# Anisotropic Scattering Characteristics of a Radially Multilayered Gyrotropic Sphere

Authors: Lei Cao; Yongpin Chen; Kai Kang
Journal: International Journal of Antennas and Propagation
 (2017)

---

## Abstract

We present a new closed-form solution to the scattering of a monochromatic plane wave by a radially multilayered gyrotropic sphere using theT-matrix method. This approach can be utilized to investigate the interactions of a plane wave and a gyrotropic spherical scatterer of multiple layers with each layer characterized by both permittivity and permeability tensors. Based on the completeness and noncoplanar properties of vector spherical wave functions (VSWFs), analytical expressions of the electromagnetic fields in each gyrotropic layer are first derived. The boundary conditions are then applied on each discontinuous interface to obtain the scattering coefficients. Validations are made by first comparing the radar cross section (RCS) values of a 2-layered gyrotropic sphere with that computed from the full-wave finite element method (FEM) simulation and then reducing the general case to uniaxial case to compare the RCS values with the published results computed by Fourier transform combined with VSWFs method; in both cases good agreements are observed. Several specific cases are fully explored to investigate how the RCS are influenced by the parameters of the multilayered spherical structure. The results show that when both electric and magnetic gyrotropy tensors are considered, the RCS of the multilayered spherical scatterer can be suppressed or enhanced, depending on proper configurations of the material parameters.


---

## Body

## 1. Introduction

With the development of material science and fabrication techniques, the electromagnetic (EM) peculiarities of anisotropic media have attracted continued interests of both physics and engineering communities [1–7]. The effects of anisotropy or gyrotropy of the media must be taken into account in many practical applications, where a number of new devices with unique characteristics are designed by applying these effects [8, 9]. For studying the interactions of EM waves and gyrotropic media, the most important and challenging task is to solve the vector wave equation characterizing EM fields in gyrotropic media. In the last several decades, a few analytical approaches have been developed based on the Cartesian coordinate expression of permittivity and permeability tensors, such as the perturbation expansion method [10, 11], the method of expanding EM fields in gyrotropic media using a complete set of VSWFs [12], the method of adopting VSWFs combined with Fourier transform [13–15], dyadic Green’s functions method [16–18], and the T-matrix method [19–23]. To study the spherical anisotropy, Qiu et al. have investigated the characteristics of radial anisotropic uniaxial and gyrotropic sphere, where the permittivity and permeability tensors are expressed in spherical coordinate [24–30]. Due to the completeness and noncoplanar properties of VSWFs, any three-dimensional vector function satisfying the vector Helmholtz equation can be expanded as a linear combinations of VSWFs [31]. In [20], Lin and Chui have developed a complete theory to solve Maxwell equations for gyromagnetic particles. This method has a good solution accuracy since it does not involve lengthy evaluations of complicated integrals. Li and Ong applied this method to study the EM scattering characteristics of a gyroelectric sphere in [21] and then generalized the scattering problem by considering both permittivity and permeability tensors in [22]. In [23], the homogeneous gyroelectric sphere considered in [20, 21] was extended to a multilayered gyroelectric sphere.In this paper, the method developed in [23] is further extended to the most general case: the gyrotropic sphere under consideration consists of multiple anisotropic layers, each of which is characterized by both permittivity and permeability tensors. A new T-matrix formulation is proposed for the spherical multilayered scatterer. It is shown from numerical results that the proposed computation procedure is robust and accurate for a wide range of material parameters. In this method, the EM fields in each layer are first expanded as an infinite summation of VSWFs based on the Mie scattering theory [20, 32]. The expansion coefficients in each layer and the scattering coefficients are then determined recursively by matching the boundary conditions at all interfaces. To validate the derived formulation, we compare two specific RCS results with those obtained by the FEM simulation and the Fourier transform combined with VSWFs method [33] for gyrotropic and uniaxial cases, respectively. Excellent agreements can be observed. The formulation developed in this paper can be used as a reference for other analytical approaches or numerical methods relying on discretizations. Finally, we calculate the RCS for several specific cases to provide some physical insights and possibilities of optimized designs for suppressing or enhancing RCS based on multilayered gyrotropic structures.


