Abstract
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.
Highlights
The following beam shaping problem from geometric optics was described in [1]; see Figure 1
A linear programming approach similar to the one we investigate here for more general situations for optimal approach was investigated by Ruschendorf and Uckelmann in 2000 [22] and more recently in two papers by Canavesi et al [23, 24] who propose a new algorithm inspired by, but distinct from, a linear programming approach for the single reflector problem treated in [8, 9]
We investigated two numerical schemes for solving an infinite-dimensional optimal transportation problem arising in reflector design, a straightforward discretization and a novel iterative scheme, which uses knowledge of the previous solution in each step to reduce the number of constraints
Summary
The following beam shaping problem from geometric optics was described in [1]; see Figure 1. The problem we are concerned with consists of transforming this input beam into an output beam of parallel light rays with a prescribed intensity distribution. This transformation is to be achieved with a system of two reflectors. The problem has some practical importance in engineering; see further literature cited in [1]. See the references [7,8,9,10,11] which deal with other beam shaping problems using related techniques
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