均匀各向异性媒质的简化波函数理论

Simplified Theory of Wavefunctions for Homogeneous Anisotropic Media

基于电磁波的角谱展开法,均匀各向异性媒质的第一、第二、第三、第四类球波函数已由第一作者十多年前定义并已用于解决各种工程问题。该文给出一种均匀各向异性媒质的简化波函数理论。简化的含义是在原来的角谱积分表达式的基础上导出有限和表达式。因而最终波函数的级数表达式不再含数值积分。从应用角度看就是最终线性方程组每一元素均为显式表达,没有数值积分。这一理论的数值计算和验证已在二维情况下完成,则给出适用于三维情况的,在圆柱、圆球坐标系下的简化矢量波函数的表达式。

In terms of the angular spectral expansion of electromagnetic waves, the spherical wave functions of the first, second, third and fourth kinds in homogeneous anisotropic media was introduced by the first author of this paper more than ten years ago. These wave functions have been applied to a variety of engineering applications. In this paper, a simplified theory of wave functions for homogeneous anisotropic is developed. The simplification means that finite summation expression of wave functions can be derived from the former angular spectral integral representation. Therefore the coefficients of the series of a wave function contains no numerical integral. From the point view of applications, the matrix element of the involved linear system of equations is an explicit expression without numerical integration. The numerical implementation and validation have been completed for the two-dimensional cases,while this paper introduces the representations of the simplified theory of vector wavefunctions in the circular cylindrical and spherical coordinates which will be suitable for applications in three-dimensional cases.