For certain index profiles, the lens will form perfect geometrical images of two given concentric spheres onto each other. There are an infinite number of refractive-index profiles that can produce this effect. The simplest such solution was proposed by Rudolf Luneburg in 1944. Luneburg's solution for the refractive index creates two conjugate foci outside of the lens. The solution takes a simple and explicit form if one focal point lies at infinity, and the other on the opposite surface of the lens. J. Brown and A. S. Gutman subsequently proposed solutions which generate one internal focal point and one external focal point. These solutions are not unique; the set of solutions are defined by a set of definite integrals which must be evaluated numerically.