Interplay between saturation and relaxation of nonlinear response in the modulational instability of various nonlinear media

We investigate the modulational instability (MI) of the optical beam propagating in the relaxing saturable nonlinear system. We identify and discuss the salient features of various functional forms of saturable nonlinear responses such as exponential, conventional and coupled type on the MI spectrum. Using suitable model, the relaxation of the nonlinear response is effectively included alongside the saturable nonlinear response (SNL) of the medium. The linear stability analysis is performed and an explicit dispersion relation is determined for various functional forms of SNL. Firstly, we analyze the impact of SNL on the MI spectrum and found that the MI gain and bandwidth is maximum for exponential nonlinearity in comparison to other types of SNLs. Latter the relaxation of the nonlinearity is included, the inclusion of the finite value of the response time extends the range of the unstable frequencies literally down to infinite frequencies. In the regime of slow response, the MI gets suppressed and it is found to be irrespective of the signs of dispersion coefficient. The saturation on the other hand inhibits the MI with increase in saturation parameter. To give better insight into the MI phenomena, the maximum MI gain and the optimum modulation frequency is drawn as a function of the delay. Thus the MI dynamics in the system of relaxing saturable nonlinear media is emphasized and the significance of various functional forms of SNL are highlighted.