IEEE Transactions on Antennas and Propagation [volume 58 number 8]

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AUGUST 2010

VOLUME 58

NUMBER 8

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Modeling, Design and Experimentation of Wearable RFID Sensor Tag .. . . C. Occhiuzzi, S. Cippitelli, and G. Marrocco Low Cost Planar Waveguide Technology-Based Dielectric Resonator Antenna (DRA) for Millimeter-Wave Applications: Analysis, Design, and Fabrication ..... ......... ........ ......... . W. M. Abdel Wahab, D. Busuioc, and S. Safavi-Naeini Integrated Leaky-Wave Antenna–Duplexer/Diplexer Using CRLH Uniform Ferrite-Loaded Open Waveguide . ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... T. Kodera and C. Caloz Arrays An Original Antenna for Transient High Power UWB Arrays: The Shark Antenna ..... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... L. Desrumaux, A. Godard, M. Lalande, V. Bertrand, J. Andrieu, and B. Jecko Resonant Effects and Near-Field Enhancement in Perturbed Arrays of Metal Dipoles .. ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... .... C. Mateo-Segura, G. Goussetis, and A. P. Feresidis Design and Implementation of Embedded Printed Antenna Arrays in Small UAV Wing Structures .... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ....... M. S. Sharawi, D. N. Aloi, and O. A. Rawashdeh Flat-Top Footprint Pattern Synthesis Through the Design of Arbitrary Planar-Shaped Apertures ....... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ....... A. Aghasi, H. Amindavar, E. L. Miller, and J. Rashed-Mohassel Synthesis of Unequally Spaced Antenna Arrays by Using Differential Evolution ....... ... C. Lin, A. Qing, and Q. Feng Multi-Frequency Synthetic Thinned Array Antenna for the Hurricane Imaging Radiometer .... ........ ......... ......... .. .. M. C. Bailey, R. A. Amarin, J. W. Johnson, P. Nelson, M. W. James, D. E. Simmons, C. S. Ruf, W. L. Jones, and X. Gong Electronically Reconfigurable Transmitarray at Ku Band for Microwave Applications . ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ...... P. Padilla, A. Muñoz-Acevedo, M. Sierra-Castañer, and M. Sierra-Pérez Optimal Wideband Beamforming for Uniform Linear Arrays Based on Frequency-Domain MISO System Identification . .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ B. H. Wang, H. T. Hui, and M. S. Leong Design and Performance of Frequency Selective Surface With Integrated Photodiodes for Photonic Calibration of Phased Array Antennas ......... ........ ......... ......... ........ . W. M. Dorsey, C. S. McDermitt, F. Bucholtz, and M. G. Parent Imaging A Through-Dielectric Radar Imaging System .... ........ ......... ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ........ G. L. Charvat, L. C. Kempel, E. J. Rothwell, C. M. Coleman, and E. L. Mokole Breast Lesion Classification Using Ultrawideband Early Time Breast Lesion Response ......... ........ ......... ......... .. .. ........ ......... ......... ........ ........ J. Teo, Y. Chen, C. B. Soh, E. Gunawan, K. S. Low, T. C. Putti, and S.-C. Wang

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(Contents Continued from Front Cover) Complex Media Artificial Magnetic Materials Using Fractal Hilbert Curves ...... ......... ........ ......... ... L. Yousefi and O. M. Ramahi Measurements Theory and Practice of the FFT/Matrix Inversion Technique for Probe-Corrected Spherical Near-Field Antenna Measurements With High-Order Probes ........ ........ ...... T. Laitinen, S. Pivnenko, J. M. Nielsen, and O. Breinbjerg Resilience to Probe-Positioning Errors in Planar Phaseless Near-Field Measurements .. ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ....... S. F. Razavi and Y. Rahmat-Samii Numerical FDTD Discrete Planewave (FDTD-DPW) Formulation for a Perfectly Matched Source in TFSF Simulations . ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .... T. Tan and M. Potter FDTD Analysis of Periodic Structures With Arbitrary Skewed Grid ..... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ...... K. ElMahgoub, F. Yang, A. Z. Elsherbeni, V. Demir, and J. Chen Cartesian Shift Thin Wire Formalism in the FDTD Method With Multiwire Junctions .. ...... C. Guiffaut and A. Reineix Efficient Current-Based Hybrid Analysis of Wire Antennas Mounted on a Large Realistic Aircraft .... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ....... W.-J. Zhao, J. L.-W. Li, and L. Hu Enhancement of Efficiency of Integral Equation Solutions of Antennas by Incorporation of Network Principles – Part II . .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ........ A. W. Schreiber and C. M. Butler A Calderón Multiplicative Preconditioner for Coupled Surface-Volume Electric Field Integral Equations ...... ......... .. .. ........ ......... ......... ........ ......... ......... .. H. Ba˘gcı, F. P. Andriulli, K. Cools, F. Olyslager, and E. Michielssen Comparison of Interpolating Functions and Interpolating Points in Full-Wave Multilevel Green’s Function Interpolation Method ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .. Y. Shi and C. H. Chan Propagation Modes and Temporal Variations Along a Lift Shaft in UHF Band . ..... X. H. Mao, Y. H. Lee, and B. C. Ng Physical Meaning of Perturbative Solutions for Scattering From and Through Multilayered Structures With Rough Interfaces ....... ......... ........ ......... ....... ... ........ ......... ......... ........ . P. Imperatore, A. Iodice, and D. Riccio A Novel Integration Method for Weak Singularity Arising in Two-Dimensional Scattering Problems .. ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ....... Y.-F. Jing, T.-Z. Huang, Y. Duan, S.-J. Lai, and J. Huang

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COMMUNICATIONS

E-Textile Conductors and Polymer Composites for Conformal Lightweight Antennas .. ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ... Y. Bayram, Y. Zhou, B. S. Shim, S. Xu, J. Zhu, N. A. Kotov, and J. L. Volakis A Novel Dipole Antenna Design With an Over 100% Operational Bandwidth .. ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ .... F.-Y. Kuo, H.-T. Chou, H.-T. Hsu, H.-H. Chou, and P. Nepa Frequency Reconfigurable Quasi-Yagi Folded Dipole Antenna .. ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ... P.-Y. Qin, A. R. Weily, Y. J. Guo, T. S. Bird, and C.-H. Liang Band-Rejected Ultrawideband Planar Monopole Antenna With High Frequency Selectivity and Controllable Bandwidth Using Inductively Coupled Resonator Pairs .... ........ ......... ......... ........ ......... ......... T.-G. Ma and J.-W. Tsai Reconfigurable Circularly-Polarized Patch Antenna With Conical Beam ........ ......... ...... J.-S. Row and M.-C. Chan Design and Characterization of 60-GHz Integrated Lens Antennas Fabricated Through Ceramic Stereolithography .... .. .. ........ ......... ......... ........ ...... N. T. Nguyen, N. Delhote, M. Ettorre, D. Baillargeat, L. Le Coq, and R. Sauleau Millimeter Wave Circularly Polarized Fresnel Reflector for On-Board Radar on Rescue Helicopters .. ......... ......... .. .. ........ ......... ......... ........ .. K. Mazouni, J. Lanteri, N. Yonemoto, J.-Y. Dauvignac, C. Pichot, and C. Migliaccio PCB Slot Based Transformers to Avoid Common-Mode Resonances in Connected Arrays of Dipoles . ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... D. Cavallo, A. Neto, and G. Gerini Reducing Mutual Coupling for an Extremely Closely-Packed Tunable Dual-Element PIFA Array Through a Resonant Slot Antenna Formed In-Between ..... ......... ........ ......... ......... ........ ......... S. Zhang, S. N. Khan, and S. He The Sources Reconstruction Method for Amplitude-Only Field Measurements ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... Y. Álvarez, F. Las-Heras, and M. R. Pino Reducing Complexity in Indoor Array Testing ... ........ ......... ......... ....... A. Buonanno, M. D’Urso, and G. Prisco Generation of Free Space Radiation Patterns From Non-Anechoic Measurements Using Chebyshev Polynomials ..... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ...... Z. Du, J. I. Moon, S. Oh, J. Koh, and T. K. Sarkar The ADI-FDTD Method for Transverse-Magnetic Waves in Conductive Materials ..... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... .... J. A. Pereda, A. Grande, O. González, and A. Vegas On the FDTD Near-to-Far-Field Transformations for Weakly Scattering Objects ........ ......... ........ ......... T. Martin Spectral Properties of Modulated Signal in the Doppler Domain in Urban Radio Channels With Fading ....... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ..... N. Blaunstein, D. Katz, and M. Hayakawa

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Digital Object Identifier 10.1109/TAP.2010.2063056

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 8, AUGUST 2010

Modeling, Design and Experimentation of Wearable RFID Sensor Tag Cecilia Occhiuzzi, Stefano Cippitelli, and Gaetano Marrocco

Abstract—Design of effective wearable tags for UHF RFID applications involving persons is still an open challenge due to the strong interaction of the antenna with the human body which is responsible of impedance detuning and efficiency degradation. A new tag geometry combining folded conductors and tuning slots is here discussed through numerical analysis and extensive experimentation also including the integration of a passive motion detector. The achieved designs, having size comparable with a credit card, may be applied to any part of the body. The measured performance indicates a possible application of these body-worn tags for the continuous tracking of human movements in a conventional room. Index Terms— Antennas, biomedical applications of electromagnetic radiation, biomedical telemetry, transponders.

I. INTRODUCTION

T

HE possibility to monitor and identify people by means of low-power and low-cost technology is nowadays one of the most interesting and promising features of radio frequency identification (RFID) techniques. Thanks to the advances in low-power electronics, it is now feasible to envisage sophisticated RFID-like devices integrating sensing and signal processing ability [1] able to provide real-time biomonitoring (temperature, blood pressure, heartbeat, glucose content, human behavior) and location of people within hospitals or domestic environment [2]–[5]. The UHF (860–960 MHz) standard is particularly attractive in passive low-cost applications due to the permitted high datarate and large reading distances potentially comparable with the size of typical indoor environments. The requirements of wearable antennas are small dimensions and lightweight as well as high immunity to the human body interaction which may otherwise sensibly change the radiation diagram and degrade the antenna efficiency. Some of these issues are also common to the design of tags for metal objects whose presence strongly affects the radiation diagrams of the attached antenna and prevents the use of dipole like layouts. In active and semi-active architectures, as in the case of bodycentric communication systems [6], the overall radiation performance is enhanced by additional battery-assisted electronics. In Manuscript received July 24, 2009; revised November 23, 2009; accepted February 01, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. C. Occhiuzzi and G. Marrocco are with the DISP, University of Roma “Tor Vergata,” 00133 Roma, Italy (e-mail: [email protected]; [email protected]). S. Cippitelli was with the DISP, University of Roma “Tor Vergata,” 00133 Roma, Italy. He is now with SIA, 10146 Torino, Italy. Digital Object Identifier 10.1109/TAP.2010.2050435

case of passive tags instead, where the energy to produce the response comes from a remote query unit, the antenna design is much more challenging. Several solutions have been recently investigated for the design of passive tags over metals, mainly based on the use of high permittivity slabs and of metallic shields, integrated in the antennas as ground planes. Typical antennas are the patch-like family comprising PIFA and IFA layouts, [7] (maximum gain up to 6 dB using the parasistic constructive effect of the surronding objects), [8] (gain max: 2 dB) [9] (gain max: 6.4 dB). The design of wearable passive UHF tags has up to now received much less attention. In a previous paper [10], the authors considered a family of slot antennas over a suspended patch, partly decoupled from the body by a silicone slab. The study was mostly oriented to define tuning mechanisms for the required conjugate impedance matching to a great variety of microchip impedances and to understand the dependence of the antennas’s bandwidth on the body placement. The antenna layout was intended to host additional electronics and contacting or non-contacting sensors. The maximum size of these antennas was of the order of 4–6 cm and the typical gain was rather poor (gain max: 7 dB) due to the bidirectional radiation of the slot. The expected activation ranges were therefore modest, even if it was demonstrated that the gain may be improved by enlarging the overall size. In [11] the use of automatically optimized slot-line transformers was further investigated for miniaturization and multi-band purposes. The rich study in [12] (and herein included references) considers some solutions partly decoupled from the body such as multi-folded dipole antennas over a shielding plate and regular patch and PIFA configurations. These devices are specifically designed for wearable applications and experimentally evaluated for what concerns the monitoring of runners in open areas and of personnel inside buildings. Some interesting effects are characterized, such as the influence of the tilt of the transmitting and received antennas and the mutual shadowing among people in the same area. The various antennas are not intended to host sensors but only to identify the person. The dominant size in all the cases was around 15 cm and the measured on-body gain ranges between 0 dB and 5 dB in the largest configurations. Very recently, new magnetic materials have been considered as a shielding plate for an RFID tag [13]. The innovative ferritesilicone (BaCo) composite promises to achieve very low-profile miniaturized and flexible structures potentially useful for wearable applications. The measured maximum gain in air is of the . order of This contribution proposes a planar layout which combines the tuning agility of the shaped-slot based tags and the decoupling from the body achieved by grounded antennas. The basic

0018-926X/$26.00 © 2010 IEEE

OCCHIUZZI et al.: MODELING, DESIGN AND EXPERIMENTATION OF WEARABLE RFID SENSOR TAG

Fig. 1. Layout of the proposed tag family. The H-slot acts as tuning impedance. The sensors may be allocated over the top conductor.

configuration comprises a folded patch sourced by an embedded H-slot whose main features are: on-body gain higher than previous examples in [10] and comparable with that of tags over metal, approximately constant radiation performances regardless of the different body positions, reduced sizes and the predisposition to host passive sensors. A general design procedure is here described to apply the proposed antenna configuration to RFID microchips of given input impedance by the help of an equivalent circuit model (Section II) useful to better understand the electromagnetic role of the antenna’s geometrical parameters and to provide a starting guess in the final tag design. The real performances of the tags are then evaluated (Section III) by means of an articulated experimental campaign comprising the input impedance measurement of some prototypes and the read-region characterization when the antenna plus the RFID microchip is attached onto the human body. Finally, the paper describes (Section IV) how the tag design procedure may also account for the electrical features of the sensor in the conjugate impedance matching with reference to the integration of a simple motion sensor. The performance of the resulting integrated antenna is in conclusion analyzed in the detection of typical body movements, in comparison with more accurate accelerometric data.

II. ANTENNA LAYOUT AND DESIGN PROCEDURES A rectangular plate is folded (Fig. 1) around a dielectric slab and the longest face is placed over the body through of height an optional dielectric insulator slab of thickness . Unlike the shunt-fed conventional PIFA, this geometry can be viewed as a series-fed “L”-patch. An optional strategy to further improve the decoupling with the body, could be the design of a lower and ). plate slightly wider than the upper one ( The RFID microchip will be attached in the middle of the slot’s central gap. The radiation (Fig. 2) is produced mostly by the slot and of the the patch’s open edge. Assuming that the thickness inner dielectric is small compared with the wavelength, the radiation from the folding may be considered negligible and the gain and matching features of the antenna are mainly related to the slot and to the transmission line truncation. The polarization

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Fig. 2. Near field distribution of the proposed wearable antenna. The radiation is maximum in correspondence of the central slot and the open edge (in opposite phase) and minimum close to the folding. As for conventional patches the fields along the external vertical sides (x axis in the figure) are in opposite phase, thus do not greatly contribute to the radiation.

is linear, parallel to the antenna main-direction ( axis in the figure). As for conventional patches, the increase in the horizontal produces a gain enhancement. Depending on the posize sition of the tag on the body, and on the available space, it is possible to increase that dimension in order to achieve better radiation performance. The length of the patch is chosen ap, where is the effective wavelength in proximately equal to the dielectric substrate. While the size of the slot’s central gap is mainly fixed by the microchip packaging and by the eventual sensing electronics, different shape-factors and positions may be instead considered for the matching slot. The maximization of the read distance requires the antenna to match the conjugate microchip impedance impedance . To understand the role of the many geometrical variables on the antenna impedance and to achieve a starting guess for the design, an equivalent circuit and a parametric study are here presented. A. Circuit Model Under the hypothesis that the antenna’s lower plate is considered as an ideal infinite ground plane the input impedance of the wearable antenna can be predicted by the equivalent circuit in Fig. 3. The above assumption is reasonable if the lower plate is a little larger than the upper antenna face hosting the microchip transponder. It is worth anticipating that both simulative and experimental considerations, to be presented later on, will demonstrate that the antenna’s performance is very little sensitive to the placement on different parts of the body thanks to the previously discussed decoupling mechanisms. The structure is therefore analyzed as a microstrip transmission line truncated by a non-ideal open circuit at the first termination, by a short circuit at the other one and loaded in series by a complex-impedance element: the H-slot. A transformer’s accounts for the coupling of the H-slot to the rectturn ratio angular plate. The non ideal open circuit produces fringing field, as in conventional patch antennas and can be accounted for by an open-

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Fig. 3. Equivalent transmission line model of folded patch loaded by the H-slot. Each part of the antenna is modeled as an equivalent impedance/admittance: Z for the short circuit, Z for the non ideal open circuit, Y and G for the slot, coupled by the transformer’s turn ratio n .

ended slot with equivalent parallel admittance [14] given by Fig. 4. Variation of the transformer’s turn ratio n of the circuit model with respect to the slot dimension a and its position p, having fixed (size in [mm]) L : ,W ,b ,d ,g .

= 46 5

= 80 = 3 = 10 = 2

(1) The other short-circuit truncation of the antenna can be roughly modeled as an inductance [15] (2) where is the thickness of the conductive sheet. The H-slot could be viewed as the combination of three porslot is mainly tions of slot-lines. The horizontal associated with the coupling and the radiation through a conductance [14] (3) The two identical vertical longitudinal short-circuit slot-lines of width and length , as described in [16], host phase-reversal aperture fields, and hence they mainly store reactive energy. The slot is accounted for by the series effect of each vertical of two short-circuit stubs of length , e.g. admittance (4) are the characteristic impedance and the wave where and number of the slot-line with width calculated as in [17]. Denoting with and the admittance of the microstrip’s shorted- and open-ended termination, after transfer up to the the vertical-slot admittance microchip connection, and again transferred at the center of the slot, the total input impedance of the antenna is finally given by (5) with , . The transformer’s turn ratio is related to various antenna’s parameters such as the slot size and its position along the upper patch. is roughly equal to the fraction of the current intercepted by the aperture to the total antenna current and can be calculated numerically, for instance

as described in [18], or by means of best fitting of the numerically computed input impedance to the circuital expression in in (5). Just for example, Fig. 4 shows the dependence the case of PTFE inner dielectric and having fixed the other sizes deduced by an FDTD-simulation [19] of the whole structure. As expected, the amount of current intercepted by the H-slot, and thus the ratio, increase for large slots; moreover it is maxand imum when the slot is close to the left folding minimum in proximity of the metal plate’s open-circuit trunca. The variation of the turn ratio is well aption proximated by a bilinear polynomial fitting (with respect to and [mm]) (6) B. Parametric Analysis Fig. 5 and Fig. 6 show the variation of the tag’s input impedance versus the position and versus the shape factor of the matching slot (modified by acting only on the parameter ) when the inner dielectric is again the PTFE with the same thickness as before. The antenna reactance is inductive before the first resonance and hence this configuration is suited to achieve conjugate matching to the capacitive impedance of the microchip. Moreover, the resistance and reactance change in an opposite way with respect to and , e.g., the antenna impedance increases (the resonance moves to the lower frequencies) as the slot moves closer to the folding ( reduces) while it reduces (the resonance moves to the higher frequencies) as the H-slot becomes narrower (parameter reduces). The tag design may therefore concentrate on the optimization of the parameters having fixed the remaining ones. only The insets of Fig. 5 and Fig. 6 show the good agreement in the European RFID band of the impedance estimations from circuit model with fullwave FDTD simulation of the planar antenna. The expression in (5) can be therefore used to give a

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TABLE I PARAMETERS OF THE SIMPLIFIED LIMB MODEL AT 870 MHz

TABLE II PARAMETERS OF THE TAG PROTOTYPES IN [mm]

Fig. 5. Parametric exploration of input impedance for various slot positions in x direction (Fig. 1), having fixed (size in [mm]) L : ,W ,a , b ,d ,g . Continuous lines tag the circuit data while the dashed lines indicate the fullwave results.

= 3 = 10 = 2

= 46 5

= 80 = 20

Fig. 7. Fabricated TAG-1 (left) and TAG-2 (right) prototypes of body-worn antenna matched to Zin j microchip.

= 15 0 135

III. PROTOTYPES AND PERFORMANCES IN REAL CONFIGURATIONS

Fig. 6. Parametric exploration of the input impedance for various slot form :, factors (selected by the parameter a), having fixed (size in [mm]) L W ,p ,b ,d ,g . Continuous lines tag the circuit data while the dashed lines indicate the fullwave results.

= 80 = 25 = 3 = 10 = 2

= 46 5

first approximation for slot’s sizes and position such to achieve the impedance matching with the microchip, in view of using a fullwave electromagnetic solver to refine the geometrical parameters.

Two prototypes of this class of tags have been designed, fabricated and tested in real conditions. The antennas’ matching is referred to a low impedance NXP microchip transponder with . The final antenna design has impedance been refined by including into the FDTD simulation also a rough model of human limb consisting of a stratified box of height 40 cm (parameters in Table I). The resulting fabricated prototypes, of overall size 6 6 cm (TAG-1) and 6 9 cm (TAG-2), (other parameters in Table II) are shown in Fig. 7. The insulating dielectric, contacting the body, is a thin adhesive PVC film. TAG-2 is expected to have a higher gain in comparison with v.s. , as estimated by TAG-1 ( FDTD) thanks to the larger , and to the wider ground plane which prevents the antenna radiation to be absorbed into the highly-dissipative human body. Two different experimental characterizations of the tags’ performances are here presented. The antenna design is first checked in chipless modality by the measurement of input used to calculate the power transmission impedance coefficient (7)

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Fig. 10. Simulated and measured power transmission coefficient  for the two antennas matched to Z j . Left: TAG-1. Right: TAG-2.

= 15 0 135

Fig. 8. Antenna plus balun connection. The external conductor of the coaxial line, coming out of the balun, is soldered to the antenna face up to the slot, while the internal conductor is soldered to the other slot’s edge.

Fig. 11. Measurement setup comprising the short-range reader, the measurement trick and the absorbing panels. The antenna is here placed in the center of the human torso.

Fig. 9. FDTD simulated input impedance of TAG-1 with and without the presence of the balun.

The RFID link performance is instead fully analyzed having attached the microchip at the antenna port and by estimating the maximum read-distance in controlled conditions, as explained in details later on. A. Chipless Measurements: Matching Features The tags’ impedance has been measured by means of a Vector Network Analyzer, VNA (Anritsu MS2024A) probe connected to the slot mid-point through a bazooka balun having the purpose to prevent unbalanced currents from the probe to flow on the outer surface of the coaxial cable [14]. An approximately metal sleeve, shorted at one termination encapsulates the coaxial probe (Fig. 8). The input impedance measured by the VNA will be hence the tag impedance itself without artifacts. As a proof, Fig. 9, shows a comparison between the simulated tag without cable and balun, and the also simulated impedance in the measurement condition. As visible the input impedance of the tag plus the balun, estimated by FDTD, is practically unchanged, at least in the RFID band, with respect to the standalone antenna. The antenna has been attached over the leg of a volunteer and the measurement of impedance, after de-embedding of the coaxial connector, gives a power transmission coefficient (at 869 (TAG-1) and (TAG-2) MHz) of the order of

(Fig. 10). It is worth mentioning that nearly identical results are obtained when the tags are placed onto different body segments, such as the torso and the arm, as shown in the next paragraph concerning the measurement of the realized gain. B. Realized Gain The realized gain of the tags, e.g. the radiation gain of the antenna reduced by the impedance mismatch, has been indirectly estimated for TAG-1 and TAG-2 by using the set-up in Fig. 11 comprising a short-range, remotely controlled reader CAEN A528, and a quarter-lambda patch (PIFA) with maximum gain 3.3 dB, as reader’s antenna. Under the free-space assumption, the power delivered by the reader to the tag, placed at mutual distance , is given by the Friis formula (8) where is the gain of the reader antenna, is the gain of the is the power accepted tag’s antenna, placed on the target. by the antenna of the reader unit. The polarization mismatch between the reader and the tag is here considered unitary since they have been properly aligned in all the measurements. is the realized gain of the tag. Equation (8) has been verified to hold also in a real environment if the measurement set-up is far from the side walls, the and absorbing panels are distance is small enough placed on the ground to reduce multipath. In this case by increasing the reader’s power until the tag starts to respond, the collected power at turn-on equals the chip sensitivity,

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Fig. 14. MS24M motion-vibration sensor design. (a) Bottom view; (b) Side view; (c) Longitudinal section with the inner conductive structure and the switch mass [21].

G

Fig. 12. Measured ^ for the antenna placed on the torso of the volunteer (top view schematically represented in the origin of the polar graph).

The radiation performance is hence nearly the same for the two positions, confirming that the antenna is very little sensitive to the body position. It is worth mentioning that the tags still retain similar performance in free space conditions. The experienced maximum read distance, by using the shortrange reader (emitting not more than 0.5 W EIRP), and tags’ , was 1.5 m for the microchip with typical TAG-1 and 2.1 m for the TAG-2. However, by using a longrange reader (emitting up to 3.2 W EIRP) the maximum read distance estimated from (8) could reach 4 m for the smaller design and 5.5 m for the larger one. IV. SENSOR INTEGRATION It is here shown how the tag design procedure can be modified to take the presence of a specific sensor into account. As an example, a very simple mechanical motion sensor is considered, and a fully integrated wearable sensor RFID tag is designed, fabricated and hence experimentally evaluated. The detection of the motion, in particular, is of great interest in medical application, to assist the diagnosis of some neurological diseases, involving compulsory arms movements [20], in domestic healthcare, to track the behavior of elderly, but also in logistic and security to control limited-access areas. A. Omnidirectional Motion Sensor

G

Fig. 13. Measured ^ for the antenna placed on the left arm of the volunteer (top view schematically represented in the origin of the polar graph).

, and hence the realized gain can be estimated by inverting (8), when all the other parameters are known. is shown in Fig. 12 and Fig. 13 for the tags The measured placed onto two different body regions, the torso and the left arm. The tags are attached onto the body such that the antenna polarization ( axis in Fig. 1) is parallel to the body’s longituhas been evaluated along the two principals didinal axis. rections ( - and -axis in the figures) by body rotation of 90 , 180 and 270 . As expected, the realized gain is maximum in front of the antenna while it is minimum in the rear side, due to the human body absorption. However both tags are readable in the back direction when placed on the arms. The maximum effective gain for the TAG-1 ranges between 4 dB and 3 dB depending on the body positions, while better performances are achieved by TAG-2, thanks to its larger size, with maximum realized gain ranging between 2 dB and 1 dB. These results are in full agreement with the design data.

The sensor used here is a two terminal omnidirectional switch (Fig. 14) especially designed for the detection of movements and vibrations [21]. When disturbed from its quite condition, it produces fleeting changes of its equivalent impedance state, e.g. open to close or vice versa, and if properly conditioned to the antenna microchip, it may consequently enable or deny the RFID communication. One of the two pins of the steel-gold plated capsule is connected to the external case of the sensor while the other one is isolated from the outer part of the capsule and connected to the inner switching structure (Fig. 14(c)). The switching structure comprises a dumbbell-like conductive element connected to the central pin and a conductive sphere, free to move inside the capsule. The sensor has two possible states. In state A the internal sphere touches at a same time the inner and the outer conductors of the sensor thus shorting the output pins. In state B, the sphere does not connect the structures and the circuit remains approximately open. State A is stable while state B is instable: at rest the switch is preferably in state A and during the movement it randomly changes between A and B varying its input impedance.

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Fig. 16. Particular of the prototype of the TAG-2 Motion Sensor. The inertial switch has been soldered in series to the microchip within the slot central gap. Fig. 15. Measured input Impedance of the motion sensor measured with a VNA probe connected to the switch by means of a modified SMA connector.

The sensor will be connected in series with the microchip and hence the antenna design requires to properly account for the presence of the sensor, e.g., the conjugate matching condition becomes (9) In this choice, the reader will receive the tag ID when the tag is at rest and does not receive anything if the tag is subjected to motion. The basic principle is a form of ID-modulation introduced in [22]. The RF impedance of the switch is not provided by the manufacturer and hence it has been measured with a VNA probe connected to the switch by means of a modified SMA connector (Fig. 15 inset). To avoid the unbalancing effects of the VNA coaxial cable, the capsule has been soldered directly all along the connector flange and its central pin has been inserted in the SMA inner conductor. At rest (ideally a short circuit), the sensor’s measured , thereimpedance at 870 MHz is fore showing a practically inductive reactance. The switch’s impedance in state B is not easily measurable. During the motion, the sphere randomly moves inside the capsule varying the sensor’s impedance without regularity. Basically when the sphere does not touch the sensor walls the resulting impedance is expected to be capacitive with value depending on the instantaneous sphere-wall distance. B. RFID Motion Sensor Prototype A prototype of the wearable Motion Sensor (Fig. 16) comprises a modified version of the TAG-2, with a slightly different slot size in order to achieve the matching condition in (9) having considered the sensor in series to the chip. It is worth noticing that the slot tuning has been accomplished by varying the only has been slot shape factor (the vertical dimension changed from 18 mm to 16 mm), leaving unaltered all the other parameters. In order to easily solder the inertial switch, a pack-

aged version of the microchip used instead of the strap version.

has been

C. Experimentation and Results The proposed RFID motion sensor has been tested in real conditions in order to verify the effective communication and sensing performances. The movements have been also recorded by 3-axis MEMS motion sensor (LIS302DL [23]), able to measure the accelerations on the three orthogonal axis up to (with gravitational acceleration) with a sampling rate up to 400 Hz. The MEMS sensor has been placed behind the RFID tag in order to be affected by the same acceleration of the RFID device. The measurement setup is visible in Fig. 17. Both MEMS sensor and RFID Motion sensor have been placed on the arm and a sixteen-movements sequence has been executed moving the arm randomly. Fig. 18 shows the module of the recorded MEMS sensor vectorial data (a) and the on/off data received at the reader (b), where the bars indicate the state B (motion). The reader-tag distance and the interrogation power are such that the RFID link may be in principle established for any position of the arm. A significant correlation is visible between the two motion sensors, in term of number of movements, time and duration. In particular, the RFID Motion Sensor is able to monitor every body event, regardless its standing or magnitude and, when placed onto the chest, it revealed also sensitive to very weak movements such as those produced by deep breath and couch. V. CONCLUSIONS The analytic model and the detailed experimentations have demonstrated that the proposed family of wearable tags is a good candidate to the monitoring of people in conventional indoor and outdoor area. Thanks to the particular folded geometry, the structure is not much influenced by the detuning and by the absorbing effects produced by the human body. Thanks to the slot, it offers some degree of freedom in the impedance matching, useful to integrate passive sensors. Further improvements will concern the realization of flexible conformal prototypes based on the textile technology and the in-

OCCHIUZZI et al.: MODELING, DESIGN AND EXPERIMENTATION OF WEARABLE RFID SENSOR TAG

Fig. 17. Measurement setup comprising the short-range reader, the RFID tag and the LIS302DL accelerometer placed beside the tag. Both the RFID motion sensor and the accelerometer are placed on the arm. The MEMS accelerometer data is transmitted via a WIFI module.

Fig. 18. Comparison of data returned by the LIS302DL accelerometer (a) with the tag response received from the TAG2 Motion Sensor (b): the motion events (state B), for which the microchip does not respond, are indicated by bars.

tegration of a second “control” microchip, whose ID should be received in any condition, revealing the presence and the identity of the tag, leaving to the sensor-conditioned microchip the only duty to communicate the “state” of the tagged object. ACKNOWLEDGMENT The authors wish to thank CAEN for technical support with the reader programming and NXP for providing RFID dies. Special thanks to F. Amato and S. Caizzone for their enthusiastic and valuable support in performing experiments and tests.

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[4] L. Yang, R. Vyas, A. Rida, J. Pan, and M. M. Tentzeris, “‘Wearable RFID-enabled sensor nodes for biomedical application’,” presented at the Electronic Components and Technology Conf., Lake Buena Vista, FL, 2008. [5] J. Park, J. Seol, and Y. Oh, “Design and implementation of an effective mobile healthcare system using mobile and RFID technology,” in Proc. 7th Int. Symp. HEALTCOM, 2205, 2005, pp. 263–266. [6] P. S. Hall and Y. Hao, Antennas and Propagation for Body-Centric Wireless Communications, 1st ed. Norwood, MA: Artech House, 2006. [7] L. Ukkonen, M. Schaffrath, D. W. Engels, L. Sydänheimo, and M. Kivikoski, “Operability of folded microstrip patch-type tag antenna in the UHF RFID bands within 865–928 MHz,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 414–417, 2006. [8] M. Hirvonen, K. Jaakkola, P. Pursula, and J. Saily, “Dual-band platform tolerant antennas for radio-frequency identification,” IEEE Trans. Antennas Propag., vol. 54, no. 9, p. 2632, Sep. 2009. [9] S. L. Chen and K. H. Lin, “A slim RFid tag antenna design for metallic object applications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 729–732, 2008. [10] G. Marrocco, “RFID antennas for the UHF remote monitoring of human subjects,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1862–1680, June 2007. [11] C. Calabrese and G. Marrocco, “Meandered-slot antennas for sensorRFID tags,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 5–8, 2008. ˇ [12] M. Polívka, M. Svanda, and P. Hudec, “UHF RFID of people,” in Development and Implementation of RFID Technology. Vienna: I-Tech Education and Publishing, 2009, ch. 4. [13] L. Yang, L. Martin, D. Staiculescu, C. P. Wong, and M. M. Tentzeris, “Conformal magnetic composite RFID for wearable RF and BIO-monitoring applications,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 3223–3230, Dec. 2008. [14] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [15] R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip Antenna Design Handbook. Boston, MA: Artech House, 2001. [16] M. El Yazidi, M. Himdi, and J. P. Daniel, “Transmission line analysis of nonlinear slot coupled microstrip antenna,” Electron. Lett., vol. 28, no. 15, pp. 1406–1408, 1992. [17] M. Himdi and J. P. Daniel, “Characteristics of sandwich slot lines in front of parallel metallic strip,” Electron. Lett., vol. 21, no. 5, pp. 455–457, 1991. [18] J. P. Kim and W. S. Park, “‘Analysis and network modeling of an aperture-coupled microstrip patch antenna’,” IEEE Trans. Antennas Propag., vol. 49, no. 6, pp. 849–854, 2001. [19] G. Marrocco and F. Bardati, “BEST: A finite-difference solver for time electromagnetics,” Simul. Practice Theory, no. 7, pp. 279–293, 1999. [20] W. Hening, “The clinical neurophysiology of the restless legs syndrome and periodic limb movements. Part I: Diagnosis, assessment, and characterization,” Clin. Neurophys., vol. 115, pp. 1965–1974, 2004. [21] MS24M Product Data Sheet Comus Group of Companies [Online]. Available: www.comus-intl.com [22] M. Philipose, J. R. Smith, B. Jiang, A. Mamishev, S. Roy, and K. Sundara-Rajan, “Battery-free wireless identification and sensing,” IEEE Pervasive Comput., vol. 4, no. 1, pp. 37–45, 2005. [23] “LIS302DL, MEMS Motion Sensor Data Sheet,” STMicroelectronics [Online]. Available: www.st.com

REFERENCES [1] S. Nambi, S. Nyalamadugu, S. M. Wentworth, and B. A. Chin, “Radio frequency identification sensors,” in Proc. 7th World Multiconf. Systemics, Cybernetics and Informatics (SCI 2003), 2003, pp. 386–390. [2] L. Cheng-Ju, L. Li, C. Shi-Zong, W. C. Chen, H. Chun-Huang, and C. Xin-Mei, “Mobile healthcare service system using RFID,” in Proc. IEEE Int. Conf. Networking Sensing and Control, 2004, vol. 2, pp. 1014–1019. [3] R. S. Sangwan, R. G. Qiu, and D. Jessen, “Using RFID tags for tracking patients, charts and medical equipment within an integrated health delivery network,” in Proc. IEEE Int. Conf. Networking Sensing and Control, 2004, pp. 1070–1074.

Cecilia Occhiuzzi received the M.Sc. degree in medical engineering from the University of Rome “Tor Vergata” where she is currently working toward the Ph.D. degree. In 2008, she was at the School of Engineering, University of Warwick, U.K., as a Postgraduate Student. Her research is mainly focused on wireless health monitoring by means of radiofrequency identification (RFID) and UWB techniques.

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Stefano Cippitelli received the Laurea degree in Telecommunications Engineering from the University of Rome “Tor Vergata,” in 2009. His main scientific interest concerns the design of antenna systems for RFID applications. In spring 2008, he was at Tampere University of Technology, for the advanced course “Design and Characterization of Passive RFID Systems.” He is currently employed at SIA, Torino, Italy, working on BTS design.

Gaetano Marrocco was born on August 19, 1969, in Teramo, Italy. He received the Laurea degree in electronic engineering and the Ph.D. degree in applied electromagnetics from the University of L’Aquila, Italy, in 1994 and 1998, respectively. He has been a Researcher at the University of Rome “Tor Vergata” since 1997 where he currently teaches antenna design and bioelectromagnetics. In summer 1994, he was at the University of Illinois at Urbana Champaign, as a Postgraduate Student. In autumn 1999, he was a Visiting Scientist at Imperial College in London. His research is mainly directed to the modelling and design of broadband and ultrawideband antennas and arrays as well as of miniaturized antennas for RFID applications. He has been involved in several space, avionic, naval and vehicular programs of the European Space Agency, NATO, Italian Space Agency, and the Italian Navy. He holds two patents on broadband naval antennas and one patent on sensor RFID systems. Prof. Marrocco currently serves as an Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

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Low Cost Planar Waveguide Technology-Based Dielectric Resonator Antenna (DRA) for Millimeter-Wave Applications: Analysis, Design, and Fabrication Wael M. Abdel Wahab, Student Member, IEEE, Dan Busuioc, and Safieddin Safavi-Naeini, Member, IEEE

Abstract—A compact, low cost and high radiation efficiency antenna structure, planar waveguide, substrate integrated waveguide (SIW), dielectric resonator antennas (DRA) is presented in this paper. Since SIW is a high Q- waveguide and DRA is a low loss radiator, then SIW-DRA forms an excellent antenna system with high radiation efficiency at millimeter-waveband, where the conductor loss dominates. The impact of different antenna parameters on the antenna performance is studied. Experimental data for SIW-DRA, based on two different slot orientations, at millimeter-wave band are introduced and compared to the simulated HFSS results to validate our proposed antenna model. A good agreement is obtained. The measured gain for SIW-DRA single element showed a broadside gain of 5.51 dB, 19 dB maximum cross polarized radiation level, and overall calculated (simulated using HFSS) radiation efficiency of greater than 95%. Index Terms—Dielectric resonator antenna (DRA), millimeterwave (mmW) , printed circuit board (PCB), radiation efficiency, substrate integrated waveguide (SIW), waveguide.

I. INTRODUCTION

A

S THE DEMAND for wireless communications increases, cheaper and more reliable systems have to be developed. At the same time, constraints placed on the systems by spectrum allocation issues increase. A considerable amount of research is being devoted to the development of systems designed to work in the millimeter-wave (mmW) frequency range. This frequency band (30–300 GHz [1], [2]) is vastly unused compared to lower frequency bands which are heavily populated. Developing the communication systems at mmW frequency band allows compact system components, which are critical for mobile and portable systems. Furthermore, the mmW bands provide a greater bandwidth, meaning large amounts of data can be transmitted at higher speed with better reliability. In addition, Wireless Local Area Networks (WLANs) and other high-speed multimedia delivery services

Manuscript received August 31, 2009; revised December 27, 2009; accepted February 02, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported in part by the ERI (Egypt), RIM (Canada), and in part by NSERC (Canada). The authors are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050443

require large amounts of bandwidth to operate effectively. An important application for this frequency band is satellite communications. A portable system with fixed beam around 20/30 GHz for receive and transmit respectively has been investigated in [3]. Emerging low cost 60 GHz technologies and standard (IEEE 802.15 TG3c) have paved the way for mass production of mmW in a number of innovative mass market applications such as live HD video streaming and multi Gbps WAN. Conventional planar antennas made of thin film metallic patterns in a multi-layer medium suffer problems with regards to power loss, which degrades radiation efficiency, radiated power capabilities and fabrication difficulties when reduced to the sizes necessary to operate in this band. Dielectric resonator antennas (DRAs) have been shown to radiate efficiently at high frequency [4], making them more attractive for wireless applications operating at mmW. They exhibit less conductor loss (high radiation efficiency), larger bandwidth than microstrip patch antennas (MPAs), and have a lower profile than reflector and horn antennas. As compared to planar (multilayer) antennas, using current technologies DRAs are more difficult and costly to fabricate. However, in this paper authors are proposing a simple and low cost printed board fabrication method for DRA element at mmW frequency range. Many different feeding mechanisms can be used for DRA such as coaxial-probe feed [5], [6], aperture- coupling associated with microstrip line feed [7], [8], direct microstrip-line feed [9], [10], coplanar waveguide feed [11], [12], and other feeding structures. However, feeding losses of theses excitation methods are considerable at mmW. A new and low loss excitation scheme that employs rectangular waveguide is reported in [13], [14]. However, it is bulky, expensive not compatible with monolithic microwave integrated circuits (MMIC) technology and low profile applications. On the other hand, substrate integrated waveguide (SIW) [15], [16] which is a promising approach for future design and development of low cost millimeter wave RF circuits using conventional printed circuit board (PCB) technology, provides a compact, high-Q, and low profile feeding structure. SIW is a good compromise between air field rectangular waveguide and microstrip line. Accordingly, it can minimize radiation loss and parasitic radiation. Antennas based on SIWs with operation frequency range up to Ka- band have been realized using standard PCB processes [17]–[19].

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TABLE I RDRA POSSIBLE DIMENSIONS (mm) FOR "

= 10:20 AT f = 30 GHz.

Fig. 1. Rectangular dielectric resonator antenna (RDRA) resides on infinite ground plane.

In this paper, the SIW concept is utilized as a novel feeding mechanism for dielectric resonator antenna (DRA) for low cost and high radiation efficiency antenna systems for mmW applications. Modeling, characterization, design and fabrications of SIW-based DRA are presented. A rectangular dielectric resonator antenna (RDRA) shown in Fig. 1 is designed to resonate for radiation at mmW frequency. A theoretical model combined with a numerical analysis of the modes, using HFSS Eigen Mode Solver, has been utilized to model and design this DRA. Then, a complete model of the proposed antenna consisting of RDRA excited by a narrow slot cut on the broad wall of SIW is proposed along with parametric analysis of the effect of each parameter on the antenna performance. II. RDRA THEORETICAL MODEL AND MODAL ANALYSIS The mode resonant frequency of the RDRA can be easily determined using dielectric waveguide model (DWGM) described in [20]. For convenience, these equations are written as (1) (2) (3) where is the dielectric constant of the resonator, is the speed of light, and the lengths , , are shown in Fig. 1. Alternatively, these equations may be used to determine the parameters of the resonator for a desired frequency of operation. Based on the solutions of (1)–(3), different possible RDRA dimensions can be obtained for the same reso, Table I lists some of possible nant frequency of RDRA dimensions for commercial available standard heights for . The HFSS software package eigenmode solver is used to verify the natural resonance behavior of RDRA. Modal analysis is carried out for different RDRA dimensions. The model consists of RDRA, which resides on infinite ground plane. As shown in Table I, there is a good agreement between the DWGM result and that of the HFSS eigen-

Fig. 2. Vector field distribution plots of isolated RDRA resides on infinite ground plane. (a) Electric field vector plot. (b) Magnetic field vector plot.

mode solution with the PML boundary condition in predicting the mode resonance frequency. For our reference, Fig. 2

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Fig. 4. The impact of the slot position = 2 mm, = 0 40 mm, and

L

Fig. 3. SIW- based RDRA driven model. (a) Using horizontal polarized (transverse) slot. (b) Using vertical polarized (longitudinal) slot.

shows the field’s vector plots of this mode. In order to strongly excite this mode, a source (slot) should be located at a strong field (magnetic) to ensure a good coupling III. SIW FED RDRA MODEL A narrow transverse slot cut on the SIW broad wall forming two different polarizations (horizontal and vertical) is used to excite the RDRA for radiation as shown in Fig. 3(a) and (b). The presented SIW structure consists of two integrated rows of and separated by (SIW metallized vias of diameter apart. The SIW width) and each two neighboring vias are and a thickness . The substrate has a dielectric constant of SIW parameters are chosen to minimize the guided wave transis chosen for mission losses [16]. Besides, the SIW width mode operation. The SIW width is single fundamental calculated by using the equivalent rectangular waveguide concept whose width is given by [21]:

(4)

W

:

X (mm) variation on S y = 0.

(dB),

The SIW substrate thickness and the dielectric constant are selected to provide a compact design. Furthermore, the , , and the position, , slot’s dimensions, should be optimized in order to efficiently couple the energy , is chosen to be from the SIW to the RDRA. Initially, for Fig. 3(a), and ( ( for Fig. 3(b), which correspond to a maximum standing wave field intensity (maximum coupling). Rogers RT5870 dielectric substrate with a is used for substrate in this design. thickness of . The metallic The SIW has a width of vias have a diameter of and are separated by . The RDRA element is made of Rogers with a thickness RT6010 . The RDRA has a length , and a width , respectively and symmetrically . centered with respect to slot IV. CHARACTERIZATION AND PARAMETRIC STUDY Some design parameters may affect the coupling between the , slot and the RDRA. These parameters are slot position , slot width and the RDRA relative posislot length tion to slot . These parameters have to be optimized to obtain maximum energy coupling between both the slot and RDRA. To demonstrate the impact of this effect, we studied the case in Fig. 3(a), whereas, the optimized parameters in Fig. 3(b) will be implicitly included in the final design. A. Slot Position

Characterization

The position of the transverse slot on an SIW board wall controls the amount of coupled power to the RDRA. The center position (along y-axis) on the SIW broad wall, and at (along x-axis) guarantee a maximum excitation to RDRA’s fundamental mode (maximum-coupled power). Since the effective dielectric constant seen by the slot is not known accurately, the guided wave length was calculated . This based on the average dielectric constant . The slot position from the short cirmeans that cuit was varied with an initial slot width , slot

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Fig. 5. The impact of slot length L variation on 2:50 mm, W = 0:25 mm, and y = 0.

S

(dB),

X

=

length , and DRA offset .Fig. 4 shows the effect of changing the on the Reflection coefficient frequency response (28 GHz–40 GHz). The (coupling) position changes the amount of coupling to the RDRA, . where the maximum coupling occurs at This means that the RDRA resonates (minimum reflection coefficient at . This resonance frequency differs from the designed frequency 30 GHz. This frequency shift can be accounted for as follows: The DWGM that is used on our analysis assumed that the RDRA resided on an infinite ground plane and it did not take the slot effect into consideration. In other words, we have replaced the isolated RDRA’s height with its half. However, due to the slot effect, the assumption of using half of RDRA’s to as frequency height will not be accurate and it may shift (the RDRA resonance frequency is height dependent). B. Slot Length

S

(dB),

X

=

Fig. 7. The impact of the slot- RDRA relative position y variation on S (dB), L = 3:20 mm, W = 0:30 mm, and X = 3:50 mm.

Characterization

The effect of slot length was studied by varying the length in the vicinity of , while keeping , , and constant. As the slot length was of the DRA increased. The effect of placing shortened, the RDRA on the top of slot is to change the slot impedance so that it is matched at frequency other than its natural resonance frequency as shown in Fig. 5. Although the resonance is noticeable for each slot length, the match is best when .A significant portion of energy coupled into RDRA at different frequencies which give a good indication of its broadband behavior and the slot length tuning property. C. Slot Width

variation on Fig. 6. The impact of slot width W : W = 0:25 mm, and y = 0.

2 50 mm,

Characterization

The effect of varying the slot width was studied using slot length of . Increasing the slot width has the effect of increasing both the resonance of RDRA and the bandwidth . Namely, The RDRA resonance frequency changes from low frequency 36.64 GHz when the slot width is 0.20 mm to a high frequency 37.43 GHz when the slot width is 0.40 mm. The simulated bandwidth increases from 5% at 35.64 GHz to 11.5% at 37.43 GHz, as shown in Fig. 6.

D. RDRA-Slot Relative Position

Characterization

Finally, the RDRA offset was studied using , and to determine the optimum placement of RDRA above the slot. The simulated results are presented in Fig. 7. Offsetting the RDRA from the center of of the the slot not only shifted the resonance frequency antenna but also increased the impedance bandwidth. When it was offset by in either direction has been increased to approximately 38.14 GHz which corresponds to reflection and bandwidth of . This means that of for any potential error introduced by not perfectly centering RDRA, we still have a good matching performance. Further offset to 1 mm will degrade the impedance bandwidth due to the inefficient coupling. On the hand, a very good coupling and impedance matching was obtained at (no offset) as expected. V. SIW-TO MICROSTRIP LINE TRANSITION (SIW-MSL) A transition from microstrip (MSL) to SIW which can be integrated on the same substrate [22] has been designed and optimized using HFSS software to in-band insertion loss of better

ABDEL WAHAB et al.: LOW COST PLANAR WAVEGUIDE TECHNOLOGY-BASED DRA FOR mmW APPLICATIONS

Fig. 8. Reflection coefficient and insertion loss (dB) of SIW-MSL transition.

TABLE II PHYSICAL OPTIMIZED SIW-DRA PARAMETERS (DIM. IN mm).

than 0.60 dB. The advantages of SIW-MSL transition is its planar form and wide frequency band. The designed SIW-MSL transition is used to couple the energy from the SMA edge connector to the SIW-DRA antenna for measurements. The simulated return and insertion losses for SIW-MSL transition back to back is shown in Fig. 8. They have been calculated and were (simulated) from 28 to 40 GHz. Its parameters optimized in order to match MSL line fundamental mode to the fundamental mode. waveguide VI. ANTENNA PERFORMANCE: SIMULATIONS AND MEASUREMENTS The proposed SIW-fed RDRA antenna models shown in Fig. 3 are simulated and optimized using Ansoft HFSS. Table II

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shows the optimized physical parameters for both slot based cases. The antenna modules designed previously, have been fabricated and measured in order to verify and validate the proposed model. They are fabricated using standard low cost PCB technique using a low loss and low cost material is used for SIW substrates. Rogers 5870 is used for substrate, and Rogers 6010 RDRA substrate. The manufacturing of RDRA antenna is done using a multistep, multilayer printed circuit board (PCB) manufacturing process. The board is treated as a three-layer PCB; with two dielectric layers comprising the SIW fed layer, and DR elements, respectively. First, the bottom layer which will contains the SIW feed is fabricated from a double- sided 31- mil Roger/5870 . Copper plating is 0.5 oz Duroid material which is the minimal necessary to allow thin metallization and allow for good slot coupling. The planar features are etched and the substrate is drilled through and plated to create the SIW feed. Next, a one- sided , with 0.5 oz 50 mil Roger RT/6010 substrate copper metallization is bonded using 2- mil thermally stabi. This is an irradiated polyolefin lized bonding film co-polymer developed to achieve excellent bond strength with low- flow characteristics and low temperature bonding. The top metal layer is etched away leaving a complete sheet of dielectric material. The majority of the area is milled away and what is left, are the RDRA blocks which remain bonded to the underlying layer. The only downside of this process is that there is a 2- mil substrate between the top metal layer of SIW feed and bottom of the RDRA which can affect the performance. Measurements were performed to determine the Reflection coefficient and the radiation pattern (gain) in both E-plane and H-plane of the antenna. The obtained results show a good agreement has been achieved taking into account the manufacturing tolerances as well as the SMA connector misalignment and poor contact used. The fabricated antenna prototypes are shown in Figs. 9 and 10 for both slot orientations. 1) Antenna Reflection Coefficient: The Reflection coefficient of the fabricated SIW-RDRA was obtained experimentally and compared with the simulated results from 30 GHz to 40 GHz as provided in Fig. 11(a) and (b), for both slot orientations respectively. Fig. 11(a) (for antenna prototype in Fig. 9) shows a good agreement between the measured and simulated for simularesults. The resonance which occurs at primarily due to fabrication toltions is shifted to erances and the addition of bonding material between the feed and the DRA layers. This novel multi-layer fabrication method affects the coupling to the DRA and will be the subject of a future paper discussion. Furthermore, the reduction in bandwidth is attributable to connector contact and alignment issues. This will be addressed in a future manufacturing revision of the antenna designs. The , measured minimum reflection coefficient, which is occurs at 35 GHz. On the other hand, the measured Reflection coeffi(for antenna prototype in Fig. 10) shown in cient

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Fig. 9. SIW -Based RDRA using vertical polarized (transverse) slot. (a) Front view. (b) Rear view.

Fig. 11. Simulated and measured reflection coefficient S (dB) of SIW-based RDRA. (a) Antenna prototype in Fig. 9. (b) Antenna prototype in Fig. 10.

Fig. 10. SIW-based RDRA using horizontal polarized (longitudinal) slot. (a) Front view. (b) Rear view.

Fig. 11(b) agrees well with the simulated result. The simulated antenna input impedance (referenced to the MSL) shown in Fig. 12 resonates at frequency 37.28 GHz where (A matching circuit is needed to input resistance is achieve a good coupling between the antenna and the SMA connector). Table III summarizes the antennas Reflection coefficient characteristics. 2) Antenna Radiation Pattern (Gain) and Efficiency: Radiation patterns (gain) measurements for the two antenna prototypes shown in Figs. 9 and 10 respectively were carried out in an anechoic chamber. Measured and simulated radiation patterns (gain) for the proposed antenna prototypes at frequencies 35 GHz and 37 GHz within scanning angle range in two orthogonal planes, are plotted in Figs. 13 and 15 respectively. Both the simulated and the measured results are in a very

Fig. 12. Calculated (simulated) input impedance of SIW-DRA single element based vertical polarized SIW slot arrangement.

good agreement and show a boresight gain of 5.51 dB (antenna prototype in Fig. 10) and 4.75 dB (antenna prototype in Fig. 11).

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TABLE III SIW-BASED RDRA BASIC CHARACTERISTICS

Fig. 14. Calculated (HFSS) antenna gain (dB) and radiation efficiency of SIWbased RDRA variation with frequency (GHz) (antenna prototype in Fig. 9).

Fig. 15. Simulated and measured radiation pattern of SIW based-RDRA (an(a) H-plane (x z plane), (b) E- plane tenna prototype in Fig. 10) f (y z plane).

@ = 35 GHz

Fig. 13. Simulated and measured radiation pattern of SIW based -RDRA (an. (a) E-plane (y z plane). (b) H- plane tenna prototype in Fig. 9) f (x z).

@ = 35 GHz

The measured far fields co-polarized (ECO, HCO) and cross polarized (EX, HX) radiation of the proposed prototype plane) and H in Fig. 10 @ 37 GHz in two planes, E ( ( -plane), shown in Fig. 15, show a very low level crosspolarized radiation .

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Fig. 16. Calculated (HFSS) antenna gain (dB) and radiation efficiency of SIWbased RDRA variation with frequency (GHz) (antenna prototype in Fig. 10).

Thanks to SIW guiding structure The SIW-DRA prototypes provides a very high simulated overall efficiency (greater than 95%) within a wide frequency bandwidth. Furthermore, the SIW-DRA antenna is a compact and small size that is quite suitable for portable applications. The antenna gain as well as radiation efficiency variation with frequency are shown in Figs. 14 and 16. Table III summarizes the common overall antenna radiation pattern (gain) and efficiency characteristics.

VII. CONCLUSION This work presents modeling, characterizations, and design procedure of a low manufacturing cost and high radiation efficiency planar waveguide based dielectric resonator antenna for mmW application. The SIW feed is proposed for the RDRA excitation. This idea is demonstrated by using two different coupling slot orientations to excite the RDRA in its fundamental for radiation. A parametric study (characterizamode tion) for all antenna parameters has been conducted to optimize the coupling between the SIW and RDRA. Then, SIW based -RDRA antenna modules are fabricated using a low cost multi-layer printed circuit board (PCB) technology. The measured results for the antenna prototypes are compared to the simulated ones. A good agreement is achieved.

[6] S. Long, M. McAllister, and S. Liang, “The resonant cylindrical dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. 31, pp. 406–412, 1983. [7] K. W. Leung, “Analysis of aperture-coupled hemispherical dielectric resonator antenna with a perpendicular feed,” IEEE Trans. Antennas Propag., vol. 48, pp. 1005–1007, 2000. [8] A. A. Kishk, A. Ittipiboon, Y. M. M. Antar, and M. Cuhaci, “Slot excitation of the dielectric disk radiator,” IEEE Trans. Antennas Propag., vol. 43, pp. 198–201, 1995. [9] R. A. Kranenburg and S. A. Long, “Microstrip transmission line excitation of dielectric resonator antennas,” Electron. Lett., vol. 24, pp. 1156–1157, 1988. [10] K. W. Leung, K. Y. Chow, K. M. Luk, and E. K. Young, “Low profile circular disk DR antenna of very high permittivity excited by microstrip antenna,” Electron. Lett., vol. 33, pp. 1004–1005, Jun. 1997. [11] R. A. Kranenburg, S. A. Long, and J. T. Williams, “Coplanar waveguide excitation of dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 39, pp. 119–122, 1991. [12] M. S. A. Salameh, Y. M. M. Antar, and G. Seguin, “Coplanar-waveguide-fed slot-coupled rectangular dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 50, pp. 1415–1419, 2002. [13] K. W. Leung and K. K. So, “Waveguide-excited dielectric resonator antenna,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2001, vol. 2, pp. 132–135. [14] K. W. Leung and K. K. So, “Rectangular waveguide excitation of dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 51, pp. 2477–2481, 2003. [15] K. Wu, D. Desland, and Y. Cassive, “The substrate integrated waveguide—A new concept for high-frequency electronics and optoelectronics,” presented at the 6th Int. Telecommunication Modern Satellite, Cable, Broadcast Service, Oct. 1–3, 2003. [16] X. Feng and W. Ke, “Guided-wave and leakage characteristics of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 66–73, 2005. [17] Y. Li, H. Wei, H. Guang, C. Jixin, W. Ke, and C. T. Jun, “Simulation and experiment on SIW slot array antennas,” IEEE Microw. Wireless Compon. Lett., vol. 14, pp. 446–448, 2004. [18] H. Wei, L. Bing, G. Q. Luo, Q. H. Lai, J. F. Xu, Z. C. Hao, F. F. He, and X. X. Yin, “Integrated microwave and millimeter wave antennas based on SIW and HMSIW technology,” in Proc. Antenna Technology Int. Workshop, 2007, pp. 69–72. [19] D. Stephens, P. R. Young, and I. D. Robertson, “W-band substrate integrated waveguide slot antenna,” Electron. Lett., vol. 41, pp. 165–167, 2005. [20] R. K. Mongia and A. Ittipiboon, “Theoretical and experimental investigations on rectangular dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 45, pp. 1348–1356, 1997. [21] L. Yan, W. Hong, K. Wu, and T. J. Cui, “Investigations on the propagation characteristics of the substrate integrated waveguide based on the method of lines,” Inst. Elect. Eng. Proc. Microw. Antennas Propag., vol. 152, pp. 35–42, 2005. [22] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wireless Compon. Lett., vol. 11, pp. 68–70, 2001.

REFERENCES [1] Radio Spectrum Allocations in Canada. Canada: Industry Canada, 2005. [2] T. Teshirogi and E. T. Yoneyama, Modern Millimeter-Wave Technologies. Tokyo, Japan: Ohmasha, Ltd, 2001. [3] D. J. Roscoe, J. Carrie, M. Cuhaci, A. Ittipiboon, L. Shafai, and A. Sebak, “A 30 GHz transmit array for portable communications terminals,” in Antennas and Propagation Society Int. Symp. Digest, 1996, vol. 2, pp. 1116–1119. [4] R. K. Mongia, A. Ittipiboon, and M. Cuhaci, “Measurement of radiation efficiency of dielectric resonator antennas,” IEEE Microw. Guided Wave Lett., vol. 4, pp. 80–82, 1994. [5] K. W. Leung, K. M. Luk, K. Y. A. Lai, and D. Lin, “Theory and experiment of a coaxial probe fed hemispherical dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 41, pp. 1390–1398, 1993.

Wael M. Abdel Wahab (S’10) received the B.Sc. degree in electrical and computer engineering from Zagazig University, Zagazig, Egypt, in 1998 and the M.Sc. degree in electrical and computer engineering from Cairo University, Cairo, Egypt, in 2004. He is currently working toward the Ph.D. degree from the University of Waterloo (UW), ON, Canada. From 200 to 2006, he was a Researcher Assistant in the Microstrip Department, Electronic Research Institute (ERI), Cairo, Egypt. His current research interests focus on low-loss planar waveguide technologies for low cost efficient circuits and antennas in microwave and millimeterwave frequency band, including passive components modeling, design, and fabrication. Mr. Abdel Wahab was a recipient of an international graduate scholarship from The Egyptian Government (Canada 2006–2010).

ABDEL WAHAB et al.: LOW COST PLANAR WAVEGUIDE TECHNOLOGY-BASED DRA FOR mmW APPLICATIONS

Dan Busuioc received the B.S.E.E., M.S.E.E., and Ph.D. degrees in electrical and computer engineering from University of Waterloo, Waterloo, ON, Canada, in 2001, 2002, and 2005, respectively, and the MBA degree from Boston University, Boston, MA, in 2007. From 1996 to 2009, he has held a number of positions in engineering, applications, and marketing, with renowned companies in Germany, Canada, Sweden, and United States. He was one of the original founders of MASSolutions Inc., a Canadian-based company focusing on advanced sensors, microwave packaging, and antenna systems for microwave and mm-wave frequencies. He is now with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada. His research interests include novel antenna systems and high-frequency circuitry, including miniaturized, low-cost feed systems. He has further interest in wireless systems and semiconductor test equipment industries. D. Busuioc was the recipient of the National Science and Engineering Research Council of Canada (NSERC) Industrial Postgraduate Scholarship in 2001

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(with Ericsson Radio Access, Sweden) and in 2003 (with Winegard Company, Burlington, IA), and was a recipient of the Governor of Canada Medal.

Safieddin Safavi-Naeini (M’79) received the B.Sc. degree from the University of Tehran, Tehran, Iran, in 1974 and the M.Sc. and Ph.D. degrees from the University of Illinois at Urbana-Champaign, in 1975 and 1979, respectively, all in electrical engineering. He was a faculty member of the School of Engineering, University of Tehran, from 1980 to 1995. He joined the University of Waterloo, Waterloo, ON, Canada, in 1996, where he is now a Professor in the Department of Electrical and Computer Engineering, holds the RIM/NSERC Industrial Research Chair in Intelligent Radio/Antenna and Photonics, and is also the Director of a newly established Center for Intelligent Antenna and Radio System (CIARS).

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Integrated Leaky-Wave Antenna–Duplexer/Diplexer Using CRLH Uniform Ferrite-Loaded Open Waveguide Toshiro Kodera, Member, IEEE, and Christophe Caloz, Fellow, IEEE

Abstract—A novel integrated leaky-wave antenna–duplexer/ diplexer, based on the CRLH uniform ferrite-loaded open radiating waveguide is introduced, demonstrated and characterized both numerically and experimentally. The duplexing operation, and frequencies, is performed characterized by equal directly within the antenna thanks to the non-reciprocity of the ferrite, thereby suppressing the need for a circulator external to the antenna. The diplexing function, where the and frequencies are different, is achieved by inclining the plane of the antenna structure with respect to the normal of the radiation direction. The duplexer may be seen as the particular case of the diplexer with an inclination angle of zero. Virtually unlimited isolation is provided by the leaky-wave nature of the device, which avoids typical problems of demodulation and detection/ranging errors and possible destruction of the receiver. Another advantage of this duplexer/diplexer is the possibility to tune the operation frequency by the applied magnetic bias field, whereas such tuning is prevented both by the antenna and by the circulator in conventional designs. The fabricated prototype exhibits a gain of 2.3 dBi with isolation of more than 15 dB at all ports. The diplexing frequency range reaches 400 MHz.

Tx

Rx

Tx

Tx

Rx

Rx

Index Terms—Composite right/left-handed (CRLH) transmission line metamaterial, diplexer, dispersive medium, duplexer, ferrite device, leaky-wave antenna.

I. INTRODUCTION OMBINATIONS of antennas and duplexers/diplexers1, realized either by ferrite circulators or directional couplers, are ubiquitous in communication and radar systems

C

Manuscript received September 18, 2009; revised February 04, 2010; accepted February 17, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported in part by the NSERC Strategic Project under Grant 1014 and in part by Apollo Microwaves Company. T. Kodera was with the Department of Electrical Engineering, École Polytechnique de Montréal, Montréal, QC H2T 1J3, Canada. He is now with the Department of Electrical Engineering, Yamaguchi University, Yamaguchi 7558611, Japan (e-mail: [email protected]). C. Caloz is with the Poly-Grames Research Center, Department of Electrical Engineering and the CREER, École Polytechnique de Montréal, Montréal, QC H2T 1J3, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050449 1The terminologies “duplexer” and “diplexer” are often used interchangeably in the open literature. However, stricto sensu, the two terms are distinct. Microwave101 (www.microwaves101.com) provides the following definitions. A duplexer is a three-port network that allows the transmitter and receiver in a radar or communications system to use the same antenna at the same frequency or very close frequencies for the uplink and downlink. A diplexer is a three-port network that splits incoming signals from a common port into two paths, also called “channels,” generally with different frequencies; it is the simplest form of a multiplexer, which can split signals from one common port into many different channels.

[1]–[4]. A recurrent issue in such antenna-duplexer systems is (radiated field) leakage due to signal reflection the from antenna, which may introduce demodulation or detection/ranging errors, or even destroy the receiver. A typical solution to mitigate this problem is to insert power limiters in front of the receiver [2]. However, this solution suffers of harmonic generation from the diodes of the device, which may cause intermodulation, spurious re-radiation, and other undesirable effects. This paper presents a novel integrated leaky-wave antenna–duplexer/diplexer, which offers unlimited isolation, and combines in addition in one single component the two operations of duplexing/diplexing and radiation. Moreover, this component offers a frequency tuning capability, not achievable in conventional systems. This integrated leaky-wave antenna–duplexer/diplexer is based on the uniform ferrite-loaded waveguide composite right/left-handed (CRLH) [5] full-space scanning leaky-wave antenna [5]–[7] recently reported in [8]. The novelties in [8] are (i) the uniformity of the structure, in contrast to the periodicity required in [9] for the generation of space harmonics, and (ii) the purity of the CRLH response, in contrast to the forward/backward mixed branches in [10]. This duplexer was introduced in [11], and is further investigated and numerically confirmed in this paper. In addition, as full-duplex systems typically require different and frequencies, the paper proposes and characterizes a diplexer obtained by inclining the CRLH antenna structure. II. CRLH UNIFORM FERRITE-LOADED OPEN WAVEGUIDE LEAKY-WAVE ANTENNA The CRLH uniform ferrite-loaded open waveguide leaky-wave antenna structure serving as the core of the proposed duplexer/diplexer was introduced in [8] and is shown in Fig. 1. It consists of a rectangular waveguide fully loaded by a ferrite material and open at one of its lateral sides for radiation. The structure is similar to the conventional ferrite edge-mode isolator [12], [13]. However, it uses for antenna operation a lower-frequency band which inherently exhibits a CRLH response in terms of the propagation constant or , namely backward-wave dispersion relation and forward-wave at lower and higher frequencies, respectively, with continuous non-zero positive group velocity . This structure was shown in [8] at the spectral origin to provide full-space scanning leaky-wave radiation. , only the broadside In the proposed duplexer radiation frequency, achieved at the transition frequency

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. Uniform ferrite-loaded open waveguide leaky-wave structure with CRLH response [8].

, is used, and . In contrast, the prouses a backward-wave frequency posed diplexer and forward-wave frequency for the and frequencies, with the conditions that these frequencies exhibit an equal propagation constant magnitude . Due to the relatively small leakage factor of leaky-wave structures [14], radiation leakage constitutes only a perturbation in first approximation as far as guidance is concerned, and the waveguide may then be considered as a closed waveguide, with PEC walls at that three metal sides and a PMC wall at the ferrite-open side, assuming a high permittivity contrast between the ferrite and air, which is a safe assumption due to the high permittivity of ferrites. This leads to the simple dispersion relation [8]

Fig. 2. Dispersion diagram for the structure of Fig. 1 computed by (1) for the is the applied DC bias field and is parameters given in the inset, where the saturation magnetization of the ferrite.

H

M

(1) and with the transverse wavenumber the bulk birefringent effective permeability , where and are the usual elements of the Polder tensor permeability [15]. The approximation of (1) was demonstrated in [8] to be excellent for typical parameter values of the ferrite waveguide structure diagram for an apThe corresponding dispersion plied DC bias field of , ferrite magnetization and waveguide width of of is plotted in Fig. 2, where the edge-mode and CRLH bands are indicated on the right-hand vertical axis. The CRLH mode extends from 5.0 to 6.3 GHz (range not fully shown in Fig. 2). This mode is non-reciprocal, a key property in the duplexer proposed in this paper, since the dispersion curve exhibits a strictly for a positive slope or group velocity . Within the fast-wave region given bias direction of the dispersion diagram, radiation occurs, and therefore the structure operates as a leaky-wave antenna. According to the , well-known leaky-wave scanning law plane (Fig. 1) as this antenna scans the entire space in the varies from to , with broadside radiation at the allowed by the unusual traveling-wave transition frequency regime available at this frequency. This leaky-wave antenna has three advantages over artificial CRLH leaky-wave antennas [5]–[7]: i) it is perfectly uniform and therefore much easier to design, particularly if tapering is used for sidelobe level minimization [14]; ii) it is intrinsically gap-less, irrespectively to the design parameters, as may be shown from (1) and was verified in [8], and thereby avoids the difficulty of resonance balancing [5]; iii) it can be scanned by

Fig. 3. Operation principle of the proposed integrated leaky-wave antenna– duplexer.

bias field tuning, thereby avoiding the necessity for incorporation of varactors [16] or other tuning chips. It is to be noted that, in contrast to lumped-element CRLH transmission line structures, this structure does not exhibit negative parameters and , but owes its in the backward band backward-wave regime to the anisotropy of the ferrite material. III. INTEGRATED LEAKY-WAVE ANTENNA–DUPLEXER PRINCIPLE The proposed integrated leaky-wave antenna–duplexer and its operation principle are shown in Fig. 3 [11]. The duplexer is based on the uniform ferrite-loaded open waveguide leaky-wave structure (Fig. 1) presented in Section II, where the transmit port is set at one end and the receive port is set at the other end of the structure. signal is radiated by the leaky-wave antenna like in The a conventional CRLH-broadside leaky-wave antenna. By virtue of the leaky-wave radiation mechanism, the structure may be designal power has radisigned long enough such that all of the ated out of the structure before reaching the port on the other leakage side, thereby automatically preventing any isolation. Moreover, the inand providing large signal picked up by the antenna can only propagate coming toward the port due to the non-reciprocity of the structure isolation is automati(Fig. 2), and therefore infinite cally achieved. The antenna can thus simultaneously transmit and receive without any interference or leakage between the

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Fig. 4. Limitation of isolation between Tx and Rx in a conventional antennacirculator system due to signal reflection from antenna.

and signals. Thus, it constitutes an integrated (or combined) antenna-duplexer device with excellent duplexing operaand tion, since a single antenna simultaneously performs the operations with perfect isolation. isolation property of the proThe virtually infinite posed integrated leaky-wave antenna—duplexer is very appreciable in practical communication or radar front-end systems to avoid demodulation or detection/ranging errors, or even receiver destruction, caused by the reflected signal from antenna. This problem is illustrated in Fig. 4. For instance, in an application using a typical antenna with a return loss of 15 dB and power of 50 dBm, a power of 35 dBm leaks into the a (neglecting the circulator’s losses). Although a power limiter may be used to mitigate this problem, this technique has its own limitations, including additional insertion loss, harmonic generation and power handling limitations. The proposed integrated leaky-wave antenna–duplexer solves these problems in a simple and elegant manner. IV. DUPLEXER PROTOTYPE DESIGN This section presents the experimental validation and characterization of the proposed integrated leaky-wave antenna– duplexer. Fig. 5 shows the detailed configuration of the prototype, including the matching sections and all the dimensions, while the actual prototype is shown in Fig. 6. The length of the ferrite waveguide is 10 cm, which corresponds to about at the broadside operation frequency of 5.9 GHz (Fig. 2). The ferrite material, offered by muRata, exhibits the following characteristics: , , . The magnetic bias field is provided by an NdFeB N42-class permanent magnet of 150 12 6.4 mm placed at the back of the structure underneath the ground plane and providing an internal field of 0.184 T. The matching section is constituted of two separate parts, to ensure a smooth transition between the microstrip line and the ferrite waveguide and thereby maximize the radiation efficiency of the antenna: an open stub microstrip section on a Rexolite 2200 substrate with permittivity of and thickness of 0.8 mm and a small opening in the lateral metal wall of the ferrite waveguide at its input. Matching is performed at 5.9 GHz, using traditional conjugate matching technique. The synthesis of the duplexer is essentially the same as that of the antenna which constitutes its core, except for the additional port excitation and matching, and is presented in [8].

Fig. 5. Integrated leaky-wave antenna–duplexer design with dimensions, including two identical matching sections.

Fig. 6. Integrated leaky-wave antenna–duplexer prototype corresponding to Fig. 5 with zoomed views.

V. FULL-WAVE AND EXPERIMENTAL RESULTS OF DUPLEXING ANTENNAS The leaky-wave radiation patterns of the CRLH full-space scanning antenna used in the duplexer are shown in Fig. 7. Good agreement is observed between the numerical (FEM Ansoft HFSS) and experimental results. Broadside radiation is achieved at the design frequency of 5.9 GHz. The waveguide structure is same as the one shown in [8], where good agreement is observed between the experimental and numerical dispersion characteristics. The measurement setup and ports definition for the numerical and experimental characterization of the duplexer are shown in and #2 are set so that Fig. 8. Ports #1 is the transmission direction from the viewpoint of non-recihorn antenna is placed at 400 mm procity. A from the duplexer in the broadside position, and is labeled port #3 for subsequent definition of scattering parameters. Fig. 9 shows the HFSS (FEM) numerical model reproducing the test setup of Fig. 8 for the evaluation of the transmission parameters of the duplexer. The experimental horn antenna is modeled as a waveguide port. This makes sense as the width of this horn antenna (147 mm) is comparable to the length of antenna including matching sections (134 mm). Fig. 10 shows the full-wave simulated and measured transmission characteristics of the duplexer. As predicted in Fig. 2, is observed. strong non-reciprocity This non-reciprocal property is beneficial to sensitive transmitleakage may otherwise introduce ters where

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Fig. 9. HFSS (FEM) numerical model reproducing the test setup of Fig. 8 for the evaluation of the transmission parameters of the duplexer. The air region between port #3 and the leaky-wave antenna is surrounded by perfect matched layers (PMLs).

Fig. 7. Frequency beam-scanning (H-plane) for the antenna of the duplexer of Figs. 5 and 6, including the broadside angle used for the duplexing operation. (a) Experiment ( = 2 3 dBi). (b) FEM (HFSS) ( = 4 6 dBi, = 38%, = 8 8 dB). The performances given are evaluated at broadside.

D

:

G

:

G

:



Fig. 10. Comparison of the transmission parameters between ports #1 and #2 for the experimental (solid line) and full-wave simulated (dotted line) results. Fig. 8. Measurement setup and port definitions for the characterization of the duplexer.

parasitic intermodulation in the signal. As explained in is crucial. The isolaSection III, the isolation tion of the current prototype is moderate, 25 dB, but it may be straightforwardly increased to an arbitrary level by increasing either the length of the structure or the leakage factor, since power has been radiated beyond a given length most of the port. If we simply out of the antenna before it may reach the cascade a second identical leaky-wave antenna, the isolation will increase to . From a practical point of view, ultimate limitations in isolation may be set by the non-availability of strong permanent magnet which would be sufficiently long and capable to produce a sufficiently uniform field over the required area. is of , which corThe transmission coefficient responds to 99% of energy dissipation by combined radiation and material losses. However, these two types of losses can be discriminated by noting that the radiation efficiency of the pro, as given in the caption of Fig. 7. This totype is of loss (61%) is due to material indicates that most of the dissipation, which is thus the main cause for the small gain

achieved. Naturally, this problem could be remedied by using a lower-loss ferrite material. Of note is that the directivity, which , is relatively large, due to also contributes to the gain the continuity of the radiation aperture, despite the non-uniform (exponentially decaying) distribution of the field. For comparcollinear half-waveison, an array antenna constituted of length dipole elements exhibits a maximal directivity of (corresponding to the limiting case of a zero inter-element spacing or ) [17], while the directivity of the an array period of proposed antenna is of 8.8 dB for long. Fig. 11 shows the reflection coefficient of the prototype at (#1) and (#2) ports. Excellent matching of around the is achieved at the port. The relatively poor return loss at the port does not represent the actual matching of mode since, in the prevailing situation the antenna in the matching is the reflecof non-reciprocity, the meaningful to the port, labeled tion coefficient measured from the in Fig. 8, which is not directly accessible by measurement. port is identical However, since the matching section of the port, the port is automatically matched at to that of the port, , by rathe same level as the diation reciprocity.

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Fig. 11. Comparison of the reflection coefficient at ports #1 and #2 for the experimental (solid lines) and numerical (dotted lines) results.

Fig. 12 presents the performances of the duplexer in terms of the scattering parameters related to the horn antenna—duplexer system of Fig. 8. All the measured results are normalized in order to exclude the coaxial-horn transition and free-space propagation radiation (high ) and losses. Fig. 12(a) shows that the isolation (low ), which differ by 15 dB. Fig. 12(b) the reception (high ) and non-radiation shows (low ), with a difference of 17 dB. Finally, Fig. 12(c) shows signal is received only by the port (high ) and that the port (low ), with a difference of 15 dB. The not by the isolation suppresses the problem of spurious signal radiation in direct-conversion receivers. VI. INTEGRATED LEAKY-WAVE ANTENNA-DIPLEXER PRINCIPLE The duplexer presented in the previous sections, may be transformed into a diplexer by inclining the plane of the antenna with respect to the normal of the radiation direction under an angle equal to both the backward and forward radiation angles with respect to the normal of the plane of the antenna, as illustrated in Fig. 13. In this configuration, and under the condiwith , the an links tion and . The separation range operate at two frequencies , is controlled by between these frequencies, and the inclination angle . The center frequency between , which approximately equals to , is controlled by the bias [8]. The duplexer may be seen as the particmagnetic field ular case of the diplexer with an inclination angle of zero. This inclined configuration is identical to that proposed in [18] for the case of a non-magnetic lumped-element LWA, which is capable only of diplexing due to its reciprocity, ex. In the present cluding the limiting duplexing case with structure, the operation frequencies can be simply tuned by . Moreover, unwanted symvarying the magnetic bias field with respect to ) beams caused be termination metric ( and , reflection are automatically suppressed both in which avoids the typical problems of conventional diplexers discussed in correlation with Fig. 4. Fig. 14 shows the required inclination angle versus the sepand frequencies, . aration between the This inclination angle, which is computed from the dispersion

Fig. 12. Comparison of experimental (solid lines) and numerical (dotted lines) results for the normalized scattering parameters of the horn antenna–duplexer system (a) between ports #1 and #3, (b) between ports #2 and #3, (c) and between ports #3 and #1, and between ports #3 and #2.

diagram of Fig. 2, is a function of all the physical and geometrical parameters of the structure. The maximum frequency separation for the presented prototype is of 0.4 GHz, as shown in Fig. 14. If the required frequency separation exceeds the achievfrequency difference, a conventional duplexer able based on filters would be required. However, in practical applications, the frequency separation is rather small and in this case high isolation is hard to realize. The presented device offers a solution to this issue by providing duplexing with high isolation for small and even zero frequency separations.

KODERA AND CALOZ: INTEGRATED LEAKY-WAVE ANTENNA–DUPLEXER/DIPLEXER USING CRLH

Fig. 13. Principle of the proposed diplexer. (a) Dispersion relation and its linearization in the radiation range, for the case f > f . (b) Inclined antenna at an angle  =k , where .

= sin (

)

=0

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Fig. 16. Comparison of the full-wave normalized transmission characteristics port), #2 ( port) and #3 (radiation port) for different between ports #1 ( , 0 , 20 in Fig. 15. inclination angles, 

Tx = 010

Rx

, computed

Fig. 17. Measurement setup and port definitions for the characterization of the diplexer.

Fig. 15. HFSS (FEM) numerical model of the diplexer corresponding to Fig. 13. The numerical boundaries used are the same as in Fig. 9.

Fig. 18. Comparison of the measured transmission characteristics between ports #1 ( port) and #2 ( port) and #3 (radiation port) for different inclination angles,  , 0 , 20 in Fig. 17.

Fig. 14. Required inclination angle  versus from the dispersion diagram of Fig. 2.

1f =

f

0f

VII. FULL-WAVE AND EXPERIMENTAL RESULTS OF DIPLEXING ANTENNAS Fig. 15 shows the HFSS (FEM) numerical model of the diplexer corresponding to Figs. 13 and 16 shows the correand frequensponding full-wave results. The diplexing cies are found to be in good agreement with the dispersion diagram predictions of Fig. 14. It should be noted that the isolation, computed as the difference between and at the and frequencies, respectively, in conventional

Tx

= 010

Rx

(reciprocal) diplexers, is here much larger than this difference, due to non-reciprocity. The isolation is represented by dotted line in Fig. 10, where it was noted that a much larger isolation may be obtained by simply increasing the length of the structure. Fig. 17 shows the measurement setup of the diplexer corresponding to Figs. 13 and 18 shows the corresponding experimental results. As in the case of the duplexer, good agreement is observed between the experiment and the full-wave predictions (Fig. 16).

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VIII. CONCLUSION A novel integrated leaky-wave antenna—duplexer/diplexer, based on the CRLH uniform ferrite-loaded open radiating waveguide reported in [8], has been introduced, demonstrated and characterized both numerically and experimentally. The duplexing operation is performed directly within the antenna thanks to the non-reciprocity of the ferrite, thereby suppressing the need for a circulator external to the antenna. The diplexing function is achieved by inclining the plane of the antenna structure with respect to the normal of the radiation direction. isolation is provided by the Virtually unlimited leaky-wave nature of the device, which avoids typical problems of demodulation and detection/ranging errors and possible destruction of the receiver in conventional duplexers/diplexers. Another advantage of the proposed duplexer/diplexer is the possibility to tune the operation frequency by the applied magnetic bias field, whereas such tuning is prevented both by the antenna and by the circulator in conventional designs. The fabricated prototype exhibits a gain of 2.3 dBi with isolation of more than 15 dB at all ports. This new device may find applications in various wireless communication and radar systems. ACKNOWLEDGMENT The authors would like to acknowledge Murata Manufacturing for their generous donation of ferrite materials, Ansoft Corporation for their generous donation of HFSS software licenses. REFERENCES [1] C. Muehe, “High-power duplexers,” IRE Trans. Microw. Theory Tech., vol. MTT-9, no. 6, pp. 506–512, Nov. 1961. [2] D. Blattner, W. Siekanowicz, and T. Walsh, “A 40 kilowatt X-band ferrite duplexer,” in IEEE Electron Devices Meeting Dig., Jan. 1967, vol. 13, pp. 158–160. [3] J. Adam, L. Davis, G. Dionne, E. Schloemann, and S. Stitzer, “Ferrite devices and materials,” IEEE Trans. Microw. Theory Tech., vol. MTT-50, pp. 721–737, Mar. 2002. [4] K. Xie and L. Davis, “Radiative optical isolator and circulator,” IEEE J. Lightw. Tech., vol. 19, no. 7, pp. 1028–1035, Jul. 2001. [5] C. Caloz and T. Itoh, Electromagnetic Metamaterials, Transmission Line Theory and Microw. Applications. Hoboken/Piscataway, NJ: Wiley/IEEE Press, 2005. [6] L. Liu, C. Caloz, and T. Itoh, “Dominant mode leaky-wave antenna with backfire-to-endfire scanning capability,” Electron. Lett., vol. 38, no. 23, pp. 1414–1416, Nov. 2002. [7] C. Caloz, T. Itoh, and A. Rennings, “CRLH traveling-wave and resonant metamaterial antennas,” IEEE Antennas Propag.. Mag., vol. 50, no. 5, pp. 25–39, Oct. 2008. [8] T. Kodera and C. Caloz, “Uniform ferrite-loaded open waveguide structure with CRLH response and its application to a novel backfire-to-endfire leaky-wave antenna,” IEEE Trans. Microw. Theory Tech., vol. 57, pp. 784–795, Apr. 2009. [9] H. Maheri, M. Tsutsumi, and N. Kumagai, “Experimental studies of magnetically scannable leaky-wave antennas having a corrugated ferrite slab/dielectric layer structure,” IEEE Trans. Antennas Propag., vol. 36, no. 7, pp. 911–917, Jul. 1988. [10] P. Baccarelli, C. Nallo, F. Frezza, A. Galli, and P. Lampariello, “Attractive features of leaky-wave antennas based on ferrite-loaded open waveguides,” in IEEE AP-S Int. Symp. Dig., 1997, vol. 2, pp. 1442–1445.

[11] T. Kodera and C. Caloz, “Leaky-Wave antenna integrated duplexer using CRLH uniform ferrite-loaded open waveguide,” presented at the IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, Jun. 2009. [12] M. E. Hines, “Reciprocal and nonreciprocal model of propagation in ferrite stripling and microstrip devices,” IEEE Trans. Microw. Theory Tech., vol. MTT-19, pp. 442–451, May 1971. [13] K. Araki, T. Koyama, and Y. Naito, “A new type of isolator using the edge-guided mode,” IEEE Trans. Microw. Theory Tech., vol. MTT-23, pp. 321–321, Mar. 1975. [14] A. Oliner and D. Jackson, Leaky-wave Antennas, in Antenna Engineering Handbook, J. Volakis, Ed., 4th ed. New York: McGraw Hill, 2007. [15] B. Lax and K. J. Button, Microw. Ferrites and Ferrimagnetics. New York: McGraw-Hill, 1962. [16] S. Lim, C. Caloz, and T. Itoh, “Electronically-scanned composite right/ left-handed microstrip leaky-wave antenna,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 277–279, May 2004. [17] C. Balanis, Antenna Theory, Analysis and Design, 3rd ed. New York: Wiley-Interscience, 2005. [18] S. Gupta, H. Nguyen, T. Kodera, S. Abielmona, and C. Caloz, “CRLH leaky-wave antenna based frequency division diplexing transceiver,” in Proc. Asia Pacific Microw. Conf., Dec. 2009, pp. 2014–2017.

Toshiro Kodera (M’01) received the B.E., M.E., and Dr.Eng. degrees from Kyoto Institute of Technology, Kyoto Japan, in 1996, 1998, and 2001, respectively. He developed some numerical program and devices using ferrite media. In 2001, he joined the Faculty of Engineering, Osaka Institute of Technology, as a Lecturer. In 2005, he joined Wave Engineering Laboratories, ATR international, Kyoto Japan, as a Visiting Researcher, and in 2006 he joined as a Researcher. At ATR, he engaged in R&D of GaAs MMICs for 802.15.3c Gbps wireless LAN system and microwave power amplifier. In 2008, he joined the Department of Electrical Engineering, École Polytechnique of Montréal, Canada, where he developed some microwave/millimeter wave radiative structure by magnetic material as a Research Associate. In 2010, he joined Department of Electrical Engineering, Yamaguchi University, Yamaguchi, Japan, where he is now a Associate Professor. His current research is on the microwave devices utilizing magnetic material including nanostructure.

Christophe Caloz (F’10) received the Diplôme d’Ingénieur en Électricité and the Ph.D. degree from the École Polytechnique Fédérale de Lausanne (EPFL), Switzerland, in 1995 and 2000, respectively. From 2001 to 2004, he was a Postdoctoral Research Engineer at the Microwave Electronics Laboratory, University of California at Los Angeles (UCLA). In June 2004, he joined École Polytechnique of Montréal, Canada, where he is now a Full Professor, a member of the Poly-Grames Microwave Research Center, and the holder of a Canada Research Chair (CRC). He has authored and coauthored over 360 technical conference, letter and journal papers, three books and eight book chapters, and holds several patents. His research interests include all fields of theoretical, computational and technological electromagnetics engineering, with strong emphasis on emergent and multidisciplinary topics. Prof. Caloz is a Member of the Microwave Theory and Techniques Society (MTT-S) Technical Committees MTT-15 (Microwave Field Theory) and MTT-25 (RF Nanotechnology). He is a Speaker of the MTT-15 Speaker Bureau, and the Chair of the Commission D (Electronics and Photonics) of the Canadian Union de Radio Science Internationale (URSI). He is a member of the Editorial Board of the International Journal of Numerical Modelling (IJNM), of the International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE), of the International Journal of Antennas and Propagation (IJAP), and of the journal Metamaterials of the Metamorphose Network of Excellence. He received the UCLA Chancellors Award for Postdoctoral Research in 2004 and the MTT-S Outstanding Young Engineer Award in 2007. He is an IEEE Fellow.

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An Original Antenna for Transient High Power UWB Arrays: The Shark Antenna Laurent Desrumaux, Adrien Godard, Michèle Lalande, Member, IEEE, Valérie Bertrand, Joël Andrieu, and Bernard Jecko

Abstract—A novel ultrawideband (UWB) antenna, called the Shark antenna, and designed especially for transient applications is proposed in this paper. A Shark antenna array is also conceived in order to obtain a high power UWB pulse radiation source through the frequency band [800 MHz–8 GHz]. For this application, the elementary antenna must be compact, non-dispersive, and the array must have a high transient front to back ratio. The geometry of the Shark antenna and its radiation characteristics are detailed. Moreover, an approach which evaluates the transient front to back ratio of a square array is presented.

fact, optical systems with ultrafast laser sources generate, with a small jitter (2 ps typically), ultra short electrical waveforms. The optical control of the sources allows to sum the radiated power and to easily steer the transient radiation beam [4]. Besides, another advantage of the architecture N generators/N antennas is the working continuity even if an elementary source is defective. The specifications of the high power application and the design of the elementary Shark antenna are presented in the next section.

Index Terms—High power radiations, miniature ultrawideband (UWB) antenna, transient array.

II. SHARK ANTENNA A. Transient High Power Application Specifications

I. INTRODUCTION RANSIENT radar cross section (RCS) measurements [1], ultrawideband (UWB) synthetic aperture radar (SAR) system [2], and high power UWB radiation source [3] are some applications for which the radiation of transient waveforms is interesting through bandwidths exceeding one decade. Typically, the rise time of the radiated pulses is around 100 ps and their duration is a few nanoseconds. This paper describes the Shark antenna and its main characteristics. This original antenna is used to conceive an array, allowing the obtainment of a high power UWB radiation source. In accordance with this application, the radiation system appears in the N generators/N antennas architecture. Indeed, it is difficult to consider the design of an array with more than 16 elements with the 1 generator/N antennas configuration, which requires a power divider. However, the N generators/N antennas configuration allows avoiding this problem, where any number of antennas can be considered to design the array. This radiation system presents the advantage of increasing the radiation power on one hand and offering the agility to the array on other hand. A major difficulty of limiting the synchronization of the radiating source jitters yet exists. A solution which consists in using impulse optoelectronic devices permits to bypass this difficulty. In

T

Manuscript received February 12, 2009; revised August 27, 2009; accepted January 15, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by the French armament procurement agency DGA/MRIS. L. Desrumaux, A. Godard, M. Lalande, and J. Andrieu are with the XLIM/ OSA, IUT GEII, 19100 Brive, France (e-mail: [email protected]). V. Bertrand is with the CISTEME, 87069 Limoges, France (e-mail: valerie. [email protected]). B. Jecko is with the XLIM/OSA, Université de Limoges, 87060 Limoges Cedex, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050418

In the context of electronic warfare, there is considerable interest in radiating very short pulses, which provide a large discretion. Indeed, the very short pulses are hard to detect. In order to be used in an application with UWB high power radiation and considering an architecture including N generators/N antennas, the elementary antenna must present the following specifications. — It has to be matched (reflection coefficient lower than dB) over a very wide frequency band (at least one decade); — It must be the least possible dispersive in order to radiate discrete waves and to make the system stealthy; — In order to conceive the array, the radiation pattern of the antenna must be either directive or sectoral. Moreover, the elementary antenna must be miniature (dimenwhere is the wavelength corresponding to the sions lower frequency of the bandwidth), at least in two dimensions, in order to increase the source number on a surface area as small as possible. With this point, there is a compromise between having a high field range product (rE ) [3] to radiate a power as high as possible, having a transient front to back ratio as high as possible which limits the fratricidal effects, and having a miniature antenna. In addition to this compromise, the difficulty consists in conceiving an antenna respecting the four following criteria at once: — miniaturization in two dimensions; — dispersion as low as possible; — matching through a very large bandwidth; — sector-based beaming. Table I shows the existing technologies [5]–[13], with regards to small dimensions antennas. In this table, the bandwidth and the dispersion depend on the dimensions of each antenna. The wavelength reported in this table corresponds to the lower frequency of the bandwidth. A few antennas have better performances than expected in the requirement concerning one of the criterions but do not reach the four wanted points (Archimedean

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TABLE I EXISTING TECHNOLOGIES

Fig. 1. Bi-cone at the starting point of the design. Fig. 2. Cones inclined in the same direction.

spiral, bicone, or ridged horn). Two antennas reach the four criterions (TEM Horn and Vivaldi) but their bandwidth is only just higher than a decade. The following part presents the design of the Shark antenna, which reaches all the desired criterions and has particularly a higher bandwidth than the TEM Horn and the Vivaldi. B. Design of the Shark Antenna A bi-cone is a good starting point of the Shark antenna design, thanks to its matching over a very wide frequency band. The frequency spectrum [800 MHz–8 GHz] is chosen to illustrate the design of this antenna. Its height is 84 mm and the distance between the two cones is 4 mm. The angle represented in Fig. 1 is one of the parameters that permit to optimize the matching. Its value will be given at the end of this part, thanks to a parametric study on the designed antenna. The disadvantage of this bi-cone is that its radiation pattern is omnidirectionnal in the H-plane. In order to favor the radiation in one direction, the two cones have been inclined in the chosen direction (Fig. 2), which is designated as the front of the

Fig. 3. Structure truncated at the bottom, the top, and the back.

antenna, with an angle , which is another parameter permitting to optimize the matching. As shown in Fig. 3, to keep the height h, the inclined bi-cone has been truncated at the bottom and the top of the structure. Moreover, the truncating at the back of the structure permits to increase the radiations in front of the antenna. As the radiations in front of the antenna must be higher than the radiations at the back of it, a reflecting plane has been introduced at the back of the structure. This addition is shown in

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Fig. 4. Addition of a reflecting plane.

Fig. 6. Simulated reflection coefficient.

Fig. 5. Shark antenna.

Fig. 4, and the represented distance d is also a parameter that permits to optimize the matching. The last step of the design consists in connecting the inclined and truncated bi-cone to the reflector plane. In this way, conductive planes have been positioned below and above the structure. The designed antenna is shown in Fig. 5. The feeding of this antenna is made with a coaxial cable, which is introduced inside the cone situated at the bottom. This cable is connected with the cone situated at the top with an adequate transition. The parameters permitting to vary the reflection coefficient are the angle , the angle , and the distance d, represented respectively in Figs. 1, 2, and 4. A parametric study on the final structure was done with the CST Microwaves Studio software to evaluate these parameters, with regards to the matching. The result shows that the Shark antenna is well matched through the desired frequency band for the following values: ; — ; — mm. — Indeed, the reflection coefficient is lower than or equal to dB between 800 MHz and 8 GHz. Actually, this antenna is well matched from 800 MHz up to a frequency higher than 20 GHz, as shown in Fig. 6. This matching over a very wide frequency band is due to the shape taken from the bi-cone, as far as low frequencies are concerned, and to the adequate transition between the two cones, concerning high frequencies.

Fig. 7. Dimensions of the Shark antenna.

As mentioned in the introduction, the antenna must have a very large bandwidth. This criterion is reached thanks to a bandwidth higher than 25:1. The maximum wavelength of the spectrum that has to be covmm. As shown in Fig. 7, the general size ered is of the Shark antenna is: for the width; — for the height; — — for the length. Thus, this antenna is miniature as far as the width and the height are concerned. As this antenna is intended to be used in an array, the miniaturization constraint in the length has been released. Indeed, it is not worth miniaturizing the length of this antenna because the global height and the global width are intended to be higher than it. This travelling wave antenna reaches all the criterions expressed in part A. Section III presents its radiation characteristics.

III. RADIATION CHARACTERISTICS The field range product per Volt accepted (rEd ) [3] of a system (generator antenna) is defined as the ratio of the peak

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Fig. 8. Monocycle pulse in the time domain.

Fig. 10. Transient peak radiations in the H-plane: field range product per Volt accepted.

Fig. 9. Monocycle pulse in the frequency domain.

level of the transient radiated farfield put back at a distance of 1 m out of the peak level of the voltage accepted by the antenna

Fig. 11. Transient peak radiations in the E-plane: field range product per Volt accepted.

(1) To obtain the field range product of the Shark antenna, the pulse delivered by the generator is a monocycle pulse which bandwidth corresponds to [800 MHz–8 GHz] (Figs. 8 and 9). This field range product is presented in the H-plane (Fig. 10) and in the E-plane (Fig. 11). In each of these figures, the plotted transient field corresponds to the transient field peak at each angle (taken at different time for each angle). The two last figures show that the antenna has a sectoral radiation pattern because the radiations are concentrated in a half plane. Indeed, the radiation beamwidth is 252 in the H-plane and 88 in the E-plane. The fact that the antenna is not very directive is an advantage in order to make the array agile. Moreover, these figures give the following performances: — the field range product rEd in front of the antenna is equal to 1,72; — the front to back ratio in the time domain is equal to 8,42, which is equivalent to 18,51 dB. Fig. 12 presents the behavior of the radiated fields all around the antenna in the H-plane. The delivered pulse is the one preV/m to 15 sented in Fig. 8, the vertical scale stretches from V/m with a 5 V/m step, and the horizontal scale stretches from 4 ns to 6,5 ns with a 1 ns step.

Fig. 12. Behavior of the radiated field all around the Shark antenna.

Fig. 12 shows the low dispersion of the antenna, it also shows how high the peak level of the radiated field in front of the antenna is, compared to the one at the back of the structure.

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Fig. 14. The two different antenna arrays. Fig. 13. Reflection coefficient of the two Shark antennas.

The Shark antenna respects all the criterions mentioned in Section II. Indeed, the antenna is miniature, not too dispersive, and well-matched through a very large bandwidth. Section IV is dedicated to the performances of the Shark antenna array, which must have a high field range product and a high transient front to back ratio. IV. SQUARE SHARK ANTENNA ARRAY The Shark antenna is dedicated to be used in an array structure. In that way, this section presents a radiation optimization on UWB multielement array and the results of a study concerning the evolution of the transient front to back ratio. A. Radiation Optimization on UWB Multielement Array Two Shark antennas have been designed with respect to the bandwidth [800 MHz–8 GHz]. — The Shark antenna presented in Section II has a general size of: for the width; for the height; for the length. — A Shark antenna with higher dimensions, obtained by using the “scaling” method, has a general size of: for the width; for the height; for the length. The first miniature Shark antenna is well matched from 800 MHz and the second Shark antenna, with higher dimensions, is well matched from 600 MHz. Thus, these two antennas are matched through the common frequency bandwidth [800 MHz–8 GHz], as shown in Fig. 13. From these two Shark antennas, two different Shark antenna arrays can be considered, associating one generator per antenna. These two arrays have the same surface area, as shown in Fig. 14. With a same surface area, which involves a same gain, and a same field range product (rE ) of 1 kV in front of each array,

TABLE II COMPARISON BETWEEN THE TWO ARRAYS (WITH A SAME SURFACE AREA AND A FIELD RANGE PRODUCT EQUAL TO 1 KV)

the two arrays are compared with regards to the peak level delivered by each generator and the front to back ratio in the time domain (Table II). For this comparison, the pulse delivered by each generator is the monocycle pulse presented in Fig. 8. Table II shows that the level delivered by each generator decreases with the increase of the source number. Indeed, the number of generators is different so the voltage per generator is different. The result of this comparison is that, with a same surface area, a same gain, and a maximum peak radiated field set in front of the structure, it is interesting to choose a configuration with a higher antenna number. However, the transient front to back ratio is the same in the two configurations. The fact that the cost of the generators decreases with the level they can deliver justifies the solution which consists in increasing the source number. Moreover, the use of generators which deliver low levels facilitates the synchronization between the sources. The graph that represents the transient electric field as function of time and angles is a suitable UWB radiation source descriptor [14]. Fig. 15 represents this type of characteristic for the two arrays, in the case where all the antennas are simultaneously fed and the delay between feeding allows steering angles of 10 and 30 . The maximum level of the transient electric field is the same due to the same value of the field-range-product rE . The graphs given in Fig. 15 show the following. — The radiation beam can be easily and correctly steered as well for 10 as for 30 . This point is validated for the two arrays. — The 4 4 array allows a higher beam steering than the 3 3 array. Indeed, there are more “streaks” in the case of the 4 4 array than in the case of the 3 3 array but they are less significant as far as the level of the radiated field is concerned. Moreover, the different number of streaks

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Fig. 15. Transient radiation patterns.

TABLE III EVOLUTION OF THE FRONT TO BACK RATIO WITH THE ANTENNA NUMBER

Fig. 16. Suppression of the slits at the back of the array.

between the two arrays in the beam steering diagrams is explained by the fact that each streak is the contribution of each column of an array. in the case of — The level of the radiated field at the 3 3 array is between 50% and 60% of the maximum. This level could be too important for a high power radiation source, for which it is essential to deliver a weak radiated peak field outside the azimuth 0 . B. Evolution of the Transient Front to Back Ratio This part presents the results of a study concerning the evaluation of the transient peak front to back ratio with the increase of the antenna number in a square array. Each antenna is firmly attached to the others along the horizontal axis. However, it is not the case in the vertical axis due to the presence of the coaxial cable under each antenna. So, a slit appearing with the design is filled in, as shown in Fig. 16. This practical problem of feeding was not considered in the comparison between the 3 3 and the 4 4 array. The fact of increasing the antenna number involves a rise of the peak level of the transient radiated farfield, as expected. This level grows up linearly with the antenna number, which involves a linear increase of the field range product in front of the structure.

The most important point in the conception of a high power UWB radiation source is the front to back ratio, to limit the fratricidal effects. Table III shows the evolution of the transient front to back ratio (as far as the peak level is concerned) with the increase of the antenna number, from 1 to 64. It is clearly visible that the rise is not linear. The higher the antenna number is, the less significant the transient front to back ratio increase is. From these results, the idea is to deduce the transient front to back ratio which can be estimated with a higher number of antennas, using an extrapolation method. It is first necessary to evaluate the evolution presented in Table III with an analytical function. Fig. 17 presents the fact that the radiated field at the back of an array is due to the peripheral antennas. The radiated peak field in front of the array is the sum of the high frequencies contributions of each antenna. Conversely, the radiated field at the back of each antenna is essentially made of low frequencies and, therefore, as the antennas are very close

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Fig. 19. Evolution of the transient front to back ratio as a function of the n array: extrapolation. number of antennas N in an N = n

2

The analytic curve is able to give the size of the transient front to back ratio of a square array, whatever the antenna number is. Thus, Fig. 19 presents the F/B ratio that could be obtained with a high number of antennas. For example, if a 16 16 array (= 256 antennas) is considered, the estimated transient front to back ratio is around 43 dB, according to Fig. 19.

Fig. 17. Radiations of the peripheral antennas.

V. CONCLUSION

Fig. 18. Evolution of the transient front to back ratio as a function of the n array: F/B ratio obtained with CST number of antennas N in an N = n and with (2).

2

together and coupled, the sum of the back radiations does not increase as fast as the sum of the front radiations. This fact involves a non linear evolution of the transient front to back ratio. This approach is the same for all the possible configurations. In the particular case of a square array, with regards to Fig. 17, it can be concluded that the transient front to back ratio evolves according to the following empirical relation (2), which is expressed linearly as follows: (2) where N is the antenna number. As far as the transient front to back ratio is concerned, Fig. 18 superposes both the evolution obtained with CST and the evolution obtained thanks to (2): Despite the two representative curves are the same size, the F/B simulated with CST is higher than the analytic one. Indeed, at the back of the structure, the radiated field calculated with CST is lower than the analytic one, due to the existence of coupling between antennas.

The original Shark antenna presented in this paper is suitable for transient UWB applications such as high power UWB radiation source. It presents the following advantages: — miniature in two dimensions (height and width lower than where is the wavelength corresponding to the lower frequency of the bandwidth); — low dispersive; — well-matched through a bandwidth higher than a decade; — sectoral radiation pattern. An array constituted of more than 16 Shark antennas presents: — a high field range product; — a high transient front to back ratio. This traveling wave antenna is dedicated to be used in an array with an N generators/N antennas architecture, which permits to use any number of antennas (there is no difficulty concerning the power divider) and presents the advantage that the array is agile. In order to deliver high power radiations with an array which dimensions are set, a solution consists in increasing the source number by miniaturization of the elementary antennas in order to: — reduce the radiated peak field outside the azimuth 0 ; — facilitate the synchronization between the sources; — reduce the cost of each generator. In order to limit fratricidal effects, the presented approach permits to estimate the transient front to back ratio that can be obtained according to the antenna number. REFERENCES [1] I. I. Immoreev and J. D. Taylor, “Optimal short pulse ultra-wideband radar signal detection,” in Ultra-Wideband Short-Pulse Electromagnetics 5, P. D. Smith, Ed., R. S. Cloude, Ed. New York: Springer, 2002, pp. 207–214.

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[2] M. A. Ressler, “The army research laboratory ultra wideband Boom SAR,” in Proc. Geoscience and Remote Sensing Symp., 1996, vol. 3, pp. 1886–1888. [3] C. E. Baum, W. L. Baker, W. D. Prather, J. M. Lehr, J. P. O’Loughlin, D. V. Giri, I. D. Smith, R. Altes, J. Fockler, D. McLemore, M. D. Abdalla, and M. C. Skipper, “JOLT: A highly directive, very intensive, impulse-like radiator,” Proc. IEEE, vol. 92, no. 7, Jul. 2004. [4] M. Lalande, J. C. Diot, S. Vauchamp, J. Andrieu, V. Bertrand, B. Beillard, B. Vergne, V. Couderc, A. Barthélémy, D. Gontier, R. Guillerey, and M. Brishoual, “An ultra wideband impulse optoelectronic radar: RUGBI,” PIER B 11, pp. 205–222, 2009. [5] J. C. Diot, P. Delmote, J. Andrieu, M. Lalande, V. Bertrand, B. Jecko, S. Colson, R. Guillerey, and M. Brishoual, “A novel ultra-wide band antenna for transient UWB applications in the frequency band 300 MHz–3 GHz: The valentine antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 987–990, Mar. 2007. [6] P. Delmote, C. Dubois, J. Andrieu, M. Lalande, V. Bertrand, B. Beillard, B. Jecko, T. Largeau, R. Guillerey, and S. Colson, “Two original UWB antennas: The dragonfly antenna and the valentine antenna,” in Proc. Radar, Toulouse, France, Oct. 2004. [7] Y. Chen, W. T. Joines, Z. Xie, G. Shi, Q. H. Liu, and L. Carin, “Double sided exponentially tapered GPR antenna ant its transmission line feed structure,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2615–2623, Sep. 2006. [8] X. Begaud, C. Roblin, S. Bories, A. Sibille, and A. C. Lepage, “Antenna design, analysis and numerical modeling for impulse UWB,” presented at the WPMC, Italy, Sep. 2004. [9] V. I. Koshelev, V. P. Gubanov, A. M. Efremov, S. D. Korovin, B. M. Kovalchuk, V. V. Plisko, A. S. Stepchenko, and K. N. Sukhushin, “High-Power ultrawideband radiation source with multielement array antenna,” presented at the 13th Int. Symp. on High Current Electronics, 2004. [10] Yu. A. Andreev, Yu. I. Buyanov, and V. I. Koshelev, “Combined antennas for high-power ultrawideband pulse radiation,” presented at the 14th Int. Symp. on High Current Electronics, 2006. [11] D. Ghosh, A. De, M. C. Taylor, T. K. Sarkar, M. C. Wicks, and E. L. Mokole, “Transmission and reception by ultrawideband (UWB) antennas,” IEEE Trans. Antennas Propag., vol. 48, no. 5, pp. 67–99, Oct. 2006. [12] A. Hizal and U. Kazak, “A broadband coaxial ridged horn antenna,” in Proc. 19th Eur. Microwave Conf., Oct. 1989, pp. 247–252. [13] Q. Liu, C. L. Ruan, L. Peng, and W. X. Wu, “A novel compact archimedean spiral antenna with gap-loading,” Progr. Electromagn. Res. Lett., vol. 3, pp. 169–177, 2008. [14] G. R. Salo and J. S. Gwynne, “UWB Antenna Characterization and Optimization Methodologies,” in Ultra-Wideband Short-Pulse Electromagnetics 6, E. K. Mokole, M. Kragalott, and K. F. Gerlach, Eds. New York: Springer, 2003, pp. 329–336.

Adrien Godard was born in Le Mans, France, in 1982. He received the Engineering and Masters degrees in electrical engineering from the University of Nantes, Nantes, France, in 2006. He is currently a Ph.D. student in electronics in the Département Ondes et Systèmes Associés (OSA), XLIM Research Institute, University of Limoges/National Center for Scientific Research (CNRS), Brive, France. His main research interests include high power UWB antenna array and transient metrology.

Michèle Lalande (M’92) was born in Noth, France, in 1962. She received the Ph.D. degree in electronics from the University of Limoges, Limoges, France, in 1986. She is currently a Professor with the University of Limoges. She is a member of the Département Ondes et Systèmes Associés (OSA), XLIM Research Institute, University of Limoges/National Center for Scientific Research (CNRS), Brive, France. Her research interests is in the area of antennas and transient measurement applications.

Valérie Bertrand was born in Bellac, France, in 1970. She received the Ph.D. degree in electronics from the University of Limoges, Limoges, France, in 1996. She is currently an Engineer with the Center of Technology Transfer CISTEME, Limoges. She works in the area of optoelectronics device conception, antennas, and transient measurement applications.

Joël Andrieu was born in Figeac, France, in 1964. He received the Ph.D. degree and the Research Directorship Habilitation from the University of Limoges, Limoges, France, in 1990 and 2008, respectively. He has been a Lecturer with the University of Limoges since 1991. His research is performed at the Département Ondes et Systèmes Associés (OSA), XLIM Research Institute, University of Limoges/National Center for Scientific Research (CNRS), Brive, France. His research interests include ultrawideband metrology for various applications, radars, EMCs, and HPMs.

Laurent Desrumaux was born in St-Priest, France, in 1985. He received the Engineering degree in electrical engineering from the University of Limoges, Limoges, France, in 2008. He is currently a Ph.D. student in electronics in the Département Ondes et Systèmes Associés (OSA), XLIM Research Institute, University of Limoges/National Center for Scientific Research (CNRS), Brive, France. His research interests include UWB antennas and propagation for radar and HPMs applications.

Bernard Jecko was born in Trelissac, France, in 1944. He received the Ph.D. degree in electronics from the University of Limoges, Limoges, France, in 1979. He is currently a Professor with the University of Limoges, where he manages the Département Ondes et Systèmes Associés (OSA), XLIM Research Institute. His research interests include communication networks, antennas and propagation, UWB systems (metrology, radar, and HPM), EMC problems, and bioelectromagnetism.

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Resonant Effects and Near-Field Enhancement in Perturbed Arrays of Metal Dipoles Carolina Mateo-Segura, Graduate Student Member, IEEE, George Goussetis, Member, IEEE, and Alexandros P. Feresidis, Senior Member, IEEE

Abstract—The equivalent self impedance of a perturbed array of metal dipoles is derived and the resonant effects upon plane wave illumination are studied. It is revealed that as a result of the perturbation, the scattering within a frequency range is dominated by the excitation of the odd mode. This corresponds to significant deviation compared to the unperturbed case. It is demonstrated that within this frequency range, very strong near- fields are excited in the vicinity of the array. Following a careful calculation of the near-fields using the periodic method of moments, the near-fields for a number of perturbed array designs are calculated and an increase in the near-field strength of more than 70 times compared with the incidence is demonstrated. The results are corroborated with HFSS. Index Terms—Frequency selective surfaces (FSS), near fields, perturbed arrays.

I. INTRODUCTION

P

ASSIVE periodic arrays of metallic elements on dielectric substrates have been extensively investigated in the past few decades. Due to their filtering properties on incident plane waves, they have been widely employed as frequency selective surfaces (FSS) at radomes and other antenna systems [1], [2]. Following a configuration proposed in [3] that produces an equivalent high surface impedance, several other geometries have been employed in the past few years to realize other electromagnetic band gap structures [3]–[6], artificial magnetic conductors [6]–[8] as well as left-handed metamaterials [9], [10]. As means to address the inherently narrowband performance associated with their resonance, periodic arrays with multiband response have been proposed in the past [11]–[20]. Multiple resonant bands are introduced by unit cells that include either perturbed discrete elements or fractal and other complex geometries. Manuscript received January 20, 2009; revised May 06, 2009; accepted February 16, 2010. Date of publication May 18, 2010; Date of current version August 05, 2010. C. Mateo-Segura was with the Institute for Integrated Systems, Heriot-Watt University, Edinburgh EH14 4AS, U.K. She is now with the Wireless Communications Research Group, Department of Electronic and Electrical Engineering, Loughborough University, Leicestershire LE11 3TU, U.K. (e-mail: [email protected]). G. Goussetis was with the Institute for Integrated Systems, Heriot-Watt University, Edinburgh EH14 4AS, U.K. He is now with the Institute of Electronics Communications and Information Technology (ECIT), Queen’s University Belfast, Northern Ireland, BT3 9DT, U.K. (e-mail: [email protected]). A. P. Feresidis is with the Wireless Communications Research Group, Department of Electronic and Electrical Engineering, Loughborough University, Leicestershire LE11 3TU, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050416

In [12] it was recognized that there are limitations associated with the far-field performance of multiband arrays. In particular, it was argued that for plane wave incidence, dual-band arrays at a certain frequency exhibit the so-called “modal interaction null”. For a dual band capacitive screen consisting of a periodic array of metallic dipoles, this refers to a frequency point where the array appears transparent to incident plane waves in the far-field. This point always emerges between two full reflection points, thus posing limitations in the bandwidth of the stopbands. However, a detailed investigation into the resonant effects and associated near-fields of perturbed arrays has yet to appear [21]. The electromagnetic properties of near-fields are currently receiving increased attention in the physics and engineering communities [22]–[27]. Near-fields allow overcoming limits that in the far-field are fundamental. Theoretical and experimental results in the past decade have demonstrated the possibilities to produce artificial magnetism at THz and optical frequencies [22], [23], transfer subwavelength images [24], focus electromagnetic waves beyond the diffraction limit [25], and enhance near-fields [24]–[26] for imaging or sensor applications. Recently, a perturbed array was proposed as a near-field chemical or biochemical sensor [27]. In this paper, we investigate the resonance effects in perturbed planar periodic arrays of metal dipoles and demonstrate that small perturbations give rise to strong near-field enhancement within a frequency range. In particular, we investigate the structure with unit cell as shown in Fig. 1 and derive the equivalent self impedance for the perturbed case. Using this analytical expression as well as full-wave results, we study the resonant effects that occur upon plane wave excitation. It is demonstrated that as a result of the perturbation, the underlying physics within a frequency range is dominated by the odd mode, which is a drastic departure from the unperturbed case. Moreover, we show that within this frequency range, very strong currents are excited on the array elements. Following a careful convergence study related to the calculation of the far and near-fields using the method of moments (MoM) [2], we demonstrate a longitudinal magnetic field in the vicinity of the array more than 70 times stronger compared to the incidence. Our results are corroborated with HFSS results to validate the accuracy. II. RESONANT EFFECTS OF THE PERTURBED ARRAY In this section we commence by deriving the equivalent self impedance of a perturbed array. The pole and zeros of the equivalent reactance are identified and the resonant effects exhibited by the perturbed array are discussed. Subsequently, by means of an example the nature of the resonant effects is demonstrated.

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dance with the above, the impedance matrix for the combined perturbed array can be written as

Fig. 1. Unit cell and side view of a perturbed planar periodic array of metal dipoles.

(2) where the subscript is employed to denote the mutual reactance of the two arrays. The reflection coefficient can be calculated as the ratio of the scattered over the incident field. The former can be expressed as a function of the currents excited on the dipole elements while the latter determines the equivalent voltage applied. The definition of the voltage and how this can be obtained is found in [1, ch. 4] and not repeated here for brevity. The reflection coefficient of a normally incident plane wave by the entire array in terms of the self and mutual impedances of the sub-arrays is given by [12] (3) Simple algebraic manipulation yields

Fig. 2. Equivalent circuit for the scattering of plane waves by the periodic array shown in Fig. 1 in the absence of grating lobes.

(4) where

A. Equivalent Impedance of a Perturbed Array Consider a planar periodic array of metal dipoles with self . An equivalent circuit for this array impedance is presented in [1] and is reproduced in Fig. 2 when . The reflection coefficient, , from this array upon plane wave illumination can be calculated as

(5) An equivalent circuit that yields the reflection in (4) is shown in Fig. 2. By simple comparison of (1) and (4), the equivalent impedance of the perturbed array is

(1) At resonance, the imaginary part of the self impedance goes to zero, corresponding to a full reflection for incident plane waves. Next, a perturbation is introduced by changing the length of every other dipole element in the array. Consider an array with a unit cell consisting of two dipoles (Fig. 1). Commencing from the nominal length of the dipoles, one is lengthened, and therefore inductively loaded, and the other is shortened, and therefore capacitively loaded. The modification of the length in both cases is such that the magnitude of the inductive and capaci. The combined structive loading respectively is equal to ture can be conveniently modeled as two coupled unperturbed sub-arrays, an inductively loaded (sub-array 1) and a capacitively loaded (sub-array 2) [1]. can be assumed independent of For small perturbations, the frequency. Moreover, the real part of the self and mutual impedance of the sub-arrays can, to a good approximation, be [1]. Let assumed equal to that of the nominal array, and be the self and mutual impedances of the two sub-arrays. For simplicity, the analysis is restricted to the case of plane waves normally incident on the array. In this case the two mutual . In accorimpedances/admittances are equal, i.e.,

(6) The reactive part of (6) yields two zeros (7) and a pole when (8) Considering the equivalent circuit of Fig. 2, one can conclude that at the two reactance zeros, the reflection coefficient is equal , while at the pole, , i.e., the array appears to transparent to an incident plane wave. The zeros and poles of correspond to the resonances of the perturbed array. be the amplitude of the current flowing in each subLet array element. The vector can then be obtained upon solution of the matrix equation (9)

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For small perturbations, we can assume that the voltage applied to each sub-array is equal to the nominal value, i.e., . Simple calculations produce the currents at the two reactance zeros

(10) (11) and at the reactance pole

(12)

Fig. 3. Reflection coefficient from a free-standing planar periodic array of metallic dipoles (Fig. 1) upon normally incident plane wave as calculated from the impedance matrix and the method of moments for the perturbed (L = 8:2, L = 7:5, W = W = 0:5, D = 7:7, D = 10:6) and unperturbed (L = L = 7:9) case. In both cases d = 2d . Dimensions in mm.

(13) is typically negative and is assumed small [1], Since the second term in the square brackets in (10)–(11) is, to a first approximation, negligible for the top sign of the square roots. To this approximation, the current flowing in the perturbed array . A simple comparison shows that is equal to this is the current at the resonance of the unperturbed array. Therefore for the positive value of the square root in (7), the underlying physics for the resonances of the perturbed and unand excited in perturbed array is similar. The currents the two sub-arrays are flowing in parallel and, to a first approximation, are equal to those excited on the elements of the unperturbed array at resonance. On the contrary, the negative sign of the square root in (7) as well as the reactance pole (8) emerge as a result of the perturbation and correspond to significantly different physics. From (12)–(13) it is evident that the currents in the two sub-arrays from the incidence, flow in opposite directions and at i.e., the sub-arrays at the reactance pole exhibit purely inductive and capacitive behavior. Moreover, significantly larger magniis small tudes of the currents are expected in this case since ). Similar observation of (10)–(11) for the (e.g., compared to bottom sign of the square root reveals significant deviation from the resonance of the unperturbed array. B. Resonant Effects in a Perturbed Array Next, we probe into the resonant effects discussed above by means of an example involving a plane wave normally incident on a free-standing periodic array of metallic dipoles. Referring to Fig. 1, the dimensions of the unperturbed array are , , , . Unless otherwise mentioned, the dipoles are assumed symmetrically located in the unit cell, so that . The reflection characteristics upon a normally incident plane wave have been calculated using periodic spectral domain analysis based on the electric field integral equation and the MoM [2]. Fig. 3 shows the magnitude of the reflection coefficient versus frequency, which peaks at 22.15 GHz, corresponding to the resonance of the unperturbed array. Next, a perturbation is introduced by respectively lengthening and shortening the dipoles within the unit cell. The reflection characteristics from an array

with , are shown in Fig. 3 with a black solid line. In the vicinity of 22.15 GHz, the response matches well that of the unperturbed array. Nevertheless, in accordance with the discussion above and [12], another full reflection point emerges at 17.2 GHz followed by a reflection null at 17.34 GHz. The responses of Fig. 3 have also been reproduced using (4)–(5). A rigorous description of the procedure that yields the self and mutual impedances for the sub-arrays is given in chapter 4 of [1]. For the structure under consideration the following , values have been used (in Ohm): for the array self resisfor the array self reactance, for the array mutual reactance and tance, for the perturbation. The response obtained for the unperturbed . The reflection rearray can be calculated setting sponse obtained by these parameters is shown with a dotted line in Fig. 3. The good agreement with the full-wave validates the impedance matrix model within a range of frequencies from about 15 GHz to about 28 GHz, where the grating lobes emerge and manifest as a sudden dip in the MoM results. Fig. 4 shows the magnitude and phase of the current induced on the dipoles, as calculated by (9) and with MoM upon a normally incident plane wave for the perturbed as well as the unperturbed array. The good agreement further validates the accuracy of both methods. In the vicinity of 22.15 GHz, the currents excited on the elements of the perturbed array converge to those of the unperturbed case, revealing analogous underlying physics in both cases. In particular, the currents in both dipoles flow in parallel with comparable magnitudes and are in phase with the incidence at exactly 22.15 GHz. This frequency is associated to the resonance of the array, indicated by the purely ohmic behavior as well as the full reflection of the incident field. In the case of the perturbed array, the currents induced on the capacitively loaded sub-array 2 are zero at 16.58 GHz and increase rapidly with frequency. Similarly, the currents induced on the inductively loaded sub-array 1 reduce rapidly to a zero value at 18.10 GHz. In contrast with the unperturbed case, between these frequencies the currents in the two sub-arrays are out-ofphase and obtain very large values. At 17.34 GHz, where the perturbed array appears transparent to an incident plane wave, the currents on the two sub-arrays are equal in magnitude and

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Fig. 4. Currents induced on the free-standing planar periodic array of metallic dipoles (Fig. 1) upon normally incident plane wave with E = 1 V=m as calculated from the impedance matrix and the method of moments for the perturbed (SA1: inductive sub-array 1 and SA2: capacitive sub-array 2) and unperturbed case. Dimensions as in Fig. 3.

in phase difference from the incident field (purely reactive). As a result their contributions to the total far-field cancel out. Below this frequency, the current on sub-array 1 is stronger than that on sub-array 2 while above this frequency the situation swaps. At 17.20 GHz, the currents on sub-array 1 are real, positive and higher in magnitude than those on sub-array 2, which are real and negative. As a result, the perturbed array appears fully reflective to the incident plane wave. Away from the 16.58–18.10 GHz frequency range the currents on the two sub-arrays converge to those of the unperturbed case. This study reveals that within the frequency range 16.58–18.10 GHz the physical effects are dominated by the excitation of the anti-symmetric (odd) resonant mode. The excitation of the anti-symmetric mode is known for the case of horizontally spaced layers of vertically stacked pairs of metallic strips [28], [29], where it gives rise to an effective magnetic permeability. However, this is a drastically different case, since the uniplanar features of the array suggest excitation of this mode by the electric rather than the magnetic field. Excitation of the odd mode in this case is possible due to the asymmetry introduced in the perturbed array. This corresponds to drastically different physics compared to the unperturbed case. For small perturbations, the coupling of the odd mode to the incident plane wave is weak. As a result, the external quality factor is large, leading to narrowband and strong resonant effects, manifested here as strong induced currents. These are expected to give rise to strong near-fields, which are calculated in the following section.

