球矢量波函数在各向异性介质电磁散射中的应用
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摘要
本文用球矢量波函数、Fourier变换和用各向同性介质球矢量波函数展开的本征矢量和场平面波因子乘积解析表达式来研究各向异性介质球结构电磁散射的解析解，主要对两种各向异性介质——各向异性单轴介质和等离子体介质球结构电磁散射开展相应的理论研究，并实现数值计算，具体如下：
一、从各向异性单轴介质的无源麦克斯韦方程组出发，导出各向异性单轴介质电场矢量满足的微分方程，并引入Fourier变换来求解电场矢量微分方程，在电场的平面波谱表示式及未知角谱傅立叶展开的基础上，利用各向同性球矢量波函数展开的本征矢量与平面波因子乘积的解析表达式，我们可得到无源麦克斯韦方程组在各向异性单轴介质中电场矢量的解析表达式，在此基础上，通过谱域方法求出各向异性单轴介质的磁场表达式，该电磁场的解析表达式是由两组待定系数和对球矢量波函数的—重积分构成，利用电场、磁场在各向异性单轴介质球边界上切向连续的边界条件研究了均匀各向异性单轴介质球对平面波的电磁散射特性，数值计算的结果和各项同性介质球的Mie理论以及矩量法-共扼梯度-傅立叶变换(MOM-CG-FFT)所计算各向异性单轴介质球的数值结果进行了比较，两者符合地较好，可得出各向同性介质球电磁散射解析解是本文特例的结论。同时还给出了有损耗和电大尺寸的各向异性单轴介质球对平面波电磁散射的数值计算结果。
二、在建立均匀各向异性单轴介质球矢量波函数理论的基础上，利用二阶线性偏微分方程的性质和第一、第二、第三和第四类球Bessel函数满足相同的微分方程和递推关系，我们分别研究了单轴介质球壳和单轴介质涂覆导体球对平面波的电磁散射特性，首先给出了各个区域的电磁场用球矢量波函数来表示的解析表达式，进而利用电磁场在边界上满足电磁场切向连续的边界条件和球谐函数的正交性，得出了各向异性单轴介质球结构中电磁场用球矢量波函数表示的系数所满足的矩阵方程。通过矩阵方程的求解和球Bessel函数的大宗量渐进表达式，我们给出了各向异性单轴介质球结构对平面波电磁散射的数值计算结果，并和其他的数值计算结果进行了比较，其结果正如我们所预料的一样，达到了预期的结果。同时还给出了电大尺寸备向异性单轴介质球结构对平面波电磁散射的数值计算结果。
三、从各向异性等离子体介质的无源麦克斯韦方程组出发，导出各向异性等离子体介质的矢量电场方程，并引入傅立叶变换来求解该矢量场方程，在场的平面波谱表示式及未知角谱的傅立叶展开的基础上，利用本征波矢量与平面波因子乘积的各向同性介质球矢量波函数展开的解析表示式，我们可得到无源麦克斯韦方程组在各向异性等离子体介质的电磁场解析表示式。在此基础上，利用电磁场在各向异性等离子体球面上切向连续的边界条件，研究平面波入射情况下，均匀各向异性等离子体球的电磁散射特性，并对其进行了相应的数值计算，将本文数值计算的结果和MOM-CG-FFT所计算的结果进行了比较，两者符合的比较好。给出了谐振区各向异性等离子体球对平面波电磁散射的数值计算结果。
四、在建立均匀各向异性等离子体介质球矢量波函数基础上，分别研究了两层、多层各向异性等离子体的电磁散射特性，和各向异性单轴介质的相似，利用第一、第二类各向异性等离子体介质的球矢量波函数满足相同微分方程和迭代关系，我们分别给出了两层和多层各向异性等离子体的各个区域电磁场用各向异性等离子体球矢量波函数的表示形式，近而利用边界上电磁场切向连续的边界条件，可分别求出两层和多层各向异性等离子体中电磁场用球矢量波函数展开的展开系数，进而给出了两层和多层各向异性等离子体球的雷达散射截面，并实现数值计算。
五、在完成上述工作的基础上，我们还对各向异性单轴负折射介质球和球壳对平面波的电磁散射特性开展了数值计算，并和各向同性负折射介质球的数值计算结果进行了比较，验证了我们所得的理论结果不仅可计算自然界所存在的各向异性右手介质，而且也能计算目前国际上研究比较多的热点问题——负折射介质。
在文章的最后，我们给出了进一步所需要做的工作。
From the source-free Maxwell equations, using the Fourier transform and plane wave factors expansion with spherical vector wave functions in three-dimensional isotropic medium, the analytical solution of electromagnetic fields in anisotropic medium in terms of spherical vector wave functions can be derived. With the spherical vector wave functions in isotropic medium, the solution to electromagnetic scattering by spherical uniaxial anisotropic medium and plasma anisotropic medium with spherical vector wave functions have been obtained in this paper. The results of this dissertation are as follows:
First, using the source-free Maxwell's equations in uniaxial anisotropic media and making the Fourier transform of the field quantities, the electromagnetic fields in spectral domain in uniaxial anisotropic media are assumed to have the similar form to the plane wave expanded also in terms of the spherical vector wave functions. Applying the continuous boundary conditions of electromagnetic fields on the surface between the air region and uniaxial anisotropic sphere, the coefficients of scattered fields in free space and the transmitted fields in uniaxial anisotropic medium can be obtained analytically in the expansion form of spherical vector wave eigenfunctions. Numerical results for some special cases are obtained, and compared with those of the classical Mie theory and the Method of Moments (MoM) accelerated with the Conjugate-Gradient Fast-Fourier-Transform (CG-FFT) approach. We also present some new numerical results for the more general uniaxial dielectric material media.
