A New T-Matrix Formulation for Electromagnetic

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

A New T-Matrix Formulation for ElectromagneticScattering by a Radially Multilayered

Gyroelectric SphereLei 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 characteris-tics of electromagnetic scattering by a radially multilayered gyro-electric sphere is proposed. The spherical geometry is divided into

regions (where can be an arbitrary integer), and eachlayer is characterized by a scalar permeability and a gyrotropicpermittivity tensor. Electromagnetic fields inside and outside thesphere are theoretically formulated based on the eigenfunction ex-pansion technique in terms of vector spherical wave functions. Nu-merical calculations are subsequently performed using those de-rived formulas. The derived formulas and the developed sourcecodes are partially verified by the good agreement between ournumerical results of radar cross sections with those obtained byGeng et al. using the Fourier transform method. After the valida-tions, some new specific examples are further considered and theirresults are presented, so as to investigate specific characteristics ofthese electromagnetic scattering problems.

Index Terms—Anisotropic media, eigenvalues and eigen-functions, electromagnetic scattering, electromagnetic theory,gyroelectric media, radar cross sections (RCSs), vector wavefunction.

I. INTRODUCTION

E LECTROMAGNETIC scattering by anisotropic mediahas always been a topic of interests due to its vast appli-

cations in areas such as radar cross section controls for variousobjects, electromagnetic (EM) or microwave cloaking, antennaradome designs, and development of radar absorbers. Severalnumerical and analytical techniques have been developed inthe literature to tackle these electromagnetic problems, basedeither on the Maxwell partial differential equation model orits equivalent surface integral equation reformulations [1]–[5].Based on the Lorenz–Mie scattering theory, field solutionsto the problem of electromagnetic or light scattering froman anisotropic sphere can be derived using some useful, an-alytical techniques, such as, the eigen-expansion method interms 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; acceptedSeptember 19, 2012. Date of publication October 16, 2012; date of currentversion January 30, 2013. The work of L. Cao and J. L.-W. Li was supported bythe Special Talent Program at University of Electronic Science and Technologyof China (UESTC) and the Chinese Government’s 1000-Talent Plan viaUESTC, Chengdu, China.The authors are with the Institute of Electromagnetics and School of Elec-

tronic 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 havebeen studied based on the vector spherical wave functions com-bined with T-matrix method [9], [10]. Also, the vector wavefunction methodology and the Fourier transform techniques areadopted by Geng et al. to produce rigorous analytical solutionsto the electromagnetic scattering by a plasma anisotropic sphere[11], an uniaxial anisotropic sphere [12], a plasma anisotropicspherical shell [13], multilayered plasma anisotropic sphericalshells [14], and impedance sphere coated with an uniaxialanisotropic 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 mostregion being free space. Fields in each layer and the scatteredfields are obtained theoretically based on the vector sphericalwave functions, and numerical calculations are performed usingthose derived formulations. In Section IV, we validate the al-gorithm proposed in this article for some examples, by imple-menting the method in Mathematica and comparing our numer-ical radar cross section (RCS) results with those obtained by theFourier transform method in [14]. For the axial incident planewaves, some new numerical results are also presented for thefirst time to further investigate the scattering characteristic for amultilayered 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 consistsof regions with spherical layers and we denotethe outer-most region and the inner-most region respectively asthe 0th and th region. Each layer of the sphere is assumed tobe homogeneous and denoted by where .

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CAO et al.: NEW T-MATRIX FORMULATION FOR EM SCATTERING BY RADIALLY MULTILAYERED GYROELECTRIC SPHERE 837

For , let denote the radius of the th sphericalstructure. To make the problem more practical, Region-0 is as-sumed to be the free space whose permittivity and permeabilityare represented by and , respectively. Region- is charac-terized by a scalar permeability and a permittivity tensorin Cartesian coordinate, which are expressed as follows:

(1)

where , and a time dependence of is as-sumed subsequently herein.The multilayered gyroelectric sphere is a natural extension

of a homogeneous sphere discussed in [17]. Based on the au-thors’ previous work, electromagnetic fields in everyregion are expanded by vector spherical wavefunctions with unknown expansion coefficients, which are de-termined by applying boundary conditions at the interfaces.

