Rays in gradient-index media: separable systems

As an aid to methods of designing conventional optical systems incorporating gradient-index elements, it is useful to know particular index functions that lead to simple solutions of the differential ray equations. This paper deals with the problem of finding such functions systematically. To make it well defined, certain criteria are first specified that objectively determine whether or not a given solution is to be called “simple.” After that only “separable” systems are admitted, i.e., those whose index functions are such that there exists at least one choice of coordinate system for which the Hamilton–Jacobi equation is separable. The generic forms of particular index distributions that lead to simple solutions (in the sense already defined) may then be specified. The method is illustrated by five explicit examples. The paper concludes with some remarks on (i) the finite ray equations and (ii) the point characteristic.