Fig. 5. Amplitude (normalized) of the excitation of the fundamental and higher order (a) TM and (b) TE FSH upon normal plane wave incidence for the perturbed array of Fig. 3 at 17.34 GHz.

III. NEAR-FIELD ENHANCEMENT The scattering of plane waves from planar periodic arrays of metal dipoles can be efficiently modeled using the EFIE in the spectral domain and subsequently solved using the Galerkin MoM [2]. The method is well established and is commonly employed to obtain the far-field transmission and reflection characteristics. In this section we present the calculation of the far as well as near-fields upon plane wave incidence, and we demonstrate the strong near-field enhancement as a result of the odd mode excitation. A. Near-Field Calculation According to the Floquet expansion of the fields for a doubly periodic surface, electromagnetic fields can be expressed as a superposition of an infinite set of Floquet space harmonics (FSH). Each FSH satisfies the same boundary conditions at the edges of the unit cell. Therefore, along the direction of periodicity all FSH share the same group velocity and transverse field distribution. However each Floquet harmonic has a unique wavenumber (and therefore phase velocity) along each direction of periodicity. Assuming 1D periodicity, the FSH of order n is characterized by a wavenumber

(14)

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Fig. 6. z-component of the magnetic field in the unit cell of the perturbed FSS of Fig. 3 at z = 0 as calculated using MoM for normal plane wave incidence with E = 1 V=m at (a) 17.34 GHz and (b) 22.15 GHz.

Fig. 7. Vectorial representation of the magnetic field in the unit cell of the perturbed FSS of Fig. 3 at the xz-plane as calculated using MoM for normal plane wave incidence at (a) 17.34 GHz (odd mode) and (b) 22.15 GHz (even mode).

where is the length of the unit cell and is the wavenumber of the 0th order. For a doubly periodic array, FSH are charac, each corresponding to every terized by a pair of integers direction of periodicity. For low frequencies, all but the 0th order harmonic, represent slow waves in the direction of periodicity, and therefore are evanescent in the direction normal to the plane of the array. Thus, they do not contribute to the far-field. However, they need to be taken into account in a rigorous full-wave formulation of the EFIE in the spectral domain. Fig. 5 shows a graph of the relative excitation of TE and TM FSH in the example considered above at 17.34 GHz. Despite some observed oscillation, lower order harmonics in general are more strongly excited and hence contribute more to the electromagnetics of the scattering. This allows for truncation of the infinite dimensions of the problem to a finite set of space harmonics, which are typically employed to model the transmission and reflection coefficient upon plane wave incidence. This graph is instructive in determining the number of FSH that need to be considered for calculation of the near-fields. For a typical capacitive array, such as the one discussed in the previous 400 FSH (20 FSH in and direction) suffice to obtain results that have converged with an accuracy of more than 5% in the far-field. This has been confirmed by our investigations, not shown here for brevity. In order to calculate the near-fields

in the vicinity of the array, contributions by each FSH have to be summed, each weighted according to Fig. 5. According to this graph, to account for FSH that contribute up to 5% of the maximum, we need to consider 90 FSH in direction and 40FSH in direction, to the total of 3600 FSH. This conclusion has been confirmed by our simulations (not shown here for brevity), where we observed fictitious ripples and reduced amplitudes in the near-field distributions if we employ the same number of FSH that is required for convergence in the far-field. B. Enhanced Near-Fields The opposing currents flowing in the two dipoles within the unit cell can be associated with a current loop, where continuity of the current between the two open ends is provided by the displacement current. It is therefore expected that this resonance is associated with strong magnetic fields in the direction normal to the surface. To illustrate this, Fig. 6(a) shows the longitudinal component of the magnetic field at the plane of the array and frequency 17.34 GHz. The illumination is a normally incident plane wave linearly polarized along the -axis and of amplitude 1 V/m. For comparison, Fig. 6(b) also shows the same field component for the same illumination at 22.15 GHz. Interestingly, although in the former case the array appears transparent to the incident wave, significantly higher energy is stored in its vicinity in the form of reactive near- fields.

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Fig. 8. Magnitude of the z-component of the magnetic field at the center of the unit cell for normal plane wave incidence with E = 1 V=m as calculated using MoM and HFSS. Dimensions are as in Fig. 3.

A vectorial representation of the magnetic field at the -plane at both frequencies is depicted in Fig. 7. At 22.15 GHz, the parallel currents in the two dipoles produce circulation of the magnetic field in the same direction [see Fig. 7(b)]. Thus, the normal magnetic field in the center of the unit cell goes to zero. On the contrary, at 17.34 GHz, due to the antiparallel flow of the currents, the magnetic field produced by the dipoles circulates in opposite directions, leading to constructive interference in the area between the dipoles [Fig. 7(a)]. The field distributions shown in Fig. 7 have also been validated using HFSS (not shown here for sake of brevity). It is interesting to note that since a loop element provides a longer current path, the odd (anti-symmetric) mode is associated with a lower resonant frequency compared to the even mode. As a result, the odd mode resonance appears at lower frequencies than the even mode resonance. To quantify the comparison, Fig. 8 shows the magnitude of the -component of the magnetic field at the center of the unit cell as obtained with HFSS and the in-house MoM (90 FSH in and 40 FSH in have been employed for the calculation). Good agreement between the two results is observed. The discrepancy is due to the lack of conof the location of the maxima of vergence of HFSS. The incident magnetic field, polarized along the -axis, has amplitude of 2.65 mA/m. On the other hand, the magnitude of the normal magnetic field ( -component), at the center of the unit cell, shown in Fig. 8 is 35.7 mA/m, showing a more than ten-fold near-field enhancement. C. Geometrical Considerations Having demonstrated the excitation of the odd mode and validated our simulation tools, we present a more detailed study of the effect of the geometrical parameters on the excitation of the odd mode and the near-field enhancement. The effect of the periodicity along , , on the currents excited at the odd mode is summarized in Fig. 9. Fig. 9(a) depicts the amplitude of the current at the frequency where this is maximum for both dipoles , is varied. The dipoles are located symas the periodicity, (Fig. 1). The metrically within the unit cell, so that maximum of the odd mode current increases with increasing periodicity. This is also the case for the even mode and the unperturbed array (not shown here). Fig. 9(b) shows the point where from the incidence, the currents at the two dipoles are at

Fig. 9. (a) Maximum value of the current for the two dipoles at the odd mode for increasing periodicity D . (b) Current magnitude value at the point where the array appears transparent to incident coming waves (the currents on the two dipoles are equal and at phase 90 from the incidence). All dimensions except D are as in Fig. 3.

6

equal in magnitude and the array appears transparent to incident waves. Interestingly, despite the increased value of the peak current, Fig. 9(b) shows a decreasing trend for the cross-over point. This results in reduced near-fields at the cross-over point. The level of perturbation is known to affect the frequency separation between the two full reflection points as well as the modal interaction null [12]. It is therefore reasonable to expect the bandwidth of the odd mode excitation to be broader for larger levels of perturbation. This is depicted in Fig. 10(a) which shows the lower and upper frequency points where the currents excited on the two dipoles are out of phase as the length of the second dipole varies. Interestingly, the maximum frequency of the odd mode increases almost linearly with the level of perturbation, while the lower frequency remains almost constant. The broader bandwidth observed for greater perturbations suggests that the external quality factor is reduced, and therefore less reactive energy is stored at resonance. This is indeed confirmed in Fig. 10(b), which shows the common current value at the point where the array appears transparent to incident plane waves (cross-over point as in Fig. 9(b). Clearly, the excitation of the odd mode is stronger for smaller perturbation levels. However, attention should be paid to the fact that very narrow resonances will also experience very high thermal absorption. For applications that require strong and localized near-fields, our study reveals that the two dipoles of the unit cell should preferably be arranged in closely located pairs, e.g., so that . Fig. 11 shows the normal component of the magnetic field at the plane of the array and frequency 18.32 GHz, which corresponds to the frequency where the array appears transparent

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times when compared to the incidence. This observation can be particularly interesting for sensing applications, such as those proposed in [27]. IV. CONCLUSIONS The impedance of the perturbed periodic array of metal dipoles has been derived and the resonant effects have been studied. It has been demonstrated that as a result of the perturbation, the physics underlying the scattering of plane waves from such arrays is, within a frequency range, dominated by the excitation of the odd mode, with currents flowing antiparallel between successive dipoles. Corroborated by full-wave results, it was demonstrated that within this frequency range strong reactive fields are stored in the vicinity of the array. A study on the geometrical parameters on the excitation of the odd mode and the near-field enhancement was also presented. ACKNOWLEDGMENT The authors would like to express their gratitude to Profs. S. Tretyakov, C. Simovski and B.A. Munk for useful discussions and encouragement. G. Goussetis wishes to acknowledge the support by the Royal Academy of Engineering under a five-year research fellowship. Fig. 10. (a) Lower and upper frequency points where the currents excited on the two dipoles are out of phase as L varies. (b) Current value at the point where the array appears transparent to incident coming waves (the currents on the two dipoles are equal and at phase 90 from the incidence). All dimensions except L are as in Fig. 3.

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Fig. 11. Z-component of the magnetic field in the unit cell of the perturbed FSS of Fig. 3, where the distance between the dipoles, d, has been reduced to D =9, at z = 0 as calculated using MoM for normal plane wave incidence with E = 1 V=m at 18.32 GHz.

to incident plane waves. The dimensions are as in Fig. 3, except and . As shown, from the fact that significant localized field enhancement is observed. Compared to Fig. 8, the magnetic field component normal to the array at the center of the unit cell is enhanced by a factor of about 6, resulting in an overall near-field enhancement of more than 70

REFERENCES [1] B. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [2] R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces-A review,” Proc. IEEE, vol. 76, pp. 593–1615, Dec. 1988. [3] D. Sievenpiper, Z. Lijun, R. F. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [4] S. Maci, M. Caiazzo, A. Cucini, and M. Casaletti, “A pole-zero matching method for EBG surfaces composed of a dipole FSS printed on a grounded dielectric slab,” IEEE Antennas Propag., vol. 53, no. 1, pp. 70–81, Jan. 2005. [5] G. Goussetis, A. P. Feresidis, and P. Kosmas, “Efficient analysis, design and filter applications of EBG waveguide with periodic resonant loads,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 11, pp. 3885–3892, Nov. 2006. [6] G. Goussetis, A. P. Feresidis, and J. C. Vardaxoglou, “Tailoring the AMC and EBG characteristics of periodic metallic arrays printed on grounded on grounded dielectric substrate,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 82–89, Jan. 2006. [7] 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. [8] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Raisanen, and S. A. Tretyakov, “Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches,” IEEE Trans. Antenna Propag., vol. 56, no. 6, pp. 1624–1632, Jun. 2008. [9] Y. Guo, G. Goussetis, A. P. Feresidis, and J. C. Vardaxoglou, “Efficient modeling of novel uniplanar left-handed metamaterials,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1462–1468, Apr. 2005. [10] M. Beruete, I. Campillo, J. E. Rodríguez-Seco, E. Perea, M. NavarroCía, I. J. Núñez-Manrique, and M. Sorolla, “Enhanced gain by doubleperiodic stacked subwavelength hole array,” IEEE Microw. Wireless Comp. Lett., vol. 17, no. 12, pp. 831–833, Dec. 2007. [11] J. Huang, T.-K. Wu, and S. W. Lee, “Tri-band surface with frequency selective circular ring elements,” IEEE Trans. Antennas Propag., vol. 42, no. 2, pp. 166–175, Feb. 1994.

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[12] R. A. Hill and B. A. Munk, “The effect of perturbating a frequency selective surface and its relation to the design of a dual-band surface,” IEEE Trans. Antennas Propag., vol. 44, no. 3, pp. 368–374, Mar. 1996. [13] A. D. Chuprin, E. A. Parker, and J. C. Batchelor, “Convoluted double square: Single layer FSS with close band spacings,” IEE Elect. Lett., vol. 36, no. 22, pp. 1830–1831, Oct. 2000. [14] J. Romeu and Y. Rahmat-Samii, “Fractal FSS: A novel dual-band frequency selective surface,” IEEE Trans. Antennas Propag., vol. 48, no. 7, pp. 1097–1105, Jul. 2000. [15] J. P. Gianvittorio, J. Romeu, S. Blanch, and Y. Rahmat-Samii, “Selfsimilar prefractal frequency selective surfaces for multiband and dualpolarized applications,” IEEE Trans. Antennas Propag., vol. 51, no. 11, pp. 3088–3096, Nov. 2003. [16] M. Ohira, H. Deguchi, M. Tsuji, and H. Shigesawa, “Multiband singlelayer frequency selective surface designed by combination of genetic algorithm and geometry-refinement technique,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2925–2931, Nov. 2004. [17] 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. [18] M. A. Hiranandani, A. B. Yakovlev, and A. A. Kishk, “Artificial magnetic conductors realised by frequency-selective surfaces on a grounded dielectric slab for antenna applications,” IEE Proc. Microwaves Antennas Propag., vol. 153, no. 5, pp. 487–493, Oct. 2006. [19] G. Q. Luo, W. Hong, H. J. Tang, J. X. Chen, and K. Wu, “Dualband frequency-selective surfaces using substrate-integrated waveguide technology,” IET Proc. Microwaves Antenna Propag., vol. 1, no. 2, pp. 408–413, Jan. 2007. [20] G. Q. Luo, W. Hong, H. J. Tang, J. X. Chen, and L. L. Sun, “Triband frequency selective surface with periodic cell perturbation,” IEEE Microw. Wireless Comp. Lett., vol. 17, no. 6, p. 436, Jun. 2007. [21] B. Munk, Private Communication. [22] A. N. Lagarkov and A. K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys. Rev. B., vol. 53, no. 10, pp. 6318–6336, Mar. 1996. [23] T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science, pp. 1494–1496, Mar. 2004. [24] P. Belov, C. Simovski, and P. Ikonen, “Canalization of subwavelength images by electromagnetic crystals,” Phys. Rev. B, vol. 71, p. 193105, 2005. [25] 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. [26] P. Alitalo, S. Maslovski, and S. Tretyakov, “Near-field enhancement and imaging in double cylindrical polariton-resonant structures: Enlarging superlens,” Phys. Lett. A, vol. 357, no. 4-5, pp. 397–400, 2006. [27] C. Debus and P. H. Bolivar, “Frequency selective surfaces for high sensitivity terahertz sensing,” Appl. Phys. Lett., vol. 91, p. 184102, Oct. 2007. [28] V. A. Podolskiy, A. Sarychev, and V. Shalaev, “Plasmon modes and negative refraction in metal nanowire composites,” Opt. Express, vol. 11, pp. 735–745, Mar. 2003. [29] G. Shvets and Y. A. Urzhumov, “Negative index meta-materials based on two-dimensional metallic structures,” J. Opt. A: Appl. Pure Opt., vol. 8, p. S122, Mar. 2006.

Carolina Mateo-Segura (GS’08) was born in Valencia, Spain, in 1981. She received the M.Sc. degree in telecommunications engineering from the Polytechnic University of Valencia, Valencia, Spain, in 2006. She is currently working toward the Ph.D. degree jointly between the University of Edinburgh and Heriot-Watt University, Edinburgh, U.K. During the first half of 2006, she joined the Security and Defense Department of Indra Systems, Madrid, Spain, as a Junior Engineer. Currently, she is a Research Associate in the Wireless Communications Research Group, Department of Electronic and Electrical Engineering, Loughborough University, Leicestershire, U.K. Her research interests include the analysis and design of frequency selective surfaces, artificial periodic electromagnetic structures with applications on high-gain array antennas and medical imaging systems. Her research has been funded primarily by the Joint Research Institute for Integrated Systems, EPSRC, MRC and BBSRC. Dr. Mateo-Segura was awarded a prize studentship from the Edinburgh Research Partnership and the Joint Research Institute for Integrated Systems to join the RF and Microwave group at Heriot-Watt University, Edinburgh, Scotland, U.K.

George Goussetis (S’99–M’02) received the Electrical and Computer Engineering degree from the National Technical University of Athens, Athens, Greece, in 1998, the B.Sc. degree in physics (first class honors) from University College London (UCL), London, U.K., in 2002, and the Ph.D. degree from the University of Westminster, Westminster, U.K., also in 2002. In 1998, he joined Space Engineering, Rome, Italy, as a Junior RF Engineer and in 1999 the Wireless Communications Research Group, University of Westminster, as a Research Assistant. Between 2002 and 2006, he was a Senior Research Fellow at Loughborough University, Leicestershire, U.K. Between 2006 and 2009 he was a Lecturer (Assistant Professor) with the School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, U.K. He joined the Institute of Electronics Communications and Information Technology at Queen’s University Belfast, U.K., in September 2009 as a Reader (Associate Professor). He has authored or coauthored over 100 peer-reviewed papers three book chapters and two patents. His research interests include the modeling and design of microwave filters, frequency-selective surfaces and EBG structures, microwave heating as well numerical techniques for electromagnetics. Dr. Goussetis received the Onassis Foundation Scholarship in 2001. In October 2006, he was awarded a five-year research fellowship by the Royal Academy of Engineering, U.K.

Alexandros P. Feresidis (S’98–M’01–SM’08) was born in Thessaloniki, Greece, in 1975. He received the Physics degree from Aristotle University of Thessaloniki, Greece, in 1997, the M.Sc. (Eng.) degree in radio communications and high frequency engineering from the University of Leeds, Leeds, U.K., in 1998, and the Ph.D. in electronic and electrical engineering from Loughborough University, Leicestershire, U.K., in 2002 During the first half of 2002, he was a Research Associate and in the same year he was appointed Lecturer in Wireless Communications at the Department of Electronic and Electrical Engineering, Loughborough University, where, in 2006, he was promoted to a Senior Lecturer in the same department. He has published more than 100 papers in peer reviewed international journals and conference proceedings and has coauthored three book chapters. His research interests include analysis and design of artificial periodic metamaterials, electromagnetic band gap (EBG) structures and frequency selective surfaces (FSS), high-gain and base station antennas, small/compact antennas, computational electromagnetics and microwave circuits.

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Design and Implementation of Embedded Printed Antenna Arrays in Small UAV Wing Structures Mohammad S. Sharawi, Member, IEEE, Daniel N. Aloi, Senior Member, IEEE, and Osamah A. Rawashdeh, Member, IEEE

Abstract—Unmanned aerial vehicles (UAV) are extensively being used in exploration, surveillance and military applications. Such vehicles often collect data via special sensors and send the data back to the central station via wireless links. Embedded in wing structure printed antennas will eliminate the drag due to friction, and allow for extended load capability due to their extra light weight. In this work we present the design and implementation of a 4-element linear antenna array embedded in a small UAV wing structure. The antenna array operates in the 2.4 GHz ISM band. Simulations and measurements that characterize the performance of the antenna array are presented. Field measurements show the impact of utilizing beam-forming in enhancing the communication link throughput. Index Terms—Antenna arrays, beam-forming, embedded antennas, printed monopoles, unmanned aerial vehicles (UAVs), WLAN.

I. INTRODUCTION NMANNED aerial vehicles (UAVs) are widely used in military, scientific and exploration missions. They are used routinely to collect and send information back to a ground station that provides real-time information on the covered area. The data transmission from UAVs is done via wireless links. Single antennas and antenna arrays are being used for sending the data back to the ground station for analysis and decision making. NASA is supporting major UAV projects for terrestrial as well as space missions [1]–[3]. Antenna arrays are widely used in communication systems because they provide higher directivity and antenna gain, higher signal to noise ratio (SNR), and thus better data throughput. Also, antenna arrays have the capability of steering the antenna radiation pattern to track the transmit/receive antenna’s direction. Examples of antenna arrays are evident in wireless cellular base stations and the IEEE 802.11 n standard. Antenna arrays can be designed to perform beam-steering and null-steering. The former enables the antenna array to direct its radiation beam into

U

Manuscript received August 30, 2009; revised December 24, 2009; accepted February 19, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported in part by the Michigan Space Grant Consortium (MSGC) and in part by the National Science Foundation (NSF) Major Instrumentation Program, 2005. M. S. Sharawi is with the Electrical Engineering Department, King Fahd University for Petroleum and Minerals, Dhahran 31261, Saudi Arabia (e-mail: [email protected].). D. N. Aloi and O. A. Rawashdeh are with the Electrical and Computer Engineering Department, Oakland University, Rochester, MI 48309 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.2010.2050440

a pre-specified direction according to an optimization criteria, i.e., the incoming carrier to noise ratio (C/No) [4], [5]. C/No optimization can be used to create a null in the radiation pattern in the direction of an anticipated interferer. Beam/Null steering can be accomplished by using the appropriate algorithm that feeds the appropriate voltage and phase levels to the antenna array elements in order to steer the beam or create a null in the direction of interest. Although the utilization of antenna arrays in UAVs has been present in some literature [1], [6]–[9], none of them investigated the integration of such arrays as a part of the mechanical structure of the UAV or used any of the structural components as radiating elements. Also, limited previous work focused on the use of beam steering to enhance the communication quality, the range and the transmission data rates. Utilizing the high gain (high directivity) and beam steering capability of antenna arrays in the design of UAV structures is very attractive for several reasons. 1) Provide the UAV with the capability to transmit at higher data rates; 2) Antenna arrays extend the range of communication with the UAV; 3) Re-use of structural components as radiators can: • reduce weight: thus gaining more flight time and build smaller UAVs; • lower manufacturing costs; • make the system more robust by eliminating external components that can be damaged by the user or during take offs/landings; • internal structures reduce drag which provides longer flight times. 4) Beam steering provides a better communication channel via the ability to “null out” interfering sources, hide the information from enemy areas and direct the transmission to the ground station location. In [1], several antenna elements were suggested to be integrated within the high altitude long endurance (HALE) UAV wing structure. Only helical antennas were realized while structurally integrated antennas like the printed Vivaldi were only simulated. The antennas were made for the L-band, and only wing surface antennas were considered. While in [10], a mechanical tracking mono-pulse radar was realized at the ground station to track and enhance the directivity of the UAV-ground station link. This allowed for tracking ranges of 50–100 m. The antennas on the UAV were not altered, and no arrays were used. In [6], a broadband linear antenna array was used for the measurement of variations in the L-band (1.4 GHz) microwave

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emissions due to consumer electronics interference that can obscure the remote sensing of soil moisture in NASA missions. The array of patch antennas was designed to be attached to the bottom of the body/wings of the UAV with a separate substrate thus adding to its total weight, also it has not been tested on a flying vehicle. While in [7], another L-band patch array was developed for the repeat-pass interferometric synthesis aperture radar (InSAR) instrument. A planar array was developed with 48 patch elements that were to be placed at the bottom of the UAV body structure with a radome height of 1.5 cm. The paper describes the simulation results of such an array and its importance in accurately mapping the crystal deformations associated with natural hazards. No hardware prototyping results were presented. In [8], some key enabling technologies for antenna array implementations for SATCOM and UAV communications were illustrated. The use of UAVs in mobile ground terminal localization was discussed in [9]. A tracking technique for antenna arrays operating in the 2.4 GHz range for UAVs was discussed in [11] using commercially-off-the-shelf (COTS) electronics and an 8-element dipole array. The array was only simulated, and was designed to track the UAV within the ground station. In this work we present the design and implementation of a 4-element linear antenna array embedded in the structural components of a small UAV wing. The basic array element is a small size printed monopole antenna operating in the 2.4 GHz industrial, scientific, and medical (ISM) radio band. The antenna array utilizes beam-forming via digitally controlled RF electronics, and is characterized in an outdoor antenna range facility. Simulation, measurement and field test results are presented and discussed. We have used a 65% smaller size monopole antenna element compared to that in [12] and [13] in our array implementation. The application at hand requires an extremely small element to be printed on the wing structural component because there is very limited space available for the feeding mechanism. Most of the antennas presented in the literature for operation in the 2.4 GHz band will not be suitable, either because of the size limitation or the feeding mechanism. The following Section II highlights the design and modeling of the antenna array, and how we built the array in the wing structure of a small UAV. The paper is organized as follows; Section II illustrates the modeling and construction of the printed monopoles as part of the wing structure for a UAV. Section III presents and discusses the simulation, measurement and field test results. Finally, Section IV concludes the paper. II. MODELING OF PRINTED EMBEDDED ANTENNAS A printed monopole on a 0.8 mm FR-4 substrate with and was used as the basic radiating element for the array. It replaced a similar shape wing-slot as shown in Fig. 1, thus having a structural component as a radiating element. The wing-slot is the basic element in the wing structure of a small UAV. The mini-Telemaster UAV (Fig. 1) used in this work has a wing span of 114.3 cm (45 in.) and a total of 18 wing-slots. The fuselage length is 82.6 cm (32.5 in.). The UAV and can carry a 20 oz. load. is made of balsa wood

Fig. 1. UAV model used and antenna array location within the wing structure.

Fig. 2. Prototyped antenna array within the wing structure.

Fig. 1 shows the location of the antenna array on the UAV wing. Fig. 2 shows a closer look on the embedded antenna array and the feeding cables. and a short The L-shaped monopole has a long arm arm . The ground plane had a length of from . The length of a single the center of the feeding center hole wing-slot is 15.5 cm and its height is 2.08 cm. The inter-wingslot (inter-antenna) spacing is 6.5 cm. The printed embedded antennas are to be fed using RG-316 coaxial cables through the center holes. For the linear antenna array designed, the amplitude was uniform (same for all elements) while the phase was digitally controlled using RF phase shifters. For a linear array, the total gain is obtained using (1) is the linear array factor, and the where is the 3D gain pattern of the L-shaped monopole. The computed for a specific phase as

is

(2)

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Fig. 4. Reflection coefficient of embedded antenna elements. Fig. 3. Diagram of the embedded linear antenna array and its beam-forming electronics.

TABLE I MEASURED VS. SIMULATED ARRAY ELEMENT PARAMETERS

where , is the wave number , is the inter-element spacing, and is the beam steering angle. The progressive phase values in radians are computed from: (3) values are saved in a look-up table in the micro-controller in 10 increments. module for Also, the progressive phases were passed to the simulation models. The 4-element array was modeled using the full wave method-of-moments (MoM) simulation tool FEKO [14]. A was used. mesh size of The feeding cables were connected to four digitally controlled RF phase shifters (HMC647LP3) and then passed to . The RF phase a passive power combiner/splitter shifters receive their 6-bit phase control from a micro-controller board that has a program controlling the phases according to (3). The relative phase between the antennas steer the beam towards a specific elevation angle. This configuration (Fig. 3) allows for beam steering below the wing structure when the UAV is above/on the ground. In this work, it is assumed the beam steering angle is known apriori. Several algorithms can be used to detect the direction of the incoming RF signal (i.e., MUSIC [15], ESPRIT [16], continuous scanning [11], etc.), but are not discussed in this work. III. RESULTS AND DISCUSSION The dimensions of the designed single element L-shaped , , embedded antenna were, and . The GND plane length was found from the center of the hole that had a radius of 3.5 mm. These dimensions were obtained via numerical optimization within the field solver constrained with the wing-slot dimensions and thickness. The overall area of this miniaturized antenna is (from center of feeding hole to the end of the 38 20.8 antenna element), which is about 65% smaller than the ones that appeared in [12] and [13].

for each antenna element The reflection coefficient was tested inside the wing. Fig. 4 shows the simulated and meacurves. It is important to note that all feeding cables sured are of the same length and (2 ft. long with 1.1 dB of loss and 49 of phase @ 2.5 GHz) are passed through the center hole of the wing slots. For example, there are 3 cables each with an outer diameter of 2.29 mm (0.102 in.) that pass in the 7 mm diameter hole of element 1 (see Fig. 3). So, the cables are in very close proximity of the feeding point of the antenna element. The reflection coefficient curves show acceptable agreement between the simulation and measurement results. Table I shows the resvalue and the bandonance of each element, its width (BW) from simulations and measurements. The resonant frequency and BW of the elements allow the embedded array to operate within a WLAN or ZigBee frameworks in the 2.4 GHz ISM band. The UAV wing with the embedded antenna array was tested in an outdoor antenna range facility at Oakland University, Michigan, USA. The antenna was placed on a plastic stand that was 1.2 m high, and sitting on top of a 6 m diameter metallic turn table. The transmit antenna was 9.14 m away on a gantry arm. The test setup is shown in Fig. 5. The feeding RF cables for the array were connected to four digitally controlled phase shifters. Their outputs were passed to a power combiner. The combiner output was connected to the turn table return cable that went back to the measurement equipment. The array steering angle in the elevation plane was controlled using an HSC12 micro-controller based board, that passed the progressive digital phases to the RF phase shifters. Figs. 6–8 show the measured and simulated polar gain patterns for steering angles of 60 , 90 and 130 , respectively. Elevation

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Fig. 5. Embedded antenna array test setup in the outdoor range.

Fig. 8. Gain plot (dBi) in the elevation plane for a steering angle of 130 .

Fig. 6. Gain plot (dBi) in the elevation plane for a steering angle of 60 . Fig. 9. Gain plot (dBi) in the azimuth plane for a steering angle of 90 .

Fig. 7. Gain plot (dBi) in the elevation plane for a steering angle of 90 .

plane cuts ( – plane) are shown, where 0 corresponds to the outer wing edge, 90 exactly beneath the wing and 180 towards the UAV fuselage. This covers the complete area beneath the wing of the UAV. The measured array performance follows that predicted by simulations and from the theoretical values predicted by the expressions in (1)–(2). An azimuth plane cut ( – plane) for was measured and simulated (Fig. 9). Fig. 10 shows the simulated and measured rectangular gain plots for most of the steering angles. The maximum gain values as well as the steering angle direction are presented in Fig. 11 for both simulations and measurements. The results exhibit very good . The maximum gain obagreement between which was about 11 dBi (maximum tained was at gain for a single element is about 2.0 dBi). This angle correis closer sponded to the direction beneath the wing. When to 0 or 180 , the beam is indistinguishable and acts as if no

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Fig. 10. Simulated and Measured rectangular gain patterns for a collection of steering angles.

Fig. 11. (a) Maximum array gain (b) direction of maximum radiation vs. the steering angle of the array.

steering took place which is something that is predicted from the array factor of small antenna arrays. The observation that the maximum gain is not symmetric around the 90 angle may be attributed to the presence of the feeding wires towards one end of the array (going towards the fuselage) and is absent towards the other end of the array.

The gain measurements were obtained using the gain substitution method with a standard gain horn antenna (SGA-20, Seavy Inc.) as the reference antenna. The errors from this gain measurement setup (outdoor antenna range) and using this reference antenna do not exceed 1 dB. The difference between measurements and simulations can be attributed to several factors. First, the feeding mechanism along with the construction of the implemented antenna array compared to simulations. The simulations did not have cables passing in the middle of the center wing slot holes and the feeding is done via a microstrip edge feed that extend 0.8 mm out of the antenna towards the center hole. The implemented array was fed using coaxial cables, that are manually soldered and are 2 ft. long. These cables ran through the center holes of each wing-slot (Fig. 2). In the gain measurements, a 3 dB gain increase beyond the theoretical limit of 6 dB for a 4 element array was observed (maximum of 11 dBi measured compared to about 8 dBi theoretical). This is attributed to measurement errors and the feeding mechanism. The feeding cables are in very close proximity to the radiating elements. As mentioned earlier, the cables pass through the center holes, and since the outer shield is grounded, this presents an extra GND plane perpendicular to the original one beneath the antennas causing the directivity to increase, and thus boosting the gain. The cables cannot be passed to the antenna array except by using these center holes as the model of the UAV wing slots cannot be changed, and we had to utilize the relatively small hole radius to pass them. This effect is not evident in the simulations since the RF cables were not mod-

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Fig. 12. (a) HPBW (b) SLL vs. the steering angle of the array. Fig. 13. Open field satellite image at OU, Michigan (Google Earth).

eled as they were built, rather simple microstrip feeds for each antenna with the appropriate phase excitation were used. The antenna range setup might have slightly added to the measurement errors as well. It is also worth mentioning that the power combiner could have added to the error in the measurements since an imbalance of about 1.07 dB was measured between ports 1 and 4 as will be discussed later. The reference antenna was connected to port 1 of the combiner at calibration. Second, the phase shifter had a resolution of 5.625 per bit (6-bit digital control), and thus most of the bit combinations were slightly lower or higher than the target value. For example, to be able to get a 130 phase shift, a digital combination of is used, which translates to . Third, when characterizing the phase shifters alone, an imbalance of 2 was measured between the 4 samples used. The HPBW and the side lobe level (SLL) of the antenna array as a function of are shown in Fig. 12. The HPBW is almost identical for both simulations and measurements in the range . Values between 21 –27 were measured. The abrupt change in the simulated HPBW curve comes from the up until fact that the main beam will be focused towards the steering angle becomes more than 40 , after which the main beam splits into two (a linear array will have two beams, one and another at ). We believe that the narrower HPBW at in the measurements for the lower steering angles comes from the presence of the feeding cables that might have enhanced the directivity of the antenna array. An array HPBW of 10 to 20 with a beam-forming capability covering from the center angle of 90 is considered acceptable for the operational ranges and speeds of UAVs [11]. Our design satisfies this criteria. The SLL within this same range was comparable between simulations and measurements. The differences could be attributed to the difference in the feed mechanism, as well as the insertion loss difference between the various ports of the power combiner. A maximum difference of 1.07 dB was measured between ports 1 and 4 in the 2.45 GHz band. It is clear that the SLL is not that far from the maximum peak value of the array and grating lobes appear in the pattern as well. This is due to the fact that we are

only using a 4-element antenna array and the spacing between . This inter-elethe elements of the array is greater than ment spacing is predefined by the location of the wing-slots and cannot be changed without a redesign of the wing structure. Increasing the array elements as well as using a non-uniform amplitude distribution on the feed lines will lower the SLL. After the characterization of the embedded antenna array in the outdoor range, an actual field test with a closed loop data transmission was performed to test the effect of the antenna array in an actual data exchange scenario. A ZigBee transceiver (XBee-Pro) [17] board was installed on the UAV, and the output of the array combiner was connected to the RF antenna input of the board. It had an RS-232 serial data loop-back dongle, and was configured to communicate with another module connected to a laptop computer 1300 ft. (0.25 mi) away in an open field as shown in Fig. 13. The module connected to the computer had a wire monopole antenna with a maximum gain of 1.5 dBi. The UAV was placed on a 1.2 m plastic stand. Three steering angles were tested in this setup, . The angles were chosen based on our ability to rotate the plastic stand with good accuracy in the angles . For each , location pair (total of 15), 1000 data packets were sent from the computer and looped-back in the UAV transceiver and received by the computer. Two metrics were recorded for each position (combination): the percent of successful packets received and the received signal strength indicator (RSSI). Fig. 14 shows the percentage of successful packets as a function of and . The high percentage of successful received packets is achieved , or in its vicinity. For example, for , when 100% successful packets are transmitted and received. While and , the percentage drops to almost 2%, for or basically no reception. Fig. 15 shows the RSSI levels as a function of , . It is observed that the highest received power levels are obtained when as expected. This verifies that the beam is steered in the proper direction. For example, when the RSSI level

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link and reduce weight which will enhance the exploration and surveillance missions of such vehicles. ACKNOWLEDGMENT The authors would like to thank Mr. B. Sababha and Mr. H. Yang for their help in testing setup and measurements. REFERENCES

Fig. 14. Percentage of successful packets received as a function of the steering angle ( ) and the UAV wing orientation ( ).

Fig. 15. RSSI as a function of the steering angle ( ) and the UAV wing orientation ( ).

was

when , and it dropped to when . The use of antenna arrays in UAVs enhances their communication links, extends the range of operation and allows for higher data transmissions. In this work, we have designed, built and tested a 4-element linear embedded antenna array in a UAV wing structure. The measurements and field testing showed very promising results that will allow such integration in the future. This will enhance the quality of surveillance and exploration missions. More complicated embedded antenna arrays and algorithms will be the focus of future work. IV. CONCLUSIONS

The design and implementation of a 4-element linear antenna array embedded in a UAV wing structure was demonstrated and tested in an actual field environment. The simulation and measurement results of the gain patterns for different steering an. The gles below the wing structure matched for array used very small size antenna elements that fit into the wing structure. Field measurements of data transmission using ZigBee transceivers verified the correct performance of the antenna array by monitoring the presence of received packets and RSSI. The use of antenna arrays as part of the structural components of UAVs will reduce drag, improve the communication

[1] R. L. Cravey et al., “Structurally Integrated antenna concepts for Hale UAVs,” NASA Langley Research Center, 2006, NASA/TM-2006214513. [2] D. K. Jackson et al., “Evolution of an avionics system for a high-altitude UAV,” presented at the AIAA Infotech at Aerospace Conf., Sep. 2005. [3] A. Simpson et al., “Big blue II: Mars aircraft prototype with inflatablerigidizable wings,” presented at the 43rd AIAA Aerospace Sciences Meeting and Exhibit, Jan. 2005. [4] M. S. Sharawi and D. N. Aloi, “Null steering with minimized PCV and GD for large aperture antenna arrays,” IEEE Trans. Antennas Propag., vol. 55, pp. 2120–2123, Jul. 2007. [5] M. S. Sharawi and D. N. Aloi, “C/No estimation in a GPS software receiver in the presence of RF interference mitigation via null steering for the multipath limiting antenna,” in Proc. IEEE GlobeComm Conf., Nov. 2006, pp. 1–5. [6] L. M. Hilliard et al., “Lightweight linear broadband antennas enabling small UAV wing systems and space flight nanosat concept,” in Proc. IEEE Int. Geoscience and Remote Sensing Symp., Sep. 2004. [7] N. Chamberlain et al., “The UAVSAR phased array aperture,” presented at the IEEE Aerospace Conf., Mar. 2006. [8] C. Quintero et al., “MSAG based MAE-UAV active array antennas,” in Proc. IEEE Radar Conf., May 2001, pp. 393–397. [9] H. Tsuji et al., “Radio location estimation experiment using array antennas for high altitude platforms,” in Proc. 18th Annual IEEE Int. Symp. on Personal, Indoor, and Mobile Radio Communications, Sep. 2007, pp. 1–5. [10] S. Jenvey, J. Gustafsson, and F. Henriksson, “A portable monopulse tracking antenna for UAV communications,” in Proc. 22nd Int. Unmanned Air Vehicle Systems Conf., Apr. 2007, pp. 1–8. [11] L. Gezer et al., “Digital tracking array using off-the-shelf hardware,” IEEE Antennas Propag. Mag., vol. 50, no. 1, pp. 108–114, Feb. 2008. [12] Y. L. Kuo and K. L. Wong, “Printed double-T monopole antenna for 2.4/5.2 GHz dual-band WLAN operations,” IEEE Trans. Antennas Propag., vol. 51, pp. 2187–2192, Sep. 2003. [13] H. M. Chen and Y. F. Lin, “Printed monopole antenna for 2.4/5.2 GHz dual-band operation,” in Proc. IEE Int. Symp. on Antennas and Propagation, Jun. 2003, vol. 3, pp. 60–63. [14] FEKO User Manual South Africa, EM Software and Systems Ltd. (EMSS), 2005. [15] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag., vol. 34, pp. 276–280, Mar. 1986. [16] R. Roy and T. Kailath, “Estimation of signal parameters via rotational invariant techniques,” IEEE Trans. Acoust, Speech, Signal Process., vol. 37, no. 7, pp. 984–995, Jul. 1989. [17] XBee Pro. Data sheet Digi International Inc [Online]. Available: http:// www.digi.com

Mohammad S. Sharawi (S’98–M’06) received the Ph.D. degree in systems engineering from Oakland University, Rochester, MI, in 2006. He is an Assistant Professor of electrical engineering at King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. He was a Research Scientist with the Applied Electromagnetics and Wireless Laboratory in the Electrical and Computer Engineering Department, Oakland University, from 2008 to 2009. He was a faculty member in the Computer Engineering Department, Philadelphia University, Amman, Jordan, from 2007 to 2008. His research interests include RF electronics and antenna arrays, applied electromagnetics, wireless communications and hardware/embedded systems design. From 2002 to 2003, he was a hardware design Engineer with Silicon Graphics Inc., CA. Dr. Sharawi served as the Organization Chair of the IEEE Conference on Systems, Signals and Devices that was held in Jordan in July 2008.

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Daniel N. Aloi (SM’07) received the B.S.E.E. degree (1992), M.S.E.E. degree (1996), and Ph.D. degree in electrical engineering (1999), all from Ohio University, Athens. He is an Associate Professor in the Electrical and Computer Engineering Department and Founder/Director of the Applied Electromagnetic and Wireless Laboratory at Oakland University, Rochester, Michigan. His other professional assignments have included positions as a Visiting Assistant Professor in the School of Electrical Engineering and Computer Science, Ohio University, and as a Senior Project Engineer at OnStar, Inc., which is a subsidiary of General Motors Corporation. His research interests have focused on various aspects of applied electromagnetics and location technology in the aviation and automotive industries. He has been awarded 1.8 M$ in external funding as a principal investigator, published over 40 technical papers and is an inventor of five patents related to automotive navigation. In addition, he served as a key Technical Advisor to the Federal Aviation Administration’s Satellite Program Office for the Local Area Augmentation System. Prof. Aloi is currently an Associate Editor in the area of navigation for the IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS.

Osamah A. Rawashdeh (M’06) received the B.S. degree (with honors), M.S., and Ph.D. degrees in electrical engineering from the University of Kentucky, Lexington, in 2000, 2003, and 2005, respectively. Previously, he held internships at Daimler Benz AG and Siemens AG. In fall 2007, he served as a Lecturer in the Department Electrical and Computer Engineering, University of Kentucky. Currently, he is an Assistant Professor in the Department of Electrical and Computer Engineering, Oakland University, Rochester, MI. His research interests include embedded control systems and reconfiguration-based fault-tolerance particularly in the area of unmanned systems. Dr. Rawashdeh is a member of ACM, AIAA, and ARRL.

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Flat-Top Footprint Pattern Synthesis Through the Design of Arbitrary Planar-Shaped Apertures Alireza Aghasi, Graduate Student Member, IEEE, Hamidreza Amindavar, Eric Lawrance Miller, Senior Member, IEEE, and Jalil Rashed-Mohassel, Senior Member, IEEE

Abstract—The problem of generating a flat-top main beam with an arbitrary footprint for array elements placed in an arbitrary planar aperture is considered in this paper. Some simplifying properties of the Bessel functions, encourages the general framework of the paper to encompass patterns produced by circular aperture and eventually generalize it to arbitrary aperture geometries. In this regard two synthesis methods are presented. The first method is based on the use of the Rayleigh quotient to obtain constant phase array patterns, hence, a class of generally linear phase patterns can be considered. The second approach is based on power pattern synthesis where there is no restriction on the phase of the pattern, hence, it provides us with greater flexibility. The nonlinear problem is appropriately modeled and formulated for amenable performance. These two new methods can exhibit a significant reduction in the number of unknown parameters, and high flexibility in shaping the desired main beam by arbitrary lattice geometry. Index Terms—Array pattern synthesis, flat-top pattern, power patterns, shaped beam antennas.

I. INTRODUCTION HE far field radiation pattern of a planar aperture is related to the current distribution on the aperture itself. The main purpose of the synthesis techniques is to determine the current distribution which generates a desired radiation pattern. In practice, it is not possible to impose a specific current distribution on a finite sized aperture. By utilizing the spirit of sampling theory here, the aperture can be appropriately split into a lattice of uncoupled elements where every element is fed by a sampled excitation of the continuous current distribution. This combination can generate the desired radiation pattern [1]. In many planar array synthesis scenarios the problem is shaping the main radiation beam. This is the problem of interest in many mobile and satellite communication applications where the main lobe is required to specifically conform to a rather complex region of interest and reduce the radiation in

T

Manuscript received March 12, 2009; revised January 17, 2010; accepted February 08, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. A. Aghasi and E. L. Miller are with the Department of Electrical and Computer Engineering, Tufts University, Medford, MA 02155 USA (e-mail: [email protected]; [email protected]). H. Amindavar is with Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran (e-mail: [email protected]). J. Rashed-Mohassel is with Department of Electrical Engineering, University of Tehran, Tehran, Iran (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050420

other directions to prevent out of region radiation and undesired energy losses. In other words the patterns of interest have a specific footprint in which the radiation is constant, i.e., flat-top main lobe, and suppressed in the exterior. One of the classic methods which was presented for linear arrays is Woodward-Lawson (WL) method [2], which was later extended to circular apertures [3]. The major weakness of this method is the lack of control over radiation ripples in desired coverage and side lobe levels. Elliot et al. introduced synthesis methods to obtain footprint patterns from planar apertures as extensions of Taylor circular distributions [4]–[6]. Besides quasianalytical methods based on this idea [7], the rapid increase in available computing power, later laid the groundwork for extensions to more complex distributions and larger array lattices [8]–[10]. The drawback with many of these methods is that they either require the array to lie on a rectangular or circular lattice, or require an angle dependent homothetic transformation (ADHT). It is shown that ADHT is incapable of synthesizing arrays with some arbitrary footprints, and in order to remedy this shortcoming, additional array boundary optimizations should be enforced [11]. In addition to the constraints on the footprint of the pattern, there may be constraints on the lattice shape as well. This can arise when there are limitations on the space and geometry that array lattice may occupy. Although some methods are presented to find the optimum lattice geometry and footprint patterns generated by arbitrary lattices, this problem is still an open problem which needs more study to provide more efficient methods [12], [13]. Regarding the general problem of generating arbitrary footprint patterns from arbitrary array lattices, two scenarios may arise. In some applications there are constraints both on the phase and modulus of the radiation pattern. Here less flexibility is available on the shape of the lattice itself. This is because of the inter-relation between the current distribution and the pattern. In fact in such cases any attempt to fit the current into the specific aperture geometry will cause distortions in the pattern itself and the problem changes into a trade-off problem. The other case which arises in many antenna array synthesis problems is when there are no specific phase requirements for the radiation pattern and it is only required to have a modulus of a specific shape. This class of synthesis problems is known as the power pattern synthesis. This type of synthesis in fact offers an additional freedom to improve the pattern shaping and limit the current within the desired aperture at the expense of solving a nonlinear problem. Generally this problem is neither convex nor can be converted to an equivalent convex problem [14], [15].

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Power patterns and especially flat-top power patterns are more often considered for linear arrays. They have been studied in the literature from various perspectives [15]–[20]. In this paper, which is an extension of our recent works in [21], [22], we present two synthesis algorithms for flat-top footprint patterns generated by arrays lying on lattices of arbitrary geometry. In Section II we reformulate the pattern synthesis problem in terms of a finite basis expansion. In Section III, the synthesis problem is modelled as a Rayleigh quotient problem the solution of which produces constant phase flat-top patterns. Rayleigh quotient mostly inspires the idea of Eigenfilters [23]. The problems with classic Eigenfilters is the need to a reference frequency point [24]. As will be shown, the method proposed here is independent of specifying a reference point or reference pattern function. In Section IV, we consider power patterns which offer more freedom to restrict the current inside a desired aperture and better shaping of the pattern. In Section V, through some synthesis examples, we demonstrate the performance of the new algorithms and finally there are some concluding remarks provided in Section VI. II. PATTERN FORMULATION 1) The Pattern Produced by a Circular Aperture: Consider a current distribution on a planar circular aperture of radius centered at the origin. Here and are the radial and angular coordinates in the aperture. For and being the angular spherical coordinates, the far-field pattern radiated from this aperture is written as

where and denotes the wavenumber. It is shown in [25] that if the current distribution is represented as the Fourier series

be replaced by will give

and an inverse Hankel transform applied to it

(3) From the properties of the Bessel functions of integer order, we have [26]

(4) is the Hankel function of first kind and order . where is expressed as Combining (3) and (4), we see that if (5) where and then

is a positive integer varying for different values of is the positive root of order Bessel function,

(6) as the Bessel function of second kind and order . denoting expressed as in (6), It is important to note here that with all the terms in the summation vanish and produce a near continuous function which satisfies the assumption that is continuous and vanishes for . Generally speaking, if the following expansion is performed to express the desired pattern: (7) where

then the pattern can be written as (8) (1) then the current distribution could be expressed as where (2) where For a given , the quantities can be computed by inverting the Fourier series in (1). With this formulation, we to that of deterhave reduced the problem of finding by inverting (2) for each . Clearly the current mining distribution is restricted within the aperture and therefore should vanish for . If such exists and it is continuous at , the upper bound of the integration in (2) can

For notational purposes in this paper, we define two spaces; the current space with polar coordinates and which has corresponding rectangular coordinates of and , and the pattern space with polar coordinates and having the corresponding rectangular coordinates of and . Since , it is obvious

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plane. shaped region to be a simply connected region in the Also with no loss of generality, one may assume the origin of plane belonging , since can be the by . obtained by multiplying Lets start with a rather straightforward problem where is a priori assigned, the current aperture is circular and the problem is to be solved in a collocation sense, i.e., the should match at some sample summation in (7) and . By definition of the Fourier series points in (9) Fig. 1. Synthesis spaces, and related coordinates.

Fig. 2. A desired shaped region along with other defined regions and parameters.

that and . For the functions already defined in polar coordinates of either spaces (e.g., and ), the same function defined over the rectangular coordinates of that space is denoted with an additional hat above the notation. Fig. 1 illustrates aforementioned conventions which are going to be held throughout the paper. 2) Effective Thinning of Unknowns for Apertures of Finite Radius: Most synthesis problems seek a pattern which plane known as the is constant and nonzero in a subset of coverage or shaped region and zero in a blockage or unshaped region. A typical case is shown in Fig. 2, where the dark region , the lighter region represents the shaped region denoted as and the remaining represents the transient region denoted as region in which excludes and , represents the unshaped region. As already explained in the previous section, to find the cur, the rent distribution corresponding to a desired pattern unknown coefficients in (7) should be found. In practice the infinite number of unknowns in (7) should be made finite through an appropriate series truncation. In this section efficient truncation of the series in (7) is discussed. Although the method can be applied to synthesis scenarios involving multiply connected shaped regions, for sake of simplicity we consider the

At this point the following lemma is true. Lemma: Consider (5) as a collocation problem, where the collocation points are taken to be . If for where , and for some integer values , , then . , the left side Proof: By taking very close to of (5) must be zero according to the assumption. On the right in the numerator side, because of the appearance of of all summation terms, all the terms tend to zero, except the which has a nonzero limit of corresponding term for . Equating both sides results in . Although this lemma is expressed in a collocation framework, it can provide some useful information on how the series in (7) or are should be truncated. In other words if to be shaped analogous to Fig. 2, this lemma would elucidate which terms in (7) have the least impact in shaping the pattern and therefore prone to be neglected in a truncation process. To get into more details about this, we initially need to explain how the zeros of the Bessel functions are distributed and then setup the truncation process. Regarding our problem the zeros of the Bessel functions have the property that for an integer constant , as increases towards also increases towards infinity. They are in fact infinity, lower bounded as [27], [28]

Also for , as increases towards infinity, towards infinity, i.e.,

increases

For a typical pattern as Fig. 2 we define and , where is the Euclidean distance function to the origin. By choosing and and assuming relatively large aper), the following facts can form a sensible tures (i.e., truncation process for (7). depending on , such that Fact 1: There exists a value . the index in (7) can be effectively chosen to be that Discussion: There always exists a positive

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TABLE I ROOTS OF THE BESSEL FUNCTIONS,

=5

Fig. 3. A thinning example for an aperture of a  and a footprint with  : and  : . The coefficients in dark boxes remain in the basis expansion and the others are ignored. According to symmetry, only coefficients are shown. with n

=04 0

=02

As

for , using (9) one may deduce that for . Since for , for all . using the lemma will result , the corresponding Fact 2: For every , where indexes of can be effectively chosen to be smaller than a value of dependent on . Discussion: Because of the ordered structure mentioned with about the roots of the Bessel functions, for every , there exists , a positive root of order Bessel function, which

As an example of the truncation process, Fig. 3 shows a and sample truncation result for an aperture of radius a desired footprint, for which and . The roots of the Bessel functions which may be used in this example are provided in Table I. Using Fact 1 and according to the first is set to 9 as . column of the table, is found according to Fact For every value of from 0 to 8, 2 and referring to the corresponding rows in the table. Later to satisfy (11) according to the using Fact 3 and checking and exist. To rows of the table, it is revealed that only specify the most effective terms they are represented in darker boxes in the figure. Apart from Fact 3 which is applicable to some cases, based on the information we get about the shaped region, the infinite series in (7) can be efficiently truncated as

Besides reducing the dimensionality of the problem, this truncaand therefore in tion can help reduce . In other words when is expressed as

(10) Now since

for

, and

for

, using the lemma results in for and . , there may exist a Fact 3: For some , where value depending on such that the corresponding indexes . of can be chosen as , there can Discussion: For some values where exist , a positive root of order Bessel function, which (11) is flat-top, then is almost constant in the If shaped region, and therefore using (9) results in for and . Therefore if exists, for all and . It should be noted that if does not exist , it means that and therefor a positive , which means fore neither exists for . We should emphasize that this is flat-top truncation is only applicable to cases where and not in general applicable when is flat-top.

then no matter what the values

are, we are guaranteed that

which means has many zeros in . As the aperture tends to radius increases, the zeros get denser and also . This phenomenon is discussed in more details zero in in Appendix A and we have shown that if is a radius slightly , as the size of the aperture increases greater than tends to zero. We would like to emphasize here that the collocation approach based on which Facts 1–3 were formed was only used as a tool to extract the most efficient terms in the infinite series. As will be shown in next sections, having the most efficient terms of the series (7) can help desirably shape the pattern and extend the problem to arbitrary shapes of the aperture and cases is a priori known and itself is not where only and available. Also although Facts 1–3 are valid for themselves, loosening them to and can provide some degrees of freedom for later constraints added to the problem.

AGHASI et al.: FLAT-TOP FOOTPRINT PATTERN SYNTHESIS THROUGH THE DESIGN OF ARBITRARY PLANAR-SHAPED APERTURES

Fig. 4. A typical aperture geometry along with defined regions.

Of course providing this flexibility will be at the expense of increasing the number of unknowns. Summarizing the truncation process introduced in this section we have the following. and find and , and choose some • For a given and . reasonable values for and using find the value of using Fact • For 2. Note that if the desired footprint is independent this is the last step. find through Fact 1. • Using • For every where , using and find and through Facts 2 and 3. and are ex• For negative values of the quantities actly the same as and calculated in the previous and are the same. step, because the roots of to be the To avoid complexity in future notations we assume total number of terms in the truncated series. After desirably indexing the terms, the expansion can be represented over a single index as (12) where (13)

III. A RAYLEIGH QUOTIENT APPROACH TO ARBITRARY FOOTPRINTS THROUGH ARBITRARY APERTURES In previous sections it was shown that the current on a circular aperture of finite radius can be related to the pattern through a finite termed basis expansion. In this section we generalize the problem to planar apertures of arbitrary shape. Fig. 4 shows a typical aperture of arbitrary shape in dark color in the plane. The original aperture is denoted as . We as a subset of a circular aperture, radius can always consider

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. The portion of which is defined as is denoted as . of the circle which excludes Back to the synthesis problem, obtaining a desired pattern may be feasible by finding the unknown coefficients in (12), so , the that the current is as close as possible to zero outside and also the enpattern is nonzero and as flat as possible in excluding ergy of the pattern is minimized in the disk . In practice for relatively large apertures, the last criteria can be further simplified and instead of minimizing the patdefined as the tern energy in such region, minimizing it in excluding . This simplification is valid region based on what we explained in the previous section about the for large apertures. pattern already tending to zero in , the functions are conIn fact for a given and decay relatively fast after it, according centrated about to the quadratic term in the denominator. Also contributing to this decay is the asymptotic inverse square root behavior of the Bessel functions. Another advantage of modelling the synthesis problem as the basis expansion (12) is that since (6) guarantees for , then the minimization of the current plane exmagnitude does not have to be over the entire but rather . cluding Based on what was explained, for a pattern to be generated by an aperture of arbitrary shape, according to Fig. 2 and Fig. 4, the following quantities should be minimized: • The energy of the gradient of the pattern in the shaped region for being optimally flat-top, i.e., . • The energy of the pattern in the unshaped region, or equiv. alently • Some current energy minimization in the forbidden zone , where of the circular aperture, as is an area independent multiple . of A cost function to minimize all aforementioned terms can be expressed as

(14)

where and are two positive regularization parameters. To write (14) in matrix form we exploit numerical integration , strategies according to which are the samples in the region and where the quantities their corresponding weights. Consider having the following samples along with their corresponding weights: • points in denoted as and correfor ; sponding weights of • points in denoted as and corresponding weights of for ; • points in denoted as and corresponding weights of for . Assuming the number of unknown coefficients in (12) , we rewrite in the matrix form as to be

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pose operator. elements

where denotes the conjugate transis a vertical vector as for , is a diagonal matrix with diagonal and

of minimizing synthesis cost function as

and rewrite a total

(19)

where (15)

To rewrite (19) in a matrix form similar numerical integration scheme is used and analogous forms of and matrices terms. Additionally a new [(15)–(18)] are defined for matrix is defined as (20)

(16)

for

and

and for definitions, (19) is now rewritten as

, also

. Using the new (21)

(17) for

and

for

, and (22)

(18) and . for It is obvious that is the trivial minimizer for . Since in the shaped region there are no reference values for and only nonzero values are sought, with no loss of generality , which converts the minimization cost we can assume function to

This is in fact a Rayleigh quotient problem for which an optimal weight can be obtained by finding the eigenvector cor. responding to the smallest nonzero eigenvalue of

where represents a Hadamard product, an operator that multiplies the corresponding elements of and . Similarly trying to minimize (21) with no constraint on , leads to the trivial result, i.e., . To overcome this phenomenon again we use the fact that there are no reference values in and only nonzero values are conthe shaped region for sidered. Therefore without loss of generality we assume one of the unknown coefficients to be nonzero and perform the minimization over the remaining coefficients. This assumption ensures that . Writing the unknown complex weights as for , to consider the most general case we assume the real part of one of the coefficients, say , to be constant as . In other words

IV. A NONLINEAR APPROACH TO POWER PATTERNS As already mentioned seeking power patterns ignores the phase requirements for the pattern. This provides an additional degree of freedom which can improve the results at the expense in of making the problem nonlinear. Again consider (12), where now the unknown coefficients are to be found to form a desired power pattern . As is not and only is flat, necessarily flat in the region in the truncation process Fact 3 should be bypassed and only Fact 1 and Fact 2 are applicable in this problem. Therefore assume that (12) is now rewritten with a new number of terms as . For the power pattern to be optimally flat-top , but one can think of minimizing to avoid an integrand with the appearance of square roots in the denominator and making the formulation easier we think of @ f^ (u; v)=@u or @ f^ (u; v )=@v can easier performed by using the fact that (@ f^ (u; v )=@u) (@f (; )=@ )(@=@u)+(@f (; )=@)(@=@u). 1Calculation

be

=

(23) which provides unknowns. As the patterns of interest have nonzero values at the origin of the plane, it could be a good idea to choose the function as defined in coming out of the summation in (23) to be in (13) corresponding to (8). In other words having and . This is because has the highest absolute value and peak at the origin compared with other functions . Eventually for the minimization problem we seek a vector as

.. .

(24)

AGHASI et al.: FLAT-TOP FOOTPRINT PATTERN SYNTHESIS THROUGH THE DESIGN OF ARBITRARY PLANAR-SHAPED APERTURES

D

D

Fig. 5. (a) The regions and for a rectangular footprint, along with integration sample points of each region, elements positioned on an elliptical aperture and the integration samples in , . (c) Synthesis results.

D N = 997

Assuming such structure for , one may rewrite (21) as (25) where is easily obtained by taking the square root of elements of diagonal matrix . Various nonlinear least square methods can be applied to this problem. The algorithm we propose in this paper is the Levenberg-Marquardt algorithm [29], which is efficiently capable of minimizing (25). To apply this algorithm the Jacobian, is needed which using (22) can be written in matrix closed forms, as shown in Appendix B. For most nonlinear problems calculation of the Jacobian is a time consuming process, but in our case, one of the most computationally in, , , and , is only tensive steps, the calculation of performed once at the beginning of the optimization process and accordingly updating the Jacobian is preformed only through simple matrix operations. V. EXAMPLES To demonstrate the performance of proposed methods, two synthesis examples will be provided. The first and more simple one is the non-circular aperture example in [12]. The second example will consider a more complex scenario. In both examples, the unknown coefficients are initially obtained using the optimization methods addressed in the paper. Later the required current distribution is formed and sampled at the array positions using the sampling method in [1], which provides less distortion of the pattern in the sampling process. For sake of simplicity the integration scheme used for both examples is the simplest form of Riemann integral where the integrating samples are taken from a regular grid.

N = 390, N

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= 1332. (b) Array

region along with the transition region and corresponding inand , are shown in Fig. 5(a). Also tegration samples in spacing placed on the elliptical lat160 array elements of are shown in Fig. 5(b). tice and the integration samples in According to this setting, and . We set , which is sufficiently large to provide required peris set to zero, mentioning the fact that setformance. Also ting it to any other values between zero and 0.1, will not cause any reduction in the number of unknowns. Performing the series truncation process using the aforementioned values will result . in Besides the two methods proposed in this paper, the results using two other methods are also included in this example. The first method is the alternating projections method (AP) [30] based on the idea of sets intersection and the second method is the two step method proposed in [12] where the excitations for an initial pattern are obtained through an approximate inverse Hankel transform and later used as a starting point for a simulated annealing (SA) global search. The algorithms proposed in this paper are based on finding the unknown coefficients in (12) while for both AP and SA methods referenced, the unknown parameters are the excitations of the array elements. In order to make a reasonable comparison between the nonlinear approach proposed in this paper and both AP and SA methods, we initialize the optimizations from identical starting points (it should be noted that the Rayleigh quotient method is independent of a starting point). We start with an ideal satinitial pattern for which the phase is not zero and isfies the constraints of the problem. The approximate current distribution generating this pattern can be obtained using the relation given in [12]. Using this approximation the components of the corresponding current distribution are obtained as

A. Rectangular Footprint and Elliptical Aperture The purpose of this example is to generate a rectangular footand , through an elprint covering the region and . The purpose liptical array lattice with semi axes of is synthesizing flat-top patterns of ripples less than 1 dB in the shaped region and side lobe levels about 20 dB. The shaped

(26) where is related to the ideal desired pattern through (9). Sampling the current distribution formed by (26) at the array po-

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TABLE II THE RESULTS OBTAINED FOR THE RECTANGULAR FOOTPRINT USING DIFFERENT SYNTHESIS ALGORITHMS

D

D

Fig. 6. (a) The regions and for a fractal footprint, along with integration sample points of each region, positioned on a keyhole shaped aperture and the integration samples in for which .

D

sitions would result the initial excitations of the array elements for the AP and SA methods. To use this distribution as a starting in (26) should be expressed in point for our approach, form of (6) with the corresponding coefficients . We have shown in Appendix C that in a collocation sense the values are obtained as (27) Table II provides the synthesis results for this example using the mentioned methods. The simulations were performed on a 2.66 GHz Intel desktop computer. The cost functions are set to minimize the pattern ripples in the shaped region and also reduce the side lobe levels. Initially referring to the second column of Table II we can see how the truncation process has decreased the number of unknowns for our methods. In terms of the performances, the first row shows the results using SA algorithm. This algorithm in theory can converge to the global minima. The price for this convergence is obviously the processing time, which makes this method inefficient for large problems. Also if the minimization is not started form a good initial guess, this method can take an enormous amount of time to converge. When a good initialization was not included, all our attempts to obtain similar results failed. The AP algorithm on the other hand shows a good convergence speed but the results are poor since this method is very much capable of getting trapped into a local minima. The Rayleigh quotient method provides the fastest algorithm beside the low side lobe level and

N = 982

N = 971, N

= 1686. (b) Array elements

a reasonable shaped region ripple. The nonlinear approach also provides good results in the shaped region while having a reasonable side lobe level. Although the processing time for the nonlinear approach is more than the AP method, the pattern specifications are much more satisfying. The pattern resulted using the nonlinear approach is provided in Fig. 5(c). In the next example which is more complex, we have shown that besides the good performance of the Rayleigh quotient, in large problems the pattern shaping can be very much improved using the nonlinear approach. B. Fractal Footprint and a Non-Circular Aperture In many footprint pattern applications dealing with irregular and detailed coverage zones is inevitable. In this example a rather complicated and fractal coverage zone called fudgeflake [31] is considered. Also the desired aperture geometry is taken to be a keyhole geometry and the subject of interest here since it is the union of two basic aperture geometries, an ellipse and a along with rectangle. The fractal coverage zone, labelled as and the corresponding integration samples are depicted in Fig. 6(a). In Fig. 6(b) the keyhole aperture containing 2168 array spacing is depicted. Along with that, and elements of its corresponding integration samples are also shown. The circle . Based on the surrounding the aperture has a radius of and . Accordingly desired footprint, we choose and as they are sufficient to highlight the performance of proposed methods. For a better representation of each method, we separately consider them in the following.

AGHASI et al.: FLAT-TOP FOOTPRINT PATTERN SYNTHESIS THROUGH THE DESIGN OF ARBITRARY PLANAR-SHAPED APERTURES

1) Using Rayleigh Quotient Approach: After applying the truncation process, using Facts 1–3, the number of unknown co. The regularization paramefficients turns out to be eters are set to be and . is in fact a balance between the side lobes and shaped region ripples and can vary according to the application. is chosen to make sure , and that the current magnitude is sufficiently minimized in therefore later when sampled at array positions, causes the least distortion. After applying the method a current distribution as shown in Fig. 7(a) and an array pattern shown in Fig. 7(b) are obtained. As it can be seen the current is well concentrated within . For the obtained pattern, the ripples in the shaped region are less than 1.3 dB and the side lobe levels are less than 20.4 dB. To compare the resulted pattern with the desired footprint, a contour plot of Fig. 7(c) is provided. For the synthesized patterns obtained through the Rayleigh quotient method, as the first term in (14) enforces both real and to carry minimum variation, they are imaginary parts of both enforced to stay almost constant. This means the resulted patterns will be constant phase patterns. For the cases where the pattern is not required to be generally linear phase (which consists the constant phase), the nonlinear approach improves the results. 2) Using the Nonlinear Approach: To generate a power pattern, again the truncation process excluding Fact 3 is applied, reunknown coefficients. An appropriate cost sulting function should also be set through determination of the regularization parameters. Determining the regularization parameters for large nonlinear problems is an open problem of interest [32]–[35]. Various methods can be exploited to determine the regularization parameters. Among the most widely used ones are the L-curve methods [32], [36], generalized cross validation methods [37] and the discrepancy principle [38]. Despite different methods which can be used to determine the regularization parameters in this problem, the truncation process already performed here can make a significant simplification. Heuristiis not relatively cally for cases where the area occupied by large, and can be determined quite independently. By first and setting an appropriate value for , within assuming a few iterations, a pattern is obtained which is optimally flat-top in the shaped region and the related current distribution is con. The pattern energy still needs to be minicentrated within . This is the point where can be determined. An mized in appropriate cost function can now be defined by simultaneously using both parameters. and and Following this idea, by setting running the nonlinear optimization for 100 iterations, an initial pattern is obtained. The initial values used for and are all unity. Although can also be symin the opbolically set to unity, to prevent large values of . timization process, it is assumed to be Now to minimize the side lobes, we take which shows efficient in performing this task. Using the pair of regularization parameters obtained, along with the initialization used at the beginning, the final pattern can be generated. The relative current distribution is shown in Fig. 8(a) which after performing the sampling process generates the pattern depicted in Fig. 8(b). The ripples in the shaped region for this pattern are

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Fig. 7. (a) The current distribution required for the fractal footprint pattern after applying Rayleigh quotient approach. (b) Obtained pattern of the array. (c) A contour plot of the pattern.

less than 0.6 dB and the side lobe levels are less than 20.2 dB. To compare the desired footprint and the obtained footprint a contour plot of the pattern is provided in Fig. 8(c). As expected, by comparing Fig. 8(c) and Fig. 7(c) we can see that using the nonlinear approach can help shaping the pattern in more details. A cost function plot for individual terms of (19) is depicted in

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Fig. 9. A cost function plot indicating each term value in the iterative process, kr jF u; v j k dudv , C jF u; v j dudv where C and C

=

=

( ^( ) )

^ (x; y)j jG

dxdy .

=

^( )

weights in an efficient basis expansion and capable of significantly reducing the computational load. VI. CONCLUSION A still open problem of generating flat-top footprint patterns from array lattices of arbitrary geometry is considered in this paper. After sufficient simplification of the problem, it is represented as a finite basis expansion problem. This representation can significantly reduce the number of unknowns in the problem specially when the diameter of the desired shaped region is relatively small. For the purpose of shaping the pattern two methods are provided. The first approach is capable of generating patterns of constant phase and it is suitable for the applications where the array pattern is required to have equal phase in all directions of the shaped region. This problem ended up with a Rayleigh quotient problem. The second approach considered is suitable for non-specific phase requirements of the desired pattern. In this case the problem is converted to a nonlinear problem for which the Levenberg-Marquardt algorithm is used. To support the ideas presented, illustrative examples are provided. APPENDIX A BOUNDING THE PATTERN VALUES NEAR THE EXTERIOR BOUNDARIES OF Fig. 8. (a) The current distribution required for the fractal footprint pattern after applying the nonlinear approach. (b) Array power pattern obtained. (c) A contour plot of the power pattern.

We already explained how the values and are obtained based on . At this point the pattern is expressed as (28)

Fig. 9, which shows the convergence after almost 200 iterations. Again it should be noted that for both optimization algorithms provided in this example the number of unknowns are reduced significantly and opposing to most array synthesis algorithms (e.g., see [12], [39]), which are based on optimizing the weights of the array elements, our optimization is based on deciding the

where (29)

AGHASI et al.: FLAT-TOP FOOTPRINT PATTERN SYNTHESIS THROUGH THE DESIGN OF ARBITRARY PLANAR-SHAPED APERTURES

In this section we will find an upper bound on , where is a radius sufficiently larger than . Before getting into this problem, we initially provide some bounds on the roots of the Bessel functions which will be later useful. As already mentioned in the paper, the Bessel roots are lower bounded as

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. As a constraint for From (10) we already know that , consider it to be sufficiently large to have (37)

(30)

This constraint basically means that when , at least twice becomes zero. Now for every the function using (29) would result

(31)

(38)

which can be represented in a single form as where

. From (36) we have

(32) where and using (32) and (35) As the upper bound, from [40], [41] we know

and for Denoting (39) (33)

we therefore have

which may be written in a single form as (34)

(40) , the function is a positive and increasing function (it can be easily veri). As a result of these two properties we have fied that For

where

The difference of two Bessel zeros can also be bounded. We know from [28], [42] that for a positive index

A similar expression should be derived for and (33) we have

. Using (31)

(41)

and therefore a more general bound for the difference of two Bessel roots would be written as (35)

To find an upper bound for the resulted expression we do it through two steps. As the first step we prove that

where

Knowing the appropriate bounds for the Bessel roots, we now get back to the original problem. , a given radius in the pattern space and Consider . For every we can always find a positive index such that (36)

(42) To prove this, from (36) and (34) we know that (43) Plugging in the value of

from (39) gives

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At this point we can easily verify that

which finally results (51)

(44) and this constraint which after simplification requires is already satisfied according to (37). Comparing (44) and (43) will directly result (42). In the second step we prove that

(45)

From (37) we have

and hence

This result shows that as the aperture radius gets larger and tends to zero outside the disc . larger, the pattern Moreover the 1/3 inverse power in (50) is in fact a general as tends to infinity. For suffibut loose bound for ciently large , . APPENDIX B CALCULATION OF THE JACOBIAN FOR THE NONLINEAR PROBLEM Using basic differentiation rules for (22), the Jacobian can be easily written as

and (46) Also from (37) we have (47)

where the syntax

represents the

column of

and

and from (39) and (37) it is obvious that (48)

and

Finally combining (46), (47) and (48) would result (45). So far using (38), (40), (41), (42) and (45) we have (49)

Also we have

It is proved in [43] that the Bessel functions are bounded as (50) where would result

. Since

Again using (36), (37) and (30) we get

, using (28)

and

APPENDIX C FINDING THE UNKNOWN COEFFICIENTS Based on the definition of in (26) and the basis expansion (6) for , the aim is to find in the following expansions

and hence (52)

AGHASI et al.: FLAT-TOP FOOTPRINT PATTERN SYNTHESIS THROUGH THE DESIGN OF ARBITRARY PLANAR-SHAPED APERTURES

The right side expressions in (26) and (52) are identical and therefore

(53) For an auxiliary variable , multiplying both sides of (53) by and integrating with respect to from 0 to results

(54) Applying the identity

to both sides of (54) and using the fact that

which is derived from

results in

(55) can be obtained by taking very close Now for a given , in (55). Doing this would cause all the terms in the to summation at the left side to vanish, except the corresponding which in limit tends to . term for Using this fact would change the equality relation in (55) to

or equivalently

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ACKNOWLEDGMENT The authors would like to thank Dr. R. J. Mailloux for his supportive and helpful reviews and feedback. They also appreciate Dr. C. Börgers for his very useful suggestions. REFERENCES [1] R. E. Hodges and Y. Rahmat-Samii, “On sampling continuous aperture distributions for discrete planar arrays,” IEEE Trans. Antennas Propag., vol. 44, no. 11, pp. 1499–1508, Nov. 1996. [2] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Hoboken, NJ: Wiley-Interscience, 2005. [3] J. Ruze, “Circular aperture synthesis,” IEEE Trans. Antennas Propag., vol. 12, no. 6, pp. 691–694, Nov. 1964. [4] R. S. Elliott and G. J. Stern, “Shaped patterns from a continuous planar aperture distribution,” Inst. Elect. Eng. Proc. H Microw., Antennas Propag., no. 6, pp. 366–370, Dec. 1988. [5] R. S. Elliott and G. J. Stern, “Footprint patterns obtained by planar arrays,” Inst. Elect. Eng. Proc. H Microw., Antennas Propag., vol. 137, no. 2, pp. 108–112, Apr. 1990. [6] F. Ares, R. S. Elliott, and E. Moreno, “Design of planar arrays to obtain efficient footprint patterns with an arbitrary footprint boundary,” IEEE Trans. Antennas Propag., vol. 42, no. 11, pp. 1509–1514, Nov. 1994. [7] J. A. Rodriguez, A. Trastoy, F. Ares, and E. Moreno, “Arbitrary footprint patterns obtained by circular apertures,” Electron. Lett., vol. 40, no. 25, pp. 1565–1566, Dec. 9, 2004. [8] A. Trastoy, F. Ares, and E. Moreno, “Arbitrary footprint patterns from planar arrays with complex excitations,” Electron. Lett., vol. 36, no. 20, pp. 1678–1679, Sep. 28, 2000. [9] A. Trastoy, E. Ares, and E. Moreno, “Synthesis of non- -symmetric patterns from circular arrays,” Electron. Lett., vol. 38, no. 25, pp. 1631–1633, Dec. 5, 2002. [10] F. Ares, J. Fondevila-Gomez, G. Franceschetti, E. Moreno-Piquero, and J. A. Rodriguez-Gonzalez, “Synthesis of very large planar arrays for prescribed footprint illumination,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 584–589, Feb. 2008. [11] J. Fondevila-Gomez, J. A. Rodriguez-Gonzalez, A. Trastoy, and F. Ares-Pena, “Optimization of array boundaries for arbitrary footprint patterns,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 635–637, Feb. 2004. [12] A. Rodriguez, R. Munoz, H. Estevez, E. Ares, and E. Moreno, “Synthesis of planar arrays with arbitrary geometry generating arbitrary footprint patterns,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2484–2488, Sep. 2004. [13] P. O. Savenko and V. J. Anokhin, “Synthesis of amplitude-phase distribution and shape of a planar antenna aperture for a given power pattern,” IEEE Trans. Antennas Propag., vol. 45, no. 4, pp. 744–747, Apr. 1997. [14] H. Lebret and S. Boyd, “Antenna array pattern synthesis via convex optimization,” IEEE Trans. Signal Process., vol. 45, no. 3, pp. 526–532, Mar. 1997. [15] F. Wang, V. Balakrishnan, P. Y. Zhou, J. J. Chen, R. Yang, and C. Frank, “Optimal array pattern synthesis using semidefinite programming,” IEEE Trans. Signal Process., vol. 51, no. 5, pp. 1172–1183, May 2003. [16] Z. Shi and Z. Feng, “A new array pattern synthesis algorithm using the two-step least-squares method,” IEEE Signal Process. Lett., vol. 12, no. 3, pp. 250–253, Mar. 2005. [17] Y. Wen, W. S. Gan, and J. Yang, “Initial value independent optimisation for flat-top power pattern synthesis using non-uniform linear arrays,” Electron. Lett., vol. 41, no. 12, pp. 677–678, Jun. 9, 2005. [18] Y. Wen, W. S. Gan, and J. Yang, “Nonlinear least-square solution to flat-top pattern synthesis using arbitrary linear array,” Signal Processing, vol. 85, pp. 1869–1874, 2005. [19] S. Yang, Y. B. Gan, and T. P. Khiang, “A new technique for power-pattern synthesis in time-modulated linear arrays,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 1, pp. 285–287, 2003. [20] 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. [21] A. Aghasi, H. Amindavar, and E. L. Miller, “Synthesis of planar arrays with arbitrary geometry for flat-top footprint patterns,” in Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP, Apr. 19–24, 2009, pp. 2153–2156.

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[22] A. Aghasi, H. Amindavar, and E. L. Miller, “Flat-top power patterns of arbitrary footprint produced by arrays of arbitrary planar geometry,” in Proc. 13th Int. Symp. on Antenna Technology and Applied Electromagnetics and the Canadian Radio Science Meeting ANTEM/URSI, Feb. 15–18, 2009, pp. 1–4. [23] P. Vaidyanathan and T. Nguyen, “Eigenfilters: A new approach to leastsquares fir filter design and applications including nyquist filters,” IEEE Trans. Circuits Syst., vol. 34, no. 1, pp. 11–23, Jan. 1987. [24] S.-C. Pei and C.-C. Tseng, “A new eigenfilter based on total least squares error criterion,” IEEE Trans. Circuits Syst. I, vol. 48, no. 6, pp. 699–709, Jun. 2001. [25] R. S. Elliott, Antenna Theory and Design, revised ed. ed. Hoboken, NJ: Wiley, 2003. [26] G. Watson, A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge Math. Library, Cambridge Univ. Press, 1995, Reprint of the Second (1944) Edition. [27] E. Ifantis and P. Siafarikas, “A differential equation for the zeros of Bessel functions,” Applicable Analysis, vol. 20, no. 3–4, pp. 269–281, 1985. [28] A. Elbert, “Some recent results on the zeros of Bessel functions and orthogonal polynomials,” J. Comput. Appl. Math., vol. 133, no. 1–2, pp. 65–83, 2001. [29] K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2nd ed. Kgs. Lyngby: Technical Univ. Denmark, 2004. [30] O. Bucci, G. Franceschetti, G. Mazzarella, and G. Panariello, “Intersection Approach to Array Pattern Synthesis,” vol. 137, no. 6, pp. 349–357, 1990. [31] G. Edgar, Measure, Topology, and Fractal Geometry. New York: Springer, 2008. [32] M. Belge, M. E. Kilmer, and E. L. Miller, “Efficient determination of multiple regularization parameters in a generalized l-curve framework,” Inverse Problems vol. 18, no. 4, pp. 1161–1183, 2002 [Online]. Available: http://stacks.iop.org/0266-5611/18/1161 [33] C. Brezinski, M. Redivo-Zaglia, G. Rodriguez, and S. Seatzu, “Multiparameter regularization techniques for ill-conditioned linear systems,” Numerische Mathematik, vol. 94, no. 2, pp. 203–228, 2003. [34] Y. Zhang, A. Ghodrati, and D. Brooks, “An analytical comparison of three spatio-temporal regularization methods for dynamic linear inverse problems in a common statistical framework,” Inverse Problems, vol. 21, no. 1, pp. 357–382, 2005. [35] F. Bauer and O. Ivanyshyn, “Optimal regularization with two interdependent regularization parameters,” Inverse Problems, vol. 23, pp. 331–342, 2007. [36] P. Hansen, “Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms, vol. 6, no. 1, pp. 1–35, 1994. [37] G. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics, vol. 21, no. 2, pp. 215–223, 1979. [38] V. A. Morozov and , Methods for solving incorrectly posed problems, Z. Nashed, Ed. Berlin: Springer, 1984. [39] F. J. Villegas, “Parallel genetic-algorithm optimization of shaped beam coverage areas using planar 2-D phased arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1745–1753, Jun. 2007. [40] Á. Elbert and A. Laforgia, “Further results on McMahon’s asymptotic approximations,” J. Phys. A: Math. Gen, vol. 33, pp. 6333–6341, 2000. [41] A. Laforgia, “Sugli zeri delle funzioni di Bessel,” Calcolo, vol. 17, no. 3, pp. 211–220, 1980. [42] Á. Elbert and A. Laforgia, “Monotonicity properties of the zeros of Bessel functions,” SIAM J. Math. Anal., vol. 17, p. 1483, 1986. [43] L. Landau, “Bessel functions: Monotonicity and bounds,” J. London Math. Society, vol. 61, no. 01, pp. 197–215, 2000.

Alireza Aghasi (GS’09) received the B.Sc. degree from Isfahan University of Technology, Isfahan, Iran, in 2002 and the M.S. degree from Amirkabir University of Technology (Tehran Polytechnic), in 2006, both in electrical engineering. He is currently working toward the Ph.D. degree at Tufts University, Medford, MA. In 2008, he joined Tufts University. His working experiences include mobile network planning and sensor networks and his research interests are antenna arrays, signal and image processing, tomography and imaging, inverse problems and variational modeling.

Hamidreza Amindavar received the B.Sc., M.Sc., and Ph.D. degrees from the University of Washington in Seattle, in 1985, 1987, and 1991, respectively, all in electrical engineering. He also holds an M.Sc. degree in applied mathematics from the same university. He is currently an Associate Professor in the Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran. His research interests include signal and image processing, array processing, and multiuser detection. Dr. Amindavar is a member of Tau Beta Pi and Eta Kappa Nu.

Eric Lawrance Miller (S’90–M’95–SM’03) received the B.S., M.S., and Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge, in 1990, 1992, and 1994, respectively, all in electrical engineering and computer science. He is currently a Professor in the Department of Electrical and Computer Engineering and adjunct Professor of computer science at Tufts University, Medford, MA. Since September 2009, he has served as the Associate Dean of Research for Tufts’ School of Engineering. His research interests include physics-based tomographic image formation and object characterization, inverse problems in general and inverse scattering in particular, regularization, statistical signal and imaging processing, and computational physical modeling. This work has been carried out in the context of applications including medical imaging, nondestructive evaluation, environmental monitoring and remediation, landmine and unexploded ordnance remediation, and automatic target detection and classification. Dr. Miller is a member of Tau Beta Pi, Phi Beta Kappa, and Eta Kappa Nu. He received the CAREER Award from the National Science Foundation in 1996 and the Outstanding Research Award from the College of Engineering at Northeastern University in 2002. He is currently serving as an Associate Editor for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING and was in the same position at the IEEE TRANSACTIONS ON IMAGE PROCESSING from 1998–2002. He was the Co-General Chair of the 2008 IEEE International Geoscience and Remote Sensing Symposium held in Boston, MA.

Jalil Rashed-Mohassel (SM’07) received the M.Sc. degree in electronics engineering from the University of Tehran, Tehran, Iran, in 1976 and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, in 1982. Previously, he was with the University of Sistan and Baluchestan, Zahedan, Iran, where he held several academic and administrative positions. In 1994, he joined University of Tehran where he is teaching and performing research as a Professor in antennas, EM theory and applied mathematics. He served as the academic Vice-Dean, College of Engineering, General Director of Academic Affairs, University of Tehran and is currently Chairman of the ECE Department, Principal member of Center of Excellence on Applied Electromagnetic Systems and the Director of the Microwave Laboratory. Prof. Rashed-Mohassel was selected as the Brilliant National Researcher by the Iranian Association of Electrical and Electronics Engineers in 2007, and was the Distinguished Professor (2008–2009) in the 1st Education Festival, University of Tehran.

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Synthesis of Unequally Spaced Antenna Arrays by Using Differential Evolution Chuan Lin, Anyong Qing, Senior Member, IEEE, and Quanyuan Feng, Senior Member, IEEE

Abstract—Synthesis of unequally spaced linear antenna arrays is considered in this paper. A recently developed new differential evolution algorithm is applied to solve the problem. Both positiononly and position-phase synthesis have been studied. Effect of angle resolution has also been investigated. Synthesis results show that our algorithm is able to obtain better synthesis result reliably and efficiently. Index Terms—Angle resolution, antenna arrays, best of random, differential evolution, pattern synthesis.

I. INTRODUCTION

T

HE main target of antenna array synthesis is to find appropriate excitation vector and layout of the elements to generate desirable radiation pattern. It is therefore a classic optimization problem in electromagnetics. In recent years, synthesis of unequally spaced array, also termed as aperiodic array, has attracted increasing attention [1]–[11]. Unlike equally spaced array or periodic array, the lack of periodicity in the unequally spaced array enables lower sidelobe level (SLL) and reduced number of elements with given aperture size [1]. Another advantage of unequally spaced array is that a low SLL design can be obtained by using uniform amplitude excitation, while in equally spaced array the SLL is generally suppressed by using non-uniform amplitude excitation, which increases system cost and difficulties in designing feeding network [1], [2]. However, synthesis of unequally spaced array also imposes tough challenges to antenna engineers. The challenges mainly come from the nonlinear and non-convex dependency of the array factor to element positions and excitation phases [1]–[3]. The constraints placed on element positions also increase the difficulty of synthesis. Various analytical and numerical techniques have been developed to synthesize unequally spaced array [1]–[11]. In recent years, evolutionary algorithms (EAs) including genetic Manuscript received July 10, 2009; manuscript revised November 25, 2009; accepted January 25, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported in part by the China Scholarship Council, the Ph.D. Innovation Foundation of Southwest Jiaotong University (No. 2008-3), the National Natural Science Foundation of China (No. 60990320; 60990323; 10876029) and in part by the National 863 Project of China under Grant 2009AA01Z230. C. Lin and Q. Feng are with the School of Information Science and Technology, Southwest Jiaotong University, Chengdu 610031, China (e-mail: [email protected]). A. Qing is with Temasek Laboratories, National University of Singapore, Singapore 117411, Singapore . Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2048864

algorithms (GAs) [5]–[9], particle swarm optimization (PSO) [1], simulated annealing (SA) [10] and so on, have become more and more popular in this community as they are applicable to both regular and irregular arrays, with or without constraints. Moreover, they can simultaneously optimize excitations and element positions. Sometimes, the analytical procedures and stochastic optimization algorithms are hybrid to get better performance. For example, in [11], a hybrid method, which combines the most attractive features of GA and those of difference sets method, was presented to synthesize massively thinned antenna arrays. In general, unequally spaced arrays can be classified into nonuniform arrays and thinned arrays [1]. In a nonuniform array, the number of elements is fixed and the element positions are optimized in terms of real vector. On the other hand, in a thinned array, a binary string representing the on/off status of all elements in an equally spaced array has to be determined to obtain lower SLL, hence the actual number of elements is not fixed. When an evolutionary algorithm (EA) is applied to the synthesis of unequally spaced arrays, a real-number EA is usually used for the synthesis of nonuniform array, while a binary EA is suitable for the synthesis of thinned array. In this paper, we focus on the synthesis of nonuniform arrays by using a new differential evolution (DE) algorithm, which is suitable for real optimization. Differential evolution is a simple, efficient and robust evolutionary algorithm [12]–[14] and is usually marked as DE/x/y/z, where x denotes how the differential mutation base is chosen, y denotes the number of vector differences added to the base vector and z indicates the crossover method. It has been successfully applied to array synthesis problems [2], [15], [16], electromagnetic inverse problems [17]–[22] and many other scientific and engineering problems [14]. Although it has been reported that differential evolution performs better than many other algorithms, it is still a dream for differential evolution practitioners to have a strategy perfectly balancing exploration and exploitation, or equivalently, reliability and efficiency. It has been well known that the critical idea behind the success of differential evolution is the creative invention of differential mutation. Different differential mutation strategies balance exploration and exploitation differently [14], generally converges [19], [23]. For example, faster due to the guidance by the best individual but may be locally trapped because of loss of diversity, while gains diversity at the cost of efficiency. In order to simultaneously provide both diversity and guidance so that exploration and exploitation can be more effectively balanced, a new differential mutation, namely the best of

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random differential mutation, or in short , has been proposed. In this paper, a dynamic differential evolution (DDE) with the best of random differential mutation, DDE/BoR/1/bin, is applied to position-only and position-phase synthesis of unequally spaced arrays. Synthesis capability, reliability and efficiency of DDE/BoR/1/bin are tested. The simulation results show that DDE/BoR/1/bin is able to achieve lower peak sidelobe levels (PSLLs) than those reported in previous literatures [2], [3], [5], [15], and converge reliably and efficiently. The rest of this paper proceeds as follows. Section II formulates synthesis of a general linear antenna array. The differential evolution implemented to synthesize the concerned antenna arrays is briefed in section III for completeness and better readability. Numerical results of position-only and position-phase synthesis of symmetric unequally spaced linear arrays are presented in section IV. Conclusion is given in Section V.

rays with uniform amplitude excitation by adjusting its excitation phases and element positions. Thus, the objective function to be minimized can be written as [2] (5) In order to reduce mutual coupling and prevent grating lobes , is [3], the distance between adjacent elements, restricted in . In this regard, the objective function is redefined as (6) where

is the vector of distances between adjacent elements.

D. Symmetric Linear Array Elements in a symmetric linear array satisfy the following constraint:

II. PATTERN SYNTHESIS OF LINEAR ANTENNA ARRAY A. Array Factor The array factor can be expressed as [2]

(7)

of a linear antenna array at angle

(1) where is the vector of excitation amplitudes, is the vector of array element positions, is the vector of excitation phases, and is wavelength. Please note that element 0 is virtual for arrays containing even elements. B. Peak Sidelobe Level The peak sidelobe level (PSLL) of the antenna array is defined as (2) where is the space spanned by angle excluding the mainlobe with the center at . In this paper, uniform amplitude excitation is considered as it is an effective way to reduce system cost and hardware implementation complexity [1], [2]. Thus (1) becomes

Therefore, only elements to symmetry.

have to be determined due

III. DIFFERENTIAL EVOLUTION A. Classic Differential Evolution Without loss of generality, minimization of a single objective function is concerned in the following description. Classic differential evolution (CDE) optimizes an objective function with individuals. It involves two stages, namely, a population of initialization and evolution. Initialization generates initial population . Then the population evolves from one generation to the next until termination conditions are met. to , the three evolutionary operaWhile evolving from tions, namely, differential mutation, crossover and selection are executed in sequence. Differential Mutation: Differential mutation generates a mutant individual with mutant vector , for each inin according to the following formula: dividual

(3) (8)

Accordingly, (2) becomes (4) In addition, without loss of generality, in this paper,

.

C. Synthesis Objective The focus of array synthesis in this paper is to minimize the peak sidelobe level (PSLL) of the unequally spaced antenna ar-

where is the vector of optimization parameters of differ, and are integer numbers, usuential mutation base ally random, , a constant usually in [0, 1], is the mutation intensity for the vector difference , and are vectors of optimization parameters of donor individand from . uals Crossover: Crossover operator is then applied to generate . Binomial crossover and exa trial individual or child

LIN et al.: SYNTHESIS OF UNEQUALLY SPACED ANTENNA ARRAYS BY USING DIFFERENTIAL EVOLUTION

ponential crossover are two most often implemented crossover methods in DE. In this paper, binomial crossover is used. is generated as follows: In binomial crossover scheme,

otherwise

(9)

where is a real random number uniform in the range [0, 1] and , a constant in [0, 1], is the crossover probability. is required to be different from In this paper, the child any of its parents. This convention is often used in evolutionary duplicates one of its parents, a algorithms. If the child , , will randomly chosen parameter of the child be replaced by the corresponding parameter of the other parent. competes Selection: In the selection phase, the child with and the better one survives in the next generation otherwise.

In , differential mutation bases are randomly chosen from the evolving population, thus a good diversity of differential mutation bases is gained but no overall constructive search guidance is provided. On the other hand, all individuals use the same best individual of the population in as the differential mutation base. Guidance provided by the best individual leads to fast convergence but at the same time risks premature convergence due to lack of diversity of differential mutation bases. The above analysis suggests that differential mutation base providing both constructive guidance and diversity is a promising solution. Based on this idea, a new differential mutation, namely best of random differential mutation, is proposed. Mathematical Formulation: Best of random differential mutation is mathematically expressed as

(10)

B. Dynamic Differential Evolution Classic differential evolution is static in nature from the point of view of population updating and is therefore inherently inefficient. To overcome this problem, the dynamic differential evolution (DDE) was developed [18]. DDE inherits all the basic evolutionary operators of CDE. But it differs from CDE by updating the population dynamically instead of generation by generation. In DDE, if the generated trial individual is better than the corresponding target individual, it will replace the target individual and be used in the following evolution immediately instead of in the next generation. The dynamic updating of population in dynamic differential evolution leads to a larger virtual population and quicker response to change of population status. Thus in general, dynamic differential evolution outperforms classic differential evolution. For more details of dynamic differential evolution, please refer to [18]. C. Best of Random Differential Mutation Exploration and Exploitation: Most evolutionary algorithms including differential evolution try to balance two contradictory aspects: exploration and exploitation. In fact, this problem can also be understood from another point of view: the balance of diversity and guidance. Increasing population diversity leads to reliable convergence but slows down it. On the other hand, guidance provided by elite individuals accelerates convergence but increases the risk of premature convergence. Generally speaking, differential mutation in differential evolution could be regarded as a kind of local search. Differential mutation base vector acts as the local search center and scaled vector differences determine the search range or search step around the center. Thus population diversity heavily depends on the selection strategies of base vectors and scaled vector differences. On the other hand, differential mutation base provides guidance. Good base provides constructive guidance to make the search concentrate mainly on promising regions and hence speeds up convergence.

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(11) where vidual way as

is the vector of optimization parameters of indiwhich is randomly chosen from in the same and and satisfies the condition .

Advantages: Best of random differential mutation, or in short , uses the best individual among randomly chosen individuals as differential mutation base and the remaining worse individuals as donors for vector differences. Randomness of the differential mutation base guarantees its diversity. At the same time, being local elite, the best of random differential mutation base provides limited constructive guidance. Thus a better balance of exploration and exploitation is achieved. is almost the same It can be easily found that except for the additional selection of local as elite among the randomly chosen individuals. Computational cost of the additional local elite selection is negligible. In addiintroduces no extra intrinsic control paramtion, eters. Please note that the parameter y in (11) is also included in and . traditional DE, such as In this paper, the ideas of dynamic differential evolution and the best of random differential mutation are integrated to get better balance of exploration and exploitation. Accordingly, the new differential evolution algorithm is denoted as . IV. NUMERICAL RESULTS DDE/BoR/1/bin is applied to re-synthesize a 37-element symmetric linear array and a 32-element symmetric linear array. Both position-only (PO) and position-phase (PP) synthesis are considered for each concerned symmetric liner array. The synthesis of these two arrays has been studied before by other researchers [2], [3], [5], [15]. The reported PSLLs for position-only synthesis of 37-element symmetric array are 19.42 dB in [3] and 20.56 dB in [5] respectively. For 32-element symmetric array, the reported PSLL of position-only synthesis in [2] is 22.53 dB while the reported PSLLs of

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TABLE I EFFECT OF ANGLE RESOLUTION ON PSLL (dB)

position-phase synthesis are in [15] respectively.

23.34 dB in [2] and

TABLE II OBTAINED BEST PSLL

23.45 dB

TABLE III RETURNED ELEMENT POSITIONS AND PHASES FOR 37-ELEMENT AND 32-ELEMENT ANTENNA ARRAYS

A. Design of Numerical Experiments Symmetric Linear Array: Distance between adjacent eland not longer than , i.e., ements is not shorter than , . Center element of the 37-element symmetric linear array is assumed to locate at origin with . Differential Evolution: The intrinsic control parameter , , , values of DDE/BoR/1/bin are: which are set based on our experience and simulation experiments. According to our experiments, DDE/BoR/1/bin is robust with respect to its intrinsic control parameters. It also performs well with other intrinsic control parameter settings, , , . Due to the space limit, we e.g. , , in this only show the results with paper. The maximum number of objective function evaluations is 500,000. Angle Resolution: In our effort to repeat the reported patterns, it has been found that the angle resolution of plays an important role. Ripples or spines invisible under larger angle resolution may become visible under smaller angle resolution and thus results in higher PSLL. Table I shows the recalculated PSLLs using the array configurations reported in [2], [3], [5], and [15], under different angle resolutions of . It is observed from Table I that results obtained by using angle resolutions equal to or smaller than 0.2 are consistent while using larger angle resolutions may lead to lower PSLL. Therefore, 0.2 is chosen as the angle resolution in this paper unless specified otherwise. Please note that under the angle resolution 0.2 , the recalculated PSLLs for the array configurations reported in [3], [5] are consistent with the reported results while those for the array configurations reported in [2], [15] are a little bit higher than the reported results. The inconsistence may be caused by the use of different angle resolutions or the approximation in the published results. B. Position-Only Synthesis of 37-Element Symmetric Linear Array Synthesis Capability: The optimal solution and the corresponding lowest PSLL are unknown yet even if they have been studied before by other researchers. Our first priority here is to look for a better synthesis to demonstrate the synthesis capability. Therefore, the synthesis process goes on until it uses . The up the maximum objective function evaluations best result obtained by DDE/BoR/1/bin with the aforementioned settings is shown in Table II. The returned lowest PSLL, 22.62 dB, is about 2 dB lower than the known best result in

Fig. 1. Pattern of 37-element symmetric array through position-only synthesis.

[5]. The element positions are shown in Table III for verification purpose. The corresponding array pattern is depicted in Fig. 1. Synthesis Reliability: Besides synthesis capability, reliability is also of great concern. To investigate reliability, 100 independent runs are repeated. Fig. 2 shows the distribution of the returned PSLLs of the 100 runs, among which most of the returned PSLLs are below 22.6 dB. Synthesis Efficiency: Efficiency is equally important to antenna engineers. With the best synthesis at hand, the efficiency can be examined by looking at the number of objective function

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TABLE V POSITION-ONLY SYNTHESIS OF 37-ELEMENT ARRAY USING CDE/rand/1/bin UNDER DIFFERENT VTRS

Fig. 2. Distribution of PSLLs in position-only synthesis of 37-element array. TABLE IV POSITION-ONLY SYNTHESIS OF 37-ELEMENT ARRAY UNDER DIFFERENT VTRS

Fig. 3. Differences between the PSLLs with : .

evaluations taken to reach a synthesis indistinguishable to the best one within engineering tolerance, or equivalently value to reach (VTR). Similarly, 100 independent runs are repeated. A run is successful if an acceptable synthesis is obtained within objective function evaluations, otherwise it is unsuccessful. The number of objective function evaluations of a suc, where cessful run is recorded. The success rate is the number of successful runs in the 100 runs. The minimal, average and maximum number of objective function eval, and , are accorduations of all successful runs, ingly obtained. Please note that the success rate provides us an alternate for reliability. Table IV shows the statistical results under ten different VTRs. It has been observed that the success rate is more than is less than 50,000 when . 90% and In general, the success rate drops and goes up as VTR is lowered. To show the advantages of DDE/BoR/1/bin over traditional CDE/rand/1/bin, the synthesis results of 37-element array using CDE/rand/1/bin under ten different VTRs are also given in Table V. Comparing the results in Tables IV and V, it is observed that the success rates of CDE/rand/1/bin are a little bit higher than those of DDE/BoR/1/bin when the VTRs are very low. But the synthesis efficiency of DDE/BoR/1/bin is much better than that of CDE/rand/1/bin under any VTR. As a whole, a better balance of efficiency and reliability is obtained in DDE/BoR/1/bin. These results are consistent with the previous analysis on different differential mutation strategies.

05

1 = 02 :

and those with

1=

Further Investigation of Angle Resolution: The effect of angle resolution has been studied by checking the PSLL of the same configuration at different angle resolutions, as mentioned above. However, its effect on the synthesis process has not been touched before. Here, it is studied by synthesizing at larger angle resolution and comparing the returned PSLL with that recalculated at smaller angle resolution using the returned . synthesis result. Here, First, the larger angle resolution for synthesis is 0.5 while the smaller angle resolution for recalculation of PSLL is 0.2 . The results of 100 independent runs are shown in Fig. 3. The averaged PSLL difference is 0.257 dB. Now, synthesize using angle resolution 0.2 and re-calculate the PSLL at 0.1 . The corresponding PSLL differences are shown in Fig. 4. The averaged PSLL difference is 0.051 dB only. These observations further justify our choice of angle resolution 0.2 . To further study the effect of number of array elements on the selection of angle resolution, the position-only synthesis of 100-element symmetric array is also simulated here. Similarly as above, the antenna array is optimized under the angle resolutions 0.5 and 0.2 , and then recalculated with the returned configurations under smaller angle resolutions 0.2 and 0.1 , respectively. The VTR is 23.5 dB. 100 independent runs are repeated for each case.

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Fig. 4. Differences between the PSLLs with . :

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1 = 01 :

and those with

1=

Fig. 6. Radiation pattern for a position-only synthesis of 100-element symmetric array with angle resolution 0.2 .

Fig. 5. Radiation pattern for a position-only synthesis of 100-element symmetric array with angle resolution 0.5 .

The simulation results show that when the larger angle resolution for synthesis is 0.5 and the smaller angle resolution for recalculation of PSLL is 0.2 , the averaged PSLL difference is as much as 10.94 dB. When the larger angle resolution for synthesis is 0.2 and the smaller angle resolution for recalculation is 0.1 , the averaged PSLL difference is 0.90 dB. Fig. 5 and Fig. 6 show the radiation patterns responding to the same configuration under the angle resolutions 0.5 and 0.2 , respectively. By comparing Figs. 5 and 6, it is observed that some ripples or spines invisible in Fig. 5 become visible in Fig. 6, which results in great increase of PSLL. These results imply that for larger number of elements, the effect of angle resolution on the calculated PSLL may be very remarkable and a finer angle resolution is necessary.

Fig. 7. Radiation pattern for a position-phase synthesis of 37-element symmetric array.

C. Position-Phase Synthesis of 37-Element Symmetric Linear Array Synthesis Capability: As shown in Table II, when element positions and phases are optimized simultaneously, the returned lowest PSLL is 24.11 dB, about 1.5 dB lower than that of position-only synthesis. The corresponding element positions and phases, and radiation pattern are respectively shown in Table III and Fig. 7.

Fig. 8. Distribution of PSLLs in position-phase synthesis of 37-element array.

Synthesis Reliability: Fig. 8 shows the distribution of PSLLs in the 100 runs. 87 of the 100 synthesis reach PSLL below

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TABLE VI POSITION-PHASE SYNTHESIS OF 37-ELEMENT ARRAY UNDER DIFFERENT VTRS

Fig. 10. Distribution of PSLLs in position-only synthesis of 32-element array.

TABLE VII POSITION-ONLY SYNTHESIS OF 32-ELEMENT ARRAY UNDER DIFFERENT VTRS

Fig. 9. Radiation pattern for a position-only synthesis of 32-element symmetric array.

23.5 dB among which 33 synthesis even reach PSLL lower than 24.0 dB. , and Synthesis Efficiency: Table VI shows , under ten different VTRs. Obviously, a slightly relaxed synthesis requirement may significantly increase the success rate and speed up the synthesis process. D. Position-Only Synthesis of 32-Element Symmetric Linear Array The returned best PSLL is 22.65 dB. The corresponding element positions and radiation pattern are shown in Table III and Fig. 9. The distribution of PSLLs in the 100 runs is shown , and under ten in Fig. 10. Table VII shows , different VTRs. It is observed that the algorithm can easily reach 22.1 dB.

Fig. 11. Radiation pattern for a position-phase synthesis of 32-element symmetric array.

E. Position-Phase Synthesis of 32-Element Symmetric Linear Array The returned best PSLL is 23.45 dB, which is a little better than that of position-only synthesis. The corresponding optimization parameters and radiation pattern are shown in Table III and Fig. 11. The distribution of PSLLs in the 100 runs is shown , and under ten in Fig. 12. Table VIII shows , different VTRs. The simulation results show that the algorithm can converge to 23.2 dB reliably.

V. CONCLUSION A new differential evolution algorithm using best of random differential mutation is applied to the synthesis of unequally spaced antenna arrays. The best of random differential mutation uses the best individual among randomly chosen individuals as the differential mutation base while the remaining worse individuals as donors for vector differences. Hence better balance of

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Fig. 12. Distribution of PSLLs in position-phase synthesis of 32-element array.

TABLE VIII POSITION-PHASE SYNTHESIS OF 32-ELEMENT ARRAY UNDER DIFFERENT VTRS

exploration and exploitation has been achieved. Symmetric aperiodic linear antenna arrays have been synthesized. Both position-only and position-phase synthesis have been studied. Effect of angle resolution has also been investigated. Synthesis results show that our algorithm exhibits stronger synthesis capability, higher reliability and efficiency. REFERENCES [1] 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, 2007. [2] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna arrays using the differential evolution algorithm,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2210–2217, 2003. [3] B. P. Kumar and G. R. Branner, “Design of unequally spaced arrays for performance improvement,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 511–523, 1999. [4] B. P. Kumar and G. R. Branner, “Generalized analytical technique for the synthesis of unequally spaced arrays with linear, planar, cylindrical or spherical geometry,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 621–634, 2005. [5] K. Chen, Z. He, and C. Han, “A modified real GA for the sparse linear array synthesis with multiple constraints,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2169–2173, 2006. [6] S. DeLuccia and D. H. Werner, “Nature-based design of aperiodic linear arrays with broadband elements using a combination of rapid neural-network estimation techniques and genetic algorithms,” IEEE Antennas Propag. Mag., vol. 49, no. 5, pp. 13–23, 2007.

[7] L. Cen, W. Ser, Z. L. Yu, and S. Rahardja, “An improved genetic algorithm for aperiodic array synthesis,” in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing, Las Vegas, NV, 31 Mar.–4 Apr. 2008, pp. 2465–2468. [8] 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, 2002. [9] M. Donelli, S. Caorsi, F. De Natale, M. Pastorino, and A. Massa, “Linear antenna synthesis with a hybrid genetic algorithm,” Progress In Electromagnetic Research PIER 49, pp. 1–22, 2004. [10] 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, 1996. [11] S. Caorsi, A. Lommi, A. Massa, and M. Pastorino, “Peak sidelobe level reduction with a hybrid approach based on GAs and difference sets,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1116–1121, 2004. [12] R. Storn and K. Price, “Differential evolution—A Simple and efficient adaptive scheme for global optimization over continuous spaces.” International Computer Science Institute, Berkley, CA, Tech. Rep. TR-95-012, 1995. [13] K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization. Berlin: Springer, 2005. [14] A. Qing, Differential Evolution: Fundamentals and Applications in Electrical Engineering. New York: Wiley, 2009. [15] 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, 2008. [16] 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, 2005. [17] A. Qing, “Electromagnetic inverse scattering of multiple two-dimensional perfectly conducting objects by the differential evolution strategy,” IEEE Trans. Antennas Propag., vol. 51, no. 6, pp. 1251–1262, 2003. [18] A. Qing, “Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems,” IEEE Trans. Geosci. Remote Sensing, vol. 44, no. 1, pp. 116–125, 2006. [19] A. Qing, “A parametric study on differential evolution based on benchmark electromagnetic inverse scattering problem,” in Proc. IEEE Congress Evolutionary Computation, Singapore, Sep. 25–28, 2007, pp. 1904–1909. [20] A. Massa, M. Pastorino, and A. Randazzo, “Reconstruction of twodimensional buried objects by a differential evolution method,” Inverse Prob., vol. 20, no. 6, pp. 135–150, 2004. [21] I. T. Rekanos, “Shape reconstruction of a perfectly conducting scatterer using differential evolution and particle swarm optimization,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 7, pp. 1967–1974, 2008. [22] A. Massa, M. Pastorino, and A. Randazzo, “The differential evolution algorithm as applied to array antennas and imaging,” in Advances in Differential Evolution, ser. Studies in Computational Intelligence. Berlin, Heidelberg: Springer-Verlag, 2008, vol. SCI-143, pp. 239–255. [23] A. Qing, “A study on base vector for differential evolution,” in Proc. IEEE World Congress Computational Intelligence—IEEE Congress Evolutionary Computation, Hong Kong, Jun. 1–6, 2008, pp. 550–556.

Chuan Lin was born in Fujian Province, China, in 1980. He received the B.Eng. degree in automation from Southwest Jiaotong University, Chengdu, China, in 2003, where he is currently working toward Ph.D. degree. From September 2008 to September 2009, he visited Temasek Laboratories, National University of Singapore as a joint-training Ph.D. student, sponsored by China Scholarship Council (CSC). His research interests include evolutionary computation, antenna arrays, adaptive filter, and so on.

LIN et al.: SYNTHESIS OF UNEQUALLY SPACED ANTENNA ARRAYS BY USING DIFFERENTIAL EVOLUTION

Anyong Qing (SM’05) was born on May 27, 1972. He received the B.Eng. degree from Tsinghua University, China, in 1993, the M.Eng. degree from Communication University of China (formerly Beijing Broadcasting Institute), in 1995, and the Ph.D. degree from Southwest Jiaotong University, China, in 1997. He was a Lecturer and Postdoctoral Fellow at Shanghai University from September 1997 to June 1998, a Research Fellow at Nanyang Technological University, Singapore, from June 1998 to June 2000, a member of scientific staff at University of Kassel, Germany, from July 2000 to May 2001, and an RF Design Engineer at VS Electronics Pte Ltd., Singapore, from June 2001 to September 2001. He joined Temasek Laboratories, National University of Singapore, in September 2001 as a Research Scientist and is currently a Senior Research Scientist. He has involved in various areas of research in electromagnetics and evolutionary computation. He has been pioneering and leading in solving electromagnetic problems using evolutionary algorithms. He has authored two books, four book chapters, 52 peer reviewed journal papers, and 40 conference presentations. His publications have been cited by other researchers about 400 times. Dr. Qing is a member of the Material Research Society Singapore and a member of the Chinese Institute of Electronics. He is a distinguished lecturer of the National Summer School on Inversion and Imaging at the University of Electronic Science and Technology of China in July 2007. He was awarded Guest Professor of Southwest Jiaotong University in 2005. He also serves various international conferences as a committee member. He is an editorial board member for the International Journal of Communication networks and Information Security, and reviewer for different organizations, journals, and conferences.

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Feng Quanyuan (SM’07) is the head of the Institute of Microelectronics of Southwest Jiaotong University, China. In the past five years, he has presided over more than ten important and common projects of the National Natural Science Foundation of China and some of Sichuan Province’s important projects. His research interests include antennas and propagation, integrated circuits design, RFID technology, embedded system, wireless communications, microwave and millimeter wave technology, smart information processing, electromagnetic compatibility and environmental electromagnetics, microwave devices and materials, etc. At present, he has been presiding over one National Natural Science Foundation of China project and one China 863 project. In the past five years, more than 150 of his papers have been published in the IEEE TRANSACTIONS ON MAGNETICS, IEEE COMMUNICATIONS LETTERS, Acta Physica Sinica, the Chinese Journal of Semiconductors, etc., among which more than 80 were registered by SCI and EI. Prof. Feng has been honored as the “Excellent Expert” and the “Leader of Science and Technology” of Sichuan province because of his outstanding contribution. He was awarded the “First Class Award of Scientific and Technologic Progress of Sichuan Province,” “Third Class Award of Scientific and Technologic Progress of Electronic Industry Ministry,” “National Mao Yisheng Scientific and Technologic Award of Chinese Scientific and Technologic Development Foundation,” and so on. He is an advisor of Ph.D. candidates. He is a reviewer of more 20 journals such as IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, IEEE TRANSACTIONS ON MAGNETICS, IEEE MAGNETICS LETTER, Chinese physics and Acta Physica Sinica, etc.

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Multi-Frequency Synthetic Thinned Array Antenna for the Hurricane Imaging Radiometer M. C. Bailey, Life Senior Member, IEEE, Ruba A. Amarin, Member, IEEE, James W. Johnson, Senior Member, IEEE, Paul Nelson, Student Member, IEEE, Mark W. James, David E. Simmons, Christopher S. Ruf, Fellow, IEEE, W. Linwood Jones, Life Fellow, IEEE, and Xun Gong, Member, IEEE

Abstract—A C-band four-frequency resonant stacked-patch array antenna is developed for synthetic thinned aperture radiometer measurements of hurricane force wind speeds. This antenna is being integrated into an aircraft instrument referred to as the Hurricane Imaging Radiometer (HIRAD). Details of the antenna design are presented along with antenna performance tests and laboratory measurements using a full-scale prototype array with a subset model of the HIRAD instrument. Index Terms—Microstrip antennas, multifrequency antennas, remote sensing, stacked microstrip antennas, synthetic aperture imaging.

I. INTRODUCTION

T

HE airborne Hurricane Imaging Radiometer (HIRAD) is currently being developed by NASA’s Marshall Space Flight Center in collaboration with NOAA and the Universities of Michigan and Central Florida. HIRAD will provide the accurate observations of ocean surface winds in the presence of intense rain that are required for forecast and warnings of tropical cyclones. NOAA’s Stepped Frequency Microwave Radiometer (SFMR) is an aircraft borne remote sensor currently used for this purpose [1], and HIRAD will be an improvement on SFMR. The SFMR is nadir viewing and produces a surface wind profile along the ground track for each hurricane pass, whereas HIRAD will have a wide swath and, from an altitude of 20 km, image approximately 60 km of the storm on each pass. The HIRAD concept includes the multi-frequency C-band capability of the SFMR that has been performance proven and adds a form of electronic scanning with a stationary array antenna that is relatively new to earth remote sensing [2]. The

Manuscript received October 27, 2009; revised February 04, 2010; accepted February 05, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by the HIRAD Project of the NASA Marshall Space Flight Center under a grant with the University of Alabama Huntsville Award NNM05AA22A. M. C. Bailey is with the University of Central Florida, Orlando, FL and also with Applied EM, Hampton, VA 23666 USA (e-mail: [email protected]). R. A. Amarin, J. W. Johnson, P. Nelson, W. L. Jones, and X. Gong are with the University of Central Florida, Orlando, FL 32816 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). M. W. James is with NASA, Marshall Space Flight Center, Huntsville, AL 35806 USA (e-mail: [email protected]). D. E. Simmons is with the University of Alabama Huntsville, Huntsville, AL 35816 USA (e-mail: [email protected]). C. S. Ruf is with the University of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2050453

Fig. 1. Schematic of HIRAD array with active elements shown in black.

result is the need for both multi-frequency, C-band operation and a scanning array radiometer system. HIRAD operates as an interferometer, sampling the spatial Fourier components of the cross-track brightness temperature scene. The HIRAD antenna is a microstrip array that operates simultaneously at four frequencies over approximately an octave bandwidth to enable the independent retrieval of wind and rain [3], [4]. The aircraft version of HIRAD currently being developed has a 0.82 m 0.57 m array antenna with 37 16 individual 4-frequency elements as illustrated in Fig. 1. Each array element is a linearly polarized 4-frequency stacked-patch antenna oriented with the electric field vector polarized in the along-track direction. Additional terminated elements were placed around the periphery of the array in order to provide nearly the same array environment for all the elements for impedance matching purposes. The array is thinned to 10 linear elements, shown as 10 lines of small solid rectangles in Fig. 1, each producing a fan beam pattern for cross-track synthesis and along-track pushbroom surface sampling. In the flight version, each linear element is fabricated with a 16-way uneven power combiner integrated onto the back of the array ground plane. These 10 elements, configured with 10 mutually coherent radiometers, taken in all possible pair combinations, can fully sample (at the Nyquist rate)

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Fig. 2. Cross-sectional sketch of stacked-patch antenna (not drawn to scale). Fig. 4. Sketch of stacked-patch antenna with top two patches removed.

have been placed in the lower frequency patches to reduce cross-polarized radiation at the highest resonance. This paper provides a detailed description of the individual element design and its evolution, with some emphasis on crosspolarization performance, plus full-scale prototype array test results and synthesized patterns from interferometer measurements. The prototype array design is currently being duplicated in a flight array for an aircraft flight instrument. Fig. 3. Top view sketch of stacked-patch antenna.

all spatial components of the brightness temperature scene [5]. The HIRAD design produces a spatial resolution at nadir of 2.5 km at 4 GHz, from a 20 km altitude, and a field-of-view of approximately 60 degrees at all 4 frequencies, for a cross-track swath width of 60 km. The individual array elements consist of 4 probe-fed, rectangular stacked patches, which in the original design resonated at 4, 5, 6, and 7 GHz. This wide frequency range required element spacing in the array to physically accommodate the lowest frequency radiators, but at the same time limit the spacing at the highest frequency to approximately 1/2 wavelength or less to limit grating lobe encroachment into the cross-track field-ofview. Given these constraints, the center-to-center spacing of individual elements was set at 38.1 mm along-track and 22.86 mm cross-track. It was later determined that an element pattern null could occur in the along-track direction near broadside with this array geometry; therefore, the upper frequency was reduced to 6.6 GHz in order to move this “array blindness” null out of the main beam of the along-track array factor. Many papers are available in the literature describing the design of stacked-patch antennas of various configurations. These generally consist of dual-stacked or triple-stacked patches designed either for bandwidth enhancement or for multiple operating bands. In these stacked-patch concepts, the coupling between the patches of the stack becomes a significant consideration, requiring accurate computer modeling to achieve an acceptable design. Typically these designs are focussed on applications which require only a single antenna. In array applications, the coupling between stacked-patch elements of the array can also become a significant design consideration. This present effort extends the microstrip antenna technology to the design of quad-stacked patches in an array environment. Coupling between patches in the stack and also coupling between stacked-patch elements of the array are taken into consideration in the design process. Strategically located slots

II. ELEMENT DESIGN The individual array elements for this present array development are four stacked-patches with a single probe feed soldered directly to the top patch and reactively coupled to the other three patches via a 2.794 mm diameter etched circle. Each patch is designed to resonate at a specific frequency while accounting for mutual coupling between stacked patches as well as coupling to adjacent stacked-patch elements of the array. The first design utilized square stacked-patches with equal thickness dielectric layers and included circular disks between patches to add capacitance for tuning out the inductance of the feed probe. Details of that earlier design are described in [6]. A commercially available method-of-moments (MoM) software package (ie3d) was used for all designs and another commercially available finite-element-method (FEM) software package (HFSS) was used later for further verification and analysis of the test results. Earlier comparisons between ie3d impedance simulations and laboratory measurements verified that the basic design was viable. The present design differs from the earlier concept by: 1) removal of the embedded tuning disks due to fabrication difficulties; 2) change to rectangular patches with a 2:1 aspect ratio to reduce cross-pol and allow closer element array spacing cross-track; 3) use of narrow slots in the bottom two patches to further reduce cross-pol at the highest resonance; and 4) use of unequal dielectric thicknesses between patches to increase the bandwidth at the lower resonances. In order to avoid special ordered materials, standard commercially available substrate thicknesses, close to that desired, were used in the design. RT/Duroid 5880 was used as the substrate material in the earlier stages of development and the earlier laboratory proof of concept evaluations; however, RT/Duroid 6002 was selected for the final design and fabrication because of its superior thermal expansion properties. The present stacked-patch antenna concept is illustrated in Figs. 2–4 with the height of the cross-sectional view being exaggerated for visual clarity. In the sketch of Fig. 4, the top two patches are removed to illustrate the etched slots in the bottom two patches. It

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TABLE I DIMENSIONS OF STACKED-PATCH ANTENNA ELEMENTS

Fig. 5. Partial array configurations for computer simulations of embedded element impedance (3 3 cluster) and patterns (3 5 cluster).

2

2

was observed during computer simulations that, when the stacked-patch antenna is placed in the array environment, significant cross-polarized radiation was observed at the highest resonance, due to mutual coupling with cross-polarized currents excited on the bottom patches of the adjacent elements. These slots were etched on the bottom patches to disrupt those orthogonal currents in an attempt to encourage the current to flow more in the desired direction. Computer model simulations indicated that indeed a reduction of cross-polarized radiation at the highest resonance could be realized; therefore, all the stacked-patch elements in the final array were fabricated with these slots. Dimensions for the final antenna element design are listed in Table I. Because of the large computer memory and CPU time required for accurate modeling of the array, a partial array of elements was used to design the stacked-patch antenna element. This partial array allows mutual coupling between adjacent stacked-patch elements to be included in the impedance 9 impedance matching for the embedded element. The 9 and scattering matrices were used to analyze the impedance 3 cluster. The characteristics of the central element of a 3 stacked-patch element embedded radiation patterns were modeled in a similar manner using a 3 5 cluster. These partial arrays are illustrated in Fig. 5. A full-size laboratory prototype array was fabricated with SSMA connectors on all of the active elements and resistive chip terminations on the inactive elements. This array served as a test article to evaluate the design. The SSMA connectors allowed for testing each individual stacked-patch element of the 160 active array elements. The prototype array was fabricated as four panels each consisting of 40 active elements. These four panels

Fig. 6. Measured reflection coefficients for one panel (40 active elements) of the prototype array.

were joined on a common ground plane to form the complete array. The swept frequency measurement of the input reflection coefficient for each stacked-patch element on one panel of the array is shown in Fig. 6. The data in Fig. 6 are representative of measurements on the other three panels. The data in Fig. 6 shows some small variations between individual elements, which could be attributed to hand installation and soldering of SSMA connector probes. Fig. 7 shows reflection coefficient measurements on two elements located on lines 1 and 5 of the array compared with predicted results. The short dashed curve in Fig. 7 was calculated for the center element in a 3 3 cluster with dielectric layers of infinite extent. The measured resonant frequencies agree with the design predictions and, although the measurements did not exhibit the predicted deep resonances at the higher frequencies, the impedance match was considered sufficient to proceed with development of a flight model of the HIRAD instrument using this antenna design. To further examine the differences between the results, another commercially available software package (HFSS) using the finite element method was used to model the element. The results of this model for the center element of the 3 3 cluster are compared in Fig. 8 with the measured data. This model tends to predict the depth of the higher frequency resonances more

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Fig. 9. Cross-track (H-plane) element patterns at 6.6 GHz for single isolated element (co-pol solid) (cross-pol dashed).

