General Solution of the Luneberg Lens Problem

The general solution is obtained for the index of refraction of a variable-index, spherical lens which will form perfect geometrical images of the points of two given concentric spheres on each other. One conjugate sphere is assumed to be outside the lens or at its surface, while the other may be either inside, outside, or at the surface. If one of the spheres is of infinite radius, the lens will focus a parallel beam perfectly at a point on the other sphere. It is shown that the index of refraction may be specified arbitrarily, subject to two simple conditions, in an outer shell of any desired thickness less than the radius of the lens. The index of the central portion is then expressed in terms of a function which is tabulated, and of an integral involving the index of the outer shell. Some properties of the general solution are discussed, and various special solutions are derived.