Modulation instability in negative refractive materials with saturable nonlinearity

Starting directly from the nonlinear propagation equation including saturable nonlinearity, the first- and the second-order nonlinear dispersions, the dispersion relation, instable condition, gain spectra, and the dimensionless cut-off frequency and gain spectra of modulation instability (MI) in the negative refractive material are deduced by adopting the linear stability analysis and Drude electromagnetic model. And the variations of the dimensionless gain spectra with the normalized angular frequency and normalized incident power are calculated and discussed for different sign relations between the linear dispersion and the third-order nonlinear coefficients. The results show that in the negative refractive index region, MI can occur irrespective of the sign relation between the linear dispersion and the third-order nonlinear coefficients. And depending on different dimensionless angular frequencies and different sign relations, the variations of the dimensionless gain spectra with incident power take on several different forms. Namely, the peak gain and the cut-off frequency of MI may increase then decrease with the increase of the incident power, or decrease monotonously. Moreover, MI may even have a threshold incident power for some cases.