Analytical Method and Semianalytical Method for Analysis of Scattering by Anisotropic Sphere: A Review

Abstract

The history of methods for the electromagnetic scattering by an anisotropic sphere has been reviewed. Two main methods, angular expansion method and T-matrix method, which are widely used for the anisotropic sphere, are expressed in Cartesian coordinate firstly. The comparison of those and the further exploration on the scattering field are illustrated afterwards. Based on the most general form concluded by variable separation method, the coupled electric field and magnetic field of radial anisotropic sphere can be derived. By simplifying the condition, simpler case of uniaxial anisotropic media is expressed with confirmed coefficients for the internal and external field. Details of significant phenomenon are presented.

1. Introduction

Since the properties of isotropic homogeneous sphere is formulated by Lorentz [1] and Mie [2], due to plenty of corresponding significant phenomena [3–6] and increasing applications in atmospheric optics, remote sensing, computational electromagnetic and photonics, electromagnetic wave propagation and scattering for a sphere has been a subject of great interest in recent years. The amazing properties of solid anisotropic materials in nature and classes of artificial metamaterials of anisotropy would possibly produce plentiful technological applications. In the meantime, fabrication technology development has greatly enlarged the area of the anisotropic materials.

Cartesian anisotropic medium and Radial anisotropic medium are considered as two classic kinds of anisotropic materials [7]. Some novel properties of Cartesian anisotropy medium have been studied: magnetic plasma realizing the subdiffraction imaging [8], optical cavity made by indefinite medium of anisotropy [9], the sharply asymmetric reflection (SAR) effect [10], formation of tunable resonant photonic band gaps [11], negative-refractive behavior manipulation [12], and self-guiding unidirectional electromagnetics edge [13]. Based on the general anisotropic materials, those amazing properties can greatly enlarge the possibility to control light and wave in special structure and also open a new research area in the interaction between the electromagnetic wave and materials. For radial anisotropic materials, the spherical particle is of great interest [14–16]. The radial anisotropic particle scattering provides an insight into the interaction between the microparticles and the microwave or optical illumination by external devices [17–19], which could help to detect the position for some abnormal proteins [7, 20]. The classic spherical cloaks have been achieved by radial anisotropic materials [21]. The surface plasma resonance invisibility and extraordinary scattering of the RA particle provide a good opportunity to study the physical phenomena of microparticles.

For most of these published works, many properties of anisotropic material have been explored and applied. However, little work has been done so far on summaries and analysis for the methods to deal with problems of such anisotropic material. Scientists and engineers have to check the anisotropic class first and then choose a convenient numerical method or some commercial software. Either way will encounter redundant difficulties: for numerical method like FDTD, FEM, MOM, and other improved ones [22], prediction of the properties of medium is quite a difficulty, let alone the new structure design; for commercial software, the methods embedded in them are commonly standard, which are not able to deal with the possible novel materials with abnormal parameters. Therefore, modeling of those novel materials becomes extremely difficult. Fortunately, the analytical method and semianalytical method such as angular expansion method and T-matrix method will be very helpful in concluding good predictions for the anisotropic problems while keeping the accuracy [23–33].

It is of major purpose to summarize the analytical and semianalytical methods and their material parameter's effects on the scattering patterns, which will be of great use in electromagnetic wave properties for Cartesian and the Radial anisotropic spheres. In Section 2, besides two general analytical and semianalytical algorithms of anisotropy in Cartesian are reviewed, advantages and disadvantages of these two methods are also illustrated. In Section 3, the variables separation method for a single radial anisotropic sphere is demonstrated and numerical results obtained. A short conclusion is provided in Section 4.

2. Electromagnetic Wave Scattering by a Cartesian Anisotropic Sphere

2.1. Cartesian Anisotropy of Spheres

For Cartesian anisotropic medium, the constitutive relations are given by𝜀=𝜀_𝑠𝜀_𝑟−𝑖𝜀_𝜅0𝑖𝜀_𝜅𝜀_𝑟0001𝜇=𝜇_𝑠𝜇_𝑟−𝑖𝜇_𝜅0𝑖𝜇_𝜅𝜇_𝑟0001, (1) where the identity dyadic is expressed as 𝐈̂=𝐱𝐱+𝐲𝐲+𝐳̂𝐳̂. By extending Mie theory in isotropic cases, many methods such as FDTD, FEM, and MOM have been established to analyzed wave interaction issue in the medium. However, they cannot provide an explicit explanation for the physical phenomenon. Being able to deal with multilayered problems, the dyadic green's function is considered as one analytical approach [34]. However, the researchers have to encounter the singularities while they choose the integration path, then the expression turns to be rather complicated. In [35], Uzunoglu first employed the Fourier form to state the field, followed by a lot of work on this direction for its clear computation procedure [36, 37]. Using the angular expansion to present electromagnetic field, Ren successfully transferred the three-dimensional problem into a two-dimension integration [23]. The field can be treated as combination of infinite plane wave, while only four kinds of Bessel function need to be considered according to the wave direction. After that, Dr. Sarkar gives an explicit expression of the plane wave using vector spherical wave functions, where the field in the media is a form of that infinite plane wave [38, 39]. Based on these clear field and plane wave expression, Dr. Geng proposed a transpicuous procedure and made a fundamental computation [24–28]. The same result has been achieved by other researcher when compared with Geng's work [29]. Besides, the method firstly dealing with the photonic band problem can be seen as another way [40]. Instead of giving an explicit expression and using the integral form on the surface, Lin used the transfer matrix method to gather the information in the medium [41]. By using the complete matrix form, they express the wave number 𝑘 by a matrix vector instead of an expression [32, 42], and this approach obtains great success [10–13]. Herein, we mainly focus on the review and analysis of these two analytical and semianalytical methods.

