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

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

2009-08-17

arxiv.org/abs/0908.2464v1

Mathematics

Analysis of PDEs

5 Figures - pdf files attached to submission, but not shown in manuscript

Scientific paper

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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