IEEE Transactions on Antennas and Propagation [volume 59 number 3]

Table of contents :
705 с1.pdf
705 с2
706-713
714-724
725-732
733-741
742-750
751-757
758-766
767-775
776-783
784-792
793-801
802-809
810-817
818-825
826-832
833-838
839-845
846-858
859-868
869-876
877-887
888-897
898-903
904-913
914-927
928-940
941-949
950-959
960-968
969-978
979-986
987-993
994-1002
1003-1012
1013-1018
1019-1026
1027-1031
1032-1035
1036-1042
1043-1046
1047-1053
1054-1056
1057-1061
1062-1064
1065-1068
с3
с4

Citation preview

MARCH 2011

VOLUME 59

NUMBER 3

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Resonance and Q Performance of Ellipsoidal ENG Subwavelength Radiators . . . . . . . . A. Ahmadi, S. Saadat, and H. Mosallaei Multi-Functional, Magnetically-Coupled, Electrically Small, Near-Field Resonant Parasitic Wire Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.-C. Lin, P. Jin, and R. W. Ziolkowski Ultrawideband Printed Log-Periodic Dipole Antenna With Multiple Notched Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Yu, W. Hong, L. Chiu, G. Zhai, C. Yu, W. Qin, and Z. Kuai Modulated Elliptical Slot Antenna for Electric Field Mapping and Microwave Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. Abou-Khousa, M. T. Ghasr, S. Kharkovsky, D. Pommerenke, and R. Zoughi Self-Shielded High-Efficiency Yagi-Uda Antennas for 60 GHz Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. A. Alhalabi, Y.-C. Chiou, and G. M. Rebeiz On-Board Printed Coupled-Fed Loop Antenna in Close Proximity to the Surrounding Ground Plane for Penta-Band WWAN Mobile Phone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.-L. Wong, W.-Y. Chen, and T.-W. Kang Pattern Purity of Coiled-Arm Spiral Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Radway, T. P. Cencich, and D. S. Filipovic´ Composite Right/Left-Handed Substrate Integrated Waveguide and Half Mode Substrate Integrated Waveguide Leaky-Wave Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Dong and T. Itoh Study of Conformal Switchable Antenna System on Cylindrical Surface for Isotropic Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Z. Zhang, X. Gao, W. Chen, Z. Feng, and M. F. Iskander Analysis of Conformal Microstrip Antennas With the Discrete Mode Matching Method . . . . . . . . M. V. T. Heckler and A. Dreher Reducing Redundancies in Reconfigurable Antenna Structures Using Graph Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Costantine, S. al-Saffar, C. G. Christodoulou, and C. T. Abdallah Performance Assessment of Bundled Carbon Nanotube for Antenna Applications at Terahertz Frequencies and Higher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Choi and K. Sarabandi Ferrite Based Non-Reciprocal Radome, Generalized Scattering Matrix Analysis and Experimental Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Parsa, T. Kodera, and C. Caloz Arrays Bandwidth Limits of Multilayer Array of Patches Excited With Single and Dual Probes and With a Shorting Post . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. K. Bhattacharyya Bandwidth Enhancement for a 60 GHz Substrate Integrated Waveguide Fed Cavity Array Antenna on LTCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Xu, Z. N. Chen, X. Qing, and W. Hong

706 714 725 733 742 751 758 767 776 784 793 802 810

818 826

(Contents Continued on p. 705)

(Contents Continued from Front Cover) A Planar Dualband Antenna Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. K. Toh, X. Qing, and Z. N. Chen Synthesis of Planar Arrays With Elements in Concentric Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. A. R. Azevedo Sparse Antenna Array Design for MIMO Active Sensing Applications . . . . . . . . . . . . . . . . . . . W. Roberts, L. Xu, J. Li, and P. Stoica Through-Wall Opportunistic Sensing System Utilizing a Low-Cost Flat-Panel Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. E. Browne, R. J. Burkholder, and J. L. Volakis Adaptive Nulling Using Photoconductive Attenuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. L. Haupt, J. Flemish, and D. Aten Electromagnetics and Imaging Volumetric-Perturbative Reciprocal Formulation for Scattering From Rough Multilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Imperatore, A. Iodice, and D. Riccio Vertex Diffracted Edge Waves on a Perfectly Conducting Plane Angular Sector . . . . A. K. Ozturk, R. Paknys, and C. W. Trueman A Uniform Asymptotic Solution for the Diffraction by a Right-Angled Dielectric Wedge . . . . . . . . . G. Gennarelli and G. Riccio Dispersion Characteristics of a Metamaterial-Based Parallel-Plate Ridge Gap Waveguide Realized by Bed of Nails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Polemi, S. Maci, and P.-S. Kildal An FFT Twofold Subspace-Based Optimization Method for Solving Electromagnetic Inverse Scattering Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Zhong and X. Chen Near-Field Microwave Imaging Based on Aperture Raster Scanning With TEM Horn Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. K. Amineh, M. Ravan, A. Trehan, and N. K. Nikolova New Reflection Suppression Method in Antenna Measurement Systems Based on Diagnostic Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. J. Cano-Fácila, S. Burgos, F. Martín, and M. Sierra-Castañer Numerical Techniques Curl-Conforming Hierarchical Vector Bases for Triangles and Tetrahedra . . . . R. D. Graglia, A. F. Peterson, and F. P. Andriulli An Augmented Electric Field Integral Equation for Layered Medium Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. P. Chen, L. Jiang, Z.-G. Qian, and W. C. Chew A Time-Domain Volume Integral Equation and Its Marching-On-in-Degree Solution for Analysis of Dispersive Dielectric Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Shi and J.-M. Jin Lyapunov and Matrix Norm Stability Analysis of ADI-FDTD Schemes for Doubly Lossy Media . . . . . D. Y. Heh and E. L. Tan Compact FDTD Formulation for Structures With Spherical Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Mock Wireless and RFID The Linear Relationship Between Attenuation and Average Rainfall Rate for Terrestrial Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. J. Townsend and R. J. Watson On the -Factor Estimation for Rician Channel Simulated in Reverberation Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Lemoine, E. Amador, and P. Besnier Design of a Near-Field Focused Reflectarray Antenna for 2.4 GHz RFID Reader Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-T. Chou, T.-M. Hung, N.-N. Wang, H.-H. Chou, C. Tung, and P. Nepa RFID Grids: Part I—Electromagnetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Marrocco

K

833 839 846 859 869

877 888 898 904 914 928 941 950 960 969 979 987

994 1003 1013 1019

COMMUNICATIONS

The Optimum Operating Frequency for Near-Field Coupled Small Antennas . . . . . . . . . . . . . . . . . . . . . . . . Y. Tak, J. Park, and S. Nam Characterization and Reduction of Mutual Coupling Between Stacked Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. Quevedo-Teruel, Z. Sipus, and E. Rajo-Iglesias Synthesis and Design of a New Printed Filtering Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.-T. Chuang and S.-J. Chung Shaping Axis-Symmetric Dual-Reflector Antennas by Combining Conic Sections . . . . . . . . F. J. S. Moreira and J. R. Bergmann Adaptive Beamforming with Real-Valued Coefficients Based on Uniform Linear Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Zhang, W. Liu, and R. J. Langley On the Uniqueness of the Phase Retrieval Problem From Far Field Amplitude-Only Data . . . . . . . . . . . . . . . K. Inan and R. E. Diaz Scaling Factors of ID-FDTD Scheme for Dispersive Media Based on the Auxiliary Differential Equation (ADE) Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Deng, M. Zhao, and I.-S. Koh Optimal Design of a Highly Compact Low-Cost and Strongly Coupled 4 Element Array for WLAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Z. Ma, V. Volski, and G. A. E. Vandenbosch Inverted-F Laptop Antenna With Enhanced Bandwidth for Wi-Fi/WiMAX Applications . . . . . . . . . . . . . . L. Pazin and Y. Leviatan

1027 1031 1036 1042 1047 1053 1057 1061 1065

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

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Digital Object Identifier 10.1109/TAP.2011.2120931

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Resonance and Q Performance of Ellipsoidal ENG Subwavelength Radiators Akram Ahmadi, Student Member, IEEE, Soheil Saadat, Student Member, IEEE, and Hossein Mosallaei, Senior Member, IEEE

Abstract—The resonance performance and Q-factor of electrically small ellipsoidal radiators made of epsilon negative (ENG) material is investigated. It is demonstrated that the material polarization can successfully provide resonance radiation at the negative material constitutive parameters. In principle, arbitrary low resonant frequencies for a fixed antenna dimension can be achieved. The dependence of resonant frequency on the shape of the structure is determined. Special attention is devoted to the sphere, thin disk, and long rod, and physical insights into the radiation characteristics and Q (or bandwidth) are highlighted. Index Terms—Antenna theory, electrically small antennas, epsilon negative material, metamaterials, resonator.

I. INTRODUCTION

S

MALLER physical size, wider bandwidth and higher radiation efficiency are three desirable characteristics of antennas integrated into communication systems. In recent years, considerable efforts have been devoted towards antenna miniaturization. Fundamentally, the ability of any antenna to radiate (where is the wave number and effectively depends on is the radius of the smallest sphere enclosing the antenna) [1], [2]. According to Chu [2], the lower bound on the quality factor (Q) of an electrically small electric or magnetic dipole antenna is inversely related to the radius of the smallest sphere that can , so the surround it by the formula smaller the radius, the higher the Q and the narrower the bandwidth. The challenge is to make the physical size of the antenna as small as possible along with achieving a wideband impedance characteristic (Q values close to the lower-bound). Best et al. presented a comprehensive study in [3] for achieving small antennas with low Q performance with the use of novel topologies, such as spherical or cylindrical folded helix antennas. Quality factor as low as 1.5 times of the Chu limit was illustrated. Recently, there have been some efforts to produce wideband electrically small resonant antennas by utilizing negative parameters materials [4]–[6]. Stuart et al. in [4] excited an epsilon negative (ENG) sphere with a dipole feed to produce the appropriate polarization required for the resonance performance. Manuscript received February 21, 2010; revised May 20, 2010; accepted August 08, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by the U.S. Office of Naval Research (ONR), Grant N00014-10-1-0264. The authors are with the Applied Electromagnetics and Optics Laboratory, Department of Electrical Engineering, Northeastern University, Boston, MA 02115 USA (e-mail: [email protected]; [email protected], [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103022

Fig. 1. The geometry of an ellipsoid with semi-axes a , a and a .

They demonstrated that one can achieve an electrically small antenna by operating the spherical radiator in a frequency that region. The sphere-shaped struccorresponds to the tures offer wideband performance close to the Chu limit. Ziolkowski et al. have also demonstrated other novel small antenna designs utilizing combinations of epsilon negative (ENG) and mu negative (MNG) or double negative (DNG) metamaterials [6]. The focus of this work is on study of small antennas realized by unique materials. Practically, it is very useful to investigate the effect of the structural shape of the material of the antenna on the resonant frequency and Q-factor. For example, can a slab or a long rod of negative permittivity material also radiate efficiently? If so, what would be the resonant frequencies and associated material indices? How close would the Q be to the lower bound? The objectives in this paper are to investigate the resonance radiation of metamaterial-based eccentrically shaped structures with particular emphasis on the bandwidth limitations or quality factors for the antennas. Thin disks and long rods, as well as the ellipsoids, will be the special cases that allow us to address the above questions. To simplify the problem, it is assumed that the size of the antenna-proper is much smaller than the wavelength in both free-space and in the material. Hence, the time-harmonic quasi-static approximation can be applied to successfully formulate the problem and predict the physical parameters of the antenna. A full-wave numerical technique (using CST STUDIO SUITE 2009 [7]) is applied to comprehensively model the structure and validate the derived theory. We demonstrate that a volume of negative permittivity material placed on a ground plane and fed by a coaxial transmission line can produce a small antenna whose operating frequency depends on material properties and the height to width ratio of the volume. The Q-factor of the different shaped radiators are numerically studied and compared to the calculated values based on the reported equations in the recent literatures [8]–[14]. II. RESONANCE FORMULATION Fig. 1 shows the geometry of an ellipsoid with semi-axes , , and located in free-space and illuminated by an arbitrary

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AHMADI et al.: RESONANCE AND Q PERFORMANCE OF ELLIPSOIDAL ENG SUBWAVELENGTH RADIATORS

polarized electric field . It is assumed that the ellipsoid has a material permittivity of , and a size which is much smaller than the free-space wavelength . A rigorous static analysis gives the field inside the ellipsoid as [15], [16]

(1) where the mined from

are the depolarization factors deter-

(2) play a critical role As can be seen, the depolarization factors in determining the induced electric field. Note that if the applied electric field is initially uniform, the resultant field within the ellipsoid is also uniform. Another observation is that the polarization of the induced field can, in general, be different from that of the applied field. The three depolarization factors for any ellipsoid satisfy (3) The sphere has three depolarization factors, each equal to 1/3, and the internal field is aligned with the applied field, either in the same or opposite direction. Other special cases are an oblate spheroid with , and a prolate spheroid with . Closed-form expressions for the integral (2) can be derived for these cases. For oblate spheroids we have [16] (4a) (4b) where the eccentricity is spheroids we have

. For prolate

(5a) (5b) where . The practical utility of the spheroidal cases lies in the oblate spheroid degenerating into a flat disk becomes very small ; and the prolate spheroid as becomes very large approaching a rod-shaped structure as . To investigate some of the important physical implications of (1), consider a metamaterial ellipsoid located in free-space

707

-polarized applied electric under the influence of a uniform simplifies to field. For the spherical case, the internal field (6) When the permittivity, , of the sphere is larger than the permittivity of the free space in which the sphere is assumed to direction, and the reside, the sphere is depolarized along the total internal field reduces as the permittivity is increased. However, for the sphere permittivity below the outside material value (vacuum), the sphere can be polarized along the excitation ( direction), and near , one can establish a resonance with strong field intensity inside the sphere (independent of the size , the polarization of induced field of sphere). Below to and, as the permittivity becomes switches again from very large negatively, the induced field tends to cancel the external field producing nearly a zero total field inside the sphere. Changing the shape of the ellipsoid has some interesting effects on the resonance performance. For instance, if the geometry deforms from the sphere with polarization factors (1/3,1/ 3,1/3) into a flat disk with those of (0, 0, 1), (1), shows that the . Thus, the thin internal field is simplified to . Hence, altering the disk becomes resonant at around antenna shape from that of a sphere to a disk will shift the reto . It is quired permittivity for resonance from also found from (1) that by changing the shape of the antenna from a sphere to an increasingly long rod (along z), the permitto tivity required to produce a resonance shifts from . III. CALCULATION OF THE LOWER BOUNDS ON Q So far, we have concentrated on the resonance characteristics of the metamaterial-based ellipsoid. The bandwidth, which is inversely proportional to the Q of an antenna [8], is another important consideration for practical electrically small antennas. The concept of lower bounds on the Q of electrically small antennas was first introduced by Wheeler [1] and Chu [2]. According to Chu [2], the minimum Q that one can achieve for an antenna confined to a spherical volume of radius obeys , , which the relationship means that decreasing the electrical size of the resonator increases its Q and narrows its bandwidth. Recently, attention has been drawn to the subject of the lower bounds on the Q for antennas confined to arbitrarily shaped volumes. Gustafsson et al. determined physical bounds on antennas of arbitrary shape [10], [11] using an approach based on fundamental principles of causality, time-translated invariance, and reciprocity applied to a general set of linear constitutive relations via a sum rule [11]. More recently, Yaghjian et al. have shown that the minimum possible Q for an electrically small dipole antenna confined to an arbitrary volume will be the Q of a PEC scatterer filling subject to a uniform incident electric field. This lower bound Q can be expressed in terms of the direction of the electric dipole , and the electrostatic polarizability dyadic moment of the PEC volume [12], [13] (7a)

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which for a principal direction of the volume

, becomes (7b)

(7a) and (7b) apply to linear electric or magnetic dipole antennas whose exciting sources can be both electric currents and magnetic currents (polarization M) outside the “antenna-proper.” For electric-dipole antennas confined to an electrically small volume excited by electric-currents only, such as the antennas in this paper, the lower bound on the quality factor reduces to [12], [13] (8a) which for a principal direction of the volume

, becomes (8b)

Fig. 2. Characteristics of the Drude permittivity material.

It is often convenient to re-express as , where is a dimensionless “shape factor.” In Sections IV and V, we numerically compute the actual Q for simulated spherical, circular-cylindrical-disk, and circularcylindrical-rod antennas, and then compare these values of Q to the Q lower bounds given in (8b) determined from the shape factors for these antennas. IV. PERFORMANCE ANALYSIS OF ENG ANTENNAS To provide physical insight into these metamaterial-based antennas, their detailed performance characteristics will now be investigated. To form an antenna element, the resonator must be coupled to a transmission line. For each antenna on a ground plane, (half sphere, half disk, and half rod), the antenna is fed by a 50 ohm coaxial transmission line with a small monopole stub. The dimension of the stub is varied in order to find the optimum impedance match. It is assumed that the metamaterial of the antenna has a Drude-dispersive permittivity satisfying (9) where is the bulk resonant frequency deterof the material, and the damping factor mines the loss. The Drude permittivity is plotted in Fig. 2. This Drude metamaterial may be constructed in microwave frequencies with the use of array of subwavelength metallic wires [17], although it features a larger frequency dispersion than a regular wire medium affecting the desired bandwidth (in optics, a plasmonic metal can simply provide the Drude dispersive property). All simulations are performed with the finite integration method using CST [7] with an absorbing boundary condition implemented at a distance of one wavelength from the antenna element. A. Spherical Radiator Fig. 3 depicts the geometry of a hemisphere located on a ground plane. For a large ground plane, the hemisphere can be considered a sphere for modeling purposes. A probe-feed is used

Fig. 3. The geometry of the hemisphere radiator constructed from the Drude dielectric medium.

to excite the resonant mode of the sphere. Since the sphere is as, compared to sumed to have a very small radius wavelength, the above quasi-static discussions can be applied. The optimum stub length and radius are determined through simulation to be 4.5 mm and 1.2 mm, respectively. The antenna input impedance and reflection coefficient (in dB) versus frequency are shown in Fig. 4. A resonance near the frequency is determined and a which is associated with impedance matching at is obtained by tuning the . This demonstrates good agreement with the stub . quasi-static prediction for the resonance frequency at The operating wavelength is 127.1 mm and the stub length is . The diameter of the sphere radiator at the operating about . The resonant frequency of the antenna frequency is about indeed corresponds to that of the fundamental mode of the negative permittivity sphere explained in the previous section. To better understand the physical performance of the sphere, the electric field at the resonant frequency is plotted in Fig. 5(a) (line-fields at some arbitrary moment in the – plane). Note that the normal component of the electric field has different signs inside and outside the sphere due to the negative permittivity value. The negative permittivity sphere acts like an inductor which is in parallel with the dipole-feed capacitor. The dipole-feed capacitor can be used for tuning the antenna resonant characteristic. The antenna radiation pattern shown

AHMADI et al.: RESONANCE AND Q PERFORMANCE OF ELLIPSOIDAL ENG SUBWAVELENGTH RADIATORS

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Fig. 4. The performance of the hemisphere structure: (a) Input impedance, and (b) reflection coefficient and radiation efficiency.

= 2 36 GHz

Fig. 5. Radiator performance at the resonant frequency, f : . (a) E-field pattern in the y –z plane. Note to the depolarized fields inside the sphere, and (b) radiation pattern. It presents a dipole mode of the antenna as expected of the field distribution inside the radiator.

in Fig. 5(b) is that of an electric dipole, as expected from the field distribution inside the radiator. The bandwidth and Q of the antenna are also determined based on (96) of [8]. The 3 dB matched VSWR bandwidth is 6.4% at the operating frequency; therefore, the Q corresponding to the half-power VSWR bandwidth yields a value of about 31.25 at the resonant frequency of the antenna. Fig. 4(b) shows reflection coefficient versus frequency for the same antenna described above with the lossless material. The length of the stub is tuned to be 4.4 mm to improve the impedance matching performance. The Q (for 100% efficiency) corresponding to half-power VSWR bandwidth (5.9%) yields a value of around 33.9 at the resonant frequency of the antenna, which is about 1.51 times the Chu lower bound . Equation (8b) for an antenna with also predicts a Q of 1.5 times the Chu lower bound for a sphere . which has a polarizability of Since the material parameters include loss, a reduction in efficiency is expected. The radiation efficiency is plotted versus frequency in Fig. 4(b). As observed, near the operating bandwidth, the efficiency is nearly flat at a value of about 92%. B. Circular Cylindrical Disk Radiator From a practical point of view, one may be interested in the resonance performance of an ENG disk instead of the sphere,

Fig. 6. The geometry of the disk-shaped Drude permittivity radiator.

the disk being easier to construct. Fig. 6 shows the geometry and height of of a disk with radius of located above a ground plane. The disk is composed of the Drude medium given in (9). Using (1) and (4), a resonance with strong field intensity inside the disk is expected , which corresponds to to occur at based on the Drude material characteristics. (At this frequency, the of the smallest circumscribing sphere is 0.37.) Here the radius and height of the disk are approximated by the major and minor axes of an ellipse, respectively. To obtain an impedance match, the inner stub is given a radius of 0.55 mm and a length of

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Fig. 7. The performance of the disk-shaped radiator: (a) Input impedance, and (b) reflection coefficient and radiation efficiency.

Fig. 8. E-field pattern in the y –z plane for the disk at the resonant frequency, f inductive behavior.

1.5 mm. The resonance performance with impedance is obtained and shown in Fig. 7, matching near and the electric field is shown in Fig. 8. A uniform depolarized field pattern inside the disk is established. It is worth mentioning of the displacement on the that since the normal component surface of the disk is continuous, a small value of permittivity inside the radiator which is the case for this resonant disk produces a very strong internal electric field. The 3 dB matched VSWR bandwidth is about 0.88%. The lossless case of a 1.5 mm stub with a radius of 0.45 mm is also shown in Fig. 7(b). The 3 dB matched VSWR bandwidth of 0.71% at the resonant frequency gives a Q equal to 281.3. The Q of the disk is much higher than that of a circumscribing sphere because the disk occupies a much smaller volume than its circumscribing sphere. To calculate the Q factor of the disk based on (8b), one needs first, which can be estimated from the shape to calculate that computes to 1.97 for a disk with height to width factor ratio of 1/3 [14]. From (8b), this shape factor gives a Q lower bound of 83.3. Our numerical calculation for the Q of the disk is much higher than the theoretical lower-bound prediction. This can be mainly because of the antenna shape and the type of its excitation. In other words, since the elecmust be continuous on the surface of the tric flux density

= 3:42 GHz. Note to the strong field depolarization inside the disk proving large

Fig. 9. The geometry of the rod-shaped Drude permittivity radiator.

disk, and since the antenna goes to the resonance for close to zero, the field concentration inside the disk must be very high . This will result in determining a very high quality factor (much larger than the Q lower bound). Special attention should be made for proper excitation of an antenna with a specific shape to achieve the Q lower bound.

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Fig. 10. The performance of the rod-shaped radiator: (a) Input impedance, and (b) reflection coefficient and radiation efficiency.

Fig. 11. E-field pattern in the y –z plane for the rod at the resonant frequency, f

The efficiency curve plotted in Fig. 7(b) shows a nearly flat efficiency across the operating bandwidth at a value of about 72% for the lossy antenna. , By increasing the aspect ratio of the disk the resonance frequency will move up toward the region that the permittivity is close to zero. This can be obtained from (1) and (4), where the depolarization factor is increased by increasing the aspect ratio of the disk. Hence, a thin disk is expected to resonate if it is made out of an epsilon near zero (ENZ) medium. Basically, as mentioned earlier, the subwavelength structure with negative permittivity can be viewed as an inductor in parallel with the dipole feed capacitor. Since near the resonant frequency, a strong field depolarization occurs inside the radiator, a large value of equivalent inductance is produced, thereby providing an inductive input impedance behavior for the antenna that can be tailored by changing the radiator shape and optimizing the feeding system to allow successful antenna matching. The disk like the sphere radiates an electric dipole pattern. C. Circular Cylindrical Rod Radiator The performance of a Drude-material rod-shaped antenna is considered next. The geometry of a long rod with radius and height located above a ground

= 1:13 GHz. plane is shown in Fig. 9. Using quasi-static (1) and (5), a resonance with strong field intensity inside the rod is predicted to which corresponds to at occur at of 0.37. (The height and radius of the rod in this the same calculation are that of an ellipse with these major and minor axes, respectively.) By optimizing the stub to have a radius of 0.5 mm and length of 2.35 mm, the antenna is matched to 50 . The numerical performance obtained ohm near by CST is illustrated in Figs. 10 and 11. The 3 dB matched VSWR bandwidth is 2.88%. The reflection coefficient for the lossless case of a 2.05 mm stub (shown in Fig. 10(b)) gives the bandwidth of 2.16% corresponding to a Q of 92.4, about for an antenna 2.6 times the Chu lower bound with where is the radius of the smallest sphere enclosing the rod. The lower bound for the actual rod volume can be found from (8b) to be 92 after using the computed shape [14]. As obfactor of 10.8 for this rod with served, the lower-bound value is close to the actual simulated value for this rod antenna. The internal field is mostly polarized along the axis of the rod. The efficiency of the lossy antenna shown in Fig. 10(b) is nearly flat across the operating bandwidth at a value of about 77% at the operating frequency of the antenna. Comparing the near-field pattern of the disk and sphere with that of the rod clearly reveals that a stronger field is established

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configurations, although it has a Q of 1.5 times of the Chu lower bound. The Chu lower bound may be achieved by an antenna design with proper excitations of both electric and magnetic polarizations and optimal configurations. ACKNOWLEDGMENT The authors would like to thank Dr. A. D. Yaghjian for sending them preprints of his papers (co-authored with Dr. H. R. Stuart) on quality-factor lower bounds, and for constructively reviewing the manuscript. REFERENCES

Fig. 12. Required negative permittivity for radiator resonation versus ellipsoid aspect ratio.

outside the rod-shaped radiator. This is related to the higher negative permittivity of the material inside the rod compared to the disk and sphere cases. It is very instructive to plot the required negative permittivity for the resonance of an ellipsoidal radiator in terms of its aspect ratio. This can be accomplished using the derived quasi-static (1)–(5). The dependence of the required permittivity on the elis shown in Fig. 12. It is oblipsoid aspect ratio , whereas thin disks served that the sphere resonates at require small-values of negative permittivity for establishing the resonance, and the long-rod resonates at a high negative value of permittivity. The numerical results obtained for the disk, sphere, and rod are in good agreement with this curve. V. DISCUSSION In this paper, the effects of the shape and material dispersion of epsilon negative (ENG) radiators on their resonance characteristics in general, and on their quality factor Q in particular, are investigated. The quasi-static model is applied to theoretically formulate the behavior of spherical, disk-shaped, and rod-shaped resonator antennas. It is demonstrated that for a spherical geom, whereas a thin disk resetry the resonance occurs at onates at smaller negative permittivity and a long rod resonates at a larger negative permittivity. The full-wave numerical technique using CST software is applied to fully characterize the antenna radiator and match its input impedance to its feed line using a monopole stub. Numerically simulated values of the quality factor Q are compared with the Q lower bounds for these different shaped radiators calculated from recently published formulas for the Q lower bounds of electric-dipole antennas confined to an arbitrarily shaped volume. The simulated Qs for the ENG sphere and rod are almost the same as the theoretical calculation, while that for the disk is about 3.38 times of the calculated value (8b). Considering the type of electric dipole excitation, the sphere and rod use the best opportunity (regarding the excitation and the use of volume) to offer the closest Q to the lower bounds. And obviously, the sphere provides the minimum Q between all these

[1] H. A. Wheeler, “Fundamental limitations of small antennas,” Proc. IRE, vol. 35, pp. 1479–1484, 1947. [2] L. J. Chu, “Physical limitation on omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [3] S. R. Best and D. L. Hanna, “A performance comparison of fundamental small-antenna designs,” IEEE Antennas Propag. Mag., vol. 52, no. 1, Feb. 2010. [4] H. R. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1644–1653, Jun. 2006. [5] H. R. Stuart, “Bandwidth limitations in small antennas composed of negative permittivity materials and metamaterials,” presented at the URSI General Assembly, Chicago, IL, 2008. [6] R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2113–2130, Jul. 2006. [7] CST Microwave Studio Ver. 2009.08. [8] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, Apr. 2005. [9] A. D. Yaghjian, “Improved formulas for the Q of antennas with highly lossy dispersive materials,” IEEE Antennas Wireless Propag. Lett., vol. 5, 2006. [10] M. Gustafsson, C. Sohl, and G. Kristensson, “Physical limitations on antennas of arbitrary shape,” Proc. R. Soc. A, vol. 463, pp. 2589–2607, 2007. [11] M. Gustafsson, C. Sohl, and G. Kristensson, “Illustration of new physical bounds on linearly polarized antennas,” IEEE Trans. Antennas Propag., vol. 57, pp. 1319–1327, 2009. [12] A. D. Yaghjian and H. R. Stuart, “Lower bounds on Q for dipole antennas in an arbitrary volume,” presented at the IEEE-APS/URSI Symp., Toronto, 2010. [13] A. D. Yaghjian and H. R. Stuart, “Lower bounds on the Q of electrically small dipole antennas,” IEEE Trans. Antennas Propag., 2010. [14] H. R. Stuart, Privately Supplied Computations of the Shape Factors for Circularly Cylindrical Disks and Rods 2010. [15] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [16] A. Sihvola, Electromagnetic Mixing Formulas and Applications. London: The Inst. Elect. Eng., 1999. [17] H. Mosallaei and Y. Rahmat-Samii, “Composite materials with negative permittivity and permeability properties: Concept, analysis, and characterization,” presented at the IEEE AP-S Int. Symp., Boston, MA, Jul. 8–13, 2001.

Akram Ahmadi (S’06) received the B.Sc. degree in electrical engineering from the University of Tehran, Tehran, Iran and the M.Sc. degree in electrical engineering from the Iran University of Science and Technology, Tehran, in 2002 and 2004, respectively. In 2006, she has joined the Applied EM and Optical Devices Laboratory, Northeastern University, and now is a Ph.D. candidate in the Department of Electrical and Computer Engineering. Her research focuses on analysis and development of metamaterial structures and their applications in subwavelength near-field imaging and antenna devices in microwave and optics. Ms. Ahmadi was the recipient of a student prize paper award at ANTEM 2005.

AHMADI et al.: RESONANCE AND Q PERFORMANCE OF ELLIPSOIDAL ENG SUBWAVELENGTH RADIATORS

Soheil Saadat (S’07) received the B.Sc. and M.Sc. degrees in electrical engineering from Manhattan College, New York, in 2004 and 2006, respectively, and is currently working toward the Ph.D. degree at Northeastern University, Boston, MA. In 2007, he joined the Advanced EM and Optics Devices Laboratory, Northeastern University, where he is currently with the Electrical and Computer Engineering Department. His research interests include theory and applications of metamaterials, non-foster circuit design, optical nanoantennas, millimeter-wave integrated antennas and photonic bandgap structures.

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Hossein Mosallaei (S’98–SM’02) received the B.Sc. and M.Sc. degrees in electrical engineering from Shiraz University, Shiraz, Iran and the Ph.D. degree in electrical engineering from the University of California, Los Angeles (UCLA), in 1991, 1994, and 2001, respectively. From 2002 to 2005, he was a Research Scientist in the EECS Department, University of Michigan. He is currently an Assistant Professor of electrical and computer engineering at the College of Engineering, Northeastern University, Boston, MA. His research focus is on electromagnetic and optical micro/nanoscale metadevices. He has been actively involved in many multidisciplinary governmental sponsored projects such as AFOSR, AFRL, NSF, ONR, and DARPA, as well as industry sponsored projects. His group conducts research in the areas of multi-physics multi-scale computational models and functional RF & photonic components and systems. He is the holder of two U.S. patents. He has authored and coauthored over 100 technical journal articles and conference papers. Dr. Mosallaei is a full member of URSI, and member of the American Association for the Advancement of Science. He is listed in Who’s Who in Science and Engineering, in America, and in the World. He was the recipient, along with his students, of student prize paper awards at AP-S 2000, ’01, ’03 and ’05, URSI Young Scientist Award in 2001, and RMTG award in 2002. His student won the Northeastern Dissertation-Writing Fellowship Award in 2010.

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Multi-Functional, Magnetically-Coupled, Electrically Small, Near-Field Resonant Parasitic Wire Antennas Chia-Ching Lin, Student Member, IEEE, Peng Jin, Member, IEEE, and Richard W. Ziolkowski, Fellow, IEEE

Abstract—Several electrically small antenna systems that utilize the magnetic couplings between a coaxially-fed semi-loop antenna and capacitively-loaded loop (CLL)-based near-field resonant parasitic (NFRP) elements are presented. Both one and two gap CLL elements are considered; their impact on the system performance, particularly their effects of the resonance frequencies and the corresponding values, is evaluated. By integrating multiple NFRP CLL elements with the coaxially-fed semi-circular loop antenna, electrically small multi-band systems are achieved. They can be designed for a broad range of frequencies by tuning the NFRP elements separately. Dual band designs are reported that achieve operation at the GPS L1 (1.5754 GHz) and L2 (1.2276 GHz) frequencies. Their operational modes and performance characteristics are studied. These lead to additional electrically small antenna designs, which feature only one driven loop antenna and two NFRP CLL elements and which achieve circularly polarized (CP) operation. at the GPS A CP antenna whose electrical size is L1 frequency is presented in detail. Its simulated bandwidth and beamwidth, for which the axial ratio (AR) is less than 3 dB, are, respectively, 7.8 MHz and 76 .

= 0 495

Index Terms—Axial ratio, circular polarization, electrically small antenna, GPS antenna, metamaterial, factor.

I. INTRODUCTION

A

RTIFICAL materials are designed and constructed to have desired, but unusual electromagnetic properties that are not generally available in nature. The emergence of metamaterials (MTMs) and their applications to electrically small antennas (ESAs) has provided a design methodology to achieve matching, high overall efficiencies, and low values. Analytical studies of MTM-engineered electrically small antennas have been reported recently [1]–[4]. It has been demonstrated that a properly-designed homogeneous, isotropic, dispersive spherical shell which surrounds an electrically small antenna can achieve a radiating system that has nearly complete impedance matching to the source and high overall efficiencies. Manuscript received October 18, 2009; revised July 29, 2010; accepted September 09, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by DARPA Contract number HR0011-05-C-0068 and in part by ONR Contract number H940030920902. C.-C. Lin was with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721-0104 USA. He is now with Ruckus Wireless, Sunnyvale, CA 94085-2920 USA. P. Jin was with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721-0104 USA. He is now with the Signal Integrity Group, Broadcom Corporation, Irvine, CA 92617 USA. R. W. Ziolkowski is with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721-0104 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.2010.2103008

More specifically, they were achieved by surrounding small electric or magnetic dipole radiating elements with an ENG (epsilon-negative) or MNG (mu-negative) spherical shell. It has been verified that an MNG metamaterial (MTM) composed of a periodic array of split ring resonators (SRRs) built from nonmagnetic thin sheets of metal exhibits negative permeability within a certain frequency range [5]. Several varieties of electrically small resonant antennas composed of SRRs or capacitively-loaded loops (CLLs) and monopoles have been reported [6]–[14]. The main difference between the SRR and CLL elements is that the SRRs consist of a pair of concentric split annular rings and the CLLs consist of only one split ring. The pairing in an SRR produces additional capacitance to lower its resonance frequency; the gap region of the CLL can have an augmented design to enhance its capacitance. Thus, one can view the CLL as a simplification of the SRR. A CLL-based metamaterial block was designed to act as a volumetric artificial magnetic conductor (AMC) [6] and then was used as the enabling technology for low-profile antennas [6], [7]. Several other CLL or SRR configurations have been considered to improve various performance characteristics of antennas. For instance, the metasolenoid [15], [16] was investigated to improve the impedance bandwidth of PIFA and patch antennas. Miniaturization and bandwidth improvements of patch antennas with various MNG unit cells have been considered [17]–[19]. Dual-frequency and other multi-functionalities have been investigated with split-rings and complementary split-rings [20], [21]. Higher directivity with a split-ring-based mu-near zero substrate has been considered [22]. The 3D magnetic-based EZ antenna introduced in [23] is composed of the electrically small loop antenna integrated with an extruded CLL element. This 3D CLL structure is designed to be a near-field resonant parasitic (NFRP) element. Measured results for the 3D magnetic EZ antenna at UHF [24] and VHF [25] frequencies have demonstrated that it provides nearly complete matching to the 50 source and a high overall efficiency (OE), i.e., the ratio of the total radiated power to the total input power. These results demonstrate that CLL-based elements can work over a very wide range of frequencies. While negative permeability can not be ascribed to the CLL element itself because it is essentially the inclusion for one unit cell and not an entire periodic structure, resonant CLL structures still provide a tuning capability to match an electrically small antenna to the source and act as a parasitic to aid in the radiation process to achieve high radiation efficiencies. Because these CLL-based antennas are engineered by controlling the strong magnetic flux generated by the small driven loop antenna and turning it into the currents flowing on the CLL element, we term this excitation process a magnetic-coupling.

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Circularly polarization is another exciting issue for electrically small antenna applications. Loop antennas such as circular, square, triangular, rectangular, and rhombic loops are usually used as linearly polarized (LP) antennas [26]. There is [27] a useful way to distinguish two different types of circularly polarized (CP) electrically small antennas. The first one, Type 1, uses a superposition of two orthogonal electric dipoles or equivalently two orthogonal magnetic dipoles driven in phase quadrature. For instance, crossed dipoles, crossed slots, turnstile antennas, and square (or circular) microstrip patches with feeds that excite two orthogonal modes are Type 1 CP antennas. The other one, Type 2, utilizes a superposition of two collinear electric and magnetic dipoles. Examples of Type 2 CP antennas include dipole/loop antennas, monopole/slot antennas, Alford loop/monopole antennas, short helix antennas, and the spherical helix antenna [28]. An electrically small CP antenna based on NFRP CLL elements has not been demonstrated previously. In this paper, a directly driven semi-loop antenna, coaxially fed through a finite ground plane, is integrated with several different types of NFRP CLL elements to achieve electrically small single-band and dual-band, LP and CP systems. Several types of magnetically-coupled CLL-based antennas are introduced in Section II. Performance comparisons of these structures are given. The proposed dual-band designs, which work at the GPS L1 and L2 frequencies, will be introduced in Section III. The crucial dual-band design factor that we will exploit to achieve the CP antenna designs will be discussed. The electrically small, magnetically-coupled, CP CLL-based NFRP antenna will be discussed in Section IV. Section V will summarize our results. We note that all of the antenna designs were numerically simulated using ANSOFT’s High Frequency Structure Simulator (HFSS) version 11.1.3. Real copper was used in all of the simulations, the conductivity being set to . Additionally, the ground plane size is fixed in all of the designs by the HFSS requirement that the radiation box walls be a quarter-wavelength (for the lowest frequency considered) away from the radiating elements. We note that all of the measured results for the EZ antennas [24], [25] have demonstrated good agreement with these simulation models, even when much smaller ground planes were utilized. II. ANTENNA DESIGNS USING MAGNETIC COUPLING METHODOLOGIES A. One Gap Case The metamaterial-inspired, electrically small 3D magnetic EZ antenna has several interesting performance characteristics. sized version For instance, as demonstrated in [24], a has approximately a 95% OE value at 300.96 MHz with a fractional bandwidth (based on the half-power VSWR bandwidth) , where of 1.66% and a corresponding , equals the radiation efficiency (RE) the lower bound on , times the Chu lower bound, i.e., [28], [29]. Note that the value of is defined in terms of the , where the free space wavefree space wave number is , being the length at the resonance frequency speed of light in vacuum, and is the radius of the smallest enclosing sphere. The rule of thumb on how to achieve the

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lowest , however, is that an antenna should occupy as efficiently as possible the volume of the minimum enclosing sphere [29], [30]. In fact, it has been demonstrated mathematically [31] that the minimum is obtained with an antenna that occupies a spherical volume rather than a dipole-like or planar structure. To verify this idea, we have redesigned the original 3D EZ antenna with a cylindrical CLL element instead of the original rectangular one. A quartz slab was inserted into the gap region of the parasitic CLL element. The selection of quartz as the dielectric material is not only for its high dielectric constant value (real ), but also for its low part of the relative permittivity, dielectric loss tangent . As shown in [24], [32], [33], the resonance frequency of such as EZ antenna is given by the expression, (1) and are, respectively, the effective inductance where and capacitance of the system. Consequently, one can scale the antenna design to other frequencies simply by changing these effective values. The choice of quartz for the spacer in the capacitive gap of the CLL element thus increases its effective capacitance and, hence, lowers its resonance frequency. It also provides structural stability in the fabricated system [24], [25]. Other low-loss dielectrics could be used as the gap spacer to achieve different resonance frequencies, as well as by adjusting the stub height or the gap length [25]. The predicted at 299.91 MHz (2.76% fractional bandwidth) was decreased 34.5 % from its value in the rectangular case while maintaining a 99.5% overall efficiency and a maximum gain value of 5.78 dB. In the same manner, the corresponding spherical case was also obtained. The predicted at 300.2 MHz (3.65% fractional bandwidth) was decreased 50.3% from its value in the rectangular case and 24.2% from its value in the cylindrical EZ case. The overall efficiency was 98.7% and the maximum gain value was 4.53 dB. As anticipated, filling the enclosing sphere more efficiently reduced the Q value without significantly changing any other performance characteristic. Since an additional simplification of the design for the modified 3D magnetic EZ antenna was desired, we replaced the extruded cylindrical CLL element with a split thick wire. The proposed magnetically-coupled antenna with a one-gap CLL NFRP element is shown in Fig. 1. The quartz-filled cylindrical gap was rotated from the middle to the end of the wire-based CLL element to potentially simplify the fabrication process. The electrically small semi-circular loop antenna generates a magnetic flux in the x direction, normal to the plane of the CLL element. This time-varying magnetic flux in turn induces currents on the CLL element and produces large electric fields across the gap. These strong electric fields generate the effective capacitance of the CLL element. The total effective inductance is the sum of the inductance due to the current path in the CLL element and the inductance of the electrically small semi-circular loop antenna. Thus, the CLL element can be matched to the reactance part of the electrically-small semi-loop circular antenna. Because the simulated surface current is not symmetrical along the entire CLL element shown in Fig. 2(a) (the magnitude of the surface

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Fig. 1. Wire-based antenna which consists of a one-gap CLL NFRP element that is magnetically-coupled to the coaxially-fed semi-loop antenna. All dimensions are in mm: R1 = 36:0, R2 = 66:4, R1 wire radius = 3:5, R2 wire radius = 2:0, and gap = 1:15.

Fig. 3. HFSS-predicted values for the one-gap, magnetically-coupled CLLbased NFRP wire antenna shown in Fig. 1. (a) Complex input impedance, and (b) magnitude of S .

Fig. 2. HFSS-predicted values at the resonance frequency, 299.864 MHz, for the one-gap, magnetically-coupled CLL-based NFRP wire antenna shown in Fig. 1. (a) Magnitude of the surface current vectors on the wire elements, and (b) gain patterns (dB) in the xz-and yz-planes.

current is proportional to the marker size), the system produces the correspondingly asymmetric gain pattern shown in Fig. 2(b). and . The maximum gain is 3.49 dB at and input impedance The corresponding HFSS-predicted values are shown in Fig. 3. The input impedance exhibits an at the first critical point anti-resonance behavior of this magnetic-based antenna. However, the real

part of the input impedance is much larger than 50 at this first resonance and, consequently, the antenna is not impedance matched to the source there. This is in contrast to previous 3D magnetic EZ antennas which were operated at their first anti-resonance [33]. On the other hand, the one-gap magnetically-coupled CLL-based NFRP antenna is nearly completely matched , at to the source at its second critical frequency . The HFSS-predicted which it has a resonance . frequency for this second resonance was Consequently, the antenna is electrically small at this resonance . The HFSS-presince there its electrical size was value at the resonance frequency was dicted minimum 27.1 dB, and the corresponding overall efficiency was 97.8%. The predicted fractional bandwidth was 2.77%. This means the . We note that because the anti-resonance and resonance frequencies are very close to each other, we could have elected to tune the antenna to be matched to the source at the first, anti-resonance frequency. However, we have found that the bandwidth at an anti-resonance frequency is always less than that obtained at a resonance frequency. B. Two Gap Case An electrically small antenna system in which the CLL element has two gaps, but remains magnetically-coupled to the driven semi-circular loop, is illustrated in Fig. 4. The two gaps are introduced symmetrically at the both ends of the CLL element and are again filled purposely with quartz. Again, the

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Fig. 4. Two gaps, magnetically-coupled CLL-based NFRP wire antenna. All dimensions are in mm: R1 = 26:0, R2 = 66:4, R1 wire radius = 2:5, R2 wire radius = 2:0, and Gap = 0:165.

Fig. 5. HFSS-predicted jS j values for the two gap, magnetically-coupled CLL-based NFRP wire antenna.

electrically small semi-circular loop antenna generates a strong time-varying magnetic flux that induces a current on the CLL element and produces correspondingly large electric fields across the gaps. The current on the CLL element enhances the radiation process; the electric fields across the gaps generate the capacitance that allows impedance matching to the source. values are shown in Fig. 5 for the The HFSS-predicted two-gap, magnetically-coupled CLL-based NFRP wire antenna whose dimensions are given in Fig. 4. It has a similar impedance behavior to that of the one gap case. It could be matched either at the first anti-resonance or resonance frequencies. However, we again opt for the larger bandwidth behavior and match at the res, where the electrical onance frequency, . The HFSS-predicted size of the entire system is value at the resonance frequency was 32.9 dB. minimum The corresponding overall efficiency was 96%; the fractional . The HFSS-prebandwidth was 1.88%, giving dicted symmetric gain patterns due to the symmetric structure are shown in Fig. 6. Additionally, these patterns clearly demonstrate that this ESA system is radiating as a magnetic dipole over a finite ground plane. We further note that, as shown in Fig. 6(a), the currents in the ground plane are very localized in the neighborhoods of the wire connections to the ground plane.

Fig. 6. HFSS-predicted values at the resonant frequency, 300.4219 MHz, for the two-gap, CLL-based NFRP antenna. (a) Magnitude of the surface current on the wire elements and ground plane, and (b) gain patterns (dB) in the xz-and yz-planes.

C. Parameters Studies of the Two Gap Case An investigation of the radiation efficiency and versus value was conducted for the two-gap magnetically-coupled CLL-based NFRP wire antenna. Its radiation behavior is determined primarily by the NFRP element. Consequently, as we values, we maintained the wire radius of the changed the CLL element to be 2.0 mm to make the comparisons fair. The results are preHFSS-predicted radiation efficiency versus values at and just below 0.5, the radiasented in Fig. 7. For tion efficiency remains in excess of 95%. However, it diminishes value was decreased quickly and goes below 10% when the below 0.1. This behavior was expected because we know that the radiation resistance of a small loop antenna with is proportional to [26], where is the radius of the loop. Furthermore, the free space portion of the loop to the loop area based on its outer radius finally begins to decrease as decreases when the R2_wire_radius is fixed. Consequently, at very small values of , the driven semi-loop creates much less flux and the CLL element becomes much less capable of capturing whatever flux is produced. Thus, the radiation efficiency actually decreases more rapidly than expected. This behavior also

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Fig. 7. Radiation efficiency versus ka values for the two-gap magneticallycoupled CLL-based NFRP wire antenna shown in Fig. 4.

Fig. 8. Q values versus ka values for the two-gap, magnetically-coupled CLL-based NFRP wire antenna.

Fig. 10. HFSS-predicted jS j values versus the rotation angle  of the driven semi-loop antenna shown in Fig. 9.

Fig. 11. HFSS-predicted jS j values when the relative orientation angle between the driven semi-loop and the plane containing the CLL element is  = 45 , with and without fine tuning to achieve nearly complete matching to the 50 source. TABLE I DESIGN PARAMETER COMPARISON FOR TWO DIFFERENT VALUES OF THE RELATIVE ORIENTATION ANGLE 

Fig. 9. Configuration of the two-gap, magnetically-coupled CLL-based NFRP wire antenna when the driven semi-loop is rotated  with respect to the plane of the CLL element. (a) 3D view, and (b) top view.

impacts the . The HFSS-predicted values versus are shown in Fig. 8. The decreases as decreases until begins to increase the latter is very small and then the again. Further analysis revealed that when it is not too small, the antenna efficaciously occupies a larger percentage of the minimum spherical volume than when it is physically bigger (but initially is smaller still electrically small). Consequently, decreases. However, because the radiation efficiency deas creases much more rapidly, it eventually begins to increase. One for observes that the cross-over point occurs around the antenna under consideration. In Section II-B, the CLL element was oriented in the best way to capture the entire magnetic flux produced by the driven semi-loop. Therefore, it can provide the best near-field magnetic coupling between the driven loop and the CLL element. To illustrate the importance of this relative orientation, the driven loop with different rotation angles , as shown in Fig. 9, was studied.

As shown in Fig. 10, when the angle was increased from changed from 32.8 dB to 0 to 90 degrees, the value of , nearly all of the incident power is 0 dB, i.e., when reflected because there is no magnetic coupling between the driven semi-loop and the CLL element. At that angle, the reactance will only be inductive because no electric field will be induced across the capacitive gaps. Consequently, the overall system can not create the necessary capacitance to achieve in, the HFSS-preternal matching to the source. When at the resonance frequency 292.9 MHz was dicted value of 8.9 dB. Although it does not have a good impedance matching at that resonance frequency, there is still enough magnetic coupling that the CLL element and the driven semi-loop could be

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COMPARISON OF

Q

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TABLE II VALUES FOR THE VARIOUS

readjusted in size to achieve nearly complete matching. The simplest way to accomplish this is to change the gap size and the radius of the driven semi-loop to obtain the necessary capacitance. However, once the angle increases beyond 45 , the magnetic coupling is too small and impedance matching becomes too difficult to achieve. The HFSS-predicted values of for after fine tuning are shown in Fig. 11. The detailed design parameters are shown in Table I. and cases shows A comparison of the values are similar even though we changed the that their design parameters to re-attain impedance matching. This was not entirely expected because these design parameter changes have an impact on the effective capacitance and inductance in the system. However, this helped confirm that the CLL element, which was not changed, truly dominates the radiation mechadoes not vary much when only the nisms. As a result, the driven element parameters are varied some. Understanding how the magnetic coupling affects the performance characteristics of the system is critical to the following dual-band and circularly polarized antenna designs. The performance characteristics of the single band antenna systems are summarized in Table II. The radiation and overall efficiencies in all of these cases is very high. In the 3D cases, is clearly achieved by the 3D spherical EZ the lowest antenna case. However, if one compares the wire antennas, the smaller than the 3D cylinasymmetric one-gap case has a drical EZ case, even though it does not fill the minimum sphere as efficiently. Again looking at the current distributions on the NFRP elements, the asymmetric CLL element case is the one that has both monopole-like and loop current components on it, while only loop current distributions are generated in the symmetric CLL element cases. III. DUAL-BAND DESIGNS Electrically small dual-band antennas have been obtained by integrating multiple CLL-based NFRP elements with a coaxially-fed semi-circular loop antenna [34]. A similar approach can be applied to the single band wire-based antennas discussed in Section II. The proposed dual-band CLL-based NFRP wire antenna is shown in Fig. 12. It is designed for operation in the GPS L1 and L2 frequency bands. It produces the desired dual-band operation by introducing two two-gap NFRP CLL elements with an electrically-small semi-circular loop antenna that is coaxially-fed through a finite PEC ground plane. These two NFRP

ka = 0:43 SINGLE BAND, CLL-BASED NFRP DESIGNS

Fig. 12. Geometry of the magnetically coupled dual-band antenna that consists of two, two-gap NFRP CLL elements excited by a coaxially-fed semi-loop , , antenna. All dimensions are in mm: , , , , , and .

R1 = 4:0 R1 wire radius = 1:0 R2 = 10:0 R2 wire radius = 1:0 R3 = 18:5 R3 wire radius = 1:0 G1 = 0:0664 G2 = 0:27

CLL elements are excited directly by the magnetic flux produced by the semi-loop antenna. The larger (smaller) one is responsible for the lower (higher) frequency band of operation. However, because there is a strong flux linkage between the two CLL elements, their individual resonance frequencies are shifted from their single-band operational values. values, along with The HFSS-predicted magnitude of the the surface vector current distributions at the indicated resonance frequencies, and the corresponding gain patterns are given in Figs. 13 and 14, respectively. When the currents on the two CLL elements are in-phase, they act as two capacitors in parallel and thus produce a larger overall effective capacitance. Consequently, these in-phase current distributions create the lower resonance frequency. On the other hand, when the currents on the two CLL elements are out-of-phase, they act as two capacitors in series and thus produce a smaller overall effective capacitance. Therefore, they produce the higher resonance frequency. The gain patterns for each resonance frequency are similar to their single-band two-gap case counterparts. The maximum gains are 4.96 dB and 6.32 dB for the 1.225 GHz and 1.575 GHz resonance frequencies with 95.9% and 92.4% radiation efficiencies, respectively. Note that, as in the single-band designs, the patterns are being generated by horizontal magnetic dipoles oriented along the x-axis over the finite ground plane.

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Fig. 13. HFSS-predicted jS j values for the dual-band CLL-based NFRP wire antenna shown in Fig. 12. The surface current vector distributions are given at the indicated resonance frequencies.

Fig. 15. The geometry of the dual-band magnetically-coupled CLL-based NFRP wire antenna with the rotated driven semi-loop. (a) Side view. The two CLL elements are located in the YZ plane. All dimensions are in mm: R1 = 5:3, R1 wire radius = 1:0, R2 = 10:0, R2 wire radius = 1:0, R3 = 18:5, R3 wire radius = 1:0, G1 = 0:0663, and G2 = 0:27. (b) Top view showing the angle of rotation, , of the coax-fed semi-loop.

Fig. 14. Gain patterns (dB) in the xz- and yz-planes for the dual-band CLLbased NFRP wire antenna at the GPS L1 and L2 resonance frequencies shown in Fig. 12.

As shown in Section II, the driven semi-loop can be rotated relative to the NFRP and still provide the necessary magnetic flux to have the NFRP generate the necessary capacitance to achieve internal impedance matching and generate sufficient current strength to maintain the radiation efficiency. One can design a dual band antenna with the same approach. This structure is shown in Fig. 15. The rotation angle, optimized to pro. Note that vide the necessary impedance matching, is an increase in the rotation angle acts in the same manner as a decrease in the radius of the driven loop. The most efficient way to retune the system when the driven semi-loop is rotated is to modify its outer radius. A comparison of the dual-band cases with and without rotation of the driven semi-loop shows that the radius of the latter had to be increased from 4.0 mm to 5.3 mm, which is similar to the increase in the single band case. values are shown in The associated HFSS-predicted Fig. 16. The minimum values achieved were all below 20 dB at the predicted resonance frequencies: and . The corresponding radiation efficiencies were 95.7% and 91.7%, respectively. The surface current vector distributions on CLL elements 1 and 2 again show

in-phase behavior for the lower resonance frequency and out-ofphase behavior for the higher resonance frequency, respectively. The corresponding gain patterns are similar to that of Fig. 14 and have maximum values equal to 4.96 dB and 6.3 dB, respectively. Comparing these results to the unrotated design, it becomes clear that the overall radiation behavior is dominated by the CLL elements instead of the driven semi-loop even when the angle is varied. Nonetheless, we have found that the rotation angle of the driven loop actual provides more flexibility to tune the multiband antenna designs. Moreover, we note again note that the currents are highly localized near the wire connections to the ground planes. The rotated driven semi-loop results suggest that one can produce a horizontal magnetic dipole in a direction at an angle with respect to the source simply by rotating the parasitic CLL element. Now consider placing the two, two-gap CLL elements in orthogonal planes and exciting them with a single rotated semi-loop as shown in Fig. 17. By rotating the semi-loop, it can still produce enough flux in both CLL elements to have them go into resonance. One CLL element is designed for the GPS L1 frequency band, the other for the L2 frequency band. This dual-band configuration produces two orthogonal, horizontal magnetic dipoles, one along the x-axis and one along the y-axis. The rotation angle, optimized to provide the necessary . impedance matching of the system to the source, is values and the surface current The HFSS-predicted vector distributions at the resonance frequencies for this optimized configuration are shown in Fig. 18. The performance characteristics of this dual band design are: the minimum

LIN et al.: MULTI-FUNCTIONAL, MAGNETICALLY-COUPLED, ELECTRICALLY SMALL, NFRP WIRE ANTENNAS

Fig. 16. HFSS-predicted jS j values for the dual-band CLL-based NFRP, rotated semi-loop, wire antenna shown in Fig. 15. The surface current vector distributions are given at the indicated resonance frequencies.

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Fig. 18. HFSS-predicted jS j values for the dual-band two orthogonal, two-gap CLL-based NFRP, rotated semi-loop, wire antenna shown in Fig. 17. The surface current vector distributions are given at the indicated resonance frequencies.

one observes that each CLL element radiates in a LP magnetic . dipole mode, each having its maximum value at From the surface current vector distribution plots in Fig. 18, one observes that the current is almost entirely on only one parasitic CLL element at each resonance frequency. In particular, only the bigger CLL element radiates at 1.227 GHz and only the smaller CLL element radiates at 1.572 GHz. This basically independent behavior is different from that of the previous dual band designs in which the two CLL elements are strongly coupled to each other. This point is critical to the following CP design because a low level of coupling strength between the parasitic CLL elements leads to the capability to individually tune each NFRP element. IV. ELECTRICALLY SMALL CIRCULARLY POLARIZED DESIGNS

Fig. 17. Geometry of the dual-band magnetically-coupled two orthogonal two-gap CLL-based NFRP wire antenna whose coax-fed semi-loop being rotated  with respect to the ZY-plane. (a) 3D view. All dimensions are in mm: R1 = 5:0, R1 wire radius = 1:0, R2 = 10:6, R2 wire radius = 1:0, R3 = 14:0, R3 wire radius = 1:0, G1 = 0:105, and G2 = 0:098. (b) Top view showing the angle of rotation, , of the coax-fed semi-loop.

and the radiation efficiency equals 93.1% and the at 1.227 GHz, and the minimum radiation efficiency equals 94.7% at 1.572 GHz. The corresponding gain patterns are similar to that of Fig. 14 with the maximum gains being 4.76 dB and 5.96 dB for the GPS L2 and L1 bands, respectively. From the corresponding gain patterns,

As discussed in the introduction, one can achieve a circularly polarized electrically small antenna with one of two approaches [27]. The first one utilizes a superposition of two orthogonal electric dipoles or equivalently two orthogonal magnetic dipoles driven in phase quadrature. The second one uses a superposition of collinear electric and magnetic dipoles. We use the first approach here to obtain the desired CP design. We start with the dual band, two orthogonal, two-gap CLL NFRP wire antenna with the rotated coax-fed semi-loop design. Simply by making the two CLL elements have similar dimensions, we can generate two equivalent, orthogonal magnetic dipoles, one along the x axis and the other along the y axis. What remains to achieve the CP behavior is generating the needed phase quadrature between them. From the discussion in Section II, we know that a wire loop driven antenna can have a resonance or an anti-resonance. By designing the CLL elements and the semi-loop properly, we can design one to operate on its resonance mode and the other work on its anti-resonance mode, i.e., these elements are tuned to operate at slightly different frequencies and . Then we can bring these two resonant modes close to each other and create the impedances and , which are introduced by the anti-resonance at and the resonance at , respectively. This approach is facilitated by the fact that the resistances generally vary much but more slowly than the reactances do, i.e.,

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Fig. 20. HFSS-predicted values for the GPS L1, CLL-based NFRP CP wire j values, and (b) axial ratio. antenna shown in Fig. 19. (a) j

S

Fig. 19. Geometry of the GPS L1, CLL-based NFRP CP wire antenna whose with respect to the ZY-plane. (a) 3D coax-fed semi-loop is rotated  : , : , view. All dimensions are in mm: : , : , : , and : . (b) Top view showing the angle of rotation, , of the coax-fed semi-loop.

= 30 R1 = 5 9 R1 wire radius = 1 0 R2 = R3 = 14 0 R2 wire radius = R3 wire radius = 1 0 G1 = 0 193 G2 = 0 182

. Under this circumstance, we generate the phase difference between these two parasitic CLL elements required for CP operation. The proposed structure, which is shown in Fig. 19, is a simple and electrically small CP antenna. Only one driven loop antenna was used to excite the two parasitic orthogonal CLL elements simultaneously. These two CLL elements are the same size and are connected to each other simply because we want to efficiently occupy the radian sphere and, hence, obtain the maximum possible bandwidth. The two loops are designed to have different resonance frequencies by adjusting the sizes of their capacitive gaps. These two resonance frequencies, 1.5695 GHz and 1.5865 GHz, are closely spaced. values are shown in Fig. 20(a). The The HFSS-predicted . operating frequency is 1.5754 GHz, which means The 10 dB bandwidth is 30.8 MHz. As shown in Fig. 20(b), this design leads to an axial ratio equal to 0.66 dB at 1.5754 GHz. We note that the bandwidth can be increased further if one does not consider the axial ratio, i.e., there is a trade off between the bandwidth and axial ratio.

Fig. 21. Gain patterns (dB) in the xz- and yz-planes for the GPS L1, CLL-based NFRP CP wire antenna shown in Fig. 19.

The HFSS-predicted gain patterns at 1.5754 GHz are given in Fig. 21. The system has a 96.9% radiation efficiency with a maximum gain of 6.28 dB. The maximum gain occurs at because both of the CLL elements are radiating in a magnetic dipole mode at the operating frequency. They produce an azimuthally symmetric pattern along the xy-ground plane. The

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antenna had only one driven loop with two orthogonal, two-gap NFRP CLL elements having different but nearby resonance frequencies. The overall electrical size of the final CP design was at 1.5754 GHz. This electrically small CP antenna was shown to have a 96.5% radiation efficiency, a 10 dB bandwidth of 30.8 MHz, and a 76 axial ratio beamwidth. This bandwidth and spatial axial ratio are attractive values for many electrically small CP antenna applications. Having developed the dual-band and CP NFRP wire antennas, combinations of these electrically small configurations are currently being investigated. We hope to report on these and other multifunctional designs in the near future. Fig. 22. Spatial axial ratio for the GPS L1, CLL-based NFRP CP wire antenna shown in Fig. 19.

axial ratio as a function of the angle is shown in Fig. 22. These results demonstrate that the system can still maintain CP be, even though havior ( 3 dB axial ratio) down to it is an electrically small antenna. Consequently, the axial ratio beamwidth of the system is 76 . This means that this CP antenna has a very reasonable spatial CP coverage, which is important for any electrically small CP antenna application. V. CONCLUSION We first reported numerical evaluations of the performance of several electrically small, efficient near-field resonant parasitic (NFRP) antennas based on the magnetic coupling of their coaxially-fed semi-loop antenna with single CLL elements. These included the original 3D EZ antenna, the modified EZ antenna, and the CLL-based wire antennas having either one or two capacitive gaps. For all of these antennas, the CLL elements provided the means to achieve nearly complete impedance matching to the source and more than a 90% overall efficiency for values at or slightly above 0.4. The values obtained for these designs varied from 7+ to 4+ by filling the enclosing hemisphere more efficiently. Several electrically small, dual band antennas were also introduced. They utilized two distinct CLL NFRP elements driven by an electrically-small semi-circular loop antenna coaxially-fed through a finite ground plane. The performance of these antennas was designed to operate at the GPS L1 and L2 bands; their performance was characterized numerically. It was demonstrated that the resonance frequencies of these antenna systems can be designed for a broad range of values by tuning separately the two near field resonant parasitic CLL elements. Again, these CLL elements not only provided the tuning ability to achieve nearly complete matching to the source, but they also help produce the high overall efficiencies at both resonant frequencies. Additionally, designs based on rotating the driven semi-loop with respect to the CLL elements were shown to provide an additional flexibility to tune these multiband systems. It led to the final CP antenna design. The electrically small CP antenna design was based on two equivalent magnetic dipoles in phase quadrature. The proposed

REFERENCES [1] R. W. Ziolkowski and A. Kipple, “Application of double negative metamaterial to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag., vol. 51, pp. 2626–2640, Oct. 2003. [2] R. W. Ziolkowski and A. Kipple, “Reciprocity between the effects of resonant scattering and enhanced radiated power by electrically small antennas in the presence of nested metamaterials shells,” Phys. Rev. E., vol. 72, p. 036602, Sep. 2005. [3] R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2113–2130, Jul. 2006. [4] R. W. Ziolkowski and A. Erentok, “At and beyond the Chu limit: Passive and active broad bandwidth metamaterial-based efficient electrically small antennas,” IET Microw., Antennas Propag., vol. 1, pp. 116–128, Feb. 2007. [5] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory. Tech., vol. 47, pp. 2075–2084, Nov. 1999. [6] A. Erentok, P. L. Luljak, and R. W. Ziolkowski, “Characterization of a volumetric metamaterial realization of an artificial magnetic conductor for antenna applications,” IEEE Trans. Antennas Propag., vol. 53, pp. 160–172, Jan. 2005. [7] A. Erentok, D. Lee, and R. W. Ziolkowski, “Numerical analysis of a printed dipole antenna integrated with a 3D AMC block,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 134–136, 2007. [8] K. B. Alici and E. Ozbay, “Electrically small split ring resonator antennas,” J. Appl. Phys., vol. 101, p. 083104, Apr. 2007. [9] O. S. Kim and O. Breinbjerg, “Miniaturized self-resonant split-ring resonator antenna,” Electron. Lett., vol. 45, pp. 196–197, Feb. 2009. [10] O. S. Kim, “Low-Q electrically small spherical magnetic dipole antennas,” IEEE Trans. Antennas Propag., vol. 58, pp. 2210–2217, Jul. 2010. [11] M. Li, X.-Q. Lin, J. Y. Chin, R. Liu, and T. J. C. , “A novel miniaturized printed planar antenna using split-ring resonator,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 629–631, 2008. [12] H. R. Stuart and C. Tran, “Subwavelength microwave resonators exhibiting strong coupling to radiation modes,” Appl. Phys. Lett., vol. 87, p. 151108, Oct. 2005. [13] H. R. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antennas Propag., vol. 54, pp. 1664–1653, Jun. 2006. [14] H. R. Stuart, “Eigenmode analysis of small multielement spherical antennas,” IEEE Trans. Antennas Propag., vol. 56, pp. 2841–2851, Sep. 2008. [15] P. Ikonen, S. Maslovski, and S. Tretyakov, “PIFA loaded with artificial magnetic material: Practical example for two utilization strategies,” Microwave Opt. Technol. Lett., vol. 46, pp. 205–210, Aug. 2005. [16] 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, pp. 1654–1662, Jun. 2006. [17] 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. 135–140, Jan. 2006.

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[18] 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. [19] F. Bilotti, A. Alú, and L. Vegni, “Design of miniaturized metamaterial patch antennas with -negative loading,” IEEE Trans. Antennas Propag., vol. 56, pp. 1640–1647, Jun. 2008. [20] F. J. Herraiz-Martínez, L. E. García-Muñoz, D. González-Ovejero, V. González-Posadas, and D. Segovia-Vargas, “Dual-frequency printed dipole loaded with split ring resonators,” IEEE Wireless Propag. Lett., vol. 8, pp. 137–140, 2009. [21] H. Zhang, Y.-Q. Li, X. Chen, Y.-Q. Fu, and N.-C. Yuan, “Design of circular/dual-frequency linear polarization antennas based on the anisotropic complementary split ring resonator,” IEEE Trans. Antennas Propag., vol. 57, pp. 3352–3355, Oct. 2009. [22] Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wavelike beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett., vol. 94, p. 044107, Jan. 2009. [23] A. Erentok and R. W. Ziolkowski, “An efficient metamaterial-inspired electrically-small antenna,” Microw. Opt. Technol. Lett., vol. 49, pp. 1287–1290, Jun. 2007. [24] R. W. Ziolkowski, C.-C. Lin, J. A. Nielsen, M. H. Tanielian, and C. L. Holloway, “Design and experimental verification of a 3D magnetic EZ antenna at 300 MHz,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 989–993, 2009. [25] C.-C. Lin, R. W. Ziolkowski, J. A. Nielsen, M. H. Tanielian, and C. L. Holloway, “An efficient, low profile, electrically small, VHF 3D magnetic EZ antenna,” Appl. Phys. Lett., vol. 96, p. 104102, Mar. 2010. [26] C. A. Balanis, Antenna Theory, 3rd ed. New York: Wiley, 2005, pp. 231–269. [27] D. M. Pozar, “New results for minimum Q, maximum gain, and polarization properties of electrically small arbitrary antennas,” in Proc. 3rd Eur. Confe. on Antennas and Propagation, Berlin, Germany, Mar. 23–27, 2009, pp. 1993–1996. [28] S. R. Best, “Low Q electrically small linear and elliptical polarized spherical dipole antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 1047–1053, Mar. 2005. [29] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [30] J. S. McLean, “A re-examination of the fundamental limits on the radiation of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, pp. 672–676, May 1996. [31] H. D. Foltz and J. S. McLean, “Limits on the radiation of electrically small antennas restricted to oblong bounding regions,” in Proc. IEEE Antennas Propag. Int. Symp., 1999, vol. 4, pp. 2702–2705. [32] A. Erentok and R. W. Ziolkowski, “Two-dimensional efficient metamaterial-inspired electrically-small antenna,” Microw. Opt. Technol. Lett., vol. 49, pp. 1669–1673, Jul. 2007. [33] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficient electrically-small antennas,” IEEE Trans. Antennas Propag., vol. 56, pp. 691–707, Mar. 2008.

[34] C.-C. Lin and R. W. Ziolkowski, “Dual-band 3D magnetic EZ antenna,” Microwave Opt. Technol. Lett., vol. 52, pp. 971–975, Apr. 2010.

Chia-Ching Lin (S’07) received the B.Sc. and M.Sc. degrees in electronic engineering from National Taiwan University of Science and Technology (NTUST), in 1999 and 2003, respectively, and the Ph.D. ECE degree from the University of Arizona, Tucson, in 2010. He joined Ruckus Wireless, Sunnyvale, CA, in January 2011. His research interests include electrically small antennas and metamaterial applications to antenna designs.

Peng Jin (S’05–M’10) received the B.Sc. EE degree from the University of Science and Technology of China, HeiFei, in 1999, the M.Sc. EE degree from the North Dakota State University, Fargo, in 2004, and the Ph.D. ECE degree from the University of Arizona, Tucson, in 2010. His is currently with the Signal Integrity Group, Broadcom Corporation, Irvine, CA. His research interests include electrically small antennas and metamaterial applications to antenna designs.

Richard W. Ziolkowski (M’97–SM’91–F’94) received the Sc.B. degree in physics (magna cum laude, with honors) from Brown University, Providence, RI, in 1974, and the M.S. and Ph.D. degrees in physics from the University of Illinois at Urbana-Champaign, in 1975 and 1980, respectively. He was a member of the Engineering Research Division, Lawrence Livermore National Laboratory, from 1981 to 1990 and served as the leader of the Computational Electronics and Electromagnetics Thrust Area for the Engineering Directorate. He currently is serving as the Litton Industries John M. Leonis Distinguished Professor in the Department of Electrical and Computer Engineering, University of Arizona. He holds a joint appointment with the College of Optical Sciences. His research interests include the application of new physics and engineering ideas to linear and nonlinear problems dealing with the interaction of electromagnetic waves with complex media, metamaterials, and realistic structures. Prof. Ziolkowski is an IEEE Fellow and an OSA Fellow. He was the President of the IEEE Antennas and Propagation Society in 2005. He continues to be very active in the IEEE, OSA, and APS professional societies.

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Ultrawideband Printed Log-Periodic Dipole Antenna With Multiple Notched Bands Chao Yu, Student Member, IEEE, Wei Hong, Senior Member, IEEE, Leung Chiu, Member, IEEE, Guohua Zhai, Chen Yu, Wei Qin, and Zhenqi Kuai

Abstract—A printed log-periodic dipole antenna (PLPDA) with multiple notched bands is proposed for ultrawideband (UWB) applications. The impedance bandwidth of 3.1 GHz–10.6 GHz with VSWR less than 2 is achieved based on the wideband property of the PLPDA as well as the half mode substrate integrated waveguide (HMSIW) Balun. Different from omnidirectional UWB antennas, the end-fire radiation pattern of the PLPDA is more stable within the UWB band. Multiple notched bands are generated by integrating U-shaped slots into the PLPDA for blocking the interference from other narrow band wireless communication systems. Several antennas with the notched frequencies of 3.5 GHz, 5.5 GHz, 6.8 GHz, and 8.5 GHz are designed, fabricated, and measured. The measured results are in agreement with the predicted results. Index Terms—Half mode substrate integrated waveguide, multiple frequency notches, printed log-periodic antenna, ultrawideband.

I. INTRODUCTION

U

LTRAWIDEBAND (UWB) wireless communications, which allow low power level and high data rate transmissions, have sparked great research interests for wireless communications applications in the 3.1 GHz–10.6 GHz frequency band. High-performance UWB antennas require both good impedance matching and low signal distortion within the specified frequency bands. There have been many reported research works on UWB antenna design. Monopole antenna is one of the typical examples, which can achieve very wide frequency band with a simple geometry. However, the radiation pattern of the monopole UWB antenna changes rapidly with frequency, which limits its practical applications. Tapered slot antenna (or Vivaldi antenna) is another typical example of UWB antenna. It radiates or receives power in the end-fire direction using travelling-wave mechanism. The peak radiation is fixed and the radiation pattern is stable within the working frequency band. This property leads it to a good candidate for UWB applications; however, the shortcoming of this kind of antenna is large size, generally with the length of minimum three guided-wavelengths at the lowest working frequency. Printed log-periodic dipole antenna

Manuscript received October 26, 2009; revised April 29, 2010; accepted August 24, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by the NSFC under Grant 60921063 and in part by the National 973 project under Grant 2010CB327400. The authors are with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103010

(PLPDA) also radiates in end-fire direction within ultrawide frequency band. With the multiple resonance property, its bandwidth can be enhanced by increasing the number of the dipole elements. Some of the research works on log-periodic antenna were reported in [1]–[4], where the difficulty of the feeding network was addressed. At the very beginning, coaxial cable is used for feeding the PLPDAs at the AM/FM radio and the TV frequency bands [3]; however, it was found that the performance will be deteriorated with frequency increasing. Stripline as a feeding network for PLPDA was reported in [5] however based on multi-layer process. There are many existing wireless communications systems, which are located at the frequency band overlapping to that of the UWB systems. The UWB systems will cause interferences to these systems such as IEEE 802.11a in USA (5.150 GHz–5.350 GHz and 5.725 GHz–5.825 GHz), HIPERLAN/2 in Europe (5.150 GHz–5.350 GHz and 5.470 GHz–5.725 GHz), and World Interoperability for Microwave Access (WiMAX) (3.400 GHz–3.690 GHz and 5.250 GHz–5.825 GHz). Thus, designing UWB antennas with multiple frequency notched bands for blocking the interference between the UWB systems and the existing narrow band wireless communication systems has become a challenge research topic. Last decade, many design techniques have been proposed to realize UWB antennas with band-notched characteristic. One of the simple ways is to etch slots on the radiators or feeding structures, such as U-shaped slots [6], [7], L-shaped slots [8], and H-shaped slots [9]. Adding parasitic strip [10], [11] near the radiation elements or the ground planes is another way to create notched bands. In addition, loading resonators to the feeding line is also a good way to realize band-notched characteristics. Several types of resonators such as split ring resonators [12], complementary split ring resonators [13], coplanar waveguide resonant cells [14], and half-mode substrate integrated waveguide (HMSIW) cavities [15], were proposed for the bandnotched UWB antennas. Most of the previous works were focused on the single notched band design, while the minorities were concentrated on multiple notched bands design. Double band-notched antennas were reported in [7], [8], [16], and a triple band-notched antenna was proposed in [9]. Recent research work on multiple notched bands antenna was presented in [15]. Recently, half mode substrate integrated waveguide (HMSIW) proposed in [17] has been successfully applied to design and implement various high performance microwave and millimeter-wave components such as filters [18], [19], couplers [20], [21], and antennas [22], [23], etc.

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for the microstrip line case is used to approximate for the dipole case. The formula is (3) is the relative permittivity of the substrate, is the where substrate thickness, is the width of the longest dipole. The relationship of spacing and the length of the dipole elements is (4) where . Finally, once the dimensions of the longest dipole element are determined, the dimensions of other dipoles can be determined in terms of the following relationship

(5)

Fig. 1. Layout of the PLPDA with denoted dimensions. (a) Top layer. (b) Bottom layer.

In this paper, a UWB PLPDA with HMSIW as the wideband feeding network (Balun) is proposed, and multiple notched bands are generated by etching U-shaped slots on the antenna. II. DESIGN OF THE PROPOSED PLPDA Fig. 1 shows the layout of the proposed PLPDA. It consists of 10 dipole elements and the wideband transition between microstrip feeding port and broadside coupled dual-lines via a section of the HMSIW. This antenna is defined as reference antenna for the comparison. A. Design Issue of the Dipole Array The design method of a log-periodic dipole antenna (LPDA) was proposed in [24], [25]. Three important steps are needed for the designing procedure. In the first step, the scale factor , spacing factor , and the number of the dipole elements should be determined, where , , and are chosen in this study. Second, the length of the longest dipole , which responses to the lowest resonance frequency , should be computed by (1) where is the longest operating wavelength. Its value can be determined by (2) where is the speed of light in vacuum and is effective dielectric constant. There is no closed-form or analytical formula to determine ; therefore, the formula of determining

B. Design Issue of the HMSIW HMSIW was firstly proposed for microwave and millimeter wave applications in 2006 [17]. This structure is obtained by bisecting a substrate integrated waveguide (SIW) with the symmetric plane along the transmission direction, which is equivalent to a perfect magnetic wall for the fundamental mode (TE10 mode). The HMSIW maintains the original field distribution and preserves the merits of the SIW, but the width is approximately reduced by half and the fundamental mode operation bandwidth is greatly enhanced. In this work, the HMSIW is used for the wideband Balun or transition between input microstrip feeding port and the balanced dual-line, which is connected to the LPDA. Actually, one end of the HMSIW also works as the reflector of the dipole antenna array to force the radiated power in the end-fire direction. of the section of the HMSIW can be deThe width signed such that the cut off frequency is lower than that of the resonance frequency of the longest dipole. Therefore, the minis imum value of (6) is the effective permittivity of the HMSIW. where Table I shows the parameters of the PLPDA obtained by fine optimizing the dimensions of the PLPDA and the HMSIW with the goal of impedance matching within the standard UWB frequency band. The simulated voltage standing wave ratio (VSWR) of the PLPDA is shown in Fig. 2. C. Design Issue of the U-Shaped Slot To realize band-notched characteristic, band-stop filter should be incorporated into the antenna design. A shunt open circuit stub is the simplest way to realize the band-stop effect. However, the shunt connection will occupy large circuit area. For the multiple notched bands antenna design, several shunt open circuit stubs with the different lengths should be used, but it will result in strong mutual coupling between the stubs, which leads to the increase of design complexity.

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TABLE I PHYSICAL DIMENSIONS OF THE PLPDA WITH HMSIW FEEDING NETWORK (UNIT: MM)

Fig. 3. Simulated current density distributions of the PLPDA at different frequencies. (a) Notched frequency (5.8 GHz). (b) Working frequency (4.5 GHz).

frequencies of 4.5 GHz, the current is concentrated at a dipole element, and no standing wave is found. The relationship between the length of the U-shaped slot and the notched frequency can be approximated by (7) where is the slot resonance frequency. The effective dican be approximated by the microstrip electric constant line case as listed in (3). Fig. 4 shows the variation of the notched band with respect to length and width of the U-shaped slot. It is obvious that longer U-shaped slot results in lower frequency notch. The width has less impact on notched characteristic. The notch frequency slightly decreases with the width, while the notched bandwidth slightly increases with the width. III. SIMULATED AND EXPERIMENTAL RESULTS

Fig. 2. Simulated frequency response of VSWR of the PLPDA.

In this paper, U-shaped slots are used to create notched bands instead of shunt open circuit stubs. Compared to L-shaped slot [23], U-shaped slot has more flexibility to choose the location and more U-shaped slots can be easily inserted into the proposed PLPDA. Due to the mechanism of PLPDA, the location of U-shaped slot should be prior to the resonated dipole element in order to obtain better band-notched performance. Fig. 3 shows the proposed PLPDA incorporated with a U-shaped slot, which can be equivalent to the shunt open circuit stub but with much compact layout. The physical length of the U-shaped slot is about quarter-wavelength at the notched frequency such that the slot resonates and stops the signal. Total signal reflection is expected at the notched frequency. Simulated current distributions of the PLPDA with single notched band are shown in Fig. 3. At the notched frequency of 5.8 GHz, the current is concentrated at the region of the U-shaped slot. It is confirmed that the U-shaped slot effectively reflects the signal power back to the excitation port. A standing wave is found around the slot. In contrast, at one of the working

In this section, four PLPDAs with different notched bands are presented. The commercial electromagnetic simulation tool, Ansoft HFSS, was used to simulate and optimize the performance of all designs. All of them were fabricated by the standard printed circuit board (PCB) fabrication process. The dielectric substrate used in the designs was Rogers RT/Duroid 5880 with dielectric constant of 2.2, loss tangent of 0.0009, and substrate thickness of 1.5748 mm. SMA connectors were used as transition between microstrip and coaxial cable for the measurement purpose. Fig. 5 shows the photographs of all fabricated PLPDAs with different notched bands. Fig. 6 shows the measured and simulated frequency responses of the VSWR of four designs, where the reference antenna is the PLPDA without notched bands as shown in Fig. 1. All of them cover the frequency band of 3.1 GHz–10.6 GHz. Distinct band notched characteristics are achieved. The values of VSWR are less than 2 except at the notched frequency bands. Generally, the bandwidth ) with VSWR more than 2 (or return loss higher than could be defined as the stop-band of a frequency notch. It can be seen from Fig. 6 that the two lower frequency notches are corresponding to the 3.5 GHz and 5.8 GHz WiMAXs, and the stop-bands are roughly matching the operation bands of these narrow band wireless communication systems. The two higher frequency notches are only for validating the ability of generating multiple frequency notches by using this method. Certainly, we can also block the interferences from narrow band wireless communications systems by inserting band-stop filters

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Fig. 4. Simulated frequency responses of VSWR of the PLPDA with a U-shaped slot. (a) Responses with different lengths of the U-shaped slot. (b) Responses with different widths of the U-shaped slot.

Fig. 5. Photographs of the fabricated PLPDAs with 1, 2, 3, and 4 U-shaped slots.

between the UWB antenna and the transceiver, but which will introduce additional insertion loss. The effect of the SMA connector is included in the measurement, but it has little impact

Fig. 6. Simulated and measured frequency responses of the VSWRs. (a) PLPDA with 1 U-shaped slot. (b) PLPDA with 2 U-shaped slots. (c) PLPDA with 3 U-shaped slots. (d) PLPDA with 4 U-shaped slots.

to the overall performance. Good agreement between the simulation and measurement is obtained. The small discrepancy between simulation and measurement is acceptable and may come

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Fig. 7. Simulated and measured radiation patterns of the PLPDA with 1 U-shaped slot in both E- and H-planes at the different frequencies. (a) 4.0 GHz, (b) 5.0 GHz, (c) 7.0 GHz, (d) 8.0 GHz, (e) 9.0 GHz, (f) 10 GHz.

TABLE II MEASURED HALF POWER BEAM-WIDTH FOR BAND-NOTCHED ANTENNAS

from the permittivity fluctuation of the dielectric substrate. The mutual coupling between adjacent U-shaped slots is very weak, such that each notch is almost independent of others. The dimensions of each U-shaped slot can be tuned individually and arbitrarily, and then the notched frequencies can be easily designed to meet the different wireless standards. The simulated and measured radiation patterns in both E-plane ( -plane) and H-plane ( -plane) of the single notch

designs are shown in Fig. 7. In order to illustrate the variation of the radiation patterns, the radiation patterns of the single notched band PLPDA at six frequency points are shown in Fig. 7. End-fire radiations with stable radiation pattern within the working frequency band were experimentally confirmed. It is expected that similar radiation patterns will be obtained if the simulation and the measurement were repeated at other frequency points within the working frequency band. Similarly, the

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Fig. 8. Simulated and measured frequency responses of the antenna gains in end-fire direction. (a) PLPDA with 1 U-shaped slot. (b) PLPDA with 2 U-shaped slots. (c) PLPDA with 3 U-shaped slots. (d) PLPDA with 4 U-shaped slots.

radiation patterns of the double, triple and quadruple notched bands PLPDAs were simulated and measured. For the triple and quadruple notched bands PLPDAs, one of their notched bands is near to 7.0 GHz, hence the frequency set of 4.0 GHz, 5.0 GHz, 7.5 GHz, 8.0 GHz, 9.0 GHz, and 10 GHz is chosen for these measurements. Due to the similar radiation patterns of the four band-notched antennas, the radiation patterns of the double, triple and quadruple notched bands PLPDAs are NOT shown. The measured half power beam-width for band-notched antennas is shown in Table II. Several characteristics of the proposed PLPDAs are found in the simulation and measurement. First, all the simulated and measured radiation patterns show the main beam of the proposed PLPDA is in the end-fire direction. Second, the front-to-back ratios of all designs achieve more than 12 dB. Third, the beam widths in E- and H-planes vary from 43.7 to 87.5 and 89.1 to 178.4 within the measured frequency band, respectively. Fourth, distinct side lobes are observed at the higher frequencies. Fifth, slightly asymmetric radiation patterns are observed due to the asymmetric geometry of the proposed PLPDA. Finally, with the increase of notched

bands, the radiation patterns of different antennas obtain the similar results within the working frequencies. The measured and simulated frequency responses of the antenna gain of the four designs in end-fire direction are shown in Fig. 8. The reference antenna described in Fig. 8, which is the PLPDA without notched bands as shown in Fig. 1, is used for comparison. The measured antenna gains can be corresponded to the frequency responses of VSWR as shown in Fig. 6. The frequency points of the notches on the antenna gain curves as shown in Fig. 8 are the same as that of the peaks of the frequency responses of VSWR as shown in Fig. 6. The existence of the U-shaped slots results in effective signal reflection at the notched bands. Since non-linear phase response results in waveform distortion in the time domain, then the linear phase response of the antenna is also very important for UWB systems. In this study, two PLPDAs with one U-shaped slot were used to transmit and receive signals, and the group time delay was measured via a vector network analyzer by considering the pair of PLPDAs as a two port network. Fig. 9 shows the measured group time

YU et al.: ULTRAWIDEBAND PRINTED LOG-PERIODIC DIPOLE ANTENNA WITH MULTIPLE NOTCHED BANDS

Fig. 9. Measured group delays of the two PLPDAs with 1 U-shaped slot and different distances.

delay with two different distances between the two PLPDAs. Flat responses are achieved within the working frequency bands leading to good linear phase response. IV. CONCLUSION In this paper, UWB PLPDAs with multiple notched bands are presented. A section of HMSIW is integrated into the proposed antenna as an ultrawideband balun in the feeding network, and a reflector for the printed dipole array. The proposed PLPDAs with single and multiple notched bands were implemented simply by etching U-shaped slots on the antenna. Unlike the monopole antennas, the proposed PLPDAs radiate power in the end-fire direction. Stable radiation patterns are experimentally confirmed within the whole working frequency bands. REFERENCES [1] S. H. Kim, J. H. Choi, J. W. Baik, and Y. S. Kim, “CPW-fed logperiodic dumb-bell slot antenna array,” Electron. Lett., vol. 42, no. 8, pp. 436–438, Apr. 2006. [2] S. Y. Chen, P. H. Wang, and P. Hsu, “Uniplanar log-periodic slot antenna fed by a CPW for UWB applications,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 256–259, Dec. 2006. [3] M. M. Tajdini and M. Shahabadi, “Wideband planar log-periodic antenna,” in Proc. Int. Workshop on Antenna Technology-IWAT’07, Mar. 2007, pp. 331–334. [4] J. Mruk, M. Uhm, and D. Filipovic, “Dual-wideband log-periodic antennas,” in Proc. IEEE AP-S. Int. Symp., Jul. 2008, pp. 1–4. [5] R. Pantoja, A. Sapienza, and F. M. Filho, “A microwave printed planar log-periodic dipole array antenna,” IEEE Trans. Antennas Propag., vol. 35, no. 10, pp. 1176–1178, Oct. 1987. [6] Y. J. Cho, K. H. Kim, D. H. Choi, S. S. Lee, and S. O. Park, “A miniature UWB planar monopole antenna with 5-GHz band-rejection filter and the time-domain characteristics,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1453–1460, May 2006. [7] W. S. Lee, D. Z. Kim, K. J. Kim, and J. W. Yu, “Wideband planar monopole antennas with dual band-notched characteristics,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2800–2806, Jun. 2006. [8] Y. H. Zhao, J. P. Xu, and K. Yin, “Dual band-notched ultra-wideband microstrip antenna using asymmetrical spurlines,” Electron. Lett., vol. 44, no. 18, pp. 1051–1052, Aug. 2008. [9] J. Y. Deng, Y. Z. Yin, S. G. Zhou, and Q. Z. Liu, “Compact ultrawideband antenna with tri-band notched characteristic,” Electron. Lett., vol. 44, no. 21, pp. 1231–1233, Oct. 2008.

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[10] K. H. Kim, Y. J. Cho, S. H. Hwang, and S. O. Park, “Band-notched UWB planar monopole antenna with two parasitic patches,” Electron. Lett., vol. 41, no. 14, pp. 783–785, Jul. 2005. [11] 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. [12] Y. Zhang, W. Hong, C. Yu, Z. Q. Kuai, Y. D. Dong, and J. Y.Zhou, “Planar ultrawideband 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. [13] T. N. Chang and M. C. Wu, “Band-notched design for UWB antennas,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 636–640, 2008. [14] S. W. Qu, J. L. Li, and Q. Xue, “A band-notched ultrawideband printed monopole antenna,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 495–498, Dec. 2006. [15] Y. D. Dong, W. Hong, Z. Q. Kuai, C. Yu, Y. Zhang, J. Y. Zhou, and J. X. Chen, “Development of ultrawideband antenna with multiple bandnotched characteristics using half mode substrate integrated waveguide cavity technology,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2894–2902, Sep. 2008. [16] Q. X. Chu and Y. Y. Yang, “A compact ultrawideband antenna with 3.4/5.5 GHz dual band-notched characteristics,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3637–3644, Dec. 2008. [17] W. Hong, B. Liu, Y. Q. Wang, Q. H. Lai, H. J. Tang, X. X. Yin, Y. D. Dong, Y. Zhang, and K. Wu, “Half mode substrate integrated waveguide: A new guided wave structure for microwave and millimeter wave application,” in Proc. Joint 31st Int. Conf. Infrared Millim. Waves 14th Int. Conf. Teraherz Electron., Shanghai, China, Sep. 18–22, 2006, pp. 219–219. [18] Y. Q. Wang, W. Hong, Y. D. Dong, B. Liu, H. J. Tang, J. X. Chen, X. X. Yin, and K. Wu, “Half mode substrate integrated Waveguide (HMSIW) bandpass filter,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 4, pp. 265–267, Apr. 2007. [19] Y. J. Cheng, W. Hong, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) directional filter,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 7, pp. 504–506, Jul. 2007. [20] B. Liu, W. Hong, Y. Q. Wang, Q. H. Lai, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) 3-dB coupler,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 1, pp. 22–24, Jan. 2007. [21] G. H. Zhai, W. Hong, K. Wu, J. X. Chen, P. Chen, J. Wei, and H. J. Tang, “Folded half mode substrate integrated waveguide 3 dB coupler,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 8, pp. 512–514, Aug. 2008. [22] G. H. Zhai, W. Hong, K. Wu, and Z. Q. Kuai, “Printed quasi-Yagi antenna fed by half mode substrate integrated waveguide,” in Proc. IEEE Asia-Pacific Microw. Conf., 2008, pp. 1–4. [23] C. Yu, W. Hong, C. Yu, Z. Q. Kuai, and W. Qin, “Band-notched UWB printed log-periodic dipole antenna fed by half mode substrate integrated waveguide,” presented at the Proc. Int. Symp. Antennas and Propagation (ISAP2009), Thailand, Oct. 2009. [24] R. Carrel, “The design of log-periodic dipole antennas,” IRE International Convention Record, vol. 9, no. 1, pp. 61–75, Mar. 1961. [25] C. Campbell, I. Traboulay, M. Suthers, and H. Kneve, “Design of a stripline log-periodic dipole antenna,” IEEE Trans. Antennas Propag., vol. 25, no. 5, pp. 718–721, Sep. 1977.

Chao Yu (S’10) was born in Anhui, China. He received the B.E. and M.E. degree from School of Information Science and Technology, Southeast University, Nanjing, China, in 2007 and 2010, respectively, and is currently working toward the Ph.D. degree at University College Dublin, Dublin, Ireland. His research interests include antenna design, microwave circuits and devices.

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Wei Hong (M’92–SM’07) received the B.S. degree from the University of Information Engineering, Zhengzhou, China, in 1982, and the M.S. and Ph.D. degrees from Southeast University, Nanjing, China, in 1985 and 1988, respectively, all in radio engineering. Since 1988, he has been with the State Key Laboratory of Millimeter Waves, Southeast University, where, since 2003, he has served as the Director of the Lab. He concurrently serves as a Professor and the Associate Dean of the School of Information Science and Engineering, Southeast University. In 1993, 1995, 1996, 1997, and 1998, he was a short-term Visiting Scholar at the University of California at Berkeley and at Santa Cruz, respectively. He has been engaged in numerical methods for electromagnetic problems, millimeter wave theory and technology, antennas, electromagnetic scattering, RF technology for mobile communications, etc. He has authored and coauthored over 200 technical publications and authored two books Principle and Application of the Method of Lines (in Chinese, Southeast University Press, 1993) and Domain Decomposition Methods for Electromagnetic Problems (in Chinese, Science Press, 2005). Dr. Hong was thrice awarded the First-Class Science and Technology Progress Prizes issued by the Ministry of Education of China and the Jiangsu Province Government. In addition, he also received the Foundations for China Distinguished Young Investigators Award for “Innovation Group” issued by the NSF of China. He is a senior member of CIE, Vice-President of the Microwave Society and Antenna Society of CIE, Chairperson of IEEE MTT-S/AP-S/EMC-S Joint Nanjing Chapter, and served as a Reviewer for many technical journals such as the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IET Proc.-H, Electronics Letters, etc. He currently serves as an Associate Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and is an editorial board member of IJAP and RFMiCAE, etc.

Leung Chiu (S’05–M’08) received the B.Eng. and Ph.D. degrees in electronic engineering from the City University of Hong Kong, Hong Kong, in 2004 and 2008, respectively. His research interests include microwave circuits and antenna arrays.

Guohua Zhai was born in Jining, Shandong Province, China, in 1981. He received the B.S. degree in electrical engineering from Qufu Normal University, Jining, China, in 2003, the M.S. degree in microwave and antenna from Nanjing Research Institution of China, in 2006, and is currently working toward the Ph.D. degree at Southeast University, Nanjing, China. His current research interests include multi-beam antennas, Silicon substrate passive circuits modeling and design.

Chen Yu was born in Jiangsu, China, in 1979. She received the B.S. and M.S. degrees in radio engineering from Southeast University, Nanjing, China, in 2001 and 2004, respectively, where she is currently working toward the Ph.D. degree. Her current research interests include microwave and millimeter-wave circuit.

Wei Qin was born in Nantong, Jiangsu Province, China. He received the B.S. and M.S. degrees from the Southeast University, Nanjing, China, in 2007 and 2010, respectively. His research interests are in design of microwave circuits and devices.

Zhenqi Kuai was born in Anhui province, China, in May 1960. He received the B.S. degree in radio engineering from Nanjing Institute of Technology (now Southeast University) Nanjing, China, in 1982. From 1982 to 1985, he worked at the Anhui Television Station. From 1985 to 2001, he was with Nanjing Marine Radar Institute. His research interests include ferrite devices, microwave components, reflector and array antennas, etc. Since 2001, he has been with the State Key Laboratory of Millimeter Waves, Southeast University, Nanjing, China, and is an Associate Professor in the School of Information Science and Engineering, Southeast University. His current research interests include antennas and array technology, as well as microwave and millimeter wave components. Mr. Kuai was awarded the First-Class Science and Technology Progress Prizes of Jiangsu Province in 2008.

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Modulated Elliptical Slot Antenna for Electric Field Mapping and Microwave Imaging Mohamed A. Abou-Khousa, Senior Member, IEEE, Mohammad Tayeb Ghasr, Member, IEEE, Sergey Kharkovsky, Fellow, IEEE, David Pommerenke, Senior Member, IEEE, and Reza Zoughi, Fellow, IEEE

Abstract—Microwave and millimeter wave imaging has shown significant potential in various applications. An imaging system commonly consists of a sensitive electric field mapping array devised to measure the spatial distribution of the scattered field from an object to be imaged. One of the most prominent methods used to realize a cost-effective real-time imaging system is the modulated scatterer technique (MST). Although the conventional MST, using small loaded dipole antennas, performs well at lower microwave frequencies, its utility at high microwave and millimeter wave frequencies is limited. To improve upon the conventional MST, a novel modulated elliptical slot antenna, loaded with a PIN diode, is introduced and analyzed in this paper. The modulation-depth (effectiveness in modulating the slot), current distribution around the slot, and the influence of PIN diode bias structure are discussed based on numerical simulation results and experiments. Finally, the efficacy of the proposed slot for electric field measurements at 24 GHz is demonstrated using a prototype slot. Index Terms—Electric field measurement, elliptical slot, loaded slot, microwave imaging, modulation-depth, modulated scatterer technique (MST), PIN diode, real-time imaging.

I. INTRODUCTION N response to addressing many critical subsurface imaging needs in industrial, aerospace, transportation, and biomedical applications, microwave and millimeter wave imaging is rapidly becoming an effective and attractive tool. These imaging techniques and systems will only become readily utilized if they can possess operational attributes such as being real-time, portable and produce high-resolution images. Properly-designed imaging systems operating at or near millimeter wave frequency range can meet the resolution and portability constraints, while the ability to produce real-time images will primarily depend of the detailed design of the imaging system

I

Manuscript received March 02, 2010; revised August 03, 2010; accepted August 30, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by a grant from NASA Marshall Space Flight Center, Alabama. M. A. Abou-Khousa is with the Imaging Research Laboratories, Robarts Research Institute, The University of Western Ontario, London, ON, Canada (e-mail: [email protected]). M. T. Ghasr, S. Kharkovsky, and R. Zoughi are with the Applied Microwave Nondestructive Testing Laboratory (amntl), Electrical and Computer Engineering Department, Missouri University of Science and Technology, Rolla, MO 65409 USA (e-mail: [email protected]; [email protected]; [email protected]). D. Pommerenke is with the Electromagnetic Compatibility Laboratory, Electrical and Computer Engineering Department, Missouri University of Science and Technology, Rolla, MO 65409 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103024

collector. Nevertheless, these constraints limit the type of design methodology that can be used in developing such systems where one must rapidly and coherently measure the spatial electric field distribution scattered from an object to be imaged, in a two-dimensional (2D) space. One effective way to accomplish this is using the modulated scatterer technique (MST) in which a 2D array of small loaded antennas (commonly dipoles) is used to coherently measure the scattered electric field distribution from an object. By changing the loading condition of the dipole, as a function of time, the electric field of interest can be spatially tagged and used to reproduce the image of the object [1]–[3]. This technique not only provides for spatial multiplexing, it can also substantially increase the measurement sensitivity (via coherent averaging), and hence enable detection of weakly scattering objects. Loading is conventionally accomplished by using a PIN diode. In this way, as the PIN diode is turned ON (forward biased) and OFF (reverse biased), it changes the overall dipole impedance and hence the scattered field from the dipole is modulated accordingly. However, there are several drawbacks when using conventional MST where small PIN diode-loaded dipole antennas are used [4]–[6]. The ability of a loaded antenna to produce significantly different scattered field for each of its loading state is related to modulation depth observed at the receiver. The use of small loaded dipoles significantly limits the modulation-depth and ultimately the system signal-to-noise (SNR) ratio [5]. The modulated scattered signal level from a loaded small dipole is relatively low. Consequently, special front-end tuning procedure should be used to suppress the dominant residual carrier [1]. Since such imaging arrays usually contain several hundred dipole elements, inter-element mutual coupling, which also cause reduction in system sensitivity, must also be addressed. These limitations are further exasperated at higher microwave or millimeter wave frequencies. To overcome these limitations for operation at higher microwave frequencies, a more efficient imaging array element must be used in the place of short dipoles [5], [6]. A modulated resonant slot antenna constitutes a prominent candidate for this purpose. The choice of a resonant slot antenna is justified by the fact that microwave and millimeter wave signals can pass though it while experiencing minimal attenuation. In its simplest form, once MST is applied to a slot, it implies that the required modulation is achieved by opening and closing the slot, i.e., allowing the signal to pass or not pass through the slot, respectively. A modulation-depth close to 100% may then be achieved when the slot is switched between an “open” and a “close” state. Therefore, it is expected that the imaging system

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Fig. 2. Schematic of the designed elliptical slot (a) with, and (b) without an active load element.

Fig. 1. Generic schematic of an imaging system based on array of modulated slots.

sensitivity can be considerably enhanced compared to the conventional imaging MST arrays based on short dipoles. To illustrate the basic idea behind using modulated slots in an imaging system, consider Fig. 1, which shows a generic schematic of such system based on array of modulated slots similar to that introduced in [7]. The slots shown in Fig. 1 are cut into a conducting screen and the receiver is placed behind that screen. Each slot is then loaded with an active element, e.g., a PIN diode, whose impedance can be electronically (or optically) controlled. Subsequently, the impedance of the active device is switched between two states, corresponding to a (near) short- and an open-circuit load. This corresponds to opening and closing the slot to the incident electric field of interest, respectively. At the initial state, all slots are closed and no signal passes through to the receiver. Then, when a given slot is opened while the rest are closed, only the respective electric field at the particular location of the open slot is coupled into that slot and subsequently, re-radiated into the opposite side of the screen where it is picked up by the receiver. Each slot is modulated, by opening and closing it, at a certain modulation rate. In this way, the electric field due to each slot can be individually discriminated and measured using a single receiver. In imaging applications, the effect of reflection from the conducting screen of the array should be properly accounted for through calibration. When all slots are open, the array becomes transparent to the EM waves at frequencies where the slots operate efficiently, e.g., a spatial band-pass filter. When most of the slots are closed, however, the reflected wave from the array might interact with the source of incident field, i.e., the scattered field from the object to be imaged. Multiple reflections between the array and the source may change the field distribution of interest. In practice this situation can be resolved by increasing the distance between the array and the source. Measuring the field of interest at multiple array-source distances can also considerably reduce the effect of multiple reflections. Using absorbing materials to cover the exposed conductor on the array can also reduce these reflections to a large extent. Similar remedies are typically used with the conventional MST arrays of dipoles and can be applied to the array of slots as well [4].

In order for the imaging scheme described above to function as expected, the slot should be designed such that it can be opened and closed completely and respond fast to an electronic control signal modulating its load. To avoid aliasing, the dimensions of the slot (in both directions) should be less than for proper electric field sampling [8]. The design of an efficiently-loaded slot antenna, for use in high-frequency, real-time, high-resolution and portable imaging and electric field mapping systems is presented next. II. MODULATED SLOT DESIGN A loaded resonant elliptical slot was recently designed and implemented successfully for a real-time imaging array operating at 24 GHz [7]. The design of this slot and its various attributes are detailed in this section. A. Elliptical Resonant Slot Slot antennas, in a conducting ground plane, are relatively easy to manufacture and possess a low profile as flush-mounted antennas, and are used extensively in many applications [9], [10]. Recently, printed elliptical and circular slot designs have received considerable attention for realizing ultra-wideband antennas for wireless communication systems (comprehensive reviews of the most recent designs can be found in [11]–[13]). Here, the design of a novel, compact resonant slot for imaging system realizations is presented. The advantages of using resonant slots are multifold. First, the resonant slot can take on, through special loading, small form-factor. Second, with small resonant slots, the mutual coupling between various array slots can be significantly reduced [14]. Furthermore, once the resonant slot is used as a modulated element, the modulation-depth can be maximized, resulting in significant sensitivity enhancement of the imaging system. In concert, these advantages render a highly sensitive electric field mapping array as desired for high-frequency and high-resolution imaging [7]. As shown in Fig. 2, the designed slot has an elliptical shape with major and minor radii of and , respectively, cut into a conducting plane. The slot is linearly polarized along the direction of the ellipse’s minor axis, i.e., -axis, and it has a broadside radiation pattern with a relatively wide beamwidth. For imaging purposes, it is advantageous to have an antenna with a relatively wide radiation pattern. Slots with wide radiation pattern possess very low directional selectivity and thus ensure detection from all directions. Furthermore, for synthetic array processing using synthetic aperture focusing technique (SAFT), which is commonly implemented with such imaging systems,

ABOU-KHOUSA et al.: MODULATED ELLIPTICAL SLOT ANTENNA FOR ELECTRIC FIELD MAPPING AND MICROWAVE IMAGING

wide-beam patterns are desirable in order to attain the maximum possible spatial resolution [15], [16]. In order to realize a high-resolution and compact array, the largest dimension of . Hence, the elliptical slot by itthe slot must be less than self should be a sub-resonant inductive slot. To make the slot resonates at the desired frequency; it is loaded by a conductive circular load of radius , as shown in Fig. 2(a). In essence, the between the circular load and the bottom gap with a length of edge of the elliptical slot, adds capacitance to the impedance of the slot structure. Consequently, the inductive elliptical slot and the capacitive gap element result in a single resonant structure. This is true even though the largest linear dimension in the . Such a concept is typically overall slot structure is less than considered for designing resonant waveguide irises [14]. The resonant frequency of the elliptical slot is a function of , and . the slot dimensions; namely One can change the resonant frequency of the slot by changing and prithese (interdependent) dimensions. Reducing marily results in increasing the resonance frequency, while causes the slot to resonate at lower frequencies. increasing The operating bandwidth around the resonance frequency is a . On the other hand, since function of the slot axial ratio the resonant frequency is a function of the slot capacitance, this frequency can be electronically controlled via loading the slot with an active element such as a PIN diode, varactor diode, etc., as shown in Fig. 2(b). Changing the properties of the active element, e.g., capacitance, causes a shift in the resonant frequency of the slot. In general, when the active element is capacitive, its capacitance is added to the capacitance of the gap, and consequently the resonant frequency decreases compared to the unloaded slot shown in Fig. 2(a). An applied control dc voltage can electronically change the properties of the active element, and hence modulate the slot as a function of time, i.e., opening and closing it. In order to attain efficient control over the slot properties, the active element should be placed in a high-intensity electric field region where it can maximally affect the field. Consequently, the electric field must be “forced” to concentrate in a specific region between the perimeters of the elliptical slot and the circular load. This is accomplished by offsetting the circular load from the center of the elliptical slot by certain amount, , towards one of the boundaries, i.e., forming a small gap with a high E-field concentration, as shown in Fig. 2(a). This offset has the effect of “trapping” the electric field in this region. Thus, the slot capacitance is primarily determined by the gap and any other equivalent impedance that may be placed near that gap; in this case the active element. Consequently, the resonant frequency of the slot can be properly controlled with such an active load. For satisfying the real-time requirement of an imaging system, loading the slot with a PIN diode is of particular interest here since it allows the desired rapid control effect, i.e., PIN diode switching time in order of few nanoseconds, and consumes a relatively small amount of power. When the diode is forward-biased, e.g., turned ON, its resistive impedance “shorts” the gap between the circular load and the elliptical slot edge, and consequently, the slot (actually its dominant field mode) does not resonate at the frequency of interest (where the receiver is tuned). Ideally, in this state the slot does not allow

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TABLE I 24 GHZ RESONANT ELLIPTICAL SLOT DIMENSIONS

any signal to pass through since it is essentially “closed.” When the diode is reverse-biased, it represents a capacitive load, which adds to the gap capacitance and makes the slot resonate at the frequency of interest, and hence a signal at that frequency passes through the slot. In this state the slot is considered to be “open.” Consequently, switching the PIN diode between these two states results in the maximum possible modulation-depth (as will be defined later). In practice however, the modulation-depth may be reduced due to signal leakages, (when the slot is closed) and losses (when the slot is open). B. Field Distribution Studying the electric field and the current distributions in and around the slot area is vital for proper slot design as well as for understanding its behavior, i.e., mutual-coupling, when it is placed near another slot in an array. For imaging or electric field mapping applications where the objective is to measure the relative electric field distribution in a certain 2D spatial domain, it is important to make sure that the slot is sensitive to the field component of interest, i.e., proper polarization, and that it provides localized field measurements. Consequently, numerical electromagnetic simulations were performed to investigate the various attributes of the designed PIN diode-loaded slot, using the CST Microwave Studio commercial simulation package [17]. Although many idealistic simulation models were considered in the early stages of slot design and prototyping, i.e., using perfect electric conductors, PIN diode with zero-ohm forward resistance, etc., the results presented here are obtained from simulating a practical 24 GHz elliptical slot on a printed circuit board of a lossy conductor, i.e., copper. The PIN diode in the ON and OFF states was modeled as lumped elements with impedance of 5 Ohm and Ohm (at 24 GHz), respectively [18]. In these simulations, the slot was fed by a K-band rectangular waveguide aperture with a 21 mm by 21 mm finite-sized flange, i.e., the slot was mounted on the flange. It should be noted that a waveguide-feed was used here to simplify the simulation model and experimental measurements. In practice, the slot can be fed using any suitable feed structure, e.g., waveguide wall, microstrip line, etc. Table I lists the dimensions of the simulated slot. Fig. 3 shows typical normalized spatial distributions of the and dominant electric and magnetic field components, respectively, for the simulated resonant loaded elliptical slot at 24 GHz for both the ON and the OFF PIN diode states. Simulations showed that, in the OFF state, i.e., when the slot is open, and field components are much higher than all the remaining field components. This indicates that the slot fields are

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Fig. 4. Tangential magnetic field distributions (proportional to surface current density) around the slot for (a) OFF and (b) ON diode states. Fig. 3. Normalized distributions of and OFF diode states.

E

and

H

in the slot area for both ON

primarily linearly polarized. The slot is shown to be mostly sensitive to electric field polarized along the -axis. In this state, it is also shown that the fields are more concentrated around the circular load and the gap area between that load and edge of the elliptical slot, as described earlier. Consequently, the integral of the field over the slot area is largely determined by the fields distributed in the small area around the circular load. This basically means that the slot is more sensitive to the local incident fields on that small area rendering highly localized field measurements, as desired. It is also interesting to note that the signal couples into the slot when it is open primarily via the magnetic (slots in general provide higher magnetic field component coupling). When the PIN diode is turned ON, the electric and magnetic field intensities diminish significantly compared to the OFF state. This is especially true in the case of electric field, as shown in Fig. 3, where the dominant field intensity diminish to almost zero in the slot area. Hence, when the PIN diode is turned ON, the slot blocks the signal and does not allow it to pass through; that is, the slot is essentially “closed.” In addi, remains tion, in this state, the magnetic field component, relatively high around the area where the PIN diode is located, as shown in Fig. 3. The coupled energy into the slot due to this component is not totally radiated by the slot. A large portion of this energy is dissipated in the forward-bias resistance of the PIN diode. Nevertheless, the small remaining portion of the energy actually gets radiated through the slot, i.e., it induces a current on the slot structure. Consequently, the slot cannot be completely closed, and this in turn, slightly reduces the obtained modulation-depth compared to the ideal zero-ohm PIN diode case, as it will be shown later. Examining the surface current distribution on the slot structure gives valuable insight into its behavior when it is “opened” and “closed.” The surface current distribution follows the distribution of the tangential magnetic field components. Figs. 4(a)–(b) show the magnitude distribution of the tangential components ( and ) of the magnetic field around the slot area in the ON and OFF states of the PIN diode, respectively. When the slot is open, the signal induces a current density on the metallic structure around the slot, as shown in Fig. 4(a). In this case, the slot radiates efficiently. The high current density around the diode location indicates that most of the current

actually passes through that point. When the diode is turned ON, a small amount of energy is coupled into the slot and this energy hardly induces any radiating currents around the slot as shown in Fig. 4(b). In this case, the largest current density is observed again around the PIN diode. As pointed out earlier, in this state the diode represents a resistive load and consequently, most of the energy coupled into the slot (due to ) is dissipated in the PIN diode. In 2D imaging arrays, the slots are placed side-by-side along -axis (H-plane), and in a collinear arrangement along y-axis distance from one another. Typically, at (E-plane), within a any given time one of the slots will be open while the rest in the array are closed. The interaction (transfer of energy) between the open slot and remaining closed slots constitutes undesired mutual coupling. From the results presented in Fig. 4(a)–(b) and referring to Fig. 2, it can be seen that when two slots are placed in a collinear arrangement along -axis, they interact more than when they are placed side-by-side along x-axis since the surface current density in the former case extends out further than the latter case. Mutual coupling between similar resonant slots (similar current distributions) has been studied thoroughly via numerical simulation as well as experiments [14]. It was found dB that the mutual-coupling between such slots is less than for a typical interspacing. For imaging purposes it is also important to study the far-field radiation pattern of the slot. Fig. 5 shows typical far-field pattern of the designed loaded elliptical slot. The 3-dB beamwidths in both principle E- and H-planes are around 100 and 120 , respectively. Such large beamwidths are desired for the type of imaging applications, as mentioned earlier. Finally, the radiation efficiency of this slot was calculated to be as high as 97%, which is also desired for sensitive field measurements. The main dip in the broadside direction of the E-plane pattern is primarily attributed to the finite size of the ground plane [19], i.e., the flange size was 21 by 21 mm (the surface current reaches and interact with the flange edge in that plane as shown in Fig. 4(a).1 III. MODULATED SLOT PROTOTYPE To experimentally demonstrate the operational efficacy of the proposed PIN diode-loaded resonant slot, an elliptical slot was designed to resonate at 24 GHz in the K-band (18–26.5 GHz) when loaded with a PIN diode. The dimensions of the designed 1This was confirmed by numerical simulations where the main beam dip vanishes gradually as the flange size increases.

ABOU-KHOUSA et al.: MODULATED ELLIPTICAL SLOT ANTENNA FOR ELECTRIC FIELD MAPPING AND MICROWAVE IMAGING

Fig. 5. Typical far- field pattern of the loaded elliptical slot (diode OFF).

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Fig. 7. Magnitude of the reflection coefficient of the designed slot (cf. Fig. 6) fed by a rectangular waveguide.

A. Reflection and Transmission Measurements

Fig. 6. A magnified picture of the manufactured slot operating at 24 GHz and the biasing structure.

slot were as shown in Table I. The slot was manufactured using standard photolithographic printed circuit board (PCB) technology on a Rogers 4350 board with a thickness of 0.5 mm. Note that the free-space wavelength is 12.5 mm at 24 GHz. The mm, is a bit larger largest slot linear dimension, than one-third of the free-space wavelength. In fact, it becomes very challenging to realize a resonant slot with such dimensions without special loading, as in the case of this novel design. A magnified picture of the slot is shown in Fig. 6. A MA4GP907 high-frequency flip-chip PIN diode (0.63 mm by 0.38 mm) was used to load the slot [18]. A 0.25 mm-wide dc bias line routed to the circular load location, was used to control the PIN diode, as shown in Fig. 6. To minimize the potential interference with the desired electric field at the elliptical slot aperture, the bias line was routed such that it was orthogonal to the dominant mode electric field polarization over the slot aperture. The PIN diode required 1.45 V (10 mA) to switch between the reverse- and the forward-biased states. When reverse biased, the fF, and in the forward diode represents a capacitive load of state it is equivalent to a 5-Ohm resistive load. Additionally, two 5-pF small surface-mount (SMT) capacitors (0.5 mm by 0.25 mm) were used to create an RF return path to ground, i.e., a short-circuit, for any 24 GHz signal that might couple into the dc bias lines, and thus preventing any spurious radiation or resonances that could result from such coupling.

The fabricated slot shown in Fig. 6 was centrally placed at the aperture of (i.e., fed by) a standard K-band rectangular waveguide and the input reflection coefficient (at the other port of the , was measured using a HP8510C Vector Netwaveguide), work Analyzer (VNA). Fig. 7 shows the schematic of the measurement setup and the magnitude of the simulated2 and measured reflection coefficient for the cases when: i) the slot is not loaded with the PIN diode, ii) the slot is loaded and the diode is OFF, and iii) the slot is loaded and the diode is ON. The measurement results show that when the slot is not loaded GHz. Loading the with the PIN diode, it resonates at slot with the reverse-biased PIN diode increases the overall capacitance and consequently shifts the resonance frequency to GHz. This shift is expected since the diode, and its effective capacitive impedance in this state, is in parallel with the offset gap between the circular load and the edge of the elliptical slot. It is also important to note that the slot has a 20 dB GHz (in return-loss bandwidth of 530 MHz centered at is dB at 24 GHz and the slot is “open”). this state On the other hand, when the diode is switched ON, the reflection coefficient is relatively high at 24 GHz indicating no signal is dB and the is passing through the slot (in this state slot is “closed”), resulting in a large modulation-depth. It is also observed that the difference between the ON and OFF state over the 20 dB return-loss bandwidth is larger than 19 dB. Consequently, using this slot, the overall system becomes more robust against signal source frequency drifts. It is also worth mentioning that in general, the simulation results are in good agreement with the measurement results. The small discrepancies such as the shift in the resonance frequency are largely attributed to the manufacturing tolerance margins in material properties and dimensions, as well as non-nominal diode parameters. Although the reflection experiments discussed above validates the basic slot design, i.e., resonance frequency, it is imperative to study the response of the modulated slot in a 2The simulation model included the vias surrounding the slot and the bias structure.

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Fig. 9. Measured normalized spectrum of the signal received through the GHz. K-band slot, f

= 24

magnitude and phase format, as the slot is modulated over time between open and closed states. The magnitude difference between signals received when the slot is open and closed is as high as 15 dB, which indicates very strong modulation. B. Modulated Response The modulation-depth is the ultimate figure-of-merit which truly establishes the utility of the designed slot as a modulated element in an imaging array. The baseband signal measurements performed using the VNA, as presented above, cannot be used to estimate the modulation-depth directly.3 However, modulationdepth can be calculated from the spectrum measurements of the modulated signal where the carrier, i.e., signal at a frequency of 24 GHz, and the fundamental harmonic (in one sideband) are measured and the following is applied [20]: Fig. 8. The measured baseband signal transmitted through the slot represented in (a) polar and (b) rectangular Magnitude/Phase formats.

% typical through-transmission setup as well since the slot is designed to be used in an imaging system (cf., Fig. 1). Fig. 8 shows the measurement setup and results of a through-transmission experiment. An antenna connected to port 1 of the HP8510C VNA was used to illuminate the slot from a distance of 12.5 cm away. The slot was mounted on the aperture of a rectangular waveguide, which was then connected to port 2 of the VNA. The measured baseband complex through-transmission signal, , is shown in polar format in Fig. 8(a). The small transmission coefficient seen near zero when the slot is closed indicates a very small signal leakage through the slot, as discussed earlier. When the slot is open, maximum signal passes through the slot to the VNA receiver. The magnitude and phase of the complex signal, or the vector difference between the signals received when the slot is open and closed as depicted in Fig. 8(a), are proportional to those of the incident signal. Similar to a modulated dipole, ideal detection should consider the vector difference, , between the received signals for both modulation states. Fig. 8(b) shows the measured baseband signal, in rectangular

(1)

is the difference in the measured power (in dB) bewhere tween the carrier and the fundamental harmonic in one of the sidebands. To measure the modulation-depth, the designed slot was modulated at 20 kHz and the spectrum of the received signal was measured using a spectrum analyzer, as shown in Fig. 9. As can be seen from (1), for 100% modulation depth, the modulated signal should be 6.02 dB below the carrier when the slot is completely opened and closed in modulation. For the designed slot, the fundamental signal harmonic is 6.33 dB below the carrier (dBc), i.e., 0.31 dB deviation from the ideal case. This small discrepancy arises from small signal leakage and losses when the slot is closed and opened, respectively. From these results and using (1), the calculated modulation%. Achieving such a depth for the designed loaded slot is high modulation-depth is attributed to the unique design of the slot where it can, almost completely, be “opened” and “closed.” 3The VNA has a narrow-band tuned-receiver which cannot be used to measure the 24 GHz carrier signal and the modulation sidebands at the same time.

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As opposed to the conventional MST which is based on modulating dipole elements, detecting a modulated signal that is 6 dB below the carrier such as the one resulting from modulating the designed slot is not problematic. In this case, proper amplification can be used to increase the modulated signal level without saturating the receiver front-end, unlike in the conventional MST systems [1]. Besides improving the system sensitivity by the virtue of increasing the modulation-depth, the residual carrier suppression procedures routinely used with the conventional MST imaging arrays are not needed when such a slot is utilized. This enhances the robustness of the imaging system. C. Design Enhancement The modulated response of the slot is affected by the signals leaking from/to biasing structure, i.e., the line used to control the PIN diode. In the above design, small surface mount (SMT) decoupling capacitors were used to reduce this coupling (cf. Fig. 6), and consequently improved modulation-depth. In 2D imaging arrays where hundreds of slots may be used, mounting such capacitors on each slot may be costly and time-consuming. A simple remedy is to route the bias line on the bottom PCB conductor using vias. At high frequencies, a cross-layer via presents a large inductive impedance, and hence, it achieves the required signal decoupling without the need for SMT capacitors. To illustrate the above idea, consider Fig. 10(a), which shows schematics of the unloaded slot without biasing components, i.e., SMT capacitors, and two unloaded slot designs with different bias line routing schemes. One of designs uses bias line components, i.e., SMT capacitors, similar to the design considered earlier. The second loaded slot design instead uses vias to route the bias line. The magnitude of the measured reflection coefficient for each of these three slots is shown in Fig. 10(b). When neither SMT capacitors nor vias are used, high frequency signal couples into the bias line and it shows up as relatively low-frequency resonance in the measured response. This additional resonance is due to higher-order slot mode, and generating this mode requires energy to be taken away from the dominant mode. In this case, the minimum measured response is dB. On the other hand, using the SMT capacitors reduces the coupling to the bias line, i.e., the additional resonance disappears, and causes the minimum reflection to improve to dB. The reflection response with vias, not only reduces the bias line coupling but also further improves the slot reflection dB. This in effect improves the overall slot properties to modulation performance. Note that adding the SMT capacitors and vias change the resonance frequency slightly. IV. ELECTRIC FIELD MEASUREMENT RESULTS To demonstrate the capability of the proposed loaded resonant elliptical slot for accurate electric field distribution measurement (mapping) at high frequencies, an experiment was conducted in which the modulated slot was used to measure the y-component electric field distribution, , as a function of the distance, , from an antenna under test (AUT) at frequency of 24 GHz. The experimental setup is shown in Fig. 11. A flanged K-band open-ended rectangular waveguide was used as AUT

Fig. 10. (a) Schematics of unloaded elliptical slots with different bias line routings, and (b) the corresponding measured magnitude of the reflection coefficient, S .

in this experiment. The AUT and the probing modulated slot were placed inside small anechoic chamber. The parts of the positioning platform which extended inside the chamber were covered by absorbing material to reduce the effects of multiple reflections on the electric field measurements. The magnitude and phase (unwrapped) of the measured electric field using the modulated slot are shown in Fig. 12. The measured electric field distributions are compared to the theoretical values calculated using [21],

(2)

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Fig. 11. Setup for measuring the electric field as a function the distance from the AUT.

slot design is based on loading an elliptical resonant structure with PIN diode. The electric field passing through the slot is modulated by turning the PIN diode ON and OFF (closing and opening the slot, respectively). A prototype slot was constructed to investigate this design concept. Modulation depth close to %100 was obtained using this slot design. The efficacy of the proposed modulated slot for electric field mapping was also experimentally demonstrated at 24 GHz. It has been shown that accurate electric field mapping can be obtained using the proposed slot. The developed slot is attractive as imaging array element due to its small size and high efficiency/modulation-depth. Furthermore, it is a more efficient alternative to the small dipoles used in conventional MST-based imaging and electric field mapping systems Finally, the designed elliptical slot can be used as an electronically controlled waveguide iris for many other purposes (for instance, in constructing waveguide reflective phase shifters and multiplexers/switches). REFERENCES

Fig. 12. Measured and calculated electric field, E , for an open-ended waveguide antenna at 24 GHz.

where is the propagation constant, is the impedance of the medium, is the permeability of the medium, and are the spectral wave-numbers. The expression in (2) defines the y-component electric field radiated by an open-ended waveguide into an infinite half-space using TE and TM spectral funcand . These functions are obtained by tions matching the boundary conditions while including the higherorder modes as well [21]. The field measurements obtained using the modulated slot is in very good agreement with the theoretical results, as shown in Fig. 12. The average magnitude and phase error between the measured and calculated electric field were around 5% and 6 , respectively. Overall, these results show the significant capability and relative performance of a modulated resonant elliptical slot for electric field distribution measurement at high frequencies (i.e., field mapping). Finally, the near-field imaging resolution of the designed slot is inline with that of a typical open-ended waveguide aperture probe, i.e., half the aperture size [22]. V. SUMMARY A novel modulated resonant slot antenna, to be used for electric field mapping at or near millimeter wave frequencies, was introduced and analyzed in this paper. The proposed modulated

[1] J.-C. Bolomey and G. E. Gardiol, Engineering Applications of the Modulated Scatterer Technique. Norwood, MA: Artech House, 2001. [2] J. H. Richmond, “A modulated scattering technique for measurement of field distributions,” IEEE Trans. Microw. Theory Tech., vol. 3, no. 4, pp. 13–15, July 1955. [3] M.-K. Hu, “On measurements of microwave E and H field distributions by using modulated scattering methods,” IRE Trans. Microw. Theory Tech., vol. 8, no. 3, pp. 295–300, May 1960. [4] J.-C. Bolomey et al., “Rapid near-field antenna testing via arrays of modulated scattering probes,” IEEE Trans. Antennas Propag., vol. 36, no. 6, pp. 804–814, June 1988. [5] M. A. Abou-Khousa, “Novel modulated antennas and probes for millimeter wave imaging applications,” Ph.D. dissertation, Electrical and Comput. Eng. Dept., Missouri Univ. Science Technol., Rolla, MO 65409, Apr. 2009. [6] M. A. Abou-Khousa and R. Zoughi, “Multiple loaded scatterer method for E-field mapping applications,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 900–907, Mar. 2010. [7] M. Ghasr, M. Abou-Khousa, S. Kharkovsky, R. Zoughi, and D. Pommerenke, “A novel 24-GHz, one-shot, rapid, and portable microwave imaging system,” in Proc. IEEE Intern. Instrum. and Meas. Tech. Conf. I2MTC, May 2008, pp. 1798–1802. [8] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1999. [9] J. D. Kraus, Antennas, 2nd ed. New York: McGraw-Hill, 1988. [10] Antenna Handbook: Theory, Applications, and Design, Y. T. Lo and S. W. Lee, Eds. New York: Van Nostran Reinhold, 1988. [11] P. Li, J. Liang, and X. Chen, “Study of printed elliptical/circular slot antennas for ultra wideband applications,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1670–1675, Jun. 2006. [12] A. S. Angelopoulos et al., “Circular and elliptical CPW-fed slot and microstrip-fed antennas for ultra wideband application,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 294–297, 2006. [13] C.-Y. Hong, C.-W. Ling, I.-Y. Tarn, and S.-J. Chung, “Design of a planner ultra-wideband antenna with a new band-notch structure,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3391–3397, Dec. 2007. [14] M. A. Abou-Khousa, S. Kharkovsky, and R. Zoughi, “On the mutual-coupling between circular resonant slots,” in Proc. 3rd Int. Conf. on Electromagnetic Near-Field Charact. and Imaging (ICONIC), St. Louis, Jun. 27–29, 2007, pp. 117–122. [15] J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-D radar imaging using range migration techniques,” IEEE Trans. Antennas Propag., vol. 48, no. 5, pp. 728–737, May 2000. [16] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 9, pp. 1581–1592, Sep. 2001. [17] CST—Computer Simulation Technology, [Online]. Available: http://www.cst.com [18] Ma-com Technology Solutions, PIN Diode Data Sheet [Online], [Online]. Available: http://www.macomtech.com/DataSheets/MA4AGP 907_FCP910

ABOU-KHOUSA et al.: MODULATED ELLIPTICAL SLOT ANTENNA FOR ELECTRIC FIELD MAPPING AND MICROWAVE IMAGING

[19] C. Balanis, “Pattern distortion due to edge diffractions,” IEEE Trans. Antennas Propag., vol. 18, no. 4, pp. 561–563, Jul. 1970. [20] A. D. Skinner, “Modulation: Fundamental techniques for traceability,” Inst. Elect. Eng. Colloq. Accred. RF Meas., pp. 6/1–6/6, Feb. 1993. [21] M. Ghasr, D. Simms, and R. Zoughi, “Multimodal solution for a waveguide radiating into multilayered structures—Dielectric property and thickness evaluation,” IEEE Trans. Instrum. Meas., vol. 58, no. 5, pp. 1505–1513, May 2009. [22] N. Qaddoumi, M. Abou-Khousa, and W. Saleh, “Near-field microwave imaging utilizing tapered rectangular waveguides,” IEEE Trans. Instrum. Meas., vol. 55, no. 5, pp. 1752–1756, Oct. 2006. Mohamed A. Abou-Khousa (S’02–M’09–SM’10) was born in Al-Ain, UAE, in 1980. He received the B.S. EE degree (magna cum laude) from the American University of Sharjah (AUS), Sharjah, UAE, in 2003, the M.S. EE degree from Concordia University, Montreal, QC, Canada, in 2004, and the Ph.D. degree in electrical engineering from Missouri University of Science and Technology (Missouri S&T), Rolla, in 2009. Currently, he is an RF Research Engineer with the Imaging Research Laboratories, Robarts Research Institute, London, ON, Canada. His efforts at Robarts are focused on developing RF solutions to improve the performance of the high-field magnetic resonance imaging (MRI) scanners (7 Tesla and beyond). His research interests include high count coil array design for single echo high-field MRI, millimeter wave and microwave instrumentation, numerical electromagnetic analysis, modulated antennas, and wideband wireless communication systems.

Mohammad Tayeb Ghasr (S’01–M’10) received the B.S. degree in electrical engineering (magna cum laude) from the American University of Sharjah (AUS), Sharjah, in 2002, the M.S. degree in electrical engineering from the University of Missouri-Rolla, Rolla, in 2004, and the Ph.D. degree in electrical engineering from Missouri University of Science and Technology (Missouri S&T), Rolla, in 2009. Currently, he is a Research Assistant Professor with the Applied Microwave Nondestructive Testing Laboratory (amntl), Electrical and Computer Engineering Department, Missouri S&T. His research interests include microwave and millimeter-wave instrumentation and measurement, RF circuits, antennas, and numerical electromagnetic analysis.

Sergey Kharkovsky (M’01–SM’03–F’11) received the Diploma in electronics engineering from Kharkov National University of Radioelectronics, Kharkov, Ukraine, in 1975, the Ph.D. degree in radiophysics from Kharkov State University, in 1985, and the D.Sc. degree in radiophysics from the Institute of Radio-Physics and Electronics (IRE), National Academy of Sciences of Ukraine, in 1994. Currently he is a Research Associate Professor in the Applied Microwave Nondestructive Laboratory (amntl), Electrical and Computer Engineering Department, Missouri University of Science and Technology (Missouri S&T), formerly University of Missouri-Rolla. Prior to joining Missouri S&T in March 2003 he was a Member of the Research Staff at IRE from 1975 to 1998, and a Professor in the Electrical and Electronics Engineering Department, Cukurova University, Adana, Turkey, from December 1998 to February 2003. His current research interest is microwave and millimeter wave physics and techniques, imaging, material characterization and nondestructive evaluation of composite structures. Prof. Kharkovsky is an Associate Editor of the IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT.

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David Pommerenke (M’98–SM’03) received the Diploma in electrical engineering and the Ph.D. degree in transient fields of electrostatic discharge from the Technical University of Berlin, Berlin, Germany, in 1989 and 1995, respectively. After working at Hewlett Packard for five years he joined the Electromagnetic Compatibility (EMC) Laboratory, University Missouri-Rolla (currently Missouri University of Science and Technology), in 2001 where he is a Professor in the Electrical and Computer Engineering Department. He has published more than 100 papers and is the inventor on 10 patents. His main research interests are system level ESD, numerical simulations, EMC measurement methods and instrumentations. Prof. Pommerenke was a Distinguished Lecturer for the IEEE EMC Society in 2006/2007.

Reza Zoughi (S’85–M’86–SM’93–F’06) received the B.S.E.E., M.S.E.E., and Ph.D. degrees in electrical engineering (radar remote sensing, radar systems, and microwaves) from the University of Kansas, Lawrence. From 1981 until 1987, he was at the Radar Systems and Remote Sensing Laboratory (RSL), University of Kansas. In 1991, he joined the Missouri University of Science and Technology (Missouri S&T), formerly University of Missouri-Rolla (UMR), where he is currently the Schlumberger Endowed Professor of Electrical and Computer Engineering. Prior to joining Missouri S&T in January 2001 and since 1987, he was with the Electrical and Computer Engineering Department, Colorado State University (CSU). He is the author of the textbook Microwave Nondestructive Testing and Evaluation Principles (Kluwer Academic, 2000) and the coauthor of a chapter on microwave techniques in the undergraduate introductory textbook Nondestructive Evaluation: Theory, Techniques, and Applications (Marcel and Dekker, 2002). He is the coauthor of over 105 journal papers, 250 conference proceedings and presentations and 89 technical reports. He has ten patents to his credit all in the field of microwave nondestructive testing and evaluation. Dr. Zoughi is the Editor-in-Chief of the IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. He is serving his second term as an At-Large AdCom member of the IEEE Instrumentation and Measurement (I&M) Society and also serves as the society’s VP of Education. He is also an IEEE I&M Society Distinguished Lecturer. He has been the recipient of numerous teaching awards both at CSU and Missouri S&T. He is the recipient of the 2007 IEEE Instrumentation and Measurement Society Distinguished Service Award, the 2009 American Society for Nondestructive Testing (ASNT) Research Award for Sustained Excellence Award and the 2011 IEEE Joseph F. Keithley Award in Instrumentation and Measurement. He is also a Fellow of the American Society for Nondestructive Testing. For more information see http://amntl.mst.edu/people/zoughi.html.

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Self-Shielded High-Efficiency Yagi-Uda Antennas for 60 GHz Communications Ramadan A. Alhalabi, Student Member, IEEE, Yi-Chyun Chiou, Member, IEEE, and Gabriel M. Rebeiz, Fellow, IEEE

Abstract—A high-efficiency self-shielded microstrip-fed Yagi-Uda antenna has been developed for 60 GHz communications. The antenna is built on a Teflon substrate ( r ) with a thickness of 10 mils (0.254 mm). A 7-element design results in a measured 11 of at 56.0 – 66.4 GHz with a at 58 – 63 GHz. The antenna shows excellent performance in free space and in the presence of metal-planes used for shielding purposes. A parametric study is done with metal plane heights from 2 mm to 11 mm, and the Yagi-Uda antenna . at 58 – 63 GHz for results in a A 60 GHz four-element switched-beam Yagi-Uda array is also presented with top and bottom shielding planes, and allows for 180 angular coverage with amplitude variations. This antenna is ideal for inclusion in complex platforms, such as laptops, for point-to-point communication systems, either as a single element or a switched-beam system.

gain

S 9 5 dBi

10 dB

gain 12 dBi

= 22

h = 5 8 mm

3 dB

Index Terms—60 GHz, automotive radars, endfire antennas, millimeter-wave antennas, millimeter-wave communication systems, planar antennas, Yagi-Uda antenna.

I. INTRODUCTION HE 60 GHz frequency band presents many attractive properties for wireless communication systems such as wide bandwidth (5–7 GHz) and high atmospheric absorption which makes it ideal for Gbps short distance communication systems. In the past 2–3 years, the 60 GHz has seen a lot of activity with CMOS and SiGe transceiver chips [1]–[6]. It is therefore important to develop planar high-gain antennas for these systems which offer high efficiency and are insensitive to their surroundings. Several antennas and antenna arrays have been demonstrated in the past few years: A 60 GHz aperture-coupled microstrip antenna integrated on LTCC multilayer technology and with 7.6 dBi gain is presented in [7]. A 60 GHz uniplanar-compact electromagnetic band-gap structure on LTCC is presented in [8] and used with aperture-coupled microstrip antenna to improve its

T

gain and reduce mutual coupling. Also, an 8 8 array of patch gain and 3 GHz antennas on a Teflon substrate with bandwidth was presented in [9], and high gain active microstrip antenna arrays on alumina substrates were demonstrated in [10]. Kim et al. presented a 60 GHz CPW-fed micro-machined postsupported microstrip patch antenna which is compatible with silicon processing [11]. A 60 GHz dipole antenna integrated silicon-on-insulator with a gain of 4.5 dBi and on 0.13 8% impedance bandwidth is presented in [12]. The antennas in [7]–[12] are all of the broadside type and do not easily lead to a switched-beam system. On the other hand, endfire antennas such as endfire dipoles [13], Yagi-Uda antennas [14]–[16] or endfire horn antennas [17] can result in high gain and are compatible with a 2- or 4-element switched-beam array. In this paper, we present a microstrip-fed 60 GHz 7-element endfire Yagi-Uda antenna. The design is based on the 24 GHz antenna which was presented in [15]. Differing from [15] which only presented the antenna performance in free space, the Yagi-Uda antenna is characterized first in free space and then in the presence of two metal sheets placed above and below the antenna and with different spacings. In addition, the antenna is characterized inside a metal box (top, bottom and two side-walls) with different heights. The metal shields isolate the Yagi-Uda antenna from its surroundings and allow its insertion in complicated platforms with batteries and dense printed circuit boards (PCBs) available in laptops and mobile phones. A four-element switched-beam Yagi-Uda array is also presented with top and bottom ground planes. This array allows for 180 angular coverage and with excellent performance. To the best of our knowledge, this paper demonstrates for the first time, a 60 GHz Yagi-Uda antenna and a four-element array inside metal sheets, box and PCB boards, with wideband impedance . matching, high efficiency performance, and a II. SINGLE ELEMENT IN FREE SPACE A. Geometry

Manuscript received April 07, 2010; revised July 07, 2010; accepted August 28, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported by INTEL Corporation. The work of Y.-C. Chiou was supported by the National Science Council, Taiwan, under Grant NSC97-2917-I-564-118 for the period September 2009 to August 2010. R. A. Alhalabi was with the Electrical and Computer Engineering Department, University of California, San Diego, La Jolla, CA 92093-0407 USA. He is now with Cavendish Kinetics, Inc., San Jose, CA 95134 USA (e-mail: [email protected]). Y.-C. Chiou and G. M. Rebeiz are with the Electrical and Computer Engineering Department, University of California, San Diego, La Jolla, CA 920930407 USA (e-mail: [email protected]; [email protected]) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103032

Fig. 1 presents the geometry of the 7-element microstrip-fed Yagi-Uda antenna. The antenna is built on a Rogers RT/Duroid 5880 substrate ( ) with a thickness of 10 mils (0.254 mm). The antenna was first designed at 24 GHz on a 15 mils (0.381 mm) Teflon substrate and is scaled here to 60 GHz. The Yagi-Uda antenna utilizes 5 directors printed on the top side of the substrate with a spacing of 1.0 mm. The microstrip trunfrom the driving cated ground plane is located at dipole and acts as a reflector. The driving dipole and the balun between the microstrip feed and the dipole is built using the top and bottom-sides of the Teflon substrate. The antenna was

0018-926X/$26.00 © 2010 IEEE

ALHALABI et al.: SELF-SHIELDED HIGH-EFFICIENCY YAGI-UDA ANTENNAS FOR 60 GHZ COMMUNICATIONS

Fig. 1. 60 GHz microstrip-fed Yagi-Uda antenna geometry: d = 1:1, d = 1:1, L = 2:1, L = 0:9, L = 1:3, W = 0:3, W = 0:2, W = 0:8 (all dimensions are in mm).

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Fig. 3. Measured and simulated radiation patterns of the 60 GHz Yagi-Uda antenna in free space: E-plane (top), H-plane (bottom).

The radiation patterns of the Yag-Uda antenna are measured in the receive mode using a 40–60 GHz diode detector and a lock-in amplifier (Stanford Research Systems, SR830). The RF signal is amplitude modulated with a 1 kHz sine-wave signal and the rectified signal is measured using the lock-in amplifier. A thin absorber is used over the connector and the diode detector to reduce its scattering effects. The measured patterns agree well with HFSS simulations as shown in Fig. 3. The cross-polarization is mainly due to the vertical fields between the dipole arms placed on both sides of the substrate [13], [15], and is at most frequencies. The antenna has a measured 3-dB beamwidth of 54 and 58 at 60 GHz in the - and -plane respectively; and a simulated directivity of 11.0 dBi at 60 GHz.

C. Gain

Fig. 2. (a) Fabricated 60 GHz Yagi-Uda antenna, (b) measured and simulated S . Time domain gating is used to remove the connector effects.

optimized using Ansoft HFSS [19] including a short magnetic ground-plane edge (Fig. 1). B. Input Impedance and Radiation Patterns The input impedance of the Yagi-Uda antenna is measured with a 67 GHz network analyzer (Agilent E8361A) using a 2.4 mm Southwest microwave connector (Fig. 2(a)). The 2.4 mm connector introduces some mismatch to the feeding microstrip line, and the mismatch effects were taken out using the time domain gating. The 7-element Yagi-Uda antenna results in a from 56.0 – 66.4 GHz (Fig. 2(b)). measured A frequency shift of 2.7% is observed and could be due to a change in the substrate value, under- or over-etching of the microstrip line, and time gating effects to remove the connector reflections.

The absolute gain of the Yagi-Uda antenna is measured using a network analyzer (Agilent PNA network analyzer E8361A) and the gain transfer method. Two identical standard gain (horn) antennas are first connected to the two ports of the network analyzer and characterized. One horn antenna is then replaced by the antenna under test and its gain is obtained from the differfor both cases. ence in the measured The measured gain of the Yagi-Uda antenna is shown in ) and Fig. 4. The losses of the microstrip line ( ) at 60 GHz were measured the 2.4 mm connector ( separately and taken out, and this places the reference plane at plane 2 (Fig. 2(a)). The antenna impedance mismatch loss is included in the measured gain. The ripples in the measured gain are due to scattering effects from the connector which was not covered by absorbers during the gain measurements. The from 58 – measured gain of the Yagi-Uda antenna is 63 GHz (10.8 dBi at 60 GHz) and agrees well with simulations. The antenna efficiency, defined as the measured gain over the simulated directivity, is 95% at 60 GHz. This is reasonable since the gain is referred to Ref. plane 2 (see Fig. 2) and does not include any microstrip line loss.

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Fig. 4. Measured and simulated gain of the 60 GHz Yagi-Uda antenna. An error of = : is present in the gain measurements.

+ 0 0 5 dB

Z h L = 1 3 center-to-center spacing =

Fig. 6. HFSS simulated mutual impedance ( ) in free space and with top and bottom metal planes with different , : , : (a) magnitude, (b) phase.

1 mm

S

Fig. 5. 60 GHz Yagi-Uda antenna sandwiched between two foam pieces with thickness of ( ) mm and with metal sheets on the top and bottom planes. . Fig. 5(a) cross-section is to scale for

h

h = 2 mm

III. SINGLE ELEMENT WITH SELF-SHIELDING A. Top and Bottom Metal Planes The high-gain Yagi-Uda antenna will be embedded inside a portable device such as a laptop or a mobile phone where it will be surrounded by unknown metal layers and other electronic components. As a result, it is important to design an antenna which is self-shielded and is not affected by its surroundings. One way to achieve this is to use two metal shielding planes above and below the Yagi-Uda antenna. These metal planes are not connected to the microstrip ground plane and are separated using thin foam pieces by a distance from the Teflon substrate (Fig. 5). The Yagi-Uda antenna operation is based on the mutual coupling between the antenna elements (driving dipole, reflector and directors), and it is necessary that the metal ground planes

Fig. 7. Measured of the 60 GHz Yagi-Uda antenna for different spacing from the metal sheets ( ). (Measurements include connector effects. Simulations are done without connector effects.)

h

do not disturb the mutual coupling between the antenna elements. Fig. 6 presents the HFSS simulated magnitude and phase of the mutual impedance ( ) between two elements with a and center-to-center spacing of 1 mm (see length at Fig. 1). It can be seen that the phase variations are 56–63 GHz for . However, for , the coupling between the two elements changes considerably, and the phase at is (not shown). As a result, the metal shielding planes should be at least 2 mm from the Yagi-Uda antenna; otherwise one has to redesign the antenna for a specific ground-shielding height. of the Yagi-Uda antenna embedded beThe measured tween two metal planes is shown in Fig. 7 for different spacings. The measurements show that metal planes have negligible as long as . Measureeffect on the measured and resulted in a poor ments were also done on match ( ) and are not shown.

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Fig. 8. Measured and simulated radiation patterns with top and bottom metal planes for: (a) h = 2 mm and (b) h = 5 mm: E -plane (top), H-plane (bottom).

Fig. 9. Measured E-plane (left) and H-plane (right) radiation patterns at 60 GHz in free space and with top and bottom metal shielding planes for h = 2 and 5 mm.

The measured and simulated radiation patterns for and 5 mm are presented in Fig. 8 and agree well with simulations. The cross-polarization remains low in the principal planes ) and is slightly higher than the free space antenna ( due to a slight coupling to the shielding planes generating a small vertical electric-field component. The Yagi-Uda antenna excites currents on the shielding planes with certain magnitude and phase depending on the distance from the antenna (h). The currents on the top and bottom planes radiate, and the -plane pattern is a result of the interaction between the current on the Yagi-Uda antenna elements and , there is a the currents on the shielding planes. For significant amount of RF current on the top and bottom planes which are not in phase with the current on the Yagi-Uda, and the resulting -plane pattern is much wider than the free space case. The -plane pattern remains approximately the same since the currents on the top and bottom planes affect mostly , the excited currents on the -plane patterns. For the top and bottom planes are in phase with the current on the Yagi-Uda antenna, and result in a narrower -plane pattern and higher antenna gain than the free space case. The measured radiation patterns at 60 GHz in free space and with top and and bottom metal planes are compared in Fig. 9 for 5 mm.

Fig. 10. Measured: (a) gain versus frequency for different h, (b) gain versus h at 60 GHz of the 60 GHz Yagi-Uda antenna with top and bottom metal sheets. An error of 0:5 dB is present in the gain measurements.

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The measured gain of the Yagi-Uda antenna versus frequency , 5 mm and in free space are shown in Fig. 10(a). for Fig. 10(b) presents the measured and simulated gain at 60 GHz for ( at 60 versus , and shows a GHz). For , the antenna gain and impedance match are greatly affected due to the change in the mutual coupling parameters between the director elements. The design results in an increase in antenna gain (over the free space design) at due to a sharpening of the -plane. B. Self-Shielding Box The antenna can also be shielded from the sides by adding metal walls on each side as shown in Fig. 11(a). The side-walls are kept 1 mm away from the Teflon substrate and do not touch

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Fig. 11. (a) Geometry of the 60 GHz Yagi-Uda antenna sandwiched between two pieces of foam with metal planes on top, bottom and on the sides (b) measured S for different h values. The measurements include the 2.4 mm connector. Fig. 11(a) front-view is to scale with h = 2 mm.

the Teflon substrate or the microstrip ground plane. The total at 60 GHz and therefore can box width is 10 mm, which is , , , , sustain several waveguide modes ( ) etc.). Also, the box height is at least 4.25 mm ( and 10.25 mm ( ) with a partially grounded substrate, and can sustain orthogonal waveguide modes. In this case, the radiation patterns can be determined by the aperture field distribution, and HFSS analysis at 56 – 64 GHz for the first 20 waveguide modes indicates that most of the power couples into mode for and 5 mm (Fig. 12). However, at the , the power is coupled into the , and modes which result in tapering in both the and -planes, and thus wider radiation patterns and a lower gain. of the Yagi-Uda antenna with shielding The measured box is presented in Fig. 11(b) for different values, and the response is from 56 – 64 GHz for . The measured radiation patterns are shown in Fig. 13 and agree well with simulations. The 60 GHz pattern versus are compared with the free space antenna in Fig. 14. Note that for the box , the aperture field distribution results in wider with -plane pattern and narrower -plane patterns. As a result, the is very similar to the free space antenna gain for is gain as shown in Fig. 15(a). Also, the gain for

Fig. 12. Simulated waveguide modes and aperture fields at 59–64 GHz for: (a) h = 2 mm, (b) h = 4 mm, (c) h = 5 mm.

considerably higher at 56–60 GHz than the free space case due to a sharpening in both the - and -plane patterns (Fig. 15(a)). The measured gain versus shows gain enhancement over , a dip at (explained free space for due to distortion above) and a sharp drop in gain for in mutual coupling (Fig. 15(b)). The measurements agree well with simulations. The radiation patterns were also measured with two PCB boards, covered with metal and different electronic components, on top and bottom of the Yagi-Uda antenna and with a spacing (Fig. 16). The measured radiation patterns are of compared to the free space pattern measurements for two cases: not-shielded Yagi-Uda and shielded-box Yagi-Uda (Fig. 17). The -plane patterns are very similar to the free space patterns and the -plane patterns are narrower than the -plane patterns in free space and agrees with previous measurements with top and bottom metal planes. This shows that the PCBs can act as top and bottom metal planes and that highly directive Yagi-Uda antennas can be integrated in complex laptop and mobile phones.

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Fig. 13. Measured and simulated radiation patterns with metal box for: (a) h = 2 mm and (b) h = 5 mm: E-plane (top), H-plane (bottom).

Fig. 14. Measured E-plane (left) and H-plane (right) radiation patterns at 60 GHz in free space and with box shielding for h = 2 mm and 5 mm.

Fig. 16. Yagi-Uda antenna on a PCB setup: without top and bottom metal shield (left), with shielded box (right). The PCB size is 6 cm 4 cm. In the top picture, one PCB is removed for clarity.

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IV. 4-ELEMENT ARRAY

Fig. 15. Measured: (a) gain versus frequency for different h, (b) gain versus h at 60 GHz of the 60 GHz Yagi-Uda antenna with box metal shield. An error of += 0:5 dB is present in the gain measurements.

0

The single element Yagi-Uda antenna has a measured 3-dB beamwidth of 54 in the -plane as shown in Fig. 3. Therefore, four Yagi-Uda antennas with an angular spacing of 45 can be used to cover a 180 radiation angle with a and with gain variation over the whole coverage angle. A single-pole four-throw (SP4T) switch can be used to switch between the four beams [18] or independent transceivers can be placed at each beam for increased data rates. Two fabricated 4-element Yagi-Uda arrays are shown in Fig. 18(a) with the microstrip feed line connected either to Ant. 1 or to Ant. 2. In effect, four different arrays were built to measure each element in the array, but only two are shown. In each case, the other three antenna feeds are left unloaded since it is virtually impossible to use commercial lumped-element resistive loads at 60 GHz (even 0201 resistors are not 50 at 60 GHz). The open ends of the microstrip lines were covered by absorbers during the measurements to reduce their radiation effects on the antenna-under-test. Fig. 18(b) presents the and ) in simulated mutual coupling coefficients ( , ) free space and with top and bottom metal planes ( between the array elements. The mutual coupling is

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Fig. 17. Measured radiation patterns of the 60 GHz Yagi-Uda antenna with (a) two PCBs on the top and bottom of the antenna (h = 5 mm), (b) PCBs with box shielding (h = h ).

Fig. 19. (a) Fabricated 4-element switched-beam array with top and bottom shielding planes, (antennas are not visible), (b) measured S of Ant. 1 and Ant. 2. Fig. 18. (a) Fabricated 4-element switched-beam array of the 60 GHz Yagi-Uda antennas, (b) simulated mutual coupling coefficients (S ).

at 52 – 67 GHz in free space and remains at 54 – 64 GHz with top and bottom metal planes (Fig. 18(b)). of Ant. 1 and Ant. 2 are presented in The measured Fig. 19(b) in free space and with top and bottom metal sheets and 5 mm). It is seen that the over the entire array (

top and bottom ground planes have no effect on the measured impedance. Fig. 20 presents the measured and simulated -plane patterns of the 4-elements in free space and with top and bottom and agree well with simulations. metal planes at , the gain variation over the 180 range is 5 dB For which is higher than the expected value of 3 dB. This is due to the effects of the unloaded elements in the array (no absorbers

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The absolute gain of Ant. 1 and Ant. 2 is also measured in free space and next to metal sheets on top and bottom of the (Fig. 21). As expected from the single-elearray for ment antenna measurements, the gain of the Yagi-Uda antenna due to the naris enhanced by the metal sheets for rower -plane patterns. The measured gain of Ant. 2 is lower than the simulated gain and is due to the coupling to the adjacent unloaded antennas. Simulations done with Ansoft HFSS on antenna 2 and with three unloaded antennas next to antennaunder-test agree well with measurements. V. CONCLUSION This paper presented a comprehensive analysis and experimental characterization of 60 GHz Yagi-Uda antennas with shielding metal-planes. It is seen that the Yagi-Uda antenna performance can be enhanced with well designed shielding structures (metal-planes or boxes), and results in a gain improvement at 60 of 2 – 4 dB over the free-space case ( GHz). The Yagi-Uda antenna was also characterized in a practical environment with printed circuit boards, and showed no degradation in performance. Fig. 20. Measured and simulated E-plane patterns of the 4-element array at 60 GHz: (a) in free space, (b) with top and bottom metal planes (the three unused antennas are not loaded with 50 ).

ACKNOWLEDGMENT The authors thank Drs. U. Karacaoglu, M. Hiranandani, A. Konanur, and S. Yang, all from Intel Corporation, Oregon, for technical discussions and their help in the project. REFERENCES

Fig. 21. Measured and simulated gain of the 4-element array in free space and with metal sheets above and below (h = 5 mm): (a) Ant. 1 and (b) Ant. 2 (the three unused antennas are not loaded with 50 ).

were used under the metal planes to cover the open ended feeds of the antennas) and will not occur if the 4-element array is implemented with 50 terminations on all antennas. HFSS simulations with the unused elements properly terminated show that results in amplithe 4-element array with angle coverage (patterns are not tude variation for shown).

