Fredholm's alternative breaks the confinement of electromagnetic waves

J M Velázquez-Arcos

doi:10.1063/1.4821336

Abstract

The recovery of information from the so called electromagnetic evanescent waves seems to be a very well explained item. Nevertheless, the travelling waves that becomes from the evanescent waves emerge from integral or differential equations that are very different to those describing the conventional ones. Indeed, we can say that the two kinds of solutions, the travelling and evanescent waves represent a mutually discriminating problem in which we cannot have simultaneous validity of both kinds of equations even they represents the physical evolution of a the same system. But if we can describe our system with a Fredholm's equation we can relate both situations through the properties of the Fredholm's eigenvalue. When the Fredholm's eigenvalue has its values 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 some general conditions for the Fredholm's eigenvalue in order to establish general rules for the breaking of the waves’ confinement.

Highlights

We have shown how the Fredholm’s alternative (FA) acts as a trigger of switching between positive refraction indexlike conditions of broadcasting media to negative refraction index-like conditions
From general conditions we found that the eigenvalue function can be selected to have the structure of a phase factor (equation (10))
We have shown explicitly the orthogonality rule (equation (14)) for the transition between non-resonant to resonant conditions that tell us that the resonant solutions vanish at the point sources

Summary

INTRODUCTION

The two different kinds of physical conditions there are travelling waves that allow communications, their frequencies are basically different because in the positive refraction index-like conditions, the corresponding frequency values to the resonant ones are really hidden or in a practical sense are confined to the so called near zone without the possibility to bring any information because of their exponentially decaying amplitudes.[9,10] The paper is organized as follows: In order to visualize the relation between the ever changing real conditions and the academic strict dichotomy, we will show the generalized Fredholm’s equation frequency dependent with a source term (section II). In appendix B we justify to suppose that the eigenvalue can be selected as a phase factor

THE GENERALIZED FREDHOLM’S EQUATION FOR ELECTROMAGNETIC FIELD WITH SOURCES

THE HOMOGENEOUS FREDHOLM’S EQUATION AND THE FREDHOLM’S EIGENVALUE

CONCLUDING REMARKS