A New T-Matrix Formulation for Electromagnetic ... - IEEE Xplore

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 2, FEBRUARY 2013

A New T-Matrix Formulation for Electromagnetic Scattering by a Radially Multilayered Gyroelectric Sphere Lei Cao, Student Member, IEEE, Joshua Le-Wei Li, Fellow, IEEE, and Jun Hu, Senior Member, IEEE

Abstract—In this paper, a new solution for electrical characteristics of electromagnetic scattering by a radially multilayered gyroelectric sphere is proposed. The spherical geometry is divided into regions (where can be an arbitrary integer), and each layer is characterized by a scalar permeability and a gyrotropic permittivity tensor. Electromagnetic fields inside and outside the sphere are theoretically formulated based on the eigenfunction expansion technique in terms of vector spherical wave functions. Numerical calculations are subsequently performed using those derived formulas. The derived formulas and the developed source codes are partially verified by the good agreement between our numerical results of radar cross sections with those obtained by Geng et al. using the Fourier transform method. After the validations, some new specific examples are further considered and their results are presented, so as to investigate specific characteristics of these electromagnetic scattering problems. Index Terms—Anisotropic media, eigenvalues and eigenfunctions, electromagnetic scattering, electromagnetic theory, gyroelectric media, radar cross sections (RCSs), vector wave function.

I. INTRODUCTION

E

LECTROMAGNETIC scattering by anisotropic media has always been a topic of interests due to its vast applications in areas such as radar cross section controls for various objects, electromagnetic (EM) or microwave cloaking, antenna radome designs, and development of radar absorbers. Several numerical and analytical techniques have been developed in the literature to tackle these electromagnetic problems, based either on the Maxwell partial differential equation model or its equivalent surface integral equation reformulations [1]–[5]. Based on the Lorenz–Mie scattering theory, field solutions to the problem of electromagnetic or light scattering from an anisotropic sphere can be derived using some useful, analytical techniques, such as, the eigen-expansion method in terms of vector spherical wave functions and Debye potentials [6]–[8]. In recent years, electromagnetic scattering by optically

Manuscript received July 24, 2011; revised June 08, 2012; accepted September 19, 2012. Date of publication October 16, 2012; date of current version January 30, 2013. The work of L. Cao and J. L.-W. Li was supported by the Special Talent Program at University of Electronic Science and Technology of China (UESTC) and the Chinese Government’s 1000-Talent Plan via UESTC, Chengdu, China. The authors are with the Institute of Electromagnetics and School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: [email protected]). 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.2012.2225012

Fig. 1. Geometry of a radially multilayered gyroelectric sphere.

anisotropic magnetic particles and gyrotropic particles have been studied based on the vector spherical wave functions combined with T-matrix method [9], [10]. Also, the vector wave function methodology and the Fourier transform techniques are adopted by Geng et al. to produce rigorous analytical solutions to the electromagnetic scattering by a plasma anisotropic sphere [11], an uniaxial anisotropic sphere [12], a plasma anisotropic spherical shell [13], multilayered plasma anisotropic spherical shells [14], and impedance sphere coated with an uniaxial anisotropic layer [15]. In this paper, we consider a multilayered sphere with each layer characterized by different gyrotropic permittivity tensor, other than spherical shells considered in [14] with the inner most region being free space. Fields in each layer and the scattered fields are obtained theoretically based on the vector spherical wave functions, and numerical calculations are performed using those derived formulations. In Section IV, we validate the algorithm proposed in this article for some examples, by implementing the method in Mathematica and comparing our numerical radar cross section (RCS) results with those obtained by the Fourier transform method in [14]. For the axial incident plane waves, some new numerical results are also presented for the first time to further investigate the scattering characteristic for a multilayered gyroelectric sphere. II. MODEL FOR MULTILAYERED GYROELECTRIC SPHERE Consider a radially multilayered sphere depicted in Fig. 1. As shown in Fig. 1, we assume that the configuration consists regions with spherical layers and we denote of the outer-most region and the inner-most region respectively as the 0th and th region. Each layer of the sphere is assumed to . be homogeneous and denoted by where