2.2. Angular Expansion Method

Inside the sourceless and homogeneous anisotropic sphere, the constitutive relationships as used in the Maxwell equations are∇×𝐄=𝑖𝜔𝐁,∇×𝐇=−𝑖𝜔𝐃,∇⋅𝐃=0,∇⋅𝐁=0. (2)(3)(4)(5) The relations between the electric displacement vector D, the magnetic induction B, the electric field E, and the magnetic field H inside the medium are given by𝐃=𝜀⋅𝐄,𝐁=𝜇⋅𝐇. (6) The parameters are defined in Cartesian coordinates as (1). We can rewrite the Maxwell's equations in a convenient form [27]:∇×𝜇⁻¹⋅∇×𝐄−𝜔²𝜀⋅𝐄=0. (7) The solution to (7) can be written in the Fourier transform:𝐄(𝐫)=^+∞_−∞𝑑𝑘_𝑥^+∞_−∞𝑑𝑘_𝑦^+∞_−∞𝐄(𝐤)𝑒^{𝑖𝐤⋅𝐫}𝑑𝑘_𝑧, (8) where the vector wave number is expressed as 𝐤=𝑘_𝑥𝐱+𝑘_𝑦𝐲+𝑘_𝑧𝐳̂, the space vector as 𝐫=𝑥𝐱+𝑦𝐲+𝑧𝐳̂, and 𝐱,𝐲,𝐳̂ being the unit vectors in Cartesian coordinates. Here, the problem is transferred from spatial domain to spectrum domain. For each point in the medium, the electric field 𝐄 can be seen as a combination of infinite plane wave. Then, for any direction wave, we have ∇=𝑖𝐤. Substituting (8) into (7), the electric field equation is transformed into^+∞_−∞𝑑𝑘_𝑥^+∞_−∞𝑑𝑘_𝑦^+∞_−∞𝐀(𝐤)⋅𝐄(𝐤)𝑒^{𝑖𝐤⋅𝐫}𝑑𝑘_𝑧=0, (9) where𝐀(𝐤)=−𝑏₁𝑘²_𝑧−𝑏₃𝑘²_𝑧+𝑎₁𝑏₃𝑘_𝑥𝑘_𝑦−𝑖𝑏₂𝑘²_𝑧−𝑖𝑎₂𝑏₁𝑘_𝑥𝑘_𝑧+𝑖𝑏₂𝑘_𝑦𝑘_𝑧𝑏₃𝑘_𝑥𝑘_𝑦−𝑖𝑏₂𝑘²_𝑧−𝑖𝑎₂−𝑏₁𝑘²_𝑧−𝑏₃𝑘²_𝑧+𝑎₁𝑏₁𝑘_𝑦𝑘_𝑧−𝑖𝑏₂𝑘_𝑥𝑘_𝑧𝑏₁𝑘_𝑥𝑘_𝑧−𝑖𝑏₂𝑘_𝑦𝑘_𝑧𝑏₁𝑘_𝑦𝑘_𝑧+𝑖𝑏₂𝑘_𝑥𝑘_𝑧−𝑏₁𝑘²_𝑥+𝑘²_𝑦+𝑎₃ (10)with𝑎₁=𝜔²𝜀_𝑠𝜀_𝑟𝑎₂=𝜔²𝜀_𝑠𝜀_𝜅𝑎₃=𝜔²𝜀_𝑠,𝑏₁=𝜇_𝑟𝜇_𝑠𝜇²_𝑟−𝜇²_𝜅𝑏₂=𝜇_𝜅𝜇_𝑠𝜇²_𝑟−𝜇²_𝜅𝑏₃=1𝜇_𝑠. (11) In order to make (9) exist all the time, we write the eigenvalue equation asDet𝐀(𝐤)=0. (12) Here, the eigenvalue 𝑘 is a function of the angular (𝜃_𝑘,𝜙_𝑘), the equation turns to be𝐴𝜃_𝑘,𝜙_𝑘𝑘⁴−𝐵𝜃_𝑘,𝜙_𝑘𝑘²+𝐶=0, (13) where𝐴𝜃_𝑘,𝜙_𝑘=𝑏₁𝑏₃sin²𝜃_𝑘+𝑏²₁−𝑏²₂cos²𝜃_𝑘×𝑎₁sin²𝜃_𝑘+𝑎₃cos²𝜃_𝑘,𝐵𝜃_𝑘,𝜙_𝑘=𝑏₁𝑎²₁−𝑎²₂+𝑏₃𝑎₁𝑎₃sin²𝜃_𝑘+2𝑎₃𝑏₁𝑎₁+𝑏₂𝑎₂cos²𝜃_𝑘,𝐶=𝑎₃𝑎²₁−𝑎²₂,𝑘²=𝑘²_𝑥+𝑘²_𝑦+𝑘²_𝑧,𝜃_𝑘=tan⁻¹𝑘²_𝑥+𝑘²_𝑦𝑘_𝑧,𝜙_𝑘=tan⁻¹𝑘_𝑦𝑘_𝑥,𝑘²_1,3=𝐵+√𝐵²−4𝐴𝐶,𝑘2𝐴_22,4=𝐵−√𝐵²−4𝐴𝐶,2𝐴 (14) and the eigenvectors can also be obtained from (10) at the same time and are given as follows:𝐄_𝑞=𝐅^𝑒_𝑞𝑓_𝑞𝜃_𝑘,𝜙_𝑘=𝐹^𝑒_𝑞𝑥𝜃_𝑘,𝜙_𝑘𝐱+𝐹^𝑒_𝑞𝑦𝜃_𝑘,𝜙_𝑘𝐲+𝐹^𝑒_𝑞𝑧𝜃_𝑘,𝜙_𝑘𝐳̂𝑓_𝑞𝜃_𝑘,𝜙_𝑘, (15) where q = 1, 2, 3, or 4, here 𝐄_𝑞 is a plane wave, and𝐹^𝑒_𝑞𝑥Δ=−¹Δsin𝜙_𝑘+Δ₂Δcos𝜙_𝑘,𝐹^𝑒_𝑞𝑦=Δ₁Δcos𝜙_𝑘+Δ₂Δsin𝜙_𝑘,𝐹^𝑒_𝑞𝑧=1 (16) withΔ₁=𝑖𝑏₁𝑎₂+𝑏₂𝑎₁𝑘²_𝑞sin𝜃_𝑘cos𝜃_𝑘,Δ₂=𝑏₁𝑏₃𝑘²_𝑞sin²𝜃_𝑘+𝑏²₁−𝑏²₂𝑘²_𝑞cos²𝜃_𝑘×𝑘²_𝑞sin𝜃_𝑘cos𝜃_𝑘−𝑏₁𝑎₁+𝑏₂𝑎₂Δ=−𝑏₂𝑘²_𝑞cos²𝜃_𝑘+𝑎₂²+𝑏₁𝑘²_𝑞cos²𝜃_𝑘−𝑎₁×𝑏₁𝑘²_𝑞cos²𝜃_𝑘+𝑏₃𝑘²_𝑞sin²𝜃_𝑘−𝑎₁. (17) Here, all the angular and eigenvalues should be considered. The E-field in (7) is then given as𝐄(𝑟)=²_𝑞=1^𝜋₀₀^2𝜋𝐹^𝑒_𝑞𝜃_𝑘,𝜙_𝑘𝑓_𝑞𝜃_𝑘,𝜙_𝑘𝑒^{𝑖𝐤𝑞⋅𝐫}𝑘²_𝑞sin𝜃_𝑘𝑑𝜃_𝑘𝑑𝜙_𝑘,𝐤_𝑞=𝑘_𝑞sin𝜃_𝑘cos𝜙_𝑘𝐱+𝑘_𝑞sin𝜃_𝑘sin𝜙_𝑘𝐲+𝑘_𝑞cos𝜃_𝑘𝐳̂. (18)(19) Only two roots need to be considered, while the periodic function 𝑓_𝑞(𝜃_𝑘,𝜙_𝑘) denoting the angular spectrum amplitude can be expressed as𝑓_𝑞𝜃_𝑘,𝜙_𝑘=_{𝑚′,𝑛′}𝐺_{𝑚′𝑛′𝑞}𝑃^𝑚′_𝑛′cos𝜃_𝑘𝑒^{𝑖𝑚′𝜙𝑘}, (20) where 𝑃^𝑚_𝑛(𝑥) is the associated Legendre function, and ∑_{𝑚′,𝑛′} means that 𝑛′ is from 0 to +∞ and 𝑚′ is from −𝑛′ to 𝑛′. Substituting (20) to (18), we obtain𝐄(𝑟)=²_{𝑞=1𝑚′,𝑛′}𝐺_{𝑚′𝑛′𝑞}^𝜋₀₀^2𝜋𝐹^𝑒_𝑞𝜃_𝑘,𝜙_𝑘𝑃^𝑚′_𝑛′×cos𝜃_𝑘𝑒^{𝑖𝑚′𝜙𝑘}𝑒^{𝑖𝐤𝑞⋅𝐫}𝑘²_𝑞sin𝜃_𝑘𝑑𝜃_𝑘𝑑𝜙_𝑘. (21) Using the identity to stand for the part 𝑒^{𝑖𝐤𝑞⋅𝐫},𝑒^{𝑖𝐤⋅𝐫}=^∞_𝑛=0𝑖^𝑛(2𝑛+1)𝑗_𝑛()×𝑘𝑟_𝑛𝑚=0(𝑛−𝑚)!()!𝑃𝑛+𝑚^𝑚𝑛cos𝜃_𝑘𝑃^𝑚_𝑛(cos𝜃)𝑒^{𝑖𝑚(𝜙−𝜙𝑘)}+^𝑛_𝑚=0(𝑛−𝑚)!()!𝑃𝑛+𝑚^𝑚𝑛cos𝜃_𝑘𝑃^𝑚_𝑛(cos𝜃)𝑒^{−𝑖𝑚(𝜙−𝜙𝑘)}. (22) Substituting (22) into (21), we achieve the solution of 𝐄(𝐫) for a general homogeneous gyrotropic anisotropic media. Then, the vector spherical wave functions is employed to express the eigenvector, which is one key step. After this, the internal field can be in the form of VSWFs:𝐅^𝑒_𝑞𝜃,𝜙𝑒^{𝑖𝐤𝑞⋅𝐫}=_𝑚,𝑛𝐴^𝑒_𝑚𝑛𝑞𝜃_𝑘𝐌⁽¹⁾_𝑚𝑛𝑟,𝑘_𝑞+𝐵^𝑒_𝑚𝑛𝑞𝜃_𝑘𝐍⁽¹⁾_𝑚𝑛𝑟,𝑘_𝑞+𝐶^𝑒_𝑚𝑛𝑞𝜃_𝑘𝐋⁽¹⁾_𝑚𝑛𝑟,𝑘_𝑞𝑒^{−𝑖𝑚𝜙𝑘}. (23) The other internal field parameters, 𝐴^𝑒_𝑚𝑛𝑞(𝜃_𝑘), 𝐵^𝑒_𝑚𝑛𝑞(𝜃_𝑘), and 𝐶^𝑒_𝑚𝑛𝑞(𝜃_𝑘) can be obtained and details are from [27, 33].