[1] A. Tomkins, R. A. Aroca, T. Yamamoto, S. T. Nicolson, Y. Doi, and S. P. Voinigescu, “A zero-IF 60 GHz 65 nm CMOS transceiver with direct BPSK modulation demonstrating up to 6 Gb/s data rates over a 2 m wireless link,” IEEE J. Solid-State Circuits, vol. 44, pp. 2085–2099, Aug. 2009. [2] S. Reynolds, B. A. Floyd, U. R. Pfeiffer, T. Beukema, J. Grzyb, C. Haymes, B. Gaucher, and M. Soyuer, “A silicon 60 GHz receiver and transmitter chipset for broadband communications,” IEEE J. Solid-State Circuits, vol. 41, pp. 2820–2831, Dec. 2006. [3] M. Tanomura, Y. Hamada, S. Kishimoto, M. Ito, N. Orihashi, K. Maruhashi, and H. Shimawaki, “TX and RX front-ends for 60 GHz band in 90 nm standard bulk CMOS,” in IEEE ISSCC Dig. Tech. Papers, Feb. 2008, pp. 558–559. [4] T. Mitomo, R. Fujimoto, N. Ono, R. Tashibana, H. Hoshino, Y. Yoshihara, Y. Tsutsumi, and I. Seto, “A 60 GHz CMOS receiver front-end with frequency synthesizer,” IEEE J. Solid-State Circuits, vol. 43, Apr. 2008. [5] W. Shin, M. Uzunkol, and G. M. Rebeiz, “Ultra low power 60 GHz ASK receiver with 3–6 GBPS capabilities,” in Proc. IEEE Compound Semiconductor Integrated Circuit Symp., Oct. 2009, pp. 1–4. [6] J. Lee, Y. Chen, and Y. Huang, “A low-power low cost fully-integrated 60-GHz transceiver system with OOK modulation and on-board antenna assembly,” IEEE J. Solid-State Circuits, vol. 45, pp. 264–275, Feb. 2010. [7] J.-H. Lee, N. Kidera, S. Pinel, J. Laskar, and M. Tentzeris, “60 GHz high-gain aperture-coupled microstrip antennas using soft-surface and stacked cavity on LTCC multilayer technology,” in Proc. IEEE Int. Symp. on Antennas and Propagation, Jul. 2006, pp. 1621–1624. [8] A. Lamminen, A. Vimpari, and J. Saily, “UC-EBG on LTCC for 60-GHz frequency band antenna applications,” IEEE Trans. Antennas Propag., vol. 57, pp. 2904–2912, Oct. 2009. [9] S. Holzwarth and L. Baggen, “Planar antenna design at 60 GHz for high date rate point-to-point connections,” in Proc. IEEE Int. Symp. on Antennas and Propagation, Jul. 2005, vol. 1A, pp. 346–349. [10] C. Karnfelt, P. Hallbjorner, H. Zirath, and A. Alping, “High gain active microstrip antenna for 60-GHz WLAN/WPAN applications,” IEEE Trans. Microwave Theory Tech., vol. 54, pp. 2593–2603, Jun. 2006.

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[11] K. Jeong-Geun, L. H. Suk, L. Ho-Seon, A. J.-B. Yoon, and S. Hong, “60-GHz CPW-fed post-supported patch antenna using micromachining technology,” IEEE Microw. Wireless Compon. Lett., vol. 15, pp. 635–637, 2005. [12] M. Barakat, C. Delaveaud, and F. Ndagijimana, “Performance of a 0.13  SOI integrated 60 GHz dipole antenna,” in Proc. IEEE Int. Symp. on Antennas and Propagation, Jun. 2007, pp. 2526–2529. [13] R. A. Alhalabi and G. M. Rebeiz, “High-efficiency angled-dipole antennas for millimeter-wave phased array applications,” IEEE Trans. Antennas Propag., vol. 56, pp. 3136–3142, Oct. 2008. [14] D. Neculoin, G. Konstantinidis, L. Bary, D. Vasilache, A. Stavrinidis, Z. Hazopulos, A. Pantazis, R. Plana, and A. Muller, “Yagi-Uda antennas fabricated on thin GaAs membrane for millimeter wave applications,” in Proc. IEEE Int. Workshop on Antenna Technology, Mar. 2005, pp. 418–421. [15] R. A. Alhalabi and G. M. Rebeiz, “High-gain Yagi-Uda antennas for millimeter-wave switched-beam systems,” IEEE Trans. Antennas Propag., vol. 57, pp. 3672–3676, Nov. 2009. [16] R. A. Alhalabi and G. M. Rebeiz, “Differentially-fed millimeter-wave Yagi-Uda antennas with folded dipole feed,” IEEE Trans. Antennas Propag., vol. 58, pp. 966–969, Mar. 2010. [17] B. Pan, Y. Li, G. Ponchak, J. Papapolymerou, and M. Tentzeris, “A 60 GHz CPW-fed high-gain and broadband integrated horn antenna,” IEEE Trans. Antennas Propag., vol. 57, pp. 1050–1056, Apr. 2009. [18] Y. A. Atesal, B. Cetinoneri, and G. M. Rebeiz, “Low-loss 0.13  CMOS 50–70 GHz SPDT and SP4T switches,” in Proc. IEEE Radio Frequency Integrated Circuits Symp., Atlanta, GA, Jun. 2009, pp. 43–46. [19] Ansoft Corporation, Pittsburgh, PA.

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Yi-Chyun Chiou (S’04–M’08) received the B.S. and M.S. degrees in electronic engineering from Feng Chia University, Taichung, Taiwan, in 2001 and 2003, respectively, and the Ph.D. degree in communication engineering from National Chiao Tung University (NCTU), HsinChu, Taiwan, in 2008. He was a Lecturer at Nan Kai University of Technology, Nanto, Taiwan, from 2003 to 2004, and an Assistant Researcher as well as Adjunct Assistant Professor in the Department of Communication Engineering, NCTU, from 2008 to 2009. He is currently a Postdoctoral Fellow in the Electrical and Computer Department, University of California, San Diego. His research interests include the design of microwave devices and RF modules for microwave and millimeter-wave applications. Dr. Chiou is currently an Editorial Board member of the IEEE TRANSACTIONS MICROWAVE THEORY AND TECHNIQUES and the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He was a corecipient of the APMC (Asia Pacific Microwave Conference) Prize, Bangkok, Thailand, December 2007.

Gabriel M. Rebeiz (F’97) received the Ph.D. degree from the California Institute of Technology. He is a Professor of electrical and computer engineering at the University of California, San Diego. Prior to this appointment, he was at the University of Michigan from 1988 to 2004. He has contributed to planar mm-wave and THz antennas and imaging arrays from 1988 to 1996, and his group has optimized the dielectric-lens antennas, which is the most widely used antenna at mm-wave and THz frequencies. His group also developed 6–18 GHz and 30–50 GHz 8- and 16-element phased arrays on a single silicon chip, making them one of the most complex RFICs at this frequency range. His group also demon) and the new strated high- RF MEMS tunable filters at 1–6 GHz ( > angular-based RF MEMS capacitive and metal-contact switches. As a consultant, he helped develop the USM/ViaSat 24 GHz single-chip automotive radar, phased arrays operating at X, Ku-Band and W-band for defense and commercial applications, the RFMD RF MEMS switch and the Agilent RF MEMS switch. He is the Director of the UCSD/DARPA Center on RF MEMS Reliability and Design Fundamentals. He is the author of the book, RF MEMS: Theory, Design and Technology (Wiley, 2003). Prof. Rebeiz is an IEEE Fellow, an NSF Presidential Young Investigator, a URSI Koga Gold Medal Recipient, the IEEE MTT 2003 Distinguished Young Engineer, and is the recipient of the IEEE MTT 2000 Microwave Prize and the IEEE MTT 2010 Distinguished Educator Award. He also received the 1998 Eta-Kappa-Nu Professor of the Year Award, the 1998 Amoco Teaching Award given to the best undergraduate teacher at the University of Michigan, and the 2008 Teacher of the Year Award at the Jacobs School of Engineering, UCSD. His students have won a total of 19 best paper awards at IEEE MTT, RFIC and AP-S conferences. He has been an Associate Editor of the IEEE TRANSACTIONS MICROWAVE THEORY AND TECHNIQUES and a Distinguished Lecturer for IEEE MTT and AP Societies. He leads a group of 20 Ph.D. students and five postdoctoral fellows in the area of mm-wave RFIC, microwaves circuits, RF MEMS, planar mm-wave antennas and terahertz systems.

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Ramadan A. Alhalabi (S’07) received the B.S. degree in electrical engineering from the Islamic University of Gaza, Palestine, in 2003, and the M.S. and Ph.D. degrees in electrical engineering from the University of California, San Diego, in 2008 and 2010, respectively. He is currently with Cavendish Kinetics, Inc., San Jose, CA, where he is involved in the design and characterization of high performance digital variable RF-MEMS capacitors. His research interests are in the design of RF/microwave and millimeter-wave devices and planar antennas.

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On-Board Printed Coupled-Fed Loop Antenna in Close Proximity to the Surrounding Ground Plane for Penta-Band WWAN Mobile Phone Kin-Lu Wong, Fellow, IEEE, Wei-Yu Chen, and Ting-Wei Kang, Student Member, IEEE

Abstract—A small-size coupled-fed loop antenna suitable to be printed on the system circuit board of the mobile phone for pentaband WWAN operation (824–960/1710–2170 MHz) is presented. The loop antenna requires only a small footprint of 15 25 mm2 on the circuit board, and it can also be in close proximity to the surrounding ground plane printed on the circuit board. That is, very small or no isolation distance is required between the antenna’s radiating portion and the nearby ground plane. This can lead to compact integration of the internal on-board printed antenna on the circuit board of the mobile phone, especially the slim mobile phone. The loop antenna also shows a simple structure; it is formed by a loop strip of about 87 mm with its end terminal short-circuited to the ground plane and its front section capacitively coupled to a feeding strip which is also an efficient radiator to contribute a resonant mode for the antenna’s upper band to cover the GSM1800/1900/UMTS bands (1710–2170 MHz). Through the coupling excitation, the antenna can also generate a 0.25-wavelength loop resonant mode to form the antenna’s lower band to cover the GSM850/900 bands (824–960 MHz). Details of the proposed antenna are presented. The SAR results for the antenna with the presence of the head and hand phantoms are also studied. Index Terms—Handset antennas, loop antennas, mobile antennas, multi-band antennas, WWAN antennas.

I. INTRODUCTION N-BOARD internal printed antennas for penta-band WWAN operation (824–960/1710–2170 MHz) are attractive for slim mobile phone applications. A variety of the on-board printed antennas such as the printed loop antennas [1]–[5], printed monopole antennas [6]–[8], printed shorted monopole antennas or PIFAs (planar inverted-F antennas) [9]–[13] for the WWAN operation have also been reported in the published papers. In order to achieve wideband operation, these antennas are printed on the no-ground region of the system circuit board of the mobile phone and generally require an additional isolation board space to obtain negligible coupling effects [14]–[19] between the antenna’s radiating portion and the nearby ground plane. Hence, these internal printed WWAN antennas in the mobile phone are usually required to occupy the

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Manuscript received February 11, 2010; revised July 25, 2010; accepted August 09, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Department of Electrical Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (e-mail: [email protected]. edu.tw; [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.2010.2103020

entire top edge or bottom edge of the system circuit board. This behavior is similar to that of many traditional three-dimensional internal WWAN mobile phone antennas [20]–[29] and limits the compact integration of the internal antenna with associated electronic components inside the mobile phone. Further, in order to achieve decreased SAR (specific absorption rate) [30], [31], it is preferred that these printed WWAN antennas be positioned at the bottom edge of the mobile phone [8], [11], [32]. In this case, the integration of the antenna with associated electronic components such as a USB (universal serial bus) connector [33] that is usually mounted at the bottom edge of the system circuit board as a data port for the mobile phone becomes a challenging design issue. To solve the problem, the on-board printed antenna should not occupy the entire bottom edge of the circuit board and can also be in close proximity to a protruded ground which is connected with the main ground and has a size large enough to accommodate the USB connector. That is, the on-board printed antenna should have a small size for penta-band WWAN operation and can also be in close proximity to the surrounding ground plane (the protruded and main grounds) without affecting the antenna performances. In this paper, we present a promising small-size on-board printed WWAN antenna to closely integrate with the surrounding ground plane in the mobile phone. The antenna is a coupled-fed loop antenna, which shows a simple structure of comprising a 0.25-wavelength loop strip capacitively coupled by a feeding strip. The coupled-fed loop strip can generate a 0.25-wavelength loop resonant mode at about 900 MHz to cover the GSM850/900 operation. Owing to the 0.25-wavelength loop resonant mode excitation, which is an advantage over the traditional printed loop antenna with the 0.5-wavelength mode excitation as the antenna’s fundamental or lowest resonant mode excitation for mobile phone applications [4], [5], [34]–[43], the antenna requires a small footprint or 375 on the circuit board. In addition, of 15 the coupling feed structure used in the proposed antenna for the 0.25-wavelength loop resonant mode excitation is simpler than the designs used in [1], [2] which applied an internal printed LC matching circuit or an external LC matching circuit with lumped chip elements. Furthermore, wideband excitation of the 0.25-wavelength loop resonant mode is obtained in the proposed antenna to cover both the GSM850 (824–896 MHz) and GSM900 (880–960 MHz) operation. In the designs in [1], [2], the obtained bandwidth of the excited 0.25-wavelength loop resonant mode can cover only either the GSM850 or GSM900 operation.

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In addition, the feeding strip in the proposed antenna is also an efficient radiator to contribute a resonant mode centered at about 1900 MHz to cover the GSM1800/1900/UMTS operation. That is, the proposed antenna can cover penta-band WWAN operation. The proposed antenna is expected to be printed on the bottom edge of the system circuit board of the mobile phone. Owing to its small footprint required, the antenna does not occupy the entire bottom edge of the circuit board and can also be in close proximity to the surrounding ground plane formed by the protruded ground and the main ground printed on the circuit board. Also, the protruded ground has a size of at least to accommodate associated electronic component 10 [33]). The such as a USB connector (typical size 9 7 proposed antenna is fabricated and tested. The obtained results are presented. The antenna’s radiation characteristics including the SAR values for the antenna held by a hand phantom and attached onto a head phantom are also studied. II. PROPOSED QUARTER-WAVELENGTH COUPLED-FED LOOP ANTENNA Fig. 1 shows the geometry of the on-board printed coupled-fed loop antenna in close proximity to the surrounding ground plane for penta-band WWAN operation in the mobile phone. The antenna is printed on a small no-ground board and does not occupy the entire edge space of 15 of the system circuit board of the mobile phone, which is a 0.8-mm thick FR4 substrate of relative permittivity 4.4, loss tangent 0.024, length 115 mm and width 60 mm in this study. A 1-mm thick plastic casing of relative permittivity 3.0 and loss tangent 0.02 also encloses the FR4 substrate to simulate the mobile phone casing. Notice that a protruded ground connected to the main ground of size of size 10 , both printed on the back side of the FR4 sub100 strate as the system ground plane, is in close proximity to the loop strip of the antenna; the distance between the protruded ground and the loop strip is 0.8 mm only (the thickness of the FR4 substrate or the system circuit board of the mobile phone). One edge of the protruded ground is also flushed to the edge of the system circuit board, and the protruded ground has a size large enough to accommodate a USB connector (see the fabricated antenna photo with a USB connector shown in Fig. 5), which can serve as a data port for the mobile phone. Note that on the other side of the protruded ground shown in the figure, there is a second no-ground board space of size of , which can be used to accommodate associated 15 electronic components or other internal antennas. For the former case, extended ground can be added in this board space to accommodate some electronic components, and the extended ground added generally shows small effects on the performances of the proposed antenna. Detailed effects of adding the extended ground of various sizes are studied in Section III. The antenna is formed by a narrow loop strip of length about 87 mm with its end terminal (point B) short-circuited to the main ground as shown in the figure and its front section [coupling strip of length (11 mm in this study)] capacitively coupled to the feeding strip of length (21.5 mm as the preferred length here) through a coupling gap of width (1.5 mm here).

Fig. 1. Geometry of the on-board printed coupled-fed loop antenna in close proximity to the surrounding ground plane for penta-band WWAN mobile phone.

With the coupling feed, the antenna can generate a 0.25-wavelength loop resonant mode to form the antenna’s lower band to cover the GSM850/900 bands (824–960 MHz). Furthermore, the feeding strip can also function as an efficient radiator to contribute a resonant mode for the antenna’s upper band to cover the GSM1800/1900/UMTS bands (1710–2170 MHz). By varying the length of the feeding strip, the antenna’s upper band can be adjusted. On the other hand, the antenna’s lower band can be controlled by tuning the length of the coupling strip and the width of the coupling gap. Detailed effects of the parameters , and on the antenna’s two wide operating bands are analyzed in Section III. III. RESULTS AND DISCUSSION Fig. 2 shows the photo of the fabricated antenna. Results of the measured and simulated return loss are presented in Fig. 3. The simulated results are obtained using the Ansoft HFSS (High Frequency Structure Simulator) version 12 [44], and similar results of the simulation and measurement are obtained. The simulated input impedance on the Smith chart is also shown in Fig. 4 to provide more impedance information for the antenna. The antenna shows two wide operating bands to cover respectively the GSM850/900 and GSM1800/1900/UMTS operations. The impedance matching for frequencies over the two operating bands is better than 6-dB return loss, which is widely used as the design specification for the internal WWAN mobile phone antennas. Result of the measured return loss for the case with a practical USB connector mounted on the protruded ground is presented in Fig. 5. The photo of the fabricated antenna with a practical USB connector is also shown in the figure. Similar results of the measured return loss of the proposed antenna and the case with a USB connector (denoted as Ref1) are observed. The presence

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Fig. 2. Photo of the fabricated antenna; casing not included in the photo.

Fig. 5. Measured return loss for the proposed antenna and the case with a USB connector (size 9 7 4 mm ) mounted on the protruded ground (Ref1).

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Fig. 3. Measured and simulated return loss for the proposed antenna.

Fig. 4. Simulated input impedance on the Smith chart for the proposed antenna.

of a practical USB connector on the protruded ground shows very small effects on the antenna. The proposed antenna is hence promising to integrate with a USB connector in its close proximity for practical applications. The operating principle of the proposed antenna is also analyzed. Fig. 6(a) shows the comparison of the simulated return loss for the proposed antenna, the corresponding antenna with a direct feed (Ref2), and the case with the feeding strip only (Ref3). For both Ref2 and Ref3, no resonant modes at about 900 MHz for the antenna’s lower band can be excited. As seen in Fig. 6(b) in which the simulated input impedance for the three antennas studied in Fig. 6(a) is shown, there is no resonance (zero reactance) seen at about 900 MHz for Ref3. For Ref2, although a resonance at about 700 MHz is seen, the impedance level is very high to prevent the excitation of a resonant mode; that is, this resonance is actually an anti-resonance. By applying the proposed coupling feed, a resonance with its input resistance

close to 50 occurs at about 900 MHz [see Fig. 6(b))]. This leads to the good excitation of a resonant mode for the antenna’s lower band; in addition, this resonant mode has a wide operating band to cover both the GSM850 and GSM900 operations. Since the loop strip has a length of about 87 mm only (close to a quarter-wavelength at about 900 MHz), it can be concluded that a quarter-wavelength loop resonant mode is excited for the antenna [1], [2]. On the other hand, the three antennas all generate a wideband resonant mode at about 1900 MHz. It is especially interesting for Ref3 with the feeding strip only. A resonant mode at about 1900 MHz is generated and its bandwidth can cover the desired frequency range of 1710–2170 MHz. This result indicates that the feeding strip not only serves as a coupling feed for the excitation of the quarter-wavelength loop resonant mode at about 900 MHz, but also functions as an efficient radiator or more specifically a monopole antenna to contribute its fundamental resonant mode at about 1900 MHz. Simulated surface current distributions on the antenna and ground plane at 900 and 1900 MHz are shown in Fig. 7. Stronger currents on the feeding strip at 1900 MHz than at 900 MHz is seen, which supports that the feeding strip is an efficient radiator at about 1900 MHz. Also, strong currents on the loop strip at 900 MHz are seen. This supports that the loop strip is the major radiator at 900 MHz for the antenna. On the ground plane, it is seen that the currents are much smaller at the edge opposite to where the antenna is located. Hence, it can be expected that by mounting the antenna at the bottom edge of the mobile phone as shown in Fig. 13, decreased SAR values can be obtained, allowing the antenna to meet the SAR limit for practical applications. A parametric study of the major parameters on tuning the antenna’s lower and upper bands is also conducted. Fig. 8(a) shows the simulated return loss for the length of the feeding strip varied from 17.5 to 21.5 mm, while the results for the length of the coupling strip varied from 9 to 13 mm are presented in Fig. 8(b). In Fig. 8(a), very small effects on the antenna’s lower band are seen; on the other hand, there are significant effects

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Fig. 8. Simulated return loss for the proposed antenna as a function of (a) the length t of the feeding strip and (b) the length s of the coupling strip. Other dimensions are the same as in Fig. 1.

Fig. 6. Comparison of (a) the simulated return loss and (b) the input impedance for the proposed antenna, the corresponding antenna with a direct feed (Ref2), and the case with the feeding strip only (Ref3). Other dimensions are the same as in Fig. 1.

Fig. 9. Simulated return loss for the proposed antenna as a function of the width g of the coupling gap between the feeding strip and coupling strip. Other dimensions are the same as in Fig. 1.

Fig. 7. Simulated surface current distributions on (a) the antenna and (b) the ground plane.

on the upper band, and the excited resonant mode is shifted to lower frequencies with an increase in the length . This is reasonable, since the length controls the resonant length of the excited mode contributed by the feeding strip. For the results shown in Fig. 8(b), almost no variations are seen for the antenna’s upper band; this is also reasonable, because the length is fixed as 21.5 mm. Conversely, large effects on the antenna’s

lower band are seen. This also confirms that the lower band is related to the loop resonant mode, and hence the variations in the length will cause the shifting of the lower band. Effects of the coupling gap are studied in Fig. 9, in which the results of the simulated return loss for the width of the coupling gap varied from 0.5 to 1.5 mm are presented. Again, small effects on the excited resonant mode at about 1900 MHz are seen, since the dimensions of the feeding strip are not varied. However, it can be seen that the excited resonant mode at about 900 MHz is significantly affected; the impedance matching is quickly degraded when the width is not properly selected. This is because the width greatly controls the coupling excitation of the loop strip. Effects of the total width of the protruded ground and the extended ground added in the second no-ground board space on the antenna performances are studied in Fig. 10. The extended ground can be used to accommodate other associated electronic components in the mobile phone. The simulated results of the return loss for the width varied from 10 to 35 mm are presented.

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Fig. 10. Simulated return loss for the proposed antenna as a function of the total width w of the protruded ground and the extended ground in the second no-ground board space. Other dimensions are the same as in Fig. 1.

Note that when the width is 35 mm, it indicates that the extended ground covers all the second no-ground board space (size ). Small variations in the obtained return loss for 25 various widths of are seen, indicating that the size variations in the surrounding ground plane cause negligible effects on the antenna performances. The radiation characteristics of the proposed antenna are also studied. The measured three-dimensional (3-D) total-power radiation patterns are plotted in Fig. 11. At each frequency, four radiation patterns seen in different directions (front, back, top and bottom) are shown. For lower frequencies at 850 and 925 MHz, dipole-like radiation patterns are seen. For higher frequencies at 1795, 1920 and 2045 MHz, the radiation patterns become more rapidly varied, and more dips or nulls in the radiation patterns are observed. This behavior is related to the surface currents excited on the system ground plane, which is also a part of the radiator [45], [46]. For higher frequencies, since their operating wavelengths are comparable to the length of the system ground plane, there will be surface current nulls excited on the system ground plane, which leads to nulls or dips seen in the obtained radiation patterns. Fig. 12 shows the measured radiation efficiency for the proposed antenna. The radiation efficiency varies from about 50–93% for frequencies over the lower band for GSM850/900 operation, while that over the upper band for GSM1800/1900/UMTS operation varies from about 65–90%. The radiation efficiency is all better than 50%, which is acceptable for practical mobile phone applications. Fig. 13 shows the SAR simulation model and the simulated SAR values for 1-g and 10-g tissues for the proposed antenna. The SAR simulation model including the head and hand phantoms provided by SEMCAD [47] is used. The mobile phone in this study is with the proposed antenna placed at the bottom edge of the system circuit board and is held by the hand phantom and attached to the head phantom. The grip of the hand phantom is shown in the figure. The distance between the palm center and the mobile phone casing is 30 mm, which is reasonable as studied in [31]. The simulated SAR values for 1-g (10-g) head tissue and 1-g (10-g) head and hand tissue are listed in the table in the figure. The return loss given in the table shows the impedance matching level at the testing frequency. The SAR values are obtained using input power of 24 dBm for the GSM850/900 operation (859, 925 MHz) and 21 dBm for the GSM1800/1900/UMTS operation (1795, 1920, 2045 MHz). The SAR values at lower frequencies are about the same

Fig. 11. Measured three-dimensional (3-D) total-power radiation patterns for the proposed antenna.

Fig. 12. Measured radiation efficiency for the proposed antenna.

for the cases with and without the hand phantom. On the other hand, large effects of the hand phantom on the obtained SAR values are seen at higher frequencies. This behavior may be related to the smaller wavelengths at higher frequencies, which become comparable to the dimensions of the fingers of the hand phantom. In this study, however, the obtained SAR values for the proposed antenna are below the SAR limit of 1.6 W/kg for 1-g tissue [30] and 2.0 W/kg for 10-g tissue [48]. IV. CONCLUSION An on-board printed coupled-fed loop antenna with a small footprint of 15 on the system circuit board and

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Fig. 13. SAR simulation model and the simulated SAR values for 1-g and 10-g tissue for the proposed antenna. The return loss given in the table shows the impedance matching level at the testing frequency.

two wide operating bands for penta-band WWAN operation (824–960/1710–2170 MHz) has been proposed for slim mobile phone applications. With a small footprint required, the antenna is suitable to be placed in close proximity to the surrounding ground plane, leading to compact integration of the antenna inside the mobile phone. The antenna has been fabricated and tested. Good radiation characteristics for frequencies over the five operating bands have been observed. The SAR results for the antenna with the presence of the user’s hand and head phantoms meet the 1.6 W/kg limit (1-g tissue) and 2.0 W/kg limit (10-g tissue) for practical applications. REFERENCES [1] Y. W. Chi and K. L. Wong, “Quarter-wavelength printed loop antenna with an internal printed matching circuit for GSM/DCS/PCS/UMTS operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, pp. 2541–2547, Sep. 2009. [2] Y. W. Chi and K. L. Wong, “Very-small-size printed loop antenna for GSM/DCS/PCS/UMTS operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 184–192, Jan. 2009. [3] H. W. Hsieh, Y. C. Lee, K. K. Tiong, and J. S. Sun, “Design of a multiband antenna for mobile handset operations,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 200–203, 2009. [4] K. L. Wong and W. Y. Chen, “Small-size printed loop antenna for penta-band thin-profile mobile phone application,” Microwave Opt. Technol. Lett., vol. 51, pp. 1512–1517, Jun. 2009. [5] Y. W. Chi and K. L. Wong, “Internal compact dual-band printed loop antenna for mobile phone application,” IEEE Trans. Antennas Propag., vol. 55, pp. 1457–1462, May 2007. [6] K. L. Wong and S. C. Chen, “Printed single-strip monopole using a chip inductor for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 58, Mar. 2010. [7] K. L. Wong and T. W. Kang, “GSM850/900/1800/1900/UMTS printed monopole antenna for mobile phone application,” Microwave Opt. Technol. Lett., vol. 50, pp. 3192–3198, Dec. 2008. [8] T. W. Kang and K. L. Wong, “Chip-inductor-embedded small-size printed strip monopole for WWAN operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 966–971, Apr. 2009. [9] T. W. Kang and K. L. Wong, “Simple small-size coupled-fed uniplanar PIFA for multiband clamshell mobile phone application,” Microwave Opt. Technol. Lett., vol. 51, pp. 2805–2810, Dec. 2009.

[10] J. H. Kim, W. W. Cho, and W. S. Park, “A small printed dual-band antenna for mobile handsets,” Microwave Opt. Technol. Lett., vol. 51, pp. 1699–1702, Jul. 2009. [11] C. H. Chang and K. L. Wong, “Printed /8-PIFA for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, pp. 1373–1381, May 2009. [12] C. T. Lee and K. L. Wong, “Uniplanar coupled-fed printed PIFA for WWAN/WLAN operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 1250–1257, May 2009. [13] Y. Yu and J. Choi, “Compact internal inverted-F antenna for USB dongle applications,” Electron. Lett., vol. 45, pp. 92–93, Jan. 2009. [14] C. M. Su, K. L. Wong, C. L. Tang, and S. H. Yeh, “EMC internal patch antenna for UMTS operation in a mobile device,” IEEE Trans. Antennas Propag., vol. 53, pp. 3836–3839, Nov. 2005. [15] K. L. Wong and C. H. Chang, “An EMC foam-base chip antenna for WLAN operation,” Microwave Opt. Technol. Lett., vol. 47, pp. 80–82, Oct. 2005. [16] K. L. Wong and C. H. Chang, “Surface-mountable EMC monopole chip antenna for WLAN operation,” IEEE Trans. Antennas Propag., vol. 54, pp. 1100–1104, Apr. 2006. [17] C. M. Su, K. L. Wong, B. Chen, and S. Yang, “EMC internal patch antenna integrated with a U-shaped shielding metal case for mobile device application,” Microwave Opt. Technol. Lett., vol. 48, pp. 1157–1161, Jun. 2006. [18] C. I. Lin, K. L. Wong, S. H. Yeh, and C. L. Tang, “Study of an L-shaped EMC chip antenna for UMTS operation in a PDA phone with the user’s hand,” Microwave Opt. Technol. Lett., vol. 48, pp. 1746–1749, Sep. 2006. [19] C. H. Wu, K. L. Wong, and J. S. Row, “EMC internal GSM/DCS patch antenna for thin PDA phone application,” Microwave Opt. Technol. Lett., vol. 49, pp. 403–408, Feb. 2007. [20] N. Takemura, “Inverted FL antenna with self-complementary structure,” IEEE Trans. Antennas Propag., vol. 57, pp. 3029–3034, Oct. 2009. [21] R. A. Bhatti, Y. T. Im, and S. O. Park, “Compact PIFA for mobile terminals supporting multiple cellular and non-cellular standards,” IEEE Trans. Antennas Propag., vol. 57, pp. 2534–2540, Sep. 2009. [22] A. Cabedo, J. Anguera, C. Picher, M. Ribo, and C. Puente, “Multiband handset antenna combining a PIFA, slots, and ground plane modes,” IEEE Trans. Antennas Propag., vol. 57, pp. 2526–2533, Sep. 2009. [23] X. Zhang and A. Zhao, “More stabilized triple-band antenna with a rolled radiating arm and a metallic rod for mobile applications,” Microwave Opt. Technol. Lett., vol. 51, pp. 891–894, Apr. 2009. [24] C. L. Liu, Y. F. Lin, C. M. Liang, S. C. Pan, and H. M. Chen, “Miniature internal penta-band monopole antenna for mobile phones,” IEEE Trans. Antennas Propag., vol. 58, pp. 1008–1011, Mar. . [25] S. Hong, W. Kim, H. Park, S. Kahng, and J. Choi, “Design of an internal multiresonant monopole antenna for GSM900/DCS1800/USPCS/S-DMB operation,” IEEE Trans. Antennas Propag., vol. 56, pp. 1437–1443, May 2008. [26] M. Martinez-Vazquez, O. Litschke, M. Geissler, D. Heberling, A. M. Martinez-Gonzalez, and D. Sanchez-Hernandez, “Integrated planar multiband antennas for personal communication handsets,” IEEE Trans. Antennas Propag., vol. 54, pp. 384–391, Feb. 2006. [27] M. Z. Azad and M. Ali, “A miniaturized Hilbert PIFA for dual-band mobile wireless applications,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 59–62, 2005. [28] Y. X. Guo, M. Y. W. Chia, and Z. N. Chen, “Miniature built-in multiband antennas for mobile handsets,” IEEE Trans. Antennas Propag., vol. 52, pp. 1936–1944, Aug. 2004. [29] S. Y. Lin, “Multiband folded planar monopole antenna for mobile handset,” IEEE Trans. Antennas Propag., vol. 52, pp. 1790–1794, Jul. 2004. [30] American National Standards Institute (ANSI), Safety Levels With Respect to Human Exposure to Radio-Frequency Electromagnetic Field, 3 kHz to 300 GHz, , Apr. 1999, ANSI/IEEE standard C95.1. [31] C. H. Li, E. Ofli, N. Chavannes, and N. Kuster, “Effects of hand phantom on mobile phone antenna performance,” IEEE Trans. Antennas Propag., vol. 57, pp. 2763–2770, Sep. 2009. [32] M. R. Hsu and K. L. Wong, “Seven-band folded-loop chip antenna for WWAN/WLAN/WiMAX operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 543–549, Feb. 2009. [33] Universal Serial Bus (USB) [Online]. Available: http://www.usb.org/ [34] B. K. Yu, B. Jung, H. J. Lee, F. J. Harackiewwicz, and B. Lee, “A folded and bent internal loop antenna for GSM/DCS/PCS operation of mobile handset applications,” Microwave Opt. Technol. Lett., vol. 48, pp. 463–467, Mar. 2006.

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[35] B. Jung, H. Rhyu, Y. J. Lee, F. J. Harackiewwicz, M. J. Park, and B. Lee, “Internal folded loop antenna with tuning notches for GSM/ GPS/DCS/PCS mobile handset applications,” Microwave Opt. Technol. Lett., vol. 48, pp. 1501–1504, Aug. 2006. [36] Y. W. Chi and K. L. Wong, “Half-wavelength loop strip fed by a printed monopole for penta-band mobile phone antenna,” Microwave Opt. Technol. Lett., vol. 50, pp. 2549–2554, Oct. 2008. [37] C. I. Lin and K. L. Wong, “Internal multiband loop antenna for GSM/DCS/PCS/UMTS operation in the small-size mobile phone,” Microwave Opt. Technol. Lett., vol. 50, pp. 1279–1285, May 2008. [38] K. L. Wong and C. H. Huang, “Printed loop antenna with a perpendicular feed for penta-band mobile phone application,” IEEE Trans. Antennas Propag., vol. 56, pp. 2138–2141, Jul. 2008. [39] Y. W. Chi and K. L. Wong, “Compact multiband folded loop chip antenna for small-size mobile phone,” IEEE Trans. Antennas Propag., vol. 56, pp. 3797–3803, Dec. 2008. [40] W. Y. Li and K. L. Wong, “Seven-band surface-mount loop antenna with a capacitively coupled feed for mobile phone application,” Microwave Opt. Technol. Lett., vol. 51, pp. 81–88, Jan. 2009. [41] Y. W. Chi and K. L. Wong, “Very-small-size folded loop antenna with a band-stop matching circuit for WWAN operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 808–814, Mar. 2009. [42] W. Y. Li and K. L. Wong, “Small-size WWAN loop chip antenna for clamshell mobile phone with hearing-aid compatibility,” Microwave Opt. Technol. Lett., vol. 51, pp. 2327–2335, Oct. 2009. [43] C. I. Lin and K. L. Wong, “Internal meandered loop antenna for GSM/ DCS/PCS multiband operation in a mobile phone with the user’s hand,” Microwave Opt. Technol. Lett., vol. 49, pp. 759–765, Apr. 2007. [44] Ansoft Corporation HFSS [Online]. Available: http://www.ansoft.com/products/hf/hfss/ [45] P. Vainikainen, J. Ollikainen, O. Kivekas, and I. Kelander, “Resonatorbased analysis of the combination of mobile handset antenna and chassis,” IEEE Trans. Antennas Propag., vol. 50, pp. 1433–1444, Oct. 2002. [46] K. L. Wong, Planar Antennas for Wireless Communications. New York: Wiley, 2003. [47] SEMCAD, Schmid & Partner Engineering AG (SPEAG) [Online]. Available: http://www.semcad.com [48] “Human Exposure to Radio Frequency Fields From Hand-Held and Body-Mounted Wireless Communication Devices—Human Models, Instrumentation, and Procedures—Part 1: Procedure to Determine the Specific Absorption Rate (SAR) for Hand-Held Devices Used in Close Proximity to the Ear (Frequency Range of 300 MHz to 3 GHz),” 2005, IEC 62209-1.

NSYSU. He also served as Chairman of the Electrical Engineering Department from 1994 to 1997, Dean of the Office of Research Affairs from 2005 to 2008, and now as Vice President for Academic Affairs, NSYSU (2007–). He has published more than 480 refereed journal papers and 240 conference articles and has personally supervised 48 graduated Ph.D.s. He also holds over 100 patents, including U.S., Taiwan, China, EU patents, and has many patents pending. He is the author of Design of Nonplanar Microstrip Antennas and Transmission Lines (New York: Wiley, 1999), Compact and Broadband Microstrip Antennas (New York: Wiley, 2002), and Planar Antennas for Wireless Communications (New York: Wiley, 2003). Dr. Wong is an IEEE Fellow and received the Outstanding Research Award three times from National Science Council of Taiwan in 1995, 2000 and 2002, and was elevated to be a Distinguished Research Fellow of National Science Council in 2005. He also received the Outstanding Research Award from NSYSU in 1995, the ISI Citation Classic Award for a published paper highly cited in the field in 2001, the Outstanding Electrical Engineer Professor Award from Institute of Electrical Engineers of Taiwan in 2003, and the Outstanding Engineering Professor Award from Institute of Engineers of Taiwan in 2004. In 2008, the research achievements of handheld wireless communication devices antenna design of the NSYSU Antenna Lab that he led by was selected to be one of the top 50 scientific achievements of National Science Council of Taiwan in past 50 years (1959-2009). He was awarded the 2010 Outstanding Research Award of the Pan Wen Yuan Foundation in June 2010 and selected as top 100 honor of Taiwan by Global Views Monthly in August 2010 for his contribution in mobile communication antenna researches. He is a member of the National Committee of Taiwan for URSI, Institute of Antenna Engineers of Taiwan (IAET), Microwave Society of Taiwan, Institute of Electrical Engineers of Taiwan, and Institute of Engineers of Taiwan. He is listed in Who’s Who of the Republic of China (Taiwan) and Marquis Who’s Who in the World.

Kin-Lu Wong (M’91–SM’97–F’07) received the B.S. degree in electrical engineering from National Taiwan University, Taipei, and the M.S. and Ph.D. degrees in electrical engineering from Texas Tech University, Lubbock, in 1981, 1984, and 1986, respectively. From 1986 to 1987, he was a Visiting Scientist with Max-Planck-Institute for Plasma Physics, Munich, Germany. Since 1987, he has been with the Department of Electrical Engineering, National Sun Yat-Sen University (NSYSU), Kaohsiung, Taiwan, where he became a Professor in 1991. From 1998 to 1999, he was a Visiting Scholar with the ElectroScience Laboratory, The Ohio State University, Columbus. In 2005, he was elected to be Sun Yat-sen Chair Professor of

Ting-Wei Kang (S’08) was born in Kaohsiung, Taiwan, in 1985. He received the B.S. degree in electrical engineering from National Taiwan Ocean University, Keelung, Taiwan, in 2007. He is now working toward the Ph.D. degree at National Sun Yat-Sen University, Kaohsiung, Taiwan. His main research interests are in internal antennas for mobile communication devices, especially for small-size multiband antennas in the mobile phones and laptop computers. Mr. Kang won the first prize at the National Mobile Handset Antenna Design Competition, in Taiwan, in 2008. He also won the Student Paper Award in the 2009 and 2010 International Symposium on Antennas and Propagation (ISAP).

Wei-Yu Chen was born in Taipei, Taiwan, in 1986. He received the B.S. degree in electrical engineering from National Central University, Chungli, Taiwan, in 2008 and the M.S. degree in electrical engineering from National Sun Yat-sen University, Kaohsiung, Taiwan, in 2010. He is currently an Antenna Engineer with System Application Division, MediaTek Inc., Hsinchu, Taiwan. His main research interests are in WWAN and LTE antennas for mobile phone applications.

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Pattern Purity of Coiled-Arm Spiral Antennas Matthew J. Radway, Member, IEEE, Thomas P. Cencich, Senior Member, IEEE, and Dejan S. Filipovic´, Senior Member, IEEE

Abstract—The consequences of the coil-type arm treatment on the far-field characteristics of spiral antennas excited in the broadside mode (also referred to as Mode 1) are discussed. Computational analysis of the coiled-arm spiral antenna with two and four arms is performed using the moment method, and the effects on modal content, 3-dB beamwidth, azimuthal gain uniformity, off-axis axial ratio, and co-polarized broadside gain are reported. Predictions of the numerical calculations are verified experimentally through far-field measurements. It is found that the coil-type arm treatment produces significant modal contamination of the far-field pattern when compared to a conventional smooth spiral (with no arm coiling). It is shown that these effects can be mitigated by employing an antenna design that uses four spiral arms. For instance, the measured cross-polarization discrimination of the four-armed spiral at 47 degrees from broadside is improved by up to 15 dB with respect to a comparable two-armed spiral. Index Terms—Antenna radiation patterns, antennas, four-arm spiral antennas, loaded antennas, log spiral antennas, spiral antennas.

I. INTRODUCTION PIRAL antennas have been studied extensively in the engineering literature since their invention in the 1950s [1], [2]. These antennas have been used traditionally in monopulse direction finding systems [3] and in radar warning receivers because of their inherently wide circularly-polarized pattern and impedance bandwidths. In order to realize these desirable characteristics, the spiral typically must be operated above the so-called “cutoff” frequency, dictated by the antenna’s circumference and mode of excitation [3]. Over the years, many efforts have been undertaken in reducing the spiral’s cut-off frequency while maintaining the same size of the spiral. One frequently-employed miniaturization method seeks to increase the electrical circumference of the antenna structure through the use of various loading strategies. These include dielectric [4], magneto-dielectric [5], artificial media [6], arm-shaping [7]–[10], discrete-element loading [11], or a combination of these [12]–[14]. Although many have studied this concept from

S

Manuscript received April 15, 2010; revised July 27, 2010; accepted September 24, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by Lockheed Martin Space Systems Company and in part by the Office of Naval Research, United States Navy, under Grant N00014-07-1-1161. M. J. Radway and D. S. Filipovic´ are with the Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO 803090425 USA (e-mail: [email protected]; [email protected]). T. P. Cencich is with Lockheed Martin Space Systems Company, Littleton, CO 80125 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.2010.2103033

the standpoint of input impedance and broadside gain, little information is available concerning the effects on the spiral’s pattern purity. Pattern purity, as used in this paper, is the degree to which a radiation pattern is free from the detrimental effects of modal contamination, such as cross-polarization, azimuthal gain variation, and beamwidth variation, for example. Modal contamination is the presence of undesired terms in the modal decomposition of the radiation pattern, discussed in Section II. In this paper we thoroughly discuss the effects of the arm-coiling technique on the far-field characteristics of the spiral antenna. The two-armed spiral antenna has been studied far more extensively in the open literature than the four-armed spiral. It is well-known that it is a simple matter to feed a two-armed spiral with a Dyson (infinite) balun [2] in order to achieve broadband broadside operation, also referred to as Mode 1. In contrast, four-armed spirals require more complicated feed arrangements. However, the discussion to follow will show that four-armed spiral antennas possess important performance advantages over the two-armed spiral when combined with the arm-coiling treatment. For instance, it is shown that the measured cross-polarization discrimination of the four-armed spiral at 47 from broadside is improved by up to 15 dB with respect to a comparable two-armed spiral. This paper is organized as follows. In Section II, the theory of spiral modes is briefly reviewed, which can provide useful insight into the physical processes that affect the far field. The geometry under study is discussed in detail in Section III. In Section IV, the effect that the arm coiling has on the modal content of the far field is examined, and the physical features of the arm coiling that affect the modal content are identified. Using the modal decomposition data, the effects on pattern azimuthal nonuniformity (WoW), axial ratio, beamwidth, and broadside gain are discussed. The effects of the beam-forming network (BFN) are also discussed. Finally, the fabrication of a coiled-arm spiral and comparisons between measurements and simulations performed using FEKO [15] are given in Section V.

II. SPIRAL MODE THEORY The far field of an antenna has the general form (1) where is the free space wavenumber, is the radial coordi, is an amplitude nate in the spherical system distribution, is a phase distribution, and is the spherical wave function. For an infinite-armed spiral of infinite

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Fig. 1. Angle off broadside patterns for various spiral modes computed using (2) and normalized to the pattern maximum. Note that all modes except for Mode 1 have nulls at broadside.

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extent, the far-field amplitude and phase distributions have the form [16], [17]

(2) and

Fig. 2. Instantaneous phase distribution on Mode 1-excited two- and fourarmed smooth (non-coiled) spiral apertures, showing positive (lightly-shaded) and negative (darkly-shaded) phase. The active regions are highlighted with their corresponding mode indices, demonstrating the superior mode-rejection of the four-armed spiral. (a) Two-armed. (b) Four-armed.

possible for the spiral to radiate in this condition at reduced efficiency. When the spiral is in cutoff, the outward traveling Mode has not decayed by the radiation mechanism, and instead a is produced, which propagates reflected mode with index inward. The energy radiated by this spurious mode is predominantly cross-polarized to the desired mode. For any given far field it is possible to decompose each field component with respect to azimuth using the integral [19]

(3) is the spiral growth rate. In the terminology of [18], is the mode number and the resulting field constitutes a mode of the spiral antenna. Such a mode has the following features: 1) For any constant angle off broadside the phase advances as the azimuth is swept from 0 to . linearly by 2) The far field is circularly polarized with the sense of polarization dependent on the sign of . have nulls at broadside. 3) All modes with 4) The far field has zero magnitude in the plane of the spiral. The amplitude distributions for various modes are shown in Fig. 1. Despite being strictly applicable only to the infinite-armed spiral of infinite spatial extent, it turns out that these modes are sufficient to accurately describe the operation of spirals that possess a finite number of arms [18], provided that the arms are . Here excited with equal amplitude and with phases is the number of arms and is the arm number. Since an arbitrary excitation can be represented as weighted sum of terms with this form, the operation of spiral antennas is often described in terms of this so-called spiral mode theory. In this paper the far-field performance of spiral antennas is described on the fundamental level in terms of these spiral modes. Additionally, it has been noted that each mode radiates from an annular-shaped region (called the active region) on the spiral (Fig. 2) [3]. When the spiral cirwith circumference , the current decays rapidly as cumference is greater than it travels through this region. When the spiral circumference is the spiral is said to be “in cutoff”, although it is less than Where

(4) where is a given vector component of the far field (e.g., , , is the mode field. While this expresRHCP, or LHCP), and sion obviously yields Fourier series coefficients, this procedure also resembles the partial (azimuthal) decomposition of a field into spherical modes, and therefore the term modal decomposition is often used to describe the procedure. We call the entire collection of mode fields the modal content, and any undesired mode fields the modal contamination. The corresponding remains associated with that vector component, so that various interpretations of the modes in terms of polarization are pos-type excitation typically applied to sible. Due to the spirals, the fields associated with the desired mode will often dominate all other terms. This efficiency of representation makes these mode fields practical for performing analysis of far-field errors. The analysis of Section IV makes frequent use of this technique to gain insight into physical processes affecting the far field. In addition to their use in pattern analysis, the procedure is used to analyze errors in the beam-forming network (BFN). Often such analysis reveals that a particular component of the BFN error is responsible for a given far-field parameter of interest (e.g., axial ratio (AR)). The technique is used in the BFN analysis of Section IV. III. COMPUTATIONAL MODEL DESCRIPTION The concept of using various types of meandering as a miniaturization strategy for spiral antennas has been advanced by many researchers ( [7], [8], [14], [20]). Here we explore the effects of a simple type of volumetric meandering (arm coiling)

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Fig. 3. Two-armed, seven-turn, right-handed spiral with arm coiling. The center portion is a conventional smooth self-complementary spiral with 51/2 turns, and the outer portion is a coiled wire with 11/2 turns. The adopted coordinate system is shown at the center.

Fig. 5. Three-dimensional co-polarized (RHCP) gain patterns for the coiled-arm spiral antennas at the “cutoff” frequency f . The patterns are similar to those of self-complementary, smooth spiral antennas (not shown). (a) Two-armed. (b) Four-armed.

IV. COMPUTATIONAL STUDIES A. 3D Patterns

Fig. 4. Four-armed, seven-turn, right-handed spiral with arm coiling. The center portion is a conventional smooth self-complementary spiral with 51/2 turns, and the outer portion is a coiled wire with 11/2 turns. The arm length is identical to that of the two-armed spiral. The adopted coordinate system is shown at the center.

on the performance of spiral antennas with two and four arms. For this study of arm coiling, an arm geometry was chosen that features two distinct regions. First is an inner self-complementary non-coiled equiangular spiral with a growth rate of and with 51/2 turns. This is in contrast with [13], [14], where a wire (non-self-complementary) inner region was used. Second is an outer wire portion with volumetric coiling featuring 168 coils per spiral turn, with 11/2 turns, and with the same growth rate as the inner portion. This portion follows the coiling used in [13]. The upper surface of the coiled portion coincides with the plane of the inner spiral, while the lower surface grows exponentially downward with increasing radius. The exponential coil growth rate preserves the log-periodic nature of the structure to some extent. The arm lengths were kept constant among the two- (Fig. 3) and four-armed (Fig. 4) cases to make the results easily comparable. The overall effect of the coiling is to slow down the outward-traveling wave in the coiled portion of the spiral, thus decreasing the radius of the active region. The expected result is increased realized gain of the desired mode below the spiral’s cut-off.

An investigation into the co-polarized gain of four-armed spirals is performed using the commercially-available method-ofmoments code FEKO. For the study a free-standing (free-space) coiled-arm spiral antenna with a radius of 75 mm is used, with other parameters as previously described. We note that the dielectric loading and absorptive backing are not modeled (as often done in the literature [4], [21], [22]) to reduce the computational complexity; however, their effects will be assessed in Section V. Also, the free-space models used herein simplify the physical interpretation of the spiral’s performance. The results, given in the following sections, are shown normalized with refor convenience spect to the Mode 1 cutoff frequency ( is the speed of light and is the spiral’s circumference). Far below cutoff, the two- and four-armed spiral patterns resemble those of a small horizontal dipole ( in the -plane ), and 360 in the -plane) and crossed dipole ( respectively. Well above cutoff the patterns take on the Mode 1 shape ( ). At , the patterns show a superposition of these characteristics (Fig. 5). It is important to note that as frequency decreases below the polarization of the two-armed spiral becomes increasingly linear, while the four-armed spiral retains excellent circular polarization. Also, it is well-known that the spiral antenna pattern rotates with frequency [16], owing to its scaling and rotation properties. For the two-armed spiral operated below cutoff, this presents a significant challenge in practical systems. Additionally, the orientation of the polarization ellipse at broadside rotates with frequency, exacerbating the pattern rotation problem. By virtue of its circular polarization at broadside and symmetric pattern below cutoff, the four-armed spiral is far less susceptible to these problems. ) co-polarized raThe shape of the low-frequency ( diation patterns (Fig. 5) are practically unchanged compared to those for the free-standing smooth self-complementary spiral. It is apparent that the four-armed spirals have retained their advantage over the two-armed spiral.

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Fig. 7. Mode gains versus frequency at 47 off broadside for the Mode 1-excited four-armed coiled-arm spiral antenna. Note that the even-ordered modes, as well as modes 3, 1, and 5, are suppressed by the spiral geometry as well as the mode of excitation.

0

Fig. 6. Mode gains versus frequency at 47 off broadside for Mode 1-excited two-armed (a) smooth and (b) coiled spiral antennas.

0

the arm coiling is that the coils act as energy-storage elements, preventing Mode 1 from radiating as efficiently from the coiled arms as it would if the arms were made smooth. This causes . 0.4 dB gain reduction at Fig. 7 shows the modal decomposition of the Mode 1-excited four-armed coiled spiral shown in Fig. 4. In this configuration unradiated Mode 1 energy reflects from the arm ends instead of Mode as for the two-armed spiral. as Mode contamination never reaches a high level However, Mode ), the because once that mode can radiate efficiently (above energy has already been radiated from the Mode 1 active region located well within the smooth inner spiral. This clearly demonstrates the motivation for including a non-coiled inner spiral.

B. Modal Content

C. 3-dB Beamwidths

As explained in Section II, it is possible to decompose the pattern for a given angle off broadside into its constituent modes, since the spiral modes are orthogonal with respect to azimuth. Fig. 6(a) shows the mode gains of the two-armed spiral at an maximum angle off broadside of 47 , the theoretical Mode for small growth rates. Above , the Mode 1 active region is predominantly inside the spiral , allowing Mode 1 to radiate effiradiation ciently and producing less efficient spurious Mode ( at ). Above , energy not radiated from the Mode 1 active region is free to radiate from the Mode 3 active region, producing the spurious Mode 3. If the frequency becomes high enough that the active region lies within the feed region, then will again increase if the feed region cannot discrimMode inate that mode. Below , the Mode 1 active region is larger than the spiral , and therefore that mode radiates inefficiently. radiates at about the same level as the The reflected Mode desired Mode 1, so the overall polarization can be expected to be nearly linear. Fig. 6(b) shows the modal decomposition of the Mode 1-exand cited coiled two-armed spiral shown in Fig. 3. Modes appear to radiate more efficiently because their active region the smooth inner diameters have also been reduced. Above portion of the spiral radiates Mode 1 efficiently, causing Mode to decrease. Between and approximately the radiation occurs primarily from the coiled portion. A side-effect of

Using (2), the 3 dB beamwidth of an infinite spiral with small growth rate can be estimated to be approximately 76 . However, the presence of spurious modes can produce significant deviation from this value. Fig. 8 shows the minimum, maximum, and average 3 dB beamwidths for the three spirals discussed earlier. the two-armed spirals’ beamwidths Below approximately vary between 85 to 360 , consistent with a dipole-like radiathe smooth two-armed spiral shows tion pattern. Above little azimuthal variation of the beamwidth since the undesired modes have fallen to more than 20 dB below Mode 1. The coiled two-armed spiral continues to have large beamwidth variation due to Mode contamination. In contrast, the fouruntil armed spiral shows little beamwidth variation throughout due to its very low level of modal contamination. The beamwidth tran) below sitions smoothly from that of a crossed dipole ( to that of an infinite spiral ( ) well above . D. WoW Another parameter of frequent interest in direction finding [23] is the azimuthal co-polarized gain variation (Wobble of the Wave – WoW), defined as the ratio of maximum co-polarized gain to the minimum co-polarized gain, at a given angle off to the broadside. In Fig. 9 we see that in the range of arm coiling degrades the WoW compared to the smooth spiral, due to increased presence of Mode 3. By increasing the number

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Fig. 8. Co-polarized minimum, maximum, and average 3-dB beamwidths (with respect to azimuth, sampled in 6 increments) versus frequency. The shaded regions indicate the range between the minimum and maximum 3-dB beamwidths, while the solid lines indicate the average beamwidth. The presence of inductive loading has introduced a noticeable amount of ripple in the beamwidth.

Fig. 9. Co-polarized azimuthal gain variation at 47 from broadside. Performance is degraded compared to the non-coiled case. Shown in the inset are azimuthal gains at f .

of arms to four, the level of WoW can be reduced to a low level throughout. Additionally, from Fig. 10 it is seen that WoW vs. angle off broadside has also been reduced significantly by the use of the four-armed spiral. E. Axial Ratio By coiling the spiral arms, the axial ratio (Fig. 11) below has been improved somewhat compared to the smooth spiral, and the inbut is still at a very high level. Between from the arm coiling causes the creased presence of Mode AR to remain high in that region. The AR for the four-armed spiral remains low throughout, owing to the reduced modal contamination afforded by the four-armed spiral and excitation. The

Fig. 10. Co-polarized azimuthal gain variation at the cutoff frequency versus angle off broadside.

Fig. 11. Minimum, maximum, and average (with respect to azimuth, sampled in 6 increments) axial ratios at 47 from broadside. Performance is degraded substantially compared to the non-coiled case, being unacceptable for frequencies below approximately 2:5f .

main contributor to AR for the four-armed spiral is Mode , which exists due to the reflection of residual Mode 1 from the arm ends. Fig. 12 shows a plot of the AR at versus angle off broadside. Here one can see that the axial ratio of the four-armed spiral ) is perfect at broadside, and remains in a useful range ( out to 50 with very little variation over all azimuthal angles. F. Co-Polarized Broadside Gain Fig. 13 shows that the broadside gain below is improved by implementing arm coiling. Above that frequency, the arm coiling reduces the gain due to the increased modal contaminato tion. Some of this gain is recovered over the range through the use of a four-armed spiral; however, addition of a

RADWAY et al.: PATTERN PURITY OF COILED-ARM SPIRAL ANTENNAS

Fig. 12. Minimum, maximum, and average (with respect to azimuth, sampled in 6 increments) axial ratios versus angle off broadside evaluated at the cutoff frequency f . The four-armed coiled spiral shows excellent performance compared to two-armed spirals.

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Fig. 14. Comparison of broadside axial ratio for the ideal-beamformed twoarmed spiral and the four-armed spiral with realistic beam-forming network (see inset) having misbalances of 0:5 dB amplitude and 2 phase (Modes 1 and 23 dB). 3 cross-modal discrimination CMD

6



6

of the Marchand baluns, and assuming an amplitude error of and phase error of , then the cross-modal discrimination between Mode 1 and Mode 3 ( ) will be approximately 23 dB. Fig. 14 compares the two-armed spiral with an ideal BFN to the four-armed spiral with this realistic BFN. It shows that, even considering errors due to a realistic BFN, the four-armed spiral has performance below far superior to the two-armed spiral. V. FABRICATED ANTENNAS

Fig. 13. Broadside co-polarized gain versus frequency. While the gain has improved at the lower end of the frequency range, the mid-range and high-end gain have been reduced somewhat.

physical four-armed Mode 1 beamformer is likely to reduce this advantage somewhat. G. Beam-Forming Network Errors Throughout the preceding analysis, the excitation was provided by an ideal BFN. However, physically realizable BFNs will invariably have errors, which can be characterized by the procedure in Section II. A typical four-armed Mode 1 BFN is shown in the Fig. 14 inset, which consists of a quadrature hybrid followed by two Marchand baluns. The amplitude and phase errors of this network arrange themselves in such a way to produce spurious Mode 3. If the spiral can not radiate or dissipate this mode efficiently, the mode will reflect from the ends , which of the spiral arms with the same phasing as Mode produces cross-polarization at broadside. Assuming that the errors of the quadrature hybrid are much larger than the errors

Using the knowledge gained from the studies of the previous section, two 141 mm-diameter antennas with two and four arms are constructed (the four-armed spiral is shown in Fig. 15 and the fully assembled two-armed spiral in Fig. 16). Both antennas feature a combination laminate consisting of 508 -thick RT/Duroid 5880 bonded to 3.175 mm-thick FR-4. The center region consists of a non-coiled equiangular spiral printed on the RT/Duroid material. The metal-to-slot ratio ) of this portion is linearly tapered from at the ( center to (self-complementary) at the outside of the center region. Since no miniaturization is required in the center region, a 63.5 mm-diameter portion of the FR-4 is removed to ). help preserve pattern purity at high frequencies ( The outside region consists of an equiangular coiled-arm spiral printed on the combination portion of the laminate. The arm width is 1.016 mm and the coil pitch is linearly tapered from 38 coils/turn at the inside to 80 coils/turn at the outside. The greater thickness of the outside region substrate allows larger coils to be formed, increasing the path length, and thus allowing a higher degree of miniaturization. A polyvinyl chloride (PVC) sidewall (Fig. 16) with 141 mm outer diameter and 6.35 mm wall thickness is used to mechanically support the spiral. If a conducting ground plane is used, then the low-frequency gain is reduced by the tendency of reactive energy to couple to the

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Fig. 15. Fabricated four-armed spiral with arm coiling applied to the final 11/2 turns. The two-armed spiral is fabricated similarly, but with 21/4 turns of arm coiling. (a) FEKO top. (b) FEKO bottom. (c) Fabricated top. (d) Fabricated bottom.

Fig. 17. Comparison of the measured and simulated realized gains of the twoand four-armed coiled-arm spirals, averaged with respect to azimuth. (a) Broadside ( ). (b) Off-broadside ( ).

=0

Fig. 16. Fabricated two-armed spiral antenna with PVC sidewall and ferrite tile backing.

ground plane. This problem is mitigated through the application of ferrite tile absorber (TDK IB011), which predominantly absorbs the backward radiation over the frequency range of the antenna. The spiral is excited at the center using a phase-matched and soldered bundle of 2.159 mm-diameter semi-rigid coaxial cables. To simulate an ideal Mode 1 BFN (IBFN), the measured far field of each arm is given the appropriate complex weighting and then the far fields are superimposed. The use of the IBFN allows evaluation of the spiral and feed cable assembly in isolation from the BFN, which is useful when assessing the physics of the antenna element. Each spiral is modeled in FEKO, using MoM with multilayer Green’s function (see Fig. 15(a) and (b)). In order to simulate the antenna using the available computational resources, the following simplifications were made to the computational models.

= 47

• The contributions of the dielectric sidewall, the squareshaped ferrite-coated backing, and the feed structure are neglected; • The combination laminate is laterally infinite, as required by the multilayer Green’s function implementation; • The material properties of the bonding layer are assumed to be that of FR-4; • The entire coiled-arm portion is modeled as thin wires; • The removal of FR-4 from beneath the center region is not modeled. Fig. 17 shows the simulated and measured broadside and off-broadside co- and cross-polarized (i.e., RHCP and LHCP) gains of the two- and four-armed spirals, and it is found that FEKO correlates well with measurements. In both the two- and four-armed cases the measured co-polarized gain is higher than the models predict in the range 0.3–0.6 GHz, likely due to the combined unmodeled effects of the PVC sidewall and the ferrite backing. In the FEKO model, the LHCP gain at broadside is that ); however, the measured of the numerical noise (

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in radar warning receivers. In contrast, four-armed spiral antennas were shown to have a high degree of immunity from the contaminating effects of the miniaturization treatment. For many applications the performance improvement afforded by the four-armed spiral justifies the additional cost and complexity of the beam-forming network. ACKNOWLEDGMENT The authors would like to thank Messrs. J. Burford and N. Kefauver for performing the antenna measurements. REFERENCES

Fig. 18. FEKO-simulated and measured elevation pattern overlays at 475 MHz, 675 MHz (f ), and 1000 MHz for the fabricated (a) two-armed and (b) fourarmed spirals. The four-armed spiral shows consistently improved cross-polarization discrimination compared to the two-armed spiral. (a) Two arms. (b) Four arms.

cross-polarized gain does not vanish at broadside. One potential source of this discrepancy is the circularly-asymmetric antenna backing, and another is the range imperfections, including multipath, alignment error, tower bounce, and range axial ratio. Overall, the pattern of the four-armed spiral shows significant improvement in cross-polarization discrimination over the two-armed spiral, thus validating our earlier results and conclusions. For example, at 475 MHz, the discrimination of the four-armed spiral at 47 from broadside has increased to beyond 15 dB, a huge improvement over the two-armed spiral. Nearing broadside, the improvement becomes even more sigwith respect to nificant, due to the rapid decrease of Mode the desired Mode 1 (see Fig. 18). As stated above, range imperfections and asymmetry in the antenna backing are the causes of the increased cross-polarization at broadside. VI. CONCLUSIONS This paper has compared the performance of coiled-arm twoand four-armed spiral antennas. It was found that the four-armed spiral outperforms the two-armed spiral in nearly every respect when the spirals are operated near or below cutoff. In some instances the two-armed spiral was shown to have undesirable characteristics that make it an unsuitable candidate for practical antenna applications. On the other hand, the four-armed spiral retains its far-field performance characteristics in a robust way, even down to frequencies far below cutoff. Due to its flexibility and relatively low cost, arm coiling is an attractive miniaturization strategy for spiral antennas. However, it was shown that the use of arm coiling to improve low-frequency gain of the two-armed spiral produces mixed results when pattern purity effects are considered. While the low-frequency gain is improved, it occurs at the expense of other farfield parameters that are important in applications where spirals are frequently used, such as in radio direction finding and

[1] J. A. Kaiser, “The archimedean two-wire spiral antenna,” IRE Trans. Antennas Propag., vol. 8, pp. 312–323, May 1960. [2] J. D. Dyson, “The equiangular spiral antenna,” IRE Trans. Antennas Propag., vol. AP-7, pp. 181–187, Apr. 1959. [3] R. G. Corzine and J. A. Mosko, Four-Arm Spiral Antennas. Norwood, MA: Artech House, 1990. [4] B. A. Kramer, M. Lee, C.-C. Chen, and J. L. Volakis, “Design and performance of an ultrawideband ceramic-loaded slot spiral,” IEEE Trans. Antennas Propag., vol. 53, pp. 2193–2199, Jul. 2005. [5] B. A. Kramer, S. Koulouridis, C.-C. Chen, and J. L. Volakis, “A novel reflective surface for an UHF spiral antenna,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 32–34, Dec. 2006. [6] J. L. Volakis, G. Mumcu, K. Sertel, C.-C. Chen, M. Lee, B. Kramer, D. Psychoudakis, and G. Kisiltas, “Antenna miniaturization using magnetic-photonic and degenerate band-edge crystals,” IEEE Antennas Propag. Mag., vol. 48, no. 5, pp. 12–28, Oct. 2006. [7] H. Nakano and J. Yamauchi, “Sunflower spiral antenna,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 1980, pp. 709–712. [8] D. S. Filipovic´ and J. L. Volakis, “Broadband meanderline slot spiral antenna,” IEE Proc. – Microw. Antennas Propag., vol. 149, no. 2, pp. 98–105, Apr. 2002. [9] D. S. Filipovic´ and J. L. Volakis, “A flush-mounted multifunctional slot aperture (combo-antenna) for automotive applications,” IEEE Trans. Antennas Propag., vol. 52, pp. 563–571, Feb. 2004. [10] D. S. Filipovic´, A. U. Bhobe, and T. P. Cencich, “Low-profile broadband dual-mode four-arm slot spiral antenna with dual Dyson balun feed,” IEE Proc. Microw. Antennas Propag., vol. 152, no. 6, pp. 527–533, Dec. 2005. [11] M. Lee, B. A. Kramer, C.-C. Chen, and J. L. Volakis, “Distributed lumped loads and lossy transmission line model for wideband spiral antenna miniaturization and characterization,” IEEE Trans. Antennas Propag., vol. 55, pp. 2671–2678, Oct. 2007. [12] D. S. Filipovic, M. Nurnberger, and J. L. Volakis, “Ultra wideband slot spiral with dielectric loading: measurements and simulations,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2000, pp. 1536–1539. [13] B. A. Kramer, S. Koulouridis, C.-C. Chen, and J. L. Volakis, “Miniature UWB conformal aperture with volumetric inductive loading,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2006, pp. 3693–3696. [14] B. A. Kramer, C.-C. Chen, and J. L. Volakis, “Size reduction of a low-profile spiral antenna using inductive and dielectric loading,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 22–25, 2008. [15] FEKO 5.5 User Manual. Stellenbosch, S.A., EMSS: EM Software and Systems, 2009. [16] R.-S. Cheo, V. H. Rumsey, and J. W. Welch, “A solution to the frequency-independent antenna problem,” IRE Trans. Antennas Propag., vol. 9, pp. 527–534, Nov. 1961. [17] V. H. Rumsey, Frequency Independent Antennas. New York: Academic Press, 1966. [18] R. Sivan-Sussman, “Various modes of the equiangular spiral antenna,” IEEE Trans. Antennas Propag., vol. 11, pp. 533–539, Sep. 1963. [19] T. P. Cencich and J. A. Huffman, “The analysis of wideband spiral antennas using modal decomposition,” IEEE Antennas Propag. Mag., vol. 46, no. 4, pp. 20–26, Aug. 2004. [20] H. Nakano, “A meander spiral antenna,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 2004, vol. 3, pp. 2243–2246. [21] H. Nakano, K. Nogami, S. Arai, H. Mimaki, and J. Yamauchi, “A spiral antenna backed by a conducting plane reflector,” IEEE Trans. Antennas Propag., vol. 34, no. 6, pp. 791–796, Jun. 1986. [22] R. T. Gloutak and N. G. Alexopoulous, “Two-arm eccentric spiral antenna,” IEEE Trans. Antennas Propag., vol. 45, no. 4, pp. 723–730, Apr. 1997. [23] S. E. Lipsky, Microwave Passive Direction Finding. Raleigh, NC: SciTech Publishing, 2004.

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Matthew J. Radway (S’01–M’05) was born in 1977 in Sturgis, SD. After service with the U.S. Marine Corps, he received the B.S. degree in electrical engineering from the South Dakota School of Mines and Technology, Rapid City, in 2005. From 2005 to 2006, he was an Electrical Engineer with Rockwell Collins, Cedar Rapids, IA, where he was assigned to the JTRS Ground Mobile Radio Program. He is currently a Graduate Research Assistant at the University of Colorado at Boulder, where his interests are in broadband antennas, artificial electromagnetic materials, and computational electromagnetics. Mr. Radway is a member of Eta Kappa Nu and Tau Beta Pi.

Thomas P. Cencich (M’99–SM’08) was born in Michigan in 1960. He received the B.S.E.E. degree from the University of Colorado, in 1983. That same year he joined Martin Marietta (now Lockheed Martin), Littleton, CO, and as a Senior Staff Engineer has been involved with antenna test, design and analysis. His primary research interests are in broadband and multimode antennas, where he holds several patents.

Dejan S. Filipovic´ (S’98–M’03–SM’07) received the Dipl. Eng. degree in electrical engineering from the University of Nis, Nis, Serbia and Montenegro, in 1994, and the M.S.E.E. and Ph.D. degrees from the University of Michigan, Ann Arbor, in 1999 and 2002, respectively. From 1994 to 1997, he was a Research Assistant at the School of Electrical Engineering, University of Nis. From 1997 to 2002, he was a Graduate Student Research Assistant at the University of Michigan. Currently, he is Assistant Professor at the University of Colorado, Boulder. His research interests are in antenna theory and design, as well as in computational and applied electromagnetics. Dr. Filipovic´ received the prestigious Nikola Tesla Award for his outstanding graduation thesis and he won first place in the student paper competition at the IEEE AP/URSI Symposium held in San Antonio, TX.

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Composite Right/Left-Handed Substrate Integrated Waveguide and Half Mode Substrate Integrated Waveguide Leaky-Wave Structures Yuandan Dong, Student Member, IEEE, and Tatsuo Itoh, Life Fellow, IEEE

Abstract—Composite right/left-handed (CRLH) substrate integrated waveguide (SIW) and half mode substrate integrated waveguide (HMSIW) leaky-wave structures for antenna applications are proposed and investigated. Their propagation properties and radiation characteristics are studied extensively. Their backfire-toendfire beam-steering capabilities through frequency scanning are demonstrated and discussed. These metamaterial radiating structures are realized by etching interdigital slots on the waveguide surface and the ground. The slot behaves as a series capacitor as well as a radiator leading to a CRLH leaky-wave application. Four antennas are fabricated, measured, and analyzed, including two balanced CRLH SIW designs characterized by single-side or doubleside radiation, and two unbalanced HMSIW designs characterized by different boundary conditions. Antenna parameters such as return loss, radiation patterns, gain, and efficiency are all provided. Measured results are consistent with the simulation. All these proposed antennas possess the advantages of low profile, low cost, and low weight, while they are also showing their own unique features, like high directivity, quasi-omnidirectional radiation, miniaturized size, continuous beam-steering capabilities covering both the backward and forward quadrants, etc., providing much design flexibility for the real applications. Index Terms—Composite right/left-handed (CRLH), half mode substrate integrated waveguide (HMSIW), leaky-wave antennas, substrate integrated waveguide (SIW).

I. INTRODUCTION UBSTRATE integrated waveguide (SIW) and half mode substrate integrated waveguide (HMSIW) have been very popular types of planar guided-wave structures over the past decade [1]–[5]. They are synthesized on a planar substrate with linear periodic arrays of metallic vias. They have desirable features such as low profile, low cost, and easy integration with planar circuits while maintaining the advantageous characteristics of conventional rectangular waveguide. They have enabled numerous applications on high-performance planar components [6]–[13]. The concepts of HMSIW and folded substrate integrated waveguide (FSIW) were proposed aiming at a further reduction of the transverse size of the SIW [4], [5]. And their propagation properties have been studied systematically in [14], [15].

S

Manuscript received March 02, 2010; revised July 27, 2010; accepted October 12, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Electrical Engineering Department, University of California at Los Angeles, Los Angeles, CA 90095 USA (e-mail: yddong@ee. ucla.edu; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103025

Metamaterials are defined as effectively homogeneous structures with unusual properties, which have been under extensive investigation recently [16]–[18]. Metamaterial-based transmission lines (TLs) possess some unique features such as backward-wave and infinite wavelength propagation. Several types of composite right/left-handed (CRLH) TL metamaterials using SIW have been proposed and studied in [19]–[22]. Based on the preliminary study shown in [23], this paper presents an extensive investigation of a family of CRLH SIW and HMSIW leaky-wave structures. Four types of leaky-wave antennas are implemented and analyzed using the proposed planar waveguide structures. Up to now various CRLH leaky-wave antennas based on different technologies have already been investigated and presented [24]–[29]. They all exhibit the backward-to-forward beam-steering capability through frequency scanning. It is also noted that the conventional SIW and HMSIW leaky-wave antennas are introduced and discussed in [30], [31], which achieve edge-radiation by increasing the period length between the vias or taking advantage of the open boundary of the HMSIW. In [32], a transverse slot array antenna fed by an HMSIW is proposed and developed. A novel SIW leaky-wave antenna with CRLH behavior is also demonstrated and discussed in [33], which requires a multi-layer PCB process and is rather complicated to realize. Compared with the designs in the previous literature, the antennas developed in this paper are able to offer a backfire-to-endfire beam-scanning performance as well as an extremely easy way for implementation. They are achieved simply by etching interdigital slots on the top metal surface and the ground of the waveguide. The slot acts like a series capacitor, which, along with the waveguide inherent shunt inductor provided by the vias, creates the necessary condition to support the backward-wave radiation. Miniaturization can be achieved by making the antennas operated below the waveguide cutoff as well as adopting a half mode structure or a modified HMSIW configuration which will be shown later. This paper is organized as follows. The geometry of the proposed leaky-wave structures is illustrated in Section II, where four types of CRLH SIW and HMSIW unit cells are introduced. Section III gives a discussion on the equivalent circuit and the dispersion properties of the unit cells. Sections IV and V present their applications to CRLH SIW and HMSIW leaky-wave antennas. Four types of antennas are analyzed, constructed and measured, which exhibit various features such as balanced or unbalanced operating schemes, quasi-omnidirectional radiation patterns, and miniaturized size while keeping a high gain. Their

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Fig. 2. Configuration of the proposed CRLH HMSIW leaky-wave structures (a) initial unit cell, (b) modified unit cell with a folded ground, and (c) overall HMSIW leaky-wave antenna prototypes.

Fig. 1. Configuration of the proposed CRLH SIW leaky-wave structures (a) single-side radiating element, (b) double-side radiating element, and (c) overall SIW leaky-wave antenna prototype.

backfire-to-endfire beam- scanning performance is confirmed by comparing the radiation patterns at different frequencies. Finally, a conclusion is drawn in Section VI.

Fig. 3. Equivalent circuit models for (a) conventional SIW and HMSIW TL unit cells and (b) CRLH SIW and HMSIW leaky-wave unit cells as shown in Fig. 1(a) and Fig. 2(a).

B. CRLH HMSIW Leaky-Wave Structures II. CONFIGURATIONS A leaky-wave antenna is a radiating transmission line structure, either in uniform or periodic configurations. In this section, the proposed leaky-wave transmission lines will be described. All the prototypes are built on the normally used substrate of Rogers 5880 with a permittivity of 2.2, a loss tangent of 0.001 and a thickness of 0.508 mm. The vias used in the models share a common diameter of 0.8 mm and a center-to-center spacing around 1.45 mm. A. CRLH SIW Leaky-Wave Structures Fig. 1(a) and (b) show the configurations of the one period CRLH SIW element, while the prototype of the whole transmission line with its orientation in the coordinate systems is illustrated in Fig. 1(c). For the first resonator shown in Fig. 1(a), the slot is etched on the top surface and it is grounded by a solid metallic plane. For the second resonator shown in Fig. 1(b), both the top surface and the ground are incorporated with interdigital slots with a period distance of . The slots also provide the radiation. As indicated in Fig. 1(c) the radiation angle of the main beam is straightforwardly determined by (1) which shows that a full space scanning ( varies throughout the range achieved if

to 90 ) can be .

Fig. 2(a) and (b) show the configurations of the one period CRLH HMSIW element, while their overall transmission line structures with 15 unit cells are depicted in Fig. 2(c). For the conventional HMSIW, because of the large width-to-height ratio and the metallic via array, only the quasimodes can propagate in the waveguide [15]. Here the slots are embedded on the waveguide surface, leading to a CRLH HMSIW TL structure. Under this configuration wave can propagate and radiate both below and above the cutoff frequency of HMSIW while still keeping the half mode field distribution. For the second unit cell shown in Fig. 2(b), another via-wall covered by a strip on the top is placed beside the open boundary of the HMSIW. This via-wall is used to reduce the energy leakage from the open boundary. It can be viewed as a folded ground which can miniaturize the transverse size of the HMSIW, as well as restrict the radiation to go to the broadside. These leaky-wave TLs shown in Fig. 2(c) can be easily mounted on the metal surface. III. MODEL ANALYSIS AND DISPERSION RELATION A CRLH TL is an artificial TL structure constituted by the repetition of series capacitance and shunt inductance into a host conventional TL medium exhibiting a left-handed (LH) band at low frequencies and a right-handed (RH) band at higher frequencies. Fig. 3(a) presents the equivalent circuit model for the original SIW or HMSIW unit cell without the slots, which is similar to the traditional rectangular waveguide. The top metal

DONG AND ITOH: CRLH SUBSTRATE INTEGRATED WAVEGUIDE AND HALF MODE SIW LEAKY-WAVE STRUCTURES

surface and the ground are modeled as a two-wire TL with distributed series inductance and distributed shunt capacitance, which are associated with the permeability and permittivity of the substrate, respectively. It is noted that the vias, as a short-circuited stub, provide the shunt inductance. Compared with the circuit model of the CRLH structures, only the series capacitance is absent and needs to be added. Fig. 3(b) depicts the circuit model of the proposed CRLH unit cells shown in Fig. 1(a), which are symmetrical. The interdigital capacitor has been inin the center to obtain a CRLH betroduced into the model as represents the inductance generated by the via-wall. havior. They make the LH contribution. The RH contribution comes and the distributed from the distributed shunt capacitance series inductance . Note that the series slot also plays the role of a radiating element for the leaky-wave antenna. Increasing the width and the length of the slots will make the radiation more efficient. Also bear in mind that increasing the slot width leads while increasing the slot length results in an to a decrease of . Thus enhancing the radiation does not conflict increase of with achieving a balanced case. The dispersion diagrams for the proposed four unit cells are then investigated in detail by using Ansoft’s High Frequency Structure Simulator (HFSS) software package. Usually two approaches are adopted to calculate the dispersion curve for a single unit cell. One is obtained based on the -parameters from driven mode simulation [34], [35]. The other one is based on the Eigen-mode simulation by applying periodic boundary conditions [35]. The Eigen-mode simulation method is more accurate but time-consuming. To give a comparison, Fig. 4(a) plots the dispersion curves of a balanced CRLH SIW unit cell using both of the two approaches. A good agreement is obtained. It is also observed from the Eigen-mode simulation that a very small bandgap (0.1 GHz) actually exists between the LH and RH regions. Fig. 4(b) presents the dispersion diagram and Bloch impedance obtained from driven mode simulation for a balanced CRLH SIW double-side radiating unit cell. The dispersion curves for an unbalanced CRLH HMSIW unit cell and an unbalanced modified CRLH HMSIW unit cell are shown in Fig. 4(c). The main parameter values for these unit cells are shown in the caption. In all these cases the dispersion curve traverses four distinct regions as frequency increases, named LH-guidance, LH-radiation, RH-radiation and RH-guidance here, where the radiation regions are characterized by a phase velocity larger than the speed of light (airline). For the unbalanced case shown in Fig. 4(b), a bandgap region is generated between the LH and RH regions. By moving the LH region far below the waveguide cutoff, miniaturization can be obtained. In Fig. 4(a) it is seen that a balanced case is almost achieved with the balancing point located at about 10 GHz, which ensures a seamless transition from the LH to the RH band. It happens when the series resonance frequency and shunt resonance frequency are equal, or (2) Under this condition the dispersion relation splits into additive positive linear RH and negative hyperbolic LH terms [16] (3)

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Fig. 4. (a) Dispersion diagram calculated from driven mode and Eigen- mode simulations for the single-side CRLH SIW unit cell shown in Fig. 1(a), the parameter values are: w : ,w : ,w : ,n , p : ,l : ; (b) dispersion curve and Bloch impedance obtained using driven mode simulation for the double-side CRLH SIW unit cell shown in Fig. 1(b), the parameter values are: w : ,w : , w : ,n ,p : ,l : ; (c) dispersion curve using Eigen-mode simulation for the CRLH HMSIW unit cell and modified CRLH HMSIW unit cell shown in Fig. 2, the parameter values are: w : , w : ,w : ,n ,p : ,l : , d : c : .

= 0 33 mm = 8 2 mm = 3 3 mm

= 0 45 mm

= 9 2 mm = 9

= 0 32mm = 8 6 mm = 9 = 7 45mm = 2 6 mm

= 0 36 mm = 4 1 mm = 4 4 mm = 0 6 mm

=5

= 0 33 mm

= 0 27 mm = 9 8 mm = 3 3 mm

where is the length of the unit cell. This expression exhibits a null at the frequency (4)

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Fig. 6. Photograph of the two fabricated CRLH SIW leaky-wave antennas.

Fig. 5. Calculated different losses for the single-side CRLH SIW unit cell described in Fig. 4(a).

which is the transition frequency shown in Fig. 4(a). It should be noted that at this frequency group velocity is nonzero despite the infinite phase velocity, which allows leaky-wave broadside radiation. For a specified antenna design requirement, the width of the waveguide can be first chosen to approximately locate the RH region. Then we determine the size of the interdigital slot to roughly get the LH region. Due to the difficulty in extracting , and ), some numerical those lumped values ( , optimization is necessary to obtain a seamless transition for a balanced case or a required bandgap for an unbalanced case. It is also noted that the impedance matching should also be taken into account during the design process. However this matching can be designed in the last step using a taper line at the two ends of a leaky-wave antenna [2]. As the Bloch impedance shown in Fig. 4(b), the average value (real part) from 9.5 GHz to 13 GHz is around 36 Ohm, which can be easily matched to 50 Ohm using a taper line. However for the HMSIW type the average Bloch impedance is close to 50 Ohm therefore no taper line is used. The different losses for the unit cell are also an important issue. Fig. 5 presents a loss analysis for the single-side CRLH SIW unit cell. It is seen that compared with the radiation, the other losses are not very significant. By a similar analysis it is found that both the dielectric loss and conductor loss for the HMSIW case are much smaller compared with the CRLH SIW unit cell, which is due to the weaker resonating field because of its one open boundary [15]. Based on these unit cells discussed above, four CRLH leakywave antennas are designed and fabricated, which are operated at the X-band, including two balanced SIW designs as shown in Section IV and two unbalanced HMSIW designs which will be discussed in Section V. IV. CRLH SIW LEAKY-WAVE ANTENNAS Here two CRLH SIW leaky-wave antennas are designed and fabricated using the substrate of Rogers 5880 with a thickness of 0.508 mm and a relative permittivity of 2.2. The first leakywave antenna is one-side radiating while the second one is a double-side radiating antenna which is realized by etching slots on both the top surface and the ground as indicated by Fig. 1. Fig. 6 shows the photograph of the fabricated antennas. The full-wave simulation is performed using the CST Microwave

Fig. 7. Measured and simulated S -Parameters for the one-side radiating CRLH SIW leaky-wave antenna.

studio. Their performance will be discussed in the following part of this section. A. One-Side Radiating Leaky-Wave Antenna The first antenna has 15 identical elementary cells. The dispersion relation and parameter values for the unit cell have already been presented in Fig. 4 of the above section. Fig. 7 shows the measured and simulated -parameters of this leaky-wave antenna, which are in good agreement. A satisfactory return loss above 10 dB in the band of interest (from 8.5 GHz to more than 12 GHz) is achieved. The insertion loss is almost below 10 dB, which indicates good leakage radiation. The curve also shows that in the LH region, the radiation is more effective compared between the with that in the RH region. The discrepancy of simulation and measurement is due to the increase of reflection, the loss from the SMA connectors and probably the increased conductor loss. Its CRLH behavior is verified by the field distribution along the structure as shown in Fig. 8. In the LH band, the phase and group velocities are anti-parallel, and the wave propagation is backward. At the transition frequency, infinite guided wavelength is observed and there are no field variations. However the , thus the wave is group velocity is nonzero still propagating and radiating. In the RH band, the phase and group velocities are parallel, and the wave propagation is forward. Also note that when the frequency is close to the balanced frequency as shown in Fig. 8(a), the guided wavelength is larger corresponding to less field variations compared with the case shown in Fig. 8(c).

DONG AND ITOH: CRLH SUBSTRATE INTEGRATED WAVEGUIDE AND HALF MODE SIW LEAKY-WAVE STRUCTURES

Fig. 8. Electric field distribution at different frequencies for the CRLH SIW leaky-wave antenna. (a) 9.8 GHz in LH region, (b) 10.0 GHz at the transition point, and (c) 12.8 GHz in the RH region.

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Fig. 10. Measured radiation patterns of the single-side SIW leaky-wave antenna (a) E-plane (x z plane) in the LH region, (b) E-plane (x z plane) in the RH region.

0

0

Fig. 11. Measured radiation patterns at the balanced frequency for the singleside SIW leaky-wave antenna (a) in E-plane (x z plane), (b) in H-plane (y z plane).

0

0

Fig. 9. Simulated 3-D radiation patterns at different frequencies for the CRLH SIW leaky-wave antenna. (a) 9.2 GHz in LH region, (b) 11.2 GHz in the RH region, and (c) 10.0 GHz at the transition point.

Fig. 9 shows the simulated 3-D radiation patterns. It is seen that, when the frequency is increased, the main beam moves from the backfire towards the endfire direction. At the transition frequency, the radiation goes exactly to the broadside. Fig. 10 and Fig. 11 show the normalized radiation patterns measured at different frequencies. The E-plane radiation patterns at 8.6 GHz and 9.3 GHz in the LH region are given by Fig. 10(a). We find that at 8.6 GHz the beam angle is about , very close to backfire. Fig. 10(b) presents the E-plane radiation patterns in the RH region. It is found that at 12.8 GHz the beam angle switches to approximately 60 . Fig. 11 displays the measured broadside radiation patterns for both the co-polarization and cross-polarization in the E-plane and H-plane. We see that the cross-polarization level is very low and is almost negligible. Fig. 12 shows antenna gain response. The simulated radiation efficiency is also plotted in this figure. We find there is a discrepancy around 1.8 dB between the simulated and measured gains. The backfire-to-endfire beam-steering capability by the way of

Fig. 12. Gain and the simulated radiation efficiency of the single-side CRLH SIW leaky-wave antenna.

frequency scanning is confirmed with a maximum gain of approximately 10.8 dBi and an average efficiency of 82% for this antenna. B. Double-Side Radiating Leaky-Wave Antenna The double-side radiating leaky-wave antenna has a total number of 25 interdigital slots etched on the top surface and the ground of the SIW. The simulated and measured transmission response is shown in Fig. 13. Still a balanced case is realized in order to obtain a continuous beam-steering function. The and are quite low indicating a good matching observed

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Fig. 16. Antenna gain and the simulated radiation efficiency of the double-side radiating CRLH SIW leaky-wave antenna. Fig. 13. Measured S -Parameters of the double-side radiating CRLH SIW leaky-wave antenna. Parameter values are shown in Fig. 4.

Fig. 14. Measured radiation patterns of the double-side radiating CRLH SIW leaky-wave antenna (a) E-plane (x z plane) in the LH region, (b) E-plane (x z plane) in the RH region.

0

0

omnidirectional radiation in H-plane with a low cross-polarization level. We also find that the main beam for this antenna is sharper compared with the first antenna. This is due to the reason that the distance between the unit cells on one side is increased for this second antenna, which results in the decrease of the beamwidth according to the array theory. This also explains that the beamwidth in the LH region (at low frequencies) is larger than that in the RH region (at high frequencies corresponding to a smaller wavelength). Fig. 16 shows the measured and simulated antenna gains, as well as the simulated radiation efficiency. Its double-side radiating nature would decrease the antenna gain by 3 dB compared with the first antenna in theory. However, this antenna has a larger aperture size with respect to the single-side radiating antenna. Therefore, the observed gain difference between in simulation is only around 1 dB. The measured gain is 2–3 dB lower than that from the simulation. C. Discussion on Antenna Losses

Fig. 15. Measured radiation patterns at the balanced frequency for the doubleside SIW leaky-wave antenna (a) in E-plane (x z plane), (b) in H-plane (y z plane).

0

0

and a good radiation performance. The connector loss, increased reflection and conductor loss are also responsible for . the visible discrepancy of Fig. 14 shows the measured E-plane radiation patterns in the LH and RH regions, respectively. Fig. 15 gives the radiation patterns for the co-polarization and cross- polarization at the transition frequency in both the E-plane and H-plane. The E-plane patterns are similar to the inline element arrays while for the latter beam-steering is usually achieved by phase control which requires a complicated feeding network. This antenna has quasi-

In both of the above two cases, the measured gain and are lower than those obtained from simulation using CST microwave studio. There are several reasons: 1) The conductor loss and dielectric loss in the measurement should be higher than that in the simulation, especially the conductor loss. The antennas were fabricated by us based on chemical etching and the observed conductor surface is not very smooth which would lead to an increase on the conductor loss; 2) The loss from the SMA connectors is not included in the simulation; 3) The measured and terminated power in gain takes the reflection to account; 4) We found that our chamber is not big enough. In the measurement our antennas are located close to the far field region but not exactly in the far field area, especially for the double-side antenna. This would lead to some inaccuracy for the gain measurement. These factors could all cause the decrease of the measured antenna gain. V. CRLH HMSIW LEAKY-WAVE ANTENNAS Fig. 17 shows the photograph of the two CRLH HMSIW leaky-wave antennas. They are fabricated using the same substrate at the High Frequency Center of the Electrical Engineering Department of UCLA. They have the same dimensions except that an extra via-wall is placed near the open side for the

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Fig. 17. Photograph of the two fabricated CRLH HMSIW leaky-wave antennas.

Fig. 19. Measured radiation patterns of the initial CRLH HMSIW leaky-wave antenna (a) E-plane (x z plane) in the LH region, (b) E-plane (x z plane) in the RH region, and (c) H-plane (y z plane) at 10.6 GHz.

0

0

0

0

Fig. 20. Measured E-plane (x z plane) radiation patterns of the modified CRLH HMSIW leaky-wave antenna (a) in LH region and (b) in RH region. Fig. 18. Measured and simulated S -Parameters for the (a) CRLH HMSIW leaky-wave antenna and (b) Modified HMSIW leaky-wave antenna. The dimensions for the unit cells are shown in Fig. 4.

second antenna (see the modified unit-cell shown in Fig. 2(b)), which can be viewed as a folded ground and leads to a miniaturization on the transverse size. This via-wall can also reduce the wave leakage from the open boundary, thus improving the gain. Fig. 18 shows the simulated and measured transmission responses of the two leaky-wave antennas. They are unbalanced and the dispersion diagrams for the unit cells are shown in Fig. 4(c). A bandgap region is observed in both the simulation and the measurement for these two antennas. It is important to bear in mind that by changing the slot size and the position of the vias we can easily control the position of the LH band. Balanced condition can also be obtained by some optimization as shown in [21]. Fig. 19 shows the measured radiation patterns for the first CRLH HMSIW antenna, while Fig. 20 plots the measured

E-plane patterns for the second CRLH HMSIW leaky-wave antenna. Beam scanning capability in E-plane for these two antennas is clearly observed. Since this is an unbalanced case and there is no balanced point which gives broadside radiation, it is difficult to obtain the H-plane radiation patterns which are plane. We still give the H-plane pattern observed in the measured at 10.6 GHz for the first antenna. It is noted that this measured pattern does not corresponds to the maximum value as indicated by the E-plane pattern shown in Fig. 19(b). The cross-polarization level is slightly higher compared with the CRLH SIW antennas, which is mainly due to the edge radiation caused by the open boundary. Fig. 21 shows the simulated and measured gains, as well as the radiation efficiencies for these two antennas. It is noted that the average radiation efficiency for HMSIW antennas is around 87%, which is higher than that of the CRLH SIW leaky-wave antennas. This is because the HMSIW antennas have less conductor and dielectric losses. Especially at low frequencies, they are less lossy than the SIW [15]. A gain decrease around 2 dB

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Fig. 21. Measured and simulated antenna gains of the two CRLH HMSIW leaky-wave antennas.

Fig. 23. Simulated antenna gains for the two CRLH HMSIW leaky-wave antennas with varied ground configurations.

the influence of the ground plane. And it is observed that the gain of the modified CRLH HMSIW antenna is not very sensitive to the distance between the via-wall and the open edge. VI. CONCLUSION

Fig. 22. Simulated radiation patterns in terms of gain at 11 GHz for the CRLH HMSIW antennas with varied configurations. (a) Initial design with the ground : , (b) initial design with the ground extension d extension d , and (c) modified design with c : . :

1 0 mm

= 4 4 mm

= 0 4 mm

=

is also observed in the measurement compared with the simulation. We also find that the gain for the second antenna with the modified structure is a little larger than the gain of the first one, although the second antenna has a smaller ground. To explain this difference, Fig. 22 plots the simulated patterns in plane viewed towards direction with varied ground conditions. It is found that when the ground extension from the open boundary of HMSIW is small ( is small), the antenna has a low gain as the case shown in Fig. 22(b). However, when the extended ground is larger ( is large), the gain is increased as shown in Fig. 22(a), approaching the gain of the antenna with a folded ground as shown in Fig. 22(c). Fig. 23 shows a parametric study about the peak gain for the antennas with different ground configurations. It is observed that if the ground extension at the open side of the HMSIW antenna is very small, the edge leakage is substantial resulting in a small gain. To reduce this undesired radiation a large ground is required. However, we can use the modified design to minimize

A detailed investigation on a family of CRLH SIW and HMSIW leaky-wave antennas is presented in this study. Four antennas with different radiation characteristics are developed. Their circuit models are analyzed. Their dispersion relation and radiation mechanism are discussed. Their full space beam-scanning performance is confirmed by simulation and measurement. The effect of the ground for the HMSIW leaky-wave antenna is discussed and a miniaturization technique is proposed. Antenna transmission response, radiation patterns, gain and efficiency are all provided, giving designers with in-depth understanding and some useful design information. These antennas exhibit advantages in their low fabrication complexity, full space beam-scanning capability, low profile, low cost, and easy integration with other planar circuits. They are promising antenna candidates for integrated microwave and millimeter wave systems. REFERENCES [1] J. Hirokawa and M. Ando, “Single-layer feed waveguide consisting of posts for plane TEM wave excitation in parallel plates,” IEEE Trans. Antennas Propag., vol. 46, no. 5, pp. 625–630, May 1998. [2] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 2, pp. 68–70, Feb. 2001. [3] Y. Huang and K. L. Wu, “A broad-band LTCC integrated transition of laminated waveguide to air-filled waveguide for millimeter-wave applications,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 5, pp. 1613–1617, May 2003. [4] N. Grigoropoulos, B. Sanz-Izquierdo, and P. R. Young, “Substrate integrated folded waveguides (SIFW) and filters,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 12, pp. 829–831, Dec. 2005. [5] W. Hong, B. Liu, Y. Q. Wang, Q. H. Lai, and K. Wu, “Half mode substrate integrated waveguide: A new guided wave structure for microwave and millimeter wave application,” presented at the Joint 31st Int. Infrared Millimeter Waves Conf. and 14th Int. Terahertz Electron. Conf., Shanghai, China, Sep. 18–22, 2006. [6] Y. Cassivi and K. Wu, “Low cost microwave oscillator using substrate integrated waveguide cavity,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 2, pp. 48–50, Feb. 2003. [7] D. Stephens, P. R. Young, and I. D. Robertson, “W-band substrate integrated waveguide slot antenna,” Electron. Lett., vol. 41, no. 4, pp. 165–167, Feb. 2005.

DONG AND ITOH: CRLH SUBSTRATE INTEGRATED WAVEGUIDE AND HALF MODE SIW LEAKY-WAVE STRUCTURES

[8] S. Park, Y. Okajima, J. Hirokawa, and M. Ando, “A slotted post-wall waveguide array with interdigital structure for 45/SPL deg/ linear and dual polarization,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2865–2871, Sep. 2005. [9] J. H. Lee, N. Kidera, G. DeJean, S. Pinel, J. Laskar, and M. Tentzeris, “A v-band front-end with 3D integrated cavity filters/duplexers and antenna in LTCC technologies,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 7, pp. 2925–2936, Jul. 2006. [10] B. Liu, W. Hong, Y. Q. Wang, Q. H. Lai, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) 3 db coupler,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 1, pp. 22–24, Jan. 2007. [11] Y. Cheng, W. Hong, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) directional filter,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 7, pp. 504–506, Jul. 2007. [12] X. P. Chen, K. Wu, and Z. L. Li, “Dual-band and triple-band substrate integrated waveguide filters with chebyshev and quasi-elliptic responses,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 12, pp. 2569–2577, Dec. 2007. [13] T. M. Shen, C. F. Chen, T. Y. Huang, and R. B. Wu, “Design of vertically stacked waveguide filters in LTCC,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 8, pp. 1771–1779, Aug. 2008. [14] W. Che, L. Geng, K. Deng, and Y. L. Chow, “Analysis and experiments of compact folded substrate-integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 1, pp. 88–93, Jan. 2008. [15] Q. H. Lai, C. Fumeaux, W. Hong, and R. Vahldieck, “Characterization of the propagation properties of the half-mode substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 8, pp. 1996–2004, Aug. 2009. [16] C. Caloz and T. Itoh, Electromagnetic Metamaterials Transmission Line Theory and Microwave Applications. New York: Wiley-IEEE Press, 2005. [17] G. Eleftheriades and K. Balmain, Negative-Refraction Metamaterials Fundamental Principles and Applications. New York: Wiley-IEEE Press, 2005. [18] N. Engheta and R. W. Ziolkowski, Electromagnetic Metamaterials: Physics and Engineering Explorations. New York: Wiley-IEEE Press, 2006. [19] H. Zhao, T. J. Cui, X. Q. Lin, and H. F. Ma, “The study of composite right/left handed structure in substrate integrated waveguide,” in Proc. Int. Symp. Biophotonics, Nanophotonics Metamaterials, Oct. 2006, pp. 547–549. [20] T. Cui, X. Lin, Q. Cheng, H. Ma, and X. Yang, “Experiments on evanescent-wave amplification and transmission using metamaterial structures,” Phys. Rev. B, vol. 73, pp. 2451191–2451198, Jun. 2006. [21] Y. Dong and T. Itoh, “Composite right/left-handed substrate integrated waveguide and half-mode substrate integrated waveguide,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, 2009, pp. 49–52. [22] K. Okubo, M. Kishihara, A. Yamamoto, J. Yamakita, and I. Ohta, “New composite right/left-handed transmission line using substrate integrated waveguide and metal-patches,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, 2009, pp. 41–44. [23] Y. Dong and T. Itoh, “Composite right/left-handed substrate integrated waveguide leaky-wave antennas,” in Proc. Eur. Microw. Conf., Rome, Italy, Sep. 2009. [24] S. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 12, pp. 2678–2690, Dec. 2004. [25] F. C. Miranda, C. C. Penalosa, and C. Caloz, “High-gain active composite right/left-handed leaky-wave antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2292–2300, Aug. 2006. [26] T. Ueda, N. Michishita, M. Akiyama, and T. Itoh, “Dielectric-resonator-based composite right/left-handed transmission lines and their application to leaky wave antenna,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 10, pp. 2259–2268, Oct. 2008. [27] T. Ikeda, K. Sakakibara, T. Matsui, N. Kikuma, and H. Hirayama, “Beam-scanning performance of leaky-wave slot-array antenna on variable stub-loaded left-handed waveguide,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3611–3618, Dec. 2008.

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[28] S. Paulotto, P. Baccarelli, F. Frezza, and D. Jackson, “Full-wave modal dispersion analysis and broadside optimization for a class of microstrip CRLH leaky-wave antennas,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 12, pp. 2826–2837, Dec. 2008. [29] 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. Microwave Theory Tech., vol. 57, no. 4, pp. 784–795, Apr. 2009. [30] D. Deslandes and K. Wu, “Substrate integrated waveguide leaky-wave antenna: Concept and design considerations,” in Proc. Asia-Pacific Microw. Conf., Suzhou, China, 2005, pp. 346–349. [31] J. Xu, W. Hong, H. Tang, Z. Kuai, and K. Wu, “Half-mode substrate integrated waveguide leaky-wave antenna for millimeter-wave applications,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 85–88, 2008. [32] Q. H. Lai, W. Hong, Z. Q. Kuai, Y. S. Zhang, and K. Wu, “Half-mode substrate integrated waveguide transverse slot array antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1064–1072, Apr. 2009. [33] Y. Weitsch and T. Eibert, “A left-handed/right-handed leaky-wave antenna derived from slotted rectangular hollow waveguide,” in Proc. Eur. Microw. Conf., Munich, Germany, Oct. 2007, pp. 917–920. [34] D. M. Pozar, “Microwave filters,” in Microwave Engineering, 3rd ed. Hoboken, NJ: Wiley, 2005, ch. 8. [35] “Left-Handed Metamaterial Design Guide,” Ansoft Corporation, 2007.

Yuandan Dong (S’09) received the B.S. and M.S. degrees from Southeast University, Nanjing, China, in 2006 and 2008, respectively. He is currently working toward the Ph.D. degree at the University of California at Los Angeles (UCLA). From September 2005 to August 2008, he was studying at the State Key Lab. of Millimeter Waves, Southeast University. His research interests include the characterization and development of RF and microwave components, circuits, antennas and metamaterials.

Tatsuo Itoh (S’69–M’69–SM’74–F’82–LF’06) received the Ph.D. degree in electrical engineering from the University of Illinois, Urbana, in 1969. After working for the University of Illinois, SRI, and the University of Kentucky, he joined the faculty at The University of Texas at Austin, in 1978, where he became a Professor of electrical engineering in 1981. In September 1983, he was selected to hold the Hayden Head Centennial Professorship of Engineering at The University of Texas. In January 1991, he joined the University of California, Los Angeles, as a Professor of electrical engineering and holder of the TRW Endowed Chair in Microwave and Millimeter Wave Electronics (currently Northrop Grumman Endowed Chair). He has published 375 journal papers, 775 refereed conference presentations and has written 43 books/book chapters in the area of microwaves, millimeter-waves, antennas and numerical electromagnetics. He generated 70 Ph.D. students. Dr. Itoh is a member of the Institute of Electronics and Communication Engineers of Japan, and Commissions B and D of USNC/URSI. He received a number of awards including the IEEE Third Millennium Medal in 2000, and the IEEE MTT Distinguished Educator Award in 2000. He was elected to a member of National Academy of Engineering in 2003. He served as an Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES for 1983–1985. He was President of the Microwave Theory and Techniques Society in 1990. He was the Editor-in-Chief of the IEEE MICROWAVE AND GUIDED WAVE LETTERS from 1991 through 1994. He was elected as an Honorary Life Member of the MTT Society in 1994. He was the Chairman of Commission D of International URSI for 1993–1996. He serves on advisory boards and committees of a number of organizations. He served as Distinguished Microwave Lecturer on Microwave Applications of Metamaterial Structures of IEEE MTT-S for 2004–2006.

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Study of Conformal Switchable Antenna System on Cylindrical Surface for Isotropic Coverage Zhijun Zhang, Senior Member, IEEE, Xu Gao, Wenhua Chen, Member, IEEE, Zhenghe Feng, Senior Member, IEEE, and Magdy F. Iskander, Fellow, IEEE

Abstract—A conformal switchable antenna system mounted on a cylindrical surface is proposed, simulated, prototyped and measured. The antenna system is composed of several tri-polarization antennas operating at 2.4 GHz band. It is shown that this antenna system realizes a quasi-isotropic pattern without any nulls over a complete spherical surface. Specifically, it is shown that by using three tri-polarization antennas and switching among them, a complete spherical coverage could be achieved without any nulls and with a power gain larger than 2.5 dBi over more than 90 percent of a spherical surface surrounding the antenna system. Furthermore, it is shown that the performance of the antenna system does not deteriorate with the increase in the diameter of the mounting cylinder, thus facilitating the implementation of the proposed design in many applications including telemetry, satellites, and aircraft. Both simulation and experimental results of the radiation patterns are presented in this paper. Index Terms—Cylinder surface, isotropic coverage, radiation pattern, tri-polarization antenna.

I. INTRODUCTION

T

HE research on antennas mounted on conducting cylindrical surface is significant to aerospace applications such as telemetry, telecommand of satellite, radio guidance and control of airborne craft. In most of these applications, a radiation pattern without a null covering an entire spherical surface around the antenna is desired. However, due to the blockage effect of mounting structures which are often made of metal, e.g., spacecraft, it seems extremely difficult to achieve that goal with simple antenna system structures. One common approach to obtain omnidirectional patterns is to use a number of discrete antennas arrayed along the circumference of the cylinder. A conformal cylindrical microstrip array was proposed in [1]

Manuscript received May 21, 2010; revised July 08, 2010; accepted August 30, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work is supported in part by the National Basic Research Program of China under Contract 2007CB310605, by the National High Technology Research and Development Program of China (863 Program) under Contract 2007AA01Z284, by the National Natural Science Foundation of China under Contract 60771009, and in part by the National Science and Technology Major Project of the Ministry of Science and Technology of China 2010ZX03007001-01. Z. Zhang, X. Gao, W. Chen, and Z. Feng are with State Key Lab of Microwave and Communications, Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). M. F. Iskander is with HCAC, University of Hawaii at Manoa, Honolulu, HI 96822 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2103041

for producing such a pattern. Patch antenna arrays mounted on cylindrical surface in both axial and circumferential modes were analyzed. An array of 4 to 16 printed dipoles mounted on a cylindrical surface was studied for obtaining an omnidirectional pattern [2]. A study of different kinds of cylindrical conformal antennas employed on cylindrical bodies with different diameters was also reported in [3]. Several studies of circularly polarized antenna elements on cylindrical structures are discussed in [4]–[6]. In most of these studies, however, the isotropic radiation property is considered only for one plane or several planes, but not for the complete spherical space around the antenna system. Many other investigations were carried out focusing on the realization of quasi-isotropic radiation patterns. This includes the use of a combined dipole and two element slot antenna array in [7], the use of a large spherical slot antenna [8] and the use of a miniature antenna for circularly polarized quasi-isotropic coverage in [9]. As mentioned earlier, such full spherical coverage radiation pattern is important in many applications in order to maintain a communication link at all times. In these applications, therefore, it is important that the antenna system mounted on a conducting cylinder does not have any radiation nulls over a complete spherical surface surrounding it. In this paper, we describe an alternative approach for achieving a quasi-isotropic radiation pattern using the recently designed low-profile tri-polarization antenna described in [10]. Although several studies on isotropic antenna designs have been reported in literature, data on antenna systems on conducting cylinders with full isotropic spherical coverage is rather sparse. There are some reported isotropic radiators [7]–[9], but only a few of those reported provide full coverage when antennas are mounted on conducting cylinders [1]–[6]. Even in these cases, the reported designs do not achieve full omnidirectional coverage over a spherical surface; instead, omnidirectional radiation patterns in a plane perpendicular to the axis of the cylinder were reported. In this paper, a conformal version of the low-profile tri-polarization antenna proposed in [10] is used as a basic element and building block to achieve full coverage over a complete spherical surface surrounding the conducting cylinder. It is shown that by using three tri-polarization antennas and switching among them, 100 percent coverage (0 dBi gain) can be obtained over a complete spherical surface. Furthermore, higher gains, 2.5 dBi, could be achieved with no null over more than 90 percent of the spherical surface. The proposed design compares favorably with earlier ones [1], [2] where 6 to 8 microstrip antennas were used to obtain omni-directional radiation patterns in a plane perpendicular to the axis of the cylinder.

0018-926X/$26.00 © 2010 IEEE

ZHANG et al.: STUDY OF CONFORMAL SWITCHABLE ANTENNA SYSTEM ON CYLINDRICAL SURFACE FOR ISOTROPIC COVERAGE

Fig. 1. Geometry of the tri-polarization antenna: (a) Top view; (b) side view.

Simulation results and experimental verification of the proposed conformal switchable tri-polarization antenna system on a cylinder surface are described in the following sections. Specifically, the tri-polarization antenna is briefly introduced in Section II, while simulation results are presented and discussed in the Section III and IV. Measurement procedure and obtained experimental results are shown in Section V. II. ANTENNA SYSTEM DESIGN The conformal and low profile tri-polarization antenna as proposed in [10] is a fundamental building block in the paper and is briefly introduced here. The configuration of the tri-polarization antenna is shown in Fig. 1, depicting two ring patch antennas and a disk-loaded monopole which compose the tri-polarization antenna. With the three ports of this antenna working independently, its far-field has three orthogonal linear polarizations. Specifically, the E-field radiated by the ring patch is parallel to the ground plane and can provide two orthogonal polarizations excited through P1 and P2, while the monopole provides the vertical polarization component and has an isotropic radiation in the azimuth plane. The operating frequency of the tri-polarization antenna was chosen to be 2.4 GHz, and its total height is 5.8 mm. The . Although volume of the tri-polarization is here a planar structure was used for the convenience of fabrication, it is easy to design and make similar antennas which are totally conformal to cylindrical bodies. Fig. 2 shows the simulated 3D radiation pattern of the tripolarization antenna. It further shows that the radiation pattern of the monopole mode (port M3) and patterns of the patch mode (port P1 and port P2) have complementary properties.

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The low profile advantage and the ease of its conformal integration on a cylindrical surface make the tri-polarization antenna an ideal choice of antenna systems for cylindrical structures. The ground plane of the tri-polarization antenna makes it particularly suitable for use on cylindrical metal structures. Besides, due to the fact that the three ports of this antenna radiate three polarized fields that are orthogonal to one another, this antenna can receive electromagnetic waves with any kind of polarization, thus avoiding instances of polarization mismatch. Even more important than the above-mentioned advantages of the tri-polarization antenna is the fact that the patch antenna mode and the monopole antenna mode of the tri-polarization have complementing radiation properties. By switching among the three ports of the tri-polarization antenna, it is easy to obtain the full radiation coverage for a hemisphere. Therefore, for the antenna system mounted on cylinder body, just two tri-polarization antennas are required to realize the full coverage over the whole sphere. To investigate the performance of the proposed switchable antenna system, different numbers of antenna elements were adopted to build switchable antenna systems. As shown in Fig. 3, two to four tri-polarization antennas were equally spaced along the circumference of cylinder body. Another situation described in this paper is the mounting of a switchable antenna on a cylinder surface with different diameters. The simulation tool IE3D was employed in calculating radiation patterns of each element and a program written in Matlab was used to synthesize the final radiation pattern of the complete antenna system.

III. ANTENNA SYSTEM WITH DIFFERENT NUMBERS OF ELEMENTS The proposed antenna system mounted on the surface of a conducting cylinder with a diameter of 0.5 m and a height of 1 m was studied. The reason of choosing 1 m, which is 8 wavelengths at 2.4 GHz, is that this length is sufficient when considering the influence of a cylindrical body on the performance of tri-polarization antennas. Even if tri-polarization antennas were mounted on a longer cylindrical structure, the resulting difference would be rather negligible. Simulation results demonstrate this finding. A comparison between gain patterns of 3-element antenna systems mounted on a cylinder with a height of 1 m and that on another cylinder with a height of 0.5 m was realized in a computer simulation. This result shows that gain patterns of these two situations are very similar in both shape and value. The antenna systems with two, three and four elements were analyzed and compared, respectively. As each tri-polarization antenna element has three ports, there are six, nine and twelve ports in antenna systems with two, three and four elements respectively. Since the main concern of this design is full radiation coverage of a complete spherical surface, the method of selecting maximum values among all ports was used to evaluate the performance of the antenna system, which means to firstly calculate the radiation patterns of each port and then select the maximum gain among patterns of all ports at each solid angle

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Fig. 2. 3D gain patterns of tri-polarization antenna. (a) Gain total—Port P1; (b) Gain total—Port P2; (c) Gain total—Port M3.

phi in the X-Y plane, while the main polarization of the P1 port is E theta in the X-Y plane. Therefore, in the horizontal plane, the radiation patterns of E phi are averagely better than those of E theta. To better illustrate the gain pattern of antenna system over a complete spherical surface, the contour lines pattern of gain total of 3-element antenna system is also shown in Fig. 5. There are usually several ways to evaluate the isotropic performance of an antenna system. First, the radiation pattern of an isotropic antenna should not have nulls over a complete spherical surface. Another method used with considerable success in evaluating the quality of an antenna system’s isotropic coverage is to calculate the coverage factor [8]. The coverage factor is the ratio of the surface by which the gain exceeds a given threshold value over total antenna radiation field surface. The coverage factor (CF) is defined as

The surface of integration value.

Fig. 3. Antenna arrangement around a cylindrical surface.

as the system gain. This radiation pattern could then be conveniently realized by switching among all ports to obtain the maximum received power value in real application. Thus, the gain of the antenna system can be expressed as

(1) Here, N is the number of tri-polarization antennas in a given antenna system, and is the gain obtained by only exciting port . The simulated 3D gain patterns of antenna systems with 2 to 4 elements are shown in Fig. 4. The gain pattern of an antenna system with 2 elements shows that there are no nulls and that gain values are mostly above 0 dBi. Thus, even with just two tripolarization antenna elements a quasi-isotropic radiation pattern could be obtained. It is also clearly shown in Fig. 4 that the performance of the antenna system continued to improve with the increase in the number of elements, as the radiation pattern continued to closely resemble those of an isotropic source. Here, axis z follows the axis of the cylinder. The main polarizations of the M3 and P2 ports of the tri-polarization antenna are both E

(2) is determined by a given threshold

Here, is the gain function (i.e., relative to isotropic is the given threshold value. In this source) and paper, the coverage factor method is used to evaluate the performance of radiation coverage of the proposed antenna system. The calculated coverage curve of the gain of E-total, E-theta and E-phi are shown in Fig. 6, Fig. 7 and Fig. 8, respectively. The coverage curves are plotted with the coverage factor defined by (2) as y-axis, and with given thresholds as x-axis. As shown in Fig. 6, for the threshold of E total larger than 0 dBi, the antenna system with 2 elements can obtain nearly full coverage (98%) over the whole sphere, and 100 percent coverage is achieved for the antenna system employing 3 or 4 elements. Furthermore, not only full coverage of the whole sphere is achieved, but also gains larger than 2.5 dBi were obtained for more than 90 percent of the sphere when the antenna system employed 3 or 4 elements. For the gains of E-theta and E-phi patterns shown in Fig. 7 and Fig. 8, it may be noted that using 3 elements compares very favorably to an antenna system with only 2 elements. For the criteria of 0 dBi, the coverage values of E-theta are 61%,

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Fig. 4. 3D gain patterns of antenna system with different numbers of elements. (a) Gain total—2 elements; (b) Gain theta—2 elements; (c) Gain phi—2 elements; (d) Gain total—3 elements; (e) Gain theta—3 elements; (f) Gain phi—3 elements; (g) Gain total—4 elements; (h) Gain theta—4 elements; (i) Gain phi—4 elements.

Fig. 5. Gain total contour lines—3 elements. Fig. 6. Coverage curves of gain total for the proposed switchable antenna systems with different numbers of elements.

88% and 96% for antenna system with 2, 3 and 4 elements, respectively. For E-phi, they are 83%, 97% and 97%, respectively. Therefore, antenna systems with more than 3 elements can achieve nearly full coverage over the whole sphere for both polarizations. From the results of the coverage curves, it is clear that better performance of the antenna system can be obtained by

increasing the number of elements used. However, antenna systems with more elements require more complex switching systems and higher switching speeds. It is this tradeoff between performance and complexity of the switching structure that will ultimately determine the final design of the antenna system.

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Fig. 7. Coverage curves of gain theta for the proposed switchable antenna systems with different numbers of elements.

Fig. 8. Coverage curves of gain phi for the proposed switchable antenna systems with different numbers of elements.

Considering the distinct advantages of a 3 elements antenna system as compared with 2 elements, and with a relatively small improvement when employing 4 elements, it is possible to conclude that antenna systems with 3 elements is preferable and may be considered best choice. Therefore, an antenna system with 3 elements will be used in Section IV to investigate the effect of different diameters of cylindrical bodies on the performance of the antenna system. IV. ANTENNA SYSTEM MOUNTED ON CYLINDERS WITH DIFFERENT DIAMETERS To study the influence of the diameter of the cylinder on the performance of the antenna system, an antenna system with three elements is analyzed when mounted on cylinders with diameters of 0.3 m, 0.5 m and 0.7 m, respectively. The simulation results show that when mounted on cylinder bodies with different diameters, the shape of 3D radiation patterns are similar. The radiation pattern resulting for a cylinder with a diameter of 0.5 m has been presented in Fig. 4(d), (e) and (f). As shown in the radiation patterns, no distinct null exists over a complete spher-

Fig. 9. Coverage curves of gain total for the proposed switchable antenna systems for cylinder with different diameters.

Fig. 10. Coverage curves of gain theta for the proposed switchable antenna system for cylinders with different diameters.

ical surface, which means full radiation coverage is achieved. For the purpose of saving space, the 3D radiation patterns for the other two situations are not presented here. The calculated coverage curve results for cylinders with different diameters are depicted in Figs. 9–11. Fig. 9 shows the coverage curves of gain of E total, and Fig. 10 and Fig. 11 show the coverage curves of gain of E theta and E phi, respectively. From these three figures, it is safe to draw the conclusion that there is no distinctive influence of the diameter of cylinders on performance of the proposed antenna system. The proposed antenna system with 3 elements can achieve full coverage over the whole sphere for cylinders with different diameters. The performance of the antenna system does not deteriorate when the diameter of the cylinder increases. This conclusion is meaningful in real application, because in traditional solutions for omnidirectional coverage by using antenna arrays mounted on a cylinder surface, usually more antenna elements are required when the diameter of the cylinder becomes larger. However, even without adding more elements, the proposed antenna system can be conveniently used on cylinder carriers with different diameters, such as aircraft and satellite.

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Fig. 11. Coverage curves of gain phi for the proposed switchable antenna system for cylinders with different diameters. Fig. 13. Measured and simulated radiation patterns of the M3 port.

Fig. 12. Photograph of the proposed antenna system.

Fig. 14. Measured and simulated radiation patterns of the P1 port.

It is also important to emphasize that focus in this study was placed on achieving a complete spherical coverage using the proposed design. This is clearly important in wireless communications type of applications and, to this end, the design is considered satisfactory. In other radar and navigation-type applications, however, the phase center identification and its possible shifting as a result of switching needs to be further considered and carefully investigated. This will be addressed in a future publication as communications-type applications have been the focus of this study.

When comparing the measured radiation patterns in the circumferential plane of the cylinder with the simulated results, we can still verify the validity of the results in the above sections. Therefore, the radiation pattern of each port of the tri-polarization antenna mounted on the cylinder surface was measured. Fig. 13, Fig. 14 and Fig. 15 show the comparison of the simulation results and the measurement results of three ports of the tri-polarization antenna. Figs. 13, 14, and 15, show both simulated and measured radiation patterns of each port of one tri-polarization antenna element mounted on the cylinder surface at 2.4 GHz. The radiation patterns were measured by exciting one port while leaving all other ports open-circuit. Because these patterns were measured in the circumferential plane of the cylinder body and refer to the axis of the cylinder, the main polarizations of port M3 and port P2 are both E phi, and the main polarization of port P1 is E theta. As shown by these figures, the measurement results corroborate the simulation results. Small discrepancies are a result of the manufacturing tolerance.

V. EXPERIMENTAL VERIFICATION Experiments have been conducted to verify the above simulation results. A metal cylinder body with a diameter of 0.5 m and a height of 1 m was fabricated. As shown in Fig. 12, the antenna system was mounted on the cylinder surface. Due to the large volume of the cylinder, it is difficult to measure the 3D radiation pattern with our measurement equipment. However, the radiation patterns in the circumferential plane can be measured.

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[6] D. I. Wu, “Omnidirectional circularly-polarized conformal microstrip array for telemetry applications,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 1995, vol. 2, pp. 998–1001. [7] S. Long, “A combination of linear and slot antennas for quasi-isotropic coverage,” IEEE Trans. Antennas Propag., vol. AP-23, pp. 572–576, Jul. 1975. [8] D. Bugnolo, “A quasi-isotropic antenna in the microwave spectrum,” IRE Trans. Antennas Propag., vol. 10, pp. 377–383, Jul. 1962. [9] M. Huchard, C. Delaveaud, and S. Tedjini, “Miniature antenna for circularly polarized quasi isotropic coverage,” in Eur. Conf. Antennas Propagation, EuCAP, Edinburgh, Nov. 2007, pp. 1–5. [10] X. Gao, H. Zhong, Z. Zhang, Z. Feng, and M. F. Iskander, “Low-profile planar tri-polarization antenna for WLAN communications,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 83–86, 2010.

Fig. 15. Measured and simulated radiation patterns of the P2 port.

VI. CONCLUSION In this paper, a conformal switchable antenna system mounted on cylinder surface is proposed, designed, simulated and experimentally tested for isotropic coverage. This antenna system consists of several tri-polarization antennas. Studies of this antenna system show that full coverage over a complete spherical surface can be achieved by using even just two elements. Considering the performance of the antenna system and the complexity of the switching structure, a design with three elements is recommended. Specifically, this design could provide 100 percent coverage with a gain larger than 0 dBi over a complete spherical surface. Gain values as high as 2.5 dBi could be achieved over 90% of a spherical surface when using a three elements design antenna system. Furthermore, it is shown that the performance of the antenna system does not deteriorate with an increase in the diameter of the cylinder. Therefore, without adding more elements, the proposed antenna system can be conveniently used on different cylindrical carriers with different diameters, such as aircraft and satellite. The proposed antenna system is, therefore, very useful and could be used in many aerospace applications. REFERENCES [1] I. Jayakumar, R. Garg, B. Sarap, and B. Lal, “A conformal cylindrical microstrip array for producing omnidirectional radiation pattern,” IEEE Trans. Antennas Propag., vol. AP-34, no. 10, pp. 1258–1261, Oct. 1986. [2] J. Shen, “A printed dipole array for omni directional application,” in Proc. Antennas, Propagation EM Theor. Int. Symp., Nov. 2008, pp. 182–184. [3] J. Qiu, L. Zhong, H. Du, and W. Li, “Analysis and simulation of cylindrical conformal omnidirectional antenna,” in Proc. APMC’2005, Dec. 2005, vol. 4. [4] G. Dubost, J. Samson, and R. Frin, “Large-bandwidth flat cylindrical array with circular polarisation and omnidirectional radiation,” Electron. Lett., vol. 15, pp. 102–103, Feb. 15, 1979. [5] R. C. Hall and D. I. Wu, “Modeling and design of circularly-polarized cylindrical wraparound microstrip antennas,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 1996, vol. 1, pp. 672–675.

Zhijun Zhang (M’00–SM’04) received the B.S. and M.S. degrees from the University of Electronic Science and Technology of China, in 1992 and 1995, respectively, and the Ph.D. degree from Tsinghua University, Beijing, China, in 1999. In 1999, he was a Postdoctoral Fellow with the Department of Electrical Engineering, University of Utah, where he was appointed a Research Assistant Professor in 2001. In May 2002, he was an Assistant Researcher with the University of Hawaii at Manoa, Honolulu. In November 2002, he joined Amphenol T&M Antennas, Vernon Hills, IL, as a Senior Staff Antenna Development Engineer and was then promoted to the position of Antenna Engineer Manager. In 2004, he joined Nokia Inc., San Diego, CA, as a Senior Antenna Design Engineer. In 2006, he joined Apple Inc., Cupertino, CA, as a Senior Antenna Design Engineer and was then promoted to the position of Principal Antenna Engineer. Since August 2007, he has been with Tsinghua University, where he is a Professor in the Department of Electronic Engineering.

Xu Gao received the B.S. degree from Shandong University, Jinan, China, in 2007, and the M.S. degree from Tsinghua University, Beijing, China, in 2010. He is currently working toward the Ph.D. degree at the Missouri University of Science and Technology, Rolla. He is currently working in the EMC Lab, Missouri University of Science and Technology. His research interests include antenna design, wave propagation, electromagnetic compatibility, RF design and computational electromagnetics.

Wenhua Chen (M’07) received the B.S. degree from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2001 and the Ph.D. degree from Tsinghua University, Beijing, China, in 2006. He is currently an Assistant Professor with the State Key Laboratory on Microwave and Digital Communications, Tsinghua University. His research interests include computational electromagnetics, reconfigurable and smart antennas, and high-efficiency power amplifiers. He has authored and coauthored over 30 journal and conference papers.

Zhenghe Feng (M’00–SM’08) received the B.S. degree in radio and electronics from Tsinghua University, Beijing, China, in 1970. Since 1970, he has been with Tsinghua University, as an Assistant, Lecturer, Associate Professor, and Full Professor. His main research areas include numerical techniques and computational electromagnetics, RF and microwave circuits and antenna, wireless communications, smart antenna, and spatial temporal signal processing.

ZHANG et al.: STUDY OF CONFORMAL SWITCHABLE ANTENNA SYSTEM ON CYLINDRICAL SURFACE FOR ISOTROPIC COVERAGE

Magdy F. Iskander (F’91) is the Director of the Hawaii Center for Advanced Communications (HCAC), College of Engineering, University of Hawaii at Manoa, Honolulu, http://hcac.hawaii.edu. He is also a Co-Director of the NSF Industry/University joint Cooperative Research Center between the University of Hawaii and four other universities in the US. From 1997 to 1999, he was a Program Director at the National Science Foundation, where he formulated and directed a “Wireless Information Technology” Initiative in the Engineering Directorate. He spent sabbaticals and other short leaves at Polytechnic University of New York; Ecole Superieure D’Electricite, France; UCLA; Harvey Mudd College; Tokyo Institute of Technology; Polytechnic University of Catalunya, Spain; University of Nice-Sophia Antipolis, and Tsinghua University, China. He authored the textbook Electromagnetic Fields and Waves, (Prentice Hall, 1992 and Waveland Press, 2001), edited the book CAEME Software Books (Vol. I, 1991, and Vol. II, 1994), and edited four other books on the microwave processing of materials (Materials Research Society, 1990–1996). He has published over 200 paperers in technical journals, has eight patents, and has made numerous presentations in International conferences. He is the Founding Editor of the journal Computer Applications in Engineering Education (CAE) (Wiley).

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His research focus is on antenna design and propagation modeling for wireless communications and radar systems, and in computational electromagnetics. Dr. Iskander received the 2010 University of Hawaii Board of Regents’ Medal for Excellence in Teaching, and the University of Utah Distinguished Teaching Award in 2000. He also received the 1985 Curtis W. McGraw ASEE National Research Award, 1991 ASEE George Westinghouse National Education Award, 1992 Richard R. Stoddard Award from the IEEE EMC Society. He was a member of the 1999 WTEC panel on “Wireless Information Technology-Europe and Japan,” and chaired two International Technology Institute panels on “Asian Telecommunication Technology” sponsored by the DoD in 2001 and 2003. He edited two special issues of the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION ON WIRELESS COMMUNICATIONS TECHNOLOGY, 2002 and 2006, co-edited a special issue of the IEICE Journal in Japan in 2004. He is the 2002 President of the IEEE Antennas and Propagation Society, and was a member of the IEEE APS AdCom from 1997 to 1999, and 2003 to 2006. He was the General Chair of the 2000 IEEE AP-S Symposium and URSI Meeting, and the 2003, 2005, 2007, and 2010 IEEE Wireless Communications Technology Conferences in Hawaii. He was also a Distinguished Lecturer for the IEEE AP-S (1994–97) and during this period he gave lectures in Brazil, France, Spain, China, Japan, and at a large number of US universities and IEEE chapters.

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Analysis of Conformal Microstrip Antennas With the Discrete Mode Matching Method Marcos V. T. Heckler, Member, IEEE, and Achim Dreher, Senior Member, IEEE

Abstract—An extension of the discrete mode matching (DMM) method is used to analyze microstrip antennas conformed to cylinders. The structure needs to be discretized only along the azimuthal and the axial directions by sampling the fields and current densities in the interfaces between dielectric layers. The solution in the radial direction is obtained analytically with the dyadic Green’s function, which is calculated using the full-wave equivalent circuit (FWEC) in the spectral domain. The structure is truncated in the axial direction by employing absorbing boundary conditions (ABCs). The mathematical formulation is given in this paper, followed by its application to compute the main properties of conformal microstrip antennas, such as resonant frequency, radiation patterns and input impedance. Validation with the commercial software HFSS has been done and good agreement between the results has been observed. Index Terms—Conformal antennas, conformal structures, cylindrical microstrip antennas, discrete mode matching (DMM).

I. INTRODUCTION ONFORMAL microstrip structures have received increasing attention in the last decades, due to the need of antennas and devices that can be mounted on curved surfaces. One interesting property of these antennas is their low profile, which introduces only low disturbances in the aerodynamics of the vehicles where they are mounted on [1], [2]. This is particularly important in aeronautical and space applications. With the continuous improvement in computing power, the analysis of electromagnetic problems by means of numerical techniques became nearly unavoidable, since they allow reducing time and costs in comparison to a purely experimental optimization. Several techniques have been suggested for the analysis of microstrip antennas conformed on cylindrical surfaces. The simplest ones have an empirical nature, like the transmission line (TL) and the cavity model. They can be used for the analysis of simple radiators. These two methods can be employed as a first-step approach in the design of antennas and arrays [3]. For more complex geometries, more sophisticated techniques must be employed. Some examples of full-wave

C

Manuscript received December 09, 2009; revised July 30, 2010; accepted August 07, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. M. V. T. Heckler was with the German Aerospace Center (DLR), Institute of Communications and Navigation, Oberpfaffenhofen, D-82234 Wessling, Germany. He is now with the Universidade Federal do Pampa (UNIPAMPA), 97546-550 Alegrete-RS, Brazil. A. Dreher is with the German Aerospace Center (DLR), Institute of Communications and Navigation, Oberpfaffenhofen, D-82234 Wessling, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103014

techniques are the method of moments (MoM), and the finite-element method (FEM), finite-difference methods like the finite-difference time domain (FDTD) and the method of lines (MoL), and the discrete mode matching method (DMM). FDTD and FEM have been receiving increasingly attention from the research community due to their flexibility, since they allow the analysis of problems involving complex geometries [4]–[6]. For the specific case of microstrip antennas, these methods may require large computational effort, especially if the structure is composed of very thin dielectric layers. Such a problem can be overcome by employing techniques based on the Green’s function of multilayer media formulated in the spectral domain. Some authors already employed the MoM successfully for the analysis of conformal microstrip antennas [1], [7]–[13]. The discrete mode matching (DMM) has been first introduced for the analysis of multilayer structures in [14], where the case of partially filled rectangular waveguides has been considered. It has been shown that the eigenvalues of the discretized Helmholz equation computed with the MoL converge to the analytical values, if the number of discretization lines tends to infinity. In the DMM, the wave equation is not discretized and the eigenvalues used are exact and independent of the discretization. They only depend on the type of the lateral boundary conditions. The method also employs a 2D discretization for the analysis of radiating structures, which reduces the memory requirements in comparison to the 3D meshes needed by FEM and FDTD. Moreover, due to the consideration of a laterally bounded domain and the use of mode matching, no extra integration is needed for the inverse Fourier transformation. This makes the method work faster than other integral equation methods. The application of the DMM in the computation of electromagnetic fields in stratified media has been already described in the open literature for planar [14], [15] and quasi-planar cases [16]. However, no effort has been reported on its application to conformal antennas presenting circular cross sections. Therefore, this paper intends to present the extension of the DMM to allow the analysis of microstrip antennas installed on cylindrical surfaces. The technique has been employed for the analysis of microstrip lines in [17], where the modal expansion along the -direction has been applied to transform fields and current densities from the space into the spectral domain. In the present approach, the inverse Fourier transform is done both in azimuth and in the axial directions. In order to truncate the computational domain along the cylinder axis, absorbing boundary conditions (ABCs) are used. In the following section, the theory of the DMM in cylindrical coordinates is given. Then, the formulations to compute the complex resonant frequency, radiation pattern and input

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impedance are presented. In order to show the accuracy of the method, microstrip antennas are analyzed and the results obtained with the DMM are compared with data obtained using other numerical techniques, such as the cavity model [18] and FEM [19]. II. THEORY A. The Discrete Mode Matching Formulation A typical multilayer conformal microstrip structure is shown in Fig. 1. In the proposed formulation, the layers and the ground cylinder are supposed to be infinite along the -direction. The dielectric layers are assumed to be isotropic and homogeneous. Metallizations may only exist in the interfaces between dielectric layers. Considering time harmonic variations, the wave equation for the fields inside every layer is given, in the spectral domain, by (normalized to ) (1) represents each of the independent components or is the relative permittivity, and is the relative permeability. Equation (1) has the form of the classical Bessel’s differential equation. Therefore, the general and can be described by linear combinations solution for of Bessel functions. The general solution for (1) is given by

Fig. 1. Microstrip antennas mounted on a cylindrical surface.

where

(2) where and are the Bessel functions of the first and second kind, respectively. Special care must be taken to for the outer unbounded region extending from , where is the radius of the interface between the outermost dielectric layer and the unbounded medium. In this case, (2) must be modified to (3) where is the Hankel function of the second with kind, or , so as to fulfill the radiation boundary condition for outwards propagating waves. Using (2) and (3), a full-wave equivalent circuit (FWEC) can be constructed. From the formulation presented in [10], and by using circuit analysis, one can derive the Green’s functions to establish the relation between the currents and fields at the interfaces as (4) where and are diagonal matrices that compose the dyadic Green’s function in the spectral domain, and are vectors that contain the modal components of the surface current density and of the electric field in the interis the intrinsic faces between different dielectric layers, and impedance of free space. In order to impose the boundary condition stating that the electric field must vanish on the metallizations, (4) must be

transformed into the space domain. For fields written in cylindrical coordinates, the inverse Fourier transform is given by

(5) which is valid for a cylinder with circular cross section and with infinite dimension along . If the structure under analysis is bounded in some way along the azimuth and axial directions, then only discrete modes are allowed to exist in the closed computational domain. In this case, the integral existing in (5) becomes discrete and must be replaced by a summation, where only the allowed modes must be included. Therefore, (5) can be rewritten as (6) is the angle (in radians) formed by the two walls limwhere iting the domain along the -direction. The terms and represent wavefunctions and are dependent on the combination of lateral walls that bound the domain along and , respectively. Tables I and II summarize the terms that are to be used for different combinations, where is the distance between the two walls placed along . In order to impose the boundary condition on the metallization, the computational domain must be discretized to sample the and points fields and current densities in the interfaces at along and , respectively. The adopted discretization scheme is presented in Fig. 2, where a multilayer structure composed of two metallizations is depicted. The discretization is composed of two along kinds of lines, shifted by an angular distance of 0.5 and by 0.5 along . The e-lines are related to the points in the is computed, whereas is sampled at the interfaces where h-lines. The discretization lines are used to sample the fields and the current densities in all the interfaces between the dielectric

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TABLE I WAVEFUNCTIONS FOR DIFFERENT LATERAL BOUNDARY COMBINATIONS ALONG THE AZIMUTH

work, both and have been considered to be 0.25. This value has been proven to allow good convergence. The number of modes to be included in the expansions is and along the respective directions, according equal to to the sampling theorem. Therefore, one can write

(7)

(8) TABLE II WAVEFUNCTIONS FOR DIFFERENT LATERAL BOUNDARY COMBINATIONS ALONG THE AXIAL DIRECTION

After sampling the fields, (7) and (8) can be written in matrix form (9) (10) where and are the transformation matrices to be used to perform the inverse Fourier transformation. The elements of these two matrices are computed by (11) (12) The other tangential field components are related to by

and

(13) Using (9)–(13), it can be shown that and are trans, whilst the same formed into the space domain employing for . The inverse Fourier operation must be done using transformation of (4) is achieved by using these matrices. By doing so, it results that (14) where

(15) Equation (14) is the system of equations that has to be solved in order to determine the current densities that flow on the metallizations. The dimensions of the matrices are directly governed by the number of discretization lines. For instance, matrices and present dimensions and , respectively. Fig. 2. Discretization scheme showing the points where fields and currents are sampled: (a) Cross-sectional view; (b) top view. The parameters p and q are the edge parameters, as explained in the text.

layers. Moreover, the factors and , indicated in Fig. 2 are the edge parameters and state how far the edge of the metallization is located to the nearest or -lines, respectively. In the present

B. The Higdon’s Operator for the Absorbing Boundary Condition given in The wavenumbers and wavefunctions along Table II are only valid for the cases where the computational domain is closed by electric and magnetic walls. This formulation is adequate for the analysis of closed structures. The

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TABLE III WAVENUMBERS AND WAVEFUNCTIONS FOR DIFFERENT LATERAL BOUNDARY CONDITIONS IN COMBINATION WITH ABCS

wavefunctions are trigonometric sines or cosines with purely . real wavenumbers and Since antennas are supposed to radiate in an unbounded medium, absorbing boundary conditions (ABCs) should be employed to bound the computational domain artificially. By placing ABCs along the axial direction, a structure that extends can be modelled, whilst still keeping a to infinity along “closed” computational domain. This is an important feature in order to allow the use of the inverse Fourier transformation described in (6). There are several ways to implement such a boundary condition. In [20], perfect matched layers (PMLs) have been suggested, which have been extensively used for finite-difference time domain (FDTD) computations. Another technique is based on the annihilation of the fields at the ABCs by means of an appropriate operator [21]. The approach used in the present formulation is based on the Higdon’s operator [22], [23], which considers that the incident wave is absorbed perfectly in specific angles of incidence . The Higdon’s operator is given by [15] (16) where ord represents the number of the angles of perfect absorption and is the dielectric constant of the medium at the location of the ABCs. In the case of multilayer structures, one value for must be chosen. Provided that the ABCs are placed far enough, the solutions for different values of converge to the same value. For the present investigations, the dielectric constant has been always considered to be the mean value of all the in the structure, given by (17) Equation (16) must be satisfied at both ABC’s bounding the domain. The sign becomes “ ” for waves propagating in the -direction; otherwise, “ ” must be chosen. The general field response along is given by (18)

Fig. 3. Geometry for the computation of the eigenvalues and eigenmodes using ABCs.

for the case of an infinite structure along the axial direction. By and , according to the applying (18) at geometry shown in Fig. 3, and after some analytical work, the following characteristic equation is obtained (19) The values of that satisfy (19) are the wavenumbers along that are correspondent to the modes that are allowed to exist inside the computational domain bounded by the ABCs. If and (perfect absorption at normal incidence) are chosen, then (19) exhibits the same form as the equation developed by Sommerfeld for a domain enclosed by boundaries that obey the radiation condition [24]. . If is introduced in (18), From (19), then (20) whereas if

is considered, it comes out that (21)

For structures presenting electric or magnetic symmetry along the axial direction, or -walls may be placed at while keeping the ABC located at . In this case, depending on the kind of symmetry, only one set of modes exists. For an -ABC combination, the wavenumbers calculated for must be used, whereas the other set of is to be considered for an -ABC combination. The wavefunctions must be also chosen so that the boundary conditions at the or at the -wall are respected. Table III summarizes the wavenumbers and wavefunctions to be used along the -direction according to the boundary combination that delimits the computational domain.

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C. Determination of the Resonant Frequency of Rectangular Patches Once the system of equations has been obtained in the space and have to be endomain, the electric field components forced to vanish on the metallizations. For the case of finding the resonance frequency of rectangular patches, no impressed electric field (excitation) is applied. Therefore, a reduced version of the system of equations described in (14) can be obtained and is given by (22) and are reduced versions of and which where contain only the elements calculated at the sampling points located on the rectangular patch. The solution of (22) is found when the determinant of vanishes. This is done by varying the frequency until this condition is fulfilled. The result is a complex value composed of a real part , which represents the resonance frequency, and an imaginary part , being a measure for the losses in the antenna (including power lost by means of surface waves and radiation). Based on the complex resonance frequency, the quality factor is [1]. The surface current density on the given by patch can be calculated as an eigensolution of (22) at the resois computed located on the nant frequency. The component -lines, whereas is sampled on the -lines.

Fig. 4. Geometry for the computation of the input impedance.

D. Calculation of the Radiation Pattern For the purpose of far-field computation, the current densities can be considered to be an array of uncoupled Hertzian dipoles that are oriented along the and -directions and are located on the corresponding discretization points on the metallizations. The radiated fields of Hertzian dipoles can be obtained using the Green’s function in the spectral domain combined with the method of the stationary phase (MSP) [25]. The radiated fields for microstrip antennas can be calculated by adding the radiated fields produced by the several Hertzian dipoles, which model the current flowing on the metallization.

Fig. 5. Network model for the computation of the input impedance.

where is the tensor that represents the Green’s function of the multilayer structure and is the area where the surface current flows. Since the fields and currents are sampled only on the and -lines in the DMM formulation, the current density on a given line can be considered to be equivalent to a Hertzian dipole. Under this assumption, and applying (23) in every discretization point on the metallizations, the following system of equations is obtained

E. Formulation for the Input Impedance

(24)

For the calculation of the input impedance of an antenna, a geometry as shown in Fig. 4 is considered, where a voltage gap source (also known as delta-gap generator) [26]–[29] is used to excite the antenna. To make the analysis easier, this structure may be modelled by an equivalent transmission line model as shown in Fig. 5, where the antenna is treated as a load terminating the transmission line. The generator is loaded on one side by the impedance of an open-ended stub and by the impedance resulting from the input impedance of the patch . and from the feed line In the space domain, one can write the equation relating the to the induced currents on the metscattered electric field as allizations (23)

where the matrix is composed of terms that are dependent on the Green’s function of the structure. The elements of assume non-zero values only at the location of the generator. Depending on the orientation of the excitation, the impressed electric field is described by (25) if the excitation is -oriented, and by (26) if the excitation is -oriented, where and indicate the locations of the impressed field in the gap source, in cylindrical coordinates.

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Fig. 6. Discretization scheme in the region of the gap source: (a) Gap-source along z ; (b) Gap source along . Fig. 7. The use of symmetry to compute the resonant frequency and the current densities.

As shown in Fig. 6, it is interesting to note that the gap sources are treated differently depending on its orientation. If the impressed field is -oriented, then it must be located on -lines, is also sampled on -lines. Accordingly, the impressed since field must be imposed on -lines, if the gap source is -oriented, since is sampled on -lines. by The term is used to compute the impressed voltage (27) is the length of the delta-gap and V.1 As where was considered to be equal to or illustrated in Fig. 6, , depending whether is axially or circumferentially oriented, respectively. Since the current in the location of the gap generator is now known, the total impedance seen by the generator is given by

Fig. 8. Convergence of the DMM computations on the real part of the resonance frequency for different discretization densities on the -polarized patch.

(28) where

III. VALIDATION

is calculated by

A. Resonance Frequency of Rectangular Patches

(29) if the gap is oriented in the -direction, and by

(30) if is oriented along . The terms and stand for the number of and -lines existing in the gap, respectively. still contains the The result obtained in (28) for impedance introduced by the open-ended stub, which should be removed in order to obtain the correct value of the antenna input impedance by doing (31) where is the impedance obtained by simulating separately a section of line with the same length as the open-ended stub. 1The term e is given in volts instead of volts/meter as a consequence of the normalization of (1) to k .

Since a rectangular patch is a symmetric structure, the computational effort may be decreased using the image theory. The equivalent structure is shown in Fig. 7, where the choice of the correct type (E or H-wall) for the vertical and horizontal walls allows only the fundamental mode to exist in the desired direction. With this approach, the currents on only one quarter of the patch must be computed. A numerical example is considered for the case of mm and mm, with a conformal rectangular patch with 40 mm along and 25 mm along the -direction. Computations have been conducted to assess the convergence behavior of the resonance frequency under the variation of the discretization density on the patch. A two-dimensional discretization has been considered, where the number of -lines has been varied only along the resonant direction. Numerical assessment showed that the results are not significantly affected by the number of -lines placed along the nonresonant dimension, provided that at least some few are used. In the present calculation, therefore, we considered 4 -lines along the nonresonant direction. The numerical results are presented in Figs. 8–11 for the resonance along and . A maximum of 11 -lines have been used along the resonant direction, which results in in a total of 44 -lines on one quarter of the patch. The graphics show that a smooth convergence behavior is obtained with the increase of

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Fig. 9. Convergence of the DMM computations on the imaginary part of the resonance frequency for different discretization densities on the -polarized patch. Fig. 12. Normalized radiation patterns for different cylinder radii for a z -polarized rectangular patch.

Fig. 10. Convergence of the DMM computations on the real part of the resonance frequency for different discretization densities on the z -polarized patch.

Fig. 13. Normalized radiation patterns for different cylinder radii for a -polarized rectangular patch.

niques are lower than 2%, if the extrapolated result is considered. For thicker substrates, the cavity model becomes inaccurate, while the formalism based on the DMM still continues to be rigorous. B. Radiation Patterns for Different Radii

Fig. 11. Convergence of the DMM computations on the imaginary part of the resonance frequency for different discretization densities on the z -polarized patch.

the discretization density. By means of extrapolation, the result for an infinite number of discretization points can be assessed. The computed results have been validated using the cavity model. The observed deviations in the predictions of both tech-

The variation of the radiation pattern for the case of resonance along the axial and the azimuth directions is presented in Figs. 12 and 13, respectively, for . Validation has been undertaken with the code Cylindrical [18], which is based on the cavity model. Good agreement between the two techniques has been obtained. C. Input Impedance Calculation In the previous cases, the antennas were simply rectangular patches without any kind of feeding technique. For such structures, the analysis becomes an eigenvalue problem which, by

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for each frequency point for DMM and 3.2 minutes for HFSS. It should be pointed out that the DMM simulations have been carried out in MATLAB-environment, so that the code has not been compiled. In terms of memory usage, HFSS needed 827 MB for the simulation with a three-dimensional mesh composed of 45185 tetrahedra. The memory usage for the DMM simulation could not be directly assessed. An amount of 1210 -lines and the same number of -lines have been used to discretize the interface between the substrate and the air, which resulted in a matrix containing the elements of the Green’s function in spectral domain with 2420 2420 elements. After the transformation into the space domain, the final system of equations prewith 191 191 elements. sented a matrix Fig. 14. Dimensions of the simulated -polarized antenna.

IV. CONCLUSION

Fig. 15. Comparison between results computed with DMM and HFSS for the input impedance.

proper setting up of the system of equations, gives the possibility to calculate the natural resonances existing in the rectangular patch. However, in practical cases, the energy must be fed to the antenna, in the case of a transmitting system, or received from it, if the antenna works in the receiving mode. In order to achieve a proper power transfer, an appropriate impedance matching between the antenna and the feeding system has to be ensured. A -polarized antenna is depicted in Fig. 14. The ground mm and is covered by a dicylinder presents a radius of mm of thickness and . electric layer of Due to the symmetry of the structure along , only half of its geometry has been considered for the calculations using the DMM by the inclusion of a horizontal -wall, whereas the complete structure has been considered in the simulation with HFSS. The computed input impedance plotted in the Smith chart is shown in Fig. 15, where the symbols on the curves indicate the marks of every 50 MHz. The results obtained with the DMM formulation are compared with the ones computed with HFSS. One can see that good agreement between the predictions of both techniques has been obtained. Both DMM and HFSS computations have been performed on a Intel dual-core computer with 2.2 GHz clock frequency and 2 GB RAM. The elapsed computing times were about 1.5 minutes

An extension of the discrete mode matching method (DMM) that allows the analysis of conformal microstrip antennas has been presented. The method is based on the dyadic Green’s function in the spectral domain, where the inverse Fourier transformation is achieved by means of a discrete field expansion. Numerical results have been presented and validations with the finite-element based commercial package Ansoft HFSS have been undertaken. It has been demonstrated that the proposed method can provide accurate results in a shorter computing time. Calculations of the resonance frequency of conformal rectangular patches have been performed. It has been shown that the results converge smoothly with an increase in the discretization density. This fact can be of advantage, since a more accurate value for the resonance frequency can be obtained by performing calculations placing only a few number of -lines on the patch and a further extrapolation. In its current version, the method can only deal with microstrip structures conformed on cylindrical structures with circular cross sections. Further extension of the method can be the consideration of quasi-cylindrical problems with the problem formulated directly in the space domain. Moreover, the structures analyzed in this paper considered only isotropic dielectrics. A formulation including anisotropy is also a further possible extension of the method proposed in this contribution. REFERENCES [1] K.-L. Wong, Design of Nonplanar Microstrip Antennas and Transmission Lines. New York: Wiley, 1999. [2] L. Josefsson and P. Persson, Conformal Array Antenna Theory and Design. Piscataway: IEEE Press, 2006. [3] M. V. T. Heckler, M. Bonadiman, J. C. S. Lacava, and L. Cividanes, “Analysis of cylindrical circumferential array with circular polarization for space application,” in IEEE Antennas Prop. Symp. Dig., 2004, pp. 117–120. [4] T. Itoh, Numerical Techniques For Microwave and Millimeter-Wave Passive Structures. New York: Wiley, 1989. [5] K. L. Shlager and J. B. Schneider, “A selective survey of the finitedifference time-domain,” IEEE Antennas Propag. Mag., vol. 37, pp. 39–56, Aug. 1995. [6] D. H. Werner and R. Mitra, Frontiers in Electromagnetics. Piscataway: IEEE Press, 1999. [7] S. Raffaelli, Z. Sipus, and P.-S. Kildal, “Effect of element spacing and curvature on the radiation patterns of patch antenna arrays mounted on cylindrical multilayer structures,” in IEEE Antennas Prop. Symp. Dig., Orlando, FL, 1999, pp. 2474–2477. [8] S. Raffaelli, Z. Sipus, and P.-S. Kildal, “Analysis and measurements of conformal patch array antennas on multilayer circular cylinder,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1105–1113, Mar. 2005.

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[9] D. Löffler, G. Chakam, and W. Wiesbeck, “Analysis of conformal aperture coupled microstrip patch elements on cylindrical surfaces,” in 29th Eur. Microwave Conf., Munich, Germany, Oct. 1999, pp. 394–397. [10] M. Thiel, “Die Analyse von zylinderkonformen und quasi-zylinderkonformen Antennen in Streifenleitungstechnik,” Ph.D. dissertation, Tech. Univ. Munich,, Munich, Germany, 2002, (Oberpfaffenhofen: Forschungsbericht DLR-FB 2002-25). [11] C. M. da Silva, “Microstrip antenna arrays with optoelectronic interface conformed on cylindrical surfaces,” (in Portuguese) Ph.D. dissertation, Instituto Tecnológico de Aeronáutica, São José dos Campos, Brazil, 1992. [12] T. V. B. Giang, “A systematic approach to the analysis of spherical multilayer structures and its applications,” Ph.D. dissertation, University of Hamburg-Harburg, Hamburg, Germany, 2006. [13] Z. Sipus, N. Burum, and J. Bartolic, “Theoretical and experimental study of spherical rectangular microstrip patch arrays,” presented at the 3rd Eur. Workshop Conformal Antennas, Bonn, Germany, Oct. 2003. [14] A. Dreher and T. Rother, “New aspects of the method of lines,” IEEE Microw. Guided Wave Lett., vol. 5, no. 11, pp. 408–410, Nov. 1995. [15] A. Ioffe and A. Dreher, “Verwendung von absorbierenden Randbedingungen hoher Ordnung für die Discrete Mode Matching Methode,” Kleinheubacher Berichte, pp. 133–140, 1999. [16] A. Ioffe, M. Thiel, and A. Dreher, “Analysis of microstrip patch antennas on arbitrarily shaped multilayers,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1929–1935, Aug. 2003. [17] M. V. T. Heckler and A. Dreher, “Analysis of cylindrical microstrip lines using the discrete mode matching method,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 7, pp. 392–394, Jul. 2006. [18] A. F. T. Salazar, M. V. T. Heckler, D. C. Nascimento, R. Schildberg, and J. C. S. Lacava, “Cylindrical: An effective CAD package for designing probe-fed rectangular microstrip antennas conformed onto cylindrical structures,” IEEE Antennas Propag Mag., vol. 50, no. 1, pp. 164–169, Feb. 2008. [19] “Ansoft HFSS version 9 users guide,” Ansoft Corporation 2003. [20] J.-P. Bérenger, “Perfectly matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antennas Propagat., vol. 44, pp. 110–117, Jan. 1996. [21] T. G. Moore, J. G. Blaschak, A. Taflove, and G. A. Kriegsmann, “Theory and application of radiation boundary operators,” IEEE Trans. Antennas Propag., vol. 36, no. 12, pp. 1797–1812, Dec. 1988. [22] R. L. Higdon, “Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation,” Math Comp, vol. 47, no. 176, pp. 437–459, Oct. 1986. [23] R. L. Higdon, “Numerical absorbing boundary conditions for the wave equation,” Math Comp, vol. 49, no. 179, pp. 65–90, Jul. 1987. [24] A. Sommerfeld, Partielle Differentialgleichungen der Physik. Frankfurt: Verlag Harri Deutsch, 1978. [25] R. E. Collin and F. J. Zucker, Antenna Theory, Part 1. New York: McGraw-Hill, 1969. [26] I. E. Rana and N. G. Alexopoulos, “Current distribution and input impedance of printed dipoles,” IEEE Trans. Antennas Propag., vol. AP-29, pp. 99–105, Jan. 1981.

[27] P. B. Katehi and N. G. Alexopoulos, “Frequency-dependent characteristics of microstrip discontinuities in millimeter-wave integrated circuits,” IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp. 1029–1035, Oct. 1985. [28] M. Davidovitz and Y. T. Lo, “Rigorous analysis of a circular patch antenna excited by a microstrip transmission line,” IEEE Trans. Antennas Propag., vol. 37, no. 8, pp. 949–958, Aug. 1989. [29] J. Heinstadt, Effiziente numerische Strategien zur Berechnung von Einzelstrahlern und Gruppenantennen in Streifenleitungstechnik. Oberpfaffenhofen, Germany: Forschungsbericht DLR 95-16, 1995.

Marcos V. T. Heckler (M’06) was born in Rio Grande, Brazil, in 1978. He received the B.S. degree in electrical engineering (emphasis in electronics) from the Universidade Federal de Santa Maria (UFSM), Brazil, in 2001, the M.Sc. degree in electronic engineering (microwaves and optoelectronics) from Instituto Tecnológico de Aeronáutica (ITA), Brazil, in 2003, and the Dr.-Ing. (Ph.D.) degree from the Technische Universität München, Munich, Germany, in 2010. From April to August 2003, he worked as a Research Assistant with the Antennas and Propagation Laboratory, ITA, Brazil. From October 2003 to June 2010, he worked as a Research Associate with the Antenna Group, Institute of Communications and Navigation, German Aerospace Center (DLR), Oberpfaffenhofen, Germany. He is currently an Assistant Professor at Universidade Federal do Pampa, Alegrete, Brazil. His current research interests are the design of microstrip antennas and arrays, and numerical techniques for conformal microstrip antennas.

Achim Dreher (M’92–SM’99) was born in Hermannsburg, Germany, in 1955. He received the Dipl.-Ing. (M.S.) degree from the Technische Universität Braunschweig, Braunschweig, Germany, in 1983 and the Dr.-Ing. (Ph.D.) degree from the FernUniversität, Hagen, Germany, in 1992, both in electrical engineering. From 1983 to 1985, he was a Development Engineer with Rohde & Schwarz GmbH & Co. KG, München, Germany. From 1985 to 1992 he was a Research Assistant, and from 1992 to 1997, he was a Senior Research Engineer with the Department of Electrical Engineering, FernUniversität. Since 1997, he has been with the Institute of Communications and Navigation, German Aerospace Center (DLR), Oberpfaffenhofen, Germany, where he is currently Head of the Antenna Research Group. His current research interests include analytical and numerical techniques for conformal antennas and microwave structures, smart antennas for satellite communications and navigation, and antenna technology for radar applications.

