A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source. (English) Zbl 1187.78067

Glimm, Tilmann

Inverse Probl. 26, No. 4, Article ID 045001, 16 p. (2010).

Summary: 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.

MSC:

78M50	Optimization problems in optics and electromagnetic theory
78A05	Geometric optics
78M25	Numerical methods in optics (MSC2010)

Keywords:

optimization; geometric optics; other numerical methods