Inserting (23) into (21), and integrating with respect to 𝜙_𝑘, we have𝐄(𝑟)=²_{𝑞=1𝑚,𝑛𝑛′}2𝜋𝐺_{𝑚𝑛′𝑞}×^𝜋₀𝐴^𝑒_𝑚𝑛𝑞𝜃_𝑘𝐌⁽¹⁾_𝑚𝑛𝑟,𝑘_𝑞+𝐵^𝑒_𝑚𝑛𝑞𝜃_𝑘𝐍⁽¹⁾_𝑚𝑛𝑟,𝑘_𝑞+𝐶^𝑒_𝑚𝑛𝑞𝜃_𝑘𝐋⁽¹⁾_𝑚𝑛𝑟,𝑘_𝑞×𝑃^𝑚′_𝑛′cos𝜃_𝑘𝑒^{𝑖𝑚′𝜙𝑘}𝑘²_𝑞sin𝜃_𝑘𝑑𝜃_𝑘𝑑𝜙_𝑘, (24) which is an eigenfunction representation of the E-field in gyrotropic anisotropic media. We can get the H-field in a similar form [27, 33].

From the result in (24), it is found that we can use the VSWFs to present the field, and this procedure is also suitable for the external area. Afterwards, we take the plane wave as an example. Assuming that an incident plane wave is 𝐄=𝐱𝐸₀𝑒^{𝑖𝑘0𝑧}, the incident electromagnetic fields can be expanded by an infinite series of spherical vector wave functions for an isotropic medium as follows:𝐄^inc=𝐸_0𝑛,𝑚𝛿_𝑚,1+𝛿_𝑚,−1×𝑎^𝑥_𝑚𝑛𝐌⁽¹⁾_𝑚𝑛𝑘₀,𝐫+𝑏^𝑥_𝑚𝑛𝐍⁽¹⁾_𝑚𝑛𝑘₀,𝐇,𝐫_inc=𝑘₀𝑖𝜔𝜇₀𝐸_0𝑛,𝑚𝛿_𝑚,1+𝛿_𝑚,−1×𝑎^𝑥_𝑚𝑛𝐌⁽¹⁾_𝑚𝑛𝑘₀,𝐫+𝑏^𝑥_𝑚𝑛𝐍⁽¹⁾_𝑚𝑛𝑘₀,𝐫, (25) where𝑎^𝑥_𝑚𝑛=𝑖^𝑛+12𝑛+1()𝑖2𝑛𝑛+1,𝑚=1,_𝑛+12𝑛+12𝑏,𝑚=−1,_𝑥𝑚𝑛=𝑖^𝑛+12𝑛+12𝑛(𝑛+1),𝑚=1,−𝑖^𝑛+12𝑛+12𝛿,𝑚=−1,_𝑠,𝑙=1,𝑠=𝑙,0,𝑠≠𝑙, (26) the scattering fields are expanded as𝐄^𝑠=_𝑛,𝑚𝐴^𝑠_𝑚𝑛𝐌⁽³⁾_𝑚𝑛𝑘₀,𝐫+𝐵^𝑥_𝑚𝑛𝐍⁽³⁾_𝑚𝑛𝑘₀,𝐇,𝐫_𝑠=𝑘₀𝑖𝜔𝜇_0𝑛,𝑚𝐴^𝑠_𝑚𝑛𝐌⁽³⁾_𝑚𝑛𝑘₀,𝐫+𝐵^𝑥_𝑚𝑛𝐍⁽³⁾_𝑚𝑛𝑘₀,𝐫, (27) where 𝐴^𝑠_𝑚𝑛 and 𝐵^𝑠_𝑚𝑛 are unknown coefficients and stand for the external field information, and 𝑘₀=𝜔√𝜇₀𝜀₀, 𝜇₀ and 𝜀₀ denote the free space, wave number, permeability, and permittivity. Until here, this method is being discussed in an unbounded material. We should match the boundary condition on the surface when we limit it to an anisotropic sphere. The tangential components of the electromagnetic field continues at 𝑟=𝑎 and we have^2∞_{𝑞=1𝑛′}2𝜋𝐺_{𝑚𝑛′𝑞}^𝜋₀𝑄_𝑚𝑛𝑞𝑃^𝑚′_𝑛′cos𝜃_𝑘𝑘²_𝑞sin𝜃_𝑘𝑑𝜃_𝑘=𝐸₀𝛿_𝑚,1+𝛿_𝑚,−1𝑎^𝑥_𝑚𝑛⋅𝑖𝑘₀𝑎²_{2∞𝑞=1𝑛′},2𝜋𝐺_{𝑚𝑛′𝑞}^𝜋₀𝑅_𝑚𝑛𝑞𝑃^𝑚′_𝑛′cos𝜃_𝑘𝑘²_𝑞sin𝜃_𝑘𝑑𝜃_𝑘=𝐸₀𝛿_𝑚,1+𝛿_𝑚,−1𝑏^𝑥_𝑚𝑛⋅𝑖𝑘₀𝑎², (28) where𝑄_𝑚𝑛𝑞=𝐴^𝑒_𝑚𝑛𝑞1𝑘₀𝑟𝑑𝑑𝑟𝑟ℎ_𝑛⁽¹⁾𝑘₀𝑟𝑗_𝑛𝑘_𝑞𝑟−𝑖𝜔𝜇₀𝑘₀𝐵^ℎ_𝑚𝑛𝑞1𝑘_𝑞𝑟𝑑𝑑𝑟𝑟𝑗_𝑛𝑘_𝑞𝑟+𝐶^ℎ_𝑚𝑛𝑞𝑗_𝑛𝑘_𝑞𝑟𝑟⋅ℎ_𝑛⁽¹⁾𝑘₀𝑟_𝑟=𝑎,𝑅_𝑚𝑛𝑞=𝑖𝜔𝜇₀𝑘₀𝐴^ℎ_𝑚𝑛𝑞1𝑘₀𝑟𝑑𝑑𝑟𝑟ℎ_𝑛⁽¹⁾𝑘₀𝑟𝑗_𝑛𝑘_𝑞𝑟−𝐵^𝑒_𝑚𝑛𝑞1𝑘_𝑞𝑟𝑑𝑑𝑟𝑟𝑗_𝑛𝑘_𝑞𝑟+𝐶^𝑒_𝑚𝑛𝑞𝑗_𝑛𝑘_𝑞𝑟𝑟⋅ℎ_𝑛⁽¹⁾𝑘₀𝑟_𝑟=𝑎. (29) The scattering coefficients 𝐴^𝑠_𝑚𝑛 and 𝐵^𝑠_𝑚𝑛, are thus expressed as𝐴^𝑠_𝑚𝑛=1ℎ_𝑛⁽¹⁾𝑘₀𝑎^∞2_{𝑛′=0𝑞=1}2𝜋𝐺_𝑚𝑛𝑞^𝜋₀𝐴^𝑒_𝑚𝑛𝑞𝑗_𝑛𝑘_𝑞𝑎×𝑃^𝑚′_𝑛′′cos𝜃_𝑘𝑘²_𝑞sin𝜃_𝑘𝑑𝜃_𝑘−𝐸₀𝛿_𝑚,1+𝛿_𝑚,1𝑎^𝑥_𝑚𝑛𝑗_𝑛𝑘₀𝑎,𝐵^𝑠_𝑚𝑛=1ℎ_𝑛⁽¹⁾𝑘₀𝑎𝑖𝜔𝜇₀𝑘₀^∞2_{𝑛′=0𝑞=1}2𝜋𝐺_𝑚𝑛𝑞^𝜋₀𝐴^ℎ_𝑚𝑛𝑞𝑗_𝑛𝑘_𝑞𝑎×𝑃^𝑚′_𝑛′cos𝜃_𝑘𝑘²_𝑞sin𝜃_𝑘𝑑𝜃_𝑘−𝐸₀𝛿_𝑚,1+𝛿_𝑚,1𝑏^𝑥_𝑚𝑛𝑗_𝑛𝑘₀𝑎. (30) Then the internal and external field can be expressed in the form of the coefficients.

