Hidden symmetry and potential group of the Maxwell fish-eye

Abstract

The Maxwell fish-eye is an exceptional optical system that shares with the Kepler problem and the point rotor (mass point on a sphere) a hidden, higher rotation symmetry. The Hamiltonian is proportional to the Casimir invariant. The well-known stereographic map is extended to canonical transformations between of the phase spaces of the constrained rotor and the fish-eye. Their dynamical group is a pseudoorthogonal one that permits a succint ‘‘4π’’ wavization of the constrained system. The fish-eye exhibits, unavoidably, chromatic dispersion. Further, a larger conformal dynamical group contains the potential group, that relates the closed, inhomogeneous fish-eye system to similar, scaled ones. Asymptotically, it is related to free propagation in homogenous media.