Uniqueness in reflector mappings and the Monge-Ampère equation

Abstract

Abstract A uniqueness theorem is presented for reflector mappings in the complex plane. These arise in geometrical optics, in the synthesis of a reflector surface to produce a ray beam with specified angular distribution of energy when illuminated by a non-isotropic point-source. The mappings may be represented as solutions of a nonlinear boundary-value problem involving the Monge-Ampère equation. When the equation is elliptic there are at most two solutions, provided that either the incident or far-field ray cone satisfies a convexity condition, which is always the case if one cone is circular. The result has practical application to the design of single and offset dual reflectors used for radar and communications purposes.