## 2. Basic Formulations

Throughout the paper, a time dependencee-iωt is assumed but always suppressed. The model of a radially multilayered gyrotropic sphere is depicted in Figure 1, where the radii of the spherical layers are denoted by a1,a2,…,aN. Region-0 is the free space characterized by scalar permittivity (ϵ0) and permeability (μ0). Region-p is homogeneous and characterized by both permittivity and permeability tensors, which are explicitly expressed in Cartesian coordinate as(1)ϵ¯p=ϵpsϵpr-iϵpκ0iϵpκϵpr0001,μ¯p=μpsμpr-iμpκ0iμpκμpr0001,where p=1,2,…,N.Figure 1
Geometry of a radially multilayered gyrotropic sphere.


### 2.1. EM Fields in Multilayered Gyrotropic Sphere

In general, the EM fields in Region-p of a multilayered gyrotropic sphere can be expressed as [22, 23] (2a)Hp=-∑n,mE¯mn∑J∑lαplJkplωμpsd¯mn,lpMmnJkpl,r+c¯mn,lpNmnJkpl,r+w¯mn,lpLmnJkpl,r-∑J∑lαplJkplωμpsw¯00,lpL00Jkpl,r,(2b)Ep=-i∑n,mE¯mn∑J∑lαplJcmn,lpMmnJkpl,r+dmn,lpNmnJkpl,r+w¯¯mn,lpλplLmnJkpl,r-i∑J∑lαplJw¯¯00,lpλplL00Jkpl,r.When p=1,2,…,N-1, the summation subscript J must take both 1 and 3 to obtain the complete solution of the vector wave functions, since the first-kind spherical Bessel function and the third-kind spherical Bessel function (or sometimes referred to as the first-kind spherical Hankel function) are regular in these regions; however, when p=N, the summation subscript J only takes 1 due to the singularity of the second-kind spherical Bessel function (and hence the third-kind spherical Bessel function) at the origin. A detailed derivation for obtaining the expansion coefficients αpl(J) and kpl in (2a) and (2b) can be found in [22, 23] and will not be repeated here for simplicity.


