Modulation instability in nonlinear negative-index material.

We investigate modulation instability (MI) in negative-index material (NIM) with a Kerr nonlinear polarization based on a derived (3+1)-dimensional nonlinear Schrödinger equation for ultrashort pulse propagation. By a standard linear stability analysis, we obtain the expression for instability gain, which unifies the temporal, spatial, and spatiotemporal MI. It is shown that negative refraction not only brings some new features to MI, but also makes MI possible in ordinary material in which it is otherwise impossible. For example, spatial MI can occur in the defocusing regime, while it only occurs in the focusing regime in ordinary material. Spatiotemporal MI can appear in NIM in the case of anomalous dispersion and defocusing nonlinearity, while it cannot appear in ordinary material in the same case. We believe that the difference between the MI in NIM and in ordinary material is due to the fact that negative refraction reverses the sign of the diffraction term, with the signs of dispersion and nonlinearity unchanged. The most notable property of MI in NIM is that it can be manipulated by engineering the self-steepening effect by choosing the size of split-ring resonator circuit elements. To sum up the MI in ordinary material and in NIM, MI may occur for all the combinations of dispersion and nonlinearity.