Singular geometrical optics for differential operators on surfaces

Abstract

Inspired by the optical phenomenon of conical refraction, discovered by Hamilton in 1832, we study the existence of singular optical phenomena associated with linear differential operators acting on vector fields on a surface. We do this by studying the singularities of the Fresnel hyper-surface associated with the differential operator and show that the existence of these singularities can be accounted for from purely topological considerations. We associate a topological number (an integer) to the space of singularities of the Fresnel hyper-surface, which can be used to “count” the number points at which singular optical phenomena occur.