Mean path length in refractive regular polygons and prisms

Abstract

We analytically derive the mean path length of light rays diffusely incident on refractive regular polygons of $n$ sides ($n$-gons), analyzed from the dynamical billiards perspective and from ray optics. In polygons with sufficiently low refractive index, the mean path length is found to be equal to that in the invariant scattering case, i.e., the product of the mean chord length with the refractive index. If the refractive index is higher than some critical value, the mean path length is lower than the scattering value due to inaccessibility of trapped modes. Regular odd $n$-gons are found to have the same mean path length as $2n$-gons, when normalized against their mean chord length. There is a discontinuity between the mean path length in high $n$-gons and that in a circle, attributed to quasitrapped modes in high $n$-gons. This difference is removed with a small amount of absorption. We extend the results to refractive prisms with a regular polygon base, for example, hexagonal ice crystals. For prisms with any ergodic shaped base, we derive a simple formula for the mean path length.