Fourier theory of linear gain media

TL;DR

This paper develops a Fourier--Laplace integral framework to analyze wave propagation in gain media, revealing non-commuting limits and predicting media with simultaneous positive and negative refraction.

Contribution

It introduces a comprehensive method for correctly taking monochromatic and plane wave limits in gain media, and establishes criteria for absolute instabilities and novel refractive properties.

Findings

Non-commuting limits in gain media analysis.

Existence of media with simultaneous positive and negative refraction.

General criterion for absolute instabilities.

Abstract

The analysis of wave propagation in linear, passive media is usually done by considering a single real frequency (the monochromatic limit) and also often a single plane wave component (plane wave limit), separately. For gain media, we demonstrate that these two limits generally do not commute; for example, one order may lead to a diverging field, while the other order leads to a finite field. Moreover, the plane wave limit may be dependent on whether it is realized with a rect function excitation or gaussian excitation of infinite widths. We consider wave propagation in gain media by a Fourier--Laplace integral in time and space, and demonstrate how the correct monochromatic limit or plane wave limit can be taken, by deforming the integration surface in complex frequency--complex wavenumber space. We also give the most general criterion for absolute instabilities. The general theory is applied in several cases, and is used to predict media with novel properties. In particular, we show the existence of isotropic media which in principle exhibit simultaneous refraction, meaning that they refract positively and negatively at the same time.