2.3. T-Matrix Method

Different from the angular expansion method, here we use magnetic induction 𝐁 to rewrite the Maxwell equation [32]:∇×𝜀_𝑠𝜀⁻¹⋅∇×𝜇_𝑠𝜇⁻¹⋅𝐁−𝑘²_𝑠𝐁=0. (31) The divergenceless property (31) suggests that 𝐁 be expanded in terms of the vector spherical wave functions 𝐌⁽¹⁾_𝑚𝑛(𝑘,𝐫) and 𝐍⁽¹⁾_𝑚𝑛(𝑘,𝐫) [32, 41, 42]:𝐁=_𝑛,𝑚𝐸_𝑚𝑛𝑑_𝑚𝑛𝐌⁽¹⁾_𝑚𝑛(𝑘,𝐫)+𝑐_𝑚𝑛𝐍⁽¹⁾_𝑚𝑛(𝑘,𝐫), (32) where 𝑘 is not undetermined. As mentioned above, there are three kinds of VSWF's 𝐌⁽¹⁾_𝑚𝑛(𝑘,𝐫), 𝐍⁽¹⁾_𝑚𝑛(𝑘,𝐫) and 𝐋⁽¹⁾_𝑚𝑛(𝑘,𝐫). For this reason, (31) does not involve 𝐋⁽¹⁾_𝑚𝑛(𝑘,𝐫), because its curl is 0, for details of the parameters see [32, 41, 42]. Since the VSWFs is a complete frame and can stand for all vectors in the space, we can use them to present the elements in (31), written as𝜀_𝑠𝜀⁻¹⋅𝐌_𝑚𝑛=^+∞𝑣_{𝑣=0𝑢=−𝑣}𝑜̃^𝑚𝑛_𝑢𝑣𝐌_𝑢𝑣+𝑝^𝑚𝑛_𝑢𝑣𝐍_𝑢𝑣+𝑞^𝑚𝑛_𝑢𝑣𝐋_𝑢𝑣,𝜀_𝑠𝜀⁻¹⋅𝐍_𝑚𝑛=^+∞𝑣_{𝑣=0𝑢=−𝑣}𝑜^𝑚𝑛_𝑢𝑣𝐌_𝑢𝑣+𝑝^𝑚𝑛_𝑢𝑣𝐍_𝑢𝑣+𝑞^𝑚𝑛_𝑢𝑣𝐋_𝑢𝑣,𝜇_𝑠𝜇⁻¹⋅𝐌_𝑚𝑛=^+∞𝑣_{𝑣=0𝑢=−𝑣}𝑔^𝑚𝑛_𝑢𝑣𝐌_𝑢𝑣+𝑒̃^𝑚𝑛_𝑢𝑣𝐍_𝑢𝑣+𝑓^𝑚𝑛_𝑢𝑣𝐋_𝑢𝑣,𝜇_𝑠𝜇⁻¹⋅𝐍_𝑚𝑛=^+∞𝑣_{𝑣=0𝑢=−𝑣}𝑔^𝑚𝑛_𝑢𝑣𝐌_𝑢𝑣+𝑒^𝑚𝑛_𝑢𝑣𝐍_𝑢𝑣+𝑓^𝑚𝑛_𝑢𝑣𝐋_𝑢𝑣, (33) with𝑜̃^𝑚𝑛_𝑢𝑣=𝛿_𝑛𝑣𝛿_𝑚𝑢+𝑛²+𝑛−𝑚²𝜀^′_𝑟+𝑚𝜀^′_𝜅𝛿_𝑛𝑣𝛿_𝑚𝑢𝑛(),𝑝𝑛+1_{𝑚𝑛𝑢𝑣}=𝑖(𝑛+𝑚)𝑚𝜀^′_𝑟−(𝑛+1)𝜀^′_𝜅𝛿_{𝑛−1,𝑣}𝛿_𝑚𝑢𝑛()+𝑖(𝑚2𝑛+1𝑛−𝑚+1)𝜀^′_𝑟+𝑛𝜀^′_𝜅𝛿_𝑛+11,𝑣𝛿_𝑚𝑢(),𝑛+1)(2𝑛+1𝑞^𝑚𝑛_𝑢𝑣=−𝑖(𝑛+𝑚)𝑚𝜀^′_𝑟−(𝑛+1)𝜀^′_𝜅𝛿_{𝑛−1,𝑣}𝛿_𝑚𝑢()+𝑖(𝑚2𝑛+1𝑛−𝑚+1)𝜀^′_𝑟+𝑛𝜀^′_𝜅𝛿_𝑛+11,𝑣𝛿_𝑚𝑢(),2𝑛+1𝑜^𝑚𝑛_𝑢𝑣𝑖(𝑚=−𝑛+𝑚)(𝑛+1)𝜀^′_𝑟+(𝑛−1)𝜀^′_𝜅𝛿_{𝑛−1,𝑣}𝛿_𝑚𝑢𝑛()−𝑖()𝑛𝑚𝑛−1)(2𝑛+1𝑛−𝑚+1𝜀^′_𝑟−(𝑛+2)𝜀^′_𝜅𝛿_𝑛+11,𝑣𝛿_𝑚𝑢(),𝑛+1)(𝑛+2)(2𝑛+1𝑝^𝑚𝑛_𝑢𝑣=𝛿_𝑛𝑣𝛿_𝑚𝑢+2𝑛²+2𝑛+3𝑚²+2𝑛²+2𝑛−3𝑛(𝑛+1)𝜀^′_𝑟+4𝑛²+4𝑛−3𝑚𝜀^′_𝜅𝛿_𝑛𝑣𝛿_𝑚𝑢𝑛()−()𝑛+1)(2𝑛−1)(2𝑛+3𝑛+1)(𝑛+𝑚−1)(𝑛+𝑚𝜀^′_𝑟𝛿_{𝑛−2,𝑣}𝛿_𝑚𝑢()−𝑛()𝑛−1)(2𝑛−1)(2𝑛+1𝑛−𝑚+1)(𝑛−𝑚+2𝜀^′_𝑟𝛿_𝑛+2,𝑣𝛿_𝑚𝑢(),𝑛+2)(2𝑛+1)(2𝑛+3𝑞^𝑚𝑛_𝑢𝑣𝑛=−²+𝑛−3𝑚²𝜀^′_𝑟−𝑚(2𝑛−1)(2𝑛+3)𝜀^′_𝜅𝛿_𝑛𝑣𝛿_𝑚𝑢()+()2𝑛−1)(2𝑛+3𝑛+1)(𝑛+𝑚−1)(𝑛+𝑚𝜀^′_𝑟𝛿_{𝑛−2,𝑣}𝛿_𝑚𝑢()−𝑛()2𝑛−1)(2𝑛+1𝑛−𝑚+1)(𝑛−𝑚+2𝜀^′_𝑟𝛿_𝑛+2,𝑣𝛿_𝑚𝑢(),2𝑛+1)(2𝑛+3𝑔^𝑚𝑛_𝑢𝑣=𝛿_𝑛𝑣𝛿_𝑚𝑢+𝑛²+𝑛−𝑚²𝜇^′_𝑟+𝑚𝜇^′_𝜅𝛿_𝑛𝑣𝛿_𝑚𝑢𝑛(),𝑛+1𝑒̃^𝑚𝑛_𝑢𝑣=𝑖(𝑛+𝑚)𝑚𝜇^′_𝑟−(𝑛+1)𝜇^′_𝜅𝛿_{𝑛−1,𝑣}𝛿_𝑚𝑢𝑛()+𝑖(𝑚2𝑛+1𝑛−𝑚+1)𝜇^′_𝑟+𝑛𝜇^′_𝜅𝛿_𝑛+11,𝑣𝛿_𝑚𝑢(),𝑓𝑛+1)(2𝑛+1_{𝑚𝑛𝑢𝑣}=−𝑖(𝑛+𝑚)𝑚𝜇^′_𝑟−(𝑛+1)𝜇^′_𝜅𝛿_{𝑛−1,𝑣}𝛿_𝑚𝑢()+𝑖(𝑚2𝑛+1𝑛−𝑚+1)𝜇^′_𝑟+𝑛𝜇^′_𝜅𝛿_𝑛+11,𝑣𝛿_𝑚𝑢(),2𝑛+1𝑔^𝑚𝑛_𝑢𝑣𝑖(𝑚=−𝑛+𝑚)(𝑛+1)𝜇^′_𝑟+(𝑛−1)𝜇^′_𝜅𝛿_{𝑛−1,𝑣}𝛿_𝑚𝑢𝑛()−𝑖()𝑛𝑚𝑛−1)(2𝑛+1𝑛−𝑚+1𝜇^′_𝑟−(𝑛+2)𝜇^′_𝜅𝛿_𝑛+11,𝑣𝛿_𝑚𝑢(),𝑛+1)(𝑛+2)(2𝑛+1𝑒^𝑚𝑛_𝑢𝑣=𝛿_𝑛𝑣𝛿_𝑚𝑢+2𝑛²+2𝑛+3𝑚²+2𝑛²+2𝑛−3𝑛(𝑛+1)𝜇^′_𝑟+4𝑛²+4𝑛−3𝑚𝜇^′_𝜅𝛿_𝑛𝑣𝛿_𝑚𝑢𝑛()−()𝑛+1)(2𝑛−1)(2𝑛+3𝑛+1)(𝑛+𝑚−1)(𝑛+𝑚𝜇^′_𝑟𝛿_{𝑛−2,𝑣}𝛿_𝑚𝑢()−𝑛()𝑛−1)(2𝑛−1)(2𝑛+1𝑛−𝑚+1)(𝑛−𝑚+2𝜇^′_𝑟𝛿_𝑛+2,𝑣𝛿_𝑚𝑢(),𝑛+2)(2𝑛+1)(2𝑛+3𝑓^𝑚𝑛_𝑢𝑣𝑛=−²+𝑛−3𝑚²𝜇^′_𝑟−𝑚(2𝑛−1)(2𝑛+3)𝜇^′_𝜅𝛿_𝑛𝑣𝛿_𝑚𝑢()+()2𝑛−1)(2𝑛+3𝑛+1)(𝑛+𝑚−1)(𝑛+𝑚𝜇^′_𝑟𝛿_{𝑛−2,𝑣}𝛿_𝑚𝑢()−𝑛()2𝑛−1)(2𝑛+1𝑛−𝑚+1)(𝑛−𝑚+2𝜇^′_𝑟𝛿_𝑛+2,𝑣𝛿_𝑚𝑢().2𝑛+1)(2𝑛+3 (34)Here, all the coefficients are obtained using the orthogonality of the VSWFs [32, 41, 42].

