Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrodinger Equation Driven by the Square of a Gaussian Field

TL;DR

This paper analyzes how propagation effects influence the breakdown of a linear amplifier model described by a complex-Mass Schrödinger equation driven by a Gaussian field, revealing conditions for divergence of moments and the impact of propagation.

Contribution

It provides an explicit expression for the divergence point of moments in a complex-Mass Schrödinger equation with Gaussian driving field, highlighting the effect of propagation on amplifier breakdown.

Findings

Divergence points of moments are identified for the equation.

Propagation effects lower the threshold for divergence compared to no propagation.

The analysis uses a distributional Feynman path-integral approach.

Abstract

Solutions to the equation $\partial_t{\cal E}(x,t)-\frac{i}{2m}\Delta {\cal E}(x,t)=\lambda| S(x,t)|^2{\cal E}(x,t)$ are investigated, where $S(x,t)$ is a complex Gaussian field with zero mean and specified covariance, and $m\ne 0$ is a complex mass with ${\rm Im}(m)\ge 0$. For real $m$ this equation describes the backscattering of a smoothed laser beam by an optically active medium. Assuming that $S(x,t)$ is the sum of a finite number of independent complex Gaussian random variables, we obtain an expression for the value of $\lambda$ at which the $q$-th moment of $| {\cal E}(x,t)|$ w.r.t. the Gaussian field $S$ diverges. This value is found to be less or equal for all $m\ne 0$, ${\rm Im}(m)\ge 0$ and $| m| <+\infty$ than for $| m| =+\infty$, i.e. when the $\Delta {\cal E}$ term is absent. Our solution is based on a distributional formulation of the Feynman path-integral and the Paley-Wiener theorem.