Electromagnetic wave propagation through inhomogeneous material layers

TL;DR

This paper derives and solves the wave equation for electromagnetic waves in inhomogeneous media with variable permeability and permittivity, providing exact solutions using Heun functions, advancing understanding of wave behavior in complex materials.

Contribution

It introduces a simplified wave equation for inhomogeneous media with variable properties and provides an exact analytic solution in terms of Heun functions, which is novel.

Findings

Derived a general wave equation for inhomogeneous media.

Obtained an exact analytic solution using Heun functions.

Confirmed the solution aligns with known results for specific cases.

Abstract

We use Maxwell's equations in a sourceless, inhomogeneous medium with continuous permeability $\mu (\mathbf{r}) $ and permittivity $% \epsilon (\mathbf{r}) $ to study the wave propagation. The general form of the wave equation is derived and by virtue of some physical assumptions, including $\mu $ and $\epsilon $ as functions of $z,$ the equation has been simplified. Finally by introducing a smooth step dielectric variable we solve the wave equation in the corresponding medium which is in conform with the well known results. Exact double-layer solution in analytic form has been given in terms of the Heun functions.