Position and momentum bases for the monochromatic Maxwell fish-eye and the sphere

In geometric optics the Maxwell fish-eye is a medium where light rays follow circles, while in scalar wave optics this medium can only 'trap' fields of certain discrete frequencies. In the monochromatic case characterized by a positive integer ℓ, there are $2\ell +1$ independent fields. We identify two bases of functions: one, known as the Sherman–Volobuyev functions, is characterized as of 'most definite' momenta; the other is new and composed of 'most definite' positions and normal derivatives for the fish-eye scalar wavefields. Their construction uses the stereographic projection of the sphere, and their identification is corroborated in the $\ell \to \infty$ contraction limit to the homogeneous Helmholtz medium. (paper)