Modulational instability in nonlocal Kerr media with a sine-oscillatory response

We discuss modulational instability (MI) in nonlocal optical Kerr media with a sine-oscillatory response which can model nematic liquid crystals with negative dielectric anisotropy. In the framework of nonlocal nonlinear Schrödinger equation, MI in this type of media is found to have two unique properties different from those in other media discussed previously. First, MI exists both when the Kerr coefficient is positive and when it is negative. Second, the maximum gain points of MI do not shift with light intensity. We also explore the physical mechanism behind MI in local and nonlocal optical Kerr media by utilizing the theory of four-wave mixing. Through introducing a phase mismatch term (Δk) and a growth factor (γ), we deduce that the necessary and sufficient condition for MI to occur is that the phase mismatching during the four-wave-mixing process should be small enough such that |Δk|<2|γ|. Based on this condition, we can uniformly and consistently explain the results of MI in optical Kerr media obtained in the current work as well as those presented in previous work by others.