On the boundary intensities in a plane parallel slab with linearly varying refractive index

TL;DR

This paper derives an exact formula for boundary intensities in a plane parallel slab with a linearly varying refractive index, showing an equivalence between diffuse reflection law and pseudo-source methods.

Contribution

It introduces a new exact expression for boundary intensities in a medium with linearly varying refractive index and compares two analytical approaches.

Findings

Exact boundary intensity expression derived

Diffuse reflection law matches pseudo-source method results

Simplifies analysis of semi-transparent media with variable refractive index

Abstract

An exact expression of the outgoing boundary intensities in the case of a gray semi-transparent medium confined in a plane parallel slab with a refractive index linearly depending on the position is proposed. It is shown that directly using the diffuse reflection law gives in a much easier way exactly the same result as the one obtained with the pseudo-source method.