On the tunneling hypothesis for ray reflection and transmission at a concave dielectric boundary

When a ray field is incident on a concave boundary confining a dielectric medium, total reflection is perturbed by leakage due to surface curvature. The resulting modification of the conventional Fresnel ray reflection coefficient, including its uniform transition through critical incidence, has previously been addressed by the so-called "tunneling hypothesis," which is based essentially on the behavior of a corresponding peripherally guided whispering gallery or leaky modal field; the former exhibits evanescent decay (tunneling) away from the boundary in the exterior region but eventually gives rise to radiation from a caustic whereon the modal phase speed equals the speed of light. It is shown here that inferring local ray field properties from global mode field properties has limited validity. The demonstration is based on a rigorous analysis of the two-dimensional Green's function for a circular boundary. Asymptotic solutions are constructed for the various ray-optical domains, and for the transition regions near caustics and especially near the critically refracted ray. Examination of the reflected and transmitted fields reveals that the tunneling hypothesis holds only near the critically refracted ray. Elsewhere, the transmitted ray field may deviate markedly from that predicted by the tunneling model. The results clarify not only the ray field behavior but also the mechanism of local energy reflection and transmission for a nonplanar interface.