Therefore, one has𝜇_𝑠𝜇⁻¹=⋅𝐁_𝑛,𝑚𝐸_𝑚𝑛𝑑_𝑚𝑛𝐌⁽¹⁾_𝑚𝑛(𝑘,𝐫)+𝑐_𝑚𝑛𝐍⁽¹⁾_𝑚𝑛(𝑘,𝐫)+𝑤_𝑚𝑛𝐋⁽¹⁾_𝑚𝑛(+𝑘,𝐫)𝑤₀₀𝐋⁽¹⁾₀₀(𝑘,𝐫), (35) where 𝑑_𝑚𝑛=_𝑣,𝑢𝐸_𝑢𝑣𝐸_𝑚𝑛𝑔^𝑢𝑣_𝑚𝑛𝑑_𝑢𝑣+𝑔^𝑢𝑣_𝑚𝑛𝑐_𝑢𝑣,𝑐_𝑚𝑛=_𝑣,𝑢𝐸_𝑢𝑣𝐸_𝑚𝑛𝑒̃^𝑢𝑣_𝑚𝑛𝑑_𝑢𝑣+𝑒^𝑢𝑣_𝑚𝑛𝑐_𝑢𝑣,𝑤_𝑚𝑛=_𝑣,𝑢𝐸_𝑢𝑣𝐸_𝑚𝑛𝑓^𝑢𝑣_𝑚𝑛𝑑_𝑢𝑣+𝑓^𝑢𝑣_𝑚𝑛𝑐_𝑢𝑣,𝑤₀₀=−23𝜇^′_𝜅𝑑₀₁+215𝜇^′_𝑟𝑐₀₂𝐸₀. (36) Repeat the procedure and reuse the properties of the VSWFs bellow until (31) can be written in the form of 𝐌⁽¹⁾_𝑚𝑛(𝑘,𝐫), 𝐍⁽¹⁾_𝑚𝑛(𝑘,𝐫), and 𝐋⁽¹⁾_𝑚𝑛(𝑘,𝐫):∇×𝐌^(𝑗)_𝑚𝑛−𝑘𝐍^(𝑗)_𝑚𝑛=0,∇×𝐍^(𝑗)_𝑚𝑛−𝑘²𝐌^(𝑗)_𝑚𝑛=0,∇×𝐋^(𝑗)_𝑚𝑛=0. (37) One gets𝐸𝑑_{𝑛,𝑚𝑚𝑛}^≈_𝑚𝑛𝐌⁽¹⁾_𝑚𝑛(𝑘,𝐫)+^≈𝑐_𝑚𝑛𝐍⁽¹⁾_𝑚𝑛(𝑘,𝐫)=0 (38) with𝑑^≈_𝑚𝑛=𝑘²_𝑣,𝑢𝐸_𝑢𝑣𝐸_𝑚𝑛𝑝^𝑢𝑣_𝑚𝑛𝑑_𝑢𝑣+𝑝^𝑢𝑣_𝑚𝑛𝑐_𝑢𝑣−𝑘²_𝑠𝑑_𝑚𝑛,_≈𝑐_𝑚𝑛=𝑘²_𝑣,𝑢𝐸_𝑢𝑣𝐸_𝑚𝑛𝑜^𝑢𝑣_𝑚𝑛𝑑_𝑢𝑣+𝑜̃^𝑢𝑣_𝑚𝑛𝑐_𝑢𝑣−𝑘²_𝑠𝑐_𝑚𝑛. (39) Then, we write the equation form:𝑃𝑃𝑂𝑂𝑑𝑐𝑑𝑐=𝜆. (40) The expression for 𝑃, 𝑃, 𝑂, 𝑂 is in [32]. Let 𝜆_𝑙 and (𝑑_{𝑚𝑛,𝑙},𝑐_{𝑚𝑛,𝑙})^𝑇, denote, respectively, the eigenvalues and the corresponding eigenvectors of eigensystem (40), with 𝑙 representing the index of eigenvalues and corresponding eigenvectors. One can then build a new set of vector 𝐕_𝑙 in space based on the eigenvectors:𝐕_𝑙𝑘=−^𝑙𝜔_𝑛,𝑚𝐸_𝑚𝑛𝑑_{𝑚𝑛,𝑙}𝐌⁽¹⁾_𝑚𝑛𝑘_𝑙,𝐫+𝑐_{𝑚𝑛,𝑙}𝐍⁽¹⁾_𝑚𝑛𝑘_𝑙,𝐫. (41) Thus, we can express 𝐁 as𝐁=_𝑙𝛼_𝑙𝐕_𝑙=−_𝑙𝛼_𝑙𝑘_𝑙𝜔_𝑛,𝑚𝐸_𝑚𝑛𝑑_{𝑚𝑛,𝑙}𝐌⁽¹⁾_𝑚𝑛𝑘_𝑙,𝐫+𝑐_{𝑚𝑛,𝑙}𝐍⁽¹⁾_𝑚𝑛𝑘_𝑙,𝐫. (42) Use the relation of (6), 𝐇 and 𝐄 fields can be written as𝐇=𝜇⁻¹⋅𝐁=−_𝑛,𝑚𝐸_𝑚𝑛𝑙𝑘_𝑙𝜔𝜇_𝑠𝛼_𝑙×𝑑_{𝑚𝑛,𝑙}𝐌⁽¹⁾_𝑚𝑛𝑘_𝑙,𝐫+𝑐_{𝑚𝑛,𝑙}𝐍⁽¹⁾_𝑚𝑛𝑘_𝑙,𝐫+𝑤_{𝑚𝑛,𝑙}𝐋⁽¹⁾_𝑚𝑛𝑘_𝑙,𝐫−_𝑙𝑘_𝑙𝜔𝜇_𝑠𝛼_𝑙𝑤_00,𝑙𝐋⁽¹⁾₀₀(𝑘,𝐫),𝐄=−𝑖_𝑛,𝑚𝐸_𝑚𝑛𝑙𝛼_𝑙𝑐_{𝑚𝑛,𝑙}𝐌⁽¹⁾_𝑚𝑛𝑘_𝑙,𝐫+𝑑_{𝑚𝑛,𝑙}𝐍⁽¹⁾_𝑚𝑛𝑘_𝑙+𝑤,𝐫_{𝑚𝑛,𝑙}𝜆_𝑙𝐋⁽¹⁾_𝑚𝑛𝑘_𝑙,𝐫−𝑖_𝑙𝛼_𝑙𝑤_00,𝑙𝜆_𝑙𝐋⁽¹⁾₀₀(𝑘,𝐫). (43) As we have discussed in the angular expansion method, here, we also give the form of the incident wave and scatted wave.

