zero elements are explicitly expressed in the Appendix. The solution of the eigenvalue problem can be obtained by

P2 ) 0 j (Q1 0 Q2 ) RT M ID = ((PP1 + 0 P ) 0 j (Q + Q ) 1

Fig. 2. Field sampling points for the ID-FDTD scheme. (a) TE mode, (b) TM mode.

the eight field sampling points as shown in Fig. 2(a). After inserting (1) into the updated equations [6], we can formulate an eigenvalue problem [1]. The characteristic equation is the determinant of an 8 2 8 matrix, whose non-zero elements are explicitly expressed in the Appendix. The solution of the eigenvalue problem can be simply expressed as 1 0 X2 + jY RT E ID = X X + X 0 jY 1

2

(2)

2 Xm = 2Smx Cmx ((fmy =m ) 0 2m 1 Sy2 Am ); and 2 2 2 Y =2 0(2"mid ID =S ) 1 St2 + m=1 (Smx fmy =m)+ 2 2 2 2Sy [ m=1 (m =4)(1 0 2Smx )Am + (~ =mid ID )] with Smx = sin(k~mx 1=2), Cmx = cos(k~mx1=2), Smy = Sy = sin(k~y 1=2), St = sin(!0 1t=2), 2 fmu = (1 0 m 1 Smu ), Am = (fmx =m) + (~=mid ID ), ~ = 1 0 ((1 + 2 )=4), and u = x and y. Here, 1t denotes the time step; m is the weighing factor for the ID-FDTD scheme for the mth medium; "mid ID and mid ID are the effective dielectric constant

where

and the relative permeability for the grid at the interface, respectively; and S is the Courant number. For lossless media, the reflection coefficient should be real, which determines "mid ID and mid ID as

2

1

2

(6)

where

P1 = a1 a2 0 b1 b2 ; P2 = c1 c2 0 d1d2 Q1 = a1 c2 0 b1 d2 ; Q2 = a2 c1 0 b2 d1: with

2 am = 0 Cmx " fmy 0 m 1 Sy2 (Dm + Em 0 F30m ) mid ID f 1y f2y bm = Cmx " 0 30m 1 Sy2 (D30m + Em 0 Fm ) mid ID 1 2 2 cm = Smx fmy "m 0 "mid ID

0 m 1 Sy2 (Dm + Em + F30m ) dm = Smx "f1y f2y 0 30m 1 Sy2 (D30m + Em + Fm ) mid ID  Dm = 1 0 2m "1 ; Em = f"mx ; and Fm = 4"m : m m m

It can be observed that both P1 and P2 reduce to zero by assuming m = 0 (Yee scheme) and (3). Hence, (6) becomes the known re-

flection coefficient for the Yee case [1]. For the ID-FDTD scheme, however, there are two choices for obtaining the real reflection coefficient, (6): Pm = 0 or Qm = 0. Since the values of Qm are much bigger than those of Pm , in this work, we assume that Pm = 0 and Qm 6= 0. For the two polarizations, (3) should be satisfied. If not, the ID-FDTD scheme cannot provide highly accurate results for a general 3D problem. It has been already reported that the ID-FDTD scheme is more accurate than the Yee scheme [6], [7]. Hence, (3) should be satisfied for a TM mode. However, P1 = 0 cannot guarantee (3). Thus, we can set P2 = 0, which is satisfied by setting "mid Id as

"mid ID = "1 +2 "2 = "mid Y ee; (3) f2x 1 0 1 mid ID = 12 1 + 1 + 21 ff1x 0 + f2x 1 2 1 2 1x 01 + 4 0101 022 f11x 0 f22x 2 2 " +" f Hm = 21+122 + O(1) 1=!0 21+122 = mid Y ee ; (4) "mid ID = 1 2 2 0 G Hm 1 0 G I +2 "my m=1 m=1 m where "mid Y ee and mid Y ee represent the effective dielectric con- where G = Sy2 "1 "2 =4f12y f22y , Hm = 30m fmy (D30m + E30m + Fm ), and I = 1 2 Sy2 (A1 0 A2 )(B1 0 B2 ). stant and relative permeability for the Yee scheme, respectively [1]. ~mu 1=2)2  1 for a Since fmu can be approximated by 1 0 m 1 (k It can be observed that the effective dielectric constant for the ID-FDTD scheme is the same as that for the Yee scheme and the fine grid, "mid ID can be approximated as "mid ID = ("1 + "2 )=2 + effective permeability can be approximated by that for the Yee scheme O(1)  ("1 + "2 )=2. Therefore, we can use (3) for both the TE and

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Fig. 3. Comparison of the reflection coefficients for the ID-FDTD scheme, Yee scheme, and the exact coefficient for " : (a) TE mode and (b) TM mode.

= 10

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Fig. 4. Comparison of the reflection coefficient for the ID-FDTD scheme, and and three PPW as 20, 40, and 80: (a) TE mode the exact coefficient for " and (b) TM mode.

= 10

TM cases, since a fine grid is usually assumed for a FDTD simulation. From this fine grid assumption, the ratio of jP1 =(Q1 + Q2 )j is found to be negligible as

P1 =(Q1 + Q2 )j = 13 k~y4 "1 "2

j



~2x "1 + k ~1x "2 ) + O (14 ) 16(k

0:

Hence, the numerical reflection coefficient expression is approximated to

RT M

ID

~ ~ Q1 0 Q2 = tan k 2 1 1 L1 0 tan k 2 1 1 L2 Q1 + Q2 tan k~ 2 1 1 L1 + tan k~ 2 1 1 L2 2 = Rexact + O (1 ) 

(7)

where

Lm = am c30m 0 bm d30m ~ tan k (30m)x 1=2

=

2

"mid ID "30m

+ m

1

O(12 ):

Equation (7) is exact up to the second order. As for the TE case, (7) reduces to that for the Yee scheme for m = 0, and the exact coefficient as 1 goes to zero.

(

0

)

Fig. 5. Difference " " between the effective permittivity for the ID-FDTD scheme and the Yee scheme for a TM mode.

IV. VERIFICATION To verify the obtained formulations, we compare the numerical results and (5) and (7) over all incidence angles [0,86:30 ] for the nonmagnetic material with "1 = 1 and "2 = 10. The Courant number and

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points per wavelength (PPW) are fixed as 0.5 and 40, respectively. In Fig. 3, the analytical results show excellent agreement with the simulation results. The imaginary part of the reflection coefficient is less than 0.005 for both the simulation and analytical results, which is negligible. Fig. 4 shows the comparison of the exact and the ID-FDTD results for three PPW as 20, 40 and 80. The simulation conditions for Fig. 3 are used. As expected, the error decreases in the second-order accuracy with increasing PPW. Fig. 5 shows the difference between "mid ID and "mid Y ee for the TM case. For this simulation, m = 1, "1 = 1, and "2 = 10 are assumed. As mentioned before, the difference is very small (< 2 1 1004 ). A similar difference can be numerically observed for the effective permeability for the TE mode. Therefore, the effective dielectric and the relative permeability for the Yee scheme can be used for the ID-FDTD scheme.

f1y e0 e = 1t1 1 ; a55 = e0 1t f2y e0jk~ 1 a57 = 2 1 1t f2y ; a64 = e0 a58 = e 2 1 1t f1y e0 a66 = e 1 1 1t f2y e0 a75 =e0 e ; a77 = 1 1 1 1t a81 = e0 4"mid ID 1 e (ejk~ 1 + RTE ID e0jk~ 1 ) ~ 2 1t a82 = e0 e0j k 1 4"mid ID 1 e 1 + 2 1t e0 a83 = 1 4 "mid ID 1 e 1t f1y e a84 = RTE ID e0 "mid ID 1 1t f2y e0 a85 = ; a88 = e0 e "mid ID 1 1

a48

0

1

1

0

0

1

0

1

0

1

The properties of the ID-FDTD scheme for a material interface are characterized. Closed-form expressions of the numerical reflection coefficients are obtained for the ID-FDTD scheme for both TE and TM waves. The coefficients reduce to those of the Yee scheme for m = 0. Also, as the grid size goes to zero, the formulations become exact. The formulated reflection coefficients are numerically verified. The second-order accuracy can be observed for the dielectric interface as theoretically expected. The effective dielectric coefficient and the relative permeability for the grid on the interface are estimated, which can be approximated as those of the Yee scheme.

