# Stable Numerical Schemes for Nonlinear Dispersive Equations with Counter-Propagation and Gain Dynamics

## TL;DR

This paper introduces a stable, efficient numerical scheme for simulating nonlinear dispersive optical cavities with complex gain dynamics, enabling accurate modeling of diode lasers and frequency comb generation.

## Contribution

A novel predictor-corrector, operator-splitting scheme that ensures stability and efficiency in modeling nonlinear dispersive cavities with semiconductor gain dynamics.

## Key findings

- The scheme is stable under von-Neumann analysis.
- It efficiently models diode laser dynamics.
- It enables simulation of frequency comb generation.

## Abstract

We develop a stable and efficient numerical scheme for modeling the optical field evolution in a nonlinear dispersive cavity with counter propagating waves and complex, semiconductor physics gain dynamics that are expensive to evaluate. Our stability analysis is characterized by a von-Neumann analysis which shows that many standard numerical schemes are unstable due to competing physical effects in the propagation equations. We show that the combination of a predictor-corrector scheme with an operator-splitting not only results in a stable scheme, but provides a highly efficient, single-stage evaluation of the gain dynamics. Given that the gain dynamics is the rate-limiting step of the algorithm, our method circumvents the numerical instability induced by the other cavity physics when evaluating the gain in an efficient manner. We demonstrate the stability and efficiency of the algorithm on a diode laser model which includes three waveguides and semiconductor gain dynamics. The laser is able to produce a repeating temporal waveform and stable optical comblines, thus demonstrating that frequency combs generation may be possible in chip scale, diode lasers.

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Source: https://tomesphere.com/paper/1906.11097