On the eikonal function

A survey of the various mathematical derivations of the eikonal equation of Bruns is given, each having its own geometrical approximation. These include the Sommerfeld-Runge ansatz and the variational principle applied to the action integral. The apparent arbitrariness of the ansatz is justified by its giving the same form of eikonal equation as does the more rigorous extremal solution of the Euler-Lagrange equations. A two-scalar representation of the optical part of the total field is presented, termed Bateman potentials, which, for the null field, also derives the eikonal equation. It is shown that the ansatz is a form of Bateman potential, and the derivation explicitly defines rays, phase fronts, and the field polarization. When applied to ray congruences in nonuniform media, the Bateman potential description is in general agreement with known results. In the case of the Maxwell fish-eye, however, a discrepancy occurs. The reasons and consequences of this suggest the possibility of using the Bateman potential method to form an improvement to the basic geometrical optics approximation.