We investigate the modulational instability (MI) of an optical beam near the zero group dispersion wavelength of a relaxing saturable nonlinear system. Considering a suitable theoretical model, we identify and discuss significant features of higher order dispersion (HOD), especially the role of the fourth-order dispersion (FOD) in the MI spectrum of relaxing saturable systems. The influence of FOD is to suppress MI in the anomalous group dispersion regime, to promote MI sidebands in the normal group dispersion regime, and the evolution of nonconventional sidebands are observed. Particularly, the inclusion of a finite value of the response time extends the range of unstable frequencies down to infinite frequencies. This happens because the finite response time in the nonlinear response is equivalent to assuming a complex nonlinearity. Therefore, in addition to the real part of the nonlinearity (which governs the parametric MI process), the contribution from the imaginary part of the nonlinear response (which models the Raman process) extends the domain of MI through the self-phase-matched process, even when the phase matching for the parametric MI process is not feasible. In the regime of the slow response, MI is suppressed regardless of the signs of the dispersion coefficients. To give a better insight into the MI phenomena, the maximum instability gain and the optimum modulation frequency are drawn as a function of the delay time. Thus, in this paper, the MI dynamics of a relaxing saturable nonlinear system is emphasized and the influence of HOD is highlighted.
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