The confinement of electromagnetic waves and Fredholm's alternative

Abstract

Nowadays, the question about the recovery of lost information due to the presence of the so called electromagnetic evanescent waves, seems to be explained enough. But in fact we know that travelling and evanescent waves both originate from very different stems of integral or differential equations and whose behavior is classified as mutually discriminating; that is, we cannot have simultaneous validity of both kinds of equations for the same physical circumstances. Nevertheless, if our system can be described by a Fredholm's equation we can at least build a bridge between these two sets of rules, based to the Fredholm's eigenvalue. When the Fredholm's eigenvalue falls into certain range then Fredholm's equation describes a normal travelling spectrum; otherwise, we are in the presence of another type of equation with abnormal or special behavior. In this work, we analyze the so-named Fredholm's alternative, which enables us to describe the change of positive refraction index-like conditions of broadcasting media to negative refraction index-like conditions. We also sketch the frequency dependence of Fredholm's eigenvalue in order to establish general rules for the breaking of the waves' confinement.