2

0

0

0

1

0

1

For the TM case, the non-zero elements are given by e = e0 1t f1x e0 a18 = "1 1 1 t f2x a29 = e0 "2 1 a11

0

1

For the TE case, the non-zero elements are given by

=(ejk~ 1 + RTE ID e0jk~ 1 ) e0 e 1t e a16 = e0 1 1 1 21 ejk~ 1 + RTE ID e0jk~ 1 +41 (ej2k~ 1 + RTE ID ej2k~ 1 ) 1 1t a18 = e0 411 e ~ a22 =e0j k 1 e0 e 1t e a27 = e0 2 1 1 22 e0jk~ 1 + 42 e0jk~ 1 2 1t a28 = e0 ; a33 = e0 421 e 1 1t a36 = e0 4mid ID 1 e (ejk~ 1 + RTE ID e0jk~ 1 ) ~ 2 1t a37 = e0 e0j k 1 4mid ID 1 e 1 + 2 1t a38 = 1 e0 4 mid ID 1 e a44 = e RTE ID e0 e0 1t f1y (ejk~ 1 + RTE ID e0jk~ 1 ) a46 = 1 1 a11

0

0

2

a33

=

a38

= "11 1t e0

0

1

a39

1

0

0

= "12 1t e0

0

0

1

=e0 1t f2y a69 = "2 1 a66

1

a77

=e0

0

0

RTM

e

ID e0

+ 42 e0

0

0

e0

0

e0

e

ID e0

ID e0

e

e

e

RTM

e

e

e

RTM

e0

0

=e0 1t f1y a58 = "1 1 e

0

0

e0

e

1 22 e0

a55

0

0

e

= e0

e

0

1

2

0

0

1

e a49

2

0

e

0

= 14"t111 e0

2

0

1

0

; a22

e

1

0

0

0

0

e

ID e0

RTM

1 21 + 41 e

0

a44

0

1

0

=e0 e0 1t 2 e0 a48 = 4"2 1

0

2

e

2

0

:

0

1

APPENDIX

e

0

1

V. CONCLUSION

0

0

0

e

e

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 9, SEPTEMBER 2011

1t 1 f1y e "mid ID 1 1t 1 f2y e0 a79 = 0 "mid ID 1 1t 1 1 e0 a81 = 40 1 a78 =

e

2

a83 =

0

1t e0 0 1 0

RT M

0

Multilevel Fast Multipole Algorithm-Based Direct Solution for Analysis of Electromagnetic Problems

ID e

Zhaoneng Jiang, Yijun Sheng, and Songge Shen 0

e

RT M 0

e

2

0

0

ID e

e

RT M

1 21 0

0

ID e

1t 1 e0 e e0 401 1t f1y (ejk~ 1 + RT M ID e0jk~ 1 ); a87 = 1t f1y a85 = 0 1 0 1 0 0 RT M ID e e e a88 = e 1t 2 e0 a92 = e e0 401 1t 2 e0 a93 = e 401 a84 =

1

0

1

0

0

0

1

2

1

0

1

Abstract—In this communication, a multilevel fast multipole algorithm (MLFMA)-based direct method is proposed for solving electromagnetic scattering problems that are formulated using the electric-field integral equation (EFIE) approach. The method is based on the multilevel compressed block decomposition (MLCBD) algorithm. Previously, the matrix filling procedure of the MLCBD is based on the matrix decomposition algorithm-singular value decomposition (MDA-SVD) method. Although the MDA-SVD is more efficient than direct filling, it requires a longer filling time for the far-field matrix than for the MLFMA. The problems are used to demonstrate that the matrix filling memory requirement of the MDA-SVD is also higher than that of the MLFMA. Hence, the MLFMA is utilized to reduce both the matrix filling time and memory of the MLCBD. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method. Index Terms—Electromagnetic scattering, fast direct solution, multilevel compressed block decomposition (MLCBD), multilevel fast multipole algorithm (MLFMA).

0

e

0

RT M

0

ID e

1t e0 0 e 1 0 22 e0 0 1 1t 1 f2y e0jk~ 1 ; a97 = 1t 1 f2y a96 = 0 0 1 0 1 a94 =

a99 =e0

3491

e0

0

e

:

Here, aij is the (i; j )th element of the eigenvalue matrix.

REFERENCES [1] A. Christ, S. Benkler, J. Fröhlich, and N. Kuster, “Analysis of the accuracy of the numerical reflection coefficient of the finite-difference time-domain method at planar material interfaces,” IEEE Trans. Electromagn. Compat., vol. 48, no. 2, May 2006. [2] M. F. Hadi and M. Piket-May, “A modified FDTD (2; 4) scheme for modeling electrically large structures with high-phase accuracy,” IEEE Trans. Antennas Propag., vol. 45, pp. 254–264, 1997. [3] J. B. Cole, “A high-accuracy realization of the Yee algorithm using nonstandard finite difference,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 991–996, 1997. [4] S. Wang and F. L. Teixeira, “Dispersion-relation-preserving FDTD schemes of large-scale three-dimensional problems,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1818–1828, 2003. [5] B. Finkelstein and R. Kastner, “Finite difference time domain dispersion reduction schemes,” J. Comp. Phys., vol. 221, no. 1, pp. 422–438, 2007. [6] I. Koh, H. Kim, J. Lee, J. Yook, and C. Pil, “Novel explicit 2-D FDTD scheme with isotropic dispersion and enhanced stability,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3505–3510, Nov. 2006. [7] W. Kim, I. Koh, and J. Yook, “3D Isotropic Dispersion (ID)-FDTD algorithm: Update equation and characteristics analysis,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1251–1259, Apr. 2010. [8] P. Deng and I. Koh, “Reflection coefficient of the Isotropic-Dispersion Finite-Difference Time-Domain (ID-FDTD) method at planar dielectric interfaces,” PIERS Online, vol. 6, no. 3, pp. 217–221, 2010. [9] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989.

I. INTRODUCTION Electromagnetic integral equations are often discretized using the method of moments (MoM) [1], [2], one of the most widespread and well established techniques for electromagnetic problems. However, the matrix associated with the resulting linear systems is usually large and dense for electrically large targets for electromagnetic scattering problems [1]. It is basically impractical to solve electric-field integral-equation (EFIE) matrix equations using direct filling because their memory requirement and computational complexity are of the orders of O(N 2 ) and O(N 3 ), respectively, where N is the number of unknowns. To alleviate this problem, many fast solution algorithms have been developed. The first kind of algorithms is the fast iterative solution. The most popular fast iterative solution includes the multilevel fast multipole algorithm (MLFMA) [3]–[6], has O(N log N ) complexity for a given accuracy. Though efficient and accurate, this algorithm is highly technical. It utilizes a large number of tools, such as partial wave expansion, exponential expansion, filtering, and interpolation of spherical harmonics. MDA-SVD is another popular iterative solution used to analyze the scattering/radiation [7], [8], which exploits the well known fact that for well separated sub-scatterers, the corresponding sub-matrices are low rank and can be compressed. The second kind of algorithms is the fast direct solution. The MLCBD algorithm is one of the fast direct solutions, which is based on a blockwise compression of the impedance matrix [9]–[11]. The numerical complexity of the algorithm is shown to be O(N 2 ) and the storage requirements scale with O(N 1:5 ). Manuscript received November 10, 2010; revised January 11, 2011; accepted March 07, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the Major State Basic Research Development Program of China (973 Program: 2009CB320201), in part by Natural Science Foundation under Contracts 60871013, 60701004, and in part by the Jiangsu Natural Science Foundation under Contract BK2008048. The authors are with the Department of Communication Engineering, Nanjing University of Science and Technology, Nanjing, China (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161560

0018-926X/$26.00 © 2011 IEEE

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Fig. 1. The form of the impedance matrix for 1-level of MLCBD, Shaded block indicates the sub-matrix Z .