Fig. 7. Reflection coefficient for two elements of the prototype array (solid = line 5) (long dash = line 1) (short dash = calc: for 3 3 array center element).

2

Fig. 10. Cross-Track (H-plane) element patterns at 6.6 GHz for the central element of 3 5 cluster (co-pol solid) (cross-pol dashed).

2

Fig. 8. Reflection coefficient for two elements of the prototype array (solid = 3 array center element).

line 5) (long dash = line 1) (dash = calc: for 3

2

accurately and provide additional verification of the laboratory measurements. The two smaller dips in the S11 curves on either side of the 5 GHz resonance in Fig. 7 and Fig. 8 are attributed to the slots in the bottom patches, which were described earlier. These artifacts were not observed on previous array test articles and simulations without the slots. The effect of the slots on cross-pol patterns is illustrated by comparing the calculated element patterns with and without the slots in Fig. 9 for a single element (1 1) and in Fig. 10 for the central element in a 3 5 cluster (3 elements in the E-plane direction and 5 elements in the H-plane direction). These patterns were obtained from the MoM model, which used the planar dielectric Green’s function. These results demonstrate that the slots had no effect on the cross-pol pattern for the single isolated element; however, the slots did reduce the maximum cross-pol level by about 8 dB in the 3 5 cluster where the active element is surrounded by other elements and mutual coupling to cross-polarized currents on adjacent patches becomes significant. Rippling in the patterns of Fig. 10 are attributed to the

Fig. 11. HIRAD prototype with the array mounted on a large ground plane.

small finite size (5 elements) of the array in the pattern direction. A larger array could not be modeled with sufficient accuracy due to the large number of unknowns required in the numerical model simulation. III. ARRAY TESTS AND CHARACTERIZATION A full-scale prototype array was fabricated and fully characterized in the Marshall Space Flight Center anechoic chamber as shown in Fig. 11. The prototype antenna is a complete 37 16 array, with a line of matching elements around the perimeter and the array was thinned to 10 linear elements as illustrated in Fig. 1. A 16-way connectorized power combiner was attached to each of the 10 linear elements. Two radiometers were mounted in a supporting structure behind the array and translated along the back of the array to mate with the 10 power combiners.

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Fig. 13. Comparison of cross-track patterns at 6.6 GHz (solid 5 cluster).

measured)(dash = calculated 3

2

=

Fig. 12. Measured cross-track patterns for linear element # 5 (co-pol dashed) (cross-pol solid).

Two sets of measurements were made, individual patterns in the 2 principal planes to characterize the 10 linear elements, and interference patterns for all combinations (i.e., pairs) of the linear elements to characterize the system as an interferometer. The pair of coherent radiometers was used in measuring the interferometer interference patterns. Each radiometer provided a power measurement for each of the linear elements, and all pairs of radiometers provided a complex multiplication of 2 respective signals for the interference patterns. The interference patterns are the impulse response of the array for each linear element pair in the interferometer, and define the spatial filtering. Synthesized array patterns over the HIRAD swath, which also characterize the array, may be computed from the interference patterns. The conventional range measurement, with a coherent source and square-law detector, was used for element pattern measurements to provide the large dynamic range required, particularly for the along-track patterns.

A. Element Patterns Fig. 12 shows co-pol and cross-pol patterns at each of the four frequencies for linear element #5 near the center of the array. One would notice that the cross-pol level for the measured patterns at 6.6 GHz is very close to that calculated in Fig. 10 for the 3 5 cluster. This demonstrates that the cross-pol reducing slots are working as expected. The measured cross-track pattern for linear element #5 at 6.6 GHz is compared in more detail in Fig. 13 with the MoM and FEM model calculations for the center element of a 3 5 cluster. Due to the much larger computer memory required, a larger array could not be accurately modeled; however, the 3 5 array model should provide sufficient interelement mutual coupling effects upon radiation patterns for elements near the center of the array. The MoM model is for dielectric layers of

Fig. 14. Measured cross-track patterns for linear element #1 (co-pol dashed) (cross-pol solid).

infinite extent; whereas, the FEM model is for truncated dielectric layers; thus causing a difference in the pattern ripple due to the finite array and finite structure. The test article was a much larger array, which results in a higher spatial frequency ripple in the measured patterns. Although there is some difference in the shape of the patterns for the two simulation models, the measured cross-pol level is very close to the predictions for both models. The co-pol and cross-pol patterns at each of the four frequencies for linear element #1 at the edge of the array are shown in Fig. 14. Patterns for linear elements in the central part of the array tend to be essentially the same as that for linear element #5, whereas the patterns for linear elements at the edge of the array tend to show an asymmetry, e.g. the cross-track patterns for #10 are the mirror images of the patterns for #1. This asymmetry in the cross-track patterns for elements near the edge of the array is characteristic of the asymmetry expected for arrays with inter-element mutual coupling. Mutual coupling for this stacked-patch array is much stronger at 4.0 GHz than for the other resonances; therefore, the asymmetric distortion in the 4.0 GHz co-pol pattern in Fig. 14 tends to be more pronounced. The along-track patterns for all 10 linear elements were essentially the same as #5, which is shown in Fig. 15. The power

BAILEY et al.: MULTI-FREQUENCY SYNTHETIC THINNED ARRAY ANTENNA FOR THE HURRICANE IMAGING RADIOMETER

Fig. 16. Measured interference patterns for baseline #5 and #6).

Fig. 15. Measured along-track Co-Pol patterns for linear element 5.

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n

= 7 (linear elements

combiners contained a tapered amplitude distribution to lower the along-track sidelobes. B. Array Patterns For a 1-D interferometer imager such as HIRAD, the complex output products (visibilities) for all element pairs in the thinned array are a function of the cross-track scene brightness temperature, , the element patterns, and the interference patterns. In general, the Fourier decomposition of the brightness temperature scene by the interferometric measurement is described by [2]

(1) where is the visibility for each interferometer baseline (element pair spacing), is the spacing in wavelengths, is the product of the 2 element voltage patterns at each baseline, and the exponential term is the interference pattern for each baseline. Equation (1) can be simplified for the one-dimensional case by eliminating the variable and using a matrix formulation, as given by

(2) where is the total number of visibilities, or baselines, and is the number of pixels in the brightness temperature scene. The matrix ‘ ’ is populated with the complete set of measured interference patterns and fully characterizes the spatial frequency response of the array. In practice, a brightness temperature image is formed from the measured visibility by operating the pseudo-

Fig. 17. Measured interference patterns for baseline #2 and #9).

n = 34 (linear elements

inverse of on to estimate . A pseudo-inverse, such as the Moore-Penrose version, is used because and are typically not equal. The conditioning of is generally not an issue during the inversion process. The matrix, viewed as a linear operator, is to first order a Discrete Fourier Transform (DFT) of the image vector, . Second order effects, such as inter-element mutual coupling, introduce small deviations from an ideal DFT. This requires the operator to be inverted using a least squares pseudoinverse method as opposed to an analytical inverse DFT. However, the conditioning of is dominated by its DFT component, which is well suited to inversion. In practice, the inversion is found to be very stable provided all necessary Fourier components of the brightness temperature are sampled, up to the maximum component determined by the physical dimensions of the synthesized array (which they are).

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Fig. 18. Synthesized patterns from radiometer measurements at 6.6 GHz.

Interior array elements #5 and #6 were used to form the interand the envelope ference patterns in Fig. 16 for baseline ripple is relatively small compared to that in Fig. 17, where the outside elements #2 and #9 were used. The envelope ripple in the 4 GHz and 6.6 GHz patterns is due to the ripple in the element patterns. Synthesized array patterns were computed from the measured G-Matrix using the method described by LeVine [7], which is based on the inversion of (2). A full 180 pattern with boresight angle at any pixel in the scene can be computed once the array is characterized by the G-matrix. Fig. 18 includes two examples showing synthesized patterns at 0 and 60 scan angles for 6.6 GHz. These demonstrate the negative sidelobes that are characteristic of complex correlation radiometers. Fig. 19 shows synthesized patterns at all four frequencies on both linear and log scales, expanded about the zero scan angle to emphasize the main beam and near-in sidelobes. The 6.6 GHz beamwidth is the narrowest at approximately 1.8 , which is expected for an array of this size, and the beamwidth varies properly in accordance with the other frequencies. Main beam half-power widths for synthesized patterns of interferometer arrays are typically 1/2 the beamwidth for real (filled) apertures of the same size [2]. The first sidelobes were symmetric about the main beam in the 0 case and at a level of approximately 6.7 dB, which is expected for a uniform array taper with synthesized patterns, as compared to 13.3 dB sidelobes for a real aperture. IV. CONCLUSIONS A four-frequency synthetic thinned array of stacked-patch elements was designed, fabricated and prototype tested for the Hurricane Imaging Radiometer. Based upon this successful laboratory evaluation, an aircraft flight qualified model of this antenna is being integrated with the HIRAD instrument for mapping of severe oceanic storms. REFERENCES

Fig. 19. Synthesized patterns from radiometer measurements at all four frequencies for zero beam scan (a) linear scale (b) log scale.

Since was a point source in the chamber tests, (1) shows that for each scan, the measured visibility (interference pattern) is a sinusoid at a spatial frequency determined by the baseline, , with an envelope equal to the product of the two element voltage patterns. For the HIRAD design, –36 and is the minimum spacing between linear elements. The maximum spacing is ; therefore, the spatial frequencies of the visibilities measured in the chamber will vary from 1 Hz to 36 Hz over the 4.0 to 6.6 GHz operating frequency range. Both co-pol and cross-pol interference patterns were measured for each of the 36 baselines, fully populating a G-matrix at each of the 4 frequencies. The patterns at all four frequencies for baselines 7 and 34 are shown in Figs. 16 and 17 as examples.

[1] E. W. Uhlhorn, P. G. Black, J. L. Franklin, M. Goodberlet, J. Carswell, and A. S. Goldstein, “Hurricane surface wind measurements from an operational stepped frequency microwave radiometer,” Monthly Weather Rev., vol. 135, no. 9, pp. 3070–3085, 2007. [2] A. B. Tanner and C. T. Swift, “Calibration of a synthetic aperture radiometer,” IEEE Trans. Geosci. Remote Sensing, vol. 31, pp. 257–267, Jan. 1993. [3] J. Johnson, R. Amarin, S. El-Nimri, L. Jones, and M. Bailey, “A wideswath, hurricane imaging radiometer for airborne operational measurements,” presented at the IGARSS, Denver, CO, 31 July–4 Aug. 2006. [4] C. Ruf, R. Amarin, M. C. Bailey, B. Lim, R. Hood, M. James, J. Johnson, L. Jones, V. Rohwedder, and K. Stephens, “The hurricane imaging radiometer—An octave bandwidth synthetic thinned array radiometer,” presented at the IGARSS, Barcelona, Spain, Jul. 23–28, 2007. [5] C. S. Ruf, “Numerical annealing of low redundancy linear arrays,” IEEE Trans. Antennas Propag., vol. 41, no. 1, pp. 85–90, 1993. [6] M. C. Bailey, Stacked-patch antenna element design for multi-frequency array applications Research Triangle Institute Tech. Rep. RTI/8348/026-01F, Sep. 2005, NASA Contract NAS1-02057, Task Order No. 7026. [7] D. M. Le Vine, A. J. Griffis, C. T. Swift, and T. J. Jackson, “ESTAR: A synthetic aperture microwave radiometer for remote sensing applications,” Proc. IEEE, vol. 82, no. 12, Dec. 1994.

BAILEY et al.: MULTI-FREQUENCY SYNTHETIC THINNED ARRAY ANTENNA FOR THE HURRICANE IMAGING RADIOMETER

M. C. Bailey (S’69–M’71–SM’81–LSM’02) received the B.S. degree in electrical engineering from Mississippi State University, Starkville, in 1964, the M.S. degree in electrical engineering from University of Virginia, Charlottesville, in 1967, and the Ph.D. degree in electrical engineering from Virginia Polytechnic Institute and State University, Blacksburg, in 1973. He is currently employed part-time by Applied EM, Hampton, VA and also serves as a Technical Consultant for the University of Central Florida, Orlando. He previously worked for Research Triangle Institute, Hampton, VA after retiring from NASA, Langley Research Center, Hampton, VA in 2000 with 36 years of federal government employment as a research engineer. He was a Visiting Professor at the University of Mississippi, Oxford, during the 1978–79 academic year. He has authored or coauthored over 110 publications in the field of antennas and electromagnetics. He coauthored the section “Antennas and Wave Propagation” in the Electronics Engineers’ Handbook (McGraw-Hill, 1982, 2nd ed), and coauthored the chapter “Radiometer Antennas” in the Antenna Engineering Handbook (McGraw-Hill, 1984, 2nd ed). Dr. Bailey has been the recipient of 27 NASA awards for recognition of technical accomplishments, including the NASA Medal for Exceptional Scientific Achievement.

Ruba A. Amarin (M’06) received the B.A. degree in electronic engineering from the Princess Sumaya University for Technology, Amman-Jordan, in 2004 and the M.S. degree in electrical engineering from the University of Central Florida, Orlando, in 2006. She is currently working toward the Ph.D. degree at the University of Central Florida, Orlando, under the supervision of Dr. Linwood Jones. She is conducting research in satellite remote sensing at the University of Central Florida.

James W. Johnson (SM’81) is a native of Virginia. He received the B.S. and M.S. degrees in electrical engineering from Virginia Polytechnic Institute, Blacksburg, in 1966 and 1969, respectively. He began a career at the NASA, Langley Research Center in Hampton, VA, in 1966 and retired in 2006. He is currently a Visiting Research Scientist with the Central Florida Remote Sensing Lab, University of Central Florida, Orlando. During his time with NASA his primary concentration was in microwave remote sensing for earth sciences. He developed techniques for measuring ocean surface and polar ice characteristics using active and passive microwave systems. This included the development of multi-frequency techniques to measure the height statistics and spectral characteristics of ocean surface waves. He served as the NASA project engineer for the development of the SeaSat Wind Field Scatterometer, which was launched in 1978. He later led conceptual design studies and technology development projects for various earth science applications, with a brief interruption for program and project management positions in the Space Station Program at NASA Headquarters and at Langley.

Paul Nelson (S’08) received the B.S. degree in electrical engineering from the University of Central Florida, Orlando (UCF), in 2008, where he is currently working toward the M.S. degree. Since, 2007, he has engaged research at the Antennas, RF and Microwave Integrated systems (ARMI) Laboratory at UCF, as an undergraduate and graduate student. His research interests include stacked-patch microstrip antenna array design and time difference of arrival direction finding systems. Mr. Nelson is a recipient of the undergraduate IEEE MTT-S Scholarship Award in 2007, as well as the Science, Mathematics, and Research for Transformation (SMART) Scholarship in 2008.

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Mark W. James received the B.S. degree in electrical engineering technology from Purdue University, West Lafayette, IN, in 1978 and the B.S. degree in electrical and computer engineering from the University of Michigan, Ann Arbor, in 1984. He is currently employed by NASA, Marshall Space Flight Center, Huntsville, AL, in the Engineering Directorate. He has worked previously at the Space Physics Research Laboratory, University of Michigan, Daedalus Enterprises, and Delco Electronics. He has coauthored in the areas of microwave and near-infrared remote sensing. He has been involved in the development of microwave radiometers, Near IR and visible spectrum airborne spectrometers, data acquisition systems and lightning detection sensors. Mr. James is a registered Professional Engineer in the State of Alabama. He has received numerous NASA awards including the Directors Commendation for technical excellence and Exceptional Service Award.

David E. Simmons received the B.S. degree in electrical engineering from Tennessee Technological University, Cookeville, in 1979. He is currently employed by the University of Alabama, Huntsville. He has previously worked for BAE Systems/MEVATEC, Jacobs/Sverdrup, Chrysler, GTE and Honeywell. He has served in various technical roles since graduation on commercial, DoD and NASA projects. He has designed and supported airborne passive radiometer system electronics, missile testing, flight experiments, automotive and telephony electronics testing and military communications equipment.

Christopher S. Ruf (S’85–M’87–SM’92–F’01) received the B.A. degree in physics from Reed College, Portland, OR, and the Ph.D. degree in electrical and computer engineering from the University of Massachusetts, Amherst. He is currently a Professor of atmospheric, oceanic, and space sciences and electrical engineering and computer science and Director of the Space Physics Research Laboratory, University of Michigan, Ann Arbor. He has worked previously at Intel Corporation, Hughes Space and Communication, the NASA Jet Propulsion Laboratory, and Penn State University. In 2000, he was a Guest Professor with the Technical University of Denmark, Lyngby. He has published in the areas of microwave radiometer satellite calibration, sensor and technology development, and atmospheric, oceanic, land surface and cryosphere geophysical retrieval algorithms. Dr. Ruf is a member of the American Geophysical Union (AGU), the American Meteorological Society (AMS), and Commission F of the Union Radio Scientifique Internationale. He has served on the Editorial Boards of the AGU Radio Science, the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (TGRS), and the AMS Journal of Atmospheric and Oceanic Technology. He is currently the Editor-in-Chief of the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. He has been the recipient of three NASA Certificates of Recognition and four NASA Group Achievement Awards, as well as the 1997 TGRS Prize Paper Award, the 1999 IEEE Resnik Technical Field Award, and the IGARSS 2006 Symposium Prize Paper Award.

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W. Linwood Jones (SM’75–F’99–LF’09) received the B.S. degree in electrical engineering from the Virginia Polytechnic Institute, Blacksburg, in 1962, M.S. degree in electrical engineering from the University of Virginia, Charlottesville, in 1965, and the Ph.D. degree in electrical engineering from the Virginia Polytechnic Institute and State University, Blacksburg, in 1971. He is currently a Professor with the School of Electrical Engineering and Computer Science at the University of Central Florida in Orlando. At UCF, he teaches undergraduate and graduate courses in communications, satellite remote sensing and radar systems. Also, he is the director of the Central Florida Remote Sensing Laboratory where he performs research in satellite microwave remote sensing technology development. Prior to becoming a college professor in 1994, he had 27 years federal government employment with NASA at the Langley Research Center in Hampton, VA; at NASA Headquarters in Wash DC and at the Kennedy Space Center, FL. Further, he spent eight years in the private aerospace industry with employment at General Electric’s Space Division in King of Prussia, PA and Harris Corp.’s Government Aerospace Systems Division in Melbourne, FL. Prof. Jones is a member of the American Geophysical Union (AGU) and Commission F of the Union Radio Scientifique Internationale. For education, he received the IEEE Orlando Section: Outstanding Engineering Educator Award 2003, the College of Engineering: Excellence in Undergraduate Teaching Award 2004, and the IEEE Florida Council: Outstanding Engineering Educator Award 2004. For his research, he received four NASA Special Achievement Awards, seven NASA Group Achievement Awards, the CNES Space Medal, the Aviation Week & Space Technology Space Program Award -1993, and the Naval Research Lab 2004 Alan Berman Research Publications Award.

Xun Gong (S’02–M’05) received the B.S. and M.S. degrees in electrical engineering from FuDan University, Shanghai, China, in 1997 and 2000, respectively, and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, in 2004. He is currently an Assistant Professor of electrical engineering and computer science at University of Central Florida, Orlando. He was with the Birck Nanotechnology Center at Purdue University, West Lafayette, IN, as a Postdoctoral Research Associate in 2005. His research interests include high-Q resonators and filters, microwave sensors, integrated RF front-end, flexible electronics, and packaging. Dr. Gong is a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), the IEEE Antennas and Propagation society (IEEE AP-S), and the International Microelectronics and Packaging Society (IMAPS). He is the recipient of the NSF Faculty Early CAREER Award in 2009. He is the recipient of the Third Place Award in the Student Paper Competition presented at the 2004 IEEE MTT-S International Microwave Symposium (IMS), Fort Worth, TX.

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Electronically Reconfigurable Transmitarray at Ku Band for Microwave Applications Pablo Padilla, Alfonso Muñoz-Acevedo, Manuel Sierra-Castañer, Member, IEEE, and Manuel Sierra-Pérez, Senior Member, IEEE

Abstract—An electronically reconfigurable transmitarray device at 12 GHz is presented in this work. This paper highlights the functioning of this kind of device and thoroughly examines the proposed reconfigurable transmitarray. The architecture is discussed along with the design and selection of all the constituting elements and the prototypes for all of them. In order to add reconfigurability to the transmitarray structure, 360 reflective phase shifters were designed, prototyped and validated for direct application. Eventually, a demonstrative prototype for an active transmitarray with phase shifters was assembled, and radiation pattern measurements were taken in an anechoic chamber to demonstrate the capabilities of this structure. Index Terms—Antenna array, constrained lens, directional coupler, Ku band, microstrip circuit, microstrip technology, patch antenna, phase shifter, phase shifting, reflective circuitry (RTPS), transmitarray, varactor.

I. INTRODUCTION N the technical world, in the area of antennas and radiating systems, there is growing interest in lens-type or reflector-type structures called transmitarrays and reflectarrays, respectively. These structures replace and, in certain cases, improve the outcomes of traditional structures, such as reflectors or lenses, depending on the considered structure. The focus of this paper is in active transmitarray structures, also called artificial or constrained lenses [1]. In particular, a transmitarray architecture is proposed and defined. All the constituting elements of the proposed transmitarray lens architecture are designed, prototyped and measured, and, for the sake of completeness, the design and manufacture of one transmitarray demonstrator is offered. The paper is organized as follows. This section (Section I) describes the scope of this work and its relationship to previously published papers. Section II offers the complete active transmitarray characterization and design, and introduces the forming elements, divided into passive and active. Section III attends to these passive transmitarray forming devices, and Section IV at-

I

Manuscript received May 11, 2009; revised November 19, 2009; accepted February 12, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by an FPU Spanish Government Grant (Ref: AP2005-0177) and the “CROCANTE” project (REF:TEC2008-06736/TEC), Spain. The authors are with the Radiation Group (GR), Technical University of Madrid (Universidad Politécnica de Madrid—UPM), Madrid, Spain (e-mail: [email protected]; [email protected], m.sierra; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050426

Fig. 1. Scheme of transmitarray general functioning.

tends to active ones (phase shifters). Section V describes the assembly and measurements of the first active transmitarray manufactured prototype, considered as a first demonstrator of this kind of artificial lens device. The clear purpose of this first prototype, and the main contribution of this work, is to demonstrate the feasibility of the proposed active device and its architecture for proper functionality, according to its expected theoretical behavior. The design of the active devices presented in this work (Section V) is another main contribution of this document. Finally, in Section VII, conclusions are drawn. The basics of transmitarray lens structures are easily understood: first, an electromagnetic wave (with specific wave-front properties) is received by the reception interface of the lens, and then the received signal is processed in a particular way (phase shift, amplification, etc.). Eventually, the processed signal is retransmitted with new wave-front properties due to the modifications introduced in the processing stage, as sketched in Fig. 1. Various transmitarray approaches have been described in the literature. Fundamental principles regarding active lenses are offered in [2] and [3]; active lenses for radar applications (HAPDAR project) have been heavily studied. Quite relevant are the models proposed in [1] and [4]. Some of these models are used in a multi-beam working scheme [4]: the main beam direction differs depending on the position of the feeder related to the lens. Regarding circuitry used to deal with received signals, lenses are classified into active lenses (if an external control signal is used for the inner circuitry configuration) [2]–[4] or passive lenses (with no circuitry reconfiguration) [6]–[8]. This paper focuses on the design and manufacture of a complete active microwave lens and also describes an electronically reconfigurable lens demonstrator. One of the main values in this kind of devices, despite the multi-beam functionality, consists of placing it in front of a particular antenna, which offers two key advantages.

0018-926X/$26.00 © 2010 IEEE

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Fig. 3. Scheme of the interconnection between the processing interface and the radiating one, with 90 change in reference plane. Fig. 2. The transmitarray main advantages. (a) Phase error correction. (b) Radiation pattern reconfiguration.

• Phase error correction due to the spherical wave front coming from the feeding antenna; • New radiation pattern configuration, modifying the phase response of each forming transmitarray cell. Fig. 2 highlights these two effects. Transmitarray lens feeders are important in the design of the final lens. Feeders with different radiation properties are used in this kind of constrained lenses, affecting the resulting radiation pattern of the lens. Some effects (such as taper) are difficult to introduce in conventional phased arrays without a high level of complexity (consider, for instance, the complexity in the introduction of a particular taper that is given by the feeder in the case of a transmitarray lens). II. COMPLETE ACTIVE TRANSMITARRAY CHARACTERIZATION AND DESIGN The transmitarray structure is divided into two principal parts: the radiating interface for reception and transmission and the processing interface for array phase conformation in each transmitarray cell. This active transmitarray demonstrator has been designed to work with linear polarization at 12 GHz and a bandwidth of 0.7 GHz. The phase shifters have been designed with a range of phase variation greater than 360 in order to enable possible electronic tilt in the H and V planes independently. For this demonstrative prototype, an existing conical corrugated horn has been used as a feeder; this implies that the spillover efficiency of the design has not been optimized. Moreover, in this prototype, due to the simplicity of the fabrication process, the radiating elements are grouped into sets of four elements, so the scan capability is limited. With a transmitarray structure with planar technology, the greatest challenge is the placement of the processing interface inside the structure, between both radiating interfaces, because of the limited space available. Both of the interfaces (processing and radiating) that form the transmitarray are designed in planar architecture over the ground plane. If the interconnection point is freed from this restriction (planar connection), making it possible to change the reference plane for each interface by means of perpendicular connection, the space constraint in one of the dimensions is considerably reduced. Thus, microstrip lines for the processing interface can be applied. However, the requirements for the other dimensional components are strictly the same. Fig. 3 illustrates this fact, with the perpendicular connection between planar interfaces.

Fig. 4. Model for patch design, in layer scheme.

In the defined architecture for the complete structure, the complexity of connecting the stacked patch array structures for transmission and for reception by means of microstrip lines is condensed in the interconnection point. Fig. 3 reveals the necessity of designing a transition between coaxial-type patch connections and microstrip transmission lines, along with the necessity of designing a 90 transition with reference plane change in the microstrip line. Thus, the structure is divided into passive elements (radiating elements, one to four distribution networks and 90 transitions), and active elements (phase shifters). III. PASSIVE TRANSMITARRAY ELEMENT DESIGN, SIMULATION AND PROTOTYPING A. Radiating Element For the radiating elements, patches were selected for their planar structure, a geometry that is especially suitable for transmitarray devices. In order to improve certain features, multilayer stacked patch elements over the ground plane were used rather than simple patch ones because of their wider working frequency band. CST Microwave Studio 2006 was used for the design stage. Fig. 4 depicts the layer distribution and materials employed. In the design process, one isolated stacked patch and one stacked patch embedded in an array are separately considered, the latter including mutual coupling effects with surrounding patches. Fig. 5 shows this prototype, the design and measured results. B. Distribution Network A one-to-four bidirectional distribution network in a microstrip line was designed, sharing the ground plane with the radiating structure (and below it). Prototypes for the distribution network exhibited adequate behavior, as seen in Fig. 6.

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Fig. 5. Stacked patch embedded in array. (a) Prototype. (b) Simulated and measured results. Fig. 8. Phase shifter complete model, with tunable circuits at ports B and C.

Fig. 9. Detail of the tunable circuit at ports B and C.

Fig. 6. Prototype of distribution network. (a) Prototype. (b) jS

Fig. 7. Reference plane change. (a) Prototype. (b) jS

j

and jS

j

and jS

j

j

.

.

C. Reference Plane Change The transition with reference plane change was characterized and designed. Fig. 7 shows the prototype results for this transition. IV. ACTIVE ELEMENT: 360 ELECTRONICALLY TUNABLE PHASE SHIFTER DESIGN AND PROTOTYPING One way to design a 360 phase shifter is to design a device with varactors as tunable elements [9]. In order to introduce phase variation by capacitance changes, two devices are placed together: a four-port directional coupler and reflective circuits (LC circuit). The resulting circuit, a combination of these two circuits, is a two-port device that combines the signal in a particular way to obtain an output signal that is the same as the input signal with a phase variation. In conventional use, all the devices connected to the directional coupler have to be properly matched

to obtain the common directional coupler behavior. However, considering port A (see Fig. 8) as the input port, if open circuits are placed in ports B and C, all the signals arising in ports B and C are reflected, re-entering the directional coupler, with ports A and D as new output ports. The configuration of the directional coupler forces in ports D and A obtain the sum in phase and the sum in opposite phase, respectively, of the reflected signals coming from ports B and C. If the open circuits in ports B and C are changed, and tunable LC circuits are placed, the phase of the reflected signal in ports B or C can be varied. As a consequence, all the input power from port A is redirected to port D and the resulting signal in port D differs in phase from the original one in port A, owing to the correction introduced by the tunable circuit, as shown in Fig. 8. It is essential to guarantee symmetry in the complete device: the same circuit (with the same phase behavior) must be placed in ports B and C in order to force the sum in phase and in opposite phase in ports D and A, respectively. As main features of the device, the phase variation range must be greater than 360 degrees, while the losses must be minimized ). (i.e., better than A. Reflective Circuit Design and Simulation The circuit connected in ports B and C behaves (for the working frequency range) as a completely mismatched circuit, reflecting the entire received signal. It is necessary to have an equivalent impedance with no real part in order to obtain the maximum value for the reflection coefficient, as shown in (1). Thus, an adequate tunable LC circuit (with a varactor controlled in voltage (V) as capacitance) could satisfy these constraints, as shown in Fig. 9

(1)

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Fig. 10. Simulation model for the equivalent printed LC circuit.

Fig. 12. Complete simulation model for the phase shifter.

Fig. 13. Simulated jS

j

parameter for the complete phase shifter.

Fig. 11. Simulated phase variation for the reflective circuit.

One of the most important constraints in the design of the device is the usage of lumped elements (inductors, capacitors) at Ku band (12 GHz in this case) with low parasitic effects. Capacitors are replaced by varactors for microwave purposes while inductances are designed using equivalent high-impedance microstrip printed circuits, working as inductances in the working frequency range [10], [11]. In this case, the varactors are the only lumped elements in the complete device. To obtain the behavior of an LC circuit, printed technology for the inductors was applied and varactors for the capacitors were used. Fig. 10 shows the simulation model in detail, with ground plane connections, printed inductive lines and varactors as the unique lumped elements. Once the model is properly designed, the simulations yield the results shown in Fig. 11 for the expected phase response for different capacitance values (taking linear capacitance variation within the valid range of the varactor selected). The range of capacitances available in the varactor yields a -degree variation in the best case. reflective circuit with As discussed later, in the complete phase shifter simulation, it is mandatory to replicate the reflector and coupler set (denoted ‘primary cell’ from now on) up to three times to reach the 360degree variation. B. Complete Phase Shifter Design and Simulation The phase shifter was designed using directional couplers. These have been designed as conventional couplers with no extra features, in microstrip technology (as for the rest of the device). Some design techniques are found in [12]–[14].

As mentioned before, the designed and simulated coupler with reflective circuits set achieves an insufficient phase variation, according to the main specifications. This constraint could be overcome with the replication of the primary cell, as exemplified in Fig. 12. The final device has two ports (input and output), with no outer access to the ports with reflective circuits. For the sake of simplicity, the capacitance values for all the varactors are the same. In this case, the control voltage for each varactor is the same (only one control voltage for the entire phase shifter). at the input port and Figs. 13 and 14 exhibit reflection transmission from the input port to the output port, respectively, in terms of S-parameters, for different capacitance values. achieved with Fig. 15 shows the phase variation the final designed phase shifter. The phase variation versus the capacitance value for the varactors is shown in Table I. C. Phase Shifter Prototype and Measurement As mentioned, the most important element of the device is the reflective circuit. The varactor selected for the variable capacitor in the reflective circuit is the M/A-COMMA46585–1209. The varactor mounting constraints for avoiding damage force a low soldering temperature: either bounding techniques or conductive epoxy soldering techniques satisfy these constraints. The latter technique is chosen. Some details are shown in Figs. 16 and 17. Transmission and reflection parameters are obtained; S parameters in terms of amplitude are shown in Fig. 18. Phase behavior is shown in Fig. 19.

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TABLE I PHASE VARIATION AT 12 GHZ VERSUS CONTROL VOLTAGE

Fig. 14. Simulated jS

j

parameter for the complete phase shifter.

Fig. 16. Complete prototype for the phase shifter.

Fig. 15. Simulated phase variation for the complete phase shifter. Edge values.

Fig. 17. Details of the prototype for the phase shifter.

A quite linear phase/voltage relation is obtained, as shown in Table I.

between groups of four patches is equal to 1.4 wavelengths. The consequence of this constraint is the reduction in the scan capability of the whole array due to the grating lobes of the structure. However, if a scan range greater than 10 deg. is required, one phase shifter per radiation element should be used. Fig. 20 shows the simulated radiation pattern for steering angles of 0, beam width remains almost con10 and 15 deg. The stant and equal to 12.1 deg in all these cases. In the last case, the grating lobes completely disrupt the pattern. For that reason, the

V. COMPLETE ACTIVE TRANSMITARRAY SCAN CAPABILITY SIMULATED RESULTS As mentioned previously, the radiation elements have been combined in groups of four in order to simplify the construction of this prototype. With this configuration, the separation

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Fig. 20. Simulated radiation pattern for different steering angles. (a) 0 deg. (b) 10 deg. (c) 15 deg.

Fig. 18. Phase shifter measurements. (a) Measured jS totype for different control voltages. (b) Measured jS totype for different control voltages.

j j

variation for the provariation for the pro-

Fig. 21. Assembly model for a four-patch transmitarray cell.

Fig. 19. Measured phase variation for different control voltages.

prototype has been characterized, for demonstrative purposes, at steering angles of 0 deg and 9 deg.

VI. COMPLETE ACTIVE TRANSMITARRAY ASSEMBLY AND MEASUREMENT RESULTS Globally, the assembly for each active transmitarray cell includes the four-to-one distribution network, the transition device between the coaxial-type feeding at each patch and the microstrip structure of the distribution network, as diagrammed in Fig. 21.

Fig. 22. Transmitarray core. (a) Radiating interfaces mounted together. (b) Distribution network integration.

Details of the assembly process are depicted in Figs. 22 and 23. Fig. 24 shows the transmitarray mounted in anechoic chamber for measurement acquisition. All the phase shifters were measured and calibrated before the lens assembly. Results are shown in Table II. According to the values in Table II, a complete set of polarization voltages for each shifter is defined, depending on the phase shift desired at each four-patch group. Analysis of Table II shows that, for the same voltage, there is a possible variation between shifters of around 30 deg. However, this error is not the one to be considered in the configuration, as each phase shifter is independently calibrated. For the sake

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TABLE II MEASURED TX PHASE RESPONSE ( ) OF PHASE SHIFTERS VS CONTROL VOLTAGE

Fig. 23. Assembly. (a) Phase shifter integration. (b) Complete transmitarray.

Fig. 24. Transmitarray in anechoic chamber, different views.

of completeness, a random phase error is introduced in the phase shifters and the radiation pattern is evaluated. The

Fig. 25. Comparison between the ideal case and a case with phase errors (worst case: Each shifter alternatively with +30 or 30 phase error), for simulated radiation patterns for steering angles of 0 deg and 9 deg.

0

radiation pattern degradation (steering direction and side lobe level) is still negligible, as it is observed in Fig. 25. For prototype validation, two working configurations for the lens measurement are defined as follows. • Pattern 1: The lens does not change the radiation pattern; it only corrects the phase error (all the phase shifters with the same polarization voltage, 0 V). • Pattern 2: The lens changes the radiation pattern in one of the main axes, applying a 9-degree tilt. The required phase and voltage (after the calibration process) for each phase shifter are shown in Table III (shifters are ordered in columns, from top left to bottom right, in the square 3 3 grid of four-cell groups).

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TABLE III REQUIRED PHASE AND APPLIED VOLTAGE TO PHASE SHIFTERS FOR PATTERN 2

Fig. 26. Transmitarray measurement results for the steering angle of 0 deg.

adequate architecture was chosen and adapted in order to allow the integration of active circuitry in the complete lens design. A complete active transmitarray was designed, manufactured, assembled and measured in an anechoic chamber. Due to some constraints (space and cost of the first prototype), patches were grouped into sets of four and each group phase behavior phase shifter that was electroniwas controlled by one cally reconfigurable. This yielded a more limited reconfigurable global device (possible problems with grating lobes and its radiation), but was valid for a first active transmitarray prototype. Design and manufacture considerations have been mentioned and a 36-element array device was manufactured. For this purpose, nine phase shifters were built, measured and characterized before being integrated into the lens. Once the first prototype was assembled, two configurations were defined in order to achieve the proper behavior of the transmitarray prototype. Prototype measurements in anechoic chamber were obtained and main features (gain, directivity, etc.) were extracted. ACKNOWLEDGMENT The simulations described in this paper were carried out using CST Microwave Studio 2006 under a cooperative agreement between CST and the Universidad Politécnica de Madrid (UPM). The NY substrate used in the prototypes was kindly provided by NELTEC S.A. REFERENCES

Fig. 27. Transmitarray measurement results for the steering angle of 9 deg.

Results for both configurations are offered in Figs. 26 and 27. These patterns are well compared with the theoretical results shown in Fig. 25. With these results, it is convenient to analyze measured gain results and expected directivity values. Measurements yield 21.6 dBi in directivity. The measured results provide 16 dBi mean value in gain. The reduction is due to the accepted horn in this case, due to spillover) and power (60% processing interface insertion losses (3 dB mean value, due to phase shifters), as expected. VII. CONCLUSION In this work, a complete electronically reconfigurable transmitarray device has been presented. Some theoretical background and architecture considerations were offered and

[1] D. McGrath, “Planar three-dimensional constrained lenses,” IEEE Trans. Antennas Propag., vol. 34, pp. 46–50, Jan. 1986. [2] P. J. Kahrilas and D. M. Jahn, “Hardpoint demonstration array radar,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-2, pp. 286–299, Nov. 1966. [3] P. J. Kahrilas, “HAPDAR—An operational phased array radar,” Proc. IEEE, vol. 56, pp. 1967–1975, Nov. 1968. [4] J. Vian and Z. Popovic, “Smart lens antenna arrays,” presented at the IEEE Microwave Symp., 2001. [5] A. Muñoz-Acevedo, P. Padilla, and M. Sierra-Castañer, “Ku band active transmitarray based on microwave phase shifters,” presented at the Eur. Conf. on Antennas and Propagation (EuCAP 2009), Berlin, Germany, Mar. 2009. [6] M. Barba, E. Carrasco, and J. A. Encinar, “Suitable planar transmit-arrays in X-band,” presented at the Eur. Conf. on Antennas and Propagation (EuCAP 2006), Nov. 2006. [7] P. Padilla and M. Sierra-Castañer, “Design and prototype of a 12 GHz transmit-array,” Microw. Opt. Technol. Lett., vol. 49, no. 12, pp. 3020–3026, Dec. 2007. [8] J. Costa, C. Fernandes, G. Godi, R. Sauleau, L. Le Coq, and H. Legay, “Compact ka-band lens antennas for LEO satellites,” IEEE Trans. Antennas Propag., vol. 56, pp. 1251–1258, May 2008. [9] S. Hopfer, “Analog phase shifter for 8–18 GHz,” Microw. J., vol. 22, pp. 48–50, Mar. 1979. [10] W. Hoefer, “Equivalent series inductivity of a narrow transverse slit in microstrip,” IEEE Trans. Microw. Theory Tech., vol. 25, pp. 822–824, Oct. 1977. [11] R. Chadha and K. Gupta, “Compensation of discontinuities in planar transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 82, pp. 2151–2156, Dec. 1982.

PADILLA et al.: ELECTRONICALLY RECONFIGURABLE TRANSMITARRAY AT KU BAND FOR MICROWAVE APPLICATIONS

[12] C. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [13] D. Pozar, Microwave Engineering, 3rd ed. Hoboken, NJ: Wiley, 2004. [14] E. O. Hammerstad, “Equations for microstrip circuit design,” in Proc. 5th Eur. Microwave Conf., Oct. 1975, pp. 268–272. [15] P. Padilla and M. Sierra-Castañer, “Transmitarray for Ku band,” presented at the Eur. Conf. on Antennas and Propagation (EuCAP 2007), Edinburgh, U.K., Nov. 2007. [16] P. Padilla and M. Sierra-Castañer, “Design of a 12 GHz transmit-array,” presented at the IEEE AP-S Int. Symp. on Antennas and Propagation (APS 2007), Hawaii, Jun. 2007. [17] E. Fotheringham, S. Rómisch, P. C. Smith, D. Popovic, D. Z. Anderson, and Z. Popovic, “A lens antenna array with adaptative optical processing,” IEEE Trans. Antennas Propag., vol. 50, May 2002. [18] D. Popovic and Z. Popovic, “Multibeam antennas with polarization and angle diversity,” IEEE Trans. Antennas Propag., vol. 50, pp. 651–657, May 2002. [19] Z. Popovic and A. Mortazawi, “Quasi-Optical transmit/receive front end,” IEEE Trans. Microw. Theory Tech., vol. 46, Nov. 1998. [20] S. Hopfer, “Analog phase shifter,” U.S. patent 4288763. [21] A. Malczewski, S. Eshelman, B. Pillans, J. Ehmke, and C. Goldsmith, “X-band RF MEMs phase shifters for phased array applications,” IEEE Microw. Guided Wave Lett., vol. 9, pp. 517–519, Dec. 1999. [22] B. Pillans, S. Eshelman, A. Malczewski, J. Ehmke, and C. Goldsmith, “Ka-band RF MEMs phase shifters,” IEEE Microw. Guided Wave Lett., vol. 9, pp. 520–522, Dec. 1999.

Pablo Padilla was born in Jaén, south of Spain, in 1982. He received the Telecommunication Engineer degree and the Ph.D. degree both from Technical University of Madrid (UPM), Spain, in 2005 and 2009, respectively. From September 2005 to September 2009, he was with the Radiation Group of the Signal, Systems and Radiocommunications Department, UPM. In 2007, he was with the Laboratory of Electromagnetics and Acoustics at Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland, as an invited Ph.D. Student. Currently, he is Assistant Professor at Universidad de Granada (UGR) and collaborates with the Radiation Group of the Technical University of Madrid. His research interests include antenna design and synthesis and the area of active microwave devices.

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Alfonso Muñoz-Acevedo was born in Toledo, Spain, in 1985. He received the Telecommunication Engineer degree from the Technical University of Madrid (UPM), Spain, in 2008, where he is currently working toward the Ph.D. degree. Since September 2007, he has been with the Radiation Group of the Signal, Systems and Radiocommunications Department of UPM. His research interests include antenna design and synthesis and the area of active microwave and millimeter wave devices.

Manuel Sierra-Castañer (M’95) was born in Zaragoza, Spain, in 1970. He received the Telecommunication Engineer degree in 1994 and the Ph.D. degree in 2000, both from Technical University of Madrid, in Madrid, Spain. Since 1997, he has been at the University “Alfonso X” as a Teaching Assistant, and since 1998, at the Technical University of Madrid as a Research Assistant, Assistant and Associate Professor. His current research interests are in planar antennas and antenna measurement systems.

Manuel Sierra-Pérez (M’78–SM’07) was born in Zaragoza, Spain, in 1952. He received the Master’s and Ph.D. degrees from Technical University of Madrid, in Madrid, Spain, in 1975 and 1980, respectively. He became a Full Professor with the Department of Signals, Systems and Radio Communications at the Technical University of Madrid in 1990. He was an Invited Professor at the National Radio-Astronomy Observatory (NRAO), VA, from 1981 to 1982, and at the University of Colorado at Boulder, from 1994 to 1995. His current research interest is in passive and active array antennas, including design theory, measurement, and applications. Dr. Sierra-Pérez was a promoter and Chairman of the IEEE joint AP/MTT Spanish Chapter. He was Treasurer of the IEEE Spain Section and has been President of the same section since 2008.

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Optimal Wideband Beamforming for Uniform Linear Arrays Based on Frequency-Domain MISO System Identification Bu Hong Wang, Member, IEEE, Hon Tat Hui, Senior Member, IEEE, and Mook Seng Leong

Abstract—Frequency-invariant (FI) beamforming for wideband antenna arrays inevitably involves array aperture loss at the higher-end frequencies of the bandwidth. In order to minimize aperture loss and to fully utilize the array aperture at different operation frequencies, an optimal wideband beamformer for uniform linear array (ULA) is designed based on Dolph-Chebyshev’s theory of beamforming. Different from the existing FI beamformers for wideband arrays, our wideband beamformer produces frequency-dependent patterns which have the narrowest mainlobe width for any given equiripple sidelobe level over a wide frequency bandwidth. These frequency-dependent patterns are obtained through using the system identification method to determine the transfer function of the beamforming network. A matrix formulation is developed to calculate the frequency-dependent optimal Riblet-Chebyshev weights for element spacings smaller than half wavelength. The transfer function of the beamforming network, which is treated as an equivalent multi-input and single-output (MISO) system, is then obtained by the method of system identification with the optimal frequency-dependent Riblet-Chebyshev weights as the input data. Numerical results are provided to verify the effectiveness and validity of the proposed method. Index Terms—Multiple-input and single-output system (MISO), Riblet-Chebyshev weight, system identification, uniform linear array, wideband beamforming.

I. INTRODUCTION REQUENCY Invariant (FI) array pattern is useful for broadband array applications. In the design of FI beamformers, a beamforming network is used to compensate for/equalize the frequency dependency of the array pattern and generate beam patterns that are approximately invariant over the frequency bandwidth. The most common approach [1]–[10] to achieving FI beamformers, as shown in Fig. 1, is to solve some optimization problems in terms of the array’s spatial-temporal weights. The optimization problems are so formulated that the targeted pattern masks are same at different operation frequencies. For the reason of realizability, the parameters of the targeted pattern masks, such as the mainlobe width, directivity, and sidelobe level, etc., are always chosen according to

F

Manuscript received June 19, 2009; revised September 21, 2009; accepted February 21, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported in part by the National University of Singapore under start-up Grant Project WBS R-263-000-469-112 and in part by the National Science Foundation of China under Grant 60601016. The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2050428

Fig. 1. A wideband beamformer and its equivalent MISO network.

the smallest array aperture at the lower-end frequencies of the frequency band. This, however, results in considerable array aperture loss at the higher-end frequencies since generally more favorable/optimal array patterns with a narrower mainlobe width for a given sidelobe level can always be synthesized by considering the corresponding larger array aperture at the higher-end frequencies. Therefore, wideband beamformers providing frequency-dependent optimal patterns over the wide bandwidth are more desirable compared to conventional FI beamformers in the sense that they can more efficiently use the array aperture at different operation frequencies. In light of this idea, an optimal wideband beamformer for uniform linear array (ULA) is designed in this paper. The optimality here means, in the Dolph-Chebyshev [11] sense, to provide the narrowest mainlobe width for a given sidelobe level. Therefore, different from existing FI beamformers for wideband arrays, our wideband beamformer provides frequency-dependent patterns with the narrowest mainlobe width for a given sidelobe level over a wide frequency bandwidth. To the best of our knowledge, this is the first attempt to design an optimal wideband beamformer in the sense of the efficient use of array aperture over the whole frequency bandwidth. There have been many significant studies in array synthesis for different array geometries and pattern performance optimization, for example [11]–[16]. The Dolph-Chebyshev pattern [17]–[19] is, however, well known as the optimal array pattern for ULAs in the sense that it provides an array pattern with the narrowest mainlobe width for a given equiripple sidelobe level. Though the beam shape of Dolph-Chebyshev pattern can

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WANG et al.: OPTIMAL WIDEBAND BEAMFORMING FOR UNIFORM LINEAR ARRAYS

not be adjusted freely (i.e., only suitable for the case of pencil beams), its compact analytical solution and easy realization (involving no optimization procedure) make it very suitable for practical applications. However it is only optimum for element , where is the wavelength of the operation spacings frequency [20]. In the practical application of wideband ULAs, to avoid the appearance of grating lobes in the visible region of the array, the element spacing is commonly selected as one half of the wavelength of the highest frequency in the bandwidth. With this element spacing, all other frequencies in the . bandwidth will satisfy the unambiguous condition of In modern communication systems, antenna arrays with small element spacings are gaining increasing significance and they are referred to as compact arrays. Compact arrays come with the advantages such as a smaller size, lighter weight, and most importantly, the possibility that they can be installed in portable devices. Consequently it is desirable to design a procedure to acquire the optimal Chebyshev weights for element spacings smaller than one half of the operating wavelength. Riblet in his excellent paper [20] proposed such an improved procedure for which can be used to calculate the optimal Chebyshev weights at different frequencies over the bandwidth. The method proposed in [20], however, involves expressing the array pattern as a polynomial whose coefficients are in terms of optimal Chebyshev weights. This polynomial is then compared with a Chebyshev polynomial of the same order, and coefficients of like power terms are then equated. This manipulation, though useful in illustrating the fundamental principles of the , becomes quite optimal Chebyshev arrays for the case involved and tedious when the number of elements is large. In this paper, a compact matrix formulation method, similar to that in [21] for the calculation of the conventional Dolph-Chebyshev weights, is introduced for the straightforward and simple calculation of the Riblet-Chebyshev weights. From the point of view of system identification, a wideband beamformer (as shown in Fig. 1), can be regarded as a multiple-input and single-output (MISO) system. Accordingly, the aim in the design of FI beamformers is basically that of finding which makes an eligible system transfer function vector the system output (a scalar function) for any input vector (i.e., the array steering vectors in the visible region of the array) approximately invariant over the frequency bandwidth. That is (1) where

Fig. 2. A ULA with

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N = 2M + 1 elements and element spacing d.

the frequency bandwidth. The least-square method [23] of frequency-domain system identification is applied in this paper to identify the optimal wideband beamformer network. Numerical results are provided to verify the effectiveness and validity of the proposed method. II. FREQUENCY-DOMAIN DATA FOR SYSTEM IDENTIFICATION In our study, data acquisition for the system identification requires to calculate the optimal Chebyshev weights at different operation frequencies. As discussed above, the Riblet-Chebyshev weights are the optimal Chebyshev weights for the case . In this section we formulate a matrix calculation method for the calculation of Riblet-Chebyshev weights at discrete sample frequencies over the bandwidth. These weights will be used as input data for the frequency-domain system identification procedure in Section III. A. Theory of the Riblet-Chebyshev Array Consider a centre-symmetric -element ULA shown in is an odd1 integer. The element Fig. 2, where spacing is and the array amplitude weights (4) are also symmetric with respect to the center element. Let the center of the array be taken as the phase reference point. The array pattern can be expressed as (5) where (6)

(2) (3)

Because of this equivalence, the design of our optimal wideband beamformer can be turned into a multi-input and singleoutput (MISO) system identification problem in the frequency domain, so long as the optimal Riblet-Chebyshev weights at different operation frequencies are available. System identification in the frequency domain [22] is a powerful tool. It can accomplish the curve-fitting function of the frequency response of a transfer function from empirical data. The frequency-domain data in our current system identification problem are the optimal Chebyshev weights calculated at discrete frequencies over

(7) Due to the fact that the Chebyshev polynomial the following condition:

satisfies (8)

the pattern in (5) is essentially a weighted sum of the Chebyshev polynomials of different orders. In the conventional formulation of the Dolph-Chebyshev weights [17], the pattern in (5) is 1The Riblet-Chebyshev weights are only applicable to the case when odd, and we assume in this paper that is always odd.

N

N is

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equated with an -scaled Chebyshev polynomial in terms of as

B. Matrix Formulation for the Calculation of the Riblet-Chebyshev Array real roots of the Chebyshev polynomial The be expressed as

can

(18) (9) th order Chebyshev polynomial where the be written in a more general form as if if

can

(10)

From (17) and (18), we can get the mapping from to corresponds to the zeros in the array pattern. That is

which

(19) Note that (16) can be rewritten in matrix form as

In (9), is a parameter determining the equiripple sidelobe level (in decibel) by the following expressions:

(20) where (21) (22)

(11)

(12) In [17], by equating the coefficients of like-power terms in the polynomial expression in (9), the optimal Dolph-Chebyshev weights for a Chebyshev array were obtained. However, due to the fact that the limited range of for element spacing cannot include an adequate portion of the graph of a polynomial to constrain the pattern for the whole field of view of the array, Dolph-Chebyshev weights are only optimum . To acquire the optimal for the element spacing weights for the case , Riblet [20] proposed an im(instead proved procedure in which the function in [17]) was used to expand the portion of of the graph of the Chebyshev polynomial that is included in the array pattern. Besides, for taking full advantage of the total , the graph of the Chebyshev polynomial in the range constraints listed below are further used

(13) (14) (15) The resulting pattern with the Riblet-Chebyshev weights is then expressed as (16) where (17)

The optimal Riblet-Chebyshev weights solving the following matrix equation:

can be obtained by

(23) where (24) (25) and then (26) To demonstrate the optimality of the Riblet-Chebyshev weights , comover the Dolph-Chebyshev weights for the case parisons of the weights and patterns between these two beam. formers are shown in Figs. 3 and 4 for an ULA with The element spacing is set to 0.2 and the required sidelobe . Fig. 4 clearly shows that the width of the level to mainlobe acquired by the Riblet-Chebyshev weights is much smaller than that acquired by the Dolph-Chebyshev weights. It should also be emphasized that the Riblet-Chebyshev weights . In the case , are only optimal for the case for some values the values of in (17) may be smaller than of falling within the field of view of the array. This will result in secondary peaks in the array pattern other than the mainlobe. The level of these secondary peaks will be larger than the given sidelobe level. For example, with the same conditions as in Figs. 3 and 4 except that the element spacing d is set to 0.6 , the values of in (17) over the field of view is depicted in Fig. 5. The corresponding Riblet-Chebyshev pattern is given in Fig. 6. It is worth mentioning that compared with the Dolph-Chebychev weighting method, the Riblet-Chebyshev weighting

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Fig. 6. The Riblet-Chebyshev pattern for the condition of x given in Fig. 5. The secondary peak level which is about higher than the given sidelobe ; : ; . and : level of between

030 dB

011 dB [090 041 8103 ] [41 8103 90 ]

Fig. 3. Riblet-Chebyshev weights and Dolph-Chebyshev weights for d : .

= 02

at higher frequencies is paid in terms of some superdirectivity characteristics at lower frequencies. However with the rapid advance in the digital beamforming technology, this shortcoming will become more and more insignificant and the efficient utilization of the array aperture over the entire bandwidth far exceeds this shortcoming. C. Frequency Data for MISO System Identification

Fig. 4. The corresponding Riblet-Chebyshev and Dolph-Chebyshev patterns for the weights in Fig. 3.

Based on the matrix formulation for the calculation of Riblet-Chebyshev weights proposed above, weights at different operation frequencies can be calculated easily. Here where is we assume that the frequency band is the highest frequency in the band, the element spacing of the ULA is chosen as one half of the wavelength at . The equivalent change of element spacing in the frequency band is thus where is the wavelength and , the at . As an example, for calculated Riblet-Chebyshev weights in the frequency band with a frequency interval are tabulated in Table I. The weights have been normalized with respect to , which therefore is set to 1. The frequency-dependence of the calculated with Riblet-Chebyshev weights , , , and frequency interval in the frequency band a is illustrated in Fig. 7. These will serve as frequency data for input to the MISO system identification procedure to be described in Section III. III. SYSTEM IDENTIFICATION FOR THE WIDEBAND BEAMFORMING NETWORK

Fig. 5. The curve of x in (17) for the case N d : .

= 06

= 9, D = 030 dB, and

method yields a narrower mainlobe width but shows the rapid phase-reversal characteristic of superdirective arrays (as shown in Fig. 3), which necessitates a very precise adjustment of element excitations and therefore complicates the system realization procedure. In other words, the increase in efficiency

Once the frequency-domain data are known, they are used as input data for the system identification of the MISO system in (2). The frequency response transfer function vector function of this transfer function vector will result in the optimal Riblet-Chebyshev patterns at different frequencies over the operation frequency band. Due to the fact that explicit mathematical relations between the Riblet-Chebyshev weights at different frequencies are unavailable, the identification of above MISO system can be regarded as a black-box identification problem in which a non-structured time-invariant rational transfer function of finite order is estimated from

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of the Riblet-Chebyshev weights, it only needs to identify SISO systems for a ULA with elements. From these SISO systems, we can assemble the corresponding MISO system for the optimal wideband beamforming network as:

(27)

Fig. 7. The frequency-dependence of the Riblet-Chebyshev weights over the normalized frequency band [0:5; 1]. TABLE I THE RIBLET-CHEBYSHEV WEIGHTS IN THE FREQUENCY BAND [0:5f

where, due to the centre-symmetric transfer function vector in (27) with of our MISO system, we have rewritten , , , etc, for the convenience of further manipulation. In the SISO system , the method we identification of these use consists in fitting the following rational polynomial with real coefficients

;f ]

(28) to the values of the frequency-domain data, i.e., the Riblet-Chebyshev weights at discrete sample frequencies that covers the whole bandwidth. In (28), are the system functions to be determined for . The estimates of parameters for the system function can be determined by solving the following optimization problem: (29) the frequency-domain data (i. e., the optimal Riblet-Chebyshev weights at the discrete sample frequencies). Existing system identification methods in frequency domain all require the frequency response of the systems to be known. In the single-input and single-output (SISO) system identification, the frequency domain data can be obtained by taking the ratio of FFT of the output data to the input data. The frequency response vector of a general MISO system is however not directly available because of the possible couplings between various input ports and the single output ports. In the case of our optimal wideband beamformer, it is found that the corresponding MISO system identification can be reduced to a sequence of SISO system identifications since the optimal Riblet-Chebyshev weights at different frequencies can be calculated separately and these optimal weights can serve as the samples of frequency transfer functions for every pair of input and output ports directly. Consequently, each transfer , in the transfer function, such as in (2), can be identified individually function vector using the SISO system identification approach. The final MISO system is then identified by a proper assemblage of the identified SISO transfer functions. Because of the centre-symmetry

(30) (31) In the case of little priori knowledge on the system characteristics (the case of a black-box system), the least square method is usually the most simple but efficient solution. The complex curve fitting method in [23] is used here to obtain the optimized parameter estimates. For the case of a ULA with and with the frequency-domain data shown in Table I or Fig. 7, the resultant polynomials of the system functions identified for are listed in (32)–(35) at the bottom of the following page. , Comparisons of the identified SISO transfer functions with the frequency-domain data set of are shown in Fig. 8. It can be seen that the frequency responses of the identified SISO systems are almost perfectly fitted to the calculated data sets. Once all the frequency response , are identified, the transfer funcfunctions tion vector in (27) for the MISO system is known and the wideband optimal beamformer can be realized.

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Fig. 9. Synthesized wideband patterns by using proposed optimal wideband beamformer over the normalized frequency band [0 5 1] for a ULA with = 9 and element spacing = 0 5 .

:

Fig. 8. The frequency response of the identified SISO systems 2 3 4.

; ;

:;

N

H (!), i =

IV. NUMERICAL RESULTS To demonstrate the effectiveness of the wideband beamformer proposed in the previous section, the frequency-dependent patterns over the normalized frequency band produced by this beamformer are depicted and element spacing in Fig. 9. The array is a ULA with . For comparison, the corresponding Dolph-Chebyshev patterns over the same normalized frequency band are depicted in the Fig. 10. It is noted from Fig. 9 that the optimal Dolph-Chebyshev patterns are obtained at different frequencies over the entire frequency band by the new wideband beamformer. Comparing Figs. 9 and 10, the array patterns of the new beamformer are much narrower (Fig. 9) than those of the conventional Dolph-Chebyshev beamformer (Fig. 10), especially at the lower-end frequencies. This is essentially due to the optimality of Riblet-Chebyshev weights for the case of . element spacing As mentioned above, the new wideband beamformer uses the Riblet-Chebyshev weights to produce frequency-dependent optimal Dolph-Chebyshev patterns for ULAs with element spac-

Fig. 10. The corresponding Dolph-Chebyshev wideband patterns of the ULA in Fig. 9.

ings . To demonstrate its performance at a smaller and element spacing, we calculate its patterns at compare them with those of the conventional Dolph-Chebyshev beamformer. The results are shown in Figs. 11 and 12. Note that the equivalent change of element spacing in the frequency band in this case is . From these two fig-

(32) (33) (34) (35)

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processors employ an approach that treats the transfer functions of the processors as finite-impulse-response (FIR) filters. However, in these existing array processors, a large number of coefficients (taps) are required to retain the desired frequency response as the bandwidth increases. In [24], it was proved that the optimal frequency-dependent array weighting functions were better approximated by infinite-impulse-response (IIR) filters with a considerable smaller number of coefficients than treating them as FIR filters. Many efforts have also been made to explore beamformers as IIR filters. The implementation issue of IIR filters is referred to [24], [25] for more details and is not elaborated here. V. CONCLUSION

Fig. 11. Synthesized wideband patterns by using proposed optimal wideband beamformer over the normalized frequency band [0:5; 1] for a ULA with N = 9 and element spacing = 0:3 .

Fig. 12. The corresponding Dolph-Chebyshev wideband patterns of the ULA in Fig. 11.

ures, it can be seen that the improvement (or the optimality) in the array patterns produced by the Riblet-Chebyshev weights is even more pronounced at this reduced element spacing. In Fig. 12, the relatively smaller array aperture has almost disabled the function of the Dolph-Chebyshev weights at the lower-end frequencies, resulting in rapidly increasing mainlobe width of the array patterns. On the other hand, as shown in Fig. 11, the new wideband beamformer still performs very well and the optimal Riblet-Chebyshev patterns are realized in the whole nor. However, due to the relatively malized frequency band smaller change of array aperture in this case, the change of the mainlobe width of the optimal Riblet-Chebyshev patterns over the whole normalized frequency band is smaller than in the case . It should be noted that the system identification procedure described in this paper for the identification of the transfer function of the wideband beamformer is based on the approach that treats the beamformer as an infinite-impulse-response (IIR) filter. On the other hand, many existing broadband array

A method for the design of optimal wideband beamformers for ULAs based on Dolph-Chebyshev’s beamforming theory has been proposed. A matrix formulation was introduced to calculate the frequency-dependent optimal Riblet-Chebyshev weights for ULAs with element spacings smaller than half wavelength. Using the technique of system identification with these optimal weights as input data, an equivalent MISO system for the wideband beamforming network was successful identified and characterized. Different from existing FI beamformers for wideband antenna arrays, our wideband beamformer produces frequency-dependent patterns which have the narrowest mainlobe width for any given equiripple sidelobe level over a wide bandwidth. Compared to the FI beamformers, this considerably improves the aperture efficiency of wideband antenna arrays and minimizes aperture loss at high frequencies. Numerical results were provided to illustrate the effectiveness and validity of the new method. This study successfully applied the system identification method to wideband beamforming and opened up a large area for future research. Practical issues such as mutual coupling or array platform effect need to be further investigated with this method. REFERENCES [1] I. D. Dotlic, “Minimax frequency invariant beamforming,” IEE Electronics Letters, vol. 40, no. 19, pp. 1230–1231, Sep. 2004. [2] D. P. Scholnik and J. O. Coleman, “Optimal design of wideband array patterns,” in Proc. IEEE Int. Radar Conf., Washington, DC, May 2000, pp. 172–177. [3] T. Sekiguchi and Y. Karasawa, “Wideband beamspace adaptive array utilizing FIR fan filters for multibeam forming,” IEEE Trans. Signal Process., vol. 48, no. 1, pp. 277–284, Jan. 2000. [4] D. B. Ward, R. A. Kennedy, and R. C. Williamson, “FIR filter design for frequency invariant beamformers,” IEEE Signal Process. Lett., vol. 3, no. 3, pp. 69–71, Mar. 1996. [5] S. F. Yan and Y. L. Ma, “Design of FIR beamformer with frequency invariant patterns via jointly optimizing spatial and frequency responses,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, Philadelphia, PA, Mar. 2005, pp. 789–792. [6] E. W. Vook and R. T. Compton, Jr, “Bandwidth performance of linear adaptive arrays with tapped delay-line processing,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, pp. 901–908, Jul. 1992. [7] S. C. Chan and H. H. Chen, “Uniform concentric circular arrays with frequency-invariant characteristics-theory, design, adaptive beamforming and DOA estimation,” IEEE Trans. Signal Process., vol. 55, pp. 165–177, Jan. 2007. [8] S. F. Yan and Y. L. Ma, “A unified framework for designing FIR filters with arbitrary magnitude and phase response,” Digital Signal Processing, vol. 14, no. 6, pp. 510–522, Nov. 2004. [9] A. Trucco, M. Crocco, and S. Repetto, “A stochastic approach to the synthesis of a robust frequency-invariant filter-and-sum beamformer,” IEEE Trans. Instrum. Meas., vol. 55, pp. 1407–1415, Aug. 2006.

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[10] B.-H. Wang, Y. Guo, and Y.-L. Wang, “Frequency-invariant pattern synthesis of conformal array with low cross-polarization,” IET Microw. Antennas Propag., vol. 2, no. 5, pp. 442–450, Aug. 2008. [11] T. Isernia and G. Panariello, “Optimal focussing of scalar fields subject to arbitrary upper bounds,” Electron. Lett., vol. 34, no. 2, pp. 162–164, Jan. 1998. [12] T. Isernia, P. Di Iorio, and F. Soldovieri, “An effective approach for the optimal focussing of scalar fields subject to arbitrary upper bounds,” IEEE Trans. Antennas Propag., vol. 48, pp. 1837–1847, Dec. 2000. [13] O. M. Bucci, L. Caccavale, and T. Isernia, “Optimal far-field focusing of uniformly spaced arrays subject to arbitrary upper bounds in nontarget directions,” IEEE Trans. Antennas Propag., vol. 50, pp. 1539–1554, Nov. 2002. [14] P. J. Bevelacqua and C. A. Balanis, “Minimum sidelobe levels for linear arrays,” IEEE Trans. Antennas Propag., vol. 55, pp. 3442–3449, Dec. 2007. [15] 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, Mar. 2004. [16] P. J. Bevelacqua and C. A. Balanis, “Geometry and weight optimization for minimizing sidelobes in wideband planar arrays,” IEEE Trans. Antennas Propag., vol. 57, pp. 1285–1289, Apr. 2009. [17] C. L. Dolph, “A current distribution for broadside arrays which optimizes the relationship between width and sidelobe level,” Proc. IRE, vol. 34, pp. 335–348, 1946. [18] A. Safaai-Jazi, “Modified Chebyshev arrays,” IEE Proc. Microw. Antennas Propag., vol. 145, no. 1, pp. 45–48, Feb. 1998. [19] A. Safaai-Jazi, “A new formulation for the design of Chebyshev arrays,” IEEE Trans. Antennas Propag., vol. 42, no. 3, pp. 439–443, 1994. [20] H. J. Riblet, “Discussion on ’A current distribution for broadside arrays which optimizes the relationship between width and sidelobe level’,” Proc. IRE, vol. 35, pp. 489–492, 1947. [21] A. Zielinski, “Matrix formulation for Dolph-Chebyshev beamforming,” Proc. IEEE, vol. 74, pp. 1799–1800, 1986. [22] R. Pintelon, P. Guillaume, Y. Rolain, J. Schoukens, and H. Van Hamme, “Parameter identification of transfer functions in the frequency domain, a survey,” IEEE Trans. Autom. Control, vol. 39, no. 11, pp. 2245–2260, Nov. 1994. [23] E. Levy, “Complex curve fitting,” IRE Trans. Autom. Control, vol. 4, no. 5, pp. 37–44, May 1959. [24] R. P. Gooch and J. J. Shynk, “Wide-band adaptive array processing using pole-zero digital filters,” IEEE Trans. Antennas Propag., vol. 34, no. 3, pp. 355–367, Mar. 1986. [25] H. Duan, B. P. Ng, and C. M. See, “A new broadband beamformer using IIR filters,” IEEE Signal Process. Lett., vol. 12, no. 11, pp. 776–779, Nov. 2005. Bu Hong Wang (M’06) received the M.S. and Ph.D. degrees in signal and information processing from the Xidian University, Xi’an, China, in 2000 and 2003, respectively. From 2003 to 2005, he was a Postdoctoral Fellow in the Postdoctoral Technical Innovation Centre, Nanjing Research Institute of Electronics Technology, Nanjing, China. From 2006 to 2008, he was an Associate Professor with the School of Electronic Engineering of Xidian University, Xi’an, China. Since January 2009, he has been a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His research interest is mainly in the area of array signal processing and its application in radar and communications. Dr. Wang’s Ph.D. thesis “On Some Crucial Aspects of High Resolution Direction of Arrival Estimation” was honored as the “Excellent Doctoral Dissertation of Shaanxi Province.” From 2003 to 2005, he received support from the National Postdoctoral Science Foundation, China.

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Hon Tat Hui (SM’04) received the B.Eng. degree (first class honors) and the Ph.D. degree from the City University of Hong Kong (City U), in 1994 and 1998, respectively. From 1998 to 2001, he worked at City U as a Research Fellow. From 2001 to 2004, he was an Assistant Professor at Nanyang Technological University, Singapore. From 2004 to 2007, he was a Lecturer in the School of Information Technology and Electrical Engineering, University of Queensland. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, National University of Singapore. His research is in antennas and wireless communications. His recent research interest also includes quantum computing, quantum information processing, and biomedical imaging (especially MRI). He has published more than 50 papers in internationally referred journals and presented many papers at international conferences. Dr. Hui has been a project reviewer of various industrial and government organizations. He was an exceptional performance reviewer for the IEEE Antennas and Propagation Society in 2008. He served as an editorial board member in various international journals. He has helped organize many local and international conferences.

Mook Seng Leong received the B.Sc.(Eng.) (Hons I) degree in electrical engineering and the Ph.D. degree in microwave engineering, from the University of London, London, U.K., in 1968 and 1972, respectively. After two years of a postdoctoral research fellowship attachment to Andrew antennas under the UK SERC, he joined the then University of Singapore in October 1973 and has been a Professor of electrical engineering at the NUS since 1989. His main research interests include advanced electromagnetics, analysis and design of microwave antennas and waveguides, and EMC control and management. Has published over 300 peer reviewed research papers and is a coauthor of the Wiley book Spherical Wave Functions in Electromagnetic Theory. Dr. Leong is a Fellow of the Institution of Engineering and Technology (IET), London. He is the founding Chairman of MTT/AP/EMC Chapter, Singapore IEEE Section. He received the MINDEF-NUS Joint R&D Award from DSO National Laboratories, Singapore, in 1966 for his contributions to their collaborative research projects. He received the Teaching Excellence Award (1996/97, 1999/2000) and was listed among NUS’s Top 100 teachers, in 2002. He is an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (2004–2007, 2007–2010), and a member of the Editorial Boards for IET Microwaves, Antennas and Propagation (2007–2009), and Microwave and Optical Technology Letters (since 1998). He has also been an the Organizing Chairman of the international conferences, hosted by Singapore, such as APMC 1999, PIERS 2003 and APMC 2009.

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Design and Performance of Frequency Selective Surface With Integrated Photodiodes for Photonic Calibration of Phased Array Antennas W. Mark Dorsey, Member, IEEE, Christopher S. McDermitt, Frank Bucholtz, Member, IEEE, and Mark G. Parent

Abstract—The design, fabrication, and integration of a frequency selective surface (FSS) with integrated photodiodes to allow for photonic calibration of phased array antennas is presented. The design includes embedding electrically short dipole antennas in each unit cell of the FSS, with a zero-biased photodiode placed across the gap of the diode. Fibers from an optical distribution network are passed through the honeycomb core of the frequency selective surface and pigtailed to the photodiodes. The RF performance of the frequency selective surface with integrated optics is investigated via simulations and measurements, and the results show that the structure maintains RF-transparency. Index Terms—Frequency selective surface (FSS), radome, calibration.

I. INTRODUCTION HASED array antennas are utilized in applications including radar, communications, and electronic warfare (EW). The performance of these phased arrays can be degraded by environmental effects including temperature, mechanical stresses, and vibration as well as long-term aging of system components [1]. A common calibration technique for large phased array apertures utilizes a calibration signal transmitted from an external source located away from the aperture of the phased array [2]–[4]. The presence of the external sources in the techniques described in [2]–[4] has the disadvantage of requiring a source that is not co-located with the array. The external source is often difficult to realize on vehicle platforms and requires additional real estate that is often not available. Conversely, the calibration technique described in [5], [6] uses an optical method consisting of photodiodes integrated within a frequency selective surface (FSS) that is placed in front of the phased array. This has the advantage of providing a compact calibration system that can be co-located with the phased array, thus minimizing the size requirements for the overall system. The calibration technique described in [5], [6] uses a photonic method consisting of photodiodes integrated within a FSS that is

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Manuscript received September 28, 2009; revised January 20, 2010; accepted February 23, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. W. M. Dorsey and M. G. Parent are with the Radar Division, U.S. Naval Research Laboratory, Washington, DC 20375 USA (e-mail: [email protected]. navy.mil). C. S. Mcdermitt and F. Bucholtz are with the Optical Sciences Division, U.S. Naval Research Laboratory, Washington, DC 20375 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.2010.2050451

placed in front of the phased array aperture. This technique has the advantage of providing a compact calibration system that can be co-located with the phased array, thus minimizing the size requirements for the overall system. The RF-modulated optical signal is detected by a zero-biased photodiode, and the RF signal excites electrically small dipole antennas (ESDAs) that are integrated in the FSS panel located in front of the phased above array elements. An ESDA element is centered each unit cell of the phased array to minimize the coupling from each ESDA to neighboring array elements. In this design, corresponds to the center frequency of the band pass response for the FSS (3 GHz). An ESDA element is present above each element of the phased-array to provide a calibration signal for each array element. The calibration signal will be monitored for changes that indicate variation in the amplitude and/or phase of the element excitation. Since the technique relies on relative change to a coupled signal, the tolerances on the registration between the FSS and the array are not critical as long as the alignment can be maintained. The performance and stability of the photonic calibration system has been described in [7], and the zero-biased photodiodes have been fully characterized [8]. In order to successfully implement this calibration system, it is essential that the FSS with integrated optical components maintain RF transparency within the operational frequency band of the phased array in order to avoid degrading RF performance. This paper will discuss the integration of a photonic calibration system into a frequency selective surface. The details of the optical calibration system integration will also be provided. This discussion will be followed by a presentation of simulated and measured results on the RF performance of the FSS with integrated optical components. These results will include simulations and measurements of the transmission coefficient through the FSS radome as well as measured radiation patterns for a phased array placed behind the FSS. Performing full calibration of a phased array with accompanying T/R modules was beyond the scope of this study. However, the performance and stability presented in [7] coupled with the performance of the FSS presented in this manuscript provide a high confidence level in the utility of the photonic calibration technique discussed in [5], [6]. II. DESIGN OF A FREQUENCY SELECTIVE SURFACE WITH INTEGRATED PHOTODIODES A frequency selective surface (FSS) provides minimal attenuation in the passband while becoming increasingly opaque outside of the pass band. This type of structure is often used in the

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DORSEY et al.: DESIGN AND PERFORMANCE OF FSS WITH INTEGRATED PHOTODIODES

Fig. 1. Top view (a) and dielectric profile (b) of a frequency selective surface (FSS) with 2.5–2.5 GHz pass band. (a) Top view of FSS unit cell, (b) Dielectric profile of FSS unit cell.

design of a hybrid radome consisting of FSS layers sandwiched between dielectric layers; the goal of the hybrid radome is to reduce the out of band radar cross section (RCS) of an antenna. At frequencies where the FSS is opaque, the incident signal is reflected back in the bi-static direction [9]. At frequencies where the radome appears transparent, minimal RCS improvements are seen. However, if the pass band of the hybrid radome coincides with the operational frequency band of the phased array, the energy incident on the array can be received with minimal attenuation. For this demonstration, a FSS with a pass band covering 2.5–3.5 GHz was designed. In FSS designs, conducting grids appear inductive to incident waves while arrays of conducting patches appear capacitive. In a design proposed by Wahid and Morris [10], a conducting grid and array of conducting patches are superimposed to result in a band pass structure. The design used in this demonstration is based on the ideas and concepts from Wahid and Morris [10]. The unit cell for the band pass FSS shown in Fig. 1(a), is a square with side length of 4.32 cm. The conducting grid consists of a rectangular grid with 0.24 cm thickness, and each corner has a 90-degree arc with a 1.27 cm radius. The center of the unit cell consists of a 1.27 cm radius circular patch that will serve as the capacitive component. The conducting pattern is printed on . cm) RO4350 substrates ( that are present above and below a 1.2 thick honeycomb core. The dielectric profile of the FSS unit cell is shown in Fig. 1(b). The optical calibration approach illustrated in Fig. 2 [5], [6] requires the installation of a zero-biased photodiode (PD) and accompanying electrically short dipole antenna (ESDA) above each element in the antenna array. Within the optical distribution network (ODN), the RF modulated signal is optically amplified and then distributed to an ESDA located within each unit cell of the FSS. Each channel of the photonic calibration network allows static adjustment over both the amplitude and phase. The ODN and fiber pigtails from the FSS photodiodes are connected via a ribbon fiber cable—allowing the ODN control unit to be remotely located on the platform and in principle, hundreds of meters away from the array aperture. After photodetection, the RF signal is placed across the ESDA antenna located at each unit cell in the FSS. The RF signal excites currents on the ESDA, and the dipole will transmit the calibration signal to the corresponding antenna element. The shape of the transmitter is shown in Fig. 3(a), and the gap where the diode will be located is circled for clarity. A photograph of the diode used in this design is

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Fig. 2. Architecture for phased-array calibrator employing an array of zerobiased photodiodes (PD) each driving an electrically short dipole (ESDA). Each PD receives RF-modulated light from a single-mode optical fiber through an optical distribution network (ODN).

Fig. 3. (a) Location of the photodiode within the ESDA and (b) photograph of the photodiode. (a) Shape of the electrically short dipole antenna (ESDA), (b) external electrical connection pads of the zero-bias photodiode.

provided in Fig. 3(b). The photodiodes used in this design were manufactured by Anadigics (Part Number PD070-001-720) and had a responsivity of 0.91 A/W. The performance of this diode was characterized in [7]. The center of the FSS unit cell is modified to include the ESDA and accompanying photodiode. Conducting solder pads are printed on the bottom side of the microwave substrate and plated through holes are added to provide electrical continuity between the ESDA and the solder pads. The photodiode is installed across the solder pads on the back side of the dielectric substrate. The unit cell of the FSS with integrated optical calibration system is detailed in Fig. 4. A detailed drawing showing the ESDA, solder pads, plated through holes, and photodiode is provided in Fig. 5. In this figure, one side of the ESDA is removed for clarity. The conducting pattern is printed on RO4350 . ) that are present substrates ( . The above and below a 3 cm thick honeycomb core bandpass for the FSS covers the 2.5–3.5 GHz operational bandwidth of the phased array. Fig. 6 shows two views of the constructed FSS panel with integrated photonics calibration system. Fig. 6(a) shows the optical fibers leaving the optical distribution network (ODN), where they pass through a channel in the honeycomb core of the FSS. The channel is machined into the honeycomb core to provide room for the optical fibers and the photodiodes. Fig. 6(b) shows the back side of the microwave substrate that

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Fig. 6. Photographs showing (a) optical fibers leaving the optical distribution network and passing through a channel in the FSS honeycomb core and (b) the optical fiber - ESDA interface. (a) Optical fibers pass through a channel in the honeycomb core of the (FSS), (b) The optical are connected to a photodiode that is integrated across two solder pads.

detailed in Fig. 5. The solder pads are electrically connected to the printed ESDA by plated through holes. The FSS design was characterized in simulations and measurements—both with and without the integrated optical calibration system—to investigate its performance. The results are presented in the following section. III. SIMULATED AND MEASURED RESULTS

Fig. 4. Dimensioned drawing of the frequency selective surface (FSS) unit cell cm). (a) Top view of FSS with integrated dipole probe and photodiode (units unit cell with integrated electically short dipole antenna (ESDA), (b) detailed view of ESDA integrated into FSS unit cell, (c) detailed view of photodiode integration.

=

Fig. 5. Detailed drawing showing the integration of the photodiode with the dipole probe.

contains the printed grid of the FSS and the solder pads that were previously illustrated in Fig. 4. The optical fibers will pass through the channel in the honeycomb core until they reach the photodiodes. One photodiode is present at each unit cell in the FSS, and they are integrated across the solder pads that are

A unit cell of the baseline FSS defined in Fig. 1 was modeled using CST Microwave Studio, a computational electromagnetic (CEM) software package employing the finite integration technique (FIT) [11]. This CEM package contains both a time domain and frequency domain solver. The frequency domain solver was used in these simulations owing to its ability to study off-axis performance for plane wave transmission through a unit-cell of the FSS. The FSS unit cell simulations focused on two orthogonal TEM modes defined as TE (y-polarized) and TM (x-polarized). The transmission coefficient for various scan angles is plane (parshown in Fig. 7. These scans were in the allel to the x-axis). This scan represents an H-plane scan for the TE mode and an E-plane scan for the TM mode. The results indicate excellent transmission in the pass band at all scan angles. The performance of the FSS deteriorates slightly at the 45 scan for both the TE and TM mode. In the TE case, the transmission coefficient degrades slightly at the low end of the pass band. Conversely, the TM case experiences a slight performance drop at the high end when scanned to 45 . However, in both cases the performance drop is small indicating that this design will appear transparent to incident waves in the pass band of 2.5–3.5 GHz. It should be noted that the transmission coefficients displayed in Fig. 7 do not indicate a sharp roll-off at high frequencies. The intention of this study was to illustrate the impact of the optical calibration system on the FSS performance, and not necessarily to show optimized FSS design. If sharper roll-off is desired in the transmission coefficient, cascaded designs consisting of multiple stacked FSS panels can be used as described in [10].

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Fig. 7. Simulated transmission coefficient of FSS unit cell for multiple scan angles.

Fig. 9. Photographs of the fabricated frequency selective surface with integrated photonic calibration system for phased array antennas. (a) Front view of the bandpass FSS radome containing the photonic calibration system for phased array antennas, (b) Detailed view showing location of the optical distribution network (ODN) and the single-mode fibers that pass through the noneycomb core of the FSS. Fig. 8. Measured transmission coefficient through FSS panels with various numbers of optical calibration probes compared to a baseline unit cell simulation.

Several FSS panels were constructed to investigate the impact of various ESDA configurations. Three sets of 10 10 panels including various numbers of ESDA elements were constructed. The first panel served as the baseline case, and it contained no ESDA elements. The second panel had ESDA elements populating one row transmitters total, and the third panel had ESDAs on all 100 elements. The transmission coefficient for the three panel types was measured using a pair of quad-ridged horn antennas and a pair of spot-focusing lenses. The measurement technique is similar to that described in [12]. Time gating was also used to further improve the fidelity of the measurement system by removing the contributions from environmental reflections that occur outside of the primary coupling. The measured results from Fig. 8 show two important results. The first observation is that the presence of the electri-

cally short dipole probes did not significantly impact the transmission properties of the FSS panels. The second observation is that the results for all three panels are in good agreement with the simulated results for the unit cell with no probes. The measured FSS show that the transmission efficiency starts to roll off slightly at the high edge of the pass band, but the measured dB across the transmission efficiency is still greater than entire band of interest. The measured transmission efficiency is shifted to slightly lower frequencies than the simulations due to the presence of a honeycomb substrate. Only an approximate value for the honeycomb substrate dielectric constant was available , and subsequently the exact value could not be used in simulations. In an actual system implementation, a transmission coefficient closer to 0 dB would be desired in the operational frequency band to minimize reflected energy that would negatively impact the performance of both the array elements and the calibration system. Overall, the measured results indicate that the optical calibration system can be integrated into the

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of the dipole array. The patterns are normalized to the baseline case with no optical calibration elements, and no noticeable gain degradation is seen in the measured data. Minimal impact is seen on wide-angle side lobe levels (SLL) at 2.75 GHz, and some nulls have filled in slightly. However, the impact on SLL and null depth is minimal. Moreover, the active element impedance match was calculated from measured s-parameters. The results—shown in Fig. 11—indicate that the presence of the photonic calibration FSS had minimal impact on the element impedance match. IV. CONCLUSION

Fig. 10. Radiation pattern of a 16-element dipole array with and without the presence of the FSS with integrated optical calibration system components. (a) 2.75 GHz, (b) 3.00 GHz, (c) 3.25 GHz.

This paper presents a method for integrating the components of an optical calibration system into a frequency selective surface (FSS). A zero-biased photodiode is placed across the gap of an integrated electrically short dipole antenna within each unit cell of the FSS. Fibers from an optical distribution network are passed through the honeycomb core of the frequency selective surface and pigtailed to the photodiodes. Measurements and simulations show that the FSS with integrated optical network remains RF-transparent. Furthermore, the radiation pattern of a phased array remained unchanged when the array was placed behind the FSS with integrated optics. Subsequently, this FSS enables placement of the calibration system in front of a phased array antenna without impacting the RF performance of the array. REFERENCES

Fig. 11. Measured active element VSWR for an array element with and without the presence of a frequency selective surface (FSS) with integrated photonic calibration network.

FSS panel without significantly degrading transmission performance of the FSS panel. After the transmission coefficient was characterized, the FSS radome with integrated ESDA elements was placed in front of a 20 20-element dipole array as shown in Fig. 9. The radiation pattern of a 16-element row of the array with integrated optical calibration network and FSS was measured in a near-field scanning facility, and the results were compared to the baseline array patterns with no FSS present. This test was completed to see if the components of the optical calibration system would interact with the antenna array elements and impact the transparency of the FSS. The measured results displayed in Fig. 10 show that the FSS with integrated optical calibration network had minimal impact on the radiation pattern

[1] P. Hughes, J. Choe, and J. Zopler, “Advanced multifunction RF system,” in GOMAC Digest, 2000, pp. 194–107. [2] W. Yao, Y. Wang, and T. Itoh, “A self-calibration antenna array system with moving apertures,” in Proc. IEEE MTT-S Int. Microwave Symp. Digest, Jun. 8–13, 2003, vol. 3, pp. 1541–1544. [3] R. X. Meyer, “Electronic compensation for structural deformations of large space antennas,” in Proc. Astrodynamics Conf., Vail, CO, Aug. 12–15, 1985, pp. 277–285. [4] E.-A. Lee and C. N. Dorny, “A broadcast reference technique for self-calibrating of large antenna phased arrays,” IEEE Trans. Antennas Propag., vol. 37, no. 8, pp. 1003–1010, Aug. 1989. [5] E. G. Paek, M. G. Parent, and J. Y. Choe, “Photonic in-situ calibration of a phased array antenna using planar lightwave circuit,” in Proc. Int. Topical Meetings on Microwave Photonics (MWP 2005), Oct. 12–14, 2005, pp. 351–354. [6] M. G. Parent, E. G. Paek, F. Bucholtz, C. McDermitt, and P. Knapp, “Phased-array calibration using radome embedded optical transducers,” in Proc. Antenna Application Symp., Monticello, IL, Sept. 21–23, 2005, pp. 345–360. [7] C. S. McDermitt, W. M. Dorsey, M. E. Godinez, F. Bucholtz, and M. G. Parent, “Performance of 16-channel, photonic, phased-array antenna calibration system,” Electron. Lett., vol. 45, no. 24, pp. 1249–1250, Nov. 19, 2009. [8] M. E. Godinez, C. S. McDermitt, A. S. Hastings, M. G. Parent, and F. Bucholtz, “RF characterization of zero-biased photodiodes,” IEEE J. Lightw. Technol., vol. 26, no. 24, pp. 3829–3834, Dec. 15, 2008. [9] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000, pp. 14–16. [10] M. Wahid and S. B. Morris, “Band pass radomes for reduced RCS,” in Proc. Inst. Elect. Eng. Colloq. on Antenna Radar Cross Section, May 7, 1991, pp. 4/1–4/9. [11] CST Microwave Studio, v.2008.04 Feb. 25, 2008. [12] D. K. Ghodgaonkar, V. V. Varadan, and V. K. Varadan, “A free-space method for measurement of dielectric constants and loss tangents at microwave frequencies,” IEEE Trans. Instrum. Meas., vol. 37, no. 3, pp. 789–793, Jun. 1989.

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W. Mark Dorsey (M’09) received the B.S. and M.S. in electrical engineering with a focus in electromagnetics from the University of Maryland, College Park, in 2002 and 2005, respectively, and the Ph.D. degree in electrical engineering from Virginia Polytechnic Institute and State University, Blacksburg, in 2009. As a doctorate student, he researched dual-band, dual-polarized antenna elements and arrays. He has worked on antenna design, measurement, and integration for the Radar Division of the U.S. Naval Research Laboratory since 1999. His primary research interests include reconfigurable and multifunction antenna design, ultrawideband (UWB) antenna design, antenna isolation studies, antenna measurement, and calibration.

Frank Bucholtz (M’82) was born in Detroit, MI, on April 4, 1953. He received the B.S. degree in physics and mathematics from Wayne State University, Detroit, MI, in 1975, and the M.S. and Ph.D. degrees in physics from Brown University, Providence, R1, in 1977 and 1981, respectively. From 1981 to 1983, he was an NRC Postdoctoral Research Associate at the U.S. Naval Research Laboratory where he conducted research in the area of ferrimagnetic devices for microwave signal processing. He is currently a member of the Optical Sciences Division at the U.S. Naval Research Laboratory, Washington, DC. His research interests include fiber-optic sensors, hyperspectral imaging, and analog microwave photonics.

Christopher S. McDermitt was born in Harrisburg, PA, on November 29, 1977. He received the B.S. degree in physics from Bloomsburg University, Bloomsburg, PA, in 2000. From 2001 to present, he has been a Research Physicist in the Optical Sciences Division, U.S. Naval Research Laboratory, Washington, DC. His research interests include photonics signal processing, fiber-optic interferometric systems, and enhancement of microwave photonic links.

Mark G. Parent received the B.S. degree in electrical engineering from Michigan Technological University in 1982 and the M.S. degree in physics from Michigan Technological University, Houghton, in 1985. In 1985, he joined the U.S. Naval Research Laboratory, Washington, DC, where he is currently Section Head in the Radar Analysis Branch of the Radar Division, Naval Research Laboratory. His research interests include phased array antenna design, optical beamforming techniques, radar cross section measurements and advanced microwave/optical system concepts.

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 8, AUGUST 2010

A Through-Dielectric Radar Imaging System Gregory L. Charvat, Member, IEEE, Leo C. Kempel, Fellow, IEEE, Edward J. Rothwell, Fellow, IEEE, Christopher M. Coleman, Member, IEEE, and Eric L. Mokole, Senior Member, IEEE

Abstract—Through-lossy-slab radar imaging will be shown at stand-off ranges using a low-power, ultrawideband (UWB), frequency modulated continuous wave (FMCW) radar system. FMCW is desirable for through-slab applications because of the signal gain resulting from pulse compression of long transmit pulses (1.926–4.069 GHz chirp in 10 ms). The difficulty in utilizing FMCW radar for this application is that the air-slab boundary dominates the scattered return from the target scene and limits the upper bound of the receiver dynamic range, reducing sensitivity for targets behind the slab. A method of range-gating out the air-slab boundary by significant band-limiting of the IF stages facilitates imaging of low radar cross section (RCS) targets behind the slab. This sensor is combined with a 1D linear rail and utilized as a rail synthetic aperture radar (SAR) imaging system. A 2D model of a slab and cylinder shows that image blurring due to the slab is negligible when the SAR is located at a stand-off range of 6 m or greater, and thus, the two-way attenuation due to wave propagation through the slab is the greatest challenge at stand-off ranges when the air-slab boundary is range-gated out of the scattered return. Measurements agree with the model, and also show that this radar is capable of imaging target scenes of cylinders and rods 15.24 cm in height and 0.95 cm in diameter behind a 10 cm thick lossy dielectric slab. Further, this system is capable of imaging free-space target scenes with transmit power as low as 5 pW, providing capability for RCS measurement. Index Terms—Dielectric slab, frequency modulated continuous wave (FMCW), linear FM, linear frequency modulation (LFM), low-power radar, pulse compresion, radar imaging, rail SAR, synthetic aperture radar, through lossy-dielectric slab imaging, ultrawideband radar.

I. INTRODUCTION

A

RADAR sensor capable of imaging targets behind a lossy dielectric slab at stand-off ranges will be shown in this paper. The motivation is to develop a sensor capable of locating targets behind a lossy dielectric slab (such as concrete, plastic, wood, ceramic, drywall, or other dielectrics) without a priori information or assumptions about the target scene. Linear frequency modulation (LFM) radar was chosen because of the

Manuscript received April 27, 2009; revised October 30, 2009; accepted January 31, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by the Office of Naval Research, ONR Code 30, the Expeditionary Maneuver Warfare & Combating Terrorism Department. G. L. Charvat is with the Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, MA 02420-9108 USA (e-mail: [email protected]). L. C. Kempel, and E. J. Rothwell are with the Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824 USA (e-mail: [email protected], [email protected]). C. Coleman is with Integrity Applications Incorporated, Chantilly, VA 20151 USA (e-mail: [email protected]). E. J. Mokole is with the Naval Research Laboratory, Washington, DC 20375 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050424

single-pulse sensitivity and dynamic range resulting from pulse compression of long time duration (10 ms) pulses. A 10 ms pulse time is too long for use in a pulsed radar in a through-slab application because the range swath is small (typically spanning 0 to 20 m or less down range) compared to the pulse length. As a result, FMCW must be utilized, where the radar transmits and receives simultaneously. Unfortunately, the scattering off of the air-slab boundary causes the greatest target return, setting the upper bound of the dynamic range in the digital radar receiver which effectively limits sensitivity. It is for this reason that the majority of through-slab radar development has focused on UWB short-pulse radar systems, where the air-slab boundary can be range-gated out in the time domain, allowing for maximum dynamic range and sensitivity to be applied to the range bins behind the air-slab boundary. Examples of this operating in the 1–3 GHz frequency range are treated in [1]–[5]. In order to achieve the average power necessary for reasonable signal-to-noise ratios (SNRs) of returns from targets behind lossy slabs these radar systems must operate at a high peak power, or alternatively, at a low peak power with a high pulse rate frequency (PRF). Coherent integration on receive is necessary to take advantage of the increased average transmit power when operating at a high PRF. UWB short-pulse radar systems rely on the latest analog to digital converter (ADC) technology because of the bandwidths required to acquire single pulses. Motivated by ability of UWB short-pulse radar to range-gate out the air-slab boundary, but with a desire to achieve high single-pulse SNR using low peak transmit power, a modified FMCW radar architecture is developed that provides an effective range gate while at the same time operating in an FMCW mode using 10 ms LFM pulses. This architecture provides high sensitivity and dynamic range for detection of targets behind a slab with easy-to-meet ADC specifications (16 bits of resolution at 200 KSPS). This radar sensor operates at S-band, where it radiates UWB S-band linear chirps from 1.926–4.069 GHz, and it is mounted on to a 1D rail where it is utilized as a rail synthetic aperture radar (SAR) imaging sensor. Most through-lossy-slab radar systems use some method of beam forming to localize targets behind a slab and place their antenna elements directly on or in close proximity to the slab in order to reduce air-to-slab path loss. When the antenna elements are located close to or directly on the slab the effects of Snell’s law severely reduce the performance of free-space beam forming algorithms by distorting wave propagation through the slab, therefor, much research focus on through-slab beam-forming algorithms has been to develop methods to counter the Snell’s law effects of the dielectric slab [6]–[11]. In this paper a 2D slab and cylinder model is developed and used to show that when SAR beamforming is utilized and the

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The slab model is based on wave matrix theory from [12], where the normalized impedance of the dielectric with a plane wave incident at an angle of from the normal is

(1)

where is the relative permittivity and the conductivity is assumed to be varying linearly with frequency from 0.142–0.186 S/m over the 2–4 GHz frequency range of the transmit chirp [13]. The electrical distance traveled by the wave inside of the dielectric slab for an oblique incidence angle is Fig. 1. Through slab imaging geometry.

imaging sensor is placed at a stand-off range (6 m or greater) very little image de-focusing occurs because the bending of the plane wave through the dielectric slab has a negligible effect on the SAR image. When range-gating is utilized the through-slab radar problem becomes one of achieving maximum sensitivity and dynamic range to overcome the stand-off distance to the slab and the two-way attenuation through the slab, rather than of re-focusing distorted radar data due to the slab, where, the single-pulse sensitivity and dynamic range of the modified FMCW architecture enables the use of this sensor at stand-off ranges. In Sections II and III a composite 2D scattering model of a lossy slab and a cylinder is developed to demonstrate the utility of operating a through-slab SAR at stand-off ranges. Although the authors have chosen to model a 2D infinite cylinder and slab, the simulated and measured results will be shown to be in agreement. From these results, a radar architecture is developed and implemented in Section IV which provides high sensitivity due to a long duration 10 ms UWB chirp and a short-duration range gate. Free-space SAR imagery of 15.24 cm tall metal rods using 10 mW and 5 pW of peak transmit power is shown in Section V, demonstrating the high sensitivity of this design for RCS measurements. In Section VI, radar imagery through a lossy-dielectric slab is shown to agree with the model, and imagery of target scenes made up of targets as small as 15.24 cm tall metal rods behind a lossy dielectric demonstrate the SAR’s sensitivity. Summary and future work are discussed in Section VII.

(2) where the free-space wave number is (3) The SAR will be using vertically polarized antennas where the incident plane wave is propagating in the direction with the electric field component in the direction and the magnetic field component in the direction. Therefor the polarization is [14] and thus the wave transverse magnetic to the axis amplitude coefficient at the dielectric boundary is (4) m is the distance from the radar system to the where surface of the dielectric slab. The simulated range-profile is calculated by solving the wave matrix equations for the complex amplitude of the field traveling in the normal direction at the air-slab interface, which is given by

(5) where the complex amplitudes ( and ) of the field traveling direction at the interfaces of the slab and cylinder, in the respectively, are related by

II. SIMULATED RANGE PROFILES The through-slab rail SAR geometry is shown in Fig. 1, where a dielectric slab of thickness d ( cm) is placed between the rail SAR and a perfect electric conductor (PEC) cylinder of cm). The radar system is located ( m) radius ( from the front of the slab, and the cylinder is located ( m) behind the slab. When simulating a range profile of this target scene it is assumed that the radar is located at a fixed location . on the rail such that the angle of incidence

(6) The cylinder oriented vertically in the direction, and thus the scattering solution of a 2D PEC cylinder from [14] is given by (7)

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range facilitates the detection of the slightest change in scattered return from the target scene behind the slab. For these reasons a range gate, high-sensitivity, and high dynamic range are required. III. SIMULATED SAR IMAGERY

Fig. 2. Simulated range profile of a 10 cm thick lossy-dielectric slab in front a 7.62 cm radius cylinder at normal incidence.

where

The model from Section II is expanded, providing a simulated SAR image of a cylinder behind a lossy slab. The geometry is shown in Fig. 1. The SAR is composed of a radar sensor m. The antenna mounted on a linear rail of length is directed toward the target scene which is made up of a dim and electric slab and a 2D PEC cylinder at ranges m from the antenna. The cylinder location along the axis is defined by the offset distance from the rail . The 2D PEC cylinder has a radius cm center and is located behind the slab. The slab has a thickness of cm. The radar sensor moves down the linear rail acquiring 48 evenly spaced LFM range profiles at incremental locations from to . The incident angle is dependent upon the on the rail relative to the location of the radar at position 2D PEC cylinder :

Since the radar system is effectively mono-static, the bi-static . observation angle is The received field, a scattered plane wave from the dielectric surface, is represented by

(9)

(8)

is the distance from the radar antenna to the surface where of the dielectric slab in the direction of the cylinder

The inverse discrete Fourier transform (IDFT) of is taken for a number of test frequencies that emulate an S-band LFM transmitted pulse from 1.929 GHz to 4.069 GHz in 1000 steps. The . The simulated range profile incident wave amplitude shows the locations of the front of the slab at approximately 41 ns and the front of the cylinder at approximately 62 ns (Fig. 2). The scattered return off of the slab-air interface is visible at approximately 42.5 ns, where the conductivity of the slab attenuates the return off of the back side of the slab. The scattered return from the slab has the greatest magnitude in this range profile, where the cylinder behind the slab is approximately 35.8 dB below the slab. If the PEC cylinder is placed directly behind and in close proximity to the slab, then the model breaks down because this model assumes that the scattering off of the cylinder is in the far zone. Consequently, the cylinder must be located at least 4 cylinder diameters away from the rear of the slab in order to achieve accurate results. From these results it is clear that if the dielectric slab is not eliminated from the range profile by range-gating, then the initial reflection from the slab will set the upper bound of the dynamic range of the radar’s ADC—thereby reducing the number of available bits for digitizing the returns from small targets behind the slab. In addition, it is important to have the greatest sensitivity possible to overcome the two-way attenuation due to the slab. Complex target scenes comprised of large and small scatterers behind the slab are expected in practical applications, therefore, it is important to be able to detect and image as many scatterers as possible. This can be achieved by coherent change detection (or coherent background subtraction), where dynamic

(10) is the distance from the opposite side of the slab to and the 2D PEC cylinder (11) depends on the variables , and The quantity , which on direct substitution into (1) through (4) yield the -dependent scattered field equation represented by (8). These calculations are represented by the frequency and rail, where position dependent scattered field matrix is the cross range radar position (in m) on the linear rail is the instantaneous radial frequency shown in Fig. 1 and at time for an LFM modulated transmit signal: (12) In this is the chirp rate of 214 GHz/s, is the radar center frequency of 3 GHz, and BW is the chirp bandwidth of 2.143 GHz. For the simulated imagery shown in this paper spans 0 to 10 ms in 256 steps. The conductivity varies from 0.140–0.188 S/m across the transmit chirp frequency range of 1.929–4.069 GHz [13]. Coherent background subtraction is utilized in order to image the cylinder behind a lossy slab. One scattered data set was simrepresented by ulated without the cylinder by letting , another was simulated with the cylinder present where

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Fig. 4. Simplified block diagram of the range-gated FMCW radar system.

degradation in radar sensitivity and dynamic range appear to be the greatest challenges to imaging behind a lossy dielectric slab when the radar imaging sensor is located at a stand-off range of 6.1 m or more. IV. SAR ARCHITECTURE

Fig. 3. Simulated SAR imagery of a 2D cylinder with radius a free-space (a) and behind a 10 cm thick lossy-dielectric slab (b).

= 7:62 cm in

is represented by (7). The difference between these two data sets is the background subtracted through slab cylinder image data set (13) The range migration algorithm (RMA) from [15] was used to process SAR images of this simulated data for the cylinder in free space at a range of 9.1 m and behind the lossy slab at the same range [Fig. 3(b)]. The resulting relative magnitude of the dB, the down range cylinder behind the dielectric slab is cm, the down range extent is approximately 8.1 location is cm, and the cross range extent is approximately 18.4 cm. The presence of the lossy slab does not significantly distort the SAR image; however, the intensity is greatly attenuated compared to the free-space image [Fig. 3(a)] where the relative magnitude is dB. The resulting down range location of the cylinder in cm, the down range extent is free-space is approximately 8.1 cm, and the cross range extent is approximately 17.1 cm. These results show that the slab causes the cylinder’s image to be slightly offset in downrange position. The cylinder image is not distorted noticeably because there is no change in down range extent and only a 1.3 cm increase in cross range extent. The return magnitude of the cylinder is significantly lower ( dB) when located behind the slab. These results indicate that a

It was shown in Section II (Fig. 2) that the air-slab boundary dominates the return amplitude in a through-slab target scene. For this reason a radar architecture is developed that provides a method for range-gating out the air-wall boundary when using long-duration 10 ms transmit chirps by utilizing a modified FMCW architecture. This architecture takes advantage of the small down-range swaths inherent in practical through-slab imaging scenarios by range gating the unwanted scatterers through the use of high-Q IF band-limiting filters which are applied to the de-correlated received radar chirp. As an added benefit this method of range gating reduces the receiver noise bandwidth fed to the digitizer. The resulting range-to-target information from a de-correlated LFM signal is in the form of low-frequency beat tones, as shown for the case of FMCW in [16], [17]. The more distant the target, the higher the frequency of the de-correlated beat tone. For this reason, it is possible to implement a short-duration range gate in a conventional FMCW radar system by simply placing a band-pass filter (BPF) at the output of the video amplifier. However, this is challenging to implement because it is difficult to design effective high circuit Q BPFs at base-band. Much higher performance BPFs are available in the form of IF communications filters centered at high frequencies which are found in two-way radios and communication receivers. Examples include crystal, ceramic, SAW, and mechanical filters. IF filters provide a high circuit Q, where Q is the filter quality [18], is the center frequency factor defined as dB bandwidth of the filter. The of the BPF, and BW is the ECS-10.7-7.5B is a 4 pole crystal filter used in this radar system dB that has an operating frequency of 10.7 MHz with a bandwidth BW of 7.5 kHz. A simplified block diagram of the radar system is shown in Fig. 4. The method of range gating by using high-Q communication filters will be described in detail (in the following discussion, amplitude coefficients are ignored). OSC1 is a high frequency oscillator which is tuned just above or below the center

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frequency of the high-Q IF filter FL1. The frequency output , and the output voltage can be represented by of OSC1 is the equation (14) The output of OSC1 is fed into the IF port of the mixer MXR1, and the local-oscillator (LO) port of MXR1 is driven by the 1.926–4.069 GHz voltage-tuned Yttrium Iron Garnet (YIG) oscillator OSC2. OSC2 is the source of the FMCW ramp modulation for this radar system. OSC2 is linear frequency modulated by a ramp input, where the output of OSC2 can be represented by the equation (15) and is the chirp rate in Hz/s. OSC1 and OSC2 are mixed together in MXR1 to produce the transmit signal, which is then amplified by power amplifier PA1. The output of PA1 is fed into the transmit antenna ANT1 and radiates towards the target scene. The transmitted signal from ANT1 is

which after some simplification becomes

(16) After propagating to the target scene, scatters from the target and propagates back to the receive antenna ANT2 in the . The received signal at ANT2 is repreround-trip time sented by

(17) The output of ANT2 is amplified by the low-noise amplifier LNA1 and is then fed into mixer MXR2. The LO port of MXR2 is fed by OSC2, which has the same LO input as MXR1. The IF output of MXR2 is the product

As a practical consideration, the high-frequency terms can be dropped, since the IF port of MXR2 cannot output microwave frequencies. Performing the indicated multiplication in the preceding equation yields

(18) As another practical consideration, the DC blocking capacitors in IF amplifier AMP1 rejects the DC phase terms, resulting in

(19)

is fed into the high-Q IF filter FL1, which has center Then and bandwidth BW. Oscillator OSC1 is set to a frequency , causing FL1 to frequency such that . Thus the output of FL1 is pass only the lower sideband of

(20) Only beat frequencies in the range, , are passed through FL1. The band-limited IF signal (which is proportional to the down-range target location) is effectively a hardware range gate, because the range to the target is directly proportional to the beat frequency in an FMCW radar system. Because increasing (decreasing) the bandwidth of FL1 increases (decreases) the range-gate duration, the range gate is adjustable if a number of different bandwidth filters are switched is increased, then FL1 in and out of the IF signal chain. If passes only signals that satisfy the equality in (20). To allow the would have to increase IF signals to pass through FL1, frequency, because is to compensate for a higher subtracted from . Thus, filter FL1 will only pass beat tones further down range but at the same range duration in length, were increased. Consequently, the range gate is adif justable in physical down-range location (physical down-range . time delay) by simply adjusting the frequency of In the last step of the signal chain in Fig. 4, the output of FL1 is down-converted to base band through MXR3. The LO port of MXR3 is driven by OSC1, and the output of MXR3 is fed through Video Amp1, which has an output represented by

Video Amp1 is an active low-pass filter that rejects the higher frequency component of the cosine multiplication, resulting in the video output signal

(21) which is a range-gated base-band video signal. This result is similar to the traditional FMCW systems described in [16], [17], except that this signal is band limited by a high-Q band-pass filter that provides a short-duration range gate for the long duration LFM chirp waveforms produced by this radar system. This radar is capable of chirping from 1.926 GHz to 4.069 GHz in 2.5 of 857 GHz/s, ms, 5 ms, and 10 ms providing chirp rates 428 GHz/s, and 214 GHz/s respectively. The narrow IF bandwidth provides a range gate of 8.75 ns for a chirp rate of 857 GHz/s, 17.5 ns for a chirp rate of 428 GHz/s, and 35 ns for a chirp rate of 214 GHz/s. This range gate rejects the flash off of the air-slab boundary, allowing full dynamic range of the digitizer to be applied to the target scene behind the air-slab boundary. The hardware implementation is much like that of [17], where the radar sensor is moved automatically down a 2.44 m linear rail to acquire range profiles of the target scene at 5.08 cm evenly spaced increments. A photograph of the radar system is shown

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Fig. 6. Target scene of 15.24 cm tall 0.95 cm diameter carriage bolts in a block ‘S’ pattern. Fig. 5. The S-band through-slab radar imaging system.

in Fig. 5. The receiver noise figure (NF) is 3.3 dB, the minimum detectable signal (MDS) was measured to be less than dBm, and the system analog dynamic range was measured to be greater than 120 dB (before pulse compression). The peak transmit power is approximately 10 mW and is adjustable down to the pW level. The transmit and receive antennas are linearly tapered slot antennas ([19]). Further details are presented in [20]. Calibration is achieved by acquiring a single range profile of a 1.5 m tall metal pole at a known location down range. This pole is assumed to be a point target. Calibration coefficients are applied to all range profiles prior to SAR processing. Coherent background subtraction is utilized in all data shown. The transmit pulse is continuously correlated with the received pulses in MXR2, the output of Video Amp1 is digitized, and the IDFT is applied to this data to determine range. This technique is known as stretch mode pulse compression processing [21]. By using a 200 KSPS 16 bit ADC, this radar is capable of ranging targets from 0–70 m down-range alias-free with a 10 ms pulse that chirps across 2.14 GHz of bandwidth. This architecture provides an equivalent 467 ps radar pulse resulting in a 7 cm range resolution, but at the same time, is as sensitive as a receiver with a 100 Hz bandwidth [22]. An equivalent UWB impulse radar architecture would transmit a 467 ps short pulse and would have to acquire the entire instantaneous bandwidth of this pulse scattered off of the target scene, requiring an ADC with 2.14 GHz of bandwidth and a sampling rate of at least 4.28 GSPS. Both architectures would provide the same range resolution; however, a 4.28 GSPS ADC is generally more expensive than a 200 KSPS ADC utilized in the modified FMCW radar architecture. Furthermore, the average power transmitted by the modified FMCW architecture is 2 mW, where the peak power is 10 mW with a 10 ms pulse width and a PRF of approximately 20 Hz (a PRF of 90 Hz should be possible with better data acquisition equipment). An equivalent 2 mW average power UWB short-pulse radar would have to transmit 2 watts of peak power at a high PRF of 2.14 MHz to achieve an alias-free maximum range of 70 m. This radar would have to integrate approximately 21 400 pulses coherently to achieve the same SNR as one 10 ms pulse from the modified FMCW radar. In addition to this, 2 watt

power amplifiers covering 2–4 GHz are generally more expensive than 10 mW power amplifiers. V. FREE-SPACE RADAR IMAGERY In this section, free-space imagery is discussed for selected scenarios to test the SAR’s sensitivity and resolution. Although the radar was exercised for target scenes with numerous objects, only the free-space images of a block-S configuration of 14 carriage bolts, with equal spacing of approximately 0.305 m between adjacent bolts, are discussed (Fig. 6). The bolts are [14]) that is parallel to mounted on a styrofoam board ( the ground and 3.7 m from the rail SAR. Furthermore, the board is essentially electromagnetically transparent at the SARs transmission frequencies (1.926–4.069 GHz). Each bolt is 15.24 cm long with a diameter of 0.95 cm. Each bolt is mounted vertically as shown in Fig. 6 providing small down range and cross range extents, allowing the bolts act like a point scatterers. Imaging these bolts allows the SAR’s resolution to be tested. The range resolution was measured by irradiating the target scene with 10 mW of peak transmit power resulting in the image shown in Fig. 7(a). The expected range resolution (without weighting) depends on the chirp bandwidth, and was calculated using the equation [15] (22) where is the chirp bandwidth ( GHz), resulting in cm. The measured down-range resolution is 8.8 cm, showing that the SAR is performing close to what theory predicts. The cross range resolution depends on the length of the array ( m) and the distance of the point target to the front of the array in both down range and cross range. This is calculated (without weighting) by using [15] (23) is the range to point target, is the angle from the where center of the aperture to the point target, and is the change in target aspect angle from 0 to across the aperture. The expected cross range resolution for all targets shown at 4.87 m down range is 10.4 cm. The measured cross range is 11.7 cm. The expected

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Fig. 7. SAR image of a group of carriage bolts in free-space using 10 mW (a), 10 nW (b), 100 pW (c), and 5 pW (d) of transmit power.

cross range resolution for all targets shown at 5.76 m is 12.2 cm. The measured cross range is 11.9 cm. These results show that the SAR is performing close to predictions. To test the SAR’s sensitivity, the configuration was irradiated at three extremely low power levels at the antenna terminals [10 nW, 100 pW, 5 pW peak power measured images shown in Fig. 7(b)–(d)]. On comparing the imagery at these power levels to the imagery at the full-power setting (10 mW), the authors observed that the apparent image signal-to-noise ratios and resolution for every power level were nearly identical, except for the 5 pW case where the point targets furthest down range are fading into the noise. This near equivalence indicates that the high sensitivity of this radar architecture is effective for low-power free-space imaging. At the high-power setting, the image is a reasonably clear rendition of the actual configuration [Fig. 7(a)] One can discern the 14-bolt pattern as the 14 fuzzy rectangles in a block-S configuration. In the interest of comparison, Fig. 7(d) displays the image at the lowest power level (5 pW), with 64 coherent integrations or less per range profile. Coherent integration is required to increase the SNR for measurements at this power level. While the bolts near the top of the block-S appear to be fading into the noise, each bolt is clearly shown in the image. By visually inspecting Fig. 7(a)–(d), one can estimate the down-range and cross-range extents. In particular, the down-range extent is 0.89 m between 4.87 m and 5.76 m, and

m and 0.5 m. The the cross-range extent is 1 m between bolts are not as well separated in the cross-range dimension as they are in the down-range dimension because the down-range resolution is better than the cross-range resolution. A highly sensitive system like this SAR could have numerous applications, including RCS measurement, automotive radar, and low probability of detection radar. VI. THROUGH-SLAB SAR IMAGERY To conduct through-slab imaging, a lossy dielectric slab was built out of solid concrete blocks that were mortared together. The resulting slab was approximately 10 cm thick, 3 m wide, and 2.4 m tall. The slab was built onto a wood structure with casters so that the ensemble could be repositioned within the laboratory space (inside a garage). A photograph of the target scene, showing the rail SAR positioned 9 m from the slab in the garage, is displayed in Fig. 8. The dielectric properties of this slab are difficult to define because they depend on the mixture of the concrete used in the blocks. Consequently, based on the and the work of Halabe et al. [13], the relative permittivity of the slab are assumed to be 5 and varying linconductivity early from 0.140–0.188 S/m across the transmit chirp frequency range of 1.929–4.069 GHz. These dielectric properties are presented here as a reference and were not used in the imaging algorithm or to focus the resulting imagery.

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Fig. 8. Through-lossy-dielectric slab image scene where the S-band rail SAR is located 9 m from the slab.

Measured images of objects behind the slab were achieved using 10 mW of peak transmit power and 32 (or less) coherent integrations per range profile. All targets imaged were made up of good electric conductors. For the purpose of comparison to the measured imagery, the simulated imagery was calculated from the model in Section III. All imagery is based on the object are scene depicted in Fig. 1, where the distances taken to be 9 m, 11.4 m, 0.34 m . When the hidden object is a cylinder of radius 7.62 cm, the simulated and measured images are in close agreement (Fig. 9). The additional features in the measured image are probably induced by interactions between the cylinder and the (subtracted) background clutter. For a second scene, three objects (the carriage bolts of Fig. 6) having much smaller RCSs than the cylinder are placed along a diagonal line behind the slab. These bolts are mounted vertically, providing small down range and cross range extents that allow the bolts to act like point scatterers, hence enabling the SAR’s resolution through a lossy dielectric to be tested. In the image (Fig. 10), the location of each bolt is clearly discernible. The fact that this group of bolts can be imaged demonstrates the SAR’s sensitivity when imaging through a lossy dielectric slab. These results show the potential for imaging other low RCS targets behind a slab, including objects made out of dielectrics. The dielectric objcts would have to scatter an RCS equal to or greater than a 15.24 cm tall metal bolt. The expected down range resolution, calculated from (22) is 7 cm. The measured down range resolution is 8.9 cm and 7.8 cm for the bolts located at 1114 cm and 1205 cm down range, respectively. The expected cross range resolutions calculated from (23) are 23.1 cm and 24.9 cm at 1114 cm and 1205 cm down range, respectively. The measured cross range resolutions are approximately 21.5 cm and 32.2 cm at 1114 cm and 1205 cm down range. When imaging through a lossy dielectric slab, the range resolution is very close to what is possible in free-space. Consequently, such a radar system could be useful in imaging diverse object scenes behind dielectric slabs, when little a priori knowledge of the obscured objects is available.

= 7 62

Fig. 9. SAR image of an a : cm radius cylinder behind a 10 cm thick lossy-dielectric slab simulated (a) and measured (b).

Fig. 10. Diagonal row of three 15.24 cm tall 0.95 cm diameter carriage bolts imaged behind a 10 cm thick lossy-dielectric slab.

VII. CONCLUSION A LFM radar solution to through-slab imaging was chosen because of the single-pulse sensitivity and dynamic range

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achievable by pulse compression of long duration pulses. Due to the small imaging geometry of practical through-slab scenarios, an FMCW radar mode had to be utilized. It was shown by simulation that the scattered field due to the air-slab boundary dominates the single-pulse scattered return from the target scene. It was also shown that the return from a target behind a lossy slab is significantly lower than in free space. It was determined that, due to the large scattered signal from the air-slab boundary, a modified FMCW architecture had to be developed that is capable of range-gating out the scattered return from the air-slab boundary, allowing low RCS target scenes to be imaged through a lossy-dielectric slab with nearly perfect resolution. In addition, low RCS target scenes in free-space could be SAR imaged at low transmit power, down to 5 pico-watts. This radar sensor is effective at SAR imaging small targets behind a lossy dielectric slab at stand-off ranges using low amounts of transmit power (10 milli-watts peak) and utilizing easy-to-meet ADC specifications (16 bits at 200 KSPS). Future work will include the real-time implementation of this SAR sensor on a multiple-input multiple-output (MIMO) antenna array.

REFERENCES [1] M. A. Barnes, S. Nag, and T. Payment, “Covert situational awareness with handheld ultra-wideband short pulse radar,” in Proc. Radar Sensor Technology VI, SPIE, 2001, vol. 4374. [2] S. Nag, M. A. Barnes, T. Payment, and G. W. Holladay, “An ultrawideband through-wall radar for detecting the motion of people in real time,” in Proc. Radar Sensor Technology and Data Visualization, PSPIE, 2002, vol. 4744. [3] R. Benjamin, I. J. Craddock, E. McCutcheon, and R. Nilavalan, “Through-wall imaging using real-aperture radar,” in Proc. Sensors, and Command, Control, Communications, and Intelligence (C3I) Technologies for Homeland Security and Homeland Defense IV, May 20, 2005, vol. 5778. [4] A. Berri and R. Daisy, “High-resolution through-wall imaging,” in Proc. Sensors, and Command, Control, Communications, and Intelligence (C3I) Technologies for Homeland Security and Homeland Defense V, SPIE, 2006, vol. 6201, no. 201J. [5] W. Zhiguo, L. Xi, and F. Yuanchun, “Moving target position with through-wall radar,” presented at the CIE’06 Int. Conf. on Radar, Oct. 2006. [6] F. Ahmad, M. G. Amin, S. A. Kassam, and G. J. Frazer, “A wideband, synthetic aperture beamformer for through-the-wall imaging,” in Proc. IEEE Int. Symp. on Phased Array Systems and Technology, Oct. 14–17, 2003, pp. 187–192. [7] F. Ahmad, M. G. Amin, and S. A. Kassam, “Through-the-wall wideband synthetic aperture beamformer,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 20–25, 2004, vol. 3, pp. 3059–3062. [8] F. Ahmad, M. G. Amin, and S. A. Kassam, “Synthetic aperture beamformer for imaging through a dielectric wall,” IEEE Trans. Aerosp. Electron. Syst., vol. 41, no. 1, pp. 271–283, Jan. 2005. [9] M. Lin, Z. Zhongzhao, and T. Xuezhi, “A novel through-wall imaging method using ultra wideband pulse system,” in Proc. IIH-MSP’06 Int. Conf. on Intelligent Information Hiding and Multimedia Signal Processing, Dec. 2006, pp. 147–150. [10] M. Dehmollaian and K. Sarabandi, “Refocusing through building walls using synthetic aperture radar,” IEEE Trans. Geosci. Remote Sensing, vol. 46, no. 6, pp. 1589–1599, Jun. 2008. [11] M. G. Amin and F. Ahmad, “Wideband synthetic aperture beamforming for through-the-wall imaging,” IEEE Signal Processing Mag., pp. 110–113, Jul. 2008. [12] R. E. Collin, Field Theory of Guided Waves, 2nd ed. Piscataway, NJ: IEEE Press, 1991.

[13] U. B. Halabe, K. Maser, and E. Kausel, “Propagation characteristics of electromagnetic waves in concrete,” Massachusetts Institute of Technol. Cambridge Dept. Civil Eng., Tech. Rep. AD-A207387, Mar. 1989. [14] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [15] W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms. Boston, MA: Artech House, 1995. [16] A. G. Stove, “Linear FMCW radar techniques,” Proc. Inst. Elect. Eng. Radar and Signal Processing, vol. 139, pp. 343–350, Oct. 1992. [17] G. L. Charvat, “Low-cost high resolution X-band laboratory radar system for synthetic aperture radar applications,” presented at the Antenna Measurement Techniques Association Conf., Austin, Texas, Oct. 22–27, 2006. [18] The American Radio Relay League, Inc71st ed. Newington, CT, 1994, The ARRL Handbook. [19] R. Janaswamy, D. H. Schaubert, and D. M. Pozar, “Analysis of the transverse electromagnetic mode linearly tapered slot antenna,” Radio Sci., vol. 21, no. 5, pp. 797–804, Sep.–Oct. 1986. [20] G. L. Charvat, “A Low-Power Radar Imaging System,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Michigan State Univ., East Lansing, MI, 2007. [21] M. I. Skolnik, Introduction to Radar Systems. New York, NY: McGraw-Hill, 1962. [22] B. L. Lewis, F. F. Kretschmer Jr, and W. W. Shelton, Aspects of Radar Signal Processing. Norwood, MA: Artech House, 1986. [23] G. L. Charvat, L. C. Kempel, and C. Coleman, “A low-power highsensitivity X-band rail SAR imaging system,” IEEE Antennas Propag. Mag., pp. 108–115, Jun. 2008. [24] U. L. Rhode, J. Whitaker, and T. T. N. Bucher, Communications Receivers, 2nd ed. New York, NY: McGraw-Hill, 1996.