### 2.2. Matching Boundary Conditions

The EM boundary conditions on the interfaces are(3a)Ep-1×er=Ep×er,(3b)Hp-1×er=Hp×er,for r=ap(p=2,3,…,N), and(4a)E1×er=Einc+Esca×er,(4b)H1×er=Hinc+Hsca×er,for r=a1. In the above equations, er is the unit vector of the radial component in spherical coordinate, (Esca,Hsca) and (Einc,Hinc) are the scattered and incident fields expanded by VSWFs (see (26) and (28) in [23] for details). Based on the boundary conditions, we can obtain the following recurrent formulation:(5)T¯p-1,p·Xp-1=T¯p,p·Xp,where(6a)T¯p,f=Ap,fA¯p,fBp,fB¯p,fCp,fC¯p,fDp,fD¯p,f,Xp=αp1αp3,forp=2,3,…,N-1,(6b)T¯N,f=AN,f0BN,f0CN,f0DN,f0,XN=αN10,forp=N.The matrix elements are defined as(7)Amn,lp,f=k0kplψn′kplafψn′k0afdmn,lp+k0kps2afψnkplafψn′k0afw¯¯mn,lp,A¯mn,lp,f=k0kplξn′kplafψn′k0afdmn,lp+k0kps2afξnkplafψn′k0afw¯¯mn,lp,Bmn,lp,f=k0kplψnkplafψnk0afcmn,lp,B¯mn,lp,f=k0kplξnkplafψnk0afcmn,lp,Cmn,lp,f=μ0μpsψnkplafψnk0afd¯mn,lp,C¯mn,lp,f=μ0μpsξnkplafψnk0afd¯mn,lp,Dmn,lp,f=μ0μpsψn′kplafψn′k0afc¯mn,lp+μ0kplμpsafψnkplafψn′k0afw¯mn,lp,D¯mn,lp,f=μ0μpsξn′kplafψn′k0afc¯mn,lp+μ0kplμpsafξnkplafψn′k0afw¯mn,lp.After a careful comparison of (7) and (6) in [23], we observe that, by adding the gyromagnetic tensors, additional terms are introduced into the matrix elements of Dmn,l(p,f) and D¯mn,l(p,f). In (7), ψn(z) and ξn(z) are the Riccati-Bessel functions given by(8)ψnz=zjnz,ξnz=zhn1z,with jn(z) and hn(1)(z) being the spherical Bessel functions of the first and the third kind, respectively. According to the recurrence relations in (5), the expansion coefficients X1 in Region-1 can be related to XN in Region-N as(9)X1=α11α13=T·XN,where T=T-1,2-1·T-2,2⋯T-N-1,N-1·T-N,N. Then we apply the EM boundary condition in (4a) and (4b) at r=a1 to solve a scattering problem as(10a)Λ¯00Λab+A1,1A¯1,1B1,1B¯1,1α11α13=pq,(10b)Λ00Λ¯ab+C1,1C¯1,1D1,1D¯1,1α11α13=pq.In (10a) and (10b), [a,b]T and [p,q]T are the scattering and incident coefficients (see detailed expressions in [23]). The matrices Λ and Λ¯ are expressed as [20](11a)Λmn,uv=ξnx1ψnx1δnvδmu,(11b)Λ¯mn,uv=ξn′x1ψn′x1δnvδmu,where x1=k0a1. By substituting (9) into (10a) and (10b), the scattering coefficients amn and bmn and the expansion coefficients αN for Region-N can all be solved numerically. After obtaining XN, the field expansion coefficients for internal layers (from Region-(N-1) to Region-1) can be readily calculated by applying the inward to outward recurrence procedure.Whenr→∞, the asymptotic forms of the third-kind VSWFs are expressed as [34](12a)Mmn3k0,r=-inπmncos⁡θeθ+iτmncos⁡θeϕeik0rk0reimϕ,(12b)Nmn3k0,r=-inτmncos⁡θeθ+iπmncos⁡θeϕeik0rk0reimϕ,where eθ and eϕ are the unit vectors of the θ and ϕ components in the spherical coordinate, respectively, and(13a)πmncos⁡θ=msin⁡θPnmcos⁡θ,(13b)τmncos⁡θ=dPnmcos⁡θdθ.Consequently, the RCS of the multilayered gyrotropic scatterer can be calculated by using the obtained scattering coefficients as(14)σ=limr→∞⁡4πr2Esca2Einc2=λ2π∑n,mE¯mnamnπmncos⁡θ+bmnτmncos⁡θeθ+iamnτmncos⁡θ+bmnπmncos⁡θeϕeimϕ2.


## 2.1. EM Fields in Multilayered Gyrotropic Sphere

In general, the EM fields in Region-p of a multilayered gyrotropic sphere can be expressed as [22, 23] (2a)Hp=-∑n,mE¯mn∑J∑lαplJkplωμpsd¯mn,lpMmnJkpl,r+c¯mn,lpNmnJkpl,r+w¯mn,lpLmnJkpl,r-∑J∑lαplJkplωμpsw¯00,lpL00Jkpl,r,(2b)Ep=-i∑n,mE¯mn∑J∑lαplJcmn,lpMmnJkpl,r+dmn,lpNmnJkpl,r+w¯¯mn,lpλplLmnJkpl,r-i∑J∑lαplJw¯¯00,lpλplL00Jkpl,r.When p=1,2,…,N-1, the summation subscript J must take both 1 and 3 to obtain the complete solution of the vector wave functions, since the first-kind spherical Bessel function and the third-kind spherical Bessel function (or sometimes referred to as the first-kind spherical Hankel function) are regular in these regions; however, when p=N, the summation subscript J only takes 1 due to the singularity of the second-kind spherical Bessel function (and hence the third-kind spherical Bessel function) at the origin. A detailed derivation for obtaining the expansion coefficients αpl(J) and kpl in (2a) and (2b) can be found in [22, 23] and will not be repeated here for simplicity.


