Stereographic projection of axisymmetric nonuniform lens media

The radial variation of refractive index in the Maxwell fish-eye can be derived from the stereographic projection of the geodesics of the uniform sphere, as shown originally by Luneburg. This produces a perfect focus at the point diametrically opposite the source point. The generalisation of this is a refractive index law that provides for repetitive refocusing at points equispaced about the circle and for a medium which is axisymmetric. It is shown that this can be derived as the stereographic projection of a nonuniform sphere on which the latitudinal refractive index is such that the same form of focusing is obtained about the equator. The method in reverse then gives the surface law on the sphere corresponding to any radial variation in the plane and is illustrated for the Luneburg lens itself.