Second, based on the spherical vector wave function in uniaxial anisotropic medium (part one), and the first, second, third and fourth spherical Bessel functions satisfy the same differential equation and recursive formula. The scattering fields in terms of spherical vector wave function from a uniaxial anisotropic spherical shell and an anisotropic uniaxial-coated conducting sphere by a plane wave are derived. The electromagnetic fields in uniaxial anisotropic medium and free space can be expressed in terms of spherical vector wave functions in uniaxial anisotropic media and isotropic medium. Applying the boundary condition in the interface between the uniaxial anisotropic medium and free space, the surface of the conducting sphere, the expansion coefficients of electromagnetic fields in terms of spherical vector wave function in uniaxial anisotropic medium are obtained, and then the expansion coefficients of scattering fields and radar cross sections can be obtained. Numerical results between this method and Mie theory are in good agreement as we expect. Some new numerical results have been given in the end of this part.
Third, an analytical solution of electromagnetic fields in homogeneous plasma anisotropic media is obtained in this part. In the source-free plasma anisotropic media, the source-free Maxwell's equations are utilized, where the expansion of plane wave factors is made in terms of the spherical vector wave functions in isotropic media, and the Fourier transformation is then applied. As a result, the field expressions in terms of spherical vector wave functions in plasma anisotropic medium represented using eigenfunctions are obtained in spectral domain. Applying boundary conditions on the spherical interface between air and plasma anisotropy, the electromagnetic fields of the plane wave scattered by a plasma anisotropic sphere are derived. Numerical results for the very general plasma dielectric material media are obtained and those in a special case are compared between the present method and the Method of Moments (MoM) speeded up with the Conjugate-Gradient
Fast-Fourier-Transform (CG-FFT) approach. Some new numerical results have been given in later of this part.
Fourth, on the base of the analytical solution of electromagnetic fields in terms of spherical vector wave functions in source-free plasma anisotropic medium (part three), the first, second spherical vector wave functions in source-free plasma anisotropic medium satisfy the same Maxwell's equations, and the electromagnetic fields in plasma anisotropic spherical medium can be expressed as an addition of the first and the second spherical vector wave functions. Applying the continue boundary conditions of electromagnetic fields in the interface of plasma anisotropic spherical medium, the coefficients of electromagnetic fields in terms of spherical vector wave function can be derived in stratified plasma anisotropic medium. Two concentric plasma anisotropic spheres and multilayer anisotropic spheres have been discussed in detail. Some numerical results has been given, the theory and numerical results show that the present method can be degenerated to a single plasma anisotropic sphere when the two plasma anisotropic media nearly have the same parameter of medium.