## 2.2. Matching Boundary Conditions

The EM boundary conditions on the interfaces are(3a)Ep-1×er=Ep×er,(3b)Hp-1×er=Hp×er,for r=ap(p=2,3,…,N), and(4a)E1×er=Einc+Esca×er,(4b)H1×er=Hinc+Hsca×er,for r=a1. In the above equations, er is the unit vector of the radial component in spherical coordinate, (Esca,Hsca) and (Einc,Hinc) are the scattered and incident fields expanded by VSWFs (see (26) and (28) in [23] for details). Based on the boundary conditions, we can obtain the following recurrent formulation:(5)T¯p-1,p·Xp-1=T¯p,p·Xp,where(6a)T¯p,f=Ap,fA¯p,fBp,fB¯p,fCp,fC¯p,fDp,fD¯p,f,Xp=αp1αp3,forp=2,3,…,N-1,(6b)T¯N,f=AN,f0BN,f0CN,f0DN,f0,XN=αN10,forp=N.The matrix elements are defined as(7)Amn,lp,f=k0kplψn′kplafψn′k0afdmn,lp+k0kps2afψnkplafψn′k0afw¯¯mn,lp,A¯mn,lp,f=k0kplξn′kplafψn′k0afdmn,lp+k0kps2afξnkplafψn′k0afw¯¯mn,lp,Bmn,lp,f=k0kplψnkplafψnk0afcmn,lp,B¯mn,lp,f=k0kplξnkplafψnk0afcmn,lp,Cmn,lp,f=μ0μpsψnkplafψnk0afd¯mn,lp,C¯mn,lp,f=μ0μpsξnkplafψnk0afd¯mn,lp,Dmn,lp,f=μ0μpsψn′kplafψn′k0afc¯mn,lp+μ0kplμpsafψnkplafψn′k0afw¯mn,lp,D¯mn,lp,f=μ0μpsξn′kplafψn′k0afc¯mn,lp+μ0kplμpsafξnkplafψn′k0afw¯mn,lp.After a careful comparison of (7) and (6) in [23], we observe that, by adding the gyromagnetic tensors, additional terms are introduced into the matrix elements of Dmn,l(p,f) and D¯mn,l(p,f). In (7), ψn(z) and ξn(z) are the Riccati-Bessel functions given by(8)ψnz=zjnz,ξnz=zhn1z,with jn(z) and hn(1)(z) being the spherical Bessel functions of the first and the third kind, respectively. According to the recurrence relations in (5), the expansion coefficients X1 in Region-1 can be related to XN in Region-N as(9)X1=α11α13=T·XN,where T=T-1,2-1·T-2,2⋯T-N-1,N-1·T-N,N. Then we apply the EM boundary condition in (4a) and (4b) at r=a1 to solve a scattering problem as(10a)Λ¯00Λab+A1,1A¯1,1B1,1B¯1,1α11α13=pq,(10b)Λ00Λ¯ab+C1,1C¯1,1D1,1D¯1,1α11α13=pq.In (10a) and (10b), [a,b]T and [p,q]T are the scattering and incident coefficients (see detailed expressions in [23]). The matrices Λ and Λ¯ are expressed as [20](11a)Λmn,uv=ξnx1ψnx1δnvδmu,(11b)Λ¯mn,uv=ξn′x1ψn′x1δnvδmu,where x1=k0a1. By substituting (9) into (10a) and (10b), the scattering coefficients amn and bmn and the expansion coefficients αN for Region-N can all be solved numerically. After obtaining XN, the field expansion coefficients for internal layers (from Region-(N-1) to Region-1) can be readily calculated by applying the inward to outward recurrence procedure.Whenr→∞, the asymptotic forms of the third-kind VSWFs are expressed as [34](12a)Mmn3k0,r=-inπmncos⁡θeθ+iτmncos⁡θeϕeik0rk0reimϕ,(12b)Nmn3k0,r=-inτmncos⁡θeθ+iπmncos⁡θeϕeik0rk0reimϕ,where eθ and eϕ are the unit vectors of the θ and ϕ components in the spherical coordinate, respectively, and(13a)πmncos⁡θ=msin⁡θPnmcos⁡θ,(13b)τmncos⁡θ=dPnmcos⁡θdθ.Consequently, the RCS of the multilayered gyrotropic scatterer can be calculated by using the obtained scattering coefficients as(14)σ=limr→∞⁡4πr2Esca2Einc2=λ2π∑n,mE¯mnamnπmncos⁡θ+bmnτmncos⁡θeθ+iamnτmncos⁡θ+bmnπmncos⁡θeϕeimϕ2.