Although both the numerical complexity and the storage requirement of the fast iterative solution are less than that of the fast direct solution, the convergence rate of iterative methods can vary in an unpredictable way. The complexity of the iterative solution method is depending on the matrix condition number. Iterative solvers may be quite satisfactory for only a few right-hand sides (RHS) such as antenna or bistatic problems, but become expensive for monostatic scattering with many required sampling angles. It is well known that the matrix condition number of EFIE for electrically large problem is large [12]. Therefore, the system has poor convergence history. The aim of this communication is to present a MLFMA-based direct method for solving electromagnetic problems. It utilizes MLFMA to efficient filling the matrices of MLCBD. Simulation results show that the MLFMA-based direct method is computationally more efficient than for the conventional MLCBD and the MLFMA. This communication is organized as follows. Section II describes the MLFMA-based direct method in more detail. Section III gives some numerical examples to demonstrate the accuracy and computation efficiency of our approach. II. THE MLFMA-BASED DIRECT SOLUTION The impedance matrix filled by MLCBD is carried out based on the same multilevel spatial decomposition of the underlying geometry [9]–[11]. The single level of CBD is presented in [13], whereas the MLCBD is shown in [9]–[11]. Suppose that the object is decomposed into 1-level of MLCBD, the form of the impedance matrix is shown in Fig. 1. The shaded block in the Fig. 1 is compressed by MDA-SVD in [9]–[11]. In the MDA-SVD implementation, the impedance matrix of the two sufficiently separated boxes can be expressed in terms of three small matrices [14]–[20]

[

Z mm 1 14 ]

= [

1 01 1 14 ] [ 14 ]

U m r [! rr V rm 1 14 ]

(1)

1 ]m m is the far field sub-matrix of impedance matrix. m1 where [Z14 1 and m2 are the dimensions of the [Z14 ]m m . The index r denotes the number of equivalent RWG sources [14]–[20] and is much smaller than m1 and m2 . The far field matrix of the impedance matrix [ZF ] can be expressed in the following form

ZF ] =

[

L M (l) Far(l(i)) l=1 i=1 j=1

Uijl ][!ijl ][Vijl ]

[

(2)

where L indicates the number of the level, M (l) is the number of nonempty group at level l and, F ar(l(i)) denotes the number of far interaction groups of the ith nonempty group for each observer group l(i) at level l. The product [Uijl ][!ijl ][Vijl ] is associated with the interaction between the observer group l(i) and the source group l(j ). For a given observer group l(i), it is needed to store the matrix [Vijl ] for

Fig. 2. The form of the impedance matrix for 2-level of MLCBD, Shaded block indicates the sub-matrix Z .

different source group l(j ), increasing the memory requirement. For each observer group l(i), there are many source groups of the far interaction part of the impedance matrix. The sub-matrices associated with the observer groups and source groups are compressed by MDA-SVD. Therefore, the matrix filling time of (2) is also very long. When the object is decomposed into more than 1-level of MLCBD, the matrix filling time of (2) is becoming longer. Therefore, the matrix filling time of the MLCBD is very long. In order to improve the matrix filling time of the conventional MLCBD, the MLFMA is used to fill the far field sub-matrix of impedance matrix. In the Fig. 1, the shaded block is expressed as follows,

Rmp k 1 q=2B m2 and n 2

1 Z14 ] =

[

1

(

1(1)

)

2 1  1pq (k;rpq )Fqn (k )d k ;

(3)

1(4)

l (k) and Rmp l (k) are the aggregation factor and disaggregawhere Fqn tion factor at level l, respectively, and lpq (k;r ) is the translation factor. The forms of the three matrices in (3) are shown in [4]. When the object is analyzed by 2-level of MLCBD, the form of the impedance matrix is shown in Fig. 2 [9]–[11]. The shaded block which is shown in Fig. 2 can be expressed by MLFMA as follows: 1 Z14 ] =

[

Rmp k 1

n2G

2

(

)

P21 (k))T eikr

(

1

q=2B

1pq (k; kpq )

2  2 eikr P21 (k)Fqn (k )d k ; m 2 1(1) and n 2 1(4)

(4)

where the operator Pll01 (k) is used to denote an interpolation operator that interpolates the discrete far field values from level l to level l 1 (k ) [4]. When the MLFMA is utilized to fill the far field sub-matrix of l ], [!ijl ] and [Vijl ] which are impedance matrix, the matrices [Uij shown in the (2) are not needed to store. It is only needed to store the aggregation factor, the disaggregation factor, and the translation factor of MLFMA. The blocks (which do not include the self-interaction blocks) representing interactions between adjacent source and observation boxes are compressed by SVD (T) compression. The details of the SVD (T) compression are shown in [13]. When the matrices are compressed by MLFMA and the SVD (T) compression, it can be applied to the procedure of the MLCBD algorithm to form the direct solution. The details of the MLCBD are shown in [9]–[11]

0

III. NUMERICAL RESULTS In this section, a number of numerical examples are presented to demonstrate the efficiency of the proposed method in solving linear systems of electromagnetic scattering problems. All the computations

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TABLE I THE FAR FIELD MATRIX BUILDING AND THE INVERSION TIME AND THE FAR FIELD MATRIX BUILDING AND THE INVERSION MEMORY FOR PLANE-CYLINDER GEOMETRY

Fig. 3. Bistatic scattering cross section of plane-cylinder geometry.

TABLE II THE FAR FIELD MATRIX BUILDING AND THE INVERSION TIME AND THE FAR FIELD MATRIX BUILDING AND THE INVERSION MEMORY FOR MISSILE GEOMETRY

Fig. 4. Bistatic scattering cross section of missile geometry.

are carried out on Intel(R) Core(TM) 2 Quad CPU at 2.83 GHz and 8 GB of RAM in double precision. A. Comparison of the Proposed Method With the Conventional MLCBD First, the accuracy of the proposed method is checked by computing the bistatic RCS of two scattering geometries. In the first example, plane-cylinder geometry with 14776 unknowns at 300 MHz is considered in Fig. 3. The edge length of square plane is 4 m, the radius of the column is 0.1 m and the height of the column is 2 m. The z -axis is used as the rotation axis. The second example is the missile geometry shown in Fig. 4. The height of the missile is 4.7 m, and the radius of the cylinder is 0.5 m. The rotation axis of missile geometry is z -axis. It consists of the missile with 20106 unknowns. It can be found that the results of the proposed method agree very well with that of MLFMA. Next, the matrix-filling efficiency of the proposed method is checked by computing the bistatic RCS of the two scattering geometries. The far field matrix building and the inversion time and the far field matrix building and the inversion memory of the conventional MLCBD and the proposed method for the two geometries are shown in Tables I and II, respectively. It can be found that the far field filling time of the conventional MLCBD is 33 times longer than that of the proposed method, whereas the far field filling memory of the conventional MLCBD is 5 times larger than that of the proposed method. The inversion memory and the inversion time of the proposed method are nearly the same as that of the conventional MLCBD. B. Comparison of the Proposed Method With the Direct Filling The complexities of the inversion memory and the inversion time of the proposed method are checked for a plane geometry. The length and width of the plane are 6 m and 5 m, respectively. The z -axis is used