For incident wave,𝐄_inc=−_𝑛,𝑚𝑖𝐸_𝑚𝑛𝑝_𝑚𝑛𝐍⁽¹⁾_𝑚𝑛(𝑘,𝐫)+𝑞_𝑚𝑛𝐌⁽¹⁾_𝑚𝑛(,𝐇𝑘,𝐫)_inc𝑘=−⁰𝜔𝜇_0𝑛,𝑚𝐸_𝑚𝑛𝑞_𝑚𝑛𝐍⁽¹⁾_𝑚𝑛(𝑘,𝐫)+𝑝_𝑚𝑛𝐌⁽¹⁾_𝑚𝑛(𝑘,𝐫). (44) For scattered wave,𝐄_𝑠=_𝑛,𝑚𝑖𝐸_𝑚𝑛𝑎_𝑚𝑛𝐍⁽³⁾_𝑚𝑛𝑘₀,𝐫+𝑏_𝑚𝑛𝐌⁽³⁾_𝑚𝑛𝑘₀,𝐇,𝐫_𝑠=𝑘₀𝜔𝜇_0𝑛,𝑚𝐸_𝑚𝑛𝑏_𝑚𝑛𝐍⁽³⁾_𝑚𝑛𝑘₀,𝐫+𝑎_𝑚𝑛𝐌⁽³⁾_𝑚𝑛𝑘₀,𝐫. (45) The relevant coefficients are in [41]. Then, match boundary conditions on the sphere surface, we can get a system of equations for the coefficients, which is𝜉^′_𝑛(𝑥)𝜓^′_𝑛(𝑥)𝑎_𝑚𝑛+_𝑙1𝑚_𝑠𝑘_𝑙𝜓^′_𝑛𝑘_𝑙𝑚_𝑠𝑥𝜓^′_𝑛(𝑥)𝑑_{𝑚𝑛,𝑙}𝛼_𝑙+_𝑙𝜇₀𝜇_𝑠𝑗_𝑛𝑘_𝑙𝑚_𝑠𝑥𝜓^′_𝑛(𝑥)𝜔_{𝑚𝑛,𝑙}𝛼_𝑙=𝑝_𝑚𝑛,𝜉_𝑛(𝑥)𝜓_𝑛(𝑥)𝑏_𝑚𝑛+_𝑙1𝑚_𝑠𝑘_𝑙𝜓_𝑛𝑘_𝑙𝑚_𝑠𝑥𝜓_𝑛(𝑥)𝑐_{𝑚𝑛,𝑙}𝛼_𝑙=𝑞_𝑚𝑛,𝜉_𝑛(𝑥)𝜓_𝑛(𝑥)𝑎_𝑚𝑛+_𝑙𝜇₀𝜆_𝑙𝜇_𝑠𝜓_𝑛𝑘_𝑙𝑚_𝑠𝑥𝜓_𝑛(𝑥)𝑑_{𝑚𝑛,𝑙}𝛼_𝑙=𝑝_𝑚𝑛,𝜉^′_𝑛(𝑥)𝜓^′_𝑛(𝑥)𝑏_𝑚𝑛+_𝑙𝜇₀𝜆_𝑙𝜇_𝑠𝜓^′_𝑛𝑘_𝑙𝑚_𝑠𝑥𝜓^′_𝑛(𝑥)𝑐_{𝑚𝑛,𝑙}𝛼_𝑙+_𝑙𝜇₀𝜇_𝑠𝑗_𝑛𝑘_𝑙𝑚_𝑠𝑥𝜓^′_𝑛(𝑥)𝜔_{𝑚𝑛,𝑙}𝛼_𝑙=𝑞_𝑚𝑛, (46) then rewrite the matrix in a clear form:0Λ0Λ𝑎𝑏+𝑉𝑉𝑝𝑞𝛼=,𝑎𝑏+Λ00Λ𝑈𝑈𝑝𝑞𝛼= (47) withΛ_{𝑚𝑛,𝑢𝑣}=𝑆_𝑛(𝑥)𝛿_𝑛𝑣𝛿_𝑚𝑢,Λ_{𝑚𝑛,𝑢𝑣}=𝑆_𝑛(𝑥)𝛿_𝑛𝑣𝛿_𝑚𝑢,𝑈_{𝑚𝑛,𝑙}=1𝑚_𝑠𝑘_𝑙𝑇_𝑛𝑥,𝑚_𝑠𝑘_𝑙𝑥𝑐_{𝑚𝑛,𝑙},𝑈_{𝑚𝑛,𝑙}=1𝑚_𝑠𝑘_𝑙𝑇_𝑛𝑥,𝑚_𝑠𝑘_𝑙𝑥𝑑_{𝑚𝑛,𝑙},𝑉_{𝑚𝑛,𝑙}=𝜇₀𝜆_𝑙𝜇_𝑠𝑇_𝑛𝑥,𝑚_𝑠𝑘_𝑙𝑥𝑑_{𝑚𝑛,𝑙},𝑉_{𝑚𝑛,𝑙}=𝜇₀𝜆_𝑙𝜇_𝑠𝑇_𝑛𝑥,𝑚_𝑠𝑘_𝑙𝑥𝑐_{𝑚𝑛,𝑙}+𝑊_{𝑚𝑛,𝑙},𝑊_{𝑚𝑛,𝑙}=𝜇₀𝜇_𝑠1𝑚_𝑠𝑘_𝑙𝑥𝑇_𝑛𝑥,𝑚_𝑠𝑘_𝑙𝑥𝐷_𝑛⁽¹⁾𝑚_𝑠𝑘_𝑙𝑥𝑤_{𝑚𝑛,𝑙},𝑆_𝑛(𝑥)=𝜉_𝑛(𝑥)𝜓_𝑛(𝑥)𝑆_𝑛(𝑥)=𝜉^′_𝑛(𝑥)𝜓^′_𝑛(𝑥),𝑇_𝑛()=𝜓𝑥,𝑧^𝑛(𝑧)𝜓_𝑛(𝑥)𝑇_𝑛()=𝜓𝑥,𝑧^′𝑛(𝑧)𝜓^′_𝑛(𝑥). (48) Afterwards, we can solve the equations and get the coefficients matrix, and the scattering information is in the matrix (^𝑎_𝑏):𝑝𝑞,𝑎𝑏𝑝𝑞,𝛼=𝑅=𝑆 (49) where𝑍=𝑌00−𝑌⁻¹𝑉−𝑈,𝑉−𝑈𝑆=𝑍𝑅,𝑌=Λ−Λ. (50)

2.4. Numerical Results and Comparison

For the two methods mentioned above, difference between them will be analyzed as follows. In angular expansion method, the plane wave expansion is employed and the wave in such complex medium is a combination of plane wave of different direction. The whole procedure is clear and easy to understand. At the same time, the wave number 𝑘 has its physical meaning, which can be expressed. Theoretically, this method can deal with any complex medium. Different from angular expansion method, the T-matrix method does not give an expression for the wave number, the properties of spherical vector wave function are used instead and all the wave information is gathered from the matrix. In computation, the T-matrix method has more advantage over angular expansion method, which does not need to consider grid and can be applied to calculate larger problem.

