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. The formalism proposed in the present work makes it possible to reduce the calculation of the eikonal function to a linear programming problem. Besides, the calculation of the “ray mapping” corresponding to the eikonal function is reduced to the solution of a linear assignment problem. The proposed approach is illustrated by examples of calculation of optical elements for focusing a circular beam into a rectangle and a beam of square section into a ring.