Modulational instability in randomly dispersion-managed optical fiber links

We study modulational instability in a dispersion-managed optical fiber system where the sign of the group-velocity dispersion is changed at uniformly distributed random distances around a reference length. We find an instability gain of stochastic origin comparable to the conventional values found in a homogeneous anomalous dispersion fiber. We develop an accurate analytical technique based on transfer matrices to estimate the instability gain from the linearized nonlinear Schrödinger equation, which is also solved numerically. The comparison of numerical and analytical results confirms the validity of our approach. Modulational instability sidebands of purely stochastic origin appear and the competition between sidebands of periodic and stochastic origin is also discussed. These results may be of interest in tailoring and controlling modulational instability sidebands for telecommunications and parametric sources. Our method can also be applied to general linear stochastic differential equations with multiplicative noise, which broadly occur in Physics.

Figure 1

Schematic representation of the GVD profile in a typical fiber realization.

Figure 2

False-color plot of MI gain for a periodic DM fiber as a function of dimensionless detuning $\omega$ and half period $\overline{L}$. The color bar shows the values of $G_1\left(\omega \right)$ (defined in the text). The red dashed horizontal line identifies $\overline{L}=1.07$ (MI threshold), while the cyan dotted line denotes the reference value $\overline{L}=1.15$, used in the inset and below in the random length fluctuation examples.

Figure 3

MI gain curves for different values of $\overline{L}$ and $\varepsilon$, for $\omega \ge 0$. (a) $\overline{L}=1$, $\varepsilon =0.2$; (b) $\overline{L}=1$, $\varepsilon =0.5$; (c) $\overline{L}=1.15$, $\varepsilon =0.2$; (d) $\overline{L}=1.15$, $\varepsilon =0.5$. Blue crosses (purple circles) represent ${\overline{G}}_2$ (${\overline{G}}_1$) from numerical data, solid yellow (dash-dotted red) lines represent the theoretical estimates $G_2$ ($G_1$), dashed blue (dotted purple) lines represent the semianalytical estimates ${\tilde{G}}_2$ (${\tilde{G}}_1$). In (c) and (d), we include the MI gain in the periodic case $\varepsilon =0$ as a thin green dashed line.

Figure 4

(a) Maximum gain values and (b) their corresponding $\omega$ as a function of $\varepsilon$. The plus (cross) markers correspond to the maxima $G_{2,\max }$ for $\overline{L}=1$ ($\overline{L}=1.15$). The dashed (solid) blue lines correspond to the maxima of $G_2$ for $\overline{L}=1$ ($\overline{L}=1.15$). The green dash-dotted lines report, for reference, the constant values found in the periodic limit, i.e., the maxima of $G_1=G_2$ for $\varepsilon =0$. The red circles show the values of $G_{1,\max }$ and ${\omega }_{1,\max }$ for $\overline{L}=1.15$, for which red dotted lines illustrate the corresponding maxima of $G_1$. In (b), the line stops at $\varepsilon \approx 0.54$ (cutoff for $G_1$).