Study of electromagnetic wave propagation in active medium and the equivalent Schrodinger equation with energy-dependent complex potential

TL;DR

This paper explores the analogy between electromagnetic wave propagation in active media and quantum mechanics by modeling the system with a complex, energy-dependent potential in the Schrödinger equation, revealing stability thresholds.

Contribution

It introduces an energy-dependent complex potential framework to represent active medium wave propagation and analyzes stability thresholds via quantum analogies.

Findings

Identification of a frequency-dependent gain threshold for stability.

Demonstration of pole crossing in the complex energy plane.

Comparison between time-dependent and stationary solutions.

Abstract

We study the massless limit of the Klein-Gordon (K-G) equation in 1+1 dimensions with static complex potentials as an attempt to give an alternative, but equivalent, representation of plane electromagnetic (em) wave propagation in active medium. In the case of dispersionless em medium, the analogy dictates that the potential in the K-G equation is complex and energy-dependent. In the non-relativistic domain we study an analogous inertial system by considering wave packet propagation through a complex potential barrier and solve the time-independent Schrodinger equation with a potential that has the same energy dependence as that of the K-G equation. The behavior of these solutions is compared with those found elsewhere in the literature for the propagation of electromagnetic plane waves in a uniform active medium with complex dielectric constant. Our study concluded that the discrepancy between the time dependent and stationary results is due to the energy poles crossing the real axis in the complex energy plane. It was demonstrated unambiguously that there is a frequency (energy) and size dependent gain threshold above which the stationary results become unstable. This threshold corresponds to the value of the gain at which the pole crosses the real axis.