椭球粒子对高斯波束的散射
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摘要
本文从理论上详细的研究了椭球粒子在高斯波束中的散射，其主要成果
在于：
1． 在椭球坐标系中，采用了一种巧妙的方法，将斜入射的高斯波束用椭
球波函数展开，提出了高斯光的波束因子的理论表述形式，它可以借
助已有的高斯光在球坐标系中的波束因子，很方便的计算出高斯光在
椭球坐标系中的波束因子。为进一步研究椭球粒子散射打下了基础。
在国际上尚无人作过此方面的工作。
2． 给出了数值计算椭球波函数的方法。在本文中，将超越方程化为矩阵
方程，采用数值方法求解椭球波函数特征值和特征函数。
3． 纠正了Shoji Asano在处理椭球介质粒子电磁散射问题的边界条件中
所出现的几个错误参数，Shoji Asano提出了的处理边界条件的理论
方法，是一种公认的研究椭球粒子平面波散射的一种重要方法。本文
经过详细的理论推导，并通过数学软件包Mathmatica验证，对其椭球
粒子电磁散射的边界条件的结果进行了修正，纠正了有错的几个参数。
4． 利用高斯光在椭球坐标系中的展开形式，采用分离变量方法，给出了
介质椭球粒子在高斯光中的散射场，椭球内部场的理论公式，数值计
算了椭球粒子在高斯光中散射强度分布。并给出了散射强度分布的二
维，三维图形，理论及其计算程序不仅适用于计算椭球粒子在高斯光
中的散射，而且适用于计算椭球粒子在平面波中的散射，以及球形粒
子在平面波和高斯波中的散射。
5． 给出了在实验室中椭球粒子对激光光束的散射的彩虹现象的理论分析
结果，并与Moebius彩虹角进行了比较。计算了大气粒子的散射强度
的角分布，极化度。
In this thesis, the analysis of electromagnetic wave scattering by spheroids
illuminated by Gaussian Beam has been done. The main contributions of the
author抯 work are as follows:
(1).We have studied the scattering of the Gaussian beam by a spheroidal particle
and presented an approach to expand the Gaussian beam in terms of the
spheroidal wavefunctions in spheroidal coordinates for oblique incidence . The
determination of the beam-shape coefficients G~ (c), F~ (c) is effected by first
expanding the wave in terms of the analogous spherical vector wave function
(which are orthogonal over the surface of a sphere) and then using the expansion
of the spherical wave function in terms of the spheroidal wave functions. Thus,
the beam-shape coefficients of the Gaussian beam in spheroidal coordinates can
be computed conveniently by using the known expression of g~ coefficients in
spherical coordinates. As so far we know, no analysis has been presented for the
beam-shape coefficients of the Gaussian beam by a spheroidal particle.
(2).We discuss the numerical computation of spheroidal wave functions and give
numerical values of the spheroidal eigenvalues.
(3).Theoretical expression for the boundary condition for electromagnetic
scattering by spheroidal particles is given by virtue of the method presented by
Shoji Asano and Giichi Yamamoto. The incorrect coefficients of expansions
given by Shoji抯 has been corrected .All above results have been verified by
Mathmatica.
(4).The unknown expansion coefficients of scattered and internal
f electromagnetic fields are determined by a system of equations derived from the
boundary conditions regarding the continuity of tangential components of the
electric and magnetic vectors across the surface of the spheroid. The numerical
values of the expansion coefficients and scattered intensities distribution for
incidence of Gaussian beam in the on-axis case are given.
(5).A theoretical analysis for the rainbow is given. The scattered intensities
distribution for particle in the atmosphere are computed.
在于：
1． 在椭球坐标系中，采用了一种巧妙的方法，将斜入射的高斯波束用椭
球波函数展开，提出了高斯光的波束因子的理论表述形式，它可以借
助已有的高斯光在球坐标系中的波束因子，很方便的计算出高斯光在
椭球坐标系中的波束因子。为进一步研究椭球粒子散射打下了基础。
在国际上尚无人作过此方面的工作。
2． 给出了数值计算椭球波函数的方法。在本文中，将超越方程化为矩阵
方程，采用数值方法求解椭球波函数特征值和特征函数。
3． 纠正了Shoji Asano在处理椭球介质粒子电磁散射问题的边界条件中
所出现的几个错误参数，Shoji Asano提出了的处理边界条件的理论
方法，是一种公认的研究椭球粒子平面波散射的一种重要方法。本文
经过详细的理论推导，并通过数学软件包Mathmatica验证，对其椭球
粒子电磁散射的边界条件的结果进行了修正，纠正了有错的几个参数。
4． 利用高斯光在椭球坐标系中的展开形式，采用分离变量方法，给出了
介质椭球粒子在高斯光中的散射场，椭球内部场的理论公式，数值计
算了椭球粒子在高斯光中散射强度分布。并给出了散射强度分布的二
维，三维图形，理论及其计算程序不仅适用于计算椭球粒子在高斯光
中的散射，而且适用于计算椭球粒子在平面波中的散射，以及球形粒
子在平面波和高斯波中的散射。
5． 给出了在实验室中椭球粒子对激光光束的散射的彩虹现象的理论分析
结果，并与Moebius彩虹角进行了比较。计算了大气粒子的散射强度
的角分布，极化度。
In this thesis, the analysis of electromagnetic wave scattering by spheroids
illuminated by Gaussian Beam has been done. The main contributions of the
author抯 work are as follows:
(1).We have studied the scattering of the Gaussian beam by a spheroidal particle
and presented an approach to expand the Gaussian beam in terms of the
spheroidal wavefunctions in spheroidal coordinates for oblique incidence . The
determination of the beam-shape coefficients G~ (c), F~ (c) is effected by first
expanding the wave in terms of the analogous spherical vector wave function
(which are orthogonal over the surface of a sphere) and then using the expansion
of the spherical wave function in terms of the spheroidal wave functions. Thus,
the beam-shape coefficients of the Gaussian beam in spheroidal coordinates can
be computed conveniently by using the known expression of g~ coefficients in
spherical coordinates. As so far we know, no analysis has been presented for the
beam-shape coefficients of the Gaussian beam by a spheroidal particle.
(2).We discuss the numerical computation of spheroidal wave functions and give
numerical values of the spheroidal eigenvalues.
(3).Theoretical expression for the boundary condition for electromagnetic
scattering by spheroidal particles is given by virtue of the method presented by
Shoji Asano and Giichi Yamamoto. The incorrect coefficients of expansions
given by Shoji抯 has been corrected .All above results have been verified by
Mathmatica.
(4).The unknown expansion coefficients of scattered and internal
f electromagnetic fields are determined by a system of equations derived from the
boundary conditions regarding the continuity of tangential components of the
electric and magnetic vectors across the surface of the spheroid. The numerical
values of the expansion coefficients and scattered intensities distribution for
incidence of Gaussian beam in the on-axis case are given.
(5).A theoretical analysis for the rainbow is given. The scattered intensities
distribution for particle in the atmosphere are computed.
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