On the use of the supporting quadric method in the problem of designing double freeform surfaces for collimated beam shaping
Albert A. Mingazov, Dmitry A. Bykov, Evgeni A. Bezus, Leonid L. Doskolovich
Abstract
We propose a version of the supporting quadric method for calculating a refractive optical element with two working surfaces for collimated beam shaping. Using optimal mass transportation theory and generalized Voronoi cells, we show that the proposed method can be regarded as a gradient method of maximizing a concave function, which is a discrete analogue of the Lagrange functional in the corresponding mass transportation problem. It is demonstrated that any maximum of this function provides a solution to the problem of collimated beam shaping. Therefore, the proposed method does not suffer from "trapping" at a local extremum, which is typical for gradient methods. We present design examples of refractive optical elements illustrating high performance of the method.