Resolving the wave-vector and the refractive index from the coefficient of reflectance

TL;DR

This paper presents a method to uniquely determine the wave-vector and refractive index in active media by analyzing the invariance of reflectance in finite films and their limits, resolving previous ambiguities.

Contribution

It introduces a novel approach that uses the invariance of reflectance in active media films to resolve the sign ambiguity of the wave-vector and determine the refractive index.

Findings

The reflectance coefficient remains invariant under sign inversion of $k_z$ in active media films.

The limit of infinite film thickness yields a unique, physically consistent wave-vector and refractive index.

The method clarifies the selection of the wave-vector sign in active media, resolving existing controversies.

Abstract

We resolve the existing controversy concerning the selection of the sign of the normal-to-the-interface component of the wave-vector $k_z$ of an electromagnetic wave in an active (gain) medium. Our method exploits the fact that no ambiguity exists in the case of a {\em film} of the active medium since its coefficient of reflectance is invariant under the inversion of the sign of $k_z$. Then we show that the limit of the infinite film thickness determines a unique and physically consistent choice of the wave-vector and the refractive index. Practically important implications of the theory are identified and discussed.