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Reducing Redundancies in Reconfigurable Antenna Structures Using Graph Models Joseph Costantine, Member, IEEE, Sinan al-Saffar, Member, IEEE, Christos G. Christodoulou, Fellow, IEEE, and Chaouki T. Abdallah, Senior Member, IEEE

Abstract—We present an approach for reducing redundancies in the design of reconfigurable antenna structures using graph models. The basics of graph models, their rules, and how they can be applied in the design of switch-based reconfigurable antennas are introduced. Based on these rules, a methodology is developed and formulated to reduce the number of switches and parts in the antenna structure, without sacrificing the desired antenna functions. This approach not only optimizes the overall structure of the antenna but it also reduces cost and overall losses. Several examples are presented and discussed to demonstrate the validity of this new approach through simulations and measurements that present good agreement. Index Terms—Graph theory, reconfigurable antennas, redundancy, switches.

Fig. 1. a. An example of an undirected graph. b. An example of a directed graph with weighted edges.

I. INTRODUCTION ECONFIGURABILITY, when used in the context of antennas, is the capacity to change an individual radiator’s fundamental operating characteristics through electrical, mechanical, or other means [1]. The reconfiguration of such an antenna is achieved through an intentional redistribution of the currents or, equivalently, the electromagnetic fields of the antenna’s effective aperture, resulting in reversible changes in the antenna impedance and/or radiation properties [2]. Many techniques can be used to achieve the reconfiguration of an antenna. Most of these techniques employ switches, diodes or capacitors. Other techniques resort to mechanical alterations like a rotation or bending of a certain antenna part. Reconfigurable antennas are mostly used on systems that require some type of change from one application to another. Reconfigurable antennas are used in multiple input multiple output (MIMO) situations, in cognitive radios, on laptops, in cellular phones and many other systems. Graph models are widely used in computer science and in the development of networking algorithms [3]. Graphs also find

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Manuscript received August 03, 2009; revised July 21, 2010; accepted August 20, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. J. Costantine is with the Electrical Engineering Department, California State University Fullerton, Fullerton, CA 92834 USA (e-mail: jcostantine@fullerton. edu). S. al-Saffar is with the Knowledge Systems Group, Pacific National Laboratory, Seattle, WA 98115 USA. C. G. Christodoulou and C. T. Abdallah are with the Electrical and Computer Engineering Department, University of New Mexico, Albuquerque, NM 87131 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.2010.2103005

applications in self-assembly robotics, where they are used to model the physics of particles by describing the outcomes of interactions among subsystems [4]. Herein we use graph models to optimize the structure of a reconfigurable antenna. We set graph modeling rules for the different types of switch-reconfigured antennas. We present several examples elaborating our modeling rules and optimization technique. II. INTRODUCTION TO GRAPHS Graphs are symbolic representations of relationships between different components of a system. They are mathematical tools used to model complex systems in order to organize them and improve their status. In [5] the authors showed, briefly, that reconfigurable antennas can be modeled using graphs. A graph is defined as a collection of vertices that are connected by lines called edges. A graph can be either directed or undirected. The edges in a directed graph have a certain determined direction, while this is not the case in an undirected graph as shown in Fig. 1. Vertices may represent physical entities while the edges between them in the graph represent the presence of a function resulting from connecting these entities. If one is proposing a set of guidelines for antenna design, then a possible modeling rule may be to create an edge between two vertices whenever their physical connection results in a meaningful antenna function. Edges may have weights associated with them, as shown in Fig. 1(b). These weights represent costs or benefits that are to be minimized or maximized. A path is an uninterrupted sequence of edges that are traveled in the same direction from an originating vertex to a destination vertex. The weight of a path is defined as the sum of the weights of its constituent edges. In some cases it is useful to find the shortest path connecting two vertices. This notion is used in graph algorithms in order to optimize a certain function. The shortest path distance in a

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non-weighted graph is defined as the minimum number of edges in any path from one vertex to another. If the graph is weighted, then the shortest path corresponds to the least sum of weights in a particular path. In a reconfigurable antenna design, a shorter path may mean a shorter current flow and thus a certain resonance associated with it. A longer path may denote a lower resonance frequency than that of a shorter path. Adjacency-Matrix Representation: The adjacency-matrix representation of a graph , assuming that the vertices are in some arbitrary manner, consists of a numbered such that [3] Fig. 2. The antenna structure in [6] with its graph model.

(1) The adjacency matrix of the graph shown in Fig. 1(a) is prebelow. The adjacency-matrix represensented in the matrix tation can also be used for weighted graphs. The corresponding weights in a graph are grouped into the adjacency matrix. For is a weighted graph with edge-weight example, if of the edge , then is simply stored as function the entry in row and column of the adjacency matrix. The lack of an edge is indicated by 0 in the adjacency matrix. The adjacency matrix of the graph shown in Fig. 1(b) is shown in the as follows: matrix

Graph Modeling Rules: There are several ways to graph model reconfigurable antennas. Here we set some rules for graph modeling the different types of switch-reconfigured antennas. These rules are required for our optimization approach. We set constraints for each rule in order to facilitate the graph modeling process. These constraints explain how to graph model each specific case of switch-reconfigured antennas. Herein an antenna is called a multi-part antenna if it is composed of an array of identical or different elements (triangular, rectangular parts). Otherwise it is called a single-part antenna. Rule 1: A multi-part antenna connected with switches is modeled as a weighted undirected graph. This graph consists of a vertex for each antenna part and connects those vertices with undirected weighted edges wherever the parts have a physical connection. Constraints: The connection between two parts has a distinctive angular direction. The designer defines a reference axis that represents the direction that the majority of parts have in relation to each other or with a main part. The connections between the parts are represented by the edges. The edges’ weights represent the angles that the connections make in relation to the is assigned to an edge reprereference axis. A weight senting a connection that has an angle 0 or 180 in relation to is assigned to the the reference axis. Otherwise a weight edge as shown in (2).

(2)

where represents the angle that the connection i,j form with the reference axis. The removal or addition of a part in the reference axis direction affects one parameter in the antenna radiation characteristics (i.e. S11). The addition or removal of a part in a direction different from the reference axis direction affects many parameters (i.e., S11, radiation pattern and polarization); thus the bigger weight. Example of rule 1: As an example, we take the antenna shown in Fig. 2(a) [6] and model it by a graph following rule 1. The basic antenna is a 130 balanced bowtie. A portion of the antenna corresponds to a two-iteration fractal Sierpinski dipole. The remaining elements are added (three on each side) to make the antenna a more generalized reconfigurable structure. Following rule 1, the vertices in the graph model represent the triangles added. The edges connecting these vertices represent the connection of the corresponding triangles by MEMS switches. The graph modeling of this antenna is shown in Fig. 2(b). If a switch is activated to connect triangle T1 to then an edge appears between the vertex T1 and the triangle , as shown in Fig. 2(b). In this design the connections vertex and , and between T1 and T2, T2 and T4, are collinear with the reference axis. As a result, the edges representing these connections are weighted with , and . the other connections are weighted with Rule 2: A single part antenna with switches bridging over slots is modeled as a non-weighted undirected graph. This graph consists of a vertex for every switch end-point and connects those vertices with non-weighted edges wherever switches are activated. Constraints: In the case of switches bridging multiple slots in one antenna structure, the graph model takes into consideration one slot at a time. Example of rule 2: As an example, we take the antenna shown in Fig. 3(a) [7]. This antenna is a triangular patch antenna with two slots. The authors suggested five switches to bridge each slot in order to achieve the required functions. The graph modeling this antenna following rule 2 is shown in Fig. 3(b), where vertices represent the end points of each switch, and edges represent the connections between these end points. When switch 1 is activated, an edge appears between N1 representing the two end-points of switch 1. The graph and model in Fig. 3(b) represents each slot at a time. For example,

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Fig. 3. The antenna structure in [7] with its graph model for all possible connections. a. Antenna structure; b. graph model.

N1 represents end-point 1 for switch 1 in slot 1 and end-point 1 for switch 1 in slot 2. III. STRUCTURE REDUNDANCY OPTIMIZATION An optimum reconfigurable antenna design yields the smallest number of redundant elements and achieves the functions required with great reliability. Our optimization approach allows the designer to identify the redundant parts in the design. These parts might be antenna topology parts that need to be removed, or electronic components such as switches. This technique removes redundancies from reconfigurable antenna structures to reduce costs and losses. Herein, a part is defined as redundant if its presence gives the antenna more functions than required and its removal does not affect the antenna’s desired performance. The removal of a part from the antenna structure may require a change in the dimensions of the remaining parts in order to preserve the antenna’s original characteristics. That is, a redundant part can be removed as long as its removal will not affect the polarization status of the antenna in a reconfigurable return loss and reconfigurable polarization antenna. For Multi-Part Switch-Reconfigured Antennas: The minimum number of edges present in any graph model according to rule 1 is equal to (N-1) and the maximum number is equal to N(N-1)/2. N represents the number of vertices in the graph model. In the case of a multi-part antenna and according to rule 1, N represents the different parts of the antenna. Eq. (3) shows the bounds of the number of edges NE in a graph model of this category

(3) where represents the maximum number of edges in a graph modeling a multi-part antenna. The number of unique paths or else idle ver(NUP) in such a graph model is always tices are present; Keeping in mind that a path is a continuous sequence of edges that are traveled in the same direction from is suffian originating vertex to a destination vertex. Then cient to be considered the necessary number of unique paths required to minimize redundancies. By decreasing the number of unique paths, the number of possible configurations is reduced, which results in reducing the number of vertices and removing redundant parts. Fig. 4 shows an example of how to identify the unique paths in a graph modeling a multi-part switch-reconfigured antenna. Equation (4.b) shows the necessary number of available configurations (NAC) where the case of no connection is added.

Fig. 4. An example of all possible unique paths in a given graph modeling a multi-part switch-reconfigured antenna.

Equation (4.c) is a direct derivation of (4.a) and (4.b) and represents the number of vertices required to achieve a certain number of configurations. The reconfigurable antenna may have more possible configurations than NAC for a given set of vertices; however, NAC represents the minimum number of antenna configurations that are necessary to achieve the maximum number of functions with the least number of components (4a) (4b) Using (4.a) and (4.b) we get

(4c) For Single-Part Switch-Reconfigured Antennas: In the case of single-part antennas, vertices represent different end-points of switches. The number of vertices N in a graph modeling a single part switch-reconfigured antenna is twice the number of all possible edges. In this case, the number of possible unique paths is equal to the number of possible edges in the graph based on rule 2. Equation (5.a) represents the minimum number of available antenna configurations to achieve an efficient design. It is the number of possible edges in addition to the case where no connection exists. As in (4), the reconfigurable antenna might have more possible configurations than NAC for a given set of vertices. By rearranging (5.a), (5.b) is obtained and represents the number of vertices necessary to achieve a certain number of configurations (5a) (5b)

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Fig. 5. Antenna in [8] and its graph model.

Fig. 6. The optimized structure with its graph model.

IV. OPTIMIZING REDUNDANCY IN MULTI-PART SWITCH-RECONFIGURED ANTENNAS Example IV.1: As an example we take the switch-reconfigured antenna shown in Fig. 5 [8]. This antenna is built out of a hexagonal main patch and six trapezoidal parts placed around it. The graph model of this antenna conforms to rule 1. The anwith tenna is designed on an FR4 epoxy substrate . In addition to its original frequencies of operation, when all the switches are off, this antenna is required to have three more configurations that resonate as follows: Configuration 1: 1 GHz, 3.5 GHz, 4.5 GHz; Configuration 2: 3.5 GHz, 4.5 GHz, 5 GHz; Configuration 3: 1 GHz, 2.5 GHz, 5 GHz; Configuration 4 (All switches OFF): 3 GHz, 3.5 GHz, 4.5 GHz. These frequencies represent practical applications such as WIMAX, WIFI, and GPS. This antenna was designed in [8] to have six switches connecting six sections to a main section. The application of (4.a), (b) to the graph model of this antenna shows that this antenna has at least 22 configurations, while just four configurations are required

The application of (4.c) reveals that we need at most three vertices in the graph model to achieve these four required configurations

To reduce the redundancy and complexity of this system and to minimize the design time and the number of simulations, the number of switches used has to be reduced to two. To preserve the radiation properties the general shape of the antenna as a six-armed hexagon cannot be disturbed, especially when all switches are OFF. The designer optimizes by simulations

Fig. 7. The simulated input reflection (S11) plot for the required configurations.

the placement of the two switches to achieve the required frequencies and configurations. The placements of these switches as well as the graph model of the optimized antenna are shown in Fig. 6. The simulated S11, the input reflection of this antenna, for all required configurations is shown in Fig. 7. By applying this technique, the design time has been reduced and, instead of determining the placement and topology of the antenna with six switches we need to do the work for only two. A comparison of the antenna’s radiation patterns with redundant switches and the one without redundant elements at 4.517 GHz is shown in Fig. 8 for the x-y and y-z plane cuts. Example IV.2: In this example the antenna [9] is a MEMS-reconfigurable pixel antenna that provides two functions: reconfiguration of its modes of radiation and reconfiguration of the operating frequency. The proposed antenna uses a 13 13 matrix of metallic pixels connected through MEMS switches in which circular patches of different radii are mapped. Each metallic pixel has dimensions 1.2 1.2 mm, and the pixels are separated by 2 mm to provide enough space to allocate the MEMS switches and connecting lines. The MEMS switches around each pixel are activated or deactivated depending on the DC voltage that is supplied to the pixels. The DC connectivity

COSTANTINE et al.: REDUCING REDUNDANCIES IN RECONFIGURABLE ANTENNA STRUCTURES USING GRAPH MODELS

Fig. 8. Comparison of the antenna’s simulated radiation patterns with and without redundant elements at 4.517 GHz for the x-y and y-z plane cuts.

is provided through bias lines that connect the pixels to the back side of the substrate. In order to connect two metallic pixels, the voltage difference between them has to be around 30 V. The metallic pixels and the bias lines are connected through RF resistive lines made of Ni-chrome alloy. The substrate used is a quartz substrate that is 2 2 in, 1.575 mm thick and has a dielectric constant of 3.78. This antenna can generate five orthogonal radiation patterns at any frequency between 6 and 7 GHz. , These patterns are those generated by the modes and , all of them with , with , [10]. At any fixed frequency bewith and tween 6 and 7 GHz, five radiation states can be selected. The , simulated flattened 3-D far field pattern for the with , , with and modes are shown in [10] This antenna exhibits frequency tuning as well as pattern/polarization diversity for fixed frequencies. The optimization approach introduced takes into consideration one reconfiguration function at a time, which in this case is the pattern/polarization. It is noted that five configurations are required. To graph model this antenna, the parts constituting its structure are treated as vertices. These vertices are connected by weighted undirected edges. The graph model of the antenna configurations required to achieve the five different modes of operation is shown in Fig. 9 and follows rule 1. Since only five configurations are required, applying (4.c) to this antenna gives us the number of parts required to achieve the desired configurations

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Fig. 9. Graph model of the required antenna configurations.

configurations. To identify the different sections of the antenna necessary to achieve the desired behavior, the adjacency matrix representation of the graph is used assuming the edges are not weighted. A part connected by an edge has a value 1, while a part that is not connected by any edge is represented by 0. The adjacency matrix representation for all possible configurations is shown in Table I. The matrices in Table I can be expressed as shown in (6)

(6a) (6b)

(6c)

(6d) (6e)

Only four configurations are required to achieve five antenna functions. The shape of the antenna with four parts will be very different from the one shown in Fig. 9 and needs to be simulated and investigated extensively. The antenna designer in [9] required a minimization of the number of switches used while keeping the same antenna topology. To preserve the same antenna topology, redundant connections have to be identified and redundant switches eliminated. By comparing the different graph models in Fig. 9, one notices that edges connect only certain vertices and the rest of the vertices remain idle in all five

where are defined in Table II. The matrices in Table II can be translated into graphs representing each case. The corresponding graphs can be translated to antenna sections. These 27 antenna sections are shown in Fig. 10. Inside each section, the square patches are connected constantly, which eliminates the need for switches. Switches will be used only to connect the sections. Parts belonging to the same sections are always connected, and there is no need for switches inside each section. Some antenna parts are never connected, to achieve polarization diversity, and they are shown in black in Fig. 10. Using this technique, the number of switches is reduced by more than 100 from 312 to 166, while preserving the antenna topology. The reduction of the number of switches does not affect the radiation characteristics in [9] because the

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TABLE I ADJACENCY MATRIX REPRESENTATION FOR ALL POSSIBLE ANTENNA CONFIGURATIONS

TABLE II THE MATRICES COMPOSING THE MATRICES OF TABLE I

topology has been preserved. Another example of reducing redundancies from a multi-part switch-reconfigured antenna can be found in [11]; where the antenna was optimally redesigned to maintain the same functionality and radiation characteristics. V. OPTIMIZING REDUNDANCY IN SINGLE-PART SWITCH-RECONFIGURED ANTENNAS Example V.1: In this example a single-part antenna is considered. This antenna [12], shown in Fig. 11 is a multi-band low-cost antenna that employs Koch fractal geometry. The antenna is fabricated on a 1.6 mm-thick FR4-epoxy substrate with dimensions 4 cm 4.5 cm, is microstrip fed, and has a partial ground plane flushed with the feed line. A trapezoidal matching section connects the feed line to a U-Koch-slotted rectangular-

Koch patch. The first-iterated Koch fractal geometry is used in the patch, and the U-slot is inserted to increase the antenna’s electrical length for operation at lower frequency bands. Five pairs of RF MEMS are mounted across the slot, as shown in Fig. 11. In [12] the symmetrically placed switches are activated two at a time to achieve the desired configurations. The first configand uration, (10000), represents the activation of N1 and the deactivation of the other eight switches. This configuration exhibits a narrow resonance at 1.9 GHz as well as a wide band operation between 2.7 and 6.6 GHz. The second configuration, (01000), represents the activation of N2 and while deactivating the other switches. This configuration achieves a resonance at 2.1 GHz and a wide band operation. from 2.4 to 6.7 GHz. The third configuration, (11111), represents the activation

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Fig. 12. Antenna with optimized number of switches and their positions, with the corresponding graph model.

Fig. 10. The 27 antenna sections: red line indicating the presence of a switch and black line indicating the presence of a permanent connection (no need for a switch).

Fig. 13. Comparison of the input reflection of the original and the antenna with the optimized number of switches for case 1.

Fig. 11. The antenna topology in [12].

of all ten switches. This configuration achieves a wide band operation from 2.5 to 6.7 GHz. The graph model of the original structure is the same as the one shown in Fig. 3(b). Investigating redundancies in this antenna structure, we can preserve the distinctive topology while removing redundant switches. Switches in this case have to be studied two at a time to preserve the antenna’s operating modes. In (5.a), N represents the end points of one side of the slots, in this case. Applying (5.a) to this antenna reveals so that the minimum number of antenna configurations that can be achieved with five pairs of switches is six

Fig. 14. The fabricated optimized prototype.

Since just three configurations are required, applying (5.b) reveals that only two switches are needed.

The antenna topology with two switch positions is shown in Fig. 12 with the corresponding graph model. A comparison between the input reflection parameters of the original antenna and the antenna with the optimized number of switches is shown in Fig. 13 for the first required configurations. The fabricated prototype is shown in Fig. 14. A comparison of the analogies be-

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Fig. 15. Comparison of simulated and measured input reflection for the (10) optimized case.

tween the measured and simulated S11 for the (10) optimized case is shown in Fig. 15.

[4] E. Klavins, “Programmable self assembly,” IEEE Control Syst. Mag., vol. 27, no. 4, pp. 43–56, Aug. 2007. [5] J. Costantine, C. G. Christodoulou, and S. E. Barbin, “Mapping reconfigurable antennas using graphs,” in Proc. NASA/ESA Conf. on Adaptive Hardware and Systems, Jun. 2008, pp. 133–140. [6] A. Patnaik, D. E. Anagnostou, C. G. Christodoulou, and J. C. Lyke, “Neurocomputational analysis of a multiband reconfigurable planar antenna,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3453–3458, Nov. 2005. [7] L. M. Feldner, C. D. Nordquist, and C. G. Christodoulou, “RFMEMS reconfigurable triangular patch antenna,” in Proc. IEEE AP/URSI Int. Symp., Jul. 2005, vol. 2A, pp. 388–391. [8] J. Costantine, C. G. Christodoulou, and S. E. Barbin, “A new reconfigurable multi band patch antenna,” in Proc. IEEE IMOC Conf., Salvador, Brazil, Oct. 2007, pp. 75–78. [9] A. Grau, L. Ming-Jer, J. Romeu, H. Jafarkhani, L. Jofre, and F. De Flaviis, “A multifunctional MEMS-reconfigurable pixel antenna for narrowband MIMO communications,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2007, pp. 489–492. [10] R. G. Vaughan, “Two-port higher mode circular microstrip antennas,” IEEE Trans. Antennas Propag., vol. 36, no. 3, pp. 309–321, Mar. 1988. [11] J. Costantine, C. G. Christodoulou, C. T. Abdallah, and S. E. Barbin, “Optimization and complexity reduction of switch-reconfigured antennas,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1072–1075, 2009. [12] A. Ramadan, K. Y. Kabalan, A. El-Hajj, S. Khoury, and M. Al-Husseini, “A reconfigurable U-Koch microstrip antenna for wireless applications,” Progr. Electromagn. Res., vol. PIER 93, pp. 355–367, 2009.

VI. CONCLUSION In this paper, graphs are introduced as a modeling tool for reconfigurable antennas. Guidelines for graph modeling different types (multi-part and single part) of switch reconfigurable antennas are presented. We also present a new methodology for removing redundancies from antenna structures. This approach is based on comparing the number of unique paths in a given graph with the number of the required antenna configurations. Equations are introduced for different types of switch reconfigurable antennas. In some cases redundant elements are simply removed from the antenna structures and in different cases a complete redesigning of the antenna is required. This optimization approach is an efficient tool for reducing costs and losses in reconfigurable antenna structures. Examples are given on multi and single-part switch reconfigured antennas, validating the discussed approach with simulations and measurement. Furthermore this easy approach does not replace the simulation in a design process however it reduces the number of iterations needed. In addition, this paper proposes a complexity reduction approach while maintaining the desired multi-functional properties of a reconfigurable antenna, and facilitates the control of such antennas by using corresponding graph algorithms.

REFERENCES [1] J. T. Bernhard, Reconfigurable Antennas. San Rafael, CA: Morgan and Claypool, 2007. [2] C. A. Balanis, Modern Antenna Handbook. Hoboken, NJ: Wiley, 2008. [3] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 2nd ed. Cambridge, MA: MIT press, 2001.

Joseph Costantine (M’10) received the B.E. degree in electrical, electronics, computer, and communications engineering from the Lebanese University, in 2004, the M.E. degree in computer and communications engineering from the American University of Beirut, Beirut, Lebanon, in 2006, and the Ph.D. degree in electrical and computer engineering from the University of New Mexico, Albuquerque, in 2009. In July 2010, he completed his Postdoctoral Fellowship in the Computer Engineering Department, University of New Mexico. Currently he is an Assistant Professor at the Electrical Engineering Department, California State University, Fullerton. He has also published many research papers and is a co-author of an upcoming book on reconfigurable antennas. His research interests are in the areas of reconfigurable systems and antennas, antennas in wireless communications, electromagnetic fields, RF Electronic Design and communication systems. Dr. Costantine was awarded a 6-month research scholarship at Munich University of Technology (TUM) in 2006 as part of the TEMPUS program. He received many awards during his studies and career.

Sinan al-Saffar (M’10) received the M.S. degree in computer science from Arizona State University, in 1998, where his research focused on parallel and high performance computing, and the Ph.D. degree in computer engineering from the University of New Mexico, Albuquerque, in 2009. His Ph.D. work focused on applying semantic graphs and computation to subjective information valuation. Previously, he spent a few years in Silicon Valley where his industry experience matured around parallel and high performance database systems, before returning to academia. Currently, he is a Senior Computer Scientist at the Pacific Northwest National Laboratory, Seattle, WA. His present research at PNNL combines data-intensive computing and semantic graph mining for automated knowledge discovery. Dr. al-Saffar is an active member in the ACM and AAAS.

COSTANTINE et al.: REDUCING REDUNDANCIES IN RECONFIGURABLE ANTENNA STRUCTURES USING GRAPH MODELS

Christos G. Christodoulou (F’08) received the Ph.D. degree in electrical engineering from North Carolina State University, Raleigh, in 1985. He served as a faculty member at the University of Central Florida, Orlando, from 1985 to 1998. In 1999, he joined the faculty of the Electrical and Computer Engineering Department, University of New Mexico, Albuquerque, where he served as the Chair of the department from 1999 to 2005. His research interests are in the areas of modeling of electromagnetic systems, FPGA reconfigurable systems, and smart RF/photonics. He has published over 350 papers in journals and conferences, has 12 book chapters and has coauthored four books. Dr. Christodoulou is a Fellow of the IEEE and a member of Commission B of USNC/URSI. He is the winner of the John Kraus 2010 antenna award. He served as the General Chair of the IEEE Antennas and Propagation Society/URSI 1999 Symposium, Orlando, FL, as the Co-Chair of the IEEE 2000 Symposium on Antennas and Propagation for Wireless Communications, Waltham, MA, and the Co-Technical Chair for the IEEE Antennas and Propagation Society/URSI 2006 Symposium, Albuquerque. Currently, he is an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE Antennas and Propagation Magazine. He was appointed as an IEEE AP-S Distinguished Lecturer (2007–now) and elected as the President for the Albuquerque IEEE Section. He served as an Associate Editor for the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION for six years, was a Guest Editor for the Applied Computational Electromagnetics Society (ACES) Journal Special Issue on “Applications of Neural Networks in Electromagnetics,” and was the Co-Editor of the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION Special issue on “Synthesis and Optimization Techniques in Electromagnetics and Antenna System Design” (March 2007).

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Chaouki T. Abdallah (SM’95) received the M.S. and Ph.D. degrees in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1982 and 1988, respectively. He joined the Electrical and Computer Engineering (ECE) Department, University of New Mexico, Albuquerque, where he is currently Professor and Department Chair. He was a Visiting Professor at the Universita Degli Studi di Roma, Tor Vergata, Rome, in 2005. He has published seven books (three as co-editor and four as coauthor) and more than 250 peer-reviewed papers. He conducts research and teaches courses in the general area of systems theory with focus on control, communications, and computing systems. His research has been funded by NSF, AFOSR, NRL, national laboratories, and by various companies. Prof. Abdallah was the first recipient of ECE Department’s Lawton Ellis Award for combined excellence in teaching, research, and student/community involvement. He also received the School of Engineering’s senior research excellence award in 2004, and was the ECE Gardner Zemke Professor between 2002 and 2005. He served as Director of ECE’s graduate program from 1999 through 2005. He has also been active in designing and implementing various international graduate programs with Latin American and European countries. In 1990, he was a co-founder of the ISTEC Consortium, which currently includes more than 150 universities in the US, Spain, and Latin America. He served as the General Chair of the 2008 CDC, which was held in Cancun, Mexico. He is a senior member of IEEE and a recipient of the IEEE Millennium medal.

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Performance Assessment of Bundled Carbon Nanotube for Antenna Applications at Terahertz Frequencies and Higher Sangjo Choi, Student Member, IEEE, and Kamal Sarabandi, Fellow, IEEE

Abstract—The performance of bundled carbon nanotubes (BCNTs) as a conducting material for the fabrication of antennas in the terahertz frequency range and above is evaluated. The performance is compared against gold film, which is usually used for antenna fabrication. The macroscopic behavior of BCNTs is modeled by an anisotropic resistive sheet model which is extracted from the discrete circuit model of a single wall carbon nanotube (SWNT). Numerical simulations using the method of moments (MoM) and the mixed potential integral equation (MPIE) are performed to quantify radiation efficiencies of resonant strip antennas composed of BCNTs and thin gold films. For accurate high frequency simulations of antennas constructed from a thin gold layer, the Drude-Smith model is used to calculate the conductivity of gold. Simulations are carried out from 1 THz to 50 THz for conventional half-wave strip antennas. It is shown that the radiation efficiency of a BCNT antenna is consistently lower than the efficiency of a gold film antenna for BCNT equivalent density values up to 104 [CNTs m]. However, if equivalent density values above 104 [CNTs m] could ever be achieved, which are approximately 103 times higher than the currently realizable density (10 [CNTs m]), BCNTs would outperform thin gold film at frequencies above 1 THz. Index Terms—Carbon nanotubes, methods of moments (MoM), radiation efficiency, thin gold film.

I. INTRODUCTION ARBON nanotubes (CNTs) have certain unique electrical properties which are used for many applications in the field of nanotechnology, electronics, and optics. A single wall carbon nanotube (SWNT) has a cylindrical geometry with an extremely high aspect ratio. Although a SWNT has a rather high conductivity, its resistance per unit length is fairly high because of its very small radius (on the order of a few nanometers) [1]. These properties limit its direct utilization for RF applications, such as electrical interconnects and radiating elements. To circumvent this problem, bundled CNTs (BCNTs) are proposed to reduce the high intrinsic resistance of a SWNT [2], [3]. Some experimental work in analyzing the electrical properties of BCNTs has already been reported. Laboratory measurements up to 50 GHz have shown a relatively large kinetic in-

C

Manuscript received February 26, 2010; revised July 01, 2010; accepted September 09, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor, Ann Arbor, MI 48105 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2103023

ductance for the SWNT, which is scalable with the number of SWNTs within the bundle [2]. In the bundle structure, the scalability of a SWNT’s resistance, which is predicted by the Luttinger liquid theory is verified in a measurement setup where a BCNT was fabricated to behave as a 50 ohm line up to 20 GHz [3]. Results of these experiments suggest that the equivalent circuit model of BCNTs, composed of N parallel SWNTs, can simply be obtained from the equivalent circuit of an individual SWNT whose components’ value are divided by N. Theoretical properties of SWNT as an antenna have also been reported. In [4], [5], the theoretical analysis of a SWNT as a dipole antenna predicts a slow propagation velocity within the tube, high input impedance, and very low radiation efficiency. Also in [6], the radiation efficiency of a dipole antenna with a nano-meter radius was numerically simulated. However, no experimental measurement of the radiation efficiency has yet been reported. From previous work, it is shown that dipole antennas with radii smaller than hundreds of nano-meter are not effective radiators in the microwave regime; hence, SWNT or nanowire is suggested for very high frequencies with nano-scale wavelength. In this paper, the performance of BCNTs as a material for fabrication of antennas at terahertz frequencies and above is investigated using a numerical technique developed for this purpose. The discrete equivalent circuit of a SWNT, proposed in [2], is used for electromagnetic modeling of BCNTs. Hence a series ) circuit model composed of the kinetic inductance (16 ) is utilized to represent a strand and the resistance (6.5 and then, parallel circuits of SWNTs are employed to obtain a macroscopic boundary condition for a planar BCNT structure. The premise for re-examining SWNTs as a material for antennas stems from the fact that the quality factor (Q) of the series resistor-inductor (RL) model of an individual SWNT is approxi. Such a high mately 100 at terahertz frequencies Q factor of a SWNT at terahertz frequencies is preserved in the bundle structure due to the parallel connection of SWNTs. Also, the one dimensional electron transport in SWNT indicates that the concept of skin depth for this conductor is not applicable. Although the thickness of a single or multiple layers of a planar BCNT strip can be extremely small compared to the wavelength, the material still shows relatively a high conductivity at very high frequencies. On the other hand, the surface resistivity of conventional conductors is a function of skin depth and monotonically increases as frequency increases. Hence, at a certain high frequency in the terahertz regime, the radiation efficiency of BCNT antennas is expected to surpass that of metallic

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antennas. To account for the high frequency properties of metals (thin gold film in this paper), the Drude-Smith model is used. The paper is organized in two parts; first, an anisotropic resistive sheet model to represent BCNTs electromagnetically is presented. Then, a numerical model based on the method of moments (MoM) for electromagnetic simulation of antennas which consist of BCNTs and gold film is developed. The radiation efficiencies of both types of antennas are computed and compared as a function of frequency and equivalent density of BCNTs, the number of SWNTs per unit width. Finally, a cross over frequency point at which the BCNT antenna outperforms its metallic counterpart is identified. II. CARBON NANOTUBE A salient feature of CNT molecules is their atomic arrangement. Carbon nanotubes consist of graphene sheets that are rolled across an axis vector to form a cylindrical geometry. The axis where the carbon hexagonal configuration of graphene sheet is rolled determines the conducting property of the CNT [8]. Among the semiconducting and metallic CNTs, the metallic [9] which allows variety is highly conductive for its use in the design of antenna radiators and interconnects. Also, CNTs can be categorized as either single wall nanotubes (SWNTs) or multi wall nanotubes (MWNTs), based on the number of layers of graphene sheets that is rolled up to form the structure. The metallic SWNT and its bundle structure are considered herein. As a fundamental element of bundled CNTs (BCNTs), the SWNT is synthesized by the chemical vapor deposition (CVD). In recent years, significant attention has been given to developing very long strands of CNTs. The synthesis methods have been developed to fabricate centimeter-long carbon nanotubes [10]. Dielectrophoresis alignment of SWNT has been used to create the bundle structure [2]. Also, parallel arrays of SWNTs were densely grown along the patterned catalyst lines by using photolithography or polydimethylsilo-xane (PDMS) on quartz wafers [11]. Both methods show a BCNT density of approx. In a BCNT, although the electrons in imately 10 each SWNT predominantly pass along the tube axis, there is also some movement in the transverse direction, which indicates that electrons can jump to adjacent SWNTs. Even though this microscopic electron transport occurs randomly, the reported experiments have shown that the overall current is axial and that the inductance of the BCNT is equal to the kinetic inductance of a SWNT divided by the number of SWNTs in the bundle [2]. This fact justifies the macroscopic equivalent circuit model of BCNTs, which is simply represented by the circuit elements of parallel connected SWNTs in the bundle structure. III. RESISTIVE SHEET MODEL OF BUNDLED CARBON NANOTUBE To examine the electromagnetic properties of BCNTs arranged in a planar fashion, a macroscopic model is needed. Since the thickness of a layer of BCNTs is much smaller than the wavelength up to optical frequencies, such material can be modeled by a resistive sheet [12]. The continuity of a tangential electric field is maintained across such sheets which can support induced electric current flowing over them. If is

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Fig. 1. Geometry of a strip antenna made up of bundled carbon nanotubes.

the unit vector normal to the resistive sheet which is drawn to the upward (positive) side and denotes the discontinuity across the sheet, the boundary conditions are [13]:

(1) where

and is the complex surface resistivity. Assuming that the scalability property of BCNTs is valid, a macroscopic surface resistivity for planar BCNT can be obtained. However, as SWNTs can predominantly support axial currents, the surface resistivity can no longer be assumed to be a scalar. A. Resistivity of Bundled Carbon Nanotubes Fig. 1 illustrates the proposed geometry of a strip dipole antenna constructed from strands of SWNTs. These strands form an extremely thin conducting layer (approximately 10 nm [1]) which is capable of supporting the electric current along the antenna axis. In the bundle, it is assumed that each single nanotube, represented by the equivalent circuit model of the SWNT, is all connected in parallel. Also, it is assumed that the electrons move only along the axis of SWNT, so that only the electric field component parallel to the strands can excite the surface currents. Following the boundary condition (1) and referring to Fig. 1, the tangential electric field is related to the surface current by: (2) is the surface resistivity along the SWNT axis. Conwhere sidering a small rectangular segment of the surface, the voltage across the length of the rectangle can be obtained from (3) Also, the current flowing through the width of the rectangle is given by (4)

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The equivalent impedance of the rectangular segment can be and the kinetic inductance obtained from the resistance per the unit length of SWNT as well as the number density of SWNTs in the bundle and is given by

(5)

an anisotropic resistive sheet is developed. Using the magnetic vector potential and the electric scalar potential (9) where

where

(10) (11)

Using (2)–(5), it can easily be shown that BCNT resistivity can be computed from

(6) As shown, the surface impedance (complex resistivity) of the BCNTs can be modified by its number density. As will be discussed later, matching the antenna to transmission lines of reasonable characteristic impedance values require a large number of SWNTs in parallel. As mentioned before, the current can flow only in one direction and as a result, the equivalent resistivity of the BCNT is not isotropic. In this case, (1) can be replaced with

(7) . This equation only models the BCNT sheet. where Usually, BCNTs are fabricated on a substrate. To reduce or eliminate dielectric loss associated with the substrate, a thin membrane can be fabricated over silicon or quartz substrates over which the BCNT antenna can be deposited. Then, the substrate under the antenna can be removed [14]. In this case, the antenna . This will be supported by a thin membrane thin dielectric layer can also be modeled by an isotropic resistive sheet whose resistivity is given by [13]

and are, respectively, the dyadic and scalar and Green’s function of the free space. After discretizing the surface of the resistive sheet into triangular facets and expanding is expressed as a summation the unknown surface current, of Rao-Wilton-Glisson (RWG) basis functions [13]. Then by testing (7) with the same basis function, the surface current deninto (7) sity can be calculated. By plugging and using (9)–(11), the following matrix equation is formed: (12) . In the strip dipole where is the excitation obtained from antenna geometry shown in Fig. 1, the impinging voltage along the gap between two BCNT sheets defines the incident field which is formulated using a delta-gap model. In the model, constant incident field only exists over the triangular facets of the rectangular-shaped feed which connects the two BCNT sheets [16]. IV. CONDUCTIVITY OF THIN GOLD FILM In the terahertz and optical regime, the Drude model provides the conductivity of a good conductor (copper or gold) as a function of frequency. Such frequency dependence of the optical conductivity is due to the fact that free electrons moving near atoms in a good conductor are not subject to the Lorentz restoring force of the atoms, in the presence of the time-varying electric field. A. Drude-Smith Model

(8) where , , , and are, respectively, the thickness, the relative dielectric constant of the membrane, the characteristic impedance, and the propagation constant of surrounding free space. This surface resistivity is capacitive and is placed in parallel with the resistivity of the BCNT surface. However, is usually a large quantity, which can be ignored, since the membrane thickness is very low. B. Method of Moments (MoM) Formulation In order to compute the characteristics of the proposed strip dipole antenna constructed from the BCNTs, MoM formulation based on the mixed potential integral equation (MPIE) for

The “Drude-Smith model”, optimized specifically for thin gold film, provides its terahertz conductivity and has been verified in an experimental setup [17]. A formulation of the complex conductivity of thin gold film at terahertz frequencies is given by

(13) where is the plasma frequency, is the scattering time, and is the persistence of velocity. The plasma frequency of thin gold film is 2,080 THz and its scattering time is 18 fs [17]. Also, represents the ratio of electron initial velocity to its velocity after the first collision to nearby atoms. However, the velocity persistence is only effective in the case of a film thinner

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Fig. 3. Strip dipole antenna geometry fed at center using a thin voltage gap.

Fig. 2. Complex conductivity and complex permittivity of thin gold film as a function of frequency.

than 20 nm. For antenna applications, the proposed thickness of a thin gold film can be chosen to be much higher than the transition thickness of 20 nm. Henceforth is set to zero in the Drude formula for calculating conductivity of thin gold film. The general complex conductivity given by

gold at terahertz frequencies corresponds to an inductive reactance as opposed to the constant permittivity of free-space assumed for good conductors at low frequencies. Equation (16) is valid when the thickness of metal is infinite and it provides a good approximation when the thickness of thin gold film used for the strip antenna is set to twice the skin depth. The skin depth . is simply the inverse of the attenuation constant Similar to the resistive sheet model, the boundary condition of thin gold film is given by

(17) where

is an equivalent surface current on thin gold film. V. ANTENNA SIMULATION

(14) Equation (14) can easily be converted to complex permittivity which is computed from

(15) The real conductivity decreases in a Lorentzian line-shape as frequency increases. The imaginary conductivity has a minimum near 10 THz and its magnitude surpasses the magnitude of the real conductivity at higher frequencies. As shown in Fig. 2, the Drude model predicts lower conductivity and higher permittivity for thin gold films when compared to bulk gold at very low ). frequencies (bulk gold conductivity at DC is B. Surface Resistivity of Thin Gold Film Since the penetration depth of thin gold film at terahertz frequencies is still much smaller than the thickness of the , the film practically used for antenna fabrication boundary condition can be expressed in terms of equivalent surface impedance given by

(16) where and are the attenuation and propagation constants in is the complex conductivity given the gold medium, while by (14). The approximation is valid when . According to the Drude model, the negative real permittivity of

To evaluate the performance of an antenna composed of BCNTs or thin gold film, the length (L) of the strip dipole antenna in Fig. 1 is varied to obtain its fundamental resonance over the frequency range of 1–50 THz. The antennas are excited by a voltage gap at the center as shown in Fig. 3. The width is set to (w) is chosen to be L/6 and the antenna feed gap . A. Strip Antenna The input impedance and the reflection coefficient, using 50 ohm port impedance, are then studied to evaluate the performance of BCNT and gold antennas operating at the same frequencies. The resonant length of BCNT antenna (L) turned out to always be less than half of the wavelength at operating frequency due to the kinetic inductance of its equivalent circuit. Two strip BCNT antennas with density values of 10 and 50 are considered. The antenna length was chosen to be 150 corresponding to an ideal dipole antenna operating at 1 THz. Such number density values are realizable in current nanofabrication facilities; however, as shown in Fig. 4, the input range) and impedance of these antennas are rather high (in cannot be matched to any realizable transmission line. It is also shown that the inductive reactance is quite large. This simulation clearly indicates that such low densities of BCNTs cannot be used as a viable antenna due to very low mismatch efficiency. Higher BCNT number densities, however, lowers the input impedance of the antenna. Fig. 5 indicates the input reflection antenna for different BCNT density coefficient of a 150 values using a 50 port (transmission line). The first resonance is approximately 0.25 THz corresponding to BCNT density of . It is also shown that a good impedance match

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m strip antenna with BCNT density of [CNTs=m].

Fig. 4. Input impedance of 150 = , (b) N:50 (a) N:10

[CNTs m]

still cannot be achieved without an external matching network due to high surface resistance and inductance of the equivalent anisotropic resistive sheet. The miniaturization effect is also notable. Basically, the antenna is resonant at almost 1/4 of a perfect conductor antenna of the same size. This is accomplished due to the fact that the inductance of the equivalent resistive sheet has compensated the capacitive reactance of a short dipole anstrip tenna. Fig. 5 also illustrates the performance of a 150 antenna for higher BCNT density values. As shown, at BCNT , the antenna can easily be density values above matched (without external matching elements). In these cases, the surface resistance and inductance are much reduced while maintaining the quality factor. Different miniaturization factors with a good impedance match can be accomplished in a high . It is important BCNT density range also to point out that most SWNTs are assumed to have diameters of approximately 1 nm in the bundle [11] and thus it is imin a possible to exceed density values beyond single sheet. The BCNT density values mentioned here can be regarded as the equivalent density values of BCNTs assuming SWNTs of almost zero diameter spread on a single sheet. To demonstrate the miniaturization effect of BCNTs as a function of frequency, a large number of BCNT antennas are simulated. For a given BCNT density, strip dipole antennas whose lengths are equal to half of wave lengths from 1 THz to 50 THz were designed and their first resonant frequencies were determined. The resonant frequency corresponds to a frequency at which the antenna input reactance is zero (not necessarily matched to the port impedance). The miniaturization factor versus resonant frequency is shown in Fig. 6 defined by for different BCNT density values. Also, the same quantity for the thin gold antenna using the conductivity of bulk gold at low frequencies and the Drude model for gold is shown. As the to , density of BCNT increases from the normalized antenna length approaches unity. This is due to the fact that the density of BCNT is inversely proportional

Fig. 5. Reflection coefficient of 150 = . densities

[CNTs m]

Fig. 6. Normalized antenna length

m strip antenna with different BCNT

(2L=) versus resonant frequencies.

to the inductance of the bundle, so that the lower density causes a lower resonant frequency and identically a higher resonance wavelength. The BCNT antenna with the density has a surface resistivity which is very of similar to thin gold film in low terahertz frequencies, whereas it still has a certain finite reactance causing further antenna miniaturization. As the thin gold film becomes more inductive at higher frequencies, the normalized length of thin gold film antenna becomes smaller. Basically, at frequencies higher than 5 THz, the normalized length is clearly lower than unity, since the inductive reactance of the surface impedance of thin gold film, as predicted by the Drude model, is dominant. Also, to effectively demonstrate the relationship between the miniaturization effect and the number density of BCNT antenna, the normalized antenna length as a function of BCNT number density is shown in Fig. 7. Each line indicates the normalized length at a specific resonant frequency of BCNT antennas from 5 THz to 25 THz. It can be seen from Fig. 7 that both lower number densities and smaller

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Fig. 7. Normalized antenna length (2L=) versus density of BCNT.

Fig. 8. Radiation resistance of BCNT (N is higher than 5 1 10 [CNTs=m]) and thin gold film at its resonant frequencies.

antenna lengths result in shorter normalized antenna lengths, hence greater overall antenna miniaturization. Fig. 8 illustrates the radiation resistance of the same strip dipole antennas used in Fig. 7 as a function of frequency and BCNT number density. The radiation resistance of dipole antennas decreases as the antenna length decreases. This expected behavior is observable in Fig. 8 for BCNT and thin gold antennas simulated in Fig. 7. It is noteworthy that BCNT antennas to have with densities in the range of input radiation resistance from 10 to 80 . That can easily be matched to conventional 50 port impedances. B. Radiation Efficiency One of the most significant features of an antenna is its radiation efficiency. As mentioned earlier, due to increasing Q factor of BCNTs with frequency, the radiation efficiency of antennas made with BCNTs is expected to be better than that of metallic antennas at a certain high frequency. In this section, the radiation efficiency of strip dipole antennas made up of BCNTs with

Fig. 9. Antenna efficiency of strip antenna of BCNTs and thin gold film (a) BCNT densities of 10 to 5 1 10 [CNTs=m], (b) a magnified plot of efficiencies for BCNT densities of 10 to 5 1 10 [CNTs=m] and a comparison to efficiencies of thin gold antenna.

different densities is calculated at their first resonant frequencies. These values are then, compared to those of thin gold film antennas at their corresponding first resonant frequencies. The far-field radiation power is calculated on a sphere whose radius is . Fig. 9 compares the antenna efficiency of BCNT and thin gold film antennas. In these calculations, the antenna efficiencies only account for power loss on the antenna material and ignore the mismatch loss. The MoM simulation indicates that the radiation efficiencies obtained by the calculation of the power loss and the far-field radiation power are consistent. The antenna efficiencies reported in Fig. 10 shows the radiation efficiency including the mismatch (to 50 line) power loss. Fig. 9(a) indicates that BCNT antennas with number density values equal or lower than do not demonstrate high radiation efficiency. Since the imaginary component of conductivity is greater than its real component at high terahertz frequencies, the thin gold film antenna whose conductivity is calculated using the Drude model shows a more

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must be considered. Fig. 10(b) shows that when the number density (N) is higher than , then the radiation efficiency of BCNT antennas outperforms thin gold film up to a certain frequency. For example, with number densities of , , and , the radiation efficiency of BCNT antennas outperforms thin gold film at frequencies up to approximately 10 THz, 16 THz, and 30 THz, respectively. This behavior can be explained by the large mismatch due to the small input resistance of BCNT antennas, which causes their radiation efficiency to be lower than that of thin gold film antennas. VI. CONCLUSIONS Utilizing the scalability of the equivalent circuit of SWNTs in its bundle structure, BCNTs are modeled electromagnetically by an anisotropic resistive sheet. The numerical simulation based on the MoM in conjunction with the resistive sheet model is used to assess the performance of the antennas. The results indicate that a BCNT antenna with a density of SWNTs less has a very poor radiation efficiency. than The fact that the equivalent number density of BCNT antenna to achieve acceptable should be greater than values of radiation efficiency suggests the necessity of fabricating much more densely aligned SWNTs. Thus, to be utilized as an effective antenna radiator in the terahertz frequency range, the density of BCNTs should be approximately three order of magnitude higher than the what can be realized today (10 ). REFERENCES

Fig. 10. Radiation efficiency (including impedance mismatch power loss to a 50 line) of strip dipole antenna of BCNTs (N : CNTs=m) and thin gold film (a) BCNT densities of 10 to 5 1 10 [CNTs=m], (b) a magnified plot of BCNT densities of 10 to 5 1 10 [CNTs=m] and a comparison to efficiencies of thin gold antenna.

gradual decrease in the efficiency than that calculated using the DC conductivity. This results in a lower surface resistance than that of DC gold. When the number density is higher than , antenna efficiencies of BCNT antennas are comparable to those of thin gold film antennas modeled by the Drude model. The number density of BCNT antenna should be approximately to surpass a thin gold film antenna. However, it is important to mention that thin gold film provides very high radiation efficiency up to tens of terahertz. It suggests that the increased efficiency of BCNTs with very high densities may not be worth the complicated fabrication processes needed to make BCNT antennas. Matching networks are usually lossy and may be complicated to fabricate at high terahertz frequencies. To evaluate the performance of antennas without an external matching network, the impedance mismatch in the calculation of antenna efficiency

[1] S. Salahuddin, M. Lundstrom, and S. Datta, “Transport effects on signal propagation in quantum wires,” IEEE Trans. Electron Devices, vol. 52, no. 8, pp. 1734–1742, Aug. 2005. [2] J. J. Plombon, K. P. O’Brien, F. Gstrein, V. M. Dubin, and Y. Jiao, “High-frequency electrical properties of individual and bundled carbon nanotubes,” Appl. Phys. Lett., vol. 90, pp. 063106-1–063106-3, Feb. 2007. [3] C. Rutherglan, D. Jain, and P. J. Burke, “RF resistance and inductance of massively parallel single walled carbon nanotubes: Direct, broadband measurements and near perfect 50 impedance matching,” Appl. Phys. Lett., vol. 93, p. 083119-083119-3, 2008. [4] P. J. Burke, S. Li, and Z. Yu, “Quantitative theory of nanowire and nanotube antenna performance,” IEEE Trans. Nanotechnol., vol. 5, pp. 314–334, Jul. 2006. [5] G. W. Hanson, “Fundamental transmitting properties of carbon nanotube antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 3426–3435, Nov. 2005. [6] G. W. Hanson, “Radiation efficiency of nanoradius dipole antennas in the microwave and far-infrared regime,” IEEE Antennas Propag. Mag., vol. 50, pp. 66–77, 2008. [7] P. J. Burke, “An RF circuit model for carbon nanotubes,” IEEE Trans. Nanotechnol., vol. 2, pp. 55–58, Mar. 2003. [8] X. Blase, J. Charlier, and S. Roche, “Electronic and transport properties of nanotubes,” Rev. Modern Phys., vol. 79, no. 2, pp. 677–732, 2007. [9] E. L. Wolf, Nanophysics and Nanotechnology. Weinhein, Germany: WIEY-VCH, 2004. [10] C. Rutherglan and P. J. Burke, “Nanoelectromagnetics: Circuit and electromagnetic properties of carbon nanotubes,” Small, vol. 5, pp. 884–906, 2009. [11] W. Zhou, C. Rutherglan, and P. J. Burke, “Wafer scale synthesis of dense aligned arrays of single-walled carbon nanotubes,” Nano. Res, vol. 1, pp. 158–165, 2008. [12] T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics. London, U.K.: IEE Press, 1995.

CHOI AND SARABANDI: PERFORMANCE ASSESSMENT OF BUNDLED CARBON NANOTUBE FOR ANTENNA APPLICATIONS

[13] T. B. A. Senior, K. Sarabandi, and F. T. Ulaby, “Measuring and modeling the backscattering cross section of a leaf,” Radio Sci., vol. 22, pp. 1109–1116, Nov. 1987. [14] M. Moallem, J. East, and K. Sarabandi, “Optimally designed membrane-supported grounded CPW structure for submillimeter-wave applications,” presented at the IEEE Antennas and Propag. Soc. Int. Symp., 2009. [15] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, pp. 409–418, May 1982. [16] D. H. Liao, “Physics-Based near-earth radiowave propagation modeling and simulation,” Ph.D. dissertation, Dept. Elect. Eng. and Com. Sci., Univ. Michigan, Ann Arbor, 2009. [17] M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hegmann, “Terahertz conductive of thin gold films at the metalinsulator percolation transition,” Phys. Rev., vol. 76, p. 125408, 2007.

Sangjo Choi (S’09) received the B.S. degrees in electrical engineering from The University of Texas at Dallas, in 2008 and the M.S. degree in electrical engineering from The University of Michigan at Ann Arbor, in 2010, where he is working toward the Ph.D. degree. He is currently a Graduate Research Assistant with the Radiation Laboratory, the University of Michigan.

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Kamal Sarabandi (S’87–M’90–SM’92–F’00) received the B.S. degree in electrical engineering from the Sharif University of Technology, Tehran, Iran, in 1980, the M.S. degree in electrical engineering in 1986, and both the M.S. degree in mathematics and the Ph.D. degree in electrical engineering from the University of Michigan at Ann Arbor, in 1989. He is currently the Director of the Radiation Laboratory and the Rufus S. Teesdale Professor of Engineering in the Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor. His research areas of interest include microwave and millimeterwave radar remote sensing, Metamaterials, electromagnetic wave propagation, and antenna miniaturization. He possesses 25 years of experience with wave propagation in random media, communication channel modeling, microwave sensors, and radar systems and leads a large research group including two research scientists, 14 Ph.D. students. He has graduated 35 Ph.D. and supervised numerous post-doctoral students. He has served as the Principal Investigator on many projects sponsored by the National Aeronautics and Space Administration (NASA), Jet Propulsion Laboratory (JPL), Army Research Office (ARO), Office of Naval Research (ONR), Army Research Laboratory (ARL), National Science Foundation (NSF), Defense Advanced Research Projects Agency (DARPA), and a large number of industries. Currently he is leading the Center for Microelectronics and Sensors funded by the Army Research Laboratory under the Micro-Autonomous Systems and Technology (MAST) Collaborative Technology Alliance (CTA) program. He has published many book chapters and more than 180 papers in refereed journals on miniaturized and onchip antennas, metamaterials, electromagnetic scattering, wireless channel modeling, random media modeling, microwave measurement techniques, radar calibration, inverse scattering problems, and microwave sensors. He has also had more than 420 papers and invited presentations in many national and international conferences and symposia on similar subjects. Dr. Sarabandi served as a member of NASA Advisory Council appointed by the NASA Administrator in two consecutive terms from 2006–2010. He is serving as a vice president of the IEEE Geoscience and Remote Sensing Society (GRSS) and is a member of the Editorial Board of the PROCEEDINGS OF THE IEEE. He was an associate editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE SENSORS JOURNAL. He is a member of Commissions F and D of URSI and is listed in American Men and Women of Science, Who’s Who in America, and Who’s Who in Science and Engineering. Dr. Sarabandi was the recipient of the Henry Russel Award from the Regent of The University of Michigan. In 1999 he received a GAAC Distinguished Lecturer Award from the German Federal Ministry for Education, Science, and Technology. He was also a recipient of the 1996 EECS Department Teaching Excellence Award and a 2004 College of Engineering Research Excellence Award. In 2005 he received the IEEE GRSS Distinguished Achievement Award and the University of Michigan Faculty Recognition Award. He also received the best paper Award at the 2006 Army Science Conference. In 2008 he was awarded a Humboldt Research Award from The Alexander von Humboldt Foundation of Germany and received the best paper award at the IEEE Geoscience and Remote Sensing Symposium. He was also awarded the 2010 Distinguished Faculty Achievement Award from the University of Michigan. The IEEE Board of Directors announced him as the recipient of the 2011 IEEE Judith A. Resnik medal. In the past several years, joint papers presented by his students at a number of international symposia (IEEE APS’95,’97,’00,’01,’03,’05,’06,’07; IEEE IGARSS’99,’02,’07; IEEE IMS’01, USNC URSI’04,’05,’06,’10, AMTA ’06, URSI GA 2008) have received best paper awards.

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Ferrite Based Non-Reciprocal Radome, Generalized Scattering Matrix Analysis and Experimental Demonstration Armin Parsa, Member, IEEE, Toshiro Kodera, Member, IEEE, and Christophe Caloz, Fellow, IEEE

Abstract—A non-reciprocal antenna radome based on the Faraday rotation effect in a ferrite slab is proposed and analyzed. This radome allows transmission in one direction and attenuates the signal in the opposite direction. It includes two layers of strip gratings on each side of the ferrite slab, consisting of highly conductive strips for proper reflection, and thin highly lossy strips for reflection/dissipation, and three dielectric layers for matching. The radome is analyzed rigorously by the generalized scattering matrix (GSM) method and its performance experimentally demonstrated between two broadband antennas. The measured results show 21 dB and 0.85 dB loss in the isolation and transmission directions, respectively. Index Terms—Faraday effect, ferrites, non-reciprocal media, radomes, scattering matrices.

I. INTRODUCTION NTENNAS are commonly covered and protected from their environment by radomes [1]. Although radomes shield and safeguard antennas, they are transparent to electromagnetic radiation at the operating frequency of the covered antenna. Electromagnetic transparency, which is provided both for the incoming and the outgoing RF signals, allows the protected antenna to operate simultaneously as a receiver and as a transmitter. In applications where the antenna is intended to operate only in the receive or transmit mode, a non-reciprocal radome (NRR), transparent only in one direction (receive or transmit), is required. Such an NRR shields the antenna from foe or interfering signals which may alter the proper operation of the antenna (e.g., de-tune it or mismatch it) or cause various other undesirable effects. For instance, an NRR may find applications in wireless sensor networks, stealthy antennas, and radiative processing (e.g., direction of arrival estimation) systems, and may also provide suppression of packaging effects

A

Manuscript received December 20, 2009; revised March 01, 2010; accepted July 27, 2010. Date of manuscript publication December 30, 2010; date of current version March 02, 2011. This work was supported by NSERC Strategic Project Grant 350403 and with the partnership of the company Apollo Microwaves. A. Parsa is with the Rutter Inc., St. John’s, NL A1E 3T9, Canada. T. Kodera is with the Department of Electrical, Electronic and Information Engineering, Yamaguchi University, Yamaguchi 755-8611, Japan. C. Caloz is with the Department of Electrical Engineering, Poly-Grames Research Center, École Polytechnique de Montréal, Montréal QC H3T 1J4, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103016

and mitigation of mutual coupling between elements of array antennas. An NRR requires a gyrotropic material, such as a ferrite operating in the Faraday rotation situation, for non-reciprocity. The gyromagnetic Faraday property, occurring when the electromagnetic wave propagates in the direction of the applied bias field, or simply in the direction of remanence magnetization in self-biased magnetic materials such as hexaferrites [2] or ferromagnetic nano-wire composites [3], has been widely used in waveguide Faraday circulators and isolators [4]. Recently, this concept has been applied to quasi-optical isolators and circulators using hexaferrites [5]. This paper presents an NRR based on the Faraday rotation effect in a ferrite slab sandwiched between different dielectric layers including highly conductive and highly lossy strip grating polarizers. The principle is similar to that used in waveguide Faraday isolators. However, this is the first time that this concept has been applied to an antenna radome, which requires a specific design, quite different from that of a waveguide and quasi-optical isolators. A preliminary design of this NRR, using a single strip grating layer, was introduced in [6]. However, this preliminary design suffered from poor isolation. Here, this design is refined to solve this problem by using a double strip grating on each side of the ferrite slab. Moreover, the NRR is rigorously analyzed and explained using the generalized scattering matrix (GSM) method. Finally, an experimental demonstration is presented to validate the concept. The remainder of the paper is organized as follows. Section II recalls the principle of operation of the preliminary NRR with single strip gratings. Section III presents the proposed high-isolation NRR based on double strip gratings. Section IV describes the GSM theory and the design of the NRR, while Section V discusses the corresponding numerical results. Section VI provides the experimental demonstration. Finally, conclusions are given in Section VII. II. PRELIMINARY NON-RECIPROCAL RADOME WITH SINGLE STRIP GRATINGS Fig. 1 depicts the preliminary single strip gratings multilayered configuration of the NRR introduced in [6]. As shown in Fig. 1(a), the ferrite slab is placed between two single-layer conductive strip gratings sandwiched between two dielectric slabs on each side. The strips are oriented along the direction on one side of the ferrite (layer ) and along the direction on is obtained the other side (layer ). The coordinate system

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PARSA et al.: FERRITE BASED NON-RECIPROCAL RADOME, GSM ANALYSIS AND EXPERIMENTAL DEMONSTRATION

Fig. 1. Preliminary single strip gratings multilayered configuration of the nonreciprocal radome (NRR) [6]. (a) Exploded view. No spacings exist between adjacent layers. (b) Longitudinal section with explanation of the leakage issue, restricting the isolation of the NRR, due to multiple reflections.

by rotating the system by 45 clockwise about the axis when looking in the positive direction. The ferrite slab is designed in terms of thickness and applied bias field so as to produce 45 Faraday rotation for a normally incident wave traveling in the -direction. The different dielectric layers are employed for matching. The strip gratings, which consist of arrays of parallel conductive strips, act as microwave polarizers. In order to show the NRR principle of operation, first condirection (transmission sider a plane wave traveling in the direction) with polarization along the direction (transmission polarization). After crossing the first strip layer (layer ), the wave is polarized along . Since the ferrite layer rotates the polarization 45 clockwise, the wave emerges at the output of the ferrite with polarization. It is then transmitted through the next polarizer (layer ), since its strips are along the direction, which is perpendicular to the wave’s electric field. Thus, the incident wave is transmitted along the direction. direction (isoConsider next a plane wave traveling in the lation direction). In this case, assume for safety the most unfavorable scenario where this wave is polarized along the direction, which produces quasi-total penetration through the first grating (layer ). In the more favorable situation of -polarized incidence, almost all of the energy would be reflected anyways at the level of this grating. In all cases, the part of the wave reaching the output of the first grating is -polarized. This wave then experiences 45 clockwise Faraday rotation in the ferrite layer, and reaches the second polarizer (layer ) with polarization. Since this polarizer has its strips along the axis, it reflects the wave back toward its provenance. At this stage, it would seem that the structure is providing direction for NRR operation. the required attenuation in the However, additional reflections, occurring next and illustrated in Fig. 1(b), compromise this goal. The wave reflected in the direction becomes -polarized after traveling back through the ferrite layer. It is then reflected again by the strip grating

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Fig. 2. Proposed double strip grating NRR. (a) Exploded view. (b) Longitudinal section with explanation of the proposed solution to the multiple-reflection issue occurring in the single strip grating configuration of Fig. 1.

and traverses the ferrite, for the third time, now in the direction to finally emerge polarized along . Since the wave now sees perpendicular strips in grating , it can penetrate through direction. Thus, deit and exits the structure to the air in the spite some losses in the ferrite and dielectric layers, the wave is direction, essentially transmitted through the structure in the which results in poor isolation. This is the fundamental issue of this single-gratings NRR design. In our endeavours to improve the isolation, a careful inspection of Fig. 1(b) led us to consider introducing an additional grating with thin (much thinner than the skin-depth) and highlylossy metal strips. Such a grating would reduce the effect of while allowing almost complete the first reflection by layer penetration of parallelly-polarized waves. An alternative solution would be to increase the strip thickness along the direction, and decrease the strip width along the direction, similarly to the resistive vanes in waveguide isolators [4]. This could be achieved by engineering narrow, deep, thin and plated grooves in the thickness direction of the dielectric slabs. However, due to fabrication complexity, the former approach, based on double strip gratings, is preferred in this work. III. PROPOSED HIGH-ISOLATION NON-RECIPROCAL RADOME USING DOUBLE STRIP GRATINGS Fig. 2 shows the proposed double strip gratings NRR. The multilayer configuration of the NRR is best visualized in the exand ploded view of Fig. 2(a). The strips of grating layers have a large conductivity, so that they completely reflect parallelly-polarized waves. In contrast, the strips of grating layers and are thin strips of low conductivity, designed in terms of their width and thickness such that they dissipate half of the incident power of parallelly-polarized waves, while the remaining power is reflected and transmitted in equal parts (one fourth).

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Fig. 2(b) illustrates the principle of operation for a plane direction (isolation direction). The wave traveling in the phenomenology in the transmitting direction is essentially the same as for the single strip gratings design, and is therefore not direction discussed here. The incident wave traveling in the and . becomes -polarized after passing through layers The polarization rotates by 45 in the ferrite and becomes parallel to the strips in layer . For a reference power level incurs reflection and of 0 dB just before it, lossy strip layer along with a dissipation level transmission levels of . Note that a small part of the transmitted power of would which would be reflected back (not shown) by layer experience an additional 6 dB decrease across layer , corresponding to a very low level of transmission that we neglect here. Consider again the wave first reflected by . This wave travels back through the ferrite layer and becomes then -polarized. A quarter of the incident power, leading to a compared to input power, is reflected by the lossy strips of . The fraction of the wave reflected back in the layer direction at layer becomes -polarized after traveling across the ferrite. Since it is polarized perpendicularly to the strips of and , this wave can leak out of the NRR to the air. layers However, the power level is 12 dB below the input power, not accounting for additional losses and more complex multiple reflection effects, which indicates a dramatic improvement of isolation compared to the single strip gratings design of Fig. 1. IV. THEORY AND DESIGN

Fig. 3. Faraday rotation angle versus frequency in an unbounded ferrite computed by (1)–(6) for the saturation magnetization 4M = 0:185 T and slab thickness h = 3:04 mm.

travels inside the ferrite medium. When a fixed Faraday rotation angle is needed, the distance can be obtained from (1) which is an estimate of the ferrite slab thickness . Fig. 3 shows the Faraday rotation angle versus frequency obtained using (1)–(6) for the saturation magnetization and slab thickness (thickness of an available commercial ferrite slab). For such parameters, 45 Faraday rotation occurs at 9.4 GHz, 10.5 GHz, and 11.7 GHz with bias fields of 0.08 T, 0.1 T, and 0.12 T, respectively. This approach, neglecting loss and more complex effects, only provides initial design, which will be adjusted later by exact electromagnetic analysis.

A. Ferrite Parameters

B. Generalized Scattering Matrix (GSM) Method

The design parameters for the ferrite slab, which are the slab thickness and the applied bias field, are chosen so as to induce a 45 Faraday polarization rotation. The Faraday rotation angle (measured in radians) in an unbounded ferrite medium with in the direction, parallel to the direction of the bias field wave propagation, reads [4]

The multilayered NRR of Fig. 2 can be analyzed exactly using the generalized scattering matrix method [7], [8]. In addition to providing insight into the operation principle of the NRR, this approach will allow us to verify HFSS results, and to perform an efficient parametric study in terms of computational speed. Due to the small period of the gratings and the relatively large spacing between the different layers, only the fundamental space harmonic is propagating along the direction, and must therefore be accounted for. In contrast, many space harmonics must be considered in the transverse plane of the gratings. The following analysis is restricted to the case of normal incidence. Although it could in principle be generalized to address oblique incidence by adding anisotropy in the approach of [7], the paper focuses on the most fundamental properties of the NRR, and these properties are essentially captured by studying the case of normal incidence. Assuming a normally incident plane wave, the scattering ma, is defined as trix of layer ,

(1) where (2) (3) (4) with (5) (6) where is the saturation magnetization, is the gyromagnetic is called the ferromagnetic resonance. Furtherratio, and more, , , and are angular frequency, ferrite permittivity, and free space permeability, respectively. Note that the angle of rotation in (1) is proportional to the distance that the wave

(7) and are the incident field expansion coeffiwhere and are the scattered field expansion coefcients and ficients, with the subscripts , denoting the wave polarization, as shown in Fig. 4. The scattering matrix of the overall structure is obtained from the corresponding transmission matrix by standard conversion

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. A conventional electric field integral equaof width tion (EFIE) is subsequently formed. The elements of the EFIE are given by [8] impedance matrix

(11)

Fig. 4. GSM modeling of the NRR with the corresponding wave parameters and layer designations.

where , is the propagation constant of the for normal incidence), incident wave in the direction ( is the free space wavenumber, and and denote the center of the th and th the parameters cell. Furthermore, we have (12) The current induced on the strips can then be obtained by solving the EFIE system of equations (13)

Fig. 5. Strip grating layer with its relevant parameters.

formulas. The overall transmission matrix of an layer structure is obtained by multiplying the transmission matrices of the individual layers as

(8)

is the incident field vector and is the vector where including the unknown coefficients of the basis function expansion of the induced current. From the induced current, the scattered field can be written as [8] (14) where (15)

where the transmission matrix of layer

,

(16)

, is defined as (17) (9) Fig. 5 shows a strip grating layer with its relevant parameters. The grating is made of parallel conductive periodic strips with thickness , width and conductivity . The periodicity of the grating is denoted . Several authors have already addressed the analysis of the strip gratings in [7]–[11]. The scattering matrix of such a layer is given by

(10)

where and are the reflection and transmission coefficients for an incident wave with polarization parallel to the metal strips. The reflection of the wave incident with polarization perpendicular to the strips is assumed negligible. The procedure for obtaining the reflection and transmission coefficients closely follows [8] with a slight modification to accells count for lossy strips. The strip width is divided into

Assuming that only the fundamental Floquet harmonic propagates, i.e., considering only the term in (14), we can write (14) for normal incidence as (18) Once the scattered field has been obtained from (18), the reflection and transmission coefficients in (10) are found using and . The reflection and transmission coefficients are calculated at since we assume an electrically thin (small ) conductor. In order to model transmission through the ferrite layer, the linearly polarized waves are first transformed into circularly polarized waves at the ferrite layer by the transmission matrix

(19)

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where the subscripts and denote left- and right-handed circularly polarized waves. Similarly, the circularly polarized waves are transformed into the linearly polarized waves at the ferrite layer by using the inverse transformation of (19). circularly polarized wave becomes cirSince the cularly polarized after reflection, the scattering matrix representing the reflection and transmission for the wave traveling from air to ferrite is given by

(20)

is the reflection coefficient for the wave traveling where from air to ferrite given as (21) where is the ferrite relative permittivity, and and are given in (3) and (4), respectively. The other elements of the scatand tering matrix in (20) can be obtained by using . Similarly, the scattering matrix representing the reflection and transmission for the wave traveling from the ferand rite to the air can be obtained by replacing the subscripts to and in (20), respectively. The phase shift due to the wave traveling inside the ferrite can be expressed as

(22) where and are given in (2). The coordinate system is to by using a rotation matrix. The 45 rorotated from tation is obtained by

(23)

Other scattering matrices which represent the reflection and transmission parameters at the boundaries of the dielectric layers can be obtained using Fresnel reflection and transmission coefficients [7]. Inserting all the transmission matrices into (8) provides the , which relates and overall transmission matrix to and as (24) Finally, this transmission matrix is converted to the overall scattering matrix of the NRR by the standard conversion formula.

V. RESULTS AND DISCUSSION The transmission and isolation characteristics of the proposed NRR (Fig. 2) are modeled using the GSM approach presented in Section IV-B. In order to validate this model, the results are compared with the solution obtained by the commercial finite element electromagnetic solver, HFSS [13]. To model the NRR in HFSS, a unit cell model of rhombic shape is inserted between two Floquet ports (ports between which the Floquet periodic boundary condition is specified). The parameters of the ferrite (corresponding to Magneton 9CHV) are: a thickness of , a relative permittivity of , an internal , resonance full width at half maxbias field of Oe at 9.4 GHz, and a saturation imum (line width) . The dielectric materials are magnetization of Rogers TMM10I with relative permittivity of , dielec. tric loss tangent of 0.0023, and thickness of and ) have a thickness of The conductive strips (layers , a width of and a conductivity of (copper), while the lossy strips (layers and ) have a thickness of , a width of and a conductivity of (titanium). The periodicity in all the gratings is . Unless mentioned otherwise, these parameters will apply to all the results below. The choice of copper for the conductive strips was motivated by the high conductivity of this metal for high reflection of parallelly incident waves, while titanium was chosen for the lossy strips so as to lead to realizably thin strips. All the strip parameters (see Fig. 5), , and , as well as the grating period , are interdependent, and the design for these parameters was performed using GSM and HFSS analysis. A parametric study was performed to study the effect of width on reflection, transmission, and dissipation for the copper and titanium strips when and period were chosen. thicknesses It was observed that a half power dissipation could be achieved for titanium strips. when the width was close to for copper strips guarFurthermore, a width of anteed high reflection for the incident wave with polarization parallel to the strips. Fig. 6 shows the reflected and transmitted electric field components for different incident fields on an 8 GHz NRR, assuming a normal plane wave incidence with 0 dB amplitude for all , , and ). The ferrite the possible polarizations ( and dielectric losses were neglected in the GSM modeling. The GSM results are compared with HFSS results, which include both ferrite and dielectric losses, leading to slightly lower levels in all cases. Fig. 6(a) and (c) show that, as expected, most of the incident wave is reflected when the incident polarization is parallel to the strips before the incident wave reaches the ferrite layer. Fig. 6(b) shows the case where the incident wave is traveling in the iso. The corresponding isolations at 8 GHz lation direction and , for the and are polarized generated waves. In this case, most of the power is dissipated on the metal strips. Fig. 6(d) shows the results for . The incident wave traveling in the transmission direction this direction with polarization perpendicular to the strips (in and ) is almost completely transmitted with a level layers

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Fig. 7. Surface loss density (dB scale) computed by (25) on the metal strips for different incident fields, (a) A , (b) A , (c) D , (d) D . The strip gratings are denoted by , , and in correspondence with Fig. 2. This figure is to be compared with Fig. 6.





the surface loss density on the strips. This density is obtained by measuring the change in the real part of the Poynting vector on the strip surfaces and may be computed by (25)

Fig. 6. Overall ([S ] matrix) reflected and transmitted electric field components for different incident fields, (a) A , (b) A , (c) D , (d) D . The reference of the incident field is 0 dB. The results computed by GSM (solid line) are compared with HFSS (dash-dotted line) results. The internal bias field is H = 0:12 T.

where is the unit vector normal to surface, and and are the Poynting vectors at the two sides of the surface. Fig. 7 shows on the metal strips on layers to . Fig. 7(a) and (c) confirm that the amount of decay is highest on the strips with orientation parallel to the incident wave polarization, i.e., layers and for incidence and layers and for incidence. For the case of incidence (isolation direction), shown in Fig. 7(b), the amount of loss is maximum on layer , as expected. Fig. 7(d) shows the case of wave traveling in the incidence), where only a very small transmission direction ( amount of loss is observed on all metallic strips. VI. EXPERIMENTAL DEMONSTRATION

of . Comparison between GSM (with no ferrite and dielectric losses) and HFSS lossy results indicate that the effects of ferrite and dielectric losses on the overall NRR’s performance are very small. In order to appreciate the relative effects of the different gratings of the NRR, it is interesting to examine the power loss they induce separately by dissipation. For this purpose, we consider

To demonstrate the non-reciprocal characteristics of the proposed NRR, the prototype and setup shown in Fig. 2 were used, with the parameters used in the previous sections. The ferrite slab is a Magneton 9CHV of dimensions 500 500 3.04 mm. The dielectric layers are Rogers TMM10I substrates of dimensions 500 500 3.175 mm. The titanium and copper strips were deposited by magnetron sputtering.

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Fig. 9. Experimental results corresponding to the NRR and setup of Fig. 8. (a) Scattering parameter S of the broadband antennas. (b) Scattering parameters of the NRR. An electromagnet was used to generate an external DC bias of 0.31 T.

Fig. 8. Experiment. (a) Photographs of the copper grating and titanate grating layers. (b) Measurement setup. (c) Top view of the measurement setup with the DUT. (d) Layout, and (e) cross sections view of the two broadband antennas [14] which were used in the experiment.

The titanium and copper strip gratings are shown in Fig. 8(a). The NRR was placed between two broadband antennas, as shown in Fig. 8(b) and (c). The antennas are similar to those reported in [14] with a reduction factor of 0.7 in order to response centered at . Referring achieve a flat to Fig. 8(d), the geometrical parameters of the antennas are: , , , , , , , , , , , and , where all the units are in millimeters. The substrate of the antennas was RT/6002 with thickness 0.508 mm and size of 2.44 2.44 cm. An electromagnet was used to generate an external DC bias of 0.31 T, which corresponds approximately to the internal bias field of 0.12 T. The distance from the antennas to the NRR and electromagnet poles are and . The antenna at Port 2 is rotated 45 with respect to the antenna at Port 1. The experimental results are presented in Fig. 9. Fig. 9(a) of the antennas, compared shows the scattering parameter with HFSS results. The simulated boresight axial ratio at 8 GHz (not shown here) was approximately 9.54 dB. Fig. 9(b)

shows the scattering parameters of the NRR using the setup of Fig. 8(b). The measured isolation, which is the difference and , is found to be 21 dB at 8 GHz. In order between to achieve a uniform bias field, the antennas were placed very close to the NRR and the (much larger) electromagnet poles. It is noted that the measurement set up does not show directly the transmission and isolation of an NRR because it includes some reflections from the poles of the electromagnet. However, the and without the NRR at around 8 comparison between GHz shows that the transmission loss is relatively small (0.85 dB), as expected. Overall, these experimental results are in fairly good agreement with the theoretical results of Fig. 6, and validate the proposed double strip grating NRR concept. VII. CONCLUSIONS A double strip grating non-reciprocal radome (NRR) allowing transmission along one direction and providing attenuation along the opposite direction has been presented. The non-reciprocal property of the NRR is provided by Faraday rotation in a ferrite slab. In order to obtain a high isolation characteristic, two layers of strip gratings are placed on each side of the ferrite layer. The strip grating layers close to the ferrite layer have lossier strips than the strip grating layers close to the outer boundaries. The generalized scattering matrix (GSM) method has been used for modeling the non-reciprocal radome. A prototype has been fabricated and tested to validate the concept. The experimental results show 21 dB and 0.85 dB loss in the isolation and transmission directions, respectively. The NRR demonstrated experimentally in this paper requires a biasing electromagnet since the magnetic material used was a

PARSA et al.: FERRITE BASED NON-RECIPROCAL RADOME, GSM ANALYSIS AND EXPERIMENTAL DEMONSTRATION

standard ferrite. The resulting structure is obviously not practical. However, self-biased magnetic materials, such as hexaferrites and ferromagnetic nanowire composites, will solve this problem and make the proposed NRR very attractive for the applications mentioned in the introduction. The ferromagnetic nanowire composite implementation is currently under development in the authors’ research group.

ACKNOWLEDGMENT The authors would like to thank ANSYS Inc. for their generous donation of HFSS licenses.

REFERENCES [1] Radome Engineering Handbook: Design and Principles, J. D. Walton Jr., Ed.. New York: Marcel Dekker, 1970. [2] V. G. Harris, A. Geiler, Y. Chen, S. D. Yoon, M. Wu, A. Yang, Z. Chen, P. He, P. V. Parimi, X. Zuo, C. E. Patton, M. Abe, O. Acher, and C. Vittoria, “Recent advances in processing and applications of microwave ferrites,” J. Mag. Mag. Mater., vol. 321, no. 14, pp. 2035–2047, Jul. 2009. [3] L.-P. Carignan, M. Massicotte, C. Caloz, A. Yelon, and D. Ménard, “Magnetization reversal in arrays of NI nanowires with different diameters,” IEEE Trans. Magn., vol. 45, no. 10, pp. 4070–4073, Oct. 2009. [4] B. Lax and K. J. Button, Microwave Ferrites and Ferrimagnetics. New York: McGraw-Hill, 1962. [5] R. I. Hunter, D. A. Robertson, P. Goy, and G. M. Smith, “Design of high-performance millimeter wave and sub-millimeter wave quasi-optical isolators and circulators,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 5, pp. 890–898, May 2007. [6] T. Kodera, A. Parsa, and C. Caloz, “Non-reciprocal ferrite antenna radome: The faradome,” presented at the IEEE AP-S Int. Symp. USNC/ URSI Nat. Radio Sci. Meeting, Charleston, SC, Jun. 2009. [7] R. C. Hall, R. Mittra, and K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag., vol. 36, no. 4, pp. 511–517, Apr. 1988. [8] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods For Electromagnetics. Piscataway, NJ: IEEE Press, 1998, pp. 264–285. [9] R. C. Hall and R. Mittra, “Scattering from a periodic array of resistive strips,” IEEE Trans. Antennas Propag., vol. AP-33, pp. 1009–1011, Sep. 1985. [10] J. H. Richmond, “On the edge mode in the theory of TM scattering by a strip or strip grating,” IEEE Trans. Antennas Propag., vol. AP-28, pp. 883–887, Nov. 1980. [11] J. Rubin and H. L. Bertoni, “Scattering from a periodic array of conducting bars of finite surface resistance,” Radio Sci., vol. 20, pp. 827–832, Jul.–Aug. 1985. [12] H.-K. Chiu, H.-C. Chu, and C. H. Chen, “Propagation modeling of periodic laminated composite structures,” IEEE Trans. Electromagn. Compat., vol. 40, pp. 218–224, Aug. 1988. [13] High Frequency Structure Simulator (HFSS). Pittsburgh, PA: ANSYS Inc. [14] A. Mehdipour, A. Parsa, A. R. Sebak, and C. W. Trueman, “Miniaturised coplanar waveguide-fed antenna and band-notched design for ultra-wideband applications,” IET Microw Antennas Propag, vol. 3, no. 6, pp. 974–986, Sep. 2009.

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Armin Parsa (S’02–M’08) received the B.Sc. degree from Amirkabir University of Technology, Tehran, Iran, in 1997, the M.Sc. degree from Tarbiat Modarres University, Tehran, in 2001, and the Ph.D. degree from Concordia University, Montréal, QC, Canada, in 2008, all in electrical engineering. From May 2008 to May 2010, he was a Postdoctoral Research Fellow with Poly-Grames Research Center, École Polytechnique de Montréal, Canada. Currently, he is a Research Engineer with Rutter Inc., St. John’s, NL, Canada. His main research interests include ultrawideband and leaky wave antennas, propagation and scattering of waves, high frequency techniques, computational electromagnetics, small-target detection in sea clutter, and remote sensing.

Toshiro Kodera (M’01) received the B.E., M.E. and Dr. Eng. 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 an Visiting Researcher, and in 2006 he joined as a Researcher. In 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 several novel radiative structures using magnetic material inspired by metamaterial concept. In 2010, he joined Yamaguchi University where he is now an 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 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 papers, letters, journal papers, three books, eight book chapters, and he holds several patents. His research interests include all fields of theoretical, computational and technological electromagnetics engineering, with strong emphasis on emergent and multidisciplinary topics. Dr. Caloz is an IEEE Fellow. He is a Member of the Microwave Theory and Techniques Society (MTT-S) Technical Committees MTT-15 (Microwave Field Theory) and MTT-25 (RF Nanotechnology), 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), the International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE), the International Journal of Antennas and Propagation (IJAP), and Metamaterials journal of the Metamorphose Network of Excellence. He received several awards, including the UCLA Chancellor’s Award for Postdoctoral Research in 2004 and the MTT-S Outstanding Young Engineer Award in 2007.

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Bandwidth Limits of Multilayer Array of Patches Excited With Single and Dual Probes and With a Shorting Post Arun K. Bhattacharyya, Fellow, IEEE

Abstract—Maximum achievable bandwidth of a multilayer patch array is investigated. It is found that while the bandwidth increases significantly from one layer to two layers, but does not increase any further with additional layers. Furthermore, the bandwidth of a dual probe-fed patch with a series stub increases significantly from that of a single probe-fed patch array. A simple circuit model is presented to explain such bandwidth characteristics. A comparison of the electrical performances of single, dual probe-fed patches and a patch with a shorting post are presented. The circular polarization performance of a four probe-fed patch array is also shown. Index Terms—Dual probe-fed, maximum bandwidth, multilayer, patch array, shorting post, single probe-fed, wideband.

I. INTRODUCTION OR space applications microstrip patch arrays offer some advantages due to their low profile and light weight characteristics. Moreover, they can be made conformal with the body of the spacecraft. In a receiving system the active-elements can be integrated or closely placed behind the patch layer in order to eliminate the RF cable loss, particularly for high frequency applications. The major disadvantage of a microstrip patch element is its narrow bandwidth. The narrow band characteristic often disqualifies microstrip antennas as radiating elements in an array. Several attempts have been made to increase the element bandwidth either by stacking an additional patch layer [1]–[6] or by shaping the patch element [7]–[14]. For stacked patches, the bandwidth of an element is increased to about 30%. In [15], Waterhouse employs layers of high and low dielectric materials for designing stacked patch arrays and achieved over 27% bandwidth with 45-degree scanning capabilities. By shaping the patch elements, the bandwidth is found to increase significantly compared to a regular shaped patch. The shapes include patch with bent strip [8], E-shaped patch with folded feed [9], patch with folded shorting walls [10]–[13] and patch with L-shaped probes [14]. The largest bandwidth reported is about

F

Manuscript received December 07, 2009; revised June 24, 2010; accepted August 08, 2010. Date of publication January 13, 2011; date of current version March 02, 2011. The author is with Northrop Grumman, Redondo Beach, CA 90278 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.2010.2103013

50%. Most of the shaped patches, however, have squint radiation patterns and cross-polarization problems. As a result, such shaped patches are not desirable for space applications where cross-polarization isolation is a critical need. The objective of this paper is twofold. The first objective is to investigate the largest achievable bandwidth from a patch array with multiple layers. Toward that goal, a single probe-fed patch array is chosen and using an efficient optimization method the optimum dimensions are obtained for each of the multilayer patch arrays varying from one to four layers. Interestingly, it is observed that while the bandwidth increases significantly from one layer to two layers, but it does not increase any further by adding more layers. The maximum achievable bandwidth is about 2 to 1. A circuit model is introduced to understand this behavior of “bandwidth limitation” of a multilayer patch arrays. It was revealed that the upper layer patches work like a low pass filter and for a given cell size, the cutoff frequency cannot be increased beyond certain value. This limits the upper end frequency of the operating band. The second objective is to increase the bandwidth further, if possible, while keeping the structure simple and symmetrical as much as possible from low cross-polarization point of view. It cavity mode is found from the equivalent circuit that the limits the lower end frequency of the operating band. This is primarily due to the large capacitive reactance offered by the mode; thus the impedance match cannot be accomplished below a certain frequency. To eliminate this limitation the mode must not be excited. If two probes with 180 degree phase difference are used instead of one probe, then the mode does not exist. As a result, the operating band can be extended to a lower frequency, yielding overall bandwidth enhancement. However, with dual probe feed, the input reactance shows a positive offset value. To neutralize this positive reactance a series stub is employed with each probe input. With this arrangement the achievable bandwidth increases to about 3 to 1 as opposed to 2 to 1. It was initially thought that a patch with a single probe feed accompanied by a shorting post could also have similar bandwidth performance because the induced negative current would suppress the mode. In reality, it was found that necessary magnitude and the phase of the induced current were not achievable over a wide band that would suppress the mode completely. Hence a bandwidth enhancement was not observed. We present the bandwidth characteristics of patch arrays of three different feed configurations. The numerical results are generated using moment method analysis and then verified

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A. Generalized Impedance Matrix Formulation of Probe Layer The impedance matrix of the probe layer is defined as

where represents the port-voltage-vector of the probes (port 1), represents the Floquet modal voltage vector at the refer(port 2) with as the probeence plane defined by is a column vector of 2 height. For a dual probe-fed patch, is equal to the number elements. The number of elements in of propagating Floquet modes considered in the analysis. and are the current vectors. Accordingly, , , and are sub-matrices. Following the procedure described in [16], the Z-matrix elements of the dual probe feed layer are obtained as Fig. 1. Single probe-fed multilayer array of patches.

with finite element analysis. Also presented are the scan performances of the arrays. Finally, the circular polarization performance of a 4 probe-fed patch array is investigated and the results are shown. The paper is organized as follows. Section II presents the analytical model of a patch array with multiple probe feeds. Section III presents the bandwidth characteristics of the arrays for bore-sight radiation and in Section IV the characteristics are explained using an equivalent circuit model. Section V compares the scan performances of the arrays. Section VI summarizes the important conclusions obtained from the investigation.

(2a)

(2b)

(2c) II. ANALYSIS Consider an infinite array of probe-fed patch elements as and the bottom layer shown in Fig. 1. The cell size is . The probe diameter is and the probe is patch size is immersed in a dielectric substrate of dielectric constant . The corresponding substrate thickness is . In general the patch size and the substrate thickness for the -th layer are denoted and , respectively. Suppose the probes are excited as with uniform amplitudes (for a dual probe-fed element two different amplitudes for the two probes are assumed) and with linear phase progression such that an infinite number of Floquet modes exist in the array. The wave numbers for the Floquet mode with respect to a rectangular grid are given by

(1) and as the phase difference between adjacent cells with along and , respectively. The analysis of this structure is invoked by characterizing the probe layer and the patch layers independently and then using a proper combination of the generalized scattering matrices (GSMs). For the probe layer it is convenient to obtain the generalized impedance matrix (GIM) first and then transform to the GSM. For the patch layer however, the GSM can be determined directly. In the following section we consider the GIM of the probe layer.

(2d) is the Bessel function of order zero, and assume where values 1 and 2, is the coordinate of the probe number and is given by

(3) In (2)

is the free space wave number,

,

, and for the mode and for the mode. Floquet mode. It should The index corresponds to the be mentioned that for associated modes, because the z-directed probe currents do with the modes. Also, for not couple with the because the Floquet modes are orthogonal. The above GIM can be converted to the GSM matrix [16] assuming that the probes are excited by coaxial lines. using (2a), It should be mentioned that for computing the number of terms should be the same as the total number of coupling modes considered in port 2. Furthermore, the same set of modes must be considered for obtaining all the sub-matrices. , If infinite number of modes is considered for computing then the summation becomes infinitely large; as a result the final result becomes unstable.

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B. Floquet Analysis of Patch Layer For the patch layer, the GSM is obtained using Galerkin’s Method of Moment procedure using periodic Green’s function. A set of Floquet modes is assumed to be incident upon the patch layer and the induced current on the patch surface is determined in terms of a finite number of basis functions. The transmitted and reflected modes produced by the patch current yield the GSM of the patch layer. The procedure is detailed in [5], [16] and will not be repeated here. The GSMs of the patch layer and the probe layer are then combined to obtain the overall GSM of the structure. For multiple patch layers, multiple GSMs are considered. The convergence of the final solution is tested by increasing the number of coupling modes and expansion modes [16, p. 213]. For the present case, the number of coupling modes turns out to be about 450 and the number of expansion modes to be about 1000. C. Active Element Pattern The active element pattern (AEP), also called embedded element pattern, is the radiation pattern of an element in the array environment under the condition that the other elements of the array are match terminated. It is deduced in [16] that the AEP can be determined from the GSM of the array structure. The active element pattern is given by

where and are unit vectors, and respectively and are the elements corresponding to the Floquet modes in the sub-matrix of the overall GSM of the array. If the incident voltage (for the probe-fed patch, the incident voltage of the coaxial feed) is normalized with respect incident power, then the AEP becomes the active element to gain pattern. The active element gain pattern is very important for an array analysis because it incorporates the mutual coupling effects of the array [16]. The co-polarization and cross-polarization patterns can be determined using Ludwig’s third definition. In the following section we present numerical results related to wide band probe-fed patch arrays. III. BANDWIDTH CHARACTERISTICS The return loss (negative reflection coefficient in dB) characteristics of multilayer patch arrays are studied. We compare three different configurations; single probe-fed patch, dual probe-fed patch excited by 180-degree hybrid and single probe-fed patch with a shorting post. We consider low dielectric substrate of dielectric constant of 1.1 to minimize the influence of the dielectric substrate on the bandwidth performance. For each case, the structure was optimized to achieve the widest possible bandwidth with respect to 10 dB return loss. In order to examine the bandwidth performance, the number of patch layers is increased progressively from one to four. We present the results in the following section. A. Single Probe-Fed Patch Array A uniformly excited array of single probe-fed patch elements was analyzed using the method described before. We consid-

Fig. 2. Reflection Coefficients of single probe-fed square patch arrays. Cell size = 0:5 cm 0:5 cm, substrate permittivity = 1:1. One-layer patch: a1 = 0:3808 cm, h1 = 0:118 cm, xp = 0:172 cm. Two-layer patch: a1 = 0:4161 cm, a2 = 0:3311 cm, h1 = 0:0921 cm, h2 = 0:0795 cm, xp = 0:1952 cm. Probe diameter = 0:04 cm. (a) Computed results of 1 to 4 layered arrays, (b) comparison of MoM and FEM for 1 and 2 layered arrays.

2

ered from one to four layers of square patches. For each case, the patch size, layer thickness and the probe location were optimized to achieve largest possible bandwidth. Fig. 2 shows the return loss behavior of the arrays. The characteristic impedance of the coaxial feed was assumed to be 50 Ohms. As can be noted, the bandwidth for one layered array is about 40%, and with an additional layer it increases to 65%, which corresponds to about 2 to 1 ratio. However, the bandwidth does not increase any further with more patch layers. This is an interesting phenomenon that will be investigated in Section IV. In order to verify the validity of our MoM results, we simulated one-and two-layer arrays using commercially available software that employs finite element method (FEM). We used periodic boundary conditions to simulate bore-sight scan. Fig. 2(b) shows the comparison between MoM and FEM results. The agreement is very satisfactory as can be seen from the plot. B. Dual Probe-Fed Patch Array Next, we studied an array of dual probe-fed patch elements. A phase difference of 180 degree between the two probe cur-

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Fig. 4. Reflection Coefficients of 2-layer square patch arrays with one probe-feed and a shorting post, post diameter = probe diameter = 0:04 cm. For center post: a1 = 0:4561 cm, a2 = 0:3528 cm, h1 = 0:0543, h2 = 0:0621 cm, xp = 0:20 cm. For symmetrically located post: a1 = 0:3645 cm, a2 = 0:4752 cm, h1 = 0:0661 cm, h2 = 0:0381 cm, xp = 0:1622 cm.

0

Fig. 3. Reflection Coefficients of dual probe-fed square patch arrays. Cell size = 0:5 cm 0:5 cm, substrate permittivity = 1:1, d = 0:04 cm. For 2-layer array: a1 = 0:4455 cm, a2 = 0:3544 cm, h1 = 0:1221 cm, h2 = 0:0675 cm, xp = 0:19 cm. (a) Computed data for 1 to 3 layered arrays, (b) reflection coefficient improvement with series stub of length 0.225 cm (shown for 2-layered case only).

2

6

0

a shorting post. Two cases were considered. In the first case, the shorting post is placed at the patch-center and the second case the shorting post is symmetrically located at the opposite side of the probe. For both cases, the probe position, patch dimensions and the layer thickness were optimized to achieve maximum possible bandwidth. Fig. 4 yields the return loss versus frequency plot. The largest achievable bandwidth for this structure was about 45% and 16%, respectively, which are significantly lower than that of a single probe-fed patch. The MoM data is compared with FEM simulation data and the agreement is satisfactory. IV. EXPLANATION OF BANDWIDTH CHARACTERISTICS

rents was assumed. The probes were located near the two opposite edges of a patch. One can view this excitation as a two-wire transmission line fed patch; the two wires being connected to the two end of a patch. Such a feed can be implemented either by using a 180-degree hybrid or by designing a coaxial balun. Such patch arrays have very good cross-polarization performance due to the symmetry of the excitation. Fig. 3(a) shows the optimum return loss of the structure. For the present analysis we assume an ideal 180-degree hybrid fed patch. The reference impedance at hybrid’s input is 100 Ohms. As can be noted, the return loss bandwidth is not very different from that of a single probe-fed element. However, it is found that the input reactance has a positive offset value (which will be explained later). An open circuit stub in series brings down the positive offset reactance; as a result the bandwidth increases significantly from that in Fig. 2. In this case, the achievable bandwidth is about 3 to 1 ratio (see Fig. 3(b)). These results are also validated by FEM simulations as shown in Fig. 3(b). For this structure also the bandwidth does not increase if the number of layers exceeds two. C. Single Probe-Fed Patch With a Shorting Post We then investigated the bandwidth performance of an array of patches excited by a single probe accompanied by

The bandwidth characteristics of three different feed configurations are explained from a simplified circuit model of the patch array. For the bore-sight scan, an element can be modeled as shown in Fig. 5. The equivalent circuit before the transformer resulted from the cavity model. The cavity model is a reasonable approximation for the feed layer because the equivalent circuit and its components can be well estimated from the cavity modes. Furthermore, the dominant cavity mode essentially includes the combined effects of all Floquet modes. The is associated with the stored energy in the capacitance cavity mode (the parallel plate capacitor) and is the inductance associated with the higher order cavity modes. The quanand are the equivalent inductance and capacitance tities cavity mode responsible respectively, associated with the for the far field radiation. The transformer essentially represents cavity modal voltage the voltage transformation from the to the modal voltage associated with the fundamental Floquet mode. The equivalent circuit beyond the transformer represents the patch layers (the equivalent circuit in Fig. 4 considers two and are due to the patch layers). The capacitances patches that are capacitive with respect to the fundamental Floquet mode. The multi-modal transmission line (thick line) represents coupling of the two patches via the Floquet modes. The

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Fig. 5. Equivalent circuit model of a two layer, single probe-fed patch array.

terminating resistance of 377 Ohms is due to radiation to the free space. The nominal resonant frequency for the cavity formed by the ground plane and the first layer patch is approximately given by , which is a good approximation for an isois non-zero, which is dominated by lated patch. For an array the opposite polarity charge accumulation at the edges of two adjacent patches. As a result, the resonance occurs at a lower frequency. For a given cell size, the resonant frequency can be adjusted by adjusting the substrate height and the patch dimensions. For a single probe-fed patch, the input reactance at low frequency has a large negative value, which is dominated by the parallel plate capacitance, . It implies that a good impedance match cannot be accomplished at the low frequency band though the patch could be designed to resonate at low frequency (resonant frequency is defined by the frequency corresponding to the peak input resistance). This is a key constraint for a single probe-fed patch design. On the other hand, at a higher frequency on the input impedance is small. Also, to minithe effect of further, the substrate thickness of the first mize the effect of layer should be kept small (because the magnitude of equivalent reactance of the capacitor is directly proportional to the substrate thickness). For a one-layer patch, a single resonance occurs and that is , and (see Fig. 6(a)). In the case of a decided by , two-layer patch, the first resonance is mostly decided by and . The capacitor has a minimal effect because due to the smaller size its value is significantly lower than of the upper layer patch. However, a second resonance occurs at a higher frequency (see Fig. 6(a)) that can be explained in the following way. The two patches together behave like a low pass filter (LPF). Near the cutoff frequency of the LPF, the input susceptance offered by the two patches together is lower (realone as depicted sulted from optimization) than that of in Fig. 6(b). As a consequence a second resonance occurs at a higher frequency as observed in Fig. 6(a). The patch dimensions and the layer thickness can be optimized to achieve good return loss covering both resonances and the intermediate frequencies. For more than two layers the overall cutoff frequency of the LPF consisting of the patch layers does not change much because it is primarily dictated by the first layer patch, which is typically the largest one. This implies that the bandwidth does not increase significantly even if the number of layers is increased as observed in Fig. 2. It is found that the lower resonant frequency can be decreased slightly by adjusting the patch dimensions, but that causes the upper resonant frequency to decrease also. It turns out that the achievable bandwidth ratio remains fairly unaltered for a given cell size and number of layers.

Fig. 6. Input impedance and admittance of 1 and 2 layer array with single feed. The dimensions are the same as in Fig. 2. (a) Impedance. (b) Admiittance.

For a dual probe-fed patch array, where the probes are symmetrically located with respect to the patch center and the probe cavity mode is currents are 180 degree out of phase, the in Fig. 4 is absent. not excited. Consequently, the capacitor As a result, the reactance at lower band is positive as shown in Fig. 7. Also, by increasing the dielectric thickness the lower end of the band can be shifted to a lower value. This is primarily and , which due to the fact that the product of essentially sets the resonant frequency, increases with the subcan be kept strate thickness. On the other hand, the value of unchanged (that is, the patch dimensions unaltered) keeping the upper cutoff frequency unaltered. In addition, at the lower end of the band the real part of the impedance is larger for dual probe-fed patch than the single probe-fed counterpart. As a result a simple reactance compensation using a series stub extends the bandwidth to the left. Notice that for the dual probe-fed case the lower end of the band is near 11 GHz (see Fig. 3(a)), whereas for the single probe-fed array it is near 16 GHz. The upper end frequency is mostly decided by the cutoff frequency of the patch-layer LPF; hence it remains almost at the same location. Because the lower end frequency reduces significantly, the 10 dB return loss bandwidth now becomes about 3 to 1 ratio as opposed to 2 to 1. As in the previous cases, the bandwidth does not increase significantly beyond two layers. Notice that the substrate thickness, , is larger for the dual probe-fed case. It was initially thought that a single probe-fed patch with a shorting post at the symmetrical location would also have a larger bandwidth than that of a single probe-fed patch. However,

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Fig. 7. Input impedance with dual feed (without stub) seen by a probe. The dimensions are the same as in Fig. 3(a).

Fig. 9. Active element pattern cuts of 2 probe-fed 2 layer array. The dimensions are the same as in Fig. 3. (a) f = 11 GHz, (b) f = 30 GHz.

V. SCAN CHARACTERISTICS

Fig. 8. (a) Scan characteristics of multilayer array of patches with one feed. The dimensions are the same as in Fig. 2. (b) Scan characteristics of multilayer array of patches with 2 feed. The dimensions are the same as in Fig. 3 for the 2 layered case.

with extensive numerical optimization, the bandwidth did not increase much (see Fig. 4). The induced current on the shorting post was not large enough, neither has the required phase to cavity mode. Further, the induced current cancel out the increases the magnetic stored energy which effectively shrinks the bandwidth. It is also observed that by placing the shorting post at the patch-center the bandwidth somewhat improves from that of the previous case because of low induced current on the post.

The return loss characteristics of single and dual probe-fed patch arrays at 30 degree scan angle are shown in Fig. 8(a) and (b), respectively. We do not consider more than two layer patches because the largest bandwidth is attainable with two layers only. Also, for small scan angles (less than 15 degree) the return loss does not change appreciably from that at bore sight; hence they are not shown in the plots. A dual probe-fed patch is excited by an ideal balun where 180-degree phase difference between the port-currents is enforced. It is worth mentioning that for a large scan angle the magnitude of the input return loss depends on the feed network. If the dual probe-fed patch were excited by a 180-degree hybrid, then the return loss could differ from that excited with a balun. Notice that near 37 GHz frequency, the return loss is near zero dB. This is due to the grazing lobe (and the surface wave mode resonance as well) that happens for 30 degree scan near that frequency. In Fig. 9(a) and (b) we have plotted the active element pattern (which is also equivalent to the scan loss plot for an array) of a dual-probe-fed, dual layered patch array. Within 30 degree scan angle, the scan loss is very low at 11 GHz, and closely agrees pattern. At 30 GHz, the scan loss deviates from with pattern because of the mismatch loss at that scan angle. The aperture efficiency estimated from the bore-sight gain is about 95% and 88% at 11 GHz and 30 GHz, respectively. At boresight, the mismatch loss reduces the aperture efficiency from its highest possible value. The cross polarization isolation is better than 23 dB within 30 degree scan. The scan-loss characteristics

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• Dual probe-fed patch element has the largest bandwidth (about 3 to 1) compared with a single probe-fed patch element. This is primarily due to the elimination of the d.c. parallel plate capacitor caused by dual feed excitation. However, a series stub needs to be added to compensate the positive offset reactance. • Patch array with a shorting post has smaller bandwidth than that with no shorting post. The bandwidth changes with the location of the shorting post. The shorting post at the patch center shows the largest bandwidth. • The scan performance of the dual probe-fed array is found to be acceptable for many applications. The cross-polarization isolation is better than 23 dB within the achievable bandwidth and within the scan angle of 30 degree. In conclusion, the dual probe-fed configuration could be a preferable candidate for a broad band phased array antenna. The extended bandwidth is accomplished at the cost of a balanced feed. However, using printed circuit technology, such a feed network can be implemented without much complexity. ACKNOWLEDGMENT The author appreciates the constructive comments of the reviewers, which helped improving the quality of the paper. REFERENCES Fig. 10. Active element pattern (CP pattern) of an array of 2 layer patches excited by 4 probes in phase quadrature for generating circular polarization. The dimensions are same as in Fig. 3. (a) f = 11 GHz. (b) f = 30 GHz.

of the other two configurations within the respective bands are very similar to that of the dual-probe-fed patch array and are not presented here. A. Circular Polarization We investigated the performance with respect to circular polarization (CP). We consider four feeds that are located at the four sides of the unit cell. The relative phases of the feeds were 0, 90, 180 and 270 degrees, respectively. This type of excitation can also be accomplished by using only two orthogonal feeds; each feed consists of a set of two probes that are excited by a 180-degree hybrid. Fig. 10(a) and (b) show the active CP element patterns at two frequencies. The patterns are plotted with respect to the received power by a spinning linearly polarized antenna, as typically done in an antenna test chamber. The difference between two envelopes gives the axial ratio of the active element pattern. The axial ratio is very good at bore sight and deteriorates with the scan angle. For 30 GHz, this deterioration is significant when the scan angle exceeds 30 degree. VI. CONCLUSIONS In this paper we presented bandwidth performances of three different configurations of multilayer patch arrays and have the following conclusions. • The bandwidth of a patch array does not increase much if the number of layers exceeds two; that is, the maximum possible bandwidth is achievable with two layers only.

[1] R. Q. Lee, K. F. Lee, and J. Bobinchak, “Characteristics of a twolayer electromagnetically coupled rectangular patch antenna,” Electron. Lett., vol. 23, no. 20, pp. 1070–1072, Sept. 1987. [2] Y. Lubin and A. Hessel, “Wide-band, wide-angle microstrip stackedpatch-element phased arrays,” IEEE Trans. Antennas Propag., vol. 39, no. 8, pp. 1062–1070, Aug. 1991. [3] F. Croq and D. M. Pozar, “Millimeter-wave design of wide-band aperture-coupled stacked microstrip antennas,” IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1770–1776, Dec. 1991. [4] S. D. Targonski, R. B. Waterhouse, and D. M. Pozar, “Design of wideband aperture stacked patch microstrip antennas,” IEEE Trans. Antennas Propag., vol. 46, pp. 1245–1251, Sep. 1998. [5] A. K. Bhattacharyya, “A modular approach for probe-fed and capacitively coupled multilayered patch arrays,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 193–202, Feb. 1997. [6] J. J. Schuss, J. D. Hanfling, and R. L. Bauer, “Design of wideband patch radiator phased arrays,” in IEEE APS Int. Symp. Dig., 1989, pp. 1220–1223. [7] K. Ghorbani and R. B. Waterhouse, “Design of large broadband patch arrays,” in IEEE APS Int. Symp. Dig., 1999, pp. 928–931. [8] P. Li, H. W. Lai, K. M. Luk, and K. L. Lau, “A wideband patch antenna with cross-polarization suppression,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 211–214, 2004. [9] C. Y. Chiu, H. Wong, and C. H. Chan, “Study of small wideband folded-patch-feed antennas,” IET Microw Antennas Propag., vol. 1, no. 2, pp. 501–505, Apr. 2007. [10] R. Chair, K. F. Lee, C. L. Mak, K. M. Luk, and A. A. Kishk, “Wideband half u-slot patch antennas with shorting pins and shorting wall,” in IEEE APS Int. Symp. Dig., 2004, pp. 4132–4135. [11] C. Y. Chiu, C. H. Chan, and K. M. Luk, “Small wideband patch antenna with double shorting walls,” in IEEE APS Int. Symp. Dig., 2004, pp. 3844–3847. [12] Y. Li, K. M. Luk, R. Chair, and K. F. Lee, “A wideband triangular shaped patch antenna with folded shorting wall,” in IEEE APS Int. Symp. Dig., 2004, pp. 3517–3520. [13] H. Wang, X. B. Huang, and D. G. Fang, “A single layer wideband U-slot microstrip patch antenna array,” IEEE Antennas Wireless Propag. Lett., vol. 7, no. 99, pp. 9–12, 2008. [14] X.-Y. Zhang, Q. Xue, B.-J. Hu, and S.-L. Xie, “A wideband antenna with dual printed L-probes for cross-polarization suppression,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 388–390, 2006. [15] R. B. Waterhouse, “Design and scan performance of large, probe-fed stacked microstrip patch arrays,” IEEE Trans. Antennas Propag., vol. 50, no. 6, pp. 893–895, Jun. 2002.

BHATTACHARYYA: BANDWIDTH LIMITS OF MULTILAYER ARRAY OF PATCHES EXCITED WITH SINGLE AND DUAL PROBES

[16] A. K. Bhattacharyya, Phased Array Antennas, Floquet Analysis, Synthesis, BFNs and Active Array Systems. Hoboken, NJ: Wiley, 2006, ch. 7. Arun K. Bhattacharyya (F’02) was born in India, in 1958. He received the B.Eng. degree in electronics and telecommunication engineering from Bengal Engineering College, University of Calcutta, India, in 1980, and the M.Tech. and Ph.D. degrees from Indian Institute of Technology, Kharagpur, in 1982 and 1985, respectively. From November 1985 to April 1987, he was with the University of Manitoba, Canada, as a Postdoctoral Fellow in the Electrical Engineering Department. From May 1987 to October 1987, he worked at Til-Tek Limited, Kemptville, ON, Canada, as a Senior Antenna Engineer. In October 1987, he joined the Electrical Engineering Department, University of Saskatchewan, Canada, as an Assistant Professor and was

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promoted to the rank of Associate Professor in 1990. In July 1991, he joined Boeing Satellite Systems (formerly Hughes Space and Communications), Los Angeles, as a Senior Staff Engineer, and was promoted to Scientist and Senior Scientist, in 1994 and 1998, respectively. He became a Technical Fellow of Boeing in 2002. In September 2003, he joined the Northrop Grumman Space Technology Group as a Staff Scientist, Senior Grade. He became a Distinguished Engineer which is a very rare and honorable recognition in Northrop Grumman. He is the author of Electromagnetic Fields in Multilayered Structures-Theory and Applications (Artech House, 1994) and Phased Array Antennas, Floquet Analysis, Synthesis, BFNs and Active Array Systems (Wiley, 2006). He authored over 95 technical papers and has 15 issued patents. His technical interests include electromagnetics, printed antennas, multilayered structures, active phased arrays and modeling of microwave components and circuits. Dr. Bhattacharyya is the recipient of numerous awards including the 1996 Hughes Technical Excellence Award, 2002 Boeing Special Invention Award for his invention of high efficiency horns, 2003 Boeing Satellite Systems Patent Awards, and 2005 Tim Hannemann Annual Quality Award, Northrop Grumman Space Technology.

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Bandwidth Enhancement for a 60 GHz Substrate Integrated Waveguide Fed Cavity Array Antenna on LTCC Junfeng Xu, Member, IEEE, Zhi Ning Chen, Fellow, IEEE, Xianming Qing, Member, IEEE, and Wei Hong, Senior Member, IEEE

Abstract—A substrate integrated waveguide fed cavity array antenna using multilayered low temperature co-fired ceramic technology is presented and designed at V-band (60 GHz). The 8 8 antenna array is designed with an enhanced bandwidth of 17.1% and a gain up to 22.1 dBi by reconfiguring radiating elements, feeding network, and the transition. The proposed array antenna also features the merits of compact size, stable performance, and high efficiency. Index Terms—Bandwidth enhancement, cavity array antenna, low temperature co-fired ceramic (LTCC), substrate integrated waveguide (SIW).

I. INTRODUCTION

Fig. 1. Side view of the array with a transition on LTCC.

OW temperature co-fired ceramic (LTCC), as a multilayered fabrication technology, has been widely used in planar antenna design [1]–[3]. It features the flexibility to integrate the feeding structure and radiators of the antennas in a multilayered substrate using the perpendicular interconnection. Compared to the multilayered printed circuit board (PCB) technology, LTCC technology is easier to realize blind vias, buried vias, and air cavity. In LTCC, conventional planar transmission lines such as microstrip, stripline, and coplanar waveguide may suffer from the undesirable radiation leakage with additional loss and poor line-to-line isolation at millimeter-wave (mmW) bands [4]. Alternatively, substrate integrated waveguide (SIW) [5], post-wall waveguide [6] or laminated waveguide (LWG) [4], as an electromagnetically closed planar guided-wave structure, is adopted for designs at mmW. The comparisons between the SIW and the open structures have been reported in [4] and [7].

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Manuscript received January 16, 2010; revised July 08, 2010; accepted July 31, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The work was supported in part by the Agency for Science, Technology and Research (A*STAR), Singapore, Terahertz Science & Technology Inter-RI Program under Grant 082 141 0040, the Chinese National 973 project under Grant 2010CB327400, and in part by the Chinese NSFC under Grant 60921063. J. Xu, Z. N. Chen, and X. Qing are with the Institute for Infocomm Research, Singapore, Agency of Science, Technology, and Research, Singapore 138642, Singapore (e-mail: [email protected]; [email protected]; [email protected]). W. Hong is with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103018

The SIW features less radiation or transmission loss than the open structures, especially at the frequencies beyond 50 GHz [4]. The antennas based on LTCC and LWG technologies achieve good performance within bandwidths of 9% and 8% at 77 GHz [8], [9] and 5% at 94 GHz [10]. However, the antennas with wider bandwidths in 60-GHz band are desired. For example, the majority of the globe has allocated 7-GHz continuous unlicensed spectrum at varying points between 57–66 GHz or 14.6% bandwidth. Therefore, this paper presents a design with an enhanced bandwidth up to 17.1% by configuring the proposed feeding structure, radiating elements, and feeding transition properly. II. ANTENNA OVERALL STRUCTURE The side view of the array with a transition is shown in Fig. 1 where the left part is the array antenna whereas the right part is the transition for measurement. The grey regions are the SIW functional blocks. As known, the asymmetrical serial-fed structures of resonant arrays lead to a severe distortion of the radiation performances and thus limit the broadband operation [11]. The array proposed here is symmetrically parallel fed. The feeding structure is implemented on multilayered LTCC where the SIW is formed by buried vias. The LTCC substrate is formed by 20 layers numbered from its top to the bottom. Each layer is . The material of with a co-fired thickness of , LTCC substrate is Ferro A6-M with at 60 GHz [12]. The conductor used for metallization and . vias is Au with the conductivity of The SIW is formed with metallic broadwalls (top and bottom walls) and closely-aligned metallic via arrays which serve as

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sidewalls and are electrically connected to the top and bottom metallic layers. Theoretically, there is no radio frequency (RF) power leaking into other portion. The arrows indicate the path of the RF power flowing through the SIW regions. The array itself consists of three regions. Region I is formed with five layers. There are 8 8 open-ended substrate integrated cavities (SICs) as shown in Fig. 2(a). Each SIC is positioned above a feeding slot on the top of Region II. The SICs serve as the broadband transition between dielectric substrate and the air. The dots indicate the metalized vias forming the sidewalls of SIW. The transverse spacing between the adjacent elements is 3.8 mm (0.76 ) whereas the vertical spacing is 3 mm (0.6 ) at 60 GHz ( is the wavelength in the free space). The 8 8 radiating apertures are the rectangular open ends of the SICs. Region II is also formed by five layers. There are 8 4 twoelement subarrays as shown in Fig. 2(b). Each subarray consists of two feeding slots indicated by the solid lines. The slots are on the top metallic surface of the sixth layer. The hollow lines indicate the coupling slots as the output of the power divider in Region III on the top metallic surface of eleventh layer. As shown in Fig. 1, Region III is formed by eight layers. There is a 4 8 way power divider in Region III as shown in Fig. 2(c). The incident power is from the transition part. The output of the power divider goes through the 4 8 coupling slots (broadwall couplers). Each slot can couple the RF power upward into the two-element subarray in Region II. Furthermore, there is a five-layer SIC in Fig. 1, serving as the broadband transition between the SIW and an external rectangular waveguide (RWG). The input power is fed from the WR-15 RWG through a feeding aperture on the bottom of the SIC (layer 20). On the top of the SIC (top of the sixteenth layer), there is a coupling slot on the metal ground through which the RF power is coupled into the upper SIW. III. ANTENNA DESIGN The array is designed with the aid of CST Microwave Studio which is based on FIM (Finite integration method). A. Radiating Elements The radiating element is a five-layer rectangular open-ended SIC with sidewalls formed by via arrays as shown in Fig. 3(a). The radiating aperture (dashed-line rectangle) is realized on the top surface of the LTCC substrate in the screen-printing process. The SIC is fed by a transverse feeding slot located on the bottom of the cavity. The SIC is analyzed and designed together with the transverse feeding slot in the simulation model. Fig. 3(b) excludes the above SIC for clearly indicating the dimensions of the transverse feeding slot located on the short-ended SIW. The slot hardly affects the radiation patterns but significantly affects the impedance matching of the antenna. The optimized dimensions of the element for impedance matching are listed in Table I, where is the minimum distance allowed between the via and the metal edge according to the LTCC design rules used here [13]. of the single element is shown in Fig. 4. The simulated is about 23%, The achieved bandwidth for much larger than the 9% bandwidth in [8].

Fig. 2. Top view of the array on LTCC. (a) open-ended cavity array (Layer 1–5) with apertures on the top of layer 1. (b) two-element subarrays (Layer 6–10) with feeding slots on the top of layer 6. (c) power divider (Layer 11–18) with coupling slots on the top of layer 11.

Simulations of the two adjacent radiating elements in the array are performed for examining the mutual coupling effects as shown in Fig. 4. Port 1 and Port 2 stand for the ports of the

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Fig. 5. Top view of the broadwall cross coupler.

Fig. 3. Radiating element. (a) rectangular open-ended SIC. (b) transverse feeding slot. Fig. 6. Effect of the additional via on the E field in Region III. TABLE I DIMENSIONS OF THE RADIATING ELEMENT (UNIT: mm)

B. SIW Broadwall Coupler

Fig. 4. jS j of the single element and the mutual couplings in two configurations.

two elements, respectively. In Fig. 4, “co-H” refers to the two elements arrayed in the H-plane with the transverse spacing of 3.8 mm whereas “co-E” refers to the two elements arrayed in the E-plane with the vertical spacing of 3 mm. The maximum coupling over the bandwidth of 50–70 GHz is 22 dB (co-H) and 16 dB (co-E), respectively. The mutual couplings slightly affect as shown in Fig. 4. The results from the two-element models are the demonstration and estimation of the actual mutual couplings in the 8 8 array environment. The mutual coupling effects can be further compensated for in the array design. The simulated gain of the single radiating element is 6.7 dBi.

The broadwall coupler is the key component for the vertical RF power transmission in a multilayered structure. A broadwall cross coupler is proposed and studied as shown in Fig. 5, where the dashed lines represent Region II and the solid lines represent Region III. The simulation model includes these two crossed SIWs, an offset longitudinal coupling slot, and three ports. In the array configuration, Port 2 and Port 3 are connected with the feeding slots, as the two-element subarray shown in Fig. 2(b). Theoretically the distance between the coupling slot and the , should be about ( is the guided end of SIW, wavelength). However, such a distance is too small to be fabricated at 60 GHz according to the LTCC design rules [13]. Alternatively, an additional via is exhibited in Fig. 5. It is and located inside the SIW in Region III, with distances from the endwall and sidewall, respectively. Fig. 6 clearly shows the effect of the additional via on the E field near the end of the SIW. Such an effect enlarges the equivalent guided wavelength increases. so that the The optimized dimensions of the cross coupler for the 8-layer thickness in Region III are listed in Table II. Some dimensions are optimized to compare the reflection characteristics of dif” is changed to 1.09 mm for the ferent layers as well. The “ 7-layer thickness. The “ ” and “ ” are changed to 1 mm and 0.47 mm respectively for the 9-layer thickness. Optimized for different layers in Region III are shown in Fig. 7. It can be observed that 8-layer case has a wider bandwidth than the 9-layer case and it has better in-band attenuation level than the 7-layer case. Therefore, the 8-layer thickness is the good trade-off between the bandwidth and in-band attenuation level in terms of the reflection characteristics so it is selected in this and are 3.38 dB and 3.49 dB, design. In Fig. 7,

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TABLE II DIMENSIONS OF THE BROADWALL CROSS COUPLER (UNIT: mm)

Fig. 9. S-parameters of the multilayer SIW-RWG vertical transition.

Fig. 7. S-parameters of the broadwall cross coupler.

Fig. 10. Photograph of antenna (left: top view, right: bottom view).

TABLE III DIMENSIONS OF THE MULTILAYER SIW-RWG TRANSITION (IN mm)

Fig. 8. Top and side views of the multilayer SIW-RWG vertical transition.

respectively at 60 GHz. The insertion loss of the coupler is less than 1.12 dB over 55–65 GHz. Compared with the design in [8], no coupling window is used and broadband performance is achieved. C. Multilayer SIW-RWG Vertical Transition A transition between the SIW and WR-15 RWG is designed for measurement purpose. Because of the high dielectric constant of LTCC substrate, it is usually difficult to achieve the broadband transition between the SIW and RWG directly. To alleviate this problem, a transition configuration is proposed. This is demonstrated in the “transition” portion of Fig. 1 and the detailed structure is exhibited in Fig. 8, where the upper portion is its top view whereas the bottom portion is its side view. A five-layer SIC with via walls indicated by solid circles in the top view forms the transition between the SIW and RWG. The SIC

has the dimensions of and . The dashed-line rectangular in the top view indicates the feeding aperture connecting with the RWG. The simulation model includes the RWG as Port 1, as Port 2. The cavity the transition, and the SIW with width thickness greatly affects the impedance bandwidth. Five layers are used here. In the top view of Fig. 8, the hollow circles outline the fiveis layered SIW. A coupling slot with dimensions of and realized on the broadwall of SIW to couple the RF power from is wider than the the SIC. The transition SIW . The wider is good for power divider SIW the transition matching. The width of 1.7 mm is the largest via spacing allowed in the power divider layout (see dashed circle is ‘A’ in Fig. 2(c) for details). Thus, the SIW with a length used for width taper. Besides, as shown in Fig. 8, there are three SIW height steps from five layers to eight layers. The distance between two steps is . The optimized dimensions are listed in Table III. is about Fig. 9 shows that the bandwidth for 19.6% with a maximum insertion loss of 0.8 dB. The previous

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j

S

j

of the array antenna (port 1 is the feeding aperture).

Fig. 13. Radiation patterns of the array at 60 GHz. (a) E-plane. (b) H-plane.

IV. EXPERIMENTAL RESULTS

Fig. 12. Radiation patterns of the array at 55 GHz. (a) E-plane. (b) H-plane.

transitions using LTCC technology without and with an embedded air cavity achieved the bandwidths of 4% and 10.8% , respectively [14]. Thus, the proposed for transition achieves a wider bandwidth without any air cavity. The process of embedded air cavity in LTCC is expensive and challenging.

A subarray with the cross coupler consisting of two symmetrical elements is shown in Fig. 2(b). Due to the symmetry of parallel-feeding structure, the element spacing is flexible, which can be larger than one guided wavelength to decrease the mutual coupling due to the high dielectric constant of the substrate. However, it should not exceed one wavelength in free space to avoid high grating lobes. The design of the two-element subarray is extended to the cavity array antenna with 8 8 elements. The SIW-RWG transition, feeding network, and radiating elements are simulated together. The effects of the mutual couplings among the radiating elements in the array can be compensated for by the fine tuning of the dimensions of the radiation-associated structure like the feeding slots and the SICs. , Optimized dimensions of the 8 8 array are: , , . Other dimensions are not changed. 8 array was fabricated using LTCC to verify the The 8 design as shown in Fig. 10. At the bottom, the while rectangular is the feeding aperture. The two bigger holes are for the screw and the four smaller ones are the positioning holes for firmly connecting and accurately aligning with the RWG flange. The rest portion is the metal ground. The overall size of the 8 8 array with the transition is 47 mm 31 mm.

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Fig. 15. Simulated and measured gain of the array.

Fig. 14. Radiation patterns of the array at 65 GHz. (a) E-plane. (b) H-plane.

The simulated and measured of the 8 8 array are shown in Fig. 11. The measured and simulated results are in good agreement. The bandwidth for is 10.26 GHz (54.86–65.12 GHz), or 17.1% which is larger than the bandwidth of 8% in [9]. Therefore, the achieved impedance bandwidth can cover the unlicensed 60-GHz band well. The ripples of may stem from the reflection between the feeding network and the SIW-RWG transition. The measured and simulated radiation patterns of the array in both the E- and H-planes are in good agreement as shown in Figs. 12–14. The measurement was conducted in a mini electromagnetic anechoic chamber in Institute for Infocomm Research, Singapore. Over the whole operating bandwidth of 17.1% the direction of the mainlobe keeps pointing to the broadside due to the use of symmetrical feeding. Another consequence of the parallel feeding is the uniform aperture distribution with a 13.5 dB sidelobe level. All of the measured and simulated sidelobe levels over 55–65 GHz are around 13 dB, which are very close to the design target. Fig. 15 shows a flat gain response with a gain variation of 2.5 dB over the operating bandwidth of 17.1%. The achieved gain bandwidth is much larger than the 2.5-dB gain bandwidth of 7% in [8]. The larger bandwidth in terms of both impedance matching and radiation performance benefits from not only the parallel feeding structure but also the broadband design of the radiating elements and the feeding transition of the antenna. The measured and simulated gain is 22.1 dBi and 23.0 dBi at 60 GHz, respectively. The slight difference between the mea-

sured and simulated gain results may be due to the tolerance of fabrication, possible deviation of the dielectric loss of LTCC or conductivity of the metallization, as well as the measurement setup. As an example of verification, if the dielectric loss of the LTCC substrate and the conductor loss of the metallization are slightly enlarged in the simulation, the simulated gain is the same as the measured one. The estimated total loss of the array is about 2.6 dB at 60 GHz by using CST. The insertion loss of the SIW-RWG transition as well as a 12-mm-long SIW feed line is 0.8 dB. The insertion loss of the broadwall coupler is 0.6 dB. The mismatch loss is 0.2 dB. The dielectric loss and the conductor loss in the feeding network are about 1 dB. The measured efficiency is 44.4% whereas the simulated efficiency is 54.7%, both at 60 GHz. The discrepancy is due to the 0.9-dB gain drop in the measured gain. V. CONCLUSION 8 60-GHz cavity array antenna has In this paper, an 8 been presented for bandwidth enhancement. Based on the multilayered LTCC, a symmetrical parallel-fed structure has been realized with stable radiation performances. Compared with the previous work, this compact antenna array has demonstrated much wider operating bandwidth in terms of impedance matching, gain, and the radiation patterns. This has been achieved by the proposed feeding structure, radiating elements, and feeding transition. The antenna array also features high efficiency due to the electromagnetically closed and low-loss SIW structure. ACKNOWLEDGMENT The authors would like to thank J. K. W. Khoo and T. S. P. See, both from Institute for Infocomm Research, Singapore, for their help, particularly in LTCC tapeout. REFERENCES [1] T. Seki, N. Honma, K. Nishikawa, and K. Tsunekawa, “A 60-GHz multilayer parasitic microstrip array antenna on LTCC substrate for system-on-package,” IEEE Microw. Wireless Comp. Lett., vol. 15, no. 5, pp. 339–341, 2005. [2] S. Wi, Y. Sun, I. Song, S. Choa, I. Koh, Y. Lee, and J. Yook, “Packagelevel integrated antennas based on LTCC technology,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2190–2197, 2006. [3] A. Lamminen, J. Saily, and A. R. Vimpari, “60-GHz patch antennas and arrays on LTCC with embedded-cavity substrates,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2865–2870, 2008. [4] H. Uchimura, T. Takenoshita, and M. Fujii, “Development of a ‘laminated waveguide’,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2438–2443, 1998.

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[5] J. F. Xu, W. Hong, P. Chen, and K. Wu, “Design and implementation of low sidelobe substrate integrated waveguide longitudinal slot array antennas,” IET Microw., Antennas Propag., vol. 3, no. 5, pp. 790–797, 2009. [6] J. Hirokawa and M. Ando, “Single-layer feed waveguide consisting of posts for plane TEM wave excitation in parallel plates,” IEEE Trans. Antennas Propag., vol. 46, no. 5, pp. 625–630, May 1998. [7] Y. Huang, K. L. Wu, D. G. Fang, and M. Ehlert, “An integrated LTCC millimeter-wave planar array antenna with low-loss feeding network,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1232–1234, 2005. [8] H. Uchimara and T. Takenoshita, “A ceramic planar 77 GHz antenna array,” in IEEE MTT-S Int. Microw. Symp. Digest, Jun. 1999, vol. 2, pp. 453–456. [9] N. Shino, H. Uchimura, and K. Miyazato, “77 GHz band antenna array substrate for short range car radar,” in IEEE MTT-S Int. Microw. Symp. Digest, Jun. 2005, pp. 2095–2098. [10] J. Aguirre, H. Y. Pao, H. S. Lin, P. Garland, D. O’Neill, and K. Horton, “An LTCC 94 GHz antenna array,” in IEEE AP-S Int. Symp. Digest, 2008, pp. 1–4. [11] M. Hamadallah, “Frequency limitations on broadband performance of shunt slot arrays,” IEEE Trans. Antennas Propag., vol. 37, no. 7, pp. 817–823. [12] A6-M Datasheet Ferro Materials Company [Online]. Available: http:// www.ferro.com/non-cms/ems/EPM/LTCC/A6-M-LTCC-System.pdf [13] LTCC Design Rules VTT Technical Research Centre of Finland [Online]. Available: http://www.vtt.fi/files/research/mel/ltcc_design_ rules.pdf [14] J. G. Lee, J. Hirokawa, and M. Ando, “Transition to a high-dielectric constant post wall waveguide from a standard waveguide by using a hollow cavity in a laminated substrate,” in IEEE AP-S Int. Symp. Digest, 2006, pp. 1613–1616.

Junfeng Xu (M’10) was born in Nanjing, Jiangsu Province, China, in 1981. He received the B.S., M.S., and Ph.D. degrees from Southeast University, Nanjing, China, in 2003, 2006, and 2009, respectively, all in the radio engineering. Since 2010, he has been with the Institute for Infocomm Research (I R), Agency of Science, Technology, and Research (A*STAR), Singapore. He is currently holding the position of Research Fellow under the RF and Optical Department. His current research interests include microwave and millimeter-wave antenna and array design, such as leaky-wave antennas, omnidirectional antennas, low sidelobe array antennas, multi-beam array antennas, as well as LTCC antennas and arrays. Dr. Xu is the first recipient of the Best Paper Award at the 15th International Symposium on Antennas and Propagation (ISAP 2010), Macau. He serves as a Reviewer of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and IEEE Antennas and Wireless Propagation Letters.

Zhi Ning Chen (F’05) received the B.Eng., M.Eng., and Ph.D. degrees from the Institute of Communications Engineering, China, and the Ph.D. degree from University of Tsukuba, Japan, respectively, all in electrical engineering. During 1988 to 1997, he worked at Institute of Communications Engineering, Southeast University, and City University of Hong Kong, China, with teaching and research appointments. In 1997, he was awarded a JSPS Fellowship to join the University of Tsukuba, Japan. In 2001 and 2004, he visited the University of Tsukuba under a JSPS Fellowship Program (senior level). In 2004, he worked at the IBM T. J. Watson Research Center, New York, as an Academic Visitor. Since 1999, he has worked with the Institute for Infocomm Research, Singapore, and his current appointments are Principal Scientist and Department Head for RF & Optical. He is concurrently holding Adjunct/Guest Professorships at Southeast University, Nanjing University, Shanghai Jiao Tong University, Tongji University, University of Science and Technology, China, Dalian Maritime University, and National University of Singapore. He has published 290 journal and conference papers as well as authored and edited the books entitled Broadband Planar Antennas, UWB Wireless Communication, Antennas for Portable Devices, and Antennas for

Base Station in Wireless Communications. He also contributed to the books of UWB Antennas and Propagation for Communications, Radar, and Imaging as well as Antenna Engineering Handbook. He holds 28 granted and filed patents with 21 licensed deals with industry. His current research interest includes applied electromagnetic engineering, RF transmission over bio-channels, and antennas for wireless systems, in particular at mmW, submmW, and THz for medical and healthcare applications. Dr. Chen is a Fellow of the IEEE for his “contribution to small and broadband antennas for wireless.” He is serving as an IEEE Antennas and Propagation Society Distinguished Lecturer and an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He has organized many international technical events as key organizer. He is the founder of International Workshop on Antenna Technology (iWAT). He is the recipient of the CST University Publication Award 2008, IEEE AP-S Honorable Mention Student Paper Contest 2008, IES Prestigious Engineering Achievement Award 2006, I2R Quarterly Best Paper Award 2004, IEEE iWAT 2005 Best Poster Award, and ISAP 2010 Best Paper Award.

Xianming Qing (M’90) was born in 1965 in China. He received the B.Eng. degree from the University of Electronic Science and Technology of China (UESTC), China, in 1985 and the Ph.D. degree from Chiba University, Japan, in 2010. During 1987–1996, he was with the UESTC for teaching and research as a Lecturer in 1990 and an Associate Professor in 1995. He joined National University of Singapore in 1997 as a Research Scientist, where he focused on study of high-temperature superconductor microwave devices. Since 1998, he has been with the Institute for Infocomm Research (I R, formerly known as CWC and ICR), Singapore. He is currently holding the position of Research Scientist and Leader of the Antenna Group under the RF and Optical Department. His main research interests are antenna design and characterization for wireless applications. In particular, his current R&D focuses on small and broadband antennas/arrays for wireless systems, such as ultra-wideband systems, radio frequency identification systems and medical imaging systems, microwave, mmW, submmW, and THz systems. He has authored and coauthored over 100 technical papers published in international journals or presented at international conferences, and five book chapters. He holds eight granted and filed patents. Dr. Qing received six awards of advancement of science and technology in China. He is also the recipient of the IES Prestigious Engineering Achievement Award 2006, Singapore, and the ISAP 2010 Best Paper Award.

Wei Hong (M’92–SM’07) received the B.S. degree from the University of Information Engineering, Zhengzhou, China, in 1982 and the M.S. and Ph.D. degrees from Southeast University, Nanjing, China, in 1985 and 1988, respectively, all in radio engineering. Since 1988, he has been with the State Key Laboratory of Millimeter Waves, serves as the Director of the lab since 2003, and is currently a Professor and the Associate Dean of the School of Information Science and Engineering, Southeast University. In 1993, 1995, 1996, 1997 and 1998, he was a short-term Visiting Scholar with the University of California at Berkeley and the University of California at Santa Cruz, respectively. He has been engaged in numerical methods for electromagnetic problems, millimeter wave theory and technology, antennas, electromagnetic scattering, RF technology for mobile communications, etc. He has authored and coauthored over 200 technical publications, as well as authored two books Principle and Application of the Method of Lines (in Chinese, Southeast University Press, 1993) and Domain Decomposition Methods for Electromagnetic Problems (in Chinese, Science Press, 2005). Dr. Hong is a senior member of CIE, Vice-President of the Microwave Society and Antenna Society of CIE, and is serving as the Associate Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUE. He is also an editorial board member for IJAP and RFMiCAE. He was thrice awarded the first-class Science and Technology Progress Prizes issued by the Ministry of Education of China and the Jiangsu Province Government. He also received the Foundations for China Distinguished Young Investigators Award and the “Innovation Group” issued by NSF of China.

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A Planar Dualband Antenna Array Wee Kian Toh, Member, IEEE, Xianming Qing, Member, IEEE, and Zhi Ning Chen, Fellow, IEEE

Abstract—A planar dual-band antenna array for point-to-point communication is presented. The proposed compact antenna array operates at 2.4–2.5 GHz (4%) and 5.1–5.5 GHz (7.5%) with 10 dB, and corresponding gains of 11–13.3 dBi and 11 10.6–14.5 dBi respectively. It consists of four branches suspended 250 mm ground plane. Each at a height of 5 mm above a 250 quarter consists of a series-fed meandered step-impedance filter connected to a L-shaped radiator, and they transverse horizontally away from the epicenter. The insertion loss method with non-redundant synthesis for filter design is applied to control the radiation aperture of the antenna array, allowing both the 2.4 GHz and 5.5 GHz radiators to co-exist and operate independently. A stable broadside radiation pattern and gain are achieved. Index Terms—Antenna array, dual-band antenna array, filter, microstrip antennas, multifunction aperture, patch antennas, WLAN.

I. INTRODUCTION

P

LANAR antenna arrays have been employed for backhaul point-to-point communication links for decades. They have a lower profile than reflector, cavity backed, and horn antennas. Suspended patch antenna array substitutes substrate material with air to reduce the -value and loss tangent. Consequently, it achieved greater bandwidth, lower dielectric and surface wave losses, at the expense of increased height from 1% to 10% of a free space wavelength . These antennas [1] are usually edge-, coupled- or aperture-fed [2] with an aspect ratio apart to minimize of 0.7–1.3 [3]. They are spaced 0.3– mutual coupling and grating lobes. Microstrip feeding network employs substrate with high relto reduce fringing [4]. Therefore ative permittivity complicated multi-layered structures allow varying -values for optimizing the feeding network and patches. The corporate-fed network [5] is selected for its broadband characteristic and ease of phase control, at the expense of multiple parallel feeding lines. The reverse is true for a series-fed network [6]. The IEEE 802.11a/b/g wireless local area network (WLAN) operates at 2.4–2.5 GHz, 5.15–5.35 GHz, 5.45–5.75 GHz and/or 5.725–5.875 GHz. Currently, the 2.4 and 5.5 GHz point-to-point antenna arrays are mounted separately with discrete ports. It is difficult to achieve a broadband planar antenna array with stable directional radiation patterns from 2.0–6.0 GHz (100%), as gain is proportional to radiation aperture. Therefore, a dual-band antenna is preferred for its simple system configuration and cost Manuscript received May 13, 2010; revised July 08, 2010; accepted August 19, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Institute for Infocomm Research (IIR), Singapore 138632, Singapore (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.2010.2103039

Fig. 1. Equivalent T -circuit of a short transmission line.

reduction. The existing planar dual-band antenna elements with stable broadside radiation patterns [7] could not be readily arranged to form an array with controlled radiation pattern. Planar dual-band antenna array designs have been achieved by employing high order modes slotted waveguide [8]. Compact dualband solutions using multifunction aperture are also achieved by using open waveguide operating in different modes [9] and dichroic properties of frequency selective surfaces [10]. In this paper, a single port planar dual-band antenna with 2 2 2.4 GHz and 4 2 5.5 GHz array elements is presented. II. DESIGN METHODOLOGY The following issues are considered for designing a planar dual-band antenna array. (a) The radiation apertures of the optimally spaced and elevated 2.4 GHz and 5.5 GHz array elements overlap. The coupling between these radiators reduces radiation efficiency and isolations. (b) The 2.4 GHz radiators are electrically large at 5.5 GHz. The 5.5 GHz high-order mode resonances on the 2.4 GHz radiators result in unwanted radiation patterns. (c) A single port broadband or dual-band feeding network is required for all radiators. The proposed solution is as follows. (a) A series-fed network is chosen to reduce the number of microstrip lines. This minimize unwanted coupling between the feeding network and radiators. (b) Microstrip filters with a passband characteristic at 2.4 GHz and stopband characteristic at 5.5 GHz are employed to feed the 2.4 GHz radiators. This prevents the 2.4 GHz radiators from resonating at 5.5 GHz. (c) The reflected 5.5 GHz signal is radiated using 5.5 GHz resonators along the microstrip filter. This prevents the reflected signal from reactance loading the antenna feed, thus allowing for a better impedance matching. Based on the considerations above, a feeding network with filtering and radiating attributes is designed and analyzed in the next section. III. FEEDING NETWORK, FILTER AND RESONATOR DESIGN Distributed elements are used for high frequency filter design, and the distances between these elements are considerable. Lump circuit model does not predict the distributed element filter response accurately. Therefore, a full wave simulator

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Fig. 2. Filter A, 2nd order maximally flat low-pass step-impedance filter (a) microstrip design, (b) equivalent ladder-type circuit model, and simulation, (c) S -parameters, (d) gain.

IE3D is used for all simulations in this paper. The -parameter for a -equivalent circuit of a short transmission line is shown in Fig. 1, using (1)–(3) [11]. A short microstrip line with large characteristic impedance has a positive reactance and negligible shunt capacitance. The reverse is true for a short microstrip line with small characteristic impedance. They are approximated as series inductance and shunt capacitance by (4) and (5) respectively. Using the scaling equations (6) and (7), the electrical length for the inductance and capacitance are approximated using (8) and (9) (1)

Fig. 3. Filter B , (a) 2nd order step-impedance filter, and simulation (b) S -parameters, (c) gain, (d) 5.0 GHz to 6.2 GHz radiation patterns.

where is the propagation constant, and is the length of microstrip line (4) (5)

(2)

(6)

(3)

(7)

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Fig. 4. Filter C , (a) 2nd order step-impedance filter, and simulation (b) S -parameters, (c) gain, (d) 5.6 GHz radiation pattern, (e) 5.8 GHz radiation pattern, (f) 6.0 GHz radiation pattern.

(8) (9) where is the inductance, is the shunt capacitance, is the admittance, and are the normalized inductance and capacis the source resistance, is the cut off frequency, itance, and are the scaled inductance and capacitance values. Using the element value of a 2nd order maximally flat low-pass filter prototype [12], , , with (6) and (7), the microstrip step-impedance low-pass filter design with a cutoff frequency at 2.4 GHz is shown in Fig. 2(a). The 6 mm and 26

mm wide microstrip line suspended at the height of 5 mm have and respectively, an impedance of as shown in Fig. 2(b). A high-low impedance ratio is maximized to achieve a good approximation. The simulation -parameters are shown in Fig. 2(c). There is a 0.3 GHz upshifting in the passband at 2.7 GHz, and transmission poles are located at 2.1 GHz and 4.1 GHz. The reflection coefficients and are not identical beyond 4.4 GHz due to the non-symmetrical structure. Fig. 2(d) depicts the increasing radiation loss and unstable radiation patterns of the step-impedance filter. Both the low impedance microstrip line resonance sections are extended to 24 26 mm to enhance the 5.5 GHz radiation, while providing a passband at 2.7 GHz, as shown in

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Fig. 7. Measured maximum gains and cross-polarization in the E- and H-planes, and simulated maximum/boresight gains.

Fig. 5. Schematic diagram.

Fig. 6. Measured and simulated reflection coefficients.

Fig. 3(a) and (b). The reflection coefficients and are identical due to the symmetrical structure. A good impedance matching , is achieved from 5.1 GHz to 6.5 GHz. Fig. 3(c) depicts the radiation loss from 5 GHz onwards. Fig. 3(d) shows the H-plane radiation patterns from 5.0 to 6.2 GHz, with radiation nulls at boresight. Half of the step-impedance filter is rotated by 180 , so that the 5.5 GHz radiators (low impedance line section) resonate in-phase to ensure a broadside radiation pattern. This increases the coupling between the low impedance lines, as shown in Fig. 4(a). Fig. 4(b) shows that the passband at 2.7 GHz is maintained while a good impedance matching is achieved from 5.5 to from 5.5 to 6.2 6.2 GHz. The low transmission coefficient GHz and reflection coefficient indicate that it has a higher radiation loss. Fig. 4(c) shows a non-squinting broadside radiation pattern from 5.1 to 5.7 GHz, and a weak squinting radiation pattern at 2.7 GHz. The simulation radiation patterns for 5.6, 5.8, and 6.0 GHz are shown in Fig. 4(d)–(f), respectively. This structure is implemented on a planar dual-band antenna array design in the next section.

Fig. 8. Measured radiation patterns in the H- and E-planes for (a) 2.4 GHz and (b) 5.3 GHz.

IV. ANTENNA DESIGN Fig. 5 shows the proposed planar dual-band antenna array design. It is symmetrical along the -axis. It is suspended at a and ) above a 250 height of 5 mm ( 250 mm ground plane and supported using Styrofoam and nylon stubs. The step-impedance filter has a passband at 2.4 GHz and stopband for frequencies 5 GHz. It consists of two 5.5 GHz step-impedance resonators. The 2.4 GHz reactance loaded L-shaped radiators are connected to the filter. spacing between the array elements are achieved for both A

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Fig. 10. Simulated gain for height from 4 mm to 6 mm.

Fig. 9. Simulated current vector distribution at (a) 2.4 GHz and (b) 5.3 GHz.

the 2.4 GHz (62.5 mm) and 5.5 GHz (27.0 mm) elements, by using the L-shaped radiators. This ensures a non-grating lobes radiation patterns. The 25 mm long microstrip line at the center provides a 180 out-of-phase signals for feeding the 5.5 GHz radiators with reference to the top 5.5 GHz radiators. However, at 2.4 GHz the 90 phase shift is not sufficient, and an addition 90 of phase shifting is achieved by extending the feeding location of the lower 5.5 GHz to 2.4 GHz radiators. V. RESULTS AND DISCUSSION Fig. 6 shows the measured and simulated reflection coefficients. There is a 0.15 GHz and 0.4 GHz upshifting in the simulated results for the 2.4–2.5 GHz and 5.1–5.5 GHz resonances compared to the measurement. Fig. 7 compares the measured and simulated gains. There is a 0.2 GHz and 0.1 GHz upshifting in the simulation gains respectively. The measured maximum gain from 2.4–2.5 GHz in the E- and H-planes vary from 11–13.3 dBi, 2 dBi lower than that of the simulated

maximum gain. This is due to the lossy step-impedance filter. The measured maximum gain at 5.1–5.5 GHz varies from 10.6–14.5 dBi, and they are in good agreement with that of the simulation. The symmetrical structure along the -axis ensures a symmetrical H-plane radiation patterns at both 2.4 GHz and 5.3 GHz, as shown in Fig. 8. There is a 5 of beam squinting in the E-plane pattern at 2.4 GHz. The E- and H-planes 3-dB beamwidths are 25 /25 and 35 /20 at 2.4 GHz and 5.3 GHz respectively. Fig. 9(a) depicts the vector current distribution at 2.4 GHz. The L-shaped radiators are resonating in phase. The magnitude of the vector currents from the top radiators is slightly stronger than that of the bottom radiators. This results in a slightly squinted E-plane radiation pattern. Fig. 9(b) depicts the vector current distribution at 5.3 GHz. The 5.3 GHz radiators are radiating in phase, while the 2.4 GHz L-shaped radiators have a weak current distribution due to the filtering effect of the step-impedance filters. The discrete variations in the radiation aperture at 2.4 GHz and 5.3 GHz ensure a stable radiation pattern without grating lobes. Fig. 10 shows the simulation results when the height of the radiators is varied from 4 mm to 6 mm. This design has a high mechanical tolerance, and there are little changes to the impedance matching and radiation patterns. VI. SUMMARY A planar dual-band antenna array design with stable radiation pattern and gain has been studied. The methodology for designing a step-impedance filter with controlled radiation loss has been highlighted. Using the step-impedance filter, the radiation aperture of the dual-band array is controlled. This design has a high mechanical tolerance. It is compact, easy to manufacture and cost effective. REFERENCES [1] Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and experiment on microstrip antennas,” IEEE Trans. Antennas Propag., vol. AP-27, pp. 137–145, Mar. 1979. [2] D. M. Pozar, “A microstrip antenna aperture coupled to a microstripline,” Electron. Lett., vol. 21, pp. 49–50, Jan. 1985.

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[3] Z. N. Chen and M. Y. W. Chia, Broadband Planar Antennas: Design and Applications. Hoboken, NJ: Wiley, Feb. 2006. [4] D. H. Schaubert, D. M. Pozar, and A. Adrian, “Effect of microstrip antenna substrate thickness and permittivity: Comparison of theories with experiment,” IEEE Trans. Antennas Propag., vol. 37, no. 6, pp. 677–682, Jun. 1989. [5] R. E. Munson, “Conformal microstrip antennas and microstrip arrays,” IEEE Trans. Antennas Propag., vol. AP-22, pp. 74–78, Jan. 1974. [6] W. Menzel, “A 40 GHz microstrip array antenna,” in MTT-S Int. Microwave Symp. Digest, May 1980, vol. 80, no. 1, pp. 225–226. [7] W. K. Toh and Z. N. Chen, “A tunable dual-band planar antenna,” Elect. Lett., vol. 4, no. 1, pp. 8–9, Jan. 2008. [8] J. Holtzman, “A dual band array,” IEEE Trans. Antennas Propag., vol. 16, no. 5, pp. 603–604, Jan. 1968. [9] W. Gregorwich, “A multipolarization dual-band array,” in Proc. Antenna and Propagation Society Int. Symp., Jan. 1975, vol. 13, pp. 189–192. [10] J. James and G. Adrasic, “Dichroic dual-band microstrip array,” Elect. Lett., vol. 22, no. 20, pp. 1040–1042, Sep. 1986. [11] D. M. Pozar, Microwave Engineering, 3rd ed. Hoboken, NJ: Wiley, 2005, ch. 8, p. 370. [12] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching, Networks, and Coupling Structures. Boston, MA: Artech House, 1980.

Wee Kian Toh (S’01–M’05) received the B.Eng. degree from the University of Edinburgh, Edinburgh, U.K., in 2000 and the Ph.D. degree from Queen Mary University of London, London, U.K., in 2004. Since 2005, he has been a Research Fellow at the Institute for Infocomm Research, Singapore. His research interests include broadband antennas design and nonlinear dynamics.

Xianming Qing (M’90) was born in May 1965, in China. He received the B.Eng. degree in electromagnetic field engineering from the University of Electronic Science and Technology of China (UESTC, formerly known as Chengdu Institute of Radio Engineering), Chengdu, China, in 1985, and the Ph.D. degree from Chiba University, Japan, in 2010. During 1987 to 1996, he was with UESTC for teaching and research and was appointed a Lecturer in 1990 and an Associate Professor in 1995. He joined the Physics Department, National University of Singapore (NUS), in 1997, as a Research Scientist, where he focused on study of high-temperature superconductor (HTS) microwave devices. Since 1998, he has been with the Institute for Infocomm Research, Singapore (I2R, formerly known as CWC and ICR). He currently holds the position of Research Scientist and is the leader of the Antenna Group under the RF and Optical Department. His main research interests are antenna design and characterization for wireless applications. In particular, his current R&D focuses on small

and broadband antennas/arrays for wireless systems, such as ultrawideband (UWB) systems, radio frequency identification (RFID) systems and medical imaging systems, microwave, mmW, submmW, and THz imaging systems. Since 1991, he has authored and coauthored over 70 technical papers published in international journals or presented at international conferences, and five book chapters. He holds eight granted and filed patents. Dr. Qing received six awards of advancement of science and technology in China. He is also the recipient of the 2006 IES Prestigious Engineering Achievement Award, Singapore, and IEEE ISAP Best Paper Award 2010. He has been a member of the IEEE Antennas and Propagation Society since 1990. He served as the organizer and Chair for special sessions on RFID antennas at the IEEE Antenna and Propagation Symposium 2007 and 2008. He also served as Guest Editor of the International Journal on Wireless & Optical Communications Special Issue on Antennas for Emerging Radio Frequency Identification (RFID) Applications. He has served as a TPC member and Session Chair for a few conferences and as a Reviewer for many prestigious journals such as IEEE T-AP, T-MTT, IEEE-AWPL, MWCL, IET-MAP, Electronic Letters, etc.

Zhi Ning Chen (F’05) received the B.Eng., M.Eng., and Ph.D. degrees from the Institute of Communications Engineering, China, and the Ph.D. degree from University of Tsukuba, Japan, respectively all in electrical engineering. During 1988 to 1997, he worked at Institute of Communications Engineering, Southeast University, and City University of Hong Kong, China, with teaching and research appointments. In 1997, he was awarded a JSPS Fellowship to join in University of Tsukuba, Japan. In 2001 and 2004, he visited the University of Tsukuba under a JSPS Fellowship Program (senior level). In 2004, he worked at the IBM T. J. Watson Research Center, New York, as an Academic Visitor. Since 1999, he has worked with Institute for Infocomm Research, Singapore and his current appointments are Principal Scientist and Department Head for RF & Optical. He is concurrently holding Adjunct/Guest Professorships at Southeast University, Nanjing University, Shanghai Jiao Tong University, Tongji University, University of Science and Technology, China, Dalian Maritime University, and National University of Singapore. He has published 290 journal and conference papers as well as authored and edited the books entitled Broadband Planar Antennas, UWB Wireless Communication, Antennas for Portable Devices, and Antennas for Base Station in Wireless Communications. He also contributed chapters to the books entitled UWB Antennas and Propagation for Communications, Radar, and Imaging as well as Antenna Engineering Handbook. He holds 28 granted and filed patents with 21 licensed deals with industry. His current research interest includes applied electromagnetic engineering, RF transmission over bio-channels, and antennas for wireless systems, in particular at mmW, submmW, and THz for medical and healthcare applications. Dr. Chen is a Fellow of the IEEE for his contribution to “small and broadband antennas for wireless.” He is serving as an IEEE Antennas and Propagation Society Distinguished Lecturer and an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He has organized many international technical events as key organizer. He is the founder of International Workshop on Antenna Technology (iWAT). He is the recipient of the CST University Publication Award 2008, IEEE AP-S Honorable Mention Student Paper Contest 2008, IES Prestigious Engineering Achievement Award 2006, I2R Quarterly Best Paper Award 2004, IEEE iWAT 2005 Best Poster Award, and IEEE ISAP Best Paper Award 2010.

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Synthesis of Planar Arrays With Elements in Concentric Rings Joaquim A. R. Azevedo

Abstract—The circular symmetry of concentric ring arrays becomes an advantage for situations where the radiation patterns are symmetric in the azimuthal direction. For planar arrays with elements in an equispaced grid, the fast Fourier transform can be applied in the calculations involved between the array factor and the source distribution. However, for circular arrays the elements are in a non-equispaced grid, making the application of fast computation algorithms difficult. Therefore, two techniques are proposed to synthesize circular arrays. The first one provides the control of the pattern by sampling the array factor. In this case, the fast Fourier transform can be applied. The windows technique is also utilized to control the sidelobe levels and the ripple structure of the shaped beam. The second technique considers the symmetric nature of concentric circular arrays and the control of the array factor is performed imposing some specific points of the pattern. Index Terms—Antenna array synthesis, Fourier transform, planar arrays, shaped beam antennas.

