Propagation of Gaussian beams in the presence of gain and loss

TL;DR

This paper derives equations describing Gaussian beam propagation in waveguides with gain and loss, highlighting how beam width influences dynamics and enabling filtering based on width in non-Hermitian systems.

Contribution

It extends classical Gaussian beam propagation models to non-Hermitian waveguides, incorporating gain and loss effects into the equations of motion.

Findings

Beam width significantly affects propagation in gain/loss waveguides.

The derived equations enable filtering of beams based on width.

Non-Hermitian effects modify classical Hamiltonian dynamics.

Abstract

We consider the propagation of Gaussian beams in a waveguide with gain and loss in the paraxial approximation governed by the Schr\"odinger equation. We derive equations of motion for the beam in the semiclassical limit that are valid when the waveguide profile is locally well approximated by quadratic functions. For Hermitian systems, without any loss or gain, these dynamics are given by Hamilton's equations for the center of the beam and its conjugate momentum. Adding gain and/or loss to the waveguide introduces a non-Hermitian component, causing the width of the Gaussian beam to play an important role in its propagation. Here we show how the width affects the motion of the beam and how this may be used to filter Gaussian beams located at the same initial position based on their width.