Modulational instability in fiber Bragg gratings with nonlinearity management

We study modulation instability (MI) in the fiber Bragg grating with nonlinearity management based on the coupled-mode theory. The role of both average Kerr nonlinearity and variance of Kerr nonlinearity between the layers of fiber grating in MI is identified. It is found that the variance of Kerr nonlinearity affect MI gain spectrum remarkably in both anomalous dispersion and normal dispersion regimes. In the anomalous dispersion regime, when the variance of Kerr nonlinearity is much smaller than the average Kerr nonlinearity, the MI gain spectrum is similar to that without the variance of Kerr nonlinearity, but the range of wave number for MI to occur is narrowed, and the amplitude of gain decreased. When the variance of Kerr nonlinearity is enhanced to be equivalent to the average Kerr nonlinearity, the role of variance of Kerr nonlinearity in MI becomes important: At low intensity, the range of wave number for MI to occur shrinks notably, and the gain gets only a single peak compared with the original one which has two symmetrical side-bands. At high intensity, there appear three MI ranges. In the normal dispersion regime, near the lower edge of photonic band gap, the amplitude of MI gain is slowed down due to the influence of variance of Kerr nonlinearity, and only two small symmetrical MI range appear, in sharp contrast to the original case without the influence of variance of Kerr nonlinearity, in which MI occurs for all wave numbers. Whereas in the case that far away from the edge of photonic band gap, we find that the range of wave number for MI to occur and the amplitude of MI gain increase as the value of variance of Kerr nonlinearity increases.