A multiscale approach to a linear programming problem for calculating a refractive optical element forming a given far-field illumination distribution

We consider an approach to the calculation of a refractive optical element generating a prescribed irradiance distribution in the far field for a plane incident beam based on a certain variational problem. We consider an explicit formulation of this problem in the form of the Monge-Kantorovich mass transfer problem. We also demonstrate a connection between the mass transfer problem and the dual linear variational problem. A numerical solution of the linear variational problem is also considered. The direct solution of this type of problem presents a huge computational complexity. To overcome this difficulty we use the so-called multiscale approach based on constructing a chain of approximations that are solutions on refining grids.