Optical design of reflector systems and the Monge-Kantorovich mass transfer problem

We consider the geometric optics problem of constructing a system consisting of two reflectors which transforms a plane wave front with given intensity into an output plane wave front with prescribed output intensity.

In this talk, we describe how this problem is deeply connected to the Monge-Kantorovich mass transfer problem (MKP) with quadratic cost function. Namely, we show that the way in which the two light fronts are transformed into each other minimizes (or maximizes) an energy transportation cost.

This connection yields a new method for solving the two-reflector problem. Conversely, the connection also gives novel insights into the geometric nature of the dual formulation of the MKP.

The techniques extend to other reflector construction problems, for example, a single reflector problem which can be linked to Monge-Kantorovich problem on the sphere. We will further present some numerical computations of reflectors based on a linear programming approach to the MKP.

This talk is based on joint work with V. Oliker.