Exploiting billiard theory to calculate the mean path length of light in refractive spheroids

Abstract

A technique is introduced to calculate the mean path length of light rays diffusely incident on a refractive object. It uses the phase portrait from billiard theory to determine the criteria for which chords are accessible for a given refractive index. The mean path length is given as an integral over the lengths of chords accessible via refraction, which implicitly accounts for total internal reflections. We demonstrate this method by calculating the mean path length in ellipses and spheroids. The mean path length is given by a double integral for the ellipse and a triple integral for the spheroid, which may be evaluated numerically, and also allows us to deduce simple series expansions for low eccentricity. These results give analytic expressions for the orientation averaged absorption cross sections in the geometric optics limit.