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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“This is an author created, un-copy edited version of an article accepted for publication in Inverse Problems. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at DOI:10.1088/0266-5611/26/4/045001
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Glimm, Tilmann, "A Rigorous Analysis Using Optimal Transport Theory for a Two-reflector Design Problem with a Point Source" (2010). Mathematics. 59.