General approach to spatiotemporal modulational instability processes

In this article, we derive the general exact solution of the modulation instability gain. The solution described here is valid for 1-D, 2-D, and 3-D cases considering any temporal response function of the medium and with possible higher order Kerr nonlinearities. In particular, we show that the gain induced by modulation instability is initial condition dependent, while the usual calculations do not lead to such a dependence. Applications for current and high-interest nonlinear propagation problems, such as 1-D optical fiber propagation with delayed Raman response and 2-D filamentation in gases, are investigated in detail. More specifically, we demonstrate that the 2-D model of filamentation based on the balance between higher order Kerr terms leads to a modulation instability window. The impact of both self-steepening and space-time defocusing effects is also highlighted. Finally, we discuss the influence of the finite-time response of the different order electronic Kerr effects on the growth of the expected modulation instability bands.