Gregory L. Charvat (M’09) was born in Lansing, MI, on March 14, 1980. He received the B.S.E.E., M.S.E.E., and Ph.D. degree electrical engineering from Michigan State University, East Lansing, in 2002, 2003, and 2007 respectively. He is currently a technical staff member at Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, since September of 2007. He has authored or coauthored four journal articles and 18 conference papers on various topics including; applied electromagnetics, synthetic aperture radar (SAR), and phased array radar systems. He has developed four rail SAR imaging sensors, one MIMO phased array radar system, an impulse radar, and holds a patent on a harmonic radar remote sensing system. Dr. Charvat is currently serving as the publications and poster session Co-Chair on the IEEE Phased Array 2010 committee and as Chair of the IEEE AP-S Boston Chapter.

Leo C. Kempel (F’09) received the B.S.E.E. degree from the University of Cincinnati, OH, in 1989 and the M.S.E.E. and Ph.D. degrees from Michigan State University, East Lansing, in 1990 and 1994, respectively. During his undergraduate studies, he participated in the Cooperative Education Program at General Dynamics Ft. Worth Division. After completion of his undergraduate degree, he joined the Radiation Laboratory, University of Michigan, where he studied electromagnetic theory and computational electromagnetics. After a brief Postdoctoral research experience at UM, he joined Mission Research Corporation in Valparaiso, FL. He has led several funded research efforts in the general areas of computational electromagnetics, antenna design, scattering analysis, and high power/ultrawideband microwaves. During this time, he coauthored a textbook on the use of finite element methods in electromagnetics. In August 1998, he joined the Department of Electrical and Computer Engineering, Michigan State University, East Lansing.

CHARVAT et al.: A THROUGH-DIELECTRIC RADAR IMAGING SYSTEM

Edward J. Rothwell (F’05) was born in Grand Rapids, MI, on September 8, 1957. He received the B.S. degree in electrical engineering from Michigan Technological University, Houghton, in 1979, the M.S. degree in electrical engineering and the degree of electrical engineer from Stanford University, Stanford, CA, in 1980 and 1982, and the Ph.D. degree in electrical engineering from Michigan State University, East Lansing, MI, in 1985, where he held the Dean’s Distinguished Fellowship. He worked for Raytheon Co., Microwave and Power Tube Division, Waltham, MA, from 1979 to 1982, on low power traveling wave tubes, and for MIT Lincoln Laboratory, Lexington, MA, in 1985. He has been at Michigan State University from 1985 to 1990 as an Assistant Professor of electrical engineering, from 1990 to 1998, as an Associate Professor, and from 1998 as Professor. He is coauthor of the book Electromagnetics (Boca Raton, FL: CRC Press, 2001; 2nd edition 2008). Dr. Rothwell is a Fellow of the IEEE, and is a member of Phi Kappa Phi, Sigma Xi, and Commission B of URSI. He received the John D. Withrow award for teaching excellence from the College of Engineering at Michigan State University in 1991, 1996 and 2006, the Withrow Distinguished Scholar Award in 2007, and the MSU Alumni Club of Mid Michigan Quality in Undergraduate Teaching Award in 2003. He was a joint recipient of the Best Technical Paper Award at the 2003 Antenna Measurement Techniques Association Symposium, and in 2005 he received the Southeast Michigan IEEE Section Award for Most Outstanding Professional.

Christopher M. Coleman (M’02) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Michigan State University, East Lansing, in 1996, 1999, and 2002, respectively. Since 2007, he has been with Integrity Applications Incorporated, Chantilly, VA. His research interests include self-structuring antennas, synthetic aperture radar, and radar cross-section measurement techniques.

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Eric L. Mokole (SM’01) received the B.S. degree in applied mathematics from New York University, New York, NY, in 1971, the M.S. degree in mathematics from Northern Illinois University, DeKalb, in 1973, dual M.S. degrees in physics and applied mathematics from the Georgia Institute of Technology, Atlanta, in 1976 and 1978, respectively, and the Ph.D. degree in mathematics from the Georgia Institute of Technology, in 1982. For the 1982 to 1983 academic year, he was an Assistant Professor of mathematics at Kennesaw College, Kennesaw, GA. From 1983 to 1986, he held a position in the Electronic Warfare Division of the Naval Intelligence Support Center (now the Office of Naval Intelligence) in Washington DC. Since 1986, he has been employed by the Radar Division of the Naval Research Laboratory (NRL), Washington DC. From February 2001 through August 2005, he was Head of the Surveillance Technology Branch, where he supervised approximately 30 R&D scientists, engineers, and mathematicians. His duties included: providing innovative technical solutions to Navy radar problems; developing new radar concepts, signal-processing/detection techniques, and electromagnetics research concepts; directing teams of researchers to pursue R&D solutions; and conducting basic and applied research. He led efforts on space radar (trans-ionospheric propagation), ultrawideband radar (antennas, propagation, mine detection, sea scatter, impulse radar), and waveform diversity (spectrally clean waveforms). From September of 2005 until February of 2008, he was the Acting Superintendent of the Radar Division, after which he resumed his role as Head of the Surveillance Technology Branch. As Acting Superintendent, he managed a 120-person research group, directed the R&D thrust of the Radar Division, and conducted R&D on radar-related topics. He has over 70 conference publications, journal articles, book chapters, and reports and is the lead Editor of Ultra-Wideband, ShortPulse Electromagnetics 6 (Kluwer Academic/Plenum Publishing, 2003), co-editor of Ultra-Wideband, Short-Pulse Electromagnetics 7 (Kluwer Academic/ Plenum Publishing, 2007), and co-author of Physics of Multiantenna Systems and Broadband Processing (Wiley, 2008) Dr. Mokole is a member of five IEEE Societies (AP, MTT, EMC, AES, GRS), the American Geophysical Union, the American Mathematical Society, the American Physical Society, and the Society for Industrial and Applied Mathematics. In addition, he is a Senior Member of the IEEE, the Navy lead of the Program Committee for the Tri-Service Radar Symposia, a member of the High-Power Electromagnetics Committee, a member of the IEEE AES Radar Systems Panel, the Vice Chair of NATOs Sensors and Electronics Technology Panel, a government liaison to USNC-URSI, and a founding member of the Tri-Service Waveform Diversity Working Group.

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Breast Lesion Classification Using Ultrawideband Early Time Breast Lesion Response Jianqi Teo, Yifan Chen, Member, IEEE, Cheong Boon Soh, Senior Member, IEEE, Erry Gunawan, Member, IEEE, Kay Soon Low, Senior Member, IEEE, Thomas Choudary Putti, and Shih-Chang Wang

Abstract—Breast lesion characterization for discriminating between localized malignant and benign lesions is important as the current breast screening techniques do not have the required specificity to be clinically acceptable. A method using ultrawideband (UWB) microwave imaging system for such classification is proposed in this paper. The early time portion of the backscatter breast response is processed for lesion discrimination. This method provides a high resolution since the early time lesion response has the largest signal strength. A correlator is used at the receiver to extract the early response and to quantify the degree of ruggedness of a lesion through several key parameters associated with the correlation operation. Subsequently, a large scale simulation study using a two-dimensional (2D) numerical breast model with an antenna array is used for the development of a lesion classification technique. It is shown that the lesion classification method is capable of discriminating between lesions with different morphologies. Index Terms—Breast cancer detection, early time response, lesion classification, lesion morphology, microwave imaging.

I. INTRODUCTION

B

REAST cancer has been one of the highest health concerns for women worldwide. In the United States, it causes the second highest cancer-related deaths [1]. Imaging techniques such as mammography, ultrasound and magnetic resonance imaging have enabled early breast cancer detection resulting in reduced breast cancer mortalities. Recently, ultrawideband (UWB) radar-based techniques have attracted a lot of interest for breast cancer detection to complement the existing imaging modalities to improve the early detection of breast cancer. The main UWB breast imaging approaches include: delay-and-sum algorithm [2], [3], generalized likelihood test Manuscript received March 31, 2009; revised December 11, 2009; accepted February 18, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. J. Teo, C. B. Soh, E. Gunawan, and K. S. Low are with the School of Electrical and Electronic Engineering, Nanyang Technology University, Singapore 639798, Singapore (e-mail: [email protected]). Y. Chen is with the School of Engineering, University of Greenwich, Chatham Maritime, Kent ME4 4TB, U.K. and also with the School of Computer, Electronics and Information, Guangxi University, Nanning, Guangxi 530004, China (e-mail: [email protected]). T. C. Putti is with the Department of Pathology, Yong Loo Lin School of Medicine, National University of Singapore, Singapore 119077, Singapore and also with the Department of Pathology, National University Hospital, Singapore 19074, Singapore (e-mail: [email protected]). S.-C. Wang is with Sydney Medical School, University of Sydney, Sydney NSW 2006, Australia and also with the Department of Radiology, Westmead Hospital and Westmead Breast Cancer Institute, Westmead NSW, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050423

detection [4], data-independent space time beamforming [5], data adaptive beamforming [6], tissue sensing adaptive radar [7], time reversal [8] and time of arrival data fusion [9]. The existing imaging techniques are capable of localizing breast lesions satisfactorily. However, the discrimination between benign and malignant lesions is subject to variability in radiological interpretation of the image. As a result, biopsies are performed but 90% of the breast lesions that undergo biopsies are confirmed to be benign. Hence imaging techniques which improve upon the discrimination of lesions are of great clinical interest. Clinically, compact rounded lesions are mostly associated with benign masses, while malignant lesions usually exhibit irregular morphology [10]–[12]. Previous studies have suggested the possibility of using the UWB backscatter signature to characterize the target [13]–[17]. The late time response of the UWB backscatter waveform from numerical breast phantoms [13]–[16] is affected by the size, shape and dielectric values of the breast lesion. Furthermore, pattern recognition techniques have been used to classify backscatter signals from different groups of lesions [17]. On the other hand, the rough dielectric profile of the lesion is reconstructed to identify the mass morphology in [18]. In this paper, the early time backscatter response obtained using an UWB radar system is shown to have the potential for lesion classification. The early time signal is shown to be affected by the target morphology. Different scattering points on the circumference of the target will alter the shape of the early time response. A rough lesion with multiple spicules has more significant scattering points than a lesion with compact shape [17]. The current work aims at providing a lesion classification method using the early time backscatter response. A large scale simulation study is conducted to demonstrate the effectiveness of the proposed classification approach. The remaining part of the paper is organized as follows: Section II discusses the data acquisition process and briefly reviews the lesion morphological model used here and in [16]. Section III describes the lesion discrimination approach to be used. Section IV presents the numerical simulations and results. Finally some concluding remarks are given in Section V. II. NUMERICAL BREAST MODEL AND BACKSCATTER DATA ACQUISITION The backscatter signals are calculated from a two-dimenmode sional (2D) finite difference time domain (FDTD) and ) simulation. The simulation model considers ( , a patient lying in the prone position, with the breast hanging pendent in an immersion tank filled with a coupling liquid of similar dielectric property as the breast medium. A simplified

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TEO et al.: BREAST LESION CLASSIFICATION USING UWB EARLY TIME BREAST LESION RESPONSE

Fig. 1. Set-up of the 2D FDTD lattice with the center sensor acting as a transceiver. The rest of the receiving sensors are placed at equal distance from the lesion at regular intervals.

cylindrical model is hence used. The skin is not modeled by assuming that the skin sensing and subtraction methods are capable of removing any reflections from the coupling medium-skin and skin-breast interfaces [2], [3], [5], [19], [20]. In the case when skin sensing and subtraction is not perfectly done, a detrimental effect is expected for all microwave breast imaging techniques, including the proposed lesion classification performance as the early-time response will include residual from the skin response. A detailed study of its effect warrants a separate investigation, and will be left for future works. The clutter model used in this paper follows the one described in [16], which proposes that tissue inhomogeneity can be identified to occur in regions, and a number of different-sized clutter sources are randomly placed in each of these regions. The schematic of the simulation set-up is shown in Fig. 1. For the FDTD calculation, 0.5 mm square grids are used to discretize the computational space. The Courant factor, , is chosen to be 0.5 for numerical stability [21]. The computational space is a square box, with 401 cells along the direction of propagation and along the direction of electric field source. A 20-layer uniaxial perfectly matched layer (UPML) is used beyond the space to minimize reflections from the boundaries of the simulation box. The update equations used for the FDTD simulation are discussed in [13]. The same set of equations can be used for the whole grid, including the UPML, by changing certain parameters to differentiate between the simulation space and the absorbing boundary. The single pole Debye model is used to simulate the average breast medium, clutter sources generated by the tissue heterogeneity and lesion target (1) where is the angular frequency, is the static peris the infinite permittivity, is the permittivity of mittivity, free space, is the conductivity of the medium and is the relaxation constant. The Debye medium is incorporated into the

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FDTD equations through the use of time domain auxiliary differential equations method [21]. The breast tissue medium has , , the following nominal values: and [5]. The clutter sources belonging to different regions of the breast model will be assigned different values for dielectric variations. The dielectric parameters of the lesion are , , and [5]. chosen as The breast is modeled as a 5 cm radius cylinder with some regions containing clutter sources as in [16]. A directed field point source (transmitter) is placed such that the center of the breast is at a distance of 6.5 cm away from the point source which also functions as a receiver. Two additional point receivers are placed to the left and to the right of the transmitter in 15-degree intervals with reference to the center line passing through the transceiver and the breast center. All 5 sensors are at equidistance from the breast center. It is assumed that prior to lesion classification, a detection step has located the lesion accurately. With the knowledge of the target position, the array of sensors can be shifted around the perimeter of the breast such that the center sensor (the transceiver) can be aligned with the lesion. The excitation signal applied to the transmitter [17] can be expressed as (2) where , , and is a constant. The excitation signal is measured in volts per meter. The parameters of the excitation signal are chosen such that numerical dispersion is kept small [13]. All the signals received are resampled at 50 GHz. This is to take into account the capability of existing time-domain measuring devices (such a sampling capability could be achieved using Agilent’s 86100 Digital Communications Analyzer with the additional 86117A sampling module). A. Lesion and Clutter Generator To simulate the different groups of lesions, the generator introduced in [16] is adopted. An ellipse is used as the baseline for the target, which is described in the polar coordinates by (3) where and are the semi-major and semi-minor axes, is the angle from the positive -axis, and defines the boundary of the ellipse. Fig. 2 shows a simplified example of the baseline ellipse and the resulting lesion, which is generated after altering the baseline as to be made clear in the following discussion. In all the simulations, the value of is assigned as 5 mm and the value of is 4 mm. The baseline ellipse will be approximated by a -sided polygon, where denotes the angle of each vertex of the -sided polygon with respect to the -axes. denotes the uniform distribution and is randomly picked. Each value of is substituted into of which denotes the (3) to obtain the corresponding value of distance of each vertex from the center of the baseline ellipse.

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Fig. 2. A simplified lesion generated from baseline ellipse.

Fig. 4. Transmitted template signal. The dotted line is 1=e of the maximum amplitude.

Fig. 3. (a) Sample of generated smooth lesion. (b) Sample of generated rough lesion.

Once the -sided polygon is obtained, each the following equation:

is altered using (4)

where satisfies . is a parameter governing is the new length the degree of roughness of the lesion and of each vertex from the origin after alteration is made. As approaches 0, will be very similar to and a smooth lesion can take on any value beis produced. As approaches 1, , leading to a much rugged lesion, In the simutween 0 to and are applied to generate lation study, smooth and rough/spiculated lesions, respectively. The value of is always set to be 50. Fig. 3 illustrates the examples for both classes of lesions. Each object is discretized using the proposed FDTD grid and staircase approximation. The clutter sources is generated using (3) and the boundary of the clutter sources is left untouched as in [16]. Seven different regions in the breast medium are allocated for the placement of the clutter sources (also refer to Fig. 1). In each of these regions, 3 to 5 different-sized ellipses with different dielectric variations are randomly scattered. III. PROPOSED LESION CLASSIFICATION TECHNIQUE It was first noted qualitatively in [17] that lesion morphology will affect the shape of the backscatter waveform. In the current work, the early time backscatter signal is used to quantitatively characterize a lesion. The approach is very simple to implement,

and offers the capability to alleviate some of the practical problems arising from inadequate system resolution at the receiver side. This is because the early time backscatter response is of higher signal strength. The concept of correlator receiver is adopted, which relies on a template signal for correlation with the backscatter response. The information of interest here is the amount of deformation of the specular returns caused by different types of lesions. Subsequently, correlation coefficient can be used as a measure to quantify the level of waveform distortion. However, the correlation value is dependent on the amplitude of the backscatter signal. Hence proper normalization of the template and backscatter signal is necessary to isolate the effect solely due to the morphological features of lesion objects. The template waveform can be derived by either passing the transmitted UWB signal through a lesion-free tissue-mimicking phantom material, designed to exhibit dielectric properties similar to the human breast adipose tissue, or through computational electromagnetic simulations with numerical breast phantoms. Fig. 4 shows the template signal taken from a separate simulation in which a homogeneous medium with the same dielectric properties of the average breast tissue is set up. A point transmitter and a point receiver, both placed in the simulated homogeneous medium, are separated 70 cells apart (0.035 m) in a direct line of sight measurement. A portion of the received signal will be used as the template signal as discussed below. Consider the backscatter response from a generated smooth be the clean impinging signal on a receiver lesion. Let at a known distance from a transmitter (from the template signal simulation, it is 0.035 m away from the receiver). The backscatter signal can be modeled as (5) is the received signal, and are the attenwhere and are the delays due to reflections uation factors, and that occur during the entry and exit of the wave from the lesion respectively. These delays are measured with respect to the is the late time known distance of 0.035 m and

TEO et al.: BREAST LESION CLASSIFICATION USING UWB EARLY TIME BREAST LESION RESPONSE

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Fig. 5. Sample received signals for (a) Sensor 3 for a smooth lesion and (b) Sensor 1 for the same lesion. In both plots, 3 portions can be identified. The 1st reflection has undergone an inversion. There is more overlapping of neighboring regions for signal received at sensor 1.

response of the lesion. The signs are due to the possibility of signal inversion caused by the reflection coefficient between the two mediums. It is assumed that this general form of (5) is only applicable to dielectric scatterers. The late time portion is a characteristic of different scatterers, which is dependent on factors such as size, shape, dielectric properties, the surrounding medium and even the angle of illumination. The techniques in [13]–[16] made use of this late time response to differentiate between lesions with different morphological features. The above phenomenon can be seen in Fig. 5(a), where 3 different regions can be identified distinctly with a small amount of overlapping between neighboring regions. Fig. 5(b) shows the received signal at Sensor 1 from the same lesion. It can be seen that there is a higher degree of overlap between the 1st and 2nd reflections. Nevertheless, the 3 different regions are still observable. To further verify (5), both the inverted and non-inverted templates are cross-correlated with the received signal shown in Fig. 5(a). Considering the 2 peak correlation coefficients indicated in Fig. 6, there exists a difference of 19 index between the 2 peaks. It is worth noting that the first 2 peaks of the dotted plot are not used because they are caused by the 2 “crests” of the 1st reflection [see, also, Fig. 5(a)] and not due to the similarity between the 2nd reflection and the non-inverted template signal. Due to the large amplitude of the “crests,” their correlation peaks gave a higher value. Given a sampling rate of 50 GHz, a time delay of 0.38 ns exists between the two reflections. The wave is propagating in the generated smooth lesion, with a relative permittivity value of 50 evaluated at the peak of the spectrum of the template signal. The average speed of wave propagation is . The distance between the two reflections hence is calculated to be 0.0159 m, which is in good agreement with the width of the generated smooth lesion along the semi-minor axis. The overlapping of the 1st and 2nd reflections is maximum at Sensors 1 and 5 as demonstrated earlier. This is due to the reduction in distance between the reflections. Hence, we will not employ the full length of the template signal for correlation operation. Alternatively, the template used is the portion of the signal that lies between the two data points corresponding to of the maximum pulse amplitude as depicted in Fig. 4.

Fig. 6. Correlation plots of the template signal with a received signal shown in Fig. 4. The full line denote the correlation between the inverted template signal with the received signal. The dotted line denote the correlation between a non-inverted template signal with the received signal.

The correlation is computed as follows: 1) initialize the correlation process by shifting the template signal to the zero-propagation-delay position, i.e., the position where the transceiver and the target are collocated; 2) Choose the portion of the template signal that is to be nonzero valued and denote the starting and ending points as and with corresponding delays of and ; 3) Normalize the template signal between and and denote the derived signal as ; with ; 4) Correlate the received signal be the propagation delay that corresponds to the 5) Let location of the lesion with reference to the impinging signal at a known distance. From the correlation plot in Step (2), record the location of the peak with a delay satisfying . It is worth noting that is obtained by calculating the delay for the signal to travel from the center of the lesion to the receiver; from to will 6) The portion of be used as an estimate to the 1st reflection;

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Fig. 7. (a) 1st scenario with homogeneous medium. Lesion with vertical displacements only. (b) 2nd scenario with clutter sources. Lesion with vertical displacements only. (c) 3rd scenario with clutter sources. Lesion with vertical and horizontal displacements. All vertical displacements are within 1 cm to 3.5 cm from center of breast. Horizontal displacements are within 5 mm to either directions. The dielectric variations of each clutter are as follows: Cluster 1: 10% to 5%, Cluster 2: 0% to 5%, Cluster 3: 0% to 5%, Cluster 4: 5% to 10%, Cluster 5: 10% to 5%, Cluster 6: 5% to 0%, Cluster 7: 10% to 5%.

0

7) Normalize the 1st reflection. Denote this signal as ; and and 8) Perform correlation between record the peak correlation value; 9) Repeat Steps 4–8 for all the 5 sensors and record the 5 peak correlation values; 10) Repeat Steps 4–9 for all the lesions. It is expected that as a lesion gets rougher, it is more likely that the early responses of the received signals at these sensors will be different. The degree of difference would increase with the level of ruggedness. This will lead to a lower peak correlation coefficient, giving rise to a wider range of peak correlations, larger variations and lower means among the 5 signals. The lesions are therefore classified according to the following measures: , denote the peak correlation coefLet , ficients obtained from the array of sensors from one lesion. 1) Mean of the peak correlations for each lesion, . 2) Range of the peak correlations for each lesion, . 3) Standard deviation of the peak correlations for each lesion, . 4) Maximum deviation of the peak correlation with respect to the mean value for each lesion, . A. System Noise Effects To study the effects of the imaging system noise, additive white Gaussian noise (AWGN) is used to corrupt the received signals. The noise variance is chosen such that the corresponding signal-to-noise ratio (SNR) satisfies a specified value for the simulated system operation condition. For comparison purpose, a reference SNR for the lesion classification is defined. Signal averaging [22] is then used to improve the SNR of the received signal to the reference SNR. It is worth noting that signal averaging is commonly used in oscilloscope measurements, and the improvement in SNR with respect to the number of averaging follows: (6)

0

0

0

0

0

0

is the number of where is the improvement factor and averaging. To estimate the number of averaging needed, the SNR of the first received signal has to be estimated, which is derived from the frequency response of the received signal. By Parseval’s theorem, the energy of the useful signal can be obtained from the magnitude of its frequency components. The range of frequency components with significant energy is identified and labelled as signal spectrum, with the rest of the frequency spectrum assumed to be noise. Assuming a flat spectrum for an ideal AWGN, the noise power spectral density can be calculated. A portion of the energy in the signal spectrum, equivalent to noise occupying the same bandwidth having the calculated power spectral density, is removed and added to the noise spectrum. The rough SNR of the environment is obtained using these 2 estimated energy values. Using (6) above, the minimum number of averaging, , can be found. IV. NUMERICAL SIMULATIONS AND RESULTS A pictorial illustration of the breast geometries used in the simulation studies is shown in Fig. 7. Three different scenarios are considered with the same set of lesions. In the first case [refer to Fig. 7(a)], a homogeneous medium is simulated. The lesion is randomly displaced vertically at a distance of 1 cm to 3.5 cm away from the breast center to simulate different depths. This serves as a benchmark for evaluating the effectiveness of lesion differentiation using the early time response. In the second case [refer to Fig. 7(b)], clutter sources are introduced as described in Section II. The values of their dielectric properties range to over 10% and the cross-sectional area varies from . The third scenario [refer to Fig. 7(c)] is the exact replica of the second scenario with the exception that the lesion is also randomly positioned horizontally with respect to the center line, in the range of 5 mm in both directions. This is to take into account slight positioning errors. A total of 60 targets are generated using the lesion generator described in Section II. The targets are comprised of 2 groups (smooth versus spiculated) of 30 lesions each defined by the value. Each target is placed in each of the 3 different simulation setups in turn and beamed with the pulse given by (2). The

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Fig. 8. Lesion characterization criterion 1. (a) Mean correlation values in scenario 1. (b) Mean correlation values in scenario 2. (c) Mean correlation values in scenario 3. Note: “ ” denote smooth lesions and “x” denote rough lesions.

+

Fig. 9. Lesion characterization criterion 2. (a) Range of correlation values in scenario 1. (b) Range of correlation values in scenario 2. (c) Range of correlation values in scenario 3. Note: “ ” denote smooth lesions and “x” denote rough lesions.

+

backscatter signals at the 5 different locations are recorded (refer to Fig. 1). System noise is added to give a 10 dB SNR. The reference SNR is chosen to be 20 dB. To increase the measured SNR to the

reference level, the estimation step described in Section III-A is performed. The minimum number of averaging is then calculated accordingly for all the simulation scenarios. The early time responses are processed according to the correlation steps

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Fig. 10. Lesion characterization criterion 3. (a) Standard deviation for correlation values in scenario 1. (b) Standard deviation for correlation values in scenario 2. (c) Standard deviation for correlation values in scenario 3. Note: “ ” denote smooth lesions and “x” denote rough lesions.

+

Fig. 11. Lesion characterization criterion 4. (a) Max. deviation for correlation values in scenario 1. (b) Max. deviation for correlation values in scenario 2. (c) Max. deviation for correlation values in scenario 3. Note: “ ” denote smooth lesions and “x” denote rough lesions.

+

presented in Section III, and the 4 criteria for lesion classification are evaluated. Fig. 8 to Fig. 11 show the results when different lesion classification measures are used. Figs. 8(a), 9(a), 10(a) and 11(a) demonstrate the scenario of a homogeneous breast medium.

Figs. 8(b), 9(b), 10(b) and 11(b) show the scenario when clutter sources are added and the center sensor is aligned with the lesion. Finally, Figs. 8(c), 9(c), 10(c) and 11(c) illustrate the case when there are some slight positioning errors with reference to the antenna array. In Fig. 8, the mean value is

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Fig. 12. Lesion characterization criteria with 3, 5 and 7 sensors. (a) Mean criterion performance. (b) Range criterion performance. (c) Standard deviation criterion performance. (d) Maximum deviation criterion performance. Note: “ ” denote 3 sensors data for smooth lesions and “ ” denote 3 sensors data for rough lesions. “ ” denote 5 sensors data for smooth lesions and “x” denote 5 sensors data for rough lesions. “ ” denote 7 sensors data for smooth lesions and “ ” denote 7 sensors data for rough lesions.

+

4

used as a tissue discrimination indicator. It is observed that overall, the data points associated with the smooth and rough lesions are well separated in each subfigure, indicating the feasibility of applying the mean value for tissue discrimination. Figs. 9 to 11 demonstrate similar trends when other measures are used. Three important observations can be made from these simulations. Firstly, the proposed correlator classifier exhibits the potential in the initial differentiation between lesions with smooth and spiculated boundaries as can be seen from the wide separation between the “smooth lesion” and “rough lesion” data points. Secondly, slight positioning errors do not significantly affect the correlation values by comparing Fig. 8(b) to (c), Fig. 9(b) and (c), Fig. 10(b) and (c) and Fig. 11(b) and (c). , standard deviation Thirdly, overall the measures of range and maximum deviation result in better classifica. This can be tion performance as compared to the mean seen by comparing the different levels of separation between the data plots corresponding to the smooth and rough lesions in all the three scenarios. Finally, two other sets of simulations are conducted with different numbers of sensors. For simplicity, we only consider the first scenario, where the breast medium is homogeneous. The system set-up is the same as that described in Section II. For the first set of simulation, only Sensors 1, 3 and 5 (see Fig. 1) are used. On the other hand, 7 antennas are used for signal reception in the second set of simulation. These antennas span the same angular sector as the original 5 sensors, and are at equidistance from the center of the breast.

r

The received signals collected from both simulations are processed and the 4 measures are applied to each set of simulated data. Fig. 12 demonstrates the performance of using these 4 parameters for lesion classification. From Fig. 12(a) and (c), it is found that increasing the number of sensors that span the same angular sector would not give better performance by applying the mean and standard deviation classifiers. However, for the range and maximum deviation classifiers, a noticeable upward shift of the data points can be seen in the case of rough lesions as shown in Fig. 12(b) and (d), leading to enhanced lesion classification. Hence, it may be more advantageous to use more sensors when the range and maximum deviation classifiers are considered. This may be due to the reason that increasing the number of sensors provides more opportunities to capture the “deformed” backscatter signals in the case of rough lesions. V. CONCLUSION We have shown that the early time backscatter response can be used to discriminate between smooth and rough lesions. A preliminary study on the number of sensors used in covering a given angular sector has also been investigated. It has been shown that increasing the number of sensors leads to improved classification performance when the measures of range and maximum deviation are considered. A 2D scenario is chosen for this initial study as there are few papers dedicated to the issue of lesion classification for microwave radar imaging, and it is still in a very preliminary stage. There have not been any widely-accepted models for either 2D or 3D random lesion

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morphological models. Hence, a simplified 2D breast model is still useful for assessing the efficacy of the proposed approach at this preliminary stage. Nevertheless, it is still of practical significance to consider a 3D phantom in future when 3D lesion models are well established. In addition, we will also look into the following important aspects related to the proposed tissue classification approach: the optimal number of sensors to be deployed, the polarization effects of antennas, the size of angular sector covered and the effect of UWB pulse shapes. REFERENCES [1] A. Jemal, R. Siegel, E. Ward, T. Murray, J. Xu, and M. J. Thun, “Cancer statistics, 2007,” CA Cancer J. Clin. vol. 57, no. 1, pp. 43–66 [Online]. Available: http://caonline.amcancersoc.org/cgi/content/abstract/57/1/43 [2] X. Li and S. Hagness, “A confocal microwave imaging algorithm for breast cancer detection,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 3, pp. 130–132, Mar. 2001. [3] E. Fear, X. Li, S. Hagness, and M. Stuchly, “Confocal microwave imaging for breast cancer detection: Localization of tumors in three dimensions,” IEEE Trans. Biomed. Eng., vol. 49, no. 8, pp. 812–822, Aug. 2002. [4] S. Davis, H. Tandradinata, S. Hagness, and B. Van Veen, “Ultrawideband microwave breast cancer detection: A detection-theoretic approach using the generalized likelihood ratio test,” IEEE Trans. Biomed. Eng., vol. 52, no. 7, pp. 1237–1250, Jul. 2005. [5] E. Bond, X. Li, S. Hagness, and B. Van Veen, “Microwave imaging via space-time beamforming for early detection of breast cancer,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1690–1705, Aug. 2003. [6] Y. Xie, B. Guo, L. Xu, J. Li, and P. Stoica, “Multistatic adaptive microwave imaging for early breast cancer detection,” IEEE Trans. Biomed. Eng., vol. 53, no. 8, pp. 1647–1657, Aug. 2006. [7] J. Sill and E. Fear, “Tissue sensing adaptive radar for breast cancer detection—Experimental investigation of simple tumor models,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3312–3319, Nov. 2005. [8] P. Kosmas and C. Rappaport, “FDTD-based time reversal for microwave breast cancer detection-localization in three dimensions,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1921–1927, Apr. 2006. [9] Y. Chen, E. Gunawan, K. S. Low, S.-C. Wang, C. B. Soh, and L. L. Thi, “Time of arrival data fusion method for two-dimensional ultrawideband breast cancer detection,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2852–2865, Oct. 2007. [10] P. T. Huynh, A. M. Jarolimek, and S. Daye, “The false-negative mammogram,” Radiographics vol. 18, no. 5, pp. 1137–1154, 1998 [Online]. Available: http://radiographics.rsna.org/content/18/5/1137.abstract [11] R. Rangayyan, N. El-Faramawy, J. Desautels, and O. Alim, “Measures of acutance and shape for classification of breast tumors,” IEEE Trans. Med. Imaging, vol. 16, no. 6, pp. 799–810, Dec. 1997. [12] E. Claridge and J. H. Richter, “Characterization of mammographic lesions,” Dig. Mammog., vol. 1069, pp. 241–250, 1994. [13] Y. Huo, R. Bansal, and Q. Zhu, “Modeling of noninvasive microwave characterization of breast tumors,” IEEE Trans. Biomed. Eng., vol. 51, no. 7, pp. 1089–1094, Jul. 2004. [14] D. Li, B. Guo, and G. Wang, “Application of sem technique to characterization of breast tumors,” in Proc. Int. Symp. on Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, Aug. 2007, pp. 887–890. [15] Y. Chen, E. Gunawan, K. S. Low, S. C. Wang, C. B. Soh, and J. Lavanya, “Effect of lesion morphology on microwave signature in ultrawideband breast imaging: A preliminary two-dimensional investigation,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2007, pp. 2168–2171. [16] Y. Chen, E. Gunawan, K. S. Low, S.-C. Wang, C. B. Soh, and T. Putti, “Effect of lesion morphology on microwave signature in 2-d ultra-wideband breast imaging,” IEEE Trans. Biomed. Eng., vol. 55, no. 8, pp. 2011–2021, Aug. 2008. [17] S. Davis, B. Van Veen, S. Hagness, and F. Kelcz, “Breast tumor characterization based on ultrawideband microwave backscatter,” IEEE Trans. Biomed. Eng., vol. 55, no. 1, pp. 237–246, Jan. 2008. [18] M. El-Shenawee and E. Miller, “Spherical harmonics microwave algorithm for shape and location reconstruction of breast cancer tumor,” IEEE Trans. Med. Imaging, vol. 25, no. 10, pp. 1258–1271, Oct. 2006. [19] T. Williams, J. Sill, and E. Fear, “Breast surface estimation for radarbased breast imaging systems,” IEEE Trans. Biomed. Eng., vol. 55, no. 6, pp. 1678–1686, Jun. 2008.

[20] D. Winters, J. Shea, E. Madsen, G. Frank, B. Van Veen, and S. Hagness, “Estimating the breast surface using UWB microwave monostatic backscatter measurements,” IEEE Trans. Biomed. Eng., vol. 55, no. 1, pp. 247–256, Jan. 2008. [21] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Second Edition. Boston, MA: Artech House, 2000. [22] P. Haughton, Acoustics for Audiologists. New York: Academic Press, 2002.

Jianqi Teo received the B.Eng. degree in electrical and electronic from Nanyang Technological University, Singapore, in 2007, where he is currently working toward the Masters degree. His current research interests includes ultrawideband (UWB) for biomedical imaging and detection algorithms.

Yifan Chen (M’06) received the B.Eng. (Hons I) and Ph.D. degrees in electrical and electronic engineering from Nanyang Technological University (NTU), Singapore, in 2002 and 2006, respectively. From 2005 to 2007, he worked as a Research Fellow under the Singapore-University of Washington Alliance (SUWA) in bioengineering at NTU. He is presently with the School of Engineering, University of Greenwich, U.K., as a Senior Lecturer. He is also with the School of Computer, Electronics and Information, Guangxi University, China, as an Adjunct Associate Professor. His current research interests involve wireless and pervasive communications for healthcare, microwave biomedical imaging, wireless channel modeling, wireless communication theory, and network localization. Dr. Chen was awarded the Promising Research Fellowship 2010 and the Early Career Research Excellence Award 2009 by the University of Greenwich in recognition of his exceptional research contributions and potential.

Cheong Boon Soh (M’84–SM’03) received the B.Eng. degree in electrical and computer systems engineering (Hons I) and the Ph.D. degree from Monash University, Victoria, Australia, in 1983 and 1987, respectively. He is an Associate Professor in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He has published more then 110 international journal papers. His current research interests are ultrawideband (UWB) for medical applications, auscultation and stroke assessment, E-medicine, robust control, system theory, nonlinear systems, coding theory, networking, mobile communication systems and intelligent systems.

Erry Gunawan (M’90) received the B.Sc. degree in electrical and electronic engineering from the University of Leeds, U.K., in 1983, and the MBA and Ph.D. degrees from Bradford University, U.K., in 1984 and 1988, respectively. He joined the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, in 1989, and currently is an Associate Professor in the same school. He has published more then 50 papers in international journals and over 50 international conference papers on error correction coding, modeling of cellular communications systems, power control for CDMA cellular systems, MAC protocols, multicarrier modulations, multiuser detections, space-time coding, powerline communications and radio-location systems.

TEO et al.: BREAST LESION CLASSIFICATION USING UWB EARLY TIME BREAST LESION RESPONSE

Kay Soon Low (M’88–SM’00) received the B.Eng. degree in electrical engineering from the National University of Singapore, in 1985 and the Ph.D. degree in electrical engineering from the University of New South Wales, Australia, in 1995. He is an Associate Professor in the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He is also the Director of the Satellite Engineering Centre. His present funded projects are in the field of wearable wireless sensor network for rehabilitation application, on-chip pulse based neural network for imaging processing and satellite system.

Thomas Choudary Putti received the M.B.B.S. degree and the Specialist Degree in Pathology from Osmania University, Hyderabad, India, in 1985 and 1989, respectively. Subsequently, he did his anatomic pathology residency at Long Island Jewish Medical Center, New York, and received American Board Certification in 1997. He did a year of fellowship in immunopathology from Bronx-Lebanon Hospital, New York, from 1997 to 1998. His research interests include fine needle aspiration biopsy of breast, immunohistochemistry and breast pathology. Dr. Putti has been a Fellow of College of American Pathologists since 1997 and a member of the Singapore Society of Pathologists since 1998.

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Shih-Chang Wang trained in Medicine at the University of Sydney, Sydney, Australia, from 1975 to 1980, with an intercalated Bachelor of Medical Science research year in 1978. In 1997, he took a position at the National University of Singapore as a visiting Senior Fellow and eventually as an Associate Professor in 1998 and was the Head of Department in 2000. In 2008, he commenced as the Foundation Chair of Radiology at the University of Sydney, as the Parker-Hughes Professor of Radiology at the Western Clinical School at Westmead Hospital. His research interest include interventional radiology, musculoskeletal imaging and intervention, musculoskeletal tumors and dysplasias and breast screening and breast MRI.

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Artificial Magnetic Materials Using Fractal Hilbert Curves Leila Yousefi, Member, IEEE, and Omar M. Ramahi, Fellow, IEEE

Abstract—Novel configurations based on Fractal Hilbert curves are proposed for realizing artificial magnetic materials. It is shown that the proposed configuration gives significant rise to miniaturization of artificial unit cells which in turn results in higher homogeneity in the material, and reduction in the profile of the artificial substrate. Analytical formulas are proposed for design and optimization of the presented structures, and are verified through full wave numerical characterization. The electromagnetic properties of the proposed structures are studied in detail and compared to square spiral from the point of view of size reduction, maximum value of the resultant permeability, magnetic loss, and frequency dispersion. To validate the analytical model and the numerical simulation results, an artificial substrate containing second-order Fractal Hilbert curve is fabricated and experimentally characterized using a microstrip-based characterization method. Index Terms—Artificial magnetic materials, Fractal Hilbert curves, metamaterials, microstrip-line based characterization, permeability.

I. INTRODUCTION

W

HEN exposed to an applied electromagnetic field, magneto-dielectric materials are polarized both electrically and magnetically, so they exhibit enhanced relative permeability and permittivity. Recently it has been shown that utilizing magneto-dielectric materials instead of dielectrics with high permittivity offers many advantages in an important class of applications [1]–[8]. In [1], it was shown that using materials with high permeability for antenna miniaturization can increase the bandwidth while materials with high permittivity would shrink the bandwidth. Furthermore, when materials with only high permittivity are used for antenna miniaturization, the high impedance mismatch between the substrate and the air decreases the efficiency of the system, while in the case of using magneto-dielectric materials, the impedance mismatch is smaller leading to the higher efficiency for the miniaturized antenna. It was shown in [2] that magneto-dielectric resonator antennas have wider impedance bandwidth than dielectric resonator antennas. In [3], a meander

Manuscript received August 27, 2009; revised November 21, 2009; accepted February 01, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by Research in Motion and the National Science and Engineering Research Council of Canada under the NSERC/RIM Industrial Research Chair Program and the NSERC Discovery Grant Program. The authors are with the Electrical and Computer Engineering Department, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050438

line type antenna using a magneto-dielectric material as a substrate was proposed for using in RFID systems. In [4] magneto-dielectric materials were used as a superstrate to increase the gain of a patch antenna leading to lower profile in comparison to antennas with dielectric superstrates. In [7] magneto-dielectric materials were used as the substrate of a mushroomtype electromagnetic band gap structure (EBG). The results in [7] showed that using magneto-dielectrics could increase the in-phase reflection bandwidth of EBGs when they are used as artificial magnetic ground planes. For low-loss applications in the microwave region, natural material choices are limited to nonmagnetic dielectrics. When requiring relatively high permeability, the choices are limited to ferrite composites which provide high levels of magnetic loss [9]–[12]. Therefore, artificial magnetic materials are designed to provide desirable permeability and permittivity with manageable loss at these frequencies [13]–[19]. In spite of double negative metamaterials which are designed to provide negative permeability and permittivity, the goal of designing artificial magnetic materials is to provide an enhanced positive value for permeability. The idea of using the split-ring as an artificial magnetic particle was introduced first in [20]. The works on realizing such a media started in the late 1990’s [13]–[15]. Since then, engineers have proposed numerous types of inclusions [16]–[19]. The single and coupled split ring resonators (SRR), modified ring resonators, paired ping resonators, metasolenoid [16], split square spiral configuration [18], are some of the most popular configurations used in previous works. Each proposed structure provides its own advantages and disadvantages in terms of resultant permeability and dissipation. For example, in [16] it was shown that the metasolenoid configuration provides higher permeability in comparison to SRR configurations, or using split square spiral configuration results in artificial magnetic materials with smaller unit cells when compared to SRR and metasolenoid [17], [18]. One of the most important applications of artificial magnetic materials is implementing miniaturized planar structures, specially miniaturized microstrip antennas [18], [19], [21]–[24]. Although it was shown that by using these materials, significant miniaturization factors could be achieved in the planar sizes of the antennas, the height of the substrate is limited by the size of the unit cell of the artificial structures. The size of the developed artificial unit cells are typically much smaller than the wave), yet they yield a large antenna profile length (smaller than (for example, for a microstrip antenna operating at 200 MHz, the smallest profile achieved is 2 cm [18]). Therefore, miniaturizing the unit cell of artificial materials not only provides better homogeneity, but also decreases the antenna profile.

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YOUSEFI AND RAMAHI: ARTIFICIAL MAGNETIC MATERIALS USING FRACTAL HILBERT CURVES

With the aim of further miniaturization, other sub-wavelength particles have been recently proposed such as spiral resonators [17], [18], capacitively loaded embedded-circuit particles [25]–[27], as well as lumped-element based metamaterials [28]. In [17], it was shown that using spiral resonators decreases the size of the inclusion by a factor of 2 when compared to SRRs (split ring resonators). In [25]–[27], using are develparallel-plate capacitors, inclusions as small as oped. (Note that in this paper, by we mean the wavelength in the host dielectric at the resonance frequency. Since by using a high-k dielectric, the size of the inclusion decreases, and therefore, it is not useful to compare different inclusions based on wavelength in the air.) However; the structures in [25]–[27] are three dimensional and cannot be realized using printed circuit board technology. Using lumped elements within the inclusions, unit cells as small as were developed in [28]. The solution proposed in [28] is promising, however, lumped elements need to be soldered into place for each unit cell which makes the fabrication process difficult and time-consuming. were In [18], using square spirals unit cells as small as developed and used for antenna miniaturization. The structure introduced in [17], can be realized by simple printed technology. In this paper, we introduce fractal curves as inclusions for artificial magnetic material to further increase the miniaturization potential. Previous works on using fractal geometries in developing artificial structures included frequency selective surfaces [29], high-impedance surfaces [30]–[35], left-handed metamaterials [36], and complementary split-ring resonators [37]. An extended class of space-filling wire structures based on grid-graph Hamiltonian paths and cycles has also been investigated in [38]. In this work, the use of Fractal curves to miniaturize artificial magnetic materials is investigated. This idea was proposed for the first time as a conference paper in our pervious work [39]. Combining the square spiral loop configuration with fractal Hilbert curves, a new configuration is proposed to realize further miniaturization for artificial particles. It is shown that by using fourth-order fractal Hilbert curves, inclusions as small as can be realized. Using higher order Hilbert curves results in even smaller unit cells. Analytical formulas for design and analysis of the proposed structures are presented. Full-wave numerical characterization and experimental testing is carried out to validate the analytical results. The electromagnetic behavior of the proposed structures are investigated from the point of view of size reduction, maximum value of the resultant permeability, magnetic loss, and frequency dispersion. The organization of this paper is as follows: In Section II, the new inclusions based on fractal Hilbert curves are introduced and analyzed. Analytical formulas for design and analysis of the proposed structures are presented, and full-wave numerical characterization is performed to verify the analytical results. Furthermore, using the analytical model and numerical simulation results, the proposed inclusions are compared to spiral inclusions. In Section III, second-order Fractal Hilbert inclusions were fabricated and characterized. For experimental characterization, we have used a new method reported in our prior work [40]. Summary and conclusion are provided in Section IV.

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Fig. 1. (a) SRR. (b) Square spiral. (c) Second-order Fractal Hilbert inclusion. (d) Third-order Fractal Hilbert inclusion. (e) Fourth-order Fractal Hilbert inclusion. Note that as the order of Hilbert curve increases, the size of inclusion decreases.

II. ARTIFICIAL MAGNETIC MATERIALS BASED ON FRACTAL HILBERT CURVES Fig. 1 shows the proposed structures for metamaterial inclusions based on Hilbert curves, along with the split ring resonators (SRR) and spiral structures. Using circuit models, it was shown in [17] that the effective capacitance of the spiral configuration is four times of that of the SRR. Therefore, spirals can reduce the resonance frequency of the inclusion by a factor of 2. This claim was verified numerically and experimentally in [17]. The structures proposed in this work combine the idea of using spiral configuration (proposed in [17]) and Fractal Hilbert curve to provide inclusions with smaller size. The dimensions of the inclusions as a fraction of the wavelength in the dielectric at the resonance frequency are shown in Fig. 1. Clearly observed is that when using a fourth-order Fractal Hilbert curve makes it possible to realize inclusion as small as . This size is 63% of the size of spiral inclusion and 32% of the size of SRR. A. Analytical Model Fig. 2 shows a unit cell of a third-order Hilbert inclusion. In what follows, a general formulation, which can be used for any order of Hilbert inclusions, is derived for the effective permeability. The unit cell in Fig. 2 has dimensions of , , and in the x, y, and z directions, respectively. An applied external in the y direction induces an electromotive magnetic field

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fields, the effective permeability will be equal to that of the host media which is unity for nonmagnetic substrates. Therefore, the artificial substrate will be anisotropic with permeability tensor of (7)

Fig. 2. A unit cell of engineered magnetic substrate composed of inclusions with 3rd order Hilbert Curve.

force in the ring. Applying Faraday’s law, one can obtain as the induced

(1) is the inwhere is the frequency of the external field, duced magnetic field, is the induced current, and is the area enclosed by the inclusion. Since the dimensions of the unit cell in artificial materials are small in comparison to the wavelength, we assume a uniform magnetic field over the area of the unit cell. The generated is related to the impedance of the rings and the induced current as (2) where is the impedance of the metallic inclusion. Equating (1) and (2) yields

From (3), it is observed that the effective inductance, inclusions can be defined as

To achieve isotropic artificial substrates with the same effective permeability in all the three directions of x, y, and z, two inclusions with the surfaces perpendicular to the x, and z directions should be added to the unit cell shown in Fig. 2. If these two inclusions have the same structure as the inclusion perpendicular to the y direction, the same effective permeability formulated in (6) will be achieved in the x, and z directions. The formula given in (6) is general and can be used for Hilbert inclusions of any order. The only difference between various , orders would be in the value of the inclusion impedance, . and the effective inductance, given in (4) is a function of the The effective inductance, area , enclosed by the inclusion; and since this area varies by the order, , of Fractal Hilbert curve, the effective inductance would be dependent on . is given as (8) where is the dimension of the inclusion in the x direction (see Fig. 2). Equation (8) can be proved using mathematical induction principle [41]. , consists of two parts: , The inclusion impedance, which models the ohmic loss of the metallic inclusions due to the finite conductivity of the strips, and the other part models the mutual impedance between external and internal loops, . , can be calculated using the Ohm’s law

(3)

(9)

of the

where and are the conductivity and skin depth, respectively, is the width of metallic strips and is the total length of is given as the metallic strips.

(4)

(10)

The magnetic polarization, defined as the average of the induced magnetic dipole moments can be expressed as (5) Using (3) and (5), the relative permeability

is obtained as (6)

It should be noted that the substrate whose unit cell is shown in Fig. 2 can provide magnetic moment vectors only in the direction perpendicular to the inclusion surface (i.e., the effective permeability formulated in (6) represents the yy component of the permeability tensor). For x-directed and z-directed magnetic

Equation (10) can be proved using mathematical induction principle [41]. In the proposed inclusions, the internal loop follows the shape of the external loop, therefore the mutual can be impedance between external and internal loops modeled as the per-unit-length mutual impedance of the strips, , times the average length of the strips, (11) where (12)

YOUSEFI AND RAMAHI: ARTIFICIAL MAGNETIC MATERIALS USING FRACTAL HILBERT CURVES

and is the gap between the metallic strips. The per-unit-length is calculated using conformal mapping techimpedance, nique for the coplanar strip lines [42]

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TABLE I COMPARISON OF THE FRACTAL SPIRAL INCLUSIONS WITH NON-FRACTAL SPIRAL INCLUSION

(13)

are the relative permittivity and the loss tanwhere and gent of the host substrate, respectively. The impedance given in (13) contains a parallel combination of a capacitance, and a conductance, to model the capacitance between the metallic strips and the loss due to the conductivity of the host substrate. It should be noted that this model considers the capacitance between two adjacent strips while it neglects the capacitance between non-adjacent strips. Therefore, the accuracy of this model increases when the space between non-adjacent strips is much larger than the space between adjacent strips. However, for the compact structures or for high order fractal curves, where the space between non-adjacent strips is comparable to the space between adjacent strips, this model gives only an approximation of the relative permeability that can be used to initiate a design. To extract exact relative permeability, one needs to use the full wave simulation method as explained below in Section II-B. The proposed analytical formula developed in (6) fits the Lorentz model [43] (14) if the following substitutions are made:

(15) Here, the term in (13) has been ignored since it is, for a typical host substrate, several orders of magnitude smaller than one. The approach introduced in this section for deriving relative permeability of Hilbert inclusions can be used for square spirals too, and results in the same formula derived in (6). The only difference would be in the value of the effective lumped elements, , , and . Equation (16) illustrates the parameters that should be replaced so that (6) can be used for spiral inclusions.

(16)

Using the above formulas, Table I compares the effective inductance, capacitance, resistance, and the resonance frequency of Hilbert inclusions with those parameters of the square spiral. The electrical parameters of the host substrate and the geometrical parameters of the unit cell and metallic strips are assumed to be the same for all the Hilbert inclusions and spiral. From Table I, using Fractal Hilbert inclusions instead of square spirals reduces the effective inductance while increasing the effective capacitance in such a way that the resonance frequency would be smaller for Fractal Hilbert curves with . Therefore, using structures with Hilbert curves of order 3 or higher results in miniaturization of inclusions. We also observe that the miniaturization factor increases with the order of Hilbert curves. Table I also gives higher value of effective resistance for Hilbert inclusions when compared to spirals. Increasing the effective resistance in the circuit model results in lower factor, which in turn results in lower rate of change in the resultant permeability with respect to frequency. Therefore, lower frequency dispersion can be achieved for the artificial medium. On the other hand, any increase in the effective resistance results in an increase in the magnetic loss of the artificial medium. Furthermore, since the effective inductance is reduced, it is expected that the maximum value of the relative permeability would be smaller for Hilbert inclusions when compared to spiral. As a conclusion, the analytical models developed here predict that using Fractal Hilbert curves with order 3 or higher results in miniaturization of the inclusions, and, furthermore, in a reduction of the frequency dispersion of the artificial medium. The drawbacks would be lower resultant permeability and higher magnetic loss. This conclusion will be verified in part C using full wave numerical simulation. B. Numerical Full Wave Analysis In order to verify the accuracy of the analytical model developed above, a full-wave numerical analysis setup is developed in this section to characterize the proposed engineered materials. The simulation setup is shown in Fig. 3. In the simulation setup, periodic boundary conditions are used in the z and y directions around a unit cell to model an infinite slab, and perfect matched layers absorbing boundaries are used in the x direction to prevent reflections from the computational domain walls. For characterization, the well known plane wave analysis method is used [44]–[46]. In this method, the effective parameters, and , are extracted from the

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Fig. 4. Real part of resultant permeability for engineered materials with inclusions shown in Fig. 1. Analytical results (solid line) are compared with numerical results (dash line).

Fig. 3. Simulation setup used for numerically characterization of metamaterials.

reflection and transmission coefficients. Equations (17)–(20) give the relationship between the effective parameters and the transmission and reflection coefficients [44]–[46]. (17) (18) (19) (20) where and are the air characteristic impedance and wave , , , and are the effective number, respectively; characteristic impedance, refractive index, permittivity, and permeability of the engineered material, respectively; and are the reflection and transmission coefficients, and d is the thickness of the engineered material. The sign in (17), and (18) is determined by the requirements that and for passive media. The numerical process was carried out based on the finite element method using Ansoft HFSS10 full-wave simulation tool. C. The Resultant Permeability Using the aforementioned analytical and numerical methods, the relative permeability of the proposed structures in Fig. 1 was derived and the results are shown in Figs. 4, 5. In these figures, the resultant permeability of Hilbert curve structures are compared to that of the square split spiral proposed in [17], [18]. The parameters and dimensions of all the structures are the same and are given as: , host dielectric of

Fig. 5. Imaginary part of resultant permeability for engineered materials with inclusions shown in Fig. 1. Analytical results (solid line) are compared with numerical results (dash line).

(

,

), , , , and the strips are assumed copper to

model the loss. Figs. 4 and 5 show the real and imaginary parts of the resultant permeability, computed analytically and numerically. As shown in these figures, for inclusions with 3rd or 4th order Hilbert curves, which resonate at lower frequencies, a strong agreement is observed between the analytical and numerical results. However; for the inclusions with spiral or 2nd order Hilbert curves, which resonate at higher frequencies, a frequency shift reaching a maximum value of 8.8% for the case of 2nd order Hilbert is observed between the numerical and analytical results. Since the analytical model has been derived based on the assumption of field uniformity throughout the unit cell, the smaller the inclusion in comparison to the wavelength, the more accurate the analytical model is. Therefore, the numerical simulations for inclusions based on the 3rd or 4th order Hilbert curves are expected to give closer agreement with the analytical model in comparison to inclusions based on lower order Hilbert curves.

YOUSEFI AND RAMAHI: ARTIFICIAL MAGNETIC MATERIALS USING FRACTAL HILBERT CURVES

Fig. 6. Imaginary part of the resultant permeability at frequencies below the resonance. The frequency is normalized to the resonance frequency.

As shown in Fig. 4, and Fig. 5, for , fractal structures exhibit lower resonance frequency compared to the split spiral with the same size. Therefore, as predicted by the circuit model, for a given resonance frequency, using Hilbert inclusions with order of 3 or higher results in a smaller size for the unit cell compared to the split spiral. Furthermore, as shown in Fig. 4, when the order of the fractal Hilbert curve increases, the response of the permeability becomes smoother. Therefore, as predicted by the circuit model, using fractal structures decreases the rate of the permeability variation with respect to frequency leading to lower dispersion in the artificial substrate. From Fig. 4, the maximum value achievable for the resultant permeability is smaller than that of the spiral, and this maximum value decreases as the order of Hilbert inclusions increases. This fact was also predicted by the circuit model as explained in Section II-A. In terms of loss comparison, as shown in Fig. 5, at the resonance frequency Hilbert inclusions yield lower value for the imaginary part, so they provide lower loss at the resonant frequency. Notice that it is difficult to compare the imaginary parts at the other frequencies using the scale shown in Fig. 5. For this, we have presented the imaginary part of the permeability at the frequencies below the resonance in Fig. 6. In this figure, the numerical results are used, and for the sake of comparison, the frequency is normalized to the resonance frequency for all inclusions. As shown in Fig. 6, for frequencies below resonance, the imaginary part of the permeability is higher for Hilbert inclusions than that of the spiral inclusion, and as the order of Hilbert increases the imaginary part increases. Therefore, as predicted by the circuit model, at the frequencies below the resonance Hilbert inclusions provide higher loss when compared to spirals. III. EXPERIMENTAL RESULTS To verify our analytical and numerical results, an artificial magnetic material composed of 2nd order Hilbert inclusions was fabricated and characterized. A unit cell of this structure, and its dimension are shown in Fig. 7. To measure the constitutive parameters of the artificial substrate, we use a novel microstrip-line based characterization

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= w = 0:180 mm, = 6:06 mm, 1y =

Fig. 7. One unit cell of the fabricated artificial substrate, s l l l l l l : ,l : , x z .

= = = = = = 3 03 mm 3 028 mm 1 = 1 = 11 mm

Fig. 8. A single strip containing 6 unit cells of inclusions fabricated using printed circuit board technology.

method reported in [40]. In this method, the permeability of the artificial substrate is extracted by measuring the input impedance of a shorted microstrip line which is implemented over the artificial substrate [40]. The advantage of this method over previously developed techniques is that the characterization can be performed by using a simple inexpensive fixture. Furthermore, no sample preparation is needed for characterization, and the same substrate designed for any microstrip device can be used for characterization as well. Using printed circuit technology, a strip of 6 unit cells of Fractal Hilbert2 inclusions was fabricated on an FR4 substrate and (See Fig. 8). Forty of these with strips were then stacked in the y direction to provide a three-dimensional substrate. Due to the thickness of the metal includevelops between the strips sions, an average air gap of 50 in the stacking process. The air space, while unavoidable in the fabrication process, is nevertheless measurable so it can be easily included in the design. A. Permittivity Measurement Using the engineered substrate and two conducting plates, a parallel-plate metamaterial capacitor was fabricated, and by measuring its capacitance, the permittivity of the artificial substrate was calculated [18]. According to the classical image theory, using only one period of the artificial unit cells, in the area between the two metallic parallel plates, can mimic the behavior of an infinite array of unit cells, which is the default assumption in the analysis and design of artificial structures. This property, therefore, makes the method presented in [18] highly robust and well-suited for metamaterial characterization. It is interesting

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Fig. 10. The fabricated fixture used for permeability measurement.

Fig. 9. Simulation results for extracted permittivity (real part).

to note, however, that the authors of [18] observed a rather large difference between the analytically estimated and measured permittivity values (a difference of more than 30%). The large discrepancy is in fact due to the approximations used in the derivation of the analytical formula. For the same artificial structure measured in [18], we have performed a full wave numerical simulation using the simulation setup discussed in Section II-B and obtained a difference between simulation and measurements of less than 7%. Using the aforementioned method, the x-directed permittivity for this artificial magnetic substrate was measured, at the low [40]. By using the simfrequency of 10 MHz, as ulation setup explained in Section II and Ansoft HFSS10, the permittivity of the artificial magnetic substrate was numerically , thus showing good agreement with calculated as air gap was included. measurement. In the simulation, the 50 However; the simulation results show that the resultant permittivity changes with frequency in such a way that at the resonance frequency of 630 MHz, the permittivity decreases to 7.6 (see Fig. 9). The method used in [40] is not suitable to measure the permittivity as a function of frequency; however, since the measurement results at low frequency are close to the simulation result, the subsequent calculation of the permeability is expected to yield a comparable level of accuracy.

Fig. 11. The measured and numerically simulated real part of the permeability for the artificial magnetic material shown in Fig. 10.

B. Permeability Measurement The same substrate that is used for permittivity measurement is used as a substrate of the shorted microstrip line to extract the permeability. The fabricated fixture used for permeability measurement is shown in Fig. 10. The fabricated substrate has dimensions of 12, 8.2, and 1.1 cm in the y, z, and x directions, and respectively. For the quasi-TEM dominant mode, the fields in the substrate will be in the y and x directions, respectively. Therefore this configuration can be used for retrieval of . Using a vector network analyzer, the complex input impedance of the shorted microstrip line shown in Fig. 10 is measured over the frequency range of 500–680 MHz. Then using the data shown in Fig. 9 for and measured impedance, is extracted by the method reported in [40]. The real and

Fig. 12. The measured and numerically simulated imaginary part of the permeability for the artificial magnetic material shown in Fig. 10.

imaginary parts of the measured y-directed permeability are shown in Fig. 11 and Fig. 12. The measurement data shown in these figures are from [40]. In these figures, the measurement results are compared with the numerical simulation results. air gap was also included in the simulation. UnforThe 50 tunately this air gap cannot be modeled within the analytical formulas presented in Section II. The analytical model, which dose not consider the air gap, results in a resonant frequency

YOUSEFI AND RAMAHI: ARTIFICIAL MAGNETIC MATERIALS USING FRACTAL HILBERT CURVES

of 520 MHz, corresponding to a 16% shift when compared to the simulation results that included the air gap. A similar shift in the resonance frequency due to the air gap has also been reported in pervious works [18]. The shaded area in Fig. 11 and Fig. 12 determines the frequencies over which the real part of the permeability is negative. As explained in [40], over this frequency range corresponding to frequencies higher than 638 MHz, the measurement results are not valid due to the restriction of the formulas used for microstrip effective permeability in [40]. As explained in [40], the formulas exist in literature for microstrip effective permeability are derived using conformal mapping technique where the permeability was assumed positive. Therefore, these equations cannot be used at over the frequency range where the permeability is negative. However, since artificial magnetic materials are designed to operate at frequencies over which the permeability is positive, characterization of the permeability behavior at those frequencies would be sufficient for application purposes. Over the frequency range where the real part of the permeability is positive, good agreement is observed between the simulation and measurement results. As briefly discussed above and more extensively in [40], in the numerical analysis, periodic boundary conditions are used to mimic an infinite number of unit cells. However, in practice we can only realize a finite number of unit cells. For example in the setup used in this work (see Fig. 10), the fabricated substrate contains 6 unit cells of inclusions in the z direction and only one unit cell in the x direction. Therefore; we do not expect a very strong agreement between simulation and measurements. By analyzing the induced magnetic field distribution within the inclusion, we observed that at the resonance frequency, a strong magnetic field is present mainly at the center of the inclusion. However, at frequencies below resonance, we observe a relatively weak field at the sides of the inclusion (the sides normal to the x-direction in Fig. 9). Accordingly, at the resonance frequency the adjacent unit cells in the x direction have a minor effect on the resultant permeability, while at the frequencies below resonance, this effect is not negligible. This observation could potentially explain the fact that we have good agreement between simulation and measurement at the resonance frequency, but the agreement becomes weaker for frequencies below resonance (see Fig. 11). It is most likely that increasing the number of unit cells in the x-direction provides higher homogeneity in the fabricated substrate which will result in a better agreement between numerical and measurement results. On the other hand, in a wide class of applications such as antenna miniaturization, only one unit cell is used in the x direction [21]–[24], and therefore, the measurement results would be of strong relevance. IV. CONCLUSIONS The use of Fractal curves to miniaturize artificial magnetic materials was investigated. Novel configurations were proposed to realize artificial magnetic materials with smaller unit cells by combining the square split loop configuration with Fractal Hilbert curves. The new designs are desirable when designing low profile miniaturized antennas in which the engineered magnetic materials are used as the substrate. Analytical models were

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introduced to design and analyze the new structures. The analytical model was validated through both full wave simulation and experimental characterization. It was shown that using forth order of Fractal Hilbert curve, it is possible to realize inclusions as small as 0.014 of the wavelength in the dielectric. This size is 63% of the size of spiral inclusion and 32% of the size of SRR. Using higher order Hilbert curves results in even further miniaturization of the unit cell. In terms of the electromagnetic properties, the new structures provide lower frequency dispersion and lower magnetic loss at the resonance frequency in comparison to the simple square spiral inclusions. This advantage comes at the expense of lower permeability and higher magnetic loss at frequencies below resonance. REFERENCES [1] R. C. Hansen and M. Burke, “Antennas with magneto-dielectrics,” Microw. Opt. Tech. Lett., vol. 26, no. 2, pp. 75–78, Jun. 2000. [2] A. Buerkle and K. Sarabandi, “A circularly polarized magneto-dielectric resonator antenna with wideband, multi-resonant response,” in Proc. IEEE AP-S Int. Symp. Antennas Propag., Jul. 2005, vol. 1B, pp. 487–490. [3] K. Min, T. V. Hong, and D. Kim, “A design of a meander line antenna using magneto-dielectric material for RFID system,” in Proc. Asia-Pacific Conf. on Microwave, Dec. 2005, vol. 4, pp. 1–4. [4] A. Foroozesh and L. Shafai, “Size reduction of a microstrip antenna with dielectric superstrate using meta-materials: Artificial magnetic conductors versus magneto-dielectrics,” in Proc. IEEE AP-S Int. Symp. Antennas Propag., July 2006, vol. 1B, pp. 11–14. [5] L. Yousefi and O. Ramahi, “Engineered magnetic materials with improved dispersion using multi-resonator structures,” in Proc. Canadian Conf. on Electrical and Computer Engineering, Apr. 2007, pp. 966–969. [6] L. Yousefi and O. Ramahi, “Miniaturized wideband antenna using engineered magnetic materials with multi-resonator inclusions,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2007, pp. 1885–1888. [7] L. Yousefi, B. Mohajer-Iravani, and O. M. Ramahi, “Enhanced bandwidth artificial magnetic ground plane for low profile antennas,” IEEE Antenna Wireless Propag. Lett., vol. 6, pp. 289–292, Jun. 2007. [8] H. Mosallaei and K. Sarabandi, “Magneto-dielectrics in electromagnetics: Concept and applications,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1558–1567, Jun. 2004. [9] K. N. Rozanova, Z. W. Li, L. F. Chen, and M. Y. Koledintseva, “Microwave permeability of co2Z composites,” J. Appl. Phys., vol. 97, pp. 013 905-1–013 905-7, Dec. 2004. [10] A. L. Adenot, O. Acher, T. Taffary, and L. Longuet, “Sum rules on the dynamic permeability of hexagonal ferrites,” J. Appl. Phys., vol. 91, pp. 7601–7603, May 2002. [11] O. Acher and A. L. Adenot, “Bounds on the dynamic properties of magnetic materials,” Phys. Rev. B, vol. 62, no. 17, pp. 11 324–11 327, Nov. 2000. [12] W. D. Callister, Materials Science and Engineering, an Introduction. New York: Wiley, 2000. [13] M. V. Kostin and V. V. Shevchenko, “Artificial magnetics based on double circular elements,” in Proc. Bian-Isotropics’94, Périgueux, May 1994, pp. 49–56. [14] 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, no. 11, pp. 2075–2084, Nov. 1999. [15] R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B, vol. 65, no. 14, pp. 44 401–44 405, Apr. 2002. [16] S. Maslovski, P. Ikonen, I. Kolmakov, and S. Tretyakov, “Artificial magnetic materials based on the new magnetic particle: Metasolenoid,” Progr. Electromagn. Res. (PIER), vol. 54, no. 9, pp. 61–81, Sept. 2005. [17] J. D. Baena, R. Marques, F. Medina, and J. Martel, “Artificial magnetic metamaterial design by using spiral resonators,” Phys. Rev. B, vol. 69, pp. 144 021–144 025, Jan. 2004. [18] K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 135–146, Jan. 2006.

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[19] P. M. T. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1654–1662, Jun. 2006. [20] S. A. Schelkunoff and H. T. Friis, Antennas: Theory and Practise. New York: Wiley, 1952. [21] M. K. Karkkainen, S. A. Tretyakov, and P. Ikonen, “Numerical study of PIFA with dispersive material fillings,” Microw. Opt. Tech. Lett., vol. 45, no. 1, pp. 5–8, Feb. 2005. [22] M. E. Ermutlu, C. R. Simovski, M. K. Karkkainen, P. Ikonen, S. A. Tretyakov, and A. A. Sochava, “Miniaturization of patch antennas with new artificial magnetic layers,” in Proc. IEEE Int. Workshop on Antenna Technology, Mar. 2005, vol. 1B, pp. 87–90. [23] P. Ikonen, S. Maslovski, and S. Tretyakov, “Pifa loaded with artificial magnetic material: Practical example for two utilization strategies,” Microw. Opt. Tech. Lett., vol. 46, no. 3, pp. 205–210, Jun. 2005. [24] M. K. Karkkainen and P. Ikonen, “Patch antenna with stacked split-ring resonators as artificial magnetodielectric substrate,” Microw. Opt. Tech. Lett., vol. 46, no. 6, pp. 554–556, Jul. 2005. [25] H. Mosallaei and K. Sarabandi, “Design and modeling of patch antenna printed on magneto-dielectric embedded-circuit metasubstrate,” IEEE Trans. Antennas Propag., vol. 55, pp. 45–52, Jan. 2007. [26] P. M. T. Ikonen and S. A. Tretyakov, “Comments on “Design and modeling of patch antenna printed on magneto-dielectric embeddedcircuit metasubstrate”,” IEEE Trans. Antennas Propag., vol. 55, pp. 2935–2936, Oct. 2007. [27] H. Mosallaei and K. Sarabandi, “Reply to comments on ’design and modeling of patch antenna printed on magneto-dielectric embedded-circuit metasubstrate,” IEEE Trans. Antennas Propag., vol. 55, pp. 2936–2937, Oct. 2007. [28] A. Erentok, R. W. Ziolkowski, J. A. Nielsen, R. B. Greegor, C. G. Parazzoli, M. H. Tanielian, S. A. Cummer, B.-I. Popa, T. Hand, D. C. Vier, and S. Schultz, “Low frequency lumped element-based negative index metamaterial,” Appl. Phys. Lett., vol. 91, pp. 1 841 041–1 841 043, Nov. 2007. [29] E. A. Parker and A. N. A. E. Sheikh, “Convoluted array elements and reduced size unit cells for frequency-selective surfaces,” in Proc. Inst. Elect. Eng. Microw., Antennas, Propag., 1991, vol. 138, pp. 19–22. [30] J. McVay, N. Engheta, and A. Hoorfar, “High-impedance metamaterial surfaces using hilbert-curve inclusions,” IEEE Microw. Wireless Comp. Lett., vol. 14, pp. 130–132, Mar. 2004. [31] J. McVay, A. Hoorfar, and N. Engheta, “Peano high-impedance surfaces,” in Proc. Radio Sci., 2005, vol. 40. [32] L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Multiband subwavelength magnetic reflectors based on fractals,” Appl. Phys. Lett., vol. 83, no. 16, pp. 3257–3259, 2003. [33] W. Zhang, A. Potts, D. Bagnall, and B. Davidson, “High-resolution electron beam lithography for the fabrication of high-density dielectric metamaterials,” J. Thin Solid Films, vol. 515, no. 7–8, pp. 3714–3717, 2007. [34] A. Mejdoubi and C. Brosseau, “Intrinsic resonant behavior of metamaterials by finite element calculations,” Phys. Rev. B, vol. 74, no. 16, p. 165424, 2006. [35] J. McVay, A. Hoorfar, and N. Engheta, “Bandwidth enhancement and polarization dependence elimination of space-filling curve artificial magnetic conductors,” in Proc. Asia-Pacific Microwave Conf. , Dec. 2007, pp. 1–4. [36] J. McVay, N. Engheta, and A. Hoorfar, “Numerical study and parameter estimation for double-negative metamaterials with Hilbert-curve inclusions,” in Proc. IEEE AP-S Int. Symp. Antennas Propag., Jul. 2005, vol. 2B, pp. 328–331. [37] V. Crnojevic-Bengin, V. Radonic, and B. Jokanovic, “Fractal geometries of complementary split-ring resonators,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 10, pp. 2312–2321, Oct. 2008. [38] V. Pierro, J. McVay, V. Galdi, A. Hoorfar, N. Engheta, and I. M. Pinto, “Metamaterial inclusions based on grid-graph Hamiltonian paths,” Microw. Opt. Tech. Lett., vol. 48, no. 12, pp. 2520–2524, Dec. 2006.

[39] L. Yousefi and O. M. Ramahi, “New artificial magnetic materials based on fractal Hilbert curves,” in Proc. IWAT07, Mar. 2007, pp. 237–240. [40] L. Yousefi, H. Attia, and O. M. Ramahi, “Broadband experimental characterization of artificial magnetic materials based on a microstrip line method,” J. Progr. Electromagn. Res. (PIER), vol. 90, pp. 1–13, Feb. 2009. [41] J. Franklin and A. Daoud, Proof in Mathematics: An Introduction. New York: Quakers Hill Press, 1996. [42] R. Schinzinger and P. A. A. Laura, A Conformal Mapping: Methods and Applications. The Netherlands: Elsevier, 1991. [43] K. E. Oughstun and N. A. Cartwright, “On the Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion,” Opt. Expr., vol. 11, no. 13, pp. 1541–1546, Jun. 2003. [44] X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E, vol. 70, no. 1, pp. 016 608.1–016 608.7, Jul. 2004. [45] K. Buell and K. Sarabandi, “A method for characterizing complex permittivity and permeability of metamaterials,” in Proc. IEEE AP-S Int. Symp. Antennas Propag., Jun. 2002, vol. 2, pp. 408–411. [46] R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1516–1529, Jul. 2003.

Leila Yousefi (M’09) was born in Isfahan, Iran, in 1978. She received the B.Sc. and M.Sc. degrees in electrical engineering from Sharif University of Technology, Tehran, Iran, in 2000 and 2003, respectively, and the Ph.D. degree in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 2009. Currently she is working as a Postdoctoral Fellow at the University of Waterloo. Her research interests include metamaterials, miniaturized antennas, electromagnetic bandgap structures, and MIMO systems.

Omar M. Ramahi (F’09) received the B.S. degrees in mathematics and electrical and computer engineering (summa cum laude) from Oregon State University, Corvallis, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Illinois at Urbana-Champaign. From 1990–to 993, he held a visiting fellowship position at the University of Illinois at Urbana-Champaign. From 1993 to 2000, he worked at Digital Equipment Corporation (presently, HP), where he was a member of the alpha server product development group. In 2000, he joined the faculty of the James Clark School of Engineering, University of Maryland at College Park, first as an Assistant Professor, later as a tenured Associate Professor, and where he was also a faculty member of the CALCE Electronic Products and Systems Center. Presently, he is a Professor in the Electrical and Computer Engineering Department and holds the NSERC/RIM Industrial Research Associate Chair, University of Waterloo, ON, Canada. He holds cross appointments with the Department of Mechanical and Mechatronics Engineering and the Department of Physics and Astronomy. Previously, he served as a consultant to several companies and was a co-founder of EMS-PLUS, LLC and Applied Electromagnetic Technology, LLC. He has authored and coauthored over 240 journal and conference papers. He is a coauthor of the book EMI/EMC Computational Modeling Handbook (Springer-Verlag, 2001). Prof. Ramahi serves as an Associate Editor for the IEEE TRANSACTIONS ON ADVANCED PACKAGING and as the IEEE EMC Society Distinguished Lecturer.

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Theory and Practice of the FFT/Matrix Inversion Technique for Probe-Corrected Spherical Near-Field Antenna Measurements With High-Order Probes Tommi Laitinen, Member, IEEE, Sergey Pivnenko, Member, IEEE, Jeppe Majlund Nielsen, and Olav Breinbjerg, Member, IEEE

Abstract—A complete antenna pattern characterization procedure for spherical near-field antenna measurements employing a high-order probe and a full probe correction is described. The procedure allows an (almost) arbitrary antenna to be used as a probe. Different measurement steps of the procedure and the associated data processing are described in detail, and comparison to the existing procedure employing a first-order probe is made. The procedure is validated through measurements. Index Terms—Antenna measurements, high-order probe, nearfield scanning, probe correction, spherical wave expansion.

I. INTRODUCTION

T

HE Technical University of Denmark (DTU) and the European Space Agency (ESA) have a 30-year history of accurate probe-corrected spherical near-field antenna measurements at the DTU-ESA Spherical Near-Field Antenna Test Facility [1]. The standard procedure for the antenna pattern characterization applied at the DTU-ESA Facility is based on the probes, since they can exuse of so-called first-order ploit the computationally efficient and stable first-order probe correction technique [2]. These probes are conical horns fed by circular waveguides excited with the dominant mode. The conical horn probes are typically operated on a relatively narrow frequency band, about 10%–15%, thus the collection of probes at the DTU-ESA Facility consists of 14 dual-polarized probes to cover the frequency range from 3–18 GHz. This type of a probe also becomes relatively heavy and large at low frequencies, for example, below 3 GHz. The capacity of computers has increased significantly during the latest 30 years. For this reason, the computational efficiency of the probe correction has become a less significant factor for many applications, whereas it has become attractive to consider, also at the expense of reduced computational efficiency, such probe correction techniques that would be applicable for probes, or, more generally, for probes that high-order Manuscript August 14, 2009; revised January 22, 2010; accepted February 12, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by the ESA/ESTEC under Contract 18222/04/NL/LvH/bj. The work of T. Laitinen was supported by the Academy of Finland (decision notification no. 129055). T. Laitinen is with the Department of Radio Science and Engineering, Aalto University, FI-00076 Aalto, Finland (e-mail: [email protected]). S. Pivnenko, J. M. Nielsen and O. Breinbjerg are with the Technical University of Denmark, Kgs. Lyngby, Denmark (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2050437

are not of the first order. In this paper, we understand high-order probes generally as those not belonging to the class of first-order probes. The use of a high-order probe correction technique allows a greater flexibility in choosing a probe that leads to an optimal compromise between the desired properties of the probe, for example, the bandwidth, the weight, the size, and the cost. In fact, several high-order probe correction techniques have been introduced recently [3]–[10]. Some of these techniques are based on the spherical wave expansion of the AUT and probe fields and involve spherical wave translations in the transmission formula [3]–[6], [8], [9]. As opposed to these, the techniques [7], [10] are based on another type of an expansion involving a plane-wave translation operator in the transmission formula. Among the techniques referred here, the techniques [5], [7], [9] and [10] are the only ones applicable for (almost) arbitrary probes. The work in [5] was carried out during 2004–2006 within a project supported by the European Space Agency [5]. The techniques [7] and [10] were published in 2008 and 2009, respectively. It has been shown in [7] and [10], that the use of an iterative solver makes it possible to solve the transmission formula in a computationally efficient manner. For example, compared to the direct matrix inversion, the use of the iterative solver has been shown to decrease the computational complexity in to in [7] solving the transmission formula from in [10]. Here is proportional to the and further to electrical radius of AUT (antenna under test) minimum sphere. For comparison, the computational complexities of the tech. By noting that the compuniques [5] and [9] are both is low enough for a major part tational complexity of of spherical near-field antenna measurement applications, the general high-order probe correction technique presented in [5], that will be referred to as the FFT/matrix inversion technique in this paper, is taken into further consideration here. It is further noted that the technique in [9] is a modification of the FFT/matrix inversion technique applicable in conjunction with another scanning technique. Naturally, the computation of the far field from the probe signals measured in the near field, including the probe correction, is only one part of a complete antenna pattern characterization procedure. The high-order probe correction techniques presented in the literature so far have, however, not touched upon the practical implementation aspects of the technique for an existing range at all, or have done that in a limited fashion, and for this reason this issue deserves further attention.

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The purpose of this paper is to present the complete antenna pattern characterization procedure developed at the DTU-ESA Facility based on the FFT/matrix inversion technique and employing high-order probes and to describe all practical steps of the procedure with the associated data processing. Comparison to the existing procedure employing first-order probes is made where relevant. The procedure is presented primarily for dual-port probes, but clarifications are provided for the case of a single-port probe. In addition to the earlier validation measurements carried out for 2.9 GHz and 3.0 GHz [5] and reported in [11], another set of validation measurements is carried out here for 1.4 GHz and 1.5 GHz, and the results are presented. The developed procedure has been tested and shown to work earlier in [5] with computer calculations for the frequency range 1–3 GHz for AUTs with the radius of the minimum sphere up to 3 m, which corresponds to approximately 30 wavelengths. The calculation results have been partly presented in [12], and are thus left outside of this paper. The theoretical background is presented in Section II. The antenna pattern characterization procedure for high-order probes is described in Section III. Test measurements for validation of the procedure are then presented in Section IV. Conclusions are given in Section V.

Fig. 1. The AUT and the probe minimum spheres, and the relation between the AUT and the probe coordinate systems.

and are the truncation numbers for the and indices, respectively. The probe is assumed to be either a dual-port probe having one fixed orientation or a single-port probe having two orientations separated by 90 . The truncated SWE of the radiated , of a reciprocal dual-port probe, electric field of one port, is expressed as

II. BACKGROUND THEORY The theory of probe-corrected spherical near-field antenna measurements with a first-order probe is presented in [2]. The part of this theory relevant to high-order probes is summarized in this section. A. Measurement Geometry The geometry for probe-corrected spherical near-field antenna measurements is presented in Fig. 1 where the AUT and the probe minimum spheres are illustrated. The and and the Cartesian coordinates of the AUT and the are the probe coordinate systems, respectively. The standard spherical coordinates of the AUT coordinate system [13]. The measurement distance is the distance between the origins of the AUT and probe coordinate systems. The angle and , the axis is the probe rotation angle; for is parallel to the and unit vectors of the AUT coordinate system, respectively. B. Spherical Wave Expansion According to [2], the radiated fields of the AUT and the probe are both expressed in terms of the spherical vector wave expansion (SWE). Assuming and suppressing the time convention of , the practical, truncated form of the SWE of the radiated , becomes electric field of the AUT, (1) where is the wave number, is the intrinsic admittance of the ambient medium, are the spherical vector wave coefficients (Q coefficients) of the AUT field, and are the power-normalized spherical vector wave functions [2, , for Ch. 2]. The triple summation is for , and for and 2, where , and

(2) , with or 2, are the Q coefficients for the where probe port or alternatively, in the case of a single-port probe, the probe orientation. The triple summation is for , for , and for and 2, , and and are the truncation where numbers for the and indices, respectively. The truncation numbers ( , , , ) may be determined from the following truncation rules [2]: (3) (4) (5) (6) and are the radii of the AUT and probe minimum where and are spheres, shown in Fig. 1, respectively, and the the radii of the AUT and probe minimum circular cylinders [2], respectively. The square brackets indicate the largest integer smaller than or equal to the number inside the brackets. The , and are chosen according values for the integers , , to the accuracy requirement, and typically is sufficient [2]. C. Transmission Formula According to the transmission formula [2], the signal at the probe port, , is

(7)

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where , and are the rotation coeffiport of the cients [2]. The probe response constants of the probe are (8)

are the translation coefficients of the spherwhere are the probe receiving coical vector wave functions and efficients [2]. , and the probe transmisThe relations between are [2, Eq. (5.68)], sion coefficients (9) (10) Here , with dimension , is the normalized complex amplitude of the excitation signal at the probe port. In the transmission formula (7) and (8), the rotation and translation coefficients are known functions, the receiving coefficients are known from a separate probe calibration measurement, and the probe port signal is known from the spherical scanning of the AUT radiated field. The formula can thus be solved for the AUT Q coefficients, and the radiated field at any larger distance, in particular, in the far field can then be obtained from (1). III. ANTENNA PATTERN CHARACTERIZATION PROCEDURE USING A HIGH-ORDER PROBE The AUT pattern characterization procedure with a high-order probe (HOP) comprises the following three steps: 1) HOP pattern calibration, 2) HOP channel balance calibration, and 3) AUT pattern measurement. In this section, these three measurements and the data processing related to each measurement are described. A. HOP Pattern Calibration In this calibration the HOP is treated as an AUT, and a highly linearly polarized antenna is used as an auxiliary probe. The auxiliary probe is aligned for orientation such that its -axis direction. Alternaradiated field is -polarized in the tively, an additional three-antenna polarization calibration and the corresponding correction can be applied for the auxiliary probe as described in [2, Sec. 5.2.3–5.2.4]. The radiation patterns of both ports of the dual-port HOP are measured with the auxiliary probe for an appropriate number for the auxiliary probe orienof measurement directions tations and 90 . The auxiliary probe is treated in this measurement as an electric Hertzian dipole, that is, no pattern correction is applied. This is a valid approach in the typical case where the far-field conditions hold for both the HOP and for the auxiliary probe. Hence, with the help of the well-known orthogonality integrals for the spherical vector wave functions [2], the Q coefficients of the HOP are now found from the transmission formula (7) by applying the known probe response constants of

Fig. 2. (a) The probe in the AUT coordinate system (x; y; z ). (b) The probe in the probe coordinate system (x ; y ; z ).

the electric Hertzian dipole. This can be done, for example, by applying the first-order probe correction technique. During the probe pattern calibration the HOP is located in are related to the the AUT coordinate system so that the patterns of the two probe ports with the HOP pointing in the -axis direction as illustrated in Fig. 2(a). However, during the AUT pattern measurement the HOP is located in the probe co-axis direction as evident ordinate system and pointed to the from Fig. 1. According to the practice adopted at the DTU-ESA Facility, the 180 rotation of the probe is performed around the axis as illustrated in Fig. 2. After this rotation, the coordinate system is changed from the AUT coordinate system to the probe coordinate system . By the change of , and , the two sets of Q cothe indices: and , for the probe pointing into efficients, -axis direction in the probe coordinate system, as illustrated in Fig. 2(b), are then obtained from [2, Eq. (5.67)], (11) The are determined using the relations (9) and (10) by assuming (without loss of generality), for example, , and by replacing in (10). The have been determined now and, thus, the probe pattern calibration has been accomplished. It is noted that probe receiving coefficients include complete information of the pattern of each port of the HOP. Thus, no separate polarization calibration, that is a part of the conventional probe calibration procedure described in [2, Sec. 5.2.3–5.2.4], is performed. It is noted, that during the HOP pattern calibration the HOP is mounted on the flange of the AUT tower of the measurement range whereas during the AUT pattern measurement it is mounted on the flange of the probe tower. The HOP receiving coefficients must, however, remain unchanged (except for the 180 rotation of the coordinate system) in the translation of the probe. This can be ensured by suppressing the influence of mounting structure on the probe pattern by a proper application of absorbers. If, in the case of a dual-port probe, the reflection coefficient of the HOP load (once placed in the AUT tower and then in the probe tower) is different, then the HOP receiving coefficients can be ensured to remain unchanged by having a high (for example 40 dB) port-to-port isolation of the probe. In the case of poor isolation, different reflection coefficient may result in a change in the HOP pattern and thus in different probe receiving coefficients.

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are the Q coefficients of the auxiliary antenna. The are obtained from (8) using known from the probe pattern calibration. The relation [2, Eq. A2.15], where is the Kronecker’s delta, is exploited here. The channel balance is now determined by means of the so-called polarization scan measurement where the auxiliary antenna is rotated around the axis, and the signals from the two probe ports at the receiver are recorded in intervals of from to . The maximum is related to the cross-section dimensions of the probe and the auxiliary antenna in wavelengths. However, it is useful to apply a smaller than the maximum in this measurement for decreasing the uncertainty, because the increase in the measurement time due to this is negligible. By writing the received signal in (14) as a Fourier expansion

where

Fig. 3. Polarization scan measurement set up for determination of channel balance.

(15) B. HOP Channel Balance Calibration The signals measured at the receiver are influenced by not only the probe pattern but also by the two channels from the probe to the receiver; the characteristics of these two channels are generally different both in magnitude and phase. This difference must be compensated, and it is performed with the so-called the channel balance calibration measurement. , the transmisAssuming (without loss of generality) sion formula (7) for a signal at the receiver from port can be written as

(12) is the channel coefficient that is independent of any where and contain the inforspherical mode of the probe. The mation of any mismatch, attenuation, and phase propagation of the signal from the two probe ports to the receiver. The channel , is defined as the ratio between the channel coefbalance, ficients of the two channels, (13) and the purpose of the channel balance calibration is to determine this ratio. Once this ratio is known, the known received and the known received signal (12) for signal (12) for multiplied by constitute the correct relative received signals. In the channel balance calibration the HOP is located in the probe coordinate system with the fixed orientation angle , and the auxiliary antenna is located in the AUT tower as illustrated in Fig. 3. The axis is pointed to the origin of the . A good choice for the probe coordinate system, thus, auxiliary antenna is an antenna with a linear polarization and the maximum radiation in the -axis direction. , the signal at the With these conditions receiver from the probe port , expressed through (12), becomes

with the Fourier coefficients (16)

it becomes obvious that the Fourier coefficients are found by the inverse discrete Fourier transformation (IDFT) of the measured signals in . The number of coefficients is the same as the number of samples. However, the coefficients are practically the only important coefficients here for for the following reasons. First, with increasing measurement distance the coupling of the signal to the probe ports occurs , which increasingly dominantly via the modes with is due to the asymptotic behavior of the translation coefficient [2, Eq. A3.22-A3.24]. Second, most (if not all) practical auxiliary antennas aligned on the axis with the maximum radi-axis direction possess a significant degree of ation in the , and this further enhances the power in modes with coupling of the signal through these modes in the channel balance measurement. Restricting the discussion now for the coefficients only, it is noted that depending on the polarization of the probe port, the amplitude of the coefficient for may . This is the case for differ significantly from that for highly non-linearly polarized probe port (with a linearly polarized auxiliary antenna). Due to noise and other uncertainties, the coefficient with the lower amplitude is also more sensitive to having a greater relative error than the one having the higher amplitude. For this reason, it is now suggested that for each port the is calculated for such index or for which is higher. Hence, conthe amplitude of the coefficient or only, sidering now the coefficients for either (13) and (16) provide

(17) (14)

LAITINEN et al.: THEORY AND PRACTICE OF THE FFT/MATRIX INVERSION TECHNIQUE

Here or and or depending on the amplitudes of the coefficients and according to above discussion. Equation (17) allows that the probe ports are different, which is a generalization compared to the traditional first-order channel balance calibration procedure. Equation (17) requires, , are in general, that the auxiliary antenna Q coefficients, known at least for the index for which the is determined. If a highly linearly polarized auxiliary antenna is used, then the electric Hertzian dipole assumption of the auxiliary antenna can typically be made. The Q coefficients of the electric Hertzian dipole are well known [2, Sec. 2.3.3]. It is furthermore noted that, in theory, such impractical probes may exist that do not possess radiated power in modes with , and hence have a zero on-axis field. Thus, if, for some reason, such an impractical HOP were used, the channel coeffior , cients should not be calculated from (17) for but for some other value of .

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and are the measured received where is signals (12) at the receiver from the two probe ports, and known from the channel balance calibration. 3) Probe Correction Calculations: The probe correction calculations include two steps. The first step consists of the IDFT of the signals at the probe ports. The second step consists of matrix inversions. These two steps are now explained. In the first step, the relative received signal at the probe ports, (18) and (19) with (12), are first written as a Fourier expansion (20) where (21)

where for

C. AUT Pattern Measurement 1) Scanning: The general high-order probe correction technique does not impose any strict requirements for the angles in , where the samples must be available. However, a good choice is to take the samples equidistantly in from 0 to 180 in steps in order to ensure that the systems of linear equations, that of are solved at a later stage of the probe correction calculations, have sufficient number of sufficiently linearly independent rows. holds, It is assumed in this paper that the condition is the total number of sampling directions in from where 0 to 180 . This condition follows from that for the traditional first-order probe correction technique, and although this may not be necessarily strictly required in the case the FFT/matrix inversion technique applied here, it is assumed in this paper. For each fixed angle, it is required that the samples in are available from 0 to with constant increments . It is further required that the condition of holds, where is the number of sampling directions in from . 0 to received signals are In each measurement direction measured from the two ports of the HOP, and the probe orientation angle is set to so that (12) becomes applicable. In the case of a single-port probe, the signals are measured for two and 90 , in each probe orientation angles, typically for measurement direction. These requirements for the measurement data are conveniently fulfilled by the -scanning scheme, where stepping is made in and scanning in . The -scanning scheme, where stepping is made in and scanning in , is not applicable. A double -step -scanning scheme, and the associated high-order probe correction technique has been presented in [9]. 2) Formation of the Signals at the Probe Ports: The probe correction calculations require that the correct relative received signals are known at the probe ports. Hence, the relative received signals at the two probe ports, and , are formed as follows:

. The Fourier coefficients are now found for each discrete value of for and 2 by the IDFT of the measured this operation signals in , and for the case with can be written as (22) an overIn the second step, for each fixed determined system of linear equations is set up from (21) for the with indices and 2, and . unknowns For each , this system of linear equations is written as (23) Here, the matrix

is .. .

where the 2

2 block matrices

.. .

are (25)

with (26) The relation between indices and is the . The for . The vectors and are

is the

, and angle

(27) (28)

(18) (19)

(24)

respectively, where

denotes transpose.

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The least-square solution to the matrix (23) is (29) where the

is (30)

denotes Hermitian transpose. The in (30) can where be found from using, for example, the PINV-function in MatLAB. Computer calculations indicate that the computational com, and plexity of the matrix inversions is of the order of it is not higher for filling in the matrices. The overall computation time is dominantly determined by the time required for matrices for . For filling and inverting the example, for and 160, with , , and , the overall computation time is approximately 6 and 30 minutes, respectively, with the current implementation of the probe correction algorithms at the DTU-ESA Facility on a normal PC of today. In these cases, the time required for the matrix fillings was approximately 2 to 3 times the time required for the matrix inversions. It is possible to apply parallel processing for reducing the computation time. matrices are typically the The condition numbers of the . A quantitative investigation carried out highest for in [5] with several different high-order probes shows that the condition numbers are steadily increasing with increasing but for all examined probes. remain below 40 with The memory requirements of the technique are dominated by the size of the matrix. At a given time it is necessary to have only one matrix in memory. The matrix is the , and then its dimension is of the largest for the indices . For example, for , that order of corresponds to the AUT with the radius of the minimum sphere , the required memory for one matrix for of more then any of the indices in the double-precision system is of the order of 64 MB. For comparison, the maximum dimension of a single array accepted by MatLAB™ in a 32-bit Windows system is more than 600 MB. IV. MEASUREMENTS AND DATA PROCESSING This section describes the measurements carried out to validate the new antenna pattern characterization procedure based on the FFT/matrix inversion technique and to illustrate its importance for a high-order probe. The probe in the earlier measurements, reported in [11], was a wideband dual-polarized probe SP800 from Satimo [14] covering the frequency range 0.8–3.2 GHz. Although, this SP800 probe must be, as noted in [5] and [11], considered a high-order probe, and hence cannot be treated as a first-order probe in accurate spherical near-field antenna measurements, the radiated relative to power in the azimuthal spherical modes with for this probe was, howthe power in the modes with ever, modest. For this reason it was decided that another test measurement is carried out with a more challenging probe pos. sessing substantial relative power in the modes with Therefore the probe for the second test measurement reported in this paper is chosen to be an offset single-port square waveguide

Fig. 4. Photograph of the AUT and its coordinate system.

probe. The offset, in particular, increases the radiated power in . Since the channel balance calibration the modes with for a dual-port HOP was verified by the earlier measurements [5], [11], it is found sufficient to perform the measurements now with a single-port probe. First, the reference measurement performed for the chosen AUT is described. Next, the HOP, its characterization, and its properties in terms of the spherical wave coefficients are presented. Then, the test measurement performed for the same AUT with the HOP are described. Finally, the comparison between the reference and test AUT patterns is made, and the significance of the high-order probe correction documented. A. Reference Measurement of the AUT A reference full-sphere measurement is performed at the DTU-ESA Facility for the AUT illustrated in Fig. 4. The AUT is a log-periodic 1–18 GHz antenna in an offset configuration attached to a metallic support arm. The offset of the log-periodic antenna is 1.6 m from the axis along the axis, i.e., the coordinates of the log-periodic antenna in the AUT , 0 m, 0 m). measurement coordinate system are ( The choice of this AUT configuration is driven by the intention to have a demanding measurement case for which the influence of the high-order modes of the probe is expected to be large. Due to the offset, the phase of the received near-field signal varies strongly with the AUT positioning angle, and this results in a wide spectrum of spherical wave modes in the expansion of the AUT field. In addition, the non-symmetric location of the radiating element tests the sensitivity of the processing algorithm to small mechanical imperfections of the setup. Obtaining a good agreement for the chosen AUT configuration is thus expected to guarantee similar or better agreement for any other AUTs, including also large and symmetrically located AUTs, with the equal or smaller radius of the AUT minimum sphere. The reference measurement of the AUT is carried out for the -polarized port of the AUT at 1.4 GHz and 1.5 GHz. The scanning scheme is chosen to be scanning in within and stepping in within with sampling intervals . This reference measurement is performed with a high-quality dual-polarized open-ended choked circular waveguide probe, that is a first-order probe. In the data processing, the traditional first-order probe correction is applied to the received probe signals and the far field is

LAITINEN et al.: THEORY AND PRACTICE OF THE FFT/MATRIX INVERSION TECHNIQUE

Fig. 5. Reference AUT pattern: co-polar directivity for  (dashed) planes and cross-polar directivity for   and  (dotted) planes.

= 90 = 90

= 0 (solid) and = 0 (dash-doted)

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Fig. 6. Measured pattern of the offset square waveguide probe (HOP): co-polar (solid) and  (dashed) planes and co-polar directivity for the  phase for the  (dash-doted) and  (dotted) planes.

=0

=0

= 90 = 90

calculated. This far field is considered as a reference in the comparisons below. The 1- uncertainty of the reference pattern is estimated to be lower than 0.05 dB around the pattern peak. The co-polar and cross-polar directivities of the AUT reference pattern at 1.5 GHz, calculated according to the Ludwig’s and third definition [15], are shown for the planes in Fig. 5. The small ripples at low levels of the co-polar plane are due to the interference of the pattern in the direct field from the radiating log-periodic element of the AUT and the diffracted field from the opposite end of the metallic support arm of the AUT. B. HOP Pattern Calibration As stated earlier the HOP is an open-ended square waveguide with the center of the aperture offset from the center of the probe axis. coordinate system. This offset is 30.5 mm along the This corresponds to about 0.15 wavelengths at 1.5 GHz. The HOP pattern calibration is performed using a calibrated first-order probe as the auxiliary probe. The auxiliary probe is the same probe that was used for the reference AUT measurement. Since the chosen HOP is a single-port antenna, only one pattern measurement is required. During the HOP pattern calibration the HOP is placed in the AUT coordinate system so that its radiation is -polarized in the -axis direction. The pattern of the HOP oriented for the polarization is found by applying a 90 rotation of the obtained HOP pattern around the axis. The probe receiving coefficients for the two polarizations, and , are finally determined according to Section III-A. The co-polar amplitude and phase patterns of the HOP at 1.5 GHz are shown in Fig. 6. It is seen that the amplitude pattern is plane. almost symmetric with a small asymmetry in the On the other hand, the phase pattern is clearly asymmetric in the plane due to the offset of the probe. The normalized spherical -mode power spectrum of the HOP is then calculated from the determined probe receiving coefficients. This spectrum, that describes the relative power of the field radiated by the probe as a function of the azimuthal , is shown in Fig. 7. For completemode index ness, the normalized spherical -mode power spectrum of the

Fig. 7. The normalized spherical -mode and  -mode power spectra of the offset square waveguide probe, and the normalized spherical -mode power spectrum of the first-order probe used in the reference AUT measurement.

HOP and the normalized spherical -mode power spectrum of the first-order probe are also presented in Fig. 7. It is seen from the -mode spectrum of the HOP that, though most of its radiated power is contained in the modes with , a significant part of the power is contained also in the other , modes. In particular, the power in the modes with 2, and 3, as compared to the modes, is in the range to . Thus, the presented spectrum of the from HOP clearly illustrates that the offset square waveguide probe is indeed a high-order probe. For comparison, in the case of the first-order probe, as seen in Fig. 7, practically all the power is . concentrated in the modes with C. Test Measurement of the AUT and Data Processing In line with the requirements to the scanning described in Section III-C, a full-sphere near-field measurement with the HOP is performed for the AUT using the -scanning scheme with two HOP polarizations. The sampling intervals in and are chosen to be and , respectively, as in the reference measurement.

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TABLE I STATISTICAL DATA FOR THE DIFFERENCE BETWEEN THE REFERENCE AND TEST CO-POLAR AUT PATTERNS WITHIN j j 

10

TABLE II STATISTICAL DATA FOR THE DIFFERENCE BETWEEN THE REFERENCE AND TEST (M AND N ) CO-POLAR PATTERNS WITHIN j j 

=1

=0

Fig. 8. Comparison between the reference and test AUT patterns in  plane: reference (solid) and test (dashed) co-polar directivity, and reference (dash-doted) and test (dotted) cross-polar directivity.

= 90

Fig. 9. Comparison between the reference and test AUT patterns in  plane: reference (solid) and test (dashed) co-polar directivity, and reference (dash-doted) and test (dotted) cross-polar directivity.

For the calculation of the Q coefficients of the AUT field, and subsequently the far field from (1), the measured probe signals are processed according to the procedure described in Section III-C. The and modes of the probe up to and , respectively, are taken into account. The truncation numbers for the probe modes are selected on the condition that only the modes above the noise floor are included in the calculations. The probe response constants are obtained from (8) using the probe receiving coefficients known from the probe pattern calibration. The channel balance is set to 1. The obtained reference and test directivity patterns of the plane in Fig. 8 AUT for 1.5 GHz are shown for the and for the plane in Fig. 9. It is seen from Figs. 8 and 9 that there is a good agreement between the co-polar patterns in both planes though the test pattern shows slightly larger ripples around in the plane. Agreement between the cross-polar patterns is also good, though some small differences can be observed. The statistics for the difference between the reference and test co-polar AUT patterns in the dB scale calculated for the region , i.e., around the pattern peak, are given in Table I.

=8

10

The mean and the standard deviation of the difference between the test and reference patterns do not exceed 0.05 dB. The observed deviations for the co-polar directivity are within the measurement uncertainty for the standard measurement procedure at the DTU-ESA Facility employing the first-order probes and the first-order probe correction technique, and the agreement is thus considered to be very good. Similar agreement between the co-polar directivities was observed also in the final measurement in [5]. The deviations for the co-polar phase seen in Table I are larger than it was observed in [5], but in view of the challenging AUT and the HOP, these values are found acceptable. The test data are also processed taking into account only modes of the HOP to study the influence of the high-order azimuthal modes on the accuracy of the obtained far-field data. The statistics for the difference between the reference co-polar pattern and test co-polar pattern processed and calculated for the same region with are given in Table II. The statistics show that in this case the difference between the test and reference patterns is several times larger as compared to the results in Table I. This was expected, and it confirms the fact that the proper high-order probe correction is clearly necessary for the employed HOP. V. CONCLUSION A complete antenna pattern characterization procedure for probe-corrected spherical near-field antenna measurements with high-order probes has been developed and described in this paper. The procedure has been verified by measurements in the DTU-ESA Spherical Near-Field Antenna Test Facility. The uncertainty of the antenna pattern determination provided by the new procedure with a high-order probe is comparable to that provided by the existing procedure based on the use of first-order probes. The results of this paper have shown that accurate antenna pattern characterization using high-order probes is possible both in theory and practice using the -scanning scheme. Future work could include similar thorough theoretical and practical investigations of the possibilities for accurate antenna pattern characterization with high-order probes with other scanning schemes, for example, with the -scanning scheme. ACKNOWLEDGMENT The authors would like to thank J. Lemanczyk from ESTEC for valuable comments and suggestions.

LAITINEN et al.: THEORY AND PRACTICE OF THE FFT/MATRIX INVERSION TECHNIQUE

REFERENCES [1] DTU-ESA Spherical Near-Field Antenna Test Facility Technical University of Denmark [Online]. Available: http://www.dtu.dk/centre/ ems/English/research/facilities.aspx [2] J. E. Hansen, Spherical Near-Field Antenna Measurements. London, U.K.: Peter Peregrinus, 1988. [3] T. A. Laitinen, S. Pivnenko, and O. Breinbjerg, “Odd-order probe correction technique for spherical near-field antenna measurements,” Radio Sci., vol. 40, no. 5, 2005. [4] T. A. Laitinen, S. Pivnenko, and O. Breinbjerg, “Iterative probe correction technique for spherical near-field antenna measurements,” IEEE Antennas Wireless Propag. Lett., vol. 4, 2005. [5] T. A. Laitinen, S. Pivnenko, J. Nielsen, and O. Breinbjerg, “Development of 1–3 GHz Probes for the DTU-ESA Spherical Near-Field Antenna Test Facility. ESTEC Contract/04/NL/LvH/bj. Final Report, Volume 1: Executive Summary,” Electromagnetic Systems, Ørsted·DTU, Tech. Univ. Denmark, Lyngby, Denmark, 2006, Rep. R 729. [6] T. A. Laitinen and S. Pivnenko, “Probe correction technique for symmetric odd-order probes for spherical near-field antenna measurements,” IEEE Antennas Wireless Propag. Lett., vol. 6, 2007. [7] C. H. Schmidt, M. M. Leibfritz, and T. F. Eibert, “Fully probe-corrected near-field to far-field transformation employing plane wave expansion and diagonal translation operators,” IEEE Trans. Antennas Propag., vol. 56, pp. 737–746, Mar. 2008. [8] T. A. Laitinen and O. Breinbjerg, “A first/third-order probe correction technique for spherical near-field antenna measurements using three probe orientations,” IEEE Trans. Antennas Propag., vol. 56, pp. 1259–1268, May 2008. [9] T. A. Laitinen, “Double -step  -scanning technique for spherical near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. 56, pp. 1633–1639, Jun. 2008. [10] C. H. Schmidt and T. F. Eibert, “Multilevel plane wave based near-field far-field transformation for electrically large antennas in free-space and above material halfspace,” IEEE Trans. Antennas Propag., vol. 57, pp. 1382–1390, May 2009. [11] T. Laitinen, J. M. Nielsen, S. Pivnenko, and O. Breinbjerg, “On the application range of general high-order probe correction technique in spherical near-field antenna measurements,” presented at the 2nd Eur. Conf. on Antennas and Propagation (EuCAP’07), Edinburgh, U.K., Nov. 2007. [12] T. A. Laitinen, S. Pivnenko, and O. Breinbjerg, “Sensitivities of various probe correction techniques to noise and inaccurate channel balance in spherical near-field antenna measurements,” in Proc. 2nd Int. Conf. on Electromagnetic Near-Field Characterization and Imaging (Iconic’05), Barcelona, Spain, 2005, pp. 411–416. [13] IEEE Standard Test Procedures for Antennas, IEEE Std 149-1979, 1979. [14] Satimo [Online]. Available: http://www.satimo.org [15] A. C. Ludwig, “The definition of cross polarization,” IEEE Trans. Antennas Propag., vol. 21, pp. 116–119, Jan. 1973. Tommi Laitinen (M’09) was born in Pihtipudas, Finland, on March 19, 1972. He received the Master of Science in Technology, the Licentiate of Science in Technology, and the Doctor of Science in Technology degrees in electrical engineering from Helsinki University of Technology (TKK), Espoo, Finland, in 1998, 2000, and 2005, respectively. He joined the Radio Laboratory at TKK as a Master’s thesis student in 1997, and continued as a doctoral student in the same place afterwards. His major research interests at TKK were small antenna measurements. From 2003 until the end of 2006, he was with the Technical University of Denmark (DTU) as a Postdoctoral Researcher and Assistant Professor. His research interests at DTU were spherical near-field antenna measurements. During these years, he mainly contributed to the development of an accurate antenna pattern characterization procedure for the DTU-ESA Spherical Near-Field Antenna Test Facility based on spherical near-field antenna measurements with a high-order probe. In the beginning of 2007 until the end of 2009, he was with the Radio Laboratory at TKK. In the beginning of 2010, he joined the Department of Radio Science and Engineering, Aalto University, Aalto, Finland, as a Senior Researcher. While still carrying on with research on spherical near-field antenna measurements, he now works also with small antenna measurements and sensor applications. His other duties include occasionally teaching master’s and postgraduate courses at Aalto University. He is the author or coauthor of approximately 50 journal and conference papers. Dr. Laitinen is the recipient of the IEEE Antennas and Propagation Society’s 2009 R. W. P. King Award for one of his papers.

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Sergey Pivnenko (M’98) was born in Kharkiv, Ukraine, in 1973. He received the M.Sc. and Ph.D. degrees in electrical engineering from Kharkiv National University, Ukraine, in 1995 and 1999, respectively. From 1998 to 2000, he was a Research Fellow at the Radiophysics Department, Kharkiv National University. In 2000, he joined the Antennas and Electromagnetics Group, Department of Electrical Engineering, Technical University of Denmark, where he now works as an Associate Professor and operates the DTU-ESA Spherical Near-Field Antenna Test Facility. Since 2000, he participated to several research projects related to design, development and characterization of satellite antennas, development of new near-field probes and probe correction techniques for near-field antenna measurements. From 2004–2007, he was a work package leader in the European Union network “Antenna Center of Excellence” where he was responsible for antenna measurement facility comparisons and participated to the other activities related to antenna measurements. He is the author or coauthor of more than 70 journal and conference papers. His research interests include antenna measurement techniques, antenna analysis and design.

Jeppe Majlund Nielsen was born on June 9, 1976. He received the M.Sc. degree in engineering from the Technical University of Denmark (DTU), Kgs. Lyngby, in 2002. Since then he has been a Research Assistant in the Department of Electrical Engineering, DTU. His work has included antenna gain calibrations by spherical near-field antenna measurements as well as planar near-field antenna measurements. He has supervised several M.Sc. projects, M.Sc. courses and special courses on antenna theory and measurements. He has participated in projects related to high-order probe correction for spherical near-field antenna measurements. He is a coauthor of nine conference papers.

Olav Breinbjerg (M’87) was born in Silkeborg, Denmark, on July 16, 1961. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Technical University of Denmark (DTU), Kgs. Lyngby, in 1987 and 1992, respectively. Since 1991 he has been on the faculty of the Department of Electrical Engineering, DTU (formerly Ørsted·DTU, Department of Electromagnetic Systems, and Electromagnetics Institute) where he is now a Full Professor and Head of the Electromagnetic Systems Group including the DTU-ESA Spherical Near-Field Antenna Test Facility. He was a Visiting Scientist at Rome Laboratory, Hanscom Air Force Base, Massachusetts, in fall 1988 and a Fulbright Research Scholar at the University of Texas at Austin, in spring 1995. His research is generally in applied electromagnetics—and particularly in antennas, antenna measurements, computational techniques and scattering—for applications in wireless communication and sensing technologies. At present, his interests focus on metamaterials, antenna miniaturization, and spherical near-field antenna measurements. He is the author or coauthor of more than 40 journal papers, 100 conference papers, and 70 technical reports, and he has been, or is, the main supervisor of 10 Ph.D. projects. He has taught several B.Sc. and M.Sc. courses in the area of applied electromagnetic field theory on topics such as fundamental electromagnetics, analytical and computational electromagnetics, antennas, and antenna measurements at DTU, where he has also supervised more than 70 special courses and 30 M. Sc. final projects. Furthermore, he has given short courses at other European universities. He is currently the coordinating teacher at DTU for the 3rd semester course 31400 Electromagnetics, and the 7–9th semester courses 31428 Advanced Electromagnetics, 31430 Antennas, and 31435 Antenna Measurements in Radio Anechoic Chambers. Prof. Breinbjerg received a US Fulbright Research Award in 1995, the 2001 AEG Elektron Foundation’s Award in recognition of his research in applied electromagnetics, and the 2003 DTU Student Union’s Teacher of the Year Award for his course on electromagnetics.

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Resilience to Probe-Positioning Errors in Planar Phaseless Near-Field Measurements Seyyed Farhad Razavi, Student Member, IEEE, and Yahya Rahmat-Samii, Fellow, IEEE

Abstract—The phaseless techniques have been discussed in the antenna measurements community and the theories behind these techniques are well explained in literature. The issue of the noise and the presence of measurement errors are not investigated in details to provide strong impetus to the importance of phaseless measurements. In this paper the near-fields of a number of different types of antennas with high, medium and low side lobes are simulated to create some realistic cases. The probe positioning error effects are investigated by implementing random errors in the position of the simulated probe samples along different axes. A novel method is also adopted to incorporate the probe-height positioning error in an actual near-field measurement of an array antenna. It is also illustrated how the positioning errors can distort the phase distributions. Through detailed characterizations of the constructed far-field patterns, robustness of the Iterative Fourier technique even at the presence of very high probe positioning errors is demonstrated. It is shown how the utilization of a phaseless technique can significantly reduce the effects of probe positioning errors. The results are compared with the results extracted from the commonly used amplitude and phase near-field measurement techniques and the clear improvements are illustrated. Index Terms—Near-field measurements, phaseless measurements, probe-positioning error, random errors.

I. INTRODUCTION HE removal of the phase measurements in phaseless techniques has some immediate advantages over the common vectorial measurements. It makes the measurement systems cost effective, well-adapted for higher frequencies and insensitive to phase instabilities. The independency of phaseless techniques from the phase data not only reduces the cost of measurement apparatus but also could increase the accuracy of the measurements when large uncompensated random position errors are present. At higher frequencies a slight fluctuation of the probe over the sampling plane can cause a very high disturbance in the detected near-field phase. Consequently the typical amplitude-phase near-field measurements are susceptible to the probe positioning errors. In contrast in the phaseless measurements only the amplitude is detected which is much less sensible to these kinds of errors [1]. The quantitative analysis of errors in near-field is previously investigated [2], [3]. This paper will demonstrate how the susceptibility of near-field measurements to the probe positioning errors will be decreased if one employs

T

Manuscript received March 20, 2009; revised February 05, 2010; accepted February 16, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. The authors are with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095-1594 USA (e-mail: [email protected], www.ee.ucla.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.2010.2050421

a phaseless technique. The study of this error for a phaseless method through practical measurements is carried out using a novel approach. To the best of the authors’ knowledge this is for the first time that this error is studied for a phaseless technique through detailed simulations and practical measurements. The probe positioning error has been extensively studied in near-field measurements. There are many solutions for compensating this type of error [4], [5]. These methods rely on the exact knowledge of the value of error. This exact knowledge is acquired by employing some optical devices to track down this error. The incorporation of these devices will certainly increase the cost of measurement system. In the planar near-field measurements it is previously shown that small fluctuation of the probe along the direction normal to the plane of measurement is similar to the injection of some random phase error [6]. The effect of this error on the other directions is more complex. This fact results to the simple implication that the phaseless methods which do not rely on the phase of the near-field can potentially do much better against this type of error. The probe positioning error is among the most important types of error in near-filed measurements [7]. Therefore the introduction of techniques which are less susceptible to them is of outmost importance. It is the goal of this paper to demonstrate this fact through some simulations and moreover through the employment of actual measurements. The phaseless method used throughout this paper is the wellknown iterative Fourier technique (IFT) which is also known to the antenna engineering community as the plane-to-plane method [8], [9]. It is well-suited to the planar near-field measurements and has a fast response due to the usage of FFT. To study the effect of the probe-positioning error a uniform random error is injected to the position vector of the samples in the planes of measurements. A high level of probe fluctuations, in the order of wavelength, is considered to account for the most extreme conditions. The response of the IFT is investigated for an aperture antenna simulated by infinitesimal dipoles. To accommodate the response to different types of antennas, various amplitude tapering is employed to form antennas with different far-field patterns. It is shown that the IFT method can determine the true far-field with a notable accuracy even for the cases where the level of the error is extremely high and the typical amplitude-phase methods fail. In a recent conference paper by the authors [10] only a limited simulation results were presented without detailed discussions and measurement results. The paper is organized in the following manner. Section II is devoted to a very brief introduction of IFT. This short introduction is followed by modeling of the effect of the probe positioning error along x-, y- and z-axis. Section III compares the susceptibility of phase versus amplitude measurements to the

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probe positioning error for different types of the antennas. The responses of simulations for antenna patterns with low, medium and high side lobes are illustrated in this section. Section IV will discuss a novel approach for incorporating this type of error in an actual near-field measurement of an array antenna. The measurements are taken in the bi-polar near-field facility of UCLA [11]. A summary of the results is then presented in the last section. II. IFT AND THE PROBE POSITIONING ERROR There are a number of approaches to retrieve phase from phaseless data, and many of them have their roots in crystallography and optics. The existing solutions can be categorized under three different approaches. The first class of methods utilize the iterative Fourier techniques [8], [12], [13] which use the near-field amplitude. The second class is based on the minimization of a functional norm [14], [15] usually based on the square of the near-field amplitude. The third class employs the holographic techniques which uses a reference source in order to detect the phase of the measured amplitude-only data [7], [16]. Although one can potentially implement any of the proposed phase retrieval methods in the literature, a modified version of the iterative Fourier technique (IFT) [9] is used in this paper due to its fast convergence and appropriateness for the planar near-field measurements. In this method initially squared amplitude measurements are taken over two planes separated a few wavelengths apart and both in the near-field region of the antenna under test (AUT). These sets of measurements can be sampled in any format, such as rectangular, plane-polar, bi-polar, etc. Then for non-rectangular planar grid formats, through the deployment of the squared amplitude optimal sampling interpolation (OSI), the measurement data are transformed onto the rectangular grids [17]. Then an initial guess for the field distribution is generated on the AUT aperture plane. Due to simplicity and yet effectiveness of the initial guess developed in [9], and also the employment of a global search mechanism ensuring the best starting point, the following initial guess is used in this paper (1) where number,

is the beam direction, is the free space wave and are the coordinate system variables and is a rough estimation of aperture field amplitude, which is unity inside the aperture area and is zero elsewhere, is the initial guess on the aperture plane. The and which searching method will try to find that direction of minimizes the error norm mentioned in [9]. This initial guess is then propagated to the first plane using the plane-wave spectrum technique. There, the calculated phase distribution is retained while the amplitude distribution is replaced by the simulated amplitude data. The resulted field will be propagated to the second plane and the same procedure is repeated, i.e., the phase is retained and the amplitude distribution is replaced by the measured amplitude data. The process of the propagation between these two planes is continued until a convergence criterion on an error metric is achieved or the number

Fig. 1. Pictorial rendition of schematics for creating the random probe positioning errors on both near-field planes.

of the iteration reaches a predetermined maximum value. Assuming the convergence is attained, the calculated phase on any of two planes can be used to find the far-field patterns by standard planar near-field techniques [11], [18]. The effect of probe positioning error is simulated using the calculated exact near-field of an array of infinitesimal dipoles. Using the infinitesimal dipoles has the advantage of comparing the exact far-field with the extracted far-field of the measured near-field. The near field is computed over a 128 128 square grid with a sample spacing of 50 (slightly less than the Nyquist sampling rate) over two planes residing at distances of 5 and 7 from the dipoles plane (AUT plane). The following steps are taken for each point on the measurement planes to get the positioning error corrupted near-fields. a) Three uniformly distributed random numbers over an interval of are generated (if the effect of x- and y-axis positioning error is studied only two numbers are generated and for the case of z-axis positioning error only one random number is needed). b) The exact tangential electric near-field is calculated at the point using available formulas for the infinitesimal dipoles. c) The calculated near-field at point is assigned as the probe positioning error corrupted near-field at point . This methodology automatically captures both the amplitude and phase error caused by the probe positioning error. Fig. 1 provides a pictorial rendition of the simulation schematics. The probe positioning error corresponds to a total simulated fluctuation of 1.2 which is clearly a large displacement. Such a high error might not occur for a probe operating at the lower frequencies like microwave region but in millimeter wave region and beyond that, this error might model a very typical phenomenon. For example at 300 GHz this fluctuation correm which could easily occur. The effect of the sponds to probe-positioning errors is studied in this section for two cases. The z-axis errors are studied separately from x- and y-axis errors due to the differences in their nature. Although these two

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Fig. 2. Simulated amplitude [dB] and phase [deg] over the first measurement plane (a) without (b) with x- and y-axis and (c) z-axis random probe-positioning errors. Similar results were also generated for the second plane 6.

6

Fig. 3. Exact and corrupted far-field of a typical aperture antenna by implementing the x- and y-axis random probe positioning error.

Fig. 4. Exact and corrupted far-field of a typical aperture antenna by implementing only the z-axis random probe positioning error.

cases are studied separately here but next section will combine both of them to simulate a more complete scenario for an actual measurement. Fig. 2 illustrates the near-field amplitude and phase of a uniform square aperture antenna modeled by an array of 21 21 infinitesimal dipoles separated half a wavelength from each other. First the exact data [Fig. 2(a)] then the corrupted data implementing only the x- and y-axis probe-positioning error [Fig. 2(b)], and finally z-axis probe-positioning error [Fig. 2(c)] are shown. It is interesting to observe a few points. First, the positioning errors cause both amplitude and phase errors. For the simulated antenna, which is modeled as a uniform aperture, the phase does not have a high deviation along x and y directions. This is usually the case for the non-scanned beam antennas. This means that a probe fluctuation along these directions cannot have noticeable

effect on the phase. This fact can be seen comparing the phase in Fig. 2(a) and (b). Therefore the implementation of IFT might not have drastic improvements in comparison to the typical nearfield amplitude-phase methods where the phase is less distorted. This is clearly depicted in Fig. 3 where the extracted far-fields of these two methods are not too different from each other. In contrast the z-axis positioning error creates a dramatically high phase error. This suggests that the implementation of IFT can be advantages. The results shown in Fig. 4 confirm this prediction. Finally the x- and y-axis positioning error can potentially corrupt amplitude data more that the z-axis positioning error. Consequently, to show the robustness of IFT to the probe positioning error one is required to account for all of these errors. In the next section these errors are considered for antennas ranging from low to high side lobe levels and the effect of implementing IFT is proved to be quite advantageous.

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Fig. 5. The far-field pattern for antenna apertures with (a) 13.3 dB SLL uniform tapering, (b) 13.3 dB SLL Chebyshev tapering, (c) 20 dB SLL quadratic tapering, (d) 20 dB SLL Chebyshev tapering, (e) 30 dB SLL quadratic tapering, and (f) 30 dB SLL Chebyshev tapering. The solid black curves are exact far-fields whereas the blue dashed and red dotted are IFT and amplitude-phase method generated far-fields using corrupted data, with probe fluctuations along x-, y- and z-axis.

III. SIMULATING THE PROBE POSITIONING ERROR FOR DIFFERENT TYPE OF ANTENNAS The far fields generated based on the error corrupted nearfield data are investigated in this section. It is shown that the IFT method as a phaseless technique is much more robust to the probe positioning error. This is done by exploring six different types of antenna apertures. The antennas have low, medium and high side lobe levels (SLL). They also either have a monotonically decreasing SLL toward the grazing angles or equal-level SLL. The near-fields are calculated using the same technique discussed in earlier section in which the aperture of the antenna is modeled by an array of 21 21 infinitesimal dipoles sepafrom each other. The different behavior of the far-field rated is modeled by employing different types of aperture tapering such as Chebyshev and Quadratic [19]. No phase tapering is introduced, henceforth all the far-fields are broad-side directed.