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CAO et al.: NEW T-MATRIX FORMULATION FOR EM SCATTERING BY RADIALLY MULTILAYERED GYROELECTRIC SPHERE

For , let denote the radius of the th spherical structure. To make the problem more practical, Region-0 is assumed to be the free space whose permittivity and permeability are represented by and , respectively. Region- is characterized by a scalar permeability and a permittivity tensor in Cartesian coordinate, which are expressed as follows:

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with equal to either for the right hand side or for the left hand side (with ) in (4). We note that all the elements in (5) of the above matrix equation (4) are also matrices associated with the indices , and . In the above recurrence matrix equation in (4), we define the scalar matrix elements associated with their corresponding matrix elements in (5) as follows:

(1) , and a time dependence of is aswhere sumed subsequently herein. The multilayered gyroelectric sphere is a natural extension of a homogeneous sphere discussed in [17]. Based on the authors’ previous work, electromagnetic fields in every region are expanded by vector spherical wave functions with unknown expansion coefficients, which are determined by applying boundary conditions at the interfaces. III. MATCHING BOUNDARY CONDITIONS

(6a)

(6b) (6c) (6d)

Using the eigen-expansion techniques (see Appendix), internal electromagnetic fields in each layer and external fields in free space have been analytically derived based on the vector spherical wave function expansions. The unknown expansion coefficients of electromagnetic fields in each region are determined using the following boundary conditions. Explicitly at (where ), they are given as (2a) (2b) where denotes the outward unit normal vector; while at , they are (3a) (3b) With this set of boundary conditions in place, and following details in Appendix, we could determine the following unknown in Regionand in Region-1 to coefficients: Region; and in Region-0 (the free space). By substituting (25) (see Appendix) into (2) and equaling corresponding vector components on both sides, the following recurrence matrix can be worked out: (4)

(6e) (6f) (6g) (6h) where the prime sign denotes its derivative with respect to the argument, and and are defined as the Riccati–Bessel functions given by (7) and being the spherical Bessel functions of with the first and the third kind, respectively. By applying the recurrence formulation, (4), from Regionto Region-1, we can associate the fields expansion coefficients of Region-1 with those of Region(8) , we have Then, matching boundary conditions further at the following coupled linear equation system to be solved:

where (9a) (5a) (9b) (5b) (9c)

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ELECTRICAL

AND

TABLE I PHYSICAL PARAMETERS SPHERICAL SHELL

OF A

5-LAYER PLASMA

ELECTRICAL

AND

TABLE II PHYSICAL PARAMETERS SPHERICAL SHELL

OF A

8-LAYER PLASMA

(9d) . where Substituting (8) into (9) and solving the linear equation system, we obtain numerically the scattering coefficients and the fields expansion coefficients for Region- . The numerical method for solving (9) and then calculating coefficients in (6) can be found in [9]. Then, (a) after is obtained, it is easy to get the expansions coefficients from Region- to Region-1 by applying an outward recurrence procedure in (4); (b) the radar cross section can be calculated based on the scattering coefficients and , which is explicitly given by [22] (10)

Fig. 2. RCS values of a 5-layer plasma spherical shell versus scattering angle in the - and -planes.

The asymptotic forms of vector spherical wave functions are given in [23]. IV. NUMERICAL RESULTS AND DISCUSSIONS A. Validations In the previous section and in Appendix, we have derived the basic formulations of electromagnetic scattering by a multilayered gyroelectric sphere. Radar cross sections of various spheres are investigated in this section. The gyroelectric spheres are assumed to be nonmagnetic (i.e., where again). To verify correctness of our theory and its corresponding codes developed, numerical computations are performed based on the formulations derived earlier. Then, we compare our numerical results with the results published by Geng et al. [14], two examples are considered here: 1) a 5-layer lossless plasma spherical shells, and 2) a 8-layer lossy plasma spherical shell. The corresponding electric and physical parameters to be used in the two examples are the same as those used previously and they are listed in Tables I and II. Numerical results of the two cases are depicted for comparison in Figs. 2 and 3. As shown, excellent agreements are observed between numerical results obtained using the -matrix method in this paper and those published by Geng et al. in [14] using Fourier transform approach together with numerical integrations. Here we use electric dimensions to depict the size of the multilayered sphere instead of the radius , where .

