Instabilities and amplification of counterpropagating waves in a Kerr nonlinear medium

We show that two linearly polarized counterpropagating waves in a Kerr nonlinear medium with linear dispersion can exhibit amplification and multimode temporal instability, which are attributed to the combined effect of nonlinear index grating and linear dispersion. At the onset of the instability, some modes become unstable, and the ensuing oscillations are periodically modulated; as pumping further increases, the motion becomes chaotic. With the pumping below the threshold, the system exhibits large amplification. In an optical fiber, appreciable gain is anticipated even with pumping much below the threshold.