The manifestation of transmission and reflection in Hamilton's principal function for rigorous ray tracing

Abstract

Hamilton's principal function for rigorous ray tracing is shown to manifest reflection and transmission at a step in a piecewise-constant index of refraction. The first and second derivatives of Hamilton's principal function are continuous across the step. This specifies the generator of the motion, Hamilton's principal function, which is sufficient to describe propagation including the bifurcation of the incoming wave function into the reflected and transmitted wave functions at the step in the index of refraction. This alternative description is consistent with the contemporary description determined by matching the wave function and its first derivative at the step in the index of refraction.