Fig. 5. The inversion memory needed for the plane geometry.

as the rotation axis. With reference to Fig. 5, the inversion memory requirement of the proposed method is much less than that of the direct filling. With reference to Fig. 6, the inversion time of the proposed method is also much less than that of the direct filling. C. Comparison of the Proposed Method With the MLFMA for Monostatic Problems The fourth example is the cylinder cavity. The length and radius of the cylinder cavity are 6.1 m and 3.3 m, respectively. The z -axis is used as the rotation axis. It consists of the missile with 10666 unknowns at 80 MHz. In implementing the MLFMA, the restarted version of the GMRES algorithm [21], [22] is used as the iterative method. The restarting number of GMRES is set to 30, whereas the stop precision for the restarted GMRES is set to 1003 . The numerical result of monostatic RCS in theta direction when  is fixed at 0 is depicted in Fig. 7. It can be found that there is an excellent agreement between the result of the proposed method and that of MLFMA. Table III shows the total memory and the total solving time of the proposed method and the MLFMA for the monostatic RCS of the cylinder cavity. The total memory of the proposed method contains the memory of the matrix building and the inversion. The total solving time of the proposed method contains the matrix building time, the inversion time and the

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Fig. 6. The inversion time needed for the plane geometry.

Fig. 7. The monostatic RCS for the cylinder cavity. TABLE III THE TOTAL MEMORY AND SOLVING TIME FOR THE CYLINDER CAVITY

solving time for monostatic RCS. The total memory of the MLFMA contains the memory of near field and far field. It can be observed that the total solving time is much less than that of MLFMA, while the memory of the proposed method is nearly the same as that of MLFMA.

IV. CONCLUSION In this communication, a MLFMA-based direct solution is proposed for solving electromagnetic problems efficiently. The method is based on the multilevel compressed block decomposition (MLCBD) algorithm. The matrix filling procedure of the conventional MLCBD is previously based on the MDA-SVD. Since either the matrix filling or memory of the MDA-SVD is much more expensive than for the MLFMA, the MLFMA-based direct solution utilizes the MLFMA instead of the MDA-SVD to fill the far field matrix of MLCBD. Our numerical results show that the proposed method is much more efficient than the conventional MLCBD. Moreover, the proposed method is much more efficient than the MLFMA for the monostatic scattering problems.

REFERENCES [1] R. F. Harrington, Field Computations by Moment Methods. New York: MacMillan, 1968. [2] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, pp. 409–418, May 1982. [3] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag., vol. 35, no. 6, pp. 7–12, Jun. 1993. [4] W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast Efficient Algorithms in Computational Electromagnetics. Boston, MA: Artech House, 2001. [5] Y. C. Pan, W. C. Chew, and L. X. Wan, “A fast multipole-methodbased calculation of the capacitance matrix for multiple conductors above stratified dielectric media,” IEEE Trans. Micro. Theory Tech., vol. 49, no. 3, pp. 480–490, Mar. 2001. [6] L. J. Jiang and W. C. Chew, “Modified fast inhomogeneous plane wave algorithm from low frequency to microwave frequency,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 2003, vol. 2, pp. 22–27. [7] J. M. Rius, J. Parron, A. Heldring, J. M. Tamayo, and E. Ubeda, “Fast iterative solution of integral equations with method of moments and matrix decomposition algorithm – Singular value decomposition,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2314–2324, Aug. 2008. [8] J. M. Rius, A. Heldring, J. M. Tamayo, and J. Parron, “The MDA-SVD algorithm for fast direct or iterative solution of discrete integral equations,” in Proc. 2nd Eur. Conf. Antennas Propag, EuCAP , 2007, pp. 1–8. [9] A. Heldring, J. M. Rius, J. M. Tamayo, and J. Parron, “Compressed block-decomposition algorithm for fast capacitance extraction,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 27, no. 2, pp. 265–271, Feb. 2008. [10] A. Heldring, J. M. Rius, J. M. Tamayo, and J. Parron, “Multilevel MDA-CBI for fast direct solution of large scattering and radiation problems,” presented at the Antennas and Propagation Society Int. Symp., 2007. [11] A. Heldring, J. M. Tamayo, J. M. Rius, J. Parron, and E. Ubeda, “Multiscale CBD for fast direct solution of MoM linear system,” presented at the Antennas and Propagation Society Int. Symp., 2008. [12] A. Heldring, “Full wave analysis of electrically large reflector antenna,” Ph.D., Delft Univ. Technol., Dept. Elect. Eng., Delft, The Netherlands, 2002. [13] A. Heldring, J. M. Rius, J. M. Tamayo, J. Parron, and E. Ubeda, “Fast direct solution of method of moments linear system,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3220–3228, Nov. 2007. [14] E. Michielssen and A. Boag, “A multilevel matrix decomposition algorithm for analyzing scattering from large structures,” IEEE Trans. Antennas Propag., vol. 44, no. 8, pp. 1086–1093, Aug. 1996. [15] J. M. Rius, J. Parron, E. Ubeda, and J. Mosig, “Multilevel matrix decomposition algorithm for analysis of electrically large electromagnetic problems in 3-D,” Microw Opt. Technol. Lett., vol. 22, no. 3, pp. 177–182, Aug. 1999. [16] J. Parron, G. Junkin, and J. M. Rius, “Improving the performance of the multilevel matrix decomposition algorithm for 2.5-D structures application to metamaterials,” in Proc. Antennas Propag. Soc. Int. Symp., Jul. 2006, pp. 2941–2944. [17] J. Parron, J. M. Rius, and J. Mosig, “Application of the multilevel decomposition algorithm to the frequency analysis of large microstrip antenna arrays,” IEEE Trans. Magn., vol. 38, no. 2, pp. 721–724, Mar. 2002. [18] J. Parron, J. M. Rius, A. Heldring, E. Ubeda, and J. R. Mosig, “Analysis of microstrip antennas by multilevel matrix decomposition algorithm,” presented at the Eur. Congress Computational Methods Applied Sciences in Engineering, Barcelona, Spain, Sep. 2000. [19] J. Parron, J. M. Rius, and J. Mosig, “Method of moments enhancement technique for the analysis of Sierpinski pre-fractal antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1872–1876, Aug. 2003. [20] J. M. Rius, A. Heldring, J. M. Tamayo, and J. Parron, “New and more efficient formulation of MLMDA for arbitrary 3D antennas and scatterers,” in Proc. 1st Eur. Conf. on Antennas Propag, EuCAP , 2006, pp. 1–6. [21] Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput., vol. 7, pp. 856–869, Jul. 1986. [22] Y. Saad, Iterative Methods for Sparse Linear System. Boston, MA: PWS Publishing Company, 1996.

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Application of Multiplicative Regularization to the Finite-Element Contrast Source Inversion Method Amer Zakaria and Joe LoVetri Abstract—Multiplicative regularization is applied to the finite-element contrast source inversion (FEM-CSI) algorithm recently developed for microwave tomography. It is described for the two-dimensional (2D) transverse-magnetic (TM) case and tested by inverting experimental data where the fields can be approximated as TM. The unknown contrast, which is to be reconstructed, is represented using nodal variables and first-order basis functions on triangular elements; the same first-order basis functions used in the FEM solution of the accompanying field problem. This approach is different from other MR-CSI implementations where the contrast variables are located on a uniform grid of rectangular cells and represented using pulse basis functions. The linear basis function representation of the -norm total variation contrast makes it difficult to apply the weighted multiplicative regularization which requires that gradient and divergence operators be applied to the predicted contrast at each iteration of the inversion algorithm; the use of finite-difference operators for this purpose becomes unwieldy. Thus, a new technique is introduced to perform these operators on the triangular mesh. Index Terms—Contrast source inversion, finite-element method, microwave tomography, multiplicative regularization.

