Laplace-Fourier analysis and instabilities of a gainy slab

TL;DR

This paper analyzes the stability of a gainy electromagnetic slab using realistic incident beams, revealing the necessity of complex frequencies or wavenumbers for describing amplified waves and clarifying earlier paradoxes.

Contribution

It introduces a realistic beam model to study gainy slabs, demonstrating the need for complex parameters and resolving previous ambiguities about instabilities in active systems.

Findings

Realistic incident beams clarify stability analysis.

Complex frequencies or wavenumbers are essential for describing amplification.

Absolute instability occurs only for normally incident plane waves.

Abstract

The idealization of monochromatic plane waves leads to considerable simplifications in the analysis of electromagnetic systems. However, for active systems this idealization may be dangerous due to the presence of growing waves. Here we consider a gainy slab, and use a realistic incident beam, which is both causal and has finite width. This clarifies some apparent paradoxes arising from earlier analyses of this setup. In general it turns out to be necessary to involve complex frequencies $\omega$ and/or complex transversal wavenumbers $k_x$. Simultaneously real $\omega$ and $k_x$ cannot describe amplified waves in a slab which is infinite in the transversal direction. We also show that the only possibility to have an absolute instability for a finite width beam, is if a normally incident plane wave would experience an instability.