Floquet stability of periodically stationary pulses in a short-pulse fiber laser

TL;DR

This paper develops a novel numerical framework using Floquet theory and a Fourier split-step method to analyze the stability of periodic pulses in short-pulse fiber lasers, addressing limitations of averaged models.

Contribution

It introduces a new Fourier split-step method and a gradient-based optimization approach for stability analysis of periodic pulses in lumped laser models.

Findings

Verified accuracy of numerical methods through simulations

Identified spectra and eigenfunctions of periodic pulses

Analyzed stability characteristics of laser pulses

Abstract

The quantitative modeling and design of modern short-pulse fiber lasers cannot be performed with averaged models because of large variations in the pulse parameters within each round trip. Instead, lumped models obtained by concatenating models for the various components of the laser are required. Since the optical pulses in lumped models are periodic, their linear stability is investigated using the monodromy operator, which is the linearization of the roundtrip operator about the pulse. A gradient-based optimization method is developed to discover periodic pulses. The computation of the gradient of the objective function involves numerical computation of the action of both the round trip operator and the adjoint of the monodromy operator. A novel Fourier split-step method is introduced to compute solutions of the linearization of the nonlinear, nonlocal, stiff equation that models optical propagation in the fiber amplifier. This method is derived by linearizing the two solution operators in a split-step method for the nonlinear equation. The spectrum of the monodromy operator consists of the essential spectrum, for which there is an analytical formula, and the eigenvalues. There is a multiplicity two eigenvalue at $\lambda=1$, which is due to phase and translation invariance. The remaining eigenvalues are determined from a matrix discretization of the monodromy operator. Simulation results verify the accuracy of the numerical methods, show examples of periodically stationary pulses, their spectra and eigenfunctions, and discuss their stability.