The aperture is relatively large and therefore the modeled antennas have relatively high gain which is typical for the antennas measured in the planar near-field facilities. Based on the discussion in the previous section the simulated probe positioning errors are due to the probe fluctuations along x-, y- and also z-axis. The maximum displacement for all of these three directions is set to be which is extremely high. A. High Side Lobe Level Antennas This subsection will demonstrate the simulation results for typical antennas of very high side lobe levels. Two types of apertures are considered, a uniform and an aperture with a ChebydB. shev tapering corresponding to a side lobe level of The results are shown in Fig. 5(a) and (b) where the far-fields of these two antennas are calculated by three methods. The black solid curves represent the exact far-field corresponding

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to the near-field data generated on errorless probe positions. The red dotted curve is the result of the typical amplitude-phase near-field method using the data corrupted with the probe positioning error along x-, y- and z-axis. The blue dashed line is the result of the IFT method with the probe positioning error corrupted amplitude data on two planes. From the plots of the uniform aperture one can observe that the response of the IFT method as a phaseless method can predict the actual far-field much better than the amplitude-phase methods. The collected data over the near-field plane corresponds to a valid . Only angle of 70 which corresponds to a (y-z) cut is shown as a representative pattern cut. The IFT response has been slightly better for the case where the side lobes are very high near the grazing angle [Fig. 5(b)]. Nonetheless it still outperforms the amplitude-phase method. B. Medium Side Lobe Level Antennas This subsection will focus on the antenna patterns with medium side lobe level. The simulation procedure is similar to the previous subsection. The only difference is that for simulating a pattern with decreasing side lobe levels-which is a condition usually occurring with high gain antennas (like reflectors) suitable for planar near-field measurements—a quadratic tapering is introduced [19]. The other pattern is generated similar to the Chebyshev tapering in previous subsection with the difference that the infinitesimal dipoles amplitude tapering dB. corresponds to a medium side lobe level of The results are shown in the Fig. 5(c) and (d) where the robustness of the IFT method is again confirmed. One can pay attention to the fact that the common amplitude and phase nearfield method is quite inaccurate in terms of a useful estimation of the far-field pattern, whereas the IFT method proves itself as a viable solution for coping with the probe positioning errors. In comparison to the high side lobe case the inaccuracy of amplitude-phase methods has even become worse. C. Low Side Lobe Level Antennas The approach is similar to the previous subsections and with the difference of decreasing the side lobe levels to dB [Fig. 5(e) and (f)]. The observations are basically similar to the previous subsections. For the quadratic tapering the far-field features are constructed again with very good accuracy up to the side lobes which are not very low. The highly low side lobes of the pattern are not constructed as accurately as the main beam and the first side lobes. We will show in the next subsection that dB for this the amplitude error standard deviation is about case which makes it difficult to extract any features of the pattern lower than this value, although it might be foreseen to use the method developed in [14] to improve the far-field patterns in this case. That method is based on the progressive enlargement of the reconstructed far-field beam span around its maximum, in another word the far-field around the main beam is first reconstructed and then the rest of the pattern is progressively extracted. In comparison to the medium and low SLL of Chebyshev distributions studied in the previous subsection here the deviation of the amplitude-phase method from the correct far-field is even more noticeable [Fig. 5(f)].

D. Quantitative Analysis of Errors The analysis of the IFT response in comparison to the amplitude-phase method up until now had been merely based on the observation of the far-field patterns. This section will provide a more quantitative analysis of the probe positioning errors. The errors as discussed before are divided into two main category of normal (z-axis) and lateral (x- and y-axis). In order to quantitatively analyze the errors; among various choices the following two error metrics, which are directly applied to amplitude and phase of each point, are selected (2)

and are the exact and error corrupted electric in which near-field at each sampling point, and are the angle and abis the maxsolute value operators, respectively, and imum value of exact near-field amplitude in the plane of measurement. It is worth mentioning that the phase of the near-field to whereas the amis a bounded number between plitude of the near-field can change based on how much power is coupled into the antenna. This means that the amplitude level in the near-field for similar antennas can be drastically different based on the excitation of the antenna where as the phase variation always stays the same. That is why the phase difference metric is not normalized whereas the amplitude is normalized. Tables I and II show the calculated statistical parameters when only normal or lateral probe-positioning errors are present. The statistical numbers in Table I are based on the average of 10 different randomly generated errors for each case. This means, for example in the uniform aperture case 10 different uniform random probe positioning error with the is generated. For each case the statistical maximum of parameters are calculated and their average is what is tabulated here. The first column of Table I has enlisted the maximum error occurred in the near-field amplitude after applying the probe positioning error. It can be seen that the level of error can be very high pointwise. This does not have significant effect on the calculated far-fields (Fig. 5), because the far-field is constructed from the collective effect of near-field samples and as shown in the third column of Table I the effective level of error (the standard deviation) is much less. Moreover the maximum phase error for both lateral and normal cases are almost 360 . This means that the probe-positioning effect is so high that it could change the phase 360 degrees which could be predicted by the probe fluctuations. A phase mean error close to zero for all the studied cases also suggests that the phase error created by the probe positioning error is completely randomly distributed around zero. For the quadratic [30 dB SLL] tapering, as a representative example, the standard deviation of amplitude error is very low for the normal error distribution (Table I); that is why the error level is merely determined by the lateral amplitude error (Table II) which is dB. This justifies the corruption of sidelobes beyond dB in Fig. 5(e).

RAZAVI AND RAHMAT-SAMII: RESILIENCE TO PROBE-POSITIONING ERRORS IN PLANAR PHASELESS NEAR-FIELD MEASUREMENTS

TABLE I STATISTICS OF NEAR-FIELD ERROR CORRESPONDING TO PROBE POSITIONING ERROR OF

TABLE II STATISTICS OF NEAR-FIELD ERROR CORRESPONDING TO PROBE POSITIONING ERROR OF

60 6 :

60 6 :





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ALONG Z-AXIS

ALONG X- AND Y-AXIS

In general the amplitude errors in Table II are higher than Table I this might suggest that IFT will not be that advantageous in comparison to the typical amplitude-phase method. Fig. 3 confirms this assessment. Nevertheless the simulations shown in Fig. 5 prove that, although the inclusion of x- and y-axis probepositioning error increases the amplitude errors in the near-field data, IFT can still predict the actual far-field much better than the regular methods when all these three types (x-, y- and z-axis) of error coexist. Therefore it can be inferred that z-axis probe positioning error has the most dramatic effect on the amplitudephase measurement results. IV. MEASUREMENTS FOR INCORPORATING THE PROBE-POSITIONING ERROR The typical antenna chambers, specifically at microwave frequencies do not have a very high level of probe-positioning error. Consequently the realization of this type of error at these frequencies is not trivial. A novel approach is adopted to create this type of error in UCLA planar bi-polar near-field facility [11]. Instead of one set of measurement for each plane, four sets of amplitude-only planar measurements are taken. The

Fig. 6. The elliptically-shaped slot array antenna operating at 9.3 GHz in UCLA bi-polar planar near-field facility. Four measurements about the planes I and II were taken to create probe positioning errors.

measurements are performed over an elliptically-shaped slot array antenna operating at 9.3 GHz at distances of 7(13/16),

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Fig. 7. Measured amplitude and phase over the second plane in the UCLA bi-polar facility; (a),(b) without implementing the probe positioning error, (c),(d) with probe-positioning error using amplitude and phase data, (e),(f) with probe-positioning error using only amplitude data and (g),(h) the extracted fields after implementing IFT over the phaseless data. One can note the clear effect of the error on the phase data. The positioning inaccuracy is 0:4 .

6

8(9/80), 8(49/80) and 8(13/16) inches for the first plane and 9(13/16), 10(1/16), 10(5/16), 10(13/16) inches for the second plane (Fig. 6). Then it is assumed that the actual planes of measurements where located at 8(9/80) and 10(1/16) inches (6.3878 and 7.9232 ) apart from the aperture of the antenna under the test.

The bi-polar samples are taken over concentric rings and each ring has a certain number of samples. The probe-positioning error at each point on the bi-polar grid is created by randomly selecting the corresponding measured data from any of the four sets of measurements. Although the number of possible heights are limited in this case (only four), the randomness of data selec-

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Fig. 8. The exact (solid black), amplitude-phase with incorporated measurement errors (dotted) and phaseless with incorporated measurement errors (dashed) reconstructed far-field patterns for (a) 8 = 0 (H-plane) cut and (b) 8 = 90 (E-plane) cut.

tion can create a high level of probe positioning error. This error is along z-axis which based on the discussions on the previous sections is the most important one among the probe-positioning errors in planar near-field measurements. It can be seen from Fig. 6 that each plane has an error level of maximum of 1 in which at the frequency of operation stands which again is a very high level of inaccufor 0.8 racy. The effect of this error is depicted in Fig. 7 where the near-field data are shown for the second plane of measurements (which is 10(1/16) inches above the AUT plane). Fig. 7(a) and (b) are the exact near-field amplitude and phase while no z-axis probe positioning error is injected using the discussed method. The near-field distribution after the implementing the errors are shown in Fig. 7(c) and (d). It can be seen that the phase data is completely corrupted while the amplitude is less affected. The data for this measurement is taken over the bi-polar grid, consequently one should use an appropriate interpolation (OSI) to find the amplitude and phase data over the rectangular grid [17]. OSI is applied over the real and imaginary parts of the measured bi-polar data and consequently this method will spread the high phase error inaccuracies of the bi-polar grid into the amplitude data inaccuracies over the rectangular grid [Fig. 7(c)]. That is why the amplitude error is much higher than the typical simulations discussed in the earlier sections [Fig. 2(c)]. This fact suggests that the typical amplitude-phase methods are even more vulnerable to the probe positioning error when they are applied over non-rectangular grids. Fig. 7(e) shows the transferred error-corrupted amplitude data from the bi-polar to the rectangular grid. One can clearly observe that the amplitude data are less affected in comparison to the amplitude-phase method used in Fig. 7(c), due to the fact that no phase data is used in the phaseless methods. The reconstructed near-field amplitude and phase using the IFT method are depicted in Fig. 7(g) and (h). Finally, Fig. 8 shows different cuts of the far-field patterns for exact (amplitude-phase without error), amplitude-phase with error and phaseless cases. It is clear that the phaseless method (IFT) is able to cope more effectively with the probe height positioning error even at the presence of

a very high level of probe positioning error created by our error incorporation approach. V. CONCLUSION This paper has focused on the study of probe positioning errors in planar near-field measurement techniques. It is shown that the amplitude data are less vulnerable to the probe positioning error in comparison to the phase data. Based on this fact and using the IFT as a phaseless method, the far-fields of a number of different antennas with low, medium and high side lobe levels are extracted at the presence of high level of probe positioning errors. This is in contrast to the common amplitude and phase methods for which the presence of this type of error is very destructive. The results extracted from the simulations are also confirmed through an actual incorporation of z-axis probe positioning error in bi-polar planar near-field measurement of a slot array antenna. The results clearly demonstrate the resiliency of the phaseless near-field measurements. REFERENCES [1] R. G. Yaccarino and Y. Rahmat-Samii, “Progress in phaseless nearfield antenna measurement research at the University of California, Los angeles,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 8–13, 2001, vol. 4, pp. 416–419. [2] A. C. Newell, “Error analysis techniques for planar near-field measurements,” IEEE Trans. Antennas Propag., vol. 36, no. 6, pp. 754–768, June 1988. [3] H. Hojo and Y. Rahmat-Samii, “Error analysis for bi-polar near-field measurement technique,” in Proc. AP-S Antennas and Propagation Society Int. Symp. Digest, June 24–28, 1991, pp. 1442–1445. [4] L. Corey and E. Joy, “Far-field antenna pattern calculation from nearfield measurements including compensation for probe positioning errors,” in Proc. Antennas and Propagation Society Int. Symp., June 1979, vol. 17, pp. 736–739. [5] O. M. Bucci, G. Schirinzi, and G. Leone, “A compensation technique for positioning errors in planar near-field measurements,” IEEE Trans. Antennas Propag., vol. 36, no. 8, pp. 1167–1172, Aug. 1988. [6] J. Romeu, P. Escobar, and S. Blanch, “Probe positioning errors in planar near field measurements. a plane wave synthesis approach,” in Proc. AP-S Antennas and Propagation Society Int. Symp. Digest, Jun. 18–23, 1995, vol. 1, pp. 264–267. [7] V. Schejbal, J. Pidanic, V. Kovarik, and D. Cermak, “Accuracy analyses of synthesized-reference-wave holography for determining antenna radiation characteristics,” IEEE Antennas Propag. Mag., vol. 50, no. 6, pp. 89–98, Dec. 2008.

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[8] A. P. Anderson and S. Sali, “New possibilities for phaseless microwave diagnostics. part1: Error reduction techniques,” Inst. Elect. Eng. Proc. H, Microw. Antennas Propag., vol. 132, no. 5, pp. 291–298, Aug. 1985. [9] S. F. Razavi and Y. Rahmat-Samii, “A new look at phaseless planar near-field measurements: Limitations, simulations, measurements, and a hybrid solution,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 170–178, Apr. 2007. [10] S. F. Razavi and Y. Rahmat-Samii, “On the robustness of planar phaseless near-field measurements to probe positioning errors,” presented at the AMTA, Nov. 2008. [11] Y. Rahmat-Samii, L. I. Williams, and R. G. Yaccarino, “The ucla bi-polar planar-near-field antenna-measurement and diagnostics range,” IEEE Antennas Propag. Mag., vol. 37, no. 6, pp. 16–35, Dec. 1995. [12] D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D: Appl. Phys., vol. 6, pp. L6–L9, 1973. [13] J. R. Fienup, “Phase retrieval algorithms: A comparison,” Appl. Opt., vol. 21, no. 15, pp. 2758–2769, Aug. 1982. [14] T. Isernia, G. Leone, and R. Pierri, “Radiation pattern evaluation from near-field intensities on planes,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 701–, May 1996. [15] F. Soldovieri, A. Liseno, G. D’Elia, and R. Pierri, “Global convergence of phase retrieval by quadratic approach,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3135–3141, Oct. 2005. [16] M. P. Leach, D. Smith, and S. P. Skobelev, “A modified holographic technique for planar near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. 56, no. 10, pp. 3342–3345, Oct. 2008. [17] O. M. Bucci, C. Gennarelli, and C. Savarese, “Fast and accurate near-field-far-field transformation by sampling interpolation of plane-polar measurements,” IEEE Trans. Antennas Propag., vol. 39, no. 1, pp. 48–55, Jan. 1991. [18] A. Yaghjian, “An overview of near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. 34, no. 1, pp. 30–45, Jan. 1986. [19] Y. Rahmat-Samii, , Y. T. Lo and S. W. Lee, Eds., “Reflector antennas,” in Antenna Handbook. New York: Van Nostrand Reinhold, 1988, ch. 15.

Seyyed Farhad Razavi (S’04) was born in Tehran, Iran. He received the B.S.E.E. degree from Sharif University of Technology, Tehran and the M.S.E.E. degree from the University of Tehran, in 2001 and 2003, respectively. He is currently working toward the Ph.D. degree at the University of California, Los Angeles (UCLA). He has been working in the Antenna Research, Analysis, and Measurement Laboratory, UCLA under the supervision of Prof. Y. Rahmat-Samii since January 2005. His research interests are in antenna design and advanced antenna measurement techniques including phaseless measurements. Mr. Razavi is the recipient of the Best AMTA Student Paper Award for three consecutive years (2006–2008) and was awarded the Best B.Sc. Project from Sharif University of Technology sponsored by the Schlumberger Institute.

Yahya Rahmat-Samii (S’73-M’75-SM’79-F’85) received the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois, UrbanaChampaign. He is a Distinguished Professor, holder of the Northrop Grumman Chair in Electromagnetics, and past Chairman of the Electrical Engineering Department, University of California, Los Angeles (UCLA). He was a Senior Research Scientist with the National Aeronautics and Space Administration (NASA) Jet Propulsion Laboratory (JPL), California Institute of Technology prior to joining UCLA in 1989. In summer 1986, he was a Guest Professor with the Technical University of Denmark (TUD). He has also been a consultant to numerous aerospace and wireless companies. He has been Editor and Guest Editor of numerous technical journals and books. He has authored and coauthored over 800 technical journal and conference papers and has written 30 book chapters. He is a coauthor of Electromagnetic Band Gap Structures in Antenna Engineering (New York: Cambridge, 2009), Implanted Antennas in Medical Wireless Communications (Morgan & Claypool Publishers, 2006), Electromagnetic Optimization by Genetic Algorithms (New York: Wiley, 1999), and Impedance Boundary Conditions in Electromagnetics (New York: Taylor & Francis, 1995). He has received several patents. He has had pioneering research contributions in diverse areas of electromagnetics, antennas, measurement and diagnostics techniques, numerical and asymptotic methods, satellite and personal communications, human/antenna interactions, frequency selective surfaces, electromagnetic band-gap structures, applications of the genetic algorithms and particle swarm optimization, etc., (visit http://www. antlab.ee.ucla.edu/). Dr. Rahmat-Samii is a Fellow of the Institute of Advances in Engineering (IAE) and a member of Commissions A, B, J and K of USNC/URSI, the Antenna Measurement Techniques Association (AMTA), Sigma Xi, Eta Kappa Nu and the Electromagnetics Academy. He was Vice-President and President of the IEEE Antennas and Propagation Society in 1994 and 1995, respectively. He was appointed an IEEE AP-S Distinguished Lecturer and presented lectures internationally. He was a member of the Strategic Planning and Review Committee (SPARC) of the IEEE. He was the IEEE AP-S Los Angeles Chapter Chairman (1987–1989); his chapter won the best chapter awards in two consecutive years. He is listed in Who’s Who in America, Who’s Who in Frontiers of Science and Technology and Who’s Who in Engineering. He has been the plenary and millennium session speaker at numerous national and international symposia. He has been the organizer and presenter of many successful short courses worldwide. He was a Vice President of AMTA for three years. He has been Chairman and Co-Chairman of several national and international symposia. He was a member of the University of California at Los Angeles (UCLA) Graduate council for three years. For his contributions, he has received numerous NASA and JPL Certificates of Recognition. In 1984, he received the Henry Booker Award from URSI, which is given triennially to the most outstanding young radio scientist in North America. Since 1987, he has been designated every three years as one of the Academy of Science’s Research Council Representatives to the URSI General Assemblies held in various parts of the world. He was also invited speaker to address the URSI 75th anniversary in Belgium. In 1992 and 1995, he received the Best Application Paper Prize Award (Wheeler Award) for papers published in 1991 and 1993 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. In 1999, he received the University of Illinois ECE Distinguished Alumni Award. In 2000, he received the IEEE Third Millennium Medal and the AMTA Distinguished Achievement Award. In 2001, he received an Honorary Doctorate in physics from the University of Santiago de Compostela, Spain. In 2001, he became a Foreign Member of the Royal Flemish Academy of Belgium for Science and the Arts. In 2002, he received the Technical Excellence Award from JPL. He received the 2005 URSI Booker Gold Medal presented at the URSI General Assembly. He is the recipient of the 2007 Chen-To Tai Distinguished Educator Award of the IEEE Antennas and Propagation Society. In 2008, he was elected to the membership of the National Academy of Engineering (NAE). In 2009, he was selected to receive the IEEE Antennas and Propagation Society highest award, Distinguished Achievement Award, for his outstanding career contributions. He is the designer of the IEEE Antennas and Propagation Society (IEEE AP-S) logo which is displayed on all IEEE AP-S publications.

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FDTD Discrete Planewave (FDTD-DPW) Formulation for a Perfectly Matched Source in TFSF Simulations Tengmeng Tan, Member, IEEE, and Mike Potter, Member, IEEE

Abstract—A technique is proposed for the generation of planewaves in the total-field scattered-field (TFSF) formulation of the FDTD method. The method is developed using the 1D properties of a planewave and optimized projection of the 3D finite difference operators to the 1D domain. The result is an efficient and accurate planewave source that can be propagated on six 1D grids concurrent with the main simulation, and that is perfectly matched to the main 2D/3D FDTD domain for any source function. Numerical simulations show that the technique is valid for any angle of propagation, and for any gridcell aspect ratio, with non-physical reflections in the scattered field domain on the order of machine precision ( 300 dB). Index Terms—Finite-difference time-domain (FDTD) methods.

I. INTRODUCTION

M

ANY practical problems of interest in electromagnetics involve the use of planewave sources. For example, one often needs to calculate or predict the radar cross-section (RCS) of an object where it is assumed that the source is far away, allowing the use of the planewave approximation. Planewaves are also often used for buried object detection in multi-layer structures. With the rapid growth of computing power, this also means that proper planewave sources are also of great interest in the electromagnetics simulation community and industry. In the finite-difference time-domain (FDTD) method, scattering problems are most often simulated by using the total-field/scattered-field (TFSF) formulation outlined in [1]. The computational domain is separated into two regions: the total field (TF) region where the scattering object, and both incident and scattered fields reside; and the scattered field (SF) region, where only the scattered fields—the output of interest—reside. An absorbing boundary condition simulates the open problem, and serves to cancel unwanted numerical reflections from the scattered field which would corrupt the solution and hence reduce the dynamic range. With the advent of PMLs [2], these reflections can be reduced to practically any desired level of accuracy. Manuscript received September 16, 2009; revised January 29, 2010; accepted February 17, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by the Natural Science and Engineering Research Council (NSERC) of Canada. The authors are with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050446

For the most accurate possible solution, ideally the planewave source would also not corrupt the outgoing scattered field. The planewave sources in the TFSF formulation are introduced at the Huygens’ surface to account for the discontinuity between the TF and SF regions. Depending on the formulation used for this source, errors are introduced as a result of interpolation and/or numerical dispersion mismatches (note: for the remainder of this manuscript we use ’dispersion’ to refer to numerical dispersion, unless otherwise stated). An efficient time-domain technique is outlined in [1] whereby an auxiliary 1D incident field array (IFA) is used to propagate a planewave source concurrent with the main simulation domain, which is used as a lookup table to populate the Huygens’ surface. Unfortunately, because of interpolation and dispersion mismatches, the scattered field is corrupted by non-physical reflections at the Huygens’ surface. Improvements to the IFA can be made with signal processing techniques [3], or by optimizing (minimizing) the dispersion mismatch (MND) [4]. Combining these techniques can amount incident field isolation in general. to approximately Other researchers have recently modified the IFA formulation in order to make it particularly applicable to problems involving multilayer structures for buried object simulation [5], [6]. The limiting factor in most of these applications is the numerical dispersion mismatch between the 1D IFA grid and the 2D/3D main grid. In [7], [8], the numerical planewave source function is constructed in the frequency domain directly from the dispersion relationship. In addition, the frequency dependent polarization projection is included in the frequency domain propagator to properly account for nonorthogonality of the fields and wave vector in the FDTD domain. Both papers reported the procedure for 2D problems, and the method became known as the Analytic Field Propagator (AFP) in the sense that the propagator is analytically known and interpolation errors are virtually nonexistent. Though able to demonstrate a virtually reflectionless source with of incident field isolation, the method is not nearly as efficient as the IFA techniques because source fields must be stored for the entire Huygens’ surface in preprocessing. Perhaps due to this inefficiency, the method has not been extended to a 3D formulation, thus somewhat limiting its application. To summarize, IFA techniques are efficient but inaccurate, whereas AFP techniques are accurate but inefficient. We have recently reported on two techniques that improve upon both. In [9], [10] we introduced the 1D-multipoint auxiliary propagator (1D-MAP), which described an efficient perfectly matched FDTD planewave source in the time domain derived directly

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from the dispersion equation for 2D problems, achieving machine level field isolation. Using the nature of numerical planewaves reported in [11], we reported on an optimized AFP (O-AFP) technique which maintains the accuracy but makes it much more efficient and extends its applicability to 3D [12], [13]. A similar methodology was reported in [14], and a fully time-domain version was briefly discussed in [15]. Recognizing the applicability of the method to other problems, Hadi recently reported [16] on an extension of the 1D-MAP methodology which uses Maxwell’s equations in their split-field form for the propagator (as opposed to the wave equation used in [10]). By doing this, he was able to demonstrate that the theory is applicable to non-standard FDTD schemes, albeit still in 2D implementations. He also highlighted the need for precise wave initiation within the auxiliary grid in order to capitalize on the benefits of the reduced levels of scattered fields. Our intention is now to report upon a formulation for planewave sources in the FDTD method that is: i) in the time domain in the 1D auxiliary grids; ii) capable of machine level field isolation in the SF region; iii) efficient and relatively straightforward in implementation; and iv) fully implementable in 3D simulations. Furthermore, the framework is set for extension to other non-standard FDTD methods. The formulation of the method is presented in Section II, including the nuances to which one must pay attention to the hard sources used as the boundary conditions in the 1D grids. Section III provides numerical examples of the technique in practice, which is followed by discussion and conclusions. The end result is a formulation that emulates exactly the propagation of planewaves at any angle, of any spectrum, in a main 2D/3D FDTD grid of any aspect ratio. II. FORMULATION As stated in [11], a true numerical planewave can be derived simply by evoking the defining properties of a planewave. Consider a field referenced by the position vector in 3D space. Since a planewave is inherently a one-dimensional object, one can index the exact . That is to say the electric field same field by and magnetic field , where the subspace projection is used and the azimuthal and define the propagation direction and polar angles . The vector field equalities stem from the fact that fields lying on a common wavefront must by definition have identical values. By using the chain-rule operation for partial derivatives, where the subscript for Stratton in [17] showed that Maxwell’s curl equations in a homogeneous electromagnetic medium

(1) for a planewave reduce to

(2)

are respectively the electric permittivity, where , , , and magnetic permeability, and electric and magnetic conductivities. Note that the curl operator since the subspace is a linear function of . Since the subspace projection is valid for all , it is also clear that a and thus spatial translation implies and . This translational invariance will greatly simplify the procedure for constructing a numerical planewave, particularly when finite difference schemes are of interest. Notice also that no temporal transformation is needed between the two coordinate systems. A. Finite Difference Scheme: FDTD Planewaves Constructing a numerical planewave in subspace that is perfectly equivalent to that in can also be obtained from the chain-rule operation. To this end, let and numerical operators denote the finite difthe tilde ference approximation of the partial derivative operators in the and it should converge to respective spaces; then . Let denote a its continuous counterpart when be a perturbation vector defined by field component and , , and . In order to have equivalent planewaves in both spaces, it is re. Keep in mind that for the quired that is given (by the difference problem at hand the LHS scheme used in the main simulation space) and only the RHS needs to be determined. Since we are initially interested in standard FDTD schemes, we concentrate on the central difference scheme where by definition

(3) The perturbation in must result in . The portions of the operator

are

and so

(4) the above forNotice that at the infinitesimal limit mulation indeed converges to its continuous counterpart. One could also arrive at (4) by applying the translational property to obtain the final directly result for . It is not hard to see that the simplicity of this direct substitution also carries through to any higher order finite difference approximation (e.g., non-standard FDTD schemes), because field values anywhere on the same . planar wavefront must satisfy We now utilize (3) and (4) to construct an FDTD planewave that is equivalent in both coordinate systems.

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For example, consider one component of Maxwell’s (1). The FDTD counterpart of this loop using the convention taken is understood to be from [1] where the subscript is given by

(5) Since is discretized by , and introducing , the subspace projection evaluates to the notation . Making use of the translational invariance, the equivalent expression for the above in the subspace is then

(6) Again, notice that the temporal discretization remains unchanged for both systems, which is important for setting up planewave sources. The geometric picture of this projection of coordinates is shown in Fig. 1, where a single Yee voxel is shown in perspective view and how it is related to field points in the auxiliary grid (the -grid) which is oriented in some arbitrary propagation direction. The particular field components that are used in (5) and (6) are indicated on the auxiliary grid. It is extremely important to recognize that the fields indicated are only for a single voxel; field components from other voxels in the main grid are also present in the auxiliary grid (though not shown here because their projected locations could be anywhere). The spacing between fields in the -grid for those consecutive fields from the shown voxel is also indicated, which will be discussed a little later. The key point to learn from this figure is that both (5) and (6) are computing exactly the same fields, and the only difference between these equations is in the grids that host their field values. The other five FDTD equations akin to (6) are obtained exactly the same way. For the space in and the space in to host identical planewaves, the formulation is complete once all six update equations are found, thus giving the basis for constructing a perfectly matched planewave source in points in the the time domain. However, there are space, where refers to the number of spatial points along one dimension of the main FDTD grid.

Fig. 1. Perspective view of the fields of a single Yee voxel, and how they are related to locations in the auxiliary source grid (r -grid). Each side of the voxel is oriented parallel with one of the Cartesian axes used to describe the main computational grid. The r -grid is oriented in an arbitrary direction specified by the propagation angles  and .

Projecting all these points onto the subspace also results in points, which would be inefficient to calculate during runtime on an auxiliary 1D grid. One more complication is that , which means that the spacing in general for the grid would become very non-uniform. With no other constraints imposed, the auxiliary grid would consist of all six field components, non-uniformly spaced with no apparent relation amongst them. Such non-uniformity makes the relation among all six components very difficult to evaluate efficiently. However, in [11] we showed that there exists a 1D auxiliary which is obtained from the ragrid with uniform spacing , where is tional angle condition an integer; in our present case this results in a uniform grid for each field component. These six uniform grids also host the proin the main grid, jection of every point which are related by , where is an integer. The advantages of adopting the uniform grid are twofold: (i) the spatial points and (ii) in the auxiliary grid reduce to the update equations become much simpler to process. , the update (6) simpliUsing the uniform grid fies to

(7) where the constants and are the updating coefficients identical to the standard FDTD values. Evidently, this planewave update equation is directly available from a standard FDTD update equation. For exis directly obtained ample, the field from

the

standard

FDTD

because , and the other components are obtained exactly the same way. Similar to the convention adopted in (5), an index is also understood to be .

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If the constant coefficients

, are defined for the magnetic material, then the full set of equations for a homogeneous FDTD planewave evaluated along the grid can be compactly written as

(8)

(9) where the addition in the subscripts is understood to be performed cyclically, i.e., , , and so forth. This set of six coupled equations allows one to efficiently propagate a planewave which is identical to one that would propagate with the standard 3D FDTD update equations at a particular angle. Although there appear to be half-integer values in the spatial subscript, this is only a result of individual field components being staggered in space in the main grid; in effect the entire 1D grid for each field component is staggered, so the indices can still be referenced logically. Fig. 2 diagramatically illustrates the resulting six auxiliary grids, and their relationship to each other. The computational stencil for (7) is also indicated. A combination of the six grids, superimposed upon each other, results in the corresponding geometric interpretation of how to relate these components to the Yee voxel, as , and indicated in Fig. 1. Note that each grid has a spacing of they are geometrically offset from each other because of how the Yee fields are uncollocated. In Fig. 1, one can see the spacing between consecutive field components in the main grid, which of . The first several nodes of some is an integer multiple of these auxiliary grids are used for initiating the plane wave via hard-sourced field components. It is important to recognize that this planewave equation is are independent valid only for a region where , , , and , , and of , which implies that the coefficients , are independent of within the homogeneous region. This does not imply that a planewave formulation cannot be extended to inhomogeneous materials. For example, if the inhomogeneity is distributed as a planar layer where the electromagnetic properties are piecewise-constant within the planar region, then a planewave can still exist in each planar region [17]. This means that (8)–(9) are valid within each planar region. A matching boundary condition at the interfaces must be applied to construct a solution which is valid for all regions. The solution in general

Fig. 2. The interrelationship between the grids hosting each of the six field components is indicated here. Each grid has a uniform spacing of r , but they are geometrically offset from each other. The fields and stencil used in the update equations for a particular E field location are also indicated. The first several nodes of some of these auxiliary grids are used for initiating the plane wave via hard-sourced field components.

1

contains a superposition of incident, reflected and transmitted planewaves. This boundary matching principle is in fact the key idea behind the techniques developed in [5], [6], or in [7], [18] where the frequency domain formulation tends to simplify the boundary analysis. This paper will not address the boundary matching associated with planar layers, but this is a natural extension of the work to be presented in the future. A few words regarding the nature of the 1D-MAP methodology as discussed in [10] may also help to understand the limitation of standard IFA techniques. Specifically, each time-step a field component in the FDTD grid is updated using four neighboring field points. Like the 1D-MAP methodology, in one time step (8)–(9) also evaluate four points but with a spatial sampling in general. On the other hand, the IFA scheme [1] including the MND procedure [4] only utilize two neighboring points in each time-step. Since the MND technique and/or ultimately amounts to modifying the grid spacing the stability factor , it is conceptually concise to state that an improved IFA scheme effectively has somehow averaged out the four spatial points into two by finding the best operating and that result in a close approximation of the multipoint FDTD planewave (8)–(9). Unfortunately, the IFA only computes one component of the electric and magnetic field, and therefore a field interpolation is required to obtain the six-uncollocated fields. Once this interpolation is employed then the benefits of MND (constructed for ) are subsequently lost, because the desired fields a specific points. In other words, are not evaluated at the discrete the field values after passing through the interpolation routine

TAN AND POTTER: FDTD-DPW FORMULATION FOR A PERFECTLY MATCHED SOURCE IN TFSF SIMULATIONS

are no longer exhibiting a minimum dispersion error—the MND but not anywhere in between. Furtheris valid exactly at more, all of the IFA techniques fail to account for the frequency dependent nature of the polarization projection. Conversely, the integer mapping proposed here guarantees that every field component and location have a value directly available from the 1D grid which has a dispersion relation identical to that of the 2D/3D-FDTD domain [11]. The storage (memory) cost of the proposed per field component, and method is in the there are six field components, compared with IFA method per field component where there are two field components. The computational cost compares in an analogous way (because there are more fields to calculate), and there are about three times as many flops per field component in the iteration in the proposed method compared to the IFA. However, at every time step the IFA requires an interpolation for every field component on the Huygens’ surface (flops depend on implementation)—something that is not required at all in the proposed method. Furthermore, the IFA requires a projection operation as part of the interpolation operation—again this issue does not exist in the present formulation. Having said all that, the main computational grid has computations . So for example in a main grid of size that scale at ), both plane wave source 200 200 200 (i.e., methods only contribute approximately 0.0025%–0.06% to the overall computational and storage requirements; this is practically negligible from a computing perspective. In contrast to the O-AFP method [13], the proposed method is evaluated completely in the time-domain and as such there is no requirement whatsoever for pre-processing and storage of the time evolution of the source. It is interesting to note that both methods will calculate identical fields (given the same reference source) for the same simulation parameters; the only difference is the domain in which fields are evaluated. This is because they share identical dispersion relationships, and are based on the same rational angle condition (a more detailed discussion on the implications of computational efficiency of both methods for arbitrary angles can be found in [11]). The drawback is that there could be points in the auxiliary grid which are not used to populate the Huygens’ surface, but needed to properly propagate the wave. In short (8)–(9) will always produce a planewave that is perfectly matched between the auxiliary 1D grids and any 2D/3D FDTD grids regardless of the nature of its source spectrum. For this reason, we call this solution the FDTD discrete planewave (FDTD-DPW). B. Boundary Conditions Another feature that sets the FDTD-DPW apart from IFA is in the source excitations or boundary points. The IFA technique uses a single source point as a reference to initiate a planewave (via hard-sourcing), because specifying one field at one location is sufficient to determine all and fields at any other locations. The FDTD-DPW scheme consists of six fields which are both coupled and uncollocated. It is therefore necessary to excite more than one point or more than one component in order to correctly launch a planewave in the FDTD-DPW scheme. Before discussing possibilities for hard-sourcing these points, we wish

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to point out that the choice of what method to use has no effect whatsoever on the level of the mismatch at the TFSF boundary: no matter what method is used, the implementation discussed in the previous section will guarantee that the field isolation is at machine precision. Ultimately the choice for the user is dictated by how closely one wants or needs to emulate the actual physical system and source that one is simulating. In the examples that follow, we choose to use two methods: i) the O-AFP for sources; and ii) an analytic expression for the source. Compared with fields in the continuous domain, FDTD field components are not collocated in space and time. For example, the functional form for the electric field update equations can be where compactly written as both and . This cyclic relation shows that in order to uniquely define its update equations, an FDTD planewave in general demands three boundary components which can be or combinations. Like in the contintaken from any of uous domain, these boundary components also need to satisfy impedance and polarization projection relations, and these relations in general should also include numerical artifacts to maintain consistency between the discrete and spaces. One further difference for the discrete planewave is that a field at a reference point will actually propagate in one single time-step (a field at propagates exactly one unit cell for all ). As a result, the maximum distance when projected on to the subspace is given by . The implication is that one boundary point or layer in general is insufficient to define the FDTD planewave uniquely since the source points within to become undefined. Specifically, by introducing the uniform grid through the rational angle condition , the maximum boundary points become . One way to populate these boundary sources, and the method that was used in [16], is to use the AFP technique [8] that was optimized in [12], [13]. This method ensures that there is consistency in terms of numerical dispersion, and there is only a very small region that needs to be pre-processed to obtain a time signature. For example, all three fields could be hard-sourced using the AFP to properly initiate the planewave. Care must be taken in bandlimiting the source, as discussed in [13], [16]. A convenient and somewhat simpler alternative is to use an analytic expression for the fields at these points, as implemented in [9], [10], [19]. The user should be advised, however, that this introduces numerical artifacts directly into the source, and the level of these artifacts may be of concern depending on what one wants to draw from the simulation as a whole. Let denote the -field polarization projection whose functional form is directly taken from [1]. For example, , with measuring the polarization angle. If is the source function referenced at , then any spatial shift corresponds to a time delay . A general 3D FDTD planewave excited by the -field thus consists of (8) but where the following hard-source is imposed:

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An almost identical form of the planewave update equations can be constructed for the boundary sources excited by the -fields. However, mixed and boundary source FDTD planewave equations may not be as compact. Also, care must be taken to make sure that the Huygens’ surface source fields in the main grid are not populated using these hard-sourced fields in the -grid: in order to achieve proper TFSF field isolation one must use fields in the -grid that are actually iterated via (8)–(9). and employed in the polarization proNote that jections should be the value evaluated from the rational angle condition. For instance, gives the trigonometric relations or for evaluating . Through the choice of and , a countably infinite set of angles at which a numerical planewave can actually propagate is defined, as reported in [11]. III. NUMERICAL EXAMPLES As a first example, Fig. 3 shows a 3D-FDTD TF/SF simuor and lation result for , , , and with a TF box (no scatterers) but surrounded by a 10 cell SF terminating time steps. A full spectrum with a PEC and run for square wave (with equal duty cycle) is employed via an anain order to demonstrate that the proposed lytic source solution will be valid for any arbitrary excitation. The top figure field for four unique points (three from the actually plots the main grid, one from the auxiliary grid) lying on the same wave; note that only front which satisfy one line is distinguishable because the fields are virtually identical. Fractional indices have been dropped for the rest of the discussion. To investigate the actual numerical discrepancy, the middle computed figure provides the plots for from the above wavefront, showing that the error is at the level of the machine precision, confirming that fields on the same wavefront have an identical value. The bottom figure shows the scattaken from the SF region at some arbitrary tered field point. Again, this scattered field value has also been suppressed field isoby up to the same finite precision limit so that lation is achieved. The sources at the Huygens’ surface are now populated with a true 3D FDTD planewave with identical dispersion relations in the main and auxiliary grids. Fig. 4 provides field in the -plane, demona cross-sectional plot of the strating that there is no field that leaks into the SF region. Notice that the wave is more distorted in the lower left corner of the figure (the trailing part of the wave) due to severe dispersion of the high frequency components of the wave. For comparitive purposes, the second example presents a more practical case where the source is bandlimited, and so the dispersion artifacts present in Fig. 3 are not nearly as pronounced. The simulation parameters are identical to the final example presented in [13], where the auxiliary grid was simulated with the O-AFP method, and here we use the fully time-domain analog developed throughout the current manuscript. A TF (surrounded by a 10 cell SF region) is region of used, at a propagation angle described by (or and ), and is run for

Fig. 3. Top figure: Four separate (though practically indistinguishable) lines are plotted for the propagated E signal at three locations on the same wavefront in the main grid, and at the analogous location in the auxiliary grid. Middle figure: The error between the auxiliary grid signal and each of the three main grid signals, overlaid. Bottom figure: The signal in the scattered field region, demonstrating effectively no spurious reflections.

Fig. 4. Cross-sectional plot of E in the first example in the yz -plane. The square wave source propagates throughout the total field domain, with no leakage at all into the scattered field region.

time-steps. For the boundary source function the O-AFP method is used with a frequency modulated Gaussian pulse with normalized peak frequency , and . Fig. 5 shows the field recorded in pairs at two locations—one near the source, and one further away. Each pair consists of a point just inside the TF regions (subscript T), and just outside in the SF region (subscript and demonstrate S). The incident waves at points

TAN AND POTTER: FDTD-DPW FORMULATION FOR A PERFECTLY MATCHED SOURCE IN TFSF SIMULATIONS

Fig. 5. Simulation domain where the source travels over several wavelengths before reaching the far side of the domain. Therefore there is an accumulation of phase distortion (compared with an analytic plane wave). Nevertheless, there is still perfect isolation from the SF region. The solid line is for a sensor point nearest the source, and the dashed line is for a sensor point in the corner furthest from the source.

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=

Fig. 7. Both the top and bottom figures show the E field in the range r r in the source grid at timestep n for the same simr to ulation parameters of example 2. Top figure: Analytic hard-sourcing is used. Bottom figure: O-AFP hard-sourcing is used.

5001

10001

= 500

with all simulation parameters being identical, but using the analytic source method rather than the O-AFP method. Fig. 7 field that is propagated in shows the Gaussian pulse for the timesteps. Our propagator formulation, the -grid after coupled with either of these hard-sourcing methods, appears to propagate a pulse that shows consistent levels of phase distortion but which maintains the general shape of the pulse. This is a qualitative comparison, but using a quantitative comparison (e.g., the difference between the two signals) would be superfluous; as mentioned at the start of this Section II-B, it is up to the user to decide what type of source is “good enough.” IV. DISCUSSION

Fig. 6. E field in a yz -plane of the computational domain where a Gaussian pulse is the incident field. The field levels in the scattered field region (the small band around the outside of the plane) are at machine precision levels.

distortion due to numerical dispersion: the plane wave effec(or cells in the main grid) tively has traveled away from the source, and phase error surely accumulates over and such a large distance. On the other hand, points demonstrate that effectively the leakage errors are still limited only by numerical precision. Fig. 6 shows a snapshot in time of field in a -plane with the Gaussian pulse propagating the across the domain. The fields in the scattered field region are at machine precision levels in this figure as well. It is worth noting that the results in these figures and the fields used to plot them are effectively identical to the results presented in [13]. This shows that the present method and the O-AFP method are completely analogous in simulating the same incident planewave. To demonstrate the effect of changing the hard-sourcing method for this more practical example, we re-ran the example

, polarizaWhen tested with arbitrary propagation angles tion reference angles , and stability factors , and over many ), the same pernon-unity aspect ratios (i.e., isolation is observed. In fact, fectly matched result of the same result even holds for a Huygens’ surface with an arbitrary staircase configuration (i.e., not necessarily a rectangular box). Our main objective was to construct a planewave source that results in a perfectly matched TFSF formulation. The question of what boundary source technique produces a waveform that best emulates the analytic source over time is open to debate. The proposed solution truly implements a matched FDTD planewave in a one dimensional grid. The price one pays is a relatively more complicated implementation than existing IFA techniques, and the use of larger 1D grids. The latter point is essentially immaterial since the 1D computational cost is a very small fraction of that required in the main grid. To date an absorbing boundary condition (ABC) for the 1D grid has not been implemented for general wide spectrum source functions. However, the O-AFP technique developed in [12], [13] effectively produces a perfect TF/SF formulation when the source function is bandlimited to below the grid cutoff . In fact a bandlimited planewave field computed frequency from the O-AFP grid is identical to a field computed using the method under current discussion, so that a small O-AFP grid could be used as an ABC if the source is bandlimited. The split-field formulation of the PML also holds promise for

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implementing an ABC, and we intend to report on the use of this for arbitrary source spectra in the near future. The technique developed in this paper is clearly not limited to the standard FDTD since obtaining a planewave between the and coordinates amounts to establishing a set of equations de. Depending on the nature rived from of the computational stencil, one may still have to determine an optimum integer mapping relation so that the result does not spatial points. In other words, in general a perevaluate fect match solution is relatively easy to construct but efficiency may become an important issue. V. CONCLUSION The proposed FDTD-DPW technique propagates a true FDTD planewave directly in the time-domain which is valid for any dimensions. The resulting leakage error is at the level benchmark) for any of the machine precision (a arbitrary source functions over any angles supported by FDTD. The resulting formulation is both efficient and accurate, and the development of the method is valid for higher-order and non-standard FDTD finite difference operators. Computing the FDTD-DPW in principle requires spatial points. Nevertheless, unless are extremely large, the computational burden of the 1D grids is negligible compared to that of the main 2D/3D simulation space.

[9] T. Tan and M. Potter, “A 1D multipoint auxiliary propagator (1DMAP) for sourcing plane waves for FDTD,” in Proc. IEEE APS Int. Symp., Honolulu, HI, Jun. 2007, pp. 1669–1672. [10] T. Tan and M. E. 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. [11] T. Tan and M. E. Potter, “On the nature of numerical plane waves in FDTD,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 505–508, 2009. [12] T. Tan and M. E. Potter, “An optimized AFP scheme for the total/scattered field FDTD formulation,” in Proc. IEEE APS Int. Symp., San Diego, CA, Jul. 2008, pp. 4–4. [13] T. Tan and M. E. Potter, “Optimized analytic field propagator (O-AFP) for plane wave injection in FDTD simulations,” IEEE Trans. Antennas Propag., to be published. [14] T. Tan and M. Potter, “Perfectly matched plane wave source in FDTD via an efficient mixed time-frequency block filter formulation,” in Proc. IEEE APS. Int. Symp., Charleston, SC, Jun. 2009, pp. 4–4. [15] T. Tan and M. Potter, “Perfectly matched plane wave source in FDTD via efficient and true time-domain update equations,” in Proc. IEEE APS. Int. Symp., Charleston, SC, Jun. 2009, pp. 4–4. [16] M. F. Hadi, “A versatile split-field 1-D propagator for perfect plane wave injection,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2691–2697, Sep. 2009. [17] J. A. Stratton, Electromagnetic Theory. New York: McGraw Hill, 1941. [18] 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. [19] T. Tan and M. Potter, “Perfectly matching plane wave Huygen’s sources in FDTD,” in Proc. 23th Int. Symp. ACES, Verona, Italy, 2007, pp. 1840–1845.

REFERENCES [1] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [2] J.-P. Berenger, “Perfectly matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antennas Propag., vol. 44, no. 1, pp. 110–117, Jan. 1996. [3] U. Oguz and L. Gurel, “An efficient and accurate technique for the incident-wave excitations in the FDTD method,” IEEE Trans. Microw. Theory Tech.., vol. 46, no. 6, pp. 869–882, Jun. 1998. [4] C. Guiffaut and K. Mahdjoubi, “A perfect wideband plane wave injector for FDTD method,” in Proc. IEEE APS. Int. Symp., Salt Lake City, UT, 2000, vol. 1, pp. 236–239. [5] S. C. Winton, P. Kosmas, and C. M. Rappaport, “FDTD simulation of TE and TM plane waves at nonzero incidence in arbitrary layered media,” IEEE Trans. Antennas Propag., vol. 53, no. 5, pp. 1721–1728, May 2005. [6] I. R. Capoglu and G. S. Smith, “A total-field/scattered-field plane-wave source for the FDTD analysis of layered media,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 158–169, Jan. 2008. [7] 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. on Antennas Propag., vol. 50, no. 9, pp. 1174–1184, Sep. 2002. [8] J. B. Schneider, “Plane waves in FDTD simulations and a nearly perfect total-field/scattered-field boundary,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3280–3287, Dec. 2004.

Tengmeng Tan (M’10) received the B.Sc. degree in electrical and computer engineering with a minor equivalence in applied mathematics from the University of Calgary, Calgary, Alberta, Canada, where he is currently working toward the Ph.D. degree. His main research interest is in applied computational electromagnetics with much of his work involving time-domain techniques. Mr. Tan was awarded an Excellent Teaching Assistant Award in 2005, and twice was a recipient of Honorable Mention in the Student Paper Competition at the AP-S/URSI, in 2008 and 2009.

Mike Potter (M’01) received the B.Eng. degree in engineering physics (electrical) from the Royal Military College of Canada, Kingston, Ontario, Canada, in 1992, and the Ph.D. degree in electrical engineering from the University of Victoria, Victoria, British Columbia, Canada, in 2001. From 1992 to 1997, he served as an officer in the Canadian Navy as a Combat Systems Engineer. After completing his service and attaining the rank of Naval Lieutentant, he completed his doctoral work in Victoria, British Columbia, Canada. He was then a Postdoctoral Fellow at the University of Arizona, Tucson, from 2001 to 2002. He currently holds the position of Associate Professor in the Department of Electrical and Computer Engineering, University of Calgary, Calgary, Alberta, Canada. His research interests include computational electromagnetics and the FDTD method. Dr. Potter is also a member of the Optical Society of America, and serves as a member of the APS Education Committee.

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FDTD Analysis of Periodic Structures With Arbitrary Skewed Grid Khaled ElMahgoub, Student Member, IEEE, Fan Yang, Senior Member, IEEE, Atef Z. Elsherbeni, Fellow, IEEE, Veysel Demir, Member, IEEE, and Ji Chen, Senior Member, IEEE

Abstract—An efficient finite-difference time-domain (FDTD) algorithm with a simple periodic boundary condition (PBC) is developed to analyze general periodic structures with arbitrary skewed grids. The algorithm is easy to implement and efficient in both memory usage and computation time. The stability criterion of the algorithm is angle independent and therefore it is suitable for implementing incidence with angle close to grazing as well as normal incidence. The validity of this algorithm is verified through several numerical examples such as dipole and Jerusalem cross frequency selective surfaces (FSS) with various skew angles. Index Terms—Finite-difference time-domain (FDTD), frequency selective surfaces (FSS), periodic boundary conditions (PBC), periodic structures, skewed grid.

I. INTRODUCTION

P

ERIODIC structures are of great importance in electromagnetics due to their wide range of applications in frequency selective surfaces (FSS), electromagnetic band gap (EBG) structures, phased antenna arrays, special periodic absorbers, and negative index metamaterials. The finite-difference time-domain (FDTD) algorithm has been utilized to analyze these structures, and various implementations of periodic boundary conditions (PBC) have been developed such that only one unit cell needs to be analyzed instead of the entire structure. In [1] these techniques are divided into two main categories: “field-transformation methods” and “direct field methods”. Field transformation methods are used to eliminate the need for time-advanced data; the transformed field equations are then discretized and solved using FDTD techniques. The spilt-field method [2] and multi-spatial grid method [3] are useful approaches in the field-transformation category. There are two main limitations with these methods. First, the transformed equations have additional terms that require special handling such as splitting the field and the use of multi-grid algorithm to implement the FDTD, which increases the complexity of the problem. Second, with the increasing of the angle of incito grazing incidence dence from normal incidence Manuscript received September 03, 2009; revised December 30, 2009; accepted February 04, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. K. ElMaghoub, F. Yang, and A. Z. Elsherbeni are with the Department of Electrical Engineering, University of Mississippi, University, MI 38677 USA (e-mail: [email protected]). V. Demir is with the Department of Electrical Engineering, Northern Illinois University, DeKalb, IL 60115 USA. J. Chen is with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204 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.2010.2050445

Fig. 1. Geometries of (a) axial and (b) skewed periodic structures.

, the stability factor decreases to zero [1]. As a result, smaller time steps are needed for oblique incidence to generate stable results, which increases the computation time in such cases. As for the direct field category, these methods work directly with Maxwell’s equations and there is no need for any field transformation. An example of these methods is the sine-cosine method [4], in which the structure is excited simultaneously with sine and cosine waveforms. The PBC for oblique incidence can be implemented using this method. The stability criteria for this technique is the same as the conventional FDTD (angle independent), which provides stable analysis for incidence near grazing. However, it is a single frequency method and loses an important property of FDTD that is the wide-band capability. In [5] a simple and efficient FDTD/PBC algorithm was introduced, which belongs to the direct field category and yet with a wideband capability. In this new approach the FDTD simulation is performed by setting a constant horizontal wavenumber instead of a specific angle of incidence. The idea of using constant wavenumber in FDTD was originated from guided wave structure analysis and eigenvalue problems in [6], and it was extended to the plane wave scattering problems in [7]–[9]. The approach offers many advantages, such as implementation simplicity, same stability condition and numerical errors as conventional FDTD, computational efficiency near the grazing incident angles, and the wide-band capability. It’s worthwhile to point out that most previous PBCs are developed to analyze axial grid periodic structures. However, there are numerous applications where the grid of the periodic structures is a general skewed grid. Fig. 1 shows the geometries of both axial and skewed grid structures. The axial periodic structures are special case of the general skewed grid structures, . where the skew angle Although the analysis of skewed grid periodic structure has been well developed using method of moment (MoM) technique [10], it has not been fully solved with FDTD. A pioneering effort

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presented in [11] utilizes the sine-cosine method in the analysis of periodic phased array with skewed grids, thus loses the wideband capability of the FDTD. Furthermore, the work presented in [11] belongs to a special case where amount of shift in the skewed direction is an integer multiple of the cell size in the same direction. This special case is referred to as “coincident” in this communication. In this paper, the constant horizontal wavenumber approach is extended to analyze the periodic structures with skewed grid. Two cases of skewed grid periodic structures are implemented. In the first case the skewed shift is coincident with the FDTD grid, and in the second case the skewed shift is non-coincident with the FDTD grid (general skewed grid periodic structure). In addition, the new algorithm is very efficient and simple, and it retains the broadband capabilities of the FDTD. The paper is organized as follows: In Section II, a brief description of the constant horizontal wavenumber approach is provided. The FDTD updating equations are derived for both the coincident and the non-coincident skewed shift cases. In Section III, several numerical examples proving the validity of the new approach are presented, including an infinite dielectric slab, a dipole FSS, and a Jerusalem cross FSS. Various incident angles, skew angles, and polarizations have been tested in these examples, and the numerical results show good agreement with the analytical results or other numerical results obtained from the frequency domain methods. Section IV provides a discussion of the new approach and conclusion. II. PERIODIC BOUNDARY CONDITIONS FOR PERIODIC STRUCTURES WITH ARBITRARY SKEWED GRID It is noticed that most of the previous FDTD approaches are used to analyze only axial periodic structures. However, the analysis of skewed grid periodic structure needs special treatment on the periodic boundary condition. In this section a new approach to implement the periodic boundary condition for such structures in FDTD is developed. The approach is an extension to the constant horizontal wavenumber approach described in [5]. A. Constant Horizontal Wavenumber Approach for Axial Case For periodic structure with periodicity along the -direction the periodic boundary condition of the electric field in frequency domain can be written as: (1) where (2) is the free space wavenumber and c is wave speed in free space. The constant horizontal wavenumber approach is to fix the value of the horizontal wavenumber in FDTD simulation instead of the angle , where is determined by both frequency is constant in (1). and angle of incidence. Thus, the term

Using direct frequency domain to time domain transformation, the field in time domain can be represented as follows: (3) It also should be pointed out that both E and H fields have complex values in the FDTD computation because of the PBC in (3) [5]. (varying incident angle with freTherefore, by fixing quency) the need for time-advanced electric field component is eliminated. An important issue related to the constant wavenumber method is the plane wave excitation. If the traditional total-field/scattered field (TF/SF) formulation described in [12] is applied, a problem arises regarding the incident angle. For example the tangential electric field component of a TM incident wave depends on the incident angle. To overcome this problem the (TF/SF) technique is modified. In the case of TM excitation, only the tangential magnetic incident field compo. The one-field nent is imposed on the excitation plane excitation allows the plane wave to propagate in both directions and ( is the excitation plane position). Thus, the entire computational domain becomes the total field region and there is no scattered field region. The scattered field can be calculated using the difference of the total and the incident fields. Similarly for TE case, only the tangential electric incident field component is imposed. In addition, there exists a problem of horizontal resonance, where fields do not decay to zero over time. To avoid such problem proper center frequency for the excitation waveform must be chosen as follows [5]: (4) is the center frequency of the Gaussian pulse and the where BW is the bandwidth of the Gaussian pulse. In this approach, conventional Yee’s scheme is used to update the E and H fields which offer several advantages, such as implementation simplicity, same stability condition and numerical errors as conventional FDTD. In addition, computational efficiency for incident angles near grazing and wideband capability are achieved as well [5]. This makes the constant horizontal wavenumber approach a good choice for the periodic structure analysis. B. The Coincident Skewed Shift The constant horizontal wavenumber approach is extended to analyze the skewed grid periodic structure. According to the ratio of the skewed shift over the FDTD cell size, the general skewed structure in Fig. 1(b) is divided into two groups: the coincident skewed shift and the non-coincident skewed shift. Fig. 2 shows the FDTD grid for the coincident skewed shift periodic structure. In this specific example the unit cell is dis; the unit A is cretized using 5 5 FDTD grid cells the one to be simulated, while unit B and unit C are the neighboring periodic units. The structure has periodicity of in the -direction and in the -direction. is the skewed shift which can be calculated as , where is the is between 0 and , the skew angle. Since the skewed shift

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and are the total number of cells in and -direcwhere tions, respectively. The two exponential terms are used to compensate the phase variations due to the oblique incidence. Using com(5) and (6) we can write the updating equation for the as ponents at the boundary

(7)

Fig. 2. FDTD grid for skewed periodic structure with skewed shift is coincident with the FDTD grid.

where for a general lossy non-dispersive medium the updating coefficients in (7) are as follows [13]: (8-a)

skewed angle is between 90 and tan . For the pe, the skew riodic structures with a square boundary angle is between 90 and 45 . For a periodic structure with a rectangular boundary, it is possible to get a small skewed angle. is multiple It can be noticed from Fig. 2 that in this case . This integer of the discretization step in the -direction configuration makes the shift coincident with the FDTD grid and this simplifies the calculation of the boundary electric fields. The magnetic field components are updated using the conventional FDTD updating equations. As for the electric field, non-boundary components will be updated using the conventional FDTD updating equations. The components at the boundaries will be updated using PBC equations based on the new approach. In this example, the skewed shift is in the -direction. A similar procedure can be used if the skewed shift is in the -direction. The updating equations for boundary electric field components are organized as follows. at and ; 1- Updating 2- Updating at and ; at , and without 3- Updating the corners; 4- Updating at the corners. at the boundary , magnetic field com(1) To update the ponents outside unit A are needed. However, due to periodicity and taking into account the skewed shift, one can use magnetic field components inside unit A to update these electric fields. For

(5)

(8-b) (8-c) The updating equation for can be written as For

components at the boundary

(9) while for

(10) components on the boundaries , (2) As for updating and , the constant horizontal wavenumber method is used with no further modification. The updating equation for at the boundary can be written as

(11) where for a general lossy non-dispersive medium the updating coefficients in (11) are as follows [13]: (12-a)

while for

(12-b) (6)

(12-c)

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at the boundary

it can be stated that (13)

components at the boundaries As for updating equation can be written as for (avoiding the corners)

and and

, the

(3) Updating components at the boundaries and will be handled in a similar manner as components, which requires taking into consideration the skewed shift. The updating equation for components at the boundaries can be written as for and (avoiding the corners) (20) (21) (4) The At

components at the corners are updated as follows: and

(14) For

(15) while for

(22) At

and

(16) where for a general lossy non-dispersive medium the updating coefficients in (14) are as follows [13]

(23) At

and

(17-a) (17-b)

(24) At

and

(17-c) The updating equation for can be written as for corners): For

components at the boundaries and (avoiding the

(18) while for

(19)

(25)

C. The Non-Coincident Skewed Shift In this section, the shift is considered to be a general shift, not a multiple integer of the discretization step in the -direction as shown in Fig. 3. In this case, two possible solutions so that the can be used. The first solution is to decrease shift becomes coincident with the new discretization and use the above formulation, but it will increase the computational time. with every In addition, one has to choose the appropriate different skew angle. The second method that will be described in this section is to use an interpolation between adjacent field components to calculate the required field component. As shown in Fig. 3 the shift is not a multiple integer of the . So the skewed shift discretization step in the -direction

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and without any further modification (11) and (13) are (20) and (21) are used to update the comused. As for the and . The components at ponents at boundaries boundaries and are updated in the same manner is calculated from interpoas in (14). Note that, the in (26). The components at lation similar to the and the corners are updated as follows. At

(28)

Fig. 3. FDTD grid for skewed periodic structure with skewed shift is non-coincident with the FDTD grid.

where is considered non-coincident with the FDTD grid. As a result, to component in cell 1 (shown in the left top corner update the in cell 2 and in cell 3 in Fig. 3) an interpolation between is needed to get the corresponding for this component. The interpolation is linear interpolation based on the two disand ( is the distance between magnetic field in tances cell 2 and position of the corresponding magnetic field and is the distance between magnetic field in cell 3 and position of the corresponding magnetic field). It should be noticed that the components in cell 2 and cell 3 are outside the unit A. two However as described in the previous section, due to periodicity and taking into account the skewed shift, one can use magnetic inside our unit of interest to derive the field components components of cell 2 and cell 3. Now the component correin cell 1 can be written as sponding to the

(29) For used. At

and

(23) and (25) can be

and

(30)

III. NUMERICAL RESULTS (26) is the ceiling function, and are two weighting where factors calculated based on distance and as: . can be updated. Similarly Using (26) and (7) the all other components on the boundary can be updated. components on the boundary , let As for the us consider the updating equation for the first component

(27) components at the boundary Similarly, all other be updated. To update components at the boundaries

can

In this section, numerical results generated using the new algorithms are presented. The FDTD code was developed in MATLAB programming language and run on a computer with an Intel Core 2, 2.66 GHz processor. These results demonstrate the validity of the new algorithm for determining reflection and transmission properties of periodic structures with arbitrary skewed grid. The first example is an infinite dielectric slab excited by TM and TE plane waves. The second example is a dipole FSS, and the third example is a JC FSS. The results are compared with results obtained from analytic solution, axial FDTD method and Ansoft Designer, which is based on MoM. The numerical results are shown in two different representations. The first representation plots results of reflection coefficient magnitude versus frequency with certain values for horizontal wavenumber. The second representation plots the results of the reflection coefficient magnitude versus frequency with certain angle of incidence, which requires multiple runs of the code to generate such results. Also, the algorithm is capable of generating the phase of the reflection coefficient, the transmission coefficient magnitude and phase, and the cross-polarization reflection and transmission coefficients.

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Fig. 5. Dipole FSS geometry with skew angle  in mm).

= 63:43

(all dimensions are

cited using cosine modulated Gaussian pulse centered at 15 GHz and with 20 GHz bandwidth. m ) and The plane wave is incident normally ( obliquely ( m ). The FDTD grid cell size is mm, and the slab is represented by 5 5 cells. In the FDTD code 2500 time steps and a Courant factor of 0.9 are used. Convolutional perfect matched layer (CPML) is used as the absorbing boundaries at the top and the bottom of the computational domain as implemented in [13]. The results are compared with analytical results in Fig. 4. From Fig. 4(b) good agreements between analytical solutions and results generated by the new algorithm for both TM and TE cases (normal and oblique incidence) can be noticed. The stability of the algorithm can be noticed even at the angles of incidence near grazing. B. A Dipole FSS

Fig. 4. (a) Geometry of the simulated dielectric slab, (b) reflection coefficient for infinite dielectric slab TM and TE case.

A. An Infinite Dielectric Slab Due to the homogeneity of an infinite dielectric slab it can be considered as a periodic structure with any skew angle. The algorithm is first used to analyze an infinite dielectric slab with thickness mm and relative permittivity . The slab is illuminated by TM and TE plane waves respectively. The skew angle of the slab is set to 60 . The slab is ex-

The algorithm is then used to analyze an FSS structure consisting of dipole elements. The dipole length is 12 mm and width is 3 mm. The periodicity is 15 mm in both and directions. The substrate has thickness of 6 mm and relative permittivity as shown in Fig. 5 [10]. The structure is first illuminated by a normally incident plane wave (with polarization along -axis), and the skew angle of the structure is set to 90 (axial case) and 63.43 (special case where the shift is half unit cell in -direction). These two cases are special cases that can be also simulated using the axial periodic boundary conditions. Fig. 6 provides results for normal incidence. The structure is excited using cosine modulated Gaussian pulse centered at 8 GHz and with 16 GHz bandwidth. In the FDTD code 2500 time steps and a Courant factor of 0.9 are used. CPML is used as the absorbing boundaries at the top and the bottom of the computational domain. The FDTD grid cell size is mm. The results are compared with results obtained from the axial FDTD code. The computational time per simulation for the skewed code is 4.28 minutes and the memory usage is 0.2 MB. For the axial code with the time is doubled due to the increase in the computational domain size as shown in Fig. 5. and Fig. 7 provides results for an oblique incidence ( ) exciting the dipole FSS structure with skew angle (general skewed grid which can’t be implemented

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Fig. 8. JC FSS geometry with skew angle  = 80 (all dimensions are in mm). Fig. 6. Reflection coefficient for dipole FSS normal incident TE plane wave with skew angle of 90 and 63.43 .

Fig. 9. Reflection coefficient co-polarization and cross polarization for JC FSS normal incident TE plane wave with skew angle of 80 .

Fig. 7. Reflection coefficient for dipole FSS oblique incident TE plane wave ( = 30 ; ' = 60 ) with skew angle of 50 .

using the axial FDTD). To generate results for specific angle of incidence multiple runs of the code are needed. The results are compared with results obtained from Ansoft Designer [14]. From Fig. 7 the good agreement between the results generated using Ansoft Designer and results generated using the new algorithm for oblique incidence can be noticed. The new algorithm results in Fig. 7 are generated using 33 different values (from 0.131 m to 83.834 m ). Simulating the oblique incidence with a fixed angle ( and ) increases the computational time to almost 120 minutes for 33 frequency points.

Fig. 10. Reflection coefficient co-polarization and cross polarization for JC FSS oblique incident TE plane wave ( = 60 ; ' = 45 ) with skew angle  = 80 .

C. A Jerusalem Cross FSS Next, the algorithm is used to analyze an FSS structure consisting of JC elements. The periodicity is 15.2 mm in both and directions [15]. The dimensions of the elements are shown in Fig. 8. The structure is illuminated by a TE plane wave (polarization along -axis). Fig. 9 provides results for normal incidence. The structure is excited using cosine modulated Gaussian pulse centered at 7 GHz and with 8 GHz bandwidth. The grid cell size is mm and mm. In the FDTD code 3000 time steps and a Courant factor of 0.9 are used. CPML

is used as the absorbing boundaries at the top and the bottom. The structure has a skew angle (general skewed grid). The results were compared with results obtained from Ansoft Designer. The computational time per simulation for skewed code is 4.53 minutes and the memory usage is 0.2 MB, while for Ansoft Designer computational time per simulation is 45 minutes for 30 frequency points and the memory usage is 21 MB using same computer. Fig. 10 provides results for oblique incidence ( and ) exciting a JC FSS structure with skew angle

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. To generate results for a specific angle of incidence multiple runs of the code are needed, which increases the computavalues (from 38.5031 m to tional time. Using 30 different 141.1781 m ) both co-polarization and cross-polarization reflection coefficients were generated. The results were compared with results obtained from Ansoft Designer. Good agreement between the results generated using Ansoft Designer and results generated using the new algorithm for both normal and oblique incidence can be noticed in Figs. 9 and 10.

[12] Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005, ch. 5. [13] A. Elsherbeni and V. Demir, The Finite Difference Time Domain Method for Electromagnetics: With MATLAB Simulations. : SciTech, Jan. 2009. [14] Ansoft Designer Software is Distributed by the Ansoft Corporation. [Online]. Available: http://www.ansoft.com/products/hf/ansoft_designer [15] I. Stevanovic´, P. Crespo-Valero, K. Blagovic´, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3688–3697, Oct. 2006.

IV. CONCLUSION This paper introduces a new FDTD approach to analyze the scattering properties of general skewed grid periodic structures. The approach is developed based on constant horizontal wavenumber technique. It is simple to implement and efficient in terms of both computational time and memory usage. In addition, the stability criterion is angle independent. Therefore, it is efficient in implementing incidence with angle close to grazing as well as normal incidence. It is capable of calculating the co-polarization and cross-polarization reflection coefficients incase of normal and oblique incidence, for both TE and TM cases, and for different skewed grid periodic structures. The numerical results show very good agreement with results from the analytical solution for dielectric slab, and the MoM solutions for dipole and JC FSS.

Khaled ElMahgoub (S’07) received the B.Sc. and M.Sc. degrees in electronics and communications engineering from Cairo University, Cairo, Egypt, in 2001 and 2006, respectively. He is currently working toward the Ph.D. degree at the University of Mississippi, University. From 2001 to 2006, he was a teaching assistant at Cairo University. Since 2007, he is a Research Assistant at the University of Mississippi. His current research interests include RFID systems, FDTD, Antenna design and Numerical techniques for electromagnetic. Mr. ElMahgoub is a member of the Applied Computational Electromagnetic Society (ACES).

REFERENCES [1] J. Maloney and M. P. Kesler, “Analysis of periodic structures,” in Computational Electrodynamics: The Finite Difference Time Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [2] J. A. Roden, S. D. Gedney, P. Kesler, J. G. Maloney, and P. H. Harms, “Time-domain analysis of periodic structures at oblique incidence: Orthogonal and nonorthogonal FDTD implementations,” IEEE Trans. Antennas Propag., vol. 46, no. 4, pp. 420–427, Apr. 1998. [3] Y. C. A. Kao and R. G. Atkins, “A finite difference-time domain approach for frequency selective surfaces at oblique incidence,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 1996, vol. 2, pp. 21–26. [4] P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propag., vol. 42, no. 9, pp. 1317–1324, Sep. 1994. [5] F. Yang, J. Chen, R. Qiang, and A. Z. Elsherbeni, “A simple and efficient FDTD/PBC algorithm for scattering analysis of periodic structures,” Radio Sci., vol. 42, July 11, 2007, RS4004. [6] S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guided Wave Lett., vol. 2, no. 5, pp. 165–167, 1992. [7] A. Aminian and Y. Rahmat-Samii, “Spectral FDTD: A novel computational technique for the analysis of periodic structures,” in Proc. IEEE Antennas and Propagation Society Symp., 2004, no. 3, pp. 3139–3142. [8] A. Aminian, F. Yang, and Y. Rahmat-Samii, “Bandwidth determination for soft and hard ground planes by spectral FDTD: A unified approach in visible and surface wave regions,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 18–28, Jan. 2005. [9] A. Aminian and Y. Rahmat-Samii, “Spectral FDTD: A novel technique for the analysis of oblique incident plane wave on periodic structures,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1818–1825, Jun. 2006. [10] B. A. Munk, Frequency Selective Surface. New York: Wiley, 2000. [11] Z. Yun and M. F. Iskander, “Implementation of Floquet Boundary Conditions in FDTD Analysis of Periodic Phased Array Antennas with Skewed Grid,” in Electromagnetics. London: Taylor and Francis Publishing, 2000, vol. 20, pp. 445–452, Issue 5.

Fan Yang (S’96–M’03–SM’08) received the B.S. and M.S. degrees from Tsinghua University, China, in 1997 and 1999, and the Ph.D. degree from the University of California, Los Angeles (UCLA), in 2002. From 1994 to 1999, he was a Research Assistant in the State Key Laboratory of Microwave and Digital Communications, Tsinghua University, China. From 1999 to 2002, he was a Graduate Student Researcher in the Antenna Lab, UCLA. From 2002 to 2004, he was a Postdoctoral Research Engineer and Instructor in the Electrical Engineering Department, UCLA. In August 2004, he joined the Electrical Engineering Department, the University of Mississippi, University, as an Assistant Professor and was promoted to Associate Professor in 2009. His research interests include antenna theory, designs, and measurements, electromagnetic band gap (EBG) structures and their applications, computational electromagnetics and optimization techniques, and applied electromagnetic systems such as the radio frequency identification (RFID) system and concentrating solar energy system. He has published over 100 technical journal articles and conference papers, five book chapters, and two books entitled Electromagnetic Band Gap Structures in Antenna Engineering and Electromagnetics and Antenna Optimization Using Taguchi’s Method. Dr. Yang was Secretary of IEEE AP Society, Los Angeles chapter, and is a Member of URSI/USNC. He serves as the Associate Editor-in-Chief of the Applied Computational Electromagnetics Society (ACES) Journal. He is also a frequent reviewer for over 20 scientific journals and book publishers, and has chaired numerous technical sessions in various international symposiums. He was a Faculty Senator and a Member of the University Assessment Committee at The University of Mississippi. For his contributions, he received several prestigious awards and recognitions. In 2004, he received the Certificate for Exceptional Accomplishment in Research and Professional Development Award from UCLA. He was awarded the Young Scientist Award at the 2005 URSI General Assembly and the 2007 International Symposium on Electromagnetic Theory. He was also appointed as The University of Mississippi Faculty Research Fellow in 2005 and 2006. In 2008, he received the Junior Faculty Research Award from The University of Mississippi. In 2009, he received the inaugural IEEE Donald G. Dudley Jr. Undergraduate Teaching Award.

ELMAHGOUB et al.: FDTD ANALYSIS OF PERIODIC STRUCTURES WITH ARBITRARY SKEWED GRID

Atef Z. Elsherbeni (S’84–M’86–SM’91–F’07) received B.Sc. degree (with honors) in electronics and communications, the B.Sc. degree (with honors) in applied physics, and the M.Eng. degree in electrical engineering, all from Cairo University, Cairo, Egypt, in 1976, 1979, and 1982, respectively, and the Ph.D. degree in electrical engineering from Manitoba University, Winnipeg, Manitoba, Canada, in 1987. He is a Professor of electrical engineering and Associate Dean for Research and Graduate Programs, the Director of the School of Engineering CAD Lab, and the Associate Director of the Center for Applied Electromagnetic Systems Research (CAESR) at the University of Mississippi, University. In 2004, he was appointed as an adjunct Professor at the Department of Electrical Engineering and Computer Science at the L.C. Smith College of Engineering and Computer Science at Syracuse University. In 2009, he was selected as Finland Distinguished Professor by the Academy of Finland and Tekes. He has conducted research dealing with scattering and diffraction by dielectric and metal objects, finite difference time domain analysis of passive and active microwave devices including planar transmission lines, field visualization and software development for EM education, interactions of electromagnetic waves with human body, sensors development for monitoring soil moisture, airports noise levels, air quality including haze and humidity, reflector and printed antennas and antenna arrays for radars, UAV, and personal communication systems, antennas for wideband applications, antenna and material properties measurements, and hardware and software acceleration of computational techniques for electromagnetics. He is a coauthor of the books The Finite Difference Time Domain Method for Electromagnetics With MATLAB Simulations (SciTech 2009), Antenna Design and Visualization Using Matlab (SciTech, 2006), MATLAB Simulations for Radar Systems Design (CRC Press, 2003), Electromagnetic Scattering Using the Iterative Multiregion Technique (Morgan & Claypool, 2007), Electromagnetics and Antenna Optimization using Taguchi’s Method (Morgan & Claypool, 2007) and the main author of the chapters Handheld Antennas and The Finite Difference Time Domain Technique for Microstrip Antennas in Handbook of Antennas in Wireless Communications (CRC Press, 2001). Dr. Elsherbeni is a Fellow member of the IEEE and the Applied Computational Electromagnetic Society (ACES). He is the Editor-in-Chief for ACES Journal and an Associate Editor to the Radio Science Journal.

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Veysel Demir (S’00–M’05) received the B.Sc. degree in electrical engineering from Middle East Technical University, Ankara, Turkey, in 1997 and the M.Sc. and Ph.D. degrees in electrical engineering from Syracuse University, Syracuse, NY, in 2002 and 2004, respectively. During his graduate studies, he worked as Research Assistant at Sonnet Software, Inc., Liverpool, NY. He worked as a Visiting Research Scholar in the Department of Electrical Engineering at the University of Mississippi from 2004 to 2007. He joined the Department of Electrical Engineering at Northern Illinois University as an Assistant Professor in August 2007. His research interests include numerical analysis techniques, finite-difference time-domain (FDTD), finite-difference frequency-domain (FDFD), and method-of-moments (MoM), as well as microwave and radiofrequency (RF) circuit analysis and design. He has coauthored more than 20 technical journal and conference papers. He is the coauthor of the books Electromagnetic Scattering Using the Iterative Multiregion Technique (Morgan & Claypool, 2007) and The Finite Difference Time Domain Method for Electromagnetics with MATLAB Simulations (Scitech 2009). Dr. Demir is a member of the Applied Computational Electromagnetic Society (ACES).

Ji Chen (S’94–M’00–SM’07) received the Bachelor’s degree from Huazhong University of Science and Technology, Wuhan, Hubei, China, in 1989, the Master’s degree from McMaster University, Hamilton, Canada, in 1994, and the Ph.D. degree from the University of Illinois at Urbana-Champaign, in 1998, all in electrical engineering. He is currently an Associate Professor with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX. His research interests include computational electromagnetic, modeling and design of biomedical instruments, stochastic analysis of periodic structure and non-periodic structures, and characterization of composite materials. Prior to joining the University of Houston, from 1998 to 2001, he was a Staff Engineer with Motorola Personal Communication Research Laboratories, Chicago, IL. Dr. Chen received the Outstanding Teaching Award and Outstanding Junior Faculty Research Award from the College of Engineering at the University of Houston. He is also the recipient of an ORISE Fellowship in 2007. His research group also received the Best Student Paper Award at the IEEE EMC Symposium in 2005 and the Best Paper Award from the IEEE APMC Conference, in 2008.

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Cartesian Shift Thin Wire Formalism in the FDTD Method With Multiwire Junctions Christophe Guiffaut, Member, IEEE, and Alain Reineix, Associate Member, IEEE

Abstract—The Holland and Simpson thin wire formalism is a versatile technique to deal with wires in the FDTD method. This paper develops an approach to correctly perform a junction between several wires at any Yee’s cell location. The application context is the wires parallel to Cartesian axes with any translation in the Yee’s Cell. The sensitivity of the thin wire current to Cartesian shift is also minimized in our enhanced formalism. The method is presented in a general framework with non-uniform FDTD spatial steps. Beside, the wire segment length can be different of the FDTD spatial steps and also the wire position does not need to coincide with any edge or node of the Yee’s cell. Index Terms—Finite difference time domain (FDTD) method, multiwire junction, thin wire.

I. INTRODUCTION

T

HIN WIRES in the finite difference time domain (FDTD) method are unavoidable structures in many applications either to model cable, via, coaxial or to model probe feed. As the meshing section of the wire leads to a very constraining refinement, some accurate models have been proposed in the past to treat wire in a coarse grid. The simpler model is to impose a null tangential electric field along cell edges in the Yee scheme. However, the wire radius becomes directly dependent on the cell size. Then, in order to control the radius effect of the wire, all serious developed models are based on static field law of an infinite wire [1]–[17] where 1/r variation of both looping magnetic field and radial electric field is used. Different approaches to incorporate static field behavior have been investigated during the two last decades. The first technique proposed by Holland et al. [1] adds two auxiliary equations to propagate charge and current along the wire and a simple FDTD-1D scheme is used to solve them. From the static field law, a cell inductance is deduced to associate the current and the average electric field surrounding the wire. Stability of this technique is approximately estimated in [4]. It has been extended to the multiwire structures [2], with linear and nonlinear circuit [3], and more recently Ledfelt [5] and Edelvik [6] have proposed a formalism for inclined wire. Moreover the formalism of the latter brings wire behavior independent of its location in the cell. Manuscript received June 04, 2009; revised December 01, 2009; accepted February 01, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by Centre d’étude de Gramat (C.E.G.), a division of the Army General Department (D.G.A.). The authors are with the XLIM Laboratory, Universite De Limoges, Lomoges 87060, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050427

A second way to a wire model for coarse grid is based on Contour-Path integral formulation of FDTD method [7]. Only the looping magnetic field components are updating with modified coefficients which take into account the integration of the static electric field distribution along radial direction. In [8], the approach of [7] is enhanced with a better open wire end condition from the near field condition. Mäkinen et al. [9], [10] are generalized this technique by integrated static field laws for all components of the electromagnetic field closed to the wire. Further, the model of wire end caps is enforced leading to a very accurate behavior. Thin wire for surge analysis and lossy medium is reported in [11], [12]. Railton et al. are studied two approaches. Firstly, a Galerkin weighted residual interpretation of the FDTD method [13] allows to naturally integrate the static field law on the basis functions in the wire vicinity. Although the general mathematical frame of this method is elegant, it is difficult to extend for wire displacement out of cell edges. So, a second technique is proposed based on the modification of assigned material parameters [14], [15]. It is an attractive approach because the FDTD algorithm is not modified and this technique is easy to extend to any edge singularities [15]. Some improvements of this last approach are performed by Taniguchi et al. [16], [17]. Among all this wire models, only the first technique [1] has seen its model extended for inclined wire and shift wire in the Yee’s cell. Besides, case of bent wires can be performed efficiently by maintaining a continuity of the current distribution on the Yee’s cells [6]. In this paper, the extension of Holland formalism aims to treat problem with wires parallels to the Cartesian axes and needing nodes of multiwire junctions in any cell location. For this we can’t reuse Edelvik’s approach [6] because the coupling between the wire and the surrounding electric field use a cylinder-shaped stencil with a section superior to the FDTD cell size and so it is too difficult to suit correctly to a junction with more than two wires. It is the reason for which in our approach, the coupling between the wire and the electric field is limited to only the cells containing a part of the wire. In this context, we show in Section II.B that the continuity of the current trace in FDTD meshing is maintained. The other capabilities of our formalism are wire shifts following lateral and longitudinal direction from cell edge, and wire segment length independent of the FDTD spatial steps. After the development of our formulation in Section II.A, the problem of a multiwire junction along with the current trace continuity in FDTD mesh are described in Section II.B. Next, the stability problem is exposed with numerical results in Section II.C. The validation of our technique is investigated on three structures (Section III) that involve emission and reception modes.

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. Thin wire of radius a, in a YEE’s cell including the notation employed in the text. Fig. 2. Integration surface to calculate the in-cell inductance L

.

II. THIN WIRE FORMALISM EXTENSION in the If the integral section is centered to the position of is related by Yee’s cell, the average inductance

A. Formulation In the following, the development is exposed for a thin wire of radius oriented following the -axis as illustrated in Fig. 1. The Holland formalism uses the static field variation in the transverse section of an infinite wire as

(6) where

(1-a) (1-b) and are the field components expressed in a cylinHere, . By inserting (1) in Maxwell’s equadrical coordinates tions and after little algebraic manipulation, one can show that (2-a) (2-b)

Here is the length between a point on the inas illustrated in tegration surface and the wire position Fig. 2. The analytical solution is not trivial so a numerical integration is applied to (6). However, only one equation is needed to update the current by finite difference. Thus to implicate the surrounding the cell , the four equations four components (5) are weighted by a bilinear interpolation rule and then are added

is the velocity in the medium surrounding where the wire. The in-cell inductance is defined by

(7)

(3)

The weighting coefficients are associated to the electric components according to the notations of the Fig. 1 with local axes centered on the wire

The fundamental change of our technique is in the manner to relate the in-cell inductance with the field . In the first time a relation like (2) is written for each of the four electric compoassociated to the four -edges of the cell noted nents

(8-a) (4) (8-b) All equations developed from (4) are expressed in a local axes centered on any segment middle noted and the cell is supposed to contain a part of this one. are inIn the following troduced to replace respectively the usual FDTD subscripts . To have a relation with the average electric field of the FDTD centered on each method, (4) is integrated over a section associated components (Fig. 2) one of the (5)

The consistency of the relation (7) is satisfying by the relation (9) Now, one can focus on the possibility to have a wire segment different to the FDTD spatial step . As a conselength quent, the nodes of the wire segments may not coincide with any Yee’s cell face or Yee’s cell corner. We denote by a segment its of the wire, is the current on the segment middle and

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The average field applied to the segment

is related by (15)

Now, the relation (12) can be compacted under the Holland’s form Fig. 3. Wire segmentation and segment cutting following the cell facets.

(16) length. The (7) is weighted with the intersection length tween the cell and the wire segment (Fig. 3)

be-

(10)

To simplified the multiwire junction treatment described in the Section II.B, the charge per unit length is substituted by the voltage and the capacitance per unit length is introduced to rely both parameters

and (17)

However the step can’t be chosen in a large scale. The 1D stability criterion gives an inferior limit. Below, time step must lead be reduced. Besides a too large value to a bad coupling between the electric field and the wire and the phenomenon is reinforced by a strong numerical dispersion don’t bring any conflict. In an other hand, a large value gain in time consuming or memory overloading so there is any convenient in this way. A reasonable limit of the segment length and then could be a value inferior to 2

Thus, the auxiliary system to be solved takes the standard form (18-a) (18-b) The FDTD solution of (18) writes

(19-a)

(11) In fact, we can consider that the optimal segment length is the spatial step parallel to the direction of the wire because the numerical dispersion becomes the same in the both spaces. However it is convenient to have segment length slowly different with the FDTD spatial steps in order to suit wire mesh to its exact length. To obtain all the field contributions to the current of the segconment , a summation of (10) is realized with all cells taining a part of the segment (Fig. 3), i.e., two cells at the most with the range (11)

(19-b) To have the reciprocity coupling with the updating equation of component, a current density associated to the current is introduced as (20) The updating of the electric field

becomes

(21) (12) B. Multiwire Junction

The scheme consistency is preserved by the condition (13) From (12) an in-cell inductance representing the relation of the wire segment with its environments can be established by the following relation (14)

A junction is placed on a wire node where the voltage is also localized. Then this problem doesn’t concern the current updating (18-a) but only the relation (18-b) is needed to modify. Three conditions are applied: the continuity of the voltage at the node (22-a), the Kirchoff current law (22-b) and the charge conservation on each segment neighbor to the node (22-c). The wire or which direction can be one of the three Cartesian axes give six wires at the most connected together (Fig. 4) (22-a)

GUIFFAUT AND REINEIX: CARTESIAN SHIFT THIN WIRE FORMALISM IN THE FDTD METHOD WITH MULTIWIRE JUNCTIONS

Fig. 4. Multiwire junction representation.

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Fig. 5. Scheme of a junction issue.

(22-b) (22-c) where represents any wire segment neighbor to the node, is the wire number connected to the node 0 is the current of the segment coming into the node 0. By conis the vention, all the currents are turned towards the node. specific charge per unit length of the segment neighbor the node 0. By adding the equations (22-c), applying (22-b) and substituting the charge per unit length by the voltage (22a), a simple relation is deduced (23) The FDTD updating solution of

is

Fig. 6. Stability of the thin wire formalism in function of its radius normalized by FDTD spatial step. Three wires positions as shown on the Fig. 7. Here ‘anal’ refers to the analytic criterion given in [4].

Then we deduce the continuity of the current traces (24)

(28)

This result was expected because all the capacitances are connected in parallel thus a unique equivalent capacitance can represent the multiwire junction

These conditions are filled for all wires taken two by two at a junction node. It involves that the continuity condition of current trace is maintained too for a multiwire junction because the Kirchoff current law (22-b) applied to the current junction distributes correctly the current on the wires and consequently on Yee’s grid edges.

(25)

C. Stability It is noted that the bilinear interpolation rule in (7) maintained the continuity of the current trace in the Yee’s cells at a junction between two perpendicular wires independently of the junction location. To simply show this assertion, let’s consider the case of Fig. 5 where two wires are in junction and a uniform current flows on these ones. First, from the (20), an equivalent current on each Yee’s grid edge coupled to the wires can be related to the current . It is so called the current trace on the Yee’s grid (26) Following the notations of Fig. 5, we can write (27-a) (27-b)

The auxiliary equations added by the thin wire formalism make the Courant-Friedrichs-Lewy (CFL) condition difficult to establish. So only approximate formulation is found as proposed in [4] only for a wire on a cell edge. To illustrate the problem of stability, numerical experiments have been carried out with a dipole in the configuration described in Section III.A. The results are showed on Fig. 6. We see that when the radius of wire the spatial step in a uniform FDTD is superior to about grid, the CFL criterion of the Yee’s scheme is no longer the stability limit condition. The more unfavorable case appears when the wire is located on a cell edge. However, our formalism enhances the stability when the wire is shifted toward inner cell or toward the middle of a cell face. Although large radius can be modeled, accuracy of the solution cannot be ensured. In particular, open-end condition has more influence when radius increases because thin wire formalism does not perform an accurate behavior on it.

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Fig. 7. 2D representation of wire shift. Cell size normalized to value 10 in each direction.

Fig. 9. Input current at dipole center in the time domain, our enhanced wire formalism.

Fig. 8. Input current at dipole center in the time domain, original Holland wire formalism.

III. NUMERICAL VALIDATION A. Dipole Antenna This first application aims to evaluate the sensitivity of the current response in the time and frequency domains when the wire is shifted in any Cartesian directions. Compared to Holland original formalism, the enhancement of the new formalism can only be due to the in-cell inductance (14). As in [6], the dipole length is 41 m, its radius is 1 cm and the FDTD grid is composed of cubic cells with 1 m edge length. The wire runs in the -direction and the wire segments are equal to m which . The corresponds to ten points per minimum wavelength ) is located at the voltage source (magnitude 1 V, wire midpoint which coincides with the midpoint of the middle segment. The waveform is Gaussian pulse defined by

(29) where

In the following, the shift of the wire in a Yee cell is referred to as shxyz. It is defined by three normalized values associated to the three Cartesian directions as illustrated on Fig. 7. Results in the time domain show the wire shift behavior with original Holland formalism (Fig. 8) and with our approach (Fig. 9). Mag-

Fig. 10. Input resistance of the dipole. Comparison between Holland formalism and MoM method.

nitude difference error is 22% with Holland formalism against 9.6% with our one. Numerical experiments show that the maximum difference is obtained with dipoles located on a cell edge compared to this one shifted at the cell center. Frequency solutions for these cases (Figs. 10–13) are compared to a method of moments (MoM) solution using FEKO software. Although impedance resonances are not relevant for practical issue, they point up the sensitivity of Holland wire formalism (Fig. 10). Our approach (Figs. 11, 12) gives solutions close to MoM’s one when the wire is shifted toward the center of the cell. The most difference with MoM corresponds to the case with the wire located on a cell edge. On the other hand, admittance resonances are in well agreement with MoM for all wire position cases (Fig. 13). Hence, at half-wavelength frequency of the dipole, i.e., 3.658 MHz, MoM resistance is 80 while our formalism gives values in the range [79.8 –81.4 ]. B. Double Rectangular Loops The great interest of the new wire formalism is the ability to treat multiwire junctions. The double rectangular loops demonstrate the achievement of this point with two multiwire junctions. Fig. 14 shows the geometry of the structure. A plane wave

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Fig. 14. Double square loops structure and plane wave polarization. Current observation points I ; I and I . Wire radius of 0.1 mm.

Fig. 11. Input resistance of the dipole. Comparison of the new formalism and MoM method.

Fig. 15. Current I of the double square loops in the time domain, four positions of the wire with the new wire formalism.

Fig. 12. Input reactance of the dipole. Comparison between the new formalism and MoM method.

Fig. 13. Input conductance of the dipole. Comparison of the new formalism and MoM method.

Fig. 16. Tail of the current solutions I (double square loops structure) in the time domain, four positions of the wire with the new wire formalism.

The FDTD spatial steps are all equal to mm. Wires are mm) in the y-axis divided in 17 uniform segments ( mm) in the x-axis. and 2 8 uniform segments ( . This Hence all values of spatial steps are close to leads to have non-integer ratio between loop size and the spatial step . Consequently, parallel wires of the loops are never at the same position in their respective cells. In the time domain, current results give very good agreement for different wire shifts (Fig. 15). Fig. 16 shows that the current return to zero at long time and also no undamped low frequency oscillation occurs. It means that continuity of the current trace on the Yee’s grid is implicitly maintained at the multiwire junctions with the bilinear interpolation rule as explained in the Section II.B. Current solutions are compared to a MoM solution with a wavelength up to 15 (Fig. 17). FDTD curves are close to MoM solution at the three current sensors. C. UHF Antenna

with grazing incidence allows to have simultaneously both magnetic and electric coupling. The waveform is Gaussian function (29) with following characteristics

To illustrate a more complex case with multiwire junctions, an application is dealt with UHF antenna composed by two dipoles and a wire reflector as illustrated in Fig. 18. The halfwavelength frequency is 750 MHz. Two FDTD configurations have been performed: a first one named FDTD-1 with uniform cm, and a second one named spatial steps

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Fig. 19. Input resistance of the UHF antenna. Curve meaning is: FDTD-1 uniform spatial step size of 1 cm, MoM-1 wire meshing of 1 cm, FDTD-2 non uniform spatial steps x y z cm, MoM-2 the two dipoles have wire meshing of 0.5 cm.

1 = 1 = 21 = 1

Fig. 17. Frequency responses on the currents I ; I and I of the double rectangular loop structure. The current is normalized by the incidence electric field. Comparison between the new thin wire formalism and MoM.

Fig. 18. UHF antenna design. All wire extremities are connected except extremities of the two dipoles. Two wire radius: 1 mm for thick line and 0.5 mm for thin line. Feedings are voltage sources distributed at the dipole middle.

FDTD-2 with non uniform spatial steps cm. The impedance resonance depends on the generator gap. Since thin wire segment size corresponds to FDTD spatial step size in each Cartesian direction, the generator gap takes following values: 2 1 cm for the first configuration and 2 0.5 cm for the second one. To have the same modeling condition, we perform simulations with MoM using the same meshes of the dipole as in FDTD method. Likewise voltage sources in MoM are distributed as in FDTD method. The effect of the voltage gap variation appears on Fig. 19. When the voltage gap size decreases, the impedance resonance shifts to the low frequencies. The case where dipoles are meshed with 1 cm step, gives a small difference between the two methods because meshing becomes too coarse. The far field responses are compared at half wavelength frequency. The far field transformation in our FDTD method uses principle of double Huygens surface [18] which provides maximum accuracy. The directivity is deduced by calculating the Poynting vector integration over the Huygens surface. The Figs. 20–21 show the directivity of the antenna in the E- and H- planes. MoM and FDTD methods give very close solutions which illustrates the close behavior of the wire reflector by the two methods.

Fig. 20. Directivity at 750 MHz in the E -plane or xz-plane. Comparison between the new thin wire formalism and MoM.

Fig. 21. Directivity at 750 MHz in the H -plane or xy-plane. Comparison between the new thin wire formalism and MoM.

IV. CONCLUSION A new formulation of the thin wire model derived from the Holland formalism is presented and successfully validated. The main advantage is the possibility to deal with multiwire junctions located at any point in the Yee’s cell. The wire current

GUIFFAUT AND REINEIX: CARTESIAN SHIFT THIN WIRE FORMALISM IN THE FDTD METHOD WITH MULTIWIRE JUNCTIONS

coupling with the electric field is limited only to the cells containing a part of the wire even if this part is very short. Wire segment length can be independent of the Yee’s cell size to well suit to the wire length or to reduce the gap of a source. The applications with multiwire junctions show that no undamped signal occurs and it has been justified that the continuity of the current trace in the Yee’s cells is maintained at any multiwire junctions. Future work will generalize this technique to multiple oblique wires in junction. REFERENCES [1] R. Holland and L. Simpson, “Finite-difference analysis of EMP coupling to thin struts and wires,” IEEE Trans. Electromagn. Compat., vol. EMC-23, no. 2, pp. 88–97, May 1981. [2] J.-P. Berenger, “A multiwire formalism for the FDTD method,” IEEE Trans. Electromagn. Compat., vol. EMC-42, no. 3, pp. 257–264, Aug. 2000. [3] J.-P. Berenger, “Introduction de circuits localisés et de charges non linéaires dans le formalisme des fils minces aux différences finies,” preColloque Int. de Compatibilité Electromagnétique, sented at the 7 Toulouse, France, 1994. [4] J. Grando, F. Issac, M. Lemistre, and J. Alliot, “Stability analysis including wires of arbitrary radius in FD-TD code,” in Proc. IEEE Trans. Antennas Propag. Soc. Int. Symposium, Jun. 28–Jul. 2 1993, vol. 1, pp. 18–21. [5] G. Ledfelt, “A stable subcell model for arbitrarily oriented thin wires for the FDTD method,” Int. J. Numerical Modeling, vol. 15, pp. 503–515, 2002. [6] F. Edelvik, “A new technique for accurate and stable modeling of arbitrarily oriented thin wires in the FDTD method,” IEEE Trans. Electromagn. Compat., vol. EMC-45, no. 2, pp. 416–423, May 2003. [7] K. Umashankar, A. Taflove, and B. Beker, “Calculation and experimental validation of induced currents on coupled wires in an arbitrary shaped cavity,” IEEE Trans. Antennas Propag., vol. AP-35, no. 11, pp. 1248–1257, Nov. 1987. [8] M. Douglas, M. Okoniewski, and M. A. Stuchly, “Accurate modeling of thin wire antennas in the FDTD method,” Microw. Opt. Technol. Lett., no. 4, pp. 261–265, May 1999. [9] R. M. Makinen, J. S. Juntunen, and M. A. Kivikoski, “An improved thin-wire model for FDTD,” IEEE Trans. Microw. Theory Tech., vol. 50, no. MTT-5, pp. 1245–1255, May 2002. [10] R. M. Makinen and M. A. Kivikoski, “A stabilized resistive voltage source for FDTD thin-wire models,” IEEE Trans. Antennas Propag., vol. AP-51, no. 7, pp. 1615–1622, Jul. 2003. [11] T. Noda and S. Yokoyama, “Thin wire representation in finite difference time domain surge simulation,” IEEE Trans. Power Del., vol. PWRD-17, no. 3, pp. 840–847, Jul. 2002. [12] Y. Baba, N. Nagaoka, and A. Ametani, “Modeling of thin wires in a lossy medium for FDTD simulations,” IEEE Trans. Electromagn. Compat., vol. EMC-47, no. 1, pp. 54–60, Feb. 2005. [13] C. J. Railton, B. P. Koh, and I. J. Craddock, “The treatment of thin wires in the FDTD method using a weighted residuals approach,” IEEE Trans. Antennas Propag., vol. AP-52, no. 11, pp. 2941–2949, Nov. 2004.

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[14] I. J. Craddock and C. J. Railton, “A new technique for the stable incorporation of static field solutions in the FDTD method for the analysis of thin wires and narrow strips,” IEEE Trans. Microw. Theory Tech., vol. MTT-46, no. 8, pp. 1091–1096, Aug. 1998. [15] C. J. Railton, D. L. Paul, and S. Dumanli, “The treatment of thin wire and coaxial structures in lossless and lossy media in FDTD by the modification of assigned material parameters,” IEEE Trans. Electromagn. Compat., vol. EMC-48, no. 4, pp. 654–660, Nov. 2006. [16] Y. Taniguchi, Y. Baba, N. Nagaoka, and A. Ametani, “Modification on a thin-wire representation for FDTD calculations in nonsquare grids,” IEEE Trans. Electromagn. Compat., vol. EMC-50, no. 2, pp. 427–431, May 2008. [17] Y. Taniguchi, Y. Baba, N. Nagaoka, and A. Ametani, “An improved thin wire representation for FDTD computations,” IEEE Trans. Antennas Propag., vol. AP-56, no. 10, pp. 3248–3252, Oct. 2008. [18] T. Martin, “An improved near- to far-zone transformation for the finitedifference time-domain method,” IEEE Trans. Antennas Propag., vol. AP-46, no. 9, pp. 1263–1271, Sep. 1998.

Christophe Guiffaut (M’04) was born on March 14, 1973 in Chateaubriant, France. He received the Master’s and Ph.D. degrees in electronics and telecommunications, in 1997 and 2000 respectively, from the University of Rennes, France. He joined the CNRS Research Center in 2002 and integrated in the same year into the XLIM Laboratory at the University of Limoges, France. His research interests include the development of numerical methods in the time domain for application areas in electromagnetic compatibility, antennas and ground penetrating radar.

Alain Reineix (A’04) was born in Limoges, France, in 1961. He received the Master’s degree in electronics and telecommunications, in 1984, and the Ph.D. degree in electronics, in 1986, both from the University of Limoges, France. In 1986, he joined CNRS (Centre National de la Recherche Scientifique) as a Researcher at the IRCOM laboratory, University of Limoges, France. Since 1991, he has been a Research Director at CNRS and also the Head of a Research Group on electromagnetic compatibility at the IRCOM laboratory and now (since 2006) XLIM laboratory, University of Limoges, France. His current research interests include improvements on numerical modelling particularly in the time domain (FDTD) and ground penetration studies in the radar domain. Dr. Reineix is an URSI correspondent and a member of the French Electric Engineering Society (SEE).

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Efficient Current-Based Hybrid Analysis of Wire Antennas Mounted on a Large Realistic Aircraft Wei-Jiang Zhao, Joshua Le-Wei Li, Fellow, IEEE, and Li Hu, Student Member, IEEE

Abstract—An efficient hybrid analysis which hybridizes surfacewire integral equations (SWIE) with physical optics (PO) approximation is presented for wire antennas attached to an electrically large aircraft platform. In the analysis, the whole surface of the platform and antennas is divided into three regions, namely, the method of moments (MoM) region, the PO region and a region referred to as the joint PO-MoM (POM) region in this paper. The MoM region, generally, the smallest one among the three regions, includes wires, wire-surface junctions and a small part of the platform surface surrounding the junctions. A large part of the remaining platform surface forms the PO region with the rest being the POM region which is situated between the MoM and PO regions. The POM region is treated as a MoM region on the one hand so that the total MoM region is sufficiently large to well characterize the platform-attached antennas, whereas it is also regarded as a PO region on the other hand so that its interaction with the PO region needs not to be precisely considered so as to drastically reduce computational requirements. Index Terms—Aircraft, antennas, computational electromagnetics, integral equation, physical optics (PO).

I. INTRODUCTION N the situation that wire antennas are mounted onto an aircraft’s surface, the aircraft body platform is an active part of the antennas, and the entire aircraft structure must be therefore included in accurate antenna analysis, characterization and design. A rigorous analysis for such problems can be provided by the method of moments (MoM) [1] or those MoM-based fast algorithms [2]–[4] in conjunction with a surface-wire integral equation (SWIE) formulation [5], [6]. However for some

I

Manuscript received September 23, 2009; accepted February 03, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported in part by the Defence Innovative Research Program (DIRP), Defence Science and Technology Agency (DSTA) of Singapore and the National University of Singapore (NUS) under Project No: DSTA-NUS-DIRP/ 2007/02, in part by the United States Air Force Office of Scientific Research (AFOSR) and in part by the Asian Office of Aerospace Research and Development (AOARD) Projects: AOARD-07-4024 and AOARD-09-4069. W.-J. Zhao was with Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore. He is now with the Department of Advanced Electronics and Electromagnetics, Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), Singapore 138632, Singapore. J. L.-W. Li is with Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore and also with the Institute of Electromagnetics, University of Electronic Science and Technology of China, Chengdu 610054, China (e-mail: [email protected]; web: http://www.ece.nus.edu.sg/lwli). L. Hu is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050450

practical applications, the aircraft platform may be very large in terms of the wavelength so that it cannot be directly handled by the numerical methods due to excessive computer requirement. Hybrid methods [7]–[9] which combine numerical methods and high-frequency (HF) asymptotic methods appear to be able to well handle such practical problems. The hybrid techniques are generally classified into two categories, field-based analysis and current-based analysis. The field-based analysis is efficient in dealing with canonical shaped geometries, however the efficiency may be lost for very complicate structures due to the difficulties of making ray tracing and of determining the required diffraction coefficients for arbitrarily shaped geometries [10]. As compared to the field-based analysis, the current-based analysis which only needs to obtain equivalent surface currents on the entire structure surface avoids the laborious ray tracing and caustic difficulties; so it is more appropriate for handling arbitrary shaped CAD models, such as the facet geometries which are capable of providing a satisfactory representation for realistic aircraft models. The current-based hybrid analysis has found great utility in characterizing properties of antennas in the presence of large structures [11]–[13]. For these applications, the MoM interaction matrix is modified by coupling into the physical optics (PO) contribution. Each MoM basis function interacts with other MoM basis functions directly and through each PO triangle indirectly, so it is necessary to have as small a MoM region as possible. Otherwise, calculating the interaction between the MoM and PO regions will lead to much longer solution time, which may make the analysis of electrically large problems become prohibitive. On the other hand, solving the problems of antennas mounted onto a large platform with only a small part of the platform surface assigned to the MoM region may result in inaccurate current distributions on antennas since the high frequency (HF) methods cannot accurately model the effects of those parts on the platform surface which are at a small distance from the junctions. In general, the current on the surface near the junctions is much more significant compared to that on other part of the platform surface, so the currents in such areas need to be carefully considered. In this paper, an efficient current-based hybrid analysis is presented for the wire antennas mounted onto an electrically large aircraft body platform. The surface of the platform and antennas are divided into three regions, the MoM region, the PO region and the POM region. The POM region is distinguished from the respective MoM and PO regions, but located between them. In the analysis, the antennas, wire-surface junctions and a small part of the platform surface in the vicinity of the junctions are assigned to the MoM region. Since the MoM region includes only a small part of the platform surface, the POM region may not be

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far away from the junctions. Hence, the interaction among different parts in the POM region should not be neglected, instead it needs to be carefully considered. Such interactions cannot be handled by the high frequency methods which obey the locality principle, and is thus described by the MoM in our analysis. On the other hand, different from either a PO region or a MoM region, the POM region decouples its interaction with the PO regions. This approach is reasonable for the aircraft platform. Assume that each basis function in the POM region acts as a source for the currents on each PO patch. The PO patches may either fall into the shadow region or be far away from the source, thus the interaction between the POM and PO regions can be ignored without much loss in the solution accuracy. In the MoM and POM regions, combined wire and surface integral equations are applied with the unknown currents being expanded by three types of triangular basis functions [5], [6]. In the PO region, the PO currents induced by the currents in the MoM region are obtained through direct use of the PO approximation on PO patches. The electric fields radiated from the PO currents are coupled into the SWIE in the MoM region but that in the POM region. The equivalent surface currents are finally obtained by solving the SWIE via the standard MoM procedure. Some numerical results are presented to illustrate the validity and efficiency of the presented hybrid analysis. II. METHOD DESCRIPTION A. SWIE for Antenna-Platform Problems Consider wire antennas mounted onto a perfectly conducting attached platform. Let the antennas be bounded by surface to the platform of surface . By requiring the tangential comand , respectively, an ponents of electric field to vanish on electric field integral equation (EFIE) formulation of Maxwell’s equations can be derived as (1) where the subscript “tan” denotes the tangential components; is the electric field due to an impressed source; the vector and represent the current densities on and functions , respectively; and and stand for the integro-differential operators associated with the EFIE, which are defined as (2a)

(2b)

are wave number, wave impedance, and where , and Green’s function of free space, respectively. In (2a), only the axial component of wire currents are considered since the radii of wires in the problem are always assumed to be small compared to the wavelength. Eq. (1) together with (2a) and (2b) represents the combined surface-wire integral equation (SWIE) for the antenna-platform problems.

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Fig. 1. The decomposition of MoM, PO and POM regions in the hybrid analysis.

B. Hybrid SWIE-PO Integral Equations The hybrid analysis presented herein starts with the platform which can be divided into three parts represented by surface , , and , respectively. is surrounding the wire anis situtennas and the smallest one among the three parts. ated between and , while is much larger than and . together with the wires and wire-surface junctions forms the MoM region, while is always assigned to the PO consists of the POM region only. Fig. 1 shows the region. decomposition of MoM, PO and POM regions. Let equivalent currents be assumed on these surfaces. Applying the EFIE on , and leads to the following integral equations:

(3a) (3b) where and denote the current densities on and , represents the PO current assumed on the respectively; while , induced by the currents and . The electric fields raare coupled into (3a) but (3b). Eq. (3a) and (3b) are diated by the hybrid SWIE-PO integral equations employed in the present analysis, which are more general than those used in the conventional hybrid techniques [12], [13]. Actually, the conventional hybrid technique can be obtained from the hybrid technique presented herein by either combining the POM and PO regions into a new PO region or combining the POM and MoM regions into a new MoM region. In practical calculation, is usually chosen as the intersection of the platform surface and a sphere with the center at the . may be chosen wire-surface junction and of radius of as large as possible so long as the total MoM region can be handled by the MoM procedure.

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C. PO Currents According to the PO approximation, we have the surface current distribution given by

plying the Galerkin’s MoM procedure result in an of linear equations as follows in (7):

system

(4) where (7)

In (7), , trices

(5)

The PO region is discretized into small triangles. The PO approximation (4) is used to calculate the PO current on each PO triangle. Since the PO current is obtained approximately, the current may not be continuous as it flows over the boundaries between the PO and MoM regions and that between the different PO regions. In [12], to ensure a continuous current flow over the entire surface to be modeled, the surface current in the PO region is expressed as linear superposition of RWG basis functions, and the PO approximation is used to calculate the expansion coefficients. Since the current on each PO triangle is related to the PO fields on its associated three edges, it is not true that the PO current, or its dependence on the PO fields at the edges, will violate the locality principle of high frequency techniques. By substituting (4) and (5) into (3a), (3a) becomes an integral , and equation satisfied by only three unknown currents . Eq. (3b) has the same three unknown currents. Hence, the integral equations in (3a) and (3b) can be solved by the standard MoM procedure.

, the elements of sub-maand are zero, and for , provided that all wire antennas are fed at the wiresurface junctions. The elements of the matrix represent the interactions among the MoM basis functions. Each MoM basis function interacts with other MoM basis functions directly or through the PO region indirectly. The matrix elements associated with the direct interaction can be calculated using the MoM formulation [5], [6], so only the expressions of those matrix elements related to the indirect interactions via the PO region need to be addressed. The expressions of the elements, and , are given as follows while the element expressions of other sub-matrices can be obtained in an analogous way:

D. Matrix Equation Let , and be expanded with triangular type of basis functions and given as follows:

(8a)

(6a)

(6b)

(8b)

(6c)

denotes triangular basis function [6], represents where the RWG basis function [5], and and are the wire and surface parts of the special wire-to-surface basis functions [6], respectively. Substituting (4)–(6c) into (3a) and (3b), and ap-

(8c)

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Fig. 3. A =2 dipole in the front of a PEC cube.

, Patch 1 is unblocked by Patch 2. Otherwise substiIf tuting (10) into (9a), the point is known. If is inside the Patch 2, then Patch 1 is blocked by Patch 2. Otherwise Patch 1 is unblocked by Patch 2.

Fig. 2. Geometry for shadowing calculation.

III. NUMERICAL RESULTS (8d)

E. Algorithm for Shadow Effects An illumination check must be conducted prior to the computation of the PO current in (4). Aircrafts generally have complicated geometries of dynamic profiles, of which some parts may be blocked by the other parts and are thus invisible to the source. Such shadow effect must be carefully considered, otherwise correct hybrid solution cannot be achieved. Exact shadowing calculation is generally time-consuming, hence it is necessary to introduce some approximations to make this part of the calculation efficient. In this paper, an approximation is adopted, that is, a facet is considered to be shadowed only if its central point is invisible to the sources. Let Patch 1 represent any of the patches in the PO region, whereas it is necessary to check if it is blocked by any other patch of the platform, Patch 2, as shown in Fig. 2. In the figure, denotes a unit vector along the direction of incidence, stands for the center of facet 1, indicates the intersection point is of the incident ray and the plane where Patch 2 is located, the surface normal unit vector of Patch 2, denotes any one of vertices of Patch 2, and O is the origin of the coordinate system. We thus have (9a) and (9b) From (9a) and (9b), the coefficient

can be determined by (10)

dipole In the first example, the directive gain patterns of a in the front of a PEC cube at a distance of 1.75 m are calculated. The geometry of the problem is shown in Fig. 3. The cube has a side-length of 2 m, and the dipole operates at a frequency of 300 MHz. In the simulation, only the wire dipole is assigned to the MoM region, with the entire cube being placed into the PO region. Hence our hybrid analysis should not be different from conventional current-based hybrid analysis, for validation purpose. The directive gain patterns in -plane and -plane are shown in Fig. 4 and Fig. 5, respectively. It can be seen that the hybrid IE-PO results are in good agreement with the MoM results, except a slight difference observed in the shadow region. The difference is due to the fact that PO can not handle the wedge diffraction from the cube. After hybridizing the UTD [14]–[16], our hybrid analysis can produce a very accurate results which are referred to as IE-PO+UTD results and also shown in these figures. In this example considered, the antenna is set off the cube, hence the current distribution can be accurately determined by the hybrid analysis with a small MoM region including only the antenna. For such problems, the solution accuracy mainly depends on how accurate the HF currents are in modeling the effects of HF region. The radiation patterns for the same geometry were presented in [14], which are similar to our results of directive gain pattern. monopole mounted onto a sphere is considA radiating ered in the second example, of which the geometry is depicted in Fig. 6. The monopole has a length of 0.25 m and operates at a frequency of 300 MHz. The radius of the sphere is assumed to be 1 m. In this example, the antenna is mounted onto the sphere, hence the sphere is also an active part of radiation resulted from the strong mutual coupling. A properly large part of the spherical surface surrounding the monopole must be assigned to the MoM region to accurately determine the distribution on the antenna. The vertical directive gain pattern is calculated and shown in Fig. 7. In the simulation, the MoM and POM regions are chosen as the intersection of the spherical surface

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Fig. 4. Directive gain pattern of the dipole in the front of a cube (E -plane).

Fig. 7. Vertical directive gain pattern of the monopole on the sphere.

Fig. 8. Lit area of the aircraft when the point source is placed at a distance of 0.025 m above the fuselage. Fig. 5. Directive gain pattern of the dipole in front of a cube (H -plane).

Fig. 6. A =4 monopole mounted on a sphere.

and a sphere centered at the wire-surface junction whose radius and , respectively. A good agreeis assumed to be ment is observed between the hybrid SWIE-PO result and the

MoM result. In this example, there should be the surface diffraction (creeping waves) from the sphere. Since the creeping waves decay exponentially along the propagation path, their main energy shall be focused on the vicinity of the wire-surface junction. A large MoM region properly chosen on the sphere surface can cover the dominant contribution from the creeping waves, so the creeping wave mode does not need to be considered separately. If the dipole is located in the front of a sphere, then the MoM region only includes the dipole with the whole sphere placed to the HF region; so the Fock currents shall need to be calculated in the current-based hybrid analysis for taking into accounted the effects of the creeping wave [17]. monopole The last example considered in this paper is a antenna mounted on a realistic aircraft model. The aircraft has a length of 13.5 m, a wing span of 9.8 m and a height of 4.2 m. The monopole is mounted on the top of the fuselage and operates at the frequency of 600 MHz. The -axis is in the vertical direction and the -axis is along the axis of the fuselage with the positive -axis directed from nose to tail. Fig. 8 shows the illuminated areas of the aircraft due to a point source located at a distance above the fuselage. In the shadowing calculation, the of algorithm described earlier in Section II-E is used. The shadow

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IV. CONCLUSION

Fig. 9. Directive gain patterns of the monopole on the aircraft in the plane of  .

=0

A new current based hybrid scheme has been proposed and presented in this paper for the analysis of wire antennas mounted on an electrically large aircraft platform. A joint PO-MoM region is introduced in the proposed scheme, which is treated respectively as a MoM region when the antenna current is determined and as a PO region when the coupling between the MoM and PO regions is computed so that accurate modeling of antenna currents can be achieved with well-controlled computational cost at the same time. While the proposed hybrid scheme can be utilized to significantly reduce the very high computational costs of the full wave approach, a good agreement is observed between the results obtained by the proposed hybrid method and the full wave method for the directive gain pattern of a wire antenna mounted on a realistic aircraft platform, except for a little difference for the backside radiation. The proposed method is currently limited, however, to smooth curved platforms such as aircrafts. After combining with fast integral equation algorithms, the proposed approach will be allowed to employ a larger full-wave region so as to handle more generally shaped platforms and to yield more accurate solutions. REFERENCES

Fig. 10. Directive gain patterns of the monopole on the aircraft in the plane of . 

= 45

effect can be clearly seen from the figure. Fig. 9 and Fig. 10 depict the vertical directive gain patterns in the planes of ( -plane) and , respectively. The results obtained by fast algorithm with the commercial software Feko are also shown in the figures for comparison. It can be seen that the hybrid SWIE-PO results agree well with the full wave results. An excellent agreement is observed in two angle ranges, from 90 to 120 and from 240 to 270 . As these regions are in the shadow region of the monopole, so the effects of the creeping wave may exist. This indicates that our hybrid analysis can really include the creeping wave effects. In the simulation, the aircraft is discretized into 115,252 triangles. The MoM and POM regions are chosen as the intersection of the sphere surface and a sphere centered at the wire-surface junction whose radius is assumed to be and 1 m , respectively. In the hybrid analysis, 0.1 m 4,549 unknowns are used and this number is much smaller than the number, 172,878, used by the full wave method.

[1] R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968. [2] W. C. Chew, J. M. Jin, C. C. Lu, E. Michielssen, and J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag., vol. AP-45, pp. 533–543, Mar. 1997. [3] X. C. Nie, L. W. Li, and N. Yuan, “Fast analysis of scattering by arbitrarily shaped three-dimensional objects using the precorrected-FFT method,” Microw. Opt. Technol. Lett., vol. 34, no. 6, pp. 438–442, 2002. [4] W. J. Zhao, L. W. Li, and Y. B. Gan, “Efficient analysis of antenna radiation in the presence of airborne dielectric radomes of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-53, pp. 442–449, Jan. 2005. [5] 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, 1982. [6] S. U. Hwu, D. R. Wilton, and S. M. Rao, “Electromagnetic scattering and radiation by arbitrary conducting wire/surface configurations,” in IEEE APS Int. Symp. Digest, 1998, pp. 890–893. [7] G. A. Thiele, “Overview of selected hybrid methods in radiating system analysis,” Proc. IEEE, vol. 80, pp. 66–78, Jan. 1992. [8] L. N. Medgyesi-Mitschang and D. S. Wang, “Hyrid methods in computational electromagnetics: A review,” Comput. Phys. Comm., vol. 68, pp. 76–94, 1991. [9] D. B. Bouche, F. A. Molinet, and R. Mittra, “Asymptotic and hybrid techniques in electromagnetic scattering,” Proc. IEEE, vol. 81, pp. 1658–1684, 1993. [10] P. E. Hussar, V. Oliker, H. L. Riggins, E. M. Smith-Rowland, W. R. Klocko, and L. Prussner, “An implementation of the UTD on facetized CAD platform models,” IEEE Antennas Propag. Mag., vol. 42, no. 2, pp. 100–106, Apr. 2000. [11] W. J. Zhao, Y. B. Gan, L. W. Li, and C. F. Wang, “Effects of an electrically large airborne radome on radiation patterns and input impedance of a dipole array,” IEEE Trans. Antennas Propag., vol. AP-55, pp. 2399–2402, Aug. 2007. [12] U. Jakobus and F. M. Landstorfer, “Improved PO-MM hybrid formulation for scattering from three-dimensional perfectly conducting bodies of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-43, pp. 162–169, Feb. 1995. [13] F. Obelleiro, J. M. Taboada, J. L. Rodrfguez, J. O. Rubinos, and A. M. Arias, “Hybrid moment-method physical-optics formulation for modeling the electromagnetic behavior of onboard antennas,” Microw. Opt. Technol. Lett., vol. 27, no. 2, pp. 88–93, Oct. 20, 2000. [14] U. Jakobus and F. M. Landstorfer, “Improvement of the PO-MM hybrid method by accounting for effects of perfectly conducting wedge,” IEEE Trans. Antennas Propag., vol. AP-43, pp. 1123–1129, Oct. 1995.

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[15] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [16] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, vol. 62, pp. 1448–1461, Nov. 1974. [17] U. Jakobus and F. M. Landstorfer, “Application of Fock-currents for curved convex surfaces within the framework of a current-based hybrid method,” in Proc. 3rd Int. Conf. on Computation in Electromagnetics, Apr. 1996, pp. 415–420.

Wei-Jiang Zhao received the B.Sc. degree in mathematics from Nankai University, Tianjin, China, in 1989 and the M.Eng. and Ph.D. degrees in electrical engineering from Xidian University, Xi’an, China, in 1992 and 1999, respectively. In 1992, he joined Xidian University, where he became an Associate Professor in 1999. In 2001, he joined Temasek Laboratories, National University of Singapore, as a Research Scientist. In 2009, he was with Department of Electrical and Computer Engineering, National University of Singapore. Currently, he is with the Department of Advanced Electronics and Electromagnetics, Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), Singapore. His current research interests include computational electromagnetics, fast algorithms for electromagnetic scattering and radiation, high-frequency methods, hybrid methods, antenna analysis, radome modeling, and EMC analysis of on-platform antennas.

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 and sponsored by the Department of Physics, La Trobe University, Melbourne, Australia, as a Research Fellow. Since 1992, he has been with the Department of Electrical and Computer Engineering, National University of Singapore where he is currently a Full Professor and Director of NUS Centre for Microwave and Radio Frequency. From 1999 to 2004, he was seconded to High Performance Computations on Engineered Systems (HPCES) Programme of Singapore-MIT Alliance (SMA) as a Course Coordinator and SMA Faculty Fellow. From May to July 2002, he was a Visiting Scientist with Research Laboratory of Electronics at the Massachusetts Institute of Technology. In October 2006, he was an Invited Visiting Professor with University of Paris VI, France. He was an Invited Visiting Professor with the Institute for Transmission, Waves and

Photonics at Swiss Federal Institute of Technology, Lausanne (EPFL) between January and June 2008; and with Department for Information Technology and Electrical Engineering at Swiss Federal Institute of Technology, Zurich (ETHZ), between July and November, 2008; both in Switzerland. He is an Advisory Professor at the State Key Laboratory of Electromagnetic Environments, Beijing since 2002; a Guest Professor at both Harbin Institute of Technology, Harbin, China, (2003–2010) and Southeast University, Nanjing, China, since 2004; and an Adjunct Professor at both Zhejiang University, Hanzhou, China, (2004–2006) and at University of Electronic Science and Technology of China, Chengdu, since 2006. 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 300 international refereed journal papers, 48 regional refereed journal papers, and over 350 international conference papers. Dr. Li was a 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 (a Member since 1998) 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 Editorial Board Member of Journal of Electromagnetic Waves and Applications (JEWA) the book series Progress In Electromagnetics Research (PIER) by EMW Publishing, International Journal of Microwave and Optical Technology and Electromagnetics journal; and an Overseas Editorial Board Member of Chinese Journal of Radio Science and Frontiers of Electrical and Electronic Engineering in China (for selected papers from Chinese Universities by Springer). He was also a Guest Editor of a Special Section on ISAP 2006 of IEICE Transactions on Communications, Japan. He also serves as a member of International Advisory Committees and/or Technical Program Committees of many international conferences or workshops, in addition to serving as General (Co-)Chairman and TPC (Co-)Chairman of a few international symposia and conferences.

Li Hu (S’07) received the B.Eng. degree in information engineering from Zhejiang University, Hangzhou, China, in 2004. He is currently working toward the Ph.D. degree at the National University of Singapore. He is currently a Research Scholar in the Department of Electrical and Computer Engineering, National University of Singapore. In 2004, he worked as a Research Assistant at Zhejiang University, Hangzhou, China. His current research interests are in general areas of computational electromagnetics.

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Enhancement of Efficiency of Integral Equation Solutions of Antennas by Incorporation of Network Principles – Part II Adam W. Schreiber, Member, IEEE, and Chalmers M. Butler, Life Fellow, IEEE

Abstract—A method is presented for enhancing the efficiency of integral equation solutions for antennas. In this paper, a folded coaxial guide junction, proposed to be part of a below the feed inductive load, is analyzed. Scattering parameters for the folded coaxial guide junction are developed to replace the coupling of the field accounted for by a coupled integral equation at the aperture joining two coaxial guides. Numerical results are shown to agree well with measurements of cascaded folded coaxial guide junctions. Index Terms—Cylindrical antennas, method of moments, monopole antennas, two-port circuits, waveguides.