## 3. Numerical Results and Discussions

The incident plane wave is assumed to be propagating along the positivez direction, where the electric field is polarized in the x direction with unit amplitude. The size parameters of the multilayered sphere are represented as xp=k0ap=2πap/λ(p=1,2,…,N), which are essentially the electrical dimensions with respect to the incident wavelength λ. In the practical computations, the summation index n in (2a) and (2b) should be truncated at nc. As suggested in Appendix A in [32] and (8) in [35], nc can be chosen to be(15)nc=x+4x1/3+2,where x denotes the size parameter of the spherical scatterer. In this paper, x is uniformly chosen as x=x1=k0a1 to make the matrices in (5) have the same dimensions. The linear equations (10a) and (10b) are solved independently of each m∈[-nc,nc] due to the Kronecker symbols δmp appearing in (11a) and (11b). In the following discussions, the numerical results of RCS are normalized by the square of the incident wavelength (λ2).The convergence criteria (15) are commonly used for homogeneous spherical scatterers [35, 36]. As for the gyrotropic case considered in this work, the validation of truncation number nc has been checked by computing the relative error of truncated series of the scattering efficiency factor Qsca=(4/x12)∑n,m‍(|amn|2+|bmn|2) [20, 36]. Explicitly, the relative error is defined as ε(N)=|Qsca(N)-Qsca|/|Qsca|, where Qsca(N) is the truncated series and Qsca is the converged value (calculated for a sufficiently large N). Four cases with different electrical dimensions are depicted in Figure 4. It can be seen that the truncated series converge well when the summation index n is truncated at the value given by (15), and the convergence accuracy is at least 10-5.To verify the correctness of the formulations derived for the multilayered gyrotropic sphere in this paper, two numerical experiments are conducted. First, the normalized RCS values of a 2-layered gyrotropic sphere are compared with that computed from the full-wave finite element method (FEM) simulation, as depicted in Figure2. The normalized root mean square deviations (NRMSD) for E-plane (xoz) and H- plane (yoz) are 0.0041 and 0.0052, respectively. In the FEM simulation, a mesh with 53294 tetrahedral elements has been used and the basis functions are constructed from polynomials of order 3. Second, the gyrotropic case is reduced to the uniaxial case (i.e., μpκ=0 and ϵpκ=0, for p=1,2,…,N), so that a direct comparison can be made with the published results (Figure 4 in [33]), as shown in Figure 3. The NRMSD for E- and H-planes are 0.007 and 0.0099, respectively. In both cases, excellent agreements of the RCS values are achieved in both E-plane and H-plane, confirming the correctness of the newly derived formulations.Figure 2
Normalized RCS of a 2-layered gyrotropic sphere versus scattering angleθs, where the electric dimensions are chosen as x1=2π and x2=1.5π. The permittivity and permeability tensor elements are assumed to be ϵ1s=ϵ2s=ϵ0, ϵ1r=1, ϵ2r=2, and ϵ1k=ϵ2k=1+i and μ1s=μ2s=μ0, μ1r=1, μ2r=2, and μ1k=μ2k=1+0.5i.Figure 3
Normalized RCS of a 6-layered uniaxial anisotropic sphere versus scattering angleθs, where the electric dimensions are chosen as x1=4.1π, x2=4π, x3=3.9π, x4=3.8π, x5=3.7π, and x6=3.6π. The permittivity and permeability tensor elements are assumed to be ϵ1sϵ1r=ϵ3sϵ3r=ϵ5sϵ5r=ϵ2s=ϵ4s=ϵ6s=3+0.15iϵ0 and ϵ2sϵ2r=ϵ4sϵ4r=ϵ6sϵ6r=ϵ1s=ϵ3s=ϵ5s=2+0.1iϵ0 and μ1sμ1r=μ3sμ3r=μ5sμ5r=μ2s=μ4s=μ6s=(1+0.05i)μ0, μ2sμ2r=μ4sμ4r=μ6sμ6r=μ1s=μ3s=μ5s=(2+0.1i)μ0.Figure 4
