The k function in geometrical optics and its relationship to the archetypal wave front and the caustic surface

The k function is the arbitrary function that arises in the general solution of the eikonal equation, a nonlinear, first-order partial differential equation, and describes completely the geometrical properties of a wave-front train and the associated caustic in a homogeneous, isotropic optical medium. The archetypal wave front in such a train is that unique wave front whose optical path length from the ultimate object point is zero. As an example, the k function of a wave-front train resulting from a plane wave front incident upon a spherical refracting surface is calculated. From it the archetype and the caustic are obtained. The results agree exactly with ray-trace data.