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On the use of the supporting quadric method in the problem of designing double freeform surfaces for collimated beam shaping.

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.