Fig. 3. RCS values of a 8-layer plasma spherical shell versus scattering angle in the - and -planes.

B. New Results and Discussions In the following discussions, we assume that the incident plane wave with unit amplitude is polarized in the -direction and propagates in the -direction. We investigate the RCS characteristics in the -plane (or plane). Four new specific examples are considered and their results are presented here. In Fig. 4, a 2-layer coated sphere of three different gyroelectric materials is considered with increasing from layer-1 to layer-3 (i.e., ). Explicitly, their values are chosen to be , and . Effects of gyroelectric ratios are examined so as to demonstrate how RCS values can be controlled by varying (where it is assumed that each layer has the same value of ). From Fig. 4, we can see

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Fig. 4. RCS of a 2-layer coated gyroelectric sphere versus scattering angle in the -plane. The electric dimensions are chosen as , , while , and and , and , respectively (where , and ).


that when increases, RCS intensity decreases in a wide range of scattering angle especially in the forward and backward directions, and the resonances shifts to the larger scattering angle side slightly, and it occurs at about . In Fig. 5, the inhomogeneous gyroelectric sphere has the same electric and physical parameters as those in Fig. 4 except that decreases from layer-1 to layer-3 (i.e., ). Specifically, their values are given by , and . It can be seen from Fig. 5 that RCS intensity decreases obviously versus scattering angle between 0 and about 135 with the increasing of in each layer, especially in the forward directions, but the back scattering value does not present obvious changes. Figs. 4 and 5 show that we can control the RCS characteristics by adjusting parameters in each layer. We predict that with this gyroelectric multilayered structure, by adjusting parameters in each layer, we can decrease or increase the RCS values as we desire. Fig. 6 shows the RCS values in the -plane for different anisotropy ratios . As is increased, the RCS has a decreased tendency in the range of 0 to 90 but changes irregu-


Fig. 7. RCS of a 4-layer coated gyroelectric sphere versus scattering angle in the - and -planes, while , and

, (where

, and ).

larly between 90 and 180 . Thus, it is hard to control the RCS intensity in the backscattering directions by adjusting . Finally, we present a 4-layer coated lossless gyroelectric sphere with a medium electric size, the RCS curves versus scattering angles in the - and -planes are depicted in Fig. 7. It is seen that (a) the forward scattering has been significantly enhanced while the backscattering has been also considerably reduced; and (b) within the scattering angle range approximately between 100 and 140 , the scattering radar cross sections reach the minimum values. V. CONCLUSION In this article, electromagnetic scattering of plane wave by a multilayered gyroelectric sphere is characterized while electromagnetic fields inside each region of the spherically layered shell and those outside of the sphere of gyroelectric materials are derived and formulated in detail. Although the approach was developed by others and applied by the authors earlier, the present work is a further extension of the approach to a more generalized multilayered geometry of more practical application potentials,

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and the associated formulations and derivations are more general and far more complicated than the available ones in literature. For some simple cases, we validated our derivations based on numerical calculations with appropriate numerical data in the literature. A few other examples are also considered to characterize the scattering features and radar cross section properties. It is seen that the gyrotropy ratios have apparent effects on the radar cross sections. Thus, by utilizing the multilayered structure and adjusting the parameters in each layer, the scattering object transparency or scattering enhancement can be somehow achieved or controlled. Valid for a sphere of gyroelectric material, the formulation procedure presented here can also be implemented in a similar fashion for analyzing electromagnetic scattering of plane wave by a gyromagnetic sphere with multilayered structure.