III. MATCHING BOUNDARY CONDITIONS

Using the eigen-expansion techniques (see Appendix), in-ternal electromagnetic fields in each layer and ex-ternal fields in free space havebeen analytically derived based on the vector spherical wavefunction expansions. The unknown expansion coefficients ofelectromagnetic fields in each region are determined using thefollowing 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 followingdetails in Appendix, we could determine the following unknowncoefficients: in Region- and in Region-1 toRegion- ; and in Region-0 (the free space).By substituting (25) (see Appendix) into (2) and equaling cor-

responding vector components on both sides, the following re-currence matrix can be worked out:

(4)

where

(5a)

(5b)

with equal to either for the right hand side or forthe left hand side (with ) in (4). We note that allthe elements in (5) of the above matrix equation (4) are alsomatrices associated with the indices , and .In the above recurrence matrix equation in (4), we define the

scalar matrix elements associated with their corresponding ma-trix elements in (5) as follows:

(6a)

(6b)

(6c)

(6d)

(6e)

(6f)

(6g)

(6h)

where the prime sign denotes its derivativewith respect to the ar-gument, and and are defined as the Riccati–Besselfunctions given by

(7)

with and being the spherical Bessel functions ofthe first and the third kind, respectively.By applying the recurrence formulation, (4), from Region-

to Region-1, we can associate the fields expansion coefficientsof Region-1 with those of Region-

(8)

Then, matching boundary conditions further at , we havethe following coupled linear equation system to be solved:

(9a)

(9b)

(9c)


TABLE IELECTRICAL AND PHYSICAL PARAMETERS OF A 5-LAYER PLASMA

SPHERICAL SHELL

(9d)

where .Substituting (8) into (9) and solving the linear equation

system, we obtain numerically the scattering coefficientsand the fields expansion coefficients for Re-

gion- . The numerical method for solving (9) and thencalculating coefficients in (6) can be found in [9]. Then, (a)after is obtained, it is easy to get the expansions coefficientsfrom Region- to Region-1 by applying an outward recurrenceprocedure in (4); (b) the radar cross section can be calculatedbased on the scattering coefficients and , which isexplicitly given by [22]

(10)

The asymptotic forms of vector spherical wave functions aregiven in [23].

IV. NUMERICAL RESULTS AND DISCUSSIONS

A. Validations

In the previous section and in Appendix, we have derived thebasic formulations of electromagnetic scattering by a multilay-ered gyroelectric sphere. Radar cross sections of various spheresare investigated in this section. The gyroelectric spheres are as-sumed to be nonmagnetic (i.e., whereagain).To verify correctness of our theory and its corresponding

codes developed, numerical computations are performed basedon the formulations derived earlier. Then, we compare ournumerical results with the results published by Geng et al. [14],two examples are considered here: 1) a 5-layer lossless plasmaspherical 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 previouslyand they are listed in Tables I and II.Numerical results of the two cases are depicted for compar-

ison in Figs. 2 and 3. As shown, excellent agreements are ob-served between numerical results obtained using the -matrixmethod in this paper and those published by Geng et al. in [14]using Fourier transform approach together with numerical in-tegrations. Here we use electric dimensions to de-pict the size of the multilayered sphere instead of the radius ,where .

TABLE IIELECTRICAL AND PHYSICAL PARAMETERS OF A 8-LAYER PLASMA

SPHERICAL SHELL

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

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

B. New Results and Discussions

In the following discussions, we assume that the incidentplane wave with unit amplitude is polarized in the -directionand propagates in the -direction. We investigate the RCScharacteristics in the -plane (or plane). Four new specificexamples are considered and their results are presented here.In Fig. 4, a 2-layer coated sphere of three different gyroelec-

tric materials is considered with increasing from layer-1 tolayer-3 (i.e., ). Explicitly, their values arechosen to be , and . Effects ofgyroelectric ratios are examined so as to demonstrate how RCSvalues can be controlled by varying (where it is assumed thateach layer has the same value of ). From Fig. 4, we can see


Fig. 4. RCS of a 2-layer coated gyroelectric sphere versus scattering anglein the -plane. The electric dimensions are chosen as ,and , while , and

, and , respectively (where , and ).



that when increases, RCS intensity decreases in a wide rangeof scattering angle especially in the forward and backward di-rections, and the resonances shifts to the larger scattering angleside 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 exceptthat 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 intensitydecreases obviously versus scattering angle between 0 andabout 135 with the increasing of in each layer, especially inthe forward directions, but the back scattering value does notpresent obvious changes. Figs. 4 and 5 show that we can controlthe RCS characteristics by adjusting parameters in each layer.We predict that with this gyroelectric multilayered structure, byadjusting parameters in each layer, we can decrease or increasethe 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 de-creased tendency in the range of 0 to 90 but changes irregu-



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

, and , (where , and ).

larly between 90 and 180 . Thus, it is hard to control the RCSintensity in the backscattering directions by adjusting .Finally, we present a 4-layer coated lossless gyroelectric

sphere with a medium electric size, the RCS curves versusscattering angles in the - and -planes are depicted in Fig. 7.It is seen that (a) the forward scattering has been significantlyenhanced while the backscattering has been also considerablyreduced; and (b) within the scattering angle range approx-imately between 100 and 140 , the scattering radar crosssections reach the minimum values.

