An optimal mass transport approach to the near-field reflector problem in optical design

In the near-field reflector (NFR) problem, one is given a point source of light with some radiation intensity and a target set at a finite distance. It is necessary to determine a reflector that intercepts the light rays emitted by the source and reflects them so that a given irradiance distribution is produced on the target. A special geometric method was developed in Kochengin and Oliker (1997 Inverse Problems 13 363–73) to solve this problem. This method was based on the representation of a reflector as an envelope of a family of confocal ellipsoids of revolution and the existence of weak solutions to the NFR problem was established in this class. In this paper, we investigate the applicability of an alternative approach to the same problem via optimal mass transport theory which was successful in several other optical design problems. Here, we show first that the above-mentioned representation of reflectors leads in a natural way to a generalized Legendre–Fenchel transform with unusual and interesting properties. This transform gives rise to a natural variational problem resembling a transportation problem with cost functional depending nonlinearly on one of the potentials. Somewhat surprisingly, it turns out that, in general, weak solutions of the NFR problem do not optimize this functional. We present an explicit example showing this. We also show that nevertheless this functional attains optimal values on reflectors in the above-mentioned class. (paper)