Perturbative Analysis of Spectral Singularities and Their Optical Realizations

TL;DR

This paper introduces a perturbative method to analyze spectral singularities in complex potentials, linking mathematical properties to optical phenomena like lasing and perfect absorption, with exact solutions for specific potential types.

Contribution

It develops a general perturbative approach for spectral singularities in complex potentials and provides exact solutions for delta-function and barrier potentials, advancing understanding of optical spectral singularities.

Findings

Established the exactness of n-th order perturbation theory for n delta-function potentials

Derived an exact transfer matrix for these potentials

Analyzed spectral singularities in complex barrier potentials of arbitrary shape

Abstract

We develop a perturbative method of computing spectral singularities of a Schreodinger operator defined by a general complex potential that vanishes outside a closed interval. These can be realized as zero-width resonances in optical gain media and correspond to a lasing effect that occurs at the threshold gain. Their time-reversed copies yield coherent perfect absorption of light that is also known as an antilaser. We use our general results to establish the exactness of the n-th order perturbation theory for an arbitrary complex potential consisting of n delta-functions, obtain an exact expression for the transfer matrix of these potentials, and examine spectral singularities of complex barrier potentials of arbitrary shape. In the context of optical spectral singularities, these correspond to inhomogeneous gain media.