V. CONCLUSION

In this article, electromagnetic scattering of plane wave bya multilayered gyroelectric sphere is characterized while elec-tromagnetic fields inside each region of the spherically layeredshell and those outside of the sphere of gyroelectric materials arederived and formulated in detail. Although the approach was de-veloped by others and applied by the authors earlier, the presentwork is a further extension of the approach to amore generalizedmultilayered geometry of more practical application potentials,


and the associated formulations and derivations are more gen-eral and far more complicated than the available ones in litera-ture. For some simple cases, we validated our derivations basedon numerical calculations with appropriate numerical data in theliterature. A few other examples are also considered to char-acterize the scattering features and radar cross section proper-ties. It is seen that the gyrotropy ratios have apparent ef-fects on the radar cross sections. Thus, by utilizing the multi-layered structure and adjusting the parameters in each layer, thescattering object transparency or scattering enhancement can besomehow achieved or controlled. Valid for a sphere of gyroelec-tric material, the formulation procedure presented here can alsobe implemented in a similar fashion for analyzing electromag-netic scattering of plane wave by a gyromagnetic sphere withmultilayered structure.

APPENDIXBASIC FORMULATIONS

Starting from theMaxwell’s equations and the constitutive re-lations, the vector wave functions which characterize EMwavesin gyroelectric media can be obtained as

(11)

where is the electric displacement in the th layer,, and

(12)

with the following given elements

(13)

A. General Solution for the Vector Wave Equation

The electromagnetic waves in gyroelectric media, modeledby (11), are first obtained in spherical coordinate as an seriessummation, and then the expansion coefficients in the layeredspherical geometry are calculated by the boundary condition(2a)–(3b). For simplicity, the subscript is omitted in this sub-section. We expand the electric displacement using the vectorspherical wave functions and [16]

(14)

where and the expansion coefficients and are yet tobe determined. The coefficient is defined in[18], [19], where

(15)

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

explicitly when , and [17], which means that theBessel functions in VSWF are chosen to be the first, the second,the third and the fourth kind. In practice, are used forthe field expressions to be derived subsequently. Unless explic-itly specified, hereinafter the summation implies that theindex runs from 1 to , while sums up from to foreach , 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)

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

(17)

where

(18a)

(18b)

(18c)

(18d)

Substituting (17) and (14) into (11), we obtain the followingequation:

(19)

where

(20a)

(20b)

From the orthogonality properties of and , weknow that and must vanish for each indices . Thusit leads from (20) to the following characteristic equation in ma-trix form:

(21)


where , and the matrix elements of submatrices, and are respectively defined as

(22a)

(22b)

(22c)

(22d)

with and denoting the row and column indices, respec-tively. Equation (21) implies an established eigen-system, witheigenvalues and eigenvectors where denotes theindices of eigenvalues and corresponding eigenvectors. We canthen construct a new set of vector basis functions, , basedon the eigenvectors, i.e.,

(23)

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

combination of :

(24)

and the electric field and magnetic field are subsequentlyderived in a general form as

(25a)

(25b)

Since the third kind of Bessel function is singular at the origin,so electromagnetic fields in Region- are obtained by justusing the first kind of VSWF , but in Region-1 toRegion- , the first and the third kind of VSWFs mustbe used to get the complete solution . And it is easyto show that , which represents the characteristic ofgyroelectric media.

B. Expansions of Incident and Scattered Fields

The incident and scattered EM fields in Region-0 (in freespace) have the same form as those in the Lorenz-Mie solution[17], [21]. The scattered fields are given explicitly as

(26a)

(26b)

where and the scattering coefficients, and, 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 withunit amplitude (i.e., ), and the unit vectors and aredefined in a spherical direction of increasing and to con-stitute a right-hand base system together with . Interms of vector spherical wave functions, the incidents electricand magnetic fields are expanded into

(28a)

(28b)

where the coefficients and of the incident wave andtheir detailed deductions can be found in [22].