I. INTRODUCTION N comparison with linear arrays, the potentialities of planar arrays for providing a better control of the radiation pattern are well known. Typically, planar arrays have rectangular or circular boundaries with elements in a rectangular or triangular grid [1]–[5]. For applications, such as in radar, acoustics and wireless communications, another type of planar arrays consists of concentric rings with elements in equispaced positions of each circular array [6]–[11]. The use of synthesis methods that permit the calculation of the array source distribution in a deterministic manner is very attractive but the control of the radiation pattern is limited. The use of numerical optimization techniques has been providing good results when the radiation pattern has many constraints. However, the time required to get the results may be very high. Tseng and Cheng [1] synthesize Chebyshev patterns with planar arrays but the method can be only applied to rectangular boundaries and the number of elements in both directions must be the same. Goto and Cheng [6] obtain concentric ring arrays sampling a circular aperture distribution. The technique uses a circular aperture with a Taylor distribution [12]. Elliot and Stern [13] extend Taylor’s circular aperture technique to produce flat-topped beams with controlled ripple and sidelobe levels. The method uses the roots of the Shelkunoff unit circle to

I

Manuscript received March 10, 2009; revised July 09, 2010; accepted September 27, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported by the CCM Research Unit. The author is with the Department of Mathematics and Engineering, University of Madeira, Funchal, Portugal (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2102999

produce a continuous aperture, with a procedure similar to the one considered for linear arrays [14]. Then, the source distribution is sampled to obtain the circular grid array. The pattern exhibits symmetry in the direction. Hodges and Rahmat-Samii [3] also deal with sampling of continuous distributions, due to the relative difficulty in synthesizing planar arrays and to the broad availability of continuous aperture synthesis techniques. The objective is to minimize the error associated with the conventional sampling. The examples are focused only on rectangular grids with circular boundary, although the authors refer that it could be extended to other grids and boundaries. Li, Ho, and Kwan [9], [15] use the expressions of continuous rings to synthesize concentric ring arrays. In his book, Elliot [16] also addresses the sampling of continuous distributions to synthesize pencil beam patterns. As the conventional sampling produces degradation of the desired patterns, perturbation techniques to improve the results are considered. In [17], Albagory et al. proposed a technique to control the sidelobe levels of concentric circular arrays. All elements in an individual ring have the same excitation and different rings are weighted with different excitations through a Gaussian window. To obtain optimal results, the study includes several graphs showing the relationship between array parameters. Keizer [5] presents a method to synthesize planar arrays with elements in a periodic grid by successively applying the fast Fourier transform (FFT). The method is iterative and has the advantage of being fast, compared with other methods that seek to control the array factor. To provide a great control of the radiation pattern, most of the published methods for the synthesis of planar arrays use optimization procedures [4], [18]–[24]. The source distribution can be optimized but it requires a lot of time to obtain the results, which is a disadvantage for real time applications. Therefore, appropriate techniques that work in conjunction with optimization methods may be considered to provide better results. In this context, some techniques use the FFT to obtain the results in a quicker manner. In this case, the array elements are usually placed in rectangular grids [4], [5], [18], [25], [26]. In a previous work [26], a method was developed for the synthesis of planar arrays that considers the periodicity of the array factor and nonuniform sampling to control each sidelobe level of a beam pattern. In this work, a procedure will be introduced that synthesizes planar arrays with circular symmetry. A concentric circular array has the advantage of being more compact and provides a rotation of the main beam by the shift of the array element excitations. However, for circular arrays the periodicity of the array factor found in arrays with equispaced elements does not exist. Therefore, an appropriate procedure must be determined if ones intends to use the FFT in the calculations

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involved. Two techniques are developed to deal with the synthesis of concentric ring arrays in order to produce pencil beams and shaped-beam patterns. II. THEORY It is well-known that for the far-field of an antenna array the source distribution is the Fourier transform of the array factor. For planar arrays, this relation was considered in [26] with apis the source distribution, the propriate variables. If array factor is given by

(1) Fig. 1. Geometry of a concentric circular array.

and, knowing the array factor, the source distribution is with and is an integer. To explore the symmetries of the array, the source distribution is defined by

(5) (2) with is the wavelength, is the angle between the direction and the point of the far-field and is the angle between the direction and the projection of the far-field point in the plane. Since the array has a limited number of elements, the summations in (1) have finite limits. For elements on equispaced grids the array factor is periodic and the sampling theory can be applied. In this case, the FFT algorithm is a powerful tool to perform the calculations between the source distribution and the array factor. Equations (1) and (2) can also be used for circular arrays, as and may be defined in any position of an aperture. However, compared with equispaced element arrays, for concentric and circular arrays the array factor is not periodic in the domain and the FFT cannot be directly applied. Nevertheless, due to the symmetry it is preferred to work in polar variables. The Fourier transform for two-dimensional systems in polar coordinates is known by Fourier-Bessel transform [27]. Considering the changing of variables

(3) and following the procedure developed in [28] and [29], the concentric rings and with array factor of a planar array with elements in the ring is given by

(4)

The term corresponds to an element in the origin. Only a few terms of are usually enough to obtain a good approximation of the array factor [29]. The main beam may be oriented to a specific direction of space by the use of the translation property of the Fourier transform. The relation between the polar coordinate variables and the spherical coordinate system, represented and . Considering (3), the source in Fig. 1, is distribution in polar coordinates is given by

(6)

As may be noted from (4), the array factor of a circular array depends on both variables of the polar coordinates, even if the source distribution is symmetric. Thus, to calculate the exact source distribution using (6) it is necessary to consider the entire domain of . However, for practical situations the desired pattern is specified inside the visible window. In this case, may be null for . Furthermore, if it is intended for the pattern to have a circular symmetry, the source distribution is . In this situation (6) becomes determined with the Hankel transform of zero order [28]

(7) Since the array factor of a non-equispaced discrete array is not limited in the variables and , and correspondingly in , the source distribution determined from (7) gives rise to an approximation of the desired array factor. For the calculation of (7) there are algorithms in literature that implement the fast Hankel transform [30], [31].

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III. SYNTHESIS TECHNIQUES Taking into account the relation between the source distribution and the array factor, a desired pattern can be approximated through the calculation of (2) or (7). However, the approximate array factor may not have the desired characteristics or the calculation procedure may be difficult. Furthermore, it is important to use an efficient computation tool to perform the calculations of a concentric circular array. In this section two synthesis techniques will be presented. The first one is based on the uniform sampling of the desired array factor and on window techniques to control some characteristics of the approximate array factor. The second one provides this control through a set of non-uniform samples of the array factor. For mutual coupling purposes, the distance between elements is . Therefore, the distance between rings is and at least each ring has a number of elements equal to or below . An element is also placed in the origin. A. Uniform Sampling of the Array Factor To consider an efficient computation tool for the synthesis problem it is important to use the properties of the Fourier transform. Taking into account that the planar array is limited in space, its coefficients can be determined by sampling . The number of samples must be large enough to avoid the aliasing effect and depends on the array dimension. In this case, the FFT can be applied to (2). In contrast to a periodic grid array, where the array factor is fully specified by the samples over a period, for circular arrays the number of samples must be high in order to obtain values around the non-equispaced element positions. If necessary the zero-padding technique is applied. When the array factor is calculated by the FFT, the determined coefficients are in a rectangular grid. The coefficients of the concentric circular array are obtained by interpolation. This procedure can be applied to any array grid and boundaries. To summarise, the synthesis technique begins with the sampling of the desired array factor in a rectangular grid to provide the application of the FFT. The coefficients determined by the FFT are also in a rectangular grid around the positions of the desired concentric circular array. The number of samples must be large enough to produce a coarse approximation of a continuous source distribution. The coefficients of the circular array are determined using an interpolation method. In this work, the cubic interpolation was considered. The result can be improved by the use of appropriate windows such as the Chebyshev and the Gaussian ones. The Chebyshev planar window is the one that produces patterns with equal sidelobe levels [1]. In the variables of the array factor, the inverse Fourier transform of the window function is

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is the length of a rectangular array with distance bewhere tween elements and the SLL is the sidelobe level of the Chebyshev pattern. The function of the Gaussian window is

(9) with the maximum radius of the array and is a parameter that controls the beamwidth and the sidelobe levels. Compared to the Chebyshev window, the Gaussian pattern is easier to manage since only one parameter is needed. The synthesis procedure consists in the application of the previously presented technique to both functions, to the desired array factor and to (8) or (9). It can be considered that the Fourier transform of a Gaussian function is also a Gaussian function. Thus, the window can be directly obtained from the corresponding transform. Then, the source distribution is the multiplication of both results. The approximate array factor is determined by (4). B. Non-Uniform Sampling on the Array Factor Radius The second technique to synthesize planar arrays of concentric rings considers the circular symmetry of the array. In this case, all elements in a ring have the same excitation and different rings may have different excitations. Taking into account this kind of source distribution, the array factor control will be performed in the variable, which corresponds to a variation in . For the variable the array factor is practically symmetrical. Considering (4), if it is intended for the array factor to pass with a value , then through the point

(10) For a given it is possible to impose values of the array equations defined by the previous factor. Therefore, with unknowns is obtained. The unexpression a system of knowns are , and are determined by matrix inversion. Therefore, the two-dimensional problem was transformed to one-dimensional one. A simple way to control the array factor is through the use are of the peaks of the function. The imposed values amplitudes of the sidelobe region and maximums and minimums of the shaped-beam zone. The initial points could be equispaced positions inside the visible window. With (4) the approximate array factor is determined. If the specifications were not verified, corrections are carried out in a similar way to what was performed in [26] for rectangular grids. IV. RESULTS

(8)

The Chebyshev and Gaussian windows permit truncation of the array coefficients in order to obtain a concentric circular array. Since in many applications the desired array factor consists of beam patterns, these functions can be used to synthesize these patterns with controlled sidelobe levels. A planar array

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Fig. 3. Beam pattern oriented to 

Fig. 2. Beam pattern: (a) array factor; (b) source distribution.

with elements in a rectangular grid is synthesized in [26] with an iterative procedure. The array has 81 elements with half a dB. If (8) wavelength spacing and sidelobe levels below is the desired array factor of a beam pattern, with dB, , and , the uniform sampling technique provides the pattern represented in Fig. 2(a). Each parameter of the window function can be used to control the array factor. The SLL value considered in (8) is the one that ensures the desired sidelobe levels in the entire region of the visible window. The FFT was applied with 16 16 points. The directivity is 23.2 dB against 22.5 dB of [26] and the beamwidths of both patterns are similar. However, with the proposed technique the calculation of the source distribution is direct. Fig. 2(b) shows the source distribution. For this example, the dynamic range ratio, , is 4.5. The next desired pattern is a pencil beam oriented to and the concentric circular array has ten rings. In this case, in (4) it is considered that and the array has 341

= 30

and 

= 45

.

and , the elements. Using the relations between can be determined. The direction of the main beam in and deinitial array factor is a Gaussian function, with layed to the beam position. Next, the first technique is applied with 32 32 points for the FFT, giving the result depicted in Fig. 3. The directivity is 28.5 dB and, for comparison, the diwould be rectivity of the beam pattern oriented to 29.2. For the following example the pattern synthesized in [22] having a rectangular contour is considered. The number of elements is 208 and they are placed in a square grid with a cir. In that cular boundary. The distance between elements is work, an iterative procedure was utilized to control the sidelobe level and the dynamic range ratio. The SLL must be below dB and the final is 24.4 for 0.3 dB of ripple amplitude. As a comparison, the first technique previously presented was applied with 32 32 points for the FFT. The desired contour region, in the variables of interest, is and . The initial concentric ring array has the same number of elements as the one considered in [22]. After the application of the uniform sampling technique, the approximate array factor has a SLL below the desired one but the dynamic range ratio is very high. To produce a dynamic range ratio of 20, all elements with small values were removed. However, the new approximate array factor has higher sidelobe levels than the desired one. Therefore, the Gaussian window was applied with , giving the result shown in Fig. 4. The peak to peak ripple is 1.3 dB and the transition region is only 1 larger the obtained before the applying the window and eliminating the elements. The source distribution is presented in Fig. 4(b) and has only 85 elements. Comparing with the most existing methods of planar array synthesis, the first technique is faster since no iterative procedure is considered and it uses the FFT. The next example is a shaped-beam pattern with circular symmetry. The second technique is applied, which corresponds to

AZEVEDO: SYNTHESIS OF PLANAR ARRAYS WITH ELEMENTS IN CONCENTRIC RINGS

Fig. 5. Cuts

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= 0 and =  for the initial circular shaped beam.

Fig. 4. Rectangular pattern: (a) array factor; (b) source distribution.

calculate the excitation for each array ring. The distance be. The beam of the desired pattween elements is at least tern is in the region , the SLL is dB and the rings ripple is 0.5 dB. The array is synthesized with corresponding to 410 elements. For these specifications, twelve points of the array factor in the variable are imposed since the desired pattern is independent of . The beam is in the region . The initial array factor, corresponding to the visible window, is depicted in Fig. 5 for two azimuth direcand . The ripple is 1.6 dB and the highest tions, sidelobe is dB. To produce the pattern with the desired characteristics, the ripple was imposed as well as the peaks of the sidelobe region. For the non-uniform sampling technique, it in (10). Fig. 6(a) was considered that shows the result after this correction to the pattern. As can be observed, most of the sidelobes have the desired amplitude for , which was the one considered to control the patthe cut tern. With one more correction the pattern has the desired ripple

=0

Fig. 6. Cuts and (b) after four iterations.

=  for circular pattern: (a) after the first correction;

structure and the sidelobe levels for are around dB. However, higher values of sidelobes can appear in other directions of the pattern. To verify the specifications, the imposed points of the higher positions of must have lower amplitudes. The result is represented in Fig. 6(b) for four corrections of the sidelobe structure. The final array factor is represented in Fig. 7 for the visible window region. Keizer [5] developed a method based on the FFT. For beam patterns with controlled sidelobe levels, the author refers to the

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Fig. 8. Beam pattern with different zones of sidelobe levels. Fig. 7. Approximate circular pattern.

great advantage of his method as being faster than several published results. Furthermore, most of them are applied to arrays with a moderate number of elements. The work presented in this paper also permits the synthesis of arrays with thousands of elements. To demonstrate that the method proposed here is also suitable for arrays with a great number of elements, the next example deals with the synthesis problem of an array with 3509 . elements. This number of elements corresponds to The non-uniform sampling technique was applied in 34 points of the array factor. Several zones of the sidelobe region have different levels in order to show the pattern control potentialities. The approximate array factor is shown in Fig. 8. To produce a sidelobe structure similar to Fig. 8 in [5], thousands of iterations were necessary, meaning it took several minutes to obtain the results. Meanwhile, using the proposed technique only five corrections on the sidelobe peaks were enough to obtain the desired , which was the direction used to control amplitude in the the pattern. For other directions the sidelobe structure follows the desired one very well. The calculations were performed on a PC with a Intel Pentium 4 processor at 3 GHz, giving 4.5 s. Therefore, in the presented technique the number of iterations to provide the results is very low. The calculated directivity is 36.4. Fig. 9 shows the amplitude of the source distribution. As can be observed, the variation between rings is relatively smooth. The method developed in [5] has the capability of providing more control of the radiation pattern in all regions of the sidelobe structure, since it deals with the excitation of each array element placed in a rectangular grid. However, for the synthesis of radiation patterns with radial symmetry, the non-uniform sampling technique has the great advantage of providing results in a quicker manner and it is also simple to implement. It is interesting to notice that even working with thousands of elements, the non-uniform sampling technique deals with a small number of unknowns. This provides fast calculations and the source distribution is very simple since all elements in a ring have the same excitation.

Fig. 9. Source distribution for the pattern of Fig. 8.

V. CONCLUSION Two techniques have been proposed for synthesizing planar arrays with concentric rings. The first one considers uniform sampling of the array factor to obtain a coarse source distribution. The control of the pattern can be performed by the use of appropriate functions or by the use of some samples of the array factor. For further control, windows functions can be applied. The fast Fourier transform permits fast calculations between the array factor and the source distribution. The second technique controls the array factor imposing some specific points of this function. The first technique also provides some control of the pattern in the region out of the visible window. This is important for scanning purposes. However, with this technique it may be difficult to impose very low sidelobe levels. The second technique provides a better control of the ripple structure of shaped-beam patterns and sidelobe levels, but this control is usually performed inside the visible window. Contrasting with

AZEVEDO: SYNTHESIS OF PLANAR ARRAYS WITH ELEMENTS IN CONCENTRIC RINGS

most published methods, where a high number of iterations is necessary to synthesize the arrays, the techniques developed in this work are direct or the number of iterations is very small, providing a significant reduction in computation time. REFERENCES [1] F. Tseng and D. K. Cheng, “Optimum scannable planar arrays with an invariant sidelobe level,” Proc. IEEE, vol. 56, pp. 1771–1778, Nov. 1968. [2] S. W. Autrey, “Approximate synthesis of nonseparable design responses for rectangular arrays,” IEEE Trans. Antennas Propag., vol. AP-35, pp. 907–912, Aug. 1987. [3] R. E. Hodges and Y. Rahmat-Samii, “On sampling continuous aperture distributions for discrete planar arrays,” IEEE Trans. Antennas Propag., vol. 44, pp. 1499–1508, Nov. 1996. [4] 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. [5] W. P. M. N. Keizer, “Fast low-sidelobe synthesis for large planar array antennas utilizing successive fast Fourier transforms of the arrays factor,” IEEE Trans. Antennas Propag., vol. 55, pp. 715–722, Mar. 2007. [6] N. Goto and D. K. Cheng, “On the synthesis of concentric-ring arrays,” Proc. Lett., vol. 58, pp. 839–840, May 1970. [7] E. S. Elliot and G. J. Stern, “Footprint patterns obtained by planar arrays,” Proc. Inst. Elect. Eng., vol. 137, pt. Pt. H., pp. 108–112, Apr. 1990. [8] B. P. Kumar and G. R. Branner, “Designs of low sidelobe circular ring arrays by element radius optimization,” in Proc. IEEE Antennas and Propagation Society Symp., Aug. 1999, pp. 2032–2035. [9] Y. Li, K. C. Ho, and C. Kwan, “Designs of broadband circular ring microphone array for speech acquisition in 3-D,” in Proc. ICASSP, Hogkong, China, Apr. 2003, pp. 221–224. [10] M. Dessousky, H. Sharshar, and Y. Albagory, “Efficient sidelobe reduction technique for small-sized concentric circular arrays,” Progr. Electromagn. Res., pp. 187–200, 2006, PIER 65. [11] K. R. Mahmoud, M. El-Adawy, and S. M. M. Ibrahem, “A comparison between circular and hexagonal array geometries for smart antenna systems using particle swarm optimization algorithm,” Progr. Electromagn. Res., pp. 75–90, 2007, PIER 72. [12] T. T. Taylor, “Design of circular apertures for narrow beamwidth and low sidelobes,” IRE Trans. Antennas and Propag., vol. 8, pp. 23–26, Jan. 1960. [13] E. S. Elliot and G. J. Stern, “Shaped patterns from a continuous planar aperture distribution,” IEE Proc., vol. 135, pt. Pt. H., pp. 366–370, Dec. 1988. [14] H. J. Orchard, R. S. Elliot, and G. J. Stern, “Optimizing the synthesis of shaped beam antenna patterns,” IEE Proc., vol. 132, pt. Pt. H., pp. 63–68, Feb. 1985. [15] Y. Li, K. C. Ho, and C. Kwan, “Beampattern synthesis for concentric circular ring array using MMSE design,” in Proc. Int. Symp. on Circuits and Systems, 2004, pp. 329–332. [16] R. S. Elliot, Antenna Theory and Design. Englewood Cliffs, NJ: Prentice-Hall, 1981, ch. 6.

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[17] Y. A. Albagory, M. Dessousky, and H. Sharshar, “An approach for low sidelobe beamforming in uniform concentric circular arrays,” Wireless Pers. Commun., vol. 43, pp. 1363–1368, Dec. 2007. [18] O. Einarsson, “Optimization of planar arrays,” IEEE Trans. Antennas Propag., vol. AP-27, pp. 86–92, Jan. 1979. [19] B. P. Ng, M. H. Er, and C. Kot, “A flexible arrays synthesis methods using quadratic programming,” IEEE Trans. Antennas Propag., vol. 41, pp. 1541–1550, Nov. 1993. [20] P. Y. Shou and M. A. Ingram, “Pattern synthesis for arbitrary arrays using adaptive array method,” IEEE Trans. Antennas Propag., vol. 47, pp. 862–869, May 1999. [21] A. Trastoy, F. Ares, and E. Moreno, “Synthesis of non-'-symmetric patterns from circular arrays,” Electron. Lett., vol. 38, pp. 1631–1633, Dec. 2002. [22] J. A. Rodriguez, R. Muñoz, H. Esteves, F. Ares, and E. Moreno, “Synthesis of planar arrays with arbitrary geometry generating arbitrary footprint patterns,” IEEE Trans. Antennas Propag., vol. 52, pp. 2484–2488, Sept. 2004. [23] J. Jin, H. L. Wang, W. M. Zhu, and Y. Z. Liu, “Array patterns synthesizing using genetic algorithm,” presented at the Progress in Electromagnetics Research Symp., Cambridge, Mar. 2006. [24] R. L. Haupt, “Optimized element spacing for low sidelobe concentric ring arrays,” IEEE Trans. Antennas Propag., vol. 56, pp. 266–268, Jan. 2008. [25] L. L. Wang, D. G. Fang, and W. X. Sheng, “Combination of genetic algorithm (GA) and fast Fourier transform (FFT) for synthesis of arrays,” Microw. Opt. Technol. Lett., vol. 37, pp. 56–59, Apr. 2003. [26] J. A. R. Azevedo, “Antenna pattern control of planar arrays for long distance communications,” J. Adv. Space Res., vol. 43, pp. 1603–1610, June 2009. [27] H. Cohen, Mathematics for Scientists & Engineers. London: Prentice-Hall Int., 1992. [28] R. N. Bracewell, The Fourier Transform and Its Applications. New York: McGraw-Hill, 2000. [29] R. E. Collin and F. J. Zucker, Antenna Theory. New York: McGrawHill, 1969. [30] L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasidiscrete Hankel transform,” Opt. Lett., vol. 23, pp. 409–411, 1998. [31] M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am., vol. 21, pp. 53–58, Jan. 2004. Joaquim A. R. Azevedo was born in 1966. He studied electrical engineering (with specialization in telecommunications) and received the Engineering degree, the Master degree, and the Ph.D. degree in telecommunications from the Universities of Oporto and Algarve, in 1990, 1994, and 2001, respectively. Currently, he is teaching in the areas of signal processing, propagation, antennas and telecommunications at the Exact Sciences and Engineering Centre, University of Madeira. His major fields of interest are unification of radiation procedures on electromagnetics, models of antennas and arrays analysis and synthesis, signal processing and wireless sensor networks. Dr. Azevedo is a member of the Engineering Order and he is also member of the CCM, a Government Research Unit.

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Sparse Antenna Array Design for MIMO Active Sensing Applications William Roberts, Member, IEEE, Luzhou Xu, Member, IEEE, Jian Li, Fellow, IEEE, and Petre Stoica, Fellow, IEEE

Abstract—In active sensing applications, MIMO systems permit increased flexibility for transmit beampattern design, via waveform diversity, compared to phased-array approaches. When uniform arrays are not mandated, additional degrees of freedom for transmit beampattern design can be obtained via sparse array design considerations. Herein, we extend the motivation behind sparse receive array methodologies to that of sparse MIMO transmit array design. We propose a cyclic approach to MIMO transmit array design that can be used to approximate desired transmit beampatterns. Furthermore, we illustrate how this iterative approach can be adapted to design sparse receive antenna arrays using both vector and matrix weighting techniques. Index Terms—Array design, MIMO, phased-array, receive beamforming, transmit beamforming, waveform diversity.

I. INTRODUCTION

F

OR a phased-array active sensing system (e.g., radar, sonar, medical imaging, etc.), the transmission of coherent waveforms allows for a narrow beamshape pattern and, thus, a high signal-to-noise ratio (SNR) upon reception [1], [2]. When the locations of targets in a scene are unknown, phase shifts can be applied to the transmitting antennas to steer the focal beam across an angular region of interest. In contrast, multiple-input multiple-output (MIMO) systems, by transmitting different, possibly orthogonal waveforms, can be used to illuminate an extended angular region over a single processing interval. Waveform diversity permits higher degrees of Manuscript received March 29, 2010; revised August 03, 2010; accepted August 31, 2010. Date of publication January 06, 2011; date of current version March 02, 2011. This material is based on research sponsored in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under Contract/Grant W911NF-07-1-0450, the National Science Foundation (NSF) under Grant CCF-0634786, the Office of Naval Research under Grant N00014-09-1-0211, the National Aeronautics and Space Administration (NASA) under Grant NNX07AO15A, the SMART Fellowship Program, the Swedish Research Council (VR) and the European Research Council (ERC). The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. W. Roberts was with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611-6130 USA. He is now with the United States Department of Defense, Washington, DC 20301-1400 USA (e-mail: [email protected]). L. Xu and J. Li are with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611-6130 USA (e-mail: [email protected]; [email protected]). P. Stoica is with the Department of Information Technology, Uppsala University, Uppsala, Sweden (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103550

freedom, which enables the MIMO system to achieve increased flexibility for transmit beampattern design (see, e.g., [3]–[17]). The transmit beampattern for any active sensing system depends, in part, on the covariance matrix (which we will denote by ) of the transmitted signals. When a phased-array is adopted, whose waveforms are fully coherent (1) Using a MIMO system with orthogonal waveforms, conversely, corresponds to a diagonal matrix whose entries represent the elemental powers of each transmitting antenna. Beampattern design using partially coherent waveforms has been extensively discussed in [12], [14]. Therein, the authors described how convex optimization can be used to determine an (under certain power constraints) which best approximates a desired beampattern. Additionally, [12] presents beampattern matching and minimum sidelobe level design approaches and further addresses how a desired beampattern can be first determined. The authors in [12], [14] assumed that the positions of the transmitting antennas, which also affect the shape of the beampattern, were fixed prior to the construction of (without loss of generality, a uniform linear array was assumed present). At the receiver, sparse, or thinned, array design has been the subject of an abundance of literature during the last 50 years (see, e.g., [18]–[31] and the references therein). The purpose of sparse array design has been to reduce the number of antennas (and thus reduce the cost) needed to produce desirable spatial receiving beampatterns. Furthermore, previous researchers sought to avoid the tapering of uniform arrays for sidelobe level control, which results in an increased mainbeam width [19], [22]. Naturally, Dolph-Chebychev design approaches have been traditionally adopted, whereby sidelobe levels are minimized with respect to mainbeam width restrictions (see, e.g., [20], [31]). Prior to the advent of high-speed computing, many previous authors attempted to model the spatial receive beampattern using series expansions, which could only approximate a desired function [18], [21]. A thorough review of older works can be found in [23]. More recent approaches have sought to design receive antenna positions and beampattern weights via optimization strategies, including a simplex-type search [32] and branch-and-bound techniques [30]. Genetic algorithms [28] and simulated annealing [29] approaches have also been proposed for receive beampattern design. Additionally, iterative optimization strategies are presented in [24]–[27]. In [24], cost function to match the author cyclically minimizes an a desired receive pattern with an actual one. Similarly, the algorithm described in [26] involves the sequential addition of antennas to a system, such that antenna weights and positions

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ROBERTS et al.: SPARSE ANTENNA ARRAY DESIGN FOR MIMO ACTIVE SENSING APPLICATIONS

are optimized at each addition. The authors in [27] instead propose starting with a full array structure and then iteratively removing the antennas which produce the highest sidelobe levels in the receive beampattern. Due to nonconvex design constraints (specifically, that the number of antennas in a system is restricted to some constant), only locally, rather than globally, optimal sparse array design approaches have been shown to exist. In this paper, we will seek to extend the ideas behind sparse receive array methodologies to that of sparse, MIMO array design. We will describe cyclic algorithms that can be used to approximate desired transmit and receive beampatterns (our cyclic methods are similar to those adopted, for a different context, in, e.g., [33], [34]) via the design of sparse antenna arrays. Our algorithms can be seen as extensions to the iterative receive beampattern designs described in [24]–[27]. In Section II, we will present our cyclic algorithm for transmit beampattern design using sparse antenna arrays. We will describe similar approaches, in Sections III and IV, to provide matrix and vector receive beamforming, respectively, via sparse arrays. We present numerical simulations in Section V and we offer conclusions in Section VI. Notation: We denote vectors and matrices by boldface lowercase and uppercase letters, respectively. denotes denotes the conjugate transpose the transpose operation, refers to the complex conjugate operation, operation, denotes the Hadamard (elementwise) matrix product, denotes the column vector containing the diagonal components and refers to an identity of some square matrix . Further, refers to the Frobenius matrix of size norm operation for matrices and the -norm for vectors. denotes the Hermitian square root of the matrix . and denote a real- and complex-valued matrix , respectively.

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denotes the carrier wavelength of the system and where denotes the steering angle, relative to array broadside. If the an, then we can colgular scanning grid is spanned by lect the set of steering vectors into the steering matrix

(3) denote the transmitted signal pulse. We let The covariance matrix of the transmitted waveforms is repre, where sented by

(4) and where refers to the expectation operator. The transmit is given by power at some angle

(5) The transmit beampattern, now for each angle represented in its vector form as

, can be

(6) Further discussion on (5) and (6) can be found in [12], [14]. In a similar way to (6), we can define a desired real-valued transmit beampattern as

(7) For sparse array design, we assume that only antennas are available at the transmitter. The beampattern then be represented by

II. TRANSMIT BEAMPATTERN DESIGN In this section, we formulate the MIMO transmit beampattern design problem. We then present our iterative approach to MIMO transmit beampattern design, which can be used to approximate a desired transmit beampattern, in a least-squares sense, by cyclically determining antenna positions and an appropriate transmit covariance matrix.

A. Problem Formulation Consider a uniform linear array (ULA) of antennas with , where refers to the separation between length transmitting antennas. Assuming that the transmit waveforms are narrowband, we can represent the transmit steering vector by

(2)

can

(8) where contains only those rows of correantennas. We let sponding to the chosen locations of the denote the steering vector, to some angle , of contains only those the sparse array. Similarly, columns and rows of corresponding to the antennas which will be used for transmission. The goal of sparse MIMO antenna transmit beampattern design is to determine a set of positions and a corresponding waveform covariance matrix such that the beampattern in (8) closely approximates the desired one in (7). B. Sparse Transmit Array Design (see (7)) and an initial value Given a desired beampattern for our synthesized beampattern (see (8)), we now propose an iterative approach to transmit beampattern design. In each of our examples, we will assume that a ULA, with antenna spacing

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and orthogonal waveforms (so that ) are used to initialize the algorithm. We will first describe an approach to cyclically update , given a set of antenna positions. Then, in the next subsection, we will explain the procedure for antenna selection. 1) Determination of via a Cyclic Approach: Assuming that a set of antenna positions (and thus a steering matrix ) has already been specified, we first describe how can be designed to approximate a desired beampattern . Consider the following optimization problem:

(9) where the positive semidefinite constraint in (9) implies that is indeed a true covariance matrix. Further, by constraining , we ensure that the beampattern has unit height at some user-determined, central angle (we assume that ). The matrix is a diagonal matrix containing user-assigned weights, denoted by

has been attained, we can directly apply a cyclic approach to minimize the optimization criterion in (12). The steps are as follows. , the columns of the minimizing • Step 1. For a given matrix of (12) are: (13) where

represents the column of (for ). A derivation of (13) is provided in

Appendix A. of (12) can be • Step 2. Given a matrix , the minimizer matrix determined as follows. Choose an , such that and is full column-rank. Then,

(14) where (15)

.. . .. .

..

(10) (16)

. and

By carefully choosing , more emphasis can be placed on matching the desired beampattern over specific angular regions to simplify (since the weights are relative, we choose are set to notation). For example, if the values unity, then equal weight will be applied to each grid point in the entire angular span. , we can reformulate By introducing a matrix the problem in (9) as

(11) Finally, we can approximate the optimization problem in (11) using the following, similar problem:

(12) where and . By (12) being simand that makes ilar to (11), we mean that any solution (12) small will lead to a small value for the optimization criterion in (11) and vice versa. For further details regarding the type of problem in (12) and its relationship to (11), see, e.g., the explanation offered in [35]. Assuming that an initial value for has been provided, we can initialize by letting . Once an initial value for

(17) , given in (14), in We motivate the expression for Appendix B. • Step 3. Repeat Steps 1 and 2 until some convergence crite(where rion is achieved, such as and where and denote the updated matrix values at the iteration). initial antenna posi2) Antenna Selection: Given a set of total antenna positions are available tions and assuming that (we assume that the antennas can be placed only at a discrete number of possible locations, where the user specifies some and maximum array length minimum spacing ), we next attempt to reposition our antennas to better model a desired transmit beampattern (see (9)). antenna in the transmit array. If we assume Consider the antennas in our array are currently that the remaining positions (in our candidate set fixed in position, then of positions) currently do not contain an antenna. Thus, we repoantenna in one of the available position the antenna), sitions (which includes the current position of the such that, upon applying the cyclic procedure in Section II-B-I , the optimization metric in (12) is best minimized. To reposition antenna, we simply imply that the row of in the (12) has been designed to reflect the antenna’s position (i.e., the transmit steering vector has been adjusted to account for the antenna’s chosen position). We perform this operation for , so that each antenna is sequentially moved

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to its best available position. Further, we repeat this entire procedure until convergence of the algorithm is attained, whereby none of the antennas are repositioned during a cycle. In practical applications, the physical size of antennas and mutual coupling effects (see, e.g., [36]) can limit the minimum permissible spacing between antennas. In lieu of using a larger , we instead seek to promote design flexibility by maintaining a dense grid of candidate positions (and thus a small ). To enforce a minimum antenna separation distance, we simply omit candidate positions that neighbor existing elements (those positions that are too close to fixed antennas) when we reposition the antennas during an iteration. In this way, we mediate design constraints while still allowing for sufficient freedom during antenna selection. For further details, see Example 3 in Section V.

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correspond to the in (18)). Similarly, we let rows and columns of the selected receive antenna positions . The receive beampattern, using the matrix weighting apin proach, can then be rewritten (22) B. Sparse Receive Array Design refers to some desired receive beampattern, then we can If formulate the beampattern design problem as follows:

III. MATRIX APPROACH TO RECEIVE BEAMPATTERN DESIGN Similar to Section II, we can also consider applying our cyclic sparse array design approach to MIMO receive beamforming. Since receive beamforming does not depend on the correlation properties of a system’s transmitted waveforms, the following discussion can furthermore be applied to any radar application involving multiple receive antennas. Traditional receive beamforming designs have adopted vector weighting approaches. However, as in the transmit beampattern formulation in Section II, we could also consider a matrix weighting approach to receive beamforming (see, e.g., [37]). A. Problem Formulation Consider an array of receive antennas designed to image a scene in the far field. We will assume that the array aperture is linear. Similar to (2), the receive steering vector for a ULA is given by (18) refers to the minimum separation between receiving where antennas, refers to the total number of receive antennas and denotes the receive array steering direction relative to the ). Additionally, we can dearray’s broadside (with by fine the steering matrix (19) If the matrix

(23) where again refers to a diagonal matrix of weight values (see represents a user-defined, central angle (10)) and of the desired beampattern. The receive beampattern design problem in (23) closely resembles the transmit problem described in (9). Thus, we can consider applying the sparse array design approach described in Section II-B to now construct a sparse receive array using matrix weighting. As in (9), we constrain to be positive semidefinite , which is implied by the definition of in (20). Since the methodology for sparse receive array design using matrix weighting directly parallels the algorithm in Section II-B, we omit the details of this approach. For details on the antenna selection scheme, please see Section IV-B-2. IV. VECTOR APPROACH TO RECEIVE BEAMPATTERN DESIGN Traditional sparse array design methods for receive beamforming have adopted vector weighting approaches (see, e.g., [18]–[31]). In fact, receive beamforming using vector weighting can be viewed as a more constrained version of the general matrix weighting approach described in Section III. Next, we show how the cyclic approach to sparse receive array design via matrix weighting can be modified to perform receive beamforming with vector weights. A. Problem Formulation

refers to a matrix beamformer, then we define as

The vector weighting approach to receive beamforming follows the matrix weighting formulation in Section III, with the additional constraint that (see (23)):

(20)

(24)

In this way, the receive beampattern can be represented

From (24), we can write

in the following form:

(21)

(25)

To obtain a sparse array, we restrict the number of available antennas for beamforming design to some . We let contain those rows of corresponding to the selected antenna positions (we also construct a sparse refrom the full array steering vector ceiving vector

. Using the result in (25), the synthesized where beampattern with vector weighting can be represented by (26)

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Remark: In Section III, we described how receive beamforming using matrix weighting can be formulated into an identical optimization problem to that of the transmit beampattern design case of Section II (without any rank constraints on the covariance matrix ). Similarly, vector weighting for receive beamforming, which includes the additional constraint in (24), is analogous to transmit beampattern design with the restriction . Under this assumption, we are reduced to a that phased-array active sensing system (see (1)). Although we neglect to address sparse transmit array design for phased-array systems (since we focus, herein, on applications in waveform diversity), certainly this formulation would directly conform with our discussion on vector weighting for receive beamforming.

B. Sparse Array Design 1) Weight Determination via a Cyclic Approach: The beampattern matching problem in (23), now for the vector weighting case, can be rewritten as

(27) represents a set of powers, Since the desired beampattern we can instead consider the following similar optimization problem [38] (see also [33], [34]):

(28) where

.. .

..

.

..

.

..

.

.. .

(29)

• Step 2. For a given matrix and through the use of a Lagrange multiplier to enforce the constraint, the minimizer of (28) is given by

(31) We provide a proof of (31) in Appendix C. • Step 3. Repeat Steps 1 and 2 until a prespecified stopping , where criterion is obtained, such as and where and denote the estimates obtained at the iteration. 2) Antenna Selection: Our algorithm for sparse receive array design using vector weighting follows just as the one for sparse transmit beampattern design in Section II-B-II. We again asinitial antenna positions are specified from a set sume that candidate positions. Assuming that the other of antennas are fixed in position, we consider placing the anavailable positions tenna in any of the available (again including the antenna’s current position). We apply the iterative procedure in Section IV-B-I for each candidate location antenna to that position which best and we then move the antenna minimizes the criterion in (28). Repositioning the row in (and thus, the corresponds to adjusting the component of each steering vector) to reflect the antenna’s current position. This operation is repeated for each of the antennas. Finally, we perform the procedure iteratively until convergence, such that the antennas are no longer repositioned. As before (see Section II-B-II), we could also consider enforcing a minimum separation distance between antennas during the selection stage (to account for mutual coupling, sufficiently small (and thus ensure a dense etc.). To keep grid of candidate positions), we simply neglect those candidate positions (when attempting to move an antenna) that are too close to the remaining fixed antennas. This constraint can be simply enforced, by the user, during the implementation of the algorithm.

and where the vector

are auxiliary variables. Further, is the elementwise square root of , such that . Note that, in (28), we restrict , rather than (a nonlinear constraint), since certainly a uniform phase change to the elements in would not affect the synthesized beampattern in (26). Assuming that we are given an initial value for the vector (for the first iteration of the algorithm, we will assume that the ), we can apply a cyclic minimizacomponents of are tion approach to the optimization criterion in (28) to obtain an updated weight vector. The algorithm can be summarized in the following steps. • Step 1. Given , the closed-form minimizer of (28) with respect to can be directly obtained

(30)

V. NUMERICAL EXAMPLES We offer, in this section, several numerical examples to demonstrate the performance of the sparse transmit and receive beampattern design approaches outlined in the previous sections. For each of the following examples, we will assume that the angular scanning region spans the array’s entire angular to 90 ) with 1 angular spacing (so that range (from ).

A. Example 1 We will first consider a transmit beampattern design problem, where the desired beampattern contains a single pulse region.

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M = 10

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M = 10, (d) sparse array transmit beampattern M = 15.

M = 15

Fig. 1. Example 1: (a) desired transmit beampattern, (b) design weights, (c) ULA transmit beampattern with with , (e) ULA transmit beampattern with , and (f) sparse array transmit beampattern with

The desired beampattern is shown in Fig. 1(a) and is given by (see (7))

otherwise.

(32)

In practice, we can select a desired beampattern by first transmitting omnidirectional waveforms, so that the approximate angular location of targets in a scene can be determined. Then, we are able to focus the transmitted energy toward the estimated target location. For further details, see [12]. of the weight matrix The diagonal components (see (10)), for this example, were chosen as

(33) otherwise. and set to ensure unit We assumed that height at (see (12)). The weight values are plotted, as a func-

tion of angle, in Fig. 1(b). Clearly, using the weights specified in (33), we sought to construct a beampattern that closely modeled the pulse in Fig. 1(a) and sidelobes further away from the mainbeam were less penalized in the cost function (see (12)). Certainly, different sparse arrays and different beampatterns can be achieved by adjusting the weight parameters in (33). antennas and inter-element Using a ULA with , we applied the cyclic approach in Section II-B-I spacing to generate an optimized covariance matrix . The transmit beampattern for the ULA, with optimized covariance, is shown in Fig. 1(c). For our sparse approach outlined in Section II-B, we candidate antenna positions were availassumed that able and that the positions were separated by (see the array steering vector definition given in (2)), so that the total . Of the candidate posiaperture length is given by antennas were used in the design approach. The tions, ULA was used to initialize the algorithm and 4 iterations of the program were performed (after 4 iterations, none of the antennas were repositioned by the algorithm). The transmit beampattern of the sparse array is shown in Fig. 1(d).

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TABLE I EXAMPLE 1: TRANSMIT BEAMPATTERN DESIGN

which is the longest array, has the smallest mainbeam width, as expected. The PSL for each approach decreases as the number antennas, the of antennas is increased. For the case of ULA’s beampattern produces a slightly lower PSL compared to the sparse array, while the sparse array has a slightly smaller . PSL than the ULA for the case that B. Example 2 Next, we will consider the problem of receive beampattern design using a sparse array. The desired receive beampattern, which is shown in Fig. 3(a), is defined

M = 10 M = 10

Fig. 2. Antenna positions for Example 1. Squares are used to indicate the positions of the antennas for the ULA with , circles show the positions of the antennas for the sparse array with , “X’s” show the positions of the antennas for the ULA with and diamonds mark the antenna . locations for the sparse array with

M = 15 M = 15

Next, in order to demonstrate the dependency of performance on the number of antennas, we repeat the previous beampattern antennas. In Fig. 1(e), we show the design steps for designed beampattern using a ULA with antennas and . Again using our sparse array design inter-element spacing candidate antenna posiapproach (and again with inter-element spacing), we show tions separated by the designed beampattern for antennas in Fig. 1(f). For this case, 5 iterations of our sparse array design approach were necessary to obtain convergence. The chosen antenna positions, along with those of the ULAs, are illustrated in Fig. 2. For clarity, we show the antennas along . We indicate the positions the full candidate aperture of the arrays’ antennas using various shapes, which are labeled in the legend of Fig. 2 (we adopt a similar representation style for each of the remaining examples). The arrays are compared in Table I, where the mean-squared is defined by error (34) Further, in Table I, corresponds to the 3 dB mainbeam width (the 3 dB mainbeam width of the desired response is 4 ) and PSL refers to the peak sidelobe level of the beampattern. We identified the PSL, for each beampattern, as the amplitude of the highest peak occurring outside of the 3 dB mainbeam. and As evidenced, the sparse arrays, for the case of antennas, attain a significantly lower compared to the corresponding ULAs. Additionally, the 3 dB mainbeam width, which largely depends on the total aperture length, is noticeably smaller using the longer sparse arrays compared , to the corresponding ULAs. The sparse array with

(35) otherwise. As evidenced, the desired receive beampattern has a triangular shape and the 3 dB width of the mainbeam is 6 . Further, we (so that unit power is maintained in the designed set beampatterns at ). The following weights were chosen for this example

(36) otherwise. Using the weights in (36), we emphasized again on achieving a good approximation to the mainbeam shape. The receive beampattern for a 10-element ULA (with element spacing) is shown in Fig. 3(c). The antenna weights for the ULA were determined using the vector approach in Section IV-B-I. For each of the remaining simulations, we maintained and . Using the ULA result as an initialization, we were able to generate a sparse antenna array using the vector beamforming algorithm in Section IV. The receive beampattern, using the vector approach, is shown in Fig. 3(d) (5 iterations of the algorithm were needed to attain convergence). For comparison, we have generated a sparse array using the method proposed in [25]. Using this sparse algorithm proposed by Kishi et al. antennas are sequentially placed into the array, given a set of candidate positions. Antennas are placed such that, along with the antennas that are currently in the system, the mean-squared error between the desired receive pattern and the generated one is minimized. Unlike our vector weighting approach, the method in [25] does not optimize the antenna weights cyclically, as in Section IV-B-I. The receive beampattern generated using the approach in [25] is shown in Fig. 3(e). Additionally, we applied the method described by Jarske et al. in [27] to our receive beampattern design problem. As mentioned in Section I, the approach described in [27] begins with a full array structure. Antennas are removed by iteratively selecting

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Fig. 3. Example 2: (a) Desired receive beampattern. (b) Design weights. (c) ULA receive beampattern. (d) Sparse array receive beampattern. (e) Kishi approach. (f) Jarske approach.

the antenna whose presence in the array yields the maximum peak sidelobe level in the resulting beampattern. The receive beampattern generated using the approach in [27] is provided in Fig. 3(f). Finally, we show each of the selected antenna positions, for all of the approaches, in Fig. 4 (we refer to the array generated using the approach in [25] as a Kishi array and the array generated using the approach in [27] as a Jarske array, for simplicity). The receive beampatterns in Figs. 3(c)–3(f) are comcorresponds to the 3 dB angular pared in Table II. Again, width of the mainbeam (the 3 dB mainbeam width of the desired response is 6 ) and PSL denotes the peak sidelobe level. Although the Kishi array achieves a more narrow 3 dB width (actually, more narrow than the desired response) owing to its longer occupied aperture, the PSL is significantly higher using this array. Not surprisingly, the Jarske array yields the lowest PSL. However, the 3 dB width of the Jarske array’s beampattern is the largest of the 4 approaches. The mean-squared error is defined similarly to (34), where and now refer to the desired and synthesized receive beampatterns, respectively.

Fig. 4. Antenna positions for Example 2. Squares are used to indicate the positions of the ULA’s antennas, our designed sparse array’s antennas are indicated by the circles, “X’s” are used to indicate the position of the antennas in Kishi’s array and diamonds are used to indicate the position of the antennas in Jarske’s array.

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TABLE II EXAMPLE 2: RECEIVE BEAMPATTERN DESIGN

Fig. 6. Antenna positions for Example 3. “X’s” are used to indicate the position , circles for the array with of the antennas in the array with and squares for the array with .

0:25

d = 0:1 d = 0:5

d =

TABLE III EXAMPLE 3: RECEIVE BEAMPATTERN DESIGN

N = 200 and d = 0:5 .

Fig. 5. Example 3. Sparse array receive beampatterns with (a) , (b) and , (c) and

d = 0:1

N = 80

d = 0:25

N = 40

As evidenced, our sparse array produces the lowest pared to the ULA, Kishi array and Jarske array.

com-

C. Example 3 For the final example, we will compare the performance of our sparse receive array design, using the vector weighting . In doing so, we seek approach, for different values of to demonstrate that a more dense set of candidate antenna positions can lead to improved beampattern performance. To maintain a fair comparison, we will restrict the minimum , to (in separation between antennas, regardless of practice, a minimum antenna separation could be necessary to avoid mutual coupling or to accommodate the physical

size of the antennas). A minimum antenna separation can be easily enforced, by the array designer, simply by restricting the available locations when the antennas are repositioned (see Section IV-B). Although we consider receive array design for this example, certainly these results can be extended to transmit array synthesis. We will again adopt the desired receive beampattern provided in (35) and the beampattern weights defined in (36). Using and (which corresponds to the result provided in Example 2), we generate the receive beampattern shown in Fig. 5(a). Five iterations of the algorithm were needed to achieve convergence. For the pattern shown in Fig. 5(b), we and . For this case, the algorithm let converged to its final result after two iterations. Finally, letting and , we generate the receive beampattern shown in Fig. 5(c); the algorithm converged after three iterations. For each of these examples, the total candidate array and (again, the length was maintained between ). As is increased, candidate length is equal to the candidate antenna positions become less closely spaced (less dense). We show the chosen antenna locations, for each case, in , PSL and values Fig. 6. Further, we compare the for each simulation in Table III, where the arrays are identified values. according to their respective (the Although each array yielded similar values for was closest to the desired value), array with and PSL values clearly decrease as the candidate the array is made more dense (as is made smaller). We reiterate, however, that for each constructed array, the minimum separa(which tion between placed antennas was no smaller than is confirmed in Fig. 6). Thus, given fixed constraints on minimum antenna separation and total aperture length, we can conclude that a more dense array of candidate antenna positions

ROBERTS et al.: SPARSE ANTENNA ARRAY DESIGN FOR MIMO ACTIVE SENSING APPLICATIONS

permits higher degrees of freedom and better matching perfor). As expected, does not seem to demance (lower pend on the density of the candidate positions (since beampattern width more closely relates to the total length of the aperture [20]). As indicated by this example, as well as the previous ones, the sparse array algorithms for the matrix and vector cases were able to converge to their final results after no more than five iterations. At least for the examples considered, the number of required iterations for convergence did not seem to increase with the number of candidate antenna locations or the total number of antennas.

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refers to the real-valued component of a complex where scalar. Using the result in (38), we can further represent the optimization problem in (37)

(39) Applying the Cauchy-Schwarz inequality, we can conclude the following:

VI. CONCLUSIONS (40) More degrees of freedom can be attained for MIMO transmit and receive beamforming design by allowing for spatial tapering via sparse antenna arrays. In this paper, we presented an iterative algorithm that can be used to design sparse MIMO transmit arrays (and a corresponding transmit waveform covariance matrix ) to approximate a desired transmit beampattern response. Further, we showed how this cyclic algorithm can be similarly extended to design sparse receive antenna arrays (to approximate a desired receive beamshape) using vector and matrix weighting approaches. Through numerical examples, we showed that our sparse array design approaches can achieve more narrow mainbeams and more closely approximate (in a least-squares sense) desired beampattern responses compared to a ULA (our approach also achieved a lower peak sidelobe level in two of our examples). Our sparse receive beamforming approach was shown to outperform the iterative designs provided in [25] and [27]. Finally, we demonstrated that by allowing for a more dense candidate array (by decreasing the separation between candidate positions), we are able to better approximate desired beampatterns while still maintaining a specified distance between placed antennas.

APPENDIX A In this appendix, we will provide justification for the update formula offered in (13). Consider the optimization problem in (12). Given a matrix , we will seek to provide a closed-form represents the column of update for . Recalling that , we first rewrite the optimization problem in (12) as

where denotes the absolute value operation. The left and right side of the inequalities in (40) are equal if

(41) where refers to an arbitrary scalar factor. To satisfy the con. Instraint in (39), we must have serting this value of into (41), we obtain the update formula in (13) (42)

APPENDIX B We describe, in this appendix, how the update formula in (14) that solves the is obtained. Given , we seek to find a matrix following optimization problem (see (12)): (43) We first determine a matrix such that the columns of span the subspace orthogonal to , i.e., and the matrix is full column-rank (hence is full rank). Then, we can decompose into two components

(44) where

and

. Then

(37)

(45) Hence, the optimization problem in (43) is equivalent to

Consider that, for some

(38)

(46)

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Solving (46) with respect to

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yields

Based on the formulation in Appendix A, the solution to the , where optimization problem in (55) is given by the vector

(47) Substituting (47) into (46), the optimization problem is reduced to

(48) where

(56) Summary: The solution can be summarized as follows: 1) Choose an matrix , such that and is full column-rank. 2) Compute

(49) (57)

(50) and

(58) (51) is the orthogonal projector on the subspace spanned ( by the matrix ). Note that the optimization criterion in (48) can be written as

is defined in (53). where 3) Compute

(59)

(52) where

(53) Omitting the constraint, represents the least-squares solution to the optimization problem in (48). when the columns Remark: Note that, in (53), of the matrix are linearly independent, which is true , in general. Consider the following explanation. If belongs to the subspace then the column vector . In other spanned by the columns of the matrix words, there exists a column vector such that

(54) If the columns of are linearly independent, then (54) , which cannot be true based on the implies that definition of . Thus, we must have that . Using the result in (52), we rewrite the optimization problem in (48) as follows:

(55)

Remark: Note that in the above algorithm, the choice of is not unique. However, the final solution is unique. This can be verified as follows. Let be another matrix such and is full column-rank. Then, we have that with being an full rank matrix. Replacing by in the above algorithm, the matrix can be canceled out and therefore we obtain the previous result. , then Remark: Consider that, if is the optimal solution to (48). Then, it can be easily shown that is given by

(60) which corresponds to the least-squares solution, again neglecting the constraint, to the optimization problem in (43). APPENDIX C We will show in this appendix how a Lagrange multiplier can be used to update , given some , in (28). The Lagrange function for the optimization problem in (28) can be written (61) where is a Lagrange multiplier. The minimization of (61) with respect to yields (62) We can then insert the result in (62) into the constraint to obtain (63)

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Solving (63) for

yields (64)

which can then be inserted into (62) to obtain the result in (31)

(65)

REFERENCES [1] A. A. Oliner and G. H. Knittel, Phased Array Antennas. Norwood, MA: Artech House, 1972. [2] E. Brookner, “Phased array radars,” Sci. Am., vol. 252, pp. 94–102, Feb. 1985. [3] D. W. Bliss and K. W. Forsythe, “Multiple-input multiple-output (MIMO) radar and imaging: Degrees of freedom and resolution,” in Proc. 37th Asilomar Conf. Signals, Syst. Comput., Pacific Grove, CA, Nov. 2003, vol. 1, pp. 54–59. [4] E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, and R. Valenzuela, “MIMO radar: An idea whose time has come,” in Proc. IEEE Radar Conf., Apr. 2004, pp. 71–78. [5] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, “Performance of MIMO radar systems: Advantages of angular diversity,” in Proc. 38th Asilomar Conf. Signals, Syst. Comput., Pacific Grove, CA, Nov. 2004, vol. 1, pp. 305–309. [6] I. Bekkerman and J. Tabrikian, “Spatially coded signal model for active arrays,” in Proc. IEEE Int. Conf. Acoustics, Speech Signal Process., Montreal, Quebec, Canada, Mar. 2004, vol. 2, pp. II/209–II/212. [7] K. Forsythe, D. Bliss, and G. Fawcett, “Multiple-input multiple-output (MIMO) radar: Performance issues,” in Proc. 38th Asilomar Conf. Signals, Syst. Comput., Pacific Grove, CA, Nov. 2004, vol. 1, pp. 310–315. [8] L. B. White and P. S. Ray, “Signal design for MIMO diversity systems,” in Proc. 38th Asilomar Conf. Signals, Syst. Comput., Pacific Grove, CA, Nov. 2004, vol. 1, pp. 973–977. [9] F. C. Robey, S. Coutts, D. D. Weikle, J. C. McHarg, and K. Cuomo, “MIMO radar theory and experimental results,” in Proc. 38th Asilomar Conf. Signals, Syst. Comput., Pacific Grove, CA, Nov. 2004, vol. 1, pp. 300–304. [10] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, “Spatial diversity in radars—models and detection performance,” IEEE Trans. Signal Process., vol. 54, pp. 823–838, Mar. 2006. [11] J. Li and P. Stoica, “MIMO radar with colocated antennas: Review of some recent work,” IEEE Signal. Process. Mag., vol. 24, no. 5, pp. 106–114, Sep. 2007. [12] P. Stoica, J. Li, and Y. Xie, “On probing signal design for MIMO radar,” IEEE Trans. Signal Process., vol. 55, no. 8, pp. 4151–4161, Aug. 2007. [13] A. H. Haimovich, R. S. Blum, and L. J. Cimini, “MIMO radar with widely separated antennas,” IEEE Signal. Process. Mag., vol. 25, no. 1, pp. 116–129, Jan. 2008. [14] D. R. Fuhrmann and G. San Antonio, “Transmit beamforming for MIMO radar systems using signal cross-correlation,” IEEE Trans. Aerosp. Electron. Syst., vol. 44, pp. 1–16, Jan. 2008. [15] B. Guo and J. Li, “Waveform diversity based ultrasound system for hyperthermia treatment of breast cancer,” IEEE Trans. Biomed. Eng., vol. 55, pp. 822–826, 2008. [16] MIMO Radar Signal Processing, J. Li and P. Stoica, Eds. Hoboken, NJ: Wiley, 2009. [17] X. Zeng, J. Li, and R. J. McGough, “A waveform diversity method for optimizing 3D power depositions generated by ultrasound arrays,” IEEE Trans. Biomed. Eng., unpublished. [18] H. Unz, “Linear arrays with arbitrarily distributed elements,” IRE Trans Antennas Propag, vol. 8, no. 2, pp. 222–223, Mar. 1960.

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[19] A. L. Maffett, “Sidelobe reduction by nonuniform element spacing,” IRE Trans Antennas Propag, vol. 9, no. 2, pp. 187–192, Mar. 1961. [20] M. G. Andreasen, “Linear arrays with variable interelement spacings,” IRE Trans Antennas Propag, vol. 10, no. 2, pp. 137–143, Mar. 1962. [21] A. Ishimaru, “Theory of unequally-spaced arrays,” IRE Trans Antennas Propag, vol. 10, no. 6, pp. 691–702, Nov. 1962. [22] A. L. Maffett, “Array factors with nonuniform spacing parameter,” IRE Trans. Antennas Propag., vol. 10, no. 2, pp. 131–136, Mar. 1962. [23] Y. T. Lo and S. W. Lee, “A study of space-tapered arrays,” IEEE Trans. Antennas Propag., vol. AP-14, no. 1, pp. 22–30, Jan. 1966. [24] R. W. Redlich, “Iterative least-squares synthesis of nonuniformly spaced linear arrays,” IEEE Trans. Antennas Propag., vol. AP-21, no. 1, pp. 106–108, Jan. 1973. [25] G. Kishi, K. Sakaniwa, and T. Uyematsu, “A design method for transversal filters with nonuniform tap spacings based on the mean square error criterion,” in Proc. 5th Int. Symp. Network Theor., Sarajevo, Yugoslavia, Sep. 1984, pp. 363–368. [26] N. M. Mitrou, “Results on nonrecursive digital filters with nonequidistant taps,” IEEE Trans. Acoust. Speech Signal Process., vol. ASSP-33, no. 6, pp. 1621–1624, Dec. 1985. [27] P. Jarske, T. Saramäki, S. K. Mitra, and Y. Neuvo, “Iterative leastsquares synthesis of nonuniformly spaced linear arrays,” IEEE Trans. Acoust. Speech Signal Process., vol. 36, no. 3, pp. 372–380, Mar. 1988. [28] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994. [29] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing,” IEEE Trans. Signal Process., vol. 44, no. 1, pp. 119–122, Jan. 1996. [30] S. Holm, B. Elgetun, and G. Dahl, “Properties of the beampattern of weight- and layout-optimized sparse arrays,” IEEE Trans. Ultrason. Ferroelect. Freq. Contr., vol. 44, no. 5, pp. 983–991, Sep. 1997. [31] M. Viberg and C. Engdahl, “Element position considerations for robust direction finding using sparse arrays,” in Proc. 33rd Asilomar Conf. Signals, Syst. Comput., Pacific Grove, CA, Nov. 1999, vol. 2, pp. 835–839. [32] R. M. Leahy and B. D. Jeffs, “On the design of maximally sparse beamforming arrays,” IEEE Trans. Antennas Propag., vol. 39, no. 8, pp. 1178–1187, Aug. 1991. [33] R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik, vol. 35, pp. 237–246, 1972. [34] S. M. Sussman, “Least-square synthesis of radar ambiguity functions,” IRE Trans. Inf. Theor., vol. 8, no. 3, pp. 246–254, Apr. 1962. [35] P. Stoica, H. He, and J. Li, “New algorithms for designing unimodular sequences with good correlation properties,” IEEE Trans. Signal Process., vol. 57, no. 4, pp. 1415–1425, Apr. 2009. [36] M. I. Skolnik, Introduction to Radar System, 3rd ed. New York: McGraw-Hill, 2002. [37] J. Li, Y. Xie, P. Stoica, X. Zheng, and J. Ward, “Beampattern synthesis via a matrix approach for signal power estimation,” IEEE Trans. Signal Process., vol. 55, pp. 5643–5657, Dec. 2007. [38] W. Roberts, H. He, J. Li, and P. Stoica, “Probing waveform synthesis and receiver filter design,” IEEE Signal. Process. Mag., vol. 27, no. 4, pp. 99–112, Jul. 2010.

William Roberts (S’10–M’11) received the B.S., M.Sc., and Ph.D. degrees in electrical engineering from the University of Florida, Gainesville, in 2006, 2007, and 2010, respectively. He is currently employed by the United States Department of Defense, Washington, DC.

Luzhou Xu, (M’06) photograph and biography not available at the time of publication.

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Jian Li (S’87–M’91–SM’97–F’05) received the M.Sc. and Ph.D. degrees in electrical engineering from The Ohio State University, Columbus, in 1987 and 1991, respectively. From July 1991 to June 1993, she was an Assistant Professor with the Department of Electrical Engineering, University of Kentucky, Lexington. Since August 1993, she has been with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, where she is currently a Professor. In Fall 2007, she was on sabbatical leave at MIT, Cambridge, MA. Her current research interests include spectral estimation, statistical and array signal processing and their applications. She is an author or coauthor of two edited books (Robust Adaptive Beamforming, MIMO Radar Signal Processing), one research monograph, four book chapters and over 210 journal and over 190 conference papers. Dr. Li is a Fellow of IEEE and a Fellow of IET. She received the 1994 National Science Foundation Young Investigator Award and the 1996 Office of Naval Research Young Investigator Award. She was an Executive Committee Member of the 2002 International Conference on Acoustics, Speech and Signal Processing, Orlando, Florida, May 2002. She was an Associate Editor of the

IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1999 to 2005, an Associate Editor of the IEEE Signal Processing Magazine from 2003 to 2005 and a member of the Editorial Board of Signal Processing, a publication of the European Association for Signal Processing (EURASIP), from 2005 to 2007. She has been a member of the Editorial Board of Digital Signal Processing—A Review Journal, a publication of Elsevier, since 2006. She is presently a member of the Sensor Array and Multichannel (SAM) Technical Committee of IEEE Signal Processing Society. She is a coauthor of papers that have received the First and Second Place Best Student Paper Awards, respectively, at the 2005 and 2007 Annual Asilomar Conferences on Signals, Systems and Computers in Pacific Grove, California. She is a coauthor of the paper that has received the M. Barry Carlton Award for the best paper published in the IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS in 2005. She is also a coauthor of the paper that has received the Lockheed Martin Best Student Paper Award at the 2009 SPIE Defense, Security and Sensing Conference in Orlando, Florida.

Petre Stoica (F’94) is a Professor of systems modeling at Uppsala University, Sweden; more details about him can be found at http://user.it.uu.se/ ps/ps.html.

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Through-Wall Opportunistic Sensing System Utilizing a Low-Cost Flat-Panel Array Kenneth E. Browne, Student Member, IEEE, Robert J. Burkholder, Fellow, IEEE, and John L. Volakis, Fellow, IEEE

Dedicated to the memory of Prof. Benedikt Munk, whose genius for wideband array theory inspired the flat-panel antenna design introduced in this paper.

Abstract—A UWB through-wall imaging system is proposed based on a planar low profile aperture array operating from 0.9 GHz to 2.3 GHz. The goal is to provide a lightweight, fixed array to serve as an alternative to synthetic aperture radars (SAR) that require continuous array movement while collecting data. The proposed system consists of 12 dual-linear printed elements arranged within a triangular lattice, each forming a “flower” shape and backed by a ground plane. The array delivers half-space radiation with wideband performance, necessary for imaging applications. UWB capability is realized by suppressing grating lobes via the introduction of virtual phase centers interwoven within the actual array feeds. The proposed system is demonstrated for through-wall imaging via a non-coherent process. Distinctively, several coherent images are forged from various fixed aperture locations (referred to as “snapshot” locations) and appropriately combined to create a composite scene image. In addition to providing a unique wideband imaging capability (as an alternative to SAR), the system is portable and inexpensive for collecting/storing scattering data. The array design and data collection system is described, and several through-wall images are presented to demonstrate functionality. Index Terms—Antenna arrays, array signal processing, beam steering, radar imaging, synthetic aperture radar.

I. INTRODUCTION

T

HERE is recent interest to detect, locate, characterize and image through visually opaque dielectric materials using electromagnetic waves. In particular, through-wall imaging (TWI) has a variety of potential applications including avalanche/earthquake rescue, non-destructive testing, mine detection, covert surveillance, and detecting objects or people behind walls. In this regard, ultrawideband (UWB) imaging has been identified as a viable approach for through-wall imaging. Microwave radiation can penetrate drywall, concrete, and bricks. By exploiting wide bandwidth and a large aperture, fine resolution can be achieved to enable detection of both stationary and moving targets [1]–[8].

Manuscript received December 14, 2009; revised July 27, 2010; accepted July 29, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with The Ohio State University, ElectroScience Laboratory, 1320 Kinnear Road, Columbus, OH 43212 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103015

Fig. 1. Illustration of the synthetic aperture radar (SAR) and proposed stationary imaging approaches. (a) SAR imaging approach; (b) proposed fixed aperture imaging approach.

Typically, a synthetic aperture radar (SAR) approach is employed for imaging purposes, including TWI applications. SAR approaches are attractive as they emulate a large aperture by moving a small one (as small as a single antenna) across a scene of interest (see Fig. 1(a)). An image is then created through coherent integration of the collected data over the movement path. Requirements for coherency implies precise measurement and calibration, making the collection process costly and time

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Fig. 2. Expanded view of the proposed snapshot imaging system. The back side of the system is connected to a PC to control the RF feed network (from the source to the antenna elements). This is accomplished via a network of RF switches and ceramic baluns. The back side integration and functionality is discussed further in Section II-C.

consuming. There are further challenges when collecting data in hostile or time sensitive environments [9], [10]. Also, a moving SAR system is difficult to apply in detecting moving targets, unless the data can be collected more quickly than target movements. In this paper, we present a lightweight wideband imaging system capable of constructing images from a few opportunistic angles/viewpoints of a scene (see Fig. 1(b)). Several fixed-aperture imaging systems have already been proposed outside of the common SAR approach [11]–[14]. However, these systems are still heavy, large, expensive, and function using Vivaldi or horn type antennas (i.e. not easily portable). Thus, they are not practical for deployment in confined spaces. The proposed fixed aperture imaging system is advantageous and unique as compared to these systems due to (a) a novel data collection method (referred to as the “snapshot” method), (b) a low cost and lightweight flat antenna array based on novel UWB dual polarized antenna elements (closely spaced over a ground plane [15], [16]), and (c) realization of virtual phase centers interwoven within actual array elements to extend the functional bandwidth. In fact, the spacing of the proposed array elements cause grating lobes at the highest operational frequency. This challenge is overcome using the concept of virtual elements (phase centers) placed midway between the physical elements at the post-processing stage. In addition, the proposed imaging system incorporates a portable computer-controlled RF multiplexing subsystem for fast automated data collection. As such, it is fully functional in an arbitrary setting. In the pursuing paragraphs, we first describe the proposed antenna array and overall system design (Section II) followed by the presentation of example images demonstrating the system’s capabilities (Section III). II. IMAGING SYSTEM CONCEPT AND DESIGN The proposed array and imaging system concept is depicted in Fig. 2. As seen, the system consists of four key components including an antenna array, RF source/receiver (network analyzer), RF multiplexing/balun circuit, and a personal computer (PC). The PC is employed to control the RF source and multiplexer, which selects a transmit/receive array configuration for

data collection. Below, we describe the fabricated system components and data collection methodology. A. Antenna Element Design The paramount component of the proposed imaging system is the antenna array. This array is relatively small (55.25 cm 45.72 cm), lightweight (less than 2 kg), low profile (4.44 cm thick), employs dual-linear polarized UWB printed elements, provides forward half space radiation (due to employing a ground plane below the elements), and can operate from 900 up to 2300 MHz (88% bandwidth). This band has been used in the past, as common concrete and cinder block walls can be penetrated without significant absorption [1], [7]. Further, we note that the individual antenna elements are matched to 100 and are fed from underneath the ground plane. For the latter, a pair of balanced 50 coaxial feed lines extending through the ground plane, are employed. Coaxial lines provided a simple feed network with minimal impact on the radiation pattern. A variety of wideband printed elements were considered to fulfill the design requirements (including circular, bow-tie, elliptical, spiral, and tapered slot, among many others). An acceptable antenna configuration was found to be a hybrid between an elliptical and bow-tie dipole (see Fig. 3). This printed element is easy to fabricate, tunable, provides wide bandwidth, maintains a constant phase center, and allows for dual-linear polarization. The latter is achieved by orienting two of the designed dipoles orthogonally around the feed points. As depicted in Fig. 4, the resulting crossed antennas take on a “flower-like” shape and are hence referred to as “flower” elements. Of course, a plethora of wideband printed dipole elements are available [17]–[19]. However, these lack the needed compact size and shape to realize the desired dual-linear polarization configuration. This 14.424 cm element is fabricated by etching the metalized surface of 4 mil thick liquid crystal polymer (LCP) sub. After the etching process, the array elements strate remain on the top surface while its bottom is free of metal. The LCP layer is then bonded onto a rigid Styrofoam spacer backed by an aluminum sheet. The sheet acts as a finite ground plane and provides forward half-space radiation. The thickness of the

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Fig. 4. Isolated dual-linear polarized prototype array element geometry and dimensions. (a) Top view of prototype element; (b) side view of prototype element.

Fig. 3. Printed hybrid dipole element configuration and its dimensions.

spacer and element dimensions are tuned to achieve the desired input impedance of 100 across the design frequency band. The final dimensions are shown in Figs. 3 and 4, and the measured performance for an isolated dipole (see Fig. 4) mounted over a 40 cm square ground plane is depicted in Fig. 5. At the center of the band (1.6 GHz), the realized gain is 8.1 dBi, and the 3 dB beamwidth is 56 and 91 in the E-plane and H-plane, respectively. In addition, the 3 dB bandwidth is greater than 88%. B. Virtual Elements The full array configuration incorporating the flower elements is depicted in Fig. 6(a). The array consists of 12 elements (12 vertical dipoles and 12 horizontal dipoles) strategically arranged in a triangular lattice and measures a total size of 55.2 cm 45.7 cm. In practice, the array is excited sequentially, cycling through one transmit/receive element pair at a time (in any combination) to collect the scattering data. The data is then post-processed to focus the beam at any point in the scene to form a 3-D image. It is known that the narrowest beam, or the best focus, occurs when the array is operated in monostatic mode [20]. In this mode, each element transmits and receives independent of one another, and the data are subsequently weighted at the post-processing stage to simulate a beam-steered array. However, be-

cause of the physical size and element shape, the spacing beat the highest design fretween elements is larger than quency of 2.3 GHz, thus, causing grating lobes. In fact, the separation distance between two adjacent element feed points at the highest operational frequency. To ensure that is grating lobes are inhibited in the entire operational band (0.9 to 2.3 GHz), the concept of virtual (or synthetic) phase centers is introduced as depicted in Fig. 6(b). To achieve any single virtual element the system operates two adjacent elements in bistatic mode (i.e., one transmits and the other receives) [21]–[23]. It is shown below that this configuration is equivalent to having a monostatic element located midway between the two adjacent real elements. Consider the point scattering problem in Fig. 7. As usual, eltransmits and element receives the bistatic scatement tered signal from the point scatterer. Of course, the total prop. Now consider agation path for bistatic scattering is the virtual element (phase center) located midway between the and elements, which has a monostatic scattering propa. We may write and in terms of gation path of and the separation distance, , between the real and virtual elements as

(1)

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Fig. 5. Measured gain and pattern of one of the dipoles integrated into a flower element as seen in Fig. 4. (a) Measured realized gain while mounted over a finite aluminum sheet ground plane; (b) measured normalized E/H radiation patterns in dB (1.6 GHz).

Fig. 6. Array dimensions and element configuration. (a) Photograph of fabricated antenna array; (b) array virtual (solid circles) and actual (dashed circles) phase center locations.

(2) In these, is the angle shown in Fig. 7 and sumed to form the approximations. It follows that,

was as-

(3) , where is the free space wavenumber and is the wavelength. This is exactly the same phase propagation observed from the virtual element location in Fig. 7 if the last term can be ignored, . To avoid grating lobes we set , i.e., if Thus,

Fig. 7. Depiction of an approximate virtual phase center location relative to actual phase center locations when utilizing bistatic element excitation.

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Fig. 8. Normalized (dB) horizontally and vertically polarized patterns of the full array (azimuth and elevation cuts). (a) Pattern cuts at 0.9 GHz; (b) pattern cuts at 1.6 GHz; (c) pattern cuts at 2.3 GHz.

making the bistatic/monostatic equivalence condition for our virtual elements to read (4) This condition is easily met for our new array as long as the imaging scene is at least 2 m from the array aperture. The validity of the bistatic/monostatic equivalence condition as derived above is independent of target size (provided that the target can be represented by a set of scattering centers). However, the radar/imaging scene separation distance and target scattering properties can also play a role. In this regard, it was shown in [23] that the target complexity can severely restrict the bistatic/monostatic validity region. Specifically, shadowing and multipath interactions typically render the assumptions associated with (4) invalid. Nevertheless, for very small bistatic angles (as is the case for the proposed snapshot imaging radar), the assumptions leading to (4) are generally valid. We remark that the virtual elements (midway between all adjacent real elements) form a new triangular lattice as depicted in Fig. 6(b). More importantly, the inter-element spacing of this new lattice is half the size of the original lattice when using only real element feeds. Concluding, the spacing between adjacent , satisfying the grating lobe “virtual” phase centers is below . Several array patterns condition [24], [25] at 2.3 GHz are depicted in Fig. 8, corresponding to the low, middle, and high ends of the frequency band. In bistatic operation, the array is excited sequentially, cycling through each adjacent pair of transmit/receive elements, one at a time. In addition, monostatic data realized from each real element are collected at the same time (as discussed below in Section II-C). The collected data can then be combined coherently and processed to create a 3-D image without distortion from grating lobes. This image formulation process is discussed in Section III. The suppression of grating lobes due to the virtual elements is also demonstrated in Section III. C. System Integration and Data Collection A significant component of any imaging radar system is the collection, storage, and processing of data to generate an image.

Fig. 9. Photograph of the imaging system components mounted on the array backplane. This makes up the feed network. The custom switch PCBs (right side) and balun PCBs (middle) are properly configured via the digital I/O PCB (left side) to connect the appropriate set of antennas in the array to the input/ output RF source ports (bottom right side).

From Fig. 2, the proposed system employs a 2-port network analyzer as the RF source and receiver. It also incorporates a serial data collection method enabled via a digitally controlled RF multiplexer constructed of printed circuit boards (PCBs) with RF switching and feed matching components located on the array backplane. The multiplexer connects any two elements of the array with the network analyzer ports via balanced feeds, which are created using integrated ceramic baluns. We note that the RF switches (manufactured by Hittite, but a variety of other RF switches can be implemented) are digitally controlled via the PC using a high speed digital I/O interface. Fig. 9 displays the backplane feed network connecting the RF source to the antenna elements. As noted, the PCB digital control lines are interfaced to a graphical user interface (GUI) (using Labview [26]) to automate the data collection process (see Fig. 10). The GUI also interfaces with the RF source, via a general purpose interface bus (GPIB) to define bandwidth, center frequency, power level, and IF bandwidth, among others. This software has the capability to

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Fig. 11. Visual interpretation of terms given in equations (5) and (7).

Fig. 10. Screen-shot of the graphical user interface for controlling the imaging system hardware and data collection.

cycle through a variety of predefined antenna transmit and receive combinations, and to store data for post-processing. A typical radar image data collection sequence involves cycling through all adjacent pairs of array elements, connecting one to port 1 and the other to port 2 of the network analyzer. The network analyzer steps through the pre-defined frequencies and stores the amplitude and phase of returned signals from each and signals are utilized for monostatic scattering, port. or values are utilized. whereas for bistatic scattering This allows data to be collected for both the real and virtual phase centers simultaneously, making this approach extremely fast to construct “snapshot” images in near real time. Vertical, horizontal, or cross-polarization images can be generated, and several of these images may be over a variety of scene viewpoints, which later can be fused into a composite scene image. This is demonstrated in the next section. III. THROUGH-WALL IMAGES A. Near Field Imaging Algorithm To examine the system functionality, images were created from measured raw scattering data using a near field back-projection approach for inverting the forward model. The model used here for the frequency domain backscattered signal rephase center located at is given by ceived by the

is the wave number, where an harmonic time convention has been assumed and suppressed. We have assumed a two-way reflection-type Green’s function rather than a two-way . point scattering Green’s function which would be This is simply because the dominant through-wall scattering mechanisms tend to be due to surface reflections rather than point scattering. It is noted that (5) is a free space model, which is applied with the understanding that some image distortion and displacement will occur due to non-free space propagation through the wall (i.e., layered/complex medium). These effects may be corrected using a more sophisticated through-wall model [27]. The imaging function we are after is an estimate of the reflec, and is found here by minimizing the error tivity density phase centers and assuming the voxel in the model over all points are linearly independent [28]. This yields the near field back-projection result,

(7)

where is an optional windowing function introduced to and are the limits shape/suppress image sidelobes, and of integration over the frequency band of interest. Note that the imaging kernel is conjugate phase-matched to the forward model of (5). This imaging function is used to generate very good 3-D images efficiently, and is relatively independent of range variations in the volume of interest. It also is expected to resolve distributed targets well [29]. B. Image Resolution Given the aperture size and frequency band, the downrange and cross-range image resolution may be estimated as [20]

(5) (8) where is the radar reflectivity density function at any voxel point in the imaging volume (see Fig. 11), and is the free space Green’s function given by

(6)

(9) where is the speed of light, is the bandwidth, is the wavelength at the center frequency, is the range, and is the aperture width. For our array, operating from 0.9 to 2.3 GHz, an ideal point target (i.e., a unit amplitude infinitesimal scatterer

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Fig. 13. Scene configuration used for completing through-wall measurements with the snapshot imaging system. Distances are in meters unless otherwise noted.

Clearly, the PSF peaks when . As depicted in Fig. 12(a), the PSF based on the actual array elements reveals significant image degradation due to grating lobes. In contrast, when the virtual phase centers are employed, grating lobes are suppressed. Thus, no distortions are present in the image. However, of interest is cross-range resolution improvement, which is discussed in the following. C. 3-D Image Creation

Fig. 12. Normalized 2-D PSF cuts in dB when utilizing actual and virtual phase centers. A bandwidth of 1.4 GHz (0.9 to 2.3 GHz) was implemented and a Hann window was applied to reduce sidelobes in down range. (a) Actual phase centers only; (b) virtual phase centers only.

independent of observation angle) positioned 1.5 m away yields a down-range resolution of 11 cm, and a cross-range resolution of 35 cm and 46 cm in the horizontal and vertical planes is achieved. To assess the importance of virtual phase centers within the array, the 3-D point spread functions (PSF) of a point scatterer positioned 1.5 m from the array center are shown in Fig. 12. These PSF plots reveal the spatial spreading of a single point scatterer for a given set of array characteristics. The PSF for a point scatterer at follows from (7) and is given by

(10)

To demonstrate the system’s imaging effectiveness, a test scene was created consisting of two 22.86 cm trihedrals placed behind a cinder block wall. Two distinct data collection “snapshots” were completed at the locations shown in Fig. 13. In general, each snapshot location should be strategically chosen with respect to each other. A large angular diversity (up to 90 ) about the scene center is desired to realize maximum cross-range resolution within the final image (this is dependent upon the angular properties of the scatterers present within the scene). As usual, system calibration was first performed to ensure that phase and amplitude variations were properly compensated. This calibration also corrects for mutual coupling effects between array elements. Figs. 14(a) and (b) depict several “snapshot” images produced from collected reflectivity density data. As originally conceived, the two coherent snapshot images were combined non-coherently to form a composite (fused) image, as seen in Fig. 14(c). Knowing the exact snapshot positions is not critical due to the nature of image fusion. Only slight image distortions (in the form of smearing) are present when snapshot positions are not exactly known. This is precisely why the fusion process is utilized in the snapshot imaging procedure. The fused images are simply an average of the amplitudes contributed from the two coherent images in Fig. 14(a) and (b). We note that several image artifacts appear behind the trihedral targets in Fig. 14(a) and (b). These are due to multipath effects caused by wall, ground, and/or other trihedral target interactions. As the fusing process de-emphasizes features not

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(a)

(b)

(c)

Fig. 14. Top and side view of 3-D normalized (relative to the maximum amplitude of the images seen in (a)) images of trihedral targets behind a cinder block wall (see Fig. 13) generated by processing data collected from 2 snapshots (i.e., locations) by the UWB flat panel array seen in Figs. 2 and 6(a). (a) First snapshot location; (b) second snapshot location; (c) fused (composite) image (created from both snapshot images in (a) and (b)).

common to both images, these artifacts are significantly reduced in Fig. 14(c). It was also observed that the individual snapshot images have poor cross-range resolution. In contrast, the composite image has better down and cross-range (azimuth) resolution since the target’s location is reinforced during the fusion process. Moreover, it reveals the locations of the two targets with a slight offset in down range. This offset is due to the slower propagation velocity within the wall, which is uncompensated for in the free space model. Nevertheless, the resolution in elevation was not improved in the final image. This is because image fusion (for this particular case) lead to angular diversity only in azimuth, and not in elevation. The images presented in this paper are highly encouraging, but more work remains to improve the system and imaging algorithms for more practical applications. This includes a study on the non-coherent integration of snapshots emerging from diverse vantage points. Improving cross-range resolution of these snapshots is also critical and can be achieved by applying coherent wideband super-resolution and direction of arrival (DOA) algorithms [13], [14], [27], [30]–[33]. Such algorithms are fairly mature and can be adapted to more precisely estimate the DOA from multiple individual scattering centers. Thus, targets can be isolated with much higher resolution. In addition, image distortion effects due to unknown wall properties will also be examined. When imaging through walls, a modified Green’s function must be used in (7) to focus and locate targets. Several approaches, including wall modeling [34], [35],

and parameter estimation (auto-focusing techniques), can be adapted to determine the proper Green’s function [27]. IV. CONCLUSION A unique UWB through-wall imaging system was demonstrated using a flat-panel fixed aperture array. This system has several advantages over traditional SAR approaches in that high resolution through-wall images can be created via a non-coherent process. A key aspect of the system is the employed fast data collection approach at individual scene locations (referred to as “snapshots”). Enhanced image clarity arises by combining snapshot images obtained from several vantage points. In addition, the system is lightweight, portable, and low cost (apart from the network analyzer used as the RF source/receiver). It is also digitally controlled to maximize operational flexibility by implementing an automated serial data collection scheme. The virtual phase centers, employed for grating lobe suppression, allow for a much wider operational bandwidth. Several through-wall images were presented to demonstrate the functionality of the system. It was shown that utilizing only 2 noncoherently integrated snapshot images led to a fairly high resolution composite image. Improved post-processing algorithms are needed to further enhance the resolution capability of the fabricated imaging system. These algorithms will allow for polarization exploitation, scene diversity, and data fusion to produce sharper composite images. Lastly, although we have shown that the proposed system gives very good through-wall images assuming only a free space

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propagation model, a through-wall propagation model should also be investigated. A non-trivial Green’s function which accounts for wall effects must be implemented so that highly focused images with less distortion result. In the course of future research, wall compensation techniques [7], [30], [36], [37] to correct for through-wall distortion and displacement effects will be extended and modified to augment the snapshot radar system’s unique functionality.

REFERENCES [1] M. Amin and K. Sarabandi, “Special issue on remote sensing of building interior,” IEEE Trans. Geosci. Remote Sensing, vol. 47, no. 5, May 2009. [2] D. Morgan, “Wideband RF detection of changes within building structures,” IEEE Geosci. Remote Sens. Lett., vol. 5, no. 4, pp. 715–719, Oct. 2008. [3] G. Hickman and J. Krolik, “A graph-theoretic approach to constrained floor plan estimation from radar measurements,” IEEE Trans. Signal Process., vol. 57, no. 5, pp. 1877–1888, May 2009. [4] F. Soldovieri, R. Solimene, A. Brancaccio, and R. Pierri, “Localization of the interfaces of a slab hidden behind a wall,” IEEE Trans. Geosci. Remote Sens. E, vol. 45, no. 8, pp. 2471–2482, Aug. 2007. [5] L.-P. Song, C. Yu, and Q. H. Liu, “Through-wall imaging (TWI) by radar: 2-D tomographic results and analyses,” IEEE Trans. Geosci. Remote Sens. E, vol. 43, no. 12, pp. 2793–2798, Dec. 2005. [6] S. Hantscher, A. Reisenzahn, and C. G. Diskus, “Through-Wall imaging with a 3D UWB SAR algorithm,” IEEE Signal Process. Lett., vol. 15, pp. 269–272, 2008. [7] M. Dehmollaian and K. Sarabandi, “Refocusing through building walls using synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. E, vol. 46, no. 6, pp. 1589–1599, Jun. 2008. [8] F. Soldovieri and R. Solimene, “Through-Wall imaging via a linear inverse scattering algorithm,” IEEE Geosci. Remote Sens. Lett., vol. 4, no. 4, pp. 513–517, Oct. 2007. [9] F. Ahmad, M. Amin, and S. Kassam, “Synthetic aperture beamformer for imaging through a dielectric wall,” IEEE Trans. Aerosp. Electron. Syst., vol. 41, no. 1, pp. 271–283, Jan. 2005. [10] F. Ahmad, G. Frazer, S. Kassam, and M. Amin, “Design and implementation of near-field, wideband synthetic aperture beamformers,” IEEE Trans. Aerosp. Electron. Syst., vol. 40, no. 1, pp. 206–220, Jan. 2004. [11] J. J. Lee, S. Livingston, and R. Koenig, “A low-profile wide-band (5:1) dual-pol array,” IEEE Antennas Propag. Lett., vol. 2, pp. 46–49, 2003. [12] J. Shin and D. Schaubert, “A parameter study of stripline-fed vivaldi notch-antenna arrays,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 879–886, May 1999. [13] M. Ghavami, “Wideband smart antenna theory using rectangular array structures,” IEEE Trans. Signal Process., vol. 50, no. 9, pp. 2143–2151, Sep. 2002. [14] M. Uthansakul and M. Bialkowski, “Fully spatial wide-band beamforming using a rectangular array of planar monopoles,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 527–533, Feb. 2006. [15] E. Pelton and B. Munk, “Scattering from periodic arrays of crossed dipoles,” IEEE Trans. Antennas Propag., vol. 27, no. 3, pp. 323–330, May 1979. [16] B. Munk and R. Luebbers, “Gain of arrays of dipoles with a ground plane,” IEEE Trans. Antennas Propag., vol. 20, no. 5, pp. 641–642, Sep. 1972. [17] S. Raut and A. Petosa, “A compact printed bowtie antenna for ultrawideband applications,” in Proc. Eur. Microwave Conf., Oct. 1–29, 2009, pp. 081–084. [18] Y.-Z. Yin, J.-P. Ma, Y.-J. Zhao, H.-L. Zheng, and Y.-M. Guo, “Wideband printed dipole antenna for wireless lan,” in Proc. IEEE Antennas Propagation Soc. Int. Symp., Jul. 2005, vol. 2B, pp. 568–571. [19] Printed Antennas For Wireless Communications, R. Waterhouse, Ed. Hoboken, NJ: Wiley, 2007. [20] D. L. Mensa, , D. K. Barton, Ed., High Resolution Radar Cross-Section Imaging. Boston, MA: Artech House, 1991.

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[21] A. Monti Guarnieri and F. Rocca, “Reduction to monostatic focusing of bistatic or motion uncompensated SAR surveys,” Radar, Sonar Navigation, IEE Proc., vol. 153, no. 3, pp. 199–207, Jun. 2006. [22] S. Gabig, K. Wilson, P. Collins, J. Terzuoli, A. J. , G. Nesti, and J. Fortuny, “Validation of near-field monostatic to bistatic equivalence theorem,” in Proc. IEEE Int. Geosci. Remote Sensing Symp, 2000, vol. 3, pp. 1012–1014. [23] C. Bradley, P. Collins, D. Falconer, J. Fortuny-Guasch, and A. Terzuoli, “Evaluation of a near-field monostatic-to-bistatic equivalence theorem,” IEEE Trans. Geosci. Remote Sens. E, vol. 46, no. 2, pp. 449–457, Feb. 2008. [24] N. Fourikis, , K. Chang, Ed., Phased Array-Based Systems and Applications. New York: Wiley-Interscience, 1997. [25] R. C. Hansen, , K. Chang, Ed., Phased Array Antennas. New York: Wiley-Interscience, 1998. [26] Labview Software National Instruments, 2009. [27] P. Chang, R. Burkholder, and J. Volakis, “Adaptive clean with target refocusing for through-wall image improvement,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 155–162, Jan. 2010. [28] L. C. Potter, P. Schniter, and J. Ziniel, “Sparse reconstruction for radar,” SPIE Algorithms Synthetic Aperture Radar Imagery XV, vol. 6970, May 2008. [29] X. Zhang, P. Willett, and Y. Bar-Shalom, “Detection and localization of multiple unresolved extended targets via monopulse radar signal processing,” IEEE Trans. Aerosp. Electron. Syst., vol. 45, no. 2, pp. 455–472, Apr. 2009. [30] Y.-S. Yoon and M. Amin, “High-Resolution through-the-wall radar imaging using beamspace MUSIC,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1763–1774, Jun. 2008. [31] T. Do-Hong and P. Russer, “Spatial signal processing for wideband beamforming,” Institute for High-Frequency Engineering Munich Univ. Technol. [32] Advances in Direction-Of-Arrival Estimation, S. Chandran, Ed. Boston, MA: Artech House, 2006. [33] P. Chang, R. Burkholder, J. Volakis, R. Marhefka, and Y. Bayram, “High-Frequency EM characterization of through-wall building imaging,” IEEE Trans. Geosci. Remote Sens. E, vol. 47, no. 5, pp. 1375–1387, May 2009. [34] R. Burkholder, P. Chang, Y. Bayram, R. Marhefka, and J. Volakis, “Model-based near-field imaging of objects inside a room,” in Proc. IEEE Antennas Propagation Soc. Int. Symp., Jun. 2007, pp. 1469–1472. [35] Y. Bayram, P. Chang, R. Burkholder, and J. Volakis, “Hybrid semianalytical technique for modeling brick walls at high frequencies,” in Proc. IEEE Antennas Propagation Soc. Int. Symp., Jun. 2007, pp. 5359–5362. [36] G. Wang and M. Amin, “Imaging through unknown walls using different standoff distances,” IEEE Trans. Signal Process., vol. 54, no. 10, pp. 4015–4025, Oct. 2006. [37] F. Ahmad, M. Amin, and G. Mandapati, “Autofocusing of through-thewall radar imagery under unknown wall characteristics,” IEEE Trans. Image Process., vol. 16, no. 7, pp. 1785–1795, Jul. 2007.

Kenneth E. Browne (S’02) was born on April 19, 1982 in Sandusky, OH. He received the B.S. (magna cum laude and with distinction) and M.S. degrees in electrical and computer engineering from The Ohio State University, Columbus, in 2005 and 2007, respectively, where he is currently working toward the Ph.D. degree. In 2005, he joined the Electronic Systems Sector, Northrop Grumman Corporation, while simultaneously working at the ElectroScience Laboratory, The Ohio State University. He has since completed research and design on a variety of application specific UWB antennas and RF systems utilized in radar calibration processes. His current research includes digital image restoration, low-profile wideband antennas/arrays, radar signal processing, compressed sensing in radar imagery, synthetic aperture radar, and radar system architectures.

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Robert J. Burkholder (S’85–M’89–SM’97–F’05) received the B.S., M.S., and Ph.D. degrees in electrical engineering from The Ohio State University, Columbus, in 1984, 1985, and 1989, respectively. Since 1989, he has been with the Ohio State University ElectroScience Laboratory, Department of Electrical and Computer Engineering, where he is a Research Professor. He has contributed extensively to the EM scattering analysis of large and complex geometries, and targets in the presence of rough surfaces and inside urban structures. His research specialties are high-frequency asymptotic techniques and their hybrid combination with numerical techniques for solving large-scale electromagnetic radiation, propagation, and scattering problems. Dr. Burkholder is an elected Full Member of URSI, Commission B, a member of the American Geophysical Union, and a member of the Applied Computational Electromagnetics Society. He is currently serving as an Associate Editor for IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

John L. Volakis (S’77–M’82–SM’89–F’96) was born on May 13, 1956 in Chios, Greece and immigrated to the U.S.A. in 1973. He received the B.E. degree (summa cum laude) from Youngstown State University, Youngstown, OH, in 1978, and the M.Sc. and Ph.D. degrees from the Ohio State University, Columbus, in 1979 and 1982, respectively. He started his career at Rockwell International (1982–1984), now Boeing Phantom Works. In 1984, he was appointed Assistant Professor at the University of Michigan, Ann Arbor, becoming a full Professor in 1994. He also served as the Director of the Radiation Laboratory from 1998 to 2000. Since January 2003, he is the Roy and Lois Chope Chair Professor of Engineering at the Ohio State University, Columbus, and also serves as the Director of the ElectroScience Laboratory. His primary research deals with antennas, computational methods, electromagnetic compatibility and interference, propagation, design optimization, RF materials, multi-physics engineering and bioelectromagnetics. He has published over 280 articles in major refereed journals, nearly 500 conference papers and 20 book chapters. He coauthored the following six books: Approximate Boundary Conditions in Electromagnetics (Institution of Electrical Engineers, London, 1995), Finite Element Method for Electromagnetics (IEEE Press, New York, 1998), Frequency Domain Hybrid Finite Element Methods in Electromagnetics (Morgan & Claypool, 2006), Computational Methods for High Frequency Electromagnetic Interference (Verlag, 2009), Small Antennas (McGraw-Hill, 2010), and edited the Antenna Engineering Handbook (McGraw-Hill, 2007). He has also written several well-edited coursepacks on introductory and advanced numerical methods for electromagnetics, and has delivered short courses on antennas, numerical methods, and frequency selective surfaces. Dr. Volakis was elected Fellow of the IEEE in 1996, and is a member of the URSI Commissions B and E. In 1998 he received the University of Michigan (UM) College of Engineering Research Excellence award, in 2001 he received the UM, Dept. of Electrical Engineering and Computer Science Service Excellence Award, and in 2010 he received the Ohio State Univ. Clara and Peter Scott award for outstanding academic achievement. He was the 2004 President of the IEEE Antennas and Propagation Society and served on the AdCom of the IEEE Antennas and Propagation Society from 1995 to 1998. He also served as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION from 1988–1992, Radio Science from 1994–1997, and for the IEEE Antennas and Propagation Society Magazine (1992–2006), the J. Electromagnetic Waves and Applications and the URSI Bulletin. He chaired the 1993 IEEE Antennas and Propagation Society Symposium and Radio Science Meeting in Ann Arbor, MI., and co-chaired the same Symposium in 2003 at Columbus, OH. He is listed by ISI among the top 250 most referenced authors. He graduated/mentored nearly 60 Ph.D. students/post-docs, and coauthored with them 14 best paper awards at conferences.

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Adaptive Nulling Using Photoconductive Attenuators Randy L. Haupt, Fellow, IEEE, Joseph Flemish, Member, IEEE, and Daniel Aten, Member, IEEE

Abstract—Traditional adaptive antennas require receivers at each element in the array in order to form the correlation matrix and derive the adaptive weights. This approach is expensive and difficult to implement on many phased array architectures. An alternative is to minimize the total array output power by making only a subset of the elements adaptive. Nulls can be placed in the sidelobes with little effect on the main beam. This paper uses the later approach coupled with new photoconductive attenuators that serve as adaptive amplitude weights. The attenuators on some the elements in a phased array serve as adaptive weights while the remaining elements are uniformly weighted. A genetic algorithm controls the infrared signals to these elements in order to minimize the total output power. Experimental and computer results demonstrate the effectiveness of this approach. Index Terms—Adaptive antennas, adaptive arrays, antenna arrays, silicon, photoconducting devices.

I. INTRODUCTION NE form of adaptive nulling uses a numerical optimization algorithm to find the element weights that minimize the total output power of the array. Random search techniques, such as the random walk, perturb the element weights until nulls form [1]. These algorithms are slow. The phase-only steepest descent algorithm sequentially toggles every element in the array and measures the output power to form a gradient vector [2]. A minimum in the output power is found by following the gradient. Not only is this approach slow, but it only finds a local minimum. Global search algorithms that simultaneously change all the element weights at once tend to be fast and do not get stuck in a local minimum [3]. Adaptive algorithms that minimize the total output power of an array cannot distinguish between the desired signal and interfering signals. If the desired signal enters the main beam while interfering signals enter the sidelobes, then the adaptive algorithm attempts to place nulls in the main beam as well as the sidelobes. One way to preclude nulls in the main beam turns the adaptive algorithm on only when the desired signal is not present. Such an approach is conceivable for a radar system that has time between transmitting and receiving a pulse. A better method limits adaptive weights to small deviations that can place nulls in the sidelobes but not in the main beam [4]. Amplitude weights should be greater than some minimum value while phase weights should be less than some maximum value. The minimum and maximum weight limits are determined from

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Manuscript received October 15, 2009; revised July 23, 2010; accepted October 13, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Applied Research Laboratory, Pennsylvania State University, State College, PA 16804 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.2010.2103007

the maximum allowable gain reduction and sidelobe level distortion while sufficiently placing nulls in the sidelobes. This approach has been successfully coupled with a genetic algorithm (GA) and experimentally demonstrated [5]. Partial adaptive nulling makes a subset of all the elements adaptive [6]. The power wielded by the subset of elements is not enough to place a null in the main beam. For instance, making half of the amplitude weights in an array zero causes a 6 dB reduction in the main beam. Thus, adjusting half the amplitude weights to place a null in the sidelobe will at most reduce the main beam by 6 dB. If half of the elements were phase shifters capable of 180 phase shifts, then a deep null can be created in the main beam that would help minimize the total output power. A successful phase-only adaptive nulling experimental linear array was reported in [7]. Nulls were adaptively placed in the low sidelobes using a steepest descent algorithm and 4 adaptive elements out of 80 total elements. This paper describes an approach to amplitude-only partial adaptive nulling using a GA that minimizes the total output power. The adaptive array is modeled using a subset of an array of 32 dipoles with a low sidelobe amplitude taper. Results indicate that nulls can be placed without significantly degrading the main beam gain. The sidelobe levels increase as the number of adaptive elements increase. In order to experimentally demonstrate this approach to adaptive nulling, a linear array of eight broadband monopoles was constructed whereby a new type of silicon-based photonic attenuator was integrated into each monopole element. The amount of attenuation at an element is a function of the current fed to an IR LED that illuminates the photoconductive silicon in the attenuator to provide a resistive shunt to ground that is a function of the illumination intensity. These attenuators have low insertion loss and phase distortion. Experimental results with two sources and 2 or 4 adaptive elements demonstrate the nulls placed in the sidelobes are much deeper than the gain lost in the main beam. II. AMPLITUDE-ONLY NULLING WITH A GENETIC ALGORITHM Speed is important for most adaptive nulling applications. If a GA is the adaptive algorithm, then the large population sizes advocated by most GA researchers is unacceptable due to the large number of function calls (antenna power measurements). GAs with small population sizes and relatively high mutation rates are very effective in quickly finding acceptable solutions [8] and are used here. Since many weight combinations produce a deep null at a designated angle, the task of finding the global minimum is not necessary, because one of the many local minima produces excellent results. A flowchart of the adaptive GA appears in Fig. 1. The population matrix has 8 rows or chromosomes. A chromosome in the GA population controls the amplitude weight at each of the adaptive elements. Assuming that the array has elements with

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Fig. 2. Diagram of the adaptive dipole array.

Fig. 1. Flow chart of the adaptive GA that minimizes the total array output power.

of those adaptive , then a chromosome has variables. The adaptive elements can be any elements in the array. Positions of the adaptive elements in the array determine the spatial Fourier components available for performing the adaptive nulling. If the array has a low sidelobe taper, then the weight of the particular element determines how much it contributes to the nulling process. For ease of notation, the adapedge elements on either tive elements are assumed to be the side of a linear array. Consequently, a chromosome contains the following attenuator settings: (1) and and . If the array has a low sidelobe amplitude taper with , then the relative signal weights, , where voltage at all the elements is the product of the array taper and the attenuator settings where

Fig. 3. Quiescent and adapted weights for a 32 element array of dipoles with 8 adaptive elements.

Since the cost function is the total output power, amplitude modulated signals as well as fading cause fluctuations in the cost which could confuse the adaptive algorithm and prevent convergence. Using time averages rather than instantaneous samples of the output power alleviates this problem. Amplitude variations plague more conventional perturbation adaptive nulling algorithms as well, so many samples are typically used to build the covariance matrix [1]. III. SIMULATION RESULTS

(2) The computer modifies the attenuator settings then measures the total output power received by the array. This process continues until each chromosome in the population matrix has an associated output power. The output power is the cost to be minimized. The GA selects the top 50% of the chromosomes (those with the lowest output power) and creates new chromosomes through a mating process (tournament selection and uniform crossover) [8]. Two sets of two chromosomes are selected and the lowest cost chromosomes from each set become parents and mate. The mating operation, uniform crossover, generates a random binary mask. In mask columns where a “0” occurs, the parents exchange their weights, otherwise, a “1” indicates they keep their weights in the corresponding columns. More details on the adaptive genetic algorithm used here can be found in [9]. After the mating process is done, 10% of the weights in the population are mutated or randomly changed. This process continues until the interfering signals are suppressed.

The adaptive array model consists of , z-oriented dipole elements spaced half a wavelength apart along the x-axis (Fig. 2). Even though the adaptive array would be a receive antenna, the model described here is a transmit array with voltage source at element . The voltage at an element is determined by (2). . Assume the desired signal is located at where the Interference signals enter the sidelobes at index indicates one of the interfering signals. The array is modeled using FEKO [10], a method of moments program that includes the mutual coupling between all the elements. Each dipole is 0.48 long and has a 0.002 radius. They long. In order to accelare broken into segments that are erate the calculations, the fast multipole option is used. The total output power is the sum of the products of the signal powers and the gains of the array pattern in the direction of the signals. Taylor The simulated array has 32 dipoles and a 30 dB low sidelobe quiescent amplitude taper. Fig. 3 shows the quiescent amplitude taper (dashed line), and Fig. 4 shows the quiescent far field pattern (dashed line). Two signals are incident

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Fig. 4. Quiescent and adapted patterns for a 32 element array of dipoles with 8 adaptive elements.

Fig. 6. Quiescent and adapted patterns for a 32 element array of dipoles with 16 adaptive elements.

the array efficiency and gain go down. Second, the sidelobe levels significantly increase as shown in Fig. 6. Not only are there more weight perturbations, but the quiescent weights have a higher value closer to the center of the array. As a result, perturbations to amplitude weights with high values have more effect than perturbations to amplitude weights with low values. Finally, the algorithm takes longer to place the nulls, because the number of optimization variables doubled. In this case, the array typically takes 7 or 8 generations to reach a minimum. Consequently, the number of adaptive elements should be large enough to place nulls in the highest sidelobes and place enough nulls to cover all the interfering sources. On the other hand, the number of adaptive weights should be minimized in order to keep the adapted antenna pattern close to the quiescent pattern. Fig. 5. Quiescent and adapted weights for a 32 element array of dipoles with 16 adaptive elements.

upon the array. A 0 dB desired signal is at , and 40 dB interference signals are at sidelobe peaks in the quiescent patand 125 . The GA quickly places nulls in the tern at desired directions—within three generations for many different independent random runs. Fig. 3 plots the adapted weights as a solid line. Adaptation did not alter the center 24 array weights. The adaptive weights reduced the quiescent sidelobe level at from dB to dB as shown in Fig. 4. went from In addition, the quiescent sidelobe level at dB to dB. The peak sidelobe level of the adaptive array pattern actually dropped almost 2 dB, but the sidelobe level went up several dB above the quiescent pattern at most angles. In this example, the main beam gain did not noticeably change even though the adaptive weights lowered the taper efat the adaptive ficiency of the array, because the values of element are small. Repeating the previous example using 16 adaptive elements gives the array more degrees of freedom in order to null more jammers but has several undesirable effects. First, the main beam gain goes down by 0.5 dB. Fig. 5 compares the quiescent array weights with the adapted array weights. The adapted weights are considerably lower than the quiescent weights, so

IV. EXPERIMENTAL RESULTS An experimental rendition of this adaptive array requires attenuators at the adaptive elements. Variable attenuators typically have high insertion loss. In addition, a change in amplitude causes an unwanted change in phase as well. Phase shifters are often used to counteract the unwanted phase change that varies with every attenuator setting. Placing the attenuators at every element, even when the attenuators are not used for adaptation, insures that each element is identical. A photoconductive attenuator was developed in conjunction with an antenna element for this application (Fig. 7) [11]. The antenna element is a planar monocone with a coplanar waveguide feed called a tab monopole in [12]. Microwave Studio [13] was used to model the antenna element and attenuator. Due to the small scale of the attenuator features and the much larger scale of the antenna element, the two components had to be modelled separately. Details of this attenuator have been reported in [11]. The attenuator consists of a meandering metal signal trace embedded within a coplanar waveguide. The metal is deposited on an oxidized high-resistivity silicon (HR-Si) substrate and is capacitively coupled to the underlying silicon. The silicon has very S/m) under dark conditions and modlow conductivity ( erate conductivity ( S/m) when illuminated by an infrared

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Fig. 7. The attenuator on the left is flipped and soldered to the antenna element on the right.

Fig. 9. Experimental measurement of S .

Fig. 8. Measured (thick solid) and simulated (thin dashed) insertion loss or attenuation when LED is on and off.

Fig. 10. Plot of S

light-emitting diode (IR-LED). The element design is optimized for low insertion loss in the on-state and maximum attenuation in the off-state. Signal attenuation occurs by illuminating the device using a single IR-LED such that the photoconductivity creates a resistive shunt between the center conductor and the ground. The oxide passivation of the HR-Si surface increases the lifetime of the photogenerated carriers thus enhances the attenuation response. In addition, silicon nitride (SiN ) antireflection coating further increases the range of attenuation by allowing more light into the silicon and by enhancing carrier lifetime due to positive fixed charge in the SiN . Such charge is well known to affect the surface recombination velocity of carriers by field-induced separation of electrons and holes at the silicon-dielectric interface [14]. Fig. 8 shows the simulated and measured characteristics under dark conditions as well as under nm, maximum illumination from a common IR LED ( mW). Experimentally, the attenuation induced is a function of the illumination intensity from the LED and can be controlled easily within the range shown. In simulations which were performed using Microwave Studio [13] the dark and illuminated conditions assumed silicon conductivity values of 0.05 and 8 S/m, respectively. The element has a measured bandwidth extending from 2.1 dB). The integrated attenuator and to 2.5 GHz ( antenna element was fabricated on FR4 circuit board material with a discontinuity in the coplanar waveguide trace centered

around a through-hole in the substrate to allow for flip-chip style mounting of the attenuator and illumination of the chip through the FR4 substrate. The of the adaptive elements was measured using the experimental setup shown in Fig. 9. A receive antenna without the attenuator was placed 5 cm in front of one of the integrated elements. The LED in back of the integrated element illuminated the attenuator. Fig. 10 is the measured as the current varies from 0 to 250 mA. Over 15 dB of signal attenuation is available at each element. Unlike most attenuators, the phase variation for the photonic attenuator is small over the range of attenuation states. The signal gains approximately of phase per 1 dB of attenuation. Fig. 11 shows that there is less than 5 degrees variation up to 50 mA. Eight integrated elements were fabricated and assembled into a linear array as shown in Fig. 12 [15]. The output from each element goes to an 8 to 1 power combiner. An Agilent spectrum analyzer receives the combiner output. A diagram of the experimental setup for testing the array is shown in Fig. 13. Two signal generators provided the desired and interfering signals. The signal generators create CW as well as modulated signals. A signal analyzer sends the received signal to the computer where the GA resides. In turn, the computer sends chromosomes to PXI current control cards that regulate the current fed to each LED. Finally, the LEDs illuminate the attenuators to control the amplitude of the signals received by the adaptive elements.

as a function of diode current for the 4 adaptive elements.

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Fig. 11. Plot of

S

as a function of diode current for the 4 adaptive elements.

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Fig. 14. Antenna pattern when the four adaptive elements are turned off.

Fig. 12. Picture of experimental array with numbered elements.

Fig. 15. Adapted pattern when a signal is incident on the array at

019

.

Fig. 13. Diagram of the adaptive array experiment.

The first test of the array was to see how much the main beam decreased when the LEDs of the four adaptive elements received 250 mA. Since the array essentially becomes a four element uniform array, the main beam should decrease by 6 dB. Fig. 14 shows a 5.2 dB decrease in the main beam of the measured far field pattern. This result demonstrates that the adaptive array reduces the desired signal entering the main beam but does not eliminate it. The gain does not decrease by 6 dB, because the elements that are essentially turned off still couple to the active elements and add to the gain. The next experiment had only one signal incident on the array . After 10 generations, the GA found the diode currents at in Table I that produce the far field pattern shown in Fig. 15. Even though there is no signal present at 0 , the main beam

Fig. 16. Adapted pattern when signals are incident on the array at 0 and

019 .

is reduced by dB. The GA lowered the sidelobe level by . A much better test of the array has a dBm 15.7 dB at signal incident at 0 , and a 15 dBm signal incident at . The resulting currents in Table I produce the adapted pattern in dB, and the sideFig. 16. Its main beam is reduced by is reduced by 17.1 dB. In this case, the relobe level at sulting adaptation with the desired signal present was approx-

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TABLE I DIODE CURRENTS TO THE ELEMENTS IN THE EXAMPLES PRESENTED

Fig. 19. Convergence of the adaptive algorithm when the signals are incident on the array at 35 and 19 for 6 independent random runs.

0

Fig. 17. Adapted pattern when a signal is incident on the array at

035

0

.

Fig. 20. Adapted pattern when a signal is incident on the array at 35 when elements 1 and 8 are adaptive.

Fig. 18. Adapted pattern when signals are incident on the array at 19 .

0

035

and

imately the same as when it was absent. As a result, further testing could be done with two interference signals entering the sidelobes while no desired signal entered the main beam. Nulls can be placed in any sidelobe of the array pattern. Fig. 17 shows the adapted pattern when only one signal is . The main beam loses 3.9 dB while the null is present at . The currents are shown 46 dB below the sidelobe leve at in Table I. Fig. 18 shows the resulting adapted pattern when two 15 dBm and . The diode currents apsignals are incident at

pear in Table I. No signal is incident upon the main beam. In this case, the main beam is reduced by 3.6 dB. The sidelobe at goes from dB to dB, and the sidelobe at goes from dB to dB. The adaptive algorithm lowered the sidelobes to approximately the same level, since the interference signals have the same power levels. Fig. 19 is a plot of the convergence curves for 6 independent random runs using this configuration. The curves are quite different after one iteration, but they are very close after 5 iterations. These convergence curves are typical of the results for all the adaptive nulling experiments. The final experiments used two adaptive elements to place nulls in the sidelobes. Fig. 20 is the adapted pattern when a signal is incident at 35 when elements 1 and 8 are adaptive. dB while the sidelobe was reThe main beam went down dBm duced by 22 dB. Fig. 21 is the adapted pattern when a signal is incident at 0 , and a 15 dBm signal incident at and elements 1 and 7 are adaptive. The main beam is reduced by dB, and the sidelobe level at is reduced by 13.1 dB. As expected, the main beam gain reduction is much less when only 2 elements are adaptive versus when 4 elements are adap-

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adaptive elements adds more degrees of freedom but results in a decrease in the main beam gain and higher sidelobe levels. The GA proved to be a relatively fast adaptive algorithm by placing the nulls within 5 generations when the population only had 8 chromosomes. REFERENCES

Fig. 21. Adapted pattern when signals are incident on the array at 0 and 18 when elements 1 and 7 are adaptive.

tive. The two additional elements were adequate to place a single null in the highest sidelobe. The diode currents for both of these cases are shown in Table I. Using only four adaptive elements in this experiment limits the number of adaptive nulls to two. This adaptive antenna is capable of suppressing more than two interference signals by lowering the sidelobes rather than placing nulls in the direction of the signals. Since no phase weights are used, the peak of the main beam stays at 0 . Adaptive phase weights tend to steer the main beam unless they are constrained. The loss in gain in this adaptive antenna may be a problem in some situations. If the array has 32 elements with 4 adaptive, then the maximum loss in main beam gain would be only 0.5 dB. but up to two nulls can still be placed. In this case, the bandwidth of the attenuators exceeds the bandwidth of the monopoles. The monopoles with attenuators from 2.11 to 2.53 GHz [15]. have a

[1] R. A. Monzingo, R. L. Haupt, and T. W. Miller, Introduction to Adaptive Arrays. Raleigh, NC: SciTech, 2010. [2] R. L. Haupt, “Adaptive nulling in monopulse antennas,” IEEE Trans. Antennas Propag., vol. 36, no. 2, pp. 202–208, 1988. [3] R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics. Hoboken, NJ: IEEE Press : Wiley-Interscience, 2007. [4] R. L. Haupt, “Phase-only adaptive nulling with genetic algorithms,” IEEE Trans. Antennas Propag., vol. 45, no. 5, pp. 1009–1015, 1997. [5] R. L. Haupt and H. L. Southall, “Experimental adaptive nulling with a genetic algorithm,” Microwave J., vol. 42, no. 1, pp. 78–89, 1999. [6] D. Morgan, “Partially adaptive array techniques,” IEEE Trans. Antennas Propag., vol. 26, no. 6, pp. 823–833, 1978. [7] R. Haupt and R. Shore, “Experimental partially adaptive nulling in a low sidelobe phased array,” in Proc. AP-S Int. Symp., 1984, pp. 823–826. [8] R. Haupt and S. Haupt, Practical Genetic Algorithms. New York: Wiley, 2004. [9] R. L. Haupt, “A mixed integer genetic algorithm for electromagnetics applications,” IEEE Trans. Antennas Propag., vol. 55, no. 3, Mar. 2007. [10] FEKO Suite 5.4, EM Software and Syst., , 2008 [Online]. Available: www.feko.info [11] J. R. Flemish, H. W. Kwan, R. L. Haupt, and M. Lanagan, “A new silicon based photoconductive microwave switch,” Microwave Opt. Technol. Lett., vol. 51, no. 1, pp. 248–252, Jan. 2009. [12] J. M. Johnson and Y. Rahmat-Samii, “The tab monopole,” IEEE Trans. Antennas Propag., vol. 45, no. 1, pp. 187–188, 1997. [13] CST Microwave Studio, Version 2009.07, Sonnet Software, Inc. [Online]. Available: www.sonnetsoftware.com Jun. 16, 2009.S [14] Dauwe, L. Mittelsta, A. Metz, and R. Hezel, “Experimental evidence of parasitic shunting in silicon nitride rear surface passivated solar cells,” Prog. Photovolt. Res. Appl., vol. 10, no. 4, pp. 271–278, 2002. [15] R. L. Haupt, J. R. Flemish, and D. W. Aten, “Broadband linear array with photoconductive weights,” IEEE AWPL, vol. 8, pp. 1288–1290, 2009.

V. CONCLUSION This paper demonstrates an array that minimizes the total output power by adaptively controlling the amplitudes of a small subset of the array elements. In this approach we employ amplitude-only partial adaptive nulling using a genetic algorithm that reconfigures the array to minimize the total output power. Modeled results for a subset of an array of 32 dipoles with a low sidelobe amplitude taper indicate that nulls can be placed without significantly degrading the main beam gain. In order to experimentally demonstrate this approach, a linear array of eight broadband monopoles with adaptive elements was constructed and tested. These elements were made adaptive by integrating a new type of silicon-based photonic attenuator into a broadband coplanar waveguide fed monocone element. The amount of attenuation at an element is a function of the current applied to an IR LED which is controlled by a genetic algorithm interfaced with appropriate detection and control hardware and software. Both numerical modeling and experiment demonstrated the viability of this approach by significantly decreasing the sidelobe level in the direction of the interference sources, while minimally decreasing the main beam gain. Increasing the number of

Randy L. Haupt (M’76–SM’81–F’00) was born in Johnstown, PA. He received the B.S.E.E. degree from the USAF Academy, CO, in 1978, the M.S. in engineering management from Western New England College, Springfield, MA, in 1981, the M.S.E.E. from Northeastern University, Boston, MA, in 1983, and the Ph.D. in EE from The University of Michigan, Ann Arbor, in 1987. He is currently a Senior Scientist and Department Head at the Applied Research Laboratory of The Pennsylvania State University, State College. He was Professor and Department Head of Electrical and Computer Engineering at Utah State University, Professor and Chair of Electrical Engineering at the University of Nevada Reno, and Professor of Electrical Engineering at the USAF Academy. Previous to his academic appointments, he was a project engineer for the OTH-B radar and a research antenna engineer for Rome Air Development Center. He is coauthor of the books Practical Genetic Algorithms (2 ed., Wiley, 2004), Genetic Algorithms in Electromagnetics (Wiley, 2007), and Antenna Arrays a Computation Approach (Wiley, 2010). Dr. Haupt was the Federal Engineer of the Year in 1993. He is a Fellow of the Applied Computational Electromagnetics Society (ACES) and is a member of Tau Beta Pi, Eta Kappa Nu, URSI Commission B, and Electromagnetics Academy. He is a member of the IEEE Antenna and Propagation Society Administrative Committee and the Antenna Standards Committee. He serves as an Associate Editor for the IEEE Antennas and Propagation Magazine and the IEEE Antennas and Wireless Propagation Letters.

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Joseph Flemish (M’99) received the B.S. degree in chemical engineering, and the M.S. and Ph.D. degrees in materials science from the Pennsylvania State University, University Park, in 1984, 1986, and 1989, respectively. He is currently a Senior Scientist at the Applied Research Laboratory and Professor of materials science and engineering at the Pennsylvania State University. His prior experience includes the positions of Senior Principal Engineer and Chief Scientist at ANADIGICS, Inc., Warren, NJ, from 1996 to 2004, and Research Engineer at the U.S. Army Research Laboratory, Ft. Monmouth, NJ, from 1989 to 1996. His professional interests are in the area of electronic and optoelectronic materials, devices and processes.

Daniel Aten (M’10) received the B.S. and M.S. degrees in electrical engineering from the Pennsylvania State University, University Park, in 2007 and 2009, respectively. He is currently an R&D Engineer 2 at the Applied Research Laboratory, The Pennsylvania State University, State College.

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Volumetric-Perturbative Reciprocal Formulation for Scattering From Rough Multilayers Pasquale Imperatore, Member, IEEE, Antonio Iodice, Senior Member, IEEE, and Daniele Riccio, Senior Member, IEEE

Abstract—We present an innovative formulation for the evaluation of the electromagnetic wave interaction with non trivial random stratifications that can include the cases of random roughnesses and volumetric inhomogeneity; the formulation is based on a volumetric perturbative approach and it is intrinsically reciprocal. The description employed to model the multilayered structure relies on a characterization of the space-variant dielectric permittivity perturbation; this approach allows us to consistently treat both interface roughness and volumetric fluctuations. Accordingly, the developed comprehensive scattering approach methodologically permits to, simultaneously and rigorously, take into account both rough-interface scattering and volume scattering. The presented first-order general formulation is then applied to the case of a layered structure with rough interfaces, but no inhomogeneities within each layer. For this case, a closed-form solution is obtained. We also demonstrate that the polarimetric solution, derived for a 3-D layered geometry and a bistatic radar configuration can be directly expressed in terms of unperturbed solutions. Our solution turns out to be formally fully consistent with the one obtained in the theoretical framework of the boundary perturbation approach. A remarkable interpretation of the analytical solution in terms of the Rumsey’s reaction concept is finally provided. Index Terms—Layered media, perturbation methods, radar polarimetry, sub-surface sensing, surface and volume scattering.

I. INTRODUCTION AND MOTIVATION

T

HE amount of data acquired by microwave sensors is continuously increasing; however, the prediction capability of the available electromagnetic (EM) models certainly not always turns out to be satisfactory: major reason for that resides in the intrinsic complexity of modeling the wave interaction with a broad class of inherent natural and man-made structures. In the perspective of overcoming this challenging difficulty, the developing of new and reliable electromagnetic models, possibly leading to closed form solutions, gains new stimulus, because of its crucial role in concretely achieving an accurate understanding and a reliable interpretation of the wide assortment of obtained experimental data.

Manuscript received July 31, 2009; revised July 29, 2010; accepted September 10, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by Agenzia Spaziale Italiana within COSMO/SkyMed AO, project 2202. The Authors are with Department of Biomedical, Electronic and Telecommunication Engineering, University of Naples “Federico II,” 80125 Napoli, Italy (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103004

Within this framework, the approaches typical of the perturbation theory can be sometimes conveniently employed: perturbation theory is introduced to deal with systems that can be regarded as obtained from a solvable system by the addition of a small effect (perturbation); this approach offers a powerful and valuable theoretical technique and allows us attaining approximate solutions of the actual system by suitably adopting some exact solutions relevant to approximate version of the system. The perturbative solution can capture as many features of the analyzed system as many terms of the perturbative development are accounted for. A variety of perturbation methods has been widely adopted in several research areas, such as acoustics, celestial mechanics, quantum mechanics, optics, atomic physics, and quantum chemistry. More specifically, in applied electromagnetics the perturbation theory formulation of Maxwell’s equations has been conveniently applied in several contexts. We underline that, even though the fundamentals of perturbation theory is very simple, however there are not general guidelines for the analytical derivation of a perturbed solution, and very often a significant amount of tedious algebraic manipulation can be required. We consider as a fundamental guideline in our perturbation approaches the use of a sound physical justification in the mathematical developments: as a matter of fact, when mechanically applied without physical justifications, the perturbative techniques can lead to final solutions that are unnecessarily involved and obscure. Generally speaking, scattering theory can be regarded as a form of perturbation analysis. Its goal is to predict the perturbation experienced by an electromagnetic wave that interacts with a medium whose properties, with respect to the ones of the original unperturbed medium, are changed. The scattered field is then the difference between the actual and the unperturbed EM wave. The problem is mathematically susceptible also of a formulation in terms of perturbation of linear operators [1]–[4]. Although several perturbation strategies have been proposed (such as small perturbation method (SPM) [5]–[9], phase perturbation method [10]–[12], self-energy perturbation method [13], [14], etc.) to cope with EM scattering, SPM remains the one widely adopted. Concerning surface scattering, SPM solution to an arbitrary order can be derived by using the Rayleigh method (also referred to as Rayleigh-Rice or Rayleigh-Fano procedure), which relies on the Rayleigh hypothesis for expanding the scattered field in power series of the surface-profile function. The same solution can be alternatively obtained by means of the extended boundary conditions, which does not require this a priori assumption, but is formally more involved (note there was some controversy on the legitimacy of the Rayleigh hypothesis, for a more comprehensive discussion see [8], [15]).

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More specifically, the analysis of scattering by layered and/or inhomogeneous structures with rough boundaries is of crucial importance for many applications. As a matter of fact, natural stratifications exhibit rough interfaces and volume inhomogeneities, that are both responsible for the scattering from the structure. A considerable effort has been devoted to study the wave scattering by stratifications and several papers have been published: within this framework, modeling in microwave remote sensing of natural structures is of interest in [16]–[25], whereas analyzing optic thin films is the subject in [26], [27]. However, due to the analytical difficulties, commonly the problem of wave scattering by random stratifications has been investigated by separately treating surface roughness and volume scattering effects. A brief discussion is now useful to understand the state of the art in this field. A closed-form solution for the volume scattering by a flat-boundaries stratification in the Born approximation may be easily found in literature for simplified geometry [5] including a very small number of layers; however, at best of our knowledge, a solution is not available for stratification with an arbitrary number of layers. More recently, a systematic formulation based on the perturbation of boundary conditions has been introduced [23], [24], to deal with the analysis of a layered structure with an arbitrary number of rough interfaces. These ensuing general first-order solutions have been obtained in closed form for a 3-D geometry and a bi-static configuration concerning scattering from [23] and through [24] layered structures; generalized reflection/transmission have been adopted to get compact solutions [23], [24]. Furthermore, these solutions enable us to express scattering amplitudes as made up of terms, each one amenable of a proper physical interpretation, that allows fully identifying the scattering mechanisms involved into the structure [25]. These formally symmetric and physically revealing solutions can be also regarded as generalization to layered media with rough interfaces of the classical SPM method originally developed for rough surfaces [5], [6]. In addition, most of the already existing perturbative approaches [16]–[19], originally developed for simplified configurations in the first-order approximation, can be regarded, in a unified framework, as special case of our solutions [20]. Although the final solution in [23], [24] is expressed in a compact form, its derivation, as presented in [23], [24], is very involved. In this paper, we show that a formal solution for scattering from rough layered media can be much more straightforwardly obtained by using a different approach, which is based on two key elements: the use of the Reciprocity Theorem [28] and an appropriate description of the scattering structure in terms of perturbation of the dielectric constant volumetric distribution; this is a formal alternative to the perturbation of the boundary conditions, which was employed in [23]–[25]. A short discussion on the newness in use of these two key elements is in order. It is well known that Reciprocity Theorem can help solving scattering problems [28]–[30], even if its use is not so popular with respect to other approaches that may lead to equivalent results. In addition, the description in terms of perturbation of the dielectric constant volumetric distribution is widely used in volumetric scattering problems, e.g., [5], [6], [15], [31], and it leads to the well known Born approximation. In some cases [27] the

description in terms of perturbation of the dielectric constant volumetric distribution has been also used to evaluate the scattering from a rough surface, although the connection of the obtained solution with the classical SPM solution has not been highlighted. However, at best of our knowledge, Reciprocity Theorem and volumetric distribution have never been used to compute scattering from a rough multi-layer. The volumetric perturbative formulation introduced in this paper is based on an intrinsically reciprocal approach; in this paper we also show how our formulation methodologically will allow us to obtain, even if in the first-order limit, a rigorous and unified treatment for both volume and interface scattering. We are here essentially interested in presenting a general theoretical formulation whose structure maintains analytical consistency with both perturbation of boundary conditions and classical Born approximation for volume fluctuation formulations. Regarding a rough multilayer, the analytical solution is then provided directly in terms of the unperturbed solutions known in closed-form. Accordingly, the formulation here presented clearly illustrates that the first-order scattered field can be formally expressed as a proper coupling of two unperturbed solutions, thus clearly revealing the intrinsic aim of use of the perturbation theory. To analytically validate the solution here obtained by following the new formulation we compare it to the solution obtained via the boundary perturbation theory [23], [24] and in particular, to the classical SPM solution for a rough surface. In addition, we demonstrate that the proposed formulation, somehow surprisingly, has a reduced mathematical complexity: This can be explained by observing that the new formulation makes only use of the vector electric field, whereas the previous formulation based on perturbation of boundary conditions requires the analysis of both magnetic and electric fields. Moreover, this new formulation allows us to avoid any cumbersome Green functions formalism. A final note is in order. Generally speaking, electromagnetic fields are generally regarded as unobservable: they can only be indirectly measured through their interactions with observable quantities. To emphasize the neat physical significance of our methodological approach, a remarkable interpretation of the scattering solution in terms of the (observable) Rumsey’s reaction concept [35] is provided in the paper. The paper is organized as follows. In Section II, we propose the new volumetric perturbative formulation. In Section III, the formulation is applied to the scattering from rough interfaces of a multilayer, and the formal consistency of the solution with the one of the boundary perturbation theory is analytically provided. A lucid interpretation of the perturbative solution is given in terms of the multi-reaction in Section IV. Section V summarizes the paper results. II. VOLUMETRIC PERTURBATIVE FORMULATION In this Section we introduce the volumetric perturbative formulation: a general scattering problem is analyzed. radiating an Let us consider a source current density in an inhomogeneous medium electromagnetic field , of its relative dieleccharacterized by a distribution, tric permittivity. The electric field satisfies the vector Helmholtz

IMPERATORE et al.: VOLUMETRIC-PERTURBATIVE RECIPROCAL FORMULATION FOR SCATTERING FROM ROUGH MULTILAYERS

equation (in the following, a factor and suppressed):

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is understood

(1) and are the propagation constant and the intrinsic where impedance of vacuum, respectively. Let us now assume that the considered medium can be seen as an unperturbed medium with to which a perturbation relative permittivity is applied, so that ; let us also define as the field radiated by in the the unperturbed field unperturbed medium (2) By subtracting (2) from (1) we get (3) is the field perturbation. Equawhere tion (3) shows that the field perturbation can be considered as radiated by an equivalent current density

(4) are assumed to be small, then in (4) medium perturbations turns out to be small with respect the field perturbation , which leaded us to replace to the unperturbed field with in (4). In order to compute the perturbed field in a generic point , we define a (fictitious) source

Fig. 1. Geometry of the scattering problem.

If the unperturbed medium have discontinuity planes orthogonal to the -axis, then it is convenient to distinguish between and longitudinal field components (i.e., the transverse field components orthogonal and parallel to the -axis)

(8) is the dielectric constant of vacuum and is the -component of the unperturbed electric flux density. Accordingly, (7) can be rewritten as where

(5) is the Dirac where is an arbitrarily oriented unit vector, A m is a unitary constant introduced delta function, and for dimensional consistency reasons. This (test) source radiates in the unperturbed medium. By applying the Recia field procity Theorem [28] we get

(9) (6)

where is a volume enclosing all the sources, as pictured in Fig. 1. Use of (4) and (5) in (6) leads to

(7)

Equation (7) allows us evaluating the field perturbation from knowledge of the medium perturbation and of the two unperturbed fields radiated by real and fictitious sources.

It is important to note that throughout this paper with a parenthesized superscript we systematically indicate the order of perturbation. On the other hand, where explicitly indicated, a subscript distinguishes the pertinent th spatial region of the medium. Indeed, we also emphasize that a field solution assumes different expressions depending on the specific region of the structure which is concerned. Accordingly, to indicate the pertinent field expression within a specific ( th) region, if necessary a proper subscript (m) is included in the field notation. Conversely, when the relevant subscript is omitted we indicate the overall field solution. For instance, denotes the relevant expression assumed by the electric field first-order perturbation in the 0th region.

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Finally, by noting that, for small medium perturbation

(10) so that

(11) we have

(12) This equation, with respect to (7), has the advantage that it is expressed in terms of unperturbed field components that are all continuous across discontinuity planes orthogonal to the z-axis. Note also that the integration in (12) is effective only over the volume involving the dielectric perturbation. III. SCATTERING FROM A ROUGH-BOUNDARIES MULTILAYER In this section we use the general formulation reported in (12) to compute the scattering from a layered medium with rough interfaces. To this end, we have to characterize the medium, explicitly compute the medium permittivity perturbation (Section III.A) and the two unperturbed fields radiated by real, , and fictitious, , sources (Section III.B).

Fig. 2. Geometry of the rough-boundaries multilayer structure.

The perturbed medium is now obtained by assuming that each interface has a roughness characterized by a zero-mean two-dimensional process, then for the th interface we have . Therefore, the perturbed permittivity distribution is

A. Layered Medium Characterization For a layered medium with rough interfaces, the unperturbed medium is provided by a stack of parallel slabs, sandwiched in between two half-spaces; the entire structure is shift invariant along and directions (infinite lateral extent in directions are assumed). Each layer is assumed to be homogeneous and characterized by deterministic parameters: the dielec, and the thickness tric relative permittivity , see Fig. 2. The parameters pertaining to th layer are and identified by a subscript ; its boundaries are . We here assume that all the layers have the same magnetic relative permeability (possibly, but not necessarily, . equal to 1). In addition, with reference to Fig. 2, we set Accordingly, the unperturbed permittivity distribution is

(14) and the perturbation of the dielectric permittivity is (15) are small enough to perWe assume that roughness heights form a series expansion of the perturbation (15) around and truncate it to its first-order. Accordingly, by using (14) in (15) and recalling that the derivative of the Heaviside’s unit step function is a Dirac delta function, we get the following first order expansion of (15):

(16) (13) is the Heaviside’s unit step function, that is zero for where negative argument and 1 for positive argument.

where is the Dirac delta function. Note that this perturba. tion is non-null only in thin regions around the planes We also assume that the perturbation has a finite extent in the

IMPERATORE et al.: VOLUMETRIC-PERTURBATIVE RECIPROCAL FORMULATION FOR SCATTERING FROM ROUGH MULTILAYERS

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TABLE I

Fig. 3. Layered structure: unperturbed geometry.

directions, i.e., fSimilarly, we can write

is zero outside region of area

.

being

(17)

(21) (22)

(18) In such a way, as the interfaces description is concerned, the actual interfaces can be regarded as volume perturbations localized around the unperturbed interfaces and, accordingly, the roughness can be replaced by discontinuous volume inhomogeneities. B. Unperturbed Field Evaluation If we assume that the field source is placed in the upper halfspace and it is in the far zone with respect to the rough interfaces, is the field present in the unthen the unperturbed field perturbed stratified medium (characterized by flat boundaries) on which a (locally) plane wave impinges. We consider an arbitrary polarized monochromatic plane wave incident from the upper half-space on the stratification at an angle with respect to the direction, as schematically shown in Fig. 3 (19) where the incident vector wavenumber direction is individuated in a spherical coordinate frame by

(20)

is the two dimensional projection Accordingly, of incident vector wave-number on the plane . It is well-known that in this case the field in each layer of the plane stratified medium is the superposition of one upward- and one downward-propagating wave. The ratio of the amplitudes of -polarized upward- and downward-propagating waves imand mediately above the interface between the regions defines the generalized reflection coefficient at the interface [15] that can be recursively expressed as (23) where the ordinary (Fresnel) reflection and transmission coefficients [5], [12], [15], at the interfaces between regions and , are given by

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with the superscripts

denoting the polarization, and . Similarly, the ratio of the amplitudes of -polarised downward-propagating waves at the top of the th layer and in the upper half-space (0th layer) at defines [23], [24] the generalized transmission coefficient

Accordingly, the unperturbed field given by

in the

th layer is

(30) (24) wherein . We stress that (generalized) reflection and transmission coefficients do not depend on the direction of . By employing above notations, see also [23], the unperturbed field in the th layer can be conveniently expressed in the following closed-form:

where and are given by (21)–(22) with superscripts replaced by superscripts , and where (31) C. Scattered Field Evaluation The integral of the (12) over the volume reduces to a multisurface one being the geometric roughness of the interfaces described by means of an appropriate volume perturbation localized around the interfaces (see, Section III.A). By substituting (16) and (18) in (12), we get

(25) where the orthonormal right-handed basis been used (see Fig. 3), and where we set:

has

(32)

(26) in the superscript on LHS represents a where the symbols given choice linked to the symbols in RHS expression. Similarly, if we assume that the test source is placed in the upper half-space and is located in the far zone with respect to the rough is the field present interfaces, then the unperturbed field in an unperturbed plane stratified medium on which a (locally) plane wave impinges; this plane wave is expressed by the electric field

(27) with a wave-vector (see also Fig. 2)

(33) wherein indicates that the longitudinal field components . Equation are evaluated immediately above the plane (33) can be concisely rewritten as

(34)

(28) wherein and amplitude

(29)

(35)

IMPERATORE et al.: VOLUMETRIC-PERTURBATIVE RECIPROCAL FORMULATION FOR SCATTERING FROM ROUGH MULTILAYERS

can be defined as a pseudo-horizontal projector, because it coincides with the classical horizontal one for perfect conductivity . In other words, the accounts for the discontinuities of the unperturbed operator is a symmetric field across the (flat) boundary. Note also that operator. At this point, we have all the elements to evaluate the field , i.e., the scattered field. As a matter of perturbation fact, by substituting (25) and (30) in (34), and noting that

for

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, and

(41)

(36)

for . In (40)–(41) superscripts and refer to the incident and scattered field directions, respectively. Equations (40)–(41) can be more concisely written in matrix form as

we get

(42) with

(43)

(37) (44) where we have set . and It is convenient to project the scattered field onto (given by (21)–(22) with superscripts replaced by superscripts ):

(45)

(38) By introducing (39) the Fourier transforms (2D-FT) of the rough interfaces, and using (29) and (31), we have

(40)

(46)

(47) Equations (42)–(47) provide a key result of our paper. Some comments are in order to illustrate major consequences from these equations. First of all, we emphasize that the proposed approach avoid somehow defining and using the Green functions, whereas our treatment directly involve the integral transform of the field. This simplifies the mathematical treatment of the problem; in addition, as it is clarified in the following, our approach

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leads to a meaningful physical interpretation of the perturbative solution. Furthermore, we highlight that formally the obtained analytical solution (42)–(47), for the scattering from rough-boundaries multilayered structures, is perfectly equivalent to the one based on the perturbation of the boundary conditions approach [23], [24]. In fact, it is easy to verify that (44)–(47) are formally coincident with (82)–(85) in [24]. In particular, it is simple to verify that, when the layered medium merely reduces to a single interface between two half-space, these coefficients exactly reduce to the classical SPM ones [5], [6], [23]. When the interfacial roughness is concerned, we emphasize that from a qualitative viewpoint, in long wavelength limit the controlling factor for the validity of our solution is not the dielectric contrast: in fact the smallness of the dielectric perturbation does not necessary requires a limitation on the dielectric contrast (whose modulus can be, and usually is, greater than 1). The relevant limitations regard the vertical extension (rms height) of the rough interface, which has to be small with respect to the wavelength of the incoming radiation. This is directly related to the role of the phase of the wave propagating inside the perturbation. In addition, regarding the roughness also a constraint on the small-slope assumption has to be considered (the gradient of the interface must be small in comparison with unit). It can be demonstrated that: this second limitation stems from the fact that, in order to represent the scattering field in the basis of the unperturbed fields, the first-order field should be solenoidal, at least approximately. Accordingly, the range of validity of the present formulation is the same as the one of the formulation of [23], [24], i.e., the height deviation of the rough interfaces, about the unperturbed interface, is everywhere small compared to the wavelength of the incoming wave and the gradient of the interface is small in comparison to unity. It should be noted that, when only first-order terms are considered, then the perturbation theory yields the Bragg scatter phenomenon referred to a multi-rough-interfaces scattering: in fact, the scattered field at a given angle turns out to be the linear combination of the amplitudes of the Fourier Transforms of the interfaces roughness at one specific vector wave-number (see (42)). Then, the scattered power at a particular angle is directly a linear combination of energies at relevant surface scales. We also emphasize that the scattering configuration we considered in (38) is compliant with the classical Forward Scattering Alignment (FSA) convention that is adopted in radar polarimetry. Concerning the azimuthal -dependence, we also note that the cos-like scattering patterns experimentally obtained for a rough surface in [38] are taken into account in our solution. Regarding this point, we underline that our method clearly indicates how this dependence is associated with the bistatic configuration geometry in which the scattering phenomenon is observed, whereas in [38] this effect is referred to as a polarization artifact. Moreover, (42)–(47) also shown how this behavior is also inherited by each polarization component. In addition, we point out that the generalized bistatic scattering matrix can be then formally expressed as

(48)

which characterizes the polarimetric response of the generic ( th) rough interface of the layered structure, to a plane wave in the direction , in a given observation direction . Lets indicate with the superscript the transpose. By means of some algebra it can be verified that the scattering matrix whose terms are in (44)–(47) satisfies the following relationship: (49) This fundamental property in the radar polarimetry was first obtained with a general purpose approach in [39]: the approach we here introduced in a different way led to coefficients that satisfy that property. This can be concisely expresses as a form of the reciprocity principle in the electromagnetic theory. It turns out that our result is invariant for an appropriate exchange between the role of transmitter and receiver. As a matter of fact, the formal exchange between the projections on the plane and is directly related to the exchange between and the incident and scattered wave-vectors . Finally, some considerations on the unperturbed-waves coupling interpretation are in order. Taking into account that , we are in the position to conveniently rewrite (34) as

(50) where,

in

each

sub-region of the layered structure, the unperturbed solution assumes the form , provided that the boundary condition on the flat interfaces are satisfied. Note that (50) is invariant when and , of the corresponding two the subscripts unperturbed fields, are exchanged. It should be also noted that the (scalar) perturbation strength (51) formally represents a perturbation operator associated with the roughness of the th interface. This operator reduces the integral in (50) to a multi-surface one. We highlight the crucial role played by the resulting wave coupling, which is intimately related to structural perturbation introduced in the first-order formulation: The exchange of energy is taking place as the roughness couples the energy of the incident wave with the one of the scattered field at the receiver. Consequently, for any fixed observation point the perturbation gives rise to a scattered field readable in terms of wave coupling of unperturbed solutions [see (50)]. In the first order approximation, from the receiver viewpoint, the electromagnetic coupling between only two unperturbed waves is observed. In other words, the signal received depends on two unperturbed fields, whereas the operators affect the coupling between these two unperturbed solutions. As a result, the perturbation operators (51) can be also thought as . coupling coefficients, with

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IV. REACTION-CONCEPT-BASED INTERPRETATION OF THE SCATTERING SOLUTION In this section, we propose a very useful and informative interpretation of the proposed solution; this is done in terms of reactions. The concept of reaction, which has to be regarded as a basic physical observable, was originally introduced by Rumsey [35] (for a systematic exposition of this subject see also [28]). We underline that the reaction concept was applied in [35] by considering surface currents, whereas volume polarization currents case were taken into account in [36]. and : in an infiLet’s consider two vectorial functions, nite-dimensional linear space for the electromagnetic fields, we can introduce the symmetric bilinear form (52) to which is given the physical meaning of reaction [15] between two field quantities; the scalar on the left-hand side of (52) is a measure of the reaction (or coupling) between the source field, , and the mediating field, . Note also that, the symbol is here adopted to mathematically represent the bilinear form (52) which is a more general concept with respect to the inner product and does not require the structure of an inner product space. Using the notion (52), (50) can be then conveniently rewritten in the form of a multi-reaction (53) can where, to the first-order be interpreted as the equivalent polarization current induced into the (localized) perturbed volume by the unperturbed field . Then, the generic th term of the summation in (53) is susceptible to be physically interpreted as the , which is produced by the unperturbed electric field , “measured” by the source . sampling source Note that if the reaction is zero, then no energy is transferred by the first-order field from the transmitter to the receiver. The right-hand side of (53) shows how the scattered field is intimately related to the multi-reaction. Note also that, since the medium is also symmetric or reciprocal, this multi-reaction is symmetric according to the reciprocity theorem. Indeed, from , it turns out that a form equivalent to (53) the symmetry of is given by (54) wherein to the first-order, can be now interpreted, as the equivalent polarization current induced into the (localized) perturbed volume by the unperturbed field produced by the test-source . Accordingly, the generic th term of the summation in (54) can be now read as the unperturbed electric field , due to the real source, “measured” by the source . This is to say that the proposed formulation is reciprocal.

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V. CONCLUSION AND FUTURE DEVELOPMENT In this paper, we have proposed a volumetric perturbative formulation to deal with EM scattering by rough multilayered structures; the formulation is intrinsically reciprocal. The comprehensive scattering formulation is based on a unified description for both interfacial and volume inhomogeneities. This formulation permits to obtain a comprehensive method to evaluate the scattering, which includes in conjunction and in rigorous manner both rough-interfaces scattering and volume scattering in complex multilayer. We have shown that this formulation applied to 3D rough multilayer leads to derive, in the first-order limit of the perturbative development, a closed form solution in a very simple way. The obtained scattering solution is expressed in terms of unperturbed solutions, offering deep analytical insight into the physics of the problem. This clearly exposes the intrinsic aim of the perturbation theory, which relies on the assumption that the unperturbed solutions, for the problem we are dealing with, are known in closed form. The comparison of the obtained solution with the one obtained in the theoretical framework of perturbation of boundary conditions [23], [24] reveals an intrinsic equivalence between the two different approaches, which evaluate the scattering from the same perturbed structure starting from two different kinds of description for the structure itself. In other words, the formal identity of the final solutions reflects the full consistency of the corresponding different perturbative formulations. Indeed, a salient feature of the proposed formulation lies in its reduced mathematical complexity: In particular, the formulation here exposed can be carried out by exclusively referring to the vector electric field; conversely, the formulation based on perturbation of boundary conditions requires the analysis of both magnetic and electric fields. Concerning the analytical complexity, a crucial point involves the use of polarization currents rather than equivalent surface currents (as in [23], [24]). Although in principle both the representation in terms of surface or volume currents can be equally employed, we underline that in the analyzed problem, in which non magnetic media are concerned (i.e., whose relative permeability is unitary), the magnetic polarization currents vanish, thus we simply have to take care of the electric field distribution only. In addition, we highlight that the conducted analysis did not require to resort to the cumbersome Green functional formalism. Therefore, we have shown that the perturbation methodology applied to rough-boundaries structure gains in generality as well in simplicity when it is considered under a different perspective of the volumetric perturbation. Therefore, the general results deduced in our previous papers [23], [24] are strengthened and, at the same time, the proposed approach offers a more complete comprehension in a conceptual perspective. We finally emphasize that when a new mathematical formulation, perhaps more general, for a physical phenomenon is conceived, the mathematical structure of the new formulation itself provides a new way of thinking about the phenomenon, especially if the results, as in our problem, are closely related to the ones obtained with a different formulation. Therefore, if applied to the case of rough-boundaries layered media, the formulation here presented is not only an (equivalent) alternative with respect to the

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one [23], [24] obtained via the perturbation of the boundary conditions, but leads to a simpler formulation with clearer interpretation. Therefore, due to the formal full-consistency of the proposed solution with the one obtained in the theoretical framework of the boundary perturbation, we can certainly refer to the numerical examples reported in [23]–[25]. We are also planning to compare our results with the ones derived via numerical methods or also with the ones provided by Kirchhoff-based models; in the second case, however, the comparison can be not easy due to the different domains of validity. On the other hand, we highlight that the mentioned validity conditions are fully consistent with the theoretical result of the boundary perturbation [23], [24]; in addition, rigorous demonstration and comparative discussion on the regime of validity lie outside of the aim of this paper, and therefore are deferred to a further publication. Furthermore, it is worth noting that if the point source is placed in far field with respect to the illuminated volume, then the plane-wave-incidence approximation can be used, and the results of our method can be used. Otherwise, a plane wave expansion of the incident field can be performed, and the presented approach can be used for each plane wave component. It should be noted that our method can remind other theoretical approaches [40], [41], because in these methods the roughness interface is also seen as a permittivity fluctuation. However, for the structure mainly considered in our paper, the perturbation is characterized by means of zero-mean processes, so that the mean scattered (far) field is null, except that in a narrow cone around the specular direction. In addition, our formulation leads to closed form solution, whereas the approaches [40], [41] are semi-analytical in as much as the multi-layer Green function has to be first computed numerically. We emphasize that the proposed formulation exhibits several appealing features, which are of interest for future developments. First of all, the approach that we have presented leads to a solution which is directly susceptible of a powerful interpretation in terms of the Rumsey’s reaction concept which allows interpreting the scattering solution in terms of multi-reaction. This furnishes a clear interpretation of the scattering problem and might be also useful to interpret the first-order perturbative approximation. Analogously, this approach opens the way to evaluate and interpret the higher-order terms of the perturbative development. Similarly, the problem of the scattering through rough boundaries multilayer can be also advantageously addressed by using the proposed approach. Finally, the comprehensive formulation presented in this paper can be used also to derive closed form solution for scattering, from and through, inhomogeneities that are possibly present in each slab of a structure with an arbitrary number of layers. This will be a matter of a paper that is currently in preparation, thus opening the way to a complete analytical solution for fully space-variant dielectric multilayered structures. REFERENCES [1] P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953. [2] T. Kato, Perturbation Theory of Linear Operators. Berlin: Springer Verlag, 1995.

[3] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non Rel-Rativistic Theory). Oxford: Pergamon, 1958. [4] W. Greiner, Quantum Mechanics: An Introduction. Berlin: Springer Verlag, 2001. [5] L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing. New York: Wiley, 1985. [6] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing. Reading, MA: Addison-Wesley, 1982, vol. I, II, II. [7] A. K. Fung, Microwave Scattering and Emission. Models and Their Application. Norwood, MA: Artech House, 1994. [8] A. G. Voronovich, Wave Scattering from Rough Surfaces, ser. Springer Series on Wave Phenomena. New York: Springer, 1994. [9] F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces. Oxford: Pergamon, 1979. [10] D. P. Winebrenner and A. Ishimaru, “Application of the phase perturbation technique to randomly rough surface,” J. Opt. Soc. America, vol. 2, pp. 2285–2294, 1985. [11] J. Shen and A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B, vol. 22, no. 9, pp. 4234–4240, Nov. 1, 1980. [12] L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves, ser. Wiley series in remote sensing. New-York: Wiley-Interscience, 2000, vol. I, II, III. [13] E. I. Chaikina, A. G. Navarrete, E. R. Méndez, A. Marti‘nez, and A. A. Maradudin, “Coherent scattering by one-dimensional randomly rough metallic surfaces,” Appl. Opt., vol. 37, no. 6, Feb. 1998. [14] A. A. Maradudin, R. E. Luna, and E. R. Mendez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media, vol. 3, pp. 51–60, 1993. [15] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: IEEE Press, 1997. [16] 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, 2000. [17] I. M. Fuks, “Wave diffraction by a rough boundary of an arbitrary plane-layered medium,” IEEE Trans. Antennas Propag., pp. 630–639, 2001. [18] A. Tabatabaeenejad and M. Moghaddam, “Bistatic scattering from three-dimensional layered rough surfaces,” IEEE Trans. Geosci. Remote Sensing, vol. 44, no. 8, Aug. 2006. [19] 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. [20] 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. [21] K. Sarabandi and T. Chiu, “Electromagnetic scattering from slightly rough surface with inhomogeneous dielectric profiles,” IEEE Trans. Antennas Propag., vol. 45, pp. 1419–1430, Sep. 1997. [22] N. P. Zhuck, “Scattering of EM waves from a slightly rough surface of a generally anisotropic plane-layered halfspace,” IEEE Trans. Antennas Propag., vol. 45, no. 12, pp. 1774–1782, Dec. 1997. [23] P. Imperatore, A. Iodice, and D. Riccio, “Electromagnetic wave scattering from layered structures with an arbitrary number of rough interfaces,” IEEE Trans. Geosci. Remote Sens. E, vol. 47, no. 4, pp. 1056–1072, Apr. 2009. [24] 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. [25] P. Imperatore, A. Iodice, and D. Riccio, “Physical meaning of perturbative solutions for scattering from and through multilayered structures with rough interfaces,” IEEE Trans. Antennas Propag., vol. 58, no. 8, pp. 2710–2724, Aug. 2010. [26] P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: Theory and experiment,” J. Opt. Soc. Amer., vol. 71, no. 9, Sep. 1981. [27] N. R. Hill, “Integral-equation perturbative approach to optical scattering from rough surfaces,” Phys. Rev. B, vol. 24, Dec. 1981. [28] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [29] K. Sarabandi and P. F. Polatin, “Electromagnetic scattering from two adjacent objects,” IEEE Trans. Antennas Propag., vol. 42, no. 4, pp. 510–517, Apr. 1994. [30] R. E. Collin, Field Theory of Guided Waves, 2nd ed. NY/NJ: WileyIEEE Press, 1990.

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[31] A. Ishimaru, Wave Propagation and Scattering in Random Media. New York: Academic, 1993. [32] G. V. Rozhnov, “Diffraction of electromagnetic waves by irregular interfaces in stratified, uniaxial anisotropic media,” J. Exp. Theor. Phys., vol. 77, no. 5, pp. 709–718, 1993. [33] O. Cmielewski, H. Tortel, A. Litman, and M. Saillard, “A two-step procedure for characterizing obstacles under a rough surface from bistatic measurements,” IEEE Trans. Geosci. Remote Sens. E, vol. 45, no. 9, pp. 2850–2858, Sep. 2007. [34] R. Carminati, J. J. Sàenz, J.-J. Greffet, and M. Nieto-Vesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the S matrix of fields containing evanescent components,” Phys Rev A, vol. 62, 2000. [35] V. H. Rumsey, “Reaction concept in electromagnetic theory,” Phys. Rev. B, vol. 94, pp. 1483–1491, Jun. 1954. [36] M. Cohen, “Application of the reaction concept to scattering problems,” IRE Trans Antennas Propag, vol. 3, no. 4, pp. 193–199, Oct. 1955. [37] 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. [38] A. Y. Nashashibi and F. T. Ulaby, “MMW polarimetric radar bistatic scattering from a random surface,” IEEE Trans. Geosci. Remote Sens. E, vol. 45, no. 6, pp. 1743–1755, Jun. 2007. [39] D. S. Saxon, “Tensor scattering matrix for the electromagnetic field,” PR, vol. 100, no. 6, Dec. 1955. [40] O. Calvo-Perez, A. Sentenac, and J.-J. Greffet, “Light scattering by a two-dimensional, rough penetrable medium: A mean-field theory,” RadioScience, vol. 34, pp. 311–335, Mar. 1999. [41] A. Sentenac, H. Giovannini, and M. Saillard, “Scattering from rough inhomogeneous media: Splitting of surface and volume scattering,” J Opt Soc Am A, vol. 19, no. 4, pp. 727–736, 2002.

Pasquale Imperatore (M’10) 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, Naples, Italy, where he worked on modeling and simulation of wave propagation, advanced ray-tracing-based prediction tool design for wireless application in urban environment, as well as simulation and processing of synthetic-aperture-radar 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.

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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, both from the University of Naples “Federico II,” Naples, Italy, in 1993 and 1999, respectively. In 1995, he received a grant from the CNR (Italian National Council of Research) to be spent at IRECE (Istituto di Ricerca per l’Elettromagnetismo e i Componenti Elettronici), Naples, Italy, 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. He is the author or coauthor of about 170 papers published in refereed journals/proceedings of international and national conferences. 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. 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 is currently a Professor of electromagnetics and remote sensing with the Department of Biomedics Electronic and Telecommunication Engineering, University of Naples Federico II. 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. 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 (including Scattering, Natural Surfaces and Fractals). and more than 240 published papers. Prof. Riccio is an Associate Editor for the journals Sensors, Remote Sensing, and The Open Remote Sensing Journal. He was the recipient of the 2009 Sergei A. Schelkunoff Transaction Prize Paper Award for the best paper published in year 2008 on the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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Vertex Diffracted Edge Waves on a Perfectly Conducting Plane Angular Sector Alper K. Ozturk, Member, IEEE, Robert Paknys, Senior Member, IEEE, and Christopher W. Trueman, Senior Member, IEEE

Abstract—Numerical diffraction coefficients are presented for vertex-diffracted edge waves induced on an infinitely-thin, perfectly conducting, semi-infinite plane angular sector. The current density on the surface of the plane angular sector is modeled using the physical theory of diffraction (PTD). The vertex-diffracted currents are defined as the difference between the exact and the PTD currents. The difference current is then modeled as a wave traveling away from the corner with unknown amplitude and phase factors. The unknown coefficients for the vertex-diffracted currents are calculated by using a least squares fit approximation. The vertex-diffracted currents are successfully modeled even for narrow angular sectors. The vertex-diffracted currents provide a substantial improvement to the accuracy of RCS patterns in off-specular directions. Index Terms—Edge waves, electromagnetic diffraction, physical theory of diffraction, vertex diffraction.

