Abstract

The time-dependent Maxwell system describing electromagnetic wave propagation in inhomogeneous isotropic media in the one-dimensional case reduces to a Vekua-type equation for bicomplex-valued functions of a hyperbolic variable, see [7]. In [5] using this reduction a representation of a general solution of the system was obtained in terms of a couple of Darboux-associated transmutation operators [8]. In [6] a Fourier–Legendre expansion of transmutation integral kernels was obtained. This expansion is used in the present work for obtaining an exact solution of the problem of the transmission of a normally incident electromagnetic time-dependent plane wave through an arbitrary inhomogeneous layer. The result can be used for efficient computation of the transmitted modulated signals. In particular, it is shown that in the classical situation of a signal represented in terms of a trigonometric Fourier series the solution of the problem can be written in the form of Neumann series of Bessel functions with exact formulas for the coefficients. The representation lends itself to numerical computation.

中文翻译：

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非均匀介质中随时间变化的一维电磁波传播：关于Trans变和贝塞尔函数的诺伊曼级数的精确解

摘要

在一维情况下，描述电磁波在非均质各向同性介质中传播的时变麦克斯韦系统简化为双曲变量双复值函数的Vekua型方程，请参见[7]。在[5]中，使用这种归约法，根据几个与Darboux相关的trans变算子[8]获得了系统的一般解。在[6]中，获得了integral变积分核的傅里叶-勒根德式展开式。在当前工作中使用这种扩展来获得对垂直入射的电磁时变平面波通过任意不均匀层传输的问题的精确解决方案。该结果可用于有效计算所传输的调制信号。尤其是，结果表明，在以三角傅里叶级数表示的信号的经典情况下，问题的解决方案可以用具有精确系数公式的贝塞尔函数的诺伊曼级数形式来表示。该表示法适合进行数值计算。