Generalizations of spherically symmetric lenses

The purpose of this paper is to generalize the solutions of some spherically symmetric lens and lens-reflector problems recently treated by Kay. The original problem was to find a variable-index structure, with a point source at its surface or at infinity, which would produce a beam of finite angular width, having a prescribed variation of intensity with angle. It is shown that a prescribed exit beam can be obtained from a point source at any given distance from the lens, and that the index of refraction may be specified more or less arbitrarily in the outer part of the lens. A special case is solved in terms of tabulated functions.