A theoretical study on threshold conditions of modulation instability in oppositely directed couplers

We theoretically investigate threshold conditions to observe modulation instability (MI) in a two-core nonlinear oppositely directed coupler (ODC) with a negative-index material (NIM) channel. Using linear stability analysis, we obtain an expression for the instability gain. The analysis shows, with two discrete instability regions, that the band at lower values off (ratio of the backward to forward-propagating waves amplitude) is a result of the nonlinear positive index material (PIM) channel while the broader range band is a consequence of the nonlinear NIM channel. Both bands are highly sensitive to system parameters. We demonstrate that MI has a threshold-like condition in the normal dispersion regime. This study also shows that gain increases proportionally to power increases and the two instability regions approach each other, thereby narrowing the stability region. Furthermore, we report that the effect of pump power and the coupling coefficient on instability gain have an opposite relationship. Similarly, study of the influence of the nonlinear coefficient on critical power (P-c) shows that P-c gradually decreases as. the nonlinear coefficient increases. Thus, a comprehensive study on the influence of various physical effects on MI is reported in this paper.