TEMPORAL-MODULATION INSTABILITIES OF COUNTERPROPAGATING WAVES IN A FINITE DISPERSIVE KERR MEDIUM - I - THEORETICAL-MODEL AND ANALYSIS

This paper presents a comprehensive analytical study of temporal modul ation instabilities in a finite, nonlinear, dispersive medium in which two counterpropagating pump beams interact through a Kerr-type nonlin earity. The analysis includes self-and cross-phase modulations, group- velocity dispersion, four-wave mixing, and reflections occurring at th e two facets of the dispersive Kerr medium. The use of a new method ba sed on a small-parameter analysis has resulted in a physically transpa rent model in terms of a doubly resonant optical parametric oscillator that allows characterization of the complicated nonlinear system in a familiar language. The effects of boundary reflections are shown to b e very important. In the low-frequency limit, in which dispersive effe cts are negligible, our results reduce to those obtained previously. A t high frequencies, dispersive effects lead to new instabilities both in the normal-and anomalous-dispersion regions of the dispersive Kerr medium. The anomalous-dispersion case is discussed in detail after inc luding weak boundary reflections. The growth rate and the threshold fo r the absolute instability are obtained in an analytical form for arbi trary pump-power ratios. Our analytic results are in agreement with pr evious numerical work done by neglecting boundary reflections and assu ming equal powers for the counterpropagating pump beams. (C) 1998 Opti cal Society of America.