Time-dependent one-dimensional electromagnetic wave propagation in inhomogeneous media: exact solution in terms of transmutations and Neumann series of Bessel functions

TL;DR

This paper presents an exact analytical solution for one-dimensional electromagnetic wave transmission through inhomogeneous media using transmutation operators and Neumann series of Bessel functions, enabling efficient numerical computation.

Contribution

It introduces a novel exact solution method for wave propagation in inhomogeneous media based on transmutation operators and Bessel function series, extending previous theoretical frameworks.

Findings

Exact solution expressed as Neumann series of Bessel functions

Efficient numerical computation of transmitted signals

Applicable to arbitrary inhomogeneous layers

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 arXiv:1001.0552. In arXiv:1410.4873 using this reduction a representation of a general solution of the system was obtained in terms of a couple of Darboux-associated transmutation operators arXiv:1111.4449. In arXiv:1508.02738 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.