A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source

TL;DR

This paper rigorously analyzes a two-reflector design problem in geometric optics using optimal transport theory, proving existence and uniqueness of solutions, and providing a practical algorithm for reflector construction.

Contribution

It introduces a novel rigorous mathematical framework for the two-reflector problem using optimal transport, including existence, uniqueness, and an algorithmic solution.

Findings

Proved existence of solutions for a broad class of data.

Established uniqueness of the reflector configuration.

Developed a practical algorithm based on linear optimization.

Abstract

We consider the following geometric optics problem: Construct a system of two reflectors which transforms a spherical wavefront generated by a point source into a beam of parallel rays. This beam has a prescribed intensity distribution. We give a rigorous analysis of this problem. The reflectors we construct are (parts of) the boundaries of convex sets. We prove existence of solutions for a large class of input data and give a uniqueness result. To the author's knowledge, this is the first time that a rigorous mathematical analysis of this problem is given. The approach is based on optimal transportation theory. It yields a practical algorithm for finding the reflectors. Namely, the problem is equivalent to a constrained linear optimization problem.