When is an Optical Design Problem a Mass Transfer Problem?

Abstract

This paper investigates whether optical design problems can be represented as Monge--Kantorovich mass transfer problems with some cost function. An optical problem is specified by a function defining a particular family of supporting surfaces for the optical element. We introduce the notion of geometric equivalence between optical and transport problems, which is directly related to the possibility of formulating an optical design problem as a mass transfer problem. A necessary condition for such equivalence is derived in the semi-discrete setting. It allows a transport representation to be ruled out without choosing a cost function in advance. The condition is applied to the near-field refractor problem and to the design of two refracting surfaces transforming a spherical beam into a spherical or collimated beam. For these problems, the necessary condition is shown to fail in physically admissible configurations. We also study an optical element with a single working surface and an arbitrary regular incident wavefront. In the far field, geometric equivalence to a mass transfer problem requires a planar or spherical wavefront, whereas in the near field with a planar target, the necessary condition fails for an arbitrary smooth wavefront.