I. INTRODUCTION

I

N THIS PAPER the authors describe a step along the path of designing a tuned frequency hopping antenna. Other authors have created such antennas by using automatic impedance tuners and switched impedance networks [1], [2]. It is proposed to realize a frequency hopping antenna by electronically varying the length of a folded coaxial cavity located below the feed of a coaxially-fed, cylindrical monopole antenna above an infinite ground plane (Fig. 1). In this investigation, the end result of the coupling of the field from one region to another in the proposed cavity, a folded coaxial guide, is accomplished by a 2-port network. Use of this approach reduces by one the number of coupled integral equations in the analysis of this antenna and thereby increases the potential for lowering the cost (CPU time, processing power, and memory use) of computing a solution, which is important in design via optimization. Similar domain decomposition techniques have been used successfully in the analysis of antennas with shielded loads [3], [4]. To this end, scattering parameters for the network representing the coax-to-coax junction are determined from the solution of an integral equation for the aperture electric field in the slot between the coaxial guides. The accuracy of the network model for the junction is demonstrated by comparing computed and measured scattering parameters for an apparatus

Manuscript received October 12, 2009; revised February 03, 2010; accepted February 07, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. A. Schreiber was with the Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634-0915 USA. He is now at Fredericksburg, VA 22407 USA (e-mail: [email protected]). C. Butler is with the Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634-0915 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2050452

Fig. 1. Cylindrical monopole antenna with folded coaxial cavity below the feed.

Fig. 2. Folded coaxial guide.

containing four cascaded junctions joined by coaxial transmission lines. The sensitivity of the scattering parameters to slot length and to mismatches in characteristic impedance is evaluated. It is important to note that the use of network analysis to increase the efficiency of integral equation solutions is not limited to the guide junction shown in Fig. 2 [5]. Such efficiency gains can be realized whenever the role of a member of a set of coupled integral equations can be replaced by a network.

0018-926X/$26.00 © 2010 IEEE

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II. INTEGRAL EQUATION The folded coaxial guide, illustrated in cross-section in Fig. 2, comprises two coaxial guides placed one within the other and joined by a circular gap in the shared conductor. As depicted in and filled by materials the figure, the guides are shorted at and . The superscripts and characterized by denote terms belonging to the inner and outer coaxial regions, respectively. The incident TEM electromagnetic waves in the inner and outer coaxial regions are (1) and

Fig. 3.

E

in aperture at 1 GHz.

(2) TABLE I DIMENSIONS OF FOLDED COAXIAL GUIDE JUNCTION

respectively, where and is the magnitude of the electric field. The short-circuit magnetic field in each region is (3) and (4) in which . The -directed magnetic field due to a -directed electric field in the gap in the side wall of either shorted coaxial region is [6], [7]

and is the norm of in the appropriate region [8]. When continuity of the -directed magnetic field through the slot in the shared conductor is enforced, the integral equation for the aperture electric field in the slot is obtained

(5) where and represent the inner and outer conductor radii, respectively, of either coaxial region in which the field is being is the aperture electric field. In (5) evaluated, and (6) in which

are the

zeros of

in each region

(8) Equation (8) is solved by typical numerical methods: the unis expanded in terms of piecewise constant pulses and known the resulting equation is point matched. The series converge at a sufficient rate that series acceleration is not needed to achieve numerical efficiency and accuracy. III. NETWORK PARAMETERS

(7)

Because coaxial guides support TEM fields, unique voltages , such that one can conveniently can be defined at

SCHREIBER AND BUTLER: ENHANCEMENT OF EFFICIENCY OF INTEGRAL EQUATION SOLUTIONS OF ANTENNAS

Fig. 4.

S

and S

for a single junction. Fig. 6. Reflection and transmission coefficients for s :  c , :  d , and :  e .

10 ( ) 15 ( )

Fig. 5.

S

and S

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20 ( )

= 0:1 (a), 0:5 (b),

for a single junction.

determine scattering parameters in terms of the traveling voltages in the two guides

(9) and denote reflected and incident voltages at port at . Port 1 and port 2 are located in the reference surface the outer and inner coaxial guides, respectively. Note that, if a negative polarity is assigned to the outer conductor of the outer guide and a positive polarity to the shared conductor, voltage in the outer guide is defined as (10)

Fig. 7. Reflection and transmission coefficients for differing ratios of Z

=Z

.

. The TEM electric field due to a sidewall ( and defined electric field in a coaxial guide, where as in (5)), is

(12) and the scattering parameters (9) are, explicitly

(13a)

and in the inner guide as (11)

(13b)

From (1) and (2) and the appropriate definitions of voltages, one finds that and

(13c)

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Fig. 8. Measurement apparatus.

and

TABLE II DIMENSIONS OF MEASUREMENT APPARATUS

(13d) IV. DATA AND DISCUSSION (at 1 GHz) in the aperture between the Sample data for two coaxial guides in the junction specified by the dimensions displays the and materials in Table I are presented in Fig. 3. expected edge condition at and appropriately does not exhibit singular behavior at . Data for and , and and for the junction (specified by Table I) are plotted as a functions of frequency in Figs. 4 and 5, respectively. For this lossless, reciprocal network, the scattering matrix is symmetric and unitary [9]. Furthermore, whenever the materials in the outer and inner guides are identical, , and , the solution of (8) for will be identical, regardless of whether (3) or(4) is considered to be the forcing function, which in(13) causes to be equal to and to be equal to . To determine the approximate location of that minimizes the reflection coefficients and maximizes the transmission coefficients, a comparison of the magnitude of reflection and transmission coefficients is presented in Fig. 6 as a function of frequency for a junction with the dimensions given in Table I and with set to , , , , and , respectively. The -axis scales chosen for Fig. 6 make it clear that selecting produces the optimal result of those presented with almost ideal reflection and transmission characteristics of 0 and 1, respectively. The sensitivity of the reflection and transmission coefficients (at 300 MHz) to a mismatch of characteristic impedances is depicted in Fig. 7, in which the characteristic impedance of the outer coaxial line is held constant at 50 and the characteristic impedance of the inner coaxial line is varied from 25 to 75 with . The coefficients become increasingly non-ideal as the mismatch of the characteristic impedances grows. Thus, it is desired to match the characteristic impedances as closely as possible, but the junction’s performance is insensitive to small mismatches of characteristic impedance due to manufacturing imprecision. The scattering parameters have been normalized by the appropriate characteristic impedances [9] to facilitate this comparison. It is important

to note that ideal reflection and transmission coefficients and a matched characteristic impedance are not necessary to tune a folded coaxial cavity. The apparatus depicted in cross-section in Fig. 8 is used to test the accuracy of the network model of the junction and is not intended for inclusion in an antenna. The coaxial lines on either end are GR874, which facilitates fabrication and the capture of measurements with standard 50 network analyzers. The inner conductors of the GR874 lines are attached to a hub supporting two thin-walled tubes. The GR874 lines and the hub are enclosed in a larger diameter tube and the outer conductors of the GR874 lines are soldered to the end caps. Retaining rings, not illustrated, hold the apparatus in compression to assure electrical contact is made between the outermost tube and the right end cap at the inner edge of the outermost tube. The dimensions of the apparatus are given in Table II. For computing the overall reflection and transmission coefficients of the apparatus, scattering parameters and the BLT equation are chosen over ABCD parameters and cascading because scattering parameters are more easily incorporated into integral equation solutions of antennas. The apparatus is modeled as five uniform transmission lines connected at four junctions as in Fig. 9 [10]–[13], where ports 1 and 2 of the apparatus are loand , respectively. The dimensions for cated at each junction and the length of each uniform transmission line are given in Tables III and IV, respectively. The propagation matrix is shown in (14) at the bottom of the next page, where in all cases because air is the dielectric material throughout the apparatus. The coefficients of the exponentials

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Fig. 9. Cascaded junctions connected by transmission lines.

Fig. 10. Measured and computed values for S

of the apparatus.

Fig. 11. Measured and computed values for S

of the apparatus.

Fig. 12. Measured and computed values for S

of the apparatus.

in (14) are a consequence of the voltage sign conventions introduced in (10) and (11). The scattering matrix is

(15) is the ratio of the reflected voltage at port to the where incident voltage at port in the junction with the other ports terminated in matching loads. The reversal of rows and columns in two of the junction scattering matrices is due to the propagation path of an electromagnetic wave through the apparatus. The

propagation path is indicated in Fig. 9 by the placement of the labels and to indicate the location of ports 1 and 2, respectively, for each junction. The propagation and scattering matrices are substituted into the voltage BLT equation (16)

(14)

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V. CONCLUSION

Fig. 13. Measured and computed values for S

A method is presented for enhancing the efficiency of integral equation solutions of cylindrical monopole antennas loaded with folded coaxial cavities below the feed. Agreement between computed and measured scattering parameters indicates that this method can be used to accurately replace the field coupling represented by a coupled integral equation with a 2-port network. The reflection and transmission coefficients of the junction appear to be relatively insensitive to small mismatches in characteristic impedance. The proposed folded coaxial guide cavity is necessary because of the impracticality of placing an inductor with a ferrite core below the feed. Because the length of a coaxial cavity below the feed can be extended in this fashion, a greater range of inductive values will be accessible at low frequencies than would ordinarily be possible. The method herein is limited to the case in which the antenna is operated below cutoff of the higher-order mode with the lowest frequency, but can be applied wherever a 2-port network is identified. Future work is planned to show the efficacy of this method in the optimization of a cylindrical monopole antenna.

of the apparatus.

TABLE III DIMENSIONS FOR JUNCTIONS

REFERENCES

TABLE IV TRANSMISSION LINE LENGTHS

in which

is the identity matrix and

is the source vector

(17)

depending on if the incident wave being considered is present at port 1 or 2, respectively. A comparison between computed and measured values for the scattering parameters of the apparatus is presented in Figs. 10–13, where computed and measured data are shown to agree well. The discrepancy between the computed and measured data increases with frequency and is likely due to modeling the thin-walled tubes in the apparatus as vanishingly thin tubes with their tube walls located at the mean of their inner and outer radii and to machining (by the authors) of the apparatus components.

[1] J. R. Moritz and Y. Sun, “Frequency agile antenna tuning and matching,” in Proc. 8th Int. Conf. on HF Radio Systems and Techniques, 2000, pp. 169–174, IEE Conf. Pub. No. 474. [2] J. D. Neal and D. J. Miley, “Broadbanding of electrically small HF antennas for frequency agile applications by use of pin diode switched matching networks,” in Proc. IEEE Military Communications Conf. MILCOM, Nov. 31–2, 1983, vol. 2, pp. 544–548. [3] F. A. Pisano, III, C. M. Butler, and J. P. Rudbeck, “Analysis of a tubular monopole loaded with a shielded helical coil,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 969–977, Apr. 2004. [4] M. D. Lockard, “Accurate modeling and design techniques for loaded monopole antennas,” Ph.D. dissertation, Clemson University, Charleston, SC, 2005. [5] A. W. Schreiber and C. M. Butler, “Increased efficiency of integral equation solutions of antennas by incorporation of network principles,” presented at the National Radio Science Meeting, Boulder, CO, 2009. [6] C. M. Butler, “Some integral equation difficulties,” presented at the IEEE International Symposium on Antennas and Propagation and USNC-URSI National Radio Science Meeting, Charleston, SC, 2009. [7] C. M. Butler, “Analytical difficulties encountered in eigen expansions of fields,” presented at the IEEE Int. Symp. on Antennas and Propagation and USNC-URSI National Radio Science Meeting, Charleston, SC, 2009. [8] M. G. Harrison and C. M. Butler, An Analytical and experimental investigation of planar discontinuities in coaxial waveguides Air Force Weapons Lab., 1981, Tech. Rep. AFWL-TR-79-187. [9] R. E. Collin, Foundations for Microwave Engineering. New York: Wiley, 2001. [10] C. E. Baum, T. K. Liu, and F. M. Tesche, “On the analysis of general multiconductor transmission-line networks,” Interaction Note, Nov. 1978. [11] F. M. Tesche and T. K. Liu, “Application of multiconductor transmission line network analysis to internal interaction problems,” Electromagnetics, vol. 6, 1986. [12] F. M. Tesche, M. Ianoz, and T. Karlsson, EMC Analysis Methods and Computational Models. New York: Wiley, 1997. [13] F. M. Tesche and C. M. Butler, “On the addition of EM field propagation and coupling effects in the BLT equation,” Interaction Note, Jun. 2004.

SCHREIBER AND BUTLER: ENHANCEMENT OF EFFICIENCY OF INTEGRAL EQUATION SOLUTIONS OF ANTENNAS

Adam W. Schreiber (S’01–M’09) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Clemson University, Clemson, SC, in 2005, 2007, and 2009, respectively. His research interests include analytical Green’s functions, penetration of fields through apertures, broadband and low-profile antennas, and electromagnetic compatibility. Dr. Schreiber was a Science, Mathematics and Research for Transformation Fellow from 2007 to 2009. He is a member of Eta Kappa Nu, Tau Beta Pi, and the National Society of Collegiate Scholars, and is an associate member of Commission E of the International Union of Radio Science (URSI).

Chalmers M. Butler (S’61–M’63–SM’75–F’83– LF’01) received the B.S. and M.S. degrees from Clemson University, Clemson, SC, and the Ph.D. degree from the University of Wisconsin, Madison. He has been a member of the faculty at Louisiana State University, the University of Houston, and the University of Mississippi where he was Chairman of Electrical Engineering (1965–74) and University Distinguished Professor (1976–83). Since 1985, he has been at Clemson University where he is presently Alumni Distinguished Professor and the Warren H. Owen Professor of Electrical and Computer Engineering. His research has focused primarily upon Green’s functions and integral equation techniques in electromagnetics and upon numerical methods for solving integral equations. His principal applications interests are in antennas and aperture penetration. Prof. Butler is a member of Sigma Xi, Tau Beta Pi, Phi Kappa Phi, and Eta Kappa Nu. He received the Western Electric Fund Award and numerous awards for excellence in teaching at the University of Mississippi and Clemson University. He has received two Best Paper Awards: Best Basic EMP NonSource Region Paper during 1975–78 from the SUMMA Foundation and the 1986 Oliver Lodge Premium Award from the Institute of Engineering and Technology (IET) London, U.K. He received the 1990 and the 2003 Editor’s Citation

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for Excellence in Refereeing in Radio Science and several awards for excellence in research and scholarship at Clemson. From the University of Wisconsin he received the Centennial Medal for Contributions to Electrical and Computer Engineering. He is a recipient of the IEEE Millennium Medal and the 2003 recipient of the IEEE AP-S Chen-To Tai Distinguished Educator Award. In 2009 he was recognized by the National Academies of Science, Engineering, and Medicine for “distinguished contributions to the field of radio science and outstanding service to both the USNC and the international scientific union.” He is a member of Commissions B and F of the International Union of Radio Science (URSI), and he is a Life Fellow of the IEEE. He has served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (1972–77 and 1981–83) and of the IEEE TRANSACTIONS ON EDUCATION, as a member of the IEEE Antennas and Propagation Society Administrative Committee (1976–78 and 1988–90), as an IEEE Antennas and Propagation Society National Distinguished Lecturer (1977–79), as Chairman of U. S. Commission B of URSI (1983–85), as International Commission B Editor of the Review of Radio Science (1978–80 and 1981–83), and as National President of Eta Kappa Nu. He has been a member of the Editorial Boards of Electromagnetics and of Computer Applications in Engineering Education, and has served as Guest Editor of two special issues of Radio Science. He served as Secretary (1985–87), Vice Chair (1987–90), and Chair (1990–93) of the U.S. National Committee for URSI. He has been Vice Chair (1994–96) and Chair (1997–99) of International Commission B of URSI. In 2008 he completed a six-year term as a Vice President of URSI. He was a U.S. delegate to the 18th – 26th General Assemblies of URSI and chaired the U.S. delegation for the 24th (Kyoto, 1993). He has served on a number of committees and panels including the IEEE Hertz Medal Committee (1996–97), the IEEE Awards Board, the National Academy of Sciences Panel for Evaluation of the Center for Electronics and Engineering of the National Bureau of Standards (now NIST) (1982–86), and the National Research Council’s Panel for the Evaluation of the U.S. Army’s Mine Detection Program (1985–90), which he chaired. He has been a member of numerous technical program committees of National Radio Science Meetings and IEEE AP-S International Symposia and served as General Chairman of the NIST Workshop on EMI/EMC Metrology Challenges for Industry. He was the Vice Chairman of the Technical Program Committee for the 1995 URSI Electromagnetic Theory Symposium in St. Petersburg, Russia, and chaired the same committee for the 1998 URSI Electromagnetic Theory Symposium held in Thessaloniki, Greece.

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A Calderón Multiplicative Preconditioner for Coupled Surface-Volume Electric Field Integral Equations Hakan Ba˘gcı, Member, IEEE, Francesco P. Andriulli, Member, IEEE, Kristof Cools, Femke Olyslager, Fellow, IEEE, and Eric Michielssen, Fellow, IEEE

Abstract—A well-conditioned coupled set of surface (S) and volume (V) electric field integral equations (S-EFIE and V-EFIE) for analyzing wave interactions with densely discretized composite structures is presented. Whereas the V-EFIE operator is well-posed even when applied to densely discretized volumes, a classically formulated S-EFIE operator is ill-posed when applied to densely discretized surfaces. This renders the discretized coupled S-EFIE and V-EFIE system ill-conditioned, and its iterative solution inefficient or even impossible. The proposed scheme regularizes the coupled set of S-EFIE and V-EFIE using a Calderón multiplicative preconditioner (CMP)-based technique. The resulting scheme enables the efficient analysis of electromagnetic interactions with composite structures containing fine/subwavelength geometric features. Numerical examples demonstrate the efficiency of the proposed scheme. Index Terms—Calderón preconditioning, multiplicative preconditioning, surface electric field integral equation, volume electric field integral equations.

I. INTRODUCTION NTEGRAL equation techniques for modeling electromagnetic interactions with composite structures comprised of perfect electrically conducting (PEC) surfaces and dielectric volumes have many practical applications. These techniques often seek the solution of a coupled set of surface (S) and volume (V) electric field integral equations (EFIEs) that enforce electric field boundary and consistency conditions on PEC surfaces and throughout dielectric volumes, respectively [1]–[3]. To permit the analysis of structures with sub-wavelength geometric features, e.g., microwave circuits and complex antenna feeds, these techniques should apply robustly to PEC

I

Manuscript received March 01, 2009; revised October 22, 2009; accepted February 09, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported in part by AFOSR MURI Grant F014432-051936 aimed at modeling installed antennas and their feeds and in part by NSF Grant DMS 0713771. H. Ba˘gcı was with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA. He is now with the Division of Physical Sciences and Engineering, King Abdullah University of Science and Engineering, Thuwal 23955, Saudi Arabia (e-mail: [email protected]). F. P. Andriulli was with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 48109, USA. He is now with the Electronics Department, Politecnico di Torino, 10129 Turin, Italy. K. Cools is with the Department of Information Technology at the Ghent University, B-9000 Ghent, Belgium. F. Olyslager (deceased) was with the Department of Information Technology, Ghent University, B-9000 Ghent, Belgium. E. Michielssen is with the Department of Electrical Engineering and Computer Science at the University of Michigan, Ann Arbor, MI 48109 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.2010.2050419

surfaces and dielectric volumes approximated by locally or globally-dense spatial meshes. While V-EFIE operators are bounded and well-posed even when applied to densely discretized volumes [4], [5], S-EFIE operators become ill-posed when applied to densely discretized surfaces [6]–[10]. As a result, the methods-of-moments (MOM) systems obtained upon discretizing coupled sets of S- and V-EFIEs applied to structures involving densely discretized PEC surfaces tend to be ill-conditioned and their iterative solution becomes prohibitively expensive. In recent years, many techniques that leverage Calderón identities to alleviate the ill-posedness of S-EFIE operators have been proposed [6]–[10]. These techniques exploit the self-regularizing property of the S-EFIE, i.e., the fact that its square has a bounded spectrum, thus giving rise to MOM matrices that are well-conditioned, independent of the surface discretization density. Unfortunately, many of these methods suffer from implementation difficulties related to the fact that the EFIE operators’ product needs to be discretized [8]. Various methods that use ad hoc integration rules and/or operational manipulations have been used for this purpose [7], [9], [10]; unfortunately none of them is easily integrated into existing MOM codes that discretize the S-EFIE using the well-known Rao-Wilton-Glisson (RWG) functions [11]. This paper presents a well-conditioned coupled set of S- and V-EFIEs. The first equation of the coupled set imposes electric field boundary conditions on PEC surfaces and is regularized by the S-EFIE operator. The resulting equation calls for the discretization of S-EFIE/S-EFIE and S-EFIE/V-EFIE operator products, which is accomplished using a Calderón multiplicative preconditioner (CMP)-based technique [8]. The second equation of the coupled set links electric fields and polarization currents throughout dielectric volumes. The proposed approach preserves the original CMP’s multiplicative nature and requires only a standard RWG and Schaubert-Wilton-Glisson (SWG) based discretization [12] of the surfaces and volumes. As a result, the proposed preconditioner is easily implemented into existing MOM codes and the resulting solver can trivially be accelerated via available fast matrix-vector multiplication methods including the adaptive integral method (AIM) [13], multilevel fast multipole algorithm (MLFMA) [14], [15], and their parallelized versions. The contributions of this paper are threefold: (i) The paper extends the CMP-based discretization technique of [8], originally developed to precondition the S-EFIE (for analyzing electromagnetic interactions with PEC structures when discretized using dense or multiscale meshes), to allow the

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˘ et al.: A CALDERÓN MULTIPLICATIVE PRECONDITIONER FOR COUPLED S-V EFIEs BAGCI

preconditioning of the coupled system of S- and V-EFIEs (for analyzing electromagnetic interactions with composite structures discretized using dense or multiscale meshes). The paper details several issues, which are key to a successful implementation of a CMP-based preconditioner for coupled S- and V-EFIEs and discusses the efficiency of the proposed method when used in conjunction with the AIM for accelerating matrix-vector multiplications. (ii) A numerical study is performed to establish the spectral properties of the coupled system (in lieu of a mathematical analysis, which is cumbersome). For a simple geometry, the spectra of the matrices arising from the standard MOM- and CMP-based discretizations of the coupled system are compared. This comparison demonstrates that the extended CMP-based discretization technique produces a much better conditioned discretized coupled set of equations than the standard MOM-based discretization technique. (iii) In addition to the tests done on simple geometries, the paper also demonstrates that the proposed CMP-based discretization technique maintains its effectiveness when used to characterize realistic composite structures with sub-wavelength features, including a dielectric-filled waveguide slot antenna on a shuttle model and dielectric antennas with fine-featured metallic feeds. It should be noted that the characterization of these structures would be impossible without the extension of the CMP-based discretization technique proposed in this paper. This paper is organized as follows. Section II formulates a coupled set of S- and V-EFIEs and details its CMP-based discretization. Sections III and IV present numerical results and conclusions and avenues for future research, respectively.

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Fig. 1. Description of scattering problem involving an abstract composite structure.

Here,

denotes the total electric field, is a unit normal to , and the operators and are

(3) and

II. FORMULATION

(4)

This section describes the proposed CMP regularization technique for the coupled set of S- and V-EFIEs. Section II-A formulates the set of S-EFIE and V-EFIE in surface current and electric flux densities and details its standard MOM-based discretization. Section II-B describes the proposed CMP regularizer.

In (3) and (4), space Green function. In

,

and

is the freeare related as

(5) A. Coupled Set of S- and V-EFIEs and Its Standard MOM-Based Discretization

is the contrast parameter [12] and is the electric flux density. Inserting (5) into (1) and (2) yields the coupled set of S- and V-EFIEs in and where

Consider a composite structure comprising PEC surfaces and potentially inhomogeneous dielectric volumes that reside and denote the permittivity of in free space (Fig. 1). Let and free-space, respectively; let denote the permeability excites of all of space. A time-harmonic electric field and ; here and in what follows, time dependence is assumed and suppressed. The surface and volume (polarization) and induced on and in gencurrent densities . Enforcing electric field erate the scattered electric field boundary and consistency conditions on and in yields

(1) (2)

(6) (7) Here, the operators , are ator

and

, which complement the oper-

(8) (9)

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with matrix

. The impedance in (12) can be decomposed as (15)

, , , and where account for surface test-surface basis, volume test-volume basis, surface test-volume basis, and volume test-surface basis interactions, respectively. Their entries are

(16) (17) (18) (19) Fig. 2. Basis functions used in the CMP-based discretization of S-EFIE. (r), (b) curl-conforming (a) Div-conforming RWG basis function, f ^ (r ) RWG basis function, n (r), (c) div- and quasicurl-conforming f (r), (d) curl- and quasidiv-conforming BC function, BC basis function f ^ (r) n f (r).

2

2

To numerically solve (6) and (7), and are approximated by meshes of planar triangles (with smallest edge size ) and and tetrahedrons (with smallest edge size ), and are approximated as (10)

(11) where , and , are unknown expansion coefficients, , are zeroth-order div-conforming RWG surface basis functions [Fig. 2(a)] [11] defined on pairs of , are zeroth-order triangles, and div-conforming SWG volume basis functions [12] defined on tetrahedron facets. To determine the coefficients and , (10) and (11) are inserted into (6) and (7), and the resulting equations are tested by curl-conforming , [Fig. 2(b)] and , ; this produces the linear system of equations of dimension (12) Here, and are vectors of expansion coefficients and tested incident fields, respectively; their entries are (13)

else else

(14)

When analyzing electrically large and/or complex structures, is large, (12) cannot be solved dii.e., when rectly and iterative solvers are called for. The computational cost of solving (12) iteratively scales multiplicatively with the cost of applying the impedance matrix to a trial solution vector and the number of iterations required to reach a specified residual. The cost of a matrix-vector multiplication always can be reduced by using AIM [13] or MLFMA [14], [15] accelerators. The required number of iterations typically scales with ’s condition number with small condition numbers guaranteeing fast convergence of the iterative solver. The coupled system’s [(6) and (7)] spectrum depends on the spectral properties of the S- and V-EFIE operators ( and ) and the non-symmetric off-diagonal (coupling) terms. While it is tempting to “guess” the conditioning properties of from those of its diagonal blocks, this guess must be verified numerically. Numerical verification is required not only because the spectral properties of the coupled operator cannot be established mathematically, but also because the discretization of a (spectrally) well-behaving operator does not necessarily produce a well-conditioned matrix. The spectral properties of the S-EFIE operator are well-documented: it is known that its singular values accumulate at zero and infinity as it contains a singular operator (the first integral in (3), i.e., the vector potential contribution) and a hypersingular operator (the second integral in (3), i.e., the scalar potential contribution) [6]; in other words, the S-EFIE operator is unbounded. As a reis increasingly ill-conditioned when sult, . Unlike the S-EFIE operator, the V-EFIE operator’s spectrum is bounded [4], [5]. The scalar potential contribution of the V-EFIE operator is Cauchy-singular (but not hypersingular) and its dominant contribution results from a volume integral. It has been shown in [4], [5] that matrices resulting from the discretization of V-EFIE operator are well-conditioned regardless of the discretization density. This means that is well-conditioned even when . (It should be noted here that the conditioning of also depends on but this dependence is weak and even for very large the is much smaller than that of condition number of for .) Unfortunately, alone

˘ et al.: A CALDERÓN MULTIPLICATIVE PRECONDITIONER FOR COUPLED S-V EFIEs BAGCI

renders ill-conditioned and the iterative solution of (12) prohibitively expensive or even impossible in the presence of dense discretizations.

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and well-conditioned. The spaces of discretized operator products are expressed as

B. Calderon Regularization and CMP-Based Discretization of Hybrid Set of S- and V-EFIEs

(23)

The unbounded nature of the S-EFIE operator can be cured using the well-known Calderón identity [16]

(24)

(20) Here, is a compact operator when acting on smooth surfaces [6]; this makes a second kind operator. Therefore, (6) can be and the set of equations regularized using

The entries of the impedance matrices and are given by (16) and (18) while those of the and Gram matrix are impedance matrix (25) and (26)

(21) (22) can be solved instead of the standard set (6) and (7). The discretization of the operator products in (21) is by no means trivial. Various methods that use ad hoc integration rules and/or operational manipulations for discretizing the operator have been proposed [7], [9], product [10]; however none of these methods is easily integrated into readily available MOM codes that discretize the standard S-EFIE using the well-known RWG basis functions [11]. In this work, the CMP approach first proposed in [8] to discretize Calderón-preconditioned S-EFIEs is used to discretize the operator products in (21). The reader is referred to [8] for a detailed formal mathematical description of the CMP concept. Consider the initial discretization of comprised of planar is expanded in terms of standard triangles on which , div-conforming RWG functions, [see (10)]. A barycentric mesh is obtained by adding the three medians to each triangle of the initial mesh; on the edges of this , barycentric mesh a new set of RWG basis functions and Buffa-Christiansen (BC) basis functions [17] , [Fig. 2(c)] are defined. BC basis functions are linear and they are div- and quasicurl-concombinations of are curl- and quasidiv-conforming. Note that forming [Fig. 2(d)] [17]. These properties render the Gram matrix, which links spaces discretized by quasicurl-conforming and curl-conforming , well condi, the tioned [8]. In the operator product source and test spaces of the right S-EFIE operator are and , respectively. discretized using The source and test spaces of the left S-EFIE operator are discretized using and , respectively. Similarly, in the operator product , are the source and the test spaces of the operator and , respectively. discretized using is discretized as above. The left S-EFIE operator These choices of basis and testing functions render the Gram to the test matrices linking the source space of

Note that if one would use div- and curl-conforming RWGs to discretize the source and test spaces of and , the Gram matrix test space of would be singular [8]. Let , , and denote the spaces spanned by , , and , respectively. Upon conand that express basis structing transformation matrices and as linear combinations of those functions in in , (23) can be rewritten using only two impedance , as matrices of the same type, viz.

(27) where and the entries of the impedance matrix are (28) Because only the barycentric mesh (and not the initial one) is supplied to the code, (24) is replaced with the discretization

(29) where the entries of the impedance matrix

are (30)

The discretization of the operator in (22) can be achieved and . However this is classically using not done directly but using and since only the barycentric mesh (and not the initial one) is supplied to the code. In other words, (31) where the entries of the impedance matrix given by

are (32)

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The discretization of the operator

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in (22) is unchanged (33)

are given by The entries of the impedance matrix (17). The impedance matrices , , , and are trivially computed using existing MOM codes that use RWG and SWG basis functions. Explicit expressions of the elements of the matrices and can be found in [8]. Inserting (27), (29), (31), and (33) into (21) and (22) discretizing the right hand sides, rearranging the resulting equations, and applying diagonal preconditioning yields (34), shown is at the bottom of the page. Here, the impedance matrix (35) the entries of the matrices

and

are (36)

else

(37)

else the entries of the right hand side vector

Fig. 3. Spectra of matrices obtained via standard MOM- and CMP-based discretizations. (a) Geometry description: Isometric, side and top views. (b) Singular values of the matrices.

are the standard MOM-based discretization, the spectrum of the matrix else (38)

and is the vector of unknown coefficients [see (13)]. Note that the multiplication with properly scales the en(so that ’s diagonal entries of renders tries are all 1’s) and multiplication with well-conditioned [8] in the presence of multiscale discretizations. The scaling constant in (37) ensures that the diagonal entries of the matrix are also all 1’s; the value of can easily be obtained by dividing any entry of by the corresponding entry of the matrix product . The regularized coupled system’s [(21) and (22)] spectrum, similar to that of the standard one [(6) and (7)], depends on the spectral properties of the regularized S-EFIE and V-EFIE operators and the off-diagonal (coupling) terms and the technique used for discretizing them. To demonstrate that the CMPbased discretization produces better-conditioned matrices than

(39) is compared to that of the standard MOM matrix. The results are presented in Section III. It should be noted here that this matrix is never computed explicitly during the solution of matrix equation (34), which is done iteratively. The number of iterations is independent of the smallest edge sizes and as demonstrated in Section III. The computational cost of solving (34) is that of performing the matrix-vector multiplications on its left hand side, times the number of iterations. The cost of multiplying by a vector scales as only the sparse matrices , , and . (Note that inversion of the sparse Gram matrix is never carried out explicitly; whenever the matrix-vector product is needed, it is computed via the iterative solution of the , which only requires a linear system

(34)

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Fig. 4. Analysis of scattering from PEC and dielectric spheres. (a) Geometry and excitation description. (b) Number of iterations required for the relative residual error of the solutions of (34) and (12) to reach 10 , versus  =  . (c) Comparison of RCS obtained on  = 0 and  = 90 planes after solving (34) and (12).

few iterations [8].) The dominant computational cost at a given with a vector, iteration is then due to the multiplication of which can be reduced using various acceleration techniques [13]–[15]. In this work, AIM [13] is used for this purpose. Let and represent the costs of multiplying and by a vector; , where then and are the numbers of nodes on the three dimensional (3-D) auxiliary AIM grid that encloses the composite structure and is used for accelerating the by a vector, respectively. Here, multiplication of and and are the AIM grid sizes (i.e., AIM node spacing); they are assumed identical along the , , and directions. As a rule and , where and of thumb, are the average edge sizes in the (combined) initial surface and volume meshes, and the (combined) barycentric surface and volume meshes, respectively. Ignoring the cost of operations associated with the multiplication of the sparse , while matrices, the total cost of solving (12) is . Here the cost of solving (34) is and are the numbers of iterations required for the relative residual error of the solutions of (12) and (34) to reach a certain to , it is concluded that the threshold. By comparing iterative solution of (34) will be faster than that of (12) as long as . is always smaller than 1 (because of the The ratio barycentric division of the surface discretization) but typically larger than 0.5 (because of the presence of the volumetric mesh). A more efficient but less trivial implementation is possible. Inserting (23), (24), (31) and (33) into (21) and (22), discretizing the right hand sides, and applying diagonal preconditioning yields

(40) Solution of (40) using an existing MOM code is far less trivial than that of (34) since now needs to be computed explicitly. Additionally, one needs to modify the AIM accelerator

by a vector. Asto allow for the fast multiplication of suming such an AIM accelerator is constructed, it can use the same auxiliary grid used for accelerating the matrix-vector multiplication associated with since the support of BC basis functions is (roughly) the same as that of RWG basis functions defined on the standard mesh (i.e., node spacing needed for auxiliary AIM grids is the same). This means that the cost of solving (40) is , where depends on what ratio of the AIM’s grid encloses the PEC surfaces; is at most 2 (This happens when the structure consists of only PEC surfaces). It is clear from this discussion that the iterative solution of (40) is faster than that of (12) as long as is satisfied. Numerical results show that this is satisfied even for moderately dense discretizations. III. NUMERICAL RESULTS In this section, the proposed method is applied to the analysis of scattering from spheres and a shuttle loaded with a dielectric-filled waveguide slot antenna, and radiation from dielectric antennas inclusive their PEC feeds. The results presented here were obtained using a parallel and AIM-accelerated MOM code that uses a transpose-free quasi-minimal residual iterative scheme [18] to solve matrix (34) and (12); a diagonal preconditioner is used for (12). All simulations were carried out on a cluster of dual-core 2.8-GHz AMD Opteron 2220 SE processors at the Center for Advanced Computing, University of Michigan. A. Scattering From Spheres 1) PEC Sphere Coated by a Half Dielectric Shell: To compare the spectra of the matrices obtained by the standard MOMand CMP-based discretizations, scattering from a PEC sphere coated by a half-shell is analyzed. The radius of the sphere and the thickness of the half-shell are 0.5 m and 0.2 m, respectively . [Fig. 3(a)]. The dielectric constant of the shell is ( ). The frequency of the excitation is The largest, average, and the smallest element sizes in the mesh ), 9.3903 cm ( ), and 2.8215 are 19.393 cm ( ), respectively; the number of RWGs and SWGs cm ( and , respectively. Fig. 3(b) are compares the spectra of the matrices [see (15)] and

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Fig. 5. Analysis of scattering from a fully dielectric-coated PEC sphere. (a) Number of iterations required for the relative residual error of the solutions of (34) and (12) to reach 10 , versus  =  . (b) Comparison of solutions of (34) and (12).

Fig. 6. Analysis of scattering from a shuttle model with a dielectric-filled slot waveguide mounted on its fuselage. (a) Geometry and excitation description. (b) The multiscale discretization of the shuttle and the waveguide surfaces. (c) Relative residual error obtained during the iterative solution of (34) and (12). (d) Current density induced on surfaces of the shuttle and the waveguide decibel scale.

[see (39)] obtained via the standard MOM- and CMP-based discretizations, respectively. The condition numbers (the ratio of the largest singular value to the smallest one) of the matrices and are 9.8562 and 744.42 respectively. It should be noted here that the discretization used in this example can still be considered “coarse” when compared to the discretizations on realistic structures with sub-wavelength structures, such as the composite antennas and resonators studied below. For these examples, the large difference between iteration counts points to a big difference between the condition numbers as well.

2) PEC and Dielectric Spheres: Consider the two adjacent spheres, one PEC and the other dielectric, shown in Fig. 4(a). The PEC and dielectric spheres are centered about the origin and (2.5 m, 0, 0), respectively; both spheres have radius 1 m. The di. The electric constant of the dielectric sphere is spheres are excited by a polarized plane wave propagating in the direction. The frequency of excitation is ( ). The simulation is repeated for seven different discretizations with smallest edge sizes ranging from to . Table I presents and

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TABLE I THE NUMBERS OF STANDARD RWGS AND SWGS FOR ALL SEVEN DISCRETIZATIONS

TABLE II THE NUMBERS OF STANDARD RWGS AND SWGS FOR ALL SEVEN DISCRETIZATIONS

for all models analyzed. Fig. 4(b) presents the number of iterations required (for the relative residual error of the solutions ) versus . As expected, of (34) and (12) to reach the number of iterations required for the solution of (34) is independent of the discretization density; it is roughly 25. For the simulation with the densest discretization density ( ) measured CPU times indicated that the iterative solution of (34) was approximately 6.9 times faster than that of (12). Fig. 4(c) presents the spheres’ radar cross sections (RCSs) comand planes after solving (34) or puted on the mesh. The relative L2 (12) using the norms of the difference between the RCS results on the and planes are 0.0221% and 0.0164%, respectively. 3) PEC Sphere Coated by a Full Dielectric Shell: The proposed method is used to analyze scattering from a dielectric-coated PEC sphere centered at the origin; the radius of the sphere and the thickness of the dielectric shell are 1 m and 0.2 m, respectively. The dielectric constant of the shell is . The sphere is excited by the plane wave used in Section II-A.I. Similarly, the simulation is repeated for seven different discretizations with smallest edge sizes changing from and . Table II and for all discretizations. Fig. 5(a) presents presents the number of iterations required (for the relative ) versus error of the solutions of (34) and (12) to reach . The number of iterations required for the solution of (34) is constant and hovers around 24. For the simulation ) with the densest discretization density ( measured CPU times indicated that the iterative solution of (34) was approximately 11.2 times faster than that of (12). Fig. 5(b) shows that the solutions of (34) and (12) for the simulation with are practically the same; the relative norm of the difference between both solutions is 0.1874%.

Fig. 7. Analysis of radiation from a hemispherical dielectric resonator. Geometry description: (a) Top and (b) bottom views and (c) the cross section (dimensions are in cm). (d) Multiscale surface mesh of the geometry (zoomed to the feed probe). (e) Relative residual error obtained during the iterative solution of (34) and (12). (f) Normalized field patterns computed on xz and yz planes.

B. Scattering From a Space Shuttle Model The proposed technique is used to analyze low-frequency scattering from a shuttle model with a dielectric-filled slot waveguide mounted on its side. The shuttle model is excited by a polarized plane wave propagating in the direction at ( ). The length, width, and height

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Fig. 8. Analysis of radiation from a dielectric rod antenna. Geometry description: Isometric views of (a) the whole geometry (b) the metallic feed, and (c) the antireflective dielectric layer and (d) the cross section of the whole geometry (dimensions are in mm). (e) Relative residual error obtained during the iterative solution of (34) and (12). (f) Normalized field patterns computed on xz plane.

of the shuttle are , , and , respectively [Fig. 6(a)]. A slot-waveguide antenna, which is filled with dielectric with , is located on the side of the shuttle [Fig. 6(b)]. The width and height of the slot and the length of the waveguide are , , and , respectively. The multiscale discretization of the shuttle and the waveguide surfaces is shown in Fig. 6(c). For this mesh the largest, average, and the smallest element sizes are 1.1400 m ( ), 0.35120 m ( ), 1.0589 cm ( ), respectively; also , and . The iterative solver required 274 and 7396 iterations for the relative residual error of the solutions of (34) and (12) to reach , respectively [Fig. 6(d)]. Measured CPU times indicated that the iterative solution of (34) was approximately 2.4 times faster than that of (12). The relative L2 norm of the difference between the two solutions is 0.2168%. Fig. 6(d) shows three different views of the magnitude of the current induced on the shuttle’s surface. C. Radiation From a Hemispherical Dielectric Resonator Consider the hemispherical dielectric resonator antenna (with an air gap) shown in Fig. 6(a)–(c). The dielectric constant of the hemispherical shell is . The resonator is excited by a feed probe at ( ). The multiscale nature of the spatial mesh around the feed probe is highlighted in Fig. 7(d). The fine discretization around the feed is called for to properly model the curvature of the feed probe and the distribution of fields around it. For this discretization, the largest, average, and the smallest element sizes are 4.46694

), 1.11553 mm ( ), and 0.124458 mm ), respectively; also and . The iterative solver required 737 and 11 675 iterations for the relative residual error of the solutions of (34) and (12) to reach , respectively [Fig. 7(e)]. Measured CPU times indicated that the iterative solution of (34) was approximately 3.6 times faster than that of (12). The relative L2 norm of the difference between the two solutions is 0.8325%. Fig. 7(e) shows the normalized radiated field patterns (on the and planes) obtained from the solutions of (12) and (34).

mm ( (

D. Radiation From a Dielectric Rod Antenna Finally, the proposed method is used to analyze radiation from . a dielectric rod antenna [Fig. 7(a)–(d) [19], with The end of the rod is coated with an antireflective dielectric layer, with . The antenna is fed by a rectangular PEC waveguide and the waveguide is excited by a feed probe at ( ) [19]. Similar to the previous example, the surface of the feed probe and the waveguide surfaces near to the probe are densely discretized. For this simulation the largest, average, and smallest element sizes are 2.89224 mm ( ), 1.48322 mm ( ), and 0.044623 mm ( ), respectively; also and . The iterative solver required 659 and 14 633 iterations for the relative residual error of the solutions of (34) and (12) to reach , respectively [Fig. 8(e)]. Measured CPU times indicated that the iterative solution of (34) was approximately 6.5 times faster than that of (12). The relative norm of the

˘ et al.: A CALDERÓN MULTIPLICATIVE PRECONDITIONER FOR COUPLED S-V EFIEs BAGCI

difference between the two solutions is 0.9152%. Fig. 8(f) shows that the normalized radiated field pattern (on plane) obtained from the solution of (34) agrees well with that computed using the imaginary-distance beam-propagation method [19]. IV. CONCLUSION This paper presented a CMP-based regularizer for a coupled set of S- and V-EFIEs pertinent to the analysis of densely discretized hybrid PEC-dielectric structures. The proposed technique combines a CMP for the S-EFIE and a diagonal preconditioner for the V-EFIE. Just like in the original CMP, the preconditioner presented herein is multiplicative and easily integrated into available MOM codes that discretize S- and V-EFIEs using RWG and SWG basis functions, respectively. The proposed preconditioner is used in conjunction with an existing parallel and AIM accelerated MOM code. The numerical results obtained using this code confirmed the effectiveness of the proposed technique and its applicability to the electromagnetic characterization of composite structures with sub-wavelength features. ACKNOWLEDGMENT The authors would like to thank Mr. F. Valdés for his help in preparing meshes for the hemispherical dielectric resonator and the dielectric rod antenna. REFERENCES [1] C. C. Lu and W. C. Chew, “A coupled surface-volume integral equation approach for the calculation of electromagnetic scattering from composite metallic and material targets,” IEEE Trans. Antennas Propag., vol. 48, pp. 1866–1868, Dec. 2000. [2] A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “A parallel FFT accelerated transient field-circuit simulator,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 2851–2865, Sep. 2005. [3] T. K. Sarkar, S. M. Rao, and A. R. Djordjevic, “Electromagnetic scattering and radiation from finite microstrip structures,” IEEE Trans. Microw. Theory Tech., vol. 22, pp. 1568–1575, Nov. 1990. [4] N. V. Budko and A. B. Samokhin, “Spectrum of the volume integral operator of electromagnetic scattering,” SIAM J. Sci. Comput., vol. 28, no. 2, pp. 682–700, 2006. [5] J. Rahola, “On the eigenvalues of the volume integral operator of electromagnetic scattering,” SIAM J. Sci. Comput., vol. 21, no. 5, pp. 1740–1754, 2000. [6] J.-C. Nedelec, Acoustic and Electromagnetic Equations. New York: Springer-Verlag, 2000. [7] R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antennas Propag., vol. 52, pp. 362–372, Feb. 2004. [8] F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative Calderon preconditioner for the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 56, pp. 2398–2412, Aug. 2008. [9] S. Borel, D. P. Levadoux, and F. Alouges, “A new well-conditioned integral formulation for Maxwell equations in three dimensions,” IEEE Trans. Antennas Propag., vol. 53, pp. 2995–3004, Sep. 2005. [10] H. Contopanagos, B. Dempart, M. Epton, J. Ottusch, V. Rokhlin, J. Visher, and S. M. Wandzura, “Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 50, pp. 1824–1930, Dec. 2002. [11] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, pp. 409–418, May 1982. [12] D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogenous dielectric bodies,” IEEE Trans. Antennas Propag., vol. 32, pp. 77–85, Jan. 1984. [13] E. Bleszynski, M. Bleszynski, and T. Jaroszewic, “AIM:Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, no. 5, pp. 1225–151, Sep./Oct. 1996.

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[14] W. C. Chew, J. M. Jin, C. C. Lu, E. Michielssen, and J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, pp. 533–543, Mar. 1997. [15] J. M. Song, C.-C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag., vol. 45, pp. 1488–1493, Oct. 1997. [16] G. C. Hsiao and R. E. Kleinman, “Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, pp. 316–328, Mar. 1997. [17] A. Buffa and S. H. Christiansen, “A dual finite element complex on the barycentric refinement,” Math. Comput., vol. 76, pp. 1743–1769, 2007. [18] R. W. Freund, “A transpose-free quasi-minimal residual algorithm for non-hermitian linear systems,” SIAM J. Sci. Stat. Comput., vol. 14, pp. 470–482, Mar. 1993. [19] T. Ando, J. Yamauchi, and H. Nakano, “Numerical analysis of a dielectric rod antenna – Demonstration of the discontinuity-radiation concept,” IEEE Trans. Antennas Propag., vol. 51, pp. 2007–2013, Aug. 2003.

Hakan Ba˘gcı (S’98-M’07) received the B.S. degree in electrical and electronics engineering from the Bilkent University, Ankara, Turkey, in June 2001 and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Illinois at Urbana-Champaign (UIUC), Urbana, in August 2003 and January 2007, respectively. From June 1999 to July 2001, he worked as an Undergraduate Researcher at the Computational Electromagnetics Group, Bilkent University. From August 2001 to December 2006, he was a Research Assistant at the Center for Computational Electromagnetics and Electromagnetics Laboratory, UIUC. From January 2007 to August 2009, he worked as a Research Fellow at the Radiation Laboratory, University of Michigan. In August 2009, he joined the Division of Physical Sciences and Engineering at the King Abdullah University of Science and Technology (KAUST) as Assistant Professor of Electrical Engineering. His research interests include various aspects of computational electromagnetics with emphasis on time-domain integral equations and their fast marching-on-in-time-based solutions, well-conditioned integral-equation formulations, development of fast hybrid methods for analyzing statistical EMC/EMI phenomena on complex and fully loaded platforms, and solution of electromagnetic inverse scattering problems using sparsity constraints. Dr. Ba˘gcı was the recipient of the 2008 International Union of Radio Scientists (URSI) Young Scientist Award and the 2004–2005 Interdisciplinary Graduate Fellowship from the Computational Science and Engineering Department, UIUC. His paper titled “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex structures loaded with coaxial cables” was one of the three finalists (with honorable mention) for the 2008 Richard B. Schulz Best Transactions Paper Award given by the IEEE Electromagnetic Compatibility Society. He authored and coauthored three finalist papers in the student paper competitions at the 2005 and 2008 IEEE Antennas and Propagation Society International Symposiums.

Francesco P. Andriulli (S’05–M’09) received the Laurea degree in electrical engineering from the Politecnico di Torino, Italy, in 2004, the M.S. degree in electrical engineering and computer science from the University of Illinois at Chicago, in 2004, and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, in 2008. Since 2008 he has been a Research Associate with the Politecnico di Torino. His research interests are in computational electromagnetics with focus on preconditioning and fast solution of frequencyand time-domain integral equations, integral-equation theory, hierarchical techniques, and single source integral equations. Dr. Andriulli was awarded the University of Michigan International Student Fellowship and the University of Michigan Horace H. Rackham Predoctoral Fellowship. He was the recipient of the best student paper award at the 2007 URSI North American Radio Science Meeting. He received the first place prize in the student paper competition of the 2008 IEEE Antennas and Propagation Society International Symposium, where he authored and coauthored two other finalist papers. He has been the recipient of the 2009 RMTG Award for junior researchers.

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Kristof Cools received the M.S. degree in physical engineering from Ghent University, Belgium, in 2004. His master’s thesis dealt with the full wave simulation of metamaterials using the low frequency multilevel fast multipole method. He received the Ph.D. degree from the University of Michigan, Ann Arbor, in May, 2008, under the supervision of Prof. Femke Olyslager and Prof. Eric Michielssen. His research focuses on the spectral properties of the boundary integral operators of electromagnetics.

Femke Olyslager (S’90–M’93–SM’99–F’05), (deceased), was born in 1966 and passed away in early 2009. She received the M.S. and Ph.D. degrees in electrical engineering from Ghent University, Ghent, Belgium, in 1989 and 1993, respectively. She was a Full Professor in electromagnetics with Ghent University. Her research concerned different aspects of theoretical and numerical electromagnetics. She authored or coauthored approximately 300 papers in journals and proceedings. She coauthored Electromagnetic and Circuit Modeling of Multiconductor Transmission Lines (Oxford, U.K.: Oxford University Press, 1993) and authored Electromagnetic Waveguides and Transmission Lines (Oxford, U.K.: Oxford University Press, 1999). Dr. Olyslager was the Assistant Secretary General of the International Union of Radio Science (URSI), an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and was an Associate Editor of Radio Science. In 1994, she became Laureate of the Royal Academy of Sciences, Literature and Fine Arts of Belgium. She received the 1995 IEEE Microwave Prize for the best paper published in the 1993 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and the 2000 Best Transactions Paper award for the her paper published in the 1999 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. In 2002, she received the Issac Koga Gold Medal from the URSI General Assembly and in 2004, became Laureate of the Royal Flemish Academy of Belgium.

Eric Michielssen (M’95-SM’99-F’02) received the M.S. degree in electrical engineering (summa cum laude) from the Katholieke Universiteit Leuven (KUL, Belgium) in 1987 and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign (UIUC), in 1992. He served as a Research and Teaching Assistant in the Microwaves and Lasers Laboratory, KUL and the Electromagnetic Communication Laboratory, UIUC, from 1987 to 1988 and 1988 to 1992, respectively. Following a postdoctoral stint at UIUC, he joined their faculty in the Department of Electrical and Computer Engineering in 1993, reaching the rank of Full Professor in 2002. In 2005, he joined the University of Michigan, Ann Arbor, as a Professor of electrical engineering and computer science. He authored or coauthored over 135 journal papers and book chapters and over 200 papers in conference proceedings. His research interests include all aspects of theoretical and applied computational electromagnetics. His principal research focus has been on the development of fast frequency and time domain integral-equation-based techniques for analyzing electromagnetic phenomena, and the development of robust optimizers for the synthesis of electromagnetic/optical devices. Dr. Michielssen served as the Technical Chairman of the 1997 Applied Computational Electromagnetics Society (ACES) Symposium (Review of Progress in Applied Computational Electromagnetics, March 1997, Monterey, CA), and served on the ACES Board of Directors (1998–2001 and 2002–2003) and as ACES Vice-President (1998–2001). From 1997 to 1999, he was as an Associate Editor for Radio Science, and from 1998 to 2005 he served as Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He received a Belgian American Educational Foundation Fellowship in 1988 and a Schlumberger Fellowship in 1990. Furthermore, he was the recipient of a 1994 International Union of Radio Scientists (URSI) Young Scientist Fellowship, a 1995 National Science Foundation CAREER Award, and the 1998 Applied Computational Electromagnetics Society (ACES) Valued Service Award. In addition, he was named 1999 URSI United States National Committee Henry G. Booker Fellow and selected as the recipient of the 1999 URSI Koga Gold Medal. He also was awarded the UIUC’s 2001 Xerox Award for Faculty Research, appointed 2002 Beckman Fellow in the UIUC Center for Advanced Studies, named 2003 Scholar in the Tel Aviv University Sackler Center for Advanced Studies, and selected as UIUC 2003 University and Sony Scholar. He is a Fellow of the IEEE (elected 2002) and a member of URSI Commission B.

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Comparison of Interpolating Functions and Interpolating Points in Full-Wave Multilevel Green’s Function Interpolation Method Yan Shi, Member, IEEE, and Chi Hou Chan, Fellow, IEEE

Abstract—The difficulty for the solution of full-wave electromagnetic problems using multilevel Green’s function interpolation method (MLGFIM) lies in applying interpolating approaches to efficiently and accurately approximate Green’s function with rapidly changing phase term. We compare various interpolating schemes when radial basis function (RBF) is employed for the interpolation of scattered data of Green’s function. We show that the infinitely smooth Gaussian (GA) RBF has the best interpolation accuracy. In order to improve the interpolation efficiency, a new kind of staggered Tartan grid is proposed. A good calculation method for the shape parameter in GA RBF is given to solve its sensitivity to the group size and the number of interpolation points. Based on the analysis of variation of the number of interpolation points with electric length of the group, adaptive choice of the types of interpolation functions and interpolation points are employed. Numerical examples show that the computational efficiency of this new interpolation scheme is much improved over the previously reported ones. Index Terms—Gaussian (GA), multilevel Green’s function interpolation method (MLGFIM), radial basis function (RBF), staggered Tartan grid.

I. INTRODUCTION

F

OR the solution of large-scale electromagnetic scattering and radiation problems, various fast iterative approaches based on the method of moments (MoM) [1] have been developed in recent years, i.e., multilevel fast multipole algorithm (MLFMA) [2]–[7], adaptive integral method (AIM) [8], [9], sparse matrix canonical grid method (SMCG) [10], [11] and pre-corrected fast Fourier transform (PFFT) [12], [13], etc. In all of these methods, the number of floating-point operations for a due to approximate matrix-vector multiplication is calculation of far-field interactions. However, multipole-based methods depend on the type of Green’s function in the specific problem and therefore are kernel dependent. On the other hand, the FFT-based methods are more suitable for densely packed structures due to the adoption of volumetric grids. Manuscript received September 18, 2009; revised January 03, 2010; accepted February 01, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by the National Nature Science Foundation of China No. 60801040, the China Scholarship Council (CSC), and the Hong Kong Research Grant Council under CERG Grant CityU110606. Yan Shi is with the School of Electronic Engineering, Xidian University, Xi’an 710071, Shaanxi, China (e-mail: [email protected]). Chi Hou Chan is with State Key Laboratory of Millimeter Waves, City University of Hong Kong, Hong Kong SAR, 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.2010.2050448

Recently, a kernel independent approach, i.e., multilevel Green’s function interpolation method (MLGFIM) [14] has been developed to solve the electro-quasi-static problem. It inherits the tree structure of MLFMA and combines the interpolation ideas of PFFT. In MLGFIM, Lagrange polynomials are adopted to approximate the low-frequency Green’s function. When the number of interpolation points in all cubes of the octary-cube-tree is chosen as a small number, e.g., 27, the desired accuracy can be satisfied. It is different from the scheme proposed in [15], which requires a softened kernel, because the tree structure and lower-to-upper interpolation scheme are employed in MLGFIM. The computational and memory . complexities of MLGFIM are With application of MLGFIM into full wave electromagnetic simulation [16]–[19], it is found that the number of interpolation points in the cubes of different sizes at different tree levels is different due to rapid variation of phase terms of Green’s function. Moreover, Lagrange interpolation can not both accurately and efficiently approximate full-wave Green’s function. This problem is alleviated by using inverse multiquadric radial basis function (IMQ) RBF and the staggered Tartan grid [16], [20]. In order to further improve the interpolation efficiency, the adaptive phase compensation (APC) technique [17] and the hybrid quasi-2D/3D multilevel partitioning approach [18] are proposed, respectively. Up to date, the full-wave MLGFIM has been successfully implemented to solve electromagnetic problems including perfect electric conducting objects, coplanar objects and composite metallic and/or dielectric objects, for which [16], [16], computational complexities are and between and [17]–[19], respectively, . Unfortunately, the and all of memory complexities are efficiency of APC technique needs further improvement in 3D case due to the large number of direction bases. In addition, the hybrid quasi-2D/3D multilevel partitioning approach is more suitable for scatterers with small size in some dimension. In this paper, the interpolation efficiency and accuracy are improved by comparison of different interpolation functions and interpolation points. Different radial basis functions (RBFs) are compared so that interpolation function with the best interpolation accuracy, i.e., infinitely smooth Gaussian (GA) RBF, is chosen. A new kind of staggered Tartan grid is proposed to improve interpolation efficiency. In order to use the GA RBF and new interpolation points to efficiently approximate full-wave Green’s function, a good computational method for shape parameter of RBF is developed to solve the sensitivity of GA RBF to the group size and the number of interpolation points. By an-

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between field to [14], we know that the interaction point and source point can be approximated using the interpolation approach as follows: (1) where and are the th and th interpolation functions in field cube and source cube , and and are the th and th interpolation points, and is the number of interpolation points. In full-wave electromagnetic problems, the expressions of the are [18] interaction

Fig. 1. A scatterer is immersed in a grid space.

(2) and (3)

Fig. 2. 2D pictorial representation of the interpolation points (black circles) and interpolated points (white circles) in the field cube and source cube .

m

n

alyzing the variation of the number of interpolation points with electric length of the group, we propose the adaptive choice of the types of interpolation functions and interpolations points in order to obtain the best computational efficiency. Good performance of the new interpolation scheme is demonstrated by numerical examples. II. MLGFIM In MLGFIM, a scatterer is first discretized into smaller eleto in ments by using MoM. Each element is about size. Then the space that contains the scatterer is partitioned into an octary-cube-tree structure, i.e., we first enclose the entire object in a large cube, and then partition the large cube into eight smaller cubes. Each subcube is recursively subdivided into smaller cubes until the finest cubes satisfy the termination criterion. For simplicity, Fig. 1 shows a three-level partitioning for a 2D problem. It is noted that in Fig. 1 some tree nodes are null, and therefore they are pruned. When element is far from element , the field due to current element at current element is calculated indirectly using the interpolation approximation of Green’s function in a multilevel fashion. Multilevel here implies three procedures and two kinds of interpolations, i.e., upward pass (lower-to-upper interpolation), translation (peer-level interpolation) and downward pass (lower-to-upper interpolation). In order to illustrate the interpolation approximation of Green’s function, we consider the scenario in Fig. 2. According

It is noted that (2) implies the interaction in the electric field integral equation (EFIE) and (3) represents the interaction in magnetic field integral equation (MFIE). Substitution of (1) into the matrix element from the integral equation discretized by MoM, we can obtain

(4) in which is the interpolation function matrix consisting of and denotes the Green’s function matrix consisting , and and are related to the of weighting function and the basis function, respectively [14]. corresponding Furthermore, the impedance submatrix to the interactions between field cube and source cube can be expressed as

.. . (5) and are the numbers of unknowns in field cube where and source cube , respectively. III. COMPARISON OF INTERPOLATION FUNCTIONS AND INTERPOLATION POINTS A. Comparison of Different RBFs From [14], we know that Lagrange polynomials as interpolation functions are good enough to approximate low frequency interaction. But Lagrange interpolation suffers from insufficient

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TABLE I INFINITE SMOOTH RBFS

Fig. 3. Source point being one cube length away from field cube. TABLE II COMPARISON OF ACCURACY BETWEEN DIFFERENT RBFS

accuracy for the solution of full-wave electromagnetic problems. RBF is a good option for the interpolation of scattered is sought data of Green’s function. Specifically, a function of the form [16] (6) for , in which such that is chosen as RBF. The interpolation conditions lead to a linear system of equations which determines the expansion coefficients . In general, unlike Lagrange polynomials RBFs do not satisfy the Kronecker delta condition, i.e., (7) Satisfaction of the above condition is very important for the construction of high quality interpolation. Hence, an orthonormalization procedure is necessary [16]. Assume that a set of basis functions which satisfies (7) is a linear combination of the RBFs, e.g., (8) According to (7), a matrix equation can be formulated as follows [16]: .. .

..

.

.. .

.. .

..

.

.. .

(9)

in which is a unitary matrix. Therefore, we can employ to approximate the function , viz., (10) Here we give several infinitely smooth RBFs [21], as shown in Table I. In Table I, is a free parameter which controls the shape of the functions. Later we will discuss the shape parameter in detail. Suppose the source point is just one cube length away from the field cube, as shown in Fig. 3. In this case, the source point is closest to the field cube and therefore the interpolation accuracy for other configurations is better than that for this configuration.

Let the length of the field group be three wavelengths. From [16], we know that the interpolation accuracy for Tartan grids is worse than that for staggered Tartan grids. Hence, different RBFs with staggered Tartan grids are compared, as shown in Table II. According to Table II, it can be seen that the interpolation using GA RBF can obtain the best interpolation accuracy for the same number of interpolation points. The shape parameter in Table II can be determined by numerical experiments and the interpolation threshold is set as 0.005. Furthermore, the continuously adjusted convergence rates [22] of RBFs in Table I are compared in the case of two wavelengths group length, as shown in Fig. 4. According to Fig. 4, we can see that GA RBF has best convergent behavior compared with other RBFs, which is consistent with error estimates of RBFs given by [23]. B. Comparison of Different Number of Interpolation Points The interpolation accuracy and efficiency tightly depend on the pattern of interpolation points. The number of interpolation points for staggered Tartan grids is smaller than that for Tartan

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Fig. 4. Continuously adjusted convergent rate of RBFs.

grids when the same interpolation functions and interpolation threshold are adopted [16]. Tartan grids are just the interpolation points which are evenly spaced in the source/field group. Assume is the number of interpolation points in one dimension. The total number of interpolation points in the group will be . The staggered Tartan grid [16] consists of Tartan grids and the points in the center of the cube of which vertexes are formed by adjacent Tartan grids. Therefore, the total number of . In this interpolation points in the group will be paper, we propose a new kind of staggered Tartan grid. The new staggered Tartan grid consists of Tartan grids and the points in the center of the sides of the cube of which vertexes are formed by adjacent Tartan grids. Hence the total number of interpolation . Fig. 5 shows the points in the group will be for three kinds of interpolation grids. The comcase of parison of interpolation efficiency between the GA RBF with the new staggered Tartan grid and the IMQ RBF with staggered Tartan grid which was employed in [16] is given, as shown in Table III. It can be seen that the number of interpolation points for the former is increasingly smaller than that for the latter with the increase of the length of the group when the interpolation threshold is chosen as 0.005. Furthermore, the Lebesgue functions [24] for three kinds of interpolation points are compared to measure the interpolation approximation accuracy, as shown in Fig. 6. Here the length of group is set as two wavelengths and the numbers of interpolation points for Tartan grid, staggered Tartan grid and new staggered Tartan grid are 1000, 559, 365, respectively. Here the number of interpolation points is chosen in order to make the interpolation accuracy satisfy the error threshold. According to Fig. 6, Lebesgue function for new staggered Tartan points is smaller than those for two other cases.

2 2 2 2 2 2 2

2 2

Fig. 5. Interpolation point pattern. (a) 2 2 2 Tartan grid, (b) 2 2 2 + 1 1 1 staggered Tartan grid, (c) 2 2 2 + 3 2 1 1 new staggered Tartan grid.

2 2

TABLE III COMPARISON OF EFFICIENCY BETWEEN DIFFERENT INTERPOLATION POINTS

C. Determination of Shape Parameter Shape parameter in RBFs is directly related to the interpolation accuracy. By now, however, no analytical formula for the shape parameter is given when a function is interpolated using the RBFs. Hence the optimized shape parameter must be solved by numerical experiment so that the acceptable interpolation accuracy is obtained using the least number of interpolation points. Table IV gives the variation of shape parameter with the group length and the number of interpolation points when GA RBF and new staggered Tartan grid are employed.

From Table IV, we know that shape parameter is very sensitive to the group length and the number of interpolation points.

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TABLE IV RELATIONSHIP BETWEEN SHAPE PARAMETER AND GROUP LENGTH AND THE NUMBER OF INTERPOLATION POINTS

TABLE V COMPARISON OF AVERAGE RANK OF GREEN’S FUNCTION MATRIX

Fig. 6. Lebesgue function for three types of interpolation points. (a) Tartan grid, (b) staggered Tartan grid, (c) new staggered Tartan grid.

By further numerical analysis, we can know that for the same number of interpolation points, the shape parameter will monotonously decrease with the increase of the group length. For the same group length, the different number of the interpolation points will cause greatly different optimized shape parameter, as shown in Table IV. Therefore, it is necessary to find the range of the group length for some number of interpolation points. In and Table IV, two separating group lengths are given, i.e., . For any group length, the shape parameter can be calculated by linear interpolation of the shape parameters of two adjacent group lengths with the same interpolation points. D. Comparison of Rank of Green’s Function In MLGFIM, the Green’s function matrix in th level is a full matrix with the dimension of , which is the number of interpolation points in th level. In order to improve the com-

putational complexity of MLGFIM, QR factorization based on the modified Gram Schmidt orthogonal method with column pivoting and vector reorthogonalization process [14] is used to compress the Green’s function matrix. This is because the Green’s function matrices between two well-separated cubes are low rank matrices, viz. (11) where and are the -byunitary matrix and the -byupper triangular matrix, respectively. Table V lists the comparison of average rank of the Green’s function matrix in each level for a given problem with the largest cube of six wavelengths per linear dimension. It can be seen that the average rank

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Fig. 7. Variation of the number of interpolation points with the electric length of the group.

of Green’s function matrices for two cases is similar. And because the number of interpolation points for GA RBF with the new staggered Tartan grid is greatly smaller than that for IMQ RBF with staggered Tartan grid, the dimensions of matrices and for the former are smaller than that for the latter. Therefore, the efficiency for the former is better than that for the latter. E. Hybrid Interpolation Functions and Interpolation Points From Table IV, we can know that for the GA RBF with the new staggered Tartan grid the number of interpolation points increases with the increase of electric length of the group, which is same as the case of the IMQ RBF with staggered Tartan grid. Fig. 7 shows the variation of the number of interpolation points with the electric length of the group for two schemes. It can be seen from Fig. 7 that the number of interpolation points for the GA RBF with the new staggered Tartan grid is less than that for the IMQ RBF with staggered Tartan grid for most of electric length of the group, which is consistent with the conclusion drawn from Table III. However, there is an interval of the electric length of group in which the former is larger than the latter. Therefore, we adaptively choose interpolation functions and interpolation points so that best computational efficiency can be obtained. IV. NUMERICAL EXAMPLES In this section, some examples are given to demonstrate good performance of the proposed algorithm. All calculations are performed on a computer with 3.0 GHz and 2 GB memory. The generalized minimal residual (GMRES) with a relative error is employed for all the simulations. norm of As the first example, we consider a plane wave scattered from a finite microstrip patch array, as shown in Fig. 8(a). The geomcm, etry parameters of the array are as follows: cm, cm, cm, cm, cm. The relative permittivity and relative of the substrate are chosen as 2.17 and 1.0, repermeability spectively. The plane wave with the frequency of 3.7 GHz is on the patch array. We use 14 normally incident grids per wavelength to discretize the patch array so that the resultant total number of unknown electric and magnetic currents

Fig. 8. Plane wave scattering from the finite patch array. (a) Geometry and (b) bistatic RCS with  polarization.

is 33544. The normalized bistatic RCS is calculated by previously reported algorithm and the proposed new algorithm, as shown in Fig. 8(b). Very good agreement is observed between the two results. In the second case, plane wave scattering from a dielectric sphere array is analyzed, as shown in Fig. 9(a). Here we con. sider a 5 by 5 dielectric sphere array with , in which is the The electrical size of the sphere is free space wavenumber and is the radius of the sphere. The distance between the centers of two adjacent spheres is . Plane wave with the frequency of 300 MHz is normally incident along axis. The bistatic RCS with polarization is calculated, as shown in Fig. 9(b). Again, good agreement is observed between the two results. In the following, the proposed algorithm is used to solve the plane wave scattering from a dielectric slab with a cone frustum array, as shown in Fig. 10(a). The length, width and height of slab are 3.5 m, 3.5 m, and 0.25 m, respectively. The top and bottom radii of the cone frustum are 0.125 m and 0.25 m, respectively, and the distance between centers of two adjacent cone frustums is 1 m. The frequency of incident plane wave is polariza300 MHz and is set as 4. The bistatic RCS with tion is calculated, as shown in Fig. 10(b). The result compares well with that obtained by previously reported algorithm. In the fourth example, we consider plane wave scattering , as shown in from a dielectric cross-rod structure with Fig. 11(a). The length, width and height of the rod are 5.37 m, 0.2 m and 0.2 m, respectively. The distance between two rods is 0.2 m. The frequency of incident plane wave is 300 MHz and

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Fig. 11. Plane wave scattering from a dielectric cross-rod structure. (a) Geometry and (b) bistatic RCS with  polarization. Fig. 9. Plane wave scattering from the finite dielectric sphere array. (a) Geometry and (b) bistatic RCS with  polarization.

Fig. 10. Plane wave scattering from the dielectric slab with the cone frustum array. (a) Geometry and (b) bistatic RCS with  polarization.

the bistatic RCS with polarization is calculated, as shown in Fig. 11(b). Very good agreement is observed between two results.

Fig. 12. Plane wave scattering from a dielectric hexagonal cylinder with a circular cylinder hole. (a) Geometry and (b) bistatic RCS with  polarization.

Next, plane wave scattering from a dielectric hexagonal cylinder with a circular hole is considered, as shown in Fig. 12(a). The radii of hexagonal cylinder and circular hole are 1.3 m and 1 m, respectively, and the height of the cylinder

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TABLE VI COMPASIRON OF CPU TIME FOR EACH MATRIX-VECTOR MULTIPLICATION

adaptive algorithm is better than that of previous algorithm for all examples. Here in first four examples, the computational efficiency of the adaptive algorithm is same as that of new algorithm. V. CONCLUSION In this paper, GA RBF, an infinitely smooth RBF, and a new kind of staggered points as the interpolation function and interpolation points are proposed to improve the computational efficiency of MLGFIM for the solution of 3D full wave complex electromagnetic problems. It is found that the accuracy of GA RBF is better than that of IMQ RBF when they are used to approximate the integral kernel of the integral equation. In addition, the number of interpolation points can be greatly reduced when the new staggered points are employed. Moreover, the rank of Green’s function keeps nearly unchanged when the QR factorization technique is adopted. According to the variation of interpolation points with the electric length of the group, the adaptive choice of the type of interpolation function and interpolations points is proposed to obtain the best computational efficiency. Several numerical examples have demonstrated good performance of the proposed algorithm. REFERENCES

is 0.15 m. The plane wave with frequency of 300 MHz is normally incident on the cylinder and is chosen as 4. Fig. 12(b) shows the normalized bistatic RCS and the results calculated by the proposed algorithms compare well with that obtained by previously reported algorithm. Finally, we compare the computational performance between the newly proposed algorithm, adaptive algorithm and previously reported algorithm. The maximum electrical lengths and of these examples are (where ), respectively. CPU times for each matrix-vector multiplication using the newly proposed algorithm, adaptive algorithm and previously reported algorithm are given in Table VI. From Table VI, we can observe that for all examples except the last one, computational efficiency of the newly proposed algorithm is better than that of previously reported algorithm. CPU times of the new algorithm are 53.9%, 34.1%, 50.7% and 21.1%, respectively, of that of the previously reported algorithm. It is noticed that in the case of Example 4 the number of levels for the newly proposed algorithm is set as 5 and the number of levels for the previously reported algorithm is chosen as 6. This is because the previously reported algorithm with the number of levels of 5 cannot be run on the current computer for the solution of Example 4. It is pointed out that in Example 4 the computer memories for new and previous reported algorithms are 1572 MB and 1655 MB, respectively. On the other hand, computational efficiency of

[1] R. F. Harrington, Field Computation by Moment Methods. New York: IEEE Press, 1993. [2] V. Rokhlin, “Rapid solution of integral equations of classic potential theory,” J. Comput. Phys., vol. 60, pp. 187–207, 1985. [3] L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., vol. 73, pp. 325–348, 1987. [4] E. Darve, “The fast multipole method: Numerical implementation,” J. Comput. Phys., vol. 160, pp. 195–240, 2000. [5] C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microw. Opt. Tech. Lett., vol. 7, pp. 456–461, 1994. [6] J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag., vol. 45, pp. 1488–1493, 1997. [7] W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Boston, MA: Artech House, 2001. [8] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, pp. 1225–1251, 1996. [9] F. Ling, C. F. Wang, and J. M. Jin, “An efficient algorithm for analyzing large-scale microstrip structures using adaptive integral method combined with discrete complex image method,” IEEE Trans. Microw. Theory Tech., vol. 48, pp. 832–837, 2000. [10] C. H. Chan, C. M. Lin, L. Tsang, and Y. F. Leung, “A sparse-matrix/canonical grid method for analyzing microstrip structures,” IEICE Trans. Electron, vol. E80-C, pp. 1354–1359, 1997. [11] S. Q. Li, Y. X. Yu, C. H. Chan, K. F. Chan, and L. Tsang, “A sparsematrix/canonical grid method for analyzing densely packed interconnects,” IEEE Trans. Microw. Theory Tech., vol. 49, pp. 1221–1228, 2001. [12] J. R. Phillips and J. K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., vol. 16, pp. 1059–1072, 1997. [13] X. C. Nie, N. Yuan, L. W. Li, Y. B. Gan, and T. S. Yeo, “A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects,” IEEE Trans. Antennas Propag., vol. 52, pp. 818–824, 2005. [14] H. G. Wang, C. H. Chan, and L. Tsang, “A new multilevel Green’s function interpolation method for large-scale low-frequency EM simulations,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., vol. 24, pp. 1427–1443, 2005. [15] A. Brandt, “Multilevel computations of integral transforms and particle interactions with oscillatory kernels,” Comput. Phys. Commun., vol. 65, pp. 24–38, 1991.

SHI AND CHAN: COMPARISON OF INTERPOLATING FUNCTIONS AND INTERPOLATING POINTS

[16] H. G. Wang and C. H. Chan, “The implementation of multilevel Green’s function interpolation method for full-wave electromagnetic problems,” IEEE Trans. Antennas Propag., vol. 55, pp. 1348–1358, 2007. [17] L. Li, H. G. Wang, and C. H. Chan, “An improved multilevel Green’s function interpolation method with adaptive phase compensation for large-scale full-wave EM simulation,” IEEE Trans. Antennas Propag., vol. 56, pp. 1381–1393, 2008. [18] Y. Shi, H. G. Wang, L. Li, and C. H. Chan, “Multilevel Green’s function interpolation method for scattering from composite metallic and dielectric objects,” J. Opt. Soc. Am. A, vol. 25, pp. 2535–2548, 2008. [19] Y. Shi and C. H. Chan, “Solution to electromagnetic scattering by bi-isotropic media using multilevel Green’s function interpolation method,” Progr. Electromagn. Res., vol. 97, pp. 259–274, 2009. [20] B. J. C. Baxter, “The interpolation theory of radial basis functions,” Ph.D. dissertation, Cambridge Univ., Cambridge, U.K., 1992. [21] G. B. Wright, “Radial basis function interpolation: Numerical and analytical development,” Ph.D. dissertation, Colorado Univ., Boulder, 2000. [22] J. P. Boyd, “Prolate spheroidal wavefunctions as an alternative to Chebyshev and Legendre polynomials for spectral element and pseudospectral algorithm,” J. Comput. Phys., vol. 199, pp. 688–716, 2004. [23] W. R. Madych and S. A. Nelson, “Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation,” J. Approx. Theory, vol. 70, pp. 94–114, 1992. [24] R. Schaback, “Reproduction of polynomials by radial basis functions,” in Wavelets, Images, and Surface Fitting, P. J. Laurent, A. LeMehaute, and L. L. Schumaker, Eds. Wellesley: A K Peters, 1994, pp. 459–466.

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Yan Shi (M’07) was born in Tianjing, China. He received the B.Eng. and Ph.D. degrees in electromagnetic fields and microwave technology from Xidian University, Xi’an, China, in 2001 and 2005, respectively. He joined the school of Electronic Engineering, Xidian University, in 2005 and was promoted to Associate Professor in 2007. From 2007 to 2008, he worked at City University of Hong Kong, Hong Kong, China, as a Senior Research Associate. He was awarded a scholarship under the China Scholarship Fund and was invited to visit the University of Illinois at Urbana-Champaign as a Visiting Postdoctoral Research Associate in 2009. His research interests include computational electromagnetics, geophysical subsurface, electromagnetic compatibility, frequency selective surface and metamaterials. Dr. Shi is a senior member of Chinese Institute of Electronics (CIE).

Chi Hou Chan (S’86–M’86–SM’00–F’02) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1987. He joined the Department of Electronic Engineering, City University of Hong Kong, in 1996 and was promoted to Chair Professor of Electronic Engineering in 1998. From 1998 to 2009, he was first Associate Dean then Dean of the College of Science and Engineering, and is currently Acting Provost of the University. His research interests cover computational electromagnetics, antennas, microwave and millimeter-wave components and systems, and RFICs. Prof. Chan received the U.S. National Science Foundation Presidential Young Investigator Award in 1991 and the Joint Research Fund for Hong Kong and Macau Young Scholars, National Science Fund for Distinguished Young Scholars, China, in 2004. He received Outstanding Teacher Awards in the Electrical Engineering Department at CityU in 1998, 1999, 2000, and 2008. Students he supervised also received numerous awards including the Third (2003) and First (2004) Prizes in the IEEE International Microwave Symposium Student Paper Contests, the IEEE Microwave Theory and Techniques Graduate Fellowship for 2004–2005, Undergraduate/Pre-Graduate Scholarships for 2006–2007 and 2007–2008, and the 2007 International Fulbright Science and Technology Fellowship offered by the U.S. Department of State.

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Propagation Modes and Temporal Variations Along a Lift Shaft in UHF Band Xiao Hong Mao, Student Member, IEEE, Yee Hui Lee, Member, IEEE, and Boon Chong Ng, Senior Member, IEEE

Abstract—The guiding of electromagnetic waves along the lift shaft is studied and presented in this paper. Wideband channel sounding in UHF band along a lift shaft located in a complex environment has been conducted. Based on the measured delay in time of arrival, large structures that cause multipath clusters can be identified. The different propagation mechanisms associated with the lift shaft is then studied in detail. Analysis of the measurement results verifies the presence of electromagnetic waves being guided along the lift shaft. This guiding effect of the lift shaft is significant and is an important propagation mechanism. Temporal variation of the guided waves caused by different components of the lift shaft such as the lift door and the lift car has also been studied. Conclusions drawn from measurement results are validated using 3-D ray tracing simulation results. This research is useful for military applications such as urban warfare. Index Terms—Lift shaft, temporal variation, urban area, waveguide effect, 3-D ray tracing.

I. INTRODUCTION HE field of indoor radio propagation has been well developed since the early 1980s. Studies of indoor radio wave propagation play an important role for system designers. For example, channel characterization in hospital [1] can provide channel information for system planner so as to ensure that hospital staffs are contactable at all times without causing any interference to medical equipments. The lift shaft and its associated lift car create an RF-harsh propagation environment [2]. In fact, the propagation environment of a lift shaft is similar to that of a tunnel or an indoor corridor; topics that are well-studied in the UHF band [3]–[6]. The waveguide mechanism for UHF communication within a tunnel has been identified since 1975 [3]. Subsequently, the mine tunnel has been modelled as a waveguide for UHF band communication in [4]. In [5], it was concluded from the small path loss obtained that the corridor can exhibit waveguide effects. In [6], the corridor was found to behave like a large waveguide since the measured loss is smaller

T

Manuscript received June 23, 2009; revised January 12, 2010; accepted February 08, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by the Advanced Communications Research Program DSOCL06271, a research grant from the Directorate of Research and Technology (DRTech), Ministry of Defence, Singapore. X. H. Mao and Y. H. Lee are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]; [email protected]). B. C. Ng is with the Advance Communication Laboratories, Defence Science Organization (DSO) National Laboratories, Singapore and also with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050429

than the free space path loss. Besides these analogues-waveguide structures, the heating, ventilation and air conditioning (HVAC) ducts in buildings have also been modelled as a multimode waveguide at high frequency [7]. In [7], an approximate propagation model for a straight HVAC duct channel is presented. It is shown that the HVAC duct can be modelled as a multimode waveguide. Due to the similarity in geometry between a lift shaft, a tunnel, an indoor corridor, and a HVAC duct, waveguide effect may also exist in the propagation path along the lift shaft. Little research work has been done on the effect of propagation within the lift shaft. In [8], the effect of placement and orientation of antenna has been examined for propagation along the lift shaft through FDTD simulations. They concluded that the main mechanism for signal propagation within the lift shaft is independent of polarization. In the following year, the same group of researcher studied the propagation of GSM signals in a lift shaft for network planning purposes [2]. They concluded that, at the higher GSM frequency band of 1800 MHz, although signal propagation is believed to be worse than that at the lower GSM frequency band of 900 MHz, it was found to be approximately 5 dB better. The authors attributed this to the lower energy absorption by the lift car and the people in the lift car in the higher GSM band. They also showed that the vertical lift shaft is an oversized waveguide and therefore, have a lower attenuation factor in the higher GSM band. However, no further study on the waveguide effect has been conducted. Application focused research works have studied the effect of the moving elevator on the number of handoffs required for effective cell planning purposes [9]. In [2], [8], [9], narrowband measurement has been conducted, with the aim of studying the variation in signal strength of the moving elevator. In [10], impulsive noise measurements for indoor buildings at UHF (918 and 2440 MHz) and microwave (4 GHz) frequencies are conducted and the elevator has been identified as one of the significant sources of impulsive noise. The study shows that as the frequency increases, the impulse noise amplitude and duration decreases. Recently, there has been interest in urban warfare. It is critical for soldiers to be able to communicate with each other reliably during war time [11]. However, there is little published literature on communication in urban areas in the military UHF band of 225 to 400 MHz. In this paper, wideband channel sounding is performed at 255.6 MHz in order to understand the propagation mechanisms over a lift shaft within an urban environment. Channel measurement is performed in the frequency domain. The time domain channel response is derived from the frequency response of the channel for analysis. This wideband sounding aims to identify the propagation mechanisms for communication along a lift shaft within a complex environment.

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More importantly, the waveguide effect associated with the lift shaft will be examined. The temporal variation on the guided waves caused by the lift door and the lift car is studied in detail. To validate the measurement results, 3-D ray-tracing simulation is performed and results are compared. This paper consists of three sections. Section II describes the measurement environment, measurement setup, as well as the simulation scenarios. In Section III, results obtained from measurement and simulation are presented, compared and analyzed. Based on the analysis, propagation modes and temporal variations are discussed. This is followed by the conclusions of the findings in Section IV.

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Fig. 1. Experimental environment. (a) Location of antenna. (b) Surrounding environment of the experimental site.

II. CHANNEL MEASUREMENT AND SIMULATION A. Measurement Sites Measurements have been conducted along a lift shaft in an education building in Nanyang Technological University, Singapore in June, 2008. This education building is 7 stories high and is known as block S2. Each level (except level 7) in block S2 is about 3.9 m in height and it consists of three main blocks. Inside each block are laboratories, except for level 6 where there are offices. There are open walk ways that runs along all the 3 blocks on both sides of the building as seen in Fig. 1(a). These open walk ways have concrete railings. The walls of the 3 main blocks are made of concrete and on the side walls facing the two walk ways are glass windows. All doors in the building are made of heavy wood. The ceilings along the walk ways are lined with thin aluminium plates. The lift shaft under study has concrete walls that spans 7 levels and is approximately 27 m in height. It is situated between the middle block and the last block in the building S2. On the opposite side of the lift shaft, there is a stairwell of the same height. Throughout the experiment, the location of the transmitter and the receiver are fixed at level 3 and level 6, respectively. The antennas are placed directly outside of the lift door as shown in Fig. 1(a). From Fig. 1(b) it can be seen that there is a building in the surrounding environment known as block S1 which is identical and in parallel to block S2 (the experimental site) at a distance of 78 m away. There is also another group of education buildings known as the Communication School (CS) which is situated between block S1 and block S2 as seen in Fig. 1(b). The material of the blocks CS and S1 are similar to that of block S2. In order to identify the reflections from surrounding buildings on the propagation paths, controlled experiments are conducted. A metallic plate of dimensions, 2.3 is placed along m by 1.3 m by 3 mm the corridor to prevent the signal from propagating out of block S2 towards the surrounding buildings. After identifying the reflections from surrounding buildings, the propagation mechanisms associated with nearby environment such as the lift shaft can be isolated for analysis. The propagation mechanism within the lift shaft is then examined through a series of experiments with the lift door opened or closed and with the lift car situated at different levels within the lift shaft. The temporal variations of the guided wave can then be studied.

Fig. 2. Schematic diagram of the measurement setup.

B. System Setup Three wideband channel sounding techniques namely; direct pulse measurements; spread spectrum sliding correlator measurements; and swept frequency measurements are reported in [12]. Of the three sounding techniques, the swept frequency measurement is used because of its ability to achieve a high resolution. The measurement system consists of an Agilent Vector Network Analyser (VNA) and two identical Discone antennas AX-71C (Fig. 1(a)). Fig. 2 shows the schematic diagram of the experiment setup. The centre frequency is fixed at 255.6 MHz and 1601 uniformly distributed continuous waves are transmitted over a bandwidth of 300 MHz. With this specification, the highest resolvable path difference is 1 m and the maximum excess delay is 5.33 . To ensure that the channel is static during a single sweep of the measurement, the minimum sweep time of 111.56 ms is used. It is noted that the minimum sweep time is proportional to the number of points (1601) and inversely proportional to the intermediate frequency bandwidth of 3 kHz for the Agilent E5062A VNA. For each measurement, a set of 50 sweeps are taken and logged via the general purpose interface bus (GPIB) onto a laptop. In order to obtain the time domain channel response, post-processing is done by taking the inverse fast fourier transform (IFFT) of the recorded frequency domain transfer function as shown in (1) and (2)

(1) (2)

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III. RESULTS AND DISCUSSION A. Overview of the Power Delay Profile

Fig. 3. 3D simulation scenario.

C. Simulation Setup In this paper, a trial version of the ray-tracing simulator “Wireless Insite [13]” is used to obtain detailed channel information. The ray-tracing scenario presented is a simplified model of the actual environment and the 3-D model is simulated to address the effect of transmission, reflection, and diffraction. The applied simulation model is developed based on a hybrid shooting bouncing ray (SBR) algorithm and geometrical theory of diffraction (GTD). The SBR method is implemented with robust ray tracing techniques where once the propagation paths are found, the field is evaluated by far-field transmission and reflection coefficients. The amplitudes of the diffracted fields are evaluated by GTD [13]. In the basic form of the SBR technique, it is assumed that the incident electromagnetic wave is planar and is partitioned perpendicularly into a large number of ray tubes such that any particular ray tube will initially intersect with a small area of the whole target. The ray tubes are required to be spaced sufficiently finely so that other simplifying assumptions i.e. ray remaining planar and the shape remaining circular can remain valid throughout the full ray path. For this simulator, the ray spacing can be specified by the user based on the application. However, fine ray spacing can lead to long computation time. The advantage of the SBR algorithm is its simplicity, but the model is less effective when the target features are complex and there are multiple scatters in more than one direction [14], [15]. Fig. 3 shows the simulation scenario with the lift shaft under investigation in block S2, and the surrounding blocks S1 and CS. The material of the walls of all the buildings is modeled using layered drywall, while the floors and ceilings are modeled using concrete. For each level, two railings with a height of 1 m are added. The lift shaft and the stairwell are modeled as a series of empty rooms which spans 7 levels, and the lift car is modeled as a hollow metallic box. Vertically polarized omnidirectional dipoles are used as the transmitting and receiving antennas. The simulated power differences between the different rays/multipaths are compared to the measured ones. For ease of comparison, all powers in the simulations are normalized to the measured power. The maximum reflection is set to 2 and the maximum transmission is set to 10. For each simulation, Wireless Insite is able to generate a list of propagation paths (up to 250 rays) and the amplitudes associated with the paths.

Fig. 4(a) shows the measured time domain channel response when the transmitter is at level 3 and the receiver is at level 6 while the lift car is kept stationary at level 3. The power delay profile can be classified into three regions indicated by the veron the time axis, tical lines. Region 1 range from 0 to 0.15 while region 3 is for a time region 2 range from 0.15 to 0.5 of 0.5 and above. Fig. 4(b) shows the simulated power delay profile with the different rays identified by the simulator. Similarly, the simulated rays are classified into 3 regions according to their time of arrival. Comparing the two figures, the propagation mechanism associated with each region can be analyzed with the help of the ray visualization shown in Fig. 4(c). It is identified that rays in region 3 and region 2 are a result of signals being reflected off obstacles in the far region (block S1) and the intermediate region (block CS and far ends of block S2). The loss-distance relationship in region 1 indicates the possible sources of signals; they are the direct signal that penetrates through the ceilings and floors; the signal that are reflected and/or diffracted by nearby objects; and more importantly, the signal that enters the lift shaft, and propagates within the lift shaft to the receiver. The propagation within each region is further analyzed in the Sections III-Bto III-E. In Section III-B, the far and intermediate reflectors are identified through measurements and simulations. In Section III-C, the nearby environment especially the temporal variations associated with the lift shaft is examined in detail. In Section III-D, a comparison of the average channel gain for different measurement scenarios are presented. Finally in Section III-E, the root-mean-square (rms) delay spread for all measurement scenarios are discussed. B. Propagation Mechanism of Signals in Region 3 and 2 By examining the time of arrival of the signals within region 3 and region 2, it can be concluded that signals in these two regions are a result of reflections by large static obstacles in the far and intermediate regions. In this measurement campaign, there are two clusters of buildings (block S1 and block CS) situated on the left-hand side of the block S2. In order to identify the reflections from these buildings, a set of controlled experiments are conducted by placing a metallic plate on the left-hand side of the corridor at the level of the transmitter. This metallic plate serves to reduce the number of signals propagating out of the corridor towards the far and intermediate reflectors. Fig. 5(a) shows a comparison of measured channel response with and without the metallic plate. It is observed that when the metallic plate is present, the strength of signals received within region 2 (except the first two clusters of peaks) and region 3 decrease by up to 7.7 dB. This verifies that region 3 and region 2 contain signals reflected from the far region of block S1 and the intermediate region of block CS. The unchanged amplitude of the first two clusters of peaks in region 2 is due to the reflections of intermediate reflectors within block S2 and the incomplete blocking due to the limited size of the metallic plate. A perfect electric conductor board is used to block the same side of the corridor in the simulation. Fig. 5(b) shows the simulated power delay profile. The dotted impulses indicate the rays

MAO et al.: PROPAGATION MODES AND TEMPORAL VARIATIONS ALONG A LIFT SHAFT IN UHF BAND

Fig. 4. (a) Measured power delay profile. (b) Simulated ray tracing power delay profile. (c) 3D visualization from simulator.

that are absent when the metallic plate is placed in the corridor. The red dotted impulse represents the signal reflected from the block CS whereas the blue dotted impulse represents the signal reflected off the block S1. Comparing Fig. 5(c) and Fig. 4(c), it is observed that rays reflected from the block CS and the block S1 are missing when the metallic plate is placed in the corridor in the simulation. As shown in Fig. 5(b), there are three remaining

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Fig. 5. Metallic plate at transmitter level. (a) Measured power delay profile. (b) Simulated ray tracing power delay profile. (c) 3D visualization from simulator.

multipath rays in region 2 when the metallic place is placed in the corridor. These three rays are a result of reflections from the wall partitions within block S2. From these measurement and simulation results, the far and the intermediate reflectors are identified. Next, the nearby propagation mechanisms are analyzed.

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Fig. 7. Simulation scenario of the lift shaft in (a) solid view and (b) transparent view.

Fig. 6. Nearby environment. (a) Simulated ray tracing power delay profile. (b) 3D visualization from simulation.

C. Propagation Mechanism of Signals in Region 1 Fig. 6(a) shows the rays within region 1 from the simulator. These rays can be further classified into eight different propagation mechanisms. As expected, region 1 includes signal penetrating through the floors and ceilings, signal reflected and/or diffracted by nearby objects such as the lift door, walls and railings, and most importantly, signal guided by lift shaft. A similar wave guiding effect is observed for the stairwell. Fig. 6(b) provides the visualization of the above mentioned propagation paths. In order to study the waveguide effect of the lift shaft, controlled experiments are conducted by keeping the lift door opened and closed at the transmitter level and the receiver level; and by varying the position of the lift car from level 1 to level 7 (with the lift door closed). For all of the three sets of experiments, the transmitter and the receiver are fixed at level 3 and level 6, respectively. These experimental results are verified through the simulation of an isolated lift shaft with a lift car within the lift shaft as shown in Fig. 7. In order to aid analysis, only simulated rays that are not common between the two simulations are shown in the PDPs and the visualizations. 1) Lift Door Effect: Fig. 8(a) shows the measurement result obtained when the lift door is opened and when the lift door is

closed at the transmitter level. It is observed that almost all the resolvable signals within region 1 are affected by the opening and closing of the lift door and the strength of these signals are approximately 14 dB higher when the lift door is opened. This indicates that region 1 contains signals that enter the lift shaft from the lift door and then propagate along the lift to the receiver. Based on the diffraction theory, when the lift door is open, the door opening is larger than the wavelength, and therefore, the waves can propagate directly into the lift shaft. When the lift door is closed, the width of the slit in the door is small compared to the wavelength, and therefore, waves are diffracted into the lift shaft via the rubber seal. This results in a diffraction loss that can be as much as 14 dB. Fig. 8(b) shows the simulation results for the lift shaft when the lift door is opened and when the lift door is closed (common rays not shown). Fig. 8(c) and (d) shows the corresponding visualization of rays that result in the impulse responses in Fig. 8(b). By comparing the impulse responses in Fig. 8(b) with the lift door opened and with the lift door closed, it is observed that, the lift door stops a significant number of rays from entering the lift shaft. This accounts for the diffraction loss observed in the measured results shown in Fig. 8(a). By examining the number of rays in Fig. 8(c), and (d), it can be seen that, there is a reduction in the number of rays reflected and guided by the lift shaft when the door is closed. With the 0lift door closed, due to the perfect electric conducting lift door in the simulator, no signal can penetrate through the lift door. The gap of the lift door is modeled using a slit (free space) in the middle of the perfect conducting lift door; signals can leak through or be diffracted by the slit and enter the lift shaft. Therefore, the amount of rays guided by the lift shaft is reduced significantly as shown in Fig. 8(d). A similar analysis has been performed in order to examine the effect of the lift door (opened and closed) at the receiver level (level 6). Fig. 9(a) shows the power delay profile obtained through experiments by opening and closing lift door at the

MAO et al.: PROPAGATION MODES AND TEMPORAL VARIATIONS ALONG A LIFT SHAFT IN UHF BAND

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Fig. 8. Open-close lift door at transmitter level. (a) Measured power delay profile. (b) Simulated ray tracing power delay profiles. (c) 3D visualization from simulator for lift door open. (d) 3D visualization from simulator for lift door closed.

Fig. 9. Open-close lift door at receiver level. (a) Measured power delay profile. (b) Simulated ray tracing power delay profiles. (c) 3D visualization from simulator for lift door open. (d) 3D visualization from simulator for lift door closed.

receiver level. Fig. 9(b) shows propagating rays (common rays not shown) from the simulation results, while Figs. 9(c) and (d) provides visualization for rays in Fig. 9(b). As expected, more signals can propagate out from the lift shaft with higher signal strengths due to the open door at the receiver level. From the measured results, the maximum difference is approximately 7.6 dB. This is significantly less than that obtained by opening and

closing the lift door at the transmitter level (14 dB). A similar conclusion can be drawn from the simulation results. When the lift door is opened at the transmitter level, fewer rays are affected, meaning there is less ray difference in Fig. 9(c) and (d) whereas when lift door is opened at the transmitter level, more rays are affected, meaning there is more ray difference in Fig. 8(c) and (d).

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Fig. 10. Measured power delay profile when the lift car is moving.

As shown in both simulation and the measurement results, the status of the lift door can cause temporal variations to the guided signals. A closed lift door at the transmitter level can result in a signal strength variation as much as 14 dB, whereas that at the receiver level results in a much lower signal strength variation of 7.6 dB. 2) Attenuation Induced by Lift Car: Fig. 10 shows the temporal variation of the signals guided by the lift shaft when the lift car is moving. These plots are obtained by averaging data files over different period of times. The data files are recorded when the lift is in use and moving along the lift shaft. As can be seen in Fig. 10, the temporal variations caused by small movement of the lift car within the propagation path can lead to significant small-scale fading. Therefore, a series of controlled experiments are conducted to examine the effect of the lift car on the lift-shaft guided waves. Fig. 11(a) shows the measurement results when the transmitter is at level 3 and the receiver is at level 6, and the position of the lift car is varied from level 1 to level 7, one level at a time and the lift door is closed at all times. It can be seen that the channel gain for the signals in region 1 is the lowest when the lift car is at level 4 and level 5, while the channel gain is the highest when the lift car is at level 1, level 2 and level 7. This is because, when the lift car is at level 1, level 2 or level 7, it is not within the propagation path, therefore, leaving the wave guiding channel empty. Thus, there is no attenuation due to the obstruction of the guided waves by the lift car within the channel. This leads to the high channel gain. When the lift car is at level 4 and level 5, it is in the middle of the propagation path, and therefore, induces significant attenuation to the guided signal. Fig. 11(b) shows the power delay profile of the two extreme cases when the lift result of the effect of the lift car position is shown in Fig. 11(c) (common rays not shown) visualization of the rays is shown in Fig. 12(a), and (b). From Fig. 11(c), it is clear that the empty path between the transmitter and the receiver allows a significant amount of rays to be reflected and guided by the lift shaft to the receiver. When the lift car is in the middle of the propagation path, the path is obstructed and signals are attenuated by the lift car. Therefore, there is a significant difference in the number of rays guided along the lift shaft as shown in Fig. 12(a), and (b). Measurement and simulation results in Figs. 11 and 12 demonstrate both the guiding effect of the lift

Fig. 11. Variation of lift car position. (a) Measured power delay profile. (b) Power delay profiles when lift car is at level 5 and at level 7. (c) Simulated ray tracing power delay profile.

shaft and the temporal variation to the guided waves due to the movement of the lift car. D. Channel Gain From the previous Sections III-A, III-B, and III-C, it can be concluded that region 1 mainly consists of signals entering the

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TABLE I CHANNEL GAIN FOR OPENING/CLOSING LIFT DOOR

Fig. 12. Variation of lift car position. (a) 3D visualization from simulator for lift car at level 5. (b) 3D visualization from simulator for lift car at level 7.

lift door and then guided through the lift shaft to the receiver. The status of the lift door and the position of the lift car will affect the propagation of the signals within region 1. Besides being guided along the lift shaft in the near region (region 1), signals can be reflected by objects in the intermediate and further regions, shown in region 2 and 3 respectively. In this section, the average channel gain in the three different regions are compared and analyzed. Different measurement scenarios are considered in order to study the effect of the lift door and the effect of the position of the lift car. Table I summarizes the average channel gain within different regions when the lift door is opened/closed at the transmitter and the receiver level [corresponding PDPs shown in Fig. 8(a) and Fig. 9(a)]. It is observed that the average channel gain in regions 2 and 3 are not affected by the status of the lift door, whereas the channel gain in region 1 dependents a lot on the status of lift door as discussed in Section III-C-1. By comparing the channel gains in the three regions in Table I, it can be concluded that, signals in region 1 has the highest channel gain (at least 4.8 dB above that of region 2) and the signals in region 3 have the lowest channel gain due to the longest propagation distance. The variation of channel gain in the three regions as the position of the lift car is varied, is shown in Fig. 13 [corresponding PDP shown in Fig. 11(a)]. It is observed that the channel gain of region 1 is the highest amongst the 3 regions regardless of the position of the lift car. The standard deviations of the average gain for the 3 regions are 4.3, 0.6, and 0.3, respectively. This shows that the channel gain of region 2 and 3 are independent of the position of the lift car, whereas the channel gain of region 1 varies a lot depending on the position of lift car as discussed in

Fig. 13. Average channel gain for different lift car level.

Section III-C-2. From this analysis it can be concluded that since the channel gain in region 1 is always high, the wave-guiding effect of the lift shaft is an important propagation mechanism in this complex environment. And it is important to understand the temporal variations induced by the lift door and the lift car. E. RMS Delay Spread The rms delay spread values for the scenario when the lift door is opened/closed at the transmitter level and the receiver level are tabulated in Table II. The mean PDPs used for delay spread calculation are obtained by taking an average in the time domain of 50 continuous sweeps at a single position. All signals in the mean PDP with amplitude of at least 5 dB above the noise floor (a threshold of 5 dB signal-to-noise ratio) are considered to be significant peaks. These significant peaks will then be accounted for towards the calculation of the rms delay spread. In Table II, it can be seen that the status of the lift door has significant effect on the rms delay spread as it is determined by the signals’ amplitudes and their corresponding time delays. When the lift door is closed, the signal strength and number of rays guided along the lift shaft decreases, while the signals in regions 2 and 3 arriving with a longer delay remain unchanged, therefore, resulting in a longer delay spread. The rms delay spread for different position of the lift car is plotted in Fig. 14 (blue solid line). The delay spread varies with the position of the lift car. When the lift car is in the middle of the propagation path (level 3, 4 and 5), the guided signals are attenuated by the lift car. Therefore, a high rms delay spread of 159 ns is obtained. When the lift car is out of the wave guide (lift shaft), the rms delay spread is smaller.

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TABLE II RMS DELAY SPREAD FOR OPENING/CLOSING LIFT DOOR

Fig. 14. RMS delay spread for different lift car level.

Taking all the measurement results into consideration, the mean rms delay spread is 122.3 ns. This delay spread is similar to those found in suburban environment [16] but larger than those reported in the literature for propagation along analogueswaveguide structures [17], [18]. In [16], approximately 80% of the rms delay spread values obtained from channel characterization in the 788–794 MHz band in suburban areas is below 200 ns. In [17], it is reported that the rms delay spread is less than 32 ns for about 50% of the measurement results in an underground mine at 2.4 GHz. In [18], delay spread for propagation in a mine environment in the frequency band of 400 to 500 MHz is within the range of 5–42 ns. The large delay spread reported in this paper is because this calculation has taken into consideration multi-paths in all 3 regions (the wave guided along the lift shaft in region 1 and the open space propagation in regions 2 and 3). If only the delay spread of region 1 is considered, the delay spread for different lift car level is presented in Fig. 14 by the red dotted line. As can be seen, the average value is approximately 31 ns, which is similar to those reported in the literature for analogous waveguide structures [17], [18]. IV. CONCLUSION In this paper, channel responses from wideband measurements along a lift shaft in a complex university environment have been presented. The corresponding ray-tracing simulations have been performed to verify the measurement results and to identify and visualize the propagation mechanisms. From the channel response obtained, signals have been classified into three regions, and propagation mechanism associated with each region has been discussed. The far and intermediate reflectors have been identified and verified via a set of controlled experiments and their corresponding simulations. Detailed

examination of propagation mechanism in nearby region is conducted. It is found that the wave guided along the lift shaft is the main mode of propagation in the urban environment. Through a series of controlled experiments and corresponding simulations, the effect of the lift door and the effect of the position of the lift car are examined. By examining the channel gain and the delay spread, it can be concluded that the lift shaft provides the main propagation mechanism in the complex urban environment. The lift door and lift car induces temporal variation to the guided signals. The opening and closing of the lift door can result in as much as 14 dB of signal variation, while the movement of the lift car can result in a temporal variation of signal strength of up to 13.5 dB. The change in position of the receiving antenna by up to one wavelength has no significant effect on the average channel gain and the rms delay spread, although the signal level for individual resolvable components varies due to the small-scale effects. However, characterization of the small-scale effect is beyond the scope of this paper. In summary, the lift shaft functions as a waveguide, and is the primary propagation channel in a complex environment. In an environment where propagation is difficult, the waveguide effect of the lift shaft can be an important communication channel. This study of military UHF signal propagation in an urban complex environment is important for military application such as urban warfare. These results and analysis can also be applied to higher frequency applications such as GSM and wireless LAN. For waveguide propagation at high frequencies, the multi-modal interaction can result in rapid fluctuation in the electric fields. However, for far region waveguide propagation at high frequencies, the dominant-propagation mode suffers a lower falling rate (attenuation per unit length) [19] and therefore, less attenuation. ACKNOWLEDGMENT The authors are grateful to Anonymous Reviewers and the Associate Editor for their constructive comments and suggestions for the paper. REFERENCES [1] T. M. Schafer, J. Manurer, J. V. Hagen, and W. Wiesbeck, “Experimental characterization of radio wave propagation in hospitals,” IEEE Trans. Electromagn. Compat., vol. 47, pp. 304–311, May 2005. [2] H. Meskanen and J. Huttunen, “Comparison of a logarithmic and a linear indoor lift car propagation model,” in IEEE Int. Conf. on Personal Wireless Communication, Jaipur, India, Feb. 1999, pp. 115–120. [3] A. G. Emslie, R. L. Lagace, and P. F. Strong, “Theory of the propagation of UHF radio wave in coal mines,” IEEE Trans. Antennas Propag., vol. AP-23, pp. 192–205, May 1973. [4] B. L. F. Daku, W. Hawkins, and A. F. Prugger, “Channel measurements in mine tunnels,” in Proc. Vehicular Technology Conf., Birmingham, May 2002, pp. 380–383. [5] K. Giannopoulou, A. Katsareli, D. Dres, D. Vouyioukas, and P. Constantinou, “Measurements for 2.4 GHz spread spectrum system in modern office buildings,” in Proc. Electrotechnical Conf., Lemesos, Cyprus, May 2000, pp. 326–329. [6] U. Dersch, J. Troger, and E. Zollinger, “Multiple reflections of radio waves in a corridor,” IEEE Trans. Antennas Propag., vol. 42, pp. 1571–1574, Nov. 1994. [7] P. V. Nikitin, D. D. Stancil, A. G. Cepni, O. K. Tonguz, A. E. Xhafa, and D. Brodtkorb, “Propagation model for the HVAC duct as a communication channel,” IEEE Trans. Antennas Propag., vol. 51, pp. 945–951, May 2003. [8] H. Meskanen and O. Pekonen, “FDTD analysis of field distribution in an elevator car by using various antenna positions and orientations,” Electron. Lett., vol. 34, pp. 534–535, Mar. 1998.

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[9] T. S. Kim, H. S. Cho, and D. K. Sung, “Moving elevator-cell system in indoor buildings,” IEEE Trans. Veh. Technol., vol. 49, pp. 1743–1751, Sep. 2000. [10] K. L. Blackard, T. S. Rappaport, and C. W. Bostian, “Measurements and models of radio frequency impulsive noise for indoor wireless communications,” IEEE J. Sel. Areas Commun., vol. 11, pp. 991–1001, Sep. 1993. [11] J. R. Hampton, N. M. Merheb, W. L. Lain, D. E. Paunil, R. M. Shuford, and W. T. Kasch, “Urban propagation measurements for ground based communication in military UHF band,” IEEE Trans. Antennas Propag., vol. 54, pp. 644–654, Feb. 2006. [12] T. S. Rappaport, Wireless Communication: Principles and Practice. Englewood Cliffs, NJ: Prentice Hall, 2002, ch. 5. [13] [Online]. Available: http://www.remcom.com/wirelessinsite [14] P. E. R. Galloway and T. D. Welsh, “Extensions to the shooting bouncing ray algorithm for scattering from diffusive and grating structures,” presented at the Proc. Int. Radar Symp., Berlin, Sep. 2005, IRS. [15] H. Ling, S. W. Lee, and R. C. Chou, “High frequency RCS of open cavities with rectangular and circular cross sections,” IEEE Trans. Antennas Propag., vol. 37, pp. 648–654, May 1989. [16] A. Semmar, J. Y. Chouinard, V. H. Pham, X. B. Wang, Y. Y. Wu, and S. Lafleche, “Digital broadcasting television channel measurements and characterization for SIMO mobile reception,” IEEE Trans. Broadcast., vol. 52, pp. 450–463, Dec. 2006. [17] C. Nerguizian, C. L. Despins, S. Affes, and M. Djadel, “Radio-channel characterization of an underground mine at 2.4 GHz,” IEEE Trans. Wireless Commun., vol. 4, pp. 2441–2453, Sep. 2005. [18] M. Lienard and P. Degauque, “Natural wave propagation in mine environments,” IEEE Trans. Antennas Propag., vol. 48, pp. 1326–1339, Sep. 2000. [19] R. E. Collin, Field Theory of Guided Waves. New York: IEEE Press, 1991, ch. 5. Xiao Hong Mao (S’09) received the B.Eng. (Hons) degree in electrical and electronics engineering from Nanyang Technological University, Singapore, in 2007, where she is working toward the Ph.D. degree. She has been a Research Engineer in the School of Electrical and Electronic Engineering, Nanyang Technological University since September 2009. Her research interest is in channel modeling and characterization in complex environments.

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Yee Hui Lee (S’96–M’02) received the B.Eng. (Hons) and M.Eng. degrees in electrical and electronics engineering from the Nanyang Technological University, Singapore, in 1996 and 1998, respectively, and the Ph.D. degree from the University of York, York, U.K., in 2002. Since July 2002, she has been an Assistant Professor at the School of Electrical and Electronic Engineering, Nanyang Technological University. Her research interest is in channel characterization, rain propagation, antenna design, electromagnetic band gap structures, and evolutionary techniques.

Boon Chong Ng (M’88–SM’06) received the B.Eng. (Hons) degree in electrical engineering from the National University of Singapore in 1988, the M.Sc. degree in electrical engineering, the M.Sc. in statistics, and the Ph.D. degree in electrical engineering, from Stanford University, Stanford, CA, in 1990, 1996, and 1998, respectively. From 1988 to 1989 and 1990 to 1993, he was a Research Engineer with the Defence Science Organization (DSO), Singapore. In 1998, he was a Research Assistant in the Information Systems Laboratory at Stanford University. Currently, he holds the appointment of Head of the Advanced Communications Laboratories and is a Principal Member of the Technical Staff at DSO National Laboratories, and is also an adjunct Associate Professor at the School of Electrical and Electronic Engineering, Nanyang Technological University. He leads research and development groups in MIMO communications, software and cognitive radios. Prof. Ng was the recipient of the Singapore Government DTTA postgraduate fellowship from 1993 to 1997.

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Physical Meaning of Perturbative Solutions for Scattering From and Through Multilayered Structures With Rough Interfaces Pasquale Imperatore, Antonio Iodice, Senior Member, IEEE, and Daniele Riccio, Senior Member, IEEE

Abstract—Theoretical formulas without a clear comprehension of their intrinsic meaning are of difficult use in the context of practical applications. In this paper, we investigate on the physical meaning of existing first-order solutions for the field scattered by layered structures with rough interfaces, which were derived by Imperatore et al. in the framework of perturbation theory. To capture the intrinsic significance of the closed-form scattering solutions, suitable expansions are rigorously performed by leveraging on local scattering descriptors. The obtained series expansions, which can be seen as ray series, can be then accurately analyzed showing that each term has a direct physical explanation. The analysis is carried out for both from- and through-layered-structure scattering configurations. As a result, analytical perturbative solutions turn out to be completely interpretable by simple physical concepts, so that the global scattering response can be interpreted as the superposition of single-scattering interaction mechanisms taking place locally, which are filtered by the layered structure. The meaning of the first-order approximation is also discussed in the layered structure context. Finally, we give a complete explanation for the scattering enhancement phenomenon contemplated in the first-order limit. Index Terms—Electromagnetic scattering, generalized reflection coefficients, layered media, perturbation methods, rough surfaces, scattering enhancement.

I. INTRODUCTION AND MOTIVATION

T

HE analysis of layered structures poses challenging questions from the electromagnetic theoretical investigation point of view and certainly is of enormous interest in the application perspective. Several modelling and design problems, encountered in SAR (Synthetic Aperture Radar) [1], GPR (Ground Penetrating Radar) sensing [2], radar altimeter for planetary exploration, microstrip antennas and MMICs (Monolithic Microwave Integrated Circuits), radio-propagation in urban environment for wireless communications, through-the-wall detection technologies, optics, biomedical diagnostic of layered biological tissues, geophysical and seismic exploration, lead to the analysis of the electromagnetic wave

Manuscript received February 10, 2009; revised January 27, 2010; accepted February 21, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. The authors are with the Department of Biomedical, Electronic and Telecommunication Engineering, University of Naples “Federico II,” 80125 Naples, Italy (e-mail: [email protected]; [email protected]; dariccio@ unina.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050417

interaction with complex multilayered structure, whose boundaries can exhibit some amount of roughness. Accurate and realistic scattering models are required for the remote sensing of natural scene, to obtain a reliable interpretation of scattering data from rough boundaries stratification. For propagation analysis, the accurate prediction of field penetration through rough layered structures with rough boundaries, that can model the buildings’ walls, can be of crucial relevance, for both outdoor and indoor scenarios. Several perturbative solutions were found for different simplified configurations in [3]–[12]. In previous papers, by analyzing the wave interaction with layered structures with rough boundaries, we have shown that the first-order perturbative solution for the scattering problem is feasible and a general systematic approach was developed. In fact, general closed forms, involving the generalized reflection/transmission formalism, were recently obtained for the scattering from [13] and through [14] three-dimensional layered structures with an arbitrary number of rough interfaces. Consequently, the compact expressions in [13] and [14] allow us to analyze polarimetrically and parametrically the general functional dependence of the scattered electromagnetic field on the electromagnetic and geometric parameters of an arbitrary layered structure. We emphasize that in the following we refer exclusively to the general formulation [13], and [14], since all of the other ones [3]–[12] can be regarded as particular cases of the one in [13], [14]. On the other hand, generally speaking, even though the manageability of the analytical solutions is an essential requirement for applications, the understanding of the physical meaning can be even more crucial. Nevertheless, in analytic derivations the final results are mostly attainable in a form that can appear illegible in the physics perspective. Furthermore, modeling real situations often leads to some suitable analytical approximations whose intuitive interpretation can be lost. Conversely, when a clear physical perspective of the meaning of the obtained solutions is viable, the implications open scenarios that could not be conceived otherwise. More in general, in the radar applications the availability of closed-form scattering solutions is even more fundamental for the comprehension and the schematic handling of the problem rather than for the actual scattering evaluation. In this perspective, the physics of the scattering mechanisms involved in the scattering from and through layered structures with rough boundaries should be better clarified. Even though an interpretation of the perturbative solution was formally obtained in [8] with reference to a specific layered configuration

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with only one rough interface, the physics of the interaction of electromagnetic waves with complex layered structures with an arbitrary number of rough interfaces has not been completely clarified yet. On other hand, despite the fact that the general solutions derived in [13] and in [14] exhibit a compact and symmetric structure, the related physical meaning is not immediately obvious and a physical interpretation has not been provided yet. Nevertheless, the relevant question one might ask now is whether, using such an analytical result, the intrinsic physical meaning of the first-order global perturbative solutions can be revealed, to shed light on the contemplated scattering processes that take place locally inside the layered structure. Furthermore, we emphasize that in many applications, such as exploration of seismic events or GPR, time-dependent wave-trains are observed, rather than spectral intensities. Therefore, in some cases a time domain characterization of the layered structures response could result more attractive than a spectral one. In view of the above considerations, in this paper considerable attention is paid to the intrinsic significance of the global scattering solution, getting more concrete insight into the physics of the problem, and a physical interpretation of the existing first-order perturbative solutions is carried out on an analytical playground. Our aim is then to show that detailed, physically revealing and mathematically useful information can be extracted from such perturbative models. For this purpose, starting from the general perturbative solutions [13], [14], firstly we suitably expand the obtained solutions. The results we obtained in [7], [8] suggest us the usefulness to base the expansions on local descriptors, in order to analyze the meaning of the global scattering response. In this paper, once the nature of the local interaction is recognized, we demonstrate that the obtained expansions can be properly seen as a ray series or a geometrical optics series; so the basic scattering mechanisms involved can be accurately visualized showing that each term of the ray series has a direct physical explanation. Consequently, the local/global scattering concepts are successfully exploited, differently from [3], [9] and [10] wherein the authors resort to the radar contrast. Therefore, the suitable reformulation of the scattered field expressions and the associated ray series sheds light on the relations among global scattering and local scattering phenomena in the layered structure: the expansions explain how global scattering, from and through the layered structure, arises from the (local) scattering that takes place when the waves propagating in the structure interact locally with the corrugated interfaces; whereas the multiple bounces on the flat boundaries, preceding and following the (local) single scattering occurrence, elucidate how the interference effects acting in the structure influence the global response of the structure. Consequently, in the first-order limit, the global scattering can be considered as the superposition of waves propagating in the layered structure, each one undergoing to a local scattering phenomenon filtered by the layered structure; whereas the filter action arises from the interferential effects due to the coherent interaction with the boundaries. As a result, the global scattering problems, which were introduced as formal mathematics in the first-order perturbative limit, turn out to be fully interpretable by simple physical concepts.

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Commonly, the scattering enhancement phenomenon is only illustrated for volume scattering or second-order scattering from rough surface [18], [19]; whereas in [9], [10], in the first-order, the backscattering enhancement from a single rough surface on the top of a layered media is pointed out. In view of that, the performed physical interpretations bring us to characterize the scattering directions, for which our first order models contemplate the enhancement scattering phenomenon, associated with the coherent effects taking place in a layered medium with rough interfaces. Consequently, Enhancement Cones are identified. This result is corroborated by, and could explain, the experimental results in [20]. In addition, note also that if the incident wave is a modulated pulse, each term of the expansions corresponds to an echo that will be received with a different time delay. It is then clear that the obtained expansions open the way to a time-domain analysis of the layered structures response. It should be noted that this point is of fundamental importance for instance if the considered model must be embedded in a SAR simulator, as well in a SAR data processing strategy or in a ray-tracing code for field levels prediction in urban environment. Finally, we remark the several advantages of obtained expansions. They let us deeply understand the physics of the problem revealing the intrinsic nature of the basic scattering mechanisms involved; they elucidate the physical meaning of the first-order approximation; and they explain the enhancement phenomena contemplated in the first-order limit. What is more, the expansions are mathematically useful since they are also addressed to a direct time-domain characterization of the structure response that can be effectively applied to several situations of interest. As a result, even though our approach is primarily theoretical, the proposed analytical expansions are meaningful from the result interpretation point of view, they have interesting implications, and they open the way to new possible applications to coherent remote sensing and to radio propagation prediction in urban environment. This paper is organized as follows. In Section II, we briefly define the used formalism for propagation in layered media. Section III addresses the models, which are based on perturbation method, that are the object of our analysis. The concept of physically-based local descriptors is introduced in Section IV. In Section V, a powerful physical interpretation of the analytic solutions is provided. In Section VI, the contemplated scattering enhancement phenomenon is discussed. In Section VII numerical results are provided. Section VIII concludes with a summary. II. NOTATION AND DEFINITIONS With reference to a multi-layer structure (Fig. 1), each th layer is assumed to be homogeneous and characterized by arbitrary and deterministic parameters: the relative permittivity , the magnetic relative permeability and the thickness . Note that the parameters pertaining to layer with boundaries and are distinguished by a subscript . In particular, it has been assumed . , for the -poThe generalized reflection coefficients larization (TE or TM), at the interface between the regions and are defined as the ratios of the amplitudes of upward- and

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III. PERTURBATION THEORY: EXISTING SCATTERING MODELS

Fig. 1. Geometry for a flat boundaries layered medium.

downward-propagating waves immediately above the interface. They can be expressed by recursive relations as [16] (1)

In this section, we consider existing perturbation-based models to deal with the electromagnetic scattering from and through layered structures with rough interfaces. Noticeable progress has been attained in the past in the investigation on the extension of the classical perturbative solution for the scattering from rough surface to layered configurations; however, in several works different authors have only analyzed simplified geometries/configurations in the first-order limit, using approaches, formalisms and final solutions that can appear of difficult comparison [3]–[12]. Besides, all the formulations, which refer to the case of scattering from a single rough interface [3], [5], [6], have been unified in [8]. On the other hand, a solution for the case of two rough boundaries has also been proposed in [12]. Nonetheless, general bistatic solutions have been developed in closed-form only recently by Imperatore et al. for the scattering from [13] and through [14] a 3-D geometry (see Fig. 2) with an arbitrary number of rough interfaces. In the following we refer exclusively to these general solutions, since all the other existing solutions [3]–[12] can be regarded as particular case of the general ones in [13], [14]. Consequently, we firstly summarize the general perturbative formulation (Section III-A), after that the relevant closed-form firstorder solutions will be considered (Section III-B). A. General Perturbative Formulation

Likewise, at the interface between the regions is given by

and

,

(2) are the ordinary reflection coefficients at the where interface between the regions and [8], [16], where , where the superscripts denote , is the the polarization, wave number for the electromagnetic medium in the th layer, and

In this section, we briefly summarize the perturbation approach employed in [13], [14]. We refer to the general structure depicted in Fig. 2; where each ( th) rough interface is assumed to be characterized by a zero-mean bi-dimensional process , whose normal vector is , with small enough deviations and slopes with respect to the reference mean plane. In addition, no constraints are imposed on the correlation degree among the rough interfaces. An arbitrary polarized monochromatic planar (infinite-extent) wave is considered to be incident on the layered medium at an angle relative to the direction from the upper half-space (8)

(3) is the two dimensional projection of where . Moreover, the folvector wave number on the plane lowing notations are used: (4) (5) (6) where the factors (4), (5), (6) take into account the multiple reflection in the th layer. In addition, with the notation , with , we indicate the ordinary transmission coeffiand [8], cients at the interface between the regions [18], We stress that (7)

where in the field expression a time factor , stood, where

is underand where (9) (10)

is a basis for the horizontal/vertical polarization vectors. In the following, the symbol denotes the projection of the corre. Here , so we sponding vector on the plane and distinguish the transverse spatial coordinates the longitudinal coordinate . To obtain a solution valid in each region of the structure, the continuity of the tangential fields have to be enforced (11) (12)

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Fig. 2. Geometry for an N-rough boundaries layered medium.

where , . We assume then that, for each mth rough interface, deviations and slopes of surface with respect to the reference mean plane are small enough, so that the fields can be expanded about the reference mean plane. The field expansions can be then injected into the boundary conditions (11), (12), so that, retaining only up to the first-order terms, the following non-uniform boundary conditions can be obtained:

(13)

layered medium. In order to perform the evaluation of perturbative development, the scattered field is then represented as the sum of up-going and down-going waves (17) (18) in which

(19)

(14)

(20)

(15) (16)

denote the polarization and where the superscripts state for the incident and scattered wave, respectively, and may or vertical polarization ; is the stand for horizontal denotes the projecintrinsic impedance of the medium ; plane of the vector wave-number. In order to tion on the solve the scattering problem in terms of the unknown expansion coefficients , their amplitudes can be arranged in a single vector according to the notation

where the field solution has been formally represented as

Therefore, the boundary conditions from each th rough interface can be transferred to the associated equivalent flat interface. In addition, the right-hand sides of (13) and (14) can be interpreted as effective magnetic and electric surface current densities, respectively, with the incident polarization; so that we can identify the first-order fluctuation fields as being excited by these effective surface current densities imposed on the unperturbed interfaces. Accordingly, the geometry randomness of each corrugated interfaces is then translated in random current sheets imposed on each reference mean plane , which radiate in an unperturbed (flat boundaries)

(21) Subsequently, the non-uniform boundary conditions (13), (14) can be formulated by employing a suitable matrix notation, so horizontal polarized scattered wave we get that for the (22)

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where

and to the -polarized scattering contribution through the structure into lower half space , originated from the rough interface ; and where is the spatial between the layers , power spectral density of th corrugated interface (27)

(23) is the term associated with the effective source distribution, and where the transfer matrix operator is given by

i.e., the Fourier transformation of correlation function

-corrugated interface auto-

(28)

(24) denoting the polarization. Eq. with the superscripts (22) states in a simpler form the problem originally set by (13), implies dealing with (14): as matter of fact, solving (22) the determination of unknown scalar amplitudes instead of working with the corresponding vector unknowns . As a result, when a structure with rough interfaces is considered, the enforcement of the non uniform boundary concan be ditions through the stratification addressed by writing down a linear system of equations with the aid of the matrix formalism (22) with . It is possible to show that the formulation of non-uniform boundary conditions in matrix notation enables a systematic approach that involves the use of matrix formalism, so that, effectively using the concept of generalized reflection/ transmission coefficients, the unknown expansion coefficients can be derived in closed-form via a recursive method [13]–[15]. B. General Closed-Form First-Order Solutions With regard to the upward scattering into the upper half-space from an arbitrary three-dimensional layered structure, it has been established that, in the first-order limit, the contribution of a generic ( th) embedded rough interface to the bi-static scattering cross section of the layered structure can be written as [13]

Note also that full closed-form expressions for the polariand are provided in [13] metric coefficients and [14], respectively. IV. LOCAL SCATTERING FROM AND THROUGH A ROUGH INTERFACE The concept of local scattering is introduced in this section. To deal with the scattering property of a corrugated interface of a layered structure, it is fruitful to emphasize the local response of a rough interface, i.e., the scattering properties exhibited by the roughness when the stratification surrounding the rough interface is brought to infinite. Therefore, from the scattering mechanism point of view, it can be assumed that the wave interaction with a rough interface embedded in a layered structure can be locally assimilated to the wave interaction with a rough surface between two half-spaces. The rationale motivating this concept will appear clear in the next section, when it will be shown that the first-order perturbative solutions are susceptible of a representation in terms of the local scattering properties of the corrugated interfaces. The local scattering cross sections of the th rough interface of the structure, for the scattering from and through the roughness respectively, are then defined as (29)

(25) is relative to the -polarized inwhere the coefficient cident wave impinging on the structure from upper half space 0 and to the -polarized scattering contribution from the structure into upper half space 0, originated from the rough interface be. tween the layers , Similarly, with regard to the downward scattering into the lower half-space through an arbitrary three-dimensional layered structure, the contribution of a generic ( th) embedded rough to the bi-static scattering cross section of the layinterface ered structure can be written as [14]

(30) where

and wherein, for the

case, we have

(31) (32) (33)

(26)

(34)

is relative to the -polarized inwhere the coefficient cident wave impinging on the structure from upper half space 0

. Therefore, the local scattering with coefficients are formally identical to the classical ones relative

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A. Scattering From an Arbitrary Layered Media With Rough Interfaces The upward scattered far-field into the upper half-space from an arbitrary layered structure with an embedded rough interface case, can be written as [13] ( th), with reference to the

Fig. 3. Reciprocity for local scattering from and through a rough interface.

