Modulational instability of polarized beams in nonlocal media with stochastic parameters

We investigate analytically and numerically the modulational instability (MI) in a nonlinear optical fiber. We use a generalized model describing the pulse propagation of waveguiding structure composed of two adjacent waveguides, where the effect of nonlocal nonlinear response as well as stochastic coefficients are taken into account. Applying the linear stability analysis and stochastic calculus, we show that the MI gain spectra reads as the maximal eigenvalue of a constant matrix. The generic properties of the MI gain spectra are then demonstrated for the rectangular response function. We observe that random inhomogeneities extend the domain of the homogeneous MI gain spectra over the whole spectrum of modulation, and the nonlocality parameter reduces drastically the growth rate and bandwidth of instability caused by stochasticity both in anomalous and normal dispersion regimes. We observe also that MI does not appears for all values of the nonlocal parameter. Numerical simulations of the full stochastic system of nonlinear Schrodinger equations describing the dynamics of the waves are carried out and lead to the generation of a train of pulses.