The evanescent waves in geometrical optics and the mixed hyperbolic-elliptic type systems

TL;DR

This paper investigates evanescent waves in geometrical optics using a mixed hyperbolic-elliptic system, deriving the Ludwig system from Helmholtz, and analyzes wave phenomena like the Goos-Hanchen effect.

Contribution

It introduces a complex-valued phase approach via the Ludwig system, derived from Helmholtz, to describe evanescent waves and wave transition phenomena.

Findings

Derivation of Ludwig system from Helmholtz equation.

Description of wave transition using Stokes phenomenon and Airy functions.

Proof of the existence of the Goos-Hanchen effect.

Abstract

In this article we describe the generation of the evanescent waves which are present in the rarer medium at total reflection by using a mixed-type system, the Ludwig system, which leads naturally to consider a complex-valued phase. The Ludwig system is derived from the Helmholtz equation by using an appropriate modification of the stationary phase procedure: the Chester, Friedman and Ursell's method. The passage from the illuminated to the shadow region is described by means of the ray switching mechanism based on the Stokes phenomenon applied to the Airy function. Finally, the transport system connected to the Ludwig eikonal system is studied in the case of linear wavefronts and the existence of the Goos-Hanchen effect is proved.