Modulation instability in the oppositely directed coupler with a quadratic nonlinearity

We investigate modulation instability(MI) in a nonlinear oppositely directed coupler with a quadratic nonlinearity, where one channel is made from a positive-index material (PIM) and another channel is fabricated from a negative-index material ( NIM), trying to identify the different MI properties from those in a conventional parametric gap system with grating. Both the analytic continuouswave (CW) solutions and dispersion relation are obtained. By using standard linear instability we in detail discuss how the ratio of the backward to forward propagating wave’s power, the phase mismatch, and the coupled coefficient influence the dynamical behavior of MI. Large stable regions are found if the fundamental harmonics (FH) falls in the normal dispersion when compared to the case in the conventional Bragg grating with a quadratic nonlinearity where the CW solutions are unstable in most cases. In addition, we also observe the large stable regimes of the CW solutions even when the coupled strength for the second harmonics (SHs) is weaker than that of the FH. These findings suggest that the oppositely directed coupler with a NIMchannel provides more ways to manipulate the MI and soliton.