(37) with

to a rough surface between two half-spaces [18]–[22]. Consequently, the reciprocity for the local scattering can be expressed as follows (see Fig. 3):

(38)

(35) (36)

V. PHYSICAL INTERPRETATION OF FIRST-ORDER PERTURBATIVE SOLUTIONS In this section, to get more insight into the meaning of the first-order solutions (25) and (26) from the concerning scattering mechanism point of view, we consider instructive to carry out a complete expansion of the scattering coefficients. In order to accomplish a satisfactory comprehension of how the interaction of the EM field with rough interfaces of an arbitrary layered structure takes place, a key-point is to recognize that the interaction with the structure can be expressible in terms of local interactions with the generic rough interface. Beside the rather intuitive viewpoint of this concept, it should be also noted that the adopted terminology “local” epitomizes the following rigorous aspect: the scattering amplitude, evaluated in the first-order limit of the perturbation theory, can be analytically written as a single space integral with a kernel that depends only on the rough interface height and on its first-order derivatives at a given point [21]. As a result, in order to phenomenologically describe the scattering from and through the structure and analyze the meaning of the global scattering response, we point out the usefulness of basing the expansions on local descriptors. Specifically, four distinct types of local interaction with an embedded rough interface can be distinguished: two of them identifiable as local scattering through the roughness and the other ones as local scattering from the roughness. Moreover, since in the limit of first-order perturbation theory the global response of a structure with all rough interfaces can be directly obtained considering the superposition of the response from each interface [13], [14], we firstly focus our attention to a generic embedded rough interface. Afterwards, the general interpretation for a layered structure with an arbitrary number of rough interfaces can be addressed. Without loss of generality, since analogous considerations hold for the other polarization combinations, the analysis is concase only. ducted for the

where

where the generalized transmission coefficients in downward direction are defined as

(39) denote the polarization, and where the superscripts where the generalized transmission coefficients in upward direction are , with the generalized transmission coefficients in upward direction for the layered slab between two half-spaces ( , 0) defined as

(40) Therefore, in order to provide a symmetrical expansion, it , which is associated is possible to explicit the factor with the multiple round trip in the th layer and included in ; so we can write (41) It should be noted that the

are distinct from the co-

efficients , because in the evaluation of the effect of all the layers under the layer is taken into account, whereas are evaluated referring to a different configuration in which the intermediate layers are bounded by is susthe half-spaces 0 and . Moreover, noting that ceptible to be written equivalently in the form [16]

(42)

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and applying (9), we obtain the following expansion for the sub factors in (38):

(43) Furthermore, we introduce the following notation: (44) (45) and recognize that these factors correspond to a complete roundtrip in the intermediate layer with coherent reflections at the layer boundaries. We consider then the series expansions (geometric power series) of sub-factors (5), (6) (46) where

may stand for

or

or

, respectively. By using (46), substituting (41), (43) in (38) and taking into account (31)–(34), we obtain the final expansion

(47) Note also that, using extensively (46) the generalized transmission coefficients (39), (40) could be as well expressed as

the product of a number of summations equal to the number of layer involved. However, for the reasons substantiated before, and we focus our attention on the two layers just above under the considered roughness. In Fig. 4, the remaining part of the structure is visualized condensed in two equivalent slabs constituted, respectively, by the intermediate layers (under the th layer) and (above the th layer). The suitably expanded solution (47), expressed as an infinite sum of contributions, is then susceptible of a straightforward physical interpretation in terms of a ray series. In particular, each individual term of the absolutely summable infinite series can be physically identified as a wave propagating in the structure that experiences a local single-interaction with the roughness. Consequently, four distinct families of rays can be recognized; each one associated to one type of local interaction, so that each term in (47) can be readily identified as follows. a) Local upward scattered waves from rough interface: each of these waves [see Fig. 4(a)] undergoes a coherent transmis, through the sion into th layer , then complete round-trips intermediate layers in the th layer with coherent reflections at , then an incoherent local scattering the incident angle , upward within the from the rough interface , subsequently complete round-trips observation plane in the th layer with coherent reflections at , and finally a coherent transmission the scattering angle in the upper half-space through the intermediate layers . b) Local upward scattered waves through rough interface: each of these waves [see Fig. 4(b)] undergoes a coherent trans, through mission into the th layer , then complete roundthe intermediate layers in the th layer with coherent reflections at trips , subsequently a coherent transmission the incident angle followed by complete round-trips in the th layer and by further bounce on the th flat interface at , and after that an incoherent local scatthe incident angle tering through the rough interface , upward within the observation plane , subsequently complete in the th layer with coherent reflecround-trips tions at the scattering angle , and finally a coherent transin the upper half-space mission through the intermediate layers . c) Local downward scattered waves through rough interface: each of these waves [see Fig. 4(c)] undergoes a coherent , transmission into the th layer , then through the intermediate layers complete round-trips in the th layer with , then an coherent reflections at the incident angle incoherent local scattering through the rough interface downward in the observation plane followed by further bounce on the th flat interwith subseface complete round-trips in the th layer quently at the scattering angle , next a coherent

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Fig. 4. Physical interpretation for the scattering from an arbitrary layered structure with an embedded rough interface.

transmission

followed by subsequently

complete round-trips in the th layer with coherent reflections at the scattering angle , and finally in the a coherent transmission upper half-space through the intermediate layers . d) Local downward scattered waves from rough interface: each of these waves [see Fig. 4(d)] undergoes a coherent , transmission into the th layer , then through the intermediate layers complete round-trips in the th layer with coherent reflections at the incident angle , next a cofollowed by comherent transmission th layer plete round-trips in the and by further bounce on the th flat interface at the incident angle , and after that an incoherent local scattering from the rough interface , downward in the obser, followed by further bounce on the th vation plane with flat interface complete round-trips in the th layer subsequently at the scattering angle , then a coherent

transmission

followed by subsequently

complete round-trips in the th layer with coherent reflections at the scattering angle , and finally in the a coherent transmission upper half-space through the intermediate layers . It should be noted that [14]

Furthermore, to give reason for the several factors appearing in the expansion (47) in the form , we observe that we obtain differentiating Snell’s law (48) identifies, within the th layer, the ray scatwhere the angle tered in observation direction. This scattered ray can be thought as the contribution in the observation direction from a spherical wave emanating from the th roughness. Therefore, the factor (48) accounts for the variation of the divergence of the locally

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scattered rays, which cross the flat boundaries stratification from is interpretable as the th to the th layer. As a result, the scattered-ray-amplitude divergence factor, associated with the varying refractive index. Note also that when the geometry reduces to the one analyzed in [8], only the first term in (47) holds [see Fig. 4(a)], so that the interpretation is fully congruent with the one furnished in [8].

we do not repeat similar examination. The reader can verify that the following expansion:

B. Scattering Through an Arbitrary Layered Media With Rough Interfaces Similarly to the analysis conducted in Section V-A, the process of scattering transmitted through the structure can be investigated as well referring to a generic rough interface of the stratification. The downward scattered far-field into the lower half-space from an arbitrary layered structure with an case, embedded rough interface ( th), with reference to the can be written as [14]

(49) where

, with

, and wherein

(50) where, the generalized transmission coefficients in downward direction are , with the generalized transmission coefficients in downward direction for the layered slab between two half-spaces defined as

(53) which is the counterpart of (47) for the scattering through the structure, can be similarly interpreted in terms a series of rays identified as pictured in Fig. 5. We emphasize that [14]

(51) denote the polarization. Simwhere the superscripts ilarly to (43), the following relation can be derived:

(52) Furthermore, the following additional notation is introduced:

Likewise, by using (46) with or or or , respectively, in place of , ; taking into account (31)–(34) and substituting (40), (43), (52) in (50), the final expansion (53) is obtained. Analogous considerations can be done as in the previous case; however to save space

Once both the expansions, for the scattering from (47) and through (53) the structure, have been formulated and illustrated (Sections V-A, V-B), some remarkable considerations are in order. First of all, it should be noted that, even though the presented physical interpretations allow us directly visualizing the physics of the problem, they are not based on an intuitive approach but are carried out analytically starting from perturbative solutions obtained in first-order limit. Specifically, the expansions have been carried out exploiting the local nature of the interaction between waves and corrugated interfaces within the layered structure; so that the global scattering response, from and through the layered structure, has been decomposed in terms of local interactions. Although the investigation leads to expansions that can appear cumbersome, from the viewpoint of the comprehension of scattering mechanisms, each term of (47) and of (53) can be directly identified as a wave propagating in the structure: each of

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Fig. 5. Physical interpretation for the scattering through an arbitrary layered structure with an embedded rough interface.

them comprises a series of coherent interactions with flat boundaries and an incoherent local single-scattering occurrence, from or through the corrugated interface. Therefore, (47), (53) can be thought of as a ray series or a geometrical optics series. Finally, note also that when the arbitrary layered structure with all rough interfaces is concerned, since in the first-order limit the multiple scattering contributions are neglected, the relative physical interpretation can be obtained effortless by superposition of the several ray contributions obtained considering separately each rough interface. The last but not the least factor distinguishing our approach is that, for the application point of view, the focus is often on the observed time-dependent wave-trains, rather than spectral intensities. As a matter of fact, propagation of the transmitted wave

through the structure causes a superposition of the echoes scattered from the interfaces that are received by a sensor located above or under the structure. Concerning this aspect, note also that if the incident wave is a modulated pulse, each term of the ray series corresponds to an echo that will be received with a different time delay. Consequently, the obtained results also open the way toward a time-domain formulation of the problem. This aspect is of fundamental importance, for instance, if the considered models have to be embedded in a SAR simulator or in a ray tracing code to predict the characteristics of the radio channel. C. Global and Local Scattering Coefficients To focus formally on the relations among local and global scattering concepts, the obtained decompositions (47) and (53)

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of the global scattering response in terms of the four types of local interactions (from and through the corrugated interface), can be expressed in a compact notation as scalar products of four-element vectors (54) (55) wherein

(56) captures the local response of the th rough interface between and , and the transfer vectors two layer of permittivity and are related to the coherent propagation inside the depends on a combination of stratification. Consequently, local scattering proprieties of the roughness between two hoand , , and of comogeneous media and ). This herent propagation inside the stratification ( as the (equivalent) global scattering brings us to refer to coefficients of a layered structure with the th embedded corrugated boundary between two layers, respectively, of permittivity and . As a particular situation, we mention the case in which the stratification under the roughness vanishes. This is the case th rough interface of the layered structure, of the last depends on only, so that whereas reduces to a scalar

D. Meaning of the First-Order Approximation In this sub-section the physical significance of the first-order perturbative approximation in the layered structure context is and are afclarified. Note that the transfer vectors fected in a global way by the unperturbed stratification properties and do not depend neither on the directions of nor on the roughness. This aspect, from a different perspective, can be elucidated by means of the following further considerations. Indeed, for the phase-matching condition, the projecand , for incident and scattions of the wave vectors, tered direction, respectively, must be invariant in the flat boundaries stratification, i.e., the propagation directions in multilayer flat-boundaries structure must be coplanar. Therefore, the round trips within and the transmission through the layers are all constrained in the incidence plane or in the observation plane, inor (and z direction), respecdividuated by the vectors tively. This clarifies that the contributions contemplated by the first-order SPM approximation are restricted within these two planes, whereas the neglected multiple scattering, associated with higher-order terms of the SPM solution, are not. In fact, it should be noted that although the considered expansions contain some infinite sums associated with multiple reflections between the interfaces, they account for a single diffuse scattering only, and only one of the surface spectral components, i.e. that spec, appears in ified by the so-called momentum transfer the scattering process in the limit of the first-order perturbation method. We underline that this consideration on the meaning of the SPM approximation basically extends to arbitrary structures the outcome we have pointed out in [8] for a less general configuration.

(57) VI. SCATTERING ENHANCEMENT PHENOMENON whereas for the

case, we have (58)

As a result, the transfer vector, which measures the influence of the stratification on the local scattering, whatever the roughness is, can be expressed in terms of the generalized transmission/reflection coefficients. It has been established that, when the observation point is located above or under the stratification, the global scattering by a generic rough interface embedded in the layered structure can be considered as a result of local scattering phenomena (from e through the embedded rough interface) filtered by the layered structure. The filtering action arises from the resulting interferential effects that take place in the layered structure, which are associated with the coherent interactions with the boundaries. Moreover, when the N-rough interfaces structure is concerned, the global scattering (from and through layered structure) can be the thought as representative of superposition of filtered scattering phenomena that take place from and through each rough interface locally. Finally, we emphasize that the presented expansions and the introduced coefficients are not mere factorizations related to some analytical convenience, but are based on physical relevance.

In this section, the focus is on one of the most interesting phenomenon, associated with coherent effects, which is perhaps a universal wave phenomenon inherent to waves of whatever physical nature; the aim of our theoretical analysis is to demonstrate that this enhancement phenomenon is contemplated by our first-order perturbative models. To this purpose, using the performed physical interpretations, we show that from each generic rough interface of a rough boundaries multilayer, due to the presence of the reflective action of the boundaries, coherent effects arise in the layered structure. These effects, contemplated in the single-scattering limit, arise in particular directions that can be clearly identified. For such a purpose, we focus our attention on waves that undergo to local scattering through a rough interface. Starting then from (54), which is a formal expression of the expansion (47), and evaluating it in , we get, for the case the directions for which

(59) Examination of the expansion (47) evaluated in the directions for which , shows that the corresponding second are identical, exand third elements of the transfer vector cept for a minus sign. Both these elements are associated with

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Fig. 6. Physical interpretation of coherent effect in a layered structure with an embedded rough interface.

the local scattering through the rough interface in opposite directions (downward and upward directed). Formally

(60) where

On the other hand, analyzing the scattering directions for which , we observe that from the reciprocity (36) directly follows that:

(61) Therefore, from (60) and (61) it follows that second and third terms in (59) are identical. This formal result is susceptible of an intuitive explanation in terms of coherent interaction between waves propagating through multi-channel reciprocal paths within the structure along scattering directions such that . Preliminarily, to clarify the phenomenology we

refer to the picture illustrates schematically in Fig. 6(a). By a solid and dashed lines, we have indicated only the propagation path of the scattering wave corresponding, respectively, to the of (47) with the summation indexes terms . Note that, in far field observation point, these two reciprocal waves interfere constructively, in spite of the randomness of the rough interface, for scattering direction such that . In general, the terms of (47) relative to the local scattering through the rough interface , for which (and ) constitute a family of reciprocal local scattered ray partners; each pair undergoes to total number of round trips in the th layer as well as to a total number of round trips in th layer. This mutual coherent wave partners, scattered locally through the interface in opposite directions, whatever be the random phase introduced by the roughness, sum-up in phase. In other words, this phenomenon arises from multi-channel wave propagation of reciprocal wave partners passing through identical channels with zero phase difference. Note also that the term reciprocal derives from the fact that the two partners cross the roughness in opposite directions. Consequently, the waves partners, resulting from such reciprocal scattering events, have the same amplitude and phase (i.e. these waves interfere constructively) if the projection on the plane of wave vectors of the incident and scattered waves have the same modulus . As a result, the sum-up in phase of all these terms exists only for directions lying in a cone defined by the incident direction, and whose axis is parallel to z direction as in Fig. 6. This brings us to refer to this family of directions as the enhancement scattering cone.

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Analogous considerations hold for the case of scattering through the structure. In Fig. 6(b), by dashed and solid lines we have indicated the propagation path of the scattering wave and of (53) with corresponding to the terms , and , respectively. In conclusion, the expansions allow us to demonstrate that the well-known enhancement phenomenon is contemplated by our perturbative models. Our analysis is accompanied by theoretical interpretations describing coherent interference between reciprocal scattered waves. In addition, we have shown that the enhancement phenomenon manifestation appears not only in backscattering [9], [10], [20] and specular [20] directions, but arises also in several directions which identify an enhanced scattering cone, disappearing as the angle of scatter deviates away from these directions since the two waves partners are no longer in phase and the coherent effect weakens. However, the factor (essentially related to the definition of polarization in the global coordinate system [25]) appearing in all of the terms, see (31)–(34), is responsible for the fact that this enhancement is more evident near back- and forward-scattering directions, in agreement with the experimental data [20]. Moreover, a similar examination can be conducted for each rough interface of the structure, pointing out the corresponding Enhancement Scattering Cone. We stress that the analysis can be obviously particularized to , and analothe backscattering configuration gously a ray interpretation can be used to visualize the coherent effect. Finally, we underline that, although the two considered components sum-up in phase when the scattering directions are along the enhancement cone, however this might give rise to enhancements or reductions of the total scattered intensity, depending on how the first and fourth terms in (59) interfere with the “enhanced” second + third term. Nonetheless, to evaluate the attractiveness of this analysis a time-domain context should appear more relevant. Finally, we emphasize that the scattering enhancement phenomenon is not accounted for by the radiative transfer theory [18]. Actually, the manifestations of these effects, that remain after ensemble averaging, could not be contemplated without employing full wave analysis which preserves phase information. Therefore, this effect should be taken into account, for instance, when data of the remote sensing of the Earth’s structures are interpreted. VII. NUMERICAL EXAMPLES In this section, we present some numerical examples aimed at better clarify the actual consequences of the coherent effects on the scattering response. To this purpose, we refer to a canonical layered medium with three rough interfaces, which is representative of several situations of interest. The considered vertical profile is characterized by the following parameters: , , , ; , . In addition, we model the roughness of the interfaces as Gaussian 2D random processes with Gaussian correlations, characterized by the surface height standard deviation and correlation length , with the subscripts referring to the th interface.

Fig. 7. Scattering coefficients hh for a three rough interfaces layered media with a fixed incidence angle:  contribution (long-dashed line),  contribution (dotted-dashed line),  contribution (dotted line), total contribution (solid line). Note that an offset of 50 dB has been conveniently added to each scattering pattern. The incidence direction is also shown.

Fig. 8. hh Scattering contribution from the rough interfaces  for prescribed incidence angles ( = 37:8) and for 1 = = 4:30 (solid line) and 3.80 (dashed line). The associate incidence direction is also shown. Note that an offset of 50 dB has been conveniently added to each scattering pattern.

In order to perform a consistent comparison, we refer to interfaces with the same roughness statistics. In addition, we suppose , no correlation between the interfaces. We assume for , 1, 2. Once this reference structure has been characterized, we first study the scattering cross section of the structure as a function of the scattering vertical angle in the upper half-space, assuming fixed the incident direction. In the polar plot of Fig. 7 the total scattered field is shown (solid line), together with the individual contributions of the different interfaces, as a function of the scattering angle . To save space, only the case is shown. It should be noted that to visualize the patterns an offset of 50 dB has been conveniently added to each scattering pattern. In addition, it has been assumed ; we emphasize that this value of incidence angle has been calculated to have all the four local contributions of (59) summing up in phase when for the interface , whereas this does not happen for the other interfaces In fact, in Fig. 7 the backscattering enhancement only appears in the individual return from . In order to better illustrate this issue, we focus our attention on the scattering contribution from the interface : in Fig. 8 we plot the scattering contribution from the interface with no change with respect to the previous example (solid line) and by to the new value 3.80 (dashed line). changing the ratio Again, it is clear that in the former case a backscattering enhancement is present, whereas it is not in the latter. It should

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be also noted that the shape of pattern responses are also affected by the coherent effects due to the unperturbed interfaces of the structure. This simple example explains that, in agreement with our analysis of Section VI, a scattering enhancement effect may appear even in the first order field, although it is not always present. VIII. CONCLUSION In this paper, we have investigated the physical meaning of the existing first-order perturbative solutions for the field scattered by layered structures with rough interfaces. In order to capture the physical significance of the analyzed formulations, suitable expansions of the closed form solutions are rigorously performed leveraging on local scattering descriptors. Our general approach can be applied to both scattering configurations (from and through the layered structure); thus the obtained expansions render a lucid interpretation of the scattering mechanisms that take place in a layered structure, whereas the series, which can be seen as a ray series or a geometrical optics series, offer a clear physical perspective of the interferential phenomena involved. Consequently, the global scattering response can be thought as the superposition of single-scattering local interactions filtered by the layered structure, whereas the filter action arises from the interferential effects due to the coherent interactions with the boundaries. Moreover, the physical meaning of the first-order perturbative approximation has been clarified in the layered structure context. It should be also noted that, despite the expansions are attained rigorously without any further approximation with respect to the solution proposed in [13], [14], the resulting interpretations turn out to be extremely intuitive and surprisingly simple. Therefore, the global scattering problems, which were introduced in [13], [14] as formal mathematics in the first-order perturbative limit, turn out to be completely interpretable by simple physical concepts. As a result, the obtained expansions also allow us to identify all the scattering directions for which the scattering enhancement phenomenon may be contemplated by our perturbative models in the first-order limit (Enhancement Scattering Cones). We want to explicitly underline that the obtained expansions primarily give insight into the perturbative analytical results, so enabling a relevant physical interpretation involving ray-series representation. However, for practical calculation purposes, the more compact notation of [13] and [14] can be more conveniently used. This is certainly true if only the frequency (or, better, phasor) domain solution is of interest (this is the case, in practice, when a sinusoidal time dependence is a sufficient approximation). However, the expansions presented in this paper open the way to a time domain characterization of the scattering response, since each ray of the series corresponds to an echo that will be received with a different time delay. As a result, the proposed expansions may be also useful in practice when a time-domain solution is required. On the other hand, validate a numerical result is always an intriguing point, worth to be discussed also in this paper. As far as scattering from complicate structures is in order, answer to this point is quite difficult because the exact solution to the scattering problem is unknown: no exact analytical solution can be derived; no exact numerical solution can be obtained.

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A comparison with elsewhere available approximate solutions can then provide the only viable way to get any validation to our model. Approximate analytical solutions are first considered. However, in this case the already available analytical solutions are derived from approaches relying on completely different basis (radiative-transfer theories, ray-tracing, and other incoherent approaches) with respect to one we selected. Then, this cross comparison between numerical results obtained by moving from different approaches cannot lead to a validation of our approach, but only leads to check, maybe only in some very specific cases, the differences between the results obtained under the different approaches. Within the class of approaches with rationale similar to the one we propose, a logic comparison can support our discussion. As a matter of fact, the approximations we assume are clearly of wider application with respect to the approximations in the model elsewhere already proposed. For instance, we consider all the interfaces to be rough, whereas in [3]–[11] only one rough interface, and in [12] only two rough interfaces (in this latter case only the HH case is detailed), are taken into account. With respect to these similar approaches, we already provided an analytical comparison (that is a much more general comparison with respect to a numerical one) between our and available approaches. In particular in [8], we first re-formulated, in a unified formulation, the analytical results from available approaches. Then [13], we showed that our approach has a more general applicability and that, in the simplified particular cases in which our and other approaches are valid, the results we get are coincident with those of the already available approaches. Some full numerical approaches are available. The most advanced in this field is [26]. However, in this case, only the 2-D case (scattering from one-dimensional profiles) is taken into account: the considered problem (and also the obtained solution) is then different from the one we took into account, which is based on a 3-D scattering problem involving two-dimensional interfaces; and the results are somehow incomparable. Our results turn to be of a wider interest also considering that they are applicable to a generic number of rough interfaces (only two rough interfaces are involved in the numerical solution provided in [26]). We stress that the results we obtain in this paper stem from the theoretical construct based on the boundary perturbation formulation [13], [14]. It is also important to note that the outcomes presented in this paper apply as well when the different approach of the volumetric perturbation formulation [27] is considered. In conclusion, the implications of the obtained expansions are twofold. In fact, not only they give us deep insight into the physics of scattering problem, and as such are crucial from a speculative investigation perspective; what’s more, they open the way toward new techniques for solving the inverse problem, for designing SAR processing algorithms, and for modelling the time-domain response of layered structures. These aspects will be a matter of further investigation. REFERENCES [1] M. Moghaddam, Y. Rahmat-Samii, E. Rodriguez, D. Entekhabi, J. Hoffman, D. Moller, L. E. Pierce, S. Saatchi, and M. Thomson, “Microwave Observatory of Subcanopy and Subsurface (MOSS): A mission concept for global deep soil moisture observations,” IEEE Trans. Geosci. Remote Sensing, vol. 45, no. 8, pp. 2630–2643, Aug. 2007.

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[2] D. J. Daniels, Ground Penetrating Radar, 2nd ed. London, U.K.: IEE, 2004. [3] A. G. Yarovoy, R. V. de Jongh, and L. P. Ligthard, “Scattering properties of a statistically rough interface inside a multilayered medium,” Radio Sci., vol. 35, no. 2, pp. 455–462, 2000. [4] A. G. Yarovoy, R. V. de Jongh, and L. P. Ligthart, “Transmission of electromagnetic fields through an air-ground interface in the presence of statistical roughness,” in Proc. IEEE IGARSS’98, Seattle, WA, Jul. 6–10, 1998, vol. 3, pp. 1463–1465. [5] R. Azadegan and K. Sarabandi, “Analytical formulation of the scattering by a slightly rough dielectric boundary covered with a homogeneous dielectric layer,” in Proc. IEEE AP-S Int. Symp., Columbus, OH, Jun. 2003, pp. 420–423. [6] A. Fuks, “Wave diffraction by a rough boundary of an arbitrary planelayered medium,” IEEE Trans. Antennas Propag., vol. 24, pp. 630–639, 2001. [7] G. Franceschetti, P. Imperatore, A. Iodice, D. Riccio, and G. Ruello, “Scattering from layered medium with one rough interface: Comparison and physical interpretation of different methods,” in Proc. IEEE IGARSS, Toulouse, France, Jul. 2003, pp. 2912–2914. [8] G. Franceschetti, P. Imperatore, A. Iodice, D. Riccio, and G. Ruello, “Scattering from layered structures with one rough interface: A unified formulation of perturbative solutions,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 6, Jun. 2008. [9] I. M. Fuks, “Radar contrast polarization dependence on subsurface sensing,” in Proc. IEEE IGARSS’98, Seattle, WA, Jul. 6–10, 1998, vol. 3, pp. 1455–1459. [10] A. Kalmykov, I. Fuks, I. Scherebinin, V. Tsymbal, A. Matveev, A. Gavrilekno, M. Fix, and V. Freilikher, “Radar observations of strong subsurface scatterers. A model of backscattering,” in Proc. IEEE IGARSS’95, 1995, vol. 3, pp. 1702–1704. [11] I. M. Fuks and A. G. Voronovich, “Interference phenomena in scattering by rough interfaces in arbitrary plane-layered media,” in Proc. IGARSS, 2000, vol. 4, pp. 1739–1741. [12] A. Tabatabaeenejad and M. Moghaddam, “Bistatic scattering from three-dimensional layered rough surfaces,” IEEE Trans. Geosci. Remote Sensing, vol. 44, no. 8, pp. 2102–2114, Aug. 2006. [13] P. Imperatore, A. Iodice, and D. Riccio, “Electromagnetic wave scattering from layered structures with an arbitrary number of rough interfaces,” IEEE Trans. Geosci. Remote Sensing, vol. 47, no. 4, pp. 1056–1072, April 2009. [14] P. Imperatore, A. Iodice, and D. Riccio, “Transmission through layered media with rough boundaries: First-order perturbative solution,” IEEE Trans. Antennas Propag., vol. 57, no. 5, pp. 1481–1494, May 2009. [15] P. Imperatore, A. Iodice, and D. Riccio, “Perturbative solution for the scattering from multilayered structure with rough boundaries,” in Proc. MICRORAD, Mar. 11–14, 2008, pp. 1–4. [16] W. C. Chew, Waves and Fields in Inhomogeneous Media. Piscataway, NJ: IEEE Press, 1997. [17] A. Ishimaru, Wave Propagation and Scattering in Random Media. New York: Academic, 1993. [18] L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing. New York: Wiley, 1985. [19] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing. Reading, MA: Addison-Wesley, 1982, vol. I,II,II. [20] Z. H. Gu, J. Q. Lu, A. A. Maradudin, A. Martinez, and E. R. Mendez, “Coherence in the single and multiple scattering of light from randomly rough surfaces,” Appl. Opt., vol. 32, no. 15, May 20, 1993. [21] T. M. Elfouhaily and C. A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media, vol. 14, no. 4, pp. R1–R40, Oct. 2004. [22] F. G. Bass and I. M. Fuks, Wave Scattering From Statistically Rough Surfaces. Oxford: Pergamon, 1979. [23] H. L. Bertoni, Radio Propagation for Modern Wireless Systems. Englewood Cliffs, NJ: Prentice-Hall. [24] M. Dehmollaian and K. Sarabandi, “Refocusing through single layer building wall using synthetic aperture radar,” in Proc. IEEE IGARSS, Jul. 2007, pp. 2558–2561. [25] A. Y. Nashashibi and F. T. Ulaby, “MMW polarimetric radar bistatic scattering from a random surface,” IEEE Trans. Geosci. Remote Sensing, vol. 45, no. 6, pp. 1743–1755, Jun. 2007. [26] N. Déchamps and C. Bourlier, “Electromagnetic scattering from a rough layer: Propagation-inside-layer expansion method combined to an updated BMIA/CAG approach,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2790–2802, Oct. 2007. [27] P. Imperatore, A. Iodice, and D. Riccio, “Volumetric-perturbative reciprocal formulation for scattering from rough multilayers,” IEEE Trans. Antennas Propag., submitted for publication.

Pasquale Imperatore received the Laurea degree (summa cum laude) in electronic engineering from the University of Naples “Federico II,” Naples, Italy, in 2001. For four years, he was a Research Engineer with WISE S.p.A., Pozzuoli, Italy, where he worked on modeling and simulation of wave propagation, advanced ray-tracing-based prediction tool design for wireless application in urban environment, and the simulation and processing of synthetic aperture radar (SAR) signals. From 2005 to 2007, he was a Senior Researcher with the international research center CREATE-NET, Trento, Italy, where he conducted research and experimentation on radio-wave propagation at 3.5 GHz and on emerging broadband wireless technologies. He is currently with the Department of Electronic and Telecommunication Engineering, University of Naples, “Federico II.” His research interests include wave scattering in layered media, perturbation methods, parallel computing in electromagnetics, as well as electromagnetic propagation modeling, simulation, and channel measurement.

Antonio Iodice (S’97–M’00–SM’04) was born in Naples, Italy, on July 4, 1968. He received the Laurea degree (cum laude) in electronic engineering and the Ph.D. degree in electronic engineering and computer science from the University of Naples “Federico II,” Naples, in 1993 and 1999, respectively. In 1995, he received a Grant from the Italian National Council of Research to be spent at the Istituto di Ricerca per l’Elettromagnetismo e i Componenti Elettronici, Naples, for research in the field of remote sensing. He was with Telespazio S.p.A., Rome, Italy, from 1999 to 2000. Since 2000, he has been with the Department of Electronic and Telecommunication Engineering, University of Naples “Federico II,” where he is currently a Professor of electromagnetics. His main research interests are in the field of microwave remote sensing and electromagnetics: modeling of electromagnetic scattering from natural surfaces and urban areas, simulation and processing of synthetic aperture radar (SAR) signals, SAR interferometry, and electromagnetic propagation in urban areas. He is the author or coauthor of about 170 papers published on refereed journals or on proceedings of international and national conferences. Prof. Iodice received the 2009 Sergei A. Schelkunoff Prize Paper Award from the IEEE Antennas and Propagation Society.

Daniele Riccio (M’91–SM’99) was born in Naples, Italy, on April 13, 1962. He received the Laurea degree (cum laude) in electronic engineering from the University of Naples “Federico II,” Naples, in 1989. He was a Research Scientist with the Institute of Research on Electromagnetics and Electronic Components, Italian National Council of Research (CNR), and with the Department of Electronic and Telecommunication Engineering, University of Naples “Federico II.” He was also a Guest Scientist with the German Aerospace Center High-Frequency Institute (JPL), Munich, Germany, in 1994 and 1995, and a Visiting Professor at Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 2006. He has won several fellowships from private and public companies (SIP, Selenia, CNR, CORISTA, and CRATI) for research work in the remote sensing field. He is currently a Professor of electromagnetics and remote sensing with the Department of Electronic and Telecommunication Engineering, University of Naples “Federico II.” His research interests are mainly focused on microwave remote sensing, synthetic aperture radar with emphasis on data simulation modeling, and information retrieval for land oceanic and urban scenes, as well as in the application of fractal geometry to remote sensing and electromagnetic scattering from natural surfaces. His research activity is witnessed by three books and more than 200 published papers. He is an Associate Editor for the journal Sensors. Prof. Riccio received the 2009 Sergei A. Schelkunoff Prize Paper Award from the IEEE Antennas and Propagation Society.

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A Novel Integration Method for Weak Singularity Arising in Two-Dimensional Scattering Problems Yan-Fei Jing, Ting-Zhu Huang, Yong Duan, Sheng-Jian Lai, and Jin Huang

Abstract—In this paper, we introduce a novel mechanical quadrature method for an efficient solution of weakly singular integral equations arising in two-dimensional electromagnetic scattering problems. This approach is based on and adapted from the recently proposed mechanical quadrature methods in [Extrapolation algorithms for solving mixed boundary integral equations of the Helmholtz equation by mechanical quadrature methods, SIAM J. Sci. Comput., vol. 31, 4115–4129, 2009]. We report experiments for solving TM-polarized induced currents and scattered fields to show its superiority to the classical method of moments when accuracy is a concern. Moreover, additional numerical experiments made with an extrapolation algorithm suggest that the accuracy of the present method can be further improved dramatically by means of the extrapolation algorithm to some extent. Index Terms—Electromagnetic scattering, extrapolation algorithm, mechanical quadrature method (MQM), method of moments (MoM).

I. INTRODUCTION UMERICAL implementation of moment methods is a relatively straightforward and an intuitively logical extensions of the basic elements of numerical analysis, as described in the remarkably well-known book by Harrington [1] and discussed and used extensively in computational electromagnetics (CEM) (see, for instance, Mittra [2], [3], and Gibson [4]). Consider a source-free region of space containing an isotropic medium characterized by constant permittivity and permeability . For two-dimensional transverse magnetic (TM) fields, suppose an infinite, perfectly conducting cylinder is illuminated with time variation . Here, by an impressed electric field is the basic imaginary unit, is called the pulsation, and is the involved time. The cross section as well as the underlying coordinate system is depicted in Fig. 1. Here, is the unit normal

N

Manuscript received August 31, 2009; revised January 29, 2010; accepted February 08, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported in part by the 973 Program (2007CB311002), the NSFC under Grants (10926190, 60973015, 10871034), the Specialized Research Fund for the Doctoral Program of Higher Education (20070614001), Sichuan Province Sci. & Tech. Research Project (2008JY0052), and in part by the Project of National Defense Key Lab. (9140C6902030906). Y.-F. Jing, T.-Z. Huang, Y. Duan and J. Huang are with the School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). S.-J. Lai is with the Institute of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, China (e-mail: cem@uestc. 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.2010.2050441

Fig. 1. The cross section of a cylinder and coordinate system.

to the smooth cylinder surface . is a filament of the induced on the conducting cylinder by the incident surface currents field. and are azimuthal angles in terms of the usual spherical coordinate convention. These two angles respectively correspond to arbitrarily located observation points and source points on . All points are defined with respect to the coordinate origin 0. Denote the wavenumber, the intrinsic impedance of free space, the Hankel function of order zero of the second kind. The distance between point and point is computed as , where is the Euclidean norm (or norm). Then the corresponding concerned integral equation in the context of the TM case for conducting cylinders can be represented as follows [1]: (1) is known and is the unknown to be determined. Here, When employing the method of moments (MoM), unbounded but integrable singularities occur in with so-called selfterms involving the same testing and source subdomains for evaluation of diagonal elements of the impedance matrix. In various numerical integral formulations for electromagnetics modeling, singular integrals are often encountered in the numerical construction of the corresponding impedance matrix elements. In order to achieve the desired accuracy of quantities, analytical and numerical evaluation of singular and nearly singular integrals (see, e.g., [5]–[7]) is increasingly a mature and challenging issue, which has attracted extensive research effort in this area. The most notable treatments developed to date for dealing with such singular integrals primarily include singularity subtraction and singularity cancellation techniques. Such approaches consist of three classes: polar and modified polar approaches [8], [9], Duffy or triangular coordinates [10]–[12], and extrapolation methods [13], [14].

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In particular, Bluck, Pocock and Walker [15] employed the singularity weakening technique and the singular components cancellation approach to propose an accurate method for the evaluation of the Cauchy principal value integrals arising in time-domain electromagnetic scattering with a boundary integral equation (BIE) method. Taylor [16] made use of the methods developed for computational mechanics to present a new method for finding the Galerkin (RWG, see [17]) solution of the electric field integral equation with high accuracy in the numerical integration of the singular surface integrals. In Taylor’s treatment, a series of coordinate transformations coupled with an interchange of order of integration followed by an appropriate Duffy coordinate transform has been carried out to remove the singularity in the integrand. A purely numerical singularity cancellation method for both two-dimensional and three-dimensional geometries was proposed by Khayat and Wilton [18], both theoretically and numerically proven to be not only advantageous over singularity substraction methods, but also more effective than other singularity cancellation methods in some aspects. Detailed analysis of both pros and cons of the singularity subtraction and singularity cancellation methods can also be noted in the above wonderful paper. Moreover, Järvenpää, Taskinen and Ylä-Oijala [19] developed a representative singularity subtraction technique for computing the impedance matrix elements of various electromagnetic surface integral equation formulations with the Galerkin method and high-order basis functions. For other treatments, refer to [20]–[22] by Tsalamengas and his colleagues for direct singular integral equation techniques while refer to [23] by Polimeridis and Yioultsis for a quite recent direct evaluation approach for weakly singular integrals in Galerkin mixed potential integral equation formulations. The focus of this paper is to explore and apply a novel mechanical quadrature method (MQM) to deal with weakly singular integrals in the practical solution of (1) for two-dimensional scattering problems. The present MQM method for the considered problems can be easily adapted from Huang and Wang’s MQM method [24], which was newly devised for solving mixed BIE of the Helmholtz equation. From the point of view of discretization of the scatterer contour, the implementation of the MQM method can be considered to be with pulse functions for a basis. Therefore, we will take the classical MoM method with pulse basis functions and point matching for testing for comparison. We consider in our numerical experiments three typical and representative types of cross sections, including circular and elliptic cross sections of perfectly metallic cylinders, and non-convex boomerang-shaped cross sections of sound-soft cylinders. By comparison, the MQM method can effectively provide higher accuracy for the solutions of the TM-polarized induced currents and scattered fields of infinite two-dimensional cylinders under impressed electric field illumination. Moreover, the CPU consuming time for the MQM method is a little less than that for the MoM method. The remainder of the paper is organized as follows. The adapted MQM method as well as an extrapolation algorithm (EA) is described in Section II. Numerical experiments will be given in Section III to demonstrate the efficiency and high accuracy of the MQM method and further improvement by the

EA formula in terms of accuracy. Finally, some conclusions will be made in the last section. II. THE MQM METHOD For the purpose of adapting the MQM method proposed in appearing in (1), [24] to treat the singularity problem of some basic knowledge is recalled first. Recall that the first and second kinds of Hankel functions of and can be defined as follows” order zero (2) (3) where and are respectively the first and second kinds of Bessel functions of order zero with the following corresponding expansions” (4)

(5) with a constant . From the comparison of (2) and (3), (1) can be expressed in by terms of (6) where the overbar (“ ”) denotes the conjugate complex of a scalar. This can transform the singularity problem of into that of . Denoting (5) together result in

as the function

, (4) and (7)

where

and

. In the following, we will illustrate the novel MQM method, most of whose similar derivation can be also found in [24]. Essential to the numerical formulation is the splitting of the involved integrands into appropriate parts, allowing effective integrations without any singular integrals but with high accuracy. Assume that the cylinder surface in Fig. 1 can be described by the parameter mapping with , where “ ” represents the derivative of the corresponding function with respect the variable . First by one-term addition and subtraction, split into two parts

where

(8) , and

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, . This splitting produces the following two integral operators on (9)

(10) Letting gral operator on

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can be obtained with appropriate induced surface currents linear solvers for the solution of the resultant system of linear equations. Furthermore, secondary quantities of interest such as the scattering cross section can be calculated correspondingly. In order to clearly show the differences between the MQM method and the classical MoM method with pulse basis functions and point matching for testing, we list the evaluation of the impedance matrix elements generated by the employed MoM method as follows (see, e.g., [1] and [4]):

gives rise to another intefor for

(11)

(15) With the help of the above three integral operators, (6) leads to (12) where is the unknown and is the . right-hand side with As stated by Huang and Wang in [24], the integral operator has a weakly singular kernel in (9) while the integral kernels in (10) and in (11) are both sufficiently smooth with respect to the smooth cylinder surface in Fig. 1. , the quadrature For the weakly singular integral operator formula by Sidi and Israrli [25] can be employed to construct the Fredholm approximation. For the two regular integral operand , numerical integrations with the midpoint or the ators trapezoidal rule can be implemented to construct the Nyström approximation [26]. (More detailed considerations concerning the involved integral operators can be found in [24].) segments Discretizing the scatterer contour into as the mesh width and as the filament with of , (12) results in the following system of linear equations: (13) , and with nodes . For the existence and uniqueness of the solution to (13), see [24]. If we denote the sum of , and as the assembled matrix , i.e., , by th element of careful derivation and calculation, the can be evaluated as in (14), shown at the bottom of the page, where . Once the elements of the impendance matrix and the forcing right-hand side in (13) are determined, an approximation to the where

,

. where From (14) and (15) together with (2) and (3), it is obviously observed that the evaluation of the non-self terms is treated the same way in both the MQM method and the MoM method. The difference between the two methods rests with different ways in evaluating the self terms. At the end of this section, the extrapolation algorithm (EA) established in [24] is restated as the following formula (16) and are the solutions of (13) with the correwhere and , respectively. The sponding filaments of taken to be above EA formula, which can be considered as a post-treatment technique, will be utilized to improve the accuracy of approximations obtained by the MQM method to certain extent. III. NUMERICAL EXPERIMENTS In this section, far from being exhaustive, the applicability and capacity of the MQM method with respect to accuracy are demonstrated in three typical and representative circumstances. The high accuracy of the present method is evaluated and verified in comparison of simulation results as well as numerical errors with the classical MoM method with pulse functions for a basis and point matching for testing. In addition, numerical comparisons are also made between the present MQM method and the EA formula mentioned in the end of the previous section. Numerical experiments are performed for solutions of the TM-polarized induced surface currents and scattering cross secfor cylinders under imtion with the incident angle pressed uniform plane wave illumination. The three types of cross sections of cylinders are circular, elliptic, and non-convex boomerang-shaped. See [1], [4] and [27] for more considerations of these circumstances. All the experiments have been

for for

(14)

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carried out with machine precision in double precision floating point arithmetic in MATLAB 7.0.4 with a PC-AMD Turion(tm) 64 2 Mobile, 2.10 GHz, 2.93 GB of RAM. For numerical solution of (12), the scatterer contours of segments with the cylinders are divided into filament of , separately making the MQM and MoM methods systems of linear equations with unknowns. result in It is noted that the resultant systems of linear equations are not so large to become prohibitive for direct methods to solve [28]. Therefore, Gaussian elimination is employed for the solution of the obtained systems of linear equations in convenience of comparison between the MQM and MoM methods. Example 1 Scattering by an Infinite Conducting Cylinder With a Circular Cross Section: TM Polarization [4]: The first example is one in which the modal solutions for the induced exist and can be taken as a function of the currents on the two-dimensional cylinder in the azimuthal angle following [29]: (17) are Hankel functions of the first kind of order , , and , otherwise. In such circumstances, besides comparing simulation results obtained by the MQM and MoM methods with the modal solutions in figures, we can also make comparisons between the MQM method and the MoM method in terms of errors listed in tables. We report three aspects of error data. The first two are the maximal and minimal entries of the absolute error vector. The last one is the relative error, calculated by means of dividing the Euclidean norm of the absolute error vector by the Euclidean norm of the corresponding modal solutions . For numerical comparison between the present MQM method and the MoM method, five different values for the circular cross , , , and have been section radius , i.e., . The chosen with the discretization size equal to process for the modal solutions is terminated after 20, 40, 60, 80 , , , and , respectively. and 100 modes for and scattering cross Plots of the induced surface currents , where is the angle at which is section ( evaluated) are respectively displayed on the left-hand-side and the right-hand-side of Fig. 2. In terms of reduced surface currents, no distinct comparison with respect to convergence performance of the MQM and MoM methods should be inferred from the left top two plots of Fig. 2 since the results of both the MQM and MoM methods are in good agreement with the modal solutions. However, with the increase of the circular cross section radius , as shown by the left last three plots of Fig. 2, undesired oscillations in the induced surface currents computed via the MoM method can be observed more and more severely while the calculated induced surface currents by the MQM method in general show good correspondence with the modal solutions in almost all cases. Such anomalies for the results obtained by the MoM method may be attributed to the low accuracy acquired, which can be reflected in Table I. The term “Ratio” there is obtained by means of dividing the error data of the MoM method by the counterparts of the MQM method. Such terms in the latter numerical reports can be explained in a similar way. where

Fig. 2. Infinite cylinders with different circular cross sections: TM polarization (MQM and MoM). Left: induced surface currents (jJ j). Right: scattering cross section  (0    2 ).

It is found in Table I that the accuracy of approximations obtained by the MQM method is almost always higher than that obtained by the MoM method. In most cases, the accuracy of approximations computed by the former is at least a factor of one higher than the accuracy of approximations computed by the latter. Especially for the case of , the accuracy obtained by the MQM method is a factor of two more than that by the MoM method. And it may be taken as a possible explanation for why the value of the relative error of the MoM method is greater than unity in this case. All in all, the higher accuracy of the MQM method observed in Table I may account for the

JING et al.: A NOVEL INTEGRATION METHOD FOR WEAK SINGULARITY ARISING IN 2-D SCATTERING PROBLEMS

TABLE I NUMERICAL COMPARISON RESULTS IN TERMS OF ERRORS OF INDUCED SURFACE CURRENTS FOR INFINITE CYLINDERS WITH DIFFERENT CIRCULAR CROSS SECTIONS: TM POLARIZATION (MoM AND MQM)

wonderful agreement between the MQM results and the modal solutions in terms of induced surface currents as reflected in the left five plots of Fig. 2. In addition, looking at Table III, the CPU consuming time of the MQM method is a little less than that of the MoM method. By the way, numerical experiments have been carried out between the EA formula represented as in (16) and the MQM method. For comparison, the discretization size is chosen to be for the EA formula while two discretization sizes are taken for the MQM method. Denote MQM720 in the case of and MQM1440 in the case of for the MQM method. Other relevant denotation has the same correspondence. We found no distinguishing differences in plots for the induced surface currents and scattering cross section computed by the latter two approaches in general. But for the case of , the plot of the induced surface currents by the EA formula behaves a little smoother than that by the MQM method. Reports of comparison in terms of errors for the EA and MQM methods are shown in Table II. It is observed that the accuracy of approximations calculated by the MQM method can be improved dramatically by means of the EA formula, particularly for cases of , and . For these three cases, the EA method can achieve the accuracy of the MQM method with discretization size equal to . Moreover, as shown in Table III, the EA method takes much less CPU consuming time than the MQM method with discretization size equal to in the achievement of such high accuracy. Therefore, when accuracy is a predominant concern, the EA formula is recommended. Example 2 Scattering by an Infinite Conducting Cylinder With an Elliptic Cross Section: TM Polarization [1]: In this situation, the elliptic cross section of an infinite cylinder with short radius and long radius will be considered to further illustrate the wonderful capacity of the MQM method for handling two-dimensional scattering problems. Here, for purposes of numerical comparison, two pairs of cross sections, i.e.,

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TABLE II NUMERICAL COMPARISON RESULTS IN TERMS OF ERRORS OF INDUCED SURFACE CURRENTS FOR INFINITE CYLINDERS WITH DIFFERENT CIRCULAR CROSS SECTIONS: TM POLARIZATION (MQM AND EA)

TABLE III CPU CONSUMING TIME IN SECONDS FOR INFINITE CYLINDERS WITH DIFFERENT CIRCULAR CROSS SECTIONS: TM POLARIZATION (MoM, MQM AND EA)

, , and , have been chosen. For the MQM method, the discretization size is . For the MoM method, , 720, 1440 will be used as the discretization sizes. It is noted that in this example as well as in the next one, we cannot find the analytical solutions. So we only plot in Fig. 3 the induced surface currents and scattering cross section ( , where is the angle at which is evaluated) for comparison. It is of quite interest to observe in the left two plots of Fig. 3 that, the larger the discretization size for the MoM method is, the much closer to the counterparts of the MQM method the induced surface currents computed via the MoM method become. Hence this phenomenon can provide a good demonstration of the superior performance of the MQM method to the MoM method. In addition, no dramatically discernible differences between the scattering cross sections as computed by the MQM and MoM methods can be observed in the right two plots of Fig. 3. This may be due to the fact that the scattering

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Fig. 3. Infinite cylinders with different elliptic cross sections: TM polarization (MQM and MoM). Left: induced surface currents (jJ j). Right: scattering cross section  (0    2 ).

Fig. 4. Sound-soft cylinders with different non-convex boomerang-shaped cross sections: TM polarization (MQM and MoM). Left: induced surface currents (jJ j). Right: scattering cross section  (0    2 ).

TABLE IV CPU CONSUMING TIME IN SECONDS FOR INFINITE CYLINDERS WITH DIFFERENT ELLIPTIC CROSS SECTIONS: TM POLARIZATION (MQM AND MoM)

TABLE V CPU CONSUMING TIME IN SECONDS FOR SOUND-SOFT CYLINDERS WITH DIFFERENT NON-CONVEX BOOMERANG-SHAPED CROSS SECTIONS: TM POLARIZATION (MQM AND MoM)

cross section is obtained by an integration of the induced surface currents on the surface. By observation of the left two plots of Fig. 3 and Table IV, we can get the conclusion that if the accuracy obtained by the MoM method in this example is expected to reach the accuracy obtained by the MQM method, a larger discretization size for the MoM method is necessary and, hence, much more CPU consuming time is required by the MoM method compared to the MQM method. Example 3 Scattering by a Sound-Soft Cylinder With a Non-Convex Boomerang-Shaped Cross Section: TM Polarization [27].: In the final example, we choose sound-soft cylinders with two different non-convex boomerang-shaped cross sections with boundaries described by the following two parametric representations . Case 1) ; Case 2) , where, . Fig. 4 depicts plots of the induced surface currents and scattering cross section ( , where is the angle at which is evaluated). CPU consuming time is reported in Table V. Similar phenomena with respect to the advantages of the MQM method can be investigated as those illustrated in the previous example.

cylinders under impressed electric field illumination. The presented MQM method takes the essential strategy of splitting the integrands into appropriate parts. These parts allow effective integrations without any singular integrals but with high accuracy to deal with weak singularities encountered in the integral formulation. The higher accuracy obtained by the MQM method compared to the MoM method is not only demonstrated by the excellent correspondence in terms of induced surface currents between the MQM computed results and the modal solutions, but also verified in accordance by the numerical comparative reports in terms of errors. The results presented in this study will contribute to assess the potential of the adapted MQM method for dealing with singularity problems enriching the database of this technology in the scientific computing community in electromagnetism. Furthermore, this elegant method is applied for perfectly metallic cylinders and sound-soft cylinders. More complex scattering problems are of great interest and are currently under investigation. Solutions of the obtained systems of linear equations by iterative methods with preconditioning techniques instead of Gaussian elimination are also taken into consideration. ACKNOWLEDGMENT

IV. CONCLUSION In this paper, a novel mechanical quadrature method (MQM) is exploited to the solutions of two-dimensional TM-polarized induced currents and scattered fields of infinite two-dimensional

The authors would like to gratefully thank Editor-in-Chief Prof Dr. Trevor Bird, the anonymous reviewers and Associate Editor for their insightful comments and constructive and valuable suggestions, which contributed substantially to the quality

JING et al.: A NOVEL INTEGRATION METHOD FOR WEAK SINGULARITY ARISING IN 2-D SCATTERING PROBLEMS

of the paper and added nicely to the presentation and readability of the paper. Their suggested numerous improvements are deeply appreciated, which led to both a clearer presentation and a more extensive analysis of the numerical experiments. REFERENCES [1] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [2] R. Mittra, Computer Techniques for Electromagnetics. New York: Pergamon, 1973. [3] R. Mittra, Numerical and Asymptotic Techniques in Electromagnetics. New York: Springer Verlag, 1975. [4] W. C. Gibson, The Method of Mometns in Electromagnetics. London: Chapman & Hall/CRC: Taylor & Francis, 2008. [5] M. Tanaka, V. Sladek, and J. Sladek, “Regulatization techniques applied to BEM,” Appl. Mech. Rev., vol. 47, pp. 457–499, 1994. [6] V. Sladek and J. Sladek, Singular Integrals in Boundary Element Methods. Southampton: Computational Mechan. Pub., 1998. [7] M. Bonnet, G. Maier, and C. Polizzotto, “Symmetric Galerkin boundary element methods,” Appl. Mech. Rev., vol. 51, pp. 669–704, 1998. [8] C. Schwab and W. L. Wendland, “Numerical integration of singular and hypersingular integrals in boundary element methods,” in Proc. NATO Advanced Research Workshop on Numerical Integration: Recent Developments, Software and Applications, Dordrecht, Netherlands, Jun. 17–21, 1991, pp. 203–218, Kluwer. [9] R. Klees, “Numerical calculation of weakly singular surface integrals,” J. Geodesy., vol. 70, pp. 781–797, 1996. [10] A. H. Stroud, Approximate Calculation of Multiple Integrals. Englewood Cliffs, NJ: Prentice-Hall, 1984, pp. 31–32. [11] M. G. Duffy, “Quadrature over a pyramid or cube of integrands with a singularity at a vertex,” SIAM J. Numer. Anal., vol. 19, pp. 1260–1262, 1982. [12] R. D. Graglia, “Static and dynamic potential integrals for linearly varying source distributions in two- and three-dimensional problems,” IEEE Trans. Antennas Propag., vol. AP-35, pp. 662–669, 1987. [13] C. B. Lin, T. Lü, and T. M. Shih, The Splitting Extrapolation Method. Singapore: World Scientific, 1995. [14] Y. S. Xu and Y. H. Zhao, “An extrapolation method for a class of boundary integral equations,” Math. Comp., vol. 65, pp. 587–610, 1996. [15] M. J. Bluck, M. D. Pocock, and S. P. Walker, “An accurate method for the calculation of singular integrals arising in time domain integral equation analysis of electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 45, pp. 1793–1798, 1997. [16] D. J. Taylor, “Accurate and efficient numerical integration of weakly singular integrals in Galerkin EFIE solutions,” IEEE Trans. Antennas Propag., vol. 51, pp. 1630–1637, 2003. [17] 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, 1982. [18] M. A. Khayat and D. R. Wilton, “Numerical evaluation of singular and near singular potential integrals,” IEEE Trans. Antennas Propag., vol. 53, pp. 3180–3190, 2005. [19] S. Järvenpää, M. Taskinen, and P. Ylä-Oijala, “Singularity subtraction technique for high-order polynomial vector basis functions on planar triangles,” IEEE Trans. Antennas Propag., vol. 54, pp. 42–49, 2006. [20] J. L. Tsalamengas and I. O. Vardiambasis, “Plane wave scattering by strip loaded circular dielectric cylinders in the case of oblique incidence and arbitrary polarization,” IEEE Trans. Antennas Propag., vol. AP-43, pp. 1099–1108, 1995. [21] J. L. Tsalamengas, “Direct singular integral equation methods in scattering from strip laded dielectric cylinders,” J. Electromag. Waves Appl., vol. 10, pp. 1331–1358, 1996. [22] J. L. Tsalamengas, “Direct singular integral equation methods in scattering and propagation in strip or slot loaded structures,” IEEE Trans. Antennas Propag., vol. AP-46, pp. 1560–1570, 1998. [23] A. G. Polimeridis and T. V. Yioultsis, “On the direct evaluation of weakly singular integrals in Galerkin mixed potential integral equation formulations,” IEEE Trans. Antennas Propag., vol. 56, pp. 3011–3019, 2008. [24] J. Huang and Z. Wang, “Extrapolation algorithms for solving mixed boundary integral equations of the Helmholtz equation by mechanical quadrature methods,” SIAM J. Sci. Comput., vol. 31, pp. 4115–4129, 2009. [25] A. Sidi and M. Israrli, “Quadrature methods for periodic singular Fredholm integral equation,” J. Sci. Comp., vol. 3, pp. 201–231, 1988.

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[26] P. Davis, Methods of Numerical Integration, 2nd ed. New York: Academic Press, 1984. [27] R. Kress and I. H. Sloan, “On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation,” Numer. Math., vol. 66, pp. 199–214, 1993. [28] I. S. Duff, A. M. Ersiman, and J. K. Reid, Direct Methods for Sparse Matrices. Oxford, U.K.: Oxford Univ. Press, 1986. [29] W. B. Gordon, “High frequency approximations to the physical optics scattering integral,” IEEE Trans. Antennas Propag., vol. 42, pp. 427–432, 1994. Yan-Fei Jing received the B.S. degree from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2005, where he is currently working toward the Ph.D. degree. He is the author or coauthor of more than 10 research papers. His research interests include iterative methods of linear systems and preconditioning techniques with applications in computational electromagnetics. Mr. Jing is a member of the International Linear Algebra Society (ILAS).

Ting-Zhu Huang received the B.S., M.S., and Ph.D. degrees in computational mathematics from Xi’an Jiaotong University, Xi’an, China, in 1986, 1992, and 2000, respectively. During 2005, he was a Visiting Scholar in the Department of Computer Science, Loughborough University, U.K. He is currently a Full Professor in the School of Mathematical Sciences, UESTC. He is currently an Editor of Advances in Numerical Analysis, Journal of Pure and Applied Mathematics: Advances and Applications, Journal of Electronic Science and Technology of China. He is the author or coauthor of more than 100 research papers. His current research interests include numerical linear algebra with applications, iterative methods of linear systems and saddle point problems, preconditioning technologies, and matrix analysis with applications.

Yong Duan was born in Sichuan, China, on October 10, 1972. He received the D.Sc. degree in applied mathematics from Fudan University, Shanghai, China, in 2005. His research interests include applied partial differential equations and numerical methods for partial differential equations as well as computational electromagnetics.

Sheng-Jian Lai was born in Sichuan, China, on July 11, 1976. He received the M.Sc. degree in physical electronic from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2003, where he is working toward the D.Sc. degree. His research interests include the computational electromagnetics and the general electromagnetics simulation platform development.

Jin Huang received the Ph.D. degree in computational mathematics from Sichuan University, Sichuan, China, in 2003. He is currently a Full Professor in the School of Mathematical Sciences, University of Electronic Science and Technology of China (UESTC), Chengdu. He is the author or coauthor of dozens of research papers. His current research interest is numerical solution of integral equations with applications.

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Communications E-Textile Conductors and Polymer Composites for Conformal Lightweight Antennas Yakup Bayram, Yijun Zhou, Bong Sup Shim, Shimei Xu, Jian Zhu, Nick A. Kotov, and John L. Volakis

Abstract—We present a conformal and lightweight antenna technology based on E-textile conductors and polymer-ceramic composites. Unique advantages of the proposed technology are its structural integrity, light weight and conformity to the platform. E-textile conductors are fabricated with single wall carbon nanotube (SWNT) and Ag coated textiles. They demonstrate good structural integrity with polymer composites due to their mechanical compatibility. Similarly, polymer composites demonstrate superior RF performance with permittivity ranging from 3 to 13. Fabrication process for E-textile conductors and integration process with polymer composites is described in detail. We also demonstrated merit of the proposed technology with a simple patch antenna whose radiation performance is measured when it was flat and conformed onto a cylindrical surface. We compared its performance with that of an ideal patch. Experiments suggested that the sample patch antenna based on the proposed technique achieved 6 dB gain, which is 2 dB below a patch which has the same dimensions and made of ideal lossless materials. When it is conformed onto a cylindrical surface, we achieved 2.5 dB less gain than that of antenna realized with a PEC surface. This clearly validates the merit of the proposed conformal antenna technique based on non-traditional materials. Index Terms—Carbon nanotube, conformal antennas, E-textile antennas, lightweight antennas, polymer composite.

I. INTRODUCTION Conformal, lightweight RF materials are critical to structurally integrated antennas for next generation unmanned aerial vehicles (UAVs) and small ground vehicles. Primary interest is to develop smaller antennas that retain their wavelength performance despite their smaller electrical size. This implies that they must take advantage of structures/ airframe and volume must therefore be fully integrated with the platform. Structural integrity with the airframe and platform requires new non-traditional materials that are also low loss and suitable for RF functionality. New material properties such as conformity, lightweight and strong shear and tensile stress ratings are critical to the structural integration as well. Polymers and polymer-ceramic composites are shown to be very flexible and low loss, thus, well suited for load bearing RF applications (see [1] and [2]). Their RF performance is very good with Manuscript received August 28, 2009; revised February 06, 2010; accepted February 12, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported in part by the U.S. Air Force Office of Scientific Research (AFOSR) under Grant FA9550-07-1-0462. Y. Bayram and J. L. Volakis are with the ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212 USA (e-mail: [email protected]; [email protected]). Y. Zhou was with the ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212 USA. He is now with Apple Inc., Cupertino, CA 95014 USA. B-S. Shim, S. Xu, J. Zhu, and N. A. Kotov are with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 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.2010.2050439

Fig. 1. Very flexible E-textile antenna printed on polymer composite.

their permittivity ranging from 3 to over 13 with low loss tangent (less than 0.02 at several GHz frequencies). Such composites can also be mixed with magnetic powders to achieve higher permeability which is critical to antenna miniaturization [3]. Thus, effective antenna miniaturization can be achieved by proportionally increasing substrate permittivity and permeability. However, the printing of the antenna geometry on polymer composites is a challenge for load bearing antennas intended for UAVs. Metals such as copper and gold are typically used as high conductive elements for their low loss and large availability. Even though such metals exhibit superior conductivity, their lack of mechanical and structural flexibility hinders their ability to effectively conform to the surfaces. They inherently do not possess mechanical compatibility with polymer composites to strongly adhere to the surface under the load/stress. Strong adhesion between the conductor and the substrate is vital to the structural integration and strong RF performance without compromising the antenna’s overall mechanical/structural performance. In this communication, we propose non-traditional materials such as polymer-ceramic composites and carbon nanotube coated E-textile for novel conformal and lightweight antennas. Such a unique combination provides us with a superb mechanical flexibility and strong, structural compatibility and integration with the UAV airframe and smart skin (see Fig. 1). Several E-textile technologies for antenna applications have already been shown in [4]–[6]. These technologies are based on metal coating of textile yarns for improved conductivity. In this communication, we propose E-textile fabrics coated with carbon nanotubes and sputtered with gold/silver particles for improved conductivity. Carbon nanotubes are of special interest due to their superior conductivity when aligned properly and strong mechanical characteristics. They have been investigated by other researchers for their RF performance (see [7], [8] and [9]) and found that they exhibit low efficiency when used as a standalone dipole antenna due to their large input impedance. However, SWNT coated textiles are very promising candidates for conformal and lightweight antennas because they are not only flexible and structurally very adhesive to the polymer-ceramic composites, but they also exhibit strong conductivity on a textile grid. This allows for planar printed antennas based on SWNT coated E-textiles. In the subsequent paragraphs, we start with describing the E-textile conductors. This is followed by polymer-ceramic composites. We subsequently proceed to describe the E-textile printing technique on polymer-ceramic composite. In the final section, we discuss performance of a sample patch antenna based on the proposed technology and compare its performance to that of an ideal patch made of lossless materials. This is followed by an additional measurement to characterize performance of an E-textile antenna conformed onto a cylindrical surface.

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Fig. 2. (i) E-textile structure comprised of a fabric and coated CNTs. (ii) Photograph image of (A) the original cotton textile, (B) 1 layer of SWNT dye, and (C) 10 layers of SWNT dye on cotton textiles.

II. E-TEXTILE CONDUCTIVE SHEET E-textile is fabricated by simple dyeing technique (see [11] and [12]). In other words, a textile was coated with SWNTs and Au particles for improved conductivity (see Fig. 2). We first dispersed single-walled carbon nanotubes (SWNTs; HiPCo SWNTs from Carbon Nanotechnologies Inc.) in diluted nafion-ethanol (see [10]). A commodity cotton textile was dipped 10 seconds in the prepared SWNT dye dispersion and dried for 1 hour at 60 C. This dying process was repeated 10 times to increase mean electrical conductivity of the E-textile and reduce its resistance (see Fig. 3(i) for resistance versus SWNT dipping) To further reduce the resistance of E-textile, Au layers were sputtered for 200 seconds, which obviously improved the conductivity but slightly compromised flexible nature of the E-textile. Therefore, we added 2 more layers of SWNT dyes. After adding 4 layers of these Au/SWNT sandwich, the E-textile loss reached to a resistance of 10 [see Fig. 3(ii)] for an E-textile patch of size 4 cm 2 4 cm [see Fig. 3(i) A-1]. We next treated the sample in a hot press overnight for 24 hours at 100 C to achieve a strong adhesion of SWNT and Au into the cotton textiles so that the sample still preserves its low loss nature even after severe bending. We later tried sputtering Ag particles to optimize the performance of the E-textile by instead of Au for higher conductivity and better adhesion. After dipping the sample 10 times in SWNT, we used 2 layers of SWNT and Ag sputtering. Overall resistance reduced to 1.1 compared to 10 resistance with Au sputtering [see Fig. 3(iii)]. As a final step, we applied hot press for further adhesion. The resistance of the final E-textile varies between 1.1 to 2.0 depending on the bending. Ag sputtering has led to lower resistance than Au sputtering and this is likely due to better wetting of SWNT by Ag as compared to Au and better spreading of evaporated Ag; thus, leading to stronger adhesion with SWNTs and ultimately higher conductivity. III. POLYMER-CERAMIC MIX FOR CONFORMAL LIGHTWEIGHT ANTENNAS

AND

Polymer-ceramic composites are novel highly flexible, lightweight materials for RF applications. With a proportional mixture of ST ce-

Fig. 3. Resistances of SWNT—Metal antenna patches. (i) Resistance changes of e-cotton fabrics by SWNT dippings. (A-1) Dimensions of antenna shape SWNT e-fabric. (ii) only front side of 10 layered SWNT dyed e-fabrics. Resistance rises after bending the fabrics. Hot press processing could regenerate the initial resistances after bending as well as prevent resistance increases by bending the fabrics. (iii) Combination of sputter Ag coating of 10 layered SWNT dyed e-fabrics with optimized hot press processing.

ramic powder with Polydimethylsiloxane (PDMS), dielectric permittivity ranging from 3 to 13 with less than loss tangent of 0.01 can be achieved (see Fig. 4). We preferred PDMS as the polymer matrix over many other available because of its low cost and low-temperature handling. Several different ceramic powders including barium titanate

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Fig. 4. Dielectric permittivity and loss tangent of PDMS-ceramic composites for various volume mix.

Fig. 5. E-textile antenna fabrication process: (i) E-textile preparation via the aforementioned process. (ii) E-textile cut based on planar antenna dimensions. (iii) E-textile mix with polymer composite. (iv) Final form of the antenna after several hours of curing.

(BaTiO3: BT), strontium titanate (SrTiO3: ST), Mg-Ca-Ti (MCT) and Bi-Ba-Nd-Titanate (BBNT). For our application, we chose SrTiO3 due to its high dielectric constant and low loss. Dielectric permittivity and loss tangent of polymer-ceramic composite for various ceramic mixing ratios is shown in Fig. 4. More detailed information in regards to mixture and fabrication process is given in [2]. Further information on the RF performance of polymer-ceramic composites can also be found in [1] and [2].

Fig. 6. (i) E-textile patch antenna platform. (ii) E-textile antenna dimensions. (iii) Return loss performance of the E-textile patch antenna versus ideal patch. (iv) Gain of the E-textile patch antenna versus ideal patch antenna made out of lossless materials.

IV. E-TEXTILE ANTENNA FABRICATION PROCESS E-textile is embedded on a polymer composite to fabricate planar conformal lightweight antennas. The process is a simple mix of polymer-ceramic composite with E-textile. Referring to the Fig. 5

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 8, AUGUST 2010

Fig. 7. Photograph of the cylindrically mounted E-textile patch antennas. (a) E-plane bending, (b) H-plane bending.

displaying the process of printing a planar E-textile antenna on a polymer-ceramic composite, we first patterned the textile according to the planar antenna geometry specifications. We subsequently mixed the polymer-ceramic composite over the E-textile fabric and waited 12 hours for curing. Our tests showed that the conductive textile adheres strongly to the polymer-ceramic composite, thus, implying strong mechanical and chemical compatibility. The final configuration was very flexible patch as shown in Fig. 5. The E-textile still preserved its conductivity while bent, implying that press process has provided strong connectivity among the SWNTs and Au/Ag particles sputtered on the textile. It is also important to note that the fabrication process described above can be extended to commercial scale and be achieved at substantially low cost. Conductive textile can be cut according to any planar antenna specifications and printed on the polymer-ceramic composite at the room temperature. V. RF PERFORMANCE OF E-TEXTILE POLYMER COMPOSITE ANTENNAS To demonstrate the performance of the proposed E-textile antenna, we fabricated a sample patch antenna operating at 2 GHz. The antenna dimensions are 50 mm 2 50 mm on a polymer substrate of 300 mil (7.62 mm) thickness and permittivity of 4 as displayed in Fig. 6(ii). E-textile patch with 35 mm 2 35 mm dimensions was embedded on a polymer substrate using the previously described fabrication technique.

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Fig. 8. Return loss and antenna gain for a E-textile patch bend in E-plane. (a) Return loss (E-plane), (b) antenna gain (E-plane).

The resulting textile had 2 resistance. The antenna was mounted on a ground plane to carry out measurements in an anechoic chamber. To compare performance of the sample antenna to that of an ideal patch, we also modeled a PEC patch with lossless substrate of 4.0 permittivity in HFSS of Ansoft. Referring to the Fig. 6(iii) and (iv) displaying return loss and gain of the sample and the ideal patch, respectively, we note that the proposed E-textile technology has RF performance of a traditional patch antenna considering that E-textile sample has 6 dB of gain at 2 GHz, 2 dB less than that of an ideal patch. The sample patch has a slightly wider bandwidth than that of ideal patch and this is likely due to the loss with the E-textile conductor. A large gain of 6 dB clearly demonstrates comparable performance of the proposed technology to that of a patch made out of traditional materials. It is also important to note that resistance of the E-textile patch plays a critical role in the gain of the antenna as expected. The E-textile patch with 10 resistance led to 0 dB antenna gain which is 6 dB less gain when Ag is used instead of Au sputtering. This also demonstrates how critical the choice of Ag over Au sputtering is. VI. CONFORMAL E-TEXTILE PATCH ANTENNAS Next, we looked into the performance of the conformal E-textile patch when mounted on a cylindrical surface [13]. Referring to the Fig. 7, where we attached an E-textile-polymer patch on a metal cylindrical surface (80 mm in diameter and 160 mm in length) both in E-plane and H-plane. E-textile patch was 30 mm 2 30 mm (conductive

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REFERENCES

Fig. 9. Return loss and antenna gain for a E-textile patch bend in H-Plane. (a) Return loss (H-plane), (b) antenna gain (H-plane).

surface) on a 2 mm thick substrate. H-plane bending implies that the current flows along the axial direction. Similarly, E-plane bending implies that the current flows in the circumferential direction. We next measured the return loss and gain of the conformal E-textile polymer patches on this platform at an anechoic chamber. Referring to the Figs. 8 and 9 where return loss and gain of the proposed E-textile patch antenna in both E-plane and H-plane are presented, we find that the E-textile patch has a broader bandwidth in both cases compared to a PEC patch on the same surface. Such a larger bandwidth is due to the conductive losses on the E-textile patch. Despite the fact that E-textile is stretched further, thus higher resistance; proposed E-textile patch suffers on average 2.5 dB gain less than PEC attached on the same cylindrical surface. VII. CONCLUSION We proposed an E-textile antenna technology based on Carbon Nanotube coated textile and polymer-ceramic composited for conformal, lightweight antenna applications. We described the fabrication process for E-textile conductors and printing on polymer composites. We showed that proposed technology is readily scalable to mass production and easily repeatable. We also found out that choice of metal sputtering (Ag versus Au) is also critical to achieving high conductivity. We also demonstrated the RF performance of the proposed technology with a sample patch antenna and achieved a gain of 6 dB, which is less than 2 dB gain for an ideal antenna with the same dimensions and lossless materials.

[1] J. 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] M. C. Dimri, S. C. Kashyap, and D. C. Dube, “Complex permittivity and permeability of Co2U (Ba4Co2Fe36O60) hexaferrite bulk and composite thick films at radio and microwave frequencies,” IEEE Trans. Magn., vol. 42, no. 11, pp. 3635–3640, 2006. [4] Y. Ouyang, E. Karayianni, and W. J. Chappell, “Effect of fabric patterns on electrotextile patch antennas,” in Proc. IEEE Antennas and Propagation Society International Symp., Jul. 3–8, 2005, vol. 2B, pp. 246–249. [5] 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. [6] 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 on Antennas and Propagation for Body-Centric Wireless Communications, Apr. 24–24, 2007, pp. 77–83. [7] G. W. Hanson, “Fundamental transmitting properties of carbon nanotube antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 3426–3435, Nov. 2005. [8] G. W. Hanson, “Current on an infinitely-long carbon nanotube antenna excited by a gap generator,” IEEE Trans. Antennas Propag., vol. 54, pp. 76–81, Jan. 2006. [9] Y. Huang, W.-Y. Yin, and Q. H. Liu, “Performance prediction of carbon nanotube bundle dipole antenna,” IEEE Trans. Nanotechnol., vol. 7, pp. 331–337, May 2008. [10] J. Wang, M. Musameh, and Y. Lin, “Solubilization of carbon nanotubes by nafion toward the preparation of amperometric biosensors,” J. Am. Chem. Soc., vol. 125, no. 9, pp. 2408–2409, 2003. [11] M. in het Panhuis, J. Wu, S. A. Ashraf, and G. G. Wallace, “Conducting textiles from single-walled carbon nanotubes,” Synth. Met., vol. 157, no. 8–9, pp. 358–362, 2007. [12] Y. Liu, J. Tang, R. Wang, H. Lu, L. Li, Y. Kong, K. Oi, and J. H. Xin, “Artificial lotus leaf structures from assembling carbon nanotubes and their applications in hydrophobic textile,”17th ed. 2007, pp. 1071–1078. [13] L. C. Kempel, J. L. Volakis, and R. Sliva, “Radiation by cavity-backed antennas on a circular cylinder,” in Proc. Inst. Elect. Eng.-H, 1995, pp. 233–239.

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A Novel Dipole Antenna Design With an Over 100% Operational Bandwidth Fang-Yao Kuo, Hsi-Tseng Chou, Heng-Tung Hsu, Hsi-Hsir Chou, and Paolo Nepa

Abstract—A novel design of a dipole antenna with ultrawideband characteristics is presented, which uses an additional shorter dipole as a matching element whose imaginary part (or susceptance) of input admittance compensates for that of the principal dipole in the frequency band of operation. Since both dipole elements efficiently radiate energy, the proposed twin-dipole antenna exhibits a higher efficiency compared to a conventional design. The measurement results have revealed that the twin-dipole is able 10 dB) from 1.6 GHz up to to achieve a bandwidth of 131% ( 11 7.7 GHz, with good radiation performance and more than 75% radiation efficiency.

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A novel antenna structure is presented here, which uses a center-fed dipole as the principal radiator. In consideration of the inductive and capacitive features of its input reactance with respect to its length or frequency changes, an additional dipole stub with a shrink in length for a higher frequency operation is properly shunt-added to the principal dipole radiator to achieve ultra wide band operation. The measurement results on a prototype reveal that this twin-dipole antenna is able to achieve a bandwidth of 131% (S 11 < 010 dB) from 1.6 GHz up to 7.7 GHz with good radiation performance and more than 75% radiation efficiency. The sections are organized as follows. The design concept, antenna structure and performance evaluation, are described in Section II. A useful equivalent circuit is developed in Section III to model the twindipole antenna’s electrical characteristics, which can be used in a circuit design tool for an overall consideration of a RF system design. A short discussion is presented in Section IV as a conclusion of this work.

Index Terms—Antenna design, twin-dipole antennas, ultrawideband antennas, wideband dipoles.

II. ANTENNA DESIGN AND PERFORMANCE EVALUATION I. INTRODUCTION

A. Design Concept and Antenna Structure

Recent trends in communications characterized by wide- or multifrequency bands of operations have driven the needs of developing antennas with not only fulfilling these bandwidth requirements, but also exhibiting characteristics of low-profile, simple feeding structure, and easy-for-production. Among all the possible antenna structures, dipole based ones are always preferred because of their unique advantageous features of omnidirectional radiation pattern, relatively high energy efficiency and low manufacturing cost. In practice, the performance of an ideal dipole has been widely used as a reference to justify the characteristics of many practical antennas because its radiation characteristics are superior in terms of polarization purity and patterns. Above characteristics are very difficult to be retained when the antenna structure is altered to broaden its bandwidth. Generally speaking, a regular dipole antenna has an impedance bandwidth of about 10% with reflection coefficient (S 11) less than 010 dB. Past efforts [1]–[5] to broaden the bandwidth of antennas have focused on implementing impedance matching networks in a wideband manner through the introduction of lumped- or distributed- elements [6], [7]. Other approaches elaborated very much on the geometry of feeding structure (mainly CPW type) to maintain almost frequency independent impedance level for wideband operation [8]–[10]. However, the performance of such antennas is very ground-dependent, which makes them very sensitive to the environment. Dipole based quasi log-periodic antenna was proposed in [11], where two coupling stubs provided additional coupling for cancellation of the reactive part of the input impedance, to expand the operation bandwidth. Furthermore, investigation of dipole antennas with an integrated balun feed for additional bandwidth has also attracted much attention [12]–[14]. Yet, an efficiency degradation has been observed in above designs. Manuscript received July 14, 2009; revised December 15, 2009; accepted February 06, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. F.-Y. Kuo, H.-T. Chou, and H.-T. Hsu are with the Department of Communication Engineering, Yuan Ze University, Chung-Li, Taiwan (e-mail: hchou@ saturn.yzu.edu.tw). H.-H. Chou is with the Communication Research Center, Yuan Ze University, Chung-Li, Taiwan (e-mail: [email protected]). P. Nepa is with the Department of Information Engineering, University of Pisa, Pisa, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2050434

The concept initiates from the characteristics of a dipole’s input impedance, whose real and imaginary parts (referred to a current maximum) for an ideal wire dipole with a radius a are given by [15]

Rm =



2

C + ln(kL) 0 Ci (kL) + +

2

1 2 1 2

i

sin(kL) [S (2kL)

0 2Si (kL)]

cos(kL)

C + ln

kL 2

i

+ C (2kL)

0 2Ci (kL)

(1)

and

Xm =



i

i

2S (kL) + cos(kL)[2S (kL)

4

0 sin(kL)

i

2C (kL)

0 Si (2kL)]

0 Ci (2kL) 0 Ci

2ka

L

2

(2)

where C = 0:5772 (Euler’s constant),  is the free-space impedance, and L is the length of the dipole. In (1) and (2), k is the free-space wavenumber, and Si and Ci are the regular sine and cosine integrals. Fig. 1(a) shows the curves of the input impedance with respect to L for various values of the radius a. As pointed out in [15], Xm 0 as L is slightly less than n=2 (n = 1; 3; . . .), or slightly greater than n=2 (n = 2; 4; . . .), which results in nature resonances for the dipole without the need to add any impedance matching. Especially as L is about =2, Rm approaches 50 , resulting in, an almost ideal match for the widely used 50 standard coaxial cables. The key design concept is to take the advantages of the alternating inductive and capacitive characteristics of a dipole input reactance as a function of L, which occurs periodically with a period of . Thus the impedance of an inductive dipole can be closely matched by using an additional dipole with capacitive impedance, and vice versa. Since this additional dipole is also a radiator itself, such a dual-dipole arrangement may also enhance the overall radiation efficiency while

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Fig. 3. Simulated susceptance of a single dipole as a function of frequency, for a set of dipole length values. (unit: = ).

1

Fig. 1. Characteristics of a dipole input impedance to illustrate the proposed an, short dipole: L ). tenna design concept. (Long dipole: L (a) Dipole’s input impedance. (b) Design concept (unit: = ).

= 72 mm

1

= 48 mm

between these two dipoles, Fig. 1(b) illustrates the basic concept, where the imaginary parts (or susceptances) of the input admittance of two dipoles with different lengths (chosen to alternate inductive and capacitive characteristics) are shown as functions of frequency. Since the dipoles are shunt-connected, the overall admittance obtained by a superposition of the individual ones shows a vanishing imaginary part and results in broadband properties. Fig. 2(a) shows the proposed twin-dipole antenna structure with strip wires as dipole arms, where  is a tilted angle to account for the mutual coupling effects between these two dipoles. Strip wires are used because of easy implementation in practice. In Fig. 2(a), L1 and L2 are adjusted such that the reactive part of the combined structure is close to zero over the desired band of operation. As a design rule, the length of the main radiating element, L1 , was set to be half-wavelength at the lowest operating frequency (f0 ) of the desired operation bandwidth. L2 was set to be half-wavelength at 1:5f0 . The angle,  , was set to be 0 to keep the two polarizations orthogonal to each other, and minimize additional couplings between the two dipoles. The widths of the strip play a minor role since fat dipole will naturally contribute to wider operation bandwidth. A photograph of the manufactured prototype showing the feeding structure of the antenna is also included in Fig. 2(b). Note that in the current design a balun was not used in the feeding structure, to further reduce the overall power loss. B. Characteristics and Performance Evaluations

Fig. 2. (a) The proposed twin-dipole antenna structure with strip lines as diploes’ arms. The tilted angle  is used to account for the mutual coupling effects. (b) The photograph of the manufactured prototype showing the feeding structure of the antenna.

maintaining a relatively omnidirectional radiation pattern with directivity close to that of a regular dipole. Ignoring the mutual coupling

Numerical analysis, based on a finite integration in time-domain (FIT), by using CST Microwave Studio [16] is performed to analyze the twin-dipole radiation characteristics. Comparisons with measurement results from an antenna prototype have been also performed to validate the proposed design. Fig. 3 shows the simulated input susceptance of a single regular dipole element, as a function of frequency and for different lengths, L (the strip width is 1 cm). It naturally alternates inductively and capacitively with respect to a frequency sweep, indicating that a compensation is possible through a proper choice of the physical lengths of the two shunt dipoles. Fig. 4(a) shows the simulation results of the input reflection coefficient (S 11) for various lengths of the additional dipole, expressed in terms of a ratio, r defined as L2 over L1 , with  = 0 and L1 = 7:3 cm. The strip widths of the principal and matching dipole arms are 1.7 cm and 0.9 cm, respectively. Fig. 4(b) shows the simulated S 11 for various values of  with r = 5:3=7:3 in comparison with that of a regular dipole antenna.

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Fig. 4. Reflection coefficient of a twin-dipole with respect to (a) matching dipole’s length ( ,L : , and r L =L ) and (b) tilted angle : and L : ,r : = : ). In the legends, “regvalues (L ular” indicates a regular dipole element without adding any stub for impedance matching L .

=0 = 7 3 cm ( = 0)

= 7 3 cm = = 5 3 cm = 5 3 7 3

It is observed from Fig. 4(b) that the mutual coupling between dipoles will affect the performance of antenna in the upper and lower parts of operation band. In the intermediate band, the return loss performance remains fairly stable. Especially when  = 0 , where two dipole elements are placed orthogonally to each other with least coupling effects, the antenna has the broadest impedance bandwidth. Also Fig. 4(a) shows that the broadest bandwidth is achieved when r = 2=3, which has been adopted in the fabrication of the final prototype on a Styrofoam substrate ("r 1). Fig. 5(a) shows the simulated input impedance in comparison with that of a regular dipole antenna. Also Fig. 5(b) shows both the simulated and measured antenna reflection coefficient. An impedance bandwidth (S 11 < 010 dB) extended from 1.6 GHz up to 7.7 GHz (131% bandwidth) is observed. Fig. 5(c) shows the measured total efficiency as a function of frequency, confirming the wideband feature of the proposed antenna. Fig. 6(a) and (b) show the measured far field radiation patterns on two different cut planes, from 2 GHz to 7 GHz, indicating a nearly omnidirectional pattern with a reasonable gain. In particular, the length of the major dipole element is roughly =2 at 2 GHz. At the low frequency limit, the maximum radiation direction is on the H-plane. As the frequency increases, the electrical length of the dipole increases, which is larger than a wavelength at 7 GHz and it makes the beam peaks moving toward the main dipole axis.

Fig. 5. Performance of the proposed twin-dipole antenna: (a) simulated input impedance in comparison with that obtained from a regular dipole antenna, (b) S , and (c) measured total radiation efficiency.

11

III. EQUIVALENT CIRCUIT MODEL An equivalent circuit model [17], [18] is developed to justify and interpret the antenna’s electrical characteristics. The equivalent circuit model can be used in conjunction with other RF components for an effective RF system design. This model is illustrated in Fig. 7, where two identical configurations of electrical networks consisting of resistors, capacitors and inductors are employed to model the two dipole

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TABLE I PARAMETER VALUES OF EQUIVALENT CIRCUIT COMPONENTS FOR THE TWO DIPOLE ELEMENTS

TABLE II PARAMETER VALUES OF EQUIVALENT CIRCUIT MODELS ACCOUNTING FOR THE NORMALIZED COUPLING COEFFICIENTS BETWEEN THE TWO DIPOLES

Fig. 6. Radiation gain patterns of the twin-dipole antenna in (a) x-z and (b) x-y planes.

Fig. 7. An equivalent circuit model for the twin-dipole antenna.

elements, respectively. Some of the mutual couplings between corresponding inductors have also been included for illustration purpose. These two equivalent circuits are then parallelly connected with an impedance transformer in order to connect them to a regular transmission line with a 50 characteristic impedance. In the first stage of modeling, the circuit parameters (i.e., the values of R1 to R5 , L1 to L6 and C1 to C6 in Fig. 7) are optimized in a way to fit the modeled input impedances with the measured dipoles’ input impedances, within the desired frequency band. In this case, the two dipoles are modeled sep-

arately such that their own characteristics can be retained in the study. The modeled values of parameters are shown in Table I. It is noted that for each dipole’s equivalent circuit model two R-L-C parallel connected circuits are employed to produce two resonance frequencies as exhibited by a regular dipole in this desired band of interest. The latter are then series connected to a capacitor and an inductor (for example, C6 and L6 for 1st dipole in Fig. 7) to get a fine tuning of the dipole input reactance at low and high portions of the frequency band, respectively. The second stage of the modeling accounts for the mutual coupling between these two dipole elements by considering the mutual inductances between the inductors of these two dipoles’ circuit models, where Mij is the normalized coupling coefficient between Li and Lj . It is noted the mutual coupling will also affect the overall input impedance and, as a result, the parameters of the impedance transformers. An optimization procedure is performed to find these parameters so that the modeled input impedance may best fit with the overall twin dipoles’ input impedance in the desired frequency band. The values of the mutual inductances and parameters of the transformers are shown in Table II. The characteristics of the equivalent circuit are compared to original measured results of the twin dipole antenna. Fig. 8(a)–(c) show the results for S 11 with various tilted angles, while Fig. 8(d) and (e) show the corresponding input resistance and reactance. The same dipole configurations as in Fig. 4 are considered here in Fig. 8. It is apparent that the equivalent circuits can accurately model the proposed antenna structure. IV. CONCLUSIONS A novel antenna structure consisting of two dipole elements of different length is proposed. This antenna is found to exhibit charac-

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efficiency is larger than 75% across the band with reasonable gain performances. ACKNOWLEDGMENT This work was partially developed within an Italy-Taiwan Bilateral Agreement between the University of Pisa, Pisa, CNR, Turin, Italy, and the Yuan-Ze University, Taiwan.

REFERENCES [1] H.-D. Chen, “Broadband CPW-fed square slot antennas with a widened tuning stub,” IEEE Trans. Antennas Propag., vol. 51, pp. 1982–1986, Aug. 2003. [2] J.-Y. Sze and K.-L. Wong, “Bandwidth enhancement of a microstripline-fed printed wide-slot antenna,” IEEE Trans. Antennas Propag., vol. 49, pp. 1020–1024, Jul. 2001. [3] T. Yang and W. A. Davis, “Planar half-disk antenna structures for ultrawideband communications,” in Proc. IEEE Antennas Propagation Soc. Int. Symp., Jun. 2004, vol. 3, pp. 2508–2511. [4] H.-D. Chen and H.-T. Chen, “A CPW-fed dual-frequency monopole antenna,” IEEE Trans. Antennas Propag., vol. 52, pp. 978–982, Apr. 2004. [5] D. B. Antonino, F. M. Cabedo, B. M. Ferrando, and N. A. Valero, “Wideband double-fed planar monopole antennas,” Electron Lett., vol. 39, pp. 1635–1636, Nov. 2003. [6] R. E. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill, 1992, ch. 5, pp. 303–386. [7] D. M. Pozar, Microwave Engineering. Hoboken, NJ: Wiley, 2004. [8] A. U. Bhobe, C. L. Holloway, M. Piket-May, and R. Hall, “Coplanar waveguide fed wideband slot antenna,” Electron. Lett., vol. 36, pp. 1340–1342, Aug. 2000. [9] J. Yeo and R. Mima, “Design of a wideband antenna package with a compact spatial notch filter for wireless applications,” in Proc. IEEE Symp. on Antennas and Propagation, Jun. 2002, vol. 2, pp. 492–495. [10] J. William and R. Nakkeeran, “A novel compact CPW-fed wideband slot antenna,” in Proc. Eur. Conf. on Antennas and Propagation, Mar. 2009, pp. 1471–1474. [11] H.-T. Hsu, J. Rautio, and S.-W. Chang, “Novel planar wideband omnidirectional quasi log-periodic antenna,” presented at the Asia Pacific Microwave Conf., Dec. 2005. [12] J. I. Kim, J. M. Kim, Y. J. Yoon, and C. S. Pyo, “Wideband printed fat dipole fed by tapered microstrip balun,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2003, vol. 3, pp. 32–35. [13] Y. X. Guo, Z. Y. Zhang, L. C. Ong, and M. Y. W. Chia, “A new balanced UWB planar antenna,” in Proc. Eur. Conf. on Wireless Technology, Oct. 2005, pp. 515–517. [14] Q.-Q. He, B.-Z. Wang, and J. He, “Wideband and dual-band design of a printed dipole antenna,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 1–4, 2008. [15] C. A. Balanis, Antenna Theory-Analysis and Design, 3rd ed. New York: Wiley, 2005, ch. 8. [16] CST Studio Suite 2009, www.cst.com. [17] T. G. Tang, Q. M. Tieng, and M. W. Gunn, “Equivalent circuit of a dipole antenna using frequency-independent lumped elements,” IEEE Trans. Antennas Propag., vol. 41, no. I, pp. 100–103, Jan. 1993. [18] M. Hamid and R. Hamid, “Equivalent circuit of dipole antenna of arbitrary length,” IEEE Trans. Antennas Propag., vol. 45, no. 11, pp. 1695–1696, Nov. 1997.

Fig. 8. Measured and modeled characteristics of the antenna: (a)–(c) the reflection coefficient S 11, (d) input resistance and (e) input reactance. The input impedances shown in (d) and (e) are upward and downward shifted by 50 for  = 0 and  = 50 , respectively for a better view. (a)  = 0 , (b)  = 30 , (c)  = 50 , (d) Resistance ( ), (e) Resistance ( ).

teristics of ultra-wide bandwidth operation from 1.6 GHz to 7.7 GHz, corresponding to 131% bandwidth. Moreover, the measured total

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Frequency Reconfigurable Quasi-Yagi Folded Dipole Antenna

TABLE I DIMENSIONS OF ANTENNA I

Pei-Yuan Qin, Andrew R. Weily, Y. Jay Guo, Trevor S. Bird, and Chang-Hong Liang

Abstract—A frequency reconfigurable planar quasi-Yagi antenna with a folded dipole driver element is presented. The center frequency of the antenna is electronically tuned by changing the effective electrical length of the folded dipole driver, which is achieved by employing either varactor diodes or PIN diodes. Two antenna prototypes are designed, fabricated and measured. The first antenna enables continuous tuning from 6 to 6.6 GHz using varactor diodes and the reflection coefficient bandwidth ( 10 dB) at each frequency is greater than 15%. The second antenna enables discrete tuning using PIN diodes to operate in either the 5.3–6.6 GHz band or the 6.4–8 GHz band. Similar end-fire radiation patterns with low cross-polarization levels are achieved across the entire tunable frequency range for the two antenna prototypes. Measured results on tuning range, radiation patterns and gain are provided, and these show good agreement with numerical simulations. Index Terms—Folded dipole, planar antennas, quasi-Yagi antenna, reconfigurable antennas.

I. INTRODUCTION It is now common practice to integrate several radios in a single wireless platform that uses one antenna or one single radio device to handle multiple air-interface standards. To this end, the antennas need to cover multiple frequency bands. Multiband, wideband and reconfigurable antennas are three potential candidates to be employed in services requiring multiple operating bands. Microstrip antennas are typically used due to their advantages of low profile, light weight and easy fabrication. In [1], a wideband (48% for VSWR < 2) microstrip planar quasi-Yagi antenna is proposed. However, if only a portion of this operating band is required at any given time, such as in a cognitive radio, then a frequency reconfigurable antenna would be a better choice. Compared to multiband and wideband antennas, one of the merits of frequency reconfigurable antennas is that the antenna can provide noise rejection in the bands that are not in use so that the filter requirements of the front-end circuits can be greatly reduced [2]–[11]. In this paper an end-fire frequency reconfigurable quasi-Yagi folded dipole antenna is proposed for the first time. Two techniques for frequency reconfigurability are presented that allow either discrete or continuous frequency tuning. Reconfigurability of the center frequency of the antenna is realized by varying the effective electrical length of the folded dipole driver, which is controlled by varactor diodes for continuous tuning or PIN diodes for discrete tuning. Similar end-fire radiation patterns with low cross-polarization levels are realized across the entire tunable frequency range. To validate the concept, two antenna prototypes were designed and measured. For the first antenna Manuscript received December 06, 2009; accepted February 02, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by the DIISR Australia-China special fund CH080270 and the China Scholarship Council (CSC). P.-Y. Qin is with the National Key Laboratory of Antennas and Microwave Technology, Xidian University, Xi’an, Shaanxi 710071, China and also with the CSIRO ICT Centre, Epping, NSW 1710, Australia (e-mail: Peiyuan.qin@csiro. au). A. R. Weily, Y. J. Guo, and T. S. Bird are with the CSIRO ICT Centre, Epping, NSW 1710, Australia. C.-H. Liang is with the National Key Laboratory of Antennas and Microwave Technology, Xidian University, Xi’an, Shaanxi 710071, China. Digital Object Identifier 10.1109/TAP.2010.2050455

(denoted as Antenna I) a continuous tuning range of 10% enabled this new reconfigurable antenna to cover a total frequency bandwidth of 25% (5.5–7.1 GHz) with about 4.2 to 6.35 dBi gain by using varactor diodes. Part of the work related to the design of Antenna I has been described in [12]. The present paper extends the work in [12] by employing a more realistic equivalent circuit of the varactor diodes and describes new simulated and measured results over the tuning range for radiation patterns and gain. For the second antenna (denoted as Antenna II), the 5.3–6.6 GHz band or the 6.4–8 GHz band can be covered separately by using PIN diodes. Compared to the original wideband antenna in [1], the reconfigurable antennas described here have an inherent bandpass characteristic that enables the rejection of interfering signals from neighboring bands. Another major difference in this work is that a reconfigurable folded dipole is employed as the driver element of the quasi-Yagi antenna instead of the half-wave dipole [13]. The folded dipole allows simple biasing of the varactor and PIN diodes through the metallization of the balun and coplanar stripline (CPS) feed network, thus eliminating the need for extra bias lines, which are used in some reconfigurable antennas, that can distort the radiation pattern. If a half-wave dipole was used as the driver element, extra bias lines would be required, which would cause pattern distortion. The complete dc bias network and equivalent circuits of the varactor and PIN diodes have been included in both antenna designs. In the Section II, we describe the two new reconfigurable quasi-Yagi antenna designs. Measured results are presented in Section III and these results are compared with simulations. The implications of the results are discussed in Section IV. The paper concludes in Section V with a summary and suggestions for further work. II. RECONFIGURABLE QUASI-YAGI ANTENNA DESIGNS The configurations of Antenna I and Antenna II are shown in Figs. 1 and 2, respectively. The basic structures of Antenna I and Antenna II are the same; the main difference lies in the placement and the type of diode used for reconfiguring the operating frequency. The top side of the substrate consists of a microstrip feed, a broad-band microstrip-to-CPS balun, a folded dipole driver element fed by the CPS and a dipole parasitic director element. The bottom side is a truncated microstrip ground, which serves as the reflector element for the antenna. The combination of the parasitic director and reflector elements directs the radiation of the antenna toward the end-fire direction. The parameters and dimensions of Antenna I and Antenna II are shown in Tables I and II, respectively. It was found that the length of the folded dipole driver L7 , the length of the director L8 and the ratio of the width of the upper and lower strip of the folded dipole driver W8 =W6 were important design parameters which affected the input impedance of the quasi-Yagi folded dipole antenna [1], [14]. In this paper only the length of the folded dipole driver is changed to reconfigure the center frequency of the antenna, since only simple bias network is required.

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TABLE II DIMENSIONS OF ANTENNA II

Fig. 2. (a) Configuration of the reconfigurable quasi-Yagi folded dipole Antenna II. (b) Orientation of the PIN diodes in the folded dipole.

Fig. 1. (a) Configuration of the reconfigurable quasi-Yagi folded dipole Antenna I. (b) Orientation of the varactor diodes in the folded dipole. (c) Equivalent circuit of the dc bias network used at pad A and pad B.

For Antenna I, the reconfigurability of the center frequency of the antenna is achieved with a pair of varactor diodes loaded on the thin strip of the folded dipole. Two low loss varactor diodes (Aeroflex Metellics MGV 125-20-0805-2) with a capacitance range of 0.1–1 pF are placed in series as shown in Fig. 1(b). Two diodes are used to achieve a larger tuning range, and they are placed at the centre of the folded dipole where the currents are the highest. By varying the capacitance of the varactor diodes, the electrical length of the folded dipole driver is changed, thus leading to a change in the center frequency of the antenna. Fig. 3 shows the series RLC equivalent circuit used to model the varactor diodes in simulations. The bias network of the varactor diodes is also shown in Fig. 1(a) and (c). Capacitor 1 is used to prevent the dc bias voltage flowing into RF source at the antenna terminal. Capacitor 2 ensures dc isolation between Pad A and Pad B while maintaining RF continuity of the balun. Pad A and Pad B are connected to the positive and negative bias voltages respectively to apply a reverse bias to the varactor diodes. The dc bias voltage is isolated from the RF signal of the antenna by using two dc bias

Fig. 3. Equivalent circuit of the varactor diode (C C : – : ).

= 0 1 1 0 pF

= 0:06 pF,

networks, each consisting of a two-element low-pass filter. Each filter is composed of a chip inductor, a chip capacitor and a via with a rejection band from 3 to 10 GHz, which covers the entire band of operation of the antenna. The equivalent circuit of the low-pass filter is shown in Fig. 1(c). For Antenna II, the folded dipole driver element is printed with six 0.5 mm gaps. Six beam lead PIN diodes (MA4AGBL912) are mounted across the gaps using electrically conductive silver epoxy. The orientation of the diodes is shown in Fig. 2(b). According to the PIN diode datasheet [15], the diode represents a forward resistance of 4 (typical value) for the ON state and a parallel circuit with a capacitance

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TABLE III SUMMARY OF SIMULATED AND MEASURED PERFORMANCE FOR THE RECONFIGURABLE ANTENNA I

Fig. 4. Simulated input reflection coefficient for the different varactor diode junction capacitances of Antenna I.

Fig. 6. Simulated and measured input reflection coefficient for the different states of Antenna II.

Fig. 5. Measured input reflection coefficient for the different values of varactor diode bias voltage of Antenna I.

of 0.025 pF and a resistance of  4 k for the OFF state. The length of the folded dipole element can be changed by switching between the different states of the diodes. When diodes 1 and 2 are on, and all the other diodes are off, the length of the folded dipole is L7 . In this case, the length of the folded dipole is short and the proposed antenna resonates at a high operating frequency (denoted as State I). Changing the polarity of the dc voltage turns diodes 1 and 2 off and all the other diodes on. In this case, the length of the folded dipole is increased to L7 + 2W10 + 2L10 and the antenna resonates at a lower frequency (denoted as State II). The bias network for Antenna II is exactly the same as the one used in Antenna I. III. SIMULATED AND MEASURED RESULTS The antennas were designed for fabrication on a single Rogers substrate (0.813 mm thick and dielectric constant "r = 3:55). The antennas were analyzed using the time domain solver of CST Microwave Studio [16], which is based on Finite Integration Technique. For Antenna I, the simulated reflection coefficients versus frequency for three different varactor diode junction capacitances Cj are given in Fig. 4. The corresponding measured results are plotted in Fig. 5. From Figs. 4 and 5 it is observed that by changing the equivalent-circuit capacitance value of the two varactor diodes from 1 pF to 0.1 pF (or increasing the varactor diode bias voltages) the center frequency can be continuously tuned from 6 to 6.6 GHz and the reflection coefficient bandwidth

(010 dB) at each frequency is greater than 15%. Table III summarizes the simulated and measured performance over the operating frequency range. For Antenna II, Fig. 6 displays the simulated and measured reflection coefficients versus frequency for State I and State II. From State II to State I, the resonant frequency shifts from 5.95 to 7.2 GHz, corresponding to a frequency ratio of 1.21. The frequency bandwidth is 22.2% and 21.8% for State I and State II, respectively. Radiation patterns were measured for Antenna I and Antenna II using a spherical near-field (SNF) system. Simulated and measured normalized radiation patterns are compared for both the E -(z -x plane) and the H -(z -y plane) planes. The orientation of the rectangular coordinate system used in all radiation pattern figures is the same as the ones shown in Figs. 1(a) and 2(a). For Antenna I, the radiation plots are as follows: Fig. 7 shows the E -plane radiation patterns at 6 GHz, 6.35 GHz and 6.6 GHz for a bias voltage of 0 V,15 V and 30 V respectively. Fig. 8 displays the H -plane radiation patterns at 6 GHz, 6.35 GHz and 6.6 GHz for a bias voltage of 0 V, 15 V and 30 V respectively. For Antenna II, Figs. 9 and 10 show the normalized radiation patterns at 7.2 GHz for State I and 5.95 GHz for State II, respectively. From Figs. 7–10, well-defined end-fire radiation patterns can be observed with a maximum cross-polarization level of 015 dB and 010 dB for E -plane and H -plane, respectively. It can be noted that the cross-polarization levels in the beam-maximum direction are typically lower. It also can be seen that the antenna has similar radiation patterns over the entire tunable frequency range. Further simulations reveal that the radiation pattern does not vary very much with the tuning frequencies and has a front-to-back ratio > 15 dB. Unfortunately, due to the blockage caused by the antenna positioner

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Fig. 7. Measured and simulated E -plane (z -x plane) normalized radiation patterns of Antenna I at (a) 6 GHz, bias voltage is 0 V, and theoretical varactor diode junction capacitance is 1 pF (b) 6.35 GHz, bias voltage is 15 V, and theoretical varactor diode junction capacitance is 0.28 pF (c) 6.6 GHz, bias voltage is 30 V, and theoretical varactor diode junction capacitance is 0.1 pF.

Fig. 8. Measured and simulated H -plane (z -y plane) normalized radiation patterns of Antenna I at (a) 6 GHz, bias voltage is 0 V, and theoretical varactor diode junction capacitance is 1 pF (b) 6.35 GHz, bias voltage is 15 V, and theoretical varactor diode junction capacitance is 0.28 pF (c) 6.6 GHz, bias voltage is 30 V, and theoretical varactor diode junction capacitance is 0.1 pF.

Fig. 9. Measured and simulated (a) E -plane (z -x plane) and (b) H -plane (z -y plane) normalized radiation patterns at 7.2 GHz for state I of Antenna II.

in the SNF chamber, the front-to-back ratio cannot be measured accurately. Realized gain has also been found by the gain comparison technique [17]. The losses of the cable and bias tees have been calibrated out of the measurement. For Antenna I, both the simulated and measured gains of the antenna have been plotted by combining the gain data across the entire tunable frequency range, which are shown in Fig. 11. From this plot, it is seen that the measured gain varies from 4.2 to 6.35 dBi, while the

simulated gain varies from 4.2 to 7.4 dBi for the entire frequency range. For Antenna II, gains in the two bands are plotted in Fig. 12. The loss included in the simulation is the approximated forward resistance of the PIN diodes. The measured gains vary from 4.4 to 5.3 dBi for State I and 3.5 to 4.5 dBi for State II. The gain of State I is higher than that of State II, because fewer diodes are used and there is less series resistance in the folded dipole. In general, reasonable agreements between the simulated and measured results can be noted from Figs. 11 and 12.

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Fig. 10. Measured and simulated (a) E -plane (z -x plane) and (b) H -plane (z -y plane) normalized radiation patterns at 5.95 GHz for state II of Antenna II.

Fig. 11. Measured and simulated gain of Antenna I.

higher than the simulated results. This is due to the uncertainties in the exact forward resistance of the PIN diodes (< 4:9 ohm specified by the manufacturer). However, reasonably good agreement between the simulated and measured results for the two reconfigurable antennas is achieved. For Antenna I, the loss of the varactor diode is a series resistance of 1.6 . Simulation results show that the gain of Antenna I increases by 0.04 dB when the resistance is reduced to zero. Therefore, the effect of the loss of the varactor diodes on the gain of Antenna I is very small. The frequency tuning ranges of Antenna I and Antenna II can be further enlarged by employing varactor diodes with wider capacitance variation and increasing the value of L10 , respectively. In addition, the length of the director could be varied to adjust the input impedance of the antenna for a good match at every operating state of the frequency reconfigurable antenna; this could be realized with additional varactor or PIN diodes. The design and testing of Antenna II is useful for another purpose. Due to the fact that the dimensions of the PIN diodes (0.6 mm by 0.15 mm) are quite small, Antenna II design can be scaled by a factor of ten to provide an antenna design that can tune between a typical millimeter wave WPAN band (57–66 GHz) and E -band (71–76 GHz). This work will be reported at a later date.

V. CONCLUSION

Fig. 12. Measured and simulated gain of Antenna II.

IV. DISCUSSION From Table III and Fig. 6, it can be noted that there are slight differences between the impedance bandwidth of the simulated and measured results for Antenna I and Antenna II. This discrepancy can mostly be attributed to the inaccuracies in the fabrication process, variations in discrete component parameters from values given in manufacturer’s data sheets, and diode parasitics. In addition, from Fig. 12 it is clear that at some frequency points the measured gains of Antenna II are

Endfire microstrip quasi-Yagi antennas with continuous and discrete frequency reconfigurability have been presented. Varactor and PIN diodes are used to alter the electrical length of the folded dipole driver, which changes the center frequency of the proposed antenna. Two antenna prototypes with reconfigurable frequency have been fabricated and measured in order to verify the design concept. The first prototype demonstrates a continuous tuning of the center frequency from 6 to 6.6 GHz by varying the capacitance of the varactor diodes from 1 pF to 0.1 pF. The second prototype demonstrates discrete tuning between dual frequency bands, with a frequency ratio of 1.21, by changing the dc bias on a series of PIN diodes. The proposed antennas can provide good out-of-band rejection, thus reducing the need for extra filters while tuning over a wide frequency range. Further advantages of the antennas are that the radiation characteristics remain essentially unaffected across the entire tunable frequency range and low levels of cross-polarized radiation are realized. The reconfigurable quasi-Yagi antennas have the further advantage of being compact, which makes them suitable for use in fixed- and steered-beam reconfigurable arrays.

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ACKNOWLEDGMENT The authors thank N. Nikolic for discussions on the folded dipole quasi-Yagi antenna, R. Shaw for assisting with fabrication of the prototype antenna and M. Shen for the attachment of the PIN diodes.

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Band-Rejected Ultrawideband Planar Monopole Antenna With High Frequency Selectivity and Controllable Bandwidth Using Inductively Coupled Resonator Pairs Tzyh-Ghuang Ma and Jyh-Woei Tsai

REFERENCES [1] W. R. Deal, N. Kaneda, J. Sor, Y. Qian, and T. Itoh, “A new quasi-Yagi antenna for planar active antenna arrays,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 6, pp. 910–918, Jun. 2000. [2] J. T. Bernhard, “Reconfigurable antennas,” in The Wiley Encyclopedia of RF and Microwave Engineering, K. Chang, Ed. New York: Wiley, 2005. [3] F. Yang and Y. Rahmat-Samii, “Patch antennas with switchable slots (PASS) in wireless communications: Concepts, designs, and applications,” IEEE Antennas Propag. Mag., vol. 47, pp. 13–29, Feb. 2005. [4] A. R. Weily, T. S. Bird, and Y. J. Guo, “A reconfigurable high-gain partially reflecting surface antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3382–3390, Nov. 2008. [5] D. Peroulis, K. Sarabandi, and L. P. B. Katehi, “Design of reconfigurable slot antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 645–653, Feb. 2005. [6] G. H. Huff, J. Feng, S. Zhang, and J. T. Bernhard, “A novel radiation pattern and frequency reconfigurable single turn square spiral microstrip antenna,” IEEE Microw. Wireless Compon. Lett., vol. 13, pp. 57–59, Feb. 2003. [7] N. Behdad and K. Sarabandi, “A varactor-tuned dual-band slot antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 401–408, Feb. 2006. [8] C. Zhang, S. Yang, S. El-Ghazaly, A. E. Fathy, and V. K. Nair, “A lowprofile branched monopole laptop reconfigurable multiband antenna for wireless applications,” IEEE. Antennas Wireless Propag. Lett., vol. 8, pp. 216–219, 2009. [9] C. J. Panagamuwa, A. Chauraya, and J. C. Vardaxoglou, “Frequency and beam reconfigurable antenna using photoconducting switches,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 449–454, Feb. 2006. [10] P. Bhartia and I. J. Bahl, “Frequency agile microstrip antennas,” Microw. J., pp. 67–70, Oct. 1982. [11] D. H. Schaubert, F. G. Farrar, A. Sindoris, and S. T. Hayes, “Microstrip antennas with frequency agility and polarization diversity,” IEEE Trans. Antennas Propag., vol. 29, no. 1, pp. 118–123, Jan. 1981. [12] P. Y. Qin, A. R. Weily, Y. J. Guo, and C. H. Liang, “A reconfigurable quasi-Yagi folded dipole antenna,” presented at the IEEE AP-S Int. Symp., 2009. [13] N. Nikolic and A. R. Weily, “Printed quasi-Yagi antenna with folded dipole driver,” presented at the IEEE AP-S Int. Symp., 2009. [14] R. W. Lampe, “Design formulas for an asymmetric coplanar strip folded dipole,” IEEE Trans. Antennas Propag., vol. 33, no. 9, pp. 1028–1031, Sep. 1985. [15] MA4AGBLP912 M/A-COM. [16] CST Microwave Studio. Ver Darmstadt, 2009. [17] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New York: Wiley, 2005.

Abstract—A novel band-rejected ultrawideband (UWB) planar monopole antenna is investigated. By utilizing the resonance nature of folded strips and a new inductive coupling scheme, the antenna demonstrates bandstop-filter-like response with high frequency selectivity and controllable bandwidth at the rejection band. The design methodology is introduced. The simulated and measured results including impedance matching, radiation patterns, and gain responses are investigated. An antenna equivalent circuit model with improved accuracy is discussed as well. When compared with our previous work, the new design demonstrates substantial improvements in many aspects. The argument will be discussed and verified carefully throughout the communication. Index Terms—Equivalent circuits, modeling, monopole antennas, notch filters, resonators, ultrawideband (UWB) antennas.

I. INTRODUCTION The ultrawideband (UWB) radio has become the core technology in wireless personal area networks (WPANs). Two coexistent systems, the direct-sequence ultrawideband (DS-UWB) and multiband orthogonal frequency division multiplexing (MB-OFDM), currently compete for potential UWB applications. For both schemes, the UWB spectra have been divided into two groups, the low band from 3–5 GHz and the high band from 6–10 GHz. The 5-GHz band, on the other hand, has been abandoned to avoid potential interference due to the coexistence of UWB and WLAN systems at 5–6 GHz. For an antenna designer, one should keep the antenna gain response flat and phase response linear over both sub-bands to reduce signal distortion. In addition, by introducing filtering properties to the radiator, the power radiated in the WLAN band could be minimized without the implementation of an additional bandstop filter [1]–[9]. These designs are generally referred to as band-rejected or band notched UWB antennas. The early works on band-rejected UWB antennas are realized by introducing a single or a single pair of resonators to the antenna radiator for signal suppression [1]–[4]. In those designs, however, the bandwidth and frequency selectivity of the rejection band can be hardly controlled, and may sometimes become unsatisfactory. To tackle the problems, various advanced UWB antennas have been proposed and analyzed [5]–[8]. In [5], a new antenna with bandstop-filter-like response was realized by two pairs of asynchronously tuned coupled resonators. A novel antenna with multiple notched bands has been developed in [6]. In [7], a band-rejected UWB antenna with on-ground slot was investigated. An advanced antenna design with steep fall-off rate and sufficient bandwidth at the rejection band has been demonstrated in [8]. In this communication, we engage in investigating a new band-rejected UWB planar monopole antenna with bandstop-filter-like Manuscript received September 01, 2009; revised December 14, 2009; accepted February 12, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by the National Science Council, R.O.C., under Grants 97-2221-E-011-019-MY2 and 98-2221-E-011029. The authors are with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei 10607, Taiwan 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.2010.2050444

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Fig. 1. Electric fields and surface current distributions on the (a) vertically and (b) horizontally arranged folded-strip resonators.

response, i.e. good frequency selectivity and controllable bandwidth, at the targeted rejection band. The new antenna is realized by utilizing two pairs of horizontally placed folded-strip resonators with a novel inductive coupling scheme. The goal of the design is to overcome several drawbacks noticed in our previous work of [5]. Some preliminary results have been introduced in [9], and in this communication the antenna will be examined and optimized in detail. The antenna configuration and design methodology are introduced in Section II. The tunability of the rejection band is discussed in Section III along with parametric studies. Section IV discussed a new equivalent circuit model with improved accuracy, and the radiation characteristics are illustrated in Section V. A brief summary will be given at the end of this communication. II. CONFIGURATION AND DESIGN METHODOLOGY The band-notched antenna in [5] suffers from the following drawbacks. First, the impedance matching is unsatisfactory at both UWB low and high bands. Second, via holes are necessarily required and significantly increases the fabrication cost. In addition, due to the strong fringing fields of the vertically placed folded-strip resonators, as shown in Fig. 1(a), the antenna is prone to nearby coupling. It inevitably increases the measuring complexity and limits the practical applications. Last but not the least, in terms of the return loss response, which is not shown in [5], the accuracy of the antenna equivalent circuit model is far from satisfactory. To deal with the difficulties, in this communication a new band-rejected ultrawideband planar monopole antenna is proposed by utilizing two pairs of inductively coupled resonators. The configuration of the antenna is illustrated in Fig. 2 along with the geometric parameters. The antenna lies in the xy-plane with its normal direction being parallel to the z axis. It is symmetric about the centerline, and consists of a microstrip feed line, a truncated ground plane, a flared metal plate, and two pairs of folded strips. The flared metal plate serves as an impedance matching network as well as a wideband radiator since the energy can transform smoothly from the feed line to free space via the tapered apertures. The folded strips function as two pairs of band-notched parallel resonators for creating the rejection band. Depending on the geometric locations, the resonators are referred to as the exterior and interior resonators, respectively, in the following discussion. The subscripts “e” and “i” are used to distinguish the parameters. The main difference between the proposed antenna and that in [5] is the realization of the folded-strip resonators. In the current design, the folded-strip resonators are horizontally placed on the top layer of the substrate. Based on the full-wave simulation, Fig. 1 illustrates the

Fig. 2. Configuration of the proposed antenna.

Fig. 3. Inductive coupling scheme and its equivalent circuit model.

fringing electric fields and surface current distributions of both vertically and horizontally placed folded-strip resonators. Referring to the figure, the uniplanar folded strip in Fig. 1(b) can significantly reduce the nearby coupling since the fringing fields are more concentrated along the coupling edges. On the positive side, it improves the antenna impedance matching and simplifies the measurement setup. It cuts down the fabrication cost and makes the antenna more suitable for practical applications as well. However, on the negative side, it also poses a new problem. Since the fringing fields are mostly trapped by the coupling edges, the coupling between any two adjacent resonators is relatively weak. Accordingly, the adjacent resonators can be hardly formed a pair of coupled resonators for synthesizing the required bandstop-filter-like response at the targeted rejection band. To deal with this issue, a novel inductive coupling scheme is proposed, as illustrated in Fig. 3. By connecting the open ends of the exterior and interior resonators with a short line section, additional current path can be introduced to provide an adequate amount of coupling between the adjacent resonators. It effectively improves the rejection band response at 5–6 GHz. The coupling by the short line is principally inductive, and can be modeled by a J -inverter, or equivalently, a  -network with three inductors. One of the inductors is positive, and the remaining two are conceptually negative. All inductors have the same absolute value of Lm [10]. The inductive coupling plays a key role in realizing the bandstop-filter-like response at the notched band. A detailed investigation of the inductive coupling scheme will be given in Sections III and IV. The design procedure of the proposed antenna can be summarized as follows. First, the resonance frequencies of the exterior and interior resonators are chosen by adjusting the lengths of the folded-strip resonators, i.e. Lce and Lci . The resonance frequencies of the two resonators can be intentionally different from each other by a small

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Fig. 4. Simulated and measured return losses.

amount to take advantage of the asynchronous tuned coupled resonators in filter theory [10]. It provides additional flexibility in shaping the rejection band response. The folded strips are then connected to the flared metal plate whose tapered profile is fine-tuned to achieve good impedance matching over the whole UWB band. The truncated ground plane definitely has a pronounced effect on the impedance matching and should be taken into consideration simultaneously [5], [11]. After achieving the in-band impedance matching, the amount of inductive coupling between the exterior and interior resonators is optimized by tuning the length Lind of the short line section until the rejection band response meets the design specifications. III. IMPEDANCE MATCHING AND NOTCHED BAND TUNABILITY A prototype antenna with a rejection band at 5.15–5.35 GHz, the spectra for HYPERLAN/2, was developed for demonstration purpose. The antenna was fabricated on a 20-mil Rogers RO4003C substrate with a dielectric constant of 3.38 and loss tangent of 0.0027. The fullwave simulator Ansoft HFSS was used throughout the development to optimize the performance. Dissimilar to the preliminary design in [9], here the finite conductivity and finite thickness of the cooper plates are taken into account in the simulation, which can conspicuously improve the simulation accuracy. The final dimensions of the folded strips are Lce = 12 mm, Wce = 0:5 mm, Lci = 11:8 mm, Wci1 = 3:05 mm, Wci2 = 2:5 mm, Gc = 0:5 mm, and Gxc = 1 mm. The inductive line, Lind 3 Wind , is 2.6 mm 3 0.5 mm, and the slit Ls is 8.4 mm. The parameters associated with the flared metal plate and the microstrip line are Linv = 2:95 mm, Winv = 6:5 mm, Ginv = 1 mm, lms = 20:35 mm, and wms = 1:1 mm. The tapered profile is described by an arc with a radius of 39 mm and a sector angle of 19.5 degrees. The occupied size is 38 by 42 mm2 . The simulated and measured return losses are illustrated in Fig. 4. The measurement was completed by mounting the antenna under test (AUT) on a Cascade Microtech MTF26 microstrip test fixture at National Taiwan University. An Agilent performance network analyzer PNA-L 5230A was used to acquire the data. Several absorbers were placed around the test fixture to reduce the undesired coupling. The reference plane was calibrated to the probe tip of the test fixture using the standard one-port SOL (short-open-load) calibration and port extension technique. Referring to the figure, the simulated and measured return losses agree reasonably well for frequencies higher than 4 GHz. The measured return loss (RL) is principally better than 9.5 dB (i.e. VSWR < 2) from 3.1 to 5 GHz and from 5.5 to more than 12 GHz except for slight degradation around 4.5 and 8.5 GHz. The proposed antenna demonstrates bandstop-filter-like response at the targeted rejection band. The signal rejection is better than 5 dB from 5.075–5.375 GHz, and the maximum rejection is 2.72 dB at 5.18 GHz. At the notched band edges, the frequency fall-off rate is 6 dB per

Fig. 5. Variations of the antenna return losses with respect to (a) the length of the inductive line and (b) the gap width of the fold-strips.

