  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.