As many discussions and comparisons published already, here we mainly show the strength of the two methods [32]. Figure 1 shows the gyroelectric influence on the Radar cross-section, here 𝜇_𝜅=0,𝜇_𝑠𝜇_𝑟=4𝜇₀𝜇_𝑠=2𝜇₀,𝜀_𝑠𝜀_𝑟=4𝜀₀,𝜀_𝑠=2𝜀₀. As we can see, when the off-diagonal parameters increase, the backscattering (180^∘) will also increase. Compared to Figure 1, Figure 2 shows the gyromagnetic influence on the RCS, here 𝜀_𝑠𝜀_𝑟=2.4𝜀₀,𝜀_𝑠=2𝜀₀, the less scattering areas is reduced from three to one. Figure 3 gives a general case, the influences of permittivity and permeability are both considered; and this is also a lossy sphere, where 𝜀_𝑠=(2+1.0𝑖)𝜀₀, 𝜇_𝑠=(2+1.0𝑖)𝜇₀. When increasing the permittivity elements of the materials, the scattering for larger than 80^∘ area will increase obviously.

Figure 1 

Normalized Radar cross-section (RCS) values versus the scattering angle. Different gyroelectric influence is considered. Here, 𝑘0𝑟=𝜋.

Figure 2 

Normalized Radar cross-section (RCS) values versus the scattering angle. Different gyromagnetic influence is considered. Here, 𝑘0𝑟=0.75𝜋.

Figure 3 

Normalized Radar Cross Section (RCS) values versus the scattering angle. Here the permeability and the permittivity are complex number. The Loss gyrotropic influence is considered. Here 𝑘0𝑟=0.75𝜋.

3. Electromagnetic Wave Scattering by a Radial Anisotropic Sphere

3.1. Radial Anisotropy of Spheres

As another type medium, the radial anisotropy materials have parameters given by [15]𝜀𝜀=^𝑟𝑟00𝜇0𝜏𝜎0−𝜎𝜏,𝜇=^𝑟𝑟000𝛾𝜉0−𝜉𝛾, (51) where the identity dyadic is expressed as 𝐈̂=𝐫̂𝐫̂+𝜽𝜽+𝝓𝝓. Different from the Cartesian anisotropic (CA), less discussion have been published, because the radial anisotropic (RA) material is difficult to be fabricated [7]. The research attracts much attention recently due to the properties of materials with novel parameters. How RA particles interact with waves is an essential topic which will provide us a more physical insight into the invisibility phenomenon [43–47], enhanced surface plasmon resonance [48, 49], and Fano resonance [50]. Although Monzon developed the variables separation method and demonstrated a perfect explanation to the solution form [15], researchers prefer the Debye potential or other potentials similar [51–53]. Recently, Novitsky has stated a general form for the material, and then extend the T-matrix method to a multilayer case [54]. When a Hamilton operator is used, the Maxwell equations will be greatly simplified. By employing the impedance matrix, the field area is both simple and clear. However, this solution is just another form of variables separation method and only suitable to the uniaxial anisotropic case, which does not satisfy the general situation [54]. Herein, we explain Monzon's method and conclude an explanation to the solution.