explicitly when , and [17], which means that the Bessel functions in VSWF are chosen to be the first, the second, the third and the fourth kind. In practice, are used for the field expressions to be derived subsequently. Unless explicitly specified, hereinafter the summation implies that the index runs from 1 to , while sums up from to for each , and so is in a similar fashion. Based on the noncoplanar and completeness properties of vector spherical wave functions, it can be worked out that [9]

(16a)

(16b) APPENDIX BASIC FORMULATIONS Starting from the Maxwell’s equations and the constitutive relations, the vector wave functions which characterize EM waves in gyroelectric media can be obtained as

where the coefficients , and can be found in [17], so they are not given herein. Adopting (16), we therefore have

(17)

(11) where

where

is the electric displacement in the th layer, , and

(18a) (12)

(18b)

with the following given elements

(18c) (13)

A. General Solution for the Vector Wave Equation The electromagnetic waves in gyroelectric media, modeled by (11), are first obtained in spherical coordinate as an series summation, and then the expansion coefficients in the layered spherical geometry are calculated by the boundary condition (2a)–(3b). For simplicity, the subscript is omitted in this subsection. We expand the electric displacement using the vector spherical wave functions and [16]

where and the expansion coefficients be determined. The coefficient [18], [19], where

and

(18d) Substituting (17) and (14) into (11), we obtain the following equation: (19) where

(14)

(20a)

are yet to is defined in

(20b)

(15) with being the amplitude of the incident electric field. In general, there are three kinds of vector spherical wave functions (VSWF), namely, , and , and they are given

From the orthogonality properties of and , we know that and must vanish for each indices . Thus it leads from (20) to the following characteristic equation in matrix form: (21)


where , and

, and the matrix elements of submatrices are respectively defined as

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B. Expansions of Incident and Scattered Fields The incident and scattered EM fields in Region-0 (in free space) have the same form as those in the Lorenz-Mie solution [17], [21]. The scattered fields are given explicitly as

(22a) (22b)

(26a)

(22c) (26b)

(22d) with and denoting the row and column indices, respectively. Equation (21) implies an established eigen-system, with eigenvalues and eigenvectors where denotes the indices of eigenvalues and corresponding eigenvectors. We can then construct a new set of vector basis functions, , based on the eigenvectors, i.e.,

(23) . where Then, the general solution for (11) can be expressed as a linear combination of :

and the scattering coefficients, and where , are to be determined by matching boundary conditions. The incident electric and magnetic fields are expressed in spherical coordinates as follows: (27a) (27b) where stands for the polarization vector with unit amplitude (i.e., ), and the unit vectors and are defined in a spherical direction of increasing and to con. In stitute a right-hand base system together with terms of vector spherical wave functions, the incidents electric and magnetic fields are expanded into

(24) and the electric field and magnetic field derived in a general form as

(28a)

are subsequently (28b) where the coefficients and of the incident wave and their detailed deductions can be found in [22]. ACKNOWLEDGMENT (25a)

The authors would like to thank Miss Ong and Mr. Wan (National University of Singapore) for useful discussions. REFERENCES

(25b) Since the third kind of Bessel function is singular at the origin, so electromagnetic fields in Region- are obtained by just using the first kind of VSWF , but in Region-1 to Region, the first and the third kind of VSWFs must be used to get the complete solution . And it is easy to show that , which represents the characteristic of gyroelectric media.

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Joshua Le-Wei Li (S’91–M’92–SM’96–F’05) received the Ph.D. degree in electrical engineering from Monash University, Melbourne, Australia, in 1992. In June 1992, he was with the Department of Electrical and Computer Systems Engineering, Monash University, and sponsored by the Department of Physics, La Trobe University, Melbourne, Australia, as a Research Fellow. From November 1992 to December 2010, he was with the Department of Electrical and Computer Engineering, National

