Stochastic modulational instability in the nonlinear Schrödinger equation with colored random dispersion

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.

Stochastic Modulational Instability in Optical Fibers with Colored Random Dispersion

“…We also perform direct numerical solution in a Monte Carlo spirit, which is nevertheless essential to assess the reliability and the limits of application of the two analytic approaches. We present a detailed analysis of the various MI regimes occurring in normal GVD [5].…”

Modulational instability in optical fibers with randomly-kicked normal dispersion

“…( 1). The study of stationary colored noise, where classic perturbative techniques apply [30], will be the subject of a future work [44].…”

Random telegraph dispersion management: modulational instability

“…In the late 90s, the white noise process (an exactly solvable model) was considered [9,[13][14][15]. Only recently different random processes were considered: localized GVD kicks [16] or coloured processes of low-pass or band-pass type [17].…”

“…This strategy is usually denoted dispersion management (DM) and consists in alternating positive and negative GVD segments along the propagation direction. A uniform distribution for fluctuation of segment length is studied in [20,21] for pulse propagation and MI, respectively.…”

“…Here we adapt part of the theory presented in Ref. [17] to DM-like random fluctuations modelled as a telegraph process, i.e., a low-pass colored random process taking only two values alternating at exponentially distributed distances. Thanks to correlation splitting formulas [22][23][24][25][26][27], the MI equations (i.e.…”