I. INTRODUCTION In microwave tomography (MWT), an object of interest (OI) is illuminated by several electromagnetic sources and data are collected at different receiver locations for the purpose of quantitatively imaging the dielectric properties of the OI. The associated nonlinear inverse scattering problem is here formulated as a minimization of the leastsquares error between the collected data and a scattering model which is a function of the variables used to discretize the dielectric contrast. Various local optimization algorithms are available to perform the minimization, e.g., Gauss-Newton inversion (GNI) and the contrast source inversion (CSI), but the ill-posedness of the electromagnetic inverse scattering problem requires that some form of regularization be explicitly, or implicitly, applied. Different regularization methods have been reported in the literature and have been successfully applied for various applications [1]. A successful regularization technique which has been used is the weighted L2 -norm total variation multiplicative regularization (MR), which has been incorporated into both GNI and CSI [2]–[9]. Not only has it been shown to enhance the outcome of the inversion algorithm, i.e., regularize the optimization, but it also has other desirable features: (i) its edge-preserving characteristic, and (ii) its capacity for suppressing noise in measured data. There are many ways to discretize the various partial differential equation (PDE) or integral equation operators which can be used to formulate the electromagnetic scattering problem. Starting with a PDE formulation, the finite-element method is a powerful and flexible discretization technique which has been coupled with GNI in the past e.g., [10]. In this technique the unknown contrast is represented on a dual Manuscript received January 14, 2011; accepted February 23, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, the University of Manitoba Graduate Fellowship, the Government of Manitoba Graduate Scholarship, and in part by the Gordon P. Osler Award. The authors are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB R3T5V6 Canada (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2161564

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FEM mesh, different from the mesh used in the FEM solution of the electromagnetic field problem, and interpolation is used between the two meshes. Recently, it has been shown how FEM can be used with the CSI method [11]. Unlike other CSI implementations, FEM-CSI offers several benefits that include: (i) performing the inversion on an arbitrary irregular grid of triangles, (ii) incorporating an inhomogeneous medium as a background reference, (iii) controlling the density of the mesh adaptively within the inversion domain, and (iv) easily incorporating radiating or arbitrarily-shaped conductive boundaries surrounding the MWT setup. In this communication, a novel technique for incorporating multiplicative regularization in the FEM-CSI algorithm is introduced. In typical MR-CSI inversion algorithms, including the finite-difference CSI method reported in [4], the unknown variables are located on a uniform grid of rectangular cells [3], [7]. In such methods, finite-difference approximations for the gradient and divergence operators used in multiplicative regularization can be easily applied. When FEM is used to discretize the electromagnetic field problem, the unknown variables are located on the nodes of an irregular triangular mesh; thus applying MR using finite-differences becomes difficult. Herein a new technique is introduced to perform the gradient and divergence operators on a triangular mesh. The communication begins by describing the formulation of the MWT problem which is used, followed by a brief overview of the FEM-CSI algorithm. Next, the method for applying MR to the triangular mesh is described. To test the algorithm, it is used to invert experimental data and its reconstructions are compared with those from FEM-CSI, without MR. II. PROBLEM STATEMENT

Consider an imaging domain D within the problem domain of an MWT setup. The domain is enclosed by boundary 0. An unknown isotropic, nonmagnetic OI is located in D and is surrounded by a background medium of known electrical properties. The complex relative permittivity of the OI is r (r ), where r is a 2D position vector. The corresponding electric contrast is defined as (r ) (r (r )0 b (r ))=b (r ) where b (r ) is the background complex relative permittivity ((r ) = 0 for r 2 = D), which may be inhomogeneous. The same configuration and notation as given in [11] is used. The imaging domain D is illuminated by a harmonic TM electromagnetic field produced by one of T point sources, producing incident field Etinc when there is no OI in D . With the OI in D the total field Et for the same source t is measured at points located on a measurement Et 0 Etinc , surface S . The scattered electric field, defined by Etsct satisfies the scalar Helmholtz equation

r2 Etsct (r) + kb2 (r)Etsct(r) = 0kb2 (r)wt(r ) where

wt (r )

(1)

kb (r) = ! 0 0 b (r ) is the background wavenumber, and (r)Et (r) is the contrast source.

When the MWT setup is enclosed within a conductive-enclosure, homogeneous Dirichlet boundary conditions (BCs) are applied to the total field on 0, corresponding to inhomogeneous BCs for the scattered field. For unbounded-region problems Sommerfeld radiating BCs are imposed on the boundary. The boundary-value problem (BVP) defined by the second-order PDE (1) and the boundary conditions is solved using FEM with the Rayleigh-Ritz formulation [12]. Thus, the problem domain ( ) is divided into a mesh of triangular elements constituted by a total number of N nodes. At each node, linear-basis functions are specified whose parameters are dependent only on the geometry of the mesh. The discretization of (1) using FEM produces a matrix equation as detailed in [11]. This equation can be solved efficiently as the associated matrices

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are sparse, symmetric and independent of OI and the location of transmitter t. III. INVERSION ALGORITHM A. Contrast Source Inversion The CSI objective functional, which is to be minimized, is formulated with respect to the contrast sources, wt , and the contrast, , variables and is constructed as the sum of the normalized data-error and normalized domain-error functionals [11], [13]. It is written as

F CSI (; wt ) = F S (wt ) + F D (; wt ) 2 t kft 0 M S L [wt ]kS = 2 t kft kS inc 0 wt +  MD L[wt ] 2D t  Et : (2) + inc 2 t k Et kD Here ft 2 R holds the measured scattered field data at the R receiver locations for each transmitter,  2 I is the vector of contrast nodal values for nodes located inside D , and Etinc 2 I is the vector of incident field corresponding to transmitter t for nodes inside D . The notation a b denotes the Hadamard (i.e., element-wise) product. In addition, L 2 N 2I is the inverse of the FEM matrix operator and it transforms contrast source variables wt 2 I in D to N nodal scattered field values, the operator MS 2 R2N transforms field values

from N nodes in to R receiver locations on the measurement surface S and the operator MD 2 I 2N transforms values from N nodes in

to I points located inside the imaging domain D The CSI objective functional F CSI (; wt ) is minimized by updating the contrast source wt and the contrast  variables sequentially. First, with the contrast variables  held constant, the contrast source variables wt are updated by a conjugate-gradient (CG) method with Polak-Ribière search directions. Next, assuming the contrast source variables wt are constant, the functional becomes quadratic in the contrast variables ; therefore, the contrast variables are updated analytically. The initial guess for the FEM-CSI algorithm, wt;0 , is not set to zero, and is taken to be the minimizer of the data-error functional F S (wt ) after one line-minimization in the method of steepest descent. After evaluating wt;0 , the initial guess for  is calculated analytically. The initial search directions are set equal to zero. Due to the use of a differential-based operator (FEM) in CSI, some changes are necessary to the original CSI algorithm. These changes along with details about the FEM-CSI algorithm are detailed in [11].

