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

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 wave number, 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.

Figure 1

MI gain as a function of detuning $\omega$ for a LP random dispersion. Comparison of numerical values for a Gaussian (blue solid lines) and dichotomous process (green dotted lines) obtained from Eq. (2) with the estimates provided by Eq. (18) (red dashed lines) and by Eq. (21) (dashed-dotted purple lines). We include also the gain corresponding to the white process (yellow dotted line with pluses). In (a) $N_0=0.005$ and $B=4\pi$, so that $\varepsilon <0.5$ over the considered $\omega$ range, (b) $N_0=0.4$ and $B=4\pi$, so that $\varepsilon \approx 1$ around the maximum gain. The inset in (a) shows a single realization of the two processes.

Figure 2

Detuning at the gain maximum as a function of $B$. Numerical results of Gaussian (blue crosses) and dichotomous (green pluses) process are compared to Eq. (18) (red dashed line) and Eq. (21) (purple dashed-dotted line). We include ${\omega }_{\max }$ for the white noise, too, as reference (yellow dotted line). The dashed vertical lines highlight the value of $B$ used in Fig. 1. (a) $N_0=0.005$, (b) $N_0=0.4$.

Figure 3

Maximum gain as a function of $B$. Same convention as in Fig. 2.

Figure 4

MI gain as a function of detuning $\omega$ for a BP random dispersion ${\kappa }_0=2\pi$. Comparison of numerical values for a Gaussian (blue solid lines) and dichotomous process (green dotted lines) obtained from Eq. (2) with the estimates provided by Eqs. (18) and (23) (red dashed lines), and by Eq. (32) (dashed-dotted purple lines). In (a) $N_0=0.005$ and $B=\pi /4$, so that $\varepsilon <0.1$ around the PR resonance, (b) $N_0=3.2$ and $B=\pi /32$, so that $\varepsilon \gg 1$ around the ${\omega }_{\max }$. In (b), we include the PR-MI gain of Eq. (29) with $\theta =\sqrt{2}{\sigma }_{\xi }=\sqrt{\pi /5}$ (yellow dotted line with circles). The insets show the numerical results for a Gaussian process on a larger $\omega$ range.

Figure 5

Maximum gain $G_{\max }$ as a function of $B$ at constant $N_{0}B$. Same convention as in Fig. 2, dichotomous process excluded. We take a constant ${\kappa }_0=2\pi$ and (a) $N_{0}B=0.0039$, (b) $N_{0}B=\pi /10=0.314$. Notice the ordinate axis is in logarithmic scale in (a). The dashed vertical lines highlight the values of $B$ used in Figs. 4 and 4, respectively.

Figure 6

MI gain curves $G\left(\omega \right)$ and their maxima $G_{\max }$ obtained for different values of ${\kappa }_0\in \{\pi ,2\pi ,4\pi ,8\pi ,16\pi ,32\pi ,64\pi \}$. Comparisons of numerical results (blue crosses and blue solid lines) with the peak of Eq. (26) (red dashed line) and of Eq. (34) [purple dashed-dotted line, not shown in (a), out of scale]. We include also $G_2\left(\omega \right)$ for the white-noise limit at the corresponding (yellow dotted line). The dashed vertical lines highlight the value of ${\kappa }_0$ used in Figs. 4 and 5. (a) $N_0=0.005, B=\pi /4$; (b) $N_0=3.2, B=\pi /32$.