The perfect lens on a finite bandwidth

TL;DR

This paper derives an explicit formula for the optimal resolution of a perfect lens within a finite bandwidth, based on the properties of susceptibility functions modeled as Hermitian functions in the upper half-plane.

Contribution

It provides a novel explicit formula for the perfect lens resolution considering susceptibility functions as Hermitian $H^2$ functions with nonnegative imaginary parts.

Findings

Derived a simple explicit formula for lens resolution

Connected lens resolution to properties of Hermitian $H^2$ functions

Quantified the distance in $L^ abla$ norm for susceptibility functions

Abstract

The resolution associated with the so-called perfect lens of thickness $d$ is $-2\pi d/\ln(|\chi+2|/2)$. Here the susceptibility $\chi$ is a Hermitian function in $H^2$ of the upper half-plane, i.e., a $H^2$ function satisfying $\chi(-\omega)=\bar{\chi(\omega)}$. An additional requirement is that the imaginary part of $\chi$ be nonnegative for nonnegative arguments. Given an interval $I$ on the positive half-axis, we compute the distance in $L^\infty(I)$ from a negative constant to this class of functions. This result gives a surprisingly simple and explicit formula for the optimal resolution of the perfect lens on a finite bandwidth.