I. INTRODUCTION HE problem of electromagnetic scattering from a perfectly-conducting plane angular sector has been of interest for many years. An exact solution based on the separation of variables in the sphero-conal coordinate system was first developed by Satterwhite [1]. In his solution, fields and currents are expressed in terms of scalar wave functions that are the solutions of a two-parameter eigenvalue problem of two coupled spherical Lamé differential equations and spherical Bessel functions. The resulting expressions are in the form of eigenfunction expansions. These expansions are slowly convergent and not suitable for high frequency scattering calculations. Based on Satterwhite’s solution, several attempts have been made to obtain a tractable approximation that can be used in the high frequency modeling of the angular sector problem [2], [3]. The eigenfunction expansions for the currents and fields are particularly difficult to evaluate when both the source and observation points are far from the corner. An efficient numerical evaluation procedure for calculating the radar cross section (RCS) of an elliptic cone was presented in [4]. This procedure is based on using Euler’s sequence transformation to accelerate the convergence of the infinite series of eigenfunction expansions. Blume and Krebs [5] used a similar

T

Manuscript received July 22, 2009; revised June 04, 2010; accepted June 24, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC H3G 1M8, 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.2010.2103002

Fig. 1. Plane angular sector and the edge fixed coordinate system.

approach to derive dyadic diffraction coefficients at the tip of an elliptic cone for nose-on incidence. The same problem was solved by Babich et al. [6] through numerical evaluation of the Fredholm integral equation that is obtained by combining the soft and hard boundary conditions on the surface of the cone [7]. Even though these developments provide certain advantages in terms of describing the behavior of the currents and fields near the tip of the cone, the resulting expressions are difficult to use in numerical calculations. As a result, it was not possible to derive a corner diffraction coefficient from the exact eigenfunction solution. In [8] and [9], Radlow proposed an exact plane wave spectrum solution to derive diffraction coefficients for a quarter plane. However, Albani [10] later demonstrated that the resulting field expressions do not satisfy the boundary conditions and this solution was incorrect. Lack of a corner diffraction coefficient is a major factor that limits the accuracy of high frequency diffraction techniques. Consider the problem geometry illustrated in Fig. 1. The current density on the surface of the plate can be approximated using the physical theory of diffraction (PTD) [11]. In the application of high-frequency techniques to 3-D scattering problems, the surface currents are often expressed in the form of equivalent edge currents. The far field is then given by the line integral of the equivalent currents along the edges. When the equivalent edge currents are integrated along a finite or semi-infinite edge, the asymptotic evaluation of the line integral results in end-point contributions. These end-point contributions are interpreted as corner diffracted fields. Based on this idea, uniform theory of diffraction (UTD) vertex diffraction coefficients describing the fields diffracted by the tip of a pyramidal structure when the source point is at a finite distance from the tip were derived in [12]. An approximate corner diffraction coefficient based on the geometrical theory of diffraction (GTD) was first

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OZTURK et al.: VERTEX DIFFRACTED EDGE WAVES ON A PERFECTLY CONDUCTING PLANE ANGULAR SECTOR

proposed by Burnside and Pathak [13]. The corner diffraction coefficient was derived by asymptotically evaluating the radiation integral containing the GTD equivalent edge currents along the edge of a plane angular sector. It was demonstrated in [14] that the corner diffraction contributions improve the accuracy of radar cross section (RCS) calculations. However, it was shown that this corner diffraction coefficient yields nonunique results for various scattering configurations [15]. Furthermore, the resulting far field is discontinuous at the false shadow boundaries. In order to overcome this discrepancy, a corner diffraction coefficient was formulated in [16] by employing the equivalent edge currents based on the PTD [17]. Since higher order interactions among the edges of the scatterer are not included in this formulation, the resulting corner diffraction coefficients provide an improvement only when the higher order contributions are negligible. A UTD solution for describing corner-diffracted fields was presented by Hill and Pathak [18], [19]. The corner diffraction coefficients were derived in the UTD format by asymptotically evaluating the plane wave spectral (PWS) representation of the total field scattered from a perfectly-conducting plane angular sector. Asymptotic evaluation of the PWS integral provides the geometrical optics (GO), edge- and corner-diffracted fields. The corner-diffracted fields compensate for the discontinuities in the edge-diffracted fields. More recently, end-point contributions were evaluated based on the incremental theory of diffraction (ITD) [20]. It was shown that the inclusion of the vertex-diffracted currents leads to an accurate representation of the total current far from the vertex. However, the total current density predicted by incorporating the end-point effects fails when nose-on incidence is approached. In this case, the vertex-diffracted edge waves make a significant contribution to the scattered field. Thus, a more complete representation of the corner diffraction phenomenon should include the effect of vertex-diffracted edge waves. Towards this goal, Hansen [21] used the difference between the Method of Moments (MoM) current density and the known high-frequency currents near the corner of a rectangular plate to characterize the behavior of corner diffracted currents. This approach resulted in approximate analytical expressions for vertex diffraction coefficients for a quarter plane that are valid over a limited angular range of forward scattering directions. In this paper, we present the derivation of numerical diffraction coefficients for the vertex-diffracted edge waves at the tip of a plane angular sector. In contrast to previous works, the expressions for the vertex-diffracted currents can be used in high-frequency modeling of finite scatterers to calculate the corner-diffracted fields. This is because the vertex-diffracted currents provide accurate results for arbitrary direction of incidence and polarization. Based on the PTD, we express the total current on the surface as the sum of the physical optics (PO) current, the fringe wave (FW) currents (edge diffracted currents) from the two edges, and the vertex-diffracted current. The vertex-diffracted current for a given direction of incidence is obtained by subtracting the known PO and edge currents from the total current. The total current is calculated using the exact eigenfunction solution. This approach was previously used by Brinkley [22] for the same purpose. However, only the first-order fringe-wave

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contributions were used in calculating the vertex-diffracted currents. As a consequence, the solution becomes inaccurate when the total current on the angular sector is dominated by higherorder edge-to-edge interactions. In our derivation, we use up to second order edge-diffracted currents. In order to obtain the double-diffracted FW currents, it is assumed that the secondorder diffraction point is illuminated by the field launched at the first-order diffraction point. Assuming plane wave incidence at the second-order diffraction point, the double-diffracted current can easily be obtained by using the PTD formulation. This method was first introduced in the form of equivalent edge currents in [23]. It should be noted that, in the present formulation we use the FW expressions that are derived from Sommerfeld’s half plane solution [24] to account for edge diffractions. Since diffraction is a local phenomenon, each edge of the angular sector can be modeled as a half plane. However, as the diffraction point approaches the vertex, the effect of the truncation on edge-diffracted currents becomes substantial. This truncation effect is equivalent to the aforementioned end-point contributions. The vertex-diffracted currents not only account for the vertex-diffracted edge waves but also include the effect of these contributions and serve as a correction to the FW currents. Once the known contributions are subtracted out from the total current, the remaining portion is interpreted as the corner current associated with the vertex diffraction. The corner current is then expressed in the form of a wave traveling away from the vertex with unknown amplitude and decay factors. Using a least squares fit approximation, these unknown factors are obtained as a function of the angular position of the observation point with respect to the edge of the plane angular sector. We found that the vertex-diffracted currents can be obtained with reasonable accuracy for an arbitrary direction of incidence and angular sector opening [25]. In order to use the vertex-diffracted currents in modeling more complex scattering structures such as aircraft wings or antennas on finite ground planes, the unknown coefficients can be tabulated for all the possible directions of incidence and observation points on the scatterer surface through a pre-processing step. Once tabulated, these coefficients can directly be used in high-frequency treatment of complex radar targets. The vertex-diffracted currents are verified by comparisons with Satterwhite’s solution. Furthermore, the RCS pattern obtained by numerically integrating the total high-frequency current over finite scattering structures are compared with the MoM results. The scattering configurations used in the numerical examples are chosen to be the same as the ones used in the previous developments ([20], [22]) to be able to assess the performance of the new derivation. Section II briefly describes the formulation of the high frequency currents from a PTD point of view. Section III investigates the behavior of the vertex-diffracted currents and summarizes the computation of the diffraction coefficients. Section IV presents the numerical results and discussion, and Section V contains the conclusions. II. CURRENT DENSITY ON THE PLANE ANGULAR SECTOR Consider an infinitely-thin, perfectly-conducting plane angular sector illuminated by a plane wave as shown in Fig. 1.

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The current density on the plane angular sector can be expressed as the sum of the physical optics (PO) current, the fringe wave (FW) current and the vertex-diffracted current. The vertex diffracted current is then defined as (1) where is the exact current and is the well-known PO and are the fringe-wave contributions due to current. and denote the the two edges of the angular sector. edge-to-edge double-diffracted current contributions. An edgefixed coordinate system is defined at each diffraction point O as depicted in Fig. 1. is the normal and is the tangent to the surface of the half plane. The edge tangent must be defined . With respect to the edge-fixed coordinate such that system, the incident plane wave is given by

Fig. 2. Ray path for double diffracted current.

in which is the magnetic field diffracted from the firstorder diffraction point, . At the second-order diffraction point , the incident and diffracted rays are and respectively. Therefore, the argument of in (4) is

where and denote the and the components with the phase referenced to the tip of the angular sector. The edge diffracted fringe wave current on a half plane is obtained by subtracting the PO current from the total current [24]. When the diffraction point O lies on the edge, the parallel and the perpendicular components of the fringe wave current can be expressed as

where is measured from as depicted in Fig. 2. Equation , (4) follows directly from (2) and (3) by substituting and . Using the PO approximation, the magnetic field incident on the second-order can be expressed as diffraction point

(2)

(5) is the FW current diffracted by the first edge at . where Using (5) in (4), the double-diffracted FW current at point P can be expressed in terms of the first-order FW currents as

(3) (6)

where

The direction of propagation for the FW surface current is defined by the intersection of the Keller cone and the plate surface. The FW current from an edge is zero if the diffraction point does not lie on the corresponding edge. Referring to for edge and Fig. 1, this corresponds to for edge , where is the angle of incidence pertinent to the corresponding edge and is the sector tip angle. In order to obtain an expression for the double-diffracted currents, (2) and (3) are used at the second-order diffraction point. Consider the diffraction path depicted in Fig. 2. The diffracted is re-diffracted by the other edge at . Assuming field at plane wave incidence, the FW current due to the diffraction at the second order diffraction point, , can be expressed as

(4)

introIt is noted that the first-order diffracted current duces a normal component to the edge along edge 2. Using (6), it can be shown that this component is canceled by the introduction of the second-order diffracted current. The normal compois canceled by the nent of the first-order diffracted current at normal component of the PO current. Therefore, the edge condition [26] on the current density that is normal to the edges of the angular sector is satisfied by the introduction of double-diffracted currents. Equation (6) is derived based on the assumption that the second order diffraction point is illuminated by a plane wave. This assumption is valid when the argument of the Fresnel function in the incident FW surface current expressions (2) and (3) is large. However, (6) becomes less accurate when the plane wave assumption fails. This occurs when distance between the first- and second-order diffraction points is small. Furthermore, the field incident on the second-order diffraction point is is not ray-optical when the second-order diffraction point in the transition region of the first-order diffracted fields. This is is close to clearly the case when the direction of incidence at edge-on. Thus, for such cases, the higher-order edge-diffracted currents cannot be modeled accurately using (6). Consequently,

OZTURK et al.: VERTEX DIFFRACTED EDGE WAVES ON A PERFECTLY CONDUCTING PLANE ANGULAR SECTOR

0

Fig. 3. Plane angular sector in the x y plane and the radial cuts. The angular ^ = e^ y^. sector is centered in the first quadrant of the x-y plane so that e^ x

1

1

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Fig. 4. (a) Magnitude and (b) Phase of J on a 60 angular sector along various radial cuts for a ^ polarized plane wave incident from the direction ( ;  ) = (80 ; 225 ).

the difference current given by (1) contains information associated with the higher-order edge-diffracted currents in addition to the vertex-diffracted currents. Nevertheless, this does not cause a significant error in the far-zone fields as will be demonstrated by the numerical examples. III. VERTEX-DIFFRACTED CURRENTS In this section, we investigate the behavior of the vertex-diffracted currents near the corner of the plane angular sector. We consider the plane angular sector shown in Fig. 3. The vertexdiffracted current is obtained by subtracting the known edge-diffracted current contributions from the total current as suggested by (1). Regardless of the tip angle, the plane angular sector is centered in the first quadrant of the x-y plane so that . The total current, , is the exact solution obtained using the eigenfunction solution [1]. The FW contributions are calculated using (2) and (3). First, we assume that a plane angular is illuminated by a polarized plane wave insector of . The vertex-difcident from the direction fracted current is calculated along various radial cuts defined by , the angular position with respect to edge . The amplitude and the phase of the component of the corner current is shown in Fig. 4. It is observed that the phase behavior is the same as the phase behavior of a wave traveling away from the corner. Furthermore, the amplitude behavior along various radial cuts shows that the functional dependence on the distance from the corner is also a function of the angle . The behavior of the is very similar to that of . Based on this observation, the vertex-diffracted current is formulated as (7) where denotes the or the component of the vertex, and are diffracted current density. and the unknown functions to be obtained numerically. are real-valued and are complex-valued. The form of the corner diffracted currents given by (7) can also be deduced by using the corner condition. The behavior of the current density can be described by requiring that the electric energy density is integrable near the corner of the plane angular sector [27]. Furthermore, based on Satterwhite’s exact solution it can be shown that, close to the vertex, the vertex-diffracted for the current must have a radial dependence of

Fig. 5. (a) Magnitude and (b) Phase of J on a 90 angular sector along the radial cut = 10 for a ^ polarized plane wave incident from the direction ( ;  ) = (80 ; 160 ).

for the component. is the propagacomponent and and are the eigenvalues associated tion constant and with the solutions of the odd Neumann and even Dirichlet problems respectively which are in turn functions of the vertex angle [2]. Therefore, the form assumed in (7) seems to be a natural choice for modeling vertex-diffracted currents. It is noted that for the results illustrated in Fig. 4, the sampling locations along the given radial cuts correspond to forward scattering directions with respect to the incidence direction. Thus, there are no multiple edge diffractions involved in , ). Consecalculating the corner currents ( quently, the difference current is dominated by the vertex-diffracted currents. When higher-order edge-to-edge diffractions are present, it is important to be able to extract the vertex-diffracted current from the total current in an accurate manner. In many cases, the corner current as defined in (1) may be dominated by the higher-order edge-diffracted currents. To demonstrate this, we consider a 90 angular sector illuminated from . The vertex-diffracted curthe direction rent calculated along the radial cut is shown in Fig. 5. The current distribution labeled “FW-1” represents the vertex current obtained without subtracting the double-diffracted edge ). “FW-2” denotes the actual vertex curcurrents(i. e. rent calculated using (1). The double-diffracted current, is also shown for comparisons with the two types of difference currents. It is observed that when the double-diffracted currents are ignored, the behavior of the difference current agrees with

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the form assumed by (7). However, by comparing the phase behavior of this difference current with that of double-diffracted current alone, we deduce that the difference current is dominated by the double-diffracted current. It is noted that the phase behaviors are very similar in the two cases. However, as depicted in Fig. 5, the magnitude of the vertex-diffracted current is in fact much smaller than the one that is obtained without incorporating the double-diffracted currents. This would definitely result in erroneous far fields. Therefore, incorporation of the double-diffracted currents is quite critical in calculating the vertex-diffracted currents. , and In order to obtain the unknowns for a given direction of incidence and polarization, the vertex-diffracted current obtained using (1) is uniformly sampled along a specific radial cut . It follows from (7) that (8) (9) The radial distance to the vertex, , denotes the sampling . The unknowns can point along the radial cut defined by be obtained by solving (8) and (9) separately using linear leastshould be squares fitting [28]. The range of the radial cut chosen sufficiently large that the unknown coefficients can be obtained accurately using the least squares fit. However, it must be pointed out that the infinite series of eigenfunction expansions used to calculate the exact current, , converges more slowly as the observation point moves away from the tip of and the angular sector. In this paper, we use with . In our formulation of the eigenfunction expansions, this requires the evaluation of the first 115 terms in the infinite series of the current expression and 15 terms for each eigenfunction expansion. Using these variables, the unknown coefficients over the entire surface for the given incidence direction can accurately be calculated. The singularity , and the magnitude of the diffraction coeffiexponent, can be expressed as cient

(10)

(11) where

The phase factor and the phase of can similarly be obtained by using (10) and (11) respectively with and . It must be noted that the phase data must be unwrapped to perform the least-square fit (LSF) and the LSF approxiapproximation. The difference current mation that is obtained using (7) for two different cases is shown in Fig. 6. It is observed that, the vertex-diffracted current can accurately be approximated for any direction of incidence. In . Fig. 6(a) the direction of incidence is

Fig. 6. Magnitude of J and the least-squares approximation on a 90 angular sector along various radial cuts for ^ polarized plane wave incidence. (a) ( ;  ) = (60 ; 225 ) and (b) ( ;  ) = (30 ; 45 ).

This corresponds to forward scattering which has no double-diffracted currents. On the other hand in Fig. 6(b), the direction , for which of incidence is chosen to be double-diffracted currents are rather significant. The LSF current density given by (7) agrees well with the difference current except around the immediate vicinity of the vertex. IV. NUMERICAL RESULTS AND DISCUSSION We first consider the current density induced on a plane anpolarized incidence from the direcgular sector of 30 for . The current density obtained using tion the vertex-diffracted currents is compared with the exact solufor tion in Fig. 7. The current density is calculated at , where is the radial distance from the tip and is defined as shown in Fig. 3. “FW-1” denotes the current density obtained using only the first-order edge-diffracted currents. The solution labeled “FW-2” represents the current density obtained by superimposing the double diffraction contributions to the first order FW currents. “FW+corner” is the current density obtained using both the first- and second-order edge- and vertex-diffracted currents. It is noted that there are no discontinuities in the FW solutions since there are no diffraction shadow boundaries on the surface of the angular sector for the given direction of incidence. For the component of the surface current density, inclusion of the second-order edge-diffracted currents considerably improves the accuracy of the solution. Note from Fig. 7(b) that the first-order only solution tends to be singular at the edges. This is a non-physical behavior. On the other tends to hand, when double-diffracted currents are included, be zero near the edges as it should. In general, the double-diffracted currents provide a significant correction to the component of the total current near the edges. However, the resulting current density is still not accurate because the corner effect is missing. It is observed that corner diffraction has a greater effect on . Inclusion of the double-diffracted currents does not seem as shown in Fig. 7(a). to have a substantial improvement on However, when the vertex-diffracted current is included, the result agrees very well with the exact solution for both the and components. It is also observed that and satisfy the edge condition. Next we consider the current density on a 90 plane angular sector for forward scattering directions. For

OZTURK et al.: VERTEX DIFFRACTED EDGE WAVES ON A PERFECTLY CONDUCTING PLANE ANGULAR SECTOR

Fig. 7. (a) r^ and (b) ^ components of the total current density on a 30 angular sector at r = 0:5 for a ^ polarized plane wave incident from the direction ( ;  ) = (60 ; 45 ).

Fig. 8. (a) r^ and (b) ^ components of the total current density on a 90 angular sector at r = 0:5 for a ^ polarized plane wave incident from the direction ( ;  ) = (45 ; 225 ) and ( ;  ) = (80 ; 225 ).

, there are two shadow boundaries for the first-order edge-diffracted currents. These shadow boundaries for edge 1 and at for edge 2. appear at the When the direction of incidence is shadow boundary lines for edges 1 and 2 move to and respectively. Fig. 8 shows the current density for and . The distance at which the current density is plotted in this example . This is rather close to the tip of the angular sector. is Thus the amount of discontinuity at the shadow boundaries is quite large. As the distance to the vertex increases, the amount of discontinuity decreases. In any case, the discontinuity is removed by the inclusion of the vertex diffracted currents. The resulting high frequency current agrees very well with the exact solution. Fig. 9 shows the and components of the current density as a function of the raon the plane angular sector of . Since the dial distance from the tip along the radial cut vertex-diffracted currents are strongly guided around the edges (Fig. 7), it is particularly important to predict the vertex-diffracted currents accurately near the edges. The angular sector is illuminated by a polarized plane wave incident from the direction . It is observed that the total high frequency current agrees well with the exact solution except at distances very close to the tip. Similar behavior was observed in general for any given direction of incidence. This behavior is expected since the vertex-diffracted currents are not expected to be

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Fig. 9. (a) r^ and (b) ^ components of the total current density on a 90 angular sector along the radial cut = 1 for a ^ polarized plane wave incident from the direction ( ;  ) = (60 ; 345 ).

Fig. 10. (a) Diffraction coefficient K ( ) and (b) Phase factor P ( ) for a 90 angular sector for a ^ polarized plane wave incident from the direction ( ;  ) = (45 ; 45 ), ( ;  ) = (60 ; 255 ) and ( ;  ) = (60 ; 225 ).

ray-optical in the immediate vicinity of the vertex. We observed that this does not cause any substantial error in the RCS pattern. In Fig. 10(a), we present the amplitude of the numerical diffracas a function of for various directions of tion coefficient polarincidence for a 90 angular sector illuminated by a ized plane wave. It is observed that the diffraction coefficient becomes singular at the two edges of the plane angular sector. The vertex-diffracted waves are strongly guided by the edges of the angular sector as suggested by the edge condition [26]. As demonstrated in the earlier numerical example, the vertex-diffracted currents correct the discontinuity of the edge-diffracted currents at the shadow boundary lines. This is the part of the vertex-diffracted currents that accounts for the truncation of the FW currents. As a consequence, the diffraction coefficients are discontinuous at the shadow boundaries. It is noted that for (continuous line) the discontinuities at and correspond to shadow boundaries for the double diffracted rays. Similarly, for the diffraction coefficient is discontinuous at and due to discontinuities associated with the first-order diffracted rays from edges and . Fig. 10(b) shows the phase factor as a function of . It is observed that the phase factor is close to 1 for the entire angular range except around the transition regions. This verifies that the vertex-diffracted fields behave like a spherical wave traveling away from the vertex. Fig. 11(a) for the same scattering conshows the singularity exponent and , is discontinfigurations. Similar to uous at the shadow boundary lines associated with the first- and

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Fig. 11. (a) Singularity exponent  ( ) for a 90 angular sector for a ^ polarized plane wave incident from the direction ( ;  ) = (45 ; 45 ), ( ;  ) = (60 ; 255 ) and ( ;  ) = (60 ; 225 ), (b)  ( ) and  ( ) for a 90 angular sector for a ^ polarized plane wave incident from the direction ( ;  ) = (60 ; 225 ). The coefficients were calculated by sampling the difference current over different regions.

second-order edge diffracted rays. The steep variation of the coefficients around the shadow boundary lines are associated with the non ray-optical behavior of the edge-diffracted currents over these regions. The form of the vertex-diffracted current in (7) is not as accurate in these transition regions. It must also be noted that, far from the vertex and away from the transition regions, the and components of the vertex current density must have a raand respectively. Theredial dependence of fore, far from the vertex and away from the transition regions, and should in fact approach 1 and 2 respectively. However, it also known from the exact solution that close to the tip of the sector, the functional dependence of the current density on the radial distance to the vertex is also a function of . Thus, we are in fact approximating different kinds of behaviors and for using a single model. Fig. 11(b) shows a 90 angular sector illuminated by a polarized plane wave incident from the direction . and were calculated by sampling the difference current over and . It is obthe regions served that far from the transition regions, as the extent of the region over which the coefficients are calculated is increased, the far-zone behavior of the coefficients start to dominate ( and increase). For it is expected that and . The total current density obtained using the vertex-diffracted currents are integrated numerically to find the RCS of finite scattering structures of various shapes. First we consider the monostatic RCS of a 30 isosceles triangular plate depicted in the inset plane. of Fig. 12(a). The RCS pattern is calculated in the This is a common scattering configuration that has been studied in previous developments [20], [22], [29] to test the performance of the high-frequency solutions. In this configuration, because , the vertex-diffracted curof the narrow tip angle rents and the higher order interactions among the edges and corners makes a significant contribution to the RCS pattern when grazing directions are approached. Fig. 12(a) shows the co-polar for a polarRCS of a triangular plate of height ized incident field. “PTD-1” denotes the RCS pattern that is obtained by using only the first order edge diffracted currents. In “PTD-2”, the double diffracted FW currents are incorporated

Fig. 12. Monostatic RCS of a triangular plate. h = 3, ' = 30 . The RCS pattern is calculated in the x z plane.  is measured from the positive x-axis. (a) RCS and (b) RCS .

0

in the solution. “PTD+corner” is the pattern obtained using the total high frequency current. It is observed that the double-diffracted currents are of minor importance when the incident field polarized. On the other hand, the vertex-difis vertically fracted currents improve the accuracy of the RCS pattern by . as much as 20 dB over the angular range This improvement is mainly due to the contributions from the vertex-excited currents at corner 1. Consider the vertex contri. In this butions when nose-on incidence is approached case, the incident E-field is perpendicular to the scattering surface. As a result, there is no contribution from corner 1 and its two edges. The main contribution near nose-on incidence is due to the back edge and corners 2 and 3. The corner diffraction contribution due to these corners is very weak in the x-z plane. Thus, near nose-on incidence the RCS is dominated by the contributions from the edge along the y-axis. The small discrepancy is attributed to the higher order interactions around among the edges. As approaches 180 , the far field is dominated by the contributions from corner 1. Fig. 12(b) shows the co-polar RCS pattern of the same triangular plate for polarized incident field. Since the incident field in this case is parallel to the surface of the plate, the contribution from corner 1 is significant when nose-on incidence is approached. The effect of the double diffracted currents is evident in the RCS pattern. In calculating the double-diffracted currents, we employ ray tracing to find the first- and second-order diffraction points along the edges. Then we assume that the second-order diffraction point is illuminated by a plane wave. This assumption is valid when the two diffraction points are well separated. However, as the corner is approached, the first-order diffraction point becomes

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=5

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Fig. 13. Monostatic RCS of a triangular plate. h , ' . The RCS pattern is calculated in the x z plane.  is measured from the positive x-axis.

0

closer to the second-order diffraction point and this assumption , the disagreeis no longer valid. For horizontal polarization ment between the MoM and the PTD patterns over the angular is mainly due to the incomplete modrange eling of the higher-order edge-diffracted currents. Nevertheless, the vertex-diffracted currents act as a correction factor for the edge-diffracted currents and improve the resulting RCS pattern. In order to demonstrate the effect of the higher order diffraction mechanisms that have not been incorporated in the solution, we consider the scattering configuration in the previous example . Fig. 13 shows the co-polar with a triangular plate of RCS pattern for a polarized incident field. The resulting pattern (“PTD+corner”) agrees better with the MoM pattern as compared to the previous example. It is noted in Fig. 13 that, the RCS pattern obtained using the MoM is almost completely reconstructed when the corner diffraction terms are included over . This indicates that the the angular range discrepancy between the MoM and the high frequency patterns in Fig. 12(a) is indeed associated with the higher-order mechanisms such as corner-to-corner and corner-to-edge diffractions. As the plate size gets larger, the effect of these interactions become weaker. Even though not included in the present formulation, corner-to-corner and corner-to-edge interactions can be calculated with reasonable accuracy as demonstrated in [30]. Fig. 14 shows the bistatic RCS of a square plate of width . and cross-polar We consider the co-polar patterns for a polarized field incident from the direction . The RCS pattern is calculated at . Up to second-order edge-diffracted fields are in, the vertex-diffracted cluded in the solution “PTD”. For currents have a significant effect on the RCS pattern around and . For the cross-polar case, the PTD approximation seems to perform much better. This indicates that corner diffraction is not as significant. Nevertheless, the resulting high-frequency pattern agrees well with the MoM result. Finally, in order to demonstrate the effect of vertex-diffracted polarized incident field, we consider the RCS currents for a plate for . The high-frequency of the pattern agrees very well with the MoM pattern for both polarizations as shown in Fig. 15. It should be noted that the RCS and . pattern is slightly overestimated near This discrepancy is possibly due to the incomplete formulation

5

( ;  ) = (45 ; 0 ),  = 60

.

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( ;  ) = (80 ; 45 ),  = 90

.

Fig. 14. Bistatic RCS of a  square plate. (a) RCS and (b) RCS .

Fig. 15. Bistatic RCS of a  square plate. (a) RCS and (b) RCS .

of the double-diffracted currents and the higher-order diffraction mechanisms. Since the direction of incidence is close to grazing, higher-order interactions among different scattering

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centers such as third-order edge diffraction and corner-to-edge diffraction, are significant in this case. V. CONCLUSION In this paper, numerical diffraction coefficients for vertex-diffracted edge waves have been derived. The current density on the surface of the plane angular sector is modeled using the PTD. The vertex-diffracted currents are then defined as the difference between the exact and PTD currents. Based on the known physical behavior of the vertex-diffracted currents, the difference current is modeled as a wave traveling away from the corner. The unknown coefficients for the vertex-diffracted currents are obtained by applying a least squares fit approximation to the difference current. In many practical situations, multiple edge diffractions play an important role in calculating the current density on a polygonal flat plate. Thus, in order to be able to identify the vertex-diffracted currents in an accurate manner for arbitrary directions of incidence, it is important to eliminate the higher order edge effects from the total current. Up to second-order edge-diffracted currents are incorporated in our derivations. Unlike the previous developments, we were able to extract the plane vertex-diffracted currents even for a narrow angular sector. The vertex-diffracted currents have been validated by direct comparisons with the exact solution on the plane angular sector. The term “corner diffraction” is often used in the literature to describe the effect of the truncation of the edge diffracted currents at the tip of the plane angular sector. In the present development, vertex-diffracted currents also include the effect of the vertex-excited edge waves. Since the vertex-diffracted currents are numerically derived, in the RCS calculations it is necessary to solve the problem of diffraction from a semi-infinite plane angular sector for the given direction of incidence and polarization. However, calculation of the exact current is computationally intensive, and this requires additional cpu time especially for monostatic RCS calculations. In order to decrease the computational requirement as much as possible, we choose the maximum range of the radial cut over which the difference current is sampled as small as possible when fitting the vertex-diffracted currents. Furthermore, we pre-process the scattering structure for monostatic RCS calculations to tabulate the corner diffraction coefficients as a function of all the possible incidence directions and observation points on the surface of the scatterer. Once the unknown coefficients are tabulated, they are used directly in the RCS computation to find the vertex-diffracted currents from a pertinent corner on the finite scatterer. In contrast to previous works, the vertex-diffracted currents provide accurate results for arbitrary direction of incidence and polarization even for narrow angular sector openings. We have demonstrated using 3-D scattering problems that the vertex-diffracted currents provide a remarkable improvement in the accuracy of RCS patterns in the low-level regions. The accuracy could further be improved by incorporating the higher-order diffraction mechanisms in the solution. REFERENCES [1] R. Satterwhite and R. G. Kouyoumjian, “Electromagnetic diffraction by a perfectly conducting plane angular sector,” ElectroSci. Lab. Ohio State Univ., Columbus, OH, Tech. Rep. 2183-2, 1970.

[2] R. Satterwhite, “Diffraction by a quarter plane, the exact solution and some numerical results,” IEEE Trans. Antennas Propag., vol. AP-22, pp. 500–503, May 1974. [3] J. N. Shalos and G. A. Thiele, “The eigenfunction solution for scattered fields and surface wave currents of a vertex,” IEEE Trans. Antennas Propag., vol. AP-31, no. 1, pp. 206–211, 1983. [4] S. Blume and U. Uschkerat, “The radar cross section of the semi-infinite elliptic cone: Numerical evaluation,” Wave Motion, vol. 22, no. 3, pp. 311–324, Nov. 1995. [5] S. Blume and V. Krebs, “Numerical evaluation of dyadic diffraction coefficients and bistatic radar cross sections for a perfectly conducting semi-infinite elliptic cone,” IEEE Trans. Antennas Propag., vol. 46, no. 3, pp. 414–424, Mar. 1998. [6] V. M. Babich, V. P. Smyshlyaev, D. B. Dement’ev, and B. A. Samokish, “Numerical calculation of the diffraction coefficients for an arbitrary shaped perfectly conducting cone,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 740–747, May 1996. [7] V. P. Smyshlyaev, “Diffraction by conical surfaces at high frequencies,” Wave Motion, vol. 12, no. 4, pp. 329–339, Jul. 1990. [8] J. Radlow, “Diffraction by a quarter-plane,” Arch. Rat. Mech. Anal., vol. 8, no. 1, pp. 139–158, 1961. [9] J. Radlow, “Note on the diffraction at a corner,” Arch. Rat. Mech. Anal., vol. 19, no. 1, pp. 62–70, 1965. [10] M. Albani, “On Radlow’s quarter-plane diffraction solution,” Radio Sci., vol. 42, no. 6, p. RS6S11, 2007. [11] P. Y. Ufimtsev, Fundamentals of the Physical Theory of Diffraction. Hoboken, NJ: Wiley, 2007. [12] M. Albani, F. Capolino, G. Carluccio, and S. Maci, “UTD vertex diffraction coefficient for the scattering by perfectly conducting faceted structures,” IEEE Trans. Antennas Propag., vol. AP-57, no. 12, pp. 3911–3925, 2009. [13] P. H. Pathak and W. D. Burnside, , J. K. Skwirzynski, Ed., “A uniform GTD for the diffraction by edges, vertices and convex surfaces,” in Theoretical Methods For Determining the Interaction of Electromagnetic Waves with Structures. New York: Sijthoff and Noordhoff, 1981, pp. 497–561. [14] F. A. Sikta, W. D. Burnside, T. T. Chu, and L. J. Peters, “First-order equivalent current and corner diffraction scattering from flat plate structures,” IEEE Trans. Antennas Propag., vol. AP-31, no. 4, pp. 584–589, 1983. [15] A. Michaeli, “Comments on first-order equivalent current and corner diffraction scattering from flat plate structures,” IEEE Trans. Antennas Propag., vol. AP-32, no. 9, pp. 1011–1012, 1984. [16] T. J. Brinkley and R. J. Marhefka, Far-Zone bistatic scattering from flat plates ElectroSci. Lab. Ohio State Univ., Columbus, OH, Tech. Rep. 718295-8, 1990. [17] A. Michaeli, “Elimination of infinities in equivalent edge currents, part I: Fringe current components,” IEEE Trans. Antennas Propag., vol. AP-34, no. 7, pp. 913–918, 1986. [18] K. C. Hill, “A UTD solution to the EM scattering by the vertex of a plane angular sector,” Ph.D. dissertation, The Ohio State Univ., Columbus, 1990. [19] K. C. Hill and P. H. Pathak, “A UTD solution for the EM diffraction by a corner in a plane angular sector,” presented at the Int. IEEE Trans. Antennas and Propagation Society and URSI Nat. Radio Sci. Meeting, London, ON, Canada, Jun. 24–28, 1991. [20] S. Maci, M. Albani, and F. Capolino, “ITD formulation for the currents on a plane angular sector,” IEEE Trans. Antennas Propag., vol. AP-46, no. 9, pp. 1318–1327, Sept. 1998. [21] T. B. Hansen, “Corner diffraction coefficients for the quarter plane,” IEEE Trans. Antennas Propag., vol. AP-39, no. 7, pp. 976–984, 1991. [22] T. J. Brinkley, “Current near the vertex of a perfectly conducting angular sector,” Ph.D. dissertation, The Ohio State Univ., Columbus, 1990. [23] O. Breinbjerg, “Higher order equivalent edge currents for fringe wave radar scattering by perfectly conducting polygonal plates,” IEEE Trans. Antennas Propag., vol. AP-40, no. 12, pp. 1543–1554, 1992. [24] G. L. James, Geometrical Theory of Diffraction For Electromagnetic Waves. London, U.K.: Peter Peregrinus Ltd., 1976. [25] A. K. Ozturk, “Vertex diffracted edge waves on a perfectly conducting plane angular sector,” Ph.D. dissertation, Concordia Univ., Canada, 2009. [26] J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag., vol. AP-20, no. 7, pp. 442–446, Jul. 1972. [27] J. Van Bladel, Singular Electromagnetic Fields and Sources. Piscataway, NJ: IEEE Press, 1991. [28] F. B. Hildebrand, Introduction to Numerical Analysis. New York: McGraw-Hill, 1956.

OZTURK et al.: VERTEX DIFFRACTED EDGE WAVES ON A PERFECTLY CONDUCTING PLANE ANGULAR SECTOR

[29] L. P. Ivrissimtzis and R. J. Marhefka, “A uniform ray approximation of the scattering by polyhedral structures including higher order terms,” IEEE Trans. Antennas Propag., vol. AP-40, no. 11, pp. 1302–1312, Nov. 1992. [30] L. P. Ivrissimtzis and R. J. Marhefka, “Edge wave vertex and edge diffraction,” Radio Sci., vol. 24, no. 6, pp. 771–784, Nov.–Dec. 1989.

Alper K. Ozturk (S’03) was born in Yozgat, Turkey, in 1978. He received the B.S. and M.S. degrees in electrical and electronics engineering from Bilkent University, Ankara, Turkey, in 2000 and 2002, respectively, and the Ph.D. degree from Concordia University, Montreal, QC, Canada, in 2009. He is currently a Postdoctoral Fellow at Concordia University. His research interests include numerical and high-frequency techniques in electromagnetic scattering and diffraction.

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Robert Paknys (S’78–M’85–SM’97) was born in Montreal, QC, Canada, on June 29, 1956. He received the B.Eng. degree from McGill University, Canada, in 1979 and the M.Sc. and Ph.D. degrees from The Ohio State University, in 1982 and 1985, respectively, all in electrical engineering. He was an Assistant Professor at Clarkson University and an Engineer at MPB Technologies before joining the ECE Department, Concordia University, Montreal, QC, Canada, as a faculty member in 1989. He served the department as the Undergraduate Program Director, Associate Chair, and Interim Chair. He was a Visiting Professor at the University of Auckland in 1996 and at the University of Houston in 2004. He has been a consultant for the government and industry. His research interest is in computational electromagnetics and high frequency asymptotic methods, with applications to antennas, propagation, scattering, and diffraction. Dr. Paknys is a Registered Professional Engineer in the province of Quebec. He is a member of the Applied Computational Electromagnetics Society (ACES) and the International Scientific Radio Union (URSI). He is presently an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

Christopher W. Trueman (SM’98) received the Ph.D. degree from McGill University, Montreal, QC, Canada, in 1979. He has applied the methods of computational electromagnetics to problems such as aircraft antenna performance, antenna-to-antenna coupling and EMC on aircraft, aircraft and ship radar cross-section at HF frequencies, suppression of scattering of the signal of a commercial radio station from power lines, dielectric resonators, unconditionally-stable formulations for the finite-difference time-domain method, and the fields of portable radios such as cellular phones held against the head. Recently, his research has investigated the radar cross-section of ships at VHF frequencies, composite materials for aircraft, indoor propagation, and EMC issues between portable radios and medical equipment. He is currently the Associate Dean for Academic Affairs in the Faculty of Engineering and Computer Science, Concordia University, Montreal, QC, Canada.

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A Uniform Asymptotic Solution for the Diffraction by a Right-Angled Dielectric Wedge Gianluca Gennarelli and Giovanni Riccio, Member, IEEE

Abstract—We propose a uniform asymptotic solution for the field diffracted by a lossless right-angled dielectric wedge illuminated by a plane wave at normal incidence. The diffraction problem is solved by splitting the observation domain in the inner region of the wedge and the surrounding free-space. The scattered electric field in each region is assumed to be originated by a set of equivalent electric and magnetic surface currents involved in the well-known radiation integral. Such currents are localized on the interior and exterior faces of the wedge, and expressed in terms of the corresponding geometrical optics field. Useful analytical manipulations and asymptotic evaluations of the resulting integrals allow one to obtain the diffraction coefficients in terms of the Fresnel’s reflection and transmission coefficients of the structure and the transition function of the uniform geometrical theory of diffraction. The related diffracted field compensates the discontinuities of the geometrical optics field and gives total field levels in good agreement with finite difference time domain results. Index Terms—Dielectric wedge, diffraction, uniform asymptotic physical optics solution.

I. INTRODUCTION

T

HE problem of diffraction by wedges has received considerable attention due to the importance of its solution in radio propagation planning, analysis and design of radiating structures, radar cross section evaluation and waveguide theory. However, the existing solutions mainly concern the response of impenetrable structures at high frequency (see [1]–[4] as interesting bibliography). The scant attention to penetrable materials is due to the complex coupling between the internal region of the wedge and the surrounding space, and exact solutions are not available to date in the case of arbitrary apex angle and dielectric constant. If an exact analytical solution represents a hard task, then the best thing is an approximate solution. The methodologies proposed in literature provide approximate analytical solutions under certain hypotheses or attempt to solve the diffraction problem in an exact sense by combining analytical and numerical techniques to evaluate multi-dimensional integral equations. Rawlins [5] presented an approximate solution for the field produced when an electromagnetic plane wave is diffracted by an arbitrary-angled dielectric wedge, whose refractive index is near unity. It was obtained from an application of the

Manuscript received March 31, 2010; revised July 21, 2010; accepted July 23, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Department of Electrical and Information Engineering, University of Salerno, Fisciano, (SA) 84084, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2103031

Fig. 1. Geometry of the problem.

Kontorovich-Lebedev transform and a formal Neumann-type expansion. The results were in agreement with those derived by the same author with reference to a right-angled wedge [6]. The diffraction of an electric polarized plane wave by a dielectric wedge was formulated as integral equations in [7]. These were transformed into Fredholm integral equations and solved by iterative methods for limited values of the dielectric constant. Joo et al. [8] proposed an asymptotic solution for the field diffracted by a right-angled dielectric wedge based on a correction to the edge diffraction of physical optics (PO) approximation. The correction in the far-field zone was calculated by solving a dual series equation amenable to simple numerical evaluation. The extension of the approach to arbitrary-angled wedges was addressed in two companion papers [9], [10]. A solution based on a PO version of the geometrical theory of diffraction (GTD) was derived in [11] for an arbitrary-angled dielectric wedge. Moreover, Rouviere et al. [12] improved the Luebbers’ heuristic solution [13] in the uniform geometrical theory of diffraction (UTD) framework [1] by adding two new terms that compensate the discontinuity created by the field transmitted through the structure. The finite difference-time domain (FDTD) method was applied in [14], [15] for obtaining numerical diffraction coefficients for conducting and dielectric wedges. A uniform asymptotic physical optics (UAPO) solution for predicting the field diffracted by a lossless right-angled dielectric wedge when illuminated by a uniform plane wave at normal incidence (see Fig. 1) is proposed in this paper. The considered problem is split into two sub-problems concerning the inner region of the wedge and the surrounding space. With reference to the outer (inner) region, equivalent electric and magnetic PO surface currents lying on the external (internal) faces of the wedge are assumed as sources in the radiation integral. Useful analytical manipulations and uniform asymptotic evaluations of the resulting integrals allow determining the diffracted field in terms of the Fresnel’s reflection (transmission) coefficients of

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the structure and the UTD transition function [1]. Numerical results show that the UAPO diffracted field compensates the discontinuities of the geometrical optics (GO) field at the shadow boundaries. Moreover, it is easy to compute and turns out to be accurate as confirmed by comparisons with FDTD results. II. DIFFRACTION PROBLEM AND PROPOSED APPROACH The diffraction phenomenon originated by a linearly polarized plane wave impinging on a lossless right-angled dielectric wedge surrounded by free-space is considered in the case of normal incidence (see Fig. 1). The wedge is non-magnetic and has relative permittivity . The wedge surfaces and according to the apex angle. A are denoted with Cartesian coordinate reference system having the -plane coincident with is adopted. The incidence direction is fixed by corresponds to the grazing incidence with the angle ( . The inrespect to ), whereas the observation point is cident electric field is arbitrarily polarized and resolved in two orthogonal components directed along and . scattered in the inner region of the wedge and The field the surrounding space can be properly represented by the wellknown radiation integral

Fig. 2. Case I: 0

Fig. 3. Case II:

<  < =2.

=2 <  <  .

diffracted by the edge can be exFor each of them, the field pressed in the GTD framework as

(1) wherein , is the Green’s function, and are the intrinsic impedance and the propagation constant corresponding to the considered region, and denote the observation and source points, is the unit vector from the radiating element at to , and is the (3 3) identity matrix. In addition, and are the equivalent electric and magnetic surface currents here determined by using a PO approximation, i.e., (2) (3) where is the phase function of the electric field impinging on . It is possible to use ( is the unit vector in diffraction direction) in (1) for evaluating the edge diffraction confined to the Keller’s cone, so that

(4)

(5) denotes the incident field and the UAPO diffraction where coefficients are here determined by means of useful analytical manipulations and uniform asymptotic evaluations of the resulting diffraction integrals. Note that surface waves are neglected in the proposed approach. III. UAPO DIFFRACTION COEFFICIENTS: OUTER REGION Two cases must be considered according to the incidence diis directly illuminated (see Fig. 2); II) both rection: I) only and are directly illuminated (see Fig. 3). The field in the inner region is assumed equal to zero in accordance with the equivalence theorem. 1) Case I: : The key steps of the approach regarding the evaluation of the diffraction coefficients and are reported as reference. The same procedure is applied for obtaining the expressions of all the needed diffraction coefficients. In this context, the first step concerns the equivalent electric and magnetic PO surface currents. They lie on the external faces of the wedge and are determined by: (6)

The original problem is now split into two sub-problems related to the inner region of the wedge and the surrounding space.

(7) (8)

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The Fresnel’s reflection coefficients involved in (6) and (7) and perpendicular polarizations. refer to the parallel Accordingly, (9) where

Fig. 4. Integration path C .

(10)

The Fresnel’s transmission coefficients involved in (16) and (17) take into account the propagation through and , and

in which is the zeroth order Hankel function of the second kind. A useful integral representation of such a function [16] and the application of the Sommerfeld-Maliuzhinets inversion formula [17] allow rewriting (10) as

(19) is real in the event that coefficients can be so determined

(11)

. The related diffraction (20) (21)

wherein is the integration path in the complex -plane shown in Fig. 4. The integral in (11) can be reduced to a typical diffraction integral and evaluated by using the steepest descent method, so obtaining the diffraction term (12) is the UTD transition function [1], and sign where . As a consequence, applies when the UAPO diffraction coefficients read as

(22) sign holds when . where If , since the incident field is totally internally reflected and the surface wave produced in such a case does not contribute to the evaluation of the UAPO coefficients. : In this circumstance, 2) Case II: and are still given by (13) and (14), whereas

(13)

(23) (14) (15) For what concerns

and

(24)

, it results

(16) (25)

(17) (18)

and

sign applies when

.

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IV. UAPO DIFFRACTION COEFFICIENTS: INNER REGION The inner problem is solved by using the same approach developed for the outer problem. The field in the space surrounding the wedge is now assumed equal to zero in accordance with the equivalence theorem, and the field scattered inside the structure is originated by the equivalent electric and magnetic PO surface currents lying on the internal faces. The intrinsic impedance and the propagation constant to be used in (4) are relevant to the lossless dielectric material forming the wedge. 1) Case I: :

(26) (27)

Fig. 5. GO field and UAPO diffracted field. Amplitude of the z -component when  .

= 30

(28)

(29)

(30)

Fig. 6. Total field. Amplitude of the z -component when 

= 30

.

V. NUMERICAL RESULTS (31) : In such a case, 2) Case II: are still given by (26) and (27), whereas

and

(32)

(33)

(34)

Numerical simulations have been carried out for testing the correctness and accuracy of the proposed approach. Figs. 5 to 10 show the electric field amplitudes evaluated over a circular path , being the free-space wavelength, in the with radius case of a dielectric medium characterized by . Our first goal in this section is to demonstrate that the UAPO diffracted field fulfils expectations by compensating the GO field discontinuities at the shadow boundaries. As shown in Fig. 5, the GO , , field is discontinuous at and when the incidence . Note that in direction is fixed by this case. The UAPO diffracted field rises to peaks in correspondence of the GO boundaries (see Fig. 5) and ensures the continuity of the total field as reported in Fig. 6. This last statement applies to all the considered test cases. The next step is to demonstrate the accuracy of the UAPO solution for the diffracted field. Figures from 7 to 10 show comparisons with numerical results obtained by running a FDTD code “ad hoc” developed. This last implements a total field/scattered

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Fig. 7. Comparison between the UAPO-based approach results and the FDTD . results. Amplitude of the z -component when 

Fig. 10. Comparison between the UAPO-based approach results and the FDTD . results. Amplitude of the -component when 

Fig. 8. Comparison between the UAPO-based approach results and the FDTD . results. Amplitude of the -component when 

Fig. 11. Comparison between the UAPO-based approach results and the FDTD ," and   . results. Amplitude of the z -component when 

Fig. 9. Comparison between the UAPO-based approach results and the FDTD . results. Amplitude of the z -component when 

Fig. 12. Comparison between the UAPO-based approach results and the FDTD ," and  results. Amplitude of the z -component when   .

= 30

= 30

= 120

field (TF/SF) technique [18] to generate an accurate incident nuand merical plane wave having center frequency a Gaussian envelope. The outer boundaries of the computational domain are terminated with a uniaxial perfectly matched layer (UPML) [19] backed with a PEC wall, and the number of square

= 120

= 50

2

= 150

=2

= 15

=8

=

grid cells per wavelength is chosen equal to 15 to limit numerical dispersion. Two incidence directions corresponding to the cases I and II are considered. Note that a dB scale is used in the next figures, as opposed to the linear one in Figs. 5 and 6. Fig. 7 shows

GENNARELLI AND RICCIO: A UNIFORM ASYMPTOTIC SOLUTION FOR THE DIFFRACTION BY A RIGHT-ANGLED DIELECTRIC WEDGE

the comparison between the FDTD results and those reported in Fig. 6. As can be seen, an excellent agreement is achieved in the inner region of the wedge and the surrounding space. This statement stays valid also when considering the -component of the , so that both field (see Fig. 8). Figs. 9 and 10 refer to the faces of the wedge are illuminated. The UAPO-based approach yields accurate field levels also in this case. Moreover, a representative example relevant to the total internal reflection is given in Fig. 11. The chosen incidence and relative permittivity of direction the wedge guarantee that this is the case. There are many real scattering problems involving dielectric targets having permittivity much higher than those used in the previous examples. Accordingly, the UAPO solution for the diffracted field has been . A representative example relevant tested also when to , and is reported in the following figure. The observation path is now very close to the edge. Although the proposed solution is asymptotic, i.e., thefrom the edge, it stays accurate oretically valid far also in the considered case as shown in Fig. 12. Obviously, if becomes smaller than the wavelength, no reliable results are to be expected. Note that the accuracy of the UAPO solution decreases also when the incidence direction is close to the grazing one because of the well-known limitations of the PO-based approaches. VI. CONCLUSION AND FUTURE WORK A UAPO solution for the field diffracted by a lossless rightangled dielectric wedge illuminated by a plane wave at normal incidence has been derived in this paper. The proposed solution compensates the GO field discontinuities, is efficient from the computational viewpoint and, although approximate, provides accurate results. Accordingly, it can be useful for many practical applications. In addition, it is authors’ opinion that the extension of the here proposed approach to lossy dielectric wedges is possible, but not straightforward, and this case could represent an opportunity for further theoretical research.

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[7] S. Berntsen, “Diffraction of an electric polarized wave by a dielectric wedge,” SIAM J. Appl. Math., vol. 43, pp. 186–211, 1983. [8] C. S. Joo, J. W. Ra, and S. Y. Shin, “Scattering by a right angle dielectric wedge,” IEEE Trans. Antennas Propag., vol. AP-32, pp. 61–69, 1984. [9] S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: Part I—physical optics approximation,” IEEE Trans. Antennas Propag., vol. 39, pp. 1272–1281, 1991. [10] S. Y. Kim, J. W. Ra, and S. Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: Part II—Correction to physical optics solution,” IEEE Trans. Antennas Propag., vol. 39, pp. 1282–1292, 1991. [11] R. E. Burge et al., “Microwave scattering from dielectric wedges with planar surfaces: A diffraction coefficient based on a physical optics version of GTD,” IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 1515–1527, Oct. 1999. [12] J. F. Rouviere, N. Douchin, and P. F. Combes, “Diffraction by lossy dielectric wedges using both heuristic UTD formulations and FDTD,” IEEE Trans. Antennas Propag., vol. 47, no. 11, pp. 1702–1708, Nov. 1999. [13] R. J. Luebbers, “Finite conductivity uniform GTD versus knife edge diffraction in prediction of propagation path loss,” IEEE Trans. Antennas Propag., vol. AP-32, no. 1, pp. 70–76, Jan. 1984. [14] G. Stratis, V. Anantha, and A. Taflove, “Numerical calculation of diffraction coefficients of generic conducting and dielectric wedges using FDTD,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1525–1529, Oct. 1997. [15] J. H. Chang and A. Taflove, “Three-dimensional diffraction by infinite conducting and dielectric wedges using a generalized total-field/scattered-field FDTD formulation,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1444–1454, Apr. 2005. [16] P. C. Clemmow, “The plane wave spectrum representation of electromagnetics fields,” IEEE/OUP Series Electromagn. Wave Theor., 1996. [17] G. D. Maliuzhinets, “Inversion formula for the sommerfeld integral,” Sov. Phys. Dokl., vol. 3, pp. 52–56, 1958. [18] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method. Norwood: Artech House, 2000. [19] S. D. Gedney, “An anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag., vol. 44, pp. 1630–1639, 1996. Gianluca Gennarelli was born in Avellino, Italy, in 1981. He received the Laurea degree (cum laude) in electronic engineering and the Ph.D. degree in information engineering from the University of Salerno, Italy, in 2006 and 2010, respectively. Since 2006, he has been with the Applied Electromagnetics Research Group, University of Salerno, where he is currently a Research Fellow. His research interests include diffraction problems, near field—far field transformation techniques and numerical methods in electromagnetics.

REFERENCES [1] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, vol. 62, no. 11, pp. 1448–1461, Nov. 1974. [2] R. Tiberio, G. Pelosi, and G. Manara, “A uniform GTD formulation for diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag., vol. AP-33, pp. 867–873, 1985. [3] R. G. Rojas, “Electromagnetic diffraction of an obliquely incident plane wave field by a wedge with impedance faces,” IEEE Trans. Antennas Propag., vol. 36, pp. 956–970, 1988. [4] C. Gennarelli, G. Pelosi, and G. Riccio, “Approximate diffraction coefficients for an anisotropic impedance wedge,” UEMG, vol. 21, pp. 165–180, 2000. [5] A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. Lond., vol. 455, pp. 2655–2686, 1999. [6] A. D. Rawlins, “Diffraction by a dielectric wedge,” J. Inst. Math. Appl., vol. 19, pp. 231–279, 1977.

Giovanni Riccio (M’95) received the Laurea degree in electronic engineering from the University of Salerno, Italy. From 1995 to 2001, he was an Assistant Professor on the Engineering Faculty of the University of Salerno, where he is currently an Associate Professor of electromagnetics. His research activity concerns nonredundant sampling representations of electromagnetic fields, near field-far field transformation techniques, radar cross section of corner reflectors, wave scattering from penetrable and impenetrable structures. He is coauthor of about 220 scientific papers, mainly in international journals and conference proceedings. He served as an Editor of the Journal of Electromagnetic Waves and Applications and Progress in Electromagnetics Research Journals. Prof. Riccio is a Fellow of the Electromagnetics Academy.

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Dispersion Characteristics of a Metamaterial-Based Parallel-Plate Ridge Gap Waveguide Realized by Bed of Nails Alessia Polemi, Member, IEEE, Stefano Maci, Fellow, IEEE, and Per-Simon Kildal, Fellow, IEEE

Abstract—The newly introduced parallel-plate ridge gap waveguide consists of a metal ridge in a metamaterial surface, covered by a metallic plate at a small height above it. The gap waveguide is simple to manufacture, especially at millimeter and sub-millimeter wave frequencies. The metamaterial surface is designed to provide a frequency band where normal global parallel-plate modes are in cutoff, thereby allowing a confined gap wave to propagate along the ridge. This paper presents an approximate analytical solution for this confined quasi-TEM dominant mode of the ridge gap waveguide, when the metamaterial surface is an artificial magnetic conductor in the form of a bed of nails. The modal solution is found by dividing the field problem in three regions, the central region above the ridge and the two surrounding side regions above the nails. The fields within the side regions are expressed in terms of two evanescent TE and TM modes obtained by treating the bed of nails as an isotropic impedance surface, and the field in the central ridge region is expanded as a fundamental TEM parallel-plate mode with unknown longitudinal propagation constant. The field solutions are linked together by equalizing longitudinal propagation constants and imposing point-continuity of fields across the region interfaces, resulting in a transcendental dispersion equation. This is solved and presented in a dispersion diagram, showing good agreement with a numerical solution using a general electromagnetic solver. Both the lower and upper cutoff frequencies of the normal global parallel-plate modes are predicted, as well as the quasi-TEM nature of the gap mode between these frequencies, and the evanescent fields in the two side regions decay very rapidly away from the ridge. Index Terms—Artificial surface, bed of nails, EBG, metamaterials, waveguide.

I. INTRODUCTION HIS paper deals with the modal analysis of a new typology of metamaterial-based waveguide, particularly suitable for millimeter and sub-millimeter realizations [1], [4]. The basic structure is shown in Fig. 1. In the above-mentioned range of frequencies, this solution presents advantages compared to existing technologies like hollow rectangular

T

Manuscript received August 06, 2009; revised February 16, 2010; accepted August 14, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. A. Polemi is with the Department of Information Engineering, University of Modena and Reggio Emilia, Modena 41100, Italy (e-mail: [email protected]). S. Maci is with the Department of Information Engineering, University of Siena, Modena 41100, Italy (e-mail: [email protected]). P.-S. Kildal is with the Department of Signals and Systems, Chalmers University of Technology, Gothenborg, Sweden (e-mail: per-simon.kildal@chalmers. se). 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.2010.2103006

Fig. 1. Geometry of the ridge gap waveguide embedded in a bed of nails.

waveguide (HRW) and microstrip lines. Indeed, HRW can be manufactured in two parts and joined together, but not without consequent problems of good electrical contacts. On the other hand, microstrip lines suffer from losses with increasing frequency, limited power handling capability, and spurious resonances when encapsulated. Other solutions like Substrate Integrated Waveguide (SIW) [5], [6] exhibit undesired losses due to substrate at increasing frequencies. Therefore, there is still needs to find new technological solutions for waveguides above 30 GHz that have low losses and are cheap to manufacture. Recently, some numerical and experimental works have also proven that the same gap waveguide principle can be used to suppress cavity modes when packaging normal microstrip circuits [7]. Also, experimental works have pointed out that ridge gap waveguides have low losses [8]. The losses are actually comparable to those of rectangular waveguides, as studied in [9]. The structure analyzed in this paper is shown in Fig. 1 and can be seen as a particular realization of the more general ridge gap waveguide introduced in [1], [2]. It is composed of a pin surface, so called “Fakir’s bed” or “bed of nails,” surrounding a metallic ridge, and everything covered by a metal plate. The bed of nails can be immersed in dielectric substrate to reduce the size and the following analysis will account for it. However, in applications at millimeter wave frequency, the better solution is avoiding any dielectric. When the wavelength is large compared with the periodicity of the lattice, the bed of nails creates an high impedance surface (HIS) [10], over a certain frequency range. With the metallic top cover, it forms a PEC-HIS parallel plate waveguide which exhibits an elec, tromagnetic bandgap ( ). This implies that a quasi-TEM mode can propagate in the gap along the ridge without spreading into the cutoff region above the pin surface. The large bandwidth of the high impedance of the pin surface

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POLEMI et al.: DISPERSION CHARACTERISTICS OF A METAMATERIAL-BASED PARALLEL-PLATE RIDGE GAP WAVEGUIDE

allows a quasi-TEM mode propagation over a large frequency band without significant dispersion. The ridge gap waveguide is a result of research on so called soft and hard surfaces [11], which in their simplest form are realized by corrugations. They can in the ideal form be represented by a PEC/HIS strip grid, as explained in [12]. A hard surface covered by a PEC was in [13] and [14] shown to suppress higher order parallel plate modes. Instead there exist several degenerate local quasi-TEM waves following the ridges, as studied in detail by numerical simulations in [15]. Related surface waves in open hard structures were detected already in [18] but these follow individual grooves of corrugated surfaces rather than individual ridges, which is the case for the gap waveguides. The new ridge waveguide is a direct consequence of the local waves studied numerically in [15]. The present paper represents the first attempt to find an analytic solution to the modal fields. The gap waveguide can also be understood as a miniaturized hard waveguide having two PEC walls and two PMC walls [16] , where the vertical PMC walls of the present ridge gap waveguide are realized by the cutoff between two parallel plates which gives a very wideband solution compared to the narrow-band dielectric slab and FSS wall realizations in [16] and [17], respectively. The main purpose of the present paper is to model the dispersion characteristics of the quasi-TEM mode in an analytical form, in order to take under control the dispersion effects and to design the structure in the appropriate frequency band. The fields associated with this mode are mainly transverse to the direction of propagation, but small longitudinal components of the fields are also anticipated and included in the formulation in Section II. The field is expected to have an exponential decay laterally away from the ridge into the region above the pins. In the present formulation, we will not describe the field inside the bed of nails itself, but instead represent the latter by a spatially dispersive anisotropic homogeneous medium characterized by an equivalent homogeneous reflection coefficient according to [10]. The assumed field expressions in the region above the ridge and the two side regions above the surrounding pin surfaces are used to establish point matching continuity at the interfaces between the regions, and thereby to obtain a dispersion equation.

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Fig. 2. Bed of nails basic geometry. (left) 3D-view; (right) top-view.

where

is the wave number of the host medium and (3)

is the plasma wave number accounting for the local spatial dispersion [19], only dependent on the geometrical lattice properties. The model described by (1)–(3) is valid whenever and . Is out of the scope of this paper to illustrate the limit of validity of the homogenization process that allow the bed of nails to be considered as a continuous medium. This limitation have been indeed well addressed in [6], [7], [16]. We only emphasize here that the property of the equivalent homogenized medium is not the result of a resonant phenomenon dependent on the lattice properties, but it is rather a consequence of the high concentration of wires, more specifically of the fact that . The mathematical model described by (1) arises from an evolution for the wire medium model, which accounts for spatial dispersion. In [10] it is shown that the homogenized medium can support three different modal solutions: a transverse electromagnetic (TEM) mode, a transverse magnetic (TM- ) mode, and a transverse electric (TE- ) mode. Imposing the continuity of all the possible fields at the air-medium interface (including vertical component of induced fields), allows one to calculate the reflection coefficients. For an impinging plane TM wave [10] we obtain

II. BED OF NAILS PHENOMENOLOGY As a fist step we analyze the phenomenology associated with the bed of nails shown in Fig. 2. The nails are constituted by small metallic cylinders of height , with radius , and spacing in both and directions. For completeness, we assume the nails to be embedded in a host medium with permittivity , also . In [10], the if in practical application, it is better to have bed of nails is regarded as a spatial and frequency dispersive slab of an anisotropic medium whose permittivity is characterized by the tensor (1) In (1) (2)

(4) where

,

, and

. The reflection coefficient in (4) accounts for the penetration of TM mode inside the bed of nail through the attenuation constant gamma TM. This constant goes is dicto infinity when the nails are densely packed. In (4), tated by the impinging direction of the plane wave on the surface. For a TE polarized incident plane wave, the reflection coefficient is that associated with a bare grounded slab, because the electric field, perpendicular to the surface of the pins, does not

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significantly interact with them. This is consistent with the assumption of very small radius; this assumption can be removed by using the approach in [20]; this will be subject of a future investigation. Within the hypothesis of TE not-perturbing nails, we obtain

(5) In (4) and (5) we have maintained the definition in [10], where the reflection coefficients are those associated with the magnetic we have . field; thus, when Finally, the validity of this coefficient is given in [10] for values comparable with . of As mentioned above, when the metallic pins are closely [21], one has packed, which means (while is kept constant), and

Fig. 3. Reflection phenomenology supported by the bed of nails surface covered by a PEC plate.

(6) which is the solution of the reflection problem when only the TEM mode is considered, as expected. Thus, when the pins are densely packed, the wire medium behaves approximately as an anisotropic material with and . Its equivalent impedance is (7) where is the free-space characteristic impedance and it be, and as a capacitance haves as an inductance when (i) when (ii) . When , the equivalent impedance represents a PMC. The same condition as in [21] is in [22] referred to as an asymptotic boundary condition when dealing with corrugations and strip grids. Therefore, (7) represents also an asymptotic boundary condition of a bed of metal nails. III. DISPERSION EQUATION FOR PEC-PLATE COVERED BED OF NAILS The guiding phenomenon on the bed of nails surface is strongly modified by the presence of a metallic plate. They form together a parallel plate waveguide where one face is PEC and the other is reactive through the homogenized surface described in Section II (see Fig. 3). For simplicity, hereinafter . The same we consider free-space embedded nails has been done in [23], but one of the parallel-plate faces was there constituted by an anisotropic hard/soft surface along the direction of propagation, obtained as a periodic structure made up of corrugations. Here, the fakir’s bed homogenized surface is completely 3D, and then the modal field configuration supported is more complicated. The wave bouncing in the gap between the two faces can still be described by the reflection and , where now the value coefficients is the solution of the eigenvalue problem obtained by imposing the vanishing of the tangential electric field at the upper wall, . In absence of the i.e., dielectric background in the bed of nails, TE mode solution is

Fig. 4. Equivalent resonant circuit associated with the dispersion equation of the bed of nails surface covered by a top metal plate at height h.

that associated with the resonance between the upper and lower . For TM modes, the equation PEC walls, i.e., to be solved can be conveniently rewritten by employing (4), which leads to

(8) The above equation can be easily interpreted as a circuit series resonance equation of the kind (9) where, referring to the equivalent circuit in Fig. 4, is the TM modal impedance seen at the pin surface towards the and PEC cover plate, while are the TEM and TM modal impedances seen from the nail’s top surface towards the short circuited nail’s medium, respectively (note that the TM mode feels the pins with a ). In (9), characteristic dispersive impedance is a parameter which weights the TM mode relative to the TEM mode inside the bed of nails, and it can be easily derived from the continuity of the magnetic field at interface, or equivalently of the current in

POLEMI et al.: DISPERSION CHARACTERISTICS OF A METAMATERIAL-BASED PARALLEL-PLATE RIDGE GAP WAVEGUIDE

Fig. 5. TM dispersion equation of the complete (8) form (solid lines) and approximated (10) (dashed lines). The eigenvalue solution is real or imaginary in three frequency regions, separated by vertical dashed lines. TE eigenvalue is also shown (constant short-dashed line). The light line is also depicted (long dashed line). These results have been obtained for a reference geometry (see Fig. 3) (height of gap), d : (height of pins), a with h : (radius of pins). (periodicity of pins) and b

= 1 mm

= 7 5 mm = 0 5 mm

a series resonant circuit. In particular, when corresponding to the densely packed bed of nails, thus, the resonance condition becomes

= 2 mm

, ;

(10) thus leading to the conventional dispersion equation where the TEM mode is the only one considered inside the bed of nails. Equation (8), or possibly (10) can be solved numerically. Here, the intrinsic Matlab FSOLVE routine is employed, which finds a root of a system of nonlinear equations, given a certain starting point [24]. Results in terms of frequency are shown in Fig. 5 for (height of a reference geometry (see Fig. 3) with (height of pins), (periodicity of gap), (radius of pins). Eigenvalues of the pins) and complete TM resonance (8) and of the approximated TEM resonance (10) are reported as solid and dashed lines, respectively. The two solutions agree well, especially in the low frequency regime, where the bed of nails appears more closely packed with respect to the wavelength, and then the surface acts like the sureigenvalue is also face impedance in (7). In Fig. 5, the TE shown (dashed line), which is constant in frequency and dic. As the tated by the double PEC resonance, i.e., frequency increases, the two solutions slightly tend to separate each other. Let us discuss the type of wave in the various fre, is found to be purely imagquency region. For inary (see Fig. 6); there, a TM surface wave propagate along the interface without being substantially affected by the top cover. In the frequency region between the two resonance frequenand , is real and a bouncing wave is cies standing in the gap as depicted in Fig. 3. In this range, we iden, is real and tify two regions. In a first greater than , that implies attenuation along any direction along the propthe surface. In a second region agation is admitted within the structure The first range is the stop

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Fig. 6. Frequency versus real part of k for TM (black dots) and TE (dashed line) solutions of the pertinent eigenvalues problems. The light line is also shown (solid line). These results have been obtained for the same geometrical parameters used for Fig. 5.

band of the structure, whose upper bound is the cutoff frequency of the first TE mode. The upper cutoff of the stop band, corresponds to the lower cutoff frequency. In order to better idenis shown tify the stop band, a dispersion diagram versus , with for TM soluin Fig. 6 where for TE solution (dashed line). tion (dotted line), and The light line is shown. From the dispersion diagram in Fig. 6 it is possible to observe the stop band (shadowed area) where no propagation is permitted. As expected, there is a surface wave propagating at the low frequency up to 10 GHz, corresponding . The dispersion diagram is in the same graph comto pared with results from a commercial software (CST Microwave Studio, diamond line), where the basic cell of the periodic structure is depicted in the inset of Fig. 6, and phase shifts along and directions have been imposed. The agreement is found to be overall good. IV. DISPERSION EQUATION FOR A BED OF NAILS-BASED RIDGE GAP WAVEGUIDE We can realize a gap waveguide by inserting a conducting ridge of width into the bed of nails structure in the geometry in Fig. 3, thus leading to the structure in Fig. 1. From an intuitive point of view, we expect that a quasi-TEM modal field propagates along the ridge, in that region where the bed of nail region exhibits a bandgap. This mode will match the evanescent modes supported by the surrounding cutoff structure, in which we assume that only the first TM and TE modes (with respect to ) will be present. In particular, in order to achieve the field confinement in the ridge region, the cutoff modes must decay laterally away from the ridge (along direction ). In order to give a physical impression, we preliminary present some results obtained through CST Microwave Studio simulations. These results are plotted for a few frequencies. The amplitude of the

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Under the approximations used above, we can now write the expressions for the fields. In particular, the TMy evanescent mode fields in the region above the bed of nails are

(11a)

(11b)

(11c)

(11d) Fig. 7. Normalized amplitude of the vertical (along y ) electric field for different frequencies calculated through CST Microwave Studio.

(11e) where (12) The reference system is centered in the middle of the width of the ridge. In (11), is an unknown coefficients while is the eigenvalue solution of (8). Thus, the dispersion relation for this mode is , where the attenuation and the propagation constant are also unknowns. constant A second -evanescent TEy mode is assumed to be excited in the bed of nails region, whose expression is give by

(13a)

(13b)

Fig. 8. Normalized amplitude of the horizontal (along x) magnetic field for different frequencies calculated through CST Microwave Studio.

(13c) component of the electric field and the -component of the magnetic field is shown in Figs. 7 and 8, respectively. It is easy to recognize that the bed of nails surface stops propagation along the lateral direction and keeps the desired confinement of the electric field in the ridge region for frequencies within the stop – . The -compoband of the bed of nail region nent of the magnetic field appears less confined, within the normalized code of the colors.

(13d)

(13e)

POLEMI et al.: DISPERSION CHARACTERISTICS OF A METAMATERIAL-BASED PARALLEL-PLATE RIDGE GAP WAVEGUIDE

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where (14) In (1), is an unknown coefficient, while is the eigenvalue solution for the associated TE problem. The , dispersion equation for this mode is where the attenuation constant and the propagation constant are unknowns. In the central region above the ridge, we assume a quasi-TEM , mode, propagating along according to the phase factor of the kind (15a) (15b) (15c) is a constant related to the incident power at the where input port, and is a propagation constant along , and . The overall unknowns are six, , , , , , . Therefore, six equations are required to uniquely solve the problem. Three of these equations are provided by the dispersion relations in the three media, reported below for convenience:

Fig. 9. Dispersion diagram for the ridge waveguide (dotted line). The geometry is the same as that reported in caption of Fig. 6, but including an inserted a 5 mm wide ridge. The dispersion diagram is compared with the one obtained through a CST Microwave Studio simulation (diamond line) of the reference structure inserted above the graph. The light line is also shown.

(16a) (16b)

that, added to the ones in (16), allow us to find the dispersion equation

(16c) To find the remaining three equations, we enforce a razor blade continuity of the three field components across the separation . The matched field components , are walls namely all those belonging to the quasi-TEM mode. The non, and (only present in the bed matched components of nails regions) are very weak, since the boundary conditions at the top cover impose vanishing of them. Thus, we obtain the three matching field equations

(17a)

(17b)

(17c)

(18)

that relates

to frequency.

A. Numerical Results The solution is found numerically and plotted as the dispersion diagram in Fig. 9, where the ridge width is . Again, the Matlab FSOLVE routine is employed [24]. The geometry is the same as that used in the previous results (see caption of Fig. 6). The curve has been successfully compared with the dispersion diagram obtained by a CST Microwave Studio simulation (diamond line). The CST reference structure is shown above the graph in Fig. 9. The CST eigenvalues solver is used, then a basic cell is enclosed in a box of periodicity along the longitudinal direction with phase shift conditions. On the top and bottom walls, a perfect electric boundary condition is imposed. While a perfect magnetic boundary is set at the lateral sides of the box, in order to absorb the vertical electric field at the truncation of the nails. Our approximated solution tends to fail when the frequency is larger than the upper parallel-plate cutoff frequency, because we have only included one fundamental mode for each region, and this is not

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Fig. 10. Dispersion diagram obtained by CST Microwave Studio including all modes due to enclosure resonances. The diamond marked lines are the same as in Fig. 9. The dash-double dotted line is relevant to the higher-order mode of the ridge.

sufficient. Higher order modes could be accounted for at higher frequencies. In the most interesting frequency band between 10 and 20 GHz, the approximations used work well, and the mode propagating along the ridge is really quasi-TEM, as expected. is found, all the remaining quantities in (16) can be Once determined as well. Notice that the CST eigenmode solver gives many more modes than the curves shown in Fig. 9 due (in to the equivalent boundary conditions set at ). This modes have been left out the present case in Fig. 9. Therefore, to avoid misunderstandings, we present the complete multimode dispersion diagram in Fig. 10 to give an idea about the multiplicity of modes that can be found in sets the lower a practical packaging. The condition frequency of the operational bandwidth, which corresponds in this case. Then, we see that there are to several spurious modes outside the bed of nails stop band. One mode has a cut off at 17.2 GHz, i.e., within the bed of nails parallel-plate stop band 10?17.5 GHz. This first mode seems to have a cutoff within the bed of nails parallel-plate stop band – . This apparently strange fact is readily explained: this mode is associated with an odd transverse variation along the ridge, which is large enough at that frequency. The cutoff frequency of this mode is actually the one which sets the upper bound of the unimodal region, for ridge width not so small. The dispersion equation of this mode, and in particular its cutoff frequency, is the subject of our recent investigation [25]. There, a formula for maximizing the unimodal bandwidth is given, that is obtained by imposing the coincidence of the cutoff frequency of the higher order mode with the upper edge frequency of the bed-of-nails stopband. It is worth noticing that the stopband predicted in Fig. 6 for the parallel-plate structure without ridge still holds approximately when looking into the dispersion diagram with ridge in Fig. 9. We stress the fact that in the lower frequency regime below the lower bound of the bed-of-nails band gap, the -infinite structure supports a slow . This means that in the transition region of the wave bed of nails, attenuation constants in (16a) and (16b) become

Fig. 11. Normalized amplitude of the vertical (along y ) electric field for different frequencies calculated by the expressions in (11), (13) and (15).

Fig. 12. Normalized amplitude (in dB) of E (solid line) and H (dashed line) field distributions at 13 GHz, computed along the middle of the gap region (y = d + h=2).

propagation constants, and, as a consequence, the surface of nails cannot confine the field in the ridge region. This can be easily inferred from Fig. 7. Fig. 13). This In order to validate the modal field expansion approximation in (11), (13) and (15), the amplitude of the verelectric field is plotted, in the same way as done in tical Fig. 7. Results are shown in Fig. 11. We have for simplicity drawn the field distribution inside the gap only, according to the assumption of homogenized surface. In Fig. 12, the normalized

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(see wall of the smooth PEC plate, and spaced apart inset of procedure is valid in the limit where only one mode is present, and it seems to hold in our case. Indeed, results are in a good agreement, and they highlight that the attenuation has a almost over the entire bandwidth. value greater than 100 The same result has not been generated for the component of the magnetic field, because of the strong ripple due to the periodicity of the pins. V. CONCLUSIONS

Fig. 13. Transverse attenuation constant in x-direction plotted as dB= versus frequency compared with the one obtained through a CST simulation where the field decaying is captured by a couple of  -spaced probes located as shown in the inset.

amplitude of and (in dB) are compared with the same (inside the components obtained through CST, at , just in the middle band gap) and at a height of the gap region. The field distribution shows, as expected, a discontinuity at the region interfaces when the fields are evaluated in the middle of the gap. The reason for this is that the fields’ continuity has been enforced only at the top cover. There, the results appear quite similar due to the small dimension of the gap, the only difference being the continuity of the approximated solution. However, the agreement between the numerical and analytical solutions looks reasonable, also in consideration and from our apof the following comments. Firstly, proximated modal solutions show a different degree of decay. This is due to the fact that the decay of those components is for , and both and for . In parassociated with ticular, and are plotted in Fig. 13 in the stop band of the parallel-plate region, corresponding to the working bandwidth of the ridge gap waveis strongly varying with frequency and has a guide. While value greater than 100 almost over the entire bandwidth, is almost a constant in the same bandwidth. Indeed, since , , and the resulting value is component smaller over the bandwidth. This implies that has a weaker decay, dictated by the attenuation factor . Seccomponent, we see oscillations in ondly, regarding the CST the nails region which are due to the actual periodicity of the pins, that we neglect in the surface impedance model. It is evcomponent does not show ident from Fig. 12 that the CST has a stronger decay along than the same ripples. Indeed, , and this prevents from being influenced by the periodicity of the pins. The results are given in , where is , the same frequency used in Fig. 12. calculated at The CST curve (dashed) in Fig. 13 has been obtained from the computed field values at a couple of field probes located at the

We have presented an analytical solution for the fundamental quasi-TEM mode of a ridge-gap waveguide surrounded by a bed of nails. A quasi-TEM solution has been obtained in the EBG of the bed of nail region, by matching plane wave modal expressions in three regions; the region above the ridge and the two side regions above the bed of nails. The dispersion diagram so obtained well matches the one numerically predicted by a full-wave eigensolver and catches the main physical features of the structure; for instance, it recovers the rapid lateral field decay away from the ridge, showing values over 100 dB per wavelength within the stop band of the normal parallel-plate modes. The single-mode bandwidth of the ridge gap waveguide is mainly limited by the EBG of the surrounding bed of nailstype parallel-plate waveguide because out of it, spurious resonances can occur due to the packaging. In practical application, the gap of the ridge should be much smaller than the length of the nails so that the EBG may approach the maximum broadness, which is one octave. It is indeed seen, by this analysis, that the lower and upper frequency of the bandgap are dictated and . if , the bandwidth is close by to an octave. This is true provided that the ridge width is small enough and that the higher order mode of the ridge has a cutoff frequency higher than the stop-band upper edge. A dedicated analysis of these aspects is carried out in [25]. The single mode in Fig. 6, but it can also bandwidth is seen to be – be larger if the gap size or ridge width are reduced. The cutoff bandwidth of parallel-plates with different metamaterial-type loadings of the lower surface are studied in [26], showing bandwidths up to two octaves. The example showed in the paper is just a sample (with realistic geometry) of an extensive validation we have carried out, not presented here for lack of space. The main conclusion of this validation is that the approximate formulas presented here hold for the dominant mode till when the bed-of-nail can be assumed as a homogenized medium. The homogeneization conditions have been investigated in various papers, most of all in [10] and references therein. REFERENCES [1] P.-S. Kildal, E. Alfonso, A. Valero-Nogueira, and E. Rajo-Iglesias, “Local metamaterial-based waveguides in gaps between parallel metal plates,” Antennas Wireless Propag. Lett. (AWPL), vol. 8, pp. 84–87, 2009. [2] P.-S. Kildal, “Three metamaterial-based gap waveguides between parallel metal plates for mm/submm waves,” presented at the Proc. 3rd Eu. Conf. on Antennas and Propagation (EuCAP 2009), Berlin, Germany, Mar. 23–27, 2009. [3] H. Kirino, K. Ogawa, and T. Ohno, “A variable phase shifter using a movable waffle iron metal and its applications to phased array antennas,” in Proc. IEICE ISAP Intl. Symp., Aug. 2007, vol. 4B3–2, pp. 1270–1273.

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[4] H. Kirino, K. Ogawa, and T. Ohno, “A variable phase shifter using a movable waffle iron metal plate and its applications to phased array antennas,” IEICE Trans. Electron., vol. E91-B, no. 6, pp. 1773–1782, Jun. 2008. [5] J. Hirokawa and M. Ando, “Single-layer feed waveguide consisting of posts for plane TEM wave excitation in parallel plates,” IEEE Trans. Antennas Propag., vol. 46, no. 5, pp. 625–630, May 1998. [6] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 2, pp. 68–70, Feb. 2001. [7] E. Rajo-Iglesias, A. U. Zaman, and P.-S. Kildal, “Parallel plate cavity mode suppression in microstrip circuit packages using a lid of nails,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 1, pp. 31–33, Dec. 2009. [8] P.-S. Kildal, A. U. Zaman, E. Rajo-Iglesias, E. Alfonso, and A. ValeroNogueira, “Design and experimental verification of ridge gap waveguides in bed of nails for parallel plate mode suppression,” IET Microw., Antennas Propag., submitted for publication. [9] E. Pucci, A. U. Zaman, E. Rajo-Iglesias, P.-S. Kildal, and A. Kishk, “Losses in ridge gap waveguide compared with rectangular waveguide and microstrip lines,” presented at the EuCAP 2010 4th Eur. Conf. on Antennas and Propagation, Barcelona, Apr. 12–16, 2010. [10] M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, “Electromagnetic characterization of textured surfaces formed by metallic pins,” IEEE Trans. Antennas Propag., vol. 56, pp. 405–415, Feb. 2008. [11] P.-S. Kildal, “Definition of artificially soft and hard surfaces for electromagnetic waves,” Electron. Lett., vol. 24, no. 3, pp. 168–170, Feb. 1988. [12] P.-S. Kildal and A. Kishk, “EM modeling of surfaces with STOP or GO characteristics—Artificial magnetic conductors and soft and hard surfaces,” Appl. Comput. Electromagn. Society J., vol. 18, no. 1, pp. 32–40, Mar. 2003. [13] A. Valero-Nogueira, E. Alfonso, J. I. Herranz, and M. Baquero, “Planar slot-array antenna fed by an oversized quasi-TEM waveguide,” Microw. Opt. Technol. Lett., vol. 49, pp. 1875–1877, Aug. 2007. [14] P. Padilla de la Torre, J. M. Fernndez, and M. Sierra-Castaer, “Characterization of artificial magnetic conductor strips for parallel plate planar antennas,” Microw. Opt. Technol. Lett., vol. 50, no. 2, pp. 498–504, Feb. 2008. [15] E. Alfonso, P.-S. Kildal, A. Valero, and J. I. Herranz, “Detection of local quasi-TEM waves in oversized waveguides with one hard wall for killing higher order global modes,” presented at the IEEE Int. Symp. on Antennas and Propagation (IEEE AP-S), San Diego, Jul. 2008. [16] M. N. M. Kehn and P.-S. Kildal, “Miniaturized rectangular hard waveguides for use in multi-frequency phased arrays,” IEEE Trans. Antennas Propag., vol. 53, pp. 100–109, Jan. 2005. [17] M. N. M. Kehn, M. Nannetti, A. Cucini, S. Maci, and P.-S. Kildal, “Analysis of dispersion in Dipole-FSS loaded hard rectangular waveguide,” IEEE Trans. Antennas Propag., vol. 54, pp. 2275–2282, Aug. 2006. [18] M. Bosiljevac, Z. Sipus, and P.-S. Kildal, “Construction of Green’s functions of parallel plates with periodic texture with application to gap waveguides—A plane wave spectral domain approach,” IET Microw., Antennas Propag., accepted for publication. [19] P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B, vol. 67, pp. 103–113, Oct. 2003. [20] S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, “Wire media with negative effective permittivity: A Quasi-stati model,” Microw. Opt. Technol. Lett., vol. 35, no. 1, pp. 47–51, Oct. 2002. [21] R. J. King, D. V. Thiel, and K. S. Park, “The synthesis of surface reactance using an artificial dielectric,” IEEE Trans. Antennas Propag., vol. 31, no. 3, pp. 471–471, May 1983. [22] P.-S. Kildal, A. Kishk, and Z. Sipus, “Asymptotic boundary conditions for strip-loaded and corrugated surfaces,” Microw. Opt. Technol. Lett., vol. 14, pp. 99–101, Feb. 1997. [23] Z. Sipus, M. Bosiljevac, and P. S. Kildal, “Local wave green’s functions of parallel plate metamaterial-based gap waveguide with one hard wall,” presented at the IEEE Int. Symp. on Antennas and Propagation (IEEE AP-S), Charleston, SC, Jun. 2009. [24] [Online]. Available: www.mathworks.com

[25] A. Polemi and S. Maci, “Closed form expressions for the modal dispersion equations and for the characteristic impedance of a metamaterial based gap waveguide,” IET Microw., Antennas Propag., accepted for publication. [26] E. Rajo-Iglesias and P.-S. Kildal, “Numerical studies of bandwidth of parallel plate cutoff realized by bed of nails, corrugations and mushroom-type EBG for use in gap waveguides,” IET Microwaves, Antennas Propag., to be published. [27] E. R. Iglesias, M. Caiazzo, L. Inclan-Sanchez, and P.-S. Kildal, “Comparison of bandgaps of mushroom-type EBG surface and corrugated and strip-type soft surfaces,” IET Microw., Antennas Propag., vol. 1, no. 1, pp. 184–189, Feb. 2007.

Alessia Polemi (S’00–M’04) was born in Casteldelpiano, Italy, on July 10, 1973. She received the Dr. Ing. degree (cum laude) in telecommunications engineering and the Ph.D. degree in information engineering from the University of Siena, Siena, Italy, in July 1999 and March 2003, respectively. From January 2003 to October 2006, she was a Postdoctoral Researcher at the University of Siena. In November 2006, she was an Assistant Professor of electromagnetic fields at the Department of Information Engineering, University of Modena and Reggio Emilia, and was also the Italian Student Adviser for the Institution of Engineering and Technology (IET). In 2008, she was a Visiting Professor at the University of Pennsylvania, Philadelphia, working on plasmonic structures. She is now a Research Scientist at Drexel University, Philadelphia, working on the electromagnetic interaction between molecules and nanoparticles, on the optimization of SERS nanosurfaces, and on nanoantennas. Her current research includes high frequency scattering theories, asymptotics electromagnetic methods, numerical electromagnetic methods, periodic structures, bandgap structures, antenna design, RFID systems, metamaterials and polaritons propagation at optical frequency.

Stefano Maci (S’98–F’04) received the Laurea degree (cum laude) in Electronic engineering from the University of Florence, Italy. Since 1998, he is with the University of Siena, Italy, where he presently is a Full Professor. His research interests include EM theory, antennas, high-frequency methods, computational electromagnetics, and metamaterials. He was a coauthor of an “incremental theory of diffraction” for the description of a wide class of electromagnetic scattering phenomena at high frequency, and of a diffraction theory for the analysis of large truncated periodic structures. He was responsible and international coordinator of several research projects funded by the European Union (EU), by the European Space Agency (ESA-ESTEC), by the European Defence Agency, and by various European industries. He was the Founder and presently is the Director of the European School of Antennas (ESoA), a postgraduate school that comprises 30 courses on antennas, propagation, and EM modeling though by 150 teachers coming from 30 European research centers. He is the principal author or coauthor of more than 100 papers published in international journals, (among which 60 on IEEE journals), 10 book chapters, and about 350 papers in international conference proceedings. Prof. Maci was Associate Editor of the IEEE TRANSACTIONS ON EMC, two times Guest Editor of IEEE TRANSACTION ON ANTENNAS AND PROPAGATION (IEEE-TAP), Associate Editor of IEEE-TAP. He is presently a member of the IEEE AP-Society AdCom, a member of the Board of Directors of the European Association on Antennas and Propagation (EuRAAP), a member of the Technical Advisory Board of the URSI Commission B, a member of the Italian Society of Electromagnetism and of the Advisory Board of the Italian Ph.D. school of Electromagnetism. He was the recipient of several national and international prizes and best paper awards.

POLEMI et al.: DISPERSION CHARACTERISTICS OF A METAMATERIAL-BASED PARALLEL-PLATE RIDGE GAP WAVEGUIDE

Per-Simon Kildal (M’82–SM’84–F’95) was born in Norway on July 4, 1951. He received the M.S.E.E., Ph.D., and Doctor Technicae degrees from the Norwegian Institute of Technology (NTH), Trondheim, Norway, in 1976, 1982, and 1990, respectively. He has been a Professor at Chalmers University of Technology, Gothenburg, Sweden, since 1989. His textbook “Foundations of Antennas” was well received. He has designed and analyzed two very large antennas using his own methods and computer codes: The EISCAT VHF parabolic cylinder, and the Gregorian dual-reflector feed of the Arecibo radiotelescope operated by Cornell University. He has three inventions related to antenna feeds: the hat

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feed (www.comhat.se), the dipole-disk feed with beam-forming ring, and the recent decade bandwidth “Eleven antenna” being a candidate for future wideband radio telescopes. The last ten years his research group has pioneered the development of the reverberation chamber into an accurate tool for OTA measurements of antennas and wireless terminals subject to Rayleigh fading (www.bluetest.se). He is the originator of the concept of soft and hard surfaces in electromagnetics through which he and his coworkers preceded some of the recent research on electromagnetic bandgap surfaces and metamaterials. Recently this research resulted in a new local so-called ridge gap waveguide appearing in the gap between parallel metal plates, useful for applications above 30 GHz. Dr. Kildal is a Fellow of the IEEE since 1995, and has received two IEEE best journal article awards.

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An FFT Twofold Subspace-Based Optimization Method for Solving Electromagnetic Inverse Scattering Problems Yu Zhong and Xudong Chen

Abstract—A fast Fourier transform (FFT) twofold subspacebased optimization method (TSOM) is proposed to solve electromagnetic inverse scattering problems. As mentioned in the original TSOM (Y. Zhong, etal, Inverse Probl., vol. 25, p. 085003, 2009), one is able to efficiently obtain a meaningful coarse result by constraining the induced current to a lower-dimensional subspace during the optimization, and use this result as the initial guess of the optimization with higher-dimensional current subspace. Instead of using the singular vectors to construct the current subspace as in the original TSOM, in this paper, we use discrete Fourier bases to construct a current subspace that is a good approximation to the original current subspace spanned by singular vectors. Such an approximation avoids the computationally burdensome singular value decomposition and uses the FFT to accomplish the construction of the induced current, which reduce the computational complexity and memory demand of the algorithm compared to the original TSOM. By using the new current subspace approximation, the proposed FFT-TSOM inherits the merits of the TSOM, better stability during the inversion and better robustness against noise compared to the SOM, and meanwhile has lower computational complexity than the TSOM. Numerical tests in the two-dimensional TM case and the three-dimensional one validate the algorithm. Index Terms—Fast Fourier transform, inverse scattering, optimization, subspace, three-dimensional inverse problems.

I. INTRODUCTION NVERSE scattering problems are usually cast into optimization problems. There are usually two types of optimization approaches to solve inverse scattering problems: the deterministic type and the stochastic type. The first type has been developed for decades, such as the contrast source inversion (CSI) method [1]–[3], the Born iterative method and distorted Born iterative method [4], [5], the level set method [6], [7], etc. Based on these deterministic techniques, some multi-resolution methods have been proposed to increase the efficiency of the inversion, such as in [8], [9] and the references therein. The stochastic type of the inversion methods usually

I

Manuscript received March 13, 2010; revised July 30, 2010; accepted November 10, 2010. Date of publication January 06, 2011; date of current version March 02, 2011. This work was supported by the Singapore Temasek Defence Systems Institute under Grant TDSI/09-001/1A. The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2103027

employs a group of initial guesses and use the stochastic optimization scheme to minimize the objective function, such as the genetic algorithm and evolutionary optimization. This increases the possibilities of finding the global minimum rather than being trapped in a local minimum as deterministic optimization techniques [13], [14]. Both techniques have been applied to solve the three-dimensional (3D) inverse scattering problems also [3], [10]–[12], [15]. Recently, a subspace-based optimization method (SOM) has been proposed for solving the two-dimensional (2D) electromagnetic inverse scattering problems in a TM scenario [16], [17], i.e., reconstructing the dielectric profile of scatterers from scattered fields. The SOM uses the spectral property of the mapping from the induced current to scattered fields. Using such spectral property, SOM first determines part of the induced current, the deterministic part, and then obtains the rest part, the ambiguous part of the induced current, via optimization. Since the optimization is carried out in a subspace of the current space, SOM actually simplifies the nonlinear optimization problem. Some extensions and improvements of the SOM have been presented in [18]–[21]. In [9], [22], [23], the radiating and non-radiating current concept is used to address the inverse source problem, in which the radiating current is obtained either as the minimum norm solution or by the truncated singular value decomposition method. Compared to this well-known concept, the concept of the deterministic part and the ambiguous part of the induced current is more convenient for the numerical inversion method in the presence of noise. Detailed discussions are found in [17], [21] and [24]. Based on the SOM, a twofold SOM (TSOM) has been proposed to solve the two-dimensional electromagnetic inverse scattering problem [25]. By utilizing the spectral information of the operator which maps the induced current to the electric fields inside the domain of interest, the TSOM uses the singular vectors of the operator to construct a small current subspace and carries out the optimization of the induced current within this current subspace. Due to such constraint on the optimization, the TSOM has better stability and better robustness against noise than the SOM. However, in the TSOM, in order to obtain the bases that span the current subspace, we need to perform a singular value decomposition (SVD) of the operator that maps the induced current to the electric fields inside the domain of interest. Since the computational cost of such SVD operation , where is the number of subdomains, it is almost is prohibitive to directly apply the TSOM to solve the three-dimensional electromagnetic inverse problems.

0018-926X/$26.00 © 2010 IEEE

ZHONG AND CHEN: AN FFT TSOM FOR SOLVING ELECTROMAGNETIC INVERSE SCATTERING PROBLEMS

It is reported that there is a close relationship between the Fourier functions and the singular functions of an integral operator. In [26], it is exhibited the behavior of Fourier functions is similar to the one of the singular functions of an integral operator in the sense that the low-frequency Fourier functions correspond to those singular functions with large singular values, while the high-frequency Fourier functions to the singular functions with small singular values. Their discrete forms also follow the same rule. Thus, based on this, we may use the discrete form of the Fourier functions, the discrete Fourier bases, to span a current subspace so that such current subspace could properly approximate the current subspace spanned by the singular vectors obtained from the SVD operation. In the TSOM, the purpose of constraining the induced current within some low-dimensional current subspace is to exclude those current components in the subspace corresponding to small singular values. In this sense, although the new current subspace spanned by the discrete Fourier bases is not exactly the same as the original one spanned by the singular vector bases, the new current subspace could still work in the same way as the original one to exclude most of the current components that are in the subspace corresponding to small singular values. Consequently, we can replace the current subspace spanned by the singular vectors with the new current subspace spanned by the discrete Fourier bases, and implement the new current subspace in the TSOM. With such replacement, we significantly reduce the computational cost from the following two aspects. 1) Since we use discrete Fourier bases to construct the current subspace, we avoid the SVD operation of the matrix operator that maps the induced current to the electric field inside the domain of interest, which is highly computationally burdensome and memory demanding when dealing with problems with a large number of unknowns. 2) Since the discrete Fourier bases are used as the new current bases, the procedure to construct the ambiguous part of the induced current is the inverse discrete Fourier transform that can be accomplished by using the fast Fourier transform (FFT) algorithm. This saves a lot of computational burden during the optimization compared to the TSOM that uses the direct multiplication, and reduces the memory demand since there is no need anymore to save the singular vector bases. Based on this idea, in this paper, we propose a new FFT-TSOM, using the new current subspace spanned by the discrete Fourier bases, to solve the two-dimensional (2D) and three-dimensional (3D) electromagnetic inverse scattering problems. The proposed FFT-TSOM inherits the merits of the TSOM, better stability during the inversion and better robustness against noise compared to the SOM, and in the meanwhile has lower computational complexity than the TSOM.

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Lippmann-Schwinger equation and thus retains the properties of the operators in the Lippmann-Schwinger equation. In this paper, the domains of interest are chosen to be rectangular in order to implement the conjugate gradient fast Fourier transform (CG-FFT) scheme and apply the new Fourier bases. For convenience, we denote the one-dimensional tensor as , two-dimensional tensor as , three-dimensional tensor as , and four-dimensional tensor as . Unless otherwise specified, the subscript of the tensors denotes the index of the element, such denotes the element in with index . We use as bold symbols to denote vector variables, such as the positions and the electric fields in 3D case. In the following, the time for field quantities is assumed dependence factor and is suppressed throughout the paper. A. 2D TM MOM-Based SOM There are incident waves from different angles onto the ( , the background rectangular domain of interest 2D homogeneous medium with permittivity , permeability , and wave number ), where nonmagnetic scatterers are located, and these incident waves are expressed as . For each incidence, the antennas located at scattered fields are detected by . With above information, including every incident field inside the domain of interest and the corresponding scattered fields at the positions of all detectors, we aim at retrieving . the dielectric profile First, the domain of interest is discretized into small rectangular subdomains whose dimensions are much smaller than the wavelength and whose centers are located at, say, and , where and are the total number of subdomains along and axes, respectively ). With (so that the total number of subdomains is incidence such discretization, we have, for the (1) where

is the contrast at

is the

is the incident electric field

relative permittivity at

at is the induced current at erator defined by

, and

is an op(2)

with being integral of the Green function . [27]. All two-dimensional tensors in (1) are with size Similarly, the integral operator relating the induced current and the scattered fields could be expressed as the summation of the contribution from all subdomains

II. THE MOM-BASED SOM For the ease of applying the Fourier functions as the bases to construct the current subspace, we use the method of moments (MOM) to numerically describe the forward problem in this paper, instead of the coupled dipole method (CDM) as done in original SOM [16]. This is due to the fact that the MOM is obtained by directly discretizing the original

(3) where is and

is

a

an

dimensional matrix

with

vector, elements for

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and

. Here,

denotes the operation

of vectorizing a tensor, i.e., in this case, with . (the SVD of From [16], with the spectral information of tells , assuming that the singular values is a non-increasing sequence), the SOM splits the induced current into two parts, the deterministic portion and the ambiguous portion, the former of which can be obtained as

and could write the mismatch of (1) as

. With this operator, we

(9)

where

.

Finally, the SOM uses this objective function

(4) (10) where with , and the superscript denotes the Hermitian operation while superscript refers to the dominant current subspace, the subspace corresponding to the dominant singular values. The ambiguous part bases of the current is constructed by the remaining (5)

where

and , the latter being the unknown coefficients. Here the superscript refers to the subordinate current subspace corresponding to those subordinate singular values. , In [21], by using the identity where is the dimensional identity matrix, the authors propose a new current construction method as follows

The objective of the optimization is to minimize the objective function (10). In [17], [21], the conjugate gradient (CG) type algorithm that is used in the contrast source inversion (CSI) method is adopted to minimize this nonlinear equation by and at each iteration of the alternatively updating the optimization. Like in the CDM-based SOM [17], [21], [24], the parameter controls the convergence rate, and there is a successive range for the value of in which the SOM could converge at the optimal speed. B. 3D MOM-Based SOM For the MOM, we adopt the scheme proposed in [3]. In 3D , and case, the cuboid domain of interest is denoted as all other notations are the same as in the 2D TM case. For the incidence, the field equation is as

(6)

(11)

where is the unknown vector to be optimized. Equation (6) is equivalent to (5), but with much less computational complexity. Besides, since only the first singular vectors are needed, a thin SVD operation (generating only those singular vectors with dominant singular values) is enough. The details of this new current construction can be found in [21]. Having expressed the induced current in the aforementioned way, it is convenient to define the objective function. First, it is natural to define the mismatch of the scattered fields as

where and are the relative permittivity, the induced current and the incident electric field at , respectively, (12) with being the 3D Green’s function for the background homogeneous medium, and (13)

(7) where and are as in (4) and (6), respectively, and denotes the norm of a tensor. The current equation (1) is another key equation to satisfy. With (6), we define an operator as (8)

Now we need to write (11) into a discretized form. To this end, we first discretize the cuboid domain of interest into small cuboid subdomains, the center of which are located at and . Here, , and are the total number of subdomains along the , and direction, and is the total number of the subdomains. With such a discretization, (11) can be written like

is the inverse operation of

where , i.e.,

with

(14)

ZHONG AND CHEN: AN FFT TSOM FOR SOLVING ELECTROMAGNETIC INVERSE SCATTERING PROBLEMS

where the subscript , and denotes the , and components of a vector. As in the 2D case, the output of can be obtained by 3D FFT algorithm. For further details of the discretization of (11) and the finite difference scheme to generate from , please refer to the Appendix of [3]. The scattered fields in this 3D scenario can be obtained by (15) is a -dimenwhere sional vector with ( denotes the component of the corredimensional vector sponding vector, respectively), is a obtained by . The vectorization operation is defined to transform a four-dimensional tensor into a , we have with vector, i.e., if . In (15), the scattering operator is defined as

(16)

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and the is the inverse operation where of . Using this operator, we can define the mismatch of the current equation as

(19) where . The objective function in the 3D case shares the same form as in the 2D case (defined in (10)). We also use the CG-type optimization algorithm mentioned in the previous subsection to minimize the objective function. III. CURRENT CONSTRUCTION USING FOURIER BASES Having introduced the MOM-based SOM in 2D and 3D scenarios, in this section, we will use the discrete Fourier bases to construct the current subspace. Firstly, we look into the original current subspace in the TSOM [25], [28]. In the original TSOM, when dealing with the 2D TM case, we use

a matrix, with ,a matrix, the mapping from the component of the induced current to the component of scattered fields (the subscripts are NOT indexed for tensor elements). The explicit expression of is as (20) as the current subspace for the ambiguous part of the induced current and thus the ambiguous part of the induced current becomes (21)

(17) where

is when in which

where the

and is 0 otherwise,

with , and . Here, denotes the component of . As mentioned in the previous subsection, the induced current in (15) could be split into two parts, the deterministic part and the ambiguous part . After operating the thin SVD onto , we still have the same expression for and as (4) and (6), respectively. The only difference is that the the thin SVD of

is obtained from

, and the dimension of now becomes

.

is . Note that the complexity of thin SVD of Thus, the mismatch of the scattering data in 3D case has the same expression as (7). For the mismatch of the current equation, we need to first define an operator as

(18)

here is an

dimensional vector and

is the current subspace spanned by the singular vectors corresponding to the largest sin, which maps the induced gular values of the operator current to the electric field inside the domain of interest in the CDM (similar as the operator defined in (2) for MOM). Compared to the SOM (6), there is an extra subspace constraint . It is found that such on the current in the TSOM, the second-fold subspace constraint on the ambiguous part of the induced current works as regularization which is able to further stabilize the optimization by excluding those current components within the subordinate subspace of the operator (the subspace spanned by the singular vectors with small singular values) [25]. However, to obtain this current subspace, we need to operate the SVD on the operator (or in the previous section), which is not only computational burdensome but also memory demanding when the dimension (the number of the subdomains) of the operator ( or ) is large. Consequently, if we directly apply the original TSOM to solve the 3D inverse scattering problem, in which the dimension of the operator is much larger than the one in 2D case, both

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the computational cost and the memory consumption will be incredibly large. To apply the TSOM in a more efficient manner, we need to find another way to construct the current subspace instead of using the SVD operation. Notice that as aforementioned, the purpose of using the second-fold subspace constraint is to further regularize or stabilize the optimization by rejecting those current components in the subordinate current subspace spanned by singular vectors with small singular values, say . Thus, we may approximate the original current subspace spanned by the singular vector bases by a new current subspace so that the new one still works as the original one, i.e., excluding most of the current components in the subordinate . current subspace Here we supply a candidate of such approximation. In [26], it is reported that the behavior of Fourier functions is similar to the one of the singular functions of the integral operator in the sense that low-frequency Fourier functions correspond to those singular functions with large singular values, and high frequencies correspond to small singular values. Although the low-frequency Fourier functions are not exactly the singular functions with large singular values of the integral operator that we investigate, we still can use them to construct a current subspace that is a good approximation to the original one spanned by the singular functions so that the approximate current subspace can still filter out most of those unwanted current components. Thus, by the disin 2D TM case, replacing the singular vectors of crete Fourier bases, we can have this new current subspace for the ambiguous part of the induced current

(22) where , are the vectorized discrete means the subspace Fourier bases. Here, the symbol approximation instead of the basis approximation. In 2D case, , in which are the two-dimensional discrete Fourier basis with elements . Here,

,

transform (FFT) to calculate the ambiguous part of the induced current as follows:

(23) in which is the two-dimensional Fourier coefficients tensor with non-zero elements corresponding to those low-frequency discrete Fourier bases and zero elements corresponding to the remaining high-frequency discrete Fourier bases, and the is the vectorization operator introduced in 2D TM case in the previous section. Note that the inverse discrete Fourier transform (IDFT) is performed by the 2D FFT algorithm, the compu. Such computatational complexity of which is tional complexity is much smaller than the one of the direct mul, since is usually much tiplication in the TSOM, according to our experience. larger than In 3D case, the counterpart of the operator in 2D TM case is the summation of and , or simply, in the continuous domain, as the

(24) with dyadic Green’s function as the integral kernel. As we know, the dyadic Green’s function can be partitioned into nine scalar elements, as in the scattering operator case in (16), which represent the mapping from the three components of the induced current to the three components of the electric fields inside the domain of interest. Following the convention used in (16), we write these scalar operators as (25) at the bottom of the page, represent the and components of a where vector, respectively, and , and . By using this expression, we can rewrite the vectorial equation (11) into three coupled scalar equations as

(26) or their discrete forms

where

is a subset of . The relationship between and , or the method to choose the low-frequency discrete Fourier bases, will be discussed in the numerical simulation section. With this new current subspace, we now can use fast Fourier

(27)

(25)

ZHONG AND CHEN: AN FFT TSOM FOR SOLVING ELECTROMAGNETIC INVERSE SCATTERING PROBLEMS

where

, and

. Note that the convolution-type operators are obtained via the same procedure as mentioned in the forward problem model (see the and in (14)). From the equation set (26) or (27), we see that each scalar equation looks almost the same as the one in the 2D case except that the former has three scalar operators (with different inputs) whereas the latter has only one scalar operator. Thus, we may use the singular functions of each scalar operator, , to construct the current subspace. By doing so, we use the 3D discrete Fourier bases to span the current subspace so as to approximate the desired current subspace. It is worth mentioning that such a practice obviously achieves the aim discussed at the beginning of this section, i.e., to exclude most of the current components in the subordinate current subspace. To this end and following the procedures in 2D TM case, the ambiguous part of the induced current in 3D case can be written as

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1) Using the background medium as the initial guess and letting the current basis coefficients ( or ) be zero, with being some optimal value [17], [24], we run the CG type optimization algorithm [17] by using a small number of discrete Fourier bases. The optimization will converge to a coarse result. 2) Using the results (both the profile and the current basis coefficients) generated in the first step as the initial guess and the same as well, we run the CG type optimization algorithm by using a larger number of discrete Fourier bases, and let the algorithm search in a larger current subspace. The optimization will converge to a finer result. As mentioned in [25], the first step ensures a fast convergence to a meaningful coarse result that could be used as initial guess in the second step. In the second step, by increasing the dimension of the current subspace, the algorithm gives us a result with better resolution. Since in the second step a good initial guess is given, the convergence rate is also very fast. For some difficult problems, the inversion scheme could be carried out in multi-step by gradually increasing the number of the discrete Fourier bases at each step. IV. NUMERICAL SIMULATIONS

(28)

in which crete Fourier basis with elements

are the 3D dis-

, and , and are 3D Fourier coefficient tensors with non-zero elements corresponding to those low-frequency discrete Fourier bases and zero elements corresponding to the remaining high-frequency discrete Fourier bases. Here, , where is a subset of . The , or the method to choose relationship between and the low-frequency discrete Fourier bases, will be discussed in is the vectorizathe numerical simulation section. The tion operator introduced in the 3D case in the previous section, and similarly as in the 2D case, the IDFT is performed by 3D FFT algorithm, the computational complexity of which is , where in 3D case. We refer to as the FFT-TSOM the twofold SOM with the new current subspace constructed by the discrete Fourier bases proposed in this section. The additional computational cost for the FFT-TSOM compared to the SOM is the IDFT operation, which can be accomplished by using the FFT algorithm. The computational cost of the 2D (3D) FFT algorithm, as aforementioned, is at the same level as the cost of the operations in 2D case ( operation in 3D case). Thus, such a current subspace constraint only introduces a small computational cost compared to the SOM. Following the inversion procedure proposed in [25], we introduce the inversion procedure of the FFT-TSOM as follows:

In this section, we present several numerical examples to test the proposed algorithm in both 2D TM case and 3D case. In all tests in this section, additive white Gaussian noise (AWGN) is added on the synthetic result, and is quantified by %, where is the synthetic result for th incidence, so that . The synthetic data are generated by MOM using finer grid meshes than the ones used in the inversions. In FFT-TSOM, the stopping criterias of the inversion optimization is controlled by the change ratio of the Fourier coefficient tensors, which is defined as

(29)

in 2D TM case for the th iteration. Once the parameter is below some predefined threshold, the optimization is stopped. . For Similarly, we define such parameter for the 3D case, the TSOM in 2D, the corresponding parameter is denoted as , and also for the original SOM both in 2D and 3D case, and , respectively. In order to observe the characteristics of the algorithms, we define the piecewise error to quantify the quality of the reconstruction result, which is as

(30)

is the estimated relative permittivity and is where the true relative permittivity. For the 3D case, we have the sim. In all the simulations in this section, ilar piecewise error unless otherwise specified, the initial guesses for the profile are

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Fig. 1. Singular values of A in 2D TM case.

the background medium, air, and initial guesses for the current basis coefficients are null. A. Numerical Tests for 2D TM Case Our domain of interest is a 2 m 2 m square centered at the origin. The scatterers are illuminated with 20 plane waves at m) incident from dif400 MHz (with the wavelength ferent angles evenly distributed in . The scattered fields are collected by an antenna array with 40 antennas uniformly distributed along a circle centered at the origin with radius . In all following simulations, a 64 64 grid mesh of the domain of interest is used for our reconstructions. As for the synthetic scattered fields, they are calculated by the MOM method using a 150 150 grid mesh. 1) Relationship Between the Singular Vectors and the Discrete Fourier Bases in 2D TM Case: First, we obtain the opfrom (2), and its right singular vectors, denoted as erator , corresponding to the singular value sequence shown in Fig. 1. Transforming every singular vector operator and operinto a two-dimensional tensor by the ating the 2D FFT on each tensorized singular vector, we obtain the coefficients of the discrete Fourier transform (DFT) of these singular vectors. Fig. 1 shows that the first 500 singular values drop fast while the remaining decrease much slower. Thus, we group the singular vectors of into two sets, the first set with index from 1 to 500 and the second with index from 501 to 4096. We sample nine vectors from each set: for set 1, we choose and , an arithmetic progression with increment 62, as shown in Fig. 2; for set 2, , and , which is also an arithmetic progression with increment 443, as shown in Fig. 3. These two figures show that the absolute value of the DFT coefficients shows high value at the four corners when the index is small, and these high-value coefficients “shrink” towards the center as the increases. Note that we use MATLAB convention for the DFT coefficient index here, thus the coefficients at the four corners correspond to those low-frequency discrete Fourier bases and the ones at center correspond to high-frequency discrete Fourier bases. The conversion from

Fig. 2. The absolute value of the DFT coefficients of the singular vectors of A in 2D TM case with indexes from j to .

= 1 500

Fig. 3. The absolute value of the DFT coefficients of the singular vectors of A in 2D TM case with indexes from j to .

= 550 4096

the MATLAB convention to the mathematical convention can be found in [29]. This phenomenon is further confirmed by the result shown in Fig. 4. We plot the normalized logarithmic scale Frobenius norm of the output of the operator when the input is each pixel of this 64 64 plot discrete Fourier basis, e.g., the

ZHONG AND CHEN: AN FFT TSOM FOR SOLVING ELECTROMAGNETIC INVERSE SCATTERING PROBLEMS

Fig. 4. Normalized Frobenius norm (in logarithmic scale) of the output of the operator A when the inputs are the discrete Fourier bases.

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Fig. 6. The “Austria” profile.

2

Fig. 5. 2D coefficient tensor with four small M M effective blocks at four corners (in black) and the remaining null elements (in white).

represents the normalized value of the with the input having only one non-null element . Every pixel of such a plot represents the significance of the corresponding discrete Fourier basis as the input of the operator . From this result, we still see that the bases at the four corners are more important than those at the center. All the results obtained above are in accordance with the statement in [26] that low-frequency Fourier functions (bases) correspond to those singular functions (vectors) with large singular values, and high frequencies correspond to small singular values. Thus, we are able to use the low-frequency Fourier bases to construct a current subspace as the approximation to the original current subspace spanned by the singular vectors with large singular values. The patterns shown in Figs. 2, 3 and 4 show that in order to choose the low-frequency discrete Fourier bases, one may simply just optimize the elements at the four corners of the two-dimensional coefficient tensor and let the remaining elto be ements of be zero. Thus, we define the parameter the size of the four effective elements blocks at the corners of the coefficient tensor , as shown in Fig. 5. Note that one may also use another method to choose the low-frequency bases. 2) Reconstruction of the 2D Profiles: In the 2D TM case, we first compare the reconstruction results obtained by the proposed FFT-TSOM, using (23), and the TSOM, using (20). The ‘Austria’ profile consists of an annular, two disks and the background air. The two disks are of the same 0.2 m radius and their m and m. The annular is cencenters are m with 0.3 m inner radius and 0.6 m outer tered at radius. All three scatterers have the same relative permittivity

Fig. 7. Comparison of reconstruction results using FFT-TSOM and original ; ; ; ; and or the equivalents. The first TSOM when choosing M and third columns are the real and imaginary part of the reconstruction results generated by the FFT-TSOM. The second and fourth columns are the real and imaginary part of the reconstruction results generated by the original TSOM. The first row to the sixth row correspond to the results generated by using M to or the equivalents. All results are reconstructed from the same synthetic data with 10% AWGN.

=45678

4 9

9

=

, as shown in Fig. 6. As in [25], we choose throughout all the tests. In this test, 10% AWGN is added to the synthetic data. The termination conditions of both algorithms for the FFT-TSOM and are set to be the same, i.e., for the TSOM. The reconstruction results are shown in Fig. 7, in which the first column and the third column are the real part and imaginary part of the reconstruction results generated by the FFT-TSOM, respectively, and the second

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M =4 9

Fig. 8. Objective function value during the optimization by using FFT-TSOM to . and original TSOM when choosing

column and the fourth column are the real part and imaginary part of the reconstruction results generated by the original TSOM, respectively. The first row to the sixth row correspond to to (for the FFT-TSOM the results generated by using results) or the equivalents (for the TSOM). Note that all these inversions are independent of each other, and the equivalent here means the same number of the bases used, e.g., when we for FFT-TSOM, we use use bases for the TSOM. These reconstruction results show that in this practical inversion example, the FFT-TSOM produces almost the same results as the TSOM. The objective function values during the optimization are shown in Fig. 8, from which we also see that the convergence speeds of both algorithms are almost the same, i.e., the trends of each pair of the lines for and , are almost each case, e.g., the same. Since the FFT-TSOM does not require a prior SVD computation for the current bases and it uses FFT to construct the ambiguous current at every iteration of the optimization, the proposed algorithm saves a lot of computational burden compared to the original TSOM while it has almost the same performance (in terms of reconstruction quality and convergence speed) as the original TSOM. In our second test, we try to reconstruct a more difficult profile. The true profile is depicted in Fig. 9(a) and (b), which consists of two disks and a coated rectangle. The two disks are cenm and (0.4, 0.6) m with the same radius 0.3 tered at . The coated recm and the same relative permittivity m. The inner rectangle has a 1 m tangle is centered at long edge and a 0.6 m short edge, while the outer rectangle has a 1.6 m long edge and a 1 m short edge. The relative permittivity while the outer one with of the inner rectangle is . Firstly, we use the SOM to reconstruct from the data with 10% AWGN, and the termination condition is set . The whole optimization costs 4662 iterato be tions to reach the above condition, and the reconstructed results are shown in Fig. 9(c) and (d). From these results, we see that the SOM failed to converge to a satisfactory result, which can also be seen from the piecewise error shown in Fig. 9(e) that the reconstruction error is increasing all through.

Fig. 9. Exact profile and the reconstruction results by the SOM. (a) and (b) are the real and imaginary part of the true profile; (c) and (d) are the real and the imaginary part of the reconstruction result by the SOM after 4662 iterations, which are reconstructed from the synthetic data with 10% AWGN; (e) is the piecewise error of the SOM.

Using the same synthetic data, we test the performance of the proposed FFT-TSOM. We set the termination condition as for all the cases below. First, we choose for the reconstruction with the initial guess being air, and after around 300 iterations, the optimization stops. We use the obtained result as the initial guess of the second round optimiza, and we carry out the third round optimization with from the initial guess being the result obtion with . The whole three-round optimization tained by using costs 750 iterations, and the trajectory piecewise error is shown in Fig. 10, while the reconstruction results are shown in Fig. 11. In Fig. 11, the first, the second, and the third row are the recon, and , restruction result using spectively. The first column and the second column are the real part and the imaginary part of the reconstruction results, respectively. These results show that the FFT-TSOM obtains a satisfactory reconstruction result. We further test the performance of the FFT-TSOM under 30% and 50% AWGN. The optimizations for both two cases cost around 750 iterations. The reconstructed results are shown in Figs. 12 and 13 for 30% AWGN and 50% AWGN cases, respectively. The piecewise errors are shown in Fig. 10. These figures reveal that even if the level of the noise increases, the proposed FFT-TSOM still obtains satisfactory results. Thus, the proposed FFT-TSOM is very robust against noise.

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Fig. 10. Piecewise error obtained by FFT-TSOM using synthetic data with 10%, 30%, and 50% AWGN.

Fig. 12. Same as Fig. 11 except synthetic data with 30% AWGN is used.

Fig. 11. Reconstruction results by FFT-TSOM using the synthetic data with 10% AWGN. The first, second, and third rows are the reconstruction results ; , and . The first and second columns are the real and imagiusing nary part of the reconstruction results.

M =5 6

7

B. Numerical Test for 3D Case In this 3D inverse scattering test, the scatterer is a coated cube m and centered at the origin with its inner edge length m, as shown in Fig. 14. The relative outer edge length while the relpermittivity of the inner layer is ative permittivity of the outer layer is . The coated cube is illuminated by 60 electric dipole antennas operm), which are along ated at 300 MHz (wavelength in air is three circles (with 20 dipole antennas evenly distributed on each circle) with same radius 3 m. The three circles are in and planes, and their centers are at m, m and m, respectively. The plane are in direction of the electric dipole sources in the the direction, while those in the and planes are in the and directions, respectively. Scattered fields are collected by 60 detectors, which are located at the same positions as the 60 dipole sources. We assume that each detector measures the vector electric field in three directions. Thus, we have

Fig. 13. Same as Fig. 11 except synthetic data with 50% AWGN is used.

60 180 data points. These synthetic data are calculated by the MOM-based algorithm proposed in [3] using 60 60 60 mesh grid of a cubic domain. This cubic domain is centered at the origin and with an edge length 3 m. In this simulation, 10% AWGN is added. During the inversion, we use a 30 30 30 mesh grid for the domain, which means we have 27000 unknown contrasts in total. Using the locations of detectors and all subdomains, we can generate

and its singular values, as shown in Fig. 15.

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=1

=2 = 2+08

Fig. 14. Coated cube with inner edge a m and outer edge b m. Relative permittivities of inner cube and outer layer are  : i and  : : i.

= 15+03

^

Fig. 15. Singular values of the operator G

.

In this simulation, we choose for both FFT-TSOM and SOM. Similarly as in the 2D case, now in 3D case, we firstly , plot the norm of the output of the operators , when the input is every or their discrete forms, pixel of discrete Fourier basis. That is to say, the this 30 30 30 plot represents the normalized value of the with the input having only one and the remaining elements non-null element zero. Every pixel of such a plot represents the importance of the corresponding discrete Fourier basis as the input of the . The results for operators and operators are shown in Figs. 16 and 17, respectively, which are chosen for the as the representatives of the 9 scalar operators, for the coupling operators. non-coupling operators and Here, the coupling means the interaction between the different components of the current and field. In these two figures, the 30 subfigures are corresponding to the cross section of the to 30, i.e., the first row from the 3D result with index to , left to the right are the cross section with index respectively, and so forth. We clearly see that the most influential bases are those lowest-frequency discrete Fourier bases, which are at the eight corners of 3D cube (due to the MATLAB index convention). According to these results, we can choose those bases with high values, as shown in these two figures, to approximate the subspace of dominant current. We follow the method used in the 2D case to just optimize the elements at the eight corners of the cubic coefficient tensor, so that the

Fig. 16. Normalized Frobenius norm (in logarithmic scale) of outputs of G when inputs are the discrete Fourier bases. First row, from left to right, shows to ; second row, from left to right, shows cross sections of 3D result with p to ; and so forth. x-axis and y -axis cross sections of the 3D result with p refer to the indexes of m and n.

=1 5 = 6 10

lowest-frequency Fourier bases are chosen to span the desired current subspace. The size of the eight small cubes is controlled . by the parameter , i.e., there are In the inversion we first choose bases used to construct the current subspace for the optimization. The termination condition for FFT-TSOM , and to reach this condition the opis set to be timization takes 300 iterations. For the SOM, we set it to be 0.0001, and this condition costs more than 6000 iterations in the optimization. The piecewise errors for both cases are shown in Fig. 18, where we see that the FFT-TSOM obtains much smaller error after 300 iterations. The retrieval result obtained by FFT-TSOM after 300 iterations and the one obtained by SOM after 1000 iterations (where the piecewise error is minimum) are shown in Fig. 19. In this figure, the first column and the third column are the real and imaginary part of the reconstruction result obtained by FFT-TSOM after 300 iterations when using , respectively. The second and the fourth columns are real and imaginary part of the reconstruction result obtained by SOM after 1000 iterations, respectively. The first, second, and third rows are corresponding to the cross-sections at m, m and m. We see that the SOM failed to retrieve the profile of the inner cube while the FFT-TSOM successfully retrieve the dielectric profile although the position of the imaginary part is slightly shifted.

ZHONG AND CHEN: AN FFT TSOM FOR SOLVING ELECTROMAGNETIC INVERSE SCATTERING PROBLEMS

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Fig. 19. Reconstruction results for the coated cube by using FFT-TSOM and SOM. The first and third columns are the real and imaginary part of the recon= struction result obtained by FFT-TSOM after 300 iterations when using 6. The second and fourth columns are real and imaginary part of the reconstruction result obtained by SOM after 1000 iterations. The first, second and third rows are corresponding to the cross sections at = 0 05 m, = 0 05 m and = 0 05 m.

M

x 0:

G

Fig. 17. Same as Fig. 16 except results are for ^

z 0:

y 0:

.

Fig. 20. Objective function values obtained when using FFT-TSOM and SOM.

Fig. 18. Piecewise error obtained when using FFT-TSOM and SOM.

The slight shift aforementioned is mainly due to the insufficiency of the number of the bases used during the optimization, which could also be noticed from the objective function value shown in Fig. 20 where the FFT-TSOM converges to a higher value than the SOM does. Thus the reconstruction results can be further improved by another round of optimization with a larger ) and with the obtained number of the bases (or the value of results as the initial guess. Consequently, we use the result obas the initial guess for the optimization tained by using (i.e., bases), and the optimizawith tion reaches the condition after 122 iterations. The final reconstruction is shown in Fig. 21, which indicates that the FFT-TSOM obtains a very satisfactory reconstruction result. The FFT-TSOM algorithm now converges to the same level as

the one the SOM converges to, as shown in Fig. 20, while the piecewise error further decreases to a low level, as shown in Fig. 18. This numerical simulation shows that the FFT-TSOM has better stability during the inversion optimization and is able to obtain satisfactory result within a small number of iterations when dealing with the complex 3D inverse scattering problem. V. CONCLUSION In this paper, we propose a new twofold subspace-based optimization method (TSOM) which is based on the fast Fourier transform (FFT). By replacing the current subspace spanned by the singular vectors with the approximate one spanned by discrete Fourier bases, the proposed FFT-TSOM inherits the advantages of the original TSOM proposed earlier by the authors [25], i.e., better stability of the inversion procedure and better robustness against noise compared to the SOM. Due to the use of the discrete Fourier bases, the computationally costly SVD operation is avoided, and in the meanwhile, thanks to the FFT

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as unknowns, such as the one used in [15]. Such a property dramatically shrinks the size of the population used in the stochastic optimization so as to yield faster convergent speed and better stability and robustness against noise. This is one of our future research topics in inverse scattering. ACKNOWLEDGMENT The authors would like to thank D. Lesselier for offering good advice during the preparation of the manuscript. REFERENCES

M = 10 M =6

Fig. 21. Reconstruction result for the coated cube obtained by FFT-TSOM with the initial guess being the reconstruction result when using obtained when using . The first and second columns are the real and imaginary part of the reconstruction result after 122 iterations. The first, : m, second, and third rows are corresponding to the cross sections at z y : m and x : m.

= 00 05

= 00 05

= 00 05

algorithm, the computational cost of the current construction procedure in every iteration of the optimization is decreased significantly compared to the original TSOM. Thus, with good stability and good robustness against noise as well as low computational complexity, the proposed FFT-TSOM is able to deal in an efficient manner with the more difficult and more computationally demanding inverse scattering problems, such as the three-dimensional inverse scattering problems. The numerical simulations validate the proposed algorithm in both 2D TM case and 3D case. From the simulations shown in this paper and those in [25], we point out that the advantages of using the FFT-TSOM type objective function with such current constraint during the inversion optimization are independent of the optimization scheme. This is due to the fact that in the objective function, we use the most dominant current bases to construct the induced current. In this sense, it is very convenient to apply other deterministic optimization schemes to minimize the objective function constructed in the framework of the proposed FFT-TSOM. Besides, the stochastic type optimization schemes are also good candidates, because of their better ability in converging to the global minimum. Since in the proposed objective function the coefficients of the current bases, rather than the induced current, are the parameters to be optimized and the number of the current bases is much smaller than the number of the subdomain, the total number of the inputs of the proposed objective function is much less than the number of the inputs of the conventional objective function considering both contrast and fields (currents)

[1] P. M. van den Berg and R. E. Kleinman, “A contrast source inversion method,” Inverse Probl., vol. 13, pp. 1607–1620, 1997. [2] P. M. van den Berg, A. L. van Broekhoven, and A. Abubakar, “Extended contrast source inversion,” Inverse Probl., vol. 15, pp. 1325–1344, 1999. [3] A. Abubakar and P. M. van den Berg, “Iterative forward and inverse algorithms based on domain integral equations for three-dimensional electric and magnetic objects,” J. Comp. Phys., vol. 195, pp. 236–262, 2004. [4] Y. Wang and W. Chew, “An iterative solution of two-dimensional electromagnetic inverse scattering problem,” Int. J. Imaging Syst. Technol., vol. 1, pp. 100–108, 1989. [5] W. Chew and Y. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Medical Imag., vol. 9, pp. 218–225, 1990. [6] O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl., vol. 22, pp. R67–R131, 2006. [7] M. Benedetti, D. Lesselier, M. Lambert, and A. Massa, “A multi-resolution technique based on shape optimization for the reconstruction of homogeneous dielectric objects,” Inverse Probl., vol. 25, p. 015009, 2009. [8] S. Caorsi, M. Donelli, and A. Massa, “Detection, location, and imaging of multiple scatterers by means of the iterative multiscaling method,” IEEE Trans. Microwave Theory Tech., vol. 52, pp. 1217–1228, 2004. [9] P. Rocca, M. Donelli, G. Gragnani, and A. Massa, “Iterative multi-resolution retrieval of non-measurable equivalent currents for the imaging of dielectric objects,” Inverse Probl., vol. 25, p. 055004, 2009. [10] J. De Zaeytijd, A. Franchois, and J.-M. Geffrin, “A new value picking regularization strategy—Application to the 3-D electromagnetic inverse scattering problem,” IEEE Trans. Antennas Propag., vol. 57, pp. 1133–1149, 2009. [11] P. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl., vol. 25, p. 024003, 2009. [12] C. Yu, M. Yuan, and Q. Liu, “Reconstruction of 3D objects from multifrequency experimental data with a fast DBIM-BCGS method,” Inverse Probl., vol. 25, p. 024007, 2009. [13] M. Pastorino, “Stochastic optimization methods applied to microwave imaging: A review,” IEEE Trans. Antennas Propag., vol. 55, pp. 538–548, 2007. [14] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Probl., vol. 25, p. 123003, 2009. [15] M. Donelli, D. Franceschini, P. Rocca, and A. Massa, “Three-dimensional microwave imaging problems solved through an efficient multiscaling particle swarm optimization,” IEEE Trans. Geosci. Remote Sens., vol. 47, pp. 1467–1481, 2009. [16] X. Chen, “Application of signal-subspace and optimization methods in roconstructing extended scatterers,” J. Opt. Soc. Am. A, vol. 26, pp. 1022–1026, 2009. [17] X. Chen, “Subspace-based optimization method for solving inverse scattering problems,” IEEE Trans. Geosci. Remote Sens., vol. 48, pp. 42–49, 2010. [18] L. Pan, K. Agarwal, Y. Zhong, S. P. Yeo, and X. Chen, “Subspace-based optimization method for reconstructing extended scatterers: Transverse electric case,” J. Opt. Soc. Am. A, vol. 26, pp. 1932–1937, 2009. [19] K. Agarwal, L. Pan, and X. Chen, “Subspace-based optimization method for reconstruction of two-dimensional complex anisotropic dielectric objects,” IEEE Trans. Microwave Theory Tech., vol. 58, pp. 1065–1074, 2010.

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[20] X. Ye, X. Chen, Y. Zhong, and K. Agarwal, “Subspace-based optimization method for reconstructing perfectly electric conductors,” Progress in Electromagnetic Research, vol. 100, pp. 119–128, 2010. [21] Y. Zhong, X. Chen, and K. Agarwal, “An improved subspace-based optimization method and its implementation in solving three-dimensional inverse problems,” IEEE Trans. Geosci. Remote Sens., vol. 48, pp. 3763–3768, 2010. [22] T. Habashy, M. Oristaglio, and A. de Hoop, “Simultaneous nonlinear reconstruction of two-dimensional permittivity and conductivity,” Radio Sci., vol. 29, pp. 1101–1118, 1994. [23] S. Caorsi and G. Gragnani, “Inverse-scattering method for dielectric objects based on the reconstruction of the nonmeasurable equivalent current density,” Radio Sci., vol. 34, pp. 1–8, 1999. [24] X. Ye and X. Chen, “The role of regularization parameter of subspace-based optimization method in solving inverse scattering problems,” in Proc. Asia-Pacific Microwave Conf., Singapore, Dec. 2009, pp. 1549–1552. [25] Y. Zhong and X. Chen, “Twofold subspace-based optimization method for solving inverse scattering problems,” Inverse Probl., vol. 25, p. 085003, 2009. [26] P. C. Hansen, M. E. Kilmer, and R. H. Kjeldsen, “Exploiting residual information in the parameter choice for discrete ill-posed problems,” Numer. Algorithms, vol. 46, pp. 41–59, 2006. [27] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New York: IEEE Press, 1998. [28] Y. Zhong, “Subspace-Based inversion methods for solving electromagnetic inverse scattering problems,” Ph.D. dissertation, National University of Singapore, Singapore, 2010. [29] The Mathworks Website Jan. 2010 [Online]. Available: http://www. mathworks.com/access/helpdesk/help/techdoc/ref/fftshift.html

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Yu Zhong was born in 1980 in Guangdong, China. He received the B.S. and M.S. degrees in electronic engineering from Zhejiang University, Hangzhou, China, in 2003 and 2006, respectively, and the Ph.D. degree from the National University of Singapore, in 2010. He is currently a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore. His research interests mainly are inverse-scattering problems and optimization algorithms.

Xudong Chen was born in 1976 in Liaoning, China. He received the B.S. and M.S. degrees in electrical engineering from Zhejiang University, Hangzhou, China, in 1999 and 2001, respectively, and the Ph.D. degree from the Massachusetts Institute of Technology, Cambridge, in 2005. Since then he joined the Department of Electrical and Computer Engineering, National University of Singapore, and he is currently an Associate Professor. His research interests include mainly electromagnetic inverse problems. He has published more than 50 journal papers on inverse-scattering problems, material parameter retrieval, and optimization algorithms. He visited the University of Paris-SUD 11 in 2010 as an invited Visiting Associate Professor. Prof. Chen was the recipient of the Young Scientist Award by the Union Radio-Scientifique Internationale (URSI) in 2010.

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Near-Field Microwave Imaging Based on Aperture Raster Scanning With TEM Horn Antennas Reza K. Amineh, Maryam Ravan, Member, IEEE, Aastha Trehan, and Natalia K. Nikolova, Fellow, IEEE

Abstract—The design, fabrication, and characterization of an ultrawideband (UWB) antenna for near-field microwave imaging of dielectric objects are presented together with the imaging setup. The focus is on an application in microwave breast tumor detection. The new antenna operates as a sensor with the following properties: 1) direct contact with the imaged body; 2) more than 90% of the microwave power is coupled directly into the tissue; 3) UWB performance; 4) excellent de-coupling from the outside environment; 5) small size; and 6) simple fabrication. The antenna characterization includes return loss, total efficiency, near-field directivity, fidelity, and group velocity. The near-field imaging setup employs planar aperture raster scanning. It consists of two antennas aligned along each other’s boresight and moving together to scan two parallel apertures. The imaged object lies between the two apertures. With a blind de-convolution algorithm, the images are de-blurred. Simulation and experimental results confirm the satisfactory performance of the antenna as an UWB sensor for near-field imaging. Index Terms—Aperture scanning, microwave imaging, ultrawideband (UWB) antenna.

I. INTRODUCTION HE contrast between the electrical properties of the components of a dielectric body enables its imaging using microwave measurements. In a typical active microwave imaging scenario, power is radiated via an antenna and the scattered power is received by one or more antennas. The scattered signals are then analyzed to detect and evaluate possible scatterers (targets). The method has been applied to biomedical imaging [1]–[7], non-destructive testing and evaluation of materials [8], through-wall imaging [9], concealed weapon detection [10], etc. The design and fabrication of high-performance antennas present significant challenges in the implementation of all categories of microwave imaging. Typical design requirements that have been considered in the literature are wide impedance bandwidth, high directivity, as well as small size. Various types of antennas have been proposed for near-field microwave imaging and in particular for tissue sensing applications. Examples include the planar monopole [11], the slot antenna [12], the Fourtear antenna [13], a microstrip patch antenna [14], the

T

Manuscript received October 23, 2009; revised March 13, 2010; accepted July 29, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Department of Electrical and Computer Engineering, McMaster University, Hamilton ON L8S 4K1, Canada (e-mail: khalajr@ mcmaster.ca; [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.2010.2103009

planar “dark-eyes” antenna [15], a ridged pyramidal horn [16], and a cross-Vivaldi antenna [17]. Following the guidelines presented in [18], an improved ultrawideband (UWB) TEM horn antenna is designed and tested in planar scanning scenarios. The major features, which distinguish this antenna from the previously proposed antennas for UWB imaging, are: (1) operating in direct contact with the imaged body without the need for coupling liquids; (2) blocking the electromagnetic interference from the surrounding medium; and (3) directing most of the radiated power toward the tissue via the front aperture. The amount of power coupled to the tissue through the front aperture is quantified by a parameter called near-field directivity (NFD). The antenna features ultrawide impedance bandwidth and flat group velocity. It is designed to operate in the UWB frequency range allowed by the Federal Communication Commission (FCC) for indoor applications (3.1 GHz to 10.6 GHz). Compared to the antenna presented in [18], the new design features three times smaller physical size of the aperture which comes into contact with the imaged objects. The reduced size enables scanning applications where sampling can be as frequent as a couple of mm. This leads to better spatial resolution in the reconstructed images. In addition, dielectric material with lower loss is used in the new design resulting in improved efficiency. Finally, the antenna components are made of solid dielectric material instead of dielectric cement which greatly simplifies the fabrication. We investigate the following parameters of the antenna in the UWB range through simulation and measurement: impedance match, NFD, group velocity, fidelity, and efficiency. Novel techniques are presented for estimating the efficiency and the group velocity from measurements suitable for the characterization of antennas operating in direct contact with tissue. The antenna shows satisfactory performance as a sensor for UWB imaging based on both frequency-sweep and pulsed-radar measurements. The application of the antenna in aperture raster scanning is also demonstrated. Microwave imaging of biological tissues through aperture raster scanning was first investigated in [19], [20] where the transmission -parameters were measured in a limited frequency range around 3 GHz. The antenna and the tissue were immersed in water. The advantage of this technique is its simplicity and ability to generate the images in real-time. The disadvantages of the raster scanning methods presented in [20] stem from: 1) limited frequency range, 2) coupling liquid (water) which introduces additional loss and would complicate the maintenance and sanitation of a clinical setup, and 3) limited resolution due to the large antenna apertures. Most of these

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Fig. 1. (a) The TEM horn with the coaxial feed and the balun. (b) Top view of the TEM horn. (c) Bottom view of the TEM horn. (d) Cross-sectional view of the three antenna pieces machined out of ECCOSTOCK HiK. (e) After mounting the copper plates of the TEM horn on the middle dielectric piece, all three dielectric parts are glued together.

disadvantages are dictated by the limitations of the antennas in the acquisition setup. Subsequently, collecting data on planar or cylindrical aperture surfaces was widely considered in the literature for breast cancer detection. Unlike the originally proposed raster scanning [20], a variety of advanced post-processing techniques have been employed to reconstruct the images. Such techniques can be classified into two major categories: optimization-based techniques using frequency-domain data (e.g., [1]), and radar-based techniques using time-domain data (e.g., [21]). The optimization-based techniques aim at complete reconstruction of the distribution of the dielectric properties in the inspected region. They are expensive in terms of memory and time [22], often ill-posed and sensitive to noise [23]. In contrast, the radar-based techniques aim to detect strong scatterers inside the inspected region [24]. They are fast but very sensitive to clutter (especially reflections from skin) [4] and the heterogeneity of the tissue [25]. Also, the proposed imaging setups in both categories require immersing the antennas and the breast tissue in a coupling liquid (similar to the aperture raster scanning in [20]). All of the above drawbacks have hindered the progress toward successful clinical trials. Here, for the first time we consider the direct raster scanning technique when measuring the transmission -parameters in the UWB. We also study its feasibility in breast-cancer detection. The properties of our novel sensors enable this methodology with the following advantages. (1) The need for coupling liquids is eliminated due to the low-loss dielectric material incorporated into the antenna structure. This significantly simplifies the maintenance and sanitation of the envisioned clinical setup. Also, this eliminates the additional power loss due to the coupling liquids. (2) The small sensor aperture leads to enhanced spatial resolution in the images. (3) The UWB properties of the antenna allow for aperture raster scanning in the UWB fre-

quency range. (4) Simple and fast post-processing algorithms facilitate real-time, reliable, and robust imaging of the breast. This is achieved by an efficient blind de-convolution algorithm [26] for image de-blurring. The study of the UWB aperture raster scanning technique presented here includes simulated and measured results for threedimensional (3D) homogeneous phantoms as well as heterogeneous breast model obtained from magnetic resonance imaging (MRI). II. ANTENNA DESIGN AND FABRICATION Following the guidelines presented in [18], the antenna is a TEM horn which is placed in a partially shielded dielectric medium. Since the focus here is on near-field imaging for breast cancer diagnostics, we design the antenna to operate best when attached to a skin layer backed by a tissue layer. This influences the selection of the permittivity of the dielectric material used in the antenna structure and ultimately the antenna dimensions. Fig. 1(a) shows the two plates of the TEM horn which is fed by a coaxial cable via a balun ensuring matched transition to the parallel plates of the horn. Fig. 1(b) and (c) show the top and bottom views of the TEM horn and the balun. The TEM horn is embedded in a solid dielectric medium with permittivity of 10 so that it matches closely the weighted average permittivity of the breast tissue of a radiologically dense breast [27] in the UWB frequency range. This ensures maximum coupling of the microwave power into the imaged body. Also, using such dielectric material eliminates the need for coupling liquids. The material of the dielectric medium is ECCOSTOCK HiK from and Emerson & Cuming Microwave Products [28] with in the frequency range from 1 GHz to 10 GHz. This material has less loss compared to the dielectric cement used in [18]. Also, this solid material simplifies the fabrication process significantly. It can be readily machined to make three

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Fig. 3. Sliced view of one-half of the simulation set-up.

Fig. 2. (a) The TEM horn placed in a dielectric medium with relative permittivity of 10 with copper sheets on all outer surfaces except the front aperture and a microwave absorbing sheet on the top surface. (b) The copper sheet pattern on the top surface. (c) The copper sheet pattern on the bottom surface.

dielectric pieces (cross-sectional view shown in Fig. 1(d)) using carbide tools or by grinding. After placing the copper sheets of the TEM horn (Fig. 1(a)–(c)) on the middle piece, all three pieces are attached together to form the final shape in Fig. 1(e). In order to de-couple the antenna from the outside environment (except the front side), the antenna is shielded with patterned copper sheets as shown in Fig. 2. The shapes of the apertures on the top copper sheet have a crucial role in matching the impedance of the antenna to that of the coaxial cable. The top plate of the TEM horn is connected to the top copper sheet with a thin wire to suppress propagating modes between these two plates. The thin wire is along the same axis as the inner wire of the coaxial feed; see Fig. 3. Then, a microwave ab, , , and sorbing sheet with (ECCOSORB FGM-40, Emerson & Cuming Microwave Products [28]) is glued on the top surface of the antenna (on top of the copper sheet) to prevent the coupling of the antenna to the outside environment.

III. ANTENNA PERFORMANCE Fig. 3 shows the simulation set-up where the antenna is assumed to operate in air while its front aperture is attached to a two-layer medium with a skin layer and a tissue layer with

Fig. 4. Measured constitutive parameters of the tissue and skin layers made of glycerin phantoms: (a) dielectric constant and (b) effective conductivity.

the properties shown in Fig. 4 (same as in [18]). The design parameters have been optimized so that the antenna is matched to the 50- coaxial cable in the whole UWB frequency range. We show the performance of the antenna in the frequency domain as well as the time domain. The frequency-domain simulations are performed with HFSS ver. 11 [29] and the time-domain simulations are performed with CST Microwave Studio [30]. The measurements have been performed on the artificial phantom emulating the human breast. Table I shows the values for the design parameters of Figs. 1 and

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TABLE I DESIGN PARAMETERS OF THE ANTENNA

Fig. 6. Simulated NFD factor of the antenna.

Fig. 7. Attaching two identical antennas front-to-front to measure efficiency and group velocity.

2. The parameters , , and , which show the overall size of the antenna, are 74 mm, 19 mm, and 30 mm, respectively.

via the front aperture of the antenna. To compute this factor, the normal component of the real part of the Poynting vector is integrated on the surfaces of a cuboid enclosing the antenna from all sides with its front surface overlapping the front aperture of the antenna as shown in Fig. 3. As we show later, the radiated power is confined to the antenna aperture. The size of this cubiod does not affect the computed NFD factor. The total radiated power is obtained from the sum of all these integrations. Then, the power integral over the front surface is divided by the total radiated power to give the NFD factor. Fig. 6 shows the computed NFD factor versus frequency. The average NFD factor over the whole band is 93%. It shows that the antenna is de-coupled electromagnetically from the outside environment while coupling most of the radiated power into the imaged tissue.

A. Reflection Coefficient

C. Efficiency

The antenna geometry is optimized first in HFSS [29] to obtain a reflection coefficient below in the whole UWB. Then, the final design is also simulated in CST Microwave Studio for validation as well as computing the time-domain parameters. The reflection coefficient of the antenna is measured when the antenna is attached to a phantom with two layers: a thin skin layer and a 5 cm thick tissue layer. These layers are made of glycerin solutions with the properties shown in Fig. 4 to emulate the breast tissue. Fig. 5 shows the comparison between the simulated and measured results for the reflection coefficient of the antenna. The impedance match is good in the UWB frequency range.

We investigate the efficiency of the antenna via simulation and measurement. In order to compute the efficiency via simulation, we divide the total radiated power (computed as described in the previous section over the cuboid surface shown in Fig. 3) by the power fed to the antenna. The power fed to the antenna is computed as the power incident on the coaxial port minus the reflected power at that port. In order to have an experimental estimation of the antenna efficiency, two identical antennas are attached together front-tofront as shown in Fig. 7. The received-to-transmitted power ratio is

Fig. 5. Reflection coefficient of the antenna obtained from HFSS and CST simulations and measurement.

(1) B. Near-Field Directivity Fig. 6 shows the computed NFD factor [18] for the antenna. This factor shows the proportion of the power that is radiated

where and are the return losses for the two antennas, and are their efficiencies, and and are the NFD factors as defined in Section III.B. In a measurement with a vector

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Fig. 8. Simulated and measured efficiencies of the antenna.

Fig. 9. Group velocity obtained from measurement compared to the wave ve0:9 10 m=s. locity in a medium with permittivity of 10, v



2

network analyzer (VNA), the two antennas’ coaxial feeds rep, , and resent the two ports where . Thus, the efficiency of one antenna is estimated as

(2)

In the above formula, while the -parameters are measured, the NFD factor is simulated and the values shown in Fig. 6 are used. This equation gives only an estimate of the antenna efficiency from the measured -parameters because: (i) the NFD factor in the case when the antenna comes in contact with tissue and in the case when it comes in contact with another antenna may may also differ differ slightly, and (ii) the efficiency factor slightly in these two cases. Fig. 8 compares the antenna efficiency obtained from simulation and measurement over the UWB frequency range. The average of the efficiency over the whole band in the case of the simulation and the measurement is 40% and 32%, respectively. Considering the efficiency together with the NFD factor, this indicates that the overall coupling efficiency of the proposed antenna to the imaged medium via the front aperture is much better than that of existing low-directivity UWB antennas whose average radiation efficiencies are of comparable values; for example, see [15]. We emphasize that such low-directivity UWB antennas direct only a fraction of the radiated power toward the imaged region.

Fig. 10. Simulated fidelity versus the distance from the center of the antenna aperture and along the aperture normal.

, , and the two antennas are assumed identical, we define as , , and . We introduce Thus, (3) becomes as the angle of , . Then the group velocity is obtained as (4) Fig. 9 shows the group velocity computed from measured results. It is compared with the velocity of the wave in the antenna dielectric medium with permittivity of 10.

D. Group Velocity

E. Fidelity

We estimate the group velocity of the antenna via measurements. The variation of the group velocity with frequency is a measure of the distortion of the transmitted pulse due to the anin the same measurement set-up as shown in tenna. We use as Fig. 7. We express

In order to investigate the distortion of the pulses by the proposed antenna into homogenized breast tissue, the antenna fidelity is computed. The fidelity is the maximum magnitude of the cross-correlation between the normalized observed response and an ideal response derived from the excitation waveform at the antenna terminals [31],

(3) (5) where and are the assumed voltages at the two-antenna terminals, and are the phase constants associated with the and are the respective attensignal delay in each antenna, and are the lengths of the antennas uation constants, and from the coaxial feed point to the center of the aperture. Since

where is the observed -field normalized to unit energy, and is the input signal at the antenna terminals also normalized to unit energy. Fig. 10 shows the computed fidelity of the antenna. It is seen that the fidelity of the signal is maximum

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Fig. 11. The proposed planar microwave imaging set-up to detect tumor simulants inside a breast phantom including the compressed phantom and two UWB antennas: (a) top view, (b) side view. Large arrows show the scanning axes.

close to the antenna and decreases with distance. The reason for the drop in the fidelity with distance is the tissue dispersion and dissipation. IV. IMAGING SET-UP AND RESULTS We test the antenna performance as a microwave sensor in an aperture scanning set-up to detect embedded targets inside a lossy dielectric body. We examine various scenarios from high-contrast targets in homogeneous phantoms to low-contrast tumor simulants in a heterogeneous breast model. The former case is investigated as a general imaging scenario for hidden targets in dielectric bodies although there are many examples in the literature for breast imaging where normal tissue is assumed to be homogeneous and the tumors have large contrasts (e.g., see [1], [3]–[7], [13], [17]). The latter case gives a realistic insight into the efficiency of the scanning technique as it considers both large and small contrasts. As shown in Fig. 11, a simulant of the breast tissue layer with the properties shown in Fig. 4 is compressed between two very thin parallel plates. Then, two antennas, one transmitting and the other receiving, perform a 2-D scan by moving together along the opposite sides of the compressed breast. A slight compression of the breast between two rigid parallel plates would prevent undesired movement during the microwave measurements. It also shapes the acquisition surface as a flat plane which simplifies the signal processing techniques. We note that tissue compression is applied in X-ray mammography (severe compression) and MRI (slight compression). Compression is employed in X-ray mammography mostly to increase sensitivity while in MRI it is used to immobilize the tissue during data acquisition. Movement may cause blurring and measurement errors. The imaging setup has the following advantages for breast imaging. (1) The imaging set-up does not require coupling liquid. Thus, the imaging system would be convenient for clinical use, i.e., it would be easy to sanitize after a patient is examined. (2) The transmitter (or receiver) antenna couples (or

measures) practically all microwave power directly in (or from) the tissue through its front aperture due to an excellent NFD. This eliminates interference from the outside environment. (3) Coupling between the antennas is optimal in such a scanning sheme. (4) The image generation is simple. The transmission -parameter between the two antennas when no scatterer , is present (background medium only) at the position , is acquired. To obtain the calibrated transmission -parameter, for any target, is subtracted , as from measured transmission -parameter (6) The images are plots of . In this example, is the same for all positions, because the background medium is homogeneous. However, the same calibration can be applied with heterogeneous would depend on the position along background where the and axes. Also, here the separation distance between the two antennas is set at 3 cm. At such distance, the coupling between the two antennas is sufficiently strong to obtain reasonable signal-to-noise ratio (SNR) with conventional test instruments, e.g., vector network analyzers. In a practical breast-scanning scenario, separation distances anywhere between 8 and 12 cm are necessary in order to avoid patient discomfort. Such a system would require low-noise amplifiers and controlled environment in order to achieve good dynamic range especially at the high-frequency end. We investigate the capability of this set-up to produce images through aperture scanning with the proposed antennas. This is first done through HFSS simulations. A flat tissue layer with the properties shown in Fig. 4 and dimensions of 140 mm 140 mm 30 mm is covered by two layers of skin on the top and the bottom sides and is scanned by two antennas. The thin support plates that are supposed to compress slightly the breast are ignored here since their effect will be eliminated through the calibration process. We embed 5 tumor simulants in the breast tissue layer with various shapes as shown in Fig. 11. Table II

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Fig. 12. The simulated jS j images from 2-D scanning at: (a) 5 GHz, (b) 7 GHz, (c) 9 GHz. The de-blurred images after applying blind de-convolution at: (d) 5 GHz, (e) 7 GHz, (f) 9 GHz.

shows their position and dielectric properties. The latter are assumed frequency-independent in this example. This allows us to monitor the tumor responses with different contrasts in elecpatrical properties with respect to the background. The rameter is acquired on an area of 100 mm 100 mm where the sampling rate is 5 mm. Fig. 12(a)–(c) show the images obtained at 5 GHz, 7 GHz, and 9 GHz. These images from show that the tumor simulants can be easily detected especially at 5 and 7 GHz. It is observed that while the tumor simulants have been detected, their shape can not be determined. This is expected because the tumor simulants are electrically too small. However, larger tumors appear as larger bright spots in the images even if their permittivity contrast with respect to the background is lower (compare tumors 4 and 5 with tumor 1). Another interesting observation is that at 7 GHz the image of tumor 4, which has only permittivity contrast but no conductivity contrast with the background, appears larger than the image of tumor 2, which has the same size and shape but has contrast in both permittivity and conductivity. Here, it is worth noting that images can be generated using only magnitude information according to

(7) However, instead of sharp spots, the tumor simulants appear as doughnut shapes. Ignoring the phase information has a negative effect on the image quality. In the next section, we improve the quality of the images using a blind de-convolution technique.

TABLE II SHAPE, SIZE, ELECTRICAL PROPERTIES AND POSITION OF THE TUMOR SIMULANTS IN FIG. 11

V. APPLYING BLIND DE-CONVOLUTION TO DE-BLUR THE IMAGES discussed in the Although the images formed by previous section clearly identify the presence of scatterers in the tissue, they suffer from significant blurring. The most important factor that causes the blurring is the non-point-wise transmitter and receiver apertures. In fact, the antenna aperture acts like an averaging window, which gives approximately the integration of the power over the antenna aperture in each central position. This integration effect is actually a convolution (filtering) in the spatial domain. This effect can be reduced with blind de-convolution.

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TABLE III ESTIMATION OF THE POSITION OF THE TUMOR SIMULANTS (IN MM) OBTAINED FROM FIG. 12(D)–(F)

Fig. 14. Measured constitutive parameters for the background medium and the two scatterers: (a) dielectric constant (b) effective conductivity. Fig. 13. Normal component (x-component) of the real part of the Poynting vector on a plane overlapping the front face of the antenna at (a) 5 GHz, (b) 7 GHz, and (c) 9 GHz. The solid lines show the projection of the antenna aperture on this plane.

In blind de-convolution, it is assumed that the original unhas been convolved with a point spread known image , to create a blurred image function (PSF), (8) where denotes two-dimensional convolution. The aim in blind image de-convolution is to estimate the original image when the blurred image is known and the PSF is unknown or partially known [26]. from Here, we have the blurred image the measurement. We assume that the integrating effect occurs non-evenly over the antenna aperture. Thus, we use the simulated distribution of the normal component ( -component) of

the real part of the Poynting vector at the antenna aperture as the weighting coefficients for the integration effect at each frequency (Fig. 13 shows such distributions). This becomes the initial guess for the PSF. It needs to be further corrected since: 1) the distribution of the -component of the Poynting vector over the antenna aperture depends on the tissue properties in front of the antenna; 2) the integration of the power entering the antenna via the front aperture will not be exactly equal to the power that reaches the antenna port; and 3) in practice, the would be slightly different from the one distribution of obtained via simulation due to inevitable differences between the simulated antenna model and the fabricated antenna. Here, we use accelerated damped Lucy Richardson algorithm [33] in the MATLAB image processing toolbox [32]. The algorithm reand the point-spread function stores the original image (PSF) simultaneously in each iteration in order to maximize the likelihood between the result of the original image being restored, when convolved with the current PSF, and the blurred image. Fig. 12(d)–(f) show the de-blurred images after applying

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Fig. 15. The measured jS j images from 2-D scanning at: (a) 5 GHz, (b) 7 GHz, (c) 9 GHz. The de-blurred images after applying blind de-convolution to the images at: (d) 5 GHz, (e) 7 GHz, (f) 9 GHz.

blind de-convolution. It is clear that in these images tumors appear with much better contrast compared to the raw images of Fig. 12(a)–(c). Also, the spatial spans of the tumors have been restored in the de-blurred images especially at 9 GHz. Table III shows the estimated position of the tumor stimulants at the position of the extracted from the peaks of tumors in Fig. 12(d)–(f). The positions have been estimated with good accuracy especially at 5 GHz and 7 GHz. VI. MEASUREMENTS OF A HOMOGENEOUS PHANTOM A homogeneous flat phantom is prepared out of glycerin with a thickness of 3 cm and two tumor simulants are placed inside. Fig. 14 shows the dispersive constitutive parameters of the two embedded scatterers, denoted as Sc1 and Sc2, as well as those of the homogeneous background phantom. Sc1 is made of alginate powder while Sc2 is made out of glycerin. The background medium is also glycerin-based but the recipe is different from that for Sc2. Table IV shows the positions, sizes, and the type of materials for Sc1 and Sc2. Note that after placing the tumor simulants in the breast-tissue phantom, some diffusion takes place and the sizes in Table IV are only approximate. Two identical antennas with the parameters shown in the previous sections are used to perform a 2D scan of the compressed phantom between two thin glass sheets. With spatial sampling rates of 5 mm in both and directions, the -parameters at

TABLE IV POSITION (IN THE y -z PLANE), SIZE, AND MATERIAL TYPE FOR THE TWO SCATTERERS EMBEDDED IN THE EXPERIMENTAL HOMOGENEOUS PHANTOM

the two antenna terminals are measured utilizing an Advantest R3770 VNA. Fig. 15(a)–(c) show the images obtained after calibration (as described by (6)) at 5 GHz, 7 GHz, and 9 GHz. Sc1, which has larger contrast in the dielectric properties than Sc2, produces a brighter larger spot in the images. We also note that diffusion of the alginate powder into the homogeneous phantom makes the effective size of Sc1 larger. After insertion into the background medium, the constitutive parameters of the center of Sc1 are as those reported in Fig. 14. However, they gradually transition into the properties of the background medium away from the center due to diffusion. In order to perform blind de-convolution on the raw images, we first simulate one antenna with the thin glass sheet and the homogeneous phantom to get the initial guess for the PSF at each frequency as described in Section V. Then, we apply the blind de-convolution algorithm to improve the image quality as shown in Fig. 15(d)–(f).

