Symmetry, stability, and computation of degenerate lasing modes

TL;DR

This paper develops a method to identify stable degenerate lasing modes in symmetric 2D lasers, revealing that circulating modes are generally stable and exhibit chiral intensity patterns due to symmetry-breaking.

Contribution

It introduces a combined semi-analytical and numerical approach to analyze stable lasing solutions in degenerate symmetric lasers, extending previous ring laser results to all 2D rotational symmetries.

Findings

Stable circulating modes are generally stable in 2D symmetric lasers.

Chiral intensity patterns emerge from spontaneous mirror symmetry-breaking.

A numerical technique is proposed to solve degenerate SALT equations above threshold.

Abstract

We present a general method to obtain the stable lasing solutions for the steady-state ab-initio lasing theory (SALT) for the case of a degenerate symmetric laser in two dimensions (2d). We find that under most regimes (with one pathological exception), the stable solutions are clockwise and counterclockwise circulating modes, generalizing previously known results of ring lasers to all 2d rotational symmetry groups. Our method uses a combination of semi-analytical solutions close to lasing threshold and numerical solvers to track the lasing modes far above threshold. Near threshold, we find closed-form expressions for both circulating modes and other types of lasing solutions as well as for their linearized Maxwell--Bloch eigenvalues, providing a simple way to determine their stability without having to do a full nonlinear numerical calculation. Above threshold, we show that a key feature of the circulating mode is its "chiral" intensity pattern, which arises from spontaneous symmetry-breaking of mirror symmetry, and whose symmetry group requires that the degeneracy persists even when nonlinear effects become important. Finally, we introduce a numerical technique to solve the degenerate SALT equations far above threshold even when spatial discretization artificially breaks the degeneracy.