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arXiv · 0908.2464

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

Tilmann Glimm
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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.

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Tilmann Glimm. 2009-08-17. A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source. https://doi.org/10.1088/0266-5611%2F26%2F4%2F045001

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