Solving scattering problems in the half-line using methods developed for scattering in the full line

TL;DR

This paper develops a method to analyze scattering problems on the half-line by extending potentials to the full line, deriving explicit reflection amplitudes, and applying these results to optical systems and laser threshold conditions.

Contribution

It introduces a novel approach to relate half-line scattering problems to full-line problems, including explicit formulas and conditions for bound states and spectral singularities.

Findings

Derived explicit reflection amplitude formula for half-line problems.

Identified conditions for bound states and spectral singularities.

Established laser threshold conditions for slab lasers with mirrors.

Abstract

We reduce the solution of the scattering problem defined on the half-line $[0,\infty)$ by a real or complex potential $v(x)$ and a general homogenous boundary condition at $x=0$ to that of the extension of $v(x)$ to the full line that vanishes for $x<0$. We find an explicit expression for the reflection amplitude of the former problem in terms of the reflection and transmission amplitudes of the latter. We obtain a set of conditions on these amplitudes under which the potential in the half-line develops bound states, spectral singularities, and time-reversed spectral singularities where the potential acts as a perfect absorber. We examine the application of these results in the study of the scattering properties of a $\delta$-function potential and a finite barrier potential defined in $[0,\infty)$, discuss optical systems modeled by these potentials, and explore the configurations in which these systems act as a laser or perfect absorber. In particular, we derive an explicit formula for the laser threshold condition for a slab laser with a single mirror and establish the surprising fact that a nearly perfect mirror gives rise to a lower threshold gain than a perfect mirror. We also offer a nonlinear extension of our approach which allows for utilizing a recently developed nonlinear transfer matrix method in the full line to deal with finite-range nonlinear scattering interactions defined in the half-line.