Spatiotemporal instabilities in dispersive nonlinear media

A spatiotemporal instability is shown to occur in a dispersive nonlinear optical medium whose refractive index varies linearly with the optical intensity (the Kerr medium). The physical origin of the spatiotemporal instability lies in the simultaneous presence of diffraction and dispersion in nonlinear media. For a self-focusing medium the instability can occur for both normal and anomalous dispersion. In contrast, the instability exists only in the normal-dispersion region of a self-defocusing medium. A linear stability analysis is used to predict the initial growth rate of the spatiotemporal instability. Numerical simulations are used to analyze the behavior in the nonlinear regime where the results of the linear stability analysis become invalid. Our results show that the spatiotemporally modulated mode has a larger growth rate than that of the spatially homogeneous mode and is expected to dominate in practice. The periodic state eventually evolves into spatiotemporal chaos with further propagation inside the nonlinear medium.