AMINEH et al.: NEAR-FIELD MICROWAVE IMAGING BASED ON APERTURE RASTER SCANNING WITH TEM HORN ANTENNAS

TABLE V ESTIMATION OF THE POSITION OF THE TUMOR SIMULANTS (IN MM) OBTAINED FROM FIG. 15(D)–(F)

Fig. 16. (a) Scanning compressed heterogeneous breast model by two antennas (b) Three tumor simulants embedded in the model at the positions ( 23, 40) mm, (0, 20) mm, and (30, 40) mm in the y -z plane in the trans-fat, fibroglandular, and fat tissues, respectively.

0

0

0 0

Table V shows the estimated position of the tumor stimulants at the position of the extracted from the peaks of tumors in Fig. 15(d)–(f). The positions have been estimated with good accuracy especially at 5 GHz and 7 GHz. VII. SIMULATION RESULTS FOR HETEROGENEOUS BREAST MODEL So far we have investigated the capability of the proposed scanning set-up in detecting targets embedded in homogeneous background media. However, in applications such as breast tumor detection, the host medium is not homogeneous. Here, we examine the raster scanning with the proposed antenna for a heterogeneous breast model. This model is obtained by converting MRI intensity images into several tissues including: fat, trans-fat, fibro-glandular, and muscle. The skin layer is then created artificially in the simulation model to envelop tightly the MRI-based model. Fig. 16(a) shows the simulation set-up containing two identical antennas and the heterogeneous flattened breast model with a thickness of 3 cm. We then insert three tumor simulants with the shapes shown in the cross-sectional view of Fig. 16(b) in various tissues. The size of all tumor simulants is approximately 1 cm. The tumor in fat ) mm in the - plane. The tumor in is at the position (30,

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, ) mm in the - plane. The trans-fat is at the position ( ) mm in tumor in fibro-glandular tissue is at the position (0, the - plane. Fig. 17 shows the electrical properties assigned to the tissues which are extracted from [27] and [34]. Also, the properties shown in Fig. 4 are assigned to the skin layer. The tumor in the fibro-glandular tissue has the lowest contrast (about 50:43 in permittivity and 4.7:4 in conductivity at 5 GHz and similar contrasts at other frequencies) while the tumor in the fat has the largest contrast (about 50:5 in permittivity and 4.7:0.45 in conductivity at 5 GHz and similar contrasts at other frequencies). As described in (6), the transmission -parameter between the two antennas in the case of the heterogeneous background without tumor simulants is subtracted from the meawhen the tumor sured transmission -parameter stimulants are present, in order to obtain the calibrated transat each scanning position . mission -parameter Then, the raw images shown in Fig. 18(a)–(c) are obtained by . The images show that all three tumor simplotting ulants can be detected at 5 GHz. At 7 GHz two of the tumors are clearly visible (that in the fat and the one in the trans-fat). However, at 9 GHz only the tumor embedded in the fat tissue can be detected. This is the case of the highest contrast. The reduced detection capability at 9 GHz is due to the degradation of the numerical SNR at high frequencies. Here, we define the numerical SNR as the ratio of the signal strength at the position of the tumor and the -parameters mesh convergence error of the simulations. In all simulations, the mesh convergence error is set at 0.001. Table VI shows the computed values for the so defined SNR for all three tumor simulants at 5 GHz, 7 GHz, and 9 GHz. It is observed that the SNR decreases drastically with increasing frequency. In particular, at 9 GHz, the SNR value is larger than 1 only for the tumor embedded in the fat. In measurements, the system SNR can be increased using low-noise amplifiers at the receiver. The blind de-convolution algorithm is applied to the raw images in this example as well. Fig. 18(d)–(f) show the reconstructed images in which the tumors appear with better contrast. Table VII shows the estimated position of the tumor stimuat the position of lants extracted from the peaks of the tumors in Fig. 18(d)–(f). The tumor positions have been estimated with good accuracy especially at 5 GHz and 7 GHz but at 9 GHz only the one in fat is detected as described earlier. VIII. CONCLUSION AND DISCUSSION We have presented an improved UWB antenna which is designed to work in direct contact with the imaged body. The investigated parameters of the antenna show that it is a good candidate for near-field microwave imaging applications. An aperture raster scanning setup was used to test the antenna in both measurements and simulations. The scanning setup clearly benefits from the excellent near-field directivity of the proposed antenna. Strong scatterers inside a homogeneous dielectric body were detected both in simulations and measurements. The detection of weak and strong scatterers in a heterogeneous breast model was also examined and the results are encouraging. A blind de-convolution technique was applied to improve the image quality.

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Fig. 17. Permittivity and conductivity of the heterogeneous breast model.

Fig. 18. The simulated jS j images from 2-D scanning of the heterogeneous breast model at: (a) 5 GHz, (b) 7 GHz, (c) 9 GHz. The de-blurred images after applying blind de-convolution at: (d) 5 GHz, (e) 7 GHz, (f) 9 GHz.

TABLE VI DEGRADATION OF SNR VERSUS FREQUENCY FOR THE THREE EMBEDDED TUMOR SIMULANTS IN FIG. 16

TABLE VII ESTIMATION OF THE POSITION OF THE TUMOR SIMULANTS (IN MM) OBTAINED FROM FIG. 18(D)–(F)

Overall, the simulation and experimental results confirm the capability of the planar raster scanning setup with the proposed antennas to provide high-quality images through near-field microwave measurements. The advantages of the proposed system include: (1) an UWB antenna with reduced aperture size which enables aperture raster

scanning in the UWB, (2) elimination of the liquid coupling medium, which simplifies the maintenance and sanitation and reduces the power loss, (3) fast blind de-convolution algorithm to de-blur the images.

AMINEH et al.: NEAR-FIELD MICROWAVE IMAGING BASED ON APERTURE RASTER SCANNING WITH TEM HORN ANTENNAS

It is worth noting that in the examples considered here, the separation distance between the two antenna apertures (tissue thickness) was 3 cm. With this tissue thickness, simulation time at each scanning step, using a Pentium 4 computer with 16 GB of RAM, exceeds 2 hours. For a 2-D scan with more than 200 samples, this leads to enormous simulation time. Also, in the example of measuring the homogeneous phantom, the tissue thickness of 3 cm has been chosen to ensure that the response due to the tumor is above the noise floor of the measurement. However, in reality, tissue thicknesses will be closer to 10 cm. Thus, the expected signal levels will be much lower and the acquisition system must have larger dynamic range with sufficient SNR. We envision using low-noise amplifiers to boost the SNR at the receiving side in conjunction with proper suppression of all electromagnetic interferences and clutter. Although, the de-blurring technique helps to improve spatial contrast and resolution, without more sophisticated processing techniques, estimation of the tumor shapes and sizes is not reliable. It is possible to further improve the quality of the acquired images while retaining real-time performance by making use of: chirp-pulse signals similarly to chirp-pulse computed tomography [35], holography-based shape identification [36], [37], as well as sensitivity-based detection [38]. Another important issue is the calibration of the measurement system. The calibration involves a measurement scan of a predetermined tissue phantom, also referred to as background medium. The choice of the background medium is expected to affect strongly the images. Several types of background mediums will be studied including: (i) high-water content homogeneous phantoms (dielectric properties close to those of malignant tissue); (ii) medium-water-content homogeneous phantoms (dielectric properties close to those of fibro-glandular tissue); (iii) low-water-content homogeneous phantoms (dielectric properties close to those of fat tissue); and (iv) patient-specific heterogeneous background (e.g., left breast serves as background for imaging of the right breast of the same patient). Since each of the above background mediums can offer certain advantages, it is expected that a successful imaging algorithm will make intelligent use of all of the above calibration measurements. ACKNOWLEDGMENT The authors would like to thank A. Johnson, undergraduate student, for his help in providing the experimental setup. Also, the authors are indebted to R. K. Zimmerman, Jr., a research engineer, for discussions about the microwave measurements. REFERENCES [1] T. Rubæk, P. M. Meaney, P. Meincke, and K. D. Paulsen, “Nonlinear microwave imaging for breast-cancer screening using Gauss–Newton’s method and the CGLS inversion algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2320–2331, Aug. 2007. [2] D. A. Woten, J. Lusth, and M. El-Shenawee, “Interpreting artificial neural networks for microwave detection of breast cancer,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 12, pp. 825–827, Dec. 2007. [3] S. K. Davis, B. D. Van Veen, S. C. Hagness, and F. Kelcz, “Breast tumor characterization based on ultrawideband microwave backscatter,” IEEE Trans. Biomed. Eng., vol. 55, pp. 237–246, Jan. 2008. [4] D. J. Kurrant and E. C. Fear, “An improved technique to predict the time-of-arrival of a tumor response in radar-based breast imaging,” IEEE Trans. Biomed. Eng., vol. 56, no. 4, pp. 1200–1208, Apr. 2009.

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[5] A. Fhager and M. Persson, “Using a priori data to improve the reconstruction of small objects in microwave tomography,” IEEE Trans. Microwave Theory Tech., vol. 55, pp. 2454–2462, Nov. 2007. [6] C. Yu, M. Yuan, J. Stang, E. Bresslour, R. T. George, G. A. Ybarra, W. T. Joines, and Q. H. Liu, “Active microwave imaging II: 3D system prototype and image reconstruction from experimental data,” IEEE Trans. Microwave Theory Tech., vol. 56, pp. 991–1000, Apr. 2008. [7] M. Klemm, I. J. Craddock, J. A. Leendertz, A. Preece, and R. Benjamin, “Radar-based breast cancer detection using a hemispherical antenna array-experimental results,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1692–1704, Jun. 2009. [8] A. Khanfar, M. Abu-Khousa, and N. Qaddoumi, “Microwave near-field nondestructive detection and characterization of disbonds in concrete structures using fuzzy logic techniques,” Compos Struct., vol. 62, no. 3–4, pp. 335–339, Oct. 2003. [9] E. J. Baranoski, “Through-wall imaging: Historical perspective and future directions,” Franklin Inst. J., vol. 345, no. 6, pp. 556–569, Sep. 2008. [10] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 9, pp. 1581–1592, Sep. 2001. [11] H. M. Jafari, M. J. Deen, S. Hranilovic, and N. K. Nikolova, “A study of ultrawideband antennas for near-field imaging,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1184–1188, Apr. 2007. [12] H. M. Jafari, J. M. Deen, S. Hranilovic, and N. K. Nikolova, “Co-polarised and cross-polarised antenna arrays for breast, cancer detection,” IET Microw Antennas Propag, vol. 1, no. 5, pp. 1055–1058, Oct. 2007. [13] D. A. Woten and M. El-Shenawee, “Broadband dual linear polarized antenna for statistical detection of breast cancer,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3576–3580, Nov. 2008. [14] R. Nilavalan, I. J. Craddock, A. Preece, J. Leendertz, and R. Benjamin, “Wideband microstrip patch antenna design for breast cancer tumour detection,” IET Microw Antennas Propag, vol. 1, no. 2, pp. 277–281, Apr. 2007. [15] H. Kanj and M. Popovic, “A novel ultra-compact broadband antenna for microwave breast tumor detection,” PIER, vol. PIER 86, pp. 169–198, 2008. [16] X. Li, S. C. Hagness, M. K. Choi, and D. Van Der Weide, “Numerical and experimental investigation of an ultrawideband ridged pyramidal horn antenna with curved launching plane for pulse radiation,” Antennas Wireless Propag. Lett., vol. 2, pp. 259–262, 2003. [17] J. Zhang, E. C. Fear, and R. Johnston, “Cross-Vivaldi antenna for breast tumor detection,” Microwave Opt. Lett., vol. 51, no. 2, pp. 275–280, Feb. 2009. [18] R. K. Amineh, A. Trehan, and N. K. Nikolova, “TEM horn antenna for ultrawideband microwave breast imaging,” PIER B, vol. 13, pp. 59–74, 2009. [19] L. E. Larsen and J. H. Jacobi, “Microwave interrogation of dielectric targets, Part I: By scattering parameters,” Med. Phys., vol. 5, no. 6, pp. 500–508, Nov. 1978. [20] L. E. Larsen and J. H. Jacobi, Medical Applications of Microwave Imaging. New York: IEEE Press, 1985. [21] E. C. Fear, X. Li, S. C. Hagness, and M. A. Stuchly, “Confocal microwave imaging for breast cancer detection: Localization of tumors in three dimensions,” IEEE Trans. Biomed. Eng., vol. 49, no. 8, pp. 812–822, Aug. 2002. [22] Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3D fields coupled to 2-D parameter estimation,” IEEE Trans. Med. Imag., vol. 23, no. 4, pp. 475–484, Apr. 2004. [23] P. Lobel, L. Blanc-Feraud, C. Pichot, and M. Barlaud, “A new regularization scheme for inverse scattering,” Inverse Probl., vol. 13, pp. 403–410, Apr. 1997. [24] X. Li, E. J. Bond, B. D. Van Veen, and S. C. Hagness, “An overview of ultra-wideband microwave imaging via space-time beamforming for early-stage breast-cancer detection,” IEEE Antennas Propag. Mag., vol. 47, no. 1, pp. 19–34, Feb. 2005. [25] M. O’Halloran, M. Glavin, and E. Jones, “Effects of fibroglandular tissue distribution on data-independent beamforming algorithms,” PIER, vol. PIER 97, pp. 141–158, 2009. [26] D. Kundur and D. Hatzinakos, “Blind image de-convolution,” IEEE Signal Proc. Mag., May 1996. [27] M. Lazebnik, M. Okoniewski, J. H. Booske, and S. C. Hagness, “Highly accurate Debye models for normal and malignant breast tissue dielectric properties at microwave frequencies,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 12, Dec. 2007.

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[28] Emerson and Cuming Microwave Products [Online]. Available: http:// www.eccosorb.com [29] Ansoft Corporation [Online]. Available: http://www.ansoft.com [30] CST Computer Simulation Technology AG [Online]. Available: http:// www.cst.com [31] D. Lamensdorf and L. Susman, “Baseband-pulse-antenna techniques,” IEEE Antennas Propag. Mag., vol. 36, no. 1, pp. 20–30, Feb. 1994. [32] Matlab7.1 ed. Natick, MA, The MathWorks, Inc., 2005. [33] D. S. C. Biggs and M. Andrews, “Acceleration of iterative image restoration algorithms,” Appl. Opt., vol. 36, no. 8, pp. 1766–1775, Mar. 10, 1997. [34] T. Wuren, T. Takai, M. Fuiji, and I. Sakagami, “Effective 2-Debye-pole FDTD model of electromagnetic interaction between whole human body and UWB radiation,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 7, pp. 483–485, Jul. 2007. [35] M. Bertero, M. Miyakawa, P. Boccacci, F. Conte, K. Orikasa, and M. Furutani, “Image restoration in chirp-pulse microwave CT (CP-MCT),” IEEE Trans. Biomed. Eng., vol. 47, no. 5, pp. 690–699, May 2000. [36] M. Ravan, R. K. Amineh, and N. K. Nikolova, “Two-dimensional nearfield microwave holography,” Inverse Probl., vol. 26, p. 055011, 2010. [37] R. K. Amineh, M. Ravan, and N. K. Nikolova, “Three-dimensional microwave holography for near-field imaging of dielectric bodies,” Inverse Probl., to be published. [38] Y. Song and N. K. Nikolova, “Memory efficient method for wideband self-adjoint sensitivity analysis,” IEEE Trans. Microwave Theory Tech., vol. 56, pp. 1917–1927, Aug. 2008.

Reza K. Amineh received the B.Sc. degree from Sharif University of Technology, Tehran, Iran, in 2001 and the M.Sc. degree from Amirkabir University of Technology, Tehran, in 2004, both in electrical engineering, and the Ph.D. degree from McMaster University, Hamilton, ON, Canada, in 2010. From 2004 to 2006, he was a Researcher with the telecommunications industry in Iran. In 2006, he joined the Department of Electrical and Computer Engineering, McMaster University, where from 2006 to 2010, he was a Research Assistant and Teacher Assistant in the Department of Electrical and Computer Engineering. During the winter of 2009 he was a Ph.D. intern with the Advanced Technology Group, Research In Motion (RIM), Waterloo, Canada. His research interests include forward and inverse solutions in electromagnetism with applications in microwave imaging and non-destructive testing, antenna design and measurements, as well as high-frequency computer-aided analysis and design. He has authored and coauthored over 30 journal and conference papers and a book chapter. Dr. Amineh is the recipient of the McMaster Internal Prestige Scholarship “Clifton W. Sherman” for two consecutive years in 2008 and 2009. He is also the recipient of an Ontario Ministry of Research and Innovation (MRI) postdoctoral fellowship for 2010 and 2011. He is the coauthor of an honorable mention paper presented at the IEEE APS/URSI, San Diego, 2008.

Maryam Ravan (M’10) received the Ph.D. degree from Amirkabir University of Technology, Tehran, Iran, in 2007. From February 2007 to May 2007, she was a Research Associate with the Department of Electrical and Computer Engineering, Ryerson University, Canada. From May 2007 to December 2007, she was a Postdoctoral Fellow with the Department of Electrical and Computer Engineering, University of Toronto, Canada. From January 2008 to April 2010, she was a Postdoctoral Fellow and Lecturer with the Department of Electrical and Computer Engineering and the School for Computational Engineering & Science, McMaster University, Hamilton, Canada. Since August 2009 she has been a Postdoctoral Fellow with the Department of Electrical Engineering, University of Toronto, Toronto, ON, Canada. Her research interests include biomedical signal and image processing, neural and wavelet networks, machine learning, optimization techniques, MIMO radar systems and space time adaptive processing, power spectrum estimation, and non-destructive testing. She has authored and coauthored over 30 journal and conference papers and a book chapter.

Aastha Trehan received the B.Eng. and M.A.Sc. degrees (with honors) in electrical engineering from McMaster University, Hamilton, ON, Canada, in 2007 and 2009, respectively. Her research interests included bio-electromagnetism, biological tissues, specific absorption rate, experimenting with phantoms, antennas for biomedical applications, system integration, system fidelity studies, high-frequency simulations, self-adjoint sensitivity analysis (SASA) and magnetic resonance imaging (MRI). Ms. Trehan was on the McMaster Engineering Dean’s Honour list from 2003 to 2006 and was awarded the Natural Sciences and Engineering Research Council Undergraduate Student Research Award (NSERC USRA) for 2005–2006.

Natalia K. Nikolova (S’93–M’97–SM’05–F’11) received the Dipl. Eng. degree from the Technical University of Varna, Bulgaria, in 1989, and the Ph.D. degree from the University of Electro-Communications, Tokyo, Japan, in 1997. From 1998 to 1999, she held a Postdoctoral Fellowship of the Natural Sciences and Engineering Research Council of Canada (NSERC), during which time she was initially with the Microwave and Electromagnetics Laboratory, DalTech, Dalhousie University, Halifax, Canada, and, later, for a year, with the Simulation Optimization Systems Research Laboratory, McMaster University, Hamilton, ON, Canada. In July 1999, she joined the Department of Electrical and Computer Engineering, McMaster University, where she is currently a Professor. Her research interests include theoretical and computational electromagnetism, inverse scattering and microwave imaging, as well as methods for the computer-aided analysis and design of high-frequency structures and antennas. Prof. Nikolova was the recipient of a University Faculty Award of NSERC from 2000 to 2005. Since 2008, she is a Canada Research Chair in High-frequency Electromagnetics. She is a member of the Applied Computational Electromagnetics Society (ACES) and the International Union of Radio Science (URSI).

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New Reflection Suppression Method in Antenna Measurement Systems Based on Diagnostic Techniques Francisco José Cano-Fácila, Sara Burgos, Fernando Martín, and Manuel Sierra-Castañer, Member, IEEE

Abstract—A new method to reduce the unwanted reflection effects in antenna measurements is presented. The proposed method can be applied to measurements performed in semi-anechoic chambers, outdoor systems or even anechoic environments with a poor reflectivity level to remove reflected components and to retrieve the results one would obtain in an ideal anechoic chamber. The method is based on spatial filtering over the plane where the antenna under test (AUT) is placed. To calculate the field in this plane, a diagnostic technique is employed. However, in contrast to classic application (error source identification), the reconstructed field has to be obtained in a zone larger than the antenna dimensions. Thus, it is possible to identify virtual sources out of the antenna aperture, which appear due to the presence of reflections (image theory). Then, by cancelling the virtual and unwanted sources, the related reflected components can be suppressed. Finally, by employing this filtered reconstructed field, a new radiation pattern similar to the one obtained in a fully anechoic environment is calculated. To verify the effectiveness of the method, three examples are presented. Index Terms—Anechoic chambers (electromagnetic), antenna diagnostic techniques, antenna measurements, reflection.

I. INTRODUCTION

NE of the most important stages in the design of an antenna is radiation characterization, because it is necessary to check whether the initial specifications are fulfilled and therefore whether the antenna may be used for the desired application. Normally, the antenna measurements are carried out in anechoic chambers to reduce unwanted contributions, such as reflections or diffractions from the environment, as much as possible. However, there are special cases in which the use of that kind of measurement setup is not possible, e.g., with large antennas, where a semi-anechoic chamber or an outdoor measurement system has to be employed. In these situations, the final results will probably be affected by the presence of undesired

O

Manuscript received April 20, 2010; revised July 23, 2010; accepted August 08, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by FPU Scholarships of the Spanish Education Ministry and Spanish Projects CROCANTE TEC200806736-C03-01/TEC and TERASENSE CSD2008-00068. F. J. Cano-Fácila is with the Technical University of Madrid, ETSI Telecomunicación, Ciudad Universitaria, 28040 Madrid, Spain (e-mail: francisco@gr. ssr.upm.es). S. Burgos and M. Sierra-Castañer are with the Technical University of Madrid, ETSI Telecomunicación, Ciudad Universitaria, 28040 Madrid, Spain. F. Martín is with Indra Sistemas, 28850 Torrejón de Ardoz, Spain. Digital Object Identifier 10.1109/TAP.2010.2103035

components coming mainly from the ground. In fact, reflection waves may also appear in anechoic chambers due to imperfections in the radiation absorbing material (RAM). In any case, when the measurement is not performed in a fully anechoic environment, the unwanted contributions could significantly alter the actual antenna properties, producing a ripple in the radiation pattern. The number of approaches to analyze and cancel the effects of unwanted contributions has increased in recent years. There are methods that are employed not to remove the reflection waves, but rather to characterize the chambers by a figure of merit, usually called the reflectivity level. In the frequency-domain approach [1], [2], the reflectivity is obtained from the interference pattern, measured by moving one of the antennas over the quiet zone. In addition, time-domain characterization techniques have been proposed [3]. In this case, a short pulse is transmitted so that there is no confusion between all of the non-negligible signals whenever two or more of them do not cover the same distance (obtained from the delay). After that, the direction of any particular reflection can be detected by changing the position of the antennas, measuring the new delay value and comparing it with the previous one. Thus, this approach not only gives the reflectivity of an anechoic chamber but also the places with the highest reflectivity. This last piece of information may be used to improve the facility performance and therefore reduce the inaccuracy in the measurements. However, in this case, complicated and expensive equipment is required. Another method that is also based on measurements at several distances was presented in [4]. A compensation of the perturbations is performed by averaging the different patterns that are recorded. However, before doing the averaging, all the patterns have to be referenced to the same reference distance. Other solutions within the category of so-called compensation methods can be applied to avoid measuring the antenna under test (AUT) several times at different distances with respect to the probe. Test zone field compensation [5] is one of these methods, and it is used to reduce the effects of undesired fields created by reflections, employing only the data from one measurement performed over a spherical surface. Theoretically, such data can be separated into two parts, the first one associated with the desired components and the other one related to extraneous contributions existing in the measurement environment. The latter component cannot be determined directly from the known data but can be compensated for by applying an iterative process based on subtracting a term (calculated from the spherical mode expansion) from the estimated pattern. Then, a

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new estimated pattern, which will be employed in the next iteration, is obtained using probe compensation. Its principal drawback is the computational complexity, which can be reduced as shown in [6], in which the unwanted reflected components are associated with plane waves with which the error patterns are estimated. Time-gating techniques can also be employed as an option to remove reflected contributions. The main idea is to separate the direct signal from the reflections using a time-domain representation. This information is not used to estimate the reflectivity of an anechoic chamber as in the time-domain characterization techniques explained above, but rather to gate the direct contribution and cancel the echo effects. In the initial approaches [7], [8], the measurement is directly performed in the time-domain by utilizing pulse generators and sampling oscilloscopes. After that, a time filtering and a time-to-frequency transformation are required to infer a reliable frequency response. However, the complexity of the pieces of equipment is lower when using frequency-domain measurements as proposed in [9], [10]. Using these measurements, the time response is immediately calculated by applying an inverse Fourier transform (IFT). Once reflections have been gated out, a Fourier transform (FT) is applied to return to the frequency-domain. The fast algorithms used to compute the FT and IFT require a constant frequency step. Thus, when there are frequency regions where the field varies quickly, a small step has to be employed over the entire bandwidth to obtain a good characterization in those regions; therefore, the measured time may increase considerably. One technique based on digital signal processing was proposed in [11] to obtain the time response from frequency-domain measurements as well but does not necessarily employ uniform sampling. This technique allows an optimum characterization of the frequency range with a minimum number of samples. Moreover, because the transfer function is modeled by a finite impulse response (FIR) filter, the time response will be causal, avoiding the negative time part, which may appear when an IFT is performed and which may hide some echoes. Other possible approaches include frequency decomposition techniques, where the signal measured in the frequency-domain is divided into a set of propagating components, including the direct and all the reflected contributions. Next, taking only the desired component, a result quite similar to the one obtained in a fully anechoic chamber may be achieved. Initially, a decomposition into a sum of complex exponentials, each representing a different component, was proposed. Polynomial methods were the first methods used to carry out such a decomposition. Among them, Prony’s algorithm was the most popular [12], [13]. However, due to the fact that accurate results were only achieved with noiseless data, other solutions were studied. One variation of the polynomial approach is the pencil-of-function method [14], which enables the reduction of the above-mentioned bottleneck. Even though a better insensitivity to noise is obtained, this method is still computationally inefficient because, a two-step process is required, which involves solving a matrix equation and finding the roots of a polynomial. Both the sensitivity to noise and this computational limitation can be solved by employing the so-called generalized pencil-of-function method [15] or its improved version, the matrix pencil

method [13], [16], which was applied to a practical case of reflections in [17] and compared with a time-gating technique in [9]. A very good agreement between processed and theoretical data can be achieved whenever the measured signal can be decomposed as the sum of complex exponential functions. If not, the propagating components will not be correctly modeled and an incorrect direct signal will be identified. A higher resolution analysis is presented in [18], where the reflected components are removed in the time-frequency plane once an oversampled Gabor transform decomposition is performed. A technique called mathematical absorber reflection suppression (MARS) has also been proposed to cancel undesired reflected signals. It is based on a special filtering of unwanted cylindrical modes [19] or spherical modes [20], depending on whether the AUT has been measured in a cylindrical near-field or spherical near-field, respectively. Thus, in this work, a new method for canceling reflection components in antenna measurements is proposed. Input data are taken over only one arbitrary surface in the frequency-domain. Moreover, compared with some time-gating techniques or frequency decomposition techniques, it is not necessary to measure more than one frequency. Therefore, the measurement time is reduced considerably and, because the frequency-domain is employed, complicated equipment is not needed. The method is based on a diagnostic technique, and, because it is possible to carry out a diagnostic analysis with any kind of measurement, the proposed method can always be applied. In addition, the computational cost is not large. The paper is organized as follows. Section II gives an overview of the existing techniques used to perform the required AUT diagnostic analysis. The reflection suppression method is described in Section III. Section IV validates the method by means of some numerical results, both from simulations and from measured data. Finally, conclusions are presented in Section V. II. DIAGNOSTIC TECHNIQUES A diagnostic technique (also called sources reconstruction technique) is a method used to obtain the field or the equivalent currents distribution over the AUT plane from the knowledge of its radiated field (near- or far-field). This information can be employed to identify errors, e.g., electrical errors in arrays or mechanical errors in reflectors. Apart from its classic application, a diagnostic process also gives a complete electromagnetic characterization of the antenna, which is the basis of other applications, like near- to far-field transformations, radome applications, radioelectric coverage, or in our case, cancellation of reflections. Basically, diagnostic techniques can be divided into two types. The first of them is based on the application of the equivalence principle and the integral equations relating fields and sources (integral equation methods), and the second one is based on modal expansions (modal expansion methods). A. Integral Equation Methods Integral equation methods have general characteristics, but from a numerical point of view they are more complex than modal expansion methods because it is necessary to solve a

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system of integral equations. Several approaches to obtain a solution of such a system (sources from the radiated field) have been studied in [21], [22] for different measurement setup geometries. In these studies, the system of integral equations (1) is solved by using a numerical method like the finite difference time domain (FDTD) method, the finite element method (FEM), or the method of moments (MoM)

(1) where is the measured electric field at the observarepresents the equivalent magnetic current tion point at the source point within the reconstruction surface , and is the three-dimensional Green function. B. Modal Expansion Methods These methods require lower computational complexity, but due to the fact that the measured field has to be expressed as a superposition of orthogonal functions (vector wave solutions), they can only be used in particular situations, i.e., when the measurement is performed over canonical surfaces; planar, cylindrical and spherical are normally used because complex mechanical scannings are not required. Depending on the coordinate system, one particular kind of expansion (plane wave expansion, cylindrical wave expansion or spherical wave expansion) is carried out [23], [24]. Then, the plane wave spectrum (PWS) is calculated from the corresponding expansion, and the field over the antenna aperture can be directly obtained once the PWS is known, as shown in (2)

(2) stands for the electric field PWS in the where direction. The last step is to explain how to obtain the PWS from the modal expansions. In planar near-field measurements, the PWS is directly obtained from the modal expansion of the samples. However, this PWS is referenced to the measurement plane, , and a back-propagation to the AUT plane [23], [25], [26] has to be used to determine the appropriate PWS

(3) where AUT.

is the distance between the acquisition plane and the

Fig. 1. Reflections in antenna measurements viewed by means of image theory.

For the spherical near-field case, a transformation from a spherical wave expansion to a plane wave expansion (SWE-to-PWE) is required. This transformation was recently developed in [27], [28] and relates spherical and plane coefficients

(4) where and are the spherical coefficients calculated from the measured data and and are the functions detailed in [27], [28]. If a measurement is to be performed in a cylindrical near-field system, up to now there have been no publications explaining the approach to calculate the PWS from the input information. In this case, the only solution is to apply a cylindrical near-field to far-field transformation [23], [24]. Next, the PWS components are obtained by using (5), shown at the bottom of the page, and are the where is the wavenumber, and known far-field components, and represent the PWS components. III. REFLECTION SUPPRESSION METHOD A. Reflections as Virtual Sources If reflected waves are present in antenna measurements, the radiation properties obtained will be perturbed. These same incorrect results are achieved with an equivalent system where reflections can be viewed as direct waves coming from virtual sources (see Fig. 1). Such a replacement is explained in detail via image theory [29], which was also used in [30] to study ground reflections as image current distributions.

(5)

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The identification of the virtual sources is not possible with a conventional diagnostic technique, where the field is only reconstructed over the antenna aperture. However, if the field reconstruction is performed over a surface larger than the antenna dimensions, the aforementioned fictitious sources can be found and cancelled with a filtering process. Thus, the basis of the proposed method is a modified diagnostic technique that provides the reconstructed field in a region whose size in each direction depends on the distance between the AUT and the possible reflective surfaces, which is at least twice as large as that distance, to ensure the correct image identification. In the case of the integral equation method, equivalent currents have to be calculated at a greater number of points. Thus, the number of unknown quantities in the system of integral equations (1) increases; therefore, more time is required to solve the system. On the other hand, modal expansion methods are based on the fast Fourier transform of the PWS, and as a result, the spatial domain, in which the reconstructed field is calculated, and the spectral domain are related in (6) and (7), at the bottom of the and are the spectral steps and page, [26] where and represent the total number of samples in each spectral or , respectively. direction, With these relationships, a straightforward procedure to obtain the reconstructed field over a larger surface is to reduce the spectral step sizes, thus increasing the spatial lengths because these parameters are inversely proportional, as deduced from (6) and (7). In contrast to integral equation methods, which do not require any modification (only a greater number of unknown quantities), if a modal expansion method is used in the proposed reflection suppression algorithm, an interpolation of the PWS and samples has to be carried out. Assuming that are the initial spectral step sizes of the PWS obtained from the measurements and taking into account the minimum required size of the reconstruction plane specified in Fig. 2, these initial steps have to be reduced by the factors indicated in (8) and (9)

Fig. 2. Image theory in a general case with reflections on the floor, the ceiling and the side walls.

where and are the distances from the center of the AUT to the floor, the ceiling, the right wall and the left wall, respectively and is the radius of the AUT. B. Reconstructed Domain Orthogonality By applying either an integral equation approach with the unknown quantities (equivalent magnetic currents) distributed over at least the minimum required surface (see Fig. 2) or a modal expansion method with an appropriate interpolation of the PWS samples, the extended reconstructed field can be found. In the reconstructed domain, real sources and virtual sources are not spatially coincident, so each of them belongs to a different or , respectively subspace,

(10) (11) where is a generic reconstructed field within the space of the reconstructed signals represented by , and represents the zone where virtual sources are placed. Taking into account the previous considerations, any extended reconstructed field can be decomposed as follows:

(8) (9)

(12)

(6)

(7)

CANO-FÁCILA et al.: NEW REFLECTION SUPPRESSION METHOD IN ANTENNA MEASUREMENT SYSTEMS

On the other hand, and spaces, as demonstrated in (13)

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are orthogonal sub-

Fig. 3. Diagram of the reflection suppression method.

IV. NUMERICAL RESULTS

(13) where is the complete region where the field is reconstructed. and are orthogonal, Therefore, and thus an orthogonal projection operator can be applied to to obtain each of those components separately. In from , this case, it is desirable to obtain to which is achieved by applying

(14) The process of suppressing virtual sources by means of orthogonal projections can also be viewed as a spatial filtering. The filter, , which is applied to the reconstructed field, is easily deduced from the orthogonal projection operator

To verify the accuracy of the method presented in the previous section, three examples were analyzed. In the first of them, input data were computationally generated using the analytical expression for the radiated field of an infinitesimal dipole array. In the second case, an experimental setup was built to locate a rectangular metallic plate inside a planar-range measurement system. The last one shows the results of applying the reflection suppression method to the measurement of a RADAR antenna in an outdoor cylindrical near-field system. Thus, the validation of the proposed method was carried out in a logical manner, employing first simulation data, then experimental measurements and finally measurements of a real case where reflections were present. A. Simulated Planar Near-Field Data As an initial example, an infinitesimal dipole array placed in the xy-plane is considered. Each of its elements is assumed to be y-polarized with a radiation expression given by [31], from which the expressions of the x- and y-components of the total radiated field, and , at an observation point can be deduced (assuming an excitation that is uniform in amplitude and phase)

(15) Because virtual sources and reflections are directly related, once the sources are canceled, reflections are also suppressed. C. Description of the Method The steps of the algorithm that implements the proposed method are depicted in detail in Fig. 3. This method is a generic approach because it can be used with any kind of measurement. The only thing to keep in mind is the proper selection of the diagnostic technique because if the measurement has been performed in an arbitrary range, only an integral equation method can be used. However, in the case of an acquisition over a canonical surface, both the option mentioned above and a modal expansion method can be applied. Regardless of the method choice, after the diagnostic stage, the field distribution over the extended AUT plane is known, and virtual sources can be identified. The presence of undesired components in the reconstructed field is suppressed by applying an orthogonal projection onto the subspace of real sources, i.e., employing a spatial filtering that sets the samples related to reflections to zero. Then, a new corrected PWS is obtained by taking the inverse Fourier transform of this filtered field distribution. Finally, the far-field pattern is calculated from the last PWS by applying (5).

(16)

(17) with (18) (19) (20)

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where is the total number of infinitesimal dipoles, stands for the wavenumber, represents a constant that depends on the intrinsic impedance, the wavenumber and the excitation current, are the coordinates of the dipole . and The previous expressions can be employed to simulate the planar near-field measurement of an infinitesimal dipole array by changing the observation point within an acquisition plane parallel to the array. In our case, because reflections are also simulated, other virtual arrays have to be placed as specular images (Section III). The expressions used to simulate the contribution of these virtual arrays are also given by (16) and (17), although a weighting factor equal to the reflection coefficient is required. In the example, the planar near-field of a 5 5 infinitesimal . The separation between samdipole array was computed at ples in the acquisition plane was , the size of the plane was , and the dipoles were spaced at distances of 0.8 . Moreover, a generic case was considered by placing reflective surfaces on the ground, the ceiling and the side walls at 0.8 m, 1.2 m and 1 m, respectively. Finally, the reflection coefficient of these surfaces was assumed to be 1 in all cases except for that of the left wall, where the value was 0.2. Taking into account all these specifications and using the analytical expressions presented above, the planar near-field was obtained. This information was used as input to the proposed reflection suppression method. First of all, the extended reconstructed field had to be calculated, in which the simulation data were employed to obtain the PWS referenced to the AUT plane (modal expansion method). Then, an interpolation of the PWS samples with a reduction factor of the spectral step equal to five was carried out to ensure the correct identification not only of the real array (centered at the origin), but also of the four virtual arrays, as shown in Fig. 4. These virtual arrays cover a region larger than the AUT dimensions, so the image theory is not completely fulfilled due to the fact that the measurement was simulated over a planar surface and a truncation error was present [25]. The AUT was correctly characterized because it was just in front of the acquisition plane, but in the case of the virtual arrays, most of the radiated field information was not considered, and a bad reconstruction was performed. Nevertheless, with an appropriate filter of these unwanted sources, as presented in Fig. 4 in white dashed lines, the reflection effects could be suppressed. This suppression is of the far-field patobserved in Fig. 6, where the cut of tern obtained from the filtered reconstructed field is depicted, presenting a very good agreement with the reference pattern and reducing the ripple in the secondary lobes. B. Experimental Measurement in Planar Near-Field In the second validation, measurements from an experimental setup were employed as input information to the proposed method. The measurements were obtained using the planar-range measurement system in the anechoic chamber at the Technical University of Madrid (UPM). For the experiment, the probe and the AUT were selected to be a corrugated conical-horn antenna and a pyramidal-horn antenna, respectively, and they were separated from each other by 1.57 m. Once both antennas were mounted on their respective positioners, a reference measurement over a 2.4 m 2.4 m acquisition plane

Fig. 4. Normalized amplitude in dB of the co-polar component of the extended reconstructed field.

Fig. 5. Comparison between the reference co-polar far-field pattern and the cut. co-polar far-field pattern with reflections for the 

=0

Fig. 6. Comparison between the reference co-polar far-field pattern and the co-polar far-field pattern after reflection suppression for the  cut.

=0

was recorded in order to have a pattern with which to compare the future results with reflections and reflection suppression. Then, a rectangular metallic plate was placed in the anechoic chamber (see Fig. 7(a)) so as to introduce reflections in the measurement with the AUT at a height of 1.3 m above the plate. Next, a planar near-field to far-field transformation was applied. Fig. 8 shows a comparison with the reference far-field, where it is possible to see a large ripple in the upper lateral lobe as a consequence of the disturbance generated by the reflective

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Fig. 9. Comparison between the reference co-polar far-field pattern and the co-polar far-field pattern after reflection suppression for the  cut.

= 90

Fig. 7. (a) Experimental setup with a metallic plate in the planar near-field system in order to generate reflections. (b) Normalized amplitude in dB of the co-polar component of the extended reconstructed field.

Fig. 10. Logarithmic difference between the reference far-field pattern and the far-field pattern before and after applying the reflection suppression method for cut. the 

= 90

can also be employed [32]. This kind of difference provides high values when the level of the pattern is small, as observed in the region of larger angles within the reliable region. The improvement introduced by the reflection suppression method is shown dB as a reduction of the maximum error from approximately dB in the zone where the level of the pattern is lower than to dB. C. Measurement of a RADAR Antenna Fig. 8. Comparison between the reference co-polar far-field pattern and the cut. co-polar far-field pattern with reflections for the 

= 90

surface. By using the acquired data, the reconstructed field was computed as shown in Fig. 7(b). The AUT is located in the center, whereas the virtual sources related to the reflections are in the lower part. As in the preceding example, because the measurement was performed in planar near-field, the virtual sources were not correctly calculated. However, this problem can be solved by applying an appropriate orthogonal projection operator. After the reflection suppression, the estimated radiation pattern was obtained, which shows a very good agreement with the reference pattern, as observed in Fig. 9. A better pattern comparison is presented in Fig. 10. This comparison is the difference between the reference pattern and the patterns obtained before and after the reflection canceling. In our case, the difference was carried out on a logarithmic scale and without weighting functions, but other patterns for differences

In the previous example, the capability of the proposed method to reduce the effect of the reflections was demonstrated with an experimental setup in an anechoic chamber. However, the method will not usually be used with this kind of measurements where reflections are practically negligible, but rather with measurements in semi-anechoic chambers or outdoor systems. The employment of these special measurement systems is necessary, for instance, when it is necessary to measure a very large antenna. Thus, the measurements of an L-band RADAR antenna were analyzed and used as input to the reflection suppression method. These measurements were obtained by employing one of the abovementioned systems, as shown in Fig. 11, to achieve the azimuthal movement with the RADAR rotation and the vertical movement with the probe slide. The acquisitions were performed on a cylinder, where the distance from the AUT to the probe tower was equal to 5 m, and the vertical path of the probe was equal to 15 m. The reason to use the proposed method in the measurements of this antenna was for the detection of negative effects (ripples in the far-field) during the measurement campaign. After transforming the

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V. CONCLUSION A simple method to reduce errors due to reflections in antenna measurements was presented. The method is based on the application of a diagnostic technique to obtain the field over the AUT plane, making the identification and the later suppression of the virtual sources related to reflections possible, as has been established by the image theory. The proposed method has several advantages compared to the existing procedures to cancel echoes in non-anechoic environments. Different measurements at several distances or at several frequencies are not needed, so the measurement time is considerably reduced. Moreover, the only measurement is performed in the frequency domain, and therefore complex equipment is not required. During the processing part of the method, a diagnostic technique is employed so that the method is generic; for any kind of measurement, one or more diagnostic techniques were proposed. The method was applied to both simulated and measured data to demonstrate its effectiveness, proving that it is possible to significantly improve the far-field pattern in a simple manner.

ACKNOWLEDGMENT The authors want to thank INDRA Sistemas for the measurements of the RADAR antenna.

REFERENCES Fig. 11. (a) Measurement of an L-Band RADAR antenna in an outdoor cylindrical near-field measurement system. (b) Normalized amplitude in dB of the co-polar component of the extended reconstructed field.

Fig. 12. Comparison between the co-polar far-field pattern before and after cut. applying the reflection suppression method for the 

= 90

cylindrical near-field to the far-field and applying a microwave holographic technique, the field on the projected surface of the AUT was obtained. The cause of those errors was the presence of ground reflections in the measurement setup because virtual sources appeared below the AUT (see Fig. 11). Then, the specular image of the AUT was cancelled, resulting in a new reconstructed field. Finally, the estimated far-field pattern was calculated by means of an IFT of the filtered field, achieving a great ripple reduction, especially in the main lobe, as shown in Fig. 12.

[1] J. Appel-Hansen, “Reflectivity level of radio anechoic chambers,” IEEE Trans. Antennas Propag., vol. AP21, no. 4, pp. 490–498, Jul. 1973. [2] A. Lehto, J. Tuovinen, and A. Räisänen, “Reflectivity level of anechoic chambers at 183 GHz,” in Proc. Antennas Propag. Soc. Int. Symp., Dallas, TX, May 7–11, 1990, pp. 1310–1313. [3] E. N. Clouston, P. A. Langsford, and S. Evans, “Measurement of anechoic chamber reflections by time-domain techniques,” IEE Proc. H, Microwaves Antennas Propag., vol. 135, no. 2, pt. H, pp. 93–97, Apr. 1988. [4] M. Nagatoshi, M. Hirose, H. Tanaka, S. Kurokawa, and H. Morishita, “A method of pattern measurement to cancel reflection waves in anechoic chamber,” in Proc. Antennas Propag. Soc. Int. Symp., San Diego, CA, Jul. 5–11, 2008, pp. 1–4. [5] D. N. Black and E. B. Joy, “Test zone field compensation,” IEEE Trans. Antennas Propag., vol. 43, no. 4, pp. 362–368, Apr. 1995. [6] D. A. Leatherwood and E. B. Joy, “Plane wave, pattern subtraction, range compensation,” IEEE Trans. Antennas Propag., vol. 49, no. 12, pp. 1843–1851, Dec. 2001. [7] J. D. Young, D. E. Svoboda, and W. D. Burnside, “A comparison of time- and frequency-domain measurement techniques in antenna theory,” IEEE Trans. Antennas Propag., vol. AP-21, no. 4, pp. 581–583, Jul. 1973. [8] G. A. Burrell and A. R. Jamieson, “Antenna radiation pattern measurement using time-to-frequency transformation (TFT) techniques,” IEEE Trans. Antennas Propag., vol. AP-21, no. 5, pp. 702–704, Sept. 1973. [9] S. Loredo, M. R. Pino, F. Las-Heras, and T. K. Sarkar, “Echo identification and cancellation techniques for antenna measurement in nonanechoic test sites,” IEEE Antennas Propag Mag, vol. 46, no. 1, pp. 100–107, Feb. 2004. [10] S. Loredo, M. R. Pino, F. Las-Heras, and T. K. Sarkar, “Cancelación de ecos en cámaras de medida no anecoicas,” presented at the XVIII Symp. Nat. De La Unión Científica De Radio, URSI, Spain, Sep. 2003. [11] M. Dadic´ and R. Zentner, “A technique for elimination of reflected rays from antenna measurements performed in echoic environment,” AEÜ—Int. J. Electron. Commun., vol. 61, no. 2, pp. 90–94, Feb. 2007. [12] M. L. Van Blaricum and R. Mittra, “A technique for extracting the poles and residues of a system directly from its transient response,” IEEE Trans. Antennas Propag., vol. AP-23, no. 6, pp. 777–781, Nov. 1975.

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[13] T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag., vol. 37, no. 1, pp. 48–55, Feb. 1995. [14] T. K. Sarkar, J. Nebat, D. D. Weiner, and V. K. Jain, “Suboptimal approximation/identification of transient waveforms from electromagnetic systems by pencil-of-function method,” IEEE Trans. Antennas Propag., vol. AP-28, no. 6, pp. 928–933, Nov. 1980. [15] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 229–234, Feb. 1989. [16] R. S. Adve, T. K. Sarkar, O. M. C. Pereira-Filho, and S. M. Rao, “Extrapolation of time-domain responses from three-dimensional conducting objects utilizing the matrix pencil technique,” IEEE Trans. Antennas Propag., vol. 45, no. 1, pp. 147–156, Jan. 1997. [17] B. Fourestié, Z. Altman, J. Wiart, and A. Azoulay, “On the use of the matrix-pencil method to correlate measurements at different test sites,” IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 1569–1573, Oct. 1999. [18] B. Fourestié and Z. Altman, “Gabor schemes for analyzing antenna measurements,” IEEE Trans. Antennas Propag., vol. 49, no. 9, pp. 1245–1253, Sep. 2001. [19] S. Gregson, A. Newell, and G. Hindman, “Reflection suppression in cylindrical near-field antenna measurement systems-cylindrical MARS,” in Proc. Antenna Meas. Techniques Assoc., AMTA, Salt Lake City, UT, Nov. 1–6, 2009, pp. 119–125. [20] G. Hindman and A. Newell, “Reflection suppression in large spherical near-field range,” in Proc. Antenna Meas. Techniques Assoc., AMTA, Newport, RI, Oct. 30–Nov. 4 2005, pp. 270–275. [21] Y. Álvarez, F. Las-Heras, and M. R. Pino, “Reconstruction of equivalent currents distribution over arbitrary three-dimensional surfaces based on integral equation algorithms,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3460–3468, Dec. 2007. [22] P. Petre and T. K. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach,” IEEE Trans. Antennas Propag., vol. 40, no. 11, pp. 1348–1356, Nov. 1992. [23] A. D. Yaghjian, “An overview of near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. AP-34, no. 1, pp. 30–44, Jan. 1986. [24] R. C. Johnson, H. A. Ecker, and J. S. Hollis, “Determination of far-field antenna patterns from near-field measurements,” Proc. IEEE, vol. 61, no. 12, pp. 1668–1694, Dec. 1973. [25] E. Martini, O. Breinbjerg, and S. Maci, “Reduction of truncation errors in planar near-field aperture antenna measurements using the gerchberg-papoulis algorithm,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3485–3493, Nov. 2008. [26] J. J. H. Wang, “An examination of the theory and practices of planar near-field measurement,” IEEE Trans. Antennas Propag., vol. 36, no. 6, pp. 746–753, Jun 1988. [27] C. Cappellin, “Antenna diagnostics for spherical near-field antenna measurements,” Ph.D. Dissertation, Dept. Elect. Eng., Danmarks Tekniske Universitet, Copenhague, Denmark, Sep. 2007. [28] C. Cappellin, A. Frandsen, and O. Breinbjerg, “Application of the SWE-to-PWE antenna diagnostics technique to an offset reflector antenna,” IEEE Antennas Propag Mag, vol. 50, no. 5, pp. 204–213, Oct. 2008. [29] C. A. Balanis, “Electromagnetic theorems and principles,” in Advanced Engineering Electromagnetic. New York: Wiley, 1989, ch. 7, sec. 4, pp. 314–323. [30] R. J. Lytle, “Ground reflection effects upon radiated and received signals as viewed via image theory,” IEEE Trans. Antennas Propag., vol. AP-20, no. 6, pp. 736–741, Nov. 1972. [31] C. A. Balanis, “Linear wire antennas,” in Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997, ch. 4, sec. 2, pp. 133–142. [32] S. Pivnenko, J. E. Pallensen, O. Breinbjerg, M. S. Castañer, P. Caballero, C. Martínez, J. L. Besada, J. Romeu, S. Blanch, C. Sabatier, A. Calderone, G. Portier, H. Eriksson, and J. Zackrisson, “Comparison of antenna measurement facilities with the DTU-ESA 12 GHz validation standard antenna within the EU antenna centre of excellence,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1863–1878, Jul. 2009.

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Francisco José Cano-Fácila was born in 1985 in Don Benito, Spain. He received the M.Sc. degree (with special distinction) in telecommunication engineering from the Technical University of Madrid (UPM), Madrid, Spain, in 2008. During 2007, he worked for Telefónica Research & Development as part of the Final Client Services Department. Since 2008, he is a Ph.D. student in the Radiation Group, Signals, Systems and Radiocommunications Department, UPM. His main research interest is antenna measurement, focusing specifically on diagnostics techniques and near-field to far-field transformations. Mr. Cano-Fácila was awarded the highest marks for the telecommunication engineering degree in 2007. He was awarded a Ph.D. scholarship from the Spanish government.

Sara Burgos was born in Madrid, Spain, in September 1980. She received the M.Sc. degree in telecommunication engineering and the Ph.D. degree from the Technical University of Madrid (UPM), in 2004 and 2009, respectively. Her current research interests are in antenna measurement techniques with an emphasis on cylindrical and spherical systems. From September 2006 to March 2007, she spent seven months abroad as a guest Ph.D. student at the Technical University Denmark, researching antenna measurement enhancement. In 2009, she spent three months in the Electronic and Telecommunication Institute of Rennes (IETR), France, studying the design and measurement of antennas at millimeter wavelengths.

Fernando Martín was born in 1978 in Madrid, Spain. He received the M.Sc. degree in telecommunication engineering and the Ph.D. degree from the Technical University of Madrid (UPM), in 2002 and 2009, respectively. He has worked in different measurement facilities development projects, first for the Radiation Group, Signals, Systems and Radio-communication Department, UPM; then for INDRA Defense Integrated Systems and now for SENER. His current research interests are in antenna measurement systems for RADAR antennas and in cylindrical near field to far field transformations.

Manuel Sierra-Castañer (S’95–M’01) was born in 1970 in Zaragoza, Spain. He received the M.Sc. degree in telecommunication engineering and the Ph.D. degree from the Technical University of Madrid (UPM), Madrid, Spain, in 19994 and 2000, respectively. He worked for Airtel, a cellular company, from 1995 to 1997. Since 1997, he worked in the University “Alfonso X” as an Assistant Professor. Since 1998, he has worked at UPM as a Research Assistant, Assistant Professor and Associate Professor. His current research interests involve planar antennas and antenna measurement systems. Dr. Sierra-Castañer was awarded the IEEE APS 2007 Schelkunoff Prize for his paper “Dual-Polarization Dual-Coverage Reflectarray for Space Applications.”

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Curl-Conforming Hierarchical Vector Bases for Triangles and Tetrahedra Roberto D. Graglia, Fellow, IEEE, Andrew F. Peterson, Fellow, IEEE, and Francesco P. Andriulli, Member, IEEE

Abstract—A new family of hierarchical vector bases is proposed for triangles and tetrahedra. These functions span the curl-conforming reduced-gradient spaces of Nédélec. The bases are constructed from orthogonal scalar polynomials to enhance their linear independence, which is a simpler process than an orthogonalization applied to the final vector functions. Specific functions are tabulated to order 6.5. Preliminary results confirm that the new bases produce reasonably well-conditioned matrices.

TABLE I CLASSIFICATION OF HIERARCHICAL CURL-CONFORMING VECTOR-BASES AVAILABLE IN THE LITERATURE

Index Terms—Basis functions, finite element methods, hierarchical basis functions, method of moments.

I. INTRODUCTION

V

ECTOR basis functions find wide application in electromagnetics for volumetric discretizations of the vector Helmholtz equation in 2D and 3D and surface discretizations of the electric and magnetic field integral equations in 3D. These basis functions can be interpolatory, with coefficients that represent specific field components at interpolation points, or they can form hierarchical sets in order to facilitate adaptive refinement procedures. In contrast to interpolatory bases, hierarchical bases often exhibit poor linear independence as the order of the representation is increased, resulting in an ill-conditioned system of equations. In the following, a new hierarchical family of vector bases is proposed that alleviates the loss of linear independence. Papers proposing hierarchical vector basis functions began appearing in the electromagnetics literature in the early 1990s. Most of the proposed basis functions are of the curl-conforming variety, which on triangles or quadrilaterals are easily converted into divergence-conforming functions. For brevity, we focus on curl-conforming bases on triangular and tetrahedral cells. Table I summarizes the existing curl-conforming hierarchical vector bases suitable for triangular or tetrahedral cells. The published basis functions can be classified into three groups: A) those that span complete polynomial vector spaces,

Manuscript received November 27, 2009; revised May 05, 2010; accepted October 09, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. R. D. Graglia is with the Dipartimento di Elettronica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, and also with ISMB-Istituto Superiore Mario Boella, Via P.C. Boggio 61, 10138 Torino, Italy (e-mail: [email protected]). A. F. Peterson is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]). F. P. Andriulli is with the Microwave Department of TELECOM Bretagne, Brest, 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.2010.2103012

B) those that span the mixed-order spaces of Nédélec [1] (sometimes known as reduced gradient spaces for curl-conforming functions), and C) those with subsets that exactly span both types of spaces. As an example, the 1997 Graglia, Wilton, and Peterson interpolatory vector basis functions [2] fall into group B, since those basis functions span the mixed-order spaces of Nédélec but do not contain subsets that exactly span polynomial-complete spaces. The new hierarchical bases also belong to group B. Although several families of bases appearing in Table I are considered to be “Nédélec” bases, here we classify them as “type A” because they do not contain subspaces that properly span the reduced-gradient spaces of [1] on triangles or tetrahedra. For instance, neither the face-based functions of [8]

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Fig. 1. The hierarchial polynomials that generate the curl-conforming functions associated with the zeroth-order vector function relative to the edge formed by the and  faces have features similar to those of the equivalent interpolatory polynomials in [2]. Vanishing regions: (a) the edge-based intersection of the  face vanish on the face  ; (c) all the polynomials polynomials are different from zero on the cell boundaries; (b) all the polynomials based on the  face vanish on the face  ; (d) the volume-based polynomials vanish on both the  and  faces. The number of hierarchical based on the  edge-based polynomials; (f, g) p p = polynomials and interpolatory edge, face and volume-based polynomials of order p is the same. There are: (e) p based on the  and the  faces; (h) p p = volume-based polynomials. The figures from (e) to (h) show in red the interpolation nodes of the . interpolatory polynomials for p

=0

=0 =0

=0

=0 =3

=0 ( 0 1) 6

nor the functions of [9] properly span the Nédélec space of order 2.5; the “type 2” element-based functions of [15] do not properly span the Nédélec spaces of order 1.5 or higher. Thus these functions are listed here as belonging to type A. Our new bases, first proposed in [17] and [18], have four distinguishing features: (a) the vector basis functions are subdivided from the outset into three different groups of edge, face, and volume-based functions; (b) each basis function is obtained by using one generating edge, face or volume-based polynomial whose analytical expression involves all the four dependent parent variables ( , , , ) that describe the tetrahedral element (notice that this holds even for the basis functions of the triangular element); (c) in each group, all the generating polynomials are mutually orthogonal independent of the definition domain of the inner product, i.e., either the volume, the face, or the edge of the tetrahedron; (d) the hierarchical vector functions are either symmetric or antisymmetric with respect to the parent variables that describe each edge and face of the cell. The four features outlined above yield the following outcomes, respectively: (a) different individual polynomial orders can be used on each edge, face, and volumetric element of a given mesh, thereby facilitating the use of vector bases of different orders together in the same mesh ( -adaption); (b) the generating polynomials for the edge, the face, and the volume-based vector functions can be implemented in routines which can be used without modification to evaluate either the tetrahedral (for 3D codes) or the triangular (for 2D codes) vector functions; this greatly simplifies the implementation of the numerical codes required to deal with 3D or 2D structures; (c) our higher-order bases maintain excellent linear independence because they are derived after an analytical orthogonalization of the generating scalar polynomials, which is done in the element parent domain; (d) the procedure to enforce the conformity of the approximation across element interfaces is drastically simplified.

=0 ( + 1)

=0

=0

=0

( + 1) 2

The outcome (c) is of importance because hierarchical bases are typically ill-conditioned at high orders and usually necessitate a cumbersome (partial) orthogonalization process to improve system conditioning. As illustrated by [19], considerable effort is required to directly orthogonalize the vector functions. In contrast, our bases are defined from orthogonal generating scalar polynomials, to enhance the conditioning of the system matrices. The outcome (d) is also of importance because enforcement of the continuity of the tangential component across adjacent elements (for the curl-conforming case) can be difficult [11], [20]; our basis functions reduce this problem to one of determining the correct sign of each basis function with respect to an arbitrarily selected reference direction along adjacent elements. Our hierarchical vector functions are obtained by a three-step process. First, we orthogonalize on a given parent element appropriate linear combinations of the interpolatory scalar polynomials given in [2], to obtain hierarchical scalar polynomials. These polynomials are then multiplied by the zeroth-order vector functions of the element under consideration to obtain a set of vector functions. Finally, using a procedure similar to the one given in [2], any redundant basis function is eliminated from the resulting vector set. II. EDGE, FACE, AND VOLUME-BASED HIERARCHICAL BASES The bases for the tetrahedral and the triangular cells are derived at the same time by simply considering the triangular cell described by the three parent variables ( , , ) as the ) face of the tetrahedral cell described by the bounding ( . Let us consider a four parent variables tetrahedral element whose faces are labeled by these four parent variables and, at the same time, the triangular element defined face of this tetrahedron (see Fig. 1). In terms by the of these parent variables, the zeroth-order curl-conforming vector-function associated with the edge in common to the

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TABLE II EDGE AND FACE-BASED HIERARCHICAL POLYNOMIALS UP TO THE SIXTH ORDER

and face reads and for the tetrahedral and the triangular element, respectively [2]. Despite of the sign difference in the previous two expressions, which can be eventually eliminated by reorienting the triangle unit normal, both functions turn out to be antisymmetric with respect to the two parent variables and , since and . Because of this property, the enforcement of the tangential continuity of the field across element boundaries is greatly simplified. The continuity of the tangential component is ensured by adjusting the basis function sign to correspond to an arbitrarily selected reference direction along the adjacent elements [2].

Higher order interpolatory bases are constructed in [2] by multiplying the zeroth-order vector functions with SilvesterLagrange interpolatory polynomials. Here, linear combinations of those interpolatory polynomials are used to obtain symmetric or antisymmetric hierarchical scalar polynomials which, within each group, are constructed a priori to be mutually orthogonal. Our hierarchical vector functions are constructed using the same technique given in [2], where we simply substitute the new scalar hierarchical polynomials for the interpolatory ones of [2]. With reference to Fig. 1, the interpolatory polynomials which in [2] are associated with the edge at the intersection of the and faces are subdivided into four different groups. The first group is formed by all the polynomials inter-

GRAGLIA et al.: CURL-CONFORMING HIERARCHICAL VECTOR BASES FOR TRIANGLES AND TETRAHEDRA

TABLE III VOLUME-BASED HIERARCHICAL POLYNOMIALS V

=

polating (up to a given order) at the edge (Fig. 1(e)) and that (in general) do not vanish on the other tetrahedral boundaries (Fig. 1(a)); a second group is formed by the polynomials that face (Fig. 1(f)) interpolate (up to a given order) at the (Fig. 1(b)); a third group and that vanish on the face face is given by polynomials which interpolate at the (Fig. 1(c)); the last group is (Fig. 1(g)) and vanish on given by the remaining interpolating polynomials which vanish and faces (Fig. 1(d), (h)). Appropriate on both the linear combinations of these interpolatory polynomials, together with extensive symmetry considerations, provide four groups of orthogonal hierarchical polynomials. These polynomials, derived and normalized in the Appendix, are explicitly reported in Tables II and III up to the sixth order. We have obtained hierarchical families up to eleventh order with this approach. In Tables II–III, the first superscript or labels symmetric or antisymmetric polynomials of the and variables, respectively; similarly, the second superscript (used only for the edgeand the volume-based polynomials) labels symmetric and antisymmetric polynomials of the and variables. of Table II All the edge-based hierarchical polynomials and variables. In Table II, are symmetric in the indicates the Legendre polynomial of order , with . The polynomials based on the edge of the simplex ) are obtained by (triangular element, with . For the simplex (tetrahedral element, with setting ), the polynomials reported are those

 U

UP TO THE

953

SIXTH ORDER

, . Along its associated edge, based on the edge behaves as the Legendre polynomial . For the face-based polynomials of Table II one has to set while dealing with the simplex; for the simplex, the face. polynomials reported are those associated with the Notice that the edge-based hierarchical polynomials of simplexes, while Table II are orthogonal on the , , and the face-based polynomials are orthogonal on both the and simplexes. The number of Degrees of Freedom (DoF) for curl-conforming bases of order on a tetrahedron is ; the number of DoFs for curl- and divergence-conforming [2]. By folbases of order on a triangle is lowing the same procedure reported in [2], the elements of an order hierarchical vector base associated with DoFs internal to the tetrahedral element are obtained by forming the product of the volume-based hierarchical polynomials of Table III with three different zeroth-order curl-conforming functions. To guarantee basis function independence, the chosen zeroth-order basis factors cannot be associated with edges bounding the same elements of a order hierface [2]. Similarly, the archical base associated with DoF internal to the triangular (possibly, the face bounding the face tetrahedral element ) are obtained by forming the product of face-based hierarchical polynomials of Table II the with two zeroth-order curl-conforming functions associated

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TABLE IV CORRESPONDENCE BETWEEN DUMMY AND PARENT VARIABLES

TABLE V CONDITION NUMBERS (CN) AND NUMBER OF ITERATIONS RELATIVE TO A DRIVEN 2D – CAVITY PROBLEM

with two edges bounding . Finally, the elements of a order hierarchical base associated with edge DoF relative to (possibly, the edge bounding the edge , or the edge bounding the the tetrahedral element ) are obtained by forming the product of triangular element edge-based hierarchical polynomials of Table II the with the zeroth-order curl-conforming function along that edge. It is here understood that, in this construction process, appearing in the the dummy parent variables polynomial expressions reported in Tables II–III are replaced that corresponds to the by the permutation of appropriate zeroth-order basis factor shown in Table IV. III. NUMERICAL RESULTS A. Triangular Element Consider the vector Helmholtz equation (1) representing a two-dimensional cavity bounded by perfectly conducting walls (a homogeneous Neumann boundary). For the magnetic field (2) expressed in terms of vector basis functions matrices and have entries of the form

, the element

(3) (4) with vector-field solution (2) linearly dependent on the expansion-coefficient array (5) and where

now refers to the global matrix; the entries of are (6)

The linear independence of the basis set is an important attribute of the vector-field of a good basis. For example, the error of each coefficient apsolution depends on the error pearing in (2) and, according to (5), the confidence in the numerical precision of the vector-field solution on equal residual clearly improves for decreasing condition number of error the -matrix. Because of the nullspace of the curl operator, the element is nonsingular and its conmatrix is singular. However, dition number ( ) provides a measure of the degree of linear independence of the basis functions, which in turn gives an indication of the performance of the basis functions in numerical applications. Similar comparisons of -matrix condition numbers were carried out in [21] for lower-order (linear tangential/ quadratic normal – LT/QN) basis functions; some higher-order comparisons are provided in [19]. Preliminary numerical results for triangular cells were previously published in [22], which compared the (element and global) -matrix condition numbers arising from several families of hierarchical vector basis functions to those of the new functions proposed here. That study, which treated the solution of (1) as an eigenvalue problem for the resonant frequencies of the cavity, concluded that the proposed basis functions of order 2.5 and 3.5 produce lower matrix condition numbers over a range of meshes than those of most other families. Those results were obtained after attempts were made to improve the condition numbers for the other families by an appropriate choice of scale factors. Additional preliminary results for triangles are provided in Table V, which considers the matrix conditioning and iterative solution of the linear system associated with one example of a driven cavity described by (1). A constant magnetic field was imposed on one edge of the mesh, which contained reasonably well-controlled cell shapes, and quadratic tangential/cubic normal (QT/CuN) bases from several hierarchical families were employed. (These tests were carried out using the original scale factors provided by the authors of [6], [7], [12]–[14], and not

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two condition numbers, versus the scale factor , for . These results show that, in case of rectilinear tetrahedral cells of good quality, and for element order 2.5, the individual element condition number CNH is expected to be four to five times larger than the individual element condition number CNI obtained with interpolatory polynomials. Conversely, for poor quality cells, the condition number obtained from the hierarchical vector bases could be much higher (say, by a factor of 7) than that obtained by using interpolatory polynomials; however, this result is still within an order of magnitude and cells of such poor quality are usually avoided whenever possible. IV. CONCLUSION

T

Fig. 2. Individual element -matrix condition numbers for the hierarchical (CNH) and the interpolatory (CNI) vector basis of order 2.5 obtained by considering tetrahedral cells of different height  6 `=3, but with the same equilateral base of edge-length `.

p

those introduced by the present authors in [22].) Table V shows that the conditioning of the overall system for the driven cavity is generally proportional to that of the global -matrix considered earlier. Table V also reports the number of iteration steps using Jacobs’ required to reduce the residual error norm to form of the biconjugate gradient algorithm [23], [24] without preconditioning. Diagonal matrix preconditioning was also considered and, for these test cases of relatively small matrix order, improved the performance of the solver for all of the basis function families (in fact, the preconditioned algorithm converged to the same residual with about the same amount of computation for each basis family except those of [6], [7], which required at least three times as many iterations). Similar results were observed as mesh quality degraded and other parameters were varied. While these results suggest that the new basis functions are at least as effective as existing bases when used with simple iterative solvers, we acknowledge that additional comparisons must be carried out before conclusions can be drawn about their performance in more realistic problems or with more sophisticated preconditioners. B. Tetrahedral Element As a preliminary evaluation of the tetrahedral bases, we compare the individual element -matrix condition number obtained using the 45 QT/CuN basis functions (of order 2.5) presented in this paper and the equivalent functions given in [2]. The best possible tetrahedral cell has equilateral shape, edge. Lower-quality cells may be length , and height obtained by scaling the height of this tetrahedron, thereby oband the same (equilateral) taining cells of different height base. The element condition numbers CNH and CNI obtained with the hierarchical and the interpolatory family, respectively, depend on the value of used to modify the cell shape; for these bases, however, the condition numbers CNH and CNI are not modified by changing the value of while keeping fixed. Fig. 2 shows the behavior of the condition numbers CNH and CNI for the two types of bases, and the ratio (CNH/CNI) of these

A new family of hierarchical vector bases has been proposed for triangles and tetrahedra. The use of orthogonal scalar polynomials in their construction is believed to offer a simpler approach for enhancing their linear independence than the partial orthogonalization of the vector functions. Preliminary numerical results, presented here and in a companion paper [22], suggest that the new bases yield reasonably well-conditioned matrices. APPENDIX ORTHOGONAL POLYNOMIALS USED IN THE CONSTRUCTION OF HIERARCHICAL VECTOR BASES In the first part of this Appendix (subsection A) we introduce and discuss auxiliary polynomials needed to construct hierarchical polynomial bases with terms subdivided into volume, face, and edge-based polynomials. The symbol used for these polynomials is related to their further use; that is, , , and are the auxiliary polynomials used to construct the volume, the face, and the edge based polynomials, respectively. Then, the other three subsections of this Appendix (from B to D) show how to use the auxiliary polynomials to construct hierarchical volume, face, and edge-based polynomials called , , and , respectively. In each group, the polynomials are mutually orthogonal independent of the definition domain of the inner product, i.e., either the volume, the face, or the edge of the element. This is a very important feature of our hierarchical bases because one can use the same polynomial bases either on tetrahedral, triangular or line elements, with no need to modify their expressions. The edge, face and volume-based polynomials are normalized as reported at the bottom of Table II–Table III; in numerical applications these polynomials can be normalized differently whenever convenient. To compact the expressions of the polynomials of this Appendix it is convenient to introduce the following new variables

(7) with dependency relation (8)

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The expressions of the four parent variables in terms of the new dependent variables are straightforwardly obtained by inversion of (7). A) Auxiliary Polynomials: Polynomials with particular symmetry properties in the dependent parent variables used to describe the , the and the simplex are obtained by linear combinations of the Silvester ( ) and the shifted Silvester ( ) interpolatory polynomials given in [2]. For example, the th order polynomials

(9) with even odd

(10)

by use of the following recurrence relations with respect to the degree (14) with (15)

Orthogonal volume-based polynomials symmetric or antisymmetric in the ( , ) and the ( , ) variables can be obtained by linearly combining the Silvester and shifted Silvester interpolatory polynomials in a way similar to that used to get (9). To obtain these polynomials, however, it is much more convenient to start with expressions that involve the appropriate Legendre polynomials from the beginning. In fact, in subsection B of this Appendix, we use the following volume-based linearly independent polynomials of order (16)

values) or are either symmetric (for even or zero antisymmetric (for odd values) in and , with . in (9) are determined to define hierarThe coefficients . They chical polynomials associated with the edge are obtained by imposing, along the edge at issue (that is, the simplex), the orthogonality of each with of lower order : respect to all the polynomials (11) where is the Kronecker delta. The normalization (11) involves a unit constant weight function and makes equal to the Shifted Legendre Polynomial . In principle, other normalizations with different weight functions are possible to make , for example, equal to Chebyshev or Jacobi polynomials. Convenient expressions for the defined in (9) can be given in terms of the two new dependent variables and given in (7). All the polynomials are implicand variables since itly symmetric in the . In subsection C, to construct face-based polynomials, we also obtained by orthogonalizing the need the polynomials , for . All Silvester polynomials these polynomials contain a common factor and do not have any symmetry or antisymmetry property in , because they are independent of ; they are normalized by setting (12) and for which yields These polynomials are obtained from the lowest order ones

.

(13)

which are symmetric in ( , ) and ( , ) for even values of and , respectively (antisymmetric otherwise), with

(17) The integral of the product of a symmetric and an antisymmetric polynomial of the kind given in (9) automatically vansimplex; similarly, the product of ishes over the , , and a symmetric and an antisymmetric polynomial of the kind given in (16) has a vanishing integral over . B) Volume-Based Hierarchical Polynomials Orthogonal : In terms of the (dummy) parent variables used in Over polynomials which in [2, the present paper, the simplex read as Eq. (28)] interpolate internal points of the follows:

(18) and faces because of and they vanish on the the presence of a common factor. According to our definition, the above polynomials are volume-based. An equivalent th order hierarchical family consisting of volume-based polynomials is obtained by applying the GramSchmidt orthogonalization process to the polynomial set (16), performed by using the Legendre inner product over the simplex. The hierarchical volume-based polynomials are thus obtained by orthogonalizing in order (from the first to the last polynomial) the list of the polynomials (16) provided by run(outer loop on the ning a three nested loop: for ); , and global order

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Q 

= E F

Fig. 3. Orthogonalization path followed by the two-nested loop used to build the face-based hierarchical polynomials . The outer loop is for k 1; 2; . . . ; p; (with k = m + n) is obtained by orthogonalizing with respect to all the the inner loop is for n = k; k 1; . . . ; 1. Thus, the k th order polynomial previously obtained up to the order | + ` = k 1 as well as, in the case of m 1, by orthogonalizing with respect face-based orthogonal polynomials to the polynomials previously obtained for | = 0; 1; . . . ; m 1 with ` = k |. The figure just shows the path from k = 1 up to k = 4. The procedure starts is known. For k = 2 we first build by orthogonalizing with respect to , then we build by orthogonalizing with with k = 1, where respect to and . For k = 3 we first build by orthogonalizing with respect to , , and ; then we build, in order, and , etc.

0

Q

Q Q Q

Q

Q

Q

Q

0

0

EF

(inner loop), with (fixed in the inner loop). The hierarchical polynomials obtained in this manner are reported in Table III up to the sixth order. The global order and equivalently of these polynomials is equal to to the sum of the subscripts appearing in their expressions. C) Face-Based Hierarchical Polynomials Orthogonal and : In terms of the (dummy) parent variables Over polynomials which used in the present paper, the in [2, Eqs. (10, 28)] interpolate the points located inside the triangular face described by the three parent variables , , read as follows

(19) case is here excluded since it yields edge-based The functions that interpolate the edge of the triangle; those functions are considered in the following subsection D. The face-based interpolatory polynomials (19) are replaced by hierarchical polynomials obtained by orthogonalizing in , within a two-nested loop for order over (outer loop) and (inner loop; see Fig. 3), the th order polynomials

0 EF

Q Q

Q

Q

E F

Q

Q

EF Q

because of the presence of a common factor. In at order to obtain a hierarchical family of face-based polynomutually orthogonal on both the mials and the simplex, it is sufficient to add to the polynomials an appropriate linear combination of the (derived in subsection B) of volume-based polynomials global order less than or equal to , and which share with respect to the the same symmetry properties of and variables. The polynomials define the following polynomials of global order (21) used in subsection D to construct edge-based polynomials and variables, and orthogonal on the symmetric in the , the , and the simplices. Table II reports, up to the sixth order, the normalized facefor the and simbased hierarchical polynomials plices. D) Edge-Based Hierarchical Polynomials Orthogonal and : In terms of the (dummy) parent variables Over , polynomials which in used in the present paper, the [2, Eqs. (10), (28)] interpolate the edge described by the two parent variables , read as follows

(20)

(22)

where and with , given in subsection A. This orthogonalization process yields a polynomial set that contains all the normalized orthogonal , with , and polynomials at . The order of is and coincides and of the and with the sum of the subscripts functions in (20). Furthermore, is symmetric in , if is even or equal to zero, whereas is antisymmetric in , if is odd. and The hierarchical polynomials obtained by this procedure are mutually and orthogonal only over the triangular simplex , respectively. The are face-based , and equal to zero and nonzero on the triangular face

and could be substituted with the hierarchical polynomials of subsection A. The hierarchical polynomials for the line element (that is, the simplex) are simply obtained from by setting , which is equivalent to set . In the line-elare easily obtained by the recurement case the functions rence relation available in [25], since coincides by or, equivaconstruction with the Legendre polynomial . lently, with the Shifted Legendre polynomial However, unfortunately, the edge-based hierarchical polynoare mutually orthogonal only over the simplex . In mials order to obtain a hierarchical family of edge-based polynomials symmetric in the and variables, and mutually orthogonal simplex, the simplex also on the

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(where ), and the simplex (where ), an approit is sufficient to add to the th order polynomial priate linear combination of the polynomials of subsection C (given in (21)), and of the volume-based polynomials of subsection B, which share the same symmetry properties of with respect to the four parent variables. All the polynomials . Notice also involved in this combination are of order that the linear combination at issue here involves only those volume-based polynomials that are symmetric with respect to the and variables. obtained in this manner are The hierarchical polynomials reported in Table II, up to the sixth order. E) Distinguishing Features of the New Polynomial Bases: As previously discussed in this Appendix, in general, a polynomial of one group (either the volume, the face, or the edge-based group) is not orthogonal to a polynomial of a different group, but all the polynomials within each group are mutually orthogonal independent of the definition domain of the Legendre inner product (i.e., either the volume, the face, or the edge of the element). This feature is readily appreciated if one considers the having coefficient equal to the Legendre Gram matrix simplex) of the th and th inner product (on the , , or polynomials of the th–order-complete family, see [18, Fig. 3]. , the condition numbers of the As shown in [18], for matrices obtained by using our hierarchical polynomial bases are lower than those obtained by using the Silvester-Legendre interpolatory polynomials. REFERENCES [1] J. C. Nédélec, “Mixed finite elements in R3,” Num. Math., vol. 35, pp. 315–341, 1980. [2] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics, special issue on “Advanced numerical techniques in electromagnetics”,” IEEE Trans. Antennas Propag., vol. 45, pp. 329–342, Mar. 1997. [3] J. P. Webb and B. Forghani, “Hierarchal scalar and vector tetrahedra,” IEEE Trans. Magn., vol. 29, pp. 1495–1498, Mar. 1993. [4] C. Carrié and J. P. Webb, “Hierarchal triangular edge elements using orthogonal polynomials,” in Proc. IEEE Int. Antennas Propagation Symp., Montreal, Jul. 1997, vol. 2, pp. 1301–1313. [5] J. Wang and J. P. Webb, “Hierarchal vector boundary elements and p-adaption for 3D electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 45, no. 12, pp. 1869–1879, Dec. 1997. [6] L. S. Andersen and J. L. Volakis, “Hierarchical tangential vector finite elements for tetrahedra,” IEEE Microw. Guided Wave Lett., vol. 8, no. 3, pp. 127–129, Mar. 1998. [7] L. S. Andersen and J. L. Volakis, “Development and application of a novel class of hierarchical tangential vector finite elements for electromagnetics,” IEEE Trans. Antennas Propag., vol. 47, no. 1, pp. 112–120, Jan. 1999. [8] J. P. Webb, “Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements,” IEEE Trans. Antennas Propag., vol. 47, no. 8, pp. 1244–1253, Aug. 1999. [9] D. K. Sun, J. F. Lee, and Z. Cendes, “Construction of nearly orthogonal Nedelec bases for rapid convergence with multilevel preconditioned solvers,” SIAM J. Sci. Comput., vol. 23, no. 4, pp. 1053–1076, 2001. [10] M. Ainsworth and J. Coyle, “Hierarchic hp-edge element families for Maxwell’s equations on hybrid quadrilateral/triangular meshes,” Comput. Methods Appl. Mech. Engng., vol. 190, pp. 6709–6733, 2001. [11] M. Ainsworth and J. Coyle, “Hierarchic finite element bases on unstructured tetrahedral meshes,” Int. J. Numer. Meth. Engng., vol. 58, no. 14, pp. 2103–2130, Dec. 14, 2003. [12] S. C. Lee, J. F. Lee, and R. Lee, “Hierarchical vector finite elements for analyzing waveguiding structures,” IEEE Trans. Antennas Propag., vol. 51, pp. 1897–1905, Aug. 2003. [13] R. S. Preissig and A. F. Peterson, “A rationale for p-refinement with vector finite elements,” Appl. Computat. Electromagn. Soc. (ACES) J., vol. 19, pp. 65–75, Jul. 2004.

[14] P. Ingelström, “A new set of h(curl)-conforming hierarchical basis functions for tetrahedral meshes,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 106–114, Jan. 2006. [15] J. Schöberl and S. Zaglmayr, “High order Nédélec elements with local complete sequence properties,” Int. J. Comput. Math. Electr. Electron. Eng. (COMPEL), vol. 24, no. 2, pp. 374–384, 2005. [16] S. Zaglmayr, “High order finite element methods for electromagnetic field computation,” Ph.D. dissertation, Johannes Kepler Universität, Linz, Austria, 2006. [17] R. D. Graglia and A. F. Peterson, “Fully conforming hierarchical vector bases for finite methods,” presented at the URSI Nat. Radio Sci. Meeting, Charleston, SC, Jun. 1–5, 2009. [18] R. D. Graglia, A. F. Peterson, and F. P. Andriulli, “Hierarchical polynomials and vector elements for finite methods,” in Proc. Int. Conf. Electromagn. Adv. Applicat. (ICEAA 2009), Torino, Italy, Sep. 2009, vol. 1, pp. 1086–1089, doi: 10.1109/ICEAA.2009.5297791. [19] R. Abdul-Rahman and M. Kasper, “Orthogonal hierarchical Nédélec elements,” IEEE Trans. Magn., vol. 44, pp. 1210–1213, Jun. 2008. [20] J. P. Webb, “Matching a given field using hierarchal vector basis functions,” Electromagnetics, vol. 24, no. 1-2, pp. 113–122, Jan. 1, 2004. [21] L. S. Andersen and J. L. Volakis, “Condition numbers for various FEM matrices,” J. Electromagn. Waves Appl., vol. 13, no. 12, pp. 1663–1679, Jan. 1, 1999. [22] A. F. Peterson and R. D. Graglia, “Scale factors and matrix conditioning associated with triangular-cell hierarchical vector basis functions,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 40–43, 2010. [23] D. A. H. Jacobs, , I. S. Duff, Ed., “The exploitation of sparsity by iterative methods,” in Sparse Matrices and Their Uses. Berlin: SpringerVerlag, 1981, pp. 191–222. [24] C. F. Smith, A. F. Peterson, and R. Mittra, “The biconjugate gradient method for electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 38, no. 6, pp. 938–940, Jun 1990. [25] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1968.

Roberto D. Graglia (S’83–M’83–SM’90–F’98) was born in Turin, Italy, on July 6, 1955. He received the Laurea degree (summa cum laude) in electronic engineering from the Polytechnic of Turin, in 1979 and the Ph.D. degree in electrical engineering and computer science from the University of Illinois at Chicago, in 1983. From 1980 to 1981, he was a Research Engineer at CSELT, Italy, where he conducted research on microstrip circuits. From 1981 to 1983, he was a Teaching and Research Assistant at the University of Illinois at Chicago. From 1985 to 1992, he was a Researcher with the Italian National Research Council (CNR), where he supervised international research projects. In 1991 and 1993, he was Associate Visiting Professor at the University of Illinois at Chicago. In 1992, he joined the Department of Electronics, Polytechnic of Turin, as an Associate Professor and has been a Professor in the Electrical Engineering Department since 1999. He has authored over 150 publications in international scientific journals and symposia proceedings. His areas of interest comprise numerical methods for high- and low-frequency electromagnetics, theoretical and computational aspects of scattering and interactions with complex media, waveguides, antennas, electromagnetic compatibility, and low-frequency phenomena. He has organized and offered several short courses in these areas. Dr. Graglia has been a Member of the editorial board of ELECTROMAGNETICS since 1997. He is a past Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. He is currently an Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS and a Reviewer for several international journals. He was the Guest Editor of a special issue on Advanced Numerical Techniques in Electromagnetics for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION in March 1997. He has been Invited Convener at URSI General Assemblies for special sessions on Field and Waves in 1996, Electromagnetic Metrology in 1999, and Computational Electromagnetics in 1999. He served the International Union of Radio Science (URSI) for the triennial International Symposia on Electromagnetic Theory as Organizer of the Special Session on Electromagnetic Compatibility in 1998 and was the co-organizer of the special session on Numerical Methods in 2004. Since 1999, he has been the General Chairperson of the International Conference on Electromagnetics in Advanced Applications (ICEAA). He has been a member of the AP-S Administrative Committee for the triennium 2006–2008 and is presently an AP-S Distinguished Lecturer (2009–2012).

GRAGLIA et al.: CURL-CONFORMING HIERARCHICAL VECTOR BASES FOR TRIANGLES AND TETRAHEDRA

Andrew F. Peterson (S’82–M’83–SM’92–F’00) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Illinois at UrbanaChampaign, in 1982, 1983, and 1986, respectively. Since 1989, he has been a member of the faculty of the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, where he is now Professor and Associate Chair for Faculty Development. He teaches electromagnetic field theory and computational electromagnetics, and conducts research in the development of computational techniques for electromagnetic scattering, microwave devices, and electronic packaging applications. He is the principal author of Computational Methods for Electromagnetics (IEEE Press, 1998). Dr. Peterson is a past recipient of the ONR Graduate Fellowship and the NSF Young Investigator Award. He has served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEE Antennas and Wireless Propagation Letters, as the General Chair of the 1998 IEEE AP-S International Symposium and URSI/USNC Radio Science Meeting, and as a member of IEEE AP-S AdCom. He also served for six years as a Director of ACES, and two years as Chair of the IEEE Atlanta Section. He was the President of the IEEE AP-S during 2006. He is a recipient of the IEEE Third Millennium Medal. He is also a Fellow of the Applied Computational Electromagnetics Society (ACES), and a member of the International Union of Radio Scientists (URSI) Commission B, the American Society for Engineering Education, and the American Association of University Professors.

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Francesco P. Andriulli (S’05–M’09) received the Laurea degree in electrical engineering from the Politecnico di Torino, Italy, in 2004, the M.S. degree in electrical engineering and computer science from the University of Illinois at Chicago in 2004, and the Ph.D. degree in electrical engineering from the University of Michigan at Ann Arbor in 2008. From 2008 to 2010, he was a Research Associate with the Politecnico di Torino. Since 2010 he has been with the Microwave Department, École nationale supérieure des télécommunications de Bretagne (TELECOM Bretagne), Brest, France, where he is currently a Maître de conférences. His research interests are in computational electromagnetics with focus on preconditioning and fast solution of frequency and time domain integral equations, integral equation theory, hierarchical techniques, and single source integral equations. Dr. Andriulli was awarded the University of Michigan International Student Fellowship and the University of Michigan Horace H. Rackham Predoctoral Fellowship. He was the recipient of the best student paper award at the 2007 URSI North American Radio Science Meeting. He received the first place prize of the student paper context of the 2008 IEEE Antennas and Propagation Society International Symposium, where he authored and coauthored other two finalist papers. He was the recipient of the 2009 RMTG Award for junior researchers and was awarded a URSI Young Scientist Award at the 2010 International Symposium on Electromagnetic Theory.

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An Augmented Electric Field Integral Equation for Layered Medium Green’s Function Yongpin P. Chen, Student Member, IEEE, Lijun Jiang, Member, IEEE, Zhi-Guo Qian, Member, IEEE, and Weng Cho Chew, Fellow, IEEE

Abstract—This paper proposes an augmented electric field integral equation (A-EFIE) for layered medium Green’s function. The newly developed matrix-friendly formulation of layered medium Green’s function is applied in this method. By separating charge as extra unknown list, and enforcing the current continuity equation, the traditional EFIE can be cast into a generalized saddlepoint system. Frequency scaling for the matrix-friendly formulation is analyzed when frequency tends to zero. Rank deficiency and the charge neutrality enforcement of the A-EFIE for layered medium Green’s function is discussed in detail. The electrostatic limit of the A-EFIE is also analyzed. Without any topological loopsearching algorithm, electrically small conducting structures embedded in a general layered medium can be simulated by using this new A-EFIE formulation. Several numerical results are presented to validate this method at the end of this paper. Index Terms—Augmented electric field integral equation, dyadic Green’s function for layered medium, low frequency.

I. INTRODUCTION OMPUTATIONAL electromagnetics becomes indispensable as a CAD methodology in various electrical engineering applications, such as in integrated circuit and wireless communication device. The operating frequency of the electrical systems keeps on increasing to several gigahertz, meanwhile fabrication process has achieved nanoscale. Hence, a broadband simulation tool is badly needed for capturing circuit physics of the tiny structures as well as wave physics for the whole package. Unfortunately, the commonly used electric field integral equation (EFIE) method solved by the method of moments (MoM) [1] with the Rao-Wilton-Glisson (RWG) basis function [2] suffers from a “low frequency breakdown” problem, where vector potential gradually looses its significance compared with the scalar potential part when the frequency decreases, and the EFIE operator becomes singular [3]. Various approaches have been proposed to overcome this

C

Manuscript received July 09, 2010; accepted September 15, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported by HKG GRF Grants 711508, 711609 (General Research Fund). Y. P. Chen and L. Jiang are with the Department of Electrical and Electronic Engineering, University of Hong Kong, China. Z.-G. Qian is with Intel Corporation, Chandler, AZ 85226 USA. W. C. Chew is with the Department of Electrical and Electronic Engineering, University of Hong Kong, China on leave from the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103042

problem in the last few years. One of the most popular remedies is the loop-tree or loop-star decomposition [4], [5], where the solenoidal and irrotational components of the unknown current can be separated due to the quasi-Helmholtz decomposition (also known as Hodge decomposition), to capture inductance physics and capacitance physics when the frequency tends to zero. However, even after frequency normalization, the matrix is still ill-conditioned. Preconditioning is necessary to improve the convergence when iterative solvers are applied. Several effective preconditioners have been proposed, either based on the basis-rearrangement, where the favorable property of electrostatic problems is utilized [6], or based on the near-field interactions, where the incomplete factorization with a heuristic drop strategy is applied [7]. By using the Calderón identity and the dual basis or Buffa-Christiansen basis function [8], [9], a more effective preconditioner has been constructed [10]–[12]. The loop-tree or loop-star method has also been implemented with the layered medium Green’s function [13]–[16], which is more versatile in the simulation of printed antenna and planarly integrated circuit [17]–[19]. However, one big issue associated with the loop-tree or loopstar method is the loop-search process. It is a bottleneck for complicated interconnecting geometries with increasing number of unknowns, where many entangle global loops may exist. Situation becomes even worse when layered medium with conducting ground plane is involved, where extra implicit global loops are introduced because of the vias. To avoid the loopsearch process, the idea of separating current and charge to construct a stable formulation has been studied recently. The current and charge integral equation (CCIE) method [20] puts charges into the extra unknown list and manipulates the equation system to be of the second kind. While in the separated potential integral equation (SPIE) method [21], the scalar potential is included as the unknowns, where resistive loss and dielectric loss are introduced to flatten the condition number when the frequency is low. In the recent developed augmented electric field integral equation (A-EFIE) method [22], [23], the similar idea of separating current and charge as independent unknowns is applied. By enforcing the current continuity equation explicitly and implementing a proper frequency scaling, the EFIE can be cast into a generalized saddle point system [24]. With the help of a constraint preconditioner for the saddle point system and the mixed-form fast multipole algorithm, a real package problem with more than one million unknowns has been successfully solved on a personal computer [23]. In this paper, we extend the A-EFIE method into the layered medium problem, to make it applicable for simulating

0018-926X/$26.00 © 2010 IEEE

CHEN et al.: AN AUGMENTED ELECTRIC FIELD INTEGRAL EQUATION FOR LAYERED MEDIUM GREEN’S FUNCTION

structures embedded in a general layered substrate. The matrix-friendly layered medium Green’s function [15], [16] is applied to achieve the A-EFIE formulation. First, the A-EFIE formulation for free space Green’s function is reviewed. Then, the frequency scaling is analyzed for both lossless and lossy media and the corresponding A-EFIE is set up for layered medium. Next, the rank deficiency and the necessity of enforcing charge neutrality condition is discussed in detail in the context of layered medium. After that, the electrostatic limit of the A-EFIE is analyzed to show the consistency with the static solver. Finally, several numerical examples are presented to validate this method. II. THE AUGMENTED EFIE FORMULA FOR FREE SPACE GREEN’S FUNCTION For a conducting object illuminated by an external excitation, electric current will be induced on the surface. This current will then generate an electric field that exactly cancels the incident electric field inside the object and will yield a scattered field in the outside region due to the extinction theorem [3]. The electric field integral equation (EFIE) can then be derived as

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After applying the Galerkin procedure [1], the EFIE can be converted into a matrix system (7) with (8) is the free space wave impedance and is the wave here number. The and correspond to the magnetic vector potential and electric scalar potential (9) (10) From (8)–(10), we can see that the vector potential block and the scalar potential one are imbalanced when the frequency is , since they are in different frequency order. low, namely The operator becomes singular because any divergence-free current is a solution to the EFIE in the quasi-static limit,

(1) where the operator connects the source current and the scattered field via the free space dyadic Green’s function

(2)

(11) To balance the system, the charge can be separated and added into the unknown list to make the system stable in an augmented fashion [22], [23]. We define the normalized pulse basis function on each triangular patch as

(3) here the scalar Green’s function is the solution to the scalar Helmholtz wave equation with a point source

otherwise

If further defining a patch-to-patch scalar potential matrix as (13)

(4) To solve the EFIE in (1), the induced current can be expanded by using the RWG basis function [2] defined on an adjacent triangular patch pair

(12)

we can obtain the relationship between the patch-pair based (in terms of divergence of RWG basis) scalar potential matrix and the patch-based one (14)

(5) otherwise is the area of the two triangles associated with the th where is the vector pointing to a point from the two basis, and the vertices. Here, the function is normalized by its edge length for convenience. The surface divergence can then be written as

where the incidence matrix basis and the patch basis,

relates the domain of the RWG

Patch is the positive part of RWG Patch is the negative part of RWG otherwise.

(15)

Due to the current continuity equation, we have (16)

(6) otherwise.

where is the light speed in vacuum and is the charge unknowns. Substituting the above equations into the EFIE matrix

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equation, and enforcing the current continuity equation explicitly, we can arrive at the following augmented-EFIE (A-EFIE) system

(27) (28) (29)

(17) This equation is the generalized saddle point system with the lower right block nearly equals to zero and various methods can be applied to solve this problem efficiently [24]. III. THE AUGMENTED EFIE FORMULA FOR LAYERED MEDIUM GREEN’S FUNCTION A. The Dyadic Green’s Function for Layered Medium In a planarly layered medium, the dyadic Green’s function has no closed-form spatial domain solution, it can only be expressed as an infinite integral (Sommerfeld integral). Various approaches have been developed to derive the layered medium Green’s function, for instance, transmission line analog [13], formulation [25], and Hertz potential approach [14], vector wave formula [26]. Here, we apply the last one since it can be cast into a matrix-friendly form [15], [16], which has the singularity with the lowest order. The dyadic Green’s function takes the form of

(30)

B. Frequency Scaling Since the primary (direct) term can be analyzed in the similar way as in free space, only secondary (reflected or transmitted) terms are considered in this section. We first assume that the layered medium is lossless. For general case, namely and , when , , the frequency scaling of the Fresnel reflection coefficient is

(31) for TE wave and for TM wave. Then we can where get the frequency scaling for other quantities: (32) where , and can be found in [26]. Then the frequency scaling for the propagation factor is

(18)

(33)

where , is the index of the layer where source is triangle resides and for observation layer. The expressed as a Sommerfeld integral,

Finally the frequency scaling for the matrix element in (21)–(25) is

(19)

(34)

where is the propagation factor [26] in direction for a given , and represents TE or TM polarization. In moment method implementation, the matrix-friendly formula is much more preferred, where the dyadic Green’s function is divided into pieces and incorporated into the matrix elements

We can separate the matrix into two parts according to the frequency scaling

(20)

(35) (36) So that we can have

where

(37) (21) (22) (23) (24) (25)

with the Green’s function components as (26)

where

(38) Equation (37) has the same form as (8) in free space, which allows us to augment the EFIE in a similar fashion as in (17). Since most material is non-magnetic, namely , we discuss this situation separately. The frequency scaling of the

CHEN et al.: AN AUGMENTED ELECTRIC FIELD INTEGRAL EQUATION FOR LAYERED MEDIUM GREEN’S FUNCTION

Fresnel reflection coefficient for TE wave is an high order term of frequency, (39) Then we have

the difficulty in defining a uniform scalar potential. In the maand manifest the diftrix friendly formula, we can see ferent response of a horizontal and a vertical dipole in a layered medium. If we asymptotically make the inhomogeneity disapand for each layer, and pear, namely, make applying the Sommerfeld identity

(40) However, we still have (41) , other four terms in Notice the fact that except for the (22)–(25) consist of TE as well as TM wave, and the TM part . This means that the frequency scaling for is still on these four terms in (34) are still valid. By careful dimensional , the leading order term analysis, we can show that even for . In a word, for non-magnetic material, (34)–(38) is still are also valid. Dielectric loss and conductor loss can be introduced to alleviate the low frequency breakdown in free space [21]. For a structure embedded in a layered medium, if we introduce dielectric loss to each layer, since the equivalent permittivity is

(42) the frequency scaling of the

becomes

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(46) and recover the polarization independent vector pothe , it goes back tential in (9). Similar situation holds for the to the scalar potential part in (10) when removing the inhomogeneity. The physical meanings of the remaining two terms and are ambiguous due to the lack of exact definition of scalar potential. There is no correspondence in free space, since the TE and TM waves cancel each other when the layered medium degrades into a free space. By appearance, we can interpret it as the cross interaction between charge and the vertical and current. According to their same frequency scaling with , we can group them together to obtain the layered medium A-EFIE, as is done in (35). 2) Half Space With PEC Layer: For a half space with a PEC layer, the image method can be applied and the dyadic Green’s function can be expressed in a closed form [27]

(47) and is the free space scalar Green’s where function with real source point and image source point , with and defined as

(43) (48) (49)

so the scalar potential matrix becomes (44) The A-EFIE can then take the alternative form (45)

C. Consistency Validation We discuss two extreme cases to analyze the consistency of the A-EFIE for layered medium Green’s function. One with homogeneous layers (free space) and the other with perfect electrical conductor (PEC) layer (half space), both of which have closed form Green’s functions. 1) Free Space: In free space, the EFIE can be separated into two parts, the vector potential part and the scalar potential part, both of which are scalar problems with scalar Green’s function, because of the homogeneity of the medium, shown in (7)–(10). However, in the layered medium, the response of a dipole is polarization dependent. A vertical electric dipole can only generate a TM wave, while a horizontal electric dipole generates TE as well as TM waves. The polarization dependency leads to

where we assume the interface is at , so , . Notice the fact that and and at the interface in the propagation factor , we can reproduce (48) and (49) by our general A-EFIE formula, with the help of Sommerfeld identity. and cannot be validated by this Although the terms two cases, we can show their significance by numerical examples, where vertical structure exists in a dielectric layered medium, and the two terms are always there with nonzero value. IV. CHARGE NEUTRALITY ISSUE Charge neutrality enforcement is very important in the A-EFIE for low-frequency problems, as stated in [23]. The motivation of enforcing the charge neutrality is because of the rank deficiency of the A-EFIE. For the A-EFIE shown in (17), the upper block is exactly the same as the traditional EFIE except that the scalar potential part is expressed in terms of patch basis and charge is separated as a set of independent unknowns. The lower block constraints the current continuity condition to make the augmented system solvable. However, rank deficiency exists in the A-EFIE matrix, due to the definition of the

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incidence matrix . Here, is an eigenvalue of the A-EFIE and it tends to zero finally when the frequency goes to DC [23]. Usually, the deflation method [28] can be applied to remove the smallest eigenvalue, for example, in the CCIE Formula [20]. Motivated by the basis rearrangement preconditioner in the loop-tree decomposition [6], we can also apply the charge neutrality enforcement to remedy this problem [23]. This is driven by the physical observation of the problem, and can be easily extended to different layered medium problems. We discuss this issue in the context of layered medium. If it is backed by a conducting ground plane, which is a common situation in the circuit problems, it acts as a “charge bath” and absorbs the extra charge of the structure, so we should distinguish situations whether there is a via connected to the ground in some parts of the structure. As will be shown in the following, the condition number is always bounded when the frequency goes to DC, by properly enforcing the charge neutrality condition.

Fig. 1. The geometrical structure of the loop inductor embedded in a sevenlayer medium, unit: mm. The central layer is a magnetic material. A delta gap excitation is applied at the center of the top arm.

A. Structures Not Connected to a Ground Plane If the structure is not connected to the “charge bath”, the total charge is always neutral. This condition shall be enforced when in low frequency domain due to the rank deficiency. By defis linearly dependent inition of the RWG basis, the matrix or singular. It is evident that the lower block sub-matrix of the ). It A-EFIE is rank deficient when the frequency is low ( is necessary to enforce the charge neutrality condition to make the lower block with full rank. Two transform matrices can be introduced to fulfill the charge neutrality enforcement [23], and the final A-EFIE system becomes:

Fig. 2. The condition number versus frequency for the rectangular loop. The condition number is unbounded when decreasing the frequency. Charge neutrality enforcement (CNE) makes the condition number constant.

(50) with the forward and backward transform as (51) where is the reduced charge unknowns and the is the reduced identity matrix. A rectangular loop embedded in a seven-layer medium is shown in Fig. 1, with its layer parameters specified in the figure. The condition numbers versus frequency are demonstrated in Fig. 2. we can see that without charge neutrality enforcement, the condition number grows unboundedly when decreasing the because of the right frequency, it increases in the order of lower block. The eigenvalue distribution of the A-EFIE matrix is shown in Fig. 3. After the charge neutrality enat forcement, the smallest eigenvalue has been removed away from the origin. We can also observe that when frequency increases, however, the lower-right block is an identity matrix scaled by , thus the lower block is no longer singular and such enforcement is no longer necessary. B. Structures Connected to a Ground Plane In this case, the charge neutrality condition cannot be guaranteed since the “charge bath” absorbs the extra charge. The incidence matrix is no longer singular. In this situation, no special

Fig. 3. The eigenvalue distribution for the rectangular loop at 1 Hz. The smallest eigenvalue is removed away from the origin after the charge neutrality enforcement (CNE).

treatment is needed since the A-EFIE system is with full rank. For a half rectangular loop connected to the ground plane shown in Fig. 4, the condition number versus frequency is shown in Fig. 5. Since the A-FEIE matrix is no longer singular because of the ground plane, the condition number remains constant when decreasing the frequency without any special treatment. Here, the forward and backward transform matrices become the identity matrix in this situation (52)

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Fig. 4. The geometrical model of the half loop embedded in a five-layer medium (including the PEC layer), unit: mm. A delta gap excitation is applied at the center of the top arm.

Fig. 6. The geometrical model of the circular parallel plate capacitor, with a : ) inserted in between. A delta gap is applied at the dielectric layer ( edge. The mesh is refined to capture the fringing effect.

= 2 65

Fig. 5. The condition number versus frequency for the half loop. Since it is connected to the ground plane, charge neutrality cannot be guaranteed. The condition number is bounded when decreasing the frequency without any special treatment.

This observation can be used as a guideline when dealing with complex structures. For a structure with independent surfaces, triangular patches, inner edges and ground each with . If there are surfaces connected to the edges, ground plane, then the total number of unknowns is

(53) One should note that though the number of unknowns in A-EFIE increases much compared to the loop-tree decomposition, where the number of unknowns is the same as the number of RWG basis, the memory requirement increases marginally since all transformation matrices such as , and are all sparse and consume marginal memory when iterative solver is applied.

is the static laywhere is the surface charge density, ered medium Green’s function [30] and is the potential gen. In this limit, a typical problem is the erated by the source parallel plate capacitor, shown in Fig. 6. Here we connect the two plates with a narrow strip so that we can apply the electrodynamic analysis. In this limit, the A-EIFE suffers from an inaccuracy problem, even though the matrix is nonsingular, because the current is a higher order term in frequency. To capture current accurate to arbitrary order, the perturbation method should be applied [29]. We will show that the charge is always stable and describes the electrostatic physics. In DC, the A-EFIE becomes (57) and are matrix evaluated at . For this where problem, the current disappears, while the charge remains constant,

(58) Since the matrix is still full rank, we have unique solution. However, the current is no longer correct due to the finite numerical term manually, we have precision. If we discard the

(59)

V. ELECTROSTATIC LIMIT In the electrostatic limit, the electric potential is expressed by the following boundary value problem [3] (54) (55)

It is shown that the electrostatic information is included in the A-EFIE. In free space, it is straightforward to check that the static Green’s function in (56) and (59) is the same one

(60)

and the integral equation becomes

(56)

In a layered medium, we will show the static form of the genin the appendix, eral matrix-friendly Green’s function in

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Fig. 7. The input reactance of the rectangular loop. A-EFIE agrees well with the loop-tree (LT) decomposition, while the traditional EFIE breaks down quickly when decreasing the frequency.

Fig. 8. The input reactance of the half loop. A-EFIE maintains the scale invariance very well while the traditional EFIE breaks down quickly when decreasing the frequency. Since the non-magnetic dielectric is transparent to the inductor, a PEC half space model is applied to validate the results.

which agrees with that of [30] applied in the electrostatic analysis except for minor differences such as the layer index, conand the sign definition in the Fresnel reflection costant efficient. Numerical results will be given to further validate our statement in the next section. VI. NUMERICAL RESULTS Several numerical results are presented in this section. The input reactance of the rectangular loop shown in Fig. 1 is calculated, where the results are compared with the loop-tree (LT) decomposition and traditional EFIE in Fig. 7. The EFIE breaks down quickly when the frequency decreases, while the A-EFIE is very stable and agrees very well with the loop-tree decomposition. We also calculate the input reactance of the half rectangular loop mentioned above. Because it is connected to a conducting ground plane, the current can flow along this half loop. The input reactance is shown in Fig. 8 compared with traditional EFIE. Similar phenomenon can be observed. Since non-magnetic dielectric is transparent to the loop inductor, the PEC half space model can be used to validate the result, which is also shown in Fig. 8. Good agreement with the A-EFIE for general layered medium can be observed. Finally, a circular parallel plate capac) is shown in Fig. 6 with itor with radius of unit length (

Fig. 9. The capacitance of the circular parallel plate capacitor. A-EFIE I represents the capacitance extracted from current, while A-EFIE Q means the capacitance extracted from charge. The A-EFIE current suffers from an inaccuracy problem while the A-EFIE charge is stable. The result agrees with the static solver. Both are further validated by the analytic solution. When the frequency is below 1 MHz, the relative error of A-EFIE Q is around 0.1%.

a dielectric layer inserted in between ( ). The distance is set to be . The capacitance extracted from current and charge and static solver are shown in Fig. 9. The current suffers from an inaccuracy problem, as mentioned in last section, while the charge is accurate and agrees with the static analysis. The analytic result from asymptotic expansion [31] is also shown to validate the numerical results. In this example, when the frequency is below 1 MHz, the relative error of the A-EFIE with charge information is around 0.1%. If the frequency is increased, wave physics begins to play a role and the parallel plate is no longer a pure capacitor. VII. CONCLUSION An augmented EFIE for layered medium Green’s function is developed in this paper. The frequency scaling is analyzed for both lossless and lossy media. The rank deficiency of the A-EFIE in layered media depends on if the charge neutrality condition is satisfied. For independent structures, the enforcement is necessary in the low frequency regime, while at midfrequencies, such implementation is no longer necessary. For structures connected to the ground, the A-EFIE matrix is full rank, and no special treatment is needed. The electrostatic limit is analyzed and compared with the static formulation. Several numerical results are presented to validate this method. APPENDIX ELECTROSTATIC LAYERED MEDIUM GREEN’S FUNCTION In the electrostatic limit, the layered medium Green’s function shown in [30] can be derived from our general matrixfriendly formulation

(61) where the Green’s function is described by several images with weight and distance .

CHEN et al.: AN AUGMENTED ELECTRIC FIELD INTEGRAL EQUATION FOR LAYERED MEDIUM GREEN’S FUNCTION

: When four image terms

, there are one primary term and

(62) (63) (64) (65) (66) : When

, there are four image terms (67) (68) (69) (70)

: When

, there are also four image terms (71) (72) (73) (74)

where

(75) (76) (77) (78)

(79) (80)

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Yongpin P. Chen (S’09) was born in Zhejiang, China, in 1981. He received the B.S. and M.S. degrees in microwave engineering from the University of Electronic Science and Technology of China, Chengdu, in 2003 and 2006, respectively. He is currently working towards the Ph.D. degree in electrical engineering in the University of Hong Kong, Hong Kong, China. His research interests include waves and fields in inhomogeneous media, integral equation methods and fast algorithms in computational electromagnetics.

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

Zhi-Guo Qian (S’07–M’09) received the B.S. and M.S. degrees in electrical engineering from Southeast University, Nanjing, China, in 2001 and 2004, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois, Urbana-Champaign, in 2009. He is now an Electrical Packaging Engineer with Intel Corporation, Chandler, AZ. Before joining Intel, he was with Apache Design Solutions, San Jose, CA. His research interests are power integrity and signal integrity analysis for package and board designs. Dr. Qian was the recipient of the Intel Best Student Paper Award presented at the IEEE 17th Topical Meeting on Electrical Performance of Electronic Packaging (EPEP) in 2008.

Weng Cho Chew (S’79–M’80–SM’86–F’93) received the B.S. degree in 1976, both the M.S. and Engineer’s degrees in 1978, and the Ph.D. degree in 1980, from the Massachusetts Institute of Technology, Cambridge, all in electrical engineering. He is serving as the Dean of Engineering at The University of Hong Kong. Previously, he was a Professor and the Director of the Center for Computational Electromagnetics and the Electromagnetics Laboratory at the University of Illinois. He was a Founder Professor of the College of Engineering, and previously, the First Y.T. Lo Endowed Chair Professor in the Department of Electrical and Computer Engineering, University of Illinois. Before joining the University of Illinois, he was a Department Manager and a Program Leader at Schlumberger-Doll Research. His research interests are in the areas of waves in inhomogeneous media for various sensing applications, integrated circuits, microstrip antenna applications, and fast algorithms for solving wave scattering and radiation problems. He is the originator several fast algorithms for solving electromagnetics scattering and inverse problems. He has led a research group that has developed parallel codes that solve dense matrix systems with tens of millions of unknowns for the first time for integral equations of scattering. He has authored the book, Waves and Fields in Inhomogeneous Media, coauthored two books, Fast and Efficient Methods in Computational Electromagnetics and Integral Equation Methods for Electromagnetic and Elastic Waves, and authored and coauthored over 300 journal publications, over 400 conference publications and over ten book chapters. Dr. Chew is a Fellow of the IEEE, OSA, IOP, Electromagnetics Academy, Hong Kong Institute of Engineers (HKIE), and was an NSF Presidential Young Investigator (USA). He received the Schelkunoff Best Paper Award from the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEE Graduate Teaching Award, UIUC Campus Wide Teaching Award, and IBM Faculty Awards. In 2008, he was elected by the IEEE AP Society to receive the Chen-To Tai Distinguished Educator Award. He served on the IEEE Adcom for the Antennas and Propagation Society as well as the Geoscience and Remote Sensing Society. From 2005 to 2007, he served as an IEEE Distinguished Lecturer. He served as the Cheng Tsang Man Visiting Professor at Nanyang Technological University in Singapore in 2006. In 2002, ISI Citation elected him to the category of Most-Highly Cited Authors (top 0.5%).He is currently the Editor-in-Chief of JEMWA/PIER journals, and is on the board of directors of the Applied Science Technology Research Institute, Hong Kong.

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A Time-Domain Volume Integral Equation and Its Marching-On-in-Degree Solution for Analysis of Dispersive Dielectric Objects Yan Shi, Member, IEEE, and Jian-Ming Jin, Fellow, IEEE

Abstract—A marching-on-in-degree (MOD)-based scheme for analyzing transient electromagnetic scattering from three-dimensional dispersive dielectric objects is proposed. A time-domain volume integral equation (TDVIE) for the electric flux density throughout the object is first formulated and then solved using the MOD scheme. With the use of weighted Laguerre polynomials as entire-domain temporal basis functions, the convolution of the electric flux density and the medium susceptibility and its derivatives can be handled analytically. By employing the Galerkin temporal testing procedure, the time variable is eliminated in the resultant recursive matrix equation so that the proposed algorithm overcomes the late-time instability problem that may occur in the conventional marching-on-in-time (MOT) approach. Some complex dispersive media, such as the Debye, Lorentz, and Drude media, are simulated to illustrate the validity of the TDVIE-MOD algorithm. Index Terms—Electric flux density, marching-on-in-degree (MOD), medium susceptibility, time-domain volume integral equation (TDVIE), weighted Laguerre polynomials.

I. INTRODUCTION

T

HE ability to analyze transient electromagnetic wave interactions with frequency dispersive materials is critical to applications ranging from analysis of the radar cross section (RCS) for an aircraft to design of antennas and microwave circuits. In particular, with the recent development in engineered metamaterials [1], for instance, photonic bandgap (PBG) structures and double negative media (DNM), the need to simulate dispersive electromagnetic systems become increasingly important. One of the most common approaches to analyzing transient electromagnetic wave interactions with dispersive media is the Manuscript received May 12, 2010; revised July 26, 2010; accepted July 31, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by the National Natural Science Foundation of China under Contracts 60801040 and 61072017, by the Natural Science Basic Research Plan in Shaanxi Province of China (Program 2010JQ8013), by the Program for New Century Excellent Talents in University of China, by National Key Laboratory Foundation, by the Fundamental Research Funds for the Central Universities, and by the China Scholarship Council (CSC). Y. Shi is with the School of Electronic Engineering, Xidian University, Xi’an 710071, Shaanxi, China and also with the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA (e-mail: [email protected]). J.-M. Jin is with the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 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.2010.2103038

time-domain differential equation (TDDE)-based techniques, for example, the finite-difference time-domain (FDTD) method and the time-domain finite element method (TDFEM). There are roughly three types of frequency-dependent TDDE-based methods [2]: the first is to implement a discrete convolution of the electric field and the permittivity; the second is to discretize a differential equation relating the electric flux density to the electric field intensity; and the third is to utilize the Z-transform method. However, when analyzing electromagnetic radiating and scattering problems, the TDDE-based techniques need to implement approximate absorbing boundary conditions (ABC) or perfectly matched layers (PML) to truncate the computational domain. In contrast, the time-domain integral equation (TDIE)-based techniques [3] do not require such a truncation and hence are often preferred for such open-region problems. The most popular method to solve the TDIE is the marching-on-in-time (MOT) scheme. Unfortunately, the MOT-based approaches often suffer from the late-time instability in the form of spurious oscillation. Various approaches to improving the numerical instability have been proposed, for instance, by employing the temporal and spatial averaging techniques [4]–[6], using better temporal and spatial basis functions [7]–[10], and adopting implicit time-stepping schemes [11], [12]. Up to now, the MOT-based schemes have been applied to analyze electromagnetic scattering from conducting [4]–[9], lossy dielectric [13], and dispersive [14] objects. Besides the MOT-based techniques, the marching-on-in-degree (MOD) scheme [15]–[20] has been proposed in recent years to solve TDIEs. In the MOD-based schemes, a set of orthonormal basis functions, such as Laguerre polynomials, is utilized to expand the temporal variation of the unknown current density. With the Galerkin temporal testing procedure, the resultant recursive equations eliminate the time variable and thus overcome the late-time instability problem. The MOD-based schemes have been successfully implemented for the solution of electromagnetic scattering from conducting objects [15]–[17] and homogeneous dielectric objects [18]–[20]. In this paper, we propose an approach that solves the time-domain volume integral equation (TDVIE) formulated in terms of the electric flux density using MOD to analyze the transient electromagnetic scattering from three-dimensional dispersive dielectric objects. Different from the approach in [14], which adopts the MOT scheme to simultaneously solve the TDVIE and the auxiliary equation relating the volume electric flux density to the electric field intensity, the proposed TDVIE is combined with the auxiliary equation and then solved

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by the MOD scheme. By expanding the electric flux density and the medium susceptibility in terms of weighted Laguerre polynomials, the convolution of the electric flux density and the medium susceptibility and its temporal derivatives can be handled analytically. With the use of the MOD scheme, the proposed algorithm eliminates the late-time instability problem. Numerical results are given to illustrate the validity of the proposed algorithm. II. FORMULATION This section presents the formulations of the TDVIE and describes its MOD solution. A. Time-Domain Volume Integral Equation (TDVIE) inciConsider a transient electromagnetic wave dent on an inhomogeneous dielectric object with a volume which resides in free space with permittivity and permeability , as illustrated in Fig. 1. The object is assumed to be isotropic, nonmagnetic and electrically dispersive, and its peris given by mittivity

Fig. 1. An inhomogeneous dispersive dielectric object illuminated by an incident impulse.

as the unknown quantity to be solved for. In this scenario, by as introducing the medium susceptibility

(1) (6) where denotes an inverse Fourier transform and is the permittivity expressed as a function of frequency . In the dispersive dielectric body, Maxwell-Ampere’s law can be expressed as

in which part of

is the frequency-domain counter, (4) can be rewritten as (7)

(2) in which the equivalent volume electric current

denotes

Substituting (3) and (7) into (5), the time-domain volume inis tegral equation (TDVIE) for the electric flux density given by

(3) and the electric flux density sity by

is related to the field inten-

(4) where and are the total electric and magnetic fields, respectively, and denotes the convolution. In is equal to the dielectric body, the total electric field the sum of the incident field and the scattered field radiated by , which is given by

(8) can be decomposed as In general, the permittivity where , and can be expressed as a rational polynomial. For instance, for the Debye medium with the static the permittivity and relaxation can be expressed as [14] permittivity (9)

(5) where denotes the retarded time, is the distance between the observation and source points, and is the speed of light. Because has a continuous normal component at a media interface, it is natural to choose

Substituting (9) into (6), the medium susceptibility the Debye medium can be obtained as

for

(10)

SHI AND JIN: A TDVIE AND ITS MOD SOLUTION FOR ANALYSIS OF DISPERSIVE DIELECTRIC OBJECTS

where

is the unit step function. For the Lorentz medium, is expressed as [14]

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is a complex coefficient and is a complex in which exponent whose real part is greater than zero. To simplify the formulation, in this paper we consider the case with , for which (17) is reduced to

(11) (18) in which denotes the resonant frequency and is the damping coefficient. For this medium, can be obtained as

The extension to more general cases is straightforward. B. Marching-on-in-Degree (MOD) Solution In order to solve the TDVIE (8), we expand the electric flux in terms of spatial and temporal basis functions density (19)

(12) For the Drude medium,

is written as [21] (13)

and

can be derived as

denote the spatial basis functions, denote where are the corresponding exthe temporal basis functions, and pansion coefficients. Here is a scaling factor used to control the support provided by the expansion [19]. According to Theorem 1 given in Appendix, (18) can also be expressed in terms as of the temporal basis functions

(20) (14) For a general dispersive medium with second-order poles first-order poles, can be given by [14] and

Using Theorem 4 in Appendix, the convolution of can be obtained as

and

(15) where as

. Thus,

can be obtained

(21) Using the recursive relation and the properties of the Laguerre polynomials (42)–(48) given in Appendix, the analytical expresand sions of the first and second derivatives of can be obtained as

(16) in which and , , , and are , , , , constants which can be expressed in terms of , and . According to (16), we can obtain a general expression of the as medium susceptibility (17)

(22)

(23)

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see (24)–(25) at the bottom of the page. Substituting (19), (21), (23) and (25) into (8) and testing the resultant equation using as the temporal testing functions and as the spatial testing functions, we have (26) in which the impedance matrix given by

at the left-hand side is (27)

where

(28)

(29)

(30)

(35)

(31) in which (32) and the excitation vector

at the right-hand side is given by (33) (34)

(36)

(24)

(25)

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(37)

(38) In the above, the integral kernel

is given by [15]

(39) Note that the number of temporal basis functions in (26) is infinite. In general, the signal that we are interested in is practically bandlimited up to a certain frequency. Therefore, a minimum can be determined. For number of temporal basis functions a signal with a time duration which is assumed to be bandlimited to a frequency , the range to sample the signal in the fre, where . Therefore, quency domain is the minimum number of temporal basis functions to approxi[19]. mate the signal becomes III. NUMERICAL EXAMPLES In this section, we use the TDVIE-MOD algorithm to solve for the volume current density in a dispersive dielectric object and its scattered far-field under a Gaussian impulse plane wave illumination. The transient waveform of the incident plane wave is defined as (40) (41) with and in which is the wave impedance of free space. Here represents the pulsewidth of the Gaussian impulse and denotes a time delay of the peak from the origin. for all the following examples The values of , , , and , 80, 4 lm, and 6 lm. The unit ’lm’ denotes light are set to time which is the unit of time taken by the electromagnetic wave to travel 1.0 m distance in free space. As the first example, we consider a plane wave scattering from a dielectric cube with the side length of 0.2 m, as shown in Fig. 2(a). The cube is modeled as a Debye medium with , and . The cube is discretized using 240 unknowns. The transient equivalent volume current at and the trandensity

Fig. 2. The discretization of four dispersive objects: (a) cube; (b) cylinder; (c) triangular cylinder; (d) slab.

sient backscatter far-field are calculated by the TDVIE-MOD algorithm, and the results are shown in Fig. 3(a) and (b). For com-

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Fig. 3. Transient Gaussian impulse plane wave scattered by a cube consisting of a Debye medium: (a) transient current density; (b) transient backscatter far-field; (c) the DFT of the transient backscatter far-field.

Fig. 4. Transient Gaussian impulse plane wave scattered by a cylinder consisting of a Lorentz medium: (a) transient current density; (b) transient backscatter far-field; (c) the DFT of the transient backscatter far-field.

parison, the results for the nondispersive case with obtained by the TDVIE-MOD algorithm are also shown in the figures. Fig. 3(c) shows the comparison between the results obtained by the discrete Fourier transform (DFT) of the TDVIE-MOD solution and the frequency-domain method-ofmoments (MoM) solution, and as can be seen, the agreement is excellent. The second example considers a transient Gaussian impulse plane wave scattered by a cylinder comprising of a Lorentz , , and material with

, as shown in Fig. 2(b). The radius and height of the cylinder are 0.25 m and 0.2 m, respectively, and the cylinder is discretized using 489 unknowns. Fig. 4(a) and (b) show the transient volume current density at and the transient backscatter far-field calculated by the TDVIE-MOD algorithm. The results for the nondispersive are also given to show the effect of the case with medium dispersion. Fig. 4(c) compares the results obtained by the DFT of the TDVIE-MOD solution and the frequency-domain MoM solution, which again shows an excellent agreement.

SHI AND JIN: A TDVIE AND ITS MOD SOLUTION FOR ANALYSIS OF DISPERSIVE DIELECTRIC OBJECTS

Fig. 5. Transient Gaussian impulse plane wave scattered by an equilateral triangular cylinder consisting of a Drude medium: (a) transient current density; (b) transient backscatter far-field; (c) the DFT of the transient backscatter far-field.

The third example concerns the transient scattering from an equilateral triangular dielectric cylinder with a side length of 0.2 m and a height of 0.2 m, as shown in Fig. 2(c). The triangular cylinder made of a Drude material with , and is discretized using 234 at unknowns. Fig. 5(a) shows the transient current density , and Fig. 5(b) and (c) show the transient backscatter far-field and the DFT result calculated by the TDVIE-MOD algorithm. As can be seen, the results obtained by the TDVIE-MOD algorithm are in good agreement with the frequency-domain MoM results.

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Fig. 6. Transient Gaussian impulse plane wave scattered by a slab consisting of a Debye medium: (a) transient current density; (b) transient backscatter far-field; (c) the DFT of the transient backscatter far-field.

The final example considers the transient scattering from a square dielectric slab with a side length of 1.5 m and a height of 0.05 m, as shown in Fig. 2(d). The slab is discretized using 930 unknowns and is made of the Debye material with , , and . Fig. 6(a) shows the at and transient current density Fig. 6(b) and (c) shows the transient backscatter far-fields and the DFT results. For comparison, the results for the nondisperare also calculated. Very good sive slab with agreement between the results calculated by the TDVIE-MOD

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algorithm and the frequency-domain MoM results can be observed.

where , Laguerre polynomials, and Proof:

, is a scaling factor.

are the

IV. CONCLUSION This paper presents a marching-on-in-degree (MOD)-based numerical algorithm to analyze transient electromagnetic scattering from a three-dimensional dispersive dielectric objects of arbitrary shape. A time-domain integral equation (TDVIE) is first formulated in terms of the volume electric flux density and then solved using the MOD scheme. With the use of weighted Laguerre polynomials as the temporal basis functions, the convolution of the volume electric flux density and the medium susceptibility of the dispersive medium and its derivatives are handled analytically. The resultant TDVIE-MOD scheme eliminates the late-time instability due to the use of the Galerkin temporal testing procedure. To validate the algorithm, the equivalent volume current density and the backscatter far-field are calculated and compared with the results of the frequency-domain MoM.

Using (45), we easily obtain

APPENDIX The Laguerre polynomials are defined for and therefore are causal. They can be calculated using the recursive relation [22] Considering that (42) in which

(43) The Laguerre polynomials have the following properties [22], [23]: (44) (45) (46) (47)

we have

Theorem 2: Given , we have . Proof:

and

(48) According to the properties of the Laguerre polynomials, we can obtain the following four theorems. and , we Theorem 1: Given have

Theorem 3: Given .

, we have

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Proof:

Theorem 4: Given , we have where

and

REFERENCES

Proof:

Using Theorem 2, we can have

Furhtermore, according to Theorem 3 and (46), we can obtain

On the other hand, considering

we have

[1] C. Caloz and T. Itoh, Electromagnetic Metamaterial: Transmission Line Theory and Microwave Applications. Hoboken, NJ: Wiley, 2006. [2] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Boston, MA: Artech House, 2005. [3] S. M. Rao, Time Domain Electromagnetics. San Diego: Academic, 1999. [4] D. A. Vechinski and S. M. Rao, “A stable procedure to calculate the transient scattering by conducting surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 40, pp. 661–665, 1992. [5] B. P. Rynne and P. D. Smith, “Stability of time marching algorithm for the electric field integral equation,” J. Electromagn. Waves Applic., vol. 4, pp. 1181–1205, 1990. [6] P. J. Davies and D. B. Duncan, “Averaging techniques for time-marching schemes for retarded potential integral equation,” Appl. Numer. Mathem., vol. 23, pp. 291–310, 1997. [7] J. L. Hu, C. H. Chan, and Y. Xu, “A new temporal basis function for the time-domain integral equation method,” IEEE Microw. Wireless Compon. Lett., vol. 11, pp. 465–466, 2001. [8] D. S. Weile, G. Pisharody, N. W. Chen, B. Shanker, and E. Michielssen, “A novel scheme for the solution of the time domain integral equations of electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, pp. 283–295, 2004. [9] G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 45, pp. 527–532, 1997. [10] R. A. Wildman, G. Pisharody, D. S. Weile, B. Shanker, and E. Michielssen, “An accurate scheme for the solution of the time-domain integral equations of electromagnetics using high-order vector bases and bandlimited extrapolation,” IEEE Trans. Antennas Propag., vol. 52, pp. 2973–2984, 2004. [11] S. M. Rao and T. K. Sarkar, “An efficient method to evaluate the time-domain scattering from arbitrary shaped conducting bodies,” Microwave Opt. Technol. Lett., vol. 17, pp. 321–325, 1998. [12] T. K. Sarkar, W. Lee, and S. M. Rao, “Analysis of transient scattering from composite arbitrarily shaped complex structures,” IEEE Trans. Antennas Propag., vol. 48, pp. 1625–1634, 2000. [13] B. Shanker, K. Aygun, and E. Michielssen, “Fast analysis of transient scattering from lossy inhomogeneous dielectric bodies,” Radio Sci, vol. 39, no. RS2007, 2004. [14] G. Kobidze, J. Gao, B. Shanker, and E. Michielssen, “A fast time domain integral equation based scheme for analyzing scattering from dispersive objects,” IEEE Trans. Antennas Propag., vol. 53, pp. 1215–1226, 2005. [15] Y. S. Chung, T. K. Sarkar, B. H. Jung, M. Salazar-Palma, Z. Ji, S. Jang, and K. J. Kim, “Solution of time domain electric field integral equation using the Laguerre polynomials,” IEEE Trans. Antennas Propag., vol. 52, pp. 2319–2328, 2004. [16] Z. Ji, T. K. Sarkar, B. H. Jung, M. T. Yuan, and M. Salazar-Palma, “Solving time domain electric field integral equation without the time variable,” IEEE Trans. Antennas Propag., vol. 54, pp. 258–262, 2006.

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[17] B. H. Jung, Z. Ji, T. K. Sarkar, M. Salazar-Palma, and M. Yuan, “A comparison of marching-on in time method with marching-on in degree method for the TDIE solver,” Progress Electromagn. Res., vol. 70, pp. 281–296, 2007. [18] B. H. Jung, T. K. Sarkar, Y. S. Chung, M. Salazar-Palma, Z. Ji, S. Jang, and K. Kim, “Transient electromagnetic scattering from dielectric objects using the electric field integral equation with Lagrerre polynomial as temporal basis functions,” IEEE Trans. Antennas Propag., vol. 52, pp. 2329–2339, 2004. [19] Y. S. Chung, T. K. Sarkar, and B. H. Jung, “Solution of time domain electric field integral equation for arbitrary shaped dielectric bodies using an unconditionally stable methodology,” Radio Sci, vol. 38, no. 2002RS002759, 2003. [20] B. H. Jung, T. K. Sarkar, and Y. S. Chung, “Solution of time domain PMCHW formulation for transient electromagnetic scattering from arbitrarily shaped 3D dielectric objects,” Progress Electromagn. Res., vol. 45, pp. 291–312, 2004. [21] M. Okoniewski and E. Okoniewska, “Drude dispersion in ADE FDTD revisited,” Electron. Lett., vol. 42, no. 9, pp. 503–504, Apr. 2006. [22] A. D. Poularikas, The Transforms and Applications Handbook. Piscataway, NJ: IEEE Press, 1996. [23] W. Furmanski and R. Petronzio, “A method of analyzing the scaling violation of inclusive spectra in hard processes,” Nucl. Phys. B, vol. 195, pp. 237–261, 1990.

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 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 Council 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.

Jian-Ming Jin (S’87–M’89–SM’94–F’01) received the B.S. and M.S. degrees in applied physics from Nanjing University, Nanjing, China, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 1989. He is currently the Y. T. Lo Endowed Chair Professor of Electrical and Computer Engineering and Director of the Electromagnetics Laboratory and Center for Computational Electromagnetics with the University of Illinois at Urbana-Champaign. He was appointed as the first Henry Magnuski Outstanding Young Scholar in the Department of Electrical and Computer Engineering in 1998 and later as a Sony Scholar in 2005. He was appointed as a Distinguished Visiting Professor in the Air Force Research Laboratory in 1999 and was an Adjunct, Visiting, or Guest Professor with the City University of Hong Kong, University of Hong Kong, Anhui University, Beijing Institute of Technology, Peking University, Southeast University, Nanjing University, and Shanghai Jiao Tong University. His name often appears in the University of Illinois at Urbana-Champaign’s List of Excellent Instructors. He was elected by ISI as one of the world’s most cited authors in 2002. He has authored or coauthored over 200 papers in refereed journals and 20 book chapters. He also authored The Finite Element Method in Electromagnetics (Wiley, 1st ed, 1993, 2nd ed, 2002) and Electromagnetic Analysis and Design in Magnetic Resonance Imaging (CRC, 1998), coauthored Computation of Special Functions (Wiley, 1996) and Finite Element Analysis of Antennas and Arrays (Wiley, 2008), and coedited Fast and Efficient Algorithms in Computational Electromagnetics (Artech, 2001). He was an Associate Editor for Radio Science and is also on the Editorial Board for Electromagnetics and Microwave and Optical Technology Letters. His current research interests include computational electromagnetics, scattering and antenna analysis, electromagnetic compatibility, high-frequency circuit modeling and analysis, bioelectromagnetics, and magnetic resonance imaging. Dr. Jin is a member of Commission B of USNC/URSI and Tau Beta Pi. He was an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He was a recipient of the 1994 National Science Foundation Young Investigator Award, the 1995 Office of Naval Research Young Investigator Award, and the 1999 Applied Computational Electromagnetics Society Valued Service Award. He was also the recipient of the 1997 Xerox Junior Research Award and the 2000 Xerox Senior Research Award presented by the College of Engineering, University of Illinois at Urbana-Champaign. He was the co-chairman and Technical Program chairman of the Annual Review of Progress in Applied Computational Electromagnetics Symposium in 1997 and 1998, respectively.

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Lyapunov and Matrix Norm Stability Analysis of ADI-FDTD Schemes for Doubly Lossy Media Ding Yu Heh, Student Member, IEEE, and Eng Leong Tan, Senior Member, IEEE

Abstract—Lyapunov and matrix norm stability analysis is applied on various alternating-direction-implicit finite-difference time-domain (ADI-FDTD) schemes for doubly lossy media. The stability analysis is performed rigorously in both time and Fourier domains. Among the schemes considered are averaging, forward-backward, backward-forward, forward-forward, exponential time differencing and backward-backward. From the analysis, it is found that all schemes except backward-backward scheme are unconditionally stable. For backward-backward scheme, the condition for stability is determined. Index Terms—Alternating-direction-implicit finite-difference time-domain (ADI-FDTD), lossy media, stability.

I. INTRODUCTION

T

HE alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method [1], [2] has been developed to remove the Courant-Friedrich-Lewy (CFL) constraint on the chosen time step, hence the unconditional stability feature. Such feature makes the ADI-FDTD method attractive for further extension into lossy media using different schemes [3]–[7]. Some stability analysis using the Von Neumann eigenvalue approach has been provided in [5] and [6] for several ADI-FDTD schemes for 1-D and 2-D electrically lossy media. The Von Neumann eigenvalue method, which converts the difference equation into Fourier domain for determining the magnitude of the eigenvalues is straightforward and popular for stability analysis. While this approach is still applicable for lossless ADI-FDTD method, or perhaps simpler specific case of electrically lossy in 1-D or 2-D, it is often tedious in 3-D doubly lossy media (both electric and magnetic conductivities are nonzero) due to the complexity of the updating matrix of the ADI-FDTD schemes. Of late, the energy-based method has been widely used as an alternative method to the Von Neumann eigenvalue approach. The unconditionally stable feature of the ADI-FDTD in lossless media has been shown in [8], while the most commonly used averaging scheme of ADI-FDTD method for lossy media is presented in [9]. The matrix norm approach for stability analysis is another viable option which has been applied in lossless Crank-Nicolson, split-step and ADI-FDTD method [10], [11]. Nevertheless, it appears that the full 3-D stability analysis of Manuscript received March 05, 2010; revised July 14, 2010; accepted August 24, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2103026

various possible ADI-FDTD schemes for the most general case of doubly lossy media are still lacking. In this paper, we shall present the Lyapunov and matrix norm stability analysis applied to various ADI-FDTD schemes for doubly lossy media. The stability analysis is performed rigorously in both time and Fourier domains. Among the schemes considered are averaging, forward-backward, backward-forward, forward-forward, exponential time differencing and backward-backward (These schemes will be elaborated further in Section IV). Averaging, exponential time differencing, forward-backward and backward-forward schemes have second order temporal accuracy. For forward-forward and backward-backward schemes, the temporal accuracy is only of first order. Generally, exponential time differencing has the highest accuracy for its closest resemblance to the solution of first-order differential equation, while averaging scheme is easier to formulate and is most commonly used. Forward-backward and backward-forward schemes, on the other hand, can be used for higher efficiency implementation due to involvement of fewer conductivity terms. The Lyapunov method, somewhat related to energy-based method, and matrix norm method are ideal stability analysis approaches for various ADI-FDTD schemes due to some unique features in its updating matrices, which will be shown later. For Lyapunov stability analysis, it will be shown that one does not need to solve the discrete Lyapunov equation to deduce the stability of the particular scheme. Instead, one would only require to determine the positive definiteness of a certain matrix. From there, the subtle relationship between the Lyapunov and energy-based methods can also be seen. On the other hand, the matrix norm stability analysis yields the same condition to ensure stability as the Lyapunov method. Hence, the unconditionally stable feature of various ADI-FDTD schemes is proven. For scheme which is found to be not unconditionally stable, the condition for stability is determined. The organization of this paper is as follows. Sections II and III present the generalized Lyapunov and matrix norm stability analysis, respectively, both in time domain. The generalized stability analysis is then applied to aforementioned various ADI-FDTD schemes for doubly lossy media in Section IV. In Section V, it is shown that the analysis can be similarly carried out in the Fourier domain using the Von Neumann method.

II. GENERALIZED LYAPUNOV STABILITY ANALYSIS Consider a discrete system

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(1)

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where is the state vector with time indices and , and is the amplification matrix. The system is said to be stable in the sense of Lyapunov if the state vector remains bounded near the . Furthermore, if the state vector converges origin for all , the system is asymtotically stable. The to the origin as Lyapunov theorem states that the above discrete system is stable in the sense of Lyapunov if there exists a positive definite to the discrete Lyapunov equation

Upon discretizing in time, the ADI-FDTD update equations for doubly lossy media generally read

(2)

(11)

is positive semidefinite. If there exists a positive such that definite to (2) such that is positive definite, the system is asymtotically stable [12]–[14]. In FDTD stability analysis, researchers are often interested in whether the solution will always remain bounded (nonincreasing) as the time marches. Therefore, it should be mentioned that throughout the paper, if the system is found to be either stable in the sense of Lyapunov, or asymtotically stable, we shall simply refer them as “stable”. In the following, the above Lyapunov theorem will be applied in the stability analysis of ADI-FDTD schemes for doubly lossy media. The discretized Maxwell equations in space describing an electromagnetics problem within a computational domain terminated by perfect electric conductor (PEC) walls can be expressed in compact matrix form as

(12)

(3) where

(10a) (10b) where

and

are composed of discretization coefficients of and

spatial derivatives matrices

, respectively. Their dimensions are also dependent on the size of computational domain. and are split from such that Note that they yield computationally efficient tridiagonal matrices in both , and are all diagonal matrices substeps of (10). which comprise , and material parameters matrices , , and . For the moment, we shall let matrices , and be general but they will be further defined accordingly for different schemes in the subsequent sections. Before the discrete Lyapunov equation is applied, the following transfomation of vector (13)

(4) is used to yield the overall ADI-FDTD update equations of (5) (6)

(14) where

(7) (8)

(15) (16)

(9) Note that , and are grid position indices within the whole computation domain along , and directions, respectively. is the curl matrix comprising spatial step , and resulting from the discetization of the curl operator through cenis the null matrix. The ditral differencing approximation. mensions of all matrices and vectors are dependent on the size are the medium of the computational domain. , , and permittivity, permeability, electric and magnetic conductivities, respectively. Henceforth, the indices , , of material parameshall be dropped for convenience. ters , , and

Note that as long as we can find a pair of positive definite (or positive semidefinite) and positive definite satisfying (2), the sufficiency for stability is guaranteed. To that end, we choose as (17) It can be shown that (Appendix I), along with given and in (17), can be determined from (2) as

in (14)

(18) where

.

HEH AND TAN: LYAPUNOV AND MATRIX NORM STABILITY ANALYSIS OF ADI-FDTD SCHEMES FOR DOUBLY LOSSY MEDIA

The first term inside the bracket in (18) involves similarity which preserves its eigenvalues (defitransformation of and are positive definite (positive niteness). Hence, if semidefinite), is positive definite (positive semidefinite). Furthermore, the specified in (17) is always positive definite for (otherwise the whole amplificanonsingular in (14) is undefined) [15]. Therefore, the suffition matrix cient condition for stability of ADI-FDTD schemes for doubly and are positive definite or lossy media is such that has no bearing over the stapositive semidefinite. Note that bility of the ADI-FDTD schemes. III. GENERALIZED MATRIX NORM STABILITY ANALYSIS For the same discrete system in (1), will always be bounded (stable) as the time marches if we can find any induced matrix norm (e.g., 1-norm, 2-norm, -norm) to the amplification matrix, such that [16]. For matrix norm analysis, we proceed from (14) and apply transformation vector of (19) to yield

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. Here, where if is positive definite or positive semidefinite. Using norm inequality of (26) we note that both and imply that , which guarantees stability. It can be seen that the condition to guarantee algorithm stability formulated from matrix norm stability analysis is exactly the same as those formulated from Lyapunov stability analysis in the previous section. Both analyses show that the positive definiteness (or positive and are paramount to ensure stasemidefiniteness) of bility of ADI-FDTD schemes for doubly lossy media. IV. STABILITY ANALYSIS OF VARIOUS ADI-FDTD SCHEMES FOR DOUBLY LOSSY MEDIA We now proceed to analyze the stability of various ADI-FDTD schemes for doubly lossy media using the above generalized stability analysis. The schemes considered in this section include averaging, forward-backward, backward-forward, exponential time differencing, forward-forward, backward-backward and lossless ADI-FDTD. A. Averaging

(20) where (21) (22)

The averaging scheme [3], [9] of ADI-FDTD method for doubly lossy media is one of the most common schemes used where the conductivity terms are averaged between two time indices in both substeps. The scheme calls for the following update procedures:

It should be noted that evaluating the matrix norm directly from in (14) is difficult. The rationale behind transforming further into in (19) is to transform in (14) into in (20). and can then be extracted and their respective norm can be evaluated separately. From the definition of induced matrix norm, the 2-norm (eucan be expressed as clidean norm) of

(27a)

(23)

(27b)

where . Since ), (23) can be reduced into

where

is skew-symmetric (

(28) (24) . We can see from where (24) that if is positive definite or positive semidefinite, , and thus, . with and taking the skewSimilarly, by replacing ( ) into consideration, we symmetric nature of obtain

(25)

Note that the averaging scheme is second-order in temporal accuracy. Comparing (27) to (10) in the generalized stability analysis, it is found that (29) (30) ( ) of all grid points (inhomogeIf , , , neous media) within the computational domain and are pos-

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itive, and are always positive definite. Note that and can also be zero at certain grid points to represent combina) tion of lossless and lossy media, or electrically lossy ( and magnetically lossy ( ) media likewise. In this case, and are always positve semidefinite. As far as Lyapunov stability analysis is concerned, we can see from (18) that is always positive definite, or positive semidefinite. On the other hand, from matrix norm stability analysis, it can be found from (24) and (25) that

C. Backward-Forward The backward-forward scheme is essentially a dual of the forward-backward scheme. In this scheme, the conductivity terms are applied at the backward time ( ) in the first substep and for) in the second. The scheme calls for the folward time ( lowing update procedures:

(31) and are positive definite or posalways hold as long as itive semidefinite. Therefore, it can be concluded concurrently from both Lyapunov and matrix norm stability analysis that the averaging scheme is unconditionally stable as there is no restriction imposed on the chosen time step . B. Forward-Backward In forward-backward scheme [7], the conductivity terms are applied at the forward time ( ) in the first substep and backward time ( ) in the second. The scheme calls for the following update procedures:

(37a)

(37b) The backward-forward scheme is second-order in temporal accuracy. Comparing (37) to (10) in the generalized stability analysis, it is found that (38) (39) (40)

(32a)

Again, and are either always positive definite or positive semidefinite. As a result, from Lyapunov stability analysis, is always positive definite or positive semidefinite. From matrix norm stability analysis, we also find that (41)

(32b) The forward-backward scheme is second-order in temporal accuracy. Comparing (32) to (10) in the generalized stability analysis, it is found that

always hold as long as and are positve definite or positive semidefinite. Therefore, backward-forward scheme is also unconditionally stable. D. Exponential Time Differencing

(34)

Apart from being adopted in explicit FDTD scheme [17], [18], the exponential time differencing (ETD) scheme can also be incorporated into the ADI-FDTD method, which calls for the following update procedures:

(35)

(42a)

(33)

Here, similar arguments as in the averaging scheme applies. and are either always positive definite or positive semidefinite. As a result, from Lyapunov stability analysis, is always positive definite or positive semidefinite. From matrix norm stability analysis, we also find that (36) and are positive definite or posalways hold as long as itive semidefinite. Therefore, forward-backward scheme is also unconditionally stable.

(42b) where is diagonal. The ETD scheme is also second-order in temporal accuracy. Note that the general stability analysis procedure in the previous sections cannot be applied directly to (42). In order to apply the analysis procedure, we have to first consider the following mapping of matrices: (43)

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Using the above mapping along with some manipulation, (42) can be rewritten as

(44a)

(51b) where Comparing (51) to (10), it is observed that

(44b)

(52) (53)

Comparing (44) to (10), we obtain (45) (46) where

.

where

can be solved from (50) as

(54)

can be solved from (43) as

(47) and hence, and are alWe can see from (47) that ways positive definite or positive semidefinite. As a result, from Lyapunov stability analysis, is always positive definite or positive semidefinite. From matrix norm stability analysis, we also find that

and hence, and Equation (54) indicates that are always positive definite or positive semidefinite. As a result, from Lyapunov stability analysis, is always positive definite or positive semidefinite. From matrix norm stability analysis, we also find that

(55) (48) and are positive definite or posialways hold as long as tive semidefinite. Therefore, the ETD scheme is unconditionally stable.

always hold as long as and are positive definite or positive semidefinite. Therefore, the forward-forward scheme is unconditionally stable. F. Backward-Backward

E. Forward-Forward In forward-forward scheme [4], the conductivity terms are applied at forward time for both substeps ( for the first for second). The scheme calls for the following update and procedures:

In backward-backward scheme, the conductivity terms are applied at backward time for both substeps ( for the first and for second). The scheme calls for the following update procedures:

(49a)

(56a)

(49b)

(56b)

Note however that forward-forward scheme is only first-order in temporal accuracy. In forward-forward scheme, we now consider the mapping of

In backward-backward scheme, we consider the following mapping of

(50) Using the above mapping along with some manipulation, (49) can be rewritten as

(51a)

(57) Note that the mapping applied to backward-backward scheme, cf. (57), as well as the mappings applied to ETD and forwardforward schemes in the previous subsections, cf. (43) and (50) are used to transform their update equations into the form of (10) so that the general stability analysis procedure in Sections II and III can be carried out easily.

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Using the above mapping along with some manipulation, (56) can be written as (65b) (58a)

It can be seen that (66)

(58b) . where Comparing (58) to (10), it is observed that

(67)

(59) (60)

Due to skew-symmetric nature of , we find that is in fact Cayley transform matrix which is orthogonal. Thus, matrix can be determined directly from the Lyapunov equation. The Lyapunov equation for lossless ADIFDTD read:

(61)

(68)

However, in this case, we can see from (61) that , and hence and are not always positive definite or positive semidefof all grid points and are inite even when , , , positive. Therefore, backward-backward scheme is not unconditionally stable from Lyapunov stability analysis as is not always positive definite or positive semidefinite. In order to en, sure positive definiteness or positive semidefiniteness of the following condition must hold:

where we have also taken into account the skew-symmetric na. In lossless case, is a null matrix which is positive ture of semidefinite. As from matrix norm stability analysis, since both and are null matrices, we find that

where

can be solved from (57) as

(69) is always true. Therefore, the ADI-FDTD scheme in lossless media is unconditionally stable.

(62)

V. VON NEUMANN METHOD

From (62), we can see that the choice of is somehow bounded by , , and within the computational domain. Solving (62) for , the condition to ensure stability for backward-backward scheme is thus given by

Instead of analyzing in time domain, the Von Neumann method is often adopted for homogeneous media where all field components are transformed into Fourier domain. Doing so, (10) in Fourier domain read:

(63)

(70a)

and are all positive. Equation (63) imwhere , , , plies that the time step should sufficiently resolve the minimum relaxation time of the medium, where the electric and magnetic and , respecrelaxation time are commonly defined as tively. It should be pointed out that unlike the explicit FDTD method, the stability criterion of backward-backward scheme of ADI-FDTD is only dependent on the relaxation time of the medium and independent of the mesh size. From matrix norm stability analysis, we now find that

(70b) where (71)

(64) (72)

only hold if the condition in (63) is satisfied. G. Lossless For completeness, the stability analysis for the lossless ADIFDTD method is performed in the realm of Lyapunov and matrix norm stability. The lossless ADI-FDTD method calls for the following update procedures:

(73)

(65a)

(74)

HEH AND TAN: LYAPUNOV AND MATRIX NORM STABILITY ANALYSIS OF ADI-FDTD SCHEMES FOR DOUBLY LOSSY MEDIA

Note that we have added tilde sign to all related matrices and , and vectors to indicate Fourier domain entries. Now, are all 6 6 diagonal matrices which comprise material parameter matrices and . The Lyapunov and matrix norm stability analysis detailed above is still applicable. This can be done by substituting all related matrices and vectors with their associated complex counterpart, and replacing all matrix and vector transpose operator with hermitian (complex). It should be pointed out that the comand are plex matrices now skew-hermitian ( , ). Using the same procedures in Sections II and III, it can be shown in Fourier domain that averaging, forward-backward, backward-forward, ETD, forward-forward and lossless schemes are all unconditionally stable. For backward-backward scheme, we arrive at the in Fourier domain from (61) as following mapped (75)

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, (78) is rewritten as

Noting that

(79) Substituting one obtains

in (17) into (79) and upon some manipulation,

(80) From ADI-FDTD update procedures, we know that (81) (82) Substituting these into the first two terms of above equation, one gets

Similar to the analysis in time domain, the positive definiteness of is required to ensure stability. In order for to be positive definite, the following condition must hold: (76) (83) Solving (76) for , the condition to ensure stability for backward-backward scheme in Fourier domain is given by

Noting that and

and are skew-symmetric ( ), (83) is reduced into

(77) (84) where , , , and are all positive. For inhomogeneous media, , , , are typically chosen such that and are the minimum among different media considered within the whole computational domain. Thus, the condition (77) in Fourier domain is similar to (63) in time domain. VI. CONCLUSION Lyapunov and matrix norm stability analysis has been applied on various alternating-direction-implicit finite-difference timedomain (ADI-FDTD) schemes for doubly lossy media. Among the schemes considered are averaging, forward-backward, backward-forward, forward-forward, exponential time differencing and backward-backward. From the analysis, it has been found that all schemes except backward-backward scheme are unconditionally stable. For backward-backward scheme, the condition for stability has been determined. APPENDIX DERIVATION OF MATRIX FROM DISCRETE LYAPUNOV EQUATION An arbitrary vector and its transpose are multiplied from the left and right at both sides of (2) to yield an energy-like equation (78)

Taking into consideration the relationship between and in (82), one finally arrives at (85) where Thus,

. is finally derived as (86) REFERENCES

[1] T. Namiki, “3-D ADI-FDTD method- unconditionally stable time-domain algorithm for solving full vector Maxwell’s equations,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 10, pp. 1743–1748, Oct. 2000. [2] F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1550–1558, Sep. 2000. [3] C. C.-P. Chen, T.-W. Lee, N. Murugesan, and S. C. Hagness, “Generalized FDTD-ADI: An unconditionally stable full-wave Maxwell’s equations solver for VLSI interconnect modeling,” in Proc. IEEE/ACM Int. Conf. on Computer Aided Design, San Jose, CA, Nov. 2000, pp. 156–163. [4] C. Yuan and Z. Chen, “Towards accurate time-domain simulation of highly conductive materials,” in IEEE MTT-S Int. Microwave Symp. Dig., Seattle, WA, Jun. 2002, pp. 1135–1138. [5] J. A. Pereda, A. Grande, O. Gonzalez, and A. Vegas, “The 1D ADIFDTD method in lossy media,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 477–480, 2008.

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[6] J. A. Pereda, A. Grande, O. Gonzalez, and A. Vegas, “Analysis of two alternative ADI-FDTD formulations for transverse-electric waves in lossy materials,” IEEE Trans. Antennas Propag., vol. 57, pp. 2047–2054, Jul. 2009. [7] D. Y. Heh and E. L. Tan, “Efficient implementation of 3-D ADI-FDTD method for lossy media,” in Proc. MTT-S Int. Microwave Symp. Dig., Boston, MA, Jun. 2009, pp. 313–316. [8] B. Fornberg, A short proof of the unconditional stability of the ADI-FDTD Scheme Univ. Colorado, Dept. Applied Mathematics, 2001, Tech. Rep. 472. [9] W. Fu and E. L. Tan, “Stability and dispersion analysis for ADI-FDTD method in lossy media,” IEEE Trans. Antennas Propag., vol. 55, pp. 1095–1102, Apr. 2007. [10] S. G. Garcia, R. G. Rubio, A. Rubio Bretones, and R. G. Lopez, “Revisiting the stability of Crank-Nicolson and ADI-FDTD,” IEEE Trans. Antennas Propag., vol. 55, pp. 3199–3203, Nov. 2007. [11] S. Ogurtsov, G. Pan, and R. Diaz, “Examination, clarification, and simplification of stability and dispersion analysis for ADI-FDTD and CNSS-FDTD schemes,” IEEE Trans. Antennas Propag., vol. 55, pp. 3595–3602, Dec. 2007. [12] W. Hahn, Theory and Application of Liapunov’s Direct Method. Englewood Cliffs, NJ: Prentice-Hall, 1963. [13] A. Bacciotti and R. Lionel, Liapunov Functions And Stability In Control Theory, 2nd ed. Berlin, NY: Springer, 2005. [14] P. J. Antsaklis and A. N. Michel, Linear Systems. New York: McGraw-Hill, 1997. [15] F. Ayres Jr., Schaum’s Outline Series Of Theory and Problems Of Matrices. London, UK: McGraw-Hill, 1962. [16] G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed. Oxford, UK: Clarendon, 1985. [17] P. G. Petropoulos, “Analysis of exponential time-differencing for FDTD in lossy dielectrics,” IEEE Trans. Antennas Propag., vol. 45, pp. 1054–1057, Jun. 1997.

[18] D. Y. Heh and E. L. Tan, “Dispersion analysis of FDTD schemes for doubly lossy media,” Progr. Electromagn. Res. B, vol. 17, pp. 327–342, 2009. Ding Yu Heh (S’08) received the B.Eng. degree (first class honors) from Multimedia University, Cyberjaya, Malaysia, in 2006, and the M.Sc. degree from Nanyang Technological University, Singapore, in 2007, where he is currently working toward the Ph.D. degree. His research areas are focused mainly on computational electromagnetics, especially the finite-difference time-domain method and its application on bioelectromagnetics.

Eng Leong Tan (SM’00) received the B.Eng. (electrical) degree (first class honors) from the University of Malaya, Malaysia and the Ph.D. degree in Electrical Engineering from Nanyang Technological University, Singapore. From 1991 to 1992, he was a Research Assistant at the University of Malaya. From 1991 to 1994, he worked part time at Commercial Network Corporations Sdn. Bhd., Malaysia. From 1999 to 2002, he was a Member of Technical Staff at the Institute for Infocomm Research, Singapore. Since 2002, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, where he is currently an Associate Professor. His research interests include electromagnetic and acoustic simulations, RF and microwave circuit design.

