Geometrical theory of diffracted rays, orbiting and complex rays

TL;DR

This paper extends classical geometrical optics by modeling rays as geodesics in a Riemannian manifold, deriving new quantization and diffraction properties, and applying these to phenomena like orbiting, creeping waves, and evanescent waves.

Contribution

It introduces a geometrical framework for analyzing diffracted and complex rays, including new quantization rules and methods for handling caustics and shadow regions.

Findings

Derived geometrical quantization rule for orbiting states

Modified stationary phase method near caustics

Extended eikonal equation to complex and mixed systems

Abstract

In this article, the ray tracing method is studied beyond the classical geometrical theory. The trajectories are here regarded as geodesics in a Riemannian manifold, whose metric and topological properties are those induced by the refractive index (or, equivalently, by the potential). First, we derive the geometrical quantization rule, which is relevant to describe the orbiting bound-states observed in molecular physics. Next, we derive properties of the diffracted rays, regarded here as geodesics in a Riemannian manifold with boundary. A particular attention is devoted to the following problems: (i) modification of the classical stationary phase method suited to a neighborhood of a caustic; (ii) derivation of the connection formulae which enable one to obtain the uniformization of the classical eikonal approximation by patching up geodesic segments crossing the axial caustic; (iii) extension of the eikonal equation to mixed hyperbolic-elliptic systems, and generation of complex-valued rays in the shadow of the caustic. By these methods, we can study the creeping waves in diffractive scattering, describe the orbiting resonances present in molecular scattering beside the orbiting bound-states, and, finally, describe the generation of the evanescent waves, which are relevant in the nuclear rainbow.