Variational approach to eikonal function computation

The problem of calculating the eikonal function from the condition of focusing into a prescribed region is formulated as a variational problem and as a Monge-Kantorovich mass transportation problem. It is found that the cost function in the Monge-Kantorovich problem corresponds to the distance between a point of the original region (in which the eikonal function is defined) and a point of the focal region. The formalism proposed in this work makes it possible to reduce the calculation of the eikonal function to a linear programming problem. In this case, the calculation of the “ray mapping” corresponding to the eikonal function is reduced to the solution of a linear assignment problem. The proposed variational approaches are illustrated by examples of calculation of optical elements for focusing a circular beam into a rectangular region.