Generalization of a Theorem Proved by Abelès and Lekner Relating to Two Juxtaposed Slabs with Inhomogeneous Refractive Indices

Abstract

Abelès and Lekner have proved a simple theorem relating to reflectance and transmittance through (i) an absorbing slab, and (ii) the same absorbing slab to which is juxtaposed a nonabsorbing slab. This theorem is completely generalized, the necessary matrix algebra being formulated in terms of direct product matrices, and their Jordan canonical forms. Five cases must be distinguished in the complex plane of the transmission coefficient for the nonabsorbing slab. These are considered in detail, and some unexpected results emerge from the investigation.