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Compact FDTD Formulation for Structures With Spherical Invariance Adam Mock, Member, IEEE

Abstract—A compact, fully vectorial finite-difference time-domain algorithm is formulated for the efficient analysis of structures with spherically invariant material properties. Using this method, the full three dimensional vector solution to Maxwell’s equations can be computed using only a one dimensional computational grid. This is accomplished by expanding the electric and magnetic fields in sinusoids and associated Legendre functions to handle the azimuthal and longitudinal field variation, respectively. The method is illustrated by obtaining the electric field, resonance frequency and quality factor for different modes of a dielectric microsphere. Index Terms—Electromagnetic analysis, finite-difference timedomain (FDTD) methods, microresonators, factor, spheres.

I. INTRODUCTION HE finite-difference time-domain (FDTD) method is a widely used approach for numerically solving Maxwell’s equations. It provides the electric and magnetic fields as a function of space and time [1], [2]. It is an attractive method due to its simplicity, ease of implementation, straight-forward parallelization and generality. The FDTD method exhibits linear scaling with problem size. But even with this favorable scaling, large three dimensional computational domains can often become intractable with reasonable computing resources. For problems possessing certain spatial invariances, the dimensionality of the full three dimensional FDTD algorithm can be reduced resulting in a significant reduction in the computational analysis time. For example, some waveguiding structures are invariant along the waveguide propagation direction. If the waveguide is defined by a spatially varying electric permittivity, and the propagation direction is along , then the condition for continuous translational invariance may be expressed as where is an arbitrary real number. The electric and magnetic fields in waveguide structures with continuous translational invariance along the propagation direction have the form

T

(1) refers to the th vector component of the elecwhere tric or magnetic field, is the propagation direction, and is the

Manuscript received May 24, 2010; revised July 21, 2010; accepted July 23, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The author is with the School of Engineering and Technology, Central Michigan University, Mt. Pleasant, MI 48859 USA (e-mail: mock1ap@cmich. 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.2010.2103040

propagation constant [3]. If (1) is inserted into Maxwell’s curl equations in Cartesian coordinates, derivatives with respect to can be evaluated explicitly and replaced with . The FDTD method can then be used to solve for the two-dimensional field . In this case, the problem still incorporates quantities six vector components, but the computational domain requires only the two-dimensional cross section of the waveguide [4], [5]. The approach allows for the analysis of waveguide structures one user-specified -value at a time. In [6] this approach was termed “compact FDTD.” This method has been used to analyze the radiative losses in silicon-on-insulator microphotonic rib waveguides [7] and to obtain the electric and magnetic fields in photonic crystal fibers [8]. Another example of where dimensionality reduction is possible using the FDTD method involves structures with cylindrical invariance or bodies of revolution. Assuming a cylindrical where is coordinate system defined by the variables the radius, denotes azimuthal angular variation and refers to the axial direction, such structures are characterized by material parameters that are invariant in the direction. In this case the electric and magnetic field variation in the direction can be represented using (2) Inserting (2) into Maxwell’s curl equations in cylindrical coordinates allows derivatives with respect to to be evaluated explicitly [2], [9]. The FDTD method may then be used to obtain one user-specithe two-dimensional field distribution fied -value at a time. Three dimensional and fully vectorial field solutions to Maxwell’s equations can be obtained this way using only two-dimensional computational grids. This method has been used to analyze the radiative losses in silicon-on-insulator photonic microring resonators [7]. In addition to improving computational efficiency, taking advantage of cylindrical invariance in this way allows for the consideration of perfectly round surfaces and reduces the staircasing error associated with modeling a curved surface on a discretized grid. Cylindrical optical fiber waveguides are an example of a geometry possessing both continuous translational invariance and cylindrical invariance. In this case the two methods described in the preceding paragraphs may both be used to analyze the electrodynamics of such structures thereby reducing the computational domain dimensionality from three to only one [10]. In this work an efficient compact FDTD method is developed for the analysis of problems with spherical invariance. In these cases, the geometry under investigation has variation in the radial direction only. With this assumption, the behavior and of the electric and magnetic fields in the azimuthal directions can be represented by sinusoidal longitudinal

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and associated Legendre functions, respectively [11]. It will be shown that the three dimensional spherical problem can be analyzed using a one dimensional FDTD grid. Accurate analysis of the electromagnetics of structures with spherical invariance are important in several applications including photonics both at the microscale [12]–[15] and nanoscale [16], quantum optics [17]–[19] and modeling Schumann resonances in the ionosphere [20]. II. FORMULATION OF METHOD The analysis of simple structures with spherical invariance is a classic problem in electromagnetics [11], [21], [22]. The solution of the spherical problem with material variation along only can be classified as TM (transverse magnetic) or TE (transverse electric). The TM (TE) solutions are so named because the radial component of the magnetic (electric) field is zero. The FDTD method can be used to obtain the TM and TE solutions separately. The TM solutions are treated in what follows. The formulation for the TE solutions is analogous and appears in Appendix A. In a homogeneous, isotropic and lossless medium the fields comprising the TM solution to Maxwell’s equations are [11] (3) (4) (5) (6)

functions as in (3)–(8). Instead we write the -dependent funcwhere refers to one tions using lower case letters as in will be used for the magof , , or . A similar form netic field components. Because the FDTD method is based on a time marching scheme, the time dependence of the electric and magnetic fields is generalized and included in these lower case field variables as well. The field variables corresponding to (3)–(8) but applicable to geometries with arbitrary radial material variation are (9) (10) (11) (12) (13) (14) It should be noted that, although is allowed to take on any corresponds to a solution in non-negative integer value, which all field values are zero at the outset. The lowest order . Inserting solution for both the TE and TM solutions is (9)–(14) into the differential form of Maxwell’s equations in spherical coordinates and using the differentiation and orthogonality properties of sinusoids and associated Legendre functions [26] one obtains (15)

(7) (16) (8) (17) (where is the refractive index and In (3)–(8), is the free space wavelength), is the radial coordinate, is an th order spherical Bessel or spherical Hankel function, is a tesseral is an associated Legendre function harmonic, and of th degree and th order. and are integers, and . The prime refers to differentiation with respect to . In simple cases, the expressions for the argument of the field components in (3)–(8) can be used to obtain the electric and magnetic fields by choosing the appropriate spherical Bessel or spherical Hankel function and applying boundary conditions at material interfaces. In Section III we will use these known solutions to investigate the accuracy of the proposed FDTD method. In more complicated geometries, however, applying several boundary conditions successively can become tedious and time consuming making a fast FDTD method attractive. In geometries with a continuous spatial variation in the dielectric constant, closed form solutions in general do not exist. Furthermore, by analyzing the electromagnetic fields using the FDTD method, one may calculate loss properties [23], [24] and nonlinear [25] effects. For general material variation along the -direction only, the -dependence can no longer be written down in terms of known

(18) (19) , , and are defined by and The coefficients according to (64)–(67) in Appendix B. Equations (15)–(19) involve the quantities and which are functions of radius and time only. The next step is to discretize (15)–(19) in space and time. The FDTD method incorporates a spatially staggered arrangement of the electric and magnetic field components. Fig. 1 illustrates the staggered arrangement of the field components that will be used in calculations later in this work. In a general three dimensional space, the six field components would each occupy a unique point in space. However, when the and directions are projected onto the direction, some field components overlap. The discretized version of (15)–(19) are shown below where the spatial index is illustrated in Fig. 1(a).

(20)

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are constant, so that solutions of the form (3)–(8) apply there. the following approximation holds for spherical For small Bessel functions of the first kind [26]

(25)

Fig. 1. Illustration of spatial arrangement of the discretized electric and magnetic fields for the (a) TM solutions and (b) TE solutions.

(21)

(22)

, near the origin. For the TM soluFor tions, Fig. 1 shows that the field components that are sampled , and . From (7) and (8) with (25), at the origin are at . Therefore, and it is clear that can be set to zero. For , we use (25) in (3) to get which shows that tends to zero as for . In the case of , can be set to zero. For , converges to a constant which is not necessarily zero at the . In prinorigin, and more information is required to find ciple, one can use (20) to evaluate . However, evaluation of is complicated by the appearance of where at is not required the origin. It should be noted that obtaining to implement this FDTD method, as updating the magnetic field , for they have already been components does not require . The electric field value is required set to zero at only for postprocessing where a complete characterization of field might be required. A value for can the be obtained via interpolation from . In particular where when . Finally, one concludes that at . Similar to choosing in the compact FDTD method for in the FDTD method for cylindrical symwaveguides and at the outset in the present metry, the user chooses and values method. Individual program runs for each set of can then be performed. Structures with perfectly round spherical surfaces can be modeled using this method without any discretization effects in the azimuthal or longitudinal directions. Equations (20)–(24) incorporate five field quantities to be upfor dated with five equations. Substituting the condition TM solutions into Maxwell’s equations yields

(23)

(26) which results in

(27) (24) The origin is a special point in curvilinear coordinate sysin the differential form of tems due to the appearance of Maxwell’s equations as well as the ambiguous meaning of the and directions when . To deal with the origin within the context of this FDTD method, we return to (3)–(8) where the radial component of the electric and magnetic field components are expressed in terms of known functions. Within a small radius of the origin, one may assume that the material parameters

and Inserting into (27) the forms (10) and (11) for results in . This means that either (21) or (22) is required for the program implementation which improves efficiency. Furthermore, (16) and (17) imply that (assuming an isotropic medium), so only (23) or (24) is required for program implementation. These two observations reduce the number of field variables and update equations from five to three which significantly improves computational efficiency. For the TE solutions, a similar argument imand . plies

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TABLE I COMPARISON BETWEEN RESONANCE FREQUENCIES OBTAINED FROM THIS FDTD METHOD AND THE EXACT SOLUTION

TABLE II COMPARISON BETWEEN Q FACTORS OBTAINED FROM THIS FDTD METHOD AND AN APPROXIMATE ANALYTICAL METHOD

Fig. 2. r -component of the electric field (E (r;  ;  )) for the lowest radial order TM (l; m) = (10; 10) mode of a dielectric sphere. Only the radial field variation is shown. The position axis is normalized to the sphere radius (a). n = (= ) is the refractive index. The exact and FDTD calculated solutions are shown.

III. DEMONSTRATION To demonstrate the above compact FDTD algorithm for spherical invariance, a dielectric sphere is analyzed. The modes of a dielectric sphere have closed form solutions for the resonance frequencies and field profiles [11], which can be compared to the results of the FDTD method derived in field is plotted as a function Section II. In Fig. 2 the of radial distance normalized to the sphere radius (denoted by ) for fixed values of and . The field was obtained by initiating the field component with a value of 1 at a spatially random set of discrete points and then allowing the fields to evolve in time. This initial condition ensured that all modes of the structure were excited, as it is spectrally broadband, and the spatial distribution ensures overlap with all spatial modes. The time evolution of a representative field component was recorded. A discrete Fourier transform was used to identify the resonance frequencies, and discrete time filtering was used to isolate the modes of interest. The boundary at large radius was terminated with perfectly matched layer absorbing boundary conditions [27], [28]. The electric field shown in Fig. 2 is the lowest order TM radial field corresponding to the longitudinal and azimuthal mode . The -axis was discretized with numbers . The index of refraction of the sphere was set to 3.5 consistent with silicon at near infrared waveat the surface of the lengths [15]. The discontinuity in sphere is a result of continuity of the -component of the elec. The time step used in the tric displacement which is just below calculation was set to the maximum stable time step as discussed in Section IV. The difference between the exact result and the FDTD calculated result is indistinguishable on the scale shown in Fig. 2. Closer inspection shows differences of less than 2%. Table I displays the resonance frequencies of the five lowest frequency TM resonant modes corresponding to calculated using this FDTD method and exact resonance frequencies determined via solution to a tran-

scendental equation [11]. Further postprocessing was done on the discrete Fourier transformed time sequence with Padé interpolation to extract the center frequencies of the resonance peaks with good accuracy [24], [29]. Padé interpolation is required due to the limited resolution in the discrete Fourier transformed time sequence. Excellent agreement is seen between the exact values and the values obtained using this FDTD method. factors for the five lowest freTable II displays the quality quency TM resonant modes corresponding to calculated using this FDTD method and factors calculated using expressions derived in [30]. The factors were obtained from the FDTD time sequence data by taking a discrete Fourier transform and applying Padé interpolation [24], [29] for accurate estimation of the full width at half maximum of the resonance peaks. Disagreement between the FDTD calculations and [30] remains less than 8% for the modes analyzed in this study. Compared to Table I the larger percent errors shown in Table II are attributed to the approximate nature of the solutions in [30] as well as discretization error in the perfectly matched layer absorbing boundary conditions. IV. STABILITY ANALYSIS The maximum stable time step can be determined by substituting the closed form solutions (3)–(8) into (20)–(24) and following the steps outlined in [2]. The result is (28) where

(29) where is the free space wavelength. In (29), The function to be maximized in (29) is a monotonically decreasing function of where represents the th element

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TABLE III RESONANCE FREQUENCIES AND Q FACTORS FOR DIFFERENT REFRACTIVE INDEX PROFILES

Fig. 3. Minimum B value yielding a stable time step from (28). Values for B : for l m (triangles) and for m were calculated from (30) with i (squares). The observed minimum stable B value (circles) did not show a dependence on m.

1

= 15

=

=

in the discretized radial variable and can take on the values It is clear from (29) that is maxiand . However, this solution must mized when be ruled out, as it would imply the largest stable time step has a duration of zero. It was found that the best agreement with the and observed maximum stable time step occurred when

(30) Fig. 3 shows values for calculated using (30) with and compares them to values of determined by observing the field evolution in FDTD program runs. In these program runs was incrementally decreased until the field evolution grew without bound indicating instability. The observed values in Fig. 3 correspond to the minimum stable value for a given value. Fig. 3 shows values for calculated from (30) with and with . Program runs showed no difference in the minimum stable value when was changed for a given value. However, calculated using (30) shows a small dependence on for a given . Regardless, it is clear that (28) with (30) can be used to estimate the maximum stable time step to be used in FDTD runs. Despite the complicated appearance of the relationship between and shown in (30), Fig. 3 shows that scales approximately linearly in .

onance frequency and factor for a range of values. The results in Table III correspond to the first order radial TM mode . The same calculation parameters menwith tioned in Section III were used here. One sees that there is a frequency shift of 8.8% as the slope changes from to which corresponds to a resonance wavelength nm assuming . Even more intershift of esting is the factor of five change in the factor which suggests that linearly increasing the refractive index along the radius of a microsphere can significantly increase the factor. Physically, these trends can be attributed to an increase in the effective index of the resonance mode as increases. VI. CONCLUSION This work formulates a compact finite-difference time-domain method applicable to structures with spherically invariant material properties. The method obtains fully vectorial electromagnetic fields using only a one-dimensional computational grid. Formulations for both the TM and TE solutions are included, and the maximum stable time step was derived. The method yields results in excellent agreement with known solutions. This method can be applied to a variety of interesting problems in photonics and optics as well as in atmospheric electromagnetics. Future work looks to the possibility of formulations for structures conforming to other coordinate systems which may include elliptic, parabolic or spheroidal coordinate systems [11]. APPENDIX A FORMULATION FOR TE SOLUTIONS: The closed form solutions corresponding to (3)–(8) for the TE solutions are (31) (32) (33)

V. APPLICATION To conclude and to illustrate the utility of this method, a dielectric microsphere is analyzed with a linearly varying refractive index profile along the radial direction. Such a refractive index could be the result of radial variation in temperature, carrier concentration, doping or optical intensity. A refractive index is used. This proprofile defined by where file maintains a refractive index of 3.5 at is the radius of the sphere. The slope of the linear refractive index variation is . Table III displays the dependence on res-

(34) (35) (36) The field variables for arbitrary radial material variation corresponding to (9)–(14) for the TE solutions are (37)

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(38) (39)

, , and appearing in (15)–(19) and (20)–(24) for the TM solutions and in (43)–(47) and (48)–(52) for the TE solutions

(40)

(53)

(41)

(54)

(42)

(55)

The equations governing the functions representing the radial variation of the electric and magnetic fields ( and ) corresponding to (15)–(19) for the TE solutions are

(56) (57)

(43)

(58)

(44) (59) (45) (60) (46) (47) (61)

The discretized versions of (43)–(47) corresponding to (20)–(24) for the TE solutions are

(62)

(63)

(48)

(64) (65) (66) (67) (49) REFERENCES

(50)

(51)

(52)

APPENDIX B COEFFICENTS APPEARING IN FDTD UPDATE EQUATIONS: The following combinations of and make up the coefficients

[1] K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, pp. 302–307, 1966. [2] A. Taflove and S. C. Hagness, Computational Electrodynamics. MA: Artech House, 2000. [3] R. E. Collin, Field Theory of Guided Waves. New York: Wiley, 1990. [4] S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microwave Guided Wave Lett., vol. 2, pp. 165–167, May 1992. [5] S. Xiao and R. Vahldieck, “An efficient 2-D FDTD algorithm using real variables [guided wavestructure analysis],” IEEE Microw. Guided Wave Lett., vol. 3, pp. 127–129, 1993. [6] A. Asi and L. Shafai, “Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2D-FDTD,” IEEE Electron. Lett., vol. 28, no. 15, pp. 1451–1452, 1992. [7] A. Mock and J. D. O’Brien, “Dependence of silicon-on-insulator waveguide loss on lower oxide cladding thickness,” presented at the Integrated Photonics Nanophotonics Res. Applicat. Topical Meeting (IWG4), Boston, MA, Jul. 2008. [8] M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt Technol Lett, vol. 30, no. 5, pp. 327–330, Sep. 5, 2001.

MOCK: COMPACT FDTD FORMULATION FOR STRUCTURES WITH SPHERICAL INVARIANCE

[9] D. B. Davidson and R. W. Ziolkowski, “Body-of-revolution finite-difference time-domain modeling of space-time focusing by a three-dimensional lens,” J. Opt. Soc. Am., vol. 11, no. 4, pp. 1471–1490, Apr. 1, 1994. [10] Y. Chen and R. Mittra, “A highly efficient finite-difference time domain algorithm for analyzing axisymmetric waveguides,” Microwave Opt. Technol. Lett., vol. 15, no. 4, pp. 201–203, Jul. 1997. [11] J. A. Stratton, Electromagnetic Theory. New York: McGraw Hill, 1941. [12] M. Cai, G. Hunziker, and K. Vahala, “Fiber-optic add–drop device based on a silica microsphere-whispering gallery mode system,” IEEE Photon. Technol. Lett., vol. 11, pp. 686–687, 1999. [13] H. C. Tapalian, J.-P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett., vol. 14, no. 8, pp. 1118–1120, Aug. 2002. [14] Y. O. Yilmaz, A. Demir, A. Kurt, and A. Serpengüzel, “Optical channel dropping with a silicon microsphere,” IEEE Photon. Technol. Lett., vol. 17, no. 8, pp. 1662–1664, Aug. 2005. [15] A. Serpengüzel, K. Adnan, and U. K. Ayaz, “Microsphere photonics,” in Proc. SPIE, 2007, vol. 6593. [16] M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimonov, S. Stout, and E. Herz, “Demonstration of a spaser-based nanolaser,” Nature, vol. 460, pp. 1110–1113, 2009. [17] V. V. Datsyuk, “Gain effects on microsphere resonant emission structures,” J. Opt. Soc. Am. B, vol. 19, no. 1, pp. 142–147, 1997. [18] V. Lefèvre-Seguin and S. Haroche, “Towards cavity-QED experiments with silica microspheres,” Mater. Sci. Eng., vol. B48, pp. 53–58, 1997. [19] M. Pelton and Y. Yamamoto, “Ultralow threshold laser using a single quantum dot and a microsphere cavity,” Phys. Rev., vol. 59, no. 3, pp. 2418–2421, Mar. 1999. [20] A. Soriano, E. A. Navarro, D. L. Pual, J. A. Portí, J. A. Morente, and I. J. Craddock, “Finite difference time domain simulation of the earth-ionosphere resonant cavity: Schumann resonances,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1535–1541, Apr. 2005. [21] J. D. Jackson, Classical Electrodynamics, 3rd ed. Hoboken, NJ: Wiley, 1999. [22] M. Born and E. Wolf, Principles of Optics. New York: Cambridge Univ. Press, 2003.

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[23] M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microw. Guided Wave Lett., vol. 7, pp. 121–123, 1997. [24] S. Dey and R. Mittra, “Efficient computation of resonant frequencies and quality factors of cavities via a combination of the finite-difference time-domain technique and the Pade approximation,” IEEE Microw. Guided Wave Lett., vol. 8, no. 12, pp. 415–417, Dec. 1998. [25] M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “Highorder FDTD and auxiliary differential equation formulation of optical pulse propagation in 2D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron., vol. 40, pp. 175–182, 2004. [26] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1970. [27] J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys., vol. 114, no. 2, pp. 185–200, 1994. [28] F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guided Wave Lett., vol. 7, no. 11, pp. 371–373, Nov. 1997. [29] A. Mock and J. D. O’Brien, “Direct extraction of large quality factors and resonant frequencies from Padé interpolated resonance spectra,” Opt. Quant. Elect., vol. 40, no. 14, pp. 1187–1192, 2009. [30] C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B, vol. 9, no. 9, pp. 1585–1592, 1992. Adam Mock (S’08–M’09) received the B.S. degree from Columbia University, New York, in 2003 and the Ph.D. degree from the University of Southern California, Los Angeles, in 2009, both in electrical engineering. He is currently an Assistant Professor at Central Michigan University, Mt. Pleasant. His research interests include microphotonic device simulation and analysis. Dr. Mock is a member of the Optical Society of America.

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The Linear Relationship Between Attenuation and Average Rainfall Rate for Terrestrial Links Adrian Justin Townsend and Robert John Watson, Member, IEEE

Abstract—Attenuation measured on terrestrial links can be used to estimate surface rainfall rates along a link path. There is current interest in the use of inverse methods to estimate rainfall over areas inferred from terrestrial links. A power-law relationship can be used to estimate rainfall rate from attenuation measured by a terrestrial link. However, the inverse method is simplified with a linear relationship between attenuation and rainfall rate when determining path averaged rainfall rates. This paper investigates power-law and linear relationships between rainfall rate and attenuation. It examines their goodness of fit for a range of frequencies and rainfall rates. Six years of disdrometer data from Chilbolton in the U.K. has been used to study both linear and power-law fits to attenuation and quantify the associated errors. Index Terms—Attenuation, microwave radio propagation meteorological factors, rainfall effects.

I. INTRODUCTION EASUREMENTS of attenuation on terrestrial microwave links have been widely used to estimate surface rainfall rate. Among the first measurements were those of Atlas and Ulbrich [1]. Typically terrestrial links operate at frequencies between 10 GHz and 50 GHz and usually over a path of 1–50 km. At microwave frequencies the terrestrial link signal is attenuated by raindrops falling along the link path due to electromagnetic scattering and absorption processes. Traditionally attenuation is expressed in terms of rainfall rate using a power-law relationship [2]. Conversely, this relationship can also be used to determine rainfall rate from link attenuation. The power-law is generally considered to be a good representation of the relationship between attenuation and rainfall ( - ). However, as will be shown, since the power-law relationship is potentially non-linear it is not always possible to estimate true average rainfall from path average attenuation. The coefficients of the - relationship are also highly dependent on the raindrop size distribution (DSD), which can vary significantly. The International Telecommunication Union Radiocommunications (ITU-R) sector provides a standard definition to calculate specific attenuation from rainfall rate based on the power-law relationship, ITU-R P.838-3 [3]. In this recommendation, the listed fit coefficients are derived considering all available rainfall rates. In this paper we investigate the sensitivity of the relationship to the upper limit of the rainfall rates over which the fit is derived.

M

Manuscript received February 18, 2010; revised July 16, 2010; accepted September 02, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by the EPSRC. The authors are with the Department of Electrical and Electronic Engineering, University of Bath, Bath BA2 7AY, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2103021

The use of terrestrial links can be extremely valuable for measuring rainfall rate in areas lacking radar, rain gauges or other devices used for rain measurements [4]. Links may be most useful in urban areas or steep-sided valleys where it is difficult to place rain gauges or use radars to measure rainfall near the ground [5], [6]. The use of path averaged measurements has a number of further advantages over weather radar systems. Weather radars measure rainfall typically a few hundred metres above the Earth whereas surface rainfall is generally required for hydrology. Other factors such as fog, low-level cloud and evaporation can all affect the estimate of rainfall rate by radar [7]. Other complicating factors for radar include the sample time for a complete volume scan and the choice of radar location. For link-derived rainfall rate, the sample time may only be a few seconds compared to a few minutes for a radar volume scan. Globally there are large numbers of microwave links. Reported link densities range from 0.3 links per km to 3 links per km [5]. In the U.K. there are approaching 14 000 links in the 38 GHz frequency band alone. Furthermore, an increasing number of research experiments are also being conducted using purpose-built links to estimate rainfall rate [8]. It is reasonable to suggest that a network of links could be useful to provide new or complementary rainfall rate information over wide areas. All of these techniques require a relationship to convert the measured attenuation into a rainfall rate. This relationship should ideally be as independent of the DSD as possible and be applicable for a wide range of rainfall rates. The DSD can cause significant variability in the - relationship. The DSD is, in general, unknown and in turn can cause inaccuracy in estimating rainfall rate from path attenuation. Therefore, it is important to reduce the effects of the DSD as much as possible. Links operating at two different frequencies have been considered to reduce the impact of the DSD variability [9], [10]. The method utilizes the difference in attenuation between the two frequencies to give an estimate of rainfall rate. However, two suitable frequencies may not always be available. The effect of the variability of the DSD on single-frequency links has been considered by [11]. The authors showed that the estimated rainfall rates are sensitive to the power-law fit coefficients. The current interest in the use of inverse methods to derive surface rainfall rates requires the use of either a linear relationship or complex linearization techniques [12]. The use of links at frequencies for which the - relationship can be considered linear is restrictive, however it does imply independence of the DSD [1]. Furthermore, for a non-linear - relationship there can also be a significant difference between the true path-averaged rainfall rate and average rainfall rate inferred from path-averaged attenuation.

0018-926X/$26.00 © 2010 IEEE

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The aim of this paper is to quantify the conditions under which the - relationship can be considered linear. Specific attenuation and rainfall were calculated from DSD derived from disdrometer data. The T-Matrix method [13] was used to determine the raindrop scattering parameters over a range of frequencies between 10 and 95 GHz for both vertical and horizontal polarizations. The sensitivity of the - relationship to frequency and rainfall rate interval is analyzed. Finally, the suitability and performance of the power-law and linear relationships to express the conversion from rainfall rate to specific attenuation is evaluated. The dependence of the results on polarization is also considered although we have chosen to focus on vertical polarization since this is by far the most common for operational links (all 38 GHz links in the U.K. are vertically polarized).

II. ATTENUATION DATA ANALYSIS Six years of data collected between 2003 to 2008 from a Joss impact disdrometer was used to determine DSDs over Chilbolton, U.K. (51.14 N,1.44 W). The DSD measured by such a disdrometer can be written as: [14]

Fig. 1. Example of raindrop size distribution (DSD) measured by the Chilbolton disdrometer for different integration periods ranging from 30 seconds to one hour.

cited by a number of authors as being the best fit in a variety of climates [17], [19] (1) is the number of drops measured in the drop-size where class, is average raindrop diameter of the drop-size class is the measurement area of the disdrometer (m ), (mm), is the time interval for one measurement (s), is the fall is the bin-width of each velocity of the drop (ms ) and of the drop-size classes. In this paper, a disdrometer integration period of one minute is selected. This period balances uncertainty in the estimation of DSD parameters from measurements taken against the dynamics of the rain event. In Fig. 1 DSDs determined from the Chilbolton disdrometer are illustrated for different time integration periods. Long integration periods (e.g., one hour) reduce the uncertainty in estimating the DSD parameters but may not represent well the dynamics of rain events. Conversely, short integration periods (e.g., 30 seconds) may not yield sufficient samples to reliably estimate the DSD parameters, especially at low rainfall rates. To capture the dynamics of intense convective rain events, which may last only a few minutes, a one-minute integration time was selected as a compromise between uncertainty in the DSD parameters and rain dynamics. Joss impact disdrometers can be subject to noise caused by strong winds, splashing drops or debris landing on the device. An analytical distribution was fitted to each measured DSD in order to remove any anomalous results. A number of analytical forms of DSD have been suggested. These include the exponential distribution [15], the gamma distribution [16] and the normalized gamma distribution [17]. To test these models a goodness of fit was calculated for all measured DSD data using an metric [18]. A normalized gamma DSD was selected here because it was shown to be the best fit to the data (normalized , exponential DSD: ). gamma DSD: Furthermore, the normalized gamma distribution (2) has been

(2) where , and define the normalized gamma distribution, is the raindrop size distribution in number of drops m mm and is the drop size. Following the approach of [20], a normalized gamma distribution was fitted using a maximum likelihood method to determine and the method of moments to calculate and . Specific attenuation ( in dB km ) and rainfall rate were calculated for each fitted raindrop size distribution using (3) and (4) respectively

(3) is the (polarization dependent) total extinction cross where section, calculated using the T-Matrix method [13], with the drop shape model of Chuang and Beard [21]. It is important to note that the calculations here are dependent on temperature, drop shape model, drop fall velocity model [22], the method of scattering function calculation and polarization. Rainfall rate ( in mm hr ) can be written as

(4) where is the rainfall rate (mm hr ), (ms ) of the drop at diameter (mm).

is the fall velocity

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Fig. 2. Cumulative distribution function of rainfall intensity derived from the ITU-R P.837-5 and the Chilbolton and Sparsholt Disdrometers.

The - relationship has been extensively studied by many, such as [2]. Analytically, the - relationship can be shown to in (3) and in (4) are equal. be linear when the moments of Under these conditions the relationship becomes independent of the raindrop size distribution [1], [23]. To show where the best-fitting linear relationship region exists, we have compared the linear fit (6) to the established power-law fit (5) and the derived from disdrometer data. The comparison is repeated for - relationships over a range of frequencies between 10 to 95 GHz. The power-law fit and linear fit were determined using a least-squares fitting procedure. The power-law equation calculating specific attenuation ( in dB km ) is written as

Fig. 3. Specific Attenuation for horizontal and vertical polarization plotted against frequency for 5, 10, 20, 30 and 50 mm hr assuming a Marshall and Palmer distribution.

and 50 mm hr . The specific attenuation was calculated based on a model Marshall and Palmer exponential DSD [15]. These results show the difference between vertical and horizontal specific attenuation is relatively small, especially at low rainfall rates (e.g., 5 mm hr ). Typically at low rainfall rates the mean drop size is relatively small (below 2 mm diameter) [25]. Raindrops below 2 mm diameter can be considered approximately spherical hence the effect of polarization is minimal [26]. In this paper although we have chosen to focus on vertical polarization, selected results for both vertical and horizontal are shown. III. ERRORS CAUSED BY PATH-AVERAGING

(5) where and are coefficients determined by least-squares. The linear equation to calculate specific attenuation ( in dB km ) is given by (6) where is a coefficient and is attenuation determined by a linear fit. Fig. 2 shows the cumulative distribution function of rainfall intensity for Chilbolton disdrometer data, a second disdrometer km away) and the ITU-R P.837-5 [24]. The at Sparsholt ( Chilbolton and Sparsholt disdrometer data is broadly consistent with P.837-5. For rainfall rates above 50 mm hr the Sparsholt results differ slightly, which is likely due to insufficient data at high rainfall rates. For this reason the Sparsholt data has been excluded from this analysis. Although there are small differences below 20 mm hr the data from Chilbolton generally shows good agreement to P.837-5 for all rainfall rates and can be considered representative of the U.K. climate. Specific attenuation not only depends on frequency but also on polarization. Fig. 3 shows the affects of polarization on specific attenuation at frequencies of 10–95 GHz for 5, 10, 20, 30

Errors may occur when averaging the non-linear power-law relationship between attenuation and rainfall rate along a path. (dB km ), deFor example, average specific attenuation, is not the termined from rainfall measured along a path (dB km ), determined by the same as specific attenuation, of the path average rainfall rate (7) (8) where is link length (km) and is rainfall rate (mm hr ). In the case of this paper the average rainfall rate is wanted from average attenuation measured. Only when the - relationship , are (7) and (8) the same, then is linear, therefore . Using the synthetic storm technique [27], disdrometer data from Chilbolton were used to generate a database of rain events. A total of approximately 2.4 million rainstorms were generated using a velocity of 10 ms . The average attenuation was calculated for each synthetic rainstorm using (7) and (8), over link lengths ranging from 1.2–18 km (typical of links deployed in

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K

1K

Fig. 4. Percentage difference, , of specific attenuation averages as a function – and link length 1.2–18.0 km.

b = 0:7 1:4

K

and

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a

Fig. 5. Contours of parameter of the power-law fit as a function of frequency and maximum fitted rainfall rate. Solid contours are for vertical polarization, dashed contours are for horizontal polarization.

the U.K.). The percentage difference between the two estimates, , was calculated using

(9) expresses the error caused by nonlinearity The difference of the power-law relationship, shown in Fig. 4. It can be seen that increases. When as diverges away from 1, the error there is a big difference in error between a short link (2 km) and a long link (18 km). However, there is very little difference in error when comparing an 18 km link to a 12 km link. The error between the averages shows (8) is incorrect when used to the difference average a non-linear relationship. Where between the averages is zero. IV. RESULTS A. Power-Law Attenuation-Rainfall Relationships The power-law fits were determined for all attenuation and rainfall rates calculated from the Chilbolton disdrometer data. To investigate the sensitivity of each of the fits to the rainfall rate interval, the data used in the fitting process was conditioned by maximum rainfall rate. Each fit was calculated up to maximum rainfall rates of 5–100 mm hr in steps of 5 mm hr . To determine the effects of frequency on each fit, the process was repeated for frequencies in the range of 10–95 GHz in steps of 5 GHz. At higher rainfall rates, above 30 mm hr , the number of samples in each interval reduces, as evidenced by Fig. 2. Even considering six years of data the number of data points above 35 mm hr reduces from many thousands to tens approaching 100 mm hr . The fits remain statistically significant as they include all values up to the maximum rainfall rate. The fitting procedure weights all data points uniformly. However, the sensitivity of the fit to data points above 60 mm hr is reduced.

b

Fig. 6. Contours of parameter of the power-law fit as a function of frequency and maximum fitted rainfall rate. Solid contours are for vertical polarization, dashed contours are for horizontal polarization.

Fig. 5 shows the value of for the power-law fit. It can be seen that at is almost independent of the rainfall rate interval. At approximately 50 GHz the rate of increase in with frequency is at its highest, near 90 GHz the rate of increase declines. Fig. 6 shows , which demonstrates some dependence on rainfall rate interval. Parameter decreases with frequency, with the largest rate of decrease around 50 GHz. The decrease in is small such that the increase in is large enough to increase the resulting attenuation from the power-law equation. The region where corresponding to a linear relationship occurs for frequencies of approximately 30–40 GHz depending on the upper rainfall rate considered. To determine the goodness of fit between the disdrometer derived data and power-law fit, the percentage variance accounted has been calculated [28], [18]. for (PVAF) or R-squared PVAF was determined for all the fits, as a function of frequency

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Fig. 7. Contours of PVAF showing goodness of fit between disdrometer derived rainfall rate and rainfall rate calculated from power-law fits as a function of frequency and maximum fitted rainfall rate. Solid contours are for vertical polarization, dashed contours are for horizontal.

(10–95 GHz) and rain rate interval (5–100 mm hr given by;

Fig. 8. Contours of parameter  from linear fits as a function of frequency and maximum fitted rainfall rate. Solid contours are for vertical polarization, dashed contours are for horizontal polarization.

). PVAF is

(10) where is rainfall rate determined from the disdrometer (mm is rainfall rate determined by the power fit (dB km ) hr ), and is mean rainfall rate measured from the disdrometer (mm hr ). The PVAF for the power-law fit is at its highest around 35–38 GHz and above 15 mm hr , as shown in Fig. 7. This region is where the power-law fit accounts for the most variance in the data. As frequency increases above 38 GHz the goodness of fit reduces. Similarly, as the frequency decreases below 20 GHz the goodness of fit also degrades but more rapidly. The accuracy of the power-law fit tends to be fairly consistent over the rainfall rate interval. For frequencies above 40 GHz, fits up to a maximum rainfall rate of 10 mm hr have a marginally better PVAF than fits to higher rainfall rates. Natural variations in DSD make it difficult to estimate the - relationship exactly using a fit to the data. However, the PVAF shows that the power-law relationship consistently captures a very high percentage of variance for data in the range of 15–60 GHz. The accuracy of the power-law fit is considerably lower above 60 GHz as the PVAF decreases below 90%. B. Linear Attenuation-Rainfall Relationship Like the power-law fit, the linear fit was determined from attenuation and rainfall rate calculated from the disdrometer data. The linear relationship has only a single parameter determined by least-squares fitting and shown by the contours in Fig. 8. The single parameter limits the accuracy of the fit especially when applied to non-linear data. Obviously, while a linear fit may be applied to all frequencies it may only fit well over a limited range. Fig. 8 shows that increases with frequency, which

Fig. 9. Contours of PVAF showing goodness of fit between disdrometer derived rainfall rate and rainfall rate calculated from linear fits based on vertically polarized attenuation as a function of frequency and maximum fitted rainfall rate.

is representative of the increase in attenuation with frequency. Above 45 GHz becomes strongly dependent on the fitted maximum rainfall rate, which indicates increasing non-linearity of the - relationship. The PVAF has been calculated for all the linear fits between 10–95 GHz and for maximum rainfall rates 5 to 100 mm hr . Fig. 9 shows the PVAF for the linear fit. The highest percentage variance accounted for occurs between 30 and 38 GHz. In this frequency range the - relationship is very linear. At frequencies above 45 GHz the PVAF is larger for fits up to 10 mm hr rainfall rates. The larger PVAF implies the - relationship is more linear at low rainfall rates above 45 GHz. At low frequencies (below 15 GHz) the goodness of fit degrades irrespective of the maximum rainfall rate and at a higher rate. In summary, the linear fit cannot be universally used to represent the - relationship for all the frequencies in the range

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Fig. 10. PVAF for power-law and linear fit against maximum fitted rainfall rate for 23, 33 and 38 GHz (vertical polarization).

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Fig. 11. Difference between PVAF of power-law and PVAF of linear fit for frequencies 10–95 GHz and maximum rainfall rates of fits 5–100 mm hr (vertical polarization).

of 10–95 GHz. However, the frequency range between 25–40 GHz the - relationship becomes less sensitive to variability in the raindrop size distribution and the PVAF is at its highest value and is close to 100%. C. Error in Terms of Percentage Difference The power-law relationship has been widely accepted and the results in Fig. 7 show the power-law to be a good representation of the - relationship up to 60 GHz. Due to the credibility of the power-law fit, the linear fit is further compared to the power-law as a standard to be achieved. The PVAF is compared for 23, 33 and 38 GHz for maximum rainfall fits 5–100 mm hr in Fig. 10; 33 GHz has been chosen to show the frequency with the highest PVAF while 23 GHz and 38 GHz have been chosen due to the high number of terrestrial links operating at these frequencies. At 33 GHz the power fit better estimates the - relationship at rainfall rates below 35 mm hr , above 35 mm hr the two fits are matched in performance. The linear fit PVAF at 23 GHz is not only less than 33 GHz but is smaller in comparison to the power-law relationship (at 23 GHz). This suggests a nonlinear - relationship. In the case of 38 GHz, the linear fit PVAF is very high, matching the power fit in performance up to 50 mm hr . To illustrate the differences in the PVAF , the difference between the linear and power PVAF have been calculated as shown by (11) where is the PVAF of the power fit and is the PVAF of the linear fit. . The figure immeFig. 11 shows a contour plot of diately illustrates that the smallest difference in variance is between 30 and 45 GHz. The difference is also zero for 30–40 mm hr from 40–90 GHz. A more linear region exists below 30 mm hr and above 45 GHz as the linear fit outperforms the power fit. Above 40 mm hr and 40 GHz the power-law performs better than the linear fit as the - relationship becomes

Fig. 12. The PVAF for maximum rainfall rates used to generate each fit for 5–100 mm hr when applied to different maximum rainfall rates of 5–100 mm hr at 23 GHz. Linear fit is shown by the solid contour, power-law fit is shown by dashed contour.

increasingly varied and non-linear. The linear fit once again begins to match the performance of the power fit above 80 GHz and around 60 mm hr . Above 70 GHz the variation in the relationship is very high that neither the power-law or linear fit are representative. D. Variation in PVAF as a Function of Rainfall Rate A fit to attenuation generated up to a maximum rainfall rate (e.g., 50 mm hr ) may be used to estimate higher or lower maximum rainfall rates (e.g., up to 100 mm hr ). In the case where the maximum rainfall rate is different to the maximum rainfall rate used to generate the fit, the PVAF will be different. Therefore, the PVAF has also been calculated over maximum rainfall rates from 5–100 mm hr in steps of 5 mm hr for each maximum rainfall rate used in the fitting process from 5–100 mm hr . Figs. 12–14 show the PVAF at 23, 33 and 38 GHz. Note

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Fig. 13. The PVAF for maximum rainfall rates used to generate each fit for when applied to different maximum rainfall rates of 5–100 5–100 mm hr mm hr at 33 GHz. Linear fit is shown by the solid contour, power-law fit is shown by dashed contour.

Fig. 15. Percentage difference between attenuation calculated using power-law fit and linear fit plotted as a function frequency and maximum fitted rainfall rate for vertical polarization (solid contour) and horizontal polarization (dashed contour).

Fig. 14. The PVAF for maximum rainfall rates used to generate each fit for 5–100 mm hr when applied to different maximum rainfall rates of 5–100 mm hr at 38 GHz. Linear fit is shown by the solid contour, power-law fit is shown by dashed contour.

Fig. 16. Vertically polarized attenuation plotted against rainfall rate at 38 GHz showing a linear fit (R = 0:986) and power-law fit (R = 0:990) up to 100 mm hr .

that for clarity we have shown only vertical polarization. However, the results for horizontal polarization are broadly similar. At 23 GHz the - relationship is not linear, as demonstrated by the results in Fig. 12. The PVAF varies significantly over all fits. The variance accounted for is highest when both the maximum rainfall rate of each fit is the same as the maximum rainfall rate used in estimation. Variation unaccounted for decreases as rainfall rate increases. The linear fit PVAF is smaller than that of the power-law fit, linear fit at 33 GHz and 38 GHz. At 23 GHz the - relationship is more dependent on the raindrop size distribution and therefore not linear, hence the PVAF is smaller. The - relationship has been determined to be approximately linear over all rainfall rate intervals at 33 GHz for the disdrometer data available. This is emphasized by the results in Fig. 13. Regardless of the rainfall rate the fit is applied to, the

PVAF is very similar and very high for all rainfall rates. The dashed contours in Fig. 13 show the power-law PVAF, which is very high and performs very similar for all rainfall rate fits. The performance of the power-law fit is almost identical to the linear fit. The power-law fit is outperformed when the maximum fitted rainfall rate is below 10 mm hr and used to estimate high maximum rainfall rates (above 40 mm hr ). At 38 GHz the PVAF decreases as rainfall rate increases, therefore the - relationship is more linear at low rainfall rates, shown in Fig. 14. The maximum rainfall rate fit of 50 mm hr has the highest and most consistent variance accounted for over any rainfall rate in the range of 0–100 mm hr . The power-law fit, shown by the dashed contours in Fig. 14, has high and similar values of PVAF to the linear fit. At 38 GHz the linear fit is highly comparable to the power-law fit and is a good representation of the - relationship.

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TABLE I PARAMETER a; b;  FOR CALCULATING SPECIFIC ATTENUATION AND PERCENTAGE ERROR OF ATTENUATION E FOR VERTICAL POLARIZATION

TABLE II PARAMETER a; b;  FOR CALCULATING SPECIFIC ATTENUATION AND PERCENTAGE ERROR OF ATTENUATION E FOR HORIZONTAL POLARIZATION

E. Error in Terms of Attenuation Difference The power-law fit is once again assumed the best representation of the - relationship. The fit was used as a bench mark to and parameters of each fit for compare the linear fit. The maximum rainfall rate (5–100 mm hr ) were used to calculate attenuation from all the rainfall data derived from the disdromand were calculated as eter. The percentage difference of follows: (12) Fig. 15 shows the difference in attenuation between the power-law fit and linear fit. The region bounded by the 5% contour in Fig. 15 has been chosen to be representative of the area where the - relationship can be considered linear. For vertical polarization, this region lies between 28 GHz and 38 GHz for all rainfall rates. For frequencies above and below 33 GHz, increases as the two estimates of attenuation becomes increasingly different. This is due to the - relationship becoming increasingly nonlinear. Fig. 16 shows an example of the - relationship for specific attenuation at 38 GHz compared with rainfall rate calculated from all disdrometer data at Chilbolton. A linear and power-law

fit have been applied to the data up to 100 mm hr , shown by the solid and dashed lines on the figure respectively. Attenuation clearly increases with rainfall rate almost linearly at 38 GHz. correlation coefficients are high for both linear (0.986) The and power-law (0.990) fits. Visibly the power-law model is a better fit to data above 40 mm hr . For rainfall rates below 30 mm hr both the linear and power-law models are good fits to the - relationship. This result is expected as the PVAF is almost identical for both fits at 30 mm hr , shown by Fig. 14. and for vertically Table I summarizes the values of polarized links for a range of frequencies and maximum rainfall rate fits. Table II summaries the same values over the same ranges for horizontally polarized links. The percentage error in attenuation, , has also been included to show the accuracy of the linear fit. V. CONCLUSION Six years of disdrometer data calculating rainfall rate and attenuation were examined. Frequencies between 30 GHz and 45 GHz have been identified to have the most linear relationship between attenuation and rainfall rate. The attenuation-rainfall relationship determined by the fit at 38 GHz (a common link frequency in the U.K.) and 40 mm hr has one of lowest errors in attenuation up to a maximum rainfall rate of 50 mm hr . In

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the U.K. over 99.99% of rainfall is included up to 40 mm hr . The use of 38 GHz within the U.K. could be an ideal choice for using link measurements to infer rainfall rates. The frequency band where the power-law fit parameter is equal to one shows where the power fit estimates the - relationship is linear. This is the area of best agreement between the fits, in terms of PVAF and attenuation. It has been shown that a linear fit at 33 GHz (vertically polarized) has the highest PVAF and is consistently so over 5–100 mm hr . For horizontal polarization 35 GHz has the highest PVAF. However, other nearby frequencies such as 38 GHz for vertical and horizontal polarization have extremely similar PVAF and can also be used to estimate rainfall rate. ACKNOWLEDGMENT The authors would like to thank Rutherford Appleton Laboratory and the British Atmospheric Data Centre for providing the disdrometer data necessary for this work and the EPSRC for funding this research. REFERENCES [1] D. Atlas and C. W. Ulbrich, “Path- and area-integrated rainfall measurements by microwave attenuation in the 1–3 cm band,” J. Climate Appl. Meteor., vol. 16, no. 4, pp. 327–332, 1977. [2] R. L. Olsen, D. V. Rogers, and D. B. Hodge, “The ar relation in the calculation of rain attenuation,” IEEE Trans. Antennas Propag., vol. AP-26, no. 2, pp. 318–329, 1978. [3] Specific attenuation model for rain for use in prediction methods ITU, Geneva, Switzerland, Rec. ITU-R P.838-3, Sep. 2006, ITU-R Recommendations and Reports. [4] G. Upton, A. Holt, R. Cummings, A. Rahimi, and J. Goddard, “Microwave links: The future for urban rainfall measurement?,” Atmos. Res., 2005. [5] H. Messer, A. Zinevich, and P. Alpert, “Environmental monitoring by wireless communication networks,” Science, vol. 312, p. 713, 2006. [6] A. Zinevich, P. Alpert, and H. Messer, “Estimation of rainfall field using commercial microwave communication networks of variable density,” Adv. Water Resour., vol. 31, pp. 1470–1480, 2008. [7] P. M. Austin, “Relation between measured radar reflectivity and surface rainfall,” Mon. Weather Rev., vol. 115, no. 5, pp. 1053–11053, May 1987. [8] H. Leijnse, R. Uijlenhoet, and J. N. M. Stricker, “Hydrometeorological application of a microwave link: 2. precipitation,” Water Resour. Res., vol. 43, 2007. [9] A. R. Holt, J. W. F. Goddard, G. J. G. Upton, M. J. Willis, A. R. Rahimi, P. D. Baxter, and C. G. Collier, “Measurement of rainfall by dual-wavelength microwave attenuation,” Electron. Lett., vol. 36, no. 25, pp. 2099–2101, Dec. 2000. [10] A. R. Holt, G. G. Kuznetsov, and A. R. Rahimi, “Comparison of the use of dual-frequency and single-frequency attenuation for the measurement of path-averaged rainfall along a microwave link,” IEE Proc. —Microw., Antennas Propag., vol. 150, no. 5, pp. 315–320, 2003. [11] A. Berne and R. Uijlenhoet, “Path-averaged rainfall estimation using microwave links: Uncertainty due to spatial rainfall variability,” Geophys. Res. Lett., vol. 34, no. L07403, pp. 1–5, 2007. [12] D. Giuli, A. Toccafondi, G. B. Gentili, and A. Freni, “Tomographic reconstruction of rainfall field through microwave attenuation measurements,” J. Climate Appl. Meteor., vol. 30, pp. 1323–1340, 1991. [13] P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods. Singapore: World Scientific, 1990. [14] B. E. Sheppard and P. I. Joe, “Comparison of raindrop size distribution measurements by a Joss-Waldvogel disdrometer, a PMS 2DG spectrometer, and a POSS doppler radar,” J. Atmos. Ocean Technol., vol. 11, pp. 874–887, 1994.

[15] J. S. Marshall and W. M. Palmer, “The distribution of raindrops with size,” J. Meteorol., vol. 5, pp. 165–166, 1948. [16] C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Climate Appl. Meteorol., vol. 22, pp. 1764–1775, 1983. [17] J. Testud, S. Oury, R. A. Black, P. Amayenc, and X. Dou, “The concept of “normalized” distribution to describe raindrop spectra: A tool for cloud physics and cloud remote sensing,” J Climate Appl. Meteor., vol. 40, no. 6, pp. 1118–1140, Jun 2001. [18] P. R. Bevington and D. K. Roninson, Data Reduction and Error Analysis For the Physical Sciences. New York: McGraw-Hill, 1992. [19] P. T. Willis, “Functional fits to some observed drop size distributions and parameterization of rain,” J. Atmos. Sci., vol. 41, no. 9, pp. 1648–1661, May 1984. [20] M. Montopoli, F. S. Marzano, and G. Vulpiani, “Analysis and synthesis of raindrop size distribution time series from disdrometer data,” IEEE Trans. Geosci. Remote Sens. E, vol. 46, no. 2, pp. 466–478, Feb. 2008. [21] C. C. Chuang and K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci., vol. 47, no. 11, pp. 1374–1389, Jun 1990. [22] R. Gunn and G. D. Kinzer, “The terminal velocity of fall for water,” J. Meteorol., vol. 6, pp. 243–248, 1949. [23] R. J. Watson, A. R. Holt, V. Marécal, and J. Testud, “A rainrate-attenuation-reflectivity relation for use in the spaceborne and airborne sensing of rain,” IEEE Trans. Geosci. Remote Sens. E, vol. 37, no. 3, pp. 1447–1450, May 1999. [24] Characteristics of precipitation for propagation modelling ITU, Geneva, Switzerland, Rec. ITU-R P.837-5, Aug. 2007, ITU-R Recommendations and Reports. [25] A. J. Townsend, R. J. Watson, and D. D. Hodges, “Analysis of the variability in the raindrop size distribution and its effect on attenuation at 20–40 GHz,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1210–1213, 2009. [26] K. V. Beard and C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci., vol. 44, no. 11, pp. 1509–1524, 1986. [27] E. Matricciani, “Physical-mathematical model of the dynamics of rain attenuation based on rain rate time series and a two layer vertical structure of precipitation,” Radio Sci., vol. 31, no. 2, pp. 281–295, 1996. [28] M. A. Pitt and J. Myung, “When a good fit can be bad,” Trends Cogn. Sci., vol. 6, no. 10, pp. 421–425, 2002. Adrian Justin Townsend was born in England in March, 1985. He received the M.Eng. degree in electrical and electronic engineering (first class honours) from The University of Bath, Bath, U.K., in 2007. Upon graduation he was sponsored by the EPSRC to carry out research as a Ph.D. student in the Department of Electronic and Electrical Engineering, University of Bath. His research involves tropospheric radio-wave propagation on terrestrial and earth-space links, which has included the study of the raindrop size distribution.

Robert John Watson (M’96) was born in England in June, 1971. He received the B.Eng. and Ph.D. degrees in electronic engineering from the University of Essex, Colchester, U.K., in 1992 and 1996, respectively. He was a Senior Research Officer in the Departments of Mathematics and Electronic Systems Engineering at the University of Essex, from 1995 to 1998, where he was involved in number of propagation and weather radar projects. In October 1998, he joined the academic staff at the Department of Electronic and Electrical Engineering, University of Bath, Bath, U.K., where he is currently a Senior Lecturer. He has consulted widely for industry. His research interests include tropospheric radio-wave propagation and remote sensing. Dr. Watson is currently the Commission F representative to the U.K. panel for the International Union of Radio Science (URSI).

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On the -Factor Estimation for Rician Channel Simulated in Reverberation Chamber Christophe Lemoine, Emmanuel Amador, Student Member, IEEE, and Philippe Besnier, Senior Member, IEEE

Abstract—Reverberation chambers were recently proposed to -factor. simulate Rician radio environment with controllable The -factor is also a parameter that may tell how ideal may be a reverberation chamber when it is used for other more conventional purposes. This paper is dedicated to the problem of the correct in a reverberation chamber given a set of data estimation of measured along a stirring process. Index Terms— -factor, reverberation chamber, Rician channel, statistical estimation.

I. INTRODUCTION

I

N many radio propagation environments, the time varying envelope of the received signal can be statistically described by a Rician distribution [1]–[4]. When there is a line of sight (LOS) between the transmitter and the receiver, the received signal can be written as the sum of a complex exponential and a narrowband Gaussian process, which are known as the LOS component and the diffuse component respectively. The relative strength of the direct and scattered components of the received signal is expressed by the Rician -factor. Recently, reverberation chambers (RC) have been proposed to simulate a controllable Rician radio environment for testing wireless devices [5]. A reverberation chamber generally consists of a metallic cavity and an electrically large metallic paddle called a stirrer which enables to change the boundary conditions in the cavity (Fig. 1). The rotation of the stirrer supplies the stirring process. If this Faraday cage is overmoded enough, the field can be described as a combination of numerous modes. Stochastic field is the result of the stirring process [6]. Statistics provide appropriate methods for the evaluation of the main characteristic parameters of an RC [7], [8]. In ideal conditions, any rectangular component of the electric field follows a Rayleigh distribution [9]. Direct coupling paths between transmitting and receiving antennas must be minimized to favor this Rayleigh distribution. This is a common approach when using RC for electromagnetic compatibility (EMC) purposes. Manuscript received July 29, 2009; revised July 20, 2010; accepted November 15, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by the French Ministry of Defence DGA (Délégation Générale de l’Armement), “REI” under Grant 2008 34004. The work of E. Amador was supported by a Ph.D. Grant delivered by the DGA. The authors are with the Université Européenne de Bretagne, France, INSA, IETR, UMR CNRS 6164, F-35708 Rennes, 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.2010.2103003

Fig. 1. Reverberation chamber in IETR research laboratory (2.9 m 8.7 m). The lowest usable frequency is approximately 250 MHz.

2

2 3.7 m

In order to extend the use of RC some authors [5], [10]–[13] have investigated the potential of RCs for the emulation of controlled Rician propagation channels. Rician environment may be reproduced by adjusting levels of direct coupling paths and scattered paths in the chamber [14]. -factor is one of the key parameter of a Rician propagation channel since it represents the ratio of the first paths to the second ones. The use of mechanical stirring is analogous to a static LOS component in mobile channel; whereas the use of mechanical stirring combined with electronic stirring is analogous to a moving terminal, i.e., the phase of the LOS component is constantly changing with time. However, evaluation of Rician -factor out of RC measurements must be analyzed very carefully. A rough estimation was proposed in [5] from parameters measurements, but the estimator is biased and the statistical uncertainties of this estimation are not provided. In a recent paper [15], authors prostarting from the goodness-of-fit test posed an estimator of of the normal distributions of both real and imaginary parts of parameters between antennas. However, the transmission statistics of was not deeply investigated neither the accuracy of estimation as a function of the number of individual measurements and the number of samples. To estimate -factor, some methods use the measured power signals—amplitude only, no phase—while others use complex in-phase and quadrature signals-amplitude and phase or, components, or only the fading phase. Various approaches were done to find -factor using a maximum likelihood method. Greenwood and Hanzo [16] have proposed to compute the distributions of the envelope, then to compare the probability density function of the measured data with a set of hypothetical

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can be expressed as a sum The complex transfer function associated with the unstirred paths of a direct component associated with the stirred energy and a stirred component in the cavity (1) with the complex form (2) (3) Fig. 2. Configuration with a dominant LOS path.

parameter is the sum of Each real and imaginary part of the a deterministic term and a stochastic term (1)–(3) distributions using a suitable goodness-of-fit test. The disadvantage of this method is complexity of implementation, which is a time-consuming and computationally extensive procedure [2], [17]. In these conditions, not being suited for online implementation, this approach is more useful for testing whether the measured envelope is Rician distributed, rather than estifrom independent and mating [16]. A method to extract identically distributed complex Rician channel samples was presented in [18]. Using samples of the phase and envelope of the received signal to estimate , in [19], two estimators are proposed. The best of them will catch our attention in this paper for -factor estimation in a reverberation chamber from measurements. Other methods simpler than the alternatives menestimation. tioned above are moment-based estimators for These techniques are used to estimate -factor based on measurements of the received fading envelope [2], [16], [20]. Such methods are not optimal in coherent wireless systems because they do not take into account the additional phase information provided by complex baseband realizations. Moreover, in [21] a new Rician -factor estimator was derived using correlated channel samples in a noiseless channel, based on samples of the fading instantaneous frequency, representing the derivative of the phase oscillation of fading with time. The main disadvantage of this estimator is represented by the cost of time and estimation computational resources needed for estimation. The purpose of this paper is to provide all necessary theoretin a reverberation chamber. ical background for estimating extraction from RC measurements is revisIn Section II, ited in details. In particular, the first and second moments of estimator are derived from theoretical statistics. The proposed method is then validated through Monte-Carlo (MC) analyses and experiments in RC (Section III). II. REVISITING

(4) (5) Both stirred components follow independent zero mean normal distributions, with the same standard deviation [5], [9] (6) (7) In the simplified case where all wall reflections interact with the stirrer, the only unstirred component is the direct coupling term between antennas. Then the direct component identifies with the LOS path. On the other hand, if there is no multipath scattering only involving the paddle, the stirred component is null and . Moreover, if we assume absolutely has a direct component no reflection, we obtain the anechoic chamber (AC) situation, i.e., (8) Under the hypothesis of an ideal reverberation environment, many authors [5], [22]–[24] have shown that the scattering cofollows the same statistics as a rectangular comefficient ponent of the electric field. In the case of an overmoded cavity, is Rayleigh distributed and the phase the modulus follows a uniform distribution. In addition, the real and parts follow independent zero mean normal disimaginary tributions, with the same standard deviation . Therefore, from (1) it appears clearly that

EXTRACTION FROM RC MEASUREMENTS (9)

A. Overview of

-Factor Formulations in RC

In the paper, we consider a basic RC configuration, where two antennas are located in the cavity. A strong direct antenna coupling can be introduced for instance when decreasing the separation distance between both opposite antennas (Fig. 2). A 2-port vector network analyzer (VNA) gives access to measurements.

denoting the mean operator of data where is the with number of independent stirrer positions1. In the same way as , we will denote for the complex stirred component and respectively the real and imaginary parts of the direct . component

N is systematically implicit and not written in order to simplify the notation.

1

LEMOINE et al.: ON THE

-FACTOR ESTIMATION FOR RICIAN CHANNEL SIMULATED IN REVERBERATION CHAMBER

For a multipath environment, -factor is defined as the ratio of the unstirred energy and the stirred energy [5], [15], [25]

Moreover, as far as the estimation of is concerned, (11) leads to [15]

(11)

-factor denominator

(20)

(10) We can also write -factor as a function of the standard deviation of each real and imaginary part of the transfer function. With (see (1))

1005

is an a Priori Known Parameter: Now, 1) Assuming let be independent, normally distributed, . random variables, with mean zero and same variance be constant values. It is shown in [27] Let also is a noncentral that the distribution of distribution with degrees of freedom and noncentrality pa. Moreover, from decomposition of rameter transmitting parameter (1), (4)–(15), we have the following distribution functions:

where “Var” denotes the variance operator, we have (21) (22)

(12) and therefore2 [5] (13)

B.

Therefore, when becomes sufficiently large, in practice [28] , the central limit theorem (CLT) provides the following distribution functions [29], [30]:

(23)

Estimation in RC

As far as deterministic components are concerned, [15], [26] and shows, with (14) (15) This is equivalent to write that (2) (16) (17)

(24) follows a noncenConsequently using (19), the ratio distribution with 2 degrees of freedom and noncentrality tral parameter . distribuFurthermore, both first moments of a noncentral tion with degrees of freedom and noncentrality parameter are well-known [27]

(25)

complex parameter.3

denoting the phase of the with From (13) and (16), we have

and

Now, using the trigonometric property (14) and (15) is [15] best estimator for

(18)

(26)

, the

and the first and second moments respectively. with is the expected value4 of the Hence using (25), . As a result, for a large number of measureratio ments we find [28] the following expected value:

(19) with 2Some

denoting the estimated value of

.

(27)

papers deal with the direct-to-scattered ratio (DSR) [15], [25], [26] j j . which is very similar to -factor: 3Assuming that all wall reflections interact with the paddle, j j is equivalent to the free-space coupling term.

K

DSR = S

=

S

4And

not

v N= !

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Fig. 3. Biased estimation of dependent stirrer positions.

Now let

K -factor using K

be an estimator of

with different numbers of in-

-factor defined by5

(28) Therefore, assuming is a priori known, we show here that estimation is not the -factor, but the expected value of since the distribution of is not a centered distribution

Fig. 4. Numerical estimation of the 95% confidence interval of 30.

K

with

N=

by MC simulations give a precious idea of the goodness of the estimation of the -factor one should expect for a given and a given number of stirrer positions . is an a Priori Unknown Parameter: In the 2) Assuming is a stochastic parameter, therefore we need to real case evaluate both and in order to estimate . Consequently is more appropriate the following estimator

(31)

(29) MC simulations are conducted to simulate the estimation of the -factor for a given direct component and a given number of independent stirrer positions . Our approach is to replicate parameter according to our measurements by generating a the (21) and (22). The purpose of these simulations is to estimate the -factor and its accuracy by evaluating its confidence interval (CI). Every MC simulation is conducted using scenarios. Let be an estimator of a given scenarios, we can random variable . Using assume that the sample mean is approximately equal to the expected value of

affects Compared to Figs. 4 and 5 shows that evaluating when is relatively high. It means that even with only high values of the -factor, the accuracy of the estimation is . MC simulations show that diminished by the estimation of , the CI of is at least 3 dB. with Baddour [18] has proposed an analytical expression of -factor estimation, based on the biased estimator of variance . Since the RC community generally prefers using unbiased estimators, we provide here the appropriate -factor estimator following the same developments as Baddour in [19]. Thus, using the following unbiased estimator of the variance in (20):

(30)

(32)

Fig. 3 shows how the accuracy of the estimation of the -factor is affected by the number of independent stirrer and , the estimation is positions. For both . Moreover, the significantly biased for values under lower is the more the bias is significant. One should keep in mind that the number of independent stirrer positions in a RC at a given frequency is typically limited to several tens [30]. Fig. 4 for a given and shows the quantiles of the estimation of independent stirrer positions. The quantiles estimated 5We

would like to draw readers’ attention to the fact that although the ratio follows a noncentral distribution with 2 degrees of freedom and noncentrality parameter , does not follow a noncentral distribution with 2 degrees of freedom and noncentrality parameter .

v N=

 v N= K

K



with of mean estimator of

, being observations of the random value , one can show analytically that the unbiased -factor is the following6

(33) We denote

the correction factor and the correction term in (33). It is the first time in

6In [18], the author has used the common biased estimator of the variance, see [19]. Here we actually take into account the standard unbiased estimator of the variance (32).

LEMOINE et al.: ON THE

-FACTOR ESTIMATION FOR RICIAN CHANNEL SIMULATED IN REVERBERATION CHAMBER



1007

S

Fig. 6. Variations of the phase of the transmitting complex parameter as a function of frequency. Here the distance between horn antennas is 1 m (Fig. 2).

Fig. 5. Numerical estimation of the 95% confidence interval of 30.

K

with

N=

transmitted to the chamber can Moreover, the total power using denoting the insertion be related to the mean power loss parameter of the cavity [32]–[34]

a paper dedicated to the emulation of propagation channels in RC, that this appropriate estimator (33) is proposed. C. Increasing Accuracy With Electronic Stirring

(37) Thus we have,

Many practical situations can benefit from electronic stirring in addition to mechanical stirring, in order to improve the estimation of -factor. We have shown previously that only using mechanical stirring may not suffice to have an acceptable level of uncertainty over the evaluation of . The idea here is to add samples using electronic stirring, in order to reduce this level of uncertainty. However, it is shown in this Section that we must be very careful with the use of electronic stirring for estimating -factor. 1) Preliminary Hypothesis: The first question is how the is as a function of frequency in a RC. If the variation of -factor can be considered as a constant value in a frequency bandwidth, then applying electronic stirring is appropriate. Using as the total power transmitted by the emitter to the chamber, and assuming the transmitting antenna has a directivity , the Friis’ transmission formula provides [31], [32]

(34) where is the distance between the transmitting and receiving antennas and is the free-space impedance. On the other hand, Hill [9] provides the following relationship between the stirred and the mean power received over a stirrer component revolution

(35) leading to [5]

(36)

(38) It is well established in RC literature that the insertion loss parameter evolves in [35], [36]. Moreover, assuming a simple but realistic case where the antenna directivity is unbandwidth, we can deduce that the -factor is changed in a a function of the square root of the frequency

(39) , then variations of As shown in Appendix, if . Translated in deciBel, this the -factor are very low , then extreme means that in the bandwidth values of -factor differ only in 0.2 dB. Therefore the assumpin a reasonable frequency tion of insignificant variations of bandwidth is consistent. This conclusion remains valid only if (34) is strictly satisfied (see Section III). measurements in 2) MC Simulation Analysis: Analyzing the electronic stirring case requires particular cautions. It does and in the same way as for menot sum up to estimate chanical stirring [26]. Indeed the phase (17) changes with frefrequency bandquency and is uniformly distributed over a width (Fig. 6). Considering one sample instead of samples may lead to underestimate the -factor of more than 20 dB! This will be illustrated in Section III-B. This is the reason why we adopt the following steps to estimate a global -factor simulated in RC using electronic stirring in the frequency band . for each7 selected independent fre• First we estimate , following the approach developed in (33). quency in 7Therefore

K -factor.

we do not combine frequencies for estimating directly the

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K

Fig. 7. Confidence interval associated with the estimation of h i , with 30 independent stirrer positions and = 1, 30 and 100 independent frequencies.

N=

N

• Second, we gather all estimations to provide a global estimation of factor in . Using independent fre, we estimate the -factor as follows: quencies in

K

Fig. 8. 95% confidence interval associated with the estimation h i of -factor using a MC simulation in comparison with the CI (denoted “th.” in legend) calculated from the 2.5% and 97.5% quantiles which are analytically = 100. It is expressed respectively in (43)–(44), using = 50 and clearly shown that for estimating low values of -factor we need to increase in order to satisfy the desired CI. and/or

K N

N

N

K

N

Therefore, the 95% CI associated with the estimation of is defined by the following quantiles [28]

(40) with denoting the mean operator over data where is the number of independent frequencies used in the narrowband . The advantage of electronic stirring is to provide many estimations of the underlying -factor emulated from mechanical stirring. In comparison with the 95% confidence interval of a -factor estimation based only on mechanical stirring (Fig. 5), Fig. 7 shows the significant reduction of the uncertainty level . As when adding electronic stirring, over the estimation shown in Fig. 7, in the limit case of a Rayleigh channel situation, using only one frequency8 to estimate leads to more than 20 dB of uncertainty. But with independent frequencies (e.g., at 1 GHz [30]) we only have apin proximately 4 dB of uncertainty which is a great improvement9. As in [19], one can demonstrate that the variance of the esti(33) is mator (41) Then, using the CLT [28] the distribution of normal distribution with mean and variance

tends to a

(42) 8i.e.,

N

:

(43)

(44) Fig. 8 compares the 95% CI obtained from MC simulation with the one calculated from the 2.5% (43) and 97.5% (44) quantiles and . It shows that the analytical forwith mulations (43) and (44) match perfectly the result of MC simulations. Fig. 8 shows that if we need a more satisfying confidence interval for very low -factor, we have to select more independent stirrer positions and more independent frein . As indicated in (42), increasing is quencies to reduce the CI associated more efficient than increasing with the estimation of . Nonetheless, one must keep in mind that mechanical stirring (especially in mode-tuning) takes gennarrow erally more time than electronic stirring. So, given a bandwidth, a right method may be to first use the maximum of independent frequencies which are available in number , and second to adjust the number of independent stirrer positions in order to be consistent with the desired CI. III. EXPERIMENTAL RESULTS

=1

9About more conventional RC purposes, when we are looking for optimizing the mechanical stirring efficiency, we try to decrease -factor as most as possible in order to eliminate the entire direct component and favor as most as possible the stirred component. Without the knowledge of the associated confidence interval, the evaluation of the -factor can be strongly inaccurate.

K

K

This Section gives four different experimental results, in order to illustrate the consequence and advantages of the previous statistical analysis provided in the case of both mechanical and electronic stirring.

LEMOINE et al.: ON THE

-FACTOR ESTIMATION FOR RICIAN CHANNEL SIMULATED IN REVERBERATION CHAMBER

K

Fig. 9. Experimental estimation of -factor in RC using horn antennas (cf. Fig. 2) compared to measurements in an anechoic chamber, at 3.5 GHz, with 30 independent stirrer positions.

N=

A. Comparison With Anechoic Chamber Measurements for the Direct Path collected data for a Here the measurement consists of single frequency, where is the number of independent stirrer of the direct component is positions. The phase invariant as long as the line of sight is cleared during the rotation of the stirrer. Compared to electronic stirring, mechanical stirring is more time-consuming and the number of stirrer positions over a complete rotation should be chosen meticulously [30]. In order to confirm the results obtained by our MC simulations, we conducted measurements both in AC and in RC. Measurements in AC give us what can be interpreted as true values of the direct component for a given distance . We use two wide-band horn antennas at 3.5 GHz separated by a distance . By modifying the distance we change the direct component of parameter and thus we change the -factor the measured for each distance . in RC. In RC we estimate both and parameter consists only in an unstirred In AC the measured component. In order to compare the results of RC measurements with those issued from AC we build a -factor using the mean obtained in RC. Fig. 9 shows that for relaof the values of tively high values of mechanical stirring allows a rough estimation of the -factor whereas for low values the estimation is inaccurate. These results corroborate our MC simulations. More accuracy means more independent samples, but the number of independent stirrer positions at a given frequency is limited [30]. By adding electronic stirring, we can increase substantially the number of samples and expect a more accurate estimation of the -factor. B. Electronic Stirring With Horn Antennas The same experiment is here performed using electronic stirring in addition to mechanical stirring. We show in Fig. 10 that

1009

K :

Fig. 10. Experimental estimation of -factor in RC with horn antennas (cf. = 30 indeFig. 2), using electronic stirring in [3 45 GHz; 3 55 GHz] ( = 100 independent frequencies). The green pendent stirrer positions and asterisks correspond with a wrong estimation, highlighting that we cannot use for electronic stirring combined with mechanical stirring, the same method as for mechanical stirring only.

N

:

N

frequency stirring leads to reduce statistical fluctuations. However the improvement which is brought by both electronic stirdoes not seem greatly signifiring and the correction term cant since we can emulate only high -factors with the configuration in Fig. 2. We choose this configuration in order to have using an a reference measurement of the direct component anechoic chamber. With high -factor values, the associated CI of the estimation remains relatively small either with mechanical stirring only or with combined mechanical and electronic stirring (Fig. 7). The green asterisks correspond with a wrong estimation, highlighting that we cannot use for electronic stirring combined with mechanical stirring, the same method as for mechanical stirring only. The reason is that the phase (17) changes with frequency and is uniformly distributed over a frequency bandwidth (Fig. 6). The next experiment aims to generate lower controllable -factors in order to highlight a significant improvement in the estimation of -factor using electronic stirring combined with mechanical stirring. C. Electronic Stirring With Discone Antennas A new experiment similar to the one with horn antennas, but with discones, may simulate lower -factor values since discones are less directive than horn antennas. The result is drawn in Fig. 11. The curve related to AC measurements gives the reference -factor, and is consistent with the direct component calculated from Friis’ transmission formula [31]. However, both results coming from RC measurements do not fit the reference. The reason is clear: in RC our direct component originates in the free-space propagation but also in the numerous reflected paths which are not affected by the stirrer. Our discone antennas are indeed characterized by a very low directivity, and this explains why RC measurements cannot fit AC measurements. The evaluation of RC -factor is not wrong, as the evaluation of the direct component in AC is not biased too, but each chamber does not

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K :

Fig. 11. Experimental estimation of -factor in RC with discone antennas (cf. Fig. 2), using electronic stirring in [3 5 GHz; 3 6 GHz] ( = 50 independent stirrer positions and = 100 independent frequencies). Differences between the reference curves (Friis or AC measurements) and RC measurements are due to the fact that in RC there are multiple unstirred paths and not only the direct path observed in AC.

N

:

Fig. 12. Experimental configuration for controlling Wilkinson coupler.

N

Fig. 13. Experimental configuration leading to optimize mechanical stirring ( = 18 dB).

K 0

K -factor in RC using a Fig. 14. Experimental characterization of the influence of the correction term = 1 from RC measurements, for estimating -factor in the case of electronic stirring. One thousand frequencies have been used in [3 GHz 3 1 GHz].

C

simulate the same direct component so we cannot have comparable results. However, this result highlights the limited performances of mechanical stirring, through the -factor value which cannot be easily reduced, even if the distance between transmitting and receiving antennas is quite long. D. Experimental Validation of the Correction Term We introduced two corrections in the estimation of -factor has the main impact on the quality (33). The correction term of estimation, particularly for low values. For RC purposes, this correction term has never been used so far [5]. In order to show empirically the impact of the correction term , on the estimation of -factor we perform the following experiment. For controlling -factor, we carry out two series of measurements as shown in Fig. 12. First we measure the stirred , and second we artificially add a component direct component . The underlying goal is to have therefore a -factor reference value which we tend to recover parameter measurements using the best estimator (40). from On the one hand, we try to find the best configuration of mechanical stirring in RC with two horn antennas, decreasing as much as possible -factor, in order to have an insignificant di. We obtain rect component and therefore assume

=N

K

; :

using the configuration exposed in Fig. 13 with antennas in cross-polarization. In particular, to improve the stirring efficiency we place an additional panel in order to reflect waves toward the mechanical stirrer. By this way, we manage to reduce -factor of 5 dB with regards to the same configuration but without the additional panel. On the other hand, we control the direct component linking both transmitting and receiving cables and changing the attenuation level of the transmission (Fig. 12). Therefore, transmitting parameter issued from RC adding the complex transmitting parameter meameasurements, to the complex sured with linked cables (Fig. 12), we can control the resulting -factor [37]. Using electronic stirring with a high number of independent frequencies in a small bandwidth , we obtain an accurate evaluation of the expected value of the -factor. The effect of the correction term is clearly shown experimentally in Fig. 14. IV. CONCLUSION Following a meticulous development based on a theoretical analysis combined with simulations results, we endeavour to

LEMOINE et al.: ON THE

-FACTOR ESTIMATION FOR RICIAN CHANNEL SIMULATED IN REVERBERATION CHAMBER

Moreover, one can demonstrate that

TABLE I

VARIATIONS OF

1011

K -FACTOR AS A FUNCTION OF 1f=f

(48) Therefore, (47) provides

(49) provide all necessary elements to estimate correctly the Rician -factor using mechanical and frequency stirring in RC. Many experimental results are presented and show different important points. First, trying to relate RC measurements to AC measurements is quite easy when the direct coupling is very strong compared to all stirred paths. When the direct component is not only due to the direct coupling path, i.e., when unstirred reflections become significant, then we cannot have a reference situation in AC. Moreover, we show that in RC very different scenarios of propagation can be emulated, the lowest -factor being limited by the stirring efficiency. In our RC, we managed . On the other hand, to decrease the -factor down to it is necessary to take into account the corwhen rection term due to the non-central distribution of the direct component, in order to have the best estimation of -factor. The use of frequency stirring is highly recommended by the authors, particularly for reducing the confidence interval associ(40), ated with a -factor estimation. With the estimator using only a few number of independent stirrer positions and of independent frequencies in a narrow a few number bandwidth, we can significantly improve the accuracy of the estimation. APPENDIX Here we estimate the impact of a bandwidth centered in a frequency over the variations , and be respectively the of -factor. Let minimum, the average and the maximum of the -factor that . Starting from (39) can be emulated in

(45) , we can and supposing that the directivity is invariant in find in a good approximation the interval of variations in , defined by of

(46) With

we find

(47)

Some typical variations of

in

are indicated in Table I.

ACKNOWLEDGMENT The authors would like to thank J. Sol for assistance with the measurements. REFERENCES [1] W. C. Jakes, Microwave Mobile Communications. New York: IEEE Press, 1974. [2] L. Greenstein, D. Michelson, and V. Erceg, “Moment-method estimation of the Ricean -factor,” IEEE Commun. Lett., vol. 3, pp. 175–176, Jun. 1999. [3] J. D. Parsons, The Mobile Radio Propagation Channel, 2nd ed. Chichester, U.K.: Wiley, 2000. [4] R. Vaughan and J. B. Andersen, Channels, Propagation and Antennas For Mobile Communications. London, U.K.: Inst. Elect. Eng. Electromagnetic Wave Series, No. 50, 2003. [5] C. L. Holloway, D. A. Hill, J. M. Ladbury, P. F. Wilson, G. Koepke, and J. Coder, “On the use of reverberation chambers to simulate a Rician radio environment for the testing of wireless devices,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3167–3177, Nov. 2006. [6] P. Corona, G. Ferrara, and M. Migliaccio, “Reverberating chambers as sources of stochastic electromagnetic fields,” IEEE Trans. Electromagn. Compat., vol. 38, no. 3, pp. 348–356, Aug. 1996. [7] L. R. Arnaut, “Statistics of the quality factor of a rectangular reverberation chamber,” IEEE Trans. Electromagn. Compat., vol. 45, pp. 61–76, Feb. 2003. [8] P. Corona, G. Ferrara, and M. Migliaccio, “Generalized stochastic field model for reverberating chambers,” IEEE Trans. Electromagn. Compat., vol. 46, no. 4, pp. 655–660, Nov. 2004. [9] D. A. Hill, “Plane wave integral representation for fields in reverberation chambers,” IEEE Trans. Electromagn. Compat., vol. 40, no. 3, pp. 209–217, Aug. 1998. [10] J. Valenzuela-Valdés, M. García-Fernández, A. Martínez-González, and D. Sánchez-Hernández, “Non-isotropic scattering environments with reverberation chambers,” in Eur. Conf. Antennas Propagation, Edinburgh, U.K., Nov. 2007, pp. 1–4. [11] J. Valenzuela-Valdés, A. Martínez-González, and D. SánchezHernández, “Emulation of MIMO nonisotropic fading environments with reverberation chambers,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 325–328, 2008. [12] E. Genender, C. Holloway, K. Remley, J. Ladbury, G. Koepke, and H. Garbe, “Use of reverberation chamber to simulate the power delay profile of a wireless environment,” in Proc. EMC Eur., Hamburg, Germany, Sep. 2008, pp. 1–6. [13] J. Valenzuela-Valdés, A. Martínez-González, and D. SánchezHernández, “Diversity gain and MIMO capacity for nonisotropic environments using reverberation chamber,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 112–115, 2009. [14] N. K. Kouveliotis, P. T. Trakadas, I. Hairetakis, and C. N. Capsalis, “Experimental investigation of the field conditions in a vibrating intrinsic reverberation chamber,” Microwave Opt. Technol. Lett., vol. 40, pp. 35–38, Jan. 2004. [15] C. Lemoine, P. Besnier, and M. Drissi, “Advanced method for estimating direct-to-scattered ratio of Rician channel in reverberation chamber,” Electron. Lett., vol. 45, no. 4, pp. 194–196, Feb. 2009. [16] D. Greewood and L. Hanzo, Characterization of Mobile Radio Channels, R. Steele, Ed. London, U.K.: Mobile Radio Communications, 1992.

k

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[17] K. Talukdar and W. Lawing, “Estimation of the parameters of the rice distribution,” J. Acoust. Soc. Amer., vol. 89, no. 3, pp. 1193–1197, Mar. 1991. [18] K. E. Baddour and T. J. Willink, “Improved estimation of the Ricean k -factor from I/Q fading channel samples,” IEEE Trans. Wireless Commun., vol. 7, pp. 5051–5057, Dec. 2008. [19] K. Baddour and T. Willink, “Improved estimation of the Ricean factor from I/Q samples,” in IEEE 66th Vehicular Technology Conf., 2007, pp. 1228–1232. [20] A. Abdi, C. Tepedelenlioglu, M. Kaveh, and G. Giannakis, “On the estimation of the k parameter for the Rice fading distribution,” IEEE Commun. Lett., vol. 5, pp. 92–94, Mar. 2001. [21] G. Azemi, B. Senadji, and B. Boashash, “Ricean k -factor estimation in mobile communication systems,” IEEE Commun. Lett., vol. 8, pp. 617–619, 2004. [22] D. A. Hill, Electromagnetic theory of reverberation chambers National Institute of Standards and Technology (NIST), 1998, Tech. Note 1506. [23] P. Corona, “Reverberating chamber electromagnetic field in presence of an unstirred component,” IEEE Trans. Electromagn. Compat., vol. 42, no. 2, pp. 111–115, May 2000. [24] C. Lemoine, P. Besnier, and M. Drissi, “Investigation of reverberation chamber measurements through high power goodness of fit tests,” IEEE Trans. Electromagn. Compat., vol. 49, pp. 745–755, Nov. 2007. [25] P. Hallbjörner, “Reverberation chamber with variable received signal amplitude distribution,” Microwave Opt. Technol. Lett., vol. 35, no. 5, pp. 376–377, Dec. 5, 2002. [26] G. Lerideau, E. Amador, C. Lemoine, and P. Besnier, “Quantifying stirred and unstirred components in reverberation chamber with appropriate statistics,” presented at the IEEE Int. Symp. EMC, Austin, Aug. 2009. [27] N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions. New York: Wiley, 1995. [28] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 1965. [29] J. D. Hamilton, Time Series Analysis. Princeton, NJ: Princeton Univ. Press, 1994. [30] C. Lemoine, P. Besnier, and M. Drissi, “Estimating the effective sample size to select independent measurements in a reverberation chamber,” IEEE Trans. Electromagn. Compat., vol. 50, no. 2, pp. 227–236, May 2008. [31] C. Balanis, Antenna Theory—Analysis and Design, 2nd ed. New York: Wiley, 1997. [32] V. Fiumara, A. Fusco, V. Matta, and I. M. Pinto, “Free-space antenna field/pattern retrieval in reverberation environments,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 329–332, 2005. [33] Reverberation chamber test methods Int. Electrotech. Commission Std., 2003, IEC 61000-4-21. [34] V. Rajamani, C. Bunting, and J. West, “Sensitivity analysis of a reverberation chamber with respect to tuner speeds,” in Proc. IEEE Int. Symp. EMC, Honolulu, HI, Jul. 2007, vol. 1, pp. 1–6. [35] L. Kone, B. Démoulin, and S. Baranowski, “Application des chambres réverbérantes à brassage de modes à la caractérisation des émissions rayonnées par un équipement,” presented at the CEM 08, Paris, France, May 2008. [36] J. M. Ladbury, G. H. Koepke, and D. Camell, “Evaluation of the NASA Langley Research Center mode-stirred chamber facility,” U.S. Dept. Commerce, 1999, NIST Tech. Note 1508.

[37] E. Amador, G. Delisle, and D. Grenier, “Reverberation chamber as a synthesis instrument for Rayleigh and Rice channels,” in Proc. 5th IASTED Int. Conf. Antenna, RADAR, Wave Propagation, Baltimore, MD, Apr. 2008, pp. 65–68.

Christophe Lemoine received the Diplôme d’Ingénieur degree from Ecole Nationale Supérieure de l’Aéronautique et de l’Espace (SUPAERO), Toulouse, France, in 2004, the Master degree in financial risk management in 2005, and the Ph.D. degree in electronics from the Institut National des Sciences Appliquées (INSA), Rennes, France, in 2008. He is now an Assistant Professor at INSA of Rennes, France. His current research interest at the Institute of Electronics and Telecommunications of Rennes (IETR), Rennes, France, includes new theoretical and experimental approaches of mode-stirred reverberation chambers for EMC, propagation channels and antenna measurement applications.

Emmanuel Amador (S’10) received the Diplôme d’Ingénieur degree from the Institut National des Télécommunications (INT), Evry, France, in 2006 and the M.Sc. degree in electrical engineering from Laval University, Quebec, QC, Canada, in 2008. He is currently working toward the Ph.D. degree at the Institute of Electronics and Telecommunications of Rennes (IETR), INSA, Rennes, France.

Philippe Besnier (SM’10) received the diplôme d’ingénieur degree from Ecole Universitaire d’Ingénieurs de Lille (EUDIL), Lille, France, in 1990 and the Ph.D. degree in electronics from the University of Lille, in 1993. Following a one year period at ONERA, Meudon, as an Assistant Scientist in the EMC Division, he was with the Laboratory of Radio Propagation and Electronics, University of Lille, as a researcher at the Centre National de la Recherche Scientifique (CNRS) from 1994 to 1997. From 1997 to 2002, he was the Director of Centre d’Etudes et de Recherches en Protection Electromagnétique (CERPEM), a non-profit organization for research, expertise and training in EMC, and related activities, based in Laval, France. He co-founded TEKCEM in 1998, a private company specialized in turn key systems for EMC measurements. Since 2002, he has been with the Institute of Electronics and Telecommunications of Rennes, Rennes, France, where he is currently a Researcher at CNRS heading EMC-related activities such as EMC modeling, electromagnetic topology, reverberation chambers, and near-field probing.

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1013

Design of a Near-Field Focused Reflectarray Antenna for 2.4 GHz RFID Reader Applications Hsi-Tseng Chou, Senior Member, IEEE, Tso-Ming Hung, Nan-Nan Wang, Hsi-Hsir Chou, Chia Tung, and Paolo Nepa, Member, IEEE

Abstract—The design of a reflectarray antenna is presented when the radiated field is focused in the near-zone of the array aperture. In particular, the reflectarray antenna is implemented for RFID reader applications at 2.4 GHz. Numerical investigations on the radiation characteristics of this reflectarray, as well as an experimental validation, are presented to demonstrate its feasibility. Index Terms—Microstrip antenna, near-field focused antennas, reflectarray, RFID antennas.

I. INTRODUCTION

T

HE design and characterization of a planar reflectarray antenna are presented, when the radiated field is focused in the near-zone of array aperture, as that produced by an ellipsoidal reflector whose scattering fields focus at one of its two foci when it is illuminated by a feed located at the other focus. The near-field focused reflectarray can be viewed as a flat implementation of the ellipsoidal reflector. This work is motivated by the increasing interest in the applications relevant to RFID systems [1]–[4], vital life-detection and noncontact microwave detection [5]–[9], where the objects under detection may be located in the near zone of the antenna. A radiated field focused in the target area helps to reduce the interferences caused by the scattering from neighborhood structures, and saves the system power. This reflectarray [10]–[13] antenna can be used in the RFID applications at 2.45 GHz band whose market is expected to grow in the near future. Most of past work to produce near-field focused beams employed a phased array antenna [3]–[9], [14], [15], with only some of them using defocusing parabolic reflector [17], [18]. Array based solutions suffer from complicated beam-forming networks and relatively high power losses in the microstrip lines

Manuscript received March 25, 2010; revised July 13, 2010; accepted August 28, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by the Ministry of Economics Affairs and National Science Council, Taiwan. H.-T. Chou, T.-M. Hung, and C. Tung are with the Department of Communication Engineering and Communication Research Center, Yuan-Ze University, Chung-Li 320, Taiwan (e-mail: [email protected]). N.-N. Wang is with the Department of Microwave Enginneering, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]). H.-H. Chou is with the Communication Research Center, Yuan-Ze University, Chung-Li 320, Taiwan (e-mail: [email protected]). P. Nepa is with the Department of Information Engineering, University of Pisa, Pisa, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103030

Fig. 1. A scenario of 2.4 GHz RFID applications for a department store management.

if the number of elements is large; above issues can limit their applications as well as flexibility in a real practice. In this context, a reflectarray antenna appears to be a good candidate to reduce above shortcomings since the sophisticated beam-forming RF circuits can be omitted. A reflectarray antenna is structured by using a periodic array of reflecting elements that are illuminated by a feed antenna. The reflecting elements are properly designed to produce different phase delays that compensate for the phase differences of feed’s radiating fields propagating from the feed’s phase center to the array elements [10]. The phase compensation mechanism makes the fields scattered from the array to equip equal-phases on an aperture orthogonal to the antenna boresite direction. Since no complicated beam-forming circuits are needed, the power loss is minimized. The studies on reflectarray antennas have experienced a long history with most applied to produce highly directional beams in the far zone for satellite communications at extremely high frequencies such as Ku- and Ka- bands [10]–[13]. This paper extends their applicability to a lower frequency band for RFID reader applications as shown in the scenario in Fig. 1, which generally occurs in a department store’s product management system. In this case, the target area is typically within 2 m away from the antenna. The design implemented in this paper will make the reflected fields focused in this region to improve the radio link performance. The paper is organized as following. In Section II, the characteristics and design of a reflectarray as a NF focused antenna are developed and summarized. Section III describes the numerical and experimental comparisons to validate the antenna design.

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Concluding remarks are drawn in Section IV. A time convenis used throughout in this paper. tion of II. ANTENNA DESIGN The structure of reflectarray, feed antenna and reflecting element are summarized in this section. A. The Structure of Reflectarray The structure of the reflectarray antenna is shown in Fig. 2(a) and (b), where the side and front views are shown, respectively. The reflecting elements are located periodically along x- and y- coordinates with a circular edge boundary of 80 cm in diameter and a half-wavelength inter-element distance. The antenna feeding the reflectarray is located at 35.1 cm away. The feed’s boresite direction points to the array’s center at 30 degrees with respect to z-axis. This array resembles the characteristics of an ellipsoidal reflector that focuses its scattering at the second focal point for a near-field communication. This second focal point is located at roughly 100 cm away from the center of the array in a target area. Each reflecting element is properly designed to produce a different phase delay that is used to compensate the phase differences of fields that are radiated from the feed, experience propagation from the phase center of the feed, and scatter from the reflecting elements to reach the target area. This phase compensation mechanism results in an equal-phase superposition of scattering fields at the focal point in the target area as that produced by an ellipsoidal reflector. and being the distances As illustrated in Fig. 2(a), let reflecting element located at , and the feed’s between the is phase center and the focal point, respectively, where assumed to indicate the element located at the center of array is the distance (it is used as a phase reference). In this case, between the focal point and array center along a direction with 30 degrees deviation from the z-axis. It is assumed that the reflectarray is located in the far zone of feed antenna, which has a co-polarized radiation expressed by

Fig. 2. Reflectarray structure and its parameters for near-field focusing application. (a) Side view; (b) front view.

by referencing it to that produced at central reflecting element . Thus in (2) can be found by by assuming (3) With this phase description, the fields scattered from the reflectarray at can be expressed as

(1a) (1b) is the feed’ spherical coordinates. A is its where complex amplitude. The pattern shape is controlled by an index , which is determined by matching (1b) with the feed’s actual radiation pattern [18]. The phase of the field illuminating the reflecting element and scattered back to the focal point, can be expressed as

(2) indicates pointing to reflecting element, and where is the free space wavenumber. In (2), is the phase delay reflecting element to compensate for the field produced at phase and make an in-phase summation of fields at the focal is a constant. This phase constant is created point, i.e.,

(4) where is a real function associated to the radiation patreflecting element, with . It can be also tern of modeled with a cosine taper. In (4), the cross-polarized field components are ignored. The maximum field strength along the propagation path shown in Fig. 2 occurs at the location where the derivative of magnitude of (4) (or the square of the magnitude for the power density) with respect to distance vanishes. B. Feed Antenna Element The feed antenna shown in Fig. 3(a) is a nearly square patch and printed on a 1.6 mm thick FR4 substrate ( ) and excited by a 50 microstrip line. A circular polarization is achieved by trimming the ends of two opposite corners, which excites two orthogonal modes with a 90 time-phase

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Fig. 3. Structure and reflection coefficient of the feed antenna. (a) Layout of the circularly polarized reflectarray feeder; (b) reflection coefficient.

Fig. 5. Reflecting element and its phase variations vs. the patch size. In (c) t is the thickness of the air gaps. (a) Reflecting element (stack-up); (b) phase variation.

Fig. 4. Radiation patterns (co-polar and cross-polar components) of the feed antenna at the two principal radiation planes.

difference to produce a radiation with left-hand circular polarization (LHCP) and a half-power beamwidth of about 104 as shown in Fig. 4. The reflection coefficient shown in Fig. 3(b) exhibits an operational band equal to 2.4–2.48 GHz. C. Reflecting Elements The reflecting elements are microstrip patches whose sizes are varied to control the phases of reflected fields [13]. As illustrated in Fig. 5(a), each element consists of two FR4 substrates with square patches printed on the top surfaces of each substrate. Square patches are used because their symmetry assists to retain a satisfactory axial ratio for the scattered fields. The upper patch has a smaller size, while however the ratio of the patches’ sizes

is retained as a constant value equal to 0.7. To increase the phase variation linearity at 2.4 GHz, it is important to insert two air gaps between the FR4 substrates and the ground plane, which are realized by using Styrofoam. The phase variations with respect to patch size are shown in Fig. 5(b), where curves are relevant to different air gap thickness. They are obtained by considering the scattering from an infinite array of identical elements illuminated by a normally incident plane wave (i.e., they are relevant to the reflection coefficient phase). The interelement distance is selected to be half a wavelength in order to retain only the fundamental propagating mode and uniquely determine the required phase values. It may be observed that a thicker air gap increases the linearity of phase variation with respect to the size of metallic patches. In this case, a good linearity appears when the thickness is roughly 20 mm, which makes the overall thickness approximately half a wavelength at 2.4 GHz. Above phase curve has been used to design the reflectarray. Fig. 6 shows the co-polarized patterns of the and 90 planes when a single element scattered field at

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Fig. 6. Patterns of co-polarized scattered fields of a single reflecting element at the two principal planes.

Fig. 7. Distributions of field strength along the propagation path in Fig. 2 for various values of d . (unit: meter).

is illuminated by an incident plane wave, whose beamwidths in (4) with a cosine tapering. allow one to model III. NUMERICAL AND EXPERIMENTAL RESULTS A numerical investigation is presented in this section, which is further validated by measurements on a prototype. It is also validated by simulation results obtained by using Ansoft HFSS finite element code [19]. The index, , of cosine tapers are found and , respectively. to be 2 and 2.5 for The characteristics of near-field focusing are first examined by considering the field strengths along a propagation path are selected in shown in Fig. 2, where various values of the analysis. It is noted that an in-phase focusing results in equal phase superposition of fields, but it does not warrant a maximum field strength at the focused point, Q. As shown in Fig. 7, the field strength peak appears at a distance shorter than . To validate the present results, a comparison between analytical ((4)) and simulation results (HFSS numerical data) has , which results in a been performed by considering . Fig. 8(a) and (b) show the normalized field peak at

Fig. 8. Numerical computations and measured results of (a) normalized field strength, (b) beamwidth and (c) contoured patterns.

field amplitude and beamwidths along the propagation path. The beamwidth is defined as the distance between the points in the direction transverse to the propagation path (on the x-z plane) where the field strength is 3 dB down from the peak value. A good agreement between analytic, numerical and measurement

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the maximum field strength appears at the location where the derivative is equal to zero. With this agreement in mind, Fig. 8(c) shows the normalized contoured patterns of co-polarized field components on an plane as defined in Fig. 2(a). It is noted that the distance, r, is measured from the array center along the propagation path while is the direction orthogonal to this path on the x-z plane. It is observed in Fig. 8(c) that the field distribution along the . propagation path exhibits the field peak at around Finally the contoured patterns on the u-v planes at distances and 90 m away from the array center are shown in of Fig. 10(a) and (b), respectively. They exhibit the field focusing before and at the location where the field strength peak occurs. Fig. 9. Photo of reflectarray prototype.

IV. CONCLUSION This paper presented the design procedure of a near field focused reflectarray. An antenna for RFID reader applications was designed, prototyped and experimentally characterized. Numerical data and experimental results confirm the feasibility of the proposed reflectarray configuration as a near field focusing antenna. It is worthy to note that a reflectarray exhibits a minor complexity than phased array antennas, so representing a valuable solution in realistic applications. REFERENCES

Fig. 10. Measured contoured patterns of co-polarized components at various distances away from the array center. The patterns are plot on u-v planes or; (b) at thogonal to the propagation path, as shown in Fig. 2. (a) at r . r

= 90 cm

= 80 cm

results over the prototype in Fig. 9 has been found. The narrowest beamwidth occurs at a distance between 80 cm and 100 cm, which implies that the focused region has a diameter of around 20 cm. Also shown in Fig. 8(a) is the derivative of the power density along the above propagation path. As expected,

[1] R. Gadh, P. Chu, G. Q. Huang, K. Michael, B. S. Prabhu, and G. Roussos, “RFID—A unique radio innovation for the 21st century,” Special Issue Proc. IEEE, 2010. [2] K. V. S. Rao, P. V. Nikitin, and S. F. Lam, “Antenna design for UHF RFID tags: A review and a practical application,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 3870–3876, Dec. 2005. [3] Z.-M. Liu and R. R. Hillegas, “A 3-patch near field antenna for conveyor bottom read in RFID station application,” in Proc. IEEE Antennas Propagation Symp., Albuquerque, NM, Jul. 9–14, 2006, pp. 1043–1046. [4] A. Buffi, A. A. Serra, P. Nepa, H.-T. Chou, and G. Manara, “A focused planar microstrip array for 2.4 GHz RFID readers,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1536–1544, May 2010. [5] M. Bogosanovic and A. G. Williamson, “Antenna array with beam focused in near-field zone,” Electron. Lett., vol. 39, no. 9, pp. 704–705, May 2003. [6] M. Bogosanovic and A. G. Williamson, “Microstrip antenna array with a beam focused in the near-field zone for application in noncontact microwave industrial inspection,” IEEE Trans. Instrum. Meas., vol. 56, no. 6, pp. 2186–2195, Dec. 2007. [7] K. D. Stephan, J. B. Mead, D. M. Pozar, L. Wang, and J. A. Pearce, “A near field focused microstrip array for a radiometric temperature sensor,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1199–1203, Apr. 2007. [8] T. W. R. East, “A self-steering array for the SHARP microwave-powered aircraft,” IEEE Trans. Antennas Propag., vol. 40, no. 12, pp. 1565–1567, Dec. 1992. [9] S. Karimkashi and A. A. Kishk, “A new Fresnel zone antenna with beam focused in the Fresnel region,” presented at the XXIXth Gen. Assembly Int. Union Radio Sci., Chicago, IL, Aug. 7–16, 2008. [10] J. Huang, Reflectarray Antennas. : Wiley-IEEE, 2007. [11] S.-H. Hsu, C. Han, J. Huang, and K. Chang, “An offset linear-array-fed ku/ka dual-band reflectarray for planet cloud,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3114–3122, Nov. 2007. [12] M. Arrebola, J. A. Encinar, and M. Barba, “Multifed printed reflectarray with three simultaneous shaped beams for LMDS central station antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1518–1527, Jun. 2008. [13] J. A. Encinar, “Design of two-layer printed reflectarrays using patches of variable size,” IEEE Trans. Antennas Propag., vol. 49, no. 10, pp. 1403–1410, Oct. 2001. [14] R. C. Hansen, “Focal region characteristics of focused array antennas,” IEEE Trans. Antennas Propag., vol. AP-33, no. 12, pp. 1328–1337, Dec. 1985.

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[15] A. Badawi, A. Sebak, and L. Shafai, “Array near field focusing,” in Proc. Conf. Communications, Power Comput. (WESCANEX97), Winnipeg, MB, May 22–23, 1997, pp. 242–245. [16] L. Shafai, A. A. Kishk, and A. Sebak, “Near field focusing of apertures and reflectors antennas,” in Proc. Conf. Communications, Power Comput., Winnipeg, MB, May 22–23, 1997, pp. 246–251. [17] W. J. Graham, “Analysis and synthesis of axial field pattern of focused apertures,” IEEE Trans. Antennas Propag., vol. AP-31, no. 4, pp. 665–668, Jul. 1983. [18] Y. Rahmat-Samii and S.-W. Lee, “Directivity of planar array feeds for satellite reflector applications,” IEEE Trans. Antennas Propag., vol. AP-31, no. 3, pp. 463–470, May 1983. [19] “High Frequency Structure Simulator (HSFF) User Manual-v12,” Ansoft Corporation, Pittsburgh, PA, 2009. Hsi-Tseng Chou (S’96–M’97–SM’01) was born in Taiwan, in 1966. He received the B.S. degree from National Taiwan University in 1988, and the M.S. and Ph.D. degrees from Ohio State University (OSU), Columbus, in 1993 and 1996, respectively, all in electrical engineering. In August 1998, he joined Yuan-Ze University (YZU), Taiwan, and is currently a Professor in the Department of Communications Engineering. His research interests include wireless communication network, antenna design, antenna measurement, electromagnetic scattering, asymptotic high frequency techniques such as uniform geometrical theory of diffraction (UTD), novel Gaussian beam techniques, and UTD type solution for periodic structures. He has published more than 250 journal and conference papers. Dr. Chou is an elected member of URSI International Radio Science US commission B. He has received two awards from Taiwanese Ministry of Education and Ministry of Economic Affairs in 2003 and 2008, respectively to recognize his distinguished contributions in promoting academic researches for industrial applications, which were the highest honors these two ministries have given to university professors to recognize their industrial contributions.

Tso-Ming Hung was born in Taiepi, Taiwan, in 1984. He received the B.S. and M.S. degrees from Yuan Ze University, Taiwan, in 2007 and 2010, respectively, all in electrical engineering. His research interests mainly engage in the reflectarray antenna design for near- and far- field applications, and small sized antennas for the applications of wireless communications.

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

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

Chia Tung was born in Kaohsiung, Taiwan, in 1984. He received the B.S. degree from National Kaohsiung Normal University, Taiwan, in 2006 and the M.S. degree from Yuan Ze University, Taiwan, in 2008, in electrical and communications engineering, respectively. He is currently working toward the Ph.D. degree at Yuan Ze University. He mainly engaged in the design of reflectarray and phased array antenna designs for satellite communications.

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

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RFID Grids: Part I—Electromagnetic Theory Gaetano Marrocco

Abstract—The close displacement of UHF RFID tags can be considered as an electromagnetic interconnected system having specific properties. The so denoted RFID Grid includes single-chip tags in close mutual proximity or a single tag with a multiplicity of embedded microchips. A multi-port scattering framework is used to derive the macroscopic parameters governing the system response which could be optimized for the specific application. Moreover, unique features are introduced, such as the possibility to improve the power scavenging and the generation of analog identifiers and fingerprint. The last ones are electromagnetic responses independent on the position and orientation of the reader and on the nearby environment, with great relevance for Sensing and Security. Index Terms—Backscattering, coupling, grid, RFID, sensor.

I. INTRODUCTION

T

HE radiofrequency identification technology (RFID) for object tagging [1] is rapidly evolving toward the “Internet of Things” [2], [3]: the convergence of a number of research disciplines (identification, real-time localization, sensor networks, pervasive computing) that enable the Internet to get into the real world of physical objects interacting with web services. Measuring, labelling and timing of made things and humans and their mapping into the environment are expected to stimulate new context-aware services. Pleasant user experience will be planned at the workplaces, in public areas as well as in the home environment by embedding computational intelligence into the nearby environment and simplifying human interactions with everyday services. The pervasive interconnection with things will require to deploy a multitude [4] of RFID tags, even provided with sensing capability, and to conceive new functions. RFID tags have been also hypothesized to be randomly mixed with various bulk materials to develop amorphous computing capability [5]. Remotely powered (passive) UHF (870-960MHZ) tags are one of the most attractive option within the multiform RFID galaxy due to low-cost, no need for maintenance, room-comparable read range and high data-rate. RFID tags are tiny computers with tiny radios and merge together both digital (the microchip data generation) and analog features (antennas and propagation phenomenology). Data transmitted back to the reader during the interrogation protocol are digitally encoded, but the strength of the backscattered power is governed in an

Manuscript received February 02, 2010; revised August 03, 2010; accepted August 28, 2010. Date of publication January 13, 2011; date of current version March 02, 2011. This work was supported by the PRIN-2010 project. The author is with the DISP-University of Roma Tor Vergata, 00133 Roma, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103019

analogical manner by the interaction with nearby objects, by the propagation modality [6], and even by the mutual position and orientation among reader and tags. Sensing applications of passive multi-tags or multi-microchips devices are increasingly researched in the very last years. In [7] the use of a two-chip tag, embedding inertial accelerometers, was first introduced to transmit discrete motion information by the so called ID modulation. A similar concept was applied in [8] to encode the digital output of a sourced sensor by means of four tags. In [9] and [10] the multi-chip paradigm was extended to the continuous sensing of parameters. Here, the dependence of the tag antenna’s electrical features on the time-variation of the physical and geometrical properties of the tagged object, was fully exploited yielding the concept of self-sensing tags. Finally, multiple, closely spaced, tag were used in [11] as electromagnetic field probes. Packing many radio elements into small spaces yields an electromagnetic complex system where tags interact among themselves and with the nearby objects. In both the cases the electromagnetic properties of the data link (read distance, bit error rate, space uniformity) are affected somehow. Most of the available papers concerning the mutual proximity of tags mainly consider the dense displacement in terms of degradation of the RFID link quality and investigate on possible mitigation solutions. Back in the 1970s, it was already shown [12] that the electromagnetic coupling between antennas in transmitting and receiving modes are different and that such a coupling may be mitigated by a proper non-linear loading at the antennas’ port. More recently, the coupling mechanism involving non-linear loads has been theoretically addressed in [6] for general purpose applications, while [14] has put into evidence, by means of a combined antenna/non-linear model, the need for a multi-tag design procedure to balance the coupling effects. Rich experimentations have been presented in [15], [16] concerning stacked planes of RFID tag antennas [17]. This configuration seems to produce interference effects on readability analogous to classic reflection filters as well as phenomena of collective modulation potentially able to distort the received signals. It was moreover verified, and corroborated by theoretical models, that tags with small radar cross section are more suited for application in dense configurations. The presence of weak spots in linear ad multi-dimensional alignment of near field and long range tags was addressed in [18]–[20] by means of both experiments and modeling. Depending on the inter-tag spacing, the performances of the center antennas could be worse than the side antennas, due to the voltage standing wave across the array. Finally [21] recognizes, through experimentation and simulations, that microchips with larger impedance permit to achieve tag displacements more immune to the coupling effects. In this paper the UHF RFID grids are fully investigated as an electromagnetic interconnected system with the purpose to ex-

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Fig. 1. Interrogating reader and RFID Grid connected to microchips of impedg. The reader’s field E and the backscattered field E are both ances fZ referred to the grid’s coordinate system.

over the static field, which is instead the case of HF (13.56 MHz) and lower frequency tags. The th port is associated with an embedded gain and input resistance , both evaluated when the other ports are open-circuited. Accordingly, an embedded effective . From the relationship [22] length is also introduced among the effective length, gain and antenna input resistance , is conveniently expressed as

(1) tend the usual RFID link equations to the more complex case of multi-chip configurations, deriving the expressions of realized gain, impedance matching and backscattered power and highlighting not only the principal limitation but also useful undocumented features. The term grid is here used to indicate a generally coupled multitude of UHF tags, including standalone tags in close mutual proximity as well as tags with a multiplicity of embedded RFID microchip transponders able to achieve advanced capabilities such as redundancy and sensing. This RFID grid is here analyzed by the electromagnetic formalism of multiport scatterers with specific care to the unique feature of the asymmetric UHF RFID communication which involves energy scavenging to wake up the microchip and then the transmission of data by backscattering modulation. One of the most valuable achievement of the theoretical study is the concept of analog identifier which complements the digital identifier of a tag. This is a physical invariant deduced for an RFID grid which embeds structural properties of the specific device, as attached onto a specific object. This parameter will demonstrate to be independent on the nearby environment and may be collected by the reader from any direction. As it will be discussed next, this property could be useful to achieve non conventional sensing capabilities and to develop security fingerprints. Finally, starting from the most general configurations, interesting properties are derived for the particular class of periodic RFID grids, such as infinite and uniform array of tags, or configuration with circular symmetry, and ready to apply formulas are given for the two-port tags. II. DESCRIPTION OF THE SYSTEM A pictorial view of an RFID grid is sketched in Fig. 1. The grid’s terminals, where microchips are connected, act as the denote the equivports of the grid. Let alent impedance of the th microchip in the energy-scavenging mode. As discussed later on, this impedance will be changed during the backscattering modulation. For generality are assumed to be different. The reader unit interrogating the grid emits a field . This arrangement may represent a set of independent tags, electromagnetically coupled by proximity effects, or a unique radiating element provided with multiple independent microchips and sensors. In any case the system will be hereafter regarded as a unique electromagnetic object. Although the presented approach is of general application, the whole formulation is specified for the UHF and higher frequency RFID systems, where the radiative field are predominant

is the polarization complex unitary vector where is the vacuum impedance. It is worth noticing and that, even in case of grids with distinct single-port tags, the efaccounts for the currents over all the conducfective length tors when only the th port is fed and the others are kept in open circuit. The field radiated by the reader can be easily rewritten in of the reader’s antenna [22] terms of the effective length

(2) The dependence on link parameters such as the input power through the reader’s antenna and its gain is made explicit by means of (1) and of the relationship between current and input resistance of reader’s antenna:

(3) with

the polarization of the reader-emitted field. III. SYMBOLS

The most significant symbols used in the paper are here listed for reader’s convenience. position of the reader’s antenna with respect to the grid system. gain of the RFID grid referred to the th port. embedded realized gain of the th port. effective length of the grid related to the th port. effective length of reader’s antenna. gain of the reader’s antenna. reader’s power through the antenna. input resistance of the reader’s antenna. power sensitivity of the th microchip. impedance matrix of unloaded grid. diagonal matrix of port loading terminations grid matrix

.

.

MARROCCO: RFID GRIDS: PART I—ELECTROMAGNETIC THEORY

grid admittance matrix:

1021

.

active input impedance at the th grid’s port. power transfer coefficient at th port. electric field emitted by the reader. open circuit received voltage at th grid’s port. column vector of normalized port’s gain.

Fig. 2. Thevenin network model of the RFID-Grid where the incident field coming from the reader is accounted for by means of open circuit voltage generators V . Z and Z are respectively the impedance matrix of the terminations and of the N -port system.

polarization mismatch between reader and th grid port. power collected by the th port of the grid. power backscatterd by the th port toward the reader. field scattered by the grid. field radiated by the grid when activated the th port. backscattered field by open circuit grid. received backscatterd voltage on the reader’s load.

and the phase explicit the amplitude . The open circuit input resistance at port has been replaced by the self-resistance of the N-port network. The ports’ terminations are accounted for by a diagonal macontaining trix the microchips’ impedances. The input current and voltage at , respectively. the th port are The column of port currents are related to the open circuit by the matrix equation column vector and hence

maximum reader’s voltage due to modulation. (6)

turn on power concerning the th microchip. where and and impedance matrix, respectively.

analog identifier of the th port.

are the {grid admittance

A. Input Impedance at Grid’s Ports

IV. NETWORK REPRESENTATION An RFID grid can be considered as a multi-port loaded scatterer, whose termination loads, e.g., the microchips’ impedance status, are asynchronously changed to achieve the backscattering modulation. The electromagnetic properties of the direct link (the microchips scavenge power from the remote reader and are then ready to perform actions) and the reverse link (the microchips modulate their internal impedance to add information to the reflected wave) may be efficiently described by introducing an -port analog of the grid, according to the mathematical formalism presented in [23], here specified for an RFID system. With reference to Fig. 2, the N-port system collecting the interrogating electromagnetic wave is modelled by the matrix and by equivalent Thevenin voltage genwhich, according to (1) and (3) can erators be written as

The active input impedances of the RFID grid system, following the coupled array notation, are (7) where indicates the th row of and . The input active input impedance, e.g., is hence rewritten in the form (8)

B. Harvested Power by Grid’s Microchips The power collected at the th port is given by . By using (6) and (4), and having introduced the embedded realized gain of port

(4) (9) where the harvested power the single-port case [1]

can be formally expressed as in

(5) (10) is a normalized port’s gain which groups the parameters of the grid. The polarization mismatch between the reader’s field and the th port’s effective length has been written having made

Unlike the standard tag configuration, the roles of impedance matching, gain and polarization, are now closely confused to-

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gether into the realized gain expression, and can not be generally separated. the sensitivity of the th microchip, the Denoting with maximum activation distance for the th port is obtained from , e.g., (10) when

(11) The embedded realized gains of the grid are therefore the parameters to be taken into account within a multi-chip antenna optimization.

(see Appendix) a compact representation is achieved to separate the modulation effect from the un-modulated response

(17) As in [27], assuming modulation data encoded according to FM0 or Miller schemes and perfect-matched filter demodulation with ideal carrier recovery and the equalization at the receiver, and maximum values of input of the minimum the detector voltage can be found to be, respectively, and

V. MODULATED BACKSCATTERING (18)

According to [23], the field backscattered by the grid is with (12) where is the field scattered by the grid when all the port is the row-vector are open-circuited (structural mode) and fields radiated by the grid, considered as single port of the antenna, when the th port is sourced by a unitary current and the others are open

(19) Introducing the inverse of perturbed matrix in (17), the maximum received signal becomes

(13) The voltage collected at the reader’s load, under perfect . By impedance matching condition, is applying (1) for both reader and grids’ ports and after simple mathematical manipulations the collected signal is conveniently written as (14) The backscattering modulation imposes that the microchip impedance at the th port is switched between two states: and and hence the corresponding received signals will be and . The grid impedance matrix will be accordingly modulated as

(15) where element:

is an all-zeroes matrix except for the if if

(16)

is hence a one-element perturbation of the grid . The corresponding impedance matrix impedance matrix to be introduced in (14) can be derived from un-modulated by using the theory of Perturbed Matrix [26]. In particular, after some manipulations

(20) The above equation shows in clear way the role of the modulation impedance, the structural properties and the interrogating field (inside ). The received signal’s amplitude depends on the reader-tag distance and on the mutual reader-tag orientations since reader’s and tags’ gain are not isotropic. In the particular and very common modulation choice and (or high impedance), the signal reduces to (21) and introducing the expression of the embedded realized gain in (9), the following very compact form if found: (22) Without any loss of generality, above impedance switch modality will hereafter assumed for notation simplicity. The corresponding power at the detector is hence , where the parameter accounts for the particular demodulation used. To better appreciate the properties to be introduced in the next section, it is worth noticing are strongly dependent that the backscattered signals on the whole grid admittance as well as on mutual reader-grid and are generdistance and orientation since the gain data are not suited to remotely ally non isotropic. Hence,

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detect the variation of the chemical or physical properties of the grid, as it could be useful in sensing applications (see for instance [28]), unless the reader-grid position is kept fixed during all along the monitoring time. But even in controlled conditions, variations of the surrounding environment (e.g. a furniture is moved, a person is walking within the RFID link) will produce fluctuating data. The intrinsic properties of the RFID grid, however, will permit to overcome this issue, disclosing new opportunities as shown in the next section. VI. ANALOG IDENTIFIERS AND GRID FINGERPRINT entering into the reader’s The minimum power antenna for which the th port’s microchip wakes up and begins to perform actions is denoted as turn-on power of the th port. In that condition the power collected by such a port equals the . It is useful to combine the microchip sensitivity in (10)) and backward powers at forward ( turn-on in a way such to drop out the influence of distance and of the reader’s and port’s gains and orientation. A non-dimensional form can be hence introduced (23) The left side member includes power quantities which are known ( is declared by the microchip’s manufacturer), or , ). So the (23) gives a measurable by the reader ( unique feature of the th port and may be considered as a kind of analog identifier of the th port. is an invariant with respect to distance and to mutual orientation between reader and tag. The set of analog identifiers of the grid, together with the digital identifier of the microchips, give the electromagnetic fingerprint of the multi-port system (24) which, in case of identical port microchips, is proportional to the diagonal of the admittance matrix of the loaded grid. The fingercollected in the reader’s bandwidth, is a structural print property, independent on the particular interrogation modality and on the nearby environment since, in case a scattering object was present in the reader’s zone, it would affect both the direct and the reverse link, and the normalization in (23) would cancel such an effect. The fingerprint generalizes to multi-chip configurations a concept recently introduced in [10] for the single-chip case.

could communiConsequently the grid fingerprint cate this variation to the reader, without any requirement on the interrogation position and distance. The electromagnetic design of the grid should be therefore finalized to emphasize the sensibility of the grid admittance to the external agents to be sensed. This concept poses the basis of complex self-sensing applications where the grid acts as distributed sensor, processor and communication system at a same time. B. RFID Fingerprint for Security Implication to Security of analog identifiers is related to physical-layer certificate of authenticity (COA) [29], [30] of the tagged object or of the tag support itself. A secure code could be produced by means of joint cryptographic processing of digital and analog identifiers of the grid, the last one being object-dependent since the grid matrix is affected by the physics of the tagged item itself. The fingerprint of grids could be made unique by introducing impurities or defects into each grid in a controllable way. In this case the electromagnetic design of the grid has to be oriented to make the fingerprint as much as stable with respect to the external agents, opposite to the requirements of the sensing applications. The promising advantage with respect to other approaches as in [29], [30] will be the simplicity in fingerprint reading, e.g., identification “on the flight,” which will not require a controllable interrogation modality nor tag positioning, thanks to angular/distance invariance of the fingerprint itself. VII. PERIODIC RFID GRIDS Above general formulation takes particular simple and handy expressions when the grid’s ports have equal self impedance , and equal effective length along the grid’s normal axis. This happens, for instance, when the RFID grid exhibits some periodicity as in the case of two identical tags placed over a plane in symmetric position or when ports are placed in circular arrangements (Fig. 3). Moreover, to have invariant polarization efficiency, the reader polarization is required to be circular. For these cases the impedance matrix is a circulant matrix [24], e.g., a particular case of Toeplitz form where each row vector is rotated one element to the right relative to the preceding row vector. A circulant matrix is fully specified by a single generating vector, which appears as the first row/column. The last row/column is the generating vector grid is represented by in reverse order. For example the the following matrix

(25)

A. RFID Fingerprint for Sensing The RFID grid’s fingerprint is expected to be greatly relevant to RFID sensing and security. Just to have an idea about the possible sensing applications of the grid’s fingerprint, assume that the state of the grid is changing along with time, for instance in case geometrical or chemical properties of the surface where the grid is attached on are modified by internal or external agents, say , or if the grid itself is doped with biochemical receptors.

For such family of matrix the sum of the elements of each row or column is a constant. The same properties are shared by the and matrices in case of equal port’s impedances for any ). This feature will be useful later on to ( simplify the definition of RFID-grid parameters. If moreover the grid lays on a plane and the reader’s beam is broadside with respect to the grid the forcing term in (5)

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A. Matching Conditions It is worth discussing the maximum power which can be scavenged by the microchips. Under the assumption that the matrix is definite positive (‘ ’ means complex conjugate), the Desoer’s Theorem [25] gives the relationship between the that is required network matrix and the termination matrix ports. By the to achieve the maximum power transfer to the actual formalism, and since is a symmetric matrix, the Desoer condition specifies as (31)

Fig. 3. Examples of periodic RFID-grids. The resulting impedance matrix has a circulant form. (a) two facing equal dipoles, (b) two-port symmetric tag, (c) three-chips tag, (d) circular alignment of four tags, (e) radial alignment of eight tags, (f) line of infinite tags.

becomes a vector with identical entries and hence the active impedances in (8) reduce to (26) The embedded realized gain simplifies in (27) , are the common gain and polarization mismatch where of all the ports corresponding to the broadside incidence and (28) is the embedded power transfer coefficient associated with the th port. The power scavenged by the th port’s microchip can be therefore rewritten in complete analogy with the one-port case, e.g., (29) with full separation of impedance- and polarization-matching case, issues. As it will formally shown later on for the the embedded power transfer coefficients could exceed the unit due to the constructive electromagnetic coupling among ports. Better power scavenging properties could be hence obtained in comparison with standalone tags. In case the port impedances are all coincident the properties of as circulant matrices apply and the corresponding active input impedances for broadside illumination in (26) are independent on the particular microchips while coincide with the sum of the row element of , e.g., (30) and the realized gain and power transfer coefficient are accordingly the same for each port.

The above equation admits more than a unique solution which moreover depends also on the specific querying fields. The particular solution independent on the reader-grid mutual orientation and on the radiation pattern of the reader is trivially . However, due to the assumed form of , this solution is feasible only if is a diagonal matrix, e.g., if the ports are decoupled. So it is more useful to consider as reference solution that one occurring in case of broadside illumination and periodic grids ( is a vector of equal entries). It is easy to show that (30) accordingly reduces to thanks to the properties of circulant matrices (the sum of row or column elements is constant). So the maximum scavenged power requires equal microchips and from (30) the matching condition for broadside incidence can be finally rewritten as (32) The termination impedances should be therefore equal and such to cancel the sum of self and mutual reactance. Above condition is under-determined since there are two equations (for real and imaginary parts, respectively) and unknowns (there are independent complex matrix). Several options for the elements in the circulant geometrical topologies of the grid are hence possible to achieve the best power scavenging. The matching condition, as expected, reduces to the standard conjugate matching for the single port case. Interesting consequences on the maximum power transfer will be shown later on for the particular case of two-port grids. It is worth noticing that the electromagnetic fingerprint of a periodic grid under broadside interrogation, and optimum impedance matching as in (32), is a vector of equal entries (33) This function will give hence an indirect measure about the regularity of the grid and could be useful to emphasize the variation of the grid status due to the interaction with external agents. VIII. EXAMPLE OF TWO-PORT GRID Operative grid formulas are here specified for a two-port grid which may model the case of a two-microchip tag where a first microchip is used to transmit the identifier of the tagged object and the second to provide information about the physical status of the item. A different configuration may use both the

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microchips to communicate some discrete event according to the ID-modulation [7]. For the sake of simplicity a symmetric tag is assumed and ; and hence

(34) For identical port loading impedances the following equations hold

,

(35) (36) (37) (38) (39) Above formulas reduce to the case of one-port tag as the interport coupling vanishes. Under the Hermite matching condition as in (32) the power transfer coefficients (37) is ultimately if if if

(40)

whereas improvements over the single-port case may be theoretically achieved if the mutual resistance is made negative which means that it acts as an internal generator or in other word, as discussed in [31] and [32] in the context of MIMO systems, a portion of the power scattered by each port can be recaptured by the adjacent port, particularly when the matching network is appropriately implemented. So the overall power harvested by the two-ports RFID-grid may be in principle higher than the power collected by two non-interfering tags in the same conditions (same gain and input impedance). IX. SUMMARY AND CONCLUSIONS This paper has introduced some properties of dense displacement of RFID devices by means of formal electromagnetic tools. The RFID grid should be considered as the physical layer implementation of an interconnected system exploiting both digital and analog features. Electromagnetic coupling is the physical mean by which the individual RFID microchips are mutually correlated since the backscattered power modulated by each microchip depends on the whole grid. It has been however demonstrated that the electromagnetic granularity of the full system, e.g., the network matrix elements, can be remotely discriminated thanks to the unique features of RFID two-ways link which involves a power wake-up threshold and coded response. The most valuable achievements of the presented theoretical formulation are listed as follows:

i) the design of an RFID grid can be performed by using the general formula in (9) for the embedded realized gain which is strictly correlated to the read range of the grid. This formula is useful for implementation within an optimization tool; ii) in case of periodic grids, as a large array or a two-chips tag, very simplified and handy formulas (35) and (36) may be used to master the tag matching and to improve the read-range with respect to a single port tag; iii) the analog identifier (22) and the fingerprint (23) have been introduced for the first time as invariant of the grid with many implication in sensing and security. The RFID grid combines some of the properties of multiple-output arrays, such as the improvement of energy scavenging in comparison with coupling-free regular tags, and the distributed processing of environmental parameters. A multiplicity of sensing data, which are ambient dependent, could be collected and processed by the reader without specific requirements about reader-tag position. Based on this idea it is conceivable to develop smart self-sensing skins well suited to envelope things, plants and even body regions, which may communicate in real time their multidimensional physical history by a free sweep of the reader. In a companion part II paper, several experimentations will be presented to verify the theory and to underline the real-world limitations of the above concepts. However, a lot of work is still needed and to fully understand the so many implications of the engineered use of a multiplicity of radio identification objects, e.g., when the RFID microchips are used not as individual devices but just as an elemental electronic component of a more complex distributed system. APPENDIX A. Details on

Inversion in (17)

matrix where From [26] the inverse of a is a one-element perturbing matrix and if and , can be calculated starting from , as where (41) where and matrix symmetry

means the th column of ,

. By setting and enforcing the , (17) is obtained.

ACKNOWLEDGMENT The author wishes to thank E. Di Giampaolo, C. Occhiuzzi and S. Caizzone for valuable discussions. REFERENCES [1] D. M. Dobkin, The RF in RFID: Passive UHF RFID in Practice. Amsterdam: Elsevier, 2007. [2] D. Preuveneers and Y. Berbers, “Internet of things: A context- awareness perspective,” in The Internet of Things: From RFID to the NextGeneration Pervasive Networked Systems, L. Yan, Y. Zhang, L. T. Yang, and H. Ning, Eds. London: Auerbach Publications, 2008. [3] B. Sterling, Shaping Things. Cambridge, MA: The MIT Press, 2006.

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[4] J. Bohn and F. Mattern, “Super-distributed RFID tag infrastructures,” Lecture Notes Comput. Sci., vol. 3294/2004, pp. 1–12, 2004. [5] H. Abelson, D. Allen, D. Coore, C. Hanson, G. Homsy, T. F. Knight, F. R. Nagpal, E. Rauch, G. J. Sussman, and R. Weiss, “Amorphous computing,” Commun. ACM, vol. 43, no. 5, pp. 74–82, Mar. 2000. [6] J. D. Griffin and G. D. Durgin, “Complete link budgets for backscatterradio and RFID systems,” IEEE Antennas Propag. Mag., vol. 51, no. 2, pp. 11–25, 2009. [7] 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, Jan.–Mar. 2005. [8] L. Catarinucci, R. Colella, and L. Tarricone, “A cost-effective UHF RFID tag for transmission of generic sensor data in wireless sensor networks,” IEEE Trans Microwave Theory Tech., vol. 57, no. 5, pp. 1291–1296, May 2009. [9] G. Marrocco, L. Mattioni, and C. Calabrese, “Multi-port sensor RFIDs for wireless passive sensing—Basic theory and early simulations,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2691–2702, Aug. 2008. [10] G. Marrocco and F. Amato, “Self-sensing passive RFID: From theory to tag design and experimentation,” in Proc. Eur. Microwave Conf., Sep. 2009, pp. 1–4. [11] S. Capdevila, R. Serrano, A. Aguasca, S. Blanch, J. Romeu, and L. Jofre, “RFID-based flexible low-cost EM field probe,” in Proc. IEEE Antennas Propagat. Int. Symp., Jul. 2008, pp. 1–4. [12] J. P. Daniel, “Mutual coupling between antennas for emission or reception-Application to passive and active dipoles,” IEEE Trans. Antennas Propag., vol. 22, no. 2, pp. 347–349, 1974. [13] K.-C. Lee and T.-H. Chu, “Mutual coupling mechanisms within arrays of nonlinear antennas,” IEEE Trans. Electromagn. Compat., vol. 47, no. 4, pp. 963–970, 2005. [14] V. Rizzoli, A. Costanzo, M. Rubini, and D. Masotti, “Rigorous investigation of interactions between passive RFID tags by means of nonlinear/electromagnetic co-simulation,” in Proc. 36th European Microwave Conf., Manchester, U.K., 2006, pp. 722–725. [15] S. M. Seigand and D. M. Dobkin, “Multiple RFID tag plane array effects,” in Proc. IEEE Antennas Propag. Int. Symp., Jul. 2006, pp. 1027–1030. [16] D. M. Dobkin and S. M. Weigand, “UHF RFID and tag antenna scattering—Part I: Experimental results,” Microw. J., vol. 49, no. 5, pp. 170–190, May 2006. [17] D. M. Dobkin and S. M. Weigand, “UHF RFID and tag antenna scattering—Part II: Theory,” Microw. J., vol. 49, no. 6, pp. 88–92, Jul. 2006. [18] L. Hsueh-Jyh, H.-H. Lin, and H.-H. Wu, “Effect of antenna mutual coupling on the UHF passive RFID tag detection,” presented at the IEEE Antennas and Propag. Soc. Int. Symp., 2008. [19] F. Lu, X. Chen, and T. T. Ye, “Performance analysis of stacked RFID tags,” in Proc. IEEE Int. Conf. on RFID, 2009, pp. 330–337. [20] X. Chen, F. Lu, and T. T. Ye, “The weak spots in stacked UHF RFID tags in NFC applications,” in Proc. EEE Int. Conf. on RFID, 2010, pp. 181–186.

[21] Y. Tanaka, Y. Umeda, O. Takyu, M. Nakayama, and K. Kodama, “Change of read range for UHF passive RFID tags in close proximity,” in Proc. IEEE Int. Conf. on RFID, 2009, pp. 338–345. [22] G. Franceschetti, Electromagnetics. New York: Plenum Press, 1997. [23] J. R. Mautz and R. Harrington, “Modal analysis of loaded N-port scatterers,” IEEE Trans. Antennas Propag., vol. 21, no. 2, pp. 188–199, Mar. 1973. [24] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD: The Johns Hopkins Univ. Press, 1996. [25] C. A. Desoer, “The maximum power transfer theorem for n-ports,” IEEE Trans. Circuit Theory, vol. 20, no. 3, pp. 328–330, May 1973. [26] F. C. Chang, “Inversion of a perturbed matrix,” Appl. Math. Lett., vol. 19, pp. 169–173, 2006. [27] F. Fuschini, C. Piersanti, F. Paolazzi, and G. Falciasecca, “Analytical approach to the backscattering from UHF-RFID transponder,” IEEE Antennas Wireless Propagat. Lett., vol. 7, pp. 33–35, 2008. [28] C. Floerkemeier, R. Bhattacharyya, and S. Sarma, “Beyond the ID in RFID,” presented at the 20th Tyrrhenian Int. Workshop on Digital Communications—A CNIT Conf., Italy, 2009. [29] G. DeJean and D. Kirovski, “Making RFIDs unique—Radio frequency certificates of authenticity,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 2005, vol. 1B, pp. 393–396. [30] O. Ureten and N. Serinken, “Wireless security through RF fingerprinting,” Can. J. Elect. Comput. Eng., vol. 32, no. 1, pp. 27–33, Winter, 2007. [31] F. Broydïœ and E. Clavelier, “Taking advantage of mutual coupling in radio-communication systems using multi-port-antenna array,” IEEE Antennas Propag. Mag., vol. 49, no. 4, pp. 208–220, Aug. 2007. [32] J. W. Wallace and M. A. Jensen, “Mutual coupling in MIMO wireless systems: A rigorous network theory analysis,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1317–1325, Jul. 2004. Gaetano Marrocco was born in Teramo, Italy, on August 29. 1969. He received the Laurea degree in electronic engineering and the Ph.D. in degree in applied electromagnetics from University of L’Aquila, Italy, in 1994 and 1998, respectively. In summer 1994, he was a Postgraduate Student at the University of Illinois at Urbana Champaign. In 1997, he was appointed Researcher at the University of Rome “Tor Vergata,” where he currently teaches antenna design and bioelectromagnetics. In 2010, he achieved the the rank of Associate Professor in Electromagnetics. In autumn 1999, he was a Visiting Scientist at the Imperial College in London, London, U.K. His research is mainly directed to the modeling and design of broad band 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. Dr. Marrocco currently serves as an Associate Editor of the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS. He was the Founder and the Chair of the first two editions of the Workshop RFIDays.

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Communications The Optimum Operating Frequency for Near-Field Coupled Small Antennas Youndo Tak, Jongmin Park, and Sangwook Nam

Abstract—The optimum operating frequency of wireless power transfer is investigated for near-field coupled resonant small antennas. The existence of the frequency can be inferred from the equivalent circuit representation for coupled small antennas, based on mode-based analysis using the addition theorem, and the frequency dependence of the radiation efficiency. From the EM simulation results, it is demonstrated that there is an optimum operating frequency for maximum matched gain. In addition, it is shown that the optimum frequency can be accurately estimated using the addition theorem and the EM simulation results of a single antenna. Index Terms—Addition theorem, electrically small antenna (ESA), maximum gain, near-field coupling, optimum operating frequency, wireless power transfer.

I. INTRODUCTION In the near-field region around the antenna, the imaginary power density, which corresponds to the oscillating evanescent energy, dominates [1]. When the resonant antennas are placed closely enough, the near-fields of each antenna overlap and the antennas are reactively coupled, and then the energy can be transferred with high efficiency [2], [3]. In [2], the wireless power transfer system using the resonant coupling of the antennas was evaluated and analyzed based on the coupled mode theory (CMT). For the design of a wireless power transfer system, the power transfer ratio, or efficiency, is one of the important parameters for describing the system performance [4]. According to the previous analysis, the parameters such as the radius of the solenoid, the number of turns, and the operating frequency can affect the power transfer efficiency [5]. From the point of view of maximizing the antenna Q-factor, the optimum antenna structure and operating frequency can be determined [3]. However, if the antenna structure is given, the operating frequency needs to be determined for the optimized maximum power transfer efficiency. If the given antenna is an electrically small antenna (ESA), it predominantly generates the TE10 or the TM10 mode. Hence, the coupling between two ESAs can be analyzed by the interaction of the TE10 or the TM10 modes of each antenna, using the addition theorem [4], [6]. According to the addition theorem, the interaction of coupled antennas is dependent on the electrical distance between them. Therefore, the transmission characteristics are dependent on the operating frequency. Manuscript received April 17, 2010; revised August 03, 2010; accepted September 09, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0001958). The authors are with the School of Electrical Engineering and INMC, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103034

Fig. 1. Two coupled short dipoles with arbitrary orientation [7].

For the given antenna structure, it has its own self-resonant frequency. However, if the proper reactive element is externally connected, the antenna can be tuned in to a forced-resonance at an arbitrary frequency except the self-resonant frequency. Therefore, the analysis of the optimum operating frequency for a given antenna structure is needed.

II. MUTUAL IMPEDANCE OF NEAR-FIELD COUPLED SMALL ANTENNAS The coupling of the antennas can be considered interactions of spherical modes and analyzed using the addition theorem [7]. If two short electric dipoles generating TM10 modes are placed as shown in Fig. 1, the normalized mutual impedance is represented as (1), shown at the bottom of the next page, where  is the free space impedance, 0 = 0 u1 + 0 u2 + 0 u3 is a unit vector representing the orientation of the second antenna, and d is the physical distance between antennas. If the electrical distance is small enough, equation (1) can be expressed as

x

y

U

z

Z 12(TM10) ' sin u2 + cos u3 g  32j 2 cos fcos ' sin u1 (+kdsin )3 ' cos u2 0 sin u3 g 0 sin fcos ' cos u1 +sin (kd)3

(2)

using the asymptotic form of the Hankel function for small arguments. Hence, in the near-field region, the normalized mutual impedance of near-field coupled small antennas can be approximated as

Z 12(TM10) jnear0region / f 03 :

(3)

In the case of the small loops (or the magnetic dipoles) generating the

TE10 modes, the normalized mutual impedance is the same as in (1) due to the duality of the addition theorem.

0018-926X/$26.00 © 2010 IEEE

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III. FREQUENCY DEPENDENCE OF RADIATION RESISTANCE AND RADIATION EFFICIENCY FOR SMALL ANTENNAS The ESA that generates the TM10 or the TE10 mode can be approximately described as the short dipole or the small loop. The radiation resistance of the short dipole is given as

Rr (TM10) = 202 l

2

/ f2

(4)

where l is the total height of the dipole and  is the wavelength of the resonant frequency [1]. This means the radiation resistance is simply proportional to the square of the operating frequency for the fixed size. In the case of the small loop, the radiation resistance is given as 2 Rr(TE10)  31200 S2 / f 4 (5) where S is the area of the loop and  is the wavelength of the resonant

frequency [1]. Hence, the radiation resistance of the loop is proportional to the frequency by the fourth power. The resistive loss of the antenna is proportional to the square root of the frequency, due to the skin effect of the conducting wire, as the operating frequency is increased. Hence, the frequency dependence of the radiation efficiency for the TM10 mode and the TE10 mode can be approximately considered as

r (TM10) jLF / f r (TE10) jLF / f 3:5 : 1:5

(6) (7)

However, the approximated frequency dependences of the radiation resistance described above are valid only under the condition that the antenna is operating at low frequencies, because the radiation resistance is changed depending on the direction of the radiation current at the frequencies above the self-resonant or anti-resonant frequencies. However, the resistive loss is continuingly increased as the frequency is increased. Therefore, at the frequencies above the lowest resonant or anti-resonant frequency, the radiation efficiency needs to be found by using other numerical approaches. IV. EXISTENCE OF THE OPTIMUM OPERATING FREQUENCY AND ESTIMATION BY USING THE ADDITION THEOREM The power transfer efficiency (which is also referred to as the power gain) is related to the product of the radiation efficiency and the mutual impedance, as shown in [4]; therefore, the frequency-dependent behavior of the power gain is also different for the short dipole and the small loop, as follows:

r (TM10) 1 Z12 (TM10) jLF;near0region / f 01:5 r (TE10) 1 Z12 (TE10) jLF;near0region / f 0:5 :

(8) (9)

Conventional near-field power-transfer ESAs such as small helix antennas usually generate both the TM10 and the TE10 mode simultaneously [6]. Hence, the frequency dependence of the matched gain cannot be simply described as the short dipole or the small loop. Instead, the dominant mode at each operating frequency needs to be considered

Fig. 2. Two coupled solenoidal small antennas: (a) wide solenoidal small loop, (b) narrow solenoidal small loop, (c) wide open-ended small helix, and (d) narrow open-ended small helix.

for the analysis of the matched gain. If the dominant mode is changed from the TE10 to the TM10 mode as the operating frequency is increased, the maximum power versus frequency curve will be a convex shape, due to the frequency dependence of (9) and (8), respectively. Therefore, it is strongly implied that there is an optimum operating frequency. On the other hand, if the dominant mode is changed from the TM10 to the TE10 mode, it can be inferred that there may be a worst operating frequency. When the antenna is complex conjugate matched, the antenna is forced to be resonant at the frequency, because the port reactance is canceled out. If the structure of the antenna is fixed and the operating frequency can be varied, the optimum operating frequency is set to the frequency at which the maximum transducer power gain (which is also referred to as the matched gain) is maximized. For a given two-port network, the matched gain can be obtained from the 2-port S-parameters by using conventional relations, as shown in ([8, p. 551]). In addition, it is worth mentioning that the 2-port S-parameters for the given two-port network can be obtained either by full 2-port EM simulation of the total system or by calculation using the addition theorem with the EM simulation result of the stand-alone single antenna as shown in [6]. V. EXAMPLES As examples, the frequency-dependent characteristics of the 10-turn, open-ended small helix antennas and solenoidal small loop antennas, which are made of copper wire, are investigated. According to the ratio between the radius and the height of the antenna, the antennas are classified into the wide case and the narrow case, respectively, as shown in Fig. 2. For the solenoidal loop antenna, the wide antenna is in anti-resonance at 17 MHz, and the narrow antenna is in anti-resonance at 27 MHz. In the case of the small helix antennas, the wide antenna is in

^ (2) u1 +sin ' sin u2 +cos u3g 0 H^_ 1 (kd)sin  fcos ' cos u1 + sin ' cos u2 0 sin u3 g Z 12(TM10) = 23 2H1 (kd) cos fcos ' sin kd2!" kd (2)

(1)

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Fig. 3. Radiation efficiency of the antennas: (a) wide solenoidal small loop, (b) narrow solenoidal small loop, (c) wide open-ended small helix, and (d) narrow open-ended small helix.

Fig. 4. The power ratio between the TE and the TM modes: (a) wide solenoidal small loop, (b) narrow solenoidal small loop, (c) wide open-ended small helix, and (d) narrow open-ended small helix.

resonance at 17 MHz and in anti-resonance at 25 MHz, whereas the narrow helix is in resonance at 29 MHz and in anti-resonance at 46 MHz. The radiation efficiency, defined as the ratio between the input resistance with perfect electric conductor (PEC) and with copper, is also shown in Fig. 3. By using EM simulation, the power ratios between the TE10 and the TM10 modes in resonance for each frequency are evaluated, as shown in Fig. 4. In the case of the solenoidal loop antenna for both the wide and the narrow cases, the TE10 is the dominant mode at the low frequencies, and then the TM10 mode becomes dominant, as the operating frequency is increased. Hence, it can be supposed that there is an optimum operating frequency at which the maximum transfer efficiency is obtained. Moreover, as the operating frequency of the loop antenna is increased, the matched gain will be much reduced, due to the reduction of the radiation efficiency. On the other hand, for the wide

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Fig. 5. Maximum matched gain obtained by the full EM simulation (Method1) and estimated by the addition theorem (Method 2): (a) wide solenoidal small loop, (b) narrow solenoidal small loop, (c) wide open-ended small helix, and (d) narrow open-ended small helix.

small helix antenna, the TM10 is the dominant mode at the low frequencies, and then the TE10 mode becomes dominant as the operating frequency is increased. From the results, it can be expected that there is a worst operating frequency at low frequencies, and that the matched gain tends to be increased as the operating frequency goes up. However, as the operating frequency of the small helix antennas is increased, the matched gain will be reduced by the degradation of radiation efficiency, due to conductor loss. In the case of the narrow small helix antenna, the TM10 mode is the dominant mode; therefore, it can be expected that the matched gain will be continuously reduced as the operating frequency is increased. The matched gains of the given antennas in the collinear ( = 0 and ' = 0) and the parallel ( = =2 and ' = 0) configurations without tilt (u1 = u2 = 0 and u3 = 1) for the different distances (d = 0:5 m and d = 0:7 m) are shown in Fig. 5. The results are calculated by two different methods for each case. As mentioned above, one uses the 2-port full EM simulation and the other uses the addition theorem with the EM results for the single antenna. From the results, it can be concluded that there is an optimum resonant frequency for solenoidal loop antennas. Moreover, it is shown that, by using the addition theorem, the frequency characteristics of the matched gain for given antenna structure can be estimated without the full simulation including entire coupled antennas. From the results, the optimum operating frequencies for the small solenoidal loop antennas are about 20 MHz for the wide solenoidal loops and about 36 MHz for the narrow solenoidal loops. In the case of the small helix antennas, the matched gain is maximized at the low frequencies and then continuously reduced as the frequency is increased. For the wide helix antenna, the matched gain is somewhat increased because the dominant mode is changed from the TM10 to the TE10 mode, although the matched gain is momentarily reduced by the degradation of the radiation efficiency. VI. DISCUSSION For the above examples, the optimum operating frequencies for the small solenoidal loop antennas and the small helix antennas are investigated. From the analysis, it is shown that the optimum operating fre-

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Fig. 7. The values of the additional capacitance and inductance for matching: (a) wide solenoidal small loop, (b) narrow solenoidal small loop, (c) wide openended small helix, and (d) narrow open-ended small helix. Fig. 6. Matching network for the coupled antennas with the source impedance R and the load impedance R : (a) using the additional capacitor, and (b) using additional inductor.

quency is influenced by the dominant mode and the radiation efficiency of the given antenna. However, in the actual system implementation, the impedance of the source or load can be given by only a real value. Hence, an additional reactive element needs to be added to meet the matching conditions, as shown in Fig. 6, if the direct feeding scheme is employed. Therefore, for estimating the optimum operating frequency, the influence of the additional matching element has to be considered. Actually, the real reactive element such as a lumped inductor or capacitor is intrinsically lossy, so that it has a finite Q factor. In the case of the capacitor, the Q factor is generally high enough for the element to be considered as the ideal element. However, the Q factor of the inductor is relatively small, due to high conductor loss. Hence, the power gain drastically drops if the inductor is used, because the loss degrades the radiation efficiency. In this case, the radiation efficiency is effectively given by



= Rr + RRlr+ R

(10) ind

where Rr is the radiation resistance of the antenna, Rl is the resistive loss of the antenna, and Rind is the additional loss due to the intrinsic loss of the inductor. The additional loss of the inductor is approximately given by

Rind

= Q!L

Fig. 8. Maximum matched gain obtained estimated by the addition theorem without the consideration of the matching element (Case 1) and with the sim= 40 and = ) of plified consideration (under the assumption of the additional loss due to the matching inductor (Case 2): (a) wide solenoidal small loop, (b) narrow solenoidal small loop, (c) wide open-ended small helix, and (d) narrow open-ended small helix.

Q

Q

1

(11)

ind

where the Q-factor of the inductor is given as Qind . For the above examples, the values of the additional capacitance and inductance needed for the impedance matching are presented in Fig. 7 under the condition that the input and output impedances are given as the proper real values. In this case, the actual matched gain with the simplified inductor whose Q-factor is 40 is given as shown in Fig. 8, because the Q factor of a commercial inductor that can be used for the impedance matching for the above examples is usually less than 40 [9]. From the results for the small solenoidal loop antennas, the previously estimated optimum frequencies, which are estimated without the consideration of the additional loss due to the additional inductor for matching, may not be really optimal, because the additional inductor that reduces the radiation efficiency of the system is used for

the matching. In these cases, the physically realizable optimum operating frequency is about 16.5 MHz for the wide solenoidal loop and about 26 MHz for the narrow solenoidal loop. VII. CONCLUSION In this communication, the frequency-dependent characteristics of matched gain for resonant ESAs are investigated. In the case of the ESAs that generate the TM10 or TE10 mode, the frequency dependence of the matched gain behaves differently, according to the frequency dependence of the mutual impedance and the radiation efficiency related to the dominant mode. The existence of an optimum operating frequency can be inferred according to the different frequency-dependent behaviors of the matched gains for each mode.

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When the dominant mode of the antenna is changed from the TE10 to the TM10 mode by increasing the operating frequency, as in the case of the solenoidal small loop, it is supposed that there is an optimum frequency at which the maximum matched gain can be obtained. On the other hand, in the case of an open-ended small helix, it is supposed that the matched gain is reduced as the frequency is increased, because the dominant mode is changed from the TM10 to the TE10 mode and the radiation efficiency is degraded as the operating frequency goes up. In addition, the addition theorem is used to estimate the optimum operating frequency with the EM simulation of the isolated single antenna. However, if the additional inductor is required for the matching, the characteristics of the element of the matching network need to be considered for determining the optimum operating frequency. In the same context, it is expected that the future investigations could find additional analysis for other cases with various feeding schemes except for direct feeding.

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Characterization and Reduction of Mutual Coupling Between Stacked Patches Óscar Quevedo-Teruel, Zvonimir Sipus, and Eva Rajo-Iglesias

Abstract—A rigorous analysis of mutual coupling between stacked patches is proposed in this work. The considered stacked-patch configuration has a dielectric-air-dielectric topology, and the level of mutual coupling has been related to the working region, which is defined depending on the gain and frequency bandwidth of the antenna, as well as on the shape of the radiation pattern. For certain antenna parameters a dip appears in the mutual coupling characteristic, and detailed study of this phenomenon has been carried out. Since this dip in the mutual coupling depends on the resonant frequency of the parasitic patch, it can be used to reduce the mutual coupling in the operational frequency bandwidth of the antenna. We have developed an experimental model that has proven this concept of antenna coupling reduction. Finally, the influence of other antenna parameters on the mutual coupling level has been studied. Index Terms—Microstrip patch antennas, mutual coupling, stacked patches.

REFERENCES [1] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 1998, ch. 1. [2] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly coupled magnetic resonances,” Science, vol. 317, pp. 83–86, 2007. [3] A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Efficient wireless non-radiative mid-range energy transfer,” Ann. Phys., vol. 323, no. 1, pp. 34–48, Jan. 2008. [4] J. Lee and S. Nam, “Fundamental aspects of near-field coupling antennas for wireless power transfer,” IEEE Trans. Antennas Propag., vol. 58, pp. 3442–3449, 2010. [5] D. C. Yates, A. S. Holmes, and A. J. Burdett, “Optimal transmission frequency for ultralow-power short-range radio links,” IEEE Trans. Circuits Syst. I, vol. 51, no. 7, pp. 1405–1413, Jul. 2004. [6] Y. Tak, J. Park, and S. Nam, “Mode-based analysis of resonant characteristics for near-field coupled small antennas,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1238–1241, 2009. [7] W. K. Kahn and W. Wasylkiwskyi, “Coupling, radiation and scattering by antennas,” in Proc. Symp. Generalized Networks, Brooklyn, NY, 1966, vol. 16, pp. 83–114, Polytechnic Press. [8] D. M. Pozar, Microwave Engineering. New York: Wiley, 2005, ch. 11. [9] Murata Products 2009–2010. Kyoto, Japan: Murata Mfg. Co., Ltd., 2009.

I. INTRODUCTION Microstrip patch antennas have demonstrated to be one of the most versatile antennas in recent years [1]. Nowadays, patches (and dipoles) are probably the most used antennas in compact commercial designs. The main advantages of these antennas are high efficiency, low weight, easy manufacturing and low cost. However, as typical disadvantages, they exhibit a narrow bandwidth and high mutual coupling when used as array elements [2], [3]. Several attempts have been made to improve these disadvantages, see, e.g., [4]–[6]. Stacked patches are one of the most common solutions adopted to increase the bandwidth of patch antennas, since the structure is as compact as the original (given that the thickness of the substrate is usually thin, and this is the only dimension that is increased) and the radiation pattern can keep its characteristics over all of the working frequency band [7]–[9]. Moreover, one can vary the basic configuration of this type of antenna by adding spacers, superstrates or multiple layers providing even more versatility to it [10], [11]. On the other hand, study of the mutual coupling level between stacked patches has received little attention in the literature. We can mention only a few references dealing with this topic, such as [12]–[14]. Further studies are therefore needed in order to complete the characterization of this parameter. This motivates the work presented in this manuscript where a study of mutual coupling between stacked patch antennas is made. As a starting point we will consider a configuration that includes an air spacer (see Fig. 1; the structure is described in [8], [15]) due to its promising properties. For such case, we analyze the influence of substrate thickness on the mutual coupling level in the three different working regions (namely, broadband, abnormal and high-gain regions [8], [15]). Another important issue which will be analyzed in detail is the appearance of a dip in the mutual coupling Manuscript received November 02, 2009; revised July 28, 2010; accepted September 09, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported by projects TEC2006-13248-C04-04, TEC2009-07376-E, and CCG10-UC3M/DPI-5631 of the Regional Government of Madrid. The authors are with the Department of Signal Theory and Communications, Universidad Carlos III de Madrid, Madrid 28911, Spain (e-mail: oquevedo@tsc. uc3m.es). 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.2010.2103011

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. Side view of an isolated stacked patch with air spacer.

Fig. 3. Mutual coupling at the working frequency (dashed and solid light lines) versus x component of the electric field (solid dark line) when the thickness of the air spacer is modified. Dashed line and solid light lines represent the mutual coupling for the case of having a substrate or a superstrate as a supporting structure for the parasitic patch, respectively. Fig. 2. Front view of the complete configuration with two stacked patches that are separated by distance d.

due to the effect of the parasitic patch. The origin of such dip and its influence on antenna parameters will be studied. Furthermore, it will be demonstrated how this dip can be used to reduce the mutual coupling in the operational frequency band of antenna. Finally, the effects of other antenna parameters, like the thickness and permittivity of the dielectric substrates and the distance between patches, will be studied. The results of this work have been obtained by simulations carried out both with CST Microwave Studio and with an in-house developed software ([16]), being validated later with measurements. II. MUTUAL COUPLING LEVEL IN STACKED PATCHES WITH AN AIR SPACER In this section the level of mutual coupling between stacked patches containing an air spacer is studied as a function of the thickness of such air layer. This configuration is described in Fig. 1. This design is inspired with the one presented in [8], where the parasitic patch is placed below the upper substrate (used as a superstrate) and immediately over the air layer. This layout allows a compact design, giving certain independence from the surroundings where the antenna is placed. This is the reason why it is a common configuration in practice and why this configuration is chosen as a starting configuration in this study. In the following calculations the size of patches is 1 cm 2 1.5 cm, and the thickness and permittivity of the substrates are t = 0:0254 cm and r = 2:17, respectively. The distance between patches (d) is 1.5 cm, which is 0 =2 at 10 GHz, and the size of the ground plane is 3.75 2 3.75 cm. Fig. 2 shows a front view of the complete antenna and the used coordinate system. It is known that this type of stacked patch antennas (with a spacer in between) can be classified, according to their operation characteristics, into three regions ([8], [15]). • Broadband region: This region appears when the air thickness is small. That means that both patches have a similar resonant frequency, and consequently, there is a wideband response in frequency. • Abnormal region: Intermediate region where the radiation pattern of the patches is not broadside, since there is a strong array effect of two elements (driven and parasitic patch).

• High-Gain region: When the thickness of the air is large, the resonant frequencies of both patches are away of each other and the bandwidth is not wide, although the antenna gain increases. Fig. 3 shows the mutual coupling level as function of the thickness of the air slab (the antenna was matched in all considered cases). The regions of operation are marked with vertical lines and they have been delimited according to the described types of behaviour. In addition, the results for the case when the upper substrate is used like a substrate (instead using it as a superstrate, see [8]) are represented, both being similar except for a thin air spacers. A maximum of mutual coupling is found in the abnormal region and a minimum when the air slab is the thinnest. The explanation for this phenomenon can be found if we consider the level of the electric field in the plane of interest (i.e., in the E-plane). For this reason, Fig. 3 also includes the electric field level (Ex component placed at the centre of the patch marked with (2)), when there is only one stacked patch present (i.e., when the stacked patch (2) is removed). The considered field value corresponds to the maximum of jEx j observed at the vertical line inside the superstrate at the central position of the fictitious second patch (situated 1.5 cm away from the first patch). There is a clear correlation between both values—electric field (jEx j) and mutual coupling (jS11 j). III. STUDY

DIP IN THE MUTUAL COUPLING STACKED PATCHES

OF THE

OF

Some studies dealing with the mutual coupling between stacked patches [13], [14] have reported the existence of a dip in the frequency dependence of the S12 parameter. However, a careful analysis of such phenomenon is missing. In this section we will demonstrate the existence of such a dip in the mutual coupling centered at a frequency which is larger than the resonant frequency of the parasitic patch. To this aim, we will use a design in which there is no air spacer (shown in Fig. 4). The dimensions of the patches are 1 cm 2 1.5 cm and the thickness of both substrates (with r = 2:2) is t = 0:5 mm. The selected ground plane is 4 2 4 cm. With these dimensions the resonant frequencies of the antenna are close to 9 GHz. Fig. 5 shows the S-parameters of this particular design (simulated and measured). It can be noted that the expected dip in the mutual coupling level occurs immediately after the second resonant frequency of

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Fig. 4. Scheme of the stacked patch measured in [8].