Logarithm values of the relative errorε(N) versus the truncation number N for a 2-layered gyrotropic sphere with four different electric dimensions. The EM parameters considered here are the same as those in Figure 2 and the size parameters x2 are equal to 0.75x1. The dashed circles indicate the positions of truncation numbers given in (15).In the following, three specific cases will be presented to further investigate the scattering characteristic of the multilayered gyrotropic sphere.In Figure5, the bistatic RCS of a 2-layered lossless gyrotropic sphere with small-size parameters is presented, where two cases of different EM parameter settings are studied comparatively. The RCS results are shown in Figures 5(a) and 5(b). We observe that if the permittivity and permeability tensors in each layer satisfy the following conditions,(16)ϵpsϵ0=μpsμ0,ϵpr=μpr,ϵpκ=μpκ,the RCS in E-plane and H-plane coincide with each other. More importantly, a sharp RCS reduction in the backscattering direction can be obviously observed. Therefore, this property may have potential applications in stealth or cloaking design.Figure 5
Normalized RCS of a 2-layered lossless gyrotropic sphere versus scattering angleθs in E-plane and H-plane. The gyrotropic tensors in each layer are selected as (a) ϵ1s=4ϵ0, μ1s=2μ0, ϵ2s=2ϵ0, μ2s=4μ0, ϵpr=μpr=0.5, ϵpk=μpk=0.25, and p=1,2; (b) ϵ1s=2ϵ0, μ1s=2μ0, ϵ2s=4ϵ0, μ2s=4μ0, ϵpr=μpr=0.5, ϵpk=μpk=0.25, and p=1,2.
(a)
Case1
(b)
Case2To further understand this scattering property, another 4-layered lossy gyrotropic sphere with medium-size parameters is considered and the bistatic RCS values are depicted in Figure6. In this case, the parameters are chosen based on (16), and the property presented in Figure 5(b) still holds for this case. It can be possibly concluded that if the parameters in each layer satisfy (16), the RCS in the backward direction can be made too weak to be detected, no matter for lossless or lossy anisotropic materials.Figure 6
Normalized RCS of a 4-layered lossy gyrotropic sphere versus scattering angleθs in E-plane and H-plane. The gyrotropic tensors in each layer are selected as ϵps=5+0.2iϵ0, μps=(5+0.2i)μ0, ϵpsϵpk=ϵ0, μpsμpk=μ0 (where p=1,2,3,4), ϵ1sϵ1r=3+0.1iϵ0, μ1sμ1r=(3+0.1i)μ0, ϵ2sϵ2r=5+0.1iϵ0, μ2sμ2r=(5+0.1i)μ0, ϵ3sϵ3r=7+0.1iϵ0, μ3sμ3r=(7+0.1i)μ0, ϵ4sϵ4r=9+0.1iϵ0, and μ4sμ4r=(9+0.1i)μ0.In Figure7, the bistatic RCS of a 2-layered lossless gyrotropic sphere with large-size parameters is presented. The EM parameters are configured specifically to give another insight of the scattering property. From Figure 7, it can be observed that the RCS becomes more oscillatory than those depicted in Figures 5 and 6. When the scattering angle increases from 0° to 180°, the RCS decreases first and then increases to a high level in both E-plane and H-plane.Figure 7
Normalized RCS of a 2-layered lossless gyrotropic sphere versus scattering angleθs in E-plane and H-plane. The gyrotropic tensors in each layer are selected as ϵ1s=2ϵ0, ϵ1r=2, ϵ1k=0.5, μ1s=4μ0, μ1r=0.5, and μ1k=0.25; ϵ2s=4ϵ0, ϵ2r=0.5, ϵ2k=0.25, μ2s=2μ0, μ2r=2, and μ2k=0.5.


## 4. Conclusions

The analytical solutions of the EM scattering by a radially multilayered gyrotropic sphere are studied in this paper. First, by applying the completeness and noncoplanar properties of the VSWFs, theoretical formulations of this scattering problem have been derived. Second, we compare the numerical results of RCS values with those obtained from the full-wave simulation or from the literature to verify the validity and correctness of the proposed method. Third, specific cases are investigated to reveal some interesting physical insights of the scattering property of the multilayered gyrotropic sphere. It is found that if the gyrotropic tensors are configured properly, the RCS in the backward scattering direction can be suppressed or enhanced significantly.



---

*Source: 3205198-2017-09-28.xml*