Near Field Refraction Problem With Loss of Energy in Negative Refractive Index Material

TL;DR

This paper investigates the near field refraction problem in negative refractive index materials with energy loss, analyzing different cases of the refractive index and establishing conditions for solutions.

Contribution

It introduces a new framework for defining and analyzing the refractor in negative index materials, including existence proofs for weak solutions under various target measures.

Findings

Existence of weak solutions for discrete target measures

Analysis of Fresnel coefficients in negative index materials

Discussion of the critical case where  = -1

Abstract

This paper studies the near field refraction problem with loss of energy in negative refractive index material. Based on the relative refractive index $\kappa$, the analysis is categorized into two cases, namely $\kappa < -1$ and $-1 < \kappa < 0$. For each case, we give the definition of the refractor and discuss some crucial properties of it. The properties of Fresnel coefficients are also discussed. Based on these properties, the existence of the weak solution when the target measure is either discrete or a finite Radon measure are proved. Besides, the critical case $\kappa = -1$ is also discussed briefly at the end of this paper.