Far field refraction problem with loss of energy in negative refractive index material

TL;DR

This paper investigates the far field refraction problem in negative refractive index materials with energy loss, establishing existence of solutions and deriving inequalities related to Monge-Ampère operators to better understand this optical phenomenon.

Contribution

It introduces a Minkowski method approach to prove the existence of weak solutions in different cases of refractive index and derives key inequalities involving Monge-Ampère type operators.

Findings

Existence of weak solutions for different refractive index cases

Derivation of inequalities involving Monge-Ampère operators

Enhanced understanding of energy loss in negative refractive index materials

Abstract

This paper studies the far field refraction problem in negative refractive index material with loss of energy, which is a remaining problem in E. Stachura, Nonlinear Anal. 2017;157:76-103. The analysis is divided into two cases according to the relative refractive index $\kappa$, that is, $\kappa<-1$ and $-1<\kappa<0$. For each case, we use the Minkowski method to establish the existence of the weak solution when the target measure is either discrete or a finite Radon measure. Eventually, the inequality involving a Monge-Amp\`ere type operator satisfied by the solution of the problem is derived, which is useful to understand this complex optical phenomenon.