Hybrid design of diffractive optical elements for optical beam shaping.

Hybrid methods combining the geometrical-optics and diffraction-theory methods enable designing diffractive optical elements (DOEs) with high performance due to the suppression of stray light and speckles and, at the same time, with a regular and fabrication-friendly microrelief. Here, we propose a geometrical-optics method for calculating the eikonal function of the light field providing the generation of a required irradiance distribution. In the method, the problem of calculating the eikonal function is formulated in a semi-discrete form as a problem of maximizing a concave function.

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.

Optimal mass transportation problem in the design of freeform optical elements generating far-field irradiance distributions for plane incident beam.

We consider the problem of calculating a refracting surface generating a prescribed irradiance distribution in the far field in the case of a plane incident beam. We demonstrate that this problem can be formulated as a mass transportation problem (MTP) and obtain the cost function for the MTP. It is shown that with a special choice of coordinates, the cost function becomes quadratic.

Optimal mass transportation and linear assignment problems in the design of freeform refractive optical elements generating far-field irradiance distributions.

We consider the problem of calculating a refractive surface generating a prescribed irradiance distribution in the far field in the case of a point light source. We show that this problem can be formulated as a mass transportation problem with a non-quadratic cost function. A method for calculating the refractive surface is proposed, which is based on reducing the problem of calculating an integrable ray mapping to finding a solution to a linear assignment problem.

Linear assignment problem in the design of freeform refractive optical elements generating prescribed irradiance distributions.

We consider the problem of calculating the eikonal function defined on a certain curved surface from the condition of generating a prescribed irradiance distribution on a target surface. We show that the calculation of the "ray mapping" corresponding to the eikonal function is reduced to the solution of a linear assignment problem (LAP). We propose an iterative algorithm for calculating a refractive optical surface from the condition of generating a prescribed near-field irradiance distribution in a non-paraxial case.

Variational approach to calculation of light field eikonal function for illuminating a prescribed region.

The problem of calculation of the light field eikonal function providing focusing into a prescribed region is formulated as a variational problem and as a Monge-Kantorovich mass transportation problem. It is obtained that the cost function in the Monge-Kantorovich problem corresponds to the distance between a point of the source region (in which the eikonal function is defined) and a point of the target region. This result demonstrates that the sought-for eikonal function corresponds to a mapping, for which the total distance between the points of the original plane and the target region is minimized.