Predicting instabilities of a tuneable ring laser with an iterative map model

TL;DR

This paper introduces a nonlinear iterative map model for tuneable ring lasers, enabling prediction of stability and instabilities by analyzing pulse evolution over many cavity round trips.

Contribution

It develops a novel nonlinear iterative map approach based on a reduced nonlinear Schrödinger equation to predict instabilities in tuneable ring lasers.

Findings

Identified a sharp boundary between stable and unstable laser operation.

Observed pulse chirp saturation and shape transformation during evolution.

Provided analytical and numerical insights into the interplay of dispersion, modulation, and nonlinearity.

Abstract

Simple mathematical models have been unable to predict the conditions leading to instabilities in a tuneable ring laser. Here, we propose a nonlinear iterative map model for tuneable ring lasers. Solving a reduced nonlinear Schr\"odinger equation for each component in the laser cavity, we obtain an algebraic map for each component. Iterating through the maps gives the total effect of one round trip. By neglecting the nonlinearity, we find a linearly chirped Gaussian to be the analytic fixed point solution, which we analyze asymptotically. We then numerically solve the full nonlinear model, allowing us to probe the underlying interplay of dispersion, modulation, and nonlinearity as the pulse evolves over hundreds of round trips of the cavity. In the nonlinear case, we find the chirp saturates, and the Fourier transform of the pulse becomes more rectangular in shape. Finally, for a nominal plane in the parameter space, we uncover a rich, sharp boundary separating the stable region and the unstable region where modulation instability degrades the pulse into an unsustainable state.