We consider the geometric optics problem of finding a system of two reflectors that transform a spherical wavefront into a beam of parallel rays with prescribed intensity distribution. Using techniques from optimal transportation theory, it has been shown previously that this problem is equivalent to an infinite-dimensional linear programming (LP) problem. Here we investigate techniques for constructing the two reflectors numerically by considering the finite dimensional LP problems which arise as approximations to the infinite dimensional problem. A straightforward discretization has the disadvantage that the number of constraints increases rapidly with the mesh size, so only very coarse meshes are practical. To address this well-known issue we propose an iterative solution scheme. In each step an LP problem is solved, where information from the previous iteration step is used to reduce the number of necessary constraints. As an illustration we apply our proposed scheme to solve a problem with synthetic data, demonstrating that the method allows for much finer meshes than a simple discretization. We also give evidence that the scheme converges. There exists a growing literature for the application of optimal transportation theory to other beam shaping problems, and our proposed scheme is easy to adapt for these problems as well.
ISRN Applied Mathematics
Article ID 635263
Required Publisher's Statement
Copyright © 2013 Tilmann Glimm and Nick Henscheid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.http://dx.doi.org/10.1155/2013/635263
Glimm, Tilmann and Henscheid, Nick, "Iterative Scheme for Solving Optimal Transportation Problems Arising in Reflector Design" (2013). Mathematics. Paper 39.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.