C. Spatial Derivatives on a Triangular Mesh The calculation of the CG search directions in MR-CSI requires the accurate evaluation of the gradient and divergence of the contrast at nodal locations of the arbitrary triangular FEM mesh; this is not as straightforward as when using a uniform rectangular grid. The unknown contrast is represented using linear basis functions with a gradient which is constant over each triangle and is discontinuous between triangles. Thus, the divergence of the gradient at each node is not easily defined. If quadratic elements are used, the gradient would be a linear function over each element but the divergence would still be discontinuous between elements. Thus, some form of averaging is required. The method described here evaluates the divergence at each node by integrating the gradient over a stencil, having the centroids of the triangles associated with that node as the vertices. The spatial gradient of the contrast can be calculated numerically over each triangle in the mesh using the first-order basis functions. In FEM, the contrast within a triangle m is given by

l

Fn (; wt ) = FnMR () 2 F CSI (; wt ) where the regularization term FnMR () is given by FnMR () = bn2 (r ) jr(r)j2 + n2 drr: D

2

=1

(6)

l

(m)

(m) (r ) =

(m) + b(m) x + c(m) y

1

l

al

A(m)

l

l

:

(7)

(m) (m)

Here A(m) is the area of triangle m and the coefficients al ; bl and (m) are dependent on triangle geometry [12]. cl The spatial gradient of the contrast within triangle m is then calculated as

r(m) =

3

=1

(m) r(m) (r )

l

l

l

=

3

1

A(m)

(m)

=1

(m) x^ + c(m) y^

l

bl

l

(8)

l

where x ^ and y ^ are the Cartesian unit vectors. 2 for The spatial gradient in (8) is used to calculate the coefficients bn each triangle in D and then to evaluate the multiplicative regularization term FnMR (). To update the contrast variables , the gradient gn has to be evaluated at each node in D . For each node i let us define i;n as follows:

r 1 i;n x y = r 1 i;n x ^ + i;n y ^ x y =x ^ 1 ri;n + y ^ 1 ri;n (9) 2 2 = bi;n ri;n . Since bn and rn are calculated for each i;n =

The weighted L2 -norm regularization factor is implemented by introducing a multiplicative term to the CSI objective functional [3], [8]. Thus, the objective functional at the nth iteration becomes

bn (r ) =

(m) (m) (r )

l

where l is a local index for each node on triangle m; l is the contrast value at node l of triangle m, and the first-order linear basis function for node l is

B. Multiplicatively Regularized FEMCSI

Here

3

(m) (r ) =



(3)

(4)

where i;n triangle rather than each node, the spatial divergence in (9) needs to be approximated. Letusdefinearegion i aroundnode i asdepictedinFig.1.Thevertices of this region are the centroids of triangles sharing node i. Using vector identities and the divergence theorem, it can be shown that x ^

1

(5) 2) A (jrn01 (r )j2 + n where A is the total area of domain D and  2 = F D (CSI ; wt;n ) A01 n

n

in which A is the mean area of the mesh triangles in D . Since FnMR (n01 ) = 1, the update procedure for the contrast source variables wt remains unchanged; however, this is not the case for the contrast variables . After calculating the contrast variables in CSI, they are updated by a CG method using Polak-Ribière search directions as detailed in [7], [8].

x 1 ri;n  hx^ 1 rnx (r)i

=

1 Ai

x

0

1

n (r )^ x n ^ dli

(10)

h1i

where denotes the average value over region i ; Ai is the area of

i ; 0i is the contour (boundary) of i and n ^ is outward normal vector to 0i . Similarly the second term in (9) is approximated as y ^

y 1 ri;n  A1

i

y

0

1

n (r )^ y n ^ dli :

(11)

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Fig. 1. Region surrounding node to approximate the spatial divergence. The “ ” in the diagram represents the centroid of a triangle.

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The computational complexity of the inversion algorithm is reduced by adapting the inversion mesh such that the mesh is denser within the imaging domain D in comparison to outside D . This results in the number of unknowns in D to be 13706 nodes, while the total number of nodes N in problem domain is 19390. The ability to control the mesh density demonstrates an advantage of FEM-CSI in comparison to other CSI formulations. The inversion algorithm is allowed to run for 1024 iterations to ensure convergence. In addition, after each iteration the inversion results are constrained to lie within the region defined by 0  Re(r )  80 and 020  Im(r )  0. The real parts of inversion using FEM-CSI and MR-FEMCSI are shown in Fig. 2(c) and (d) (the imaginary parts are omitted because the background and the target are lossless). Using multiplicative regularization the shape and edges of the target is well reconstructed; however features of the “e-phantom” smaller than 8 mm (approx. 2=15, where  is the free-space wavelength) are not resolved. This result is similar to that obtained using MR-GNI on a uniform grid as reported in [16]. The MR-FEMCSI reconstruction is more homogeneous within the target contour in comparison to the FEM-CSI result; in addition the value of the permittivity is not overshot by MR-FEMCSI. V. CONCLUSION A multiplicatively regularized finite-element method contrast source inversion (MR-FEMCSI) algorithm has been presented and validated for 2D microwave tomography under the TM approximation of the fields. The algorithm retains the advantages of FEM-CSI, such as the ability to invert data on an arbitrary triangular mesh and allowing a non-uniform discretization of the problem domain. The addition of multiplicative regularization adds noise suppression to the inversion and enhances the edges of the reconstructed images while flattening regions of constant contrast. A new technique to calculate the gradient and divergence operators required for multiplicative regularization is outlined which can be used on the unknown contrast variables located on the nodes of such arbitrary triangular meshes. The performance of the algorithm is demonstrated by inverting experimental data.

=5

Fig. 2. The “e-phantom” (a) inside the imaging setup, (b) its exact profile GHz, and the reconstruction results using (c) FEM-CSI and (d) at MR-FEMCSI.

Since the values of n (r ) are known at the vertices of region i , the line integrals in (10) and (11) are evaluated numerically. Here the trapezoidal rule is used to calculate the integral over each segment in region i . IV. EXPERIMENTAL RESULTS The MR-FEMCSI algorithm is tested by inverting an experimental dataset collected using our air-filled MWT system. A full description of the MWT imaging system utilized here is outlined in [14]. This system is air-filled with 24 Vivaldi antennas used as transmitters and receivers. The Vivaldi antennas are evenly distributed on a circle of radius 0.22 m. For each transmitting antenna, 23 measurements are collected (total number of measurements for a dataset is 23 2 24 = 522). The measured data are calibrated using the procedure described in [14]. The experimental dataset is acquired at a frequency f = 5 GHz. An “e-phantom” with multiple concave features is used as a target. A side-view of the actual target is shown in Fig. 2(a), while the exact permittivity profile is depicted in Fig. 2(b). The “e-phantom” is constructed of ultrahigh-molecular-weight (UHMW) polyethylene which is a lossless material of relative permittivity r = 2:3 [15]. The inversion domain D is selected to be a square centered in the problem domain with side length equal to 0.13 m.

REFERENCES [1] P. Mojabi and J. LoVetri, “Microwave biomedical imaging using the multiplicative regularized Gauss-Newton inversion,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 645–648, 2009. [2] P. Mojabi and J. LoVetri, “Overview and classification of some regularization techniques for the Gauss-Newton inversion method applied to inverse scattering problems,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 645–648, 2009. [3] A. Abubakar, P. M. van den Berg, and J. J. Mallorqui, “Imaging of biomedical data using a multiplicative regularized contrast source inversion method,” IEEE Trans. Microwave Theory Tech., vol. 50, no. 7, pp. 1761–1777, Jul. 2002. [4] C. Gilmore, A. Abubakar, W. Hu, T. Habashy, and P. van den Berg, “Microwave biomedical data inversion using the finite-difference contrast source inversion method,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1528–1538, May 2009. [5] P. M. van den Berg and A. Abubakar, “Contrast source inversion method: State of art,” Progr. Electromagn. Res., vol. 34, pp. 189–218, 2001. [6] P. M. van den Berg, A. L. van Broekhoven, and A. Abubakar, “Extended contrast source inversion,” Inverse Prob. vol. 15, no. 5, p. 1325, 1999 [Online]. Available: http://stacks.iop.org/0266-5611/15/ i=5/a=315 [7] P. M. van den Berg, A. Abubakar, and J. Fokkema, “Multiplicative regularization for contrast profile inversion,” Radio Sci., vol. 38, no. 2, p. 23 (1–10), 2003. [8] A. Abubakar, W. Hu, P. van den Berg, and T. Habashy, “A finite-difference contrast source inversion method,” Inverse Prob., vol. 24, p. 065004 (17 pp), 2008. [9] C. Gilmore and J. LoVetri, “Enhancement of microwave tomography through the use of electrically conducting enclosures,” Inverse Prob. vol. 24, no. 3, p. 035008, 2008 [Online]. Available: http://stacks.iop. org/0266-5611/24/i=3/a=035008