3.2. Variables Separation Method

Firstly, we define new vectors to be [15]𝝃_±=𝜽±𝑗𝝓√.2sin𝜃 (52) Then, the E-field and H-field can be expressed in the new coordinates:𝐄(𝐫)=𝐸_𝑟𝐫̂+𝑒₋𝝃₊+𝑒₊𝝃₋𝑒_±=sin𝜃√2𝐸_𝜃±𝑗𝐸_𝜙,𝐇(𝐫)=𝐻_𝑟𝐫̂+ℎ₋𝝃₊+ℎ₊𝝃₋ℎ_±=sin𝜃√2𝐻_𝜃±𝑗𝐻_𝜙, (53) Similarly, the displacement vector is provided:𝜀⋅𝐄=𝜀_𝑟𝑟𝐸_𝑟𝐫̂+₊𝛼₊𝑒₋𝝃₊+𝛼₋𝑒₊𝝃₋,𝛼_±=𝜏±𝑗𝜎,𝜇⋅𝐇=𝜇_𝑟𝑟𝐻_𝑟𝐫̂+₊𝛽₊ℎ₋𝝃₊+𝛽₋ℎ₊𝝃₋,𝛽_±=𝛾±𝑗𝜁. (54) Here, two operators are defined as𝐷_±=1√𝜕2𝑟sin𝜃𝜕𝜕𝜃±𝑗𝜕𝜙,𝐷_𝑟=1𝑟𝜕𝜕𝑟⋅𝑟, (55) then the Laplace operator can be written as𝜕∇=𝑟+𝝃𝜕𝑟⁺𝐷₋+𝝃₋𝐷₊, (56) and Maxwell equations is correspondingly rewritten as𝑗𝐫̂∇×𝐄=sin²𝜃𝐷₊𝑒₋−𝐷₋𝑒₊+𝑗𝝃₊𝐷₋𝐸_𝑟−𝐷_𝑟𝑒₋+𝑗𝝃₋𝐷_𝑟𝑒₊−𝐷₊𝐸_𝑟,𝑗𝐫̂∇×𝐇=sin²𝜃𝐷₊ℎ₋−𝐷₋ℎ₊+𝑗𝝃₊𝐷₋𝐻_𝑟−𝐷_𝑟ℎ₋+𝑗𝝃₋𝐷_𝑟ℎ₊−𝐷₊𝐻_𝑟, (57) we insert (54) and (57) into (2) and (3). After some calculations including repeated operations on Maxwell’s equations, a coupled set of differential equations involving only the radial field components can be obtained [15]:𝜇_𝑟𝑟𝐷𝛾𝑟^2𝑟𝑟𝐻_𝑟+2sin²𝜃𝐷₊𝐷₋𝐻_𝑟+𝜔²𝜏𝜇_𝑟𝑟𝜏²+𝜎²𝐻_𝑟+𝑗𝜔𝜀_𝑟𝑟𝐷𝛾𝜏𝑟𝜎𝛾+𝜏𝜁_𝑟𝑟𝐸_𝑟𝜀=0,_𝑟𝑟𝐷𝜏𝑟^2𝑟𝑟𝐸_𝑟+2sin²𝜃𝐷₊𝐷₋𝐸_𝑟+𝜔²𝜏𝜀_𝑟𝑟𝛾²+𝜁²𝐸_𝑟−𝑗𝜔𝜇_𝑟𝑟𝐷𝛾𝜏𝑟𝜎𝛾+𝜏𝜁_𝑟𝑟𝐻_𝑟=0, (58) where𝐷₊𝐷₋=12𝑟²𝜕²𝜕𝜙²𝜕+sin𝜃𝜕𝜕𝜃sin𝜃.𝜕𝜃 (59) The field can be expressed in the following form:𝐸_𝑟=_𝑛,𝑚𝐸_𝑚𝑛(𝑟)𝑃^𝑚_𝑛(cos𝜃)𝑒^𝑗𝑚𝜙,𝐻_𝑟=_𝑛,𝑚𝐻_𝑚𝑛(𝑟)𝑃^𝑚_𝑛(cos𝜃)𝑒^𝑗𝑚𝜙. (60) Also when 𝐷₊𝐷₋ operate on other components, it has the following properties:𝐷₊𝐷₋=−𝑛(𝑛+1)sin²𝜃2𝑟², (61) we write the (58) in a convenient form:𝑑²𝑓+𝛼₁𝜌𝑓−𝛿𝑑𝑔=0𝑑²𝑔+𝛼₂𝜌𝑔−𝛿𝑑𝑓=0 (62) with𝑑𝑑=√𝑑𝜌𝜌=𝜅𝑟𝜅=𝜔𝜀_𝑟𝑟𝜇_𝑟𝑟𝛿=𝜎𝛾+𝜏𝜁√𝜏𝛾𝜀_𝑟𝑟𝜇_𝑟𝑟,𝛼₁𝜌=𝐴₁−𝐵₁𝜌²𝛼₂𝜌=𝐴₂−𝐵₂𝜌²,𝐴₁=𝜏𝛾𝛾²+𝜁²𝜀_𝑟𝑟𝜇_𝑟𝑟,𝐴₂=𝛾𝜏𝜏²+𝜎²𝜀_𝑟𝑟𝜇_𝑟𝑟,𝐵₁=𝜏𝜀_𝑟𝑟𝑛(𝑛+1),𝐵₂=𝛾𝜇_𝑟𝑟𝑛(𝑛+1). (63) For (62), there is an analytical solution to the differential equations only when the coupled component is 0 (the isotropic or uniaxial anisotropic case). For a general case (𝛿≠0), there is no analytical solution for this kind of differential equations in mathematics [55]. Therefore, we have to use the series to stand for the equation roots and the procedure is as follows:𝑓=^∞_𝑛=−∞𝜌^𝜆+𝑛𝐾_𝑛, (64)𝑔=^∞_𝑛=−∞𝜌^𝜆+𝑛𝑄_𝑛 (65)(𝜆+𝑛)(𝜆+𝑛−1)−𝐵₁𝐾_𝑛+𝐴₁𝐾_𝑛−2=𝛿(𝜆+𝑛−1)𝑄_𝑛−1, (66)(𝜆+𝑛)(𝜆+𝑛−1)−𝐵₂𝑄_𝑛+𝐴₂𝑄_𝑛−2=𝛿(𝜆+𝑛−1)𝐾_𝑛−1, (67)𝑄₁=𝛿𝜏₁+1/22𝜏₁+1+𝐵₁−𝐵₂,𝐾₂=𝛿𝜏₁+3/2𝑄₁−𝐴₁4𝜏₁+1. (68) When the original condition is given, the coefficients can be calculated step by step and then an expression for the radial component as we needed from (66) and (67) can be achieved. If the off-diagonal parameters are zero, 𝐸_𝑟 and 𝐻_𝑟 are not coupled and we encounter the most discussed case [51]:𝐸_𝑟=𝜔𝑖𝑘²_𝑡𝜕²𝜕𝑟²+𝑘²_𝑡Ψ_𝑇𝑀,𝐸_𝜃=𝜔𝑖𝑘²_𝑡1𝑟𝜕²Ψ_𝑇𝑀−1𝜕𝑟𝜕𝜃𝜀₀𝜀_𝑡1𝑟sin𝜃𝜕Ψ_𝑇𝐸,𝐸𝜕𝜙_𝜙=𝜔𝑖𝑘²_𝑡1𝜕𝑟sin𝜃²Ψ_𝑇𝑀+1𝜕𝑟𝜕𝜙𝜀₀𝜀_𝑡1𝑟𝜕Ψ_𝑇𝐸,𝐻𝜕𝜃_𝑟=𝜔𝑖𝑘²_𝑡𝜕²𝜕𝑟²+𝑘²_𝑡Ψ_𝑇𝐸,𝐻_𝜃=1𝜇₀𝜇_𝑡1𝑟sin𝜃𝜕Ψ_𝑇𝑀+𝜔𝜕𝜙𝑖𝑘²_𝑡1𝑟𝜕²Ψ_𝑇𝐸,𝐻𝜕𝑟𝜕𝜃_𝜙1=−𝜇₀𝜇_𝑡1𝑟𝜕Ψ_𝑇𝑀+𝜔𝜕𝜃𝑖𝑘²_𝑡1𝜕𝑟sin𝜃²Ψ_𝑇𝐸.𝜕𝑟𝜕𝜙 (69) For a plane wave, there are many discussions already; here, we only show the result for the internal and external field:𝑎_𝑛=𝜇_𝑡/𝜀_𝑡𝑗_𝑛𝑘₀𝑎𝑗^′_𝑣1𝑘_𝑡𝑎−𝑗^′_𝑛𝑘₀𝑎𝑗_𝑣1𝑘_𝑡𝑎ℎ_𝑛^′(2)𝑘₀𝑎𝑗_𝑣1𝑘_𝑡𝑎−𝜇_𝑡/𝜀_𝑡ℎ_𝑛⁽²⁾𝑘₀𝑎𝑗^′_𝑣1𝑘_𝑡𝑎𝑇_𝑛,𝑏_𝑛=𝜇_𝑡/𝜀_𝑡𝑗^′_𝑛𝑘₀𝑎𝑗_𝑣2𝑘_𝑡𝑎−𝑗_𝑛𝑘₀𝑎𝑗^′_𝑣2𝑘_𝑡𝑎ℎ_𝑛⁽²⁾𝑘₀𝑎𝑗^′_𝑣2𝑘_𝑡𝑎−𝜇_𝑡/𝜀_𝑡ℎ_𝑛^′(2)𝑘₀𝑎𝑗_𝑣2𝑘_𝑡𝑎𝑇_𝑛,𝑐_𝑛=𝑖𝜇_𝑡/𝜀_𝑡ℎ_𝑛⁽²⁾𝑘₀𝑎𝑗^′_𝑣1𝑘_𝑡𝑎−ℎ_𝑛^′(2)𝑘₀𝑎𝑗_𝑣1𝑘_𝑡𝑎𝑇_𝑛,𝑑_𝑛=𝑖𝜇_𝑡/𝜀_𝑡ℎ_𝑛⁽²⁾𝑘₀𝑎𝑗^′_𝑣2𝑘_𝑡𝑎−𝜇_𝑡/𝜀_𝑡ℎ_𝑛^′(2)𝑘₀𝑎𝑗_𝑣2𝑘_𝑡𝑎𝑇_𝑛,𝑇_𝑛=𝑖^−𝑛(2𝑛+1)𝑛().𝑛+1 (70) Using these coefficients, there are many interesting results [7], which will not be discussed here.

3.3. Numerical Results

Results of the method using the above coefficients are presented in the following. Figure 4 states the negative anisotropy (𝜇_𝑟𝑟<𝛾) influence on the backscattering when 𝜀_𝑟𝑟=𝜏=𝜀₀. Sharp changes can be observed when the radius changes. Compared to Figure 4, Figure 5 shows the positive anisotropy (𝜇_𝑟𝑟>𝛾) influence on the backscattering, from which the oscillations are more regular. Figures 6 and 7 give a general case for the absorbing sphere. Taking the influence of negative absorbing into accounting, we can see from Figure 6 that the backscattering will approach to a certain value as the radius increases. There will be more backscattering as the sphere has much loss. The positive absorbing properties are discussed in Figure 7, which behaves a similar phenomenon with Figure 6.

Figure 4 

Normalized backscattering values versus sphere radius. Different negative anisotropic value influence is considered. Here, we increase the negative values.

Figure 5 

Normalized backscattering values versus sphere radius. Different positive anisotropic value influence is considered. Here, we reduce the positive values.

Figure 6 

Normalized backscattering values versus sphere radius. Lossy negative anisotropic influence is considered. Here, the imaginary part for the permeability is bigger than 1.

Figure 7 

Normalized backscattering values versus sphere radius. Lossy positive anisotropic influence is considered. Here, the imaginary part for the permeability is smaller than 1.

4. Discussion and Conclusion

In this review, two different types of anisotropy have been discussed in Cartesian and spherical coordinates, respectively. The role of anisotropy in scattering properties is characterized analytically. Two main methods are introduced in Cartesian coordinates. As the first one, the angular expansion method can be used in the infinitely large area and present an obvious explanation for the properties of the medium. Different from the first method, the T-matrix method can avoid the complex function by using the orthogonal properties for 𝐌^(𝑙)_𝑚𝑛,𝐍^(𝑙)_𝑚𝑛, and 𝐋^(𝑙)_𝑚𝑛. There is no need for researchers to consider the mesh before the calculation. Since these two methods are analytical and semianalytical, respectively, the parameter’s influence on every part is very clear and thus gives us an insight into anisotropy materials.

For a spherical coordinate anisotropic material, no matter which operators are used, the variables separation remains the main method to settle the problem. Though the Debye potentials can be widely used based on the properties of the Maxwell equations, it is still not suitable to solve the coupled gyrotropic materials. Monzon’s work demonstrated the reason why and how the electric field and magnetic field coupled. When a specific situation is considered, for example, a uniaxial anisotropic media, the electric field and the magnetic field can be separated and be both expressed in the radial function form. This review provides the convenience for researchers who want to explore new properties of anisotropic materials and, therefore, design new structures.

Acknowledgments

The authors wish to thank Professor Li Lewei, and Ms. Ong Weeling for helpful discussions. This work is supported by CSC scholarship and Key Program of NSFC-Guangdong Union Foundation under Grant no. U0835004.