Deterministic approximations of random reflectors

TL;DR

This paper demonstrates that in 2D optics, any measure-preserving random reflector can be approximated by deterministic rough surfaces, linking microscopic roughness to probabilistic reflection models.

Contribution

It establishes a theoretical connection between deterministic microscopic roughness and macroscopic random reflection in 2D billiards.

Findings

Any measure-preserving random reflector can be approximated by deterministic rough surfaces.

The approximation respects the measure-preservation condition in billiards.

Provides a mathematical framework linking microscopic roughness to probabilistic reflection.

Abstract

Within classical optics, one may add microscopic "roughness" to a macroscopically flat mirror so that parallel rays of a given angle are reflected at different outgoing angles. Taking the limit (as the roughness becomes increasingly microscopic) one obtains a flat surface that reflects randomly, i.e., the transition from incoming to outgoing ray is described by a probability kernel (whose form depends on the nature of the microscopic roughness). We consider two-dimensional optics (a.k.a. billiards) and show that every random reflector on a line that satisfies a necessary measure-preservation condition (well established in the theory of billiards) can be approximated by deterministic reflectors in this way.