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[10] K. D. Paulsen, P. Meaney, M. Moskowitz, and J. S. , Jr., “A dual mesh scheme for finite element based reconstruction algorithms,” IEEE Trans. Med. Imaging, vol. 14, no. 3, pp. 504–514, Sep. 1995. [11] A. Zakaria, C. Gilmore, and J. LoVetri, “Finite-element contrast source inversion method for microwave imaging,” Inverse Prob. vol. 26, no. 11, p. 115010, 2010 [Online]. Available: http://stacks.iop.org/0266-5611/26/i=11/a=115010 [12] J. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 2002. [13] P. M. van den Berg and R. E. Kleinman, “A contrast source inversion method,” Inverse Prob. vol. 13, no. 6. [14] C. Gilmore, P. Mojabi, A. Zakaria, M. Ostadrahimi, C. Kaye, S. Noghanian, L. Shafai, S. Pistorius, and J. LoVetri, “A wideband microwave tomography system with a novel frequency selection procedure,” IEEE Trans. Biomed. Engrg., vol. 57, no. 4, pp. 894–904, Apr. 2010. [15] W. S. Bigelow and E. G. Farr, “Impulse propagation measurements of the dielectric properties of several polymer resins,” Measurement Notes, Note 55, 1999, pp. 1–52. [16] C. Gilmore, P. Mojabi, A. Zakaria, S. Pistorius, and J. LoVetri, “On super-resolution with an experimental microwave tomography system,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 393–396, 2010.

Statistical Characterization of Medium Wave Spatial Variability Due to Urban Factors Unai Gil, Ivan Peña, David Guerra, David de la Vega, Pablo Angueira, and Juan Luis Ordiales Abstract—The recently developed digital radio systems for the medium wave (MW) band require accurate field strength prediction methods for coverage calculi. Traditional prediction methods do not consider the influence of urban factors on MW ground-wave propagation. This influence causes signal strength time and spatial variability which in turn, provoke drop-outs below reception threshold values. In this letter the ground-wave spatial variability is statistically analyzed in urban environments by means of empirical data from four extensive field trials. The experiments were carried out in different urbanenvironmentsandatdifferentfrequenciesoftheMWband.Priortothe analysis, long-term and short-term components of the signal were separated by means of the generalized Lee method (GLM). The results show the attenuation caused by different urban factors. These attenuation values should be added to the signal strength predicted median values in order to ensure correct reception. Index Terms—Coverage prediction, medium wave (MW) , propagation models, signal variability, urban environments.

I. INTRODUCTION New digital radio broadcasting systems [1] in the medium wave (MW) band are to be fully deployed shortly. Network planning of traditional broadcasting systems in MW was based on the field strength prediction curves provided by ITU-R [2]. As the ground-wave is assumed Manuscript received August 26, 2010; revised February 22, 2011; acceptedFebruary 26, 2011. Date of publication July 12, 2011; date of current version September 02, 2011. This work was supported in part by the Spain Government under Grants TEC2005-06139 and FIT-330301-2007-1 and in part by the Government of the Basque Country This communication has been produced by the Signal and Radio Communications Group of The University of the Basque Country. This work is due to a cooperative partnership between the DRM Consortium and Cadena SER (Spain), All India Radio (India) and Radio Educación (Mexico) broadcasters. Deutsche Welle, TDF, Radio Netherlands and Fraunhofer Institute of Erlangen, Himalaya, Thomson, Transradio and Hitachi Kokusai have generously donated all the measurement equipment for these field trials. The authors are with the Electronics and Telecommunications Department, University of the Basque Country UPV/EHU, Bilbao, Bizkaia, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2161550

TABLE I TRANSMITTER FEATURES

to have no variability, these curves provide only median values [3]. However the experience from different field trials [4] showed that this assumption was optimistic in urban environments. As a consequence, the unforeseen variability could cause the signal strength to decrease below the threshold value. Up to now, time and spatial variability have been studied in rural areas [4], [5], but only time variability has been dealt with as regards urban environments. Hence, spatial variability in urban environments in the MW band needs to be analyzed [6]. This paper presents the statistical characterization of the signal variability by means of analyzing the attenuation caused by urban factors. The analysis is based on data which were recorded during four extensive field trials carried out in urban environments. Each trial included different features and studies for different frequencies, always within the MW band. First, the field trials and the features of the involved urban environments are described. Second, the methodology for processing the empirical data is explained. Third, the calculated attenuation values are presented. Finally, as a conclusion, the influence of frequency and the features of different kind of urban environments is discussed.

II. MEASUREMENT CAMPAIGNS The data analyzed in this paper was gathered during three measurement campaigns that evaluated digital radio mondiale (DRM) mobile reception in MW. An additional measurement campaign of AM signals was also included in the study. It is important to mention that considering the bandwidths involved, the broadcasted service should not affect signal propagation. The transmission features of these measurement campaigns are summarized in Table I. As can be seen, data from a complete range of frequencies within the MW band was available. The cities involved are representative of different kinds of urban environments as shown in Table II. In all three cities, the traffic density was very high when the measurements were carried out. In fact, studies in the past identified traffic as an attenuation factor for propagation in the medium wave band [4]. The measurement system was set up in a mobile unit and it was composed of a fully characterized short monopole antenna placed at 3.5 m height [7] and a professional field strength meter [8] that integrated the power within the signal bandwidth. Finally a GPS receiver and a tachometer provided ancillary data related to position, time, speed and distance. These data were captured every 400 ms (the DRM frame duration) and were stored in plain text files. More than 700 km of measurements were recorded during the measurement campaigns of Table I.

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TABLE II RECEPTION ENVIRONMENTS

III. METHODOLOGY AND PROCESSING Signal spatial variability in broadcasting services can be analyzed by considering two components. One of them is the variation of the mean value of field strength. This is usually called “long-term” variation. The long-term signal variation is caused by the alteration of transmitter-receiver path features or the urban reception environment. Superimposed, there are local instantaneous variations in the signal strength around this mean level. This local variation is usually referred as “short-term” variation and it is caused by different urban elements such as bridges, power lines, etc. In order to perform a systematic spatial variability analysis a separate study of the long and short-term components needs to be carried out [9], [10]. The generalized Lee method (GLM) [5] is the reference method for estimating the local mean values of the long-term signal along a route in the band. The short-term variation has been obtained by taking away the long-term variation from the envelope of the received signal. In parallel, the causes of variation have been identified and their attenuation has been quantified. However, the influence of the distance to the transmitter and the elevation profile features must be eliminated from the analysis. A. Data Normalization The prediction curves in [2] provide median field strength values as a function of the distance to the transmitter. In order to study and analyze the factors corresponding exclusively to urban environment features the influence of this distance should be removed for a spatial variability analysis. In addition, terrain irregularities have an impact on the attenuation [10]. The impact of this factor has also been removed from the study. Mobile reception routes have been divided into stretches that fulfill two conditions. The first condition is that the stretch is inside a square area which diagonal does not exceed 1 km. This condition eliminates possible variations associated to transmitter-receiver distance [2]. The second condition avoids influence of terrain height variations within the stretch. Numerically, it limits the difference of height between the highest and lowest point of the stretch is lower than 0:1. This way, within each stretch, the variation due to distance to the transmitter is lower than 1 dB in the worst case (diagonal of the box in radial direction from the transmitter) and the variation due to topography is negligible [11]. In consequence, the field strength variation within a stretch will be due only to urban environment characteristics. B. Application of the GLM The GLM has been applied in order to separate the long and shortterm components of the signal field strength of the stretches under analysis. The main step of the method consists of the estimation of the local mean values that form the long-term variation. The local mean at a particular location is estimated by averaging the linear values of the signal envelope inside a window of suitable width around each location. The running mean technique is applied along the stretch to obtain the successive local mean values that form the long-term signal. Three parameters are involved in the application of this technique, namely 2L, N and d. 2L is the window length and is related to the variability of the transmitter-receiver path features or the possible changes in the urban reception environment. N is the minimum number of field strength samples inside the 2L window in order to estimate one local mean value,

