Stochastic modulational instability in the nonlinear Schr\"odinger equation with colored random dispersion

TL;DR

This paper investigates how stochastic variations in group-velocity dispersion affect modulational instability in optical fibers, using analytical methods to predict MI gain and sidelobe behavior under different stochastic processes.

Contribution

It introduces analytical approaches to estimate MI gain in fibers with colored random dispersion, highlighting the effects of spectral density and correlation length on MI sidelobes.

Findings

Low-frequency MI sidelobes emerge when PSD is centered at zero wavenumber.

Parametric resonance conditions lead to MI sidelobe pairs in spatially modulated stochastic processes.

Cumulant expansion is effective for small PSD and correlation lengths, while the functional approach models a wider parameter range.

Abstract

We study modulational instability (MI) in optical fibers with random group-velocity dispersion (GVD). We consider Gaussian and dichotomous colored stochastic processes. We resort to different analytical methods (namely, the cumulant expansion and the functional approach) and assess their reliability in estimating the MI gain of stochastic origin. If the power spectral density (PSD) of the GVD fluctuations is centered at null wavenumber, we obtain low-frequency MI sidelobes which converge to those given by a white noise perturbation when the correlation length tends to 0. If instead the stochastic processes are modulated in space, one or more MI sidelobe pairs corresponding to the well-known parametric resonance (PR) condition can be found. A transition from small and broad sidelobes to peaks nearly indistinguishable from PR-MI is predicted, in the limit of large perturbation amplitudes and correlation lengths of the random process. We find that the cumulant expansion provides good analytical estimates for small PSD values and small correlation lengths, when the MI gain is very small. The functional approach is rigorous only for the dichotomous processes, but allows us to model a wider range of parameters and to predict the existence of MI sidelobes comparable to those observed in homogeneous fibers of anomalous GVD