Modulational instability in optical fibers with randomly kicked normal dispersion

We study modulational instability (MI) in optical fibers with random group-velocity dispersion (GVD) generated by sharply localized perturbations of a normal GVD fiber that are either randomly or periodically placed along the fiber and that have random strength. This perturbation leads to the appearance of low-frequency MI side lobes that grow with the strength of the perturbations, whereas they are faded by randomness in their position. If the random perturbations exhibit a finite average value, they can be compared with periodically perturbed fibers, where parametric resonance tongues (also called Arnold tongues) appear. In that case, increased randomness in the strengths of the variations tends to affect the Arnold tongues less than increased randomness in their positions.

Figure 1

Realization of a simple random-walk fiber (left panel) and a Poisson fiber (central panel) with zero-mean kick strength $\overline{\lambda }=0$, and of a simple random-walk fiber with $\overline{\lambda }=1$ (right panel). The kicks are modeled with Gaussian pulses $\delta \left(z\right)=1/{\left(2\pi w^2\right)}^{1/2}exp(-z^2/2w^2)$ of width $w=Z_{\mathrm{ref}}/20$. The position of the kicks is indicated by the vertical dotted lines; ${\lambda }_n$ are uniform random variables in $[-1,1]$.

Figure 2

Comparison of numerically computed MI gains in randomly kicked homogeneous fibers with $\overline{\lambda }=0$; ${\varepsilon }_Z$ and $\varepsilon$ are indicated in each panel. The random kicks are located at $Z_n$ as in Eq. (13) (random walk). The $\delta {\lambda }_n$ and $\delta j_n$ are taken to be uniform in $[-1,1]$. Shown are the numerically simulated sample MI gain $G\left(\omega \right)$ (dashed blue line), the mean MI gain $G_2\left(\omega \right)$ computed directly from the largest eigenvalue of $\overline{M}$ (dotted red line), the perturbative approximation of the mean MI gain (dash-dotted black line), and the perturbations' growth rates (black stars) calculated from numerical solutions of the NLSE.

Figure 3

Same as Fig. 2, but with $Z_n$ as in Eq. (14) (Poisson model). The parameter $\varepsilon$ is indicated in the panels.

Figure 4

Comparison between the mean MI gain of the white-noise model (black dots) and the mean MI gain of the random-walk (left and center panels) and Poisson models (right panel). MI gain for white-noise model is calculated from Eq. (29) from Ref. [25] with $A=1$ ($A\equiv \sqrt{P}$ in our notation) and ${\sigma }^2=0.1$. Parameters $\varepsilon$ and ${\varepsilon }_Z$ as indicated and $Z_{\mathrm{ref}}$ as in Eq. (37) with $\overline{\delta {\lambda }^2}=1/3$.

Figure 5

Sample and mean MI gains for a simple random-walk process and $\overline{\lambda }=1$ and $\delta {\lambda }_n$ uniform in $[-1,1]$. Color code is the same as in Fig. 2.

Figure 6

Sample and mean MI gains for a Poisson model with $\overline{\lambda }=1$. Color code is the same as in Fig. 2.