# Nonlinear Spectral Singularities and Laser Output Intensity

## TL;DR

This paper extends the concept of spectral singularities to nonlinear wave equations in laser physics, providing a mathematical framework to determine laser output intensity and conditions for lasing in slab lasers with Kerr nonlinearity.

## Contribution

It introduces a nonlinear spectral singularity framework for slab lasers with Kerr nonlinearity, deriving explicit formulas for laser output and conditions for lasing thresholds.

## Key findings

- Output intensity $I$ depends linearly on gain coefficient $g$
- Explicit analytic expression for the slope of $I$ versus $g$
- Lasing in oblique TM modes is forbidden beyond Brewster's angle

## Abstract

The mathematical notion of spectral singularity admits a description in terms of purely outgoing solutions of a corresponding linear wave equation. This leads to a nonlinear generalization of this notion for nonlinearities that are confined in space. We examine the nonlinear spectral singularities in arbitrary TE and TM modes of a mirrorless slab laser that involves a weak Kerr nonlinearity. This provides a computational scheme for the determination of the laser output intensity $I$ for these modes. In particular, we offer an essentially mathematical derivation of the linear-dependence of $I$ on the gain coefficient $g$ and obtain an explicit analytic expression for its slope. This shows that if the real part $\eta$ of the refractive index of the slab does not exceed 3, there is a lower bound on $\theta$ below which lasing in both its TE and TM modes requires $\eta$ to be shifted by a small amount as $g$ surpasses the threshold gain. Our results suggest that lasing in the oblique TM modes of the slab is forbidden if the incidence (emission) angle of the TM mode exceeds the Brewster's angle.