Fifth, we apply the formula of electromagnetic scattering of three-dimensional uniaxial anisotropic medium to the scattering of left-handed materials (LHMs), or negative index of refraction (NIR) materials uniaxial sphere and spherical shell, and some numerical results have been given in this part.
In the end, some works which will be done are given.
一、从各向异性单轴介质的无源麦克斯韦方程组出发，导出各向异性单轴介质电场矢量满足的微分方程，并引入Fourier变换来求解电场矢量微分方程，在电场的平面波谱表示式及未知角谱傅立叶展开的基础上，利用各向同性球矢量波函数展开的本征矢量与平面波因子乘积的解析表达式，我们可得到无源麦克斯韦方程组在各向异性单轴介质中电场矢量的解析表达式，在此基础上，通过谱域方法求出各向异性单轴介质的磁场表达式，该电磁场的解析表达式是由两组待定系数和对球矢量波函数的—重积分构成，利用电场、磁场在各向异性单轴介质球边界上切向连续的边界条件研究了均匀各向异性单轴介质球对平面波的电磁散射特性，数值计算的结果和各项同性介质球的Mie理论以及矩量法-共扼梯度-傅立叶变换(MOM-CG-FFT)所计算各向异性单轴介质球的数值结果进行了比较，两者符合地较好，可得出各向同性介质球电磁散射解析解是本文特例的结论。同时还给出了有损耗和电大尺寸的各向异性单轴介质球对平面波电磁散射的数值计算结果。
二、在建立均匀各向异性单轴介质球矢量波函数理论的基础上，利用二阶线性偏微分方程的性质和第一、第二、第三和第四类球Bessel函数满足相同的微分方程和递推关系，我们分别研究了单轴介质球壳和单轴介质涂覆导体球对平面波的电磁散射特性，首先给出了各个区域的电磁场用球矢量波函数来表示的解析表达式，进而利用电磁场在边界上满足电磁场切向连续的边界条件和球谐函数的正交性，得出了各向异性单轴介质球结构中电磁场用球矢量波函数表示的系数所满足的矩阵方程。通过矩阵方程的求解和球Bessel函数的大宗量渐进表达式，我们给出了各向异性单轴介质球结构对平面波电磁散射的数值计算结果，并和其他的数值计算结果进行了比较，其结果正如我们所预料的一样，达到了预期的结果。同时还给出了电大尺寸备向异性单轴介质球结构对平面波电磁散射的数值计算结果。
三、从各向异性等离子体介质的无源麦克斯韦方程组出发，导出各向异性等离子体介质的矢量电场方程，并引入傅立叶变换来求解该矢量场方程，在场的平面波谱表示式及未知角谱的傅立叶展开的基础上，利用本征波矢量与平面波因子乘积的各向同性介质球矢量波函数展开的解析表示式，我们可得到无源麦克斯韦方程组在各向异性等离子体介质的电磁场解析表示式。在此基础上，利用电磁场在各向异性等离子体球面上切向连续的边界条件，研究平面波入射情况下，均匀各向异性等离子体球的电磁散射特性，并对其进行了相应的数值计算，将本文数值计算的结果和MOM-CG-FFT所计算的结果进行了比较，两者符合的比较好。给出了谐振区各向异性等离子体球对平面波电磁散射的数值计算结果。
四、在建立均匀各向异性等离子体介质球矢量波函数基础上，分别研究了两层、多层各向异性等离子体的电磁散射特性，和各向异性单轴介质的相似，利用第一、第二类各向异性等离子体介质的球矢量波函数满足相同微分方程和迭代关系，我们分别给出了两层和多层各向异性等离子体的各个区域电磁场用各向异性等离子体球矢量波函数的表示形式，近而利用边界上电磁场切向连续的边界条件，可分别求出两层和多层各向异性等离子体中电磁场用球矢量波函数展开的展开系数，进而给出了两层和多层各向异性等离子体球的雷达散射截面，并实现数值计算。
五、在完成上述工作的基础上，我们还对各向异性单轴负折射介质球和球壳对平面波的电磁散射特性开展了数值计算，并和各向同性负折射介质球的数值计算结果进行了比较，验证了我们所得的理论结果不仅可计算自然界所存在的各向异性右手介质，而且也能计算目前国际上研究比较多的热点问题——负折射介质。
在文章的最后，我们给出了进一步所需要做的工作。
From the source-free Maxwell equations, using the Fourier transform and plane wave factors expansion with spherical vector wave functions in three-dimensional isotropic medium, the analytical solution of electromagnetic fields in anisotropic medium in terms of spherical vector wave functions can be derived. With the spherical vector wave functions in isotropic medium, the solution to electromagnetic scattering by spherical uniaxial anisotropic medium and plasma anisotropic medium with spherical vector wave functions have been obtained in this paper. The results of this dissertation are as follows:
First, using the source-free Maxwell's equations in uniaxial anisotropic media and making the Fourier transform of the field quantities, the electromagnetic fields in spectral domain in uniaxial anisotropic media are assumed to have the similar form to the plane wave expanded also in terms of the spherical vector wave functions. Applying the continuous boundary conditions of electromagnetic fields on the surface between the air region and uniaxial anisotropic sphere, the coefficients of scattered fields in free space and the transmitted fields in uniaxial anisotropic medium can be obtained analytically in the expansion form of spherical vector wave eigenfunctions. Numerical results for some special cases are obtained, and compared with those of the classical Mie theory and the Method of Moments (MoM) accelerated with the Conjugate-Gradient Fast-Fourier-Transform (CG-FFT) approach. We also present some new numerical results for the more general uniaxial dielectric material media.