Fig. 6. Electrical field component in x direction in a fictitious second patch for different z positions.

Fig. 5. S and S for a patch (with the disposal shown in Fig. 4). Dimensions of the patch are 1 cm 1.5 cm, and the substrates relative permittivity is  : mm. : while thickness is t

22

2

=05

=

the antenna (the one which corresponds to the parasitic patch). In addition, there is an acceptable agreement between simulation and measurements, with the exception of a slight deviation in the second resonant frequency caused by the material tolerances and by a thin air-gap that was created between the substrate layers (this air-gap was not possible to avoid due to soldering of the feeding probe). An explanation for this phenomenon can be found in the electric field level at the position of the fictitious second element (similar approach as in Section II), although this time the distance between the stacked patch elements is selected to be 0 =2 at 9 GHz. Fig. 6 shows the variation of the Ex electric field component along the z-axis (i.e., along the vertical line located at the centre of the patch). It can be seen that the electric field is very weak around f = 9:6 GHz (exactly where the dip is situated). This fact explains the appearance of the minimum in the mutual coupling. This phenomenon is present over a narrow frequency band, and for larger frequencies the level of the jEx j (and therefore the level of the mutual coupling) increases again. This effect can also be observed in Fig. 7, where the jEx j component is represented as a function of frequency at z = 1 mm (height in which the parasitic patch is located) for two different distances between patches (d = 0:5 0 and d = 0 , with 0 defined at 9 GHz). In addition, the same electric field component is also considered in the H-plane where the dip of the mutual coupling also exists. It can be seen that in principle there is no difference between the E- and H-planes (except that the coupling level is smaller in the H-plane), i.e., the dip in the field strength also occurs in the H-plane at slightly lower frequency. This dip in the mutual coupling is due to a 180 phase difference between the currents at the driven and the parasitic patch. The patches are resonant elements, and therefore there is a strong resonant-like variation in the current amplitude and the phase. Fig. 8 shows a simulation which was carried out by an in-house developed software (the moment method program with entire-domain basis functions [16], [17])

Fig. 7. Electrical field component in x direction in a fictitious second patch mm (height in which the parasitic patch is located) for E-plane and at :  and  (with  defined at 9 GHz). H-plane at

z=1

d=05

d=

where the current amplitude and the phase at these two patches (i.e., the driven and parasitic patch of the excited stacked-patch antenna) are represented. The 180 phase difference between them in the considered frequency range is clearly observed. Although the amplitude of the excited current in each patch is not the same, the cancellation of the electric fields is still present since the parasitic patch induces stronger electromagnetic fields than the driven patch (the driven patch is closer to the ground plane and therefore there is a stronger effect of the image currents of the opposite phase). Finally, although in this Section the dip in the mutual coupling was mainly defined and studied in the E-plane array configuration, the dip also exists in an H-plane configuration having the same frequency dependence as it was demonstrated in Fig. 7. IV. REDUCTION OF MUTUAL COUPLING BETWEEN STACKED PATCHES Once we have demonstrated the existence of a dip in the mutual coupling that is related to the operating frequency of the parasitic patch, our

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Fig. 8. Amplitude and phase of the currents in the patches (lower and parasitic). (a) Amplitude of the currents in the patches. (b) Phase of the currents in the patches and difference between the lower and upper patch.

Fig. 9. Scheme of a stacked patch with different permittivity substrates for reducing mutual coupling.

next step was to use the parasitic patches, not for increasing the operational bandwidth of the antennas, but for reducing the mutual coupling between them. The way of achieving this is based on decreasing the operating frequency of the parasitic patch, in order to place the dip of the mutual coupling just at the operation frequency of the driven patch.

Fig. 10. (a) Non Stacked Patches. (b) Stacked Patches. S ; S and S parameters for the proposed stacked patch antenna for reducing mutual coupling. The size of the patches is 1 cm 1.5 cm and the thicknesses of the substrates : mm and t : mm, respectively with  : and  . are t The air spacer thickness is equal to 1 mm).

=05

=05

2

=22

=3

One way is to increase the permittivity of the upper substrate as Fig. 9 shows, where an air spacer can be used for tuning the operating frequency of the upper patch. To demonstrate the reduction of the mutual coupling, an experimental model was manufactured and measured. The size of the patches was established to be 1 cm 2 1.5 cm with a 1.5 cm distance between them. The lower substrate had a thickness of 0.5 mm and a relative permittivity of 2.2. The upper substrate had a thickness of 0.5 mm and a relative permittivity of 3. Between the substrates an 1 mm thick air spacer was placed. The measured S-parameters of the developed experimental model are shown in Fig. 10. A reduction in the mutual coupling level between 5 and 18 dB is obtained in the antenna operational frequency band (where both S11 and S22 are below 010 dB) when compared to the single patch case. Therefore, this technique can be used for reducing the mutual coupling between microstrip patch antennas.

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Fig. 12. S and S parameters for different distances between patches. The size of the patches is 1 cm 1.5 cm and the thickness of both substrates is t = 0:5 mm with  = 2:2.

2

Fig. 11. Mutual coupling versus the thickness (t) of the substrates, for different permittivities ( ) and thicknesses of spacer (s). (a)  = 2:2, (b)  = 3:4.

V. INFLUENCE OF THE DISTANCE BETWEEN ELEMENTS, THE DIELECTRIC CONSTANT AND THICKNESS OF THE SUBSTRATES Two important parameters that must be studied in relation to the mutual coupling level are the thicknesses and permittivity of the substrates. To this aim, several simulations with different permittivity (r ), dielectric thickness (t), and air spacer thickness (s) have been carried out. The selected dimensions were the same as in the example studied in Section III, and the results are shown in Fig. 11. The chosen permittivities for the study were r = 2:2 and r = 3:4, which are the permittivites of typical materials used in designing microstrip patch antennas. Both substrates contribute to a rise of the mutual coupling level when the thicknesses of the air spacer (s) increases; after certain value of s the mutual coupling starts to decrease. This effect was detected also in Section II, where it was demonstrated that the mutual coupling decreases when the stacked-patches start to work in the high gain region (i.e., for high values of s). However, the value of s that corresponds to the maximum of the mutual coupling level depends also on substrate permittivity. For example, if r = 2:2, the coupling level starts to decrease between s = 0.3 cm and s = 0:6 cm, and for r = 3:4 it starts to decrease when the air layer thickness is approximately s = 1 cm.

Moreover, the thickness of the substrates (t) also has influence on the mutual coupling. When it increases the differences in mutual coupling level between antennas with different air spacers tends to be smaller. Another important parameter that has to be taken into account in the mutual coupling study is the distance between the stacked patches. In all considered cases, till now, the distance was fixed to 0 =2. Fig. 12 shows the variation of the S-parameters with the distance between patch antennas (the selected patch antenna has the same dimensions as the antenna example studied in Section III). Since the return loss (represented with the S11 parameter) did not change significantly with the increase of the distance between patches, this parameter has been drawn only once, providing a reference to the operational frequency band of the antenna. However, the mutual coupling (i.e., the S21 parameter) has been plotted for different distances between the stacked patches, being 0 =2 = 16:7 mm at 9 GHz. As expected, according to the results from Fig. 7, an increase of the distance between patch antennas will reduce the mutual coupling level. An important fact is that the dip after the second resonant frequency appears for all curves. Nevertheless, the dip is rougher for larger distances between elements. In addition, the decrease of the mutual coupling level, when the element distance is increased, is larger for frequencies that are larger than the dip frequency, compared to the frequencies that are smaller than the dip frequency. VI. CONCLUSION This manuscript presents a contribution to the mutual coupling characterization of stacked patch antennas. It was demonstrated that the mutual coupling level is related to the working region of the stacked patch antenna (the working regions were previously defined by considering the antenna gain, bandwidth and radiation pattern shape [8], [15]). The maximum of mutual coupling is located in the region known as abnormal, and the minimum in the broadband region. For certain antenna parameters the dip in the mutual coupling characteristic has been observed. It was found that the dip is related to the 180 phase difference between the induced current at the parasitic and driven patches, and it occurs at the frequency close to the resonant frequency of the parasitic patch. Moreover, a potential way of reducing the mutual coupling between microstrip patch antennas has been proposed which makes use of this dip. An experimental model was developed that has proven the proposed concept. Finally, the influence of different antenna parameters on mutual coupling level has been studied.

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ACKNOWLEDGMENT

Synthesis and Design of a New Printed Filtering Antenna

The authors thank the mobility program of the Vicerrectorado de Investigación of Carlos III University of Madrid for the help supplied. Also, they want to thank C. J. Sánchez-Fernández for manufacturing the prototypes.

Chao-Tang Chuang and Shyh-Jong Chung

REFERENCES [1] C. Martín-Pascual, E. Rajo-Iglesias, and V. González-Posadas, “Invited tutorial: Patches: The most versatile radiator?,” presented at the IASTED Int. Conf. Adv. Commun., Jul. 2001. [2] J. R. James and P. S. Hall, Handbook of Microstrip and Printed Antennas. New York: Wiley, 1997. [3] C. Balanis, Antenna Theory: Analysis and Design. Hoboken, NJ: Wiley Interscience, 2005. [4] S. Maci and G. Gentili, “Dual-frequency patch antennas,” IEEE Antennas Propag. Mag., vol. 39, no. 6, pp. 13–20, Dec. 1997. [5] E. Rajo-Iglesias, O. Quevedo-Teruel, and L. Inclan-Sanchez, “Mutual coupling reduction in patch antenna arrays by using a planar EBG structure and a multilayer dielectric substrate,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1648–1655, Jun. 2008. [6] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 397–407, Feb. 2004. [7] S. Long and M. Walton, “A dual-frequency stacked circular-disc antenna,” IEEE Trans. Antennas Propag., vol. AP-27, no. 2, pp. 270–273, Mar. 1979. [8] R. Q. Lee, K. F. Lee, and J. Bobinchak, “Characteristics of a twolayer electromagnetically coupled rectangular patch antenna,” Electron. Lett., vol. 23, pp. 1070–1072, Sep. 1987. [9] E. Rajo-Iglesias, G. Villaseca-Sánchez, and C. Martín-Pascual, “Input impedance behavior in offset stacked patches,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 28–30, Jan. 2002. [10] R. Waterhouse, “Stacked patches using high and low dielectric constant material combinations,” IEEE Trans. Antennas Propag., vol. 47, pp. 1767–1771, Dec. 1999. [11] A. Panther, A. Petosa, M. Stubbs, and K. Kautio, “A wideband array of stacked patch antennas using embedded air cavities in LTCC,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 12, pp. 916–918, Dec. 2005. [12] C. Terret, S. Assailly, K. Mahdjoubi, and M. Edimo, “Mutual coupling between stacked square microstrip antennas fed on their diagonal,” IEEE Trans. Antennas Propag., vol. 39, pp. 1049–1051, Jul. 1991. [13] R. Zentner, Z. Sipus, and J. Bartolic, “Theoretical and experimental study of mutual coupling between microstrip stacked patch antennas,” in Proc. IEEE Antennas Propagation Soc. Int. Symp., Jun. 2003, vol. 1, pp. 618–621, vol. 1. [14] K. Chung and A. Mohan, “Mutual coupling between stacked CP-EMCP antennas,” in Proc. IEEE Antennas Propagation Soc. Int. Symp., Jul. 2005, vol. 2A, pp. 246–249, vol. 2A. [15] R. Q. Lee and K. F. Lee, “Experimental study of the two-layer electromagnetically coupled rectangular patch antenna,” IEEE Trans. Antennas Propag., vol. 38, pp. 1298–1302, Aug. 1990. [16] Z. Sipus, P.-S. Kildal, R. Leijon, and M. Johansson, “An algorithm for calculating Green’s functions for planar, circular cylindrical and spherical multilayer substrates,” Appl. Computat. Electromagn. Soc. J., vol. 13, pp. 243–254, Nov. 1998. [17] N. Herscovici, Z. Sipus, and D. Bonefacic, “Circularly polarized single-fed wide-band microstrip patch,” IEEE Trans. Antennas Propag., vol. 51, pp. 1277–1280, Jun. 2003.

Abstract—Synthesis and design of a new printed filtering antenna is presented in this communication. For the requirements of efficient integration and simple fabrication, the co-design approach for the integration of filter and antenna is introduced. The printed inverted-L antenna and the parallel coupled microstrip line sections are used for example to illustrate the synthesis of a bandpass filtering antenna. The equivalent circuit model for the inverted-L antenna, which is mainly a series RLC circuit, is first established. The values of the corresponding circuit components are then extracted by comparing with the full-wave simulation results. The inverted-L antenna here performs not only a radiator but also the last resonator of the bandpass filter. A design procedure is given, which clearly indicates the steps from the filter specifications to the implementation. As an example, a 2.45 GHz third-order Chebyshev bandpass filter with 0.1 dB equal-ripple response is tackled. Without suffering more circuit area, the proposed structure provides good design accuracy and filter skirt selectivity as compared to the filter simple cascade with antenna and a bandpass filter of the same order. The measured results, including the return loss, total radiated power, and radiation gain versus frequency, agree well with the designed ones. Index Terms—Chebyshev bandpass filter, filtering antenna, filter synthesis, inverted-L antenna, skirt selectivity.

I. INTRODUCTION In many wireless communication systems, the RF filters are usually placed right after the antenna. Since the size reduction and low profile structure are a trend in the circuit design, it is desired to integrate the bandpass filter and antenna in a single module, so called filtering antenna, with filtering and radiating functions simultaneously. However, to date, there has been relatively little research conducted on an efficient integration between the filter and antenna with simple fabrication and good circuit behavior. In traditional design, the filter and the antenna are designed individually, with the common ports’ characteristic impedance Z0 , and then connected directly. The direct connection of the filter and antenna usually causes an impedance mismatch, which may deteriorate the filter’s performance (especially near the band edges) and increase the insertion loss of the circuitry. To avoid this, extra matching network should be implemented in between these two components [1]. Several investigations have been focused on adding the radiating or filtering function into an antenna or filter [2], [3]. In [2], metal posts were inserted into a horn antenna, which can generate the filtering function. And in [3], the way is to create coupled cavities into a leaky wave antenna so as to generate filtering performance on the antenna. While these proposed circuits possess the characteristics of filtering and radiating, it should be noted that they were designed without using a systematic approach nor considering much on the filter’s or antenna’s specifications. Various approaches for integrating the filter and antenna into a single microwave device have been discussed in [4], [5]. For size reduction, a Manuscript received July 20, 2009; revised July 07, 2010; accepted November 02, 2010. Date of publication January 06, 2011; date of current version March 02, 2011. This work was supported in part by the National Science Council, R.O.C., under Contract NSC 97-2221-E-009-041-MY3. The authors are with the Institute of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan 30050, R.O.C (e-mail: sjchung@cm. nctu.edu.tw). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103001

0018-926X/$26.00 © 2010 IEEE

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that, as will be seen below, the whole circuit would have the same impedance behavior as the antenna itself in a wider frequency range. This parasitic capacitance comes from the accumulation of charges around the antenna feed point due to the truncation of the ground plane. The inverted-L antenna is printed on a 0.508 mm Rogers 4003 substrate with a dielectric constant of 3.38 and loss tangent of 0.0027. The ground plane of the antenna, which is also the ground plane of the circuitry, has a fixed size of 2 = 60 mm 2 60 mm. The antenna is fed through a 50 microstrip line of width 1.17 mm. In the following, the simulated characteristics of all the structures (filters and antennas) are performed by the full-wave simulator Ansoft HFSS (High Frequency Structure Simulator based on the finite element method), while those of the antenna’s equivalent circuit are by AWR MWO (Microwave Office). For each given inverted-L antenna structure, the equivalent circuit components are extracted by first letting the resonant frequency of the circuit equal the simulated resonant frequency of the antenna. And then, we optimize the values of the circuit components so that the reflection coefficient (S11 ), as a function of frequency, of the equivalent circuit coincides with that simulated from the antenna in a frequency range as wide as possible. The difference 1S11 of the reflection coefficient between these two curves is set not beyond 3% in a 20% frequency bandwidth centered at the antenna resonant frequency. In this design, the radiation resistance A in the equivalent circuit can serve as the load impedance of the bandpass filter to be synthesized, and the series A - A circuit can be the filter’s last resonator so that

L W

Fig. 1. (a) Configuration of the proposed filtering antenna. (b) the equivalent circuit of the inverted-L antenna. (I:T: = ImpedanceTransformer).

pre-designed bandpass filter with suitable configuration was directly inserted into the feed position of a patch antenna [4]. By using an extra impedance transformation structure in between the filter and the antenna, the bandpass filter can be integrated properly with the antenna over the required bandwidth [5]. However, the transition structure needs additional circuit area, and the designs did not have good filter characteristics over the frequency range. Recently, several filtering antennas designed following the synthesis process of the bandpass filter have been presented [6], [7]. In these designs, the last resonator and the load impedance of the bandpass filter were substituted by an antenna that exhibited a series or parallel RLC equivalent circuit. Although they have been done based on the co-design approach, most of them did not show good filter performance, especially the band-edge selectivity and stopband suppression. This is due to the lack of the extraction of the antenna’s equivalent circuit over a suitable bandwidth. Only that at the center frequency was extracted and used in the filter synthesis. Moreover, the antenna gain versus frequency, which is an important characteristic of the filtering antenna, was not examined in these studies. Although in [7], the frequency response of a factor named ideal matching loss (= 1 0 jS11 j2 ) was considered, it did not take into account the circuit and antenna losses and thus missed the power transmission characteristic of the filtering antenna. In this study, a new co-design method is proposed to achieve a filtering antenna with printed structure as shown in Fig. 1(a), which con0 1 parallel half-wavelength mitains an inverted-L antenna and crostrip lines. The printed inverted-L antenna is used with its equivalent circuit completely extracted over a desired bandwidth for the synthesis of the filtering antenna. Also, to increase the fabrication tolerance, a quarter-wave admittance inverter with characteristic impedance other than 0 is introduced in the filter synthesis.

N

Z

II. EQUIVALENT CIRCUIT MODEL OF THE INVERTED-L ANTENNA Since the antenna is to be designed in the filter as the last resonator, the first step to synthesize the filtering antenna is to establish the antenna’s equivalent circuit model and extract the circuit components. Fig. 1(b) shows the equivalent circuit at the antenna feed point looking toward the antenna. Since the inverted-L antenna is a variety of a monopole antenna, the antenna exhibits a series RLC resonance near the first resonant frequency [8]. Here, A and A express the resonant inductance and capacitance of the antenna, respectively, and A corresponds to the antenna radiation resistance. It is noted that an extra shunt capacitance g is incorporated in the equivalent circuit here so

L

C

C

R

R

L C

f0 = 2pL1 C A A

f

(1)

where 0 is the center frequency of the bandpass filter and is chosen as 2.45 GHz in this work. In the process of establishing the antenna database, the antenna frequency A , which is determined by the total strip lengths ( 1 + 2  A 4), should be slightly larger than 0 due to the existence of the parasitic capacitance g . This frequency corresponds to the frequency location of the minimum value of the antenna’s reflection coefficient (S11 )

l

l

 =

f

f

C

S11 = 20 log

Yin 0 Y0 Yin + Y0

(2)

with

Yin = j 2fCg + RA + j 2fLA

0 ff02 2

1

R

01

:

L C l R ;L ;C

It is noted that the radiation resistance A and the inductance A are mainly dependent on the vertical strip length 1 due to the strongest current distribution on this strip. Also, the parasitic capacitance g is decided by the strip width and independent of the strip length 1 . It is thus observed that the values of the three components ( A A g ) in (2) are almost invariant at a fixed strip length 1 only if the antenna frequency is near 0 . Therefore, for a choosing strip length, we first extract the components of the A A , and g when the antenna is resonated at 0 , and then, taking these three components into (2), the resonant frequency A can be obtained. Various values of 1 have been analyzed in this study. It is found out that, the antenna with different dimensions of 1 has the resonant frequency A near 2.53 GHz while 0 is 2.45 GHz. Another important factor to be used for synthesizing the filtering antenna is the corresponding quality factor of the antenna, A , which is defined as

l

w

f

f

f

l

R ;L

C

f

l

l

f

Q

QA = 2fR0ALA

(3)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 3, MARCH 2011

Fig. 2. The radiation resistance R and quality factor Q as a function of the : mm, f : GHz. strip length l . w

= 1 17

= 2 45

and is that of the series RLC circuit in the antenna’s equivalent circuit, without including the effect of the parasitic capacitance Cg . It should be noticed that the QA is not the whole quality factor of the inverted-L antenna. Fig. 2 shows the variation of the radiation resistance RA and quality factor QA as a function of the strip length l1 . It can be observed that the radiation resistance RA increases substantially from 13.6 to 36.8

when the strip length l1 changes from 5 mm to 13 mm. In the same l1 variation range, the inductance LA (not shown in the figure) has only little change (from 12 nH to 14.5 nH). The radiation resistance increases much fast than the inductance as the strip length increases, which results in a decreasing quality factor due to (3). As seen, the quality factor changes from 13.7 to 6.0 when l1 increases from 5 mm to 13 mm. Although not shown here, the strip width w has minor influence on the values of equivalent circuit components. And the extracted parasitic capacitance Cg is around 0.3 pF to 0.4 pF, roughly independent of the variations of l1 and w . As to the application in the synthesis of the proposed filtering antenna, once the quality factor QA is determined by the given specifications of filter (the relationship will be derived later), the values of l1 and RA can be obtained via the QA -to-l1 and RA -to-l1 curves in Fig. 2, respectively, and then the dimensions of the antenna can be obtained. Fig. 3 shows the comparison of the impedance behaviors on the Smith chart from the full-wave simulation and the equivalent circuit calculation. The dimensions of the inverted-L antenna, which is to be used later in the synthesis of the filtering antenna, are l1 = 10 mm, w = 1:17 mm, and l2 = 17:25 mm. The extracted circuit components are LA = 14:2 nH, CA = 0:30 pF, RA = 28:6 , and Cg = 0:37 pF, with the corresponding quality factor QA = 7:65. Notice that Cg has the same level as CA so that this parasitic capacitance can not be neglected in the modeling of the antenna. It is seen from the figure that the curve of S11 for the equivalent circuit agrees well with the full-wave simulation one from 1.5 GHz to 4 GHz. Especially, they have the same value at the antenna resonant frequency (fA = 2:53 GHz), and the difference 1S11 is 0.16 dB (error of 1S11  3%) at f = 2:28 GHz and 0.148 dB (error of 1S11  3%) at f = 2:81 GHz. III. SYNTHESIS OF THE FILTERING ANTENNA Fig. 4(a) shows the proposed filtering antenna, which contains N coupled line sections and an inverted-L antenna. Note that the antenna is connected directly to the N th coupled line. The filter to be synthesized is an N th order Chebyshev bandpass filter. The N 0 1 filter resonators are provided by the coupled line sections and the last one by the inverted-L antenna. In order to match to the low antenna radiation resistance and increase the flexibility of design, here the N th coupled line

Fig. 3. Input impedances of the inverted-L antenna with l = 10 mm, w = 1:17 mm, and l = 17:25 mm. (solid line: the full-wave simulation; dotted line: the equivalent circuit calculation.

has different design as the conventional ones [9]. Consider a coupled line section with even- and odd-mode characteristic impedances Z0eN and Z0oN , respectively, as shown in Fig. 5(a), which is to be equivalent to the circuit shown in Fig. 5(b) near the center frequency f0 . The right transmission line section in Fig. 5(b) has a characteristic impedance Za , which is different from the system impedance Z0 (= 50 ) and can be selected arbitrarily. To have the same circuit performances near the center frequency, the ABCD matrices of the coupled line section and the equivalent circuit should be equal at  0 = =2, resulting into

Z0eN = Za [Z0 =Za + JN Z0 + (JN Z0 )2 ] Z0oN = Za [Z0 =Za 0 JN Z0 + (JN Z0 )2 ]: 0

0

0

0

0

0

(4a) (4b)

0 Therefore, once JN Z0 is known, the impedances and thus the dimensions of the N th coupled line section can be obtained. By using the equivalent circuits of the antenna and the coupled line sections, the filtering antenna structure shown in Fig. 4(a) can be expressed by the equivalent circuit shown in Fig. 4(b). The two transmission line sections in between the admittance inverters have lengths equal to a half wavelength near the center frequency, i.e., 2  +0   , and thus can be replaced by a parallel LC resonator as shown in the upper sub-figure of Fig. 4(c). To transfer this circuit to a typical bandpass filter topology, the antenna’s parasitic capacitance Cg should be 0 moved to the left-hand side of the admittance inverter JN as illustrated in Block 2 of Fig. 4(c). The resultant capacitance Cg0 can be derived by equalizing the input admittances of Blocks 1 and 2 in the figure. For frequencies near f0 and 0  =2, the input admittance Yin1 of Block 1 and Yin2 of Block 2 can be derived and approximated as

Yin1  JN2 Za2 j2fCg + Yin2 = j2fCg + Yin 0

0

1 j

L C

f f

0 ff + RA

(5)

(6)

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Fig. 4. (a) The structure of the proposed N-order filtering antenna. (b) Equivalent circuit of the proposed filtering antenna. (c) Modified circuit of the proposed filtering antenna. (d) A typical N-order bandpass filter circuit using shunt resonators with admittance inverters.

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1) Specify the requirements of the bandpass filter to be synthesized, including the center frequency f0 , the fractional bandwidth 1, and the type of the filter (e.g., bandpass filter with equal ripple), from which the admittance inverters Jn Z0 (n = 1; 2; . . . ; N + 1) and the parallel resonators Ln ; Cn (n = 1; 2; . . . ; N ) in Fig. 4(d) can be determined [9]. 2) Choose an antenna structure with suitable equivalent circuit that can substitute for the last resonator and load impedance of the bandpass filter. (Here in this study, the inverted-L antenna is used.) And then get a database associated with the equivalent circuit components for different antenna dimensions like those in Section II. 3) Calculate the antenna quality factor QA from (9) and then, after choosing a suitable strip width (e.g., the same width as the feed line), obtain the required strip length l1 and radiation resistance RA of the inverted-L antenna by using Fig. 2. The length l2 of the horizontal antenna strip can thus be determined by letting the antenna resonate at fA . At this step, the dimensions of the inverted-L antenna are acquired. 4) Choose suitable characteristic impedance Za and then calculate 0 Z0 by using (10). Following, the eventhe inverter constant JN 0 of the 0 and Z0oN and odd-mode characteristic impedances Z0eN N th coupled line section for the proposed filtering antenna can be attained via (4). 5) Calculate the even- and odd-mode characteristic impedances Z0en and Z0on (n = 1; 2; . . . ; N 0 1) of the coupled line sections by using the formulae in [9], and then all the dimensions of the N coupled line sections in the proposed filtering antenna can be determined. IV. DESIGN EXAMPLES AND EXPERIMENTAL VERIFICATION

Fig. 5. (a) Geometry of the Nth coupled line section and (b) the corresponding equivalent circuit.

with

Yin0 

JN2 Za2

j CL

f f

:

0 ff + R

(7)

A

By equalizing (5) and (6), one obtains

Cg0 = (JN0 Za )2 Cg :

(8)

0 Actually, the capacitance Cg0 is much smaller than CN 01 , therefore, 0 01 + Cg0 ) of the second resonator is the total capacitance CN 01 (= CN 0 01 . approximately equal to CN Finally, the equivalent circuit of Fig. 4(c) can be transferred to the conventional N th order bandpass filter circuit as shown in Fig. 4(d), and, by letting the admittance Yin0 equal Yin , one obtains

2f L =  R 2(Z J ) and J Z 2Q R Z = =

QA = JN0 Z0

0

A

A

0

N+1

1 2

N

Za

0

A



A

0

(9)

2

:

(10)

The design procedures of the proposed filtering antenna can now be summarized as follows.

In this section, an example of the proposed filtering antenna is to be presented. Following the above design procedures, a third-order Chebyshev bandpass filter with a 0.1 dB equal-ripple response, f0 = 2:45 GHz, 1 = 14%, and Z0 = 50 are firstly chosen. Based on these requirements, the inverter constants Jn Z0 (n = 1  4) of the bandpass filter can be calculated. Also, these parallel resonators in Fig. 4(d) have the values of (Ln ; Cn ) = (2:068 nH; 2:041 pF), where n = 1, 2, and 3. Then, with a strip width w = 1:17 mm, the quality factor QA = 7:37 is obtained by using (9). This corresponds to a strip length l1 = 10 mm from the QA -to-l1 relationship shown in Fig. 2. To this point, all the dimensions of the inverted-L antenna are gotten, that is, l1 = 10 mm, l2 = 17:25 mm, and w = 1:17 mm. Hence, the radiation resistance RA can be found as 28.6 via Fig. 2. Following, the inverter constant J30 Z0 of the third coupled line and 0 and Z0o3 0 thus the even- and odd-mode characteristic impedances Z0e3 can be calculated by using (10) and (4), respectively, from which the line width and the gap between lines of the third coupled line are obtained. Note that these dimensions are dependent on the characteristic impedance Za used in the synthesis of the filtering antenna. Fig. 6 depicts their variations as functions of Za . It is seen that the larger is the impedance Za , the smaller the gap size is. When Za > 50 , the gap would become smaller than 0.1 mm, which is difficult to realize. Thus, for easy fabrication, a characteristic impedance of Za = 30 is selected here. This would correspond to an inverter constant J30 Z0 = 0:5515 and even- and odd-mode characteristic impedances 0 ; Z0o3 0 ) = (75:67 ; 42:58 ). Finally, the even- and odd-mode (Z0e3 characteristic impedance of the first and second coupled line sections are calculated, and then all dimensions of each coupled line section for the proposed filtering antenna can be obtained. It should be noted that the gap size of the first coupled line section is extremely small and difficult to fabricate. To tackle this problem, the tapped structure with a

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Fig. 6. Dimensions of the third coupled line section for different proposed filtering antenna.

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Z

in the

quarter-wavelength impedance transformer [10] is utilized, as shown in Fig. 1(a). The full-wave simulated return loss of the proposed filtering antenna in comparison with that of the conventional third-order Chebyshev bandpass filter is shown in Fig. 7(a). The simulated return loss of a second-order bandpass filter directly cascaded with a 50- inverted-L antenna is also shown for reference. It is noticed that both of the proposed filtering antenna and filter directly cascaded with an antenna are third-order circuit, with two orders provided by the circuit and one by the antenna. These two structures occupy about the same circuit area. Here, the conventional third-order Chebyshev bandpass filter has the same specifications as the filtering antenna (14% bandwidth and 0.1 dB equal-ripple response). And the filter in the reference structure is a second-order Chebyshev bandpass filter with the same bandwidth and ripple level. It is observed that the bandwidth and skirt selectivity for the proposed filtering antenna agree very well with the conventional third-order Chebyshev bandpass filter, which demonstrates the design validity of the filtering antenna. On the other hand, putting an individually designed antenna after the bandpass filter via a simple cascaded 50 microstrip line, not only have no contribution to the order of the filter, but also deteriorate the original filter performance, especially resulting in bad skirt selectivity at the band edges. Fig. 7(b) compares the measured return losses of the proposed filtering antenna and the conventional third-order Chebyshev bandpass filter. The simulated return loss of the filtering antenna is also shown for comparison. Likely because of the deviations in dielectric constant and substrate thickness, the bandwidth of the measured return loss of the proposed filtering antenna is slightly narrower than the simulated one. However, it is in close agreement with the measured one of the conventional third-order Chebyshev bandpass filter. Both have the same passband poles’ positions, the selectivity at the band edge, and the return-loss behavior at the stopband. This demonstrates that the proposed filtering antenna has good selectivity in accordance with the conventional bandpass filter. Fig. 8(a) shows the full-wave simulated total radiated powers of the proposed filtering antenna and the reference structure of a 2nd -order filter directly cascaded with antenna. Here, the total radiated power has been normalized to the input power. The simulated insertion loss of the conventional third-order Chebyshev bandpass filter is also shown for comparison. As compared to the reference structure, the total radiated power of the proposed filtering antenna is flat in the passband and the bandwidth is very close to the insertion-loss bandwidth of the thirdorder Chebyshev bandpass filter. The measured total radiated power of the proposed structure, which is in close agreement with the simulated one expect for the deviations of the bandwidth, is also shown in Fig. 8(a).

Fig. 7. (a) Full-wave simulated return losses of the proposed filtering antenna, 2nd-order r BPF + antenna, and the conventional 3rd-order Chebyshev BPF. (b) Measured return loss of the proposed filtering antenna compared with the measured one of the conventional 3rd-order Chebyshev BPF.

Fig. 8(b) shows the full-wave simulated antenna gains in the +z direction versus frequency for the proposed filtering antenna and the reference structure of a 2nd -order filter directly cascaded with antenna. The simulated insertion loss of the conventional third-order Chebyshev bandpass filter is also shown for comparison. Since the inverted-L antenna has an omni-directional field pattern in the xz-plane, has only the antenna gain in the +z direction is discussed. As compared to the reference structure, the antenna gain of the proposed filtering antenna is flat in the passband and the bandwidth is very close to the insertion-loss bandwidth of the third-order Chebyshev bandpass filter. The proposed filtering antenna also provides better skirt selectivity with stopband suppression better than 22 dB. The measured antenna gains for the proposed filtering antenna is also depicted in Fig. 8(b), which shows an antenna gain, including the circuitry loss, of 01:3 dBi. It is obvious that the measurement matches well to the simulation. The amplitude noise of the measured gain in the stopbands is due to the system noise of the antenna chamber. In Fig. 8(b), a radiation null at f = 2:15 GHz in the +z direction, which makes the skirt selectivity better than that of the conventional bandpass filter, has been observed. Since an inverted-L antenna alone should exhibit a monotonous gain dropping, but not a local gain minimum when the operating frequency moves away from the antenna’s resonant frequency. Also, since it has been found that the frequency locations of these nulls depend on the observation angle, they are not caused by the circuit coupling between the first resonator and the third one (i.e., the inverted-L antenna) of the filter. It is finally found out that

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Fig. 10. Measured and simulated total-field radiation patterns in the xz, yz, and xy planes for the proposed filtering antenna. [solid line: measured results; dashed line: full-wave simulated results]. f : GHz.

= 2 45

Fig. 11. Photographs of the proposed filtering antenna (left-side) and 2ndorder BPF antenna (right-side).

+

Fig. 8. (a) Full-wave simulated and measured total radiated power compared with the simulated insertion loss of the conventional 3rd-order Chebyshev BPF. (b) Simulated and measured antenna gains versus frequency in the z direction compared with the simulated insertion loss of the conventional 3rd-order Chebyshev BPF.

+

The measured and simulated total-field radiation patterns at f0 = 2:45 GHz in the three principal planes are also presented in Fig. 10. It

is seen that the measured patterns are similar to the simulated ones, although a discrepancy occurs at  = 180 in the yz- and xz-planes (that is, the 0z direction) due to the interference of the feeding coaxial cable in the measurement. The radiation pattern in the xz-plane is nearly omnidirectional with peak gain of 0.65 dBi. Fig. 11 shows the photographs of the proposed filtering antenna compared with a second-order bandpass filter directly cascaded with an inverted-L antenna. V. CONCLUSION

=

Fig. 9. Full-wave simulated total-field radiation patterns in the xz plane at f : GHz for the proposed filtering antenna and a two-port circuit structure obtained from the filtering antenna with the inverted-L section replaced by a terminated port. [solid line: proposed filtering antenna; dashed line: two-port circuit structure.]

2 15

the last coupled line structure near the ground edge of the proposed filtering antenna induces a strong spurious ground edge current, which in turn produces extra radiation and cancels the radiation field from the antenna at some frequency. Fig. 9 shows the simulated radiation patterns in the xz-plane at f = 2:15 GHz for the proposed filtering antenna and a two-port circuit structure obtained from the filtering antenna with the inverted-L section replaced by a terminated port. It is seen that the two-port circuit produces a spurious radiation toward +z direction with peak gain about 014 dBi, which is the same level as the omni-directional field pattern of an inverted-L antenna after the attenuation of the third-order bandpass filter at f = 2:15 GHz. For field canceling, the total radiation pattern of the filtering antenna thus possesses a radiation null near the +z direction.

A filtering antenna with new co-design approach has been proposed and implemented. The design is accomplished by first extracting the circuit model of the antenna, then casting it into the synthesis of a typical parallel coupled line filter. To increase the fabrication tolerance, a quarter-wave admittance inverter with characteristic impedance other than Z0 is introduced in the filter synthesis. A design example which has the same specifications as the conventional third-order Chebyshev bandpass filter is demonstrated. The measured results agree quite well with the simulated ones. The proposed filtering antenna provides good skirt selectivity as the conventional bandpass filter. It also possesses flat antenna gain in the passband and high suppression in the stopband.

REFERENCES [1] H. An, B. K. J. C. Nauwelaers, and A. R. V. D. Capelle, “Broadband microstrip antenna design with the simplified real frequency technique,” IEEE Trans. Antennas Propag., vol. 42, no. 2, pp. 129–136, Feb. 1994. [2] B. Froppier, Y. Mahe, E. M. Cruz, and S. Toutain, “Integration of a filtering function in an electromagnetic horn,” in Proc. 33th Eur. Microw. Conf., 2003, pp. 939–942. [3] F. Queudet, B. Froppier, Y. Mahe, and S. Toutain, “Study of a leaky waveguide for the design of filtering antennas,” in Proc. 33th Eur. Microw. Conf., 2003, pp. 943–946. [4] F. Queudet, I. Pele, B. Froppier, Y. Mahe, and S. Toutain, “Integration of pass-band filters in patch antennas,” in Proc. 32th Eur. Microw. Conf., 2002, pp. 685–688.

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[5] N. Yang, C. Caloz, and K. Wu, “Co-designed CPS UWB filter-antenna system,” in Proc. IEEE AP-S Int. Symp., Jun. 2007, pp. 1433–1436. [6] T. L. Nadan, J. P. Coupez, S. Toutain, and C. Person, “Optimization and miniaturization of a filter/antenna multi-function module using a composite ceramic-foam substrate,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1999, pp. 219–222. [7] A. Abbaspour-Tamijani, J. Rizk, and G. Rebeiz, “Integration of filters and microstrip antennas,” in Proc. IEEE AP-S Int. Symp., Jun. 2002, pp. 874–877. [8] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 1998. [9] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005, ch. 8. [10] M. Matsuo, H. Yabuki, and M. Makimoto, “The design of a half-wavelength resonator BPF with attenuation poles at desired frequencies,” in IEEE MTT-S Int. Microw. Symp. Dig., 2000, pp. 1181–1184.

Shaping Axis-Symmetric Dual-Reflector Antennas by Combining Conic Sections

Fig. 1. Dual-reflector shaping by consecutively combining conic sections (ADC-like configuration).

Fernando J. S. Moreira and José R. Bergmann

Abstract—A simple procedure for the shaping of axis-symmetric dual-reflector antennas is described. The shaping procedure is based on the consecutive concatenation of local conic sections suited to provide, under geometrical optics (GO) principles, an aperture field with uniform phase, together with a prescribed amplitude distribution. The procedure has fast numerical convergence and is valid for any circularly symmetric dual-reflector configuration. To illustrate the procedure two representative configurations are investigated. The GO shaping results are validated using accurate method-of-moments analysis. Index Terms—Geometrical optics, reflector antennas, reflector shaping.

I. INTRODUCTION A procedure for the geometrical optics (GO) shaping of circularly symmetric Cassegrain and Gregorian antennas has been presented recently [1]. It is based on the combination of local dual-reflector systems to describe the generatrices of the sub- and main-reflectors, providing an aperture illumination with a uniform phase distribution together with a prescribed amplitude distribution. The procedure represents an improvement over traditional methods [2], [3] as no ordinary differential equation needs to be solved. The use of curved (biparabolic) surfaces to locally represent the reflectors together with ray tracing (i.e., GO concepts) had already been adopted in [4] to establish a nondifferential set of equations to shape offset dual-reflector antennas. In [1] the authors adopted rectangular coordinates to describe the local conic sections representing the reflectors’ generatrices, leading to a nonlinear algebraic equation, which was approximated to provide an one-step iterative solution. In the present work we improve the solution by using polar coordinates to represent the conic sections. That renders an one-step iterative procedure with simple linear algebraic Manuscript received March 25, 2010; revised July 07, 2010; accepted September 09, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported in part by the Brazilian agencies CNPq (INCT CSF), and CAPES under Grant RH-TVD-254/2008. F. J. S. Moreira is with the Department of Electronics Engineering of the Federal University of Minas Gerais, 30161-970 Belo Horizonte, MG, Brazil (e-mail: [email protected]). J. R. Bergmann is with the Center for Telecommunications Studies of the Catholic University, 22453-900 Rio de Janeiro, RJ, Brazil. Digital Object Identifier 10.1109/TAP.2010.2103028

equations, thus avoiding any approximation. Another interesting feature of the present formulation is that it is valid for any shaped axissymmetric dual-reflector configuration based on the classical axis-displaced Cassegrain (ADC), Gregorian (ADG), ellipse (ADE), or hyperbola (ADH) [5]. In Section II the GO dual-reflector shaping formulation is presented assuming a shaped Cassegrain (or ADC) geometry. In Section III, the formulation is extended to other axis-symmetric dual-reflector configurations (ADG, ADE, and ADH). Then, the shaping of two representative dual-reflector antennas (ADC and ADE configurations) is conducted to illustrate the procedure. The convergence of the shaping procedure is investigated and compared to another procedure based on the numerical solution of an ordinary differential equation [3]. The radiation characteristics of the shaped dual-reflector antennas are numerically obtained by a method-of-moments (MoM) analysis in order to validate the applicability of the proposed shaping technique. The MoM software used in the present analysis has been successfully applied in previous reflector-antenna synthesis (e.g., [6]).

II. FORMULATION OF THE SHAPING PROCEDURE The basic idea is to represent the reflector generatrices by conic sections consecutively concatenated, as depicted in Fig. 1. Notice that a shaped ADC configuration will be adopted to derive the formulation, but in Section III the procedure will be extended to the other axis-symmetric dual-reflector antennas [5], [7]. The conic sections describing the subreflector (Sn ; n = 1; . . . ; N ) have two focci. One is always at the origin O (where the feed phasecenter is assumed to be) and the other at point Pn . As n is varied from 1 to N; Pn spans the locus of the subreflector caustic. Pn is also the focus of the parabolic section Mn that describes a corresponding portion of the main reflector. The axis of Mn passes through Pn and is parallel to the symmetry axis of both reflectors (z axis), such that all the rays reflected at the main reflector arrive parallel to each other at the antenna aperture plane, thus providing a uniform phase distribution at the aperture, according to GO principles. Another GO principle used to define the conic sections is the energy conservation in the bundle of rays that departs from O and reflects at Sn and Mn before reaching the antenna aperture. In order to uniquely define the conic sections Sn and Mn at each iteration n, four parameters must be determined: the focal distance Fn

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r

n 0 1).

where Fn01 and Fn01 are known from the previous step ( The third equation is obtained from (5)

xn01 = (2an=`o) + 1 + bn 0 dn cot(Fn01=2) (8) `o dn + (bn 0 1) cot(Fn01=2) where xn01 is the main-reflector x coordinate obtained in the previous step (n 0 1) (see Fig. 2). The last equation is derived by applying the conservation of energy along the tube of rays that departs from O and arrives at the aperture plane after being reflected by Sn and Mn . The conservation of energy

is described by the following integral: Fig. 2. Conic-section parameters.

M



c e S

of the parabolic section n , the interfocal distance 2 n (i.e., the distance between and n ) and the eccentricity n of the subreflector conic section n , and the tilt angle n of the axis of n with respect to the -axis (see Fig. 2). Consequently, four equations are needed to solve the problem at each step . The iterative process marches on with Fn , which is the feed ray direction (with respect to the -axis) toward the superior extreme of n (see Fig. 2). Fn is uniformly varied from F 0 = 0 to the subreflector edge at FN = E , such that 1 F = Fn 0 Fn01 = E . In principle, the accuracy of the shaping procedure increases with . The iterations start at = 0 with to the F 0 = 0 and F 0 = S , where F is the distance from subreflector along the ray-direction F and S is the desired distance between the feed phase-center and the subreflector apex (see Fig. 1). From the polar equation of n one obtains the following relation:

z

 

O S







r

P



n S  =N N

V

S



 

r 



V

rF (F ) = bn cos F +adnn sin F 0 1 for Fn01  F  Fn , where an = cn (en 0 1=en); bn = en cos n; dn = en sin n:

z

n

O

(1)

(2)

D



n



O ;x D =

xn = (2an=`o) + 1 + bn 0 dn cot(Fn=2) `o dn + (bn 0 1) cot(Fn=2)

(4)

x

D =2

where GF (F ) is the circularly-symmetric radiated feed power density, GA(x) is the desired circularly-symmetric power density at the antenna

(3)

Equation (1) is general and may represent an ellipse (0 n 1), a hyperbola (j n j 1), or any other conic section. To ensure a uniform phase distribution at the antenna aperture, it is enforced a constant path to the aperture plane (assumed at = 0). The length ( o ) from mapping relation between F and the Cartesian coordinate of the corresponding aperture point (which is also the coordinate of the main reflector) is given by [8]

e > ` O

0

x

GF (F ) rF2 sin F dF = NF

D

x

(11)

which, together with (6)–(8), is used to determine the conic parameters

Fn; 2cn; en , and n .

A. Linear System Solution Substituting (6) and (7) into (8) and (11), one obtains the following linear system:

f1bn + g1 dn = h1 f2bn + g2 dn = h2

(12) (13)

where

f1 = xn01 cot(Fn01=2) 0 `o 0 2rFn01 cos Fn01 f2 = xn cot(Fn=2) 0 `o 0 2rFn01 cos Fn01 g1 = xn01 + `o cot(Fn01=2) 0 2rFn01 sin Fn01 g2 = xn + `o cot(Fn=2) 0 2rFn01 sin Fn01 h1 = xn01 cot(Fn01=2) + `o 0 2rFn01 h2 = xn cot(Fn=2) + `o 0 2rFn01:

(14) (15) (16) (17) (18) (19)

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The solutions of (12) and (13) are

= hf11 gg22 00 hf22gg11 f1 h2 0 f2 h1 dn = : f1 g2 0 f2 g1 bn

(20) (21)

The conic parameters are then calculated as follows. Equations (20) and (21) are substituted into (3) and (4) in order to obtain en and n . Then, (20) and (21) are substituted into (7) to obtain an . Once en and an are known, 2cn is obtained from (2). Finally, Fn is calculated from (6). With the conic parameters determined, the subreflector point at F = Fn is located by the vector

cos Fn z^ + rFn sin Fn x^ (22) where rFn is given by (1) at F = Fn . The corresponding mainrFn

reflector point is located by the vector

^ + xn x^

zn z

(23)

where the Cartesian coordinate zn is given by the parabolic equation of Mn zn

= (xn 0 24cFnnsin n ) 0 Fn + 2cn cos n: 2

(24)

Finally, the location of Pn at the subreflector caustic is given by the vector

2cn cos n z^ + 2cn sin n x^: (25) The steps are repeated until F N = E (i.e., n = N ). Once n + 1 points are obtained for each reflector, their generatrices

may be described by any standard interpolation procedure, depending on the reflector-antenna analysis method adopted. However, one should notice that the sub and main reflectors described by the conic sections are continuous and have continuous first derivatives, as two consecutive local conic sections share the same optical path (from O to the mainreflector aperture) at their common point, which means that Snell’s law is satisfied at that point. III. OTHER AXIS-SYMMETRIC DUAL-REFLECTOR CONFIGURATIONS

The formulation derived in Section II assumed a shaped Cassegrain (or ADC) antenna, as illustrated in Figs. 1 and 2. However, the shaping procedure can be extended to other dual-reflector configurations [5], [7]. For a shaped Gregorian (or ADG) antenna, as illustrated in Fig. 3(a), one just needs to change the sign of xn after its calculation from (9). For a shaped ADE, depicted in Fig. 3(b), the feed illumination toward the antenna aperture is reversed. In this case, for the calculation of xn one must replace (9) by

 0

( )

2

GF F rF

sin F dF = NF

D =2 x

Fig. 3. Other shaped axis-symmetric dual-reflector antennas based on the (a) ADG, (b) ADE, and (c) ADH configurations.

IV. RESULTS In oder to illustrate the shaping procedure, an ADC and an ADE antennas are synthesized. As the synthesis is based on GO principles, the radiation patterns are analyzed by the method of moments (MoM) for bodies of revolution [9]. A. Shaped ADC Antenna The first geometry is the Cassegrain configuration presented in [1], where the dimensions of the shaped antenna were specified with the help of a classical ADC with the following dimensions: DM = 6 m, DB = DS = 0:6 m, `o = 3 m, and E = 30 . From [5] one obtains VS = 0:409 m. The operating frequency is 5 GHz, such that DM  100 . The sub- and main-reflector generatrices of the classical ADC are illustrated with dotted lines in Fig. 4. Applying the procedure of Section II, both reflectors are shaped to provide a uniform amplitude distribution over the illuminated portion of the aperture (i.e., GA (x) is constant from x0 = DB =2 to xN = DM =2, being null elsewhere). The feed model is the same adopted in [1] 2p ( ) = cos r(2F =2)

GF F

()

GA x x dx

(26)

where, for n = 0; xn = DM =2. Finally, for a shaped ADH antenna, illustrated in Fig. 3(c), one must use (26) to calculate xn , changing its sign afterward.

F

(27)

with p = 83 to provide 025 dB edge taper [1]. The shaped reflectors’ generatrices are plotted with solid lines in Fig. 4 together with the shaped subreflector caustic. The maximum deviation of the shaped reflectors with respect to the classical configuration is approximately 7 cm for the subreflector and 4 cm for the main reflector (see Fig. 4). Both

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Fig. 6. RMS errors of the shaped ADC reflectors as functions of

.

present a comparison of surface error obtained by using different numbers ( = E 1 F ) of synthesized points. As a reference, we employ a shaped dual reflector antenna synthesized with a large number of points (  104 ), as doubling it leads to differences of less than 05 between the surfaces. Fig. 6 presents the RMS error of the 10 shaped reflector surfaces as function of log(1 ) for both schemes (i.e., the present procedure and another based on the numerical integration of a differential equation). Only points obtained at each step were considered for the error calculation. For the subreflector the RMS error is calculated from the distances Fn , while for the main reflector the error is calculated from its coordinates n . The RMS error is actually dominated by the error at the reflector rims ( = ). From Fig. 6 one observes that the shaping procedure of Section II sustains very small RMS errors even for 1 F  E 50 (i.e., with  50 conic sections used to obtain the points of each reflector generatrix). For the present ADC shaping, the formulation of [1] was also implemented and it has essentially the same convergence rate of the present formulation.

N

Fig. 4. Sub- and main-reflector generatrices of the classical (dotted lines) and shaped (solid lines) ADC antennas. The caustic refers to the shaped subreflector.

1



 =  N

=N

n

r

z



n

 =

N N

B. Shaped ADE Antenna In the second case study, an ADE is shaped to provide a tapered aperture illumination. The antenna is similar to that investigated in [10], with M = 40 64 cm, B = S = 6 6 cm, o = 21 08 cm, and E = 45 . For a classical ADE these parameters provide S = 2 39 cm [5]. The operating frequency is 14.7 GHz, such that M  20 . The feed model is still given by (27), but with = 23 5 [10]. The tapered aperture illumination is described by

 Fig. 5. Diagonal-plane radiation patterns of the classical (dashed lines) and shaped (solid lines) ADC antennas at 5 GHz. Reflectors shaped with   .

1 

10 2

classical and shaped antennas, fed by the linearly polarized feed model of (27), were analyzed by the MoM technique (the central hole of the main reflector was closed by a flat metallic disk with diameter B ). The radiation patterns in the diagonal plane ( = 45 ) are depicted in Fig. 5. As expected, the gain of the shaped antenna (49.3 dBi) is higher than that of the classical configuration (47.8 dBi). The side-lobe levels of the shaped antenna are also higher (about 7 dB) than those of the classical configuration. One also observes a 4 dB increase of the cross-polarization peak of the shaped reflector antenna. The shaping procedure of Section II is simpler and, consequently, faster than others based on the numerical integration of ordinary differential equations [2], [3]. To illustrate its numerical efficiency, we



D

D

:

D

D

:

`

p

: V D :

:



2 GA (x) = 1 0 1 0 EM2 D2xM00DDBB (28) where DB =2  x  DM =2 and EM = 0:6 [10]. The classical (dotted

lines) and shaped (solid lines) ADE reflectors’ generatrices are plotted in Fig. 7 together with the shaped subreflector caustic. The maximum deviation of the shaped reflectors with respect to the classical configuration is approximately 3 mm for both sub- and main-reflectors (see Fig. 7). The diagonal-plane radiation patterns of both classical and shaped antennas are illustrated in Fig. 8. Once again, for the MoM analysis, the main-reflector central hole was closed by a flat metallic disk. From Fig. 8 one observes that the tapered aperture illumination of (28) was able to slightly increase the antenna gain (from 34.4 dBi to

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Fig. 9. RMS errors of the shaped ADE reflectors as functions of

Fig. 7. Sub- and main-reflector generatrices of the classical (dotted lines) and shaped (solid lines) ADE antennas. The caustic refers to the shaped subreflector.

1

.

was reduced to solving linear equations embedded in a stepwise numerical procedure. The procedure is valid for any axis-symmetric dual-reflector configuration with uniform phase distribution at the antenna aperture. The shaping procedure has been successfully applied to the synthesis of a Cassegrain antenna with a uniform aperture illumination [1] and of an ADE configuration with a tapered aperture illumination [10]. All designs were further analyzed by a MoM technique for bodies of revolution to demonstrate the applicability of the shaping procedure, which is inherently based on GO concepts. The shaping procedure was also compared to another based on the numerical integration of an ordinary differential equation and it was verified that the former may converge with about 100 times less iterations.

REFERENCES

Fig. 8. Diagonal-plane radiation patterns of the classical (dashed lines) and shaped (solid lines) ADE antennas at 14.7 GHz. Reflectors shaped with   .

1 

10 2

34.7 dBi), with a small decrease of the first sidelobe level (about 1.5 dB) and a small increase of the cross-polarization peak (about 1.5 dB). The convergence of the ADE shaping procedure is illustrated in Fig. 9. Its investigation was conducted similarly to the one previously performed for the ADC antenna and illustrated in Fig. 6. From Fig. 9 one observes that the shaping procedure of Sections II and III sustains very small RMS errors even for F  E = (i.e., with N  conic sections used to obtain the points of each reflector generatrix).

1

10

10

V. CONCLUSION A method for shaping axis-symmetric dual reflectors has been presented. The synthesis is based on the consecutive combination of local conic sections to represent the reflector’s generatrices. The problem

[1] Y. Kim and T.-H. Lee, “Shaped circularly symmetric dual reflector antennas by combining local conventional dual reflector systems,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 47–56, Jan. 2009. [2] V. Galindo, “Design of dual-reflector antennas with arbitrary phase and amplitude distributions,” IEEE Trans. Antennas Propag., vol. AP-12, no. 4, pp. 403–408, Jul. 1964. [3] J. J. Lee, L. I. Parad, and R. S. Chu, “A shaped offset-fed dual-reflector antenna,” IEEE Trans. Antennas Propag., vol. AP-27, no. 2, pp. 165–171, Mar. 1979. [4] P.-S. Kildal, “Synthesis of multireflector antennas by kinematic and dynamic ray tracing,” IEEE Trans. Antennas Propag., vol. 38, no. 10, pp. 1587–1599, Oct. 1990. [5] F. J. S. Moreira and A. Prata, Jr., “Generalized classical axially symmetric dual-reflector antennas,” IEEE Trans. Antennas Propag., vol. 49, no. 4, pp. 547–554, Apr. 2001. [6] A. Prata, Jr., F. J. S. Moreira, and L. R. Amaro, “Compact high-efficiency displaced-axis axially symmetric high-gain antenna for spacecraft communications,” JPL’s IND Technol. Sci. News, no. 17, pp. 9–14, May 2003. [7] S. P. Morgan, “Some examples of generalized Cassegrainian and Gregorian antennas,” IEEE Trans. Antennas Propag., vol. AP-12, no. 6, pp. 685–691, Nov. 1964. [8] B. S. Westcott, F. A. Stevens, and F. Brickell, “GO synthesis of offset dual reflectors,” IEE Proc., vol. 128, no. 1, pp. 11–18, Feb. 1981, Pt. H. [9] J. R. Mautz and R. F. Harrington, An improved E-field solution for a conducting body of revolution Dept. Electrical and Computer Engineering, Syracuse University, Tech.Rep. TR-80-1, 1980. [10] Y.-C. Chang and M. J. Im, “Synthesis and analysis of shaped ADE reflectors by ray tracing,” in IEEE Antennas Propag. Soc. Int. Symp., Jun. 1995, pp. 1182–1185.

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Adaptive Beamforming with Real-Valued Coefficients Based on Uniform Linear Arrays Lei Zhang, Wei Liu, and Richard J. Langley

Abstract—A class of adaptive beamforming algorithms with real-valued coefficients is proposed based on the uniform linear array structure by introducing a preprocessing transformation matrix. It is derived from the beamformer with a minimum mean square error (MSE) or a maximum output signal-to-interference-plus-noise ratio (SINR), depending on the specific design criteria. The key parameter of the transformation matrix takes different values for different beamforming scenarios and three representative examples are studied: the linearly constrained minimum variance beamformer (and the generalized sidelobe canceller), the reference signal based beamformer, and the class of blind beamformers based on the constant modulus algorithm. Its advantage is twofold: 1) with real-valued coefficients, the computational complexity of the overall system is reduced significantly; 2) a faster convergence speed is achieved and given the same stepsize, the system arrives at a lower MSE (or a higher output SINR). Index Terms—Adaptive beamformer, blind beamformer, real-valued coefficient, reference signal, uniform linear array.

I. INTRODUCTION In the past decades, various algorithms have been proposed for adaptive beamforming [1]–[3], such as the well-known linearly constrained minimum variance (LCMV) beamformer [4]–[6] and its alternative implementation—the generalized sidelobe canceller (GSC) [7], [8], the reference signal based (RSB) beamformer, and the constant modulus (CM) based blind beamformers [9], [10] and its variations including the least squares constant modulus algorithm (LSCMA) [11], [12] and the recursive least squares (RLS) constant modulus algorithm (CMA) [13]–[17]. One key issue in many adaptive beamforming algorithms is the estimation of data covariance matrix. Based on the specific structure of uniform linear arrays (ULAs) and the resultant persymmetric structure of their covariance matrices, a forward-backward (FB) averaging method was proposed for covariance matrix estimation with a significantly reduced complexity [18]. It is then further exploited in different applications such as direction of arrival (DOA) estimation [19], [20], spectrum estimation [21], and adaptive beamforming [22]. It has been shown in [23], [24] that the optimum weight vector of a ULA has a generalized conjugate symmetric property, which is employed to form a constrained beamforming problem and leads to an improved performance. In this work, based on the same ULA structure and the resultant generalized conjugate symmetric property of its optimum weight vector, we will introduce a novel transformation matrix to preprocess the received array data, after which the original complex-valued optimum weight vector will be reduced to a real-valued one, so that in the following weight adaptation we can simply remove the imaginary part of the weight vector. As a result of this regularization, the same improved performance is achieved as in the case of the constrained algorithms in [23], [24], but with a much lower computational complexity. There is an undetermined phase factor in the introduced transformation matrix, the choice of this factor is studied in three different

Manuscript received May 05, 2010; revised July 18, 2010; accepted August 25, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Communications Research Group, Dept. of Electronic & Electrical Engineering, University of Sheffield, Sheffield, U.K. (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2103037

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cases: its value is known, unknown, unknown but not affecting the solution, with the LCMV beamformer (or GSC), the RSB beamformer, and the CMA-based blind beamformer as the corresponding representative examples. II. THE PROPOSED BEAMFORMING STRUCTURE WITH REAL-VALUED COEFFICIENTS We consider the same ULA model described in [23]. At the nth snapshot the received array signals can be expressed as x[n] = A 1 s[n] + n[n], where x[n] = [x0 [n]; x1 [n]; 1 1 1 ; xM 01 [n]]T 2 M 21 and s[n] = [s0 [n]; s1 [n]; 1 1 1 ; sL01 [n]]T 2 L21 are the sensor outputs and the source signals, respectively, and n[n] is the noise vector. M and L are the sensor number and signal number, respectively. f1gT denotes the transpose operation and A is the mixing matrix given by A = [a( 0 ); a( 1 ); 1 1 1 ; a( L01 )] 2 M 2L , where

a(

i

) =

; e0j ; 1 1 1 ; e0j (M 01)

1

T

(1)

is the steering vector with i = 2d sin(i )=0 . i , d, and 0 are DOA angle, adjacent sensor spacing and wavelength, respectively. By applying a set of coefficients wi , i = 0; . . . ; M 0 1 to x[n], we obtain the beamformer output y [n] as

y[n] = wH 1 x[n] = wH 1 A 1 s[n] + wH 1 n[n]

(2)

3 01 ]T is the weight vector, and f1gH where w = [w03 ; w13 ; 1 1 1 ; wM 3 and f1g denotes the Hermitian transpose and the conjugate operation, respectively. Assume the first signal s0 [n] is the desired one. Then the optimum weight vector wopt for maximizing the beamformer’s output SINR has the following generalized conjugate symmetric structure [23]: 3 wopt = ej Jwopt

(3)

where J is the exchange matrix defined as 0

J=

.. .

1

...

1

.

.. .

...

0

..

(4)

and  = 2g 0 (M 0 1) 0 , with g determined by the phase response of the beamformer to the desired signal in the following way

aH (

0 )w

=

wH a(

0)

H

g3 )3 = g = jgjej :

= (

(5)

We can see that g 3 is the resultant beamformer’s response to the desired signal and g is the angle of g . When a reference signal r[n] is available, the optimum weight vector for minimizing the MSE between the reference signal and the beamformer output follows the same relationship described in (3) [23]. The only difference is the value of , which is  = 2c 0 (M 0 1) 0 , where c is the phase of correlation between r[n] and s0 [n], i.e. E fs0 [n]r3 [n]g = c = jcjej . As a result, we can constrain the weight vector to this specific structure during adaptation, as studied in [23], [24]. In this work, we propose another way to exploit this property to improve performance and reduce computational complexity of the system. We first construct a transformation matrix T = [t0 ; 1 1 1 ; tM 01 ] 2 M 2M , with its column vector satisfying the following generalized conjugate symmetric property ti = e0j ti3 J; (i = 0; 1 1 1 ; M 0 1). Then we have the following conclusion:

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 opt = Twopt 2 : w

(6)

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T

TJ

where we have used the result 3 = ej . Transforming to  [n] by multiplying 01 at both sides of (13), we have

T

w

Fig. 1. The proposed structure with a preprocessing operation by

w[n] = T01 w [n] = 21 w[n] + ej Jw3 [n]

T.

Tw

w

Proof: Substituting (3) into (6) leads to  opt = opt = 3 is generalized conjugate symej opt . Since each column of = e0j 3 . Then metric, we have

TJw

T

TJ T 3 3  opt w opt = T3 wopt = (Twopt )3 = w  opt 2 and completes the proof. which indicates that w

(7)

I J I 0 jJ

(8)

Note such a transformation matrix is not unique. In this communication the following construction is adopted: ep 2

T=

ep 2

I

j

J

0j I p0 2

for even M

J 0

0 0j I

for odd M

T defined in this way is unitary, i.e. T01 = TH : (9)

where is the identity matrix.

w

So the norm of is not changed; moreover, it is mainly composed of 1s and j s, which will reduce the computational complexity incurred during transformation. Note that a special case of (8) with  = 0 appeared in [19], [20] to reduce the computational complexity of direction of arrival estimation algorithms. Using (9), y [n] in (2) can be changed to

wH TH Tx[n] = (Tw)H (Tx[n]) = w H x[n]: (10)  [n] = Tx[n], then the optimum w should If we transform x[n] into x y [n] =

be real-valued according to (6). The new structure is shown in Fig. 1, i [n] and w i (i = 0; 1 1 1 ; M 0 1) are the elements of the where x transformed signal vector  [n] and the new weight vector  [n], respectively. Since after the transformation, the optimum weight vector  opt is real-valued for both minimizing MSE and maximizing output SINR, we can ignore the imaginary part of  [n] during the following adaptive beamforming process. This can be achieved by  [n] = Ref ^ [n]g = (1=2)( ^ [n] + ^ 3 [n]), where ^ [n] is the intermediate result for each update of the adaptive beamforming algorithm at time index n. Since  opt is real, we can see that  [n] obtained in this way is closer to  opt than ^ [n], i.e.

x

w w

w

w

w w

w

w

w

w

w

w

kw [n] 0 w opt k  kw^ [n] 0 w opt k

(11)

where k 1 k is the Euclidean norm defined as

kw k = jw0 j2 + jw1 j2 + 1 1 1 + jwM 01 j2 :

w

Now consider the relationship between  [n] and

w[n] back

(12)

w[n]. We have

w [n] = 21 (Tw[n] + T3 w3 [n]) = 21 T w[n] + ej Jw3[n]

x

(14)

which means that preprocessing [n] and then applying a real-valued  [n] is equivalent to applying the weight vector [n] directly to [n] with the constraint that it satisfies the relationship specified in (14). This constrained approach was adopted in [23], [24] for LCMV beamforming, RSB beamforming and CMA-based beamforming, respectively. Compared to the constrained approach, the advantage of the newly proposed preprocessing approach is its significantly reduced computational complexity, which will be shown later. Moreover, we will also study its application based on the GSC, which provides an efficient and alternative implementation of the LCMV beamformer, and the RLS CMA, where a more significant reduction in computational complexity is achieved by our method. As pointed out in [24], as long as the array has a symmetric structure, such as a uniform rectangular array and a uniform circular array, its optimum weight vector will have the desired conjugate symmetric form and the proposed method can then be applied to it. The only difference is the value of the phase factor . However, in practice, this symmetric array structure is often destroyed by some unavoidable factors, such as mutual coupling, antenna position errors, discrepancies in antenna responses, etc. In this case, the performance of the proposed method will degrade and a thorough investigation of its robustness against these factors is needed as a topic of further study. In this section, we will study three different cases of the transformation matrix according to the value of :  is known,  is unknown and  is unknown but does not affect the result, which correspond to the LCMV beamformer (and GSC), RSB beamformer and the CMA-based blind beamformer, respectively.

w x

w

T

A. The LCMV Beamformer and GSC With Real-Valued Coefficients ( is Known) 1) The LCMV Beamformer: The original LCMV problem is formulated as

min J =

w

wH Rxx w

subject to

aH ( 0 )w = g = jgjej

: (15)

Here we have constrained the array response to the desired signal to a complex value g 3 . In this case, the value of g is specified in advance. So the value of  is known and can be applied to the structure in Fig. 1 directly. To implement the LCMV beamformer based on the new set of data  [n], we need to transform the constraint in (15) into the form based on  first, given by

T

x

w

aH ( 0 )w = aH ( 0 )T01 Tw = [Ta( 0 )]H w = jgjej : (16)  = jg j, or a ( 0 )H w  = Equivalently, we can use [ej Ta( 0 )]H w jgj with a( 0 ) = ej Ta( 0 ). With a( 0 ) = e0j(M 01) Ja3 ( 0 ) 3 ( 0 ) = e0j T3 a3 ( 0 ) = [23] and TJ = e0j T3 , we have a j ( 0 ), i.e. e Ta( 0 ) = a (17) a( 0 ) 2 :  = jg j. H ( 0 )w Then we have a real-valued constraint equation a

Now the new LCMV beamformer can be formulated as (13)

 J=w 2 wmin

H

Rw w w

a

subject to H (

w = jgj

0) 

(18)

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TABLE I A SUMMARY OF COMPUTATIONAL COMPLEXITIES OF LCMV BEAMFORMERS, GSCS AND RSB BEAMFORMERS ( IS EVEN)

M

Fig. 2. The GSC structure with the proposed preprocessing by

T.

where x x = E f  [n] H [n]g. Based on the original LCMV update equation, we can have the new set of updates

R

x x

w^ [n + 1] = w [n] 0 v[n] w [n + 1] = (w^ [n + 1] + w^ 3[n + 1]) =2

(19)

where  is the stepsize and

v[n] = I 0 M1 a( 0)aH ( 0) Rxx w [n]:

(20)

The second step in (19) is to take the real part of the weight vector and no multiplications are involved. x x will be replaced by its instantaneous approximation ~ x x =  [ ] H [n] in implementation. 2) The GSC: To implement the real-valued LCMV beamformer using the GSC structure, we decompose the real-valued  into two orthogonal parts  =  q 0   a , as shown in Fig. 2, where the real-valued quiescent vector  q is given as  q = ( 0 )[H ( 0 )( 0 )]01 jgj 2 ,  is in the null space of ( 0 ) with H ( 0 )  = 0, and  a is the new real-valued adaptive weight vector. The real GSC solves  a iteratively by minimizing the following cost function:

R

w w a

R xnx w

a

a

a

B

a

w

w

B

Bw w

w

min J = [w q 0 B w a ]H Rxx [w q 0 B w a ]:

 2 w

(21)

(22)

In [22], a low computational complexity algorithm was proposed by employing the FB averaging processing. Our approach is different in three aspects: 1) the derivation in [22] is based on the assumption that the phase origin of the steering vector is at the geometric center of the array and the array response to the desired signal is a real constant, which is not needed in our approach; 2) the method in [22] first transforms the weight vector into an intermediate real-valued vector, then updates it by real-valued equation to reduce computational complexity, and finally transforms it back into a complex-valued vector, while our approach keeps the weight vector real-valued at all time by preprocessing the received array signals; 3) the method in [22] is based on the SMI (sample matrix inversion) method by updating xx while we have employed a much less complex LMS (least mean square)-based adaptive algorithm. The computational complexities of the traditional LCMV beamformer (T-LCMV), Huarng’s LCMV beamformer (H-LCMV) in [22], the constrained LCMV beamformer (C-LCMV) in [23], the proposed real-valued LCMV beamformer (R-LCMV), the traditional GSC (T-GSC), Huarng’s GSC (H-GSC) in [22] and the proposed real-valued GSC (R-GSC) are listed in Table I, with complex-valued input signals in term of the number of real multiplications. According to Table I, the proposed R-LCMV has a much less computational complexity than T-LCMV, H-LCMV and C-LCMV. For the GSC

R

T.

case, the proposed R-GSC also has a much lower complexity than the other two GSCs (T-GSC and H-GSC). Notice that Table I is based on an even M , for which the prepro0j=2 cessing does not involve any multiplications, since p the factor e H can be canceled in calculating  [n] [n] and 1= 2 can be absorbed into the update stepsize. For p an odd M , since one of elements of the transformation matrix is 2, additional 2 multiplications are needed for the proposed algorithms.

x x

B. The RSB Beamformer With Real-Valued Coefficients ( is Unknown) The RSB beamforming structure with the proposed preprocessing is shown in Fig. 3. The array output y [n] is subtracted from an available reference signal r[n] to generate an error signal

We can obtain the following update equation:

w^ a [n + 1] = w a[n] + B H Rxx w q [n] 0 B w a[n] w a [n + 1] = (w^ a[n + 1] + w^ a3 [n + 1]) =2:

Fig. 3. The RSB beamformer with the proposed preprocessing by

e[n] = r[n] 0 w H [n]Tx[n] = r[n] 0 w H [n]x[n]:

(23)

 ) is usually constructed using the MSE The cost function J (w E fe[n]e3 [n]g and then replaced by the instantaneous estimate e[n]e3 [n] at a later stage [25], [26].

Using the stochastic gradient method, we can obtain the following classical LMS adaptive algorithm with a stepsize 1 [25], [26]

w^ [n + 1] = w [n] + 1e3[n]x[n]:

(24)

w

Here we use ^ [n + 1] to emphasize that it is complex-valued and it should be constrained as real-valued, by the following processing as a second step:

w [n + 1] = Re (w^ [n + 1]) = 12 (w^ [n + 1] + w^ 3[n + 1]) : (25) However, a key issue is the phase shift  in T is unknown. So we can not use the above algorithm directly. As a solution we decompose T into two parts as follows T = T1 , with  being an unknown variable with a unit magnitude and T1 defined as T1 = ej=2 T. Now T1 is independent of .  H [n]Tx[n] = The new structure is shown in Fig. 4 with y [n] = w [n]w H [n]x[n].Note here that x[n] = T1 x[n]. Now we can formulate the real-valued RSB beamformer as follows

min J (w ; ) with

 2 ;jj=1 w

(26)

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Following the same approach, the real-valued CMA with the proposed preprocessing can be described as

w^ [n + 1] = w [n] d[n]y3[n]x[n] w [n + 1] = 12 (w^ [n + 1] + w^ 3 [n + 1]) : 0

Fig. 4. The modified RSB beamformer with preprocessing.

(33)

In implementation, (33) can be written in a more compact form as 3 2 H xx H [n]w  ; ) = e[n]e [n] = jr [n]j + w  [n]  [n] J (w 3 H x [n]w [n] 0  [n]r [n] H xr3 [n]  [n] 0 [n] w

w [n + 1] = w [n] (27)

x[n] (28)

w

w^ [n + 1] = w [n] 1 xH [n]w [n] [n]r3 [n] x[n] w [n + 1] = 12 (w^ [n + 1] + w^ 3 [n + 1]) : H x[n]r3 [n]  [n] ^[n + 1] = [n] + 2 w 0

0

(29)

[n + 1] = ^ [n + 1]= j^ [n + 1]j

In Table I, we have listed the computational complexities of the proposed RSB algorithm (R-RSB), the traditional one (T-RSB), and the constrained one proposed in [23] (C-RSB). For a large M , the number of real multiplications of the R-RSB is less than half of the other two. Table I is based on an even M , and 2 additional multiplications are required for the proposed RSB with an odd M . C. The CMA-Based Beamformer With Real-Valued Coefficients ( is Unknown but Does Not Affect the Solution) In this section, we consider the third case of . i.e. its value is unknown but its choice does not affect the solution. One such an example is the CMA and we will first consider the basic CMA and then extend the result to another two well-known CMAs: the LSCMA and the RLSCMA. In CMA, its solution is ambiguous to phase response of the array to the desired signal. Then we can choose the value of g = (M 0 1) 0 =2, leading to  = 0. Thus defined in (8) has the same form as 1 used in the RSB case which is independent of . 1) The Basic-Type CMA: The cost function employed in the basictype CMA has the following form [9]

T

T

Jpq

=

E f(jy[n]jp 0  p )q g

(30)

where  is the modulus of the desired signal, and p and q are positive integers and generally chosen to be 1 or 2. A recursive update equation for can be obtained as follows:

w

w[n + 1] = w[n]

0

d[n]y [n]x[n] 3

(31)

where  is the stepsize and

d[n] = pq jy[n]jp 02 (jy[n]jp 0  p )q01 1 sgn (jy[n]jp 0  p )q : (32)

x[n]xH [n] w [n]:

(34)

w H x[nK + i]

i=1

0

1

2

(35)

x

where K is the number of samples considered and  [nK + i] = [nK + i]. Using the result of [11] and constraining the result to real value leads to the following solution:

Tx

w

For adaptive beamforming, we update  and  in the negative direction of the gradient (28), scaled by stepsizes 1 and 2 and then constrain them into real value and on the unit circle, respectively, leading to the following set of update equations:

K

LS =

min J 2

w

H  [n] 0 [n]r3 [n]  ; ) = x  [n] w w J (w H x[n]r3 [n] :  ; ) = 0 w  [n] r J (w

d[n]Re

2) The Least-Squares CMA: The real-valued LSCMA minimizes the following cost function

where we have used the result [n]3 [n] = 1. By taking the gradient of (27) with respect to  and , respectively, we have r

0

01

w [n + 1] = K1 Re R~ xx [n]

pxy [n]

(36)

 = [  [n]ye [n], X where px y [n] = (1=K )X x[nK + 1]; 1 1 1 ; x[nK + K ]], ye [n] = [y[nK + 1]=jy[nK + 1]j; 1 1 1 ; y[nK + K ]=jy[nK + K [nK + i]xH [nK + i]. When ~x K ]j]H and R x  [n] = (1=K ) i=1 x ~ K ! 1, Rxx [n] is the correlation matrix of x [n] and we have R~ xx [n] = Rxx 2 . Then w will converge to the optimum solution. ~x For a limited K , R x  [n] is generally not real-valued due to noise and ~x x error. However, it is easy to see that using R x  [n] = RefR x  [n]g ~ instead of Rxx [n] to estimate the correlation matrix will lead to a

Rxx . i.e. R xx [n] Rxx R~ xx [n] Rxx

closer approximation to

0



:

0

(37)

Define  e [n]=K p xy [n] = Re Xy K 1 = ( x[nK + i])R (x[nK + i])TR K

i=1

x

x

T I

+ ( [nK + i]) ( [nK + i])

1

I

w [n]= y[nK + i] j

x x

(38)

j

x

where ( [nK + i])R and ( [nK + i])I represent the real and imaginary part of  [nK + i], respectively. Substituting (38) into (36) leads to the following update equation

w [n + 1] = R xx [n] 01 p xy [n]:

(39)

3) The RLS CMA: RLS CMA is considered as a general case of the well-known orthogonalized CMA (OCMA) [9], [13]. The proposed RLSCMA minimizes the following cost function

RLS =

min J 2

w

K i=1

K 0i jy[i]j2 0 1

2

(40)

where 0 < < 1 is the forgetting factor. Using the result of [13] and following the same approach to constrain to real value leads to

w

w [n + 1] = w [n] + Re b[n]

y[n]j2 0 1

j

(41)

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TABLE II A SUMMARY OF COMPUTATIONAL COMPLEXITIES OF IS EVEN) THE CMA-BASED BEAMFORMERS(

M

Fig. 5. Output SINR versus number of snapshots for the LCMV beamformer and GSC: (a) LCMV beamformer; (b) GSC.

where III. SIMULATIONS

b[ n ] =

1

Simulations are performed based on a ULA with d = 0 =2. We assume that all signals have the same power with a signal-to-noise ratio (SNR) of 20 dB unless otherwise specified.

P[n 0 1]c[n]

+ 1[n] n] = cH [n]P[n 0 1]c[n]

1[

P[n] = 1 P[n 0 1] 0 b[n]cH [n]P[n 0 1] c[n] = y 3 [n]x [n]

A. The Real-Valued LCMV Beamformer and GSC (42)

~ 01 [n]. Note that the matrix inversion lemma has been with P[n] = R xx used to avoid the matrix inversion operation. P[n] is a complex-valued  [n] = RefR ~ 01 [n] g rather matrix due to limited samples. By using P xx than P[n], we can follow the same way as in LSCMA to form a realvalued RLSCMA implementation. Choosing the initial value P[0] 2  [n] instead of c[n] [n] = ( and using c xR [n]xTR [n] + x I [n]xTI [n])w into (42) will give the following update equation:  [n] jy [n]j 2 0 1  [n + 1] = w  [n] + b w

(43)

where

P[n 0 1]c[n]  n] + 1[ H  n) = c  [n]P[n 0 1] c[n] 1( b [n] =

1

P[n] = 1 P[n 0 1] 0 b [n]cH [n]P[n 0 1]  [n] : c[n] = x R [n]xTR [n] + xI [n]xTI [n] w

Our simulation is based on an 8-element ULA. There are four signals arriving from the DOA angles 0 = 010 , 1 = 20 , 2 = 70 and 3 = 040 , respectively and the first signal is the desired one. Two sets of simulations are performed for the proposed LCMV beamformer (R-LCMV) and the traditional one (T-LCMV). The result is shown in Fig. 5(a), where although the convergence rate of the two algorithms is almost the same when using the same stepsize, R-LCMV has reached a higher output SINR value. When the stepsize of the T-LCMV is reduced to 0.00015 to reach a higher steady state SINR value, its convergence becomes much slower. For the GSC, the result is shown in Fig. 5(b), and a similar conclusion can be drawn as in the LCMV case. Their convergence rate is almost the same given the same stepsize ( = 0:0002), but the proposed R-GSC has reached a higher output SINR value. We then reduce the stepsize of the T-GSC to 0.00012 to reach a higher output SINR value and this unavoidably leads to a reduced convergence rate. The number of real multiplications for each update for the T-LCMV, R-LCMV, T-GSC and R-GSC is: 132, 60, 542 and 302, respectively. B. The Real-valued RSB Beamformer

(44)

The computational complexities of the traditional CMAs (T-CMA, T-LSCMA, T-RLSCMA), the constrained CMAs (C-CMA, C-LSCMA) in [24] and the proposed algorithms (R-CMA, R-LSCMA, R-RLSCMA) are listed in Table II. For the relatively simple basic-type CMAs (T-CMA, C-CMA and R-CMA), the computational complexity of the proposed R-CMA is only half of the T-CMA for large M and even less compared with C-CMA. For the LSCMAs (T-LSCMA C-LSCMA, and R-LSCMA), since the matrix inversion operation ~ of the complex matrix R xx [n] requires O(M 3 ) real multiplications with complex-valued input signals, the proposed algorithm will save up to 75% computational load due to employing a real-valued  matrix R xx [n]. For the RLSCMA case, the proposed algorithm (R-RLSCMA) will save 75% computational load compared to the traditional one (T-RLSCMA). Notice that 2 extra multiplications are required for all of the proposed CMAs with an odd M .

There are four signals arriving at an 8-element ULA from the DOA angles 0 = 15 , 1 = 10 , 2 = 010 and 3 = 020 , respectively and the first signal is the desired one. The stepsize 1 for both the traditional RSB beamformer (T-RSB) and the proposed one (R-RSB) is 0.001, and the stepsize 2 for  changes from 0.01, 0.5 to 10. As shown in Fig. 6, an improved performance has been achieved in terms of both convergence rate and output SINR for R-RSB, as compared to T-RSB. Furthermore, we have observed that the performance of R-RSB is not sensitive to the stepsize 2 within a large dynamic range. This can be explained by the fact that only the angle of  matters to the output since its magnitude is always normalized to unity; while by checking (29), H [n]x[n]r3 [n] when 2  the angle of  will tend to be that of the part w goes to infinity, i.e. a larger 2 does not really lead to a larger change of the angle of  and there is a limit to it. For their computational complexity, 66 real multiplications are required for each update for T-RSB and 51 for R-RSB.

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transformation matrix takes different values and three representative examples have been studied. All of them have a significantly reduced overall computational complexity and a faster convergence speed compared to the traditional complex-valued beamforming algorithms.

REFERENCES

Fig. 6. Output SINR versus number of snapshots for the RSB beamformer.

Fig. 7. Output SINR versus the number of snapshots for CMA-based beamformers.

C. The Real-valued CMAs We assume that all of the CM signals are generated by the QPSK modulation scheme. There are one desired signal and three interfering signals, arriving from the DOA angles 010 , 20 , 050 and 60 , respectively. Fig. 7 shows the learning curves of the traditional CMA (T-CMA) ( = 0:001), the real-valued CMA (R-CMA) ( = 0:0015), the traditional LSCMA (T-LSCMA) (K = 30), the real-valued LSCMA (R-LSCMA) (K = 30), the traditional RLSCMA (T-RLSCMA) ( = 0:99) and the real-valued RLSCMA (R-RLSCMA) ( = 0:98). The stepsizes (forgetting factors) are chosen empirically for each pair of algorithms to reach approximately the same steady state output SINR. From the figure we can see that the proposed algorithms (R-CMA and R-RLSCMA) have a much faster convergence rate than the traditional ones. For the LSCMA pair (T-LSCMA and R-LSCMA), their convergence is not controlled by any stepsize and it seems that they have a similar convergence speed. However, the proposed R-LSCMA has achieved a higher output SINR given the same block size. We have calculated the number of real multiplications for T-CMA, R-CMA, T-LSCMA, R-LSCMA, T-RLSCMA, R-RLSCMA, which is 69, 37, 11789, 5462, 614 and 173, respectively. IV. CONCLUSION A new class of adaptive beamforming algorithms with real-valued coefficients has been proposed based on uniform linear arrays by preprocessing the received array signals by a unitary transformation matrix. With this transformation, the optimum weight vector of the beamformer with a minimum MSE or a maximum output SINR will be real-valued, so that during adaptation, we can ignore the imaginary part of the weight vector, leading to a set of real-valued coefficients. Depending on the specific beamforming scenarios, the phase factor of this

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[24] L. Zhang, W. Liu, and R. J. Langley, “A class of constant modulus algorithms for uniform linear arrays with a conjugate symmetric constraint,” Signal Processing, vol. 90, pp. 2760–2765, 2010. [25] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1985. [26] B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE Acoust., Speech, Signal Processing Mag., vol. 5, pp. 4–24, Apr. 1988.

On the Uniqueness of the Phase Retrieval Problem From Far Field Amplitude-Only Data

Fig. 1. Illustration of the problem geometry.

Kivanc Inan and Rodolfo E. Diaz

Abstract—The phase retrieval problem has gained considerable attention since the wave phase is lost or impractical to measure in several applications in Electromagnetic Engineering. Quite a few approaches have been proposed recently for the extraction of phase from amplitude-only data; however few of them investigate the uniqueness of the problem prior to the solution. In this communication, the problem of the uniqueness of the phase retrieval problem from far field amplitude-only data is examined in detail. It is shown that uniqueness is guaranteed if two experiments can be performed that differ by a quadratic term in the phase. Conversely, experiments creating only linear phase differences in the data do not guarantee uniqueness. This result is demonstrated by considering the problems of a scattering target illuminated in two experiments differing by a quadratic phase shift, corresponding to two different values of defocusing of the incident wave. Index Terms—Amplitude-only, far field, phase retrieval, uniqueness.

I. INTRODUCTION The problem of determining the phase of the function from its modulus is known as the phase retrieval problem. The loss of the phase information can lead to difficulties in data analysis. Specifically, phase information is crucial for many applications in antennas and propagation, such as the measurement of dielectric properties and inverse scattering. Similarly, in Physics, the use of scattered x-rays to determine the structure of a crystal does not lead to unique results if only intensity measurements are available. In the theory of partial coherence, the degree of coherence won’t be unique when there is only modulus information. And, in the theory of image formation, several different distributions in the Fraunhofer plane might lead to the same distribution in the Gaussian image plane when the phase of the distribution is unknown. This last example resembles our case of interest where only the far-field amplitude data exists from a scattering experiment. The problem of interest is given in Fig. 1. We are concerned with the optical inspection problem, which can be used to identify defects on the surface. The surface is assumed to be illuminated by a laser. The figure illustrates the incident field, the surface under observation and the far field distribution, and S (k), the scattered field which is found from the equivalent surface currents that constitute the radiating aperture, s(x). Manuscript received March 31, 2009; revised October 13, 2009; accepted August 24, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Department of Electrical Engineering, Ira A. Fulton School of Engineering, Arizona State University, Tempe, AZ 85287-5706 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2103000

In the following discussion we refer to the far field data in k-space and for simplicity consider the 2D case where the scattering plane is a boundary in this 2D space. There is no generality lost in this assumption because the scattered field in the principal planes of a realistic 3D surface can always be obtained by collapsing the Schelkunoff currents to the 2D plane of incidence. Thus the far field of interest can be viewed as the array factor (AF (k)) of the radiation emanating from the scattering centers on the surface under observation. It is clear that various aperture distributions might lead to the same amplitude pattern in the far-field. In other words, two measured far fields with the same amplitude distribution in general may differ by a phase function di (k)

AF (k)j exp(j1 (k)) = jAF (k)j exp(j2 (k)) exp(jdi (k)):

j

Since every assumed phase function will lead to a different aperture distribution after performing the inverse Fourier transform of the function, an amplitude pattern does not uniquely determine the aperture distribution. At first blush the amplitude information does not seem to give a unique clue to the true phase information. However, it has been shown that uniqueness can be attained by using more a priori information. In the early 1970’s, Gerchberg and Saxton proposed that the additional information required to solve the phase problem could be acquired from the intensity information obtained in both object and Fraunhofer planes [1]–[3]. The uniqueness of this approach is proven by Huiser [4], [5], assuming the analyticity of the functions and excluding the case of the symmetric objects. Following this study, a related procedure, involving the use of the intensity distributions of two slightly defocused images was put forward by Misell [6]–[8]. In [9], the author addresses the uniqueness of the phase retrieval problem of the Misell procedure. Although the author proves the sufficient condition for the uniqueness of the problem is obtained from two amplitude-only measurements with different defocusing lengths, he does not give any methodology by which the data of those two experiments can be used to actually reconstruct the missing phase information. These studies were reviewed and illustrated in [10]. Later on, Hayes focused on the similar problem and described the restrictions for the uniqueness for both one and multidimensional sequences assuming that the signals are real-valued [11]. Then he did the phase retrieval using two Fourier transform intensities as done before, assuming that the original signal is real valued and creating another signal by adding a known reference signal [12]. However, the convergence characteristics of these algorithms have been explored only partially. Following these studies, several methods have been published on the phase retrieval problem without considering the uniqueness condition

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Following Fig. 2: i. At the Fourier transform plane

 (k) = F T fT (x)g :

(2a)

ii. At the object iris plane (including defocusing)

g (k) =  (k) exp Fig. 2. Illustration of the Fraunhofer diffraction for the microscopy system in [9].

of the problem. In [13], Spectral Iteration (SI) is introduced. The results in this approach provide a new insight into the analysis of different methods for reconstructing signals from their spectral transform. Later, Sarkar presented a different method to generate the phase from amplitude-only data [14]. In this approach, a non-minimum-phase component is recovered, based on the fact that the auto-correlation of the time domain sequence is equal to a discrete cosine transformation of the square of the magnitude response. Following this study, he utilized an all-pass filter in order to reduce the optimization variables [15]. This method works well for some cases, but not for all, so it is not complete. Optimizing helps to get close results, but still it has not proven completely robust. Recently in [16], phase retrieval of the scattered fields at high frequencies using the Fast Fourier Transform is illustrated where the amplitude-only information scattering from an object placed in the near field of an antenna was used to find the corresponding phase. In [17], the uniqueness of the phase retrieval problem is re-derived as in [9] for two near field amplitude only measurements. Here, we will introduce a simpler derivation of the uniqueness condition of the problem starting with the idea in [9]. In addition to its conceptual simplicity, this proof can be applicable to other amplitude-only data problems. Thus in Section II, the proof of why two experiments differing by a quadratic phase shift are enough to guarantee the unique solution of the phase reconstruction problem is presented, followed by the conclusion. II. UNIQUENESS OF THE PROBLEM Given a scatterer defined in one dimension between 6x0 and represented by the function s(x). We assume that its scattering properties are directly proportional to s(x). By definition s(x) is a complex function bandlimited to 6x0 . Thus s(x) is a “scattering center” representation of the physical object (surface, in our case of interest). We assume that the strength of the scattering centers s(x) is invariant with angle of illumination. This is a good approximation for many radar scattering problems of interest where the radar cross section is dominated by scattering from sharp discontinuities on the object (corners, tips, gaps, etc.) and also for the optical inspection of patterned surfaces. Notice that this is also analogous to the classic microscopy problem where the object is defined by a transmission function T (x), independent of the illumination source direction or focusing as described in detail in [9]. Let us continue with the proof of why two experiments differing by a quadratic phase shift are required to guarantee the unique solution of the phase retrieval problem. The microscopy system in [9] is given in Fig. 2. Let the true object be described by the complex bandlimited function T (x), with Fourier transform properties given as

 (k) = T (x) =

+x

0x 1

2

T (x) exp(0jkx)dx +1

01

 (k) exp(jkx)dk:

(1a) (1b)

0jk2 1zf

(2b)

where 1z is the defocusing distance. iii. At the image plane

h (x; 1z ) = F T 01 [g (k)] = F T 01  (k) exp

0jk2 1zf

:

(2c) Let there be two experiments performed one using 1z1 and one using 1z2 then the results at the image plane are

h (x; 1z 1) = F T 01  (k) exp 0jk2 1z 1f 01 [g (k; 1z 1)] =FT h (x; 1z 2) = F T 01  (k) exp 0jk2 1z 2f 01 [g (k; 1z 2)] : =FT

(3)

It follows by definition that:

g (k; 1z 1) g (k; 1z 2)

0jk2 [1z1 0 1z2] f F T fh (x; 1z 1)g = : F T fh (x; 1z 2)g

= exp

(4)

The statement of the amplitude-only problem is that we can only measure jh(x; 1z 1)j and jh(x; 1z 2)j. Now suppose that we generate two complex functions: h1 (x; 1z 1), h2 (x; 1z 2) such that their amplitudes match the experimental amplitudes. Then these complex functions can only differ from the exact answer by an arbitrary error phase function

h1 (x; 1z 1) = h (x; 1z 1) exp (j 81 (x; 1z 1)) h2 (x; 1z 2) = h (x; 1z 2) exp (j 82 (x; 1z 2)) :

(5)

And in general the two error phase functions cannot be assumed to be the same. Now assume that the corresponding Fourier transforms g1 (k) and g2 (k) are bandlimited over the same domain as the original g(k). (It is further stated in [9] that the h(x) functions are also bandlimited over the same observation image plane). Since

F T fh1 (x; 1z 1)g = F T fh (x; 1z 1) exp (j 81 (x; 1z 1))g g1 (k; 1z 1) =

+1

01

g (u; 1z 1) 1 L1 (k 0 u)du;

(6)

by the convolution theorem, where

L1 (k) = F T [exp (j 81 (x; 1z 1))] :

(7)

By the band-limited assumption, g1 and g are non-zero only over the same bandlimited range. But g1 is the convolution of g with another function. The only way a convolved function, g , can have the same band-limited boundaries as it did in the first place is if the convolving function consists of the Dirac-delta function and its derivatives. That is, if by the method of construction of the solutions we guarantee solutions that only exist over the same band-limited range, the only error phase functions that we can incur are limited to the form

L1 (k) =

1

n=0

n  n (k);

(8a)

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It follows by (6) and by the Fourier Transform properties of the Dirac delta function and its derivatives that exp (j 81 (x; 1z 1)) =

1

n=0

n

an (jx)

where we have denoted the fact that 81 and 82 may be different error phase functions by using different coefficients an and bn . Now breaking apart (14) term by term on both sides we have

(8b)

where by  n we mean ;  0 ;  00 ;  iv . . . (The derivatives of the delta function). The above fact will become useful in the argument below. Now, we consider the statement in [9] that if the two test functions g1 and g2 obey the following equation: g1 (k ) = exp g2 (k )

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0jk2 [1z1 0 1z2] f

(9)

the unique solution has been found. (That is, we have the  (k) whose inverse transform gives the unique T (x) and therefore h(x) at the image plane (when perfectly focused).) The Volterra equation is invoked in [9] for the proof and arguments about the structure of zeros and poles of the analytic functions are used. Let us proceed instead by using the facts stated above to produce an alternate (simpler) proof. By the definitions given, (9) implies,

a0

1 2 h (u; 1z 2)A exp 0jq (x 0 u) du 01 +1 2 + a1 h (u; 1z 2) (ju) A exp 0jq (x 0 u) du 01 +1 2 2 + a2 h (u; 1z 2)(ju) A exp 0jq (x 0 u) du + 1 1 1 01 +

= b0 h (x; 1z 1) + b1 (ix) h (x; 1z 1) 2

+ b2 (ix) h (x; 1z 1) +

111 :

(15)

Now consider (4) in the form g (k; 1z 1) = g (k; 1z 2) exp

0jk2 [1z1 0 1z2] f

:

(16)

du:

(17)

Taking the inverse Fourier transform of both sides gives

1 2 h (u; 1z 2)A exp 0jq (x 0 u) 01 +

h (x; 1z 1) =

So that we recognize the first term in (15) allowing us to rewrite (15) as

0jk2 [1z1 0 1z2] f g1 (k ) F T fh1 (x; 1z 1)g = = g2 (k ) F T fh2 (x; 1z 2)g F T fh (x; 1z 1) exp (j 81 (x; 1z 1))g : = F T fh (x; 1z 2) exp (j 82 (x; 1z 2))g

a0 h (x; 1z 1) + + a1

exp

(10)

+ a2

1 2 h (u; 1z 2) (ju) A exp 0jq (x 0 u) du 01 +1 2 2 h (u; 1z 2)(ju) A exp 0jq (x 0 u) du 1 +1 1 1 01

= b0 h (x; 1z 1) + b1 (jx) h (x; 1z 1) 2

+ b2 (jx) h (x; 1z 1) +

Therefore

0jk2 [1z1 0 1z2] f 1 F T fh (x; 1z2) exp (j 82 (x; 1z2))g = F T fh (x; 1z 1) exp (j 81 (x; 1z 1))g :

exp

(11)

By the Fourier transform properties of Gaussians, we can guarantee that exp

0jk2 [1z1 0 1z2] f

= FT

A exp

0jqx2

;

(12)

where the constants A and q combine the Gaussian coefficients and exponents after the transform. Using (12) we recognize the left hand side of (11) as the product of Fourier Transforms. The inverse transform of both sides then leads to

1 2 h (u; 1z 2) exp (j 82 (u; 1z 2))A exp 0jq (x 0 u) 01 +

du

= h (x; 1z 1) exp (j 81 (x; 1z 1))

(13)

again by the convolution theorem. Expanding the error phase functions into their infinite sum form of (8b), we can rewrite (13) as

1 1 n h (u; 1z 2) an (ju) 01 n=0 +

A exp

0jq(x 0 u)2 1

= h (x; 1z 1)

n=0

du

n

bn (ju)

(14)

111 :

(18)

Now, since the function A exp(0jqx2 ) is a Gaussian with finite width (i.e., x is band-limited), its convolution of h(x; 1z 2) will always yield a function spanning a wider band-limited range than the original function. Since the right hand side (RHS) of (18) is band-limited to the original range, the left hand side (LHS) must also be band-limited to the same range. Therefore it is clear that the only way (18) can be satisfied is for all a’s greater than or equal to a1 to be zero. At that point we divide both sides by h(x; 1z 1) and prove that all b’s greater than or equal to b1 are zero. And so a0 = b0 . The vanishing of all these coefficients proves that the error phase function is exactly 1. Therefore we have discovered the true unique solution. Having re-derived the case in [9] using a different argument, we ask what does this argument say about our scattering problem? There are at least two possible versions of a two amplitude-only measurement protocol: In one experimental protocol the surface s(x) is illuminated at two different angles of incidence. In the other it is illuminated at the same angle of incidence but not with a plane wave, rather we use two different focused beams. The two-angle scattering problem can be described as follows. Let the true object be described by the complex bandlimited function s(x). In the far field S (k) = F T fs(x)g when illuminated at normal incidence. Illuminated off normal using kx = ka (where sin(inc ) = kx =k0 )

f f

g

S (k; ka ) = F T s (x; ka )

0

g

= F T s (x) exp ( jka x) = S (k

0 ka ) :

(19)

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Let there be two experiments performed one using ka and one using kb , then the results in the far field are given by

S (k; ka;b ) = F T fs (x; ka;b )g = F T fs (x) exp (0jka;b x)g : (20) It follows by definition that:

s (x; ka ) s (x; kb )

= exp (0j [ka

0 kb ] x) =

F T 01 [S (k; ka )] : F T 01 [S (k; kba )]

(21)

The statement of the amplitude-only problem is that we can only measure jS (k; ka )j and jS (k; kb )j. Suppose that by the method of construction we generate two complex functions, S1 (k; ka ) and S2 (k; kb ) such that their amplitudes match the experimental amplitudes. Then these complex functions can only differ from the exact answer by an arbitrary error phase function

S1;2 (k; ka;b ) = S (k; ka;b ) exp (j 81;2 (k; ka;b )) :

On the other hand, it is clear from our construction that the two test functions s1 and s2 obey the following:

0 kb ] x) =

F T 01 [S1 (k; ka )] : F T 01 [S2 (k; kb )]

(24)

The question is: Does this guarantee that we have found the unique solution as (11) above did for the case given in [9]? By the definitions given, (23) implies

exp (0j [ka

F T 01 [S1 (k; ka )] F T 01 [S2 (k; kba )] F T 01 [S (k; ka ) exp (j 81 (k; ka ))] = : (25) F T 01 [S (k; kb ) exp (j 82 (k; kb ))]

0 kb ] x) =

Therefore exp (0j [ka

0 kb ] x) F T 01 [S (k; kb ) exp (j 82 (k; kb ))] 01 [S (k; ka ) exp (j 81 (k; ka ))] : (26) = FT

0 kb ] x) = F T 01 [ (k 0 1k)]

(27)

where 1k = ka 0 kb , by the Fourier transform property of Delta functions and the shifting theorem. We recognize in the LHS of (11) a product of Fourier transforms. Therefore taking the inverse transform of both sides +1

01

S (k 0 1k; kb ) = S (k 0 1k 0 kb ) = S (k 0 ka ) = S (k; ka ) : (30) Therefore (29) simply tells us that

0 1k; kb )) = exp (j 81 (k; ka )) :

(31)

In other words, all the two-angle procedure can guarantee in this case is that the error incurred in deriving the solution at one angle maps by a shift to the error incurred at the other angle. But it does not guarantee that the error phase functions vanish. In the above procedure the difference between the two experiments is a linear phase difference, not a quadratic phase difference (due to defocusing) as assumed in [9]. Thus we surmise that the key to obtaining a unique solution with two measurements is to make sure that the two measurements differ in phase by a quadratic dependence on k, and not a linear dependence. Therefore it follows that our far field phase reconstruction method will work if it uses a similar approach given in [9]. The surface s(x) must be illuminated with a focused wave, not a plane wave and then it suffices to provide two different focal lengths at a given angle to obtain a unique solution.

III. CONCLUSION A new proof of uniqueness of the phase reconstruction problem that requires no Volterra equations is demonstrated. Two amplitude-only far field measurements differing by a quadratic phase shift, lead to a unique determination of the far field phase distribution. For the sake of simplicity and compactness, this communication has only considered the 1D scattering space problem (linear aperture). The results can be translated to the case of a two-dimensional aperture or distribution of scatterers radiating in a 3D space, which is beyond the scope of this communication. Two measurements differing by linear phase lead to non-unique answers, but if one of the measurements has a quadratic phase, the result is the unique original aperture.

ACKNOWLEDGMENT

Now we know that exp (0j [ka

Then from (19) we see that since S (k; ka ) = S (k 0 ka ), it is obvious that

exp (j 82 (k

F T 01 [S1 (k; ka;b )] = F T 01 [S (k; ka;b ) exp (j 81 (k; ka;b ))] : (23)

= exp (0j [ka

S (k 0 1k; kb ) exp (j 82 (k 0 1k; kb )) = S (k; ka ) exp (j 81 (k; ka )) : (29)

(22)

Note again that in general the two error phase functions cannot be assumed to be the same. Again we assume that the corresponding Fourier transforms s1 (x; ka ) and s2 (x; kb ) are bandlimited over the same domain as the original s(x). (Furthermore because we measure the far field only, the k -space information is also band-limited to 6k0 .) Taking the Inverse Fourier transform of the both sides of (22) will lead to

s1 (x; ka ) s2 (x; kb )

But now by the sampling property of the Delta function, the convolved function gets evaluated at u = k 0 1k

S (u; kb ) exp (j 82 (u; kb )) ((k 0 1k 0 u)) du =

S (k; ka ) exp (j 81 (k; ka )) : (28)

The writers would like to thank the reviewers for their careful work. Their comments helped create a high quality paper.

REFERENCES [1] R. W. Gerchberg and W. O. Saxton, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik, vol. 33, pp. 275–284, 1971. [2] R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane apertures,” Optik, vol. 35, pp. 237–246, 1972.

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[3] R. W. Gerchberg and W. O. Saxton, , P. W. Hawkes, Ed., “Wave phase from image and diffraction plane pictures,” in Image Processing and Computer Aided Design in Electron Optics. London: Academic, 1973. [4] A. M. J. Huiser et al., “On phase retrieval in electron microscopy from image and diffraction pattern,” Optik, vol. 45, pp. 303–316, 1976. [5] A. M. J. Huiser and H. A. Ferwerda, “On the problem of phase retrieval in electron microscopy from image and diffraction pattern, II,” Optik, vol. 46, pp. 407–420, 1976. [6] D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. test calculations,” J. Phys. D, Appl. Phys., vol. 6, pp. 2200–2216, 1973. [7] D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. sources of error,” J. Phys. D, Appl. Phys., vol. 6, pp. 2217–2225, 1973. [8] 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. [9] B. J. Hoenders, “On the solution of the phase retrieval problem,” J. Math Phys., vol. 16, pp. 1719–1725, 1975. [10] L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas. Propag., vol. AP-29, no. 2, pp. 386–391, Mar. 1981. [11] M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acs., Speech Sig. Proc., vol. ASSP-30, no. 2, pp. 140–154, Apr. 1982. [12] W. Kim and M. H. Hayes, “Phase retrieval using two Fourier-transform intensities,” J. Opt. Soc. Amer., vol. 7, no. 3, pp. 441–449, Mar. 1990. [13] Y. Shapiro and M. Porat, “Optimal signal reconstruction from spectral amplitude,” in Proc. IEEE DSP, 1997, pp. 773–776. [14] T. K. Sarkar and B. Hu, “Generation of nonminimum phase from amplitude-only data,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 8, pp. 1079–1084, Aug. 1998. [15] Y. Cho, T. K. Sarkar, and J. Koh, “Reconstruction of non-minimum phase function from amplitude only data,” Microw. Opt. Technol. Lett., vol. 35, no. 3, pp. 212–216, Nov. 2002. [16] G. Hislop, G. C. James, and A. Hellicar, “Phase retrieval of scattered fields,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2332–2341, Aug. 2007. [17] S. F. Razavi and Y. Rahmat-Samii, “On the uniqueness of planar nearfield phaseless antenna measurements based on two amplitude-only measurements,” presented at the Antennas Propag. Soc. Int. Symp., 2008.

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Scaling Factors of ID-FDTD Scheme for Dispersive Media Based on the Auxiliary Differential Equation (ADE) Method Pingping Deng, Meng Zhao, and Il-Suek Koh

Abstract—The isotropic-dispersion finite-difference time domain (ID-FDTD) scheme is modified for a dispersive medium. The proposed ID-FDTD scheme is based on the ADE method. In this correspondence, three typical dispersive media are considered: Debye, Drude, and Lorentz media, for which scaling factors are formulated. The dispersion property is then examined, and the accuracy of the proposed scheme is verified by consideration of two scattering problems. Index Terms—Dispersion, dispersive medium, finite-difference time domain (FDTD).

I. INTRODUCTION The ID-FDTD scheme was originally proposed for a non-dispersive medium [1], [2]. Two factors, known as weighting and scaling factors, are used to achieve a nearly isotropic dispersion in any non-dispersive medium [3], [4]. The optimal weighting factor can be calculated for any material parameters and cell size by the least-square-method (LMS) [4], which also can be applied to the dispersive case. Therefore, the scaling factors should be reformulated for a dispersive medium, which can be applied to 2-D and 3-D problems. In this correspondence, three typical dispersive media are considered. Two are second-order media, which are characterized by the Drude and Lorentz models, and the other is a first-order model known as a Debye medium. Several FDTD schemes have been proposed to simulate dispersive media, including the auxiliary differential equation (ADE), the bilinear transform (BT), the Z-transform (ZT), and the piecewise linear recursive convolution (PLRC) methods [5]. Among these methods, the ADE method shows good accuracy over low and high frequencies [5]. In this correspondence, use of the ID-FDTD scheme in conjunction with the ADE method is proposed for three dispersive media. In Section II, the scaling factors are formulated. In Section III, the proposed scheme is verified for two scattering problems. II. FORMULATION A. ID-FDTD Scheme Based on ADE Method The relative permittivity of a dispersive medium is a function of frequency, such as "r (!). As aforementioned, three dispersive media are considered: Debye, Drude, and Lorentz media. The Debye medium has a dielectric constant expression, given as "r (! ) = " + (1"p )=(1 + j!0 ), where 1"p = "s 0 " , and "s and " are the static permittivity and the relative permittivity at infinite frequency, respectively. 0 and ! are the relaxation time and the angular frequency, respectively. For the Drude medium, "r (! ) is given by 1 0 (!p2 )=(! 2 0 j!c ), where !p and

1 1

1

Manuscript received December 21, 2009; revised July 05, 2010; accepted July 22, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. This work was supported by an INHA University Research Grant. The authors are with the Graduate School of Information Technology & Telecommunications, Inha University, Incheon 402-751, Korea (e-mail: [email protected],). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2103017 0018-926X/$26.00 © 2010 IEEE

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c are the plasma frequency and the collision frequency, respectively. The dielectric constant of the Lorentz medium is written as "r (!) = " + (1"p !p2 )=(!p2 + j 2!p 0 ! 2 ), where !p and p are the resonance frequency and the damping coefficient of the medium, respectively. ~ is proportional In the frequency domain, the electric flux density D ~ ~ ~ to E , given as D(! ) = "0 "r (! )E (! ). This can then be transformed in the time domain by using the inverse Fourier transform relations @ 2 =@t2 as follows: @=@t and ! 2 j!

1

$0

$

a1

@2 ~ @ ~ ~ (t) (t) + a3 D D(t) + a2 D @t2 @t @2 ~ @ ~ ~ (t) (1) = a4 2 E E (t) + a6 E (t) + a5 @t @t

where aj is a constant dependent on the medium type [5]. (1) can be discretized based on the central difference equations for @=@t and @ 2 =@t2 and the approximation of F n+(1=2) = (F n+1 + F n )=2 for the first-order media. Therefore, the ADE equations for the three media are explicitly expressed as

~ n + Cb E ~ n01 + Cc D ~ n + Ce D ~ n01 (2) ~ n+1 + Cd D ~ n+1 = Ca E E

j

where n is the time step. The coefficients in (2) are given for each dispersive medium. First, for the Debye medium

Ca = BDe1 "01 Cd = (BDe2

0B

0

De2 "s ; Cc = (BDe1 +

0B

De1 )="0 ; Cb =

BDe2 )="0

Ce = 0

(3)

0 + "s0 1t; where BDe1 = 20 =ADe ; BDe2 = 1t=ADe ; ADe = 20 "1 0 = "1 sc" . "0 and 1t are the free-space "s0 = "s sc" ; and "1 permittivity and the FDTD time step, respectively. Second, for the Drude medium

1

1

Ca = 2BDu1

0B

Cb =

0B

Ce = BDu1

0B

Du3 ;

Cc = BDu1 + BDu2 ; Cd =

0B

2 Du1 ;

Du1 +

(4)

where BDu1 = 2=ADu ; BDu2 = c 1t=ADu ; BDu3 02 2 0 2! p 1t =ADu ; ADu = 2 + c 1t, and !p = !p sc! . Finally, for the Lorentz medium

1

0 BLo1 0 "0 BLo3 ; Ca = 2"1 s Cc = BLo2 + BLo1 ;

Cd = BLo3

0

2BLo1 ;

0B

=

0

0 ; J 0H

n+1=2

(7)

0

n+1=2

Hy I + 12 I 12 ; J y n+1=2 n +1 = 2 Hx I; J + 12 + Hx I; J 12 n+1=2 n+1=2 1 1 Hy I + 2 ; J +1 Hy I 2; J + 1 n+1=2 n+1=2 +Hy I + 12 ; J 1 Hy I 12 ; J 1 (8) n+1=2 n+1=2 1 Hx I +1; J + 2 + Hx I +1; J 12 n+1=2 n+1=2 Hx I 1; J + 12 + Hx I 1; J 12

0

Dzn (I; J )+ C1

0 0

+ C2

0 0 0

0

0

0 0 0 0 0 0 0

where C1 = (1 = 0:251t=1, and 0:5)1t=1; C2 C3 = ((1 0:5)1t)=(0 r 1); C4 = (0:251t)=(0 r 1) for the Debye and the Lorentz media, and C3 = 0 0 ((1 0:5)1t)=(0 r 1); C4 = (0:251t)=(0 r 1) for the Drude medium with 0r = r sc and sc defined in Subsection C. Here, 0 and r are the free-space and the relative permeability, respectively. 1 and  are the cell size and the weighting factor, respectively.

0

0

1

B. Dispersion Relation Based on the standard procedure, the dispersion relations for the three dispersive media can be obtained as follows: For the Debye medium

0 + "1

0 "10

0

1t("s

) t 1t + j 20 tan !1 2

1

2

! 1t

r sin

1+

2

s2

! 0 p 1t2 jvc 1t sin(! 1t) 4 sin2 2

1

0 + "1

where BLo1 = 1=ALo ; BLo2 = p 1t=ALo ; BLo3 = 0 (p 1t + 1); "s0 = "s sc"0 , and !p2 1t2 =ALo ; ALo = "1 0 = "1 sc"0 . In (3), (4), and (5), the scaling factors of "1 sc! ; sc" ; sc" ; sc"0 and sc"0 are formulated in Sub-section C. The ADE equations can be used in conjunction with the known 2-D or 3-D ID-FDTD update equation [1], [2]. In this correspondence, for the sake of simplicity, the update equation of the ID-FDTD scheme for a 2-D TM case is explicitly given as

1

1

Dzn+1 (I; J )

0 0 0

2

0

=

SD(kx ; ky ; kz ; 1) (9)

! 1t

2

t 0r sin2 !1 2 2 s

For the Lorentz medium

(5)

Lo2

Ezn (I +1; J ) Ezn (I; J ) 2 n n Ez (I +1; J +1) Ez (I; J +1) + C4 n n +Ez (I +1; J 1) Ez (I; J 1) 1

I + ; J + C3

=

0 BLo2 0 "0 BLo1 ; Cb = "1 1

Ce = BLo1

2

n+1=2 = Hy

where s = c1t=1 is the Courant number. For the Drude medium

BDu2 ;

Du2

1

Hyn+1=2 I + ; J

=

SD(kx ; ky ; kz ; 1): (10)

0 1

0 "1 0 )!p21t2 ("s 2 2 !p 1t + j 2p 1t sin(! 1t) 4 sin2

0

! 1t

2

2 SD k ; k ; k ; : (11) Since SD 1; 1; 1; 1 in (9), (10), and (11) depends on the spatial dift r sin2 !1 2 2 s

(

=

( x

y

z 1)

)

ferentiations, the term is independent of the ADE equation, which is expressed for 2-D and 3-D cases [1], [2]: For 2D case, SD(kx ; ky ; 1) = Kx2 + Ky2 , where Su = 2  Qu ). sin ku 1=2; Qu = Sv , and Ku = Su (1 For 3D case, SD(kx ; ky ; kz ; 1) = Kx2 + Ky2 + Kz2 , where Pu = 2 2 ; Qu = Sv2 + Sw , and Ku = Su [(Pu Qu ) Qu =2 + 1]. Sv2 Sw u is x; y or z .

0 1

0

0

C. Scaling Factors

Hxn+1=2 =

I; J +

1 2

0C E0EI; JI; J I ;J 0E I ; J I0 ; J 0E I 0 ; J

Hxn+1=2 I; J +

0C

4

1

3

2

Ezn ( +1 Ezn ( 1

+1)

+1)

n z(

n z(

+1) )

n )+ z ( +1 n 1 ) z(

(6)

Scaling factors are used to adjust the numerical phase velocity to achieve the exact velocity [3], [4]. Since each dispersive medium has different material parameters, different scaling factors should be introduced to the three media, as delineated in (3)–(5). The scaling factors are calculated at ' = 0o (2D case) or at ' = 0o & = 0o (3D case), so that identical SD( ; ; ; ) values can be used to compute the scaling

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Fig. 1. Comparison of the normalized maximum dispersion error for three dispersive media as a function of frequency. All scaling factors are calculated at 50 GHz. (a) Error for real part, (b) error for imaginary part.

factors for the 2D and 3D cases [2]. Also, the left-terms of (9)–(11) are identical for the 2D and 3D cases. Therefore, the proposed scaling factors can be applied to both the 2-D and the 3-D problems. The scaling factors are mathematically formulated by balancing (9)–(11) ~ = kexact [3]. Here, k~ and kexact are the numerical and the with k exact phase velocities, respectively. kexact is chosen as the velocity at the frequency where the dispersion error is minimized. The scaling factors for each dispersive media can be formulated as follows: For the Debye medium, see (12)–(13) at the bottom of the page, where Re[z ] and Im[z ] are the real and imaginary parts of z , respectively. For the Drude medium, see (14)–(15) at the bottom of

sc" = sc"

=

sc = sc! =

s2 Re sin2

k

s2 Re sin2

k

s2 Re sin2

k

2

1

Fig. 2. Debye medium sphere scattering problem. (a) and (b) are plots of the scattered field at 100 GHz. (a) Real part, (b) imaginary part, (c) total field in time domain.

the page. For the Lorentz medium, see (16)-(17) at the bottom of the next page.

1t 0 Im sin2 k t r "s 1t sin2 !1 2

2

1

20 tan

t 20 tan !1 + Im sin2 2 ! 1t t 2r "1 0 tan 2 sin2 !1 2 2

2

1

1

vc 1t 0 Im sin2 t r vc 1t sin2 !1 2

t Im 2s2 vc2 1t2 + 4 tan2 !1 2 2 sc r !p vc 1t3 tan

0 sin2 ! 1t 2

k

2

k

1

2

k

2

:

2

1

2 tan 1

! 1t

1t

(12) (13)

! 1t 2

(14) (15)

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= 1:726 1010 , and for the Lorentz medium, "s = 3; "1 = 1:5; r = 1; !p = 2 20 109 , and p = 0:1$p [7]. The scaling and weighting factors are calculated at 50 GHz. The dielectric constants for each media at 50 GHz are given as "r = 15:225 24:748j; "r = 0:99997 0:000003j, and "r = 3:004 0:015j for the Debye, Drude and !p

2

2

2

0

0

0

and Lorentz media, respectively. Fig. 1 shows that, as expected, the larger the dielectric constant is, the higher the error that is obtained. However, the proposed scheme can reduce the dispersion error signifo o  icantly. Since the maximum error is not happened at ' (3D case), the minimum value of the maximum error is not exactly at 50 GHz. However, the offset is not large and at 50 GHz, the error is still very small.

=0 & =0

III. NUMERICAL RESULTS To verify the proposed ID-FDTD scheme, we consider the circular dispersive medium sphere and cylinder scattering problems, which are characterized by the Debye and Lorentz. For the Drude case, however, the accuracy of the proposed scheme can numerically be observed the best among the three media, which can be expected based on Fig. 1. Due to the page limit of the correspondence, we don’t include the Drude media case. A TM wave is incident on the sphere or cylinder, whose center frequency and 0 dB bandwidth are 100 GHz and 10 GHz, respectively. The radius of the sphere or cylinder is 0 . Here, 0 is the wavelength in free space. For all simulations, the scaling and the weighting factors are calculated at the center frequency. Identical parameters of each media for Fig. 1 are used for the scattering problems. First, a pure water sphere scattering (Debye medium) is calculated. The CPW and the Courant number are assumed as 15 and 0.7, respectively. Fig. 2 presents a comparison of the results calculated by the Yee [8], the proposed ID-FDTD schemes and the exact eigen-series [9] in the frequency and time domains. The observation line is shown in Fig. 2. The frequency response is calculated at the center frequency. Next, the Lorentz medium is considered. For this simulation, the CPW is reduced to 7. Fig. 3 shows the considered problem geometry, and the scattered fields in the frequency and time domains. As seen in Figs. 2 and 3, the accuracy of the proposed ID-FDTD scheme is superior to that of the Yee scheme. Even for a small CPW, the accuracy of the ID-FDTD scheme can be maintained at a very high level.

35

5

IV. CONCLUSION Fig. 3. Lorentz medium cylinder scattering problem. (a) and (b) are the plots of the scattered field at 100 GHz. (a) Real part, (b) imaginary part, (c) total field in time domain.

The dispersion error for the three dispersive media is calculated for a 3-D case. For this simulation, we assume the CPW (cells per wavelength) is 30. Fig. 1 presents a comparison of the maximum normalized dispersion error, ;' j 0 k=kexact j, as a function of frequency for the Yee method and the proposed scheme. The material parameters are : ;" : ; r ; assumed: for the Debye medium, "s 010 , and 0 : 2 012 [6], for the Drude medium, vc 2

1 ~

max

0

= 78 2 1 = 5 5

= 8 1 10

=1 = = 3 10

In this correspondence, use of a ID-FDTD scheme based on the ADE method is proposed for three dispersive materials, including Debye, Drude, and Lorentz media. For each medium, the scaling factors are formulated, and they can be used for 2-D and 3-D problems. It is numerically shown that the dispersion error is reduced significantly as for the non-dispersive medium. Based on 3-D and 2-D scattering problems, it is shown that the proposed ID-FDTD scheme can provide very accurate results even for small CPW in time and frequency domains. ACKNOWLEDGMENT The authors thank Dr. W.-T. Kim for simulating the 3D Debye sphere scattering.

Re sin2 k 2 1 2p 1t sin(!1t) 2 k 1 [!p2 1t2 4sin2 !12 t ] 2 = +Im 2sin t r "1 p 1t sin(! 1t)sin2 !1 2 2 k 2 3 1 Re sin 2 p !p 1t s2 2 k 1 tan !12 t [!p2 1t2 4sin2 = +Im sin 2 t r "s p !p2 1t3 sin2 !1 2 1

sc0

"

sc"0

0

(16)

1

1

0

! 1t 2

0

4p2 1t2 cos

! 1t 2

]

:

(17)

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REFERENCES [1] I. Koh, H. Kim, J. Lee, J. Yook, and C. Pil, “Novel explicit 2-D FDTD scheme with isotropic dispersion and enhanced stability,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3505–3510, Nov. 2006. [2] W. Kim, I. Koh, and J. Yook, “3D isotropic dispersion (ID)-FDTD algorithm: Update equation and characteristics analysis,” IEEE Trans. Antennas Propag., to be published. [3] I. Koh, H. Kim, J. Lee, and J. Yook, “New scaling factor of 2-D isotropic-dispersion finite difference time domain (ID-FDTD) algorithm for lossy media,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 613–617, Feb. 2008. [4] Z. Meng and I. Koh, “New weighting factor of 2D isotropic-dispersion finite difference time domain (ID-FDTD) algorithm,” J. Korea Electromagn. Eng. Sci., vol. 8, no. 4, pp. 139–143, Dec. 2008. [5] Z. Lin and L. Thylén, “On the accuracy and stability of several widely used FDTD approaches for modeling Lorentz dielectrics,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3378–3381, Oct. 2009. [6] J. P. Franzen, “Wideband pulse propagation in linear dispersive biodielectrics using Fourier transforms,” Feb. 1999, U. S. Air Force Res. Lab.. [7] D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propag., vol. 44, no. 6, pp. 792–797, Jun 1996. [8] R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett., vol. 16, no. 18, pp. 1412–1414, Sep. 15, 1991. [9] N. Morita, N. Kumagai, and J. R. Mautz, Integral Equation Methods For Electromagnetics. Boston: Artech House, 1990.

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Optimal Design of a Highly Compact Low-Cost and Strongly Coupled 4 Element Array for WLAN Zhongkun Ma, Vladimir Volski, and Guy A. E. Vandenbosch

Abstract—A 4 element array of microstrip E shaped patches is designed for use in wireless local area network (WLAN) applications. The operating frequency range is from 3.4 to 3.8 GHz and the gain of the array is more than 13 dB. The main beam direction is normal to the ground plane. Considering the specifications given, the focus is on the reduction of occupied space. This implies the use of an integrated feeding technique in combination with highly coupled array elements. Although the concept was published before, no optimizers were used and the previous design did not meet the 10 dB matching criterion. It is clearly proven that, due to its complexity, this innovating topology can be fully taken advantage of only by designing it using efficient evolutionary optimizers. Both a particle swarm optimization (PSO) and a genetic algorithm (GA) technique are used. During the optimization process, the fitness is evaluated by using a full wave solver based on the method of moments (MOM). It is shown that for these types of structures, a single step optimization procedure is preferred, and PSO outperforms GA. CST EM Studio is used for validation purposes before manufacturing. A prototype array is fabricated and measured. Index Terms—Antenna array, evolutionary algorithm, genetic algorithm (GA), optimization methods, particle swarm optimization (PSO), wireless local area network (WLAN).

I. INTRODUCTION During the last decade, a rapid development of WLAN has been observed. Dedicated antennas are in urgent demand for this type of wireless application. A low-cost microstrip antenna array may seem a very promising candidate. A standard array design typically includes the consecutive design of a feeding network and array elements [1]–[3]. However, in classical designs the feeding network suffers from relatively large dissipation losses and the total size is relatively large. The size of the feeding network can be reduced considerably with in-series feeding networks. Unfortunately, then the main beam direction becomes frequency dependent. A combination of two in-series networks is capable to solve partially this problem [4], [5]. In order to function correctly the two in-series networks should be fed with 180 degrees phase difference, resulting in an extra piece of transmission line, and thus increasing the array size. In this communication, an innovative 4 element antenna array with closely spaced elements is revisited [6]. The presented topology uses radiating elements that simultaneously are also part of the feeding structure. In this way a very compact size is obtained. Due to the high and irregular mutual coupling between the elements, an advanced optimizer is absolutely needed in the design. The lack of such an optimizer was the main reason why the prototype in [6] is inferior, even though it required a huge investment of engineering time. Several optimization algorithms are considered. Genetic algorithm (GA) [7], [8], and PSO [9]–[11] are two well-known global optimization algorithms. In [12], a comparison of PSO and GA for a low dimensional problem involving a horn antenna is given (five parameters to optimize). PSO outperforms GA in [12] due to its effectiveness and easy implementation. In this communication, we compare the performance Manuscript received March 23, 2010; revised July 06, 2010; accepted October 13, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Katholieke Universiteit Leuven, ESAT, Heverlee B-3001, Belgium (e-mail: [email protected]) Digital Object Identifier 10.1109/TAP.2010.2103029

0018-926X/$26.00 © 2010 British Crown Copyright

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Fig. 1. Antenna array prototype top view.

of PSO and GA for the specific dedicated topology under consideration. This constitutes a discrete high dimensional antenna optimization problem. The optimization goal is to achieve a return loss less than 010 dB in the operating frequency range from 3.4 to 3.8 GHz, and inside this band the gain has to be larger than 13 dB. This gain requirement comes from the physical situation in which the antenna is going to be used. The purpose is to cover an angular sector up to a distance of a few kilometers. The array is fed by a 50 Ohm coaxial cable at the back side. II. THE DESIGN PROCESS A. Type of Topology The proposed topology is shown in Fig. 1. E shaped patches are selected because of their well-known wide band behavior [13]. Considering the gain of a single E shaped element, and the well-known fact that doubling the number of elements to the average rises the gain by 3 dB, the required gain implies the use of at least 4 elements. Three different types of element are used. Introducing appropriate terminology, it is proposed to name elements A and D “closing elements”. They are only radiating. They are identical, but rotated over 180 degrees. Element C is named a “tapping element”. It both taps energy, which is radiated, and transports energy to the closing element D. Element B is entitled a “feed-split” element. It contains the primary feed, takes care of some of the radiation, and further distributes the rest of the power to the other elements. This conceptual topology was a result of the design process described in [6], where a divide-and-conquer design strategy was employed. However, in [6] the three different types of element were manually designed separately for wideband behavior and connected together afterwards. The design criteria were the relative ratios of the radiated power and power forwarded to the next element, compared to the incident power. The lengths of the connections between the different patches were adjusted to achieve the required gain. For element A, this required a longer distance in order to achieve the correct phase. Without it, the gain drops considerably. It was found that, manually, it was extremely difficult, read impossible, to achieve both the 010 dB bandwidth and gain requirements simultaneously. The matching of the resulting structure in [6] was actually in the order of only 06 dB. In this communication, the feasibility of the new topology is clearly proven and demonstrated, taking into account all design criteria. For cost reasons, the array has to be produced on top of a standard FR4 substrate (relative permittivity " = 4:7, thickness 0.8 mm,

tan  = 0:014). A single-layered structure with FR4 is by far not able to deliver the necessary bandwidth. Therefore, the thickness is increased by separating the FR4 substrate from the ground plane by 4.25 mm high spacers. This height is just sufficient in order to deliver the specified bandwidth. This was checked based on common rectangular patch designs. The medium between substrate and ground plane is air. The four patches are connected in series by 4 mm wide microstrip lines. The entire structure is symmetrical with respect to the horizontal axis. The main excitation is a 50 Ohm coaxial feed connected to patch B. Although the new topology is expected to reduce feeding losses, considering the fact that the array is quite small, this effect is minor and beyond the scope of this communication. B. EM Solver Selection The evaluation of the cost function is performed using a full wave solver based on the moment of methods, designed for quasi-3D multilayered antenna structures [14], [15]. During the optimization it was assumed that the ground plane and the substrate have infinite dimensions in order to reduce the total number of unknowns. Following the advice given in [16], after the entire optimization, the candidate prototype structure was validated by another EM solver, based on another solution method: CST EM Studio 2010. C. Cost Function Five equidistant frequency points, i.e. 3.4 GHz, 3.5 GHz, 3.6 GHz, 3.7 GHz, and 3.8 GHz are considered. The S11 values at these frequencies are denoted as [X1 . . . X5 ] (in dB), and the gain values as [Y1 . . . Y5 ] (in dB), respectively. In order to avoid the effect of deep peaks in the return losses, the X values are truncated before addition. The thresholds are 010.5 dB, 012 dB, 015 dB, 012 dB, and 010.5 dB, respectively. The gain threshold is just the required gain of 13 dB. In the single step procedure (see further), the fitness function is a linear combination of these output parameters, defined by F itness Ni Mi

= =

5

Ni i=1

Xi XiT

0

5

Mi i=1

Xi > XiT Xi XiT

 13) = Y13 ((YY <  13): i

i i

(1)

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The minimum possible value of this fitness function is 0125 if all threshold values are achieved. When reaching this value, the used optimizers stop immediately. A disadvantage of this approach is that if several candidates reach the minimum value, the optimizer can’t decide what the best one is. A manual intervention is required.

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TABLE I THE TESTED OPTIMIZERS

D. Optimization 1) Multi Step PSO: First, a divide-and-conquer strategy, based on a three-step PSO optimization with small swarm size, was employed to optimize the array. The coaxial feed is connected to patch B at 5 mm from its right edge. This point has been chosen based on the experience of the design in [6], both regarding the matching of the basic single element, and the array. In the first step, the feed-split element B was optimized for a proper return loss at 3.6 GHz, without the other three elements connected. For this single frequency, the threshold was set at the value of 020 dB. Several experiments showed that this value performs best, since the S11 of element B should not be either too small, because then it behaves as a narrowband element, or too large, because then a considerable amount of energy reflects back. Five B-patch parameters [B1 to B5] were considered. After the optimization of element B element C was connected. The optimization procedure aimed at wideband behavior. It concerned the five equidistant frequency points, and involved all the six C-patch parameters [C1 to C6]. Only after achieving a 400 MHz bandwidth, the third step of the optimization was started. In case the 400 MHz bandwidth was impossible to reach, the first two steps of the optimization procedure were repeated. During the last step, both the gain and bandwidth requirements were taken into account. The pre-mentioned fitness function (1) was employed. The seven parameters of the patches A and D [A1 to A6, and D1] were optimized. In case the required fitness 0125 was not reached, the procedure turned back to the first optimization step and repeated the full three-step optimization procedure. This was repeated until the required fitness 0125 was reached, or when the number of iterations reached 10. Since the global optimization procedure was subdivided into three simpler optimization steps, only a small PSO swarm size of 20 was needed. The maximal iteration number was set to 50 in each optimization step. All other PSO settings were identical as in the single-step PSO optimization (see next section). 5 trials were executed. It has to be noted that the EM evaluation time was much shorter for the first step compared to the third one, because only one array element had to be simulated in the former case compared to four elements in the latter case. 2) Single Step PSO and GA: In the single step procedure, the global structure is controlled by 19 parameters. All of them can vary with a resolution step of 2 mm except the probe position, which can vary with a resolution of 0.5 mm, see Fig. 1. Note thus that also the probe position is optimized. For PSO, both swarm size 60 and 100 were employed. These swarm sizes are relatively large because of the large number of parameters (19) to optimize. The maximum iteration numbers were 50 and 30 for swarm size 60 and 100, respectively. The gbest network topology was chosen for fast convergence and the damping boundary condition was employed for robust performance [17], [18]. Both the cognitive rate and social rate [11] were identical and equal to 1.49. The inertia weight linearly decreased from 0.9 to 0.4 per iteration throughout the optimization [19]. For GA, the process started with the same initial populations as for PSO. Two groups of GA parameters were tested, 1) crossover rate 0.8 and mutation rate 0.05, and 2) crossover rate 0.6 and mutation rate 0.02, both with single point crossover for 1000 iterations. Uniform mutation was employed. In all cases, only a uniform random number generator was used. 5 trials were executed. A piece of memory was employed to record all evaluated antenna configurations and the corresponding fitness values.

TABLE II THE AVERAGE BEST FITNESS AND AVERAGE NUMBER OF EM EVALUATIONS OVER 5 TRIALS

For already calculated cases the EM evaluation is skipped and the fitness is directly read out from the memory. This reduces considerably the number of calls to the EM solver for all the tested optimizers. The memory can also be utilized to improve PSO learning capabilities by exploiting the “history” of the optimization for speeding up the convergence to the optimal solution and the adaptability of the control to the time-varying conditions [20], [21]. This technique was not employed. The memory was only employed to avoid duplicated EM evaluations. The tested optimizers are given in Table I. In all PSO and GA experiments the optimizers stopped when either the minimum value 0125 or the maximum iteration number was reached. III. RESULTS All results are summarized in Table II. For the three-step optimization, they concern the average best fitness in the third step, its standard deviation (STD), and the average number of EM evaluations for each step and their STD. For the single step procedures, they concern the average best fitness and the average number of EM evaluations, together with their standard deviations (STD). The averages are taken over the 5 trials. Considering the resulting fitness function value, single step PSO is superior, then multi-step PSO follows, and GA is clearly inferior. The two tested single step PSOs (with different population size) successfully produced the minimal fitness 0125 four times. However, it was found by none of the twenty GA trials. GA did not perform very well in this experiment. The major attractive idea of the three-step PSO procedure would be to accelerate the whole optimization process by employing computationally cheap EM evaluations (in the first and second

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TABLE III DIMENSIONS OF THE PROTOTYPE

Fig. 4. Broadside gain of the prototype structure simulated and measured.

Fig. 2. Photo of the prototype.

Fig. 5. Radiation pattern of the prototype simulated and measured in E-plane at 3.6 GHz.

Fig. 3. S11 of the prototype simulated and measured.

step), in this way reducing the number of computationally expensive ones. In the third step, the computational cost is identical to the ones of the one-step PSO and GA optimizations. However, unexpectedly, this was not observed. Step 3 alone already needs more EM evaluations then in the case of the single step PSO procedures. The performance of the three step procedure is compromised by the fact that the structures resulting from the first two steps to the average were not the best candidates towards the final goal. Attempts to determine the goals of the first two steps more properly proved to be problematic. In our experience, this conceptual topology is a structure which really requires the use of a global single step evolutionary optimization algorithm. Note also that keeping the probe position fixed at 5 mm from the right edge of patch B during the PSO optimization reduced the necessary number of EM evaluations by a few hundred. The resulting average best fitness is then only slightly worse. A candidate prototype structure was selected. Its dimensions are listed in Table III, according to the definitions given in Fig. 1. The probe

position was 5 mm from the right edge of patch B. After re-simulating it for validation purposes with finite ground plane in CST EM Studio, the topology was prototyped. The prototype is shown in Fig. 2. The return loss for the prototype, calculated with the MOM code, calculated with CST EM Studio, and measured, is plotted in Fig. 3. The results obtained with CST EM Studio show a minor frequency shift in comparison with the measurements and the MOM solver. Considering the results of the extensive benchmarking study of [16], this small shift is acceptable. The measurements show an S11 of 09 dB at 3.4 GHz and of 011 dB at 3.85 GHz. These results are sufficient to clearly validate this conceptual topology and the design approach. The simulated and measured gains of the prototype are plotted in Fig. 4. Both solvers produce almost the same results. The measured gain shows a 1 dB difference with the simulated gains, which is acceptable. The radiation patterns in E-plane at 3.6 GHz, simulated with the MOM solver and measured, are shown in Fig. 5. IV. CONCLUSION The concept of an innovative, compact, strongly coupled 4 element antenna array is proven. Several optimizers have been used in the design. It is clearly shown that for this topology a single step PSO opti-

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mizer is superior. The design is a very good example of a combination of physical insight and clever optimization procedures. It is shown that powerful optimization techniques are really necessary in complex antenna design. The final theoretical structure met all requirements and was very compact. Its prototype showed a little frequency shift.

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[20] M. Donelli, R. Azaro, F. G. B. De Natale, and A. Massa, “An innovative computational approach based on a particle swarm strategy for adaptive phased-arrays control,” IEEE Trans. Antennas Propag., vol. 54, no. 3, pp. 888–898, Mar. 2006. [21] M. Benedetti, R. Azaro, and A. Massa, “Memory enhanced PSO-based optimization approach for smart antennas control in complex interference scenarios,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 1939–1947, 2008.

REFERENCES [1] M. Secmen, S. Demir, L. Alatan, O. A. Civi, and A. Hizal, “A compact corporate probe fed antenna array,” in Proc. 1st Eur. Conf. on Antennas Propagation, 2006, pp. 1–4. [2] K. H. Lu and T.-N. Chang, “Circularly polarized array antenna with corporate-feed network and series-feed elements,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3288–3292, 2005. [3] C. G. Christodoulou, P. F. Wahid, M. R. Mahbub, and M. C. Bailey, “Design of a minimum-loss series-fed foldable microstrip,” IEEE Trans. Antennas Propag., vol. 48, no. 8, pp. 1264–1267, 2000. [4] K. Wincza, S. Gruszczynski, and J. Borgosz, “Microstrip antenna array with series-fed “through-element” coupled patches,” Electron. Lett., vol. 43, no. 9, Apr. 26, 2007. [5] W. Shen, J. She, and Z. Feng, “A compact high performance linear series-fed printed circuit antenna array,” presented at the Microwave Millimeter Wave Technology Conf., Apr. 18–21, 2007. [6] V. Volski, P. Delmotte, and G. A. E. Vandenbosch, “Compact lowcost 4 elements microstrip antenna array for WLAN,” in Proc. 7th Eur. Conf. Wireless Technology, Amsterdam, The Netherlands, 2004, pp. 277–280. [7] S. Kahng and J. Kim, “The s-band quadrafilar antenna on a satellite with the little back radiation and flat forward pattern designed by GA optimization,” in Proc. IEEE Antennas Propagation Soc. Int. Symp., 2009, pp. 1–4, IEEE Conferences. [8] C.-H. Chen, S.-H. Liao, M.-H. Ho, C.-C. Chiu, and K.-C. Chen, “A novel indoor UWB antenna array design by GA,” in Proc. Future Comput. Communication Conf., 2009, pp. 291–295. [9] H. Wu, J. Geng, R. Jin, J. Qiu, W. Liu, J. Chen, and S. Liu, “An improved comprehensive learning particle swarm optimization and its application to the semiautomatic design of antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pt. 2, pp. 3018–3028, Oct. 2009. [10] P. Demarcke, H. Rogier, R. Goossens, and P. De Jaeger, “Beam forming in the presence of mutual coupling based on constrained particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1655–1666, Jun. 2009. [11] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 397–407, Feb. 2004. [12] J. Robinson, S. Sinton, and Y. Rahmat-Samii, “Particle swarm, genetic algorithm, and their hybrids: Optimization of a profiled corrugated horn antenna,” in Proc. IEEE Antennas Propagation Soc. Int. Symp., San Antonio, TX, 2002, vol. 1, pp. 314–317. [13] Y. Ge, K. P. Essele, and T. S. Bird, “E-shaped patch antennas for highspeed wireless networks,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3213–3219, Dec. 2004. [14] G. A. E. Vandenbosch and A. Van de Capelle, “Mixed-potential integral expression formulation of the electric field in a stratified dielectric medium-application to the case of a probe current source,” IEEE Trans. Antennas Propag., vol. 40, no. 7, pp. 806–817, Jul. 1992. [15] M. Vrancken and G. A. E. Vandenbosch, “Semantics of dyadic and mixed potential field representation for 3D current distributions in planar stratified media,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2778–2787, Oct. 2003. [16] A. Vasylchenko, Y. Schols, W. De Raedt, and G. A. E. Vandenbosch, “Quality assessment of computational techniques and software tools for planar-antenna analysis,” IEEE Antennas Propag. Mag., vol. 51, no. 1, pp. 23–38, Feb. 2009. [17] S. Ghosh, D. Kundu, K. Suresh, S. Das, A. Abraham, B. K. Panigrahi, and V. Snasel, “On some properties of the best topology in particle swarm optimization,” in Proc. 5th Int. Conf. on Hybrid Intelligent Systems, Aug. 12–14, 2009, vol. 3, pp. 370–375. [18] S. Xu and Y. Rahmat-Samii, “Boundary conditions in particle swarm optimization revisited,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pt. 1, pp. 760–765, Mar. 2007. [19] Y. Shi and R. Eberhart, “A modified particle swarm optimizer,” in Proc. IEEE World Congr. on Computational Intell. Evol. Comput., pp. 69–73.

Inverted-F Laptop Antenna With Enhanced Bandwidth for Wi-Fi/WiMAX Applications Lev Pazin and Yehuda Leviatan

Abstract—A novel printed multiband inverted-F antenna (IFA) for laptop computers is presented. The antenna has a simple structure and it is sufficiently narrow to easily fit on each side of the housing of the display unit of the laptop. The antenna is designed to operate in all the allocated Wi-Fi and WiMAX frequency bands while providing near omnidirectional coverage in the horizontal plane. The multiband performance of the proposed antenna and its omnidirectionality are validated by measurements. Index Terms—Inverted-F antenna (IFA), IFA with slotted monopole, multiband antenna, printed IFA.

I. INTRODUCTION Various designs of inverted-F antennas (IFAs), fabricated on a thin substrate using printing technology, have been recently proposed to facilitate the Wi-Fi/WiMAX operation required by modern laptops [1]–[15]. The IFAs proposed in [2]–[8] have a simple structure and narrow ground plane, and they are small enough to fit on each side of the housing of the display (see Fig. 7 in [8]). However, even the more advanced versions of these IFAs, namely, the branched IFAs [4], [5], the IFAs with the parasitic resonant element [6], [7], and the wideband IFA [8], fall way short of covering the eleven standardized Wi-Fi and WiMAX operating bands listed in Table I. With a view toward covering a greater number of operating bands, printed IFAs of more complex structures and a much larger ground plane, including IFAs with additional branches or parasitic resonant elements [9]–[12] and IFAs with a coupled feed [13]–[15], have been proposed. These latter IFAs were designed to be mounted on the top of the display unit of the laptop, and hence their driven monopole had to protrude beyond the upper side of the display unit (see Fig. 6 in [12]), thereby requiring the housing of the display unit to be larger. Moreover, none of these IFAs appears to cover the 2.3 GHz WiMAX operating band. In this communication, we explore the possibility of covering all the standardized Wi-Fi and WiMAX frequency bands while clinging to the class of simply-structured and compact IFAs. The idea is to start with the printed flat-plate IFA described in [8], referred to hereafter as the prototype IFA, and enhance its bandwidth by cutting two slots in the antenna driven monopole while only slightly tuning the dimensions of Manuscript received April 28, 2010; revised July 26, 2010; accepted July 26, 2010. Date of publication December 30, 2010; date of current version March 02, 2011. The authors are with the Department of Electrical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel (e-mail: leviatan@ ee.technion.ac.il). 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.2010.2103036

0018-926X/$26.00 © 2010 IEEE

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TABLE I OPERATING BANDS AND CORRESPONDING FREQUENCY RANGES FOR LAPTOP COMMUNICATIONS

Fig. 1. Geometry of the proposed multiband IFA. The feed-point terminals are denoted by A and B.

its other parts. The size of the proposed IFA with the slotted monopole is very similar to that of prototype IFA. Its width of 9.3 mm is exactly the same as that of the prototype IFA, and its length is even 1.5 mm shorter than that of prototype IFA. The proposed IFA was simulated using the commercial CST Microwave Studio software. It was also fabricated and its input impedance matching and radiation characteristics were measured. The results of the simulations and measurements were found to be in good agreement. Thanks to its multiband properties, the proposed antenna can operate effectively in the 2.4/5.2/5.5/5.8 GHz Wi-Fi frequency bands and in the 2.3/2.5/3.3/3.5/3.7/5.8 WiMAX frequency bands.

Fig. 2. Matching characteristics of the proposed multiband IFA: measured (solid line) and simulated (dashed line).

II. ANTENNA DESIGN As mentioned above, the starting point for the design of the proposed bandwidth-enhanced IFA was the IFA described in [8]. The geometry of this IFA was modified and its dimensions optimized using the CST Microwave Studio simulation tool. The geometry of the proposed IFA is illustrated in Fig. 1. Just like the IFA described in [8], the proposed IFA is assumed to be printed on one side of a hard TACONIC TLY-5-0620 no-ground dielectric substrate with relative permittivity "r = 2:2 and thickness of 1.6 mm. It includes the two coupled radiating structures featured in [8], one being the planar L-shaped (90 bent) shunt-fed driven monopole, the other being the open slot formed between the monopole and the ground plane. The ground plane is connected, as usual, to the short arm of the monopole. The initial dimensions of the radiating structures and the feed-point position, which govern the lower resonant frequencies of the antenna, were selected, as in [8], so that the antenna would resonate at frequencies close to 2.5 GHz and 3.5 GHz, thereby yielding a 2.3–3.7 GHz operational bandwidth, which covers the Wi-Fi 2.4 GHz and the mobile WiMAX 2.3 GHz, 2.5 GHz, 3.3 GHz, and 3.5 GHz frequency bands. To cover the relatively wide 5 GHz frequency band, without significantly affecting the behavior of the IFA at the lower frequencies, two resonant slot radiators of slightly different lengths, one being about a quarter free-space wavelength at 5.3 GHz and the other about a quarter free-space wavelength at 5.7 GHz, were made in the driven monopole’s long arm. In the subsequent stage of the design process, the dimensions of the four resonant structures were used as design variables and were adjusted to attain the desired antenna characteristics. For example, to effectively excite the additional slots, the width of the main slot as well as that of the ground plane had to be modified. Also, the main slot was divided into two sections of roughly equal lengths but different widths. The resulting dimensions of the antenna were found to be d1 = 1 mm, h = 15 mm, l = 46:5 mm, l1 = 15 mm, l2 = 29 mm, l3 = 9:3 mm,

=82 =37

=02 =28

=17 =28

=15

=1

l4 : mm, s1 : mm, s2 : mm, s3 : mm, t mm. w1 : mm, w2 : mm, w3 : mm, and w4 : mm, The entire antenna (driven monopole and ground plane) occupies an area of 9.3 46.5 mm2 , which, as mentioned earlier, is even slightly smaller in size than that occupied by the prototype IFA presented in [8], and similar to areas occupied by IFAs commonly used in laptops. The antenna is assumed to be fed by a 50 Ohm coaxial cable, with its central conductor connected to point A and its outer conductor soldered to the ground plane at point B, just across from point A.

=53

2

III. SIMULATION AND MEASUREMENT RESULTS A graph depicting the matching characteristics of the proposed multiband IFA, obtained based on the CST Microwave Studio simulation tool, is shown in Fig. 2. From the graph it can be easily noted that the proposed IFA has four resonant frequencies, the lower ones being approximately 2.4 GHz and 3.6 GHz, and the higher ones being approximately 5.3 GHz, and 5.7 GHz. These results clearly confirm the claim made in [8] that the lower resonant frequencies of the antenna mainly depend on the length of the slot between the monopole and ground plane and on the length of the monopole’s segment extending between the antenna feed and the monopole’s open end. They also demonstrate that the proposed IFA can be effectively excited at the two additional resonant frequencies, and that these higher resonant frequencies are predominantly governed by the lengths of the additional slots made in the monopole. To gain further understanding of the way each resonance is excited, we also examined the surface current distribution on the proposed antenna. Plots of the magnitude of the surface current density at the aforementioned resonant frequencies are shown in Fig. 3. Note that at each resonant frequency, the current density is indeed higher on the very

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Fig. 4. Photo of the proposed multiband IFA and its coaxial feed line.

Fig. 3. Surface current distributions on the proposed IFA at the resonant frequencies: (a) 2.4 GHz, (b) 3.6 GHz, (c) 5.3 GHz, (d) 5.7 GHz.

same part of the antenna whose dimensions were assumed to govern that particular frequency. As for the operating bands of the proposed IFA, the graph of the simulation results shown in Fig. 2 further predicts that the antenna input impedance is well matched (jS11 j  010 dB) to the coaxial feed line in three frequency bands: the first ranging from 2.30 to 2.71 GHz, the second from 3.16 to 4.00 GHz, and the third from 5.14 to 5.88 GHz. Clearly, these three bands cover all the required frequency bands listed in Table I. The proposed IFA was fabricated. A photo of the antenna and its coaxial feed line is shown in Fig. 4. The matching characteristics of the antenna were then measured with an Agilent N5230A network analyzer for the free-standing case, and the results are plotted, for comparison purposes, alongside the simulation results in Fig. 2. The measurement results show that the fabricated antenna has two operating bands, one ranging from 2.28 to 3.85 GHz and the other from 5.15 to 5.85 GHz. These operating bands are slightly different from the simulated ones, but they equally cover all the required frequency bands listed in Table I. This discrepancy between the simulation and measurement results is probably due to the influence of the coaxial cable that was directly connected to the antenna terminals A and B during the measurement process, but, as commonly practiced, was not accounted for in the simulations. Clearly, the proposed antenna can operate at all the 10 frequency bands listed in Table I. The radiation characteristics of the proposed IFA were measured in an anechoic chamber. The radiation patterns (in the x-y; x-z , and y -z planes) are plotted for three frequencies 2.4, 3.5, and 5.5 GHz in Fig. 5. The plots show graphs of

Fig. 5. Measured polar patterns of the proposed IFA for the x-z; y -z and x-y planes at 2.4, 3.5 and 5.5 GHz. Patterns shown are for the E component (solid lines) and E component (dashed lines).

TABLE II SIMULATED DIRECTIVITY AND EFFICIENCY OF THE PROPOSED ANTENNA

the E' and E components of the radiated field. From the graphs, it is readily seen that in the y -z plane (which coincides, when the laptop’s lid housing the IFA is open and in upright position, with the horizontal plane) the E' component (which is the vertical component of the IFA’s electric field) is nearly omnidirectional at all three frequencies. Simulation results for both the directivity and efficiency of the antenna, obtained using the commercial CST Microwave Studio software, are shown in Table II. IV. CONCLUSION In this communication, a flat-plate inverted-F laptop antenna with enhanced bandwidth has been presented. The proposed antenna has a

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simple structure and narrow ground plane, and it is small enough to fit on each side of the housing of the display of the laptop screen. The multiband performance of the proposed antenna and its omnidirectionality were validated by measurements in free-standing configuration. The measurement results showed that this novel antenna operates effectively in all the required Wi-Fi and WiMAX communication bands. They also demonstrated that in free-standing configuration the antenna provides a nearly omnidirectional coverage in the horizontal plane. The performance of the antenna, when it is embedded inside the laptop, will be verified in the future. ACKNOWLEDGMENT The authors would like to thank M. Namer and Y. Komarovsky of the Technion and R. Kastner and H. Kleinman of Tel-Aviv University for assistance with the measurements.

REFERENCES [1] D. Liu, B. P. Gaucher, E. B. Flint, T. W. Studwell, H. Usui, and T. J. Beukema, “Developing integrated antenna subsystems for laptop computer,” IBM J. Res. Dev., vol. 47, pp. 355–367, 2003. [2] D. Liu, E. Flint, and B. Gaucher, “Integrated laptop antennas—design and evaluation,” in IEEE AP-S Int. Symp. Dig., 2002, vol. 4, pp. 56–59. [3] C. M. Su and K. L. Wong, “Narrow flat-plate antenna for 2.4 GHz WLAN operation,” Electron. Lett., vol. 39, no. 4, pp. 344–345, Feb. 2003. [4] D. Liu and B. Gaucher, “A branched inverted-F antenna for dual band WLAN applications,” in IEEE AP-S Int. Symp. Dig., 2004, vol. 1, pp. 2623–2626.

[5] D. Liu, B. Gaucher, and T. Hildner, “A dual-band antenna for WLAN applications,” in Proc. IEEE Ant. Techn. Int. Workshop, 2005, pp. 201–204. [6] D. Liu, B. Gaucher, and E. Flint, “A new dual-band antenna for ISM application,” in Proc. IEEE 56th Veh. Technol. Conf., Vancouver, Canada, 2002, vol. 2, pp. 937–940. [7] D. Liu and B. Gaucher, “A triband antenna for WLAN application,” in Proc. 2003 IEEE AP-S Int. Symp. USNC/URSI Nat. Radio Meeting, 2003, vol. 2, pp. 18–21. [8] L. Pazin, N. Telzhensky, and Y. Leviatan, “Wideband flat-plate inverted-F laptop antenna for WI-FI/WIMAX operation,” IET Microw Antennas Propag, vol. 2, no. 6, pp. 568–573, Sep. 2008. [9] C. M. Su, W. S. Chen, Y. T. Cheng, and K. L. Wong, “Shorted T-shaped monopole antenna for 2.4/5 GHz WLAN operation,” Microwave Opt Technol Lett, vol. 41, no. 3, pp. 202–203, May 5, 2004. [10] K. L. Wong, L. C. Chou, and C. M. Su, “Dual-band flat-plate antenna with a shorted parasitic element for laptop application,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 539–544, Jan. 2005. [11] K. L. Wong and L. C. Chou, “Internal composite monopole antenna for WLAN/WIMAX operation in a laptop computer,” Microw. Opt. Technol. Lett., vol. 48, pp. 868–871, 2006. [12] L. Pazin, N. Telzhensky, and Y. Leviatan, “Multi-band flat-plate inverted-F antenna for WI-FI/wimax operation,” IEEE Antennas Wirel. Propag. Lett., vol. 7, pp. 197–200, 2008. [13] J. Yeo, Y. J. Lee, and R. Mittra, “A novel dual-band WLAN antenna for notebook platform,” in Proc. 2004 IEEE Antennas Propag. Soc. Int. Symp. Dig., Monterey, CA, vol. 2, pp. 1439–1442. [14] S. J. Liao, K. L. Wong, and L. C. Chou, “Small-size uniplanar coupled-fed PIFA for 2.4/5.2/5.8 GHz WLAN operation in the laptop computer,” Microw. Opt. Technol. Lett., vol. 48, pp. 868–871, 2006. [15] C. T. Lee and K. L. Wong, “Uniplanar printed coupled-fed PIFA with a band-notching slit for WLAN/WIMAX operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1252–1258, Apr. 2009.

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