Propagation of Plane Electromagnetic Waves in Inhomogeneous Media

Abstract

The laws of propagation of plane electromagnetic waves along z are derived from Maxwell’s equations for inhomogeneous media in which the dielectric constant and the electric conductivity are functions of z and in which the magnetic permeability is constant. The corresponding wave equation is reduced to a Ricatti differential equation. A simple method is evolved for treating this Riccati equation without approximation. As a result, it is shown that the laws of propagation depend primarily upon two physical parameters v(z) and u(z) which have the property v(z)=n and −u(z)=nK for homogeneous media, where n is refractive index. The increase in phase retardation of the wave is always governed by the integral of v(z)dz, and v(z) is always the ratio of the phase velocity in vacuum to the phase velocity in the medium. It is shown that sufficiently continuous, single, infinite, inhomogeneous media cannot exhibit reflectance. Fresnel’s coefficients of reflectance and transmittance are derived for normal incidence upon a plane boundary between two inhomogeneous media. The theory includes homogeneous media as special cases.