University of Singapore (NUS), where he was a Full Professor and Director of the NUS Center for Microwave and Radio Frequency. From 1999 to 2004, he was seconded to High Performance Computations on Engineered Systems (HPCES) Programme of Singapore-MIT Alliance (SMA) as a Course Coordinator and SMA Faculty Fellow. From May to July 2002, he was a Visiting Scientist with the Research Laboratory of Electronics, Massachusetts Institute of Technology; and in October 2006, he was an Invited Professor with the University of Paris VI, Paris, France. He was also an Invited Visiting Professor at the Swiss Federal Institute of Technology, Lausanne (EPFL), between January and June 2008; and a Visiting Guest Professor at Swiss Federal Institute of Technology, Zurich (ETHZ), between July and November, 2008; both in Switzerland. Since 2010, he has been with University of Electronic Science and Technology of China (UESTC) as a 1000-Talent Project Chair Professor. His current research interests include electromagnetic theory, computational electromagnetics, radio wave propagation and scattering in various media, microwave propagation and scattering in tropical environment, and analysis and design of various antennas. In these areas, he has authored and coauthored two books, Spheroidal Wave Functions in Electromagnetic Theory (New York: Wiley, 2001) and Device Modeling in CMOS Integrated Circuits: Interconnects, Inductors and Transformers (London: Lambert Academic Publishing 2010), 48 book chapters, over 320 international refereed journal papers, 49 regional refereed journal papers, and over 370 international conference papers. Dr. Li is a Fellow of The Electromagnetics Academy since 2007, and was IEICE Singapore Section Chairman from 2002 to 2007. He is a regular reviewer of many archival journals, an Editor of Radio Science, and an Associate Editor of the International Journal of Numerical Modeling: Electronic Networks, Devices, and Fields, and International Journal of Antennas and Propagation; an (Overseas) Editorial Board Member of five international and regional archival journals and one book series by EMW Publishing. He is honored as an Advisory, Guest, or Adjunct Professor at one State Key Laboratory and other four leading universities in related areas of electromagnetics in China. He also serves as a member of various International Advisory Committee and/or Technical Program Committee of many international conferences or workshops, in addition to serving as a General Cochairman of ISAP2006, MRS09-Meta09, and iWEM series, since 2011; Vice General Cochairman of PIERS 2011 in Marrakech; and TPC Cochairman of PIERS2003, iWAT2006, APMC2009, ISAPE2010, and ISAPE2012. He is currently appointed as an IEEE MTT-S Commission-15 Member in 2008, IEEE AP-S Region Representative (Region 10: Asia-Pacific) in 2010, and an IEEE AP-S Distinguished Lecturer in 2011. He was a recipient of several awards, including two Best Paper Awards, the 1996 National Award of Science and Technology of China, the 2003 IEEE AP-S Best Chapter Award when he was the IEEE Singapore MTT/AP Joint Chapter Chairman, the 2004 University Excellent Teacher Award of NUS, and the 2012 University Excellent Teacher Award of UESTC.

Jun Hu (M’06–SM’01) received the B.S., M.S., and Ph.D. degrees in electromagnetic field and microwave technique from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1995, 1998, and 2000, respectively. During 2001, he was with the Center of Wireless Communications, City University of Hong Kong, Kowloon, Hong Kong, as a Research Assistant. From March to August 2010, he was a Visiting Scholar in the Electroscience Laboratory, Department of Electrical and Computer Engineering, Ohio State University. He was then a visiting Professor of the City University of Hong Kong from February to March 2011. He is currently a Full Professor with the School of Electronic Engineering, UESTC. He is the author or a coauthor of over 190 technical papers. His current research interests include integral equation methods in computational electromagnetics, domain decomposition methods for multiscale problems, and novel finite-element methods for microwave engineering. Dr. Hu is a member of the Applied Computational Electromagnetics Society. He also serves as Chairman of the Student Activities Committee of the IEEE Chengdu Section, and Vice Chairman of the IEEE Chengdu AP/EMC Joint Chapter. He received the 2004 Best Young Scholar paper prize of the Chinese Radio Propagation Society. His doctoral students were awarded the Best Student Paper Prizes in the 2010 IEEE Chengdu Section, the 2011 National Conference on Antenna, the 2011 National Conference on Microwave, and the 2012 IEEE International Workshop on Electromagnetics: Applications and Students’ Innovation Competition in Chengdu.