100 MHz, a relative steep slope as expected. On the other side, the discrepancy between the simulated and measured results becomes apparent for frequencies below 4 GHz. It is not a surprising result if the effects of the finite size ground plane and nearby coupling are taken into consideration. As indicated in [11], those nonideal effects become significant at the lower edge of the UWB low band, and could be reduced by some advanced techniques. Fig. 5(a) investigates the inductive coupling scheme. The length of the short line is increased from 2.0 to 3.2 mm with a step of 0.3 mm. The case with Lind = 0 mm is also depicted. It is obvious that without the inductive coupling, the notched band shows poor frequency selectivity at the upper band edge. As the short line lengthens, the amount of inductive coupling rises and the rejection bandwidth increases correspondingly. The observation in Fig. 5(a) verifies the idea that by utilizing the inductive coupling, the frequency selectivity at the rejected band can be significantly improved. The optimized length of the inductive line, in our experience, is around 2.2 to 2.7 mm. An over-length line cannot further improve the bandwidth with acceptable rejection level. At the meantime, an inductive line shorter than 2.0 mm will interact with the slits Ls and result in parasitic resonances. The rejection bandwidth can be also effectively control by adjusting the gap width Gc between the coupling edges of the fold strip resonators. As the gap width increases, the fields in the resonators become less bounded and tend to couple to the adjacent resonators. According to [10], the increased amount of coupling between resonators results in a wider filter bandwidth. To verify the argument, the simulated return losses with gap width varying from 0.1 to 0.9 mm are summarized in Fig. 5(b). Referring to the figure, the notched bandwidth and the maximum rejection level increase simultaneously as the gap width Gc becomes wider, which provides an effective way to control the bandwidth

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TABLE I ELEMENT VALUES OF THE LUMPED CIRCUIT MODELS FOR THE RESONATORS

The units of the inductor, capacitor, and resistor are nH, pF, and .

Fig. 6. (a) Novel equivalent circuit model for the folded-strip resonator. (b) Comparisons between the full-wave simulated and model-calculated input impedances of the exterior resonator.

of the rejection band for a specific design. For the current antenna, the rejection bandwidth with return loss < 5 dB can be adjusted from 40 MHz (0.1 mm) to 480 MHz (0.9 mm).

Fig. 7. Equivalent circuit model of the proposed antenna with improved accuracy.

IV. EQUIVALENT CIRCUIT MODEL In this section, an equivalent circuit model incorporating both lumped and distributed elements is discussed to get an even clearer understanding of the proposed antenna. For simplicity, in the following discussion, the subscripts “e” and “i” are omitted unless otherwise specifically indicated. The antenna modeling begins with the modeling of each individual folded-strip resonator. In [5], the folded-strip resonator is simply modeled by a lumped lossy parallel resonant circuit whose element values (Req ; Leq ; Ceq ) can be extracted using the standard one-port Z -parameter. The modeling is straightforward, but the accuracy suffers. To improve the accuracy, as shown in Fig. 6(a), two additional elements, Rpr and Cm , are added to the equivalent circuit model to account for the distributive nature of the folded-strip resonator. The resistance Rpr represents the parasitic radiation from the resonator, while the capacitance Cm accounts for the fringing field at the open end of the folded strip. Together with the lossy parallel resonator model, the input impedance of each individual folded-strip resonator can be approximated by the equation shown in Fig. 6(a). The element values (Req ; Leq ; Ceq ) of the lossy parallel resonator can be determined using the extraction method in [5]. It will not be repeated here to avoid redundancy. The values of the additional elements (Rpr ; Cm ) are determined by curve-fitting the calculated oneport Z -parameter to the full-wave simulated one with Agilent ADS. Table I summarizes the extracted element values of both interior and exterior resonators. For demonstration purpose, Fig. 6(b) compares the calculated input impedance of the exterior resonator using the circuit model to the full-wave simulated one using HFSS. Excellent agreement can be observed. As compared to the original model in [5], the modeling accuracy of the folded-strip resonator has shown significant improvement.

By utilizing the improved resonator modeling, a novel equivalent circuit model is presented for the proposed band-rejected UWB antenna, as illustrated in Fig. 7. Referring to the figure, each of the four foldedstrip resonators is represented by the network in Fig. 6(a). The inductive coupling sections are accounted for by the J -inverters, as suggested in Fig. 3. The J -inverters connect the open ends of the exterior and interior resonators, while the capacitors Cm;e and Cm;i again represent the fringing fields of the resonators. In contrast to [5], here the flared metal plate is described by a distributive transmission line model with two four-section stepped-impedance microstrip lines. The detailed structure of the stepped line is shown in the inset of Fig. 7. The two pairs of inductively coupled resonators are connected together with a microstrip feed line with a length of lms and characteristic impedance of Z0 . In addition to the improvement of the modeling of the folded-strip resonator and flared metal plate, the distributive radiation behavior along the strips is emphasized in the circuit model as well. An additional radiation resistance Rrad is serially connected to the feed line to account for the radiation from the flared metal plate to free space. We believe that these modifications are capable of providing a more accurate antenna modeling with physical insight. The cross coupling effect between any two adjacent resonators, on the other hand, is so weak that it is ignored in the circuit model. The extraction of the components of the antenna circuit model in Fig. 7 is more involved due to the complicated interconnection of the elements. An explicit expression for the antenna input impedance is impractical. Instead, the element values are determined by curve-fitting the calculated one-port S -parameter using the circuit model to the full-wave simulated one by HFSS with Agilent ADS. The extracted element values, including the J -inverters, stepped lines, and series radiation resistance, are summarized in Table II. To numerically verify the

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TABLE II ELEMENT VALUES OF THE ANTENNA EQUIVALENT CIRCUIT MODEL

The units of the length, inductor, capacitor, resistor, and inverter are mm, nH, pF, ohm and mho, respectively.

Fig. 9. Simulated and measured yz- and xz-plane radiation patterns at (a) 4 GHz and (b) 8 GHz.

Fig. 8. Comparisons of the calculated, simulated, and measured antenna oneport S -parameters.

equivalent circuit model, the calculated antenna one-port S -parameter is compared with the simulated and measured ones in Fig. 8(a) and (b). Reasonable agreement can be achieved over a very wide bandwidth, a major improvement which can be hardly observed with the antenna model in [5]. Referring to Fig. 8(b), the phase response is almost linear in both sub-bands, a crucial result capable of minimizing the transmission distortion in either DS-UWB or MB-OFDM systems. According to the comparisons, it is concluded that by applying a more accurate resonator model and appropriate distributed elements, the new antenna model can not only explain the complicated bandrejected behavior but lead to significantly improved agreement over the entire band. V. RADIATION CHARACTERISTICS In this section, the radiation characteristics of the proposed band-notched UWB antenna are investigated. The radiation patterns were measured in a three-dimensional spherical near-field anechoic

chamber at National Taiwan University of Science and Technology along with the Nearfield Systems Inc. (NSI) 700S-90 scanner and an Agilent E8363B network analyzer. The dimensions of the chamber are 7 3 4:9 3 4:6 m3 . An electrically switched dual-polarized log periodic antenna is used as the probe. Due to the reduced nearby coupling, the antenna is fed directly by a coaxial cable via a SMA adaptor. It significantly reduces the measurement complexity. The connecting cable is surrounded by three ferrite cores to suppress the parasitic radiation from the cable. This improves the measurement accuracy in the low frequency range. The simulated and measured radiation patterns in the yz-(H-) and xz-(E-) planes at the center frequencies, 4 and 8 GHz, of the UWB low and high bands are illustrated in Fig. 9(a) and (b), respectively. The measured radiation patterns agree reasonably well with the simulated ones in both planes. The discrepancy, especially in the cross-polarization, can be attributed to the interference from the SMA adaptor and connecting cable. The misalignment of the antenna with respect to the principle axes has some contribution as well. The radiation characteristics of the proposed antenna are, in essence, very similar to those of a conventional UWB planar monopole antenna. The details have been described in [5], and will not be repeated due to the limited space. The measured peak gain varies from 1–4 dBi over the two sub-bands. With the help of the spherical near-field system, the antenna efficiency is measured from 2 to 10 GHz, as described in Fig. 10. Apart for the degradation at 7 GHz, the antenna efficiency is higher than 65% over both sub-bands. It reaches up to 90% in several frequency ranges. The degradation around 7 GHz is likely a result of a spurious resonance in the folded strips. Meanwhile, the antenna efficiency drops dramatically at the rejection band. It reaches a minimum value of 23% at 5.13 GHz, but rises sharply at the band edges of the notched band. To achieve high performance band-rejected UWB antenna, the spatial dependence of the rejection band should be minimized as well. The gain responses, i.e. the measured antenna absolute gains versus frequencies in different reception directions [5], are examined from 3 to 11 GHz. Fig. 11 illustrates the measured gain responses in the yz-plane

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the nearby coupling. It improves the measurement accuracy and facilitates the integration with other circuitries. In addition, no via hole is required which can lower down the fabrication cost. In view of the antenna performance, the impedance matching is improved while the rejected band response is less dependent on the spatial variations.

REFERENCES

Fig. 10. Measured antenna efficiency of the proposed design.

Fig. 11. Measured antenna gain responses in the yz-plane.

at  = 0 , 45 , 90 , and 135 . Though suffering from some degradation at the boresight direction ( = 0 ), it is clear that as the reception angle varies, the signal attenuation remains better than 10 dB. At  = 90 , the rejection level can reach up to 25 dB. When compared with the design in [5], the proposed antenna demonstrates much more stable rejection-band radiation characteristics against spatial variations. In addition, at the azimuth plane ( = 90 ), the variation of the gain response is less than 5 dB at both sub-bands, a desirable result which reduces the distortion introduced by the transceiving antenna. The gain variations in other planes are a little bit higher, but remain principally lower than 10 dB in the operating bands. VI. CONCLUSION A novel UWB planar monopole antenna with bandstop-filter-like rejection response using inductively coupling scheme has been investigated thoroughly in this communication. The signal rejection is better than 5 dB from 5.075–5.375 GHz. The maximum rejection level is 2.72 dB. At the band edges, the frequency fall-off rate reaches up to 6 dB per 100 MHz. The rejection bandwidth, on a 5-dB return loss level, can be controlled from 40 to 480 MHz. The radiation patterns are nearly omnidirectional in the yz-plane with a maximum gain of 4 dBi. According to the antenna gain and phase responses, the proposed design is suitable for applications in both DS-UWB and MB-OFDM systems. An antenna equivalent circuit model with improved accuracy has been discussed as well. When compared with our previous work, the proposed design features significant improvements in many aspects. First, the horizontally arranged folded-strip resonators greatly reduce

[1] W. S. Lee, W. G. Lim, and J. W. Yu, “Multiple band-notched planar monopole antenna for multiband wireless systems,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 9, pp. 576–578, Sep. 2005. [2] Y.-C. Lin and K.-J. Hung, “Compact ultrawideband rectangular aperture antenna and band-notched designs,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3075–3081, Nov. 2006. [3] K.-H. Kim and S.-O. Park, “Analysis of the small band-rejected antenna with the parasitic strip for UWB,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1688–1692, Jun. 2006. [4] T.-G. Ma and S. J. Wu, “Ultrawideband band-notched folded strip monopole antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2473–2479, Sep. 2007. [5] T.-G. Ma, R.-C. Hua, and C.-F. Chou, “Design of a multiresonator loaded band-rejected ultrawideband planar monopole antenna with controllable notched bandwidth,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2875–2883, Sep. 2008. [6] Y. Zhang, W. Hong, C. Yu, Z.-Q. Kuai, Y.-D. Don, and J.-Y. Zhou, “Planar UWB antennas with multiple notched bands based on etched slots on the patch and/or split ring resonators on the feed line,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 3063–3068, Sep. 2008. [7] Y. D. Dong, W. Hong, Z. Q. Kuai, and J. X. Chen, “Analysis of planar UWB antennas with on-ground slot band-notched structures,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1886–1893, Jul. 2009. [8] J.-W. Jang and H.-Y. Hwang, “An improved band-rejection UWB antenna with resonant patches and a slot,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 299–302, 2009. [9] T.-G. Ma, “Band-rejected ultrawideband planar monopole antenna with bandstop-filter-like response using inductively coupling scheme,” in Proc. 3rd Eur. Conf. on Antennas Propag., Berlin, Germany, Mar. 22–27, 2009, pp. 3527–3529. [10] J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Application. New York: Wiley, 2001. [11] Z.-N. Chen, T.-S.-P. See, and X. Qing, “Small printed UWB antenna with reduced ground plane effect,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 383–388, Feb. 2007.

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Reconfigurable Circularly-Polarized Patch Antenna With Conical Beam Jeen-Sheen Row and Ming-Che Chan

Abstract—A design method is described for reconfigurable circularly-polarized (CP) microstrip array antennas with conical-beam radiation. The proposed antenna structure consists of four L-shaped patch antennas arranged in a square-ring formation. Each patch antenna is shorted to the ground plane through conducting walls. With a top-loaded monopole feed, two orthogonal resonant modes, loop mode and monopole mode, can be excited simultaneously. The effects of the shorting walls on the resonant frequencies of the two modes are investigated, and it is found that the res) of the monopole mode has a more obvious varionant frequency ( ) of the loop mode when the width or the number ation than that ( of the shorting walls is changed. In addition, if the two modes are properly coupled, the microstrip array can generate a CP radiation; meanwhile, the but is right-hand CP polarization sense is left-hand CP if if . A reconfigurable antenna prototype is fabricated, and the experimental results show that the prototype not only has the characteristic of omnidirectional radiation but also possesses the ability of switching between left-hand and right-hand CP. Simulated results carried out by HFSS are also provided and they agree well with the measured results. Index Terms—Circular polarization, conical beam, microstrip array, reconfigurable antenna.

I. INTRODUCTION For wireless mobile communication systems, the demand of circularly-polarized (CP) antennas with omnidirectional radiation is continuously increasing because they can cover a large service area and provide a stable signal quality. Based on the requirements, a considerable number of the designs related to the CP antenna with a conical beam have been proposed and studied over the past few years [1]–[11]. According to the number of radiating elements, these previous designs may be divided into three groups: one-element type, dual-element type, and array type. So far as the design using the array type is concerned, each of the array elements is first designed to be a CP antenna, and then all array elements are arranged in a specific formation to generate conical beam radiation [2], [3]. For such a design, all elements can be excited in the same amplitude and phase, and the conical-beam radiation characteristic is not required for the single element; however, the design would lead to a large antenna size and even a complicated array structure. On the contrary, for the design using the dual-element type, each element is required to have the ability of radiating a conical-beam pattern, and the two elements need to be excited with a 90 phase difference when their orientations in space are orthogonal to each other. The required phase shift can be achieved with a delay line [5] or by making the input impedances of the two elements being complex-conjugated [6]. The CP conical-beam radiation can also be generated from a single radiating element. As compared with the dual-element type and array type, the designs using the one-element type usually have a simple Manuscript received August 12, 2009; revised October 23, 2009; accepted January 25, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. 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.2010.2050436

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antenna structure but need a relatively complex mechanism to produce the CP radiation. For example, a conventional quarter-wavelength monopole antenna needs to be uniformly coated with a two-layer polarizer switching from linear to circular polarization [7]. To reduce antenna height, the designs of using microstrip antennas have also been reported. The microstrip antenna can radiate a CP conical beam from a single radiating patch when it operates at higher order modes [8], such as TM21 mode; however, a complicated feeding network is essential to symmetrically excite two orthogonal modes with a 90 phase difference [9]. Using perturbation technologies can avoid the requirement for the feeding network [10], but the consequence is that the conical beams in different  planes would be not uniform. The perturbation technology has been applied to a loop antenna as well [11]. The design described in [11] has the advantages of low profile and simple structure; moreover, its simulated results give nearly omnidirectional radiation patterns in the  plane at the elevation angle with maximum gain. However, the loop requires a circumference of approximately two free-space wavelengths and no experimental results are provided. In this communication, a new method to design a CP antenna with the conical-beam radiation is first proposed. The design concept comes from the two orthogonal linearly-polarized antenna structures respectively reported in [12] and [13]. The design in [12] indicated that four microstrip patches arrayed in a ring formation can produce a symmetric horizontally-polarized pattern with a null at broadside; on the other hand, a monopole antenna that is top loaded with a shorted cross patch can give uniform conical radiation patterns with vertical polarization [13]. Therefore, a CP antenna with the conical-beam radiation would be obtained by combining the two antenna structures and simultaneously exciting their operating modes at different frequencies. For the proposed design, it is classified into the array type in appearance but the required antenna size is much smaller than those of the previous designs; besides, the mechanism of generating CP can be considered as one kind of the perturbation technology and consequently it does not need a complicated feeding network. Furthermore, the CP sense of the proposed design can easily be changed by reconfiguring perturbation segments. A design with the ability of switching between left-hand circular polarization (LHCP) and right-hand circular polarization (RHCP) is also carried out and the obtained results are presented. II. ANTENNA STRUCTURE AND ANALYSIS The geometry of the proposed antenna is depicted in Fig. 1. A microstrip array consists of four sequentially-rotated shorted L-shaped patches distributed over the x-y plane in a square-ring formation. All of the L-shaped patches have a width of 16 mm and a total length of 36 mm, and they are joined together with a cross microstrip line of 2 mm width. A monopole top loaded with a square patch is placed below the array center, and it is used to excite the microstrip array through capacitive coupling. The microstrip array and loading patch are respectively printed on the two faces of a square FR4 substrate (thickness 1.6 mm, permittivity 4.4, side length 120 mm). The substrate is supported by shorting walls at a height of 10 mm above the ground plane. These shorting walls are classified into two groups: Group A (A1, A2, A3, A4) and Group B (B1, B2, B3, B4). Each L-shaped patch involves two shorting walls, in which one shorting wall, belonging to Group A, is connected to the array corner and the other shorting wall, belonging to Group B, is placed at a distance of s away from the corner, as shown in Fig. 1. The case without the shorting walls is first analyzed. For the single L-shaped microstrip patch antenna, the current flowing on its radiating patch presents a one-half wavelength variation when it operates at the fundamental mode. If the four L-shaped microstrip antennas are excited

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Fig. 2. Return-loss results for the microstrip array without shorting walls; antenna dimensions are shown in Fig. 1.

Fig. 1. Geometry of the proposed antenna.

in the same phase, the currents flowing on these patches can be combined to be nearly a round path, and no anti-phase current occurs along the circumference of the microstrip array. Such a current distribution is similar to that of a small loop antenna, and therefore the operation mode is named loop mode here. It is apparent that the microstrip array operating at the loop mode can radiate a conical beam with horizontal polarization, and the operating frequency is mainly determined by the total length of the L-shaped patch. When the shorting walls are introduced, besides the original loop mode, another resonant mode can be excited from the microstrip array. The resulting mode is the so-called monopolar wire-patch mode [14], and its resonant frequency is related to the width and the number of the shorting walls [15]. As the microstrip array operates at this mode, it can radiate vertically-polarized conical-beam patterns. Consequently, the microstrip array shown in Fig. 1 can provide two orthogonal modes. To explain the CP generation, the shorted microstrip array can also be considered in other aspects. Similar to the truncated corners used in a CP microstrip patch [16], the shorting walls can be regarded as the perturbation segments of the microstrip array, which have the function to produce dual orthogonal degenerated modes. By tuning the dimensions of the perturbation segments, the two orthogonal modes can be properly coupled together and a CP operating frequency can be found between the two resonant frequencies. III. EFFECTS OF SHORTING WALLS An example with the dimensions revealed in Fig. 1 is selected to demonstrate the performance of the proposed antenna. To begin with, the case without any shorting wall is explored. The simulation results obtained by HFSS are exhibited in Fig. 2 along with the experimental results. Agreement between them is satisfactory, except for the slight frequency deviation that could be due to the errors in manufacturing. From the measured results, it can clearly be seen that one resonant mode is excited from the example antenna, and the operating bandwidth, determined by 10 dB return loss, is about 4.3 % with respect to the center frequency 2760 MHz. The radiation characteristics of this mode are also examined, and both simulated and measured results are presented in Fig. 3. Conical beams with a maximum gain around  = 40 are observed as expected. In addition, the dominant polarization is E whose amplitude is at least 8 dB higher than that of E around the main beams.

Fig. 3. Radiation patterns for the microstrip array without shorting walls. (a) x-z plane, (b) y -z plane. Solid lines: E (simulated). Dashes: E (simulated). Circles: E (measured). Squares: E (measured).

Fig. 4. Simulated return-loss results of various w for the microstrip array with the shorting walls of Group A; other dimensions are shown in Fig. 1. Thin line: mm. Thick line: w mm. Dashed line: w mm. Dotted line: w mm. Circles: w w mm (measured).

=2 = 12

=6 =2

=9

A. Effects of Group A When the shorting walls of Group A are attached to the example antenna, another mode can be excited and its resonant frequency is related to the width of the shorting walls, as seen in Fig. 4. Fig. 4 presents the simulated return-loss results for the cases of w = 2, 6, 9 and 12 mm. The experimental results of w = 2 mm case are also included to validate the simulated results. Observing the measured results of w = 2 mm case, it is found that the resonant frequency of the new mode, defined at the frequency with minimum return loss, occurs at 2090 MHz, and at the same time the resonant frequency of the original mode is shifted to 2560 MHz. For distinguishing the two modes, the current distributions on the L-shaped patches are respectively inspected at 2090 and 2560 MHz, and their results are presented in Fig. 5. It is found that

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Fig. 5. Simulated current distributions for the case of (a) 2560 MHz, (b) 2090 MHz.

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w = 2 mm in Fig. 4. Fig. 7. Simulated return loss of various s for the array with the shorting walls mm, other dimensions are shown in Fig. 1. Thin line: of Group A and B; w s mm. Dotted line: s mm. Dashed line: s mm. Thick line: s mm. Circles: s mm (measured).

=0

=6

=5

=1

=3

=5

B. Effects of Group B

= 2

Fig. 6. Radiation patterns at 2090 MHz for the case of w mm in Fig. 4. (a) x-z plane. (b) y -z plane. Thick line: E (simulated). Dashed line: E (simulated). Circles: E (measured). Squares: E (measured).

the currents at 2560 MHz are mostly along the inner edges of the four L-shaped patches, and a loop current is obviously synthesized; however, plenty of currents flow into the shorting walls at 2090 MHz and the magnitude of the loop current is somewhat suppressed. It suggests that the radiation at 2090 MHz is mainly contributed by the currents on the shorting walls, and thus  -polarized conical patterns can be obtained, as shown in Fig. 6. For the cases of w = 6, 9 and 12 mm, the similar current distributions can also be observed at their two resonant frequencies. Therefore, the radiation characteristics of the two resonant modes, which are respectively classified into the loop mode and monopole mode, are the same in pattern but orthogonal in polarization. On the other hand, the resonant frequencies of the two modes, named floop and fmono , have different variations with increasing the width of the shorting walls. The simulated results in Fig. 4 indicate that as w is increased from 2 to 12 mm, fmono is obviously increased from 2100 to 3000 MHz whereas floop is only decreased by 3 %. Consequently, by selecting a specific w , the two resonant frequencies can be appropriately coupled together, which represents that the two orthogonal modes would be simultaneously excited in the same amplitude and a CP operating frequency would be found between the two resonant frequencies. The phenomenon occurs at the case of w = 6 mm, in which the corresponding fmono and floop are respectively 2280 and 2490 MHz, and the CP operating frequency with a axial ratio of 1 dB is found at 2430 MHz; besides, due to fmono < floop , which implies that E is behind E by 90 in phase, the polarization sense is LHCP with respect to a r -directional traveling wave. Under the condition that the two orthogonal modes are appropriately coupled, the polarization sense can be switched to RHCP if fmono is larger than floop . To achieve this goal, several cases using a wider shorting wall are tried but fail to obtain good RHCP performance. The reason could be that the respective polarization purity of the two orthogonal modes is deteriorated with increasing the width of the shorting walls.

An alternative method to increase the resonant frequency of the monopole mode is by means of adding the number of the shorting walls. The case of w = 6 mm in Fig. 4 is selected to study the effects of adding Group B. When the shorting walls of Group B are introduced and their widths are fixed to be 6 mm, the simulated results for various s are presented in Fig. 7. For the special case of s = 0 mm, it represents that Group A and B are joined together, and each of the resultant shorting walls can be regarded as a single folded shorting wall with a total width of 12 mm. Obviously its simulated result is different from that of the w = 12 mm case in Fig. 4. It suggests that the positions of the shorting walls would affect the two resonant frequencies, especially for fmono . As Group A and Group B are separated by a distance of s, the results in Fig. 7 indicate that both fmono and floop are increased with enlarging the separated distance but the increasing of fmono is more apparent than that of floop , leading to the change in coupling strength between them. In addition, fmono is larger than floop for all cases. So, a RHCP operation can be found by properly adjusting s, and it occurs at s = 5 mm. The experimental results of s = 5 mm case are also given in Fig. 7. According to the measured results, the RHCP operating frequency is at 2940 MHz with an axial ratio of 1.2 dB, and the corresponding fmono and floop are 3010 and 2760 MHz, respectively. IV. RECONFIGURABLE PROTOTYPE DESIGN Based on the above analyses, a straightforward method for designing a conical-beam antenna with electrically switching between LHCP and RHCP can be drawn, and its layout and associated circuits are exhibited in Fig. 8. The shorting walls of Group A and B are connected to the L-shaped patches through capacitors of 100 pF and pin diodes (SMP1320-079, Skyworks Solutions Inc.), respectively. The critical parameters are also revealed in Fig. 8, and other dimensions are the same as those of the example antenna. It is noted that because of the diode effects, the width of the shorting walls of Group B is tuned to be 4 mm. For simplifying the DC bias network that is required to control the states of the pin diodes, a thin pin is employed to connect the L-shaped patches and an isolated circular pad from the ground plane. Consequently, the positive and negative of the battery are linked to the pad and ground plane, respectively; besides, the capacitor formed by the pad and ground plane can act as a choke to prevent high-frequency signals from entering the DC bias network. Therefore, the radius of the circular pad needs to be carefully selected in order to obtain a uniform conical beam. When a forward bias (Vo = 3V) is applied to turn on the diodes, the reconfigurable microstrip array is shorted by the shorting

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Fig. 8. Layout of a reconfigurable CP microstrip array with conical beam and the photographs of the finished prototype.

Fig. 10. Radiation patterns of the prototype measured at 2475 MHz when the plane. diodes are OFF. (a) x z plane (b) y z plane (c) 

0

0

= 35

Fig. 11. Radiation patterns of the prototype measured at 2895 MHz when the plane. diodes are ON. (a) x z plane (b) y z plane (c) 

0

Fig. 9. Simulated and measured results of the prototype. (a) return loss (b) axial . Dashed line: Diodes off (simulated). Dotted line: Diodes off ratio at  (simulated). Thick line: Diodes off (measured). Thin line: Diodes on (measured)

= 35

walls of Group A and B, and thus it can radiate RHCP conical-beam patterns. On the contrary, when the diode is off (Vo = 0V), the shorting walls of Group B are disabled and the polarization sense is switched to LHCP. A prototype of the reconfigurable microstrip array is fabricated and the photographs of the finished prototype are also exhibited in Fig. 8. The measured return loss and axial ratio are shown in Fig. 9 together

0

= 35

with their simulation results. Note that the numerical analyses are obtained by employing a metal strip (1.2 2 1.2 mm2 size) and an air gap (1.2 mm length) to simulate an ohmic resistance of 0.9 and a cutoff capacitance of 0.2 pF, which are the equivalent circuits of the diodes at ON and OFF states, respectively. From the measured results, it is indicated that the LHCP operating bandwidth, determined from 3 dB axial ratio, is 1.6 % with respect to the center frequency 2475 MHz and the RHCP operating bandwidth is 2 % with respect to the center frequency 2895 MHz. It has to be mentioned that the axial ratio in Fig. 9 is determined by the maximum value within the  plane at  = 35 . Moreover, good impedance matching with a return loss of less than 10 dB is achieved across the two CP operating frequencies. The radiation patterns in two orthogonal planes as well as the plane of  = 35 are also measured for the dual CP operating frequencies, and their results are respectively plotted in Figs. 10 and 11. Both LHCP and RHCP operations have the conical-beam radiation patterns in the x0z and y 0z planes, whose maximum gain is around  = 35 , and omnidirectional

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radiations are also observed in the  = 35 plane. The measured peak gain is about 3.2 dBic for the LHCP radiation and 2.4 dBic for the RHCP radiation. The gain difference could be due to the ohmic loss of the diodes.

V. CONCLUSION A circularly-polarized microstrip array antenna with conical-beam radiation has been presented in this communication. The antenna is composed of four L-shaped patches which are connected to the ground plane through shorting walls. Dual orthogonal modes can be excited simultaneously from the antenna, and their resonant frequencies can be controlled with the shorting walls. By properly selecting the width of the shorting walls, a left-hand circular polarization operation can be found. Moreover, the polarization sense can be switched to right-hand circular polarization by increasing the number of the shorting walls. A reconfigurable prototype with the ability of electrically switching between left-hand and right-hand circular polarization has also been realized. Experimental results show that the prototype can operate at two different frequencies, and both the operating frequencies have uniform conical-beam radiation patterns with good circular polarization performance. Moreover, the size of the prototype is about 0.47 0 in side length and 0.08 0 in height, where 0 is the free-space wavelength corresponding to the lower operating frequency. The compact size and omnidirectional radiation make the antenna be suitable for the applications of present-day wireless communication systems; besides, the property of dual frequency and dual polarization may be required for the system that uses different frequencies for uplink and downlink communications.

REFERENCES [1] H. Kawakami, G. Sato, and R. Wakabayashi, “Research on circularly polarized conical-beam antennas,” IEEE Antennas Mag., vol. 39, pp. 27–39, Jun. 1997. [2] D. Zhou, R. A. Abd-Alhameed, C. H. See, N. J. McEwan, and P. S. Excell, “New circularly-polarized conical-beam microstrip patch antenna array for short-range communication systems,” Microw. Opt. Technol. Lett., vol. 51, pp. 78–81, Jan. 2009. [3] D. I. Wu, “Omnidirectional circularly-polarized conformal microstrip array for telemetry applications,” in IEEE Antennas Propag. Soc. Int. Symp. Dig., 1995, vol. 2, pp. 998–1001. [4] J. Takada, A. Tanisho, K. Ito, and N. Ando, “Circularly polarised conical beam radial line slot antenna,” Electron. Lett., vol. 30, pp. 1729–1730, Oct. 13, 1994. [5] F. R. Hsiao and K. L. Wong, “Low-profile omnidirectional circularly polarized antenna for WLAN access points,” Microw. Opt. Technol. Lett., vol. 46, pp. 227–231, Aug. 5, 2005. [6] A. Nesic, V. Brankovic, and I. Radnovic, “Circularly polarised printed antenna with conical beam,” Electron. Lett., vol. 34, pp. 1165–1167, Jun. 11, 1998. [7] J. M. Fernandez, J. L. Masa-Campos, and M. Sierra-Perez, “Circularly polarized omnidirectional millimeter wave monopole with parasitic strip elements,” Microw. Opt. Technol. Lett., vol. 49, pp. 664–668, Mar. 5, 2007. [8] J. Huang, “Circularly polarized conical patterns from circular microstrip antenna,” IEEE Trans. Antennas Propag., vol. AP-32, pp. 991–994, Sep. 1984. [9] K. L. Lau and K. M. Luk, “A wideband circularly polarized conicalbeam patch antenna,” IEEE Trans. Antennas Propag., vol. AP-54, pp. 1591–1594, May 2006. [10] H. Nakano, K. Vichien, T. Sugiura, and J. Yamauchi, “Singly-fed patch antenna radiating a circularly polarised conical beam,” Electron. Lett., vol. 26, pp. 638–640, May 10, 1990. [11] H. Nakano, K. Fujimori, and J. Yamauchi, “A low-profile conical beam loop antenna with an electromagnetically coupled feed system,” IEEE Trans. Antennas Propag., vol. AP-48, pp. 1864–1866, Dec. 2000.

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[12] N. J. McEwan, R. A. Abd-Alhameed, E. M. Ibrahim, P. S. Excell, and J. G. Gardiner, “A new design of horizontally polarized and dual-polarized uniplanar conical beam antennas for HIPERLAN,” IEEE Trans. Antennas Propag., vol. AP-51, pp. 229–237, Feb. 2003. [13] S. H. Yeh and K. L. Wong, “A broadband low-profile cylindrical monopole antenna top loaded with a shorted cross patch,” Microw. Opt. Technol. Lett., vol. 32, pp. 186–188, Feb. 5, 2002. [14] C. Delaveaud, P. Leveque, and B. Jecko, “New kind of microstrip antenna: The monopolar wire-patch antenna,” Electron. Lett, vol. 30, pp. 1–2, Jan. 6, 1994. [15] S. H. Chen, J. S. Row, and K. L. Wong, “Reconfigurable square-ring patch antenna with pattern diversity,” IEEE Trans. Antennas Propag., vol. 55, pp. 472–475, Feb. 2007. [16] Y. J. Sung, T. U. Jang, and Y. S. Kim, “A reconfigurable microstrip antenna for switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 14, pp. 534–536, Nov. 2004.

Design and Characterization of 60-GHz Integrated Lens Antennas Fabricated Through Ceramic Stereolithography Ngoc Tinh Nguyen, Nicolas Delhote, Mauro Ettorre, Dominique Baillargeat, Laurent Le Coq, and Ronan Sauleau

Abstract—Three integrated lens antennas made in Alumina and built through ceramic stereolithography are designed, fabricated and characterized experimentally in the 60-GHz band. Linear corrugations are integrated on the lens surface to reduce the effects of multiple internal reflections and improve the antenna performance. The lenses are excited by Alumina-filled WR-15 waveguides with an optimized dielectric impedance matching taper in E-plane. The main characteristics of the first two prototypes with corrugations of variable size are compared to those of a smooth lens without corrugation (third prototype). Experimentally their reflection coefficient is smaller than 10 dB between 55 GHz and 65 GHz, and their radiation characteristics (main beam, side lobe level, cross-polarization level) are very stable versus frequency. In particular, at the center frequency (60 GHz), the total antenna loss (including feed loss) is smaller than 0.9 dB and the radiation efficiency exceeds 80%. Index Terms—Broadband lens, ceramic stereolithography, integrated lens antennas, millimeter wave.

I. INTRODUCTION Two-and-one-half dimensional manufacturing techniques, like surface and volume micromachining of Silicon or Gallium-Arsenide substrates (e.g., [1], [2]), thick resist [3] and soft-polymer [4] photolithography, have been proposed for the fabrication of various kinds of microwave and millimeter wave circuits and antennas. Recently three dimensional (3-D) fabrication processes have emerged as enabling technologies for the design of compact devices Manuscript received September 10, 2009; revised January 18, 2010; accepted January 25, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by HPC resources from GENCIIDRIS (Grant 2009-050779). N. T. Nguyen, M. Ettorre, L. Le Coq, and R. Sauleau are with the Institut d’Electronique et de Télécommunications de Rennes (IETR), UMR CNRS 6164, University of Rennes 1, Rennes, France (e-mail: [email protected]; [email protected]; [email protected]). N. Delhote and D. Baillargeat are with the XLIM laboratory, UMR CNRS 6172, University of Limoges, Limoges, France (e-mail: nicolas.delhote@xlim. fr). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050447

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and interconnects, and for packaging needs. Among those, polymer [5] and ceramic [6], [7] stereolithography, as well as extrusion freeforming methods [8] are really promising since they do not require the use of molds and cutting tools. These are relatively fast and reliable techniques capable of building truly 3-D structures with a high aspect-ratio and accuracy. For instance they have been involved in the fabrication of polymer-based vertical filters and high- cavity resonators [5], periodic arrangements made in various ceramics (Zirconia, BZT, Alumina) such as bandpass filters [7], low-profile electromagnetic bandgap resonator antennas [9], and compact horn antenna arrays [10]. More precisely, stereolithography is an additive layer-by-layer process that allows forming locally solid parts by selectively illuminating polymer photoresists or photoreactive ceramic suspensions. As a consequence it becomes possible to construct all-dielectric 3-D devices with arbitrary complex geometries and/or adjustable refractive index by controlling the volumetric proportions of the composite materials. These concepts have been applied to explore the capabilities of broadband photonic-crystal waveguides [11] and design non-homo-band [6]. geneous monolithic lens antennas in On the other hand, integrated lens antennas (ILAs) are very attractive for a number of millimeter wave applications (e.g. [12] and references therein). Two main categories of ILAs are generally distinguished, namely i) the extended hemispherical ILAs ([12]–[14]), and ii) the shaped ILAs ([15]–[18]). The first ones are mainly used for beam switching and imaging/sensor applications, whereas the second ones are of particular interest for beam shaping systems. In both cases the lens is usually fabricated using computer numerically-controlled milling/lathe machines. This may result in expensive, delicate—and even challenging—tasks, especially when dealing with electrically-small or strongly shaped lenses, e.g., [19], [20]. For all these configurations, new low-cost fabrication methods are attractive alternatives and must be benchmarked. In this frame stereolithography techniques seem to be extremely promising. Therefore the purpose of this communication is to assess the feasibility of ceramic stereolithography for manufacturing ILAs at millimeter waves. As an intermediate step towards the application of such methods for the synthesis of ILAs with arbitrary shape and constitution, we have implemented this technology to manufacture simpler ILAs, namely several synthesized elliptical ILAs [13]. To our best knowledge, this work is one of the first ones studying homogeneous ILAs made by stereolithography and providing experimental results in the 60-GHz band. This communication is organized as follows. The antenna geometry and the corresponding technical choices are explained in Section II-A. The main characteristics of the proposed ILAs are given in Section II-B. The fabrication process and the experimental results are then discussed in Sections III and IV, respectively. Conclusions are finally drawn in Section V.

Q

Ka

II. ANTENNA GEOMETRY AND NUMERICAL RESULTS A. Antenna Geometry The antenna geometry is represented in 3-D view and side view in Fig. 1(a) and 1(b), respectively. It consists of a synthesized elliptical lens made in Alumina; the dielectric characteristics of Alumina have been measured at 10 GHz using a cylindrical resonant cavity: r = 9 0, tan = 5 2 1005 . The reasons why the loss tangent is so small are given in Section III. The same values are chosen at 60 GHz (note that even if the dielectric loss is ten times larger, the impact on the antenna radiation performance will be negligible). The extension length of the lens ( = 4 14 mm) and its diameter ( L = 2 = 20 mm) have been defined so that, in the Geometrical Optics approximation, all incident rays that impinge parallel on the front surface of the lens are

"



L

:

R

:

Fig. 1. Antenna geometry. (a) 3-D view. (b) Side view. (c) Zoom close to the corrugations. The lens is made in Alumina and is covered by linear corrugations serving as a transition layer to reduce the dielectric contrast at the lens interface. The antenna is excited by a rectangular waveguide with an integrated impedance matching taper in E-plane.

collected at the rear focal point [12], [13]. In contrast to the monolithic configuration previously proposed in [6], the ILAs studied here are excited by a separate external open-ended Alumina-filled WR-15 waveguide. This choice enables one to reduce possible mechanical constraints at the perpendicular junction between the lens base and the feed waveguide. The metallic waveguide and flange have been fabricated by electrical discharge machining. An impedance matching taper in E-plane is integrated inside the waveguide to match the antenna over a broad frequency band ( 1 = 10 mm, 2 = 9 6 mm). Due to fabrication and integration issues, the size of the dielectricfilled waveguide section is the same as the standard one (1.9 mm 2 3.8 mm). We have checked that manufacturing errors or possible excitation of higher order modes do not spoil the antenna performance; using a single-mode waveguide as a primary feed would produce slightly narrower beams, but at the expense of challenging fabrication issues due to the very small size of the waveguide. A rounded flange has been fabricated around the lens base ( ange = 2 mm) to facilitate the antenna mounting onto the circular ground plane whose surface has been optically-polished ( gnd = 30 mm, gnd = 5 mm). It is well known that ILAs made in dense materials (like Silicon or MgO) are desirable to favor power transfer from the feed to the lens. Nevertheless the impedance and radiation characteristics of such lenses are substantially distorted due to the excitation of multiple internal reflections, e.g., [12], [21], [22]. These limitations can be partly overcome by reducing the dielectric contrast at the lens interface, using either conventional matching layers (ML) or caps (i.e., quarter wavelength wave transformers [14], [23]), or optimized ones [16]. Such approaches require the use of—at least—two different materials that must be carefully selected (among those available commercially), fabricated and assembled. Moreover, for a number of shaped ILAs, assembling the lens core with the outer shell is even impossible due to peculiar surface curvatures, e.g., [15], [17]–[19]. To overcome these limitations an alternative solution consists in fabricating an effective material at the lens interface, for instance by drilling thin holes [24] or corrugations in the host medium. This leads to fully monolithic antenna configurations. In this work we use linear and corrugations in E-plane [Fig. 1(c)]. Their depth , width

L

L

:

t

t

P

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Fig. 3. Reflection coefficient and maximum directivity computed with the . : reflection coefficient. : directivity. FDTD method for The gray lines represent the S of a lens antenna of same size but without or coated with an ideal quarter wavelength matching corrugation

ILA

layer

(

(



).

)

ML/Free space interface shows that linear corrugations exhibit very good performance over the whole frequency band of interest. B. Antenna Performance

ILA

The antenna described in Section II-A is labeled #1 . Its characteristics have been computed with a homemade FDTD solver [12]. Fig. 3 represents its reflection coefficient S11 and directivity at broadover the whole unside. The antenna is matched S11 < 0 licensed 60-GHz band (57–64 GHz) for short range communications in the US, and its directivity varies between 20.2 dBi and 20.8 dBi. Comparison with the S11 of a lens without corrugation or with an ideal : , tM L : ) quarter wavelength matching layer ("r;M L clearly demonstrates the effectiveness of the linear corrugations to reduce the dielectric contrast at the lens interface. The co-polarization components calculated at the center frequency (60.5 GHz) and at the band edges (57 GHz and 64 GHz) are plotted in Fig. 4 in E- and H-planes. The half-power beamwidth is very stable versus frequency, and the maximum side lobe level does not exceed 016 dB [E-plane, Fig. 4(a)].

(

10 dB)

=30

Fig. 2. Three flat dielectric interfaces illuminated by a plane wave in TE mode. (a) Geometry of the problems. (b), (c) Reflection coefficients for two angles of incidence.  : alumina/free space interface. : alumina/homogeneous : alumina/corrugations/free space interface. The ML/free space interface. ML is a quarter wavelength wave transformer: " " : ,P : . Its thickness is the same as the corrugation depth  = 2" P . (a) Geometry; (b) normal incidence; (c) oblique incidence (15 ).

(4

=

) = 0 7 mm

m

m

m

=30

=

spacing S equal 700  , 300  , and 1200  , respectively. These dimensions have been chosen to obtain a relative effective permittivity close to 3. To compare the performance of such corrugations with a standard ML, let us consider the following canonical problem: an infinite flat dielectric interface between Alumina and free space illuminated by a TE-polarized plane wave [Fig. 2(a)]. This configuration constitutes a local approximation of the lens problem as explained in [25]. The magnitude of the reflection coefficient at this interface is represented in Fig. 2(b) and 2(c) (in solid line) as a function of  [normal incidence, frequency for two angles of incidence: 1  [Fig. 2(c)]. Note that the value of 2 is slightly Fig. 2(b)] and 2 :  beyond which total reflecsmaller than the critical angle c tion occurs [25]. Comparison with an ideal Alumina/Homogeneous

= 15

=0

( = 19 5 )

= 0 70 mm

III. FABRICATION PROCESS Stereolithography is a rapid prototyping process based on a space-resolved laser polymerization, and where objects are built layer by layer [6], [7]. After decomposition of the closed 3D-CAD model into a set a elementary triangles, the object is numerically sliced in layers, and the cross sectional patterns to polymerize in each layer are defined. The specific pattern, including processing parameters (laser power, scanning speed and sequence, focalization) is sent to the automated machine to physically build the object. Fig. 5 represents the experimental set-up used in this work. A thin layer of organic components (liquid photocurable resin (monomer), binders, etc.) loaded with a high percentage (>50%) of high-purity Alumina particles is firstly deposited on the working surface. A blade is used here to precisely control the layer thickness (50  ). Then the object first slice is hardened (polymerized) under UV laser exposure controlled by galvanometric X Y mirrors and according to the pre-defined patterns. Once this first slice is physically fabricated, the working surface goes down along z -axis, and another layer of ceramic suspension is deposited above the previous one by the blade. The exposure process is repeated for this second slice which is thus bonded to the previous

m

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Fig. 6. Synthesized elliptical ILA (ILA ) fabricated by ceramic stereolithography. (a) Antenna prototype (after assembly). (b) Cross-section view of the Alumina lens alone. The depth P , width W and spacing S of the corrugations are the following: 700 m, 300 m, and 1200 m for ILA , and 700 m, 400 m, and 1600 m for ILA . (c) Alumina taper.

This technology has been developed by the Centre de Transfert de Technologies Céramiques of Limoges, France [7], [27] and has an average accuracy close to 100  (after sintering). It allows fabricating arbitrarily-shaped 3-D objects (e.g., enclosures) made in ceramic (Alumina, Zirconia) that would be very difficult, or even impossible, to obtain using standard machining techniques. Such a fabrication process has been already applied to produce inhomogeneous devices like EBG woodpile structures [27] and Luneberg lenses [6]. The two major restrictions are the following: i) the enclosures must be ‘open’ so as to remove the non-polymerized paste during the cleaning step of the process, ii) only a single material can be used per fabrication; if multimaterial lenses are required, the different parts must be fabricated separately and then assembled.

m

Fig. 4. Co-polarization components computed at the center frequency (60.5 GHz) and at the band edges (57 GHz and 64 GHz). (a) E-plane. (b) H-plane. The cross-polarization level (not represented) is lower than 40 dB. : 57 GHz and 60.5 GHz. : 64 GHz.

0

IV. EXPERIMENTAL RESULTS Three antenna prototypes have been manufactured using the fabrication process described above. The lenses are stratified in height. The first one [ #1 , Fig. 6(a)] and second one #2 have corrugations of different size: their widths W equal 300  for #1 and 400  for #2 (fabricating thinner corrugations would be challenging and not reliable since the laser spot diameter is 150  ). The other corrugation dimensions are given in Fig. 6; they have been optimized to minimize the reflection coefficient at the lens interface, as already illustrated in Fig. 2. The third lens antenna #3 has a smooth surface, i.e. no corrugation. The Alumina taper has been fabricated using the same technology [Fig. 6(c)]. Considering two different corrugation widths and depths allows assessing the fabrication limitations in terms of resolution and mimimum size of small 3-D objects, whereas comparing ILA configurations with and without corrugations enables one to highlight the impact of these corrugations upon the antenna performance. Here all lenses have the same external diameter L and total height R L , and are fed by the same impedance matching taper. The measured reflection coefficients of the three lens antennas are represented in Fig. 7. In all cases it is smaller than 010 dB from 55 GHz to 65 GHz. The disagreement between experiments and simulations (Fig. 3 for #1 ) is attributed to several manufacturing issues: i) the matching taper and the lens base are slightly bent due to mechanical constraints and internal stress; this is clearly illustrated in Fig. 6(b)

ILA m ILA

(ILA ) m ILA m (ILA )

Fig. 5. Experimental set-up for fabrication through stereolithography.

layer by laser polymerization. The same process is repeated until the final device is entirely built. Finally the polymerized solid part is removed from the non-polymerized liquid. The recovered object is cleaned, debinded and sintered to obtain its final dimensions and density. The purity of the base Alumina powder and the very high density of the final object (more than 98% of the theoretical value) are the two main reasons why very low loss can be obtained [26], [27].

( + ) ILA

( )

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Fig. 7. Measured reflection coefficients of the three lens antennas. : ILA (300 m-thick corrugations). : ILA (400 m-thick corruga: ILA (no corrugation). tions).

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Fig. 9. Measured gain of the three lens antennas. : ILA (300 m-thick : ILA (400 m-thick corrugations). : ILA (no corrugations). corrugation). The lines with symbols represent the theoretical directivities of (  ), ILA (  ), and ILA (  ), respectively. ILA

asymmetry observed on the measured beams probably comes from the lens deformation mentioned above. Additional experimental results have confirmed that these patterns are very stable between 55 GHz and 65 GHz. The half power beamwidth is not the same in E- and H-planes because of the rectangular waveguide excitation. The antenna gains are plotted in Fig. 9. They have been measured with the comparison method using a 20-dBi standard gain horn in V -band. As expected this figure confirms that the gain variations versus frequency are smaller when using fine corrugations (ILA#1 ). We can also notice that there are some frequency points (e.g., at 57 or 61 GHz) where the gain of the smooth lens (ILA#3 ) is higher than the one of the corrugated lenses ILA#1 and ILA#2 . This is due to the excitation of resonant modes that increase the Q-factor of the antenna [28], thus its directivity and gain since the Alumina used here has a very low loss tangent. The phenomena would not appear where higher loss material. At 60 GHz the gain and directivity of ILA#1 equal 19 dBi and 19.9 dBi, respectively. The 0.9-dB loss can be decomposed as follows (Fig. 7): 0.2-dB metallic loss (FDTD simulations), and less than 0.1-dB return loss, 0.6-dB reflection losses (GO/PO simulations [15]). This reflection loss value is similar to the ones reported in [23], [29]. Fig. 9 also shows that the gain drop (compared to the theoretical directivity) varies experimentally between 0.7 dB (at 61 GHz) and 2.2 dB (at 63 GHz), corresponding to antenna radiation efficiencies comprised between 60% and 85% over a 10-GHz frequency band. V. CONCLUSIONS Fig. 8. Radiation patterns of ILA at 60 GHz. (a) E-plane. (b) H-plane. (solid black line): measured co-polarization component. : computed (solid gray line): measured cross-polarization co-polarization component. component.

where we can notice small deformations at the left hand side of the lens base, ii) the taper section is slightly smaller than the waveguide cross-section (3.8 mm 2 1.9 mm) to facilitate its insertion into the waveguide. Accumulation of these two defects creates unavoidable air gaps and impedance discontinuities explaining very likely the ripples observed in Fig. 7. The radiation patterns measured at 60 GHz are represented in Fig. 8. They are in good agreement with the FDTD simulations. The slight

An attractive ceramic-based stereolithography process has been developed to build 3-D monolithic all-dielectric devices at millimeter waves. Its main features and limitations have been assessed through the fabrication and experimental characterization of several 60-GHz integrated lens antennas (ILAs) made in very low-loss Alumina. As one of the prime objectives is to validate all technological steps, simple ILAs have been designed, namely synthesized elliptical lenses. The latter are fed by WR-15 metallic waveguides filled with Alumina. Optimized impedance matching tapers in E-plane are used to match the ILAs over the whole unlicensed 60-GHz band. Corrugations of various sizes have been designed to reduce the dielectric contrast at the Alumina/free space interface. Three prototypes have been manufactured and characterized in impedance and radiation. Measurements have shown that the antenna performances are very

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stable over the 55–65 GHz band. The agreement between the experimental and numerical results is very satisfactory despite some fabrication issues (mechanical stress observed along the dielectric tapers and lens bases). In particular, for 19-dBi gain antennas, the total amount of loss at 60 GHz is lower than 0.9 dB, corresponding to a radiation efficiency of 80%.

REFERENCES [1] L. P. B. Katehi, J. F. Harvey, and E. Brown, “MEMS and Si micromachined circuits for high-frequency applications,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 858–866, Mar. 2002. [2] M. Elwenspoek and H. Jansen, Silicon Micromachining. Cambridge, U.K.: Cambridge Univ. Press, 1998. [3] W. Y. Liu, D. P. Steenson, and M. B. Steer, “Membrane-supported CPW with mounted actives devices,” IEEE Microw. Wireless Comp. Lett., vol. 11, no. 4, pp. 167–169, Apr. 2001. [4] N. Tiercelin, P. Coquet, R. Sauleau, V. Senez, and H. Fujita, “Polydimethylsiloxane membranes for millimeter-wave planar ultra flexible antennas,” J. Micromec. Microeng., vol. 16, pp. 2389–2395, Sep. 2006. [5] B. Liu, X. Gong, and W. J. Chappell, “Applications of layer-by-layer polymer stereolithography for three-dimensional high-frequency components,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2567–2575, Nov. 2004. [6] K. F. Brakora, J. Halloran, and K. Sarabandi, “Design of 3-D monolithic MMW antennas using ceramic stereolithography,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 790–797, Mar. 2007. [7] N. Delhote, D. Baillargeat, S. Verdeyme, S. Delage, and C. Chaput, “Narrow Ka bandpass filters made of high permittivity ceramic by layer-by-layer polymer stereolithography,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 3, pp. 548–554, Mar. 2007. [8] X. Lu, Y. Lee, S. Yang, Y. Hao, R. Ubic, J. R. G. Evans, and C. G. Parini, “Fabrication of electromagnetic crystals by extrusion freeforming,” Metamaterials, vol. 2, pp. 36–44, 2008. [9] Y. Lee, Y. Hao, S. Yang, C. G. Parini, X. Lu, and J. R. G. Evans, “Directive millimetre-wave antennas using free-formed ceramic metamaterials in planar and cylindrical forms,” presented at the IEEE AP-S Int. Symp., San Diego, CA, Jul. 2008. [10] L. Schulwitz and A. Mortazawi, “A compact millimeter-wave horn antenna array fabricated through layer-by-layer stereolithography,” presented at the IEEE AP-S Int. Symp., San Diego, CA, Jul. 2008. [11] K. F. Brakora and K. Sarabandi, “Integration of single-mode photonic crystal waveguides to monolithic MMW subsystems constructed using ceramic stereolithography,” presented at the IEEE AP-S Int. Symp., Honolulu, HI, Jun. 2007. [12] G. Godi, R. Sauleau, and D. Thouroude, “Performance of reduced size substrate lens antennas for millimeter-wave communications,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1278–1286, Apr. 2005. [13] D. F. Filipovic, S. S. Gearhart, and G. M. Rebeiz, “Double-slot antennas on extended hemispherical and elliptical silicon dielectric lenses,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 10, pp. 1738–1749, Oct. 1993. [14] N. T. Nguyen, R. Sauleau, and C. J. Martínez Pérez, “Very broadband extended hemispherical lenses: Role of matching layers for bandwidth enlargement,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1907–1913, Jul. 2009. [15] G. Godi, R. Sauleau, L. Le Coq, and D. Thouroude, “Design and optimization of three dimensional integrated lens antennas with genetic algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 770–775, Mar. 2007. [16] J. R. Costa, C. A. Fernandes, G. Godi, R. Sauleau, L. Le Coq, and H. Legay, “Compact Ka-band lens antennas for LEO satellites,” IEEE Trans. Antennas Propag., vol. 56, no. 51, pp. 251–1258, May 2008. [17] B. Barès, R. Sauleau, L. Le Coq, and K. Mahdjoubi, “A new accurate design method for millimeter-wave homogeneous dielectric substrate lens antennas of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1069–1082, Mar. 2005. [18] R. Sauleau and B. Barès, “A complete procedure for the design and optimization of arbitrarily-shaped integrated lens antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1122–1133, Apr. 2006.

[19] B. Barès and R. Sauleau, “Electrically-small shaped integrated lens antennas: A study of feasibility in Q-band,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1038–1044, Apr. 2007. [20] A. Rolland, R. Sauleau, M. Drissi, and L. Le Coq, “Synthesis of reduced-size smooth-walled conical horns using BoR-FDTD and genetic algorithm,” IEEE Trans. Antennas Propag., submitted for publication. [21] A. V. Boriskin, A. Rolland, R. Sauleau, and A. I. Nosich, “Assessment of FDTD accuracy in the compact hemielliptic dielectric lens antenna analysis,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 758–764, Mar. 2008. [22] A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Small hemielliptic dielectric lens antenna analysis in 2-D: Boundary integral equations versus geometrical and physical optics,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 485–492, Feb. 2008. [23] M. J. M. van der Vorst, P. J. I. de Maagt, A. Neto, A. L. Reynolds, R. M. Heeres, W. Luingue, and M. H. A. J. Herben, “Effect of internal reflections on the radiation properties and input impedance of integrated lens antennas—Comparison between theory and measurements,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 6, pp. 1118–1125, Jun. 2001. [24] K. Sato and H. Ujiie, “A plate Luneberg lens with the permittivity distribution controlled by hole density,” Electron. Commun. Jpn. (Part I: Commun.), vol. 85, no. 9, pp. 1–12, Apr. 2002. [25] D. Pasqualini and S. Maci, “High-frequency analysis of integrated dielectric lens antennas,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 840–847, Mar. 2004. [26] N. M. Alforda and S. J. Penn, “Sintered alumina with low dielectric loss,” J. Appl. Phys., vol. 80, no. 10, pp. 5895–5898, Nov. 1996. [27] T. Chartier, C. Duterte, N. Delhote, D. Baillargeat, S. Verdeyme, C. Delage, and C. Chaput, “Fabrication of millimetre wave components via ceramic stereo- and microstereolithography processes,” J. Am. Ceram. Soc., vol. 91, no. 8, pp. 2469–2474, 2008. [28] A. Rolland, M. Ettorre, L. Le Coq, and R. Sauleau, “Axis-symmetric resonant shaped dielectric lens antenna with improved directivity in -band: Performance and comparison with an extended hemispherical lens,” IEEE Trans. Antennas Propag., submitted for publication. [29] X. Wu, G. V. Eleftheriades, and T. E. van Deventer-Perkins, “Design and characterization of single- and multiple-beam mm-wave circularly polarized substrate lens antennas for wireless communications,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 431–441, Mar. 2001.

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Millimeter Wave Circularly Polarized Fresnel Reflector for On-Board Radar on Rescue Helicopters Karim Mazouni, J. Lanteri, N. Yonemoto, J.-Y. Dauvignac, Ch. Pichot, and C. Migliaccio

Abstract—The increasing use of millimeter waves for civil radar application, e.g., for automotive or helicopter obstacle detection- requires the development of high gain and low cost antennas in compact form. With this aim, a Fresnel reflector with circular polarization over 5 GHz bandwidth (76–81 GHz) has been designed, fabricated and measured. The gain remains higher than 32 dBi with a peak value of 35 dBi at 79 GHz. For obtaining this performance, specific patches have been designed. They are based on circular rings, rectangular and offset patches. Each of them converts an incident linear electric field into a circular polarized one.

Fig. 1. Ray-tracing – Definition of the Fresnel zones for FZR.

Index Terms—Circular polarization, Fresnel reflector, millimeter waves.

I. INTRODUCTION Millimeter wave radar applications have been of increasing interest over the last decade. The 77 GHz automotive radar is probably one of the most common applications although W-band radars have also demonstrated their capabilities to detect power lines. A first measurement campaign has been conducted in 2006 [1] using successively a vertically and a horizontally polarized antenna. An important result collected during these in-flight measurements is that the measured responses of power lines are quite similar for both antennas. Hence, a circularly polarized antenna can be used for improving the detection provided that vertical and horizontal components from the circular polarization are separated after reception. In addition, a directive and compact antenna is required for an on-board system and printed reflectors are good candidates for this application. There are several ways for achieving circular polarization: one can use a circularly polarized primary feed and a reflector having circular and symmetrical cells, such as disks or rings. In this case, the incident circular polarization is re-radiated by the reflector [2]. This design is robust but the cost of a circularly polarized primary feed, especially in the W-band, is quite high. In addition, designing a reflector with ring-shaped cells in the W-band for covering 360 phase, would lead to a large number of cells with dimensions inferior to 100  . Moreover, they could not be realized with classical printed circuit fabrication technique although it is a key factor for keeping a low-cost antenna. In order to overcome this problem a second technique was chosen. It consists in having a linearly polarized primary feed and non-symmetrical reflector cells. The latest are designed for converting a linearly polarized wave onto a circularly polarized one. There is an additional advantage for this solution: an open-ended waveguide can be chosen as primary feed, which drastically reduces the shadowing effect.

m

Manuscript received July 09, 2009; revised December 26, 2009; accepted February 04, 2010. Date of of publication May 18, 2010; date of current version August 05, 2010. K. Mazouni, J. Lanteri, J.-Y. Dauvignac, C. Pichot, and C. Migliaccio are with the Electronics, Antennas and Telecommunications Laboratory (LEAT), University of Nice-Sophia Antipolis, CNRS, Valbonne 06560, France, (e-mail: [email protected]). N. Yonemoto is with the Electronic Navigation Research Institute, Chofu, Tokyo 180-0012, Japan. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050431

In order to simplify the reflector design, the printed reflector is of Fresnel type [3]. It combines a phase correction of 45 in its center and 90 at its periphery when the space becomes too small for maintaining a high value of the order of compensation, P . The design frequency band is 75–81 GHz. Section II describes the reflector design rules, Section III the simulations and the elementary cells printed on the reflector, and Section IV the measurements. II. REFLECTOR DESIGN A. Fresnel Reflector Design The printed Fresnel reflector [4], [5] is a quasi-optical antenna whose design is derived from the Fresnel equation giving the radius of each phase compensating zone (Fig. 1)

rn =

n P

2

+ 2nf P

(1)

where  is the free space wavelength; f is the focal length; n is the index of phase compensating zone; and P is the order of phase compensation. The design of a Fresnel reflector consists in dividing periodically the reflector surface into zones. The period is P . Each zone corresponds to a phase compensation 'k

'k = 2

(k 0 1) with k 2 [1; P ]: P

(2)

Each Fresnel zone is then divided into square elementary cells containing the patches designed to compensate for the phase of 'k . Within a given Fresnel zone, all patches are identical. Hence, the number of different patches required for making a Fresnel reflector is equal to P . This greatly simplifies the reflector design, especially if the design of one element is time consuming as in our case because of the multiple features of the elementary cell. B. Elementary Cell Design Since we aim to generate a circular polarization from a linearly polarized wave, the elementary cell requirement is twofold: Converting the polarization and achieving the Fresnel phase compensation. Let us consider a plane wave propagating along Oz whose electric field is described according to (3)

~e = E0x cos(!t + '1 )u~ x + E0y sin(!t + '2 )u~ y :

0018-926X/$26.00 © 2010 IEEE

(3)

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TABLE I PATCH SHAPES FOR EACH FRESNEL ZONE

Fig. 2. C-cell with polarization conversion.

Without any constraints on E0x , E0y and 18 = '1 0 '2 , the polarization of ~e is elliptical. The circular polarization is then obtained when

E0x = E0y and 18 = '1 0 '2 = 2 [modulo ]:

(4)

In our case, the incident field is linearly polarized (Einc ) and the polarization conversion can be achieved by choosing an elementary cell having different responses whether the incident field is polarized along ~ux or ~uy (Fig. 2). In the C-cell described in Fig. 2, the gap in the ring creates a dissymmetry in the phase responses '1 and '2 to ~ ux and ~uy polarizations respectively [3]. Without patch adjustments, the reflected wave is elliptically polarized but the dimensions can be adjusted for having a 18 of =2. Furthermore, placing the incident field at 45 from the cell ensures that E0x = E0y . Simultaneously, dimensions have to be optimized for having the Fresnel phase compensation. Below are summarized the phase requirements for the elementary cells:

'1k = 2 (kP01) with k 2 [1; P ]: '2k 0 '1k = 2

(5)

III. NUMERICAL SIMULATIONS AND REFLECTOR DESIGN In order to find the patch dimensions for fulfilling the phase requirements, numerical simulations have been conducted using Ansoft-HFSS according to the following steps. Step 1) Dimensions are obtained from a set of simulations and geometrical adjustments of the various patches using the periodic structure module with magnetic and electric walls (perfect E – perfect H) as described in [2]. The responses ux and ~uy are successively tested. to polarizations along ~ Step 2) The circular polarization of the patch is checked with the Master-Slave simulation while having a linearly polarized electric field tilted of 45 with respect to the walls. Using the perfect E – perfect H approach, a uniform plane wave impinges at normal incidence on the patch. The walls act as if this patch were placed in an infinite periodic array. The electric field is forced to be perpendicular to the electric walls because of the boundary conditions. Nevertheless, in order to check the polarization conversion, we need to tilt the incident electric field of 45 with respect to the walls. Therefore, we use the Master-Slave function. From the reflected field, we extract the values of 18 and of the axial ratio (AR) defined as the ratio between the maximum reflected field value and the minimum one

AR = kk~e~emaxkk : min

(6)

The elementary cells used in the Fresnel reflector are described in Table I. In order to understand how their shapes are determined, let

TABLE II SIMULATED AR AND 18 PATCH VALUES AT 78 GHZ

us first examine the C-cell behavior. As described in [3], the inner and outer ring diameters are the design parameters for finding the Fresnel phase compensation value ('1k ). The gap height is used to adjust the value of 18 to 90 . In principle, the C-cell should cover all the Fresnel zones but constraints when using the classical printed fabrication circuit technique lead to adopt the following strategy for the design. The minimum dimension within a patch is 100 m and the C-cell is used only for the 90 zone. Other patch shapes have been used for covering the missing Fresnel zones. Therefore, as shown in Table I, the patches are divided into three groups: • Group 1: C-cell-based patches (zones 0 , 90 , 135 ), • Group 2: Rectangular-based patches (45 , 180 , 225 ), • Group 3: Offset patches (270 , 315 ). Elements of Group 1 are obtained by successive modifications of the C-cell, when inserting more gaps and – or rectangular patches. Elements of Group 2 are derived from rectangular patches using the same kind of modifications as for C-cells. Rectangular patches are well known for being sensitive to the incident electric field polarization. This advantage has been used for folded reflectarrays [6]. Elements of Group 3 are, to our knowledge, the first examples of offset patches (offset patches) for printed reflector purpose. Although, they were used here only for two zones, recent simulations have shown that they can cover a larger phase range. The basic idea is to use the dissymmetry of the patch for obtaining the value of 18, and the offset for increasing the Fresnel phase compensation. Table II sums up the values of AR and 18 obtained using simulations with the Master-Slave method. The maximum phase deviation is 4 and the worst AR is 0.6 dB (zone 45 ). Frequency simulations were conducted accordingly within the [74–81 GHz] band. The elementary cell bandwidth, obtained in simulations for each patch, is at least 3 GHz while considering an axial ratio value less than 1 dB and 5 GHz for a value inferior to 3 dB.

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Fig. 5. Measured

Fig. 3. Photo of the Fresnel reflector with corresponding orders of compensa, . tion,

D = 156 mm f=D = 0:5

'

between 74 and 80 GHz (not normalized).

TABLE III EXTRACTED

AR AND 18 VALUES.

(18) are extracted from the measured amplitude and phase obtained while the transmitting horn is rotating in the xOy plane: this measurement will be called 'scan in the communication. Results are plotted in Fig. 5

dB = amplitudemax dB 0 amplitudemin dB and 18 = phase(' = 0) 0 phase(' = 90) AR

Fig. 4. Enlargement of the reflector centre

(P = 8).

AR and 18 extractions over frequency are shown in Table III.

IV. FABRICATION AND MEASUREMENTS The Fresnel reflector was fabricated on Duroid substrate of thickness 254  and relative permittivity 2.2. The space dedicated to each zone, defined by rn+1 0 rn , decreases as n increases. As a consequence, the phase compensation has to be reduced on the reflector periphery. in the centre, Therefore, the order of phase compensation is P in the mid-zone of the reflector and has to be reduced to 2 on P the border as shown in the Fig. 3 as well as the spherical coordinates system ( and ') used for the measurements. Fig. 4 shows an enlarged view of the center explaining how the different patches are arranged among the reflector. Several parameters have to be taken into account for the choice and design of the primary feed: Aperture blockage and spillover and taper efficiencies. A small size primary feed ensures a low aperture blockage. Therefore, a standard WR-10 open-ended waveguide is chosen. According to its radiation pattern characteristics, it can be shown by using [9] that is the best compromise between taper and aperture efficiencies is obtained for a focal length to diameter ratio of 0.5. The theoretical total efficiency (including taper and amplitude efficiencies) is 87%. Furthermore, the open-ended waveguide is tilted by an angle of 45 from the x-axis in order to have the correct incidence of the electric field on the reflector. Measurements were conducted in an anechoic chamber. The measurement system consists of a linearly polarized standard pyramidal horn antenna in transmission that faces away the antenna under test at a distance of 4.6 m [7]. The Axial ratio (AR) and phase difference

m

=4

(

(7)

)

=8

The axial ratio remains inferior to 3 dB over the investigated bandwidth with top values at 74 and 78 GHz. The latter corresponds to the central frequency of the antenna. Accordingly, the phase difference 18 remains equal to  6 (1 to 13 ). In a second step, the on-axis focusing property of the reflector has to be verified. This can be investigated by using the same measurement system with scans in the xOz or yOz planes. Due to the symmetry of the reflector, both scans should give the same results provided that the radiation pattern of the primary feed is the same in both planes which is not totally correct. The measurements are made in the yOz plane with the transmitting horn having a vertically polarized electric field plane. They are called scans and the results are plotted in Fig. 6. A slight offset of 0  is observed in the main lobe (maximum radiation direction) but one can see clearly that the antenna is directive with a 3 dB aperture of 1.5 and side lobe levels below 14 dB over the investigated frequency bandwidth. The slight offset is probably due to some alignment problems. In order to verify the circular polarization behavior in the main lobe, 'scans have been performed for different values of  . The axial ratio, AR, has been calculated for each value of  according to (7). Results are plotted in Fig. 7. AR remains inferior to 3 dB in the main lobe between 74 and 80 GHz. Finally, gain measurements were conducted up to 82 GHz. The gain values have been obtained from two different measurements. — The standard horn transmits a vertically polarized electric field, the gain is measured Gver ; — The standard horn transmits a horizontally polarized electric field, the gain is measured Ghor . The total gain, plotted in Fig. 8, is the sum of both values.

90

1

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Fig. 6. Measured 

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between 74 and 80 GHz (normalized).

Fig. 9. Measured reflection coefficient and comparison with an open-ended waveguide.

V. CONCLUSION

Fig. 7. Measured 

A wide-band (76–81 GHz) circularly polarized Fresnel reflector combining 8 and 4 correcting zones has been designed and measured. The gain remains higher than 32 dBi with a peak value of 35 dBi at 79 GHz. For this purpose, specific patches have been designed. They are based on circular rings, rectangular and offset patches. Each of them converts an incident linear electric field into a circularly polarized one. In the near future, the polarization separation in the primary feed will be included and on-ground measurements carried out for comparing the performance of the new reflector with previous ones [1] for power line detection application. In a second step, in-flight measurements will be conducted.

between 74 and 80 GHz (normalized).

REFERENCES

Fig. 8. Measured gain of the Fresnel reflector over frequency.

As expected from the scans , the gain of the antenna is high with a maximum value of 35 dBi at 79 GHz. The 3 dB gain bandwidth meets the requirements (76 to 81.5 GHz). The aperture efficiency has been calculated 

= Gmeasured 4S 

(8)

where S is the surface of the antenna. The maximum value is 27% obtained at 79 GHz. It remains higher than 15% within 76–81 GHz. This relatively low value compared to previous work [3] is balanced by the 5 GHz bandwidth achieved in circular polarization with the present Fresnel reflector antenna (FRA). The reflection coefficient measurements are plotted in Fig. 9. These results with oscillations coming from standing waves occurring between the reflector and the primary feed are typical for such reflector antenna with relatively low focal length to diameter ratio. The average value of the reflection coefficient is about 017 dB. For comparison, a WR-10 open-ended waveguide similar to the one used for the primary feed has been also measured with an average level of 013 dB. Considering a maximal value of 015 dB for the FRA, with about 3% of the input power being lost due to mismatch, remains very acceptable.

[1] C. Migliaccio et al., “Investigation of millimeter wave radar antennas for power lines detection on the ground and on helicopter flight,” presented at the Proc. 9th Int. Conf. on Control, Automation, Robotics and Vision (ICCARV2006), Singapore, Dec. 5–8, 2006. [2] C. Han, C. Rodenbeck, J. Huang, and K. Chang, “A C/Ka dual frequency dual layer circularly polarized reflect array antenna with microstrip rings elements,” IEEE Trans. Antennas Propag., vol. 52, pp. 2871–2876, Nov. 2004. [3] B. D. Nguyen, J. Lanteri, J.-Y. Dauvignac, C. Pichot, and C. Migliaccio, “94 GHz folded Fresnel reflector using C-patch elements,” IEEE Trans. Antennas Propag., vol. 55, pp. 3373–3381, Nov. 2008. [4] B. Huder and W. Menzel, “Flat printed reflector antenna for mm-wave application,” Electron. Lett., vol. 24, pp. 318–319, Mar. 1988. [5] Y. J. Guo and S. K. Barton, “Phase correcting zonal reflector incorporating rings,” IEEE Trans. Antennas Propag., vol. 43, pp. 350–354, Apr. 1995. [6] W. Menzel, D. Pilz, and R. Leberer, “A 77 GHz FM/CW radar front-end with a low-profile low-loss printed antenna,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2237–2241, Dec. 1999. [7] P. N. Betjes, “Error analysis of circular-polarization components synthesized from linearly polarized measurements,” presented at the Antenna Measurement Technique Association Conf. (AMTA), Denver, CO, 2001. [8] B. Y. Toh, R. Cahill, and V. Fusco, “Understanding and measuring circular polarization,” IEEE Trans. Edu., vol. 46, no. 3, pp. 313–319, Aug. 2003. [9] D. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarray,” IEEE Trans. Antennas Propag., vol. 45, pp. 287–296, Feb. 1997.

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PCB Slot Based Transformers to Avoid Common-Mode Resonances in Connected Arrays of Dipoles D. Cavallo, A. Neto, and G. Gerini

Abstract—The scanning performances of connected arrays are degraded by the excitation of common-mode resonances that are compatible with balanced feeding lines. Here, a strategy to avoid these resonances is outlined. The strategy involves feeding the dipoles via printed circuit board (PCB) based transformers and significantly reducing the feeding periods in the direction of the dipoles. The number of transmit/receive (T/R) modules does not have to be increased as a consequence of the increased sampling of the dipoles. Full wave simulations that validate the procedure are presented.

Fig. 1. Geometry of a two dimensional array of dipoles fed by CPS lines. The periodicity along x and y is d and d , respectively. The length of the vertical feeding lines is l .

Index Terms—Microstrip components, phased arrays, slot fed antennnas, ultrawideband antennas.

I. INTRODUCTION The realization of wideband, wide-scanning angle, phased arrays with good polarization performance has been recently receiving increasing attention. Such characteristics are required in a number of applications, such as multi-function defense and security radars in X-band and lower, communication applications in Ku-band, earth based deep space investigation (e.g., square kilometer array, [1], [2]), and satellite based sub-mm wave instruments (e.g., SPICA, [4]). Although tapered slot antennas have broad bandwidth, they are known to produce high cross-polarized components in the radiation patterns, especially in the planes at 45 to the planes containing the slots, [5]. Therefore, a relatively complicated antenna arrangement is necessary to reduce these levels in dual-polarization applications. On the other hand, conventional phased arrays based on printed radiating elements can achieve only moderate bandwidths (25%), [6]–[8]. There is a recent trend aiming at reducing cross-polarization by making arrays of long dipoles or slots periodically fed, with all the radiating parts lying in the array plane according to Wheeler’s continuous current sheet ([9]): these arrays are indicated as connected arrays of slot or dipoles. Connected arrays offer wide bandwidth, as shown in [10][11], while maintaining low cross-polarization (X-pol) levels. Scanning performance of planar connected arrays of dipoles and slots were compared in [12], where an investigation based on the Green’s function of the two structures was presented. The analysis showed that dipole arrays can achieve broader bandwidth than slots for wide scanning (up to 45 ). A conformal versions of connected array has been recently proposed and investigated, [3]. Despite the potentials, the practical implementation of the feeding network in a connected array of dipoles is a difficult problem. Balanced transmission lines should be used to feed the elements. However these lines can support both differential and common-mode propagation. This Manuscript received June 23, 2009; revised November 27, 2009, February 17, 2010; accepted February 17, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. D. Cavallo and G. Gerini are with TNO, Defense, Security and Safety, 2597 AK The Hague, Netherlands and also with the Faculty of Electrical Engineering, Eindhoven University of Technology, 5612 AZ Eindhoven, Netherlands (e-mail: [email protected]; [email protected]). A. Neto is with the Telecommunication Department, Delft University of Technology, Mekelweg 4, 2628 CD Delft, Netherlands (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.2010.2050430

latter is undesired, since it can give rise to strong resonances that ruin the array performance. An analysis of the effects of these resonances on the efficiency of a connected array was presented in [13]. The same type of resonances were also observed and investigated for arrays of differentially-fed tapered slot antennas in [14]. When the array periodicity is in the order of half wavelength, standard baluns or common-mode rejection circuits are not effective as will be shown in the following. A low cost, printed circuit board (PCB) based solution to the common-mode resonances, without resorting to active components or MMIC technology, is proposed in this communication. It is based on reducing the lengths of the resonating lines in order to shift the common-mode resonances to higher frequencies, so outside the wide operational bandwidths of the arrays. This effect is achieved by applying a denser sampling of the array, but still maintaining the same numbers of T/R modules by means of power dividers. A wideband transition between co-planar strip (CPS) lines has been designed, based on microstrip-to-slot aperture coupling, [15], to further reduce the length of continuous current paths. The The same transition can be used as a balun, when coupling a CPS line to a microstrip (MS). II. RESONANCES TYPICAL OF CONNECTED ARRAYS Let us consider the simplified case of an infinite two-dimensional array of dipoles with periodicity dx and dy , as shown in Fig. 1. Backing reflectors diminish the intrinsic bandwidth of the dipoles (theoretically infinite). For sake of generality, the array without backing reflector has been studied in order to highlight only the frequency dependence introduced by the feeding network. The array elements are fed by CPS lines, whose length is equal to l. Three types of resonant effects may occur in these, otherwise broadband, connected arrays: 1) grating lobes; 2) phase matching between Floquet waves (FW) and guided waves; 3) common-mode resonances. The first type resonance is associated with the appearance into the visible region of an higher order FW and is typical of all arrays, connected and not connected. The second type resonance only occurs in connected arrays, since in an infinitely extended dipole structure, guided waves can propagate unattenuated along the longitudinal direction. As described in Section III.B of [12], when the dipole is assumed to be in free space, these waves are represented in the spectral Green’s function (GF) as a couple of poles in kx = 6k0 . The first two types of resonance are not dependent on the feeding lines, thus they can be taken into account in an ideal model that does not include transmission lines.

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000

00

Fig. 2. Real ( ) and imaginary ( ) part of the active input impedance of an elementary cell in infinite array environment, when the array is scanning towards  = 45 and  = 45 . The periods are d = d = 15 mm and the length on the vertical lines is l = 7:5 mm.

Fig. 3. Vector surface current distribution on an elementary dipole in infinite array environment at 10 GHz. the array is scanning towards  = 45 and  = 45 . The periods are d = d = 15 mm and the length on the vertical lines is l = 7:5 mm. Common-mode propagation is visible in the vertical feeding lines.

A. Common-Mode Resonances If differential lines are included in the model, their length is critical since they may induce other resonances due to common-mode propagation. The shorter the feeding lines, the higher the associated resonant frequency. However, typically connected arrays involve the presence of a backing reflector. Accordingly, the transmission line lengths are in the order of a quarter of the free space wavelength, in order to reach the ground-plane level, where loads or source circuits are located. As an example, in the most standard design situation in which the periodicity of the array is about half wavelength, and the vertical lines are a quarter wavelength, two neighboring feeding lines together with the electrical connection via the dipole constitute a wavelength continuous electric path (dx +2l = ) that gives rise to a strongly cross-polarizing standing wave. The active input impedance for an array periodicity of dx = dy = 15 mm, and assuming l = 7:5 mm, is shown in Fig. 2. In addition to the grating lobe and the guided pole resonances (at about 14.5 and 13.5 GHz, respectively), a peak of the impedance appears at 10 GHz. By observing the vector current distribution, the resonance can be recognized as associated with common-mode propagation in the CPS lines (Fig. 3). According to the third definition of cross polarization by Ludwig, [16], common-mode currents along z radiate cross-polarized fields when scanning on the diagonal plane ( = 45 . In Fig. 4, the ratio between co-polarized and cross-polarized fields rapidly increases in proximity of the resonance at 10 GHz. Therefore, to ensure low X-pol level, the length of the path 2l + dx should be significantly shorter than a wavelength, so that the common-mode resonances

Fig. 4. Cross-polarization level as a function of frequency when scanning towards  = 45 and  = 45 . The periods are d = d = 15 mm and the length on the vertical lines is l = 7:5 mm.

Fig. 5. Layout of the CPS/GCPS transformer. The thickness of two dielectric substrates is h and the permittivity is " .

appear at higher frequencies, outside the operational bandwidth of the array. Note that the resonance depicted in Fig. 3 is the one associated with common mode only and typically occurs at the lowest frequency. Other resonances may appear at higher frequencies, which involve combinations of common and differential modes resulting in unbalanced currents. III. DESIGN OF PCB TRANSFORMERS The path of the common-mode current can be shortened by introducing a series transformer that only allows the passage of differential currents and constitutes an open circuit for the common mode. To realize such a component, one can resort to completely planar slot coupling between microstrip lines as in [17]. A. CPS/GCPS Transformer A schematic view of the component is shown in Fig. 5 where the ground plane on which the slot is etched is assumed to be of infinite extent along x. The component is divided in two parts separated by the ground plane. The part at z = h, here in after the primary circuit, comprises a transition from CPS lines to grounded CPS (GCPS) lines, then a power divider that splits the circuit in two microstrip-like equal halves, which are eventually reconnected in correspondence of a coupling slot. The secondary circuit is the same as the primary, but mirrored with respect to the slot. The initial input from the CPS lines can be associated with a differential-mode or a common-mode type of current. In both cases the equivalent detailed circuits are the same as in Fig. 6. However, from Fig. 6 it is apparent that the common-mode input corresponds to a zero of electric current in correspondence of the

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On the contrary, when the slot loading is comparable to the connected array loading, a significant part of the power received by the connected array is re-radiated by the slot before being transferred to the secondary circuit. This limits the useful bandwidth of the transition. Since the condition for high transmission levels is Zcell  Zslot , a resonant slot with high impedance, together with a smaller value of Z0cell ( Zcell ), implies larger useful bandwidths for the transformer. In order to highlight this effect, two transitions using the same slot, whose impedance is shown in Fig. 9(a), are considered. The first transition is optimized for Z0cell = 200 , while the second is optimized for Z0cell = 100 . Fig. 9 shows the S-parameters associated with these two transitions. The relative bandwidth associated with the lower impedance cell is much larger. B. CPS/MS Balun

Fig. 6. Equivalent transmission line model of the transition in Fig. 5, for (a) differential input and (b) common input.

The design of CPS/GCPS transformers described in the previous section can be easily adapted to CPS/MS balun designs. The primary circuit and the slot remain the same, while in the secondary circuit one branch of the GCPS lines becomes a quarter wavelength open stub, while the other branch constitutes the unbalanced MS input. The geometry of the balun is shown in Fig. 10. IV. EXAMPLES OF ARRAY FEEDING CIRCUIT DESIGN

Fig. 7. Common-mode rejection of the CPS/GCPS transition in Fig. 5. The transition is optimized for 100 Ohms impedance.

slot. In turn, this translates in no electric current being excited in the secondary circuit of the transformer (at z = 0h). The common-mode blocking by these type of components can be highlighted in Fig. 7, where the S12 for the common mode is reported as a function of the frequency. Note that the common mode is rejected to 017 dB’s in the worst case over a wideband. While this rejection is intrinsic in this type of transformer, the achievement of unattenuated differential signals depends on the quality of the matching. Fig. 8 shows the equivalent circuit transformation steps that can be applied to the circuit in Fig. 6(a) in order to obtain an estimate of the input impedance for the array cell and for the differential mode. The maximum power transfer for the differential mode is obtained when the connected array loading (Zcell ) is matched to the differential-mode line impedance (Z0cell ), which is realized by the series of the two microstrip lines, each of impedance Z0MS . So the condition is Z0cell = 2Z0MS = Zcell . Let us also focus on the simplifying and realistic situation in which one wants to realize unitary transforming ratios between the primary and secondary circuits and the slot access points, n1  n2  1. In this case, the load (Zcell ) is transferred at the slot level (S 0 S 0 ) and the equivalent impedance at (S 0 S 0 ) looking upward is Zk = Zcell kZslot , where Zslot is the impedance across the slot in the absence of the microstrip lines. When Zcell  Zslot , the presence of the slot is negligible and the differential signal is completely transmitted to the secondary circuit, in fact n2 Zk = n1 n2 Zk  Zcell .

The transformer described in Section III can be the key component of a periodic cell of a two-dimensional connected array. In order to minimize the number of T/R modules, the periods in x and y are maintained at 0:50 at the highest useful frequency (10 GHz). As examples, let us consider the following two configurations. 1) In each periodic cell of the array there is only one feeding point with the corresponding transformer [see Fig. 11(a)]. In this case, the periods are dx = dy = 0 =2 and the impedance of the unit cell is Zcell = 200  0 =2, where 0 is the free space characteristic impedance. Accordingly, a transformer optimized for Z0cell = 200 is considered. 2) Each periodic cell is fed at two points, with a separation distance of 0 =4 [see Fig. 11(b)]. The impedance at each feed point is Zcell = 100 and the transformer is also designed for Z0cell = 100 . A Wilkinson power divider is included to re-establish the same number of T/R modules. The two cases differ for the operational bandwidth of the transformers, as was shown in Fig. 9(b). Therefore, the active reflection coefficients for the case 2 when scanning toward  = 45 and  = 45 exhibits a wider 010 dB’s relative bandwidth, as expected [Fig. 12(a)]. However, for the 200 design, even within the frequency band in which a good matching is achieved, high losses are observed in terms of cross-polarization level. Fig. 12(b) shows the X-pol levels for the two considered cases. The difference between the two curves can be associated with the different length p of the dipole and the primary circuit of the transformer (see Fig. 11). In the case 1, the length p becomes longer than one equivalent wavelength at the highest frequency, allowing common-modes resonances to appear inside the bandwidth of the array. Case 2 instead permits having a shorter length p, shifting the resonance at frequencies higher than 10 GHz. As a consequence, a sensibly lower X-pol ratio is obtained with the second approach. A. Finite Ground Planes Note that the transformer in Section III has been introduced assuming infinite ground planes surrounding the slots. In practice, when used to feed connected arrays, the ground plane will be limited. The effect of this finiteness is not negligible, but was accounted for in the simulations presented in this section.

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Fig. 8. Equivalent circuit transformation steps for the differential-mode currents.

Fig. 10. Layout of the CPS/MS balun transformer.

Fig. 9. (a) Input impedance of the slot etched on an infinite ground plane, in the absence of the microstrip circuits. (b) Reflection and transmission coefficient of = 200 and Z = 100 , respectively. two transition optimized for Z

V. CONCLUSION A strategy to avoid common-mode resonances in arrays of connected dipoles has been presented. In order to improve the polarization purity,

Fig. 11. Geometry of a periodic cell a connected dipole array with (a) single feed per cell and with (b) double feed per cell.

an aperture-coupling-based transformer has been designed, acting as a common-mode rejection circuit. This component is not effective in an array with half wavelength spacing between the elements, because continuous current paths in the order of a wavelength are still undergoing

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[10] A. Neto and J. J. Lee, “Infinite bandwidth long slot array,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 75–78, 2005. [11] A. Neto and J. J. Lee, “Ultrawide-band properties of long slot arrays,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 534–543, Feb. 2006. [12] A. Neto, D. Cavallo, G. Gerini, and G. Toso, “Scanning performances of wide band connected arrays in the presence of a backing reflector,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3092–3102, Oct. 2009. [13] S. G. Hay J and D. O’Sullivan, “Analysis of common-mode effects in a dual-polarized planar connected-array antenna,” Radio Sci., vol. 43, Dec. 2008, RS6S04, doi:10.1029/2007RS003798. [14] E. de Lera Acedo et al., “Study and design of a differentially-fed tapered slot antenna array,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 68–78, Jan. 2010. [15] J. B. Knorr, “Slot-line transitions,” IEEE Trans. Microw. Theory Tech., pp. 548–554, May 1974. [16] A. C. Ludwig, “The definition of cross polarization,” IEEE Trans. Antennas Propag., vol. AP-21, pp. 116–119, Jan. 1973. [17] N. Herscovici and D. M. Pozar, “Full-wave analysis of aperture-coupled microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1108–1114, Jul. 1991.

Reducing Mutual Coupling for an Extremely Closely-Packed Tunable Dual-Element PIFA Array Through a Resonant Slot Antenna Formed In-Between Fig. 12. (a) Active reflection coefficient and (b) X-pol ratio for the two geometries in Fig. 11, when scanning towards  = 45 and  = 45 .

common-mode propagation. The proposed solution is splitting the periodic cell in two subcells along the longitudinal direction (x), each comprising a transformer but both connected to the same T/R module. Infinite array simulations highlight the lowering of the cross-polarization levels. Even denser sampling (24) could be possible by using high permittivity dielectrics, which allow reducing the transformer size. This may become necessary if the array is backed by a reflector for increased front to back ratio, and the overall cell impedance becomes 400 .

REFERENCES [1] M. V. Ivashina, M. N. M. Kehn, P. S. Kildal, and R. Maaskant, “Decoupling efficiency of a wideband Vivaldi focal plane array feeding a reflector antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 373–382, Feb. 2009. [2] S. G. Hay, F. R. Cooray, J. D. O’Sullivan, N.-T. Huang, and R. Mittra, “Numerical and experimental studies of a dual-polarized planar connected-array antenna for the Australian square kilometer array pathfinder,” presented at the IEEE Antennas and Propagation Symp., Charleston, SC, Jun. 1–5, 2009. [3] B. Tomasic and N. Herscovici, “Analysis of cylindrical array of infinitely-long slots fed by connected dipoles,” presented at the P IEEE Antennas and Propagation Symp., Charleston, SC, June 1–5, 2009. [4] Core Science Requirements for the European SPICA Instrument, ESIRAL-REQ-0012, Iss. 0.1.. [5] Y. S. Kim and K. S. Yngvesson, “Characterization of tapered slot antennas feeds and feed array,” IEEE Trans. Antennas Propag., vol. 38, pp. 1559–1564, Oct. 1990. [6] M. C. van Beurden et al., “Analysis of wideband infinite phased arrays of printed folded dipoles embedded in metallic boxes,” IEEE Trans. Antennas Propag., vol. 48, no. 5, pp. 784–789, May 2000. [7] M. A. Gonzalez de Aza, J. Zapata, and J. A. Encinar, “Broad-band cavity-backed and capacitively probe-fed microstrip patch arrays,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1266–1273, Sep. 2002. [8] W. S. T. Rowe, R. B. Waterhouse, and C. T. Huat, “Performance of a scannable linear array of Hi-Lo stacked patches,” IEE Proc. Microwave, Antennas Propag., vol. 150, no. 1, pp. 1–4, Feb. 2003. [9] H. Wheeler, “Simple relations derived from a phased array antenna made of an infinite current sheet,” IIEEE Trans. Antennas Propag., vol. 13, pp. 506–514, 1965.

Shuai Zhang, Salman Naeem Khan, and Sailing He

Abstract—An efficient mutual coupling reduction method is introduced for an extremely closely packed tunable dual-element planar inverted-F antenna (PIFA) array. High isolation can be achieved through a 2 folded slot antenna formed by a slot on the ground plane and the neighboring edges of the two PIFAs. Direct coupling is blocked by the slot antenna through radiating the coupling power into free space. A measured isolation of more than 36.5 dB can be achieved between the two parallel individual-element PIFAs operating at 2.4 GHz WLAN band with an inter-PIFA spacing of less than 0.063 (center to center) or 0.0147 (edge to edge). Since there is only a narrow slot antenna formed between the PIFAs in the present method, the distance can be further reduced to less than 0 0016 (edge to edge) with the maximal isolation of better than 40 dB. Both measured and simulation results show the effectiveness of the present mutual coupling reduction method. Index Terms—Diversity antennas, mutual coupling, planar inverted F antenna (PIFA), slot antenna.

I. INTRODUCTION Muliple-input and multiple-output (MIMO) communication systems have received much attention as a practical method to substantially increase wireless channel capacity without the need for additional power or spectrum in rich scattering environments [1]–[3]. However, several practical requirements exist for achieving the predicted high data rates Manuscript received July 07, 2009; revised February 02, 2010; accepted February 08, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. The work was supported in part by VINNOVA (Sweden) for “IMT advanced and beyond.” S. Zhang and S. He are with Centre for Optical and Electromagnetic Research, Zhejiang University, Hangzhou 310058, China and also with the Division of Electromagnetic Engineering, School of Electrical Engineering, Royal Institute of Technology, S-100 44 Stockholm, Sweden (e-mail: [email protected]). S. N. Khan is with the Department of Physics, COMSATS Institute of Information Technology, Lahore, Pakistan. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050432

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and coverage [4], [5]. One important requirement for a MIMO system is to minimize the mutual coupling among antennas. At the same time, for MIMO terminal applications, especially for portable devices, antenna arrays should be small enough. Planar inverted-F antennas (PIFAs), with the advantage of low profile, low cost and ease of design, have been widely utilized in communication terminal applications, such as laptop computers, and USB dongles. However, earlier work [6]–[8] indicated that no matter how two PIFAs are oriented above a common ground with air substrate, the inter-antenna spacing should be at least larger than 0:50 , where 0 is the half of the free space wavelength, in order to achieve about 20 dB isolation or more. It is therefore necessary to provide efficient mutual coupling reduction technique for an extremely closely-packed dual-element PIFA array. At the same time, in order to increase the tolerance of fabrication, it is desirable to find a simple and easy way to tune the operating frequency. Many studies have been carried out to reduce the mutual coupling between closely-spaced PIFAs [9]–[23]. Minimal mutual coupling is the fundamental limit for an array of infinite elements [9]. Most of studies are therefore based on the mutual coupling reduction among a small number of PIFAs (usually 2–3). For instance, Mushroom-like EBG structures in [10]–[12] could enhance the isolation through suppressing the surface wave propagation. However, these structures are large and difficult to fabricate. A defected ground structure (DGS) can also provide a band-stop effect and has been utilized in [13]–[16]. A compact integrated diversity antenna with two feed ports has been studied in [17], [18]. This method is not effective for PIFAs because PIFAs have no (naturally occurring) voltage null lines with well-matched feed positions [17]. In [19], a suspended neutralization strip is physically connected to the neighboring edges of two PIFAs in order to provide another field to cancel out the mutual coupling. However, this method is used for two antennas operating at different frequencies. In the past several years, the spacing between two PIFAs has been reduced rapidly while maintaining an isolation of 20 dB or more. In [20], the edge to edge distance between two PIFAs has been reduced to 0:170 with the method of separating the ground plane of two PIFAs. A space of less than 0:0780 has been achieved through a simply modified ground plane [21]. However, when the distance is reduced further, the method used in [20], [21] will not be effective. A coupling element has been introduced in [22] to enhance the isolation and a spacing of less than 0:02940 has been achieved. This method can greatly reduce the mutual coupling. Since the decoupling element occupies some areas and has to be inserted between the two PIFAs, the distance can not be reduced further. This problem is solved in [23] through removing the neutralization line in [19] from the inside edges of the two PIFAs to the outside edges. Since there are no decoupling structures between dual-element PIFAs, the inter-PIFA spacing has been reduced to 0:0270 or less. However, the port to port isolation is only around 10 dB, and the neutralization line located on the outside edges of the two PIFAs also has to occupy some area. In this communication, we present a simple and efficient way to reduce the mutual coupling between two extremely closely spaced PIFAs. We utilize a slot cut on the ground as well as the neighboring edges of two PIFAs to form a 0 =2 folded slot antenna. Different from the decoupling element in [22], this slot antenna does not produce another field to cancel out the coupling field, but forces the coupling power to be radiated directly into free space. This method has the advantages of narrower inter-PIFA spacing, lower mutual coupling, and simpler decoupling structure as compared with previous devices [10]–[23]. To demonstrate the idea, a tunable dual-element PIFA array for small MIMO terminal application is proposed at 2.4 GHz WLAN 802.11n. An inter-PIFA spacing less than 0:01470 is achieved experimentally while maintaining more than 36.5 dB isolation. A tunable device is also introduced in order to increase the fabrication tolerance.

Fig. 1. Prototype of the proposed dual-element PIFA array.

Fig. 2. Geometry of the proposed dual-element PIFA array: (a) top view, (b) side view.

II. IMPLEMENTATION OF CLOSELY PACKED TUNABLE DUAL-ELEMENT PIFA ARRAY A. An Extremely Closely Packed Tunable Dual-Element PIFA Array Design The prototype and configuration of the designed tunable dual-element PIFA array are shown in Figs. 1 and 2, respectively. This array consists of two PIFAs operating at 2.4 GHz for WLAN with MIMO under the proposed 802.11n standard. Let Lpifa , Wpifa , and Hpifa denote the length, width and height of the PIFAs, respectively. The slash part represents the standard FR4 substrate with a PCB thickness of 0.8 mm, relative permittivity of 4.7 and loss tangent of 0.03. The ground and radiation parts are made of copper sheet with thickness of 0.018 mm and 0.4 mm, respectively. A slot cut on the ground plane is made between two PIFAs and symmetrical above the center of the whole array configuration. The width of the slot or the inter-PIFA spacing is denoted by d. This slot cut plays a critical role in the present mechanism for reducing the mutual coupling. Since the slot cut is a part of the formed 0 =2 folded slot antenna, it can block the current tending to flow from one PIFA to the other on the ground plane and

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TABLE I DIMENSIONS OF PIFA ARRAYS

Fig. 3. Comparison between the measured and simulated S-parameters of the proposed dual-element PIFA array.

“1”,“2”,“3” represent our proposed antennas, conventional antennas with separated ground planes and conventional antennas with a conventional ground plane, respectively.

efficiently enhance the isolation. Each PIFA with a full shorting wall is positioned on each edge of slot and is fed with a 50- SMA coaxial probe. In order to make PIFAs matched easily, a capacitive sheet with width cf , length cf and height cf is loaded on the end of the probe, which has been widely used [8]. In [24], an operating frequency tunable device of screw type was analyzed in order to increase the tolerance of fabrication and environment effects. This device is also used in our design and the diameter of the screw is 1 mm. At the end of the screw a circular disc of radius is loaded. The detailed dimensions of the designed PIFA array are summarized in Table I. The PIFA’s resonant frequency can be estimated by:

W

L

H

r

f

r

f

= 4(L

pifa

c

c

+ Hpifa )

(1)

where r is resonant frequency and is the velocity of light in the free space. To investigate the performance of the proposed structure, computeraided design (CAD) is carried out using commercial software CST. The simulated and measured S-parameters for the tunable dual-element PIFA array are shown in Fig. 3. All the S-parameters are measured by Advantest R3765C Network Analyzer. A slightly tuning of t has been made in order to move the central frequency to 2.45 GHz. According to our measured results, the impedance bandwidth for 010 dB specification is 3.78% (from 2.39 GHz to 2.482 GHz), which covers the band of WLAN (2.4 GHz–2.48 GHz). Across the WLAN band the isolation is better than 20 dB with a maximum of 36.5 dB. From Fig. 3 one sees that the measured and simulation results agree well (particularly the impedance bandwidth and the deepest point for isolation). The measured far-field patterns are presented in Fig. 4(a) and (b) with the left PIFA in Fig. 2(a) excited and the other terminated with a 50- load. In Fig. 4(a) one sees a notch for both E-theta and E-phi along x axis ( = 90 ). This can help in enhancing the isolation of the two PIFAs in our proposed structure and explaining why the mutual coupling between two closely-packed PIFAs is so low. The measured peak gain and efficiency are 2.12 dBi and 71.5%, respectively.

H



Fig. 4. Measured radiation patterns of the proposed PIFAs at 2.45 GHz: (a) x-z plane, (b) y-z plan. Measured radiation patterns of PIFAs with separated ground planes at 2.45 GHz: (c) x-z plane, (d) y-z plane. Measured radiation patterns of PIFAs with conventional ground plane at 2.45 GHz: (e) x-z plane, (f) y-z plane.

For a multi-port antenna system, mutual coupling will also lead to the decrease of efficiency. When only one port is fed and the others are

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Fig. 5. The simulated results for the range of tunable frequency of the proposed dual-element PIFA array.

terminated with 50- load, the total efficiency can be evaluated by the following:

total mismatch+coupling

= radiation mismatch+coupling S 2 =10 S 2 j

ii j

0

j

ji j

(2) (3)

j 6=i

where total , radiation and mismatch+coupling are the total efficiency, radiation efficiency, and mismatch+ mutual coupling efficiency, respectively. Subscripts i, j represent the working and terminated ports, respectively. In this design, the PIFAs are well matched and the isolation between the antennas is better than 36.5 dB. The decrease of efficiency due to the mismatch and mutual coupling is nearly zero, i.e., the mismatch and mutual coupling are not the main reason for the efficiency loss. From the simulation results we found the loss of the FR4 substrate and the metal loss are the dominant factors for the loss of efficiency. Additional loss in connectors and cables during the prototype fabrication and measurement may also result in some reduction of efficiency.

Fig. 6. Geometry of conventional dual-element PIFA arrays: (a) antennas with separated ground planes; (b) antennas with a conventional ground plane.

B. Tunable Property Study The resonant frequency of PIFAs can be simply tuned by changing the height of the via-patch [24]. In our design, the shape of the via-patch is modified to a circular disk (instead of a square [24]), and a screw is positioned on the center of the via-patch instead of the edge. This modification can make the tuning of the operating frequency more convenient. From the simulated results, the range of the tunable frequency is from 2.36 to 2.5 GHz, as shown in Fig. 5. C. Comparison to Other Tunable Dual-Element PIFA Array Designs In order to show the effectiveness of the proposed method, comparisons are made with two conventional dual-element PIFA arrays, namely, two antennas with a conventional ground plane and two antennas with separated ground planes. The latter achieves the isolation with the same method (instead of the same structure) used in [20]. The length L of the ground plane of the latter is much longer than that of our proposed antenna. This can make the resonant frequency of the formed 0 =2 slot antenna far from 2.45 GHz so that the reduction of mutual coupling is only due to the separation of the ground plane instead of the slot antenna formed in-between. The compared structures are shown in Fig. 6. In order to make sure the compared PIFA arrays have the same operating frequency of 2.45 GHz, the parameters of the compared structures are slightly adjusted while maintaining the overall PIFA array the same. The detailed dimensions of the compared antenna arrays are given in Table I. The measured S-parameters for the compared dual-element PIFA arrays are shown in Fig. 7, and the measured radiation patterns of the two compared structures are shown in Fig. 4(c)–(f). Comparative studies of various de-

Fig. 7. Measured S-parameters for antennas with separated ground planes and antennas with a conventional ground plane. TABLE II MEASURED PERFORMANCE OF PIFA ARRAYS

“1,” “2,” and “3” represent our proposed antennas, conventional antennas with separated ground planes and conventional antennas with a conventional ground plane, respectively.

signs are summarized in Table II. The comparisons show that our proposed decoupling method is more efficient than the two conventional methods. Meanwhile, due to the strong mutual coupling the peak gains

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TABLE III SIMULATED PERFORMANCE OF ARRAYS WITH DIFFERENT INTER-PIFA SPACING

Fig. 8. Simulated distribution of surface current at 2.45 GHz.

and efficiencies of the two conventional structures are reduced significantly and thus not sufficient for MIMO applications. D. Mutual Coupling Reduction Method For far-field coupling, suppression of the coupling current on the ground is sufficient to obtain a good isolation between antennas. However, for closely spaced antennas, near-field coupling is predominant. Not only the current flowing on the ground plane but also the displacement current communicating between antenna elements will contribute to the coupling. The latter is stronger than the former in the present case. Our proposed method can reduce the mutual coupling through a 0 =2 folded slot antenna formed by the slot cut on the ground plane and the neighboring edges of the two PIFAs (see Fig. 2). The slot antenna can give a resonance and force the power that tends to couple from one antenna to the other to radiate into free space directly. When the resonance frequency of the slot is near the PIFA operating frequency the isolation is also high. Since our proposed method can suppress not only the displacement current communicating between the antenna elements but also the current on the ground plane tending to flow from one antenna to the other, the mutual coupling could be reduced more efficiently. In order to further understand the mechanism of the proposed method, the distribution of the surface current is shown in Fig. 8. The current distribution on the neighboring edges of the two PIFAs and the slot on the ground plane is very similar to that of a conventional slot antenna. One sees that the current flowing from one PIFA toward the other has been substantially concentrated to inner edges of PIFAs as well as the slot on the ground plane, and this helps to reduce the mutual coupling. III. FURTHER REDUCTION OF THE INTER-PIFA SPACING Since there is only a narrow slot between the PIFAs, the inter-PIFA spacing can be reduced significantly. In this section a further reduction of inter-PIFA spacing will be carried out.

The performance of the tunable dual-element PIFA array for different inter-PIFA spacings is listed in Table III. It shows that our proposed method is efficient even for small spacing between the PIFAs. From our simulation results an inter-PIFA spacing of less than 0:00160 can be achieved while still maintaining the isolation level of better 40 dB. However, smaller inter-PIFA spacing will cause narrower 010 dB impedance bandwidth and 20 dB isolation bandwidth, and also lower efficiency of the PIFAs. This is because the coupling of the two PIFAs is in near-field and the mutual coupling increases rapidly when the distance decreases. This can result in a narrower bandwidth of isolation. Furthermore, a smaller distance of PIFAs or narrower width of slot antenna is also an important factor for the reduction of isolation bandwidth. According to our simulation results (obtained with CST software) in Table III, when the distance of the two PIFAs becomes smaller, PIFAs’ Q factor increases and then the bandwidth of PIFAs are reduced. Thus, there is a balance between the inter-PIFA spacing and the performance of the PIFA array. Additionally, when the inter-PIFA spacing decreases, the dimensions of the proposed structure have to be slightly adjusted in order to obtain an acceptable isolation and maintain the same frequency location of maximal mutual coupling reduction. The adjusted dimensions are shown in Table III. However, for a given inter-PIFA spacing there exist optimal dimensions for maximal isolation. IV. CONCLUSION The present communication has provided and demonstrated an efficient mutual coupling reduction technique for an extremely closely packed tunable dual-element PIFA array. The proposed array with an inter-PIFA spacing of less than 0:01470 (edge to edge) has an isolation of more than 36.5 dB. High isolation can be achieved through a 0 =2 folded slot antenna formed by the slot on the ground plane and the neighboring edges of the two PIFAs. Since there is only a narrow slot between the PIFAs in the method, the distance between the antenna elements can be reduced significantly. A further reduction of the distance has also been studied and when the spacing is less than 0:00160 (edge to edge) the maximal isolation can still be better than 40 dB. ACKNOWLEDGMENT The authors sincerely appreciate the support of Lenovo Group Ltd. (Shanghai, China) for providing the anechoic chamber for measurement.

REFERENCES [1] R. D. Murch and K. B. Letaief, “Antenna systems for broadband wireless access,” IEEE Commun. Mag., vol. 40, no. 4, pp. 76–83, Apr. 2002. [2] G. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J., vol. 1, no. 2, pp. 41–59, 1996. [3] J. Wallace, M. Jensen, A. Swindlehurst, and B. Jeffs, “Experimental characterization of the MIMO wireless channel: Data acquisition and analysis,” IEEE Trans. Wireless Commun., vol. 2, no. 2, pp. 335–343, Mar. 2003.

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[4] G. Foschini and M. Ganst, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [5] S. Ko and R. Murch, “Compact integrated diversity antenna for wireless communications,” IEEE Trans. Antennas Propag., vol. 49, no. 6, pp. 954–960, Jun. 2001. [6] H. Carrasco, H. D. Hristov, R. Feick, and D. Cofre, “Mutual coupling between planar inverted-F antennas,” Microwave Opt. Technol. Lett., vol. 42, no. 3, pp. 224–227, Aug. 2004. [7] T. Taga and K. Tsunekawa, “Performance analysis of a built-in planar inverted F antenna for 800 MHz band portable radio units,” IEEE J. Select Areas Commun., vol. SAC-5, pp. 921–929, Jun. 1987. [8] C. R. Rowell and R. D. Murch, “A capacitively loaded PIFA for compact mobile telephone handsets,” IEEE Trans. Antennas Propag., vol. 45, no. 5, pp. 837–842, May 1997. [9] W. K. Kahn, “Ideal efficiency of a radiating element in an infinite array,” IEEE Trans. Antennas Propag., vol. AP-15, no. 4, Jul. 1967. [10] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [11] F. Yang and Y. Rahmat-Samii, “Microstrip antennas integrated with electromagnetic band-gap (EBG) structures: A low mutual coupling design for array applications,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2936–2946, Oct. 2003. [12] L. Li, B. Li, H. X. Liu, and C. H. Liang, “Locally resonant cavity cell model for electromagnetic band gap structures,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 90–100, Jan. 2006. [13] D. Ahn, J. S. Park, C. S. Kim, J. Kim, Y. Qian, and T. Itoh, “A design of the low-pass filter using the novel microstrip defected ground structure,” IEEE Microw. Theory Tech., vol. 49, no. 1, pp. 86–93, Jan. 2001. [14] C. Caloz, H. Okabe, T. Iwai, and T. Itoh, “A simple and accurate model for microstrip structures with slotted ground plane,” IEEE Microw. Wireless Comp. Lett., vol. 14, no. 4, pp. 133–135, Apr. 2004. [15] Y. J. Sung, M. Kim, and Y. S. Kim, “Harmonics reduction with defected ground structure for a microstrip patch antenna,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 111–113, 2003. [16] D. Guha, M. Biswas, and Y. M. M. Antar, “Microstrip patch antenna with defected ground structure for cross polarization suppression,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 455–458, 2005. [17] S. C. K. Ko and R. D. Murch, “Compact integration diversity antenna for wireless communications,” IEEE Trans. Antennas Propag., vol. 49, no. 6, pp. 954–960, Jun. 2001. [18] C. T. Song, C. K. Mak, R. D. Murch, and P. B. Wong, “Compact low cost dual polarized adaptive planar phased array for WLAN,” IEEE Trans. Antennas Propag., vol. 53, pp. 2406–2416, Aug. 2005. [19] A. Diallo, C. Luxey, P. L. Thuc, R. Staraj, and G. Kossiavas, “Study and reduction of the mutual coupling between two mobile phone PIFAs operating in the DCS1800 and UMTS bands,” IEEE Trans. Antennas Propag., vol. 54, Nov. 2006. [20] Y. Gao, X. D. Chen, Z. N. Ying, and S. He, “Design and performance investigation of a dual-element PIFA array at 2.5 GHz for MIMO terminal,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3433–3441, Dec. 2007. [21] C.-Y. Chiu, C.-H. Cheng, R. D. Murch, and C. R. Rowell, “Reduction of mutual coupling between closely-packed antenna elements,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1732–1738, Jun. 2007. [22] A. C. K. Mak, C. R. Rowell, and R. D. Murch, “Isolation enhancement between two closely packed antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3411–3419, Nov. 2008. [23] A. Chebihi, C. Luxey, A. Diallo, P. H. Thuc, and R. Staraj, “A novel isolation technique for UMTS mobile phones,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 665–668, 2008. [24] C.-Y. Chiu, K. M. Shum, and C. H. Chan, “A tunable via-patch loaded PIFA with size reduction,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 65–71, Jan. 2007.

The Sources Reconstruction Method for Amplitude-Only Field Measurements Yuri Álvarez, Fernando Las-Heras, and Marcos R. Pino

Abstract—An extension of the sources reconstruction method (SRM) for antenna diagnostics using amplitude-only field measurements is presented. While previous works limit the application to canonical domain cases, the combination of the SRM capabilities for handling with arbitrary-geometry domains jointly with phase retrieval techniques, allows antenna diagnostics extension to complex geometry antenna and field acquisition domains. The consideration of the radiation inverse problem with a general integral equation formulation using arbitrary-geometry field and currents domains, and field phaseless information, supposes a challenging ill-posed problem that is solved using iterative minimization techniques for non-linear problems. Different examples are presented, from simple to general antenna diagnostics cases. Index Terms—Antenna diagnostics, antenna measurement, integral equations, minimization methods, phaseless measurements, sources reconstruction method (SRM).

I. INTRODUCTION Antenna diagnostics methods based on the acquisition of the radiated field are becoming a key issue due to their features: first, they are non-invasive methods, so they makes possible the determination of antenna anomalies avoiding try-and-error procedures that require interacting with the antenna. In addition, they are able to work with the antenna in the final environment, allowing the knowledge of the currents distribution on the antenna and how it can be modified due to the interaction with surrounding elements. Diagnostics techniques are most based on far-field/near-field to nearfield (FF/NF-NF) transformation, in order to determine the extremely NF on a surface close to the antenna-under-test (AUT). For example [1], [2] present some applications for detecting faults in reflector antennas, based on the backward transformation of the fields using the Fourier Transform. [3], [4] extends modal wave expansion-based formulation from planar acquisition domains to spherical ones. Limitation of wave mode-based FF/NF-NF methods to canonical acquisition and diagnostics geometries is overcome by the introduction of the Sources reconstruction methods (SRM), an integral equation technique that characterizes the antenna under test (AUT) through a set of equivalent electric and/or magnetic currents distribution. The use of an integral-equation-based technique results in a computational cost increase with respect to the wave-mode methods (a computational cost comparison is presented in [5]). Nowadays, the development of computational capabilities and numerical techniques for dealing with electrically-large problems (e.g., the Fast Multipole Method [6]) makes possible to overcome these computational cost limitations. The first SRM applications for NF/FF transformation and antenna diagnostics were restricted to planar domains [7], [8]. Later, they were Manuscript received July 10, 2009; revised January 29, 2010; accepted February 21, 2010. Date of publication May 18, 2010; date of current version August 05, 2010. This work was supported by the “Ministerio de Ciencia e Innovación” of Spain/FEDER” under projects TEC2008-01638/TEC (INVEMTA) and CONSOLIDER CSD2008-00068 (TERASENSE), and by the “Cátedra Telefónica- Universidad de Oviedo.” The authors are with the Area of Signal Theory and Communications, Department of Electrical Engineering, Universidad de Oviedo, E-33203 Gijón, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050433

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acquired on the field observation domain (Sobs ) (light-gray-colored region in Fig. 1). Maxwell equations solution for free-space conditions (finite source distribution in homogeneous medium) gives the integral equations (1) and (2) [22], relating the fields radiated by electric and magnetic currents distribution

~ J (~r) = 0 j E 4k0 ~ M (~r) = 0 1 E 4 Fig. 1. Arbitrary-geometry field and sources domains.

extended to arbitrary geometries as described in [9]–[11], where the equivalent currents are reconstructed on: a) AUT radome [9], b) AUT metallic surface [10], and c) equivalent surface close to the AUT [11]. In [12], the integral-equation techniques are combined with a distributional approach-based solver to detect the faulting elements in large arrays from the knowledge of the radiated field. Finally, the SRM capabilities for super-resolution (i.e., the possibility of distinguish between elements placed closer than =2) are presented in [13] and [14]. The related antenna diagnostics and NF/FF transformation techniques are conceived to work with field amplitude and phase information. However, phase acquisition is not always possible, due to technical and/or economical measurement system restrictions. Moreover, the growing interest in submillimeter and terahertz bands makes the amplitude-only (phaseless) inverse techniques to have potential interest for applications in these bands due to existing limitations for phase measurements [15]. The problem of antenna diagnostics using amplitude-only information has been studied under different approaches. One of the significant contributions in this topic has been described in [16], [17], where a plane-to-plane iterative backpropagation method for phase retrieval is proposed both for NF-FF and antenna diagnostics applications. Other relevant contributions using an alternative formulation with a direct management of amplitude information are presented in [18]–[20]. An equivalent magnetic currents distribution is calculated by minimizing a cost function which relates the amplitude of the measured field and the contribution due to the equivalent currents. [18]–[20] propose different strategies for cost function regularization and minimization, where the amount of information and the phaseless problem constraints are also addressed in [18], [19]. Amplitude-only antenna diagnostics methods are restricted to canonical domains (planar -in most of the cases- [18]–[20], cylindrical [21]). Starting from [18]–[20], and using the SRM extension to arbitraryshape geometry domains presented in [10], it is possible to provide an amplitude-only SRM for antenna diagnostics and NF-FF transformation, which is the scope of this contribution. II. FIELD ACQUISITION OVER ARBITRARY-GEOMETRY DOMAINS The SRM is based on the electromagnetic equivalence principle [22]: given an arbitrary sources distribution, bounded by a surface Ssources , it is possible to find an equivalent electric and magnetic currents distribution (J eq (r 0 ); M eq (r 0 )) over such surface Ssources that radiate the same fields outside that surface. Thus, the knowledge of the equivalent currents makes possible the determination of the fields in any point outside the equivalent currents domain (Ssources ) (dark-gray-colored region in Fig. 1). The equivalent currents are calculated from the field

e0jk R(~r;~r ) ~ 0 J (~r ) R(~r; ~r0 ) eq e0jk R(~r;~r ) ~ 0 J (~r ) +r r1 R(~r; ~r0 ) eq

k02 S

dS 0 (1)

e0jk R(~r;~r ) ~ Meq (~r0 ) dS 0 R(~r; ~r0 )

r2 S

(2)

Where k0 is the wavenumber,  is the intrinsic impedance of the medium. Positioning vectors, r and r 0 , are defined as (3)

~r = ~r(x; y; z ) 2 Sobs ~r0 = ~r0 (x0 ; y0 ; z 0 ) 2 S 0 = Ssources R(~r; ~r0 ) = ((x 0 x0 )2 + (y 0 y0 )2 + (z 0 z 0 )2 )1=2 :

(3)

Regarding numerical solution of (1) and (2), they can be rewritten in a matrix form relating the field components (Et ; Et ) tangential to Sobs with the equivalent currents components (Jt ; Jt ; Mt ; Mt tangential to the source domain Ssources ). Pulse-basis functions are defined to expand the currents, as described in [10], [14]. Thus, the equivalent currents that characterize the AUT are obtained by solving the mentioned system of equations. Different numerical techniques are implemented: for example, [10], [14] propose the conjugate gradient [23] for solving the matrix system, by minimizing the following cost function (4)

Et Et

F= Z(Et

;It )

0

= ( Z(Et ;Jt

)

Z(Et Z(Et Z(Et

;It )

Z(Et Z(Et

;Mt )

) It =

;It )

It It

1

;It ) ;It )

Jt Mt

:

2

(4)

Depending on the acquisition and equivalent currents domains geometry, the previous system of equations can be simplified. For example, in the case of planar equivalent currents domain, the application of Images Theory makes possible the consideration of only one kind of equivalent currents, e.g., magnetic currents (Mt ; Mt ), which are determined from the acquired field. Planar reconstruction domains cover most of the practical applications for antenna diagnostics, e.g., reflector antennas, and linear and planar antenna arrays. However, for some particular complex-geometry antenna cases, the utilization of generalized formulation (involving both electric and magnetic currents on arbitrary-geometry domain) is required. III. SOURCES RECONSTRUCTION USING AMPLITUDE-ONLY DATA The knowledge of amplitude and phase information of the acquired field makes it possible the utilization of just one observation surface for recovering the equivalent currents. However, when phase information is not available, it must be retrieved from the amplitude data. Field amplitude knowledge at two or more observation domains makes possible phase recovering. For the amplitude-only case, the cost function (4) has to be reformulated (5), so the quantity to be minimized is related to the difference between the amplitude of the electric field radiated by the equivalent currents and the measured one

F=

Et Et

2

0

Z(Et Z(Et

;It ) ;It )

Z(Et Z(Et

;It ) ;It )

1

It It

2 2

: (5)

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Fig. 2. Algorithm #1: iterative scheme for recovering the cartesian components of the equivalent magnetic currents from amplitude-only field data acquired on spherical domain. Spherical field components are transformed to Cartesian components.

Fig. 3. Algorithm #2: iterative scheme for recovering the Cartesian components of the equivalent electric and magnetic currents from amplitude-only field data acquired on spherical domain.

J

While the cost function (4) is related to a linear system of equations, (5) corresponds to a non-linear cost function [24]. Here, iterative non-linear minimization methods have been introduced for solving (5): inexact Newton-Raphson [25] and Levenberg-Marquardt [25]. Two algorithms for sources reconstruction using amplitude-only field information are proposed, depending on the field components to be considered. Hereinafter, spherical field acquisition surface will be used. Algorithm #1 (Fig. 2): It is based on the decoupling of the integral equations, so that the equivalent currents component x is related to the field component y , and y to x . Thus, its application is limited to planar equivalent currents domain cases. First, the x y components are calculated from the spherical field components  ' , under an initial assumption of field phase equal to 0 , and r = 0 as initial guess. This algorithm is divided in two stages: in the first iteration block, the cost function (5) is optimized for each equivalent magnetic current component. Field phase estimation from the reconstructed x y is used at each iteration to calculate ~x ~y from  ' and an estimation of ~r , as described in [26]. In the second block of iterations, x y are calculated by minimization of cost function (4). Field phase estimation from the reconstructed x y , and measured field amplitude (  ' ), are again combined to calculate ~x ~y (see Fig. 2). Algorithm #2 (Fig. 3): It is similar to Algorithm #1, but no decoupling between integral equations is considered, being able to handle

E

M E

E

E

E ;E

E ;E

M ;M

E ;E

E ;E

M E ;E E ;E

M ;M

M

with eq and eq , and hence, allowing arbitrary-geometry equivalent currents domains. The price-to-pay is a worse system of equations conditioning, and hence, worse convergence of iterative methods. Again, the algorithm is divided in two stages: first, the cost function (5) is minimized. Second block of iterations minimizes functional (4), considering amplitude and phase information. Phase estimation is obtained, at each second stage iteration, from the field radiated by the reconstructed equivalent currents calculated in the previous iteration (see Fig. 3). Regarding iterative methods convergence, amplitude and phase 20, knowledge requires less iterations to converge (typically is the number of iterations for the second stage) than the where = 40–80 iterations, being the number amplitude-only case ( of iterations for the first stage). Convergence criterion is related to the root mean square error (RMSE) between the amplitude of the measured field and the field radiated by the reconstructed equivalent currents. A RMSE goal of 1% is chosen to consider that the method has converged to the solution [10], [14].

L

M ;M

K

K

L