ACKNOWLEDGMENT

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

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Lei Cao was born in Yunnan Province, China, in1988. He received the B.Eng. degree in electro-magnetic fields and wireless technology from theUniversity of Electronic Science and Technology ofChina (UESTC), Chengdu, China, in 2011, where heis currently working toward the M.Eng. degree.His recent research interests are electromagnetic

theory, electromagnetic composite materials, an-tennas and propagation, and nano electromagnetics.

Joshua Le-Wei Li (S’91–M’92–SM’96–F’05)received the Ph.D. degree in electrical engineeringfrom Monash University, Melbourne, Australia, in1992.In June 1992, he was with the Department of

Electrical and Computer Systems Engineering,Monash University, and sponsored by the Depart-ment of Physics, La Trobe University, Melbourne,Australia, as a Research Fellow. From November1992 to December 2010, he was with the Departmentof Electrical and Computer Engineering, National

University of Singapore (NUS), where he was a Full Professor and Directorof 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 Coor-dinator and SMA Faculty Fellow. From May to July 2002, he was a VisitingScientist with the Research Laboratory of Electronics, Massachusetts Instituteof Technology; and in October 2006, he was an Invited Professor with theUniversity of Paris VI, Paris, France. He was also an Invited Visiting Professorat the Swiss Federal Institute of Technology, Lausanne (EPFL), between Jan-uary and June 2008; and a Visiting Guest Professor at Swiss Federal Instituteof Technology, Zurich (ETHZ), between July and November, 2008; both inSwitzerland. Since 2010, he has been with University of Electronic Scienceand Technology of China (UESTC) as a 1000-Talent Project Chair Professor.His current research interests include electromagnetic theory, computationalelectromagnetics, radio wave propagation and scattering in various media,microwave propagation and scattering in tropical environment, and analysisand design of various antennas. In these areas, he has authored and coauthoredtwo books, Spheroidal Wave Functions in Electromagnetic Theory (New York:Wiley, 2001) and Device Modeling in CMOS Integrated Circuits: Intercon-nects, Inductors and Transformers (London: Lambert Academic Publishing2010), 48 book chapters, over 320 international refereed journal papers, 49regional 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 reviewerof many archival journals, an Editor of Radio Science, and an Associate Editorof the International Journal of Numerical Modeling: Electronic Networks, De-vices, and Fields, and International Journal of Antennas and Propagation; an(Overseas) Editorial Board Member of five international and regional archivaljournals and one book series by EMW Publishing. He is honored as an Advi-sory, Guest, or Adjunct Professor at one State Key Laboratory and other fourleading universities in related areas of electromagnetics in China. He also servesas a member of various International Advisory Committee and/or Technical Pro-gram Committee of many international conferences or workshops, in additionto serving as a General Cochairman of ISAP2006, MRS09-Meta09, and iWEMseries, 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-15Member 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 recipientof several awards, including two Best Paper Awards, the 1996 National Awardof Science and Technology of China, the 2003 IEEE AP-S Best Chapter Awardwhen he was the IEEE Singapore MTT/AP Joint Chapter Chairman, the 2004University Excellent Teacher Award of NUS, and the 2012 University ExcellentTeacher Award of UESTC.

Jun Hu (M’06–SM’01) received the B.S., M.S.,and Ph.D. degrees in electromagnetic field andmicrowave technique from the University of Elec-tronic Science and Technology of China (UESTC),Chengdu, China, in 1995, 1998, and 2000, respec-tively.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 VisitingScholar in the Electroscience Laboratory, Depart-

ment of Electrical and Computer Engineering, Ohio State University. He wasthen a visiting Professor of the City University of Hong Kong from Februaryto March 2011. He is currently a Full Professor with the School of ElectronicEngineering, UESTC. He is the author or a coauthor of over 190 technicalpapers. His current research interests include integral equation methods incomputational electromagnetics, domain decomposition methods for multiscaleproblems, 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 IEEEChengdu Section, and Vice Chairman of the IEEE Chengdu AP/EMC JointChapter. He received the 2004 Best Young Scholar paper prize of the ChineseRadio Propagation Society. His doctoral students were awarded the Best StudentPaper Prizes in the 2010 IEEE Chengdu Section, the 2011 National Conferenceon Antenna, the 2011 National Conference on Microwave, and the 2012 IEEEInternational Workshop on Electromagnetics: Applications and Students’ Inno-vation Competition in Chengdu.

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A New T-Matrix Formulation for Electromagnetic