Second, based on the spherical vector wave function in uniaxial anisotropic medium (part one), and the first, second, third and fourth spherical Bessel functions satisfy the same differential equation and recursive formula. The scattering fields in terms of spherical vector wave function from a uniaxial anisotropic spherical shell and an anisotropic uniaxial-coated conducting sphere by a plane wave are derived. The electromagnetic fields in uniaxial anisotropic medium and free space can be expressed in terms of spherical vector wave functions in uniaxial anisotropic media and isotropic medium. Applying the boundary condition in the interface between the uniaxial anisotropic medium and free space, the surface of the conducting sphere, the expansion coefficients of electromagnetic fields in terms of spherical vector wave function in uniaxial anisotropic medium are obtained, and then the expansion coefficients of scattering fields and radar cross sections can be obtained. Numerical results between this method and Mie theory are in good agreement as we expect. Some new numerical results have been given in the end of this part.
Third, an analytical solution of electromagnetic fields in homogeneous plasma anisotropic media is obtained in this part. In the source-free plasma anisotropic media, the source-free Maxwell's equations are utilized, where the expansion of plane wave factors is made in terms of the spherical vector wave functions in isotropic media, and the Fourier transformation is then applied. As a result, the field expressions in terms of spherical vector wave functions in plasma anisotropic medium represented using eigenfunctions are obtained in spectral domain. Applying boundary conditions on the spherical interface between air and plasma anisotropy, the electromagnetic fields of the plane wave scattered by a plasma anisotropic sphere are derived. Numerical results for the very general plasma dielectric material media are obtained and those in a special case are compared between the present method and the Method of Moments (MoM) speeded up with the Conjugate-Gradient
Fast-Fourier-Transform (CG-FFT) approach. Some new numerical results have been given in later of this part.
Fourth, on the base of the analytical solution of electromagnetic fields in terms of spherical vector wave functions in source-free plasma anisotropic medium (part three), the first, second spherical vector wave functions in source-free plasma anisotropic medium satisfy the same Maxwell's equations, and the electromagnetic fields in plasma anisotropic spherical medium can be expressed as an addition of the first and the second spherical vector wave functions. Applying the continue boundary conditions of electromagnetic fields in the interface of plasma anisotropic spherical medium, the coefficients of electromagnetic fields in terms of spherical vector wave function can be derived in stratified plasma anisotropic medium. Two concentric plasma anisotropic spheres and multilayer anisotropic spheres have been discussed in detail. Some numerical results has been given, the theory and numerical results show that the present method can be degenerated to a single plasma anisotropic sphere when the two plasma anisotropic media nearly have the same parameter of medium.
Fifth, we apply the formula of electromagnetic scattering of three-dimensional uniaxial anisotropic medium to the scattering of left-handed materials (LHMs), or negative index of refraction (NIR) materials uniaxial sphere and spherical shell, and some numerical results have been given in this part.
In the end, some works which will be done are given.
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