Determination of reflector surfaces from near-field scattering data

Abstract

Consider a reflector system consisting of a reflecting surface R, a point light source O and some object T. Suppose that the source O, reflector R and object T are positioned so that the rays reflected off R are incident on T. Depending on the geometry of R, the energy, radiated by O and redirected by R, is distributed on T producing a certain illumination pattern. We present here a solution of the inverse problem consisting of reconstructing the reflector R from the following data: the position of the source O, its radiation intensity I, the object T, and prespecified energy pattern to be achieved on T. It is shown that under assumptions of geometric optics theory the problem admits a solution, provided the total input and output energies are equal, and some other geometric conditions are satisfied. In analytic formulation, the problem leads to an equation of Monge - Ampère type in a domain on a unit sphere. In this paper we formulate the problem in terms of certain associated measures and establish the existence of weak solutions.