Fig. 1. Field strength versus the distance run along a stretch. Three graphs: whole signal (light gray continuous line), long-term component given by GLM (black line) and long-term component calculated on the basis of street width (dashed thick black line).

and d is the autocorrelation distance. The parameter d is defined as the minimum distance to obtain uncorrelated field strength samples. Given the fact that urban environments are very inhomogeneous when compared to rural ones [4], [5], the value chosen for d has been 4.5 m, which is the minimum value associated to the measurement system. In order to avoid short-term effects in the long-term component, the values of the remaining parameters were optimized following the procedure described by the GLM [5]. The 2L optimal value obtained in 94.37% of the stretches is close to one wavelength, which is lower than in rural environments [5], as expected. Finally, the value of N can be calculated as the quotient of 2L and d. C. Long-Term Component Calculation and Processing The GLM was applied to the whole set of stretches. The long-term component provided by the method is represented in black in the example. shown in Fig. 1. The long-term component variation was caused by urban factors that feature dimensions of the same order of magnitude as the optimal value of the parameter 2L given by the GLM, that is,  which ranges from 220.75 m (1359 kHz) to 370.37 m (810 kHz). The main cause of long term variation was the alteration of the width of streets, so the area where the long-term component without variation matches the length of the streets. If the width of the streets is considered the main cause of long-term attenuation due to the vicinity of buildings, the long-term component can be recalculated by means of the field strength median value in each street, as shown in the dashed black line of Fig. 1. That is, only taking into account the median value of the field strength in each street, long-term component of the received signal has been calculated, Finally, the correlation coefficient between the two long-term calculations, the proposed one and the GLM based one, has been determined. This coefficient is higher than 80% in 91.23% of the stretches under analysis so it can be assumed that the width of the streets is the main cause of long-term variation. Other urban factors, such as crossroads, squares, metallic structures, buildings, bridges, tunnels, wires and panels have been tested as causes of long-term variation but none of them nor any of their combinations have achieved a correlation coefficient value higher than 50% in the best case. D. Short-Term Component Calculation and Processing Once the long-term component based upon street width is taken away from the whole signal, the variations of the short-term component of each stretch have been characterized. This work has been done with the aid of an ortophoto cartography. In order to avoid including time variation occurrences [4] in the spatial variability analysis, only occurrences that exceeded 4 dB up or down the long-term value (the median value of the street) have been taken into account. First of all, it was necessary to calculate the portion

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Fig. 2. Relative distance affected by urban factors in percentage. TABLE III ATTENUATION DIFFERENCES WITH RESPECT TO WIDE STREETS

Fig. 3. Short-term attenuation due to urban factors.

The rest of the factors cause attenuation mean values between 5 and 8 dB with standard deviations below 2 dB. V. CONCLUSION

of the coverage area affected by short-term attenuation occurrences. According to this criterion, around a 13% of the distance over the total measured distance was affected by such occurrences in all the cities under test. These occurrences were analyzed one by one, by three experts in order to identify their causes in the cartography. Fig. 2 depicts the percentage of distance affected by each urban cause over the total measured distance for the different cities and frequencies. IV. ATTENUATION DUE TO URBAN FACTORS Wide streets with more than 8 lanes, have been chosen as reference in order to quantify long-term variation occurrences. These streets showed received values close to free-space conditions. Narrow streets will be considered the ones with 1–3 lanes and medium streets, those with 4–7 lanes. Table III shows the mean differences between the reference (wide streets) and the attenuation of the long-term component due to medium and narrow streets. The attenuation values were related to the heterogeneity degree of the urban environment, on the one hand, and to the frequency, on the other. Precisely, Madrid, which shows the most heterogeneous environment, has provided the highest attenuation values, and at 1260 kHz the effect is almost doubled with regard to 810 kHz. Fig. 3 shows the attenuation values for most frequent urban factors. As it can be seen, the short-term attenuation values caused by the factors under analysis are practically the same regardless of the urban environment. Three factors are associated to an increase of the signal strength, namely power lines [12], crossroads and squares. As for the factors that cause attenuation, bridges and tunnels have shown a wide range of attenuation values which highly depend on their actual dimensions.

The spatial variability in the MW band and urban environments has been characterized in terms of the attenuation caused by urban factors. More than 700 km of mobile measurements in a wide range of urban environments and MW frequencies have been analyzed. It has been shown that long-term variations increase as the width of streets decreases and as the frequency increases. As for short-term variability, the attenuation of most urban factor has been quantified. These results are essential in order to include spatial variability effects into field strength prediction methods for network planning in the MW band.

REFERENCES [1] ITU-R BS.1615 Recommendation, “Planning Parameters for Digital Sound Broadcasting at Frequencies Below 30 MHz,” International Telecommunication Union, 2003. [2] ITU-R P.368-8 Recommendation, “Ground-Wave Propagation Curves for Frequencies Between 10 kHz and 30 MHz,” Int. Telecommunication Union, 2006. [3] ITU-R P.1321-8 Recommendation, “Ground-Wave Propagation Curves for Frequencies Between 10 kHz and 30 MHz,” Int. Telecommunication Union, 2006. [4] D. Guerra and U. Gil et al., “Medium wave DRM field strength time variation in different reception environments,” IEEE Trans. Broadcast., vol. 52, no. 4, pp. 483–490, Dec. 2006. [5] D. de la Vega and S. Lopez et al., “Generalization of Lee method for the analysis of the signal variability,” IEEE Trans. Veh. Technol., vol. 58, no. 2, pp. 506–516, Feb. 2009. [6] J. H. Causebrook, “Medium-wave propagation in build-up areas,” Proc. IEE, vol. 125, no. 9, pp. 804–808, Sept. 1978. [7] “Active Rod Antenna HE 010,” Rohde & Schwarz, 1998, Manual. [8] “ESPI 3,” Rohde & Schwarz, 2007, Operating Manual. [9] “Field-Strength Measurements Along a Route With Geographical Coordinate Registrations,” Apr. 2005, ITU-R Rec. SM.1708. [10] J. D. Parsons, The Mobile Radio Propagation Channel, 2nd ed. Hoboken, NJ: Wiley, 2000. [11] D. de la Vega et al., “Analysis of the attenuation caused by the influence of orography in the mediumwave band,” presented at the IEEE 65th Vehicular Technology Conf., Dublin, Ireland, 2007. [12] IEEE Guide on the Prediction Measurement, and Analysis of AM Broadcast Reradiation by Power Lines, IEEE Std 1260-1996, Transmission and Distribution Committee of the IEEE Power Engineering Society, Feb. 1996.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Editorial Board Sara K. S. Hanks, Editorial Assistant Michael A. Jensen, Editor-in-Chief e-mail: [email protected] Department of Electrical and Computer Engineering (801) 422-3903 (voice) 459 Clyde Building Brigham Young University Provo, UT 84602 e-mail: [email protected] (801) 422-5736 (voice) Senior Associate Editor Karl F. Warnick Dimitris Anagnostou Hiroyuki Arai Ozlem Aydin Civi Zhi Ning Chen Jorge R. Costa Nicolai Czink George Eleftheriades Lal C. Godara

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