This paper presents a comprehensive analytical study of temporal modulation instabilities in a finite, nonlinear, dispersive medium in which two counterpropagating pump beams interact through a Kerr-type nonlinearity. The analysis includes self- and cross-phase modulations, group-velocity dispersion, four-wave mixing, and reflections occurring at the two facets of the dispersive Kerr medium. The use of a new method based on a small-parameter analysis has resulted in a physically transparent 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 be very important. In the low-frequency limit, in which dispersive effects are negligible, our results reduce to those obtained previously. At 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 including weak boundary reflections. The growth rate and the threshold for the absolute instability are obtained in an analytical form for arbitrary pump–power ratios. Our analytic results are in agreement with previous numerical work done by neglecting boundary reflections and assuming equal powers for the counterpropagating pump beams.
© 1998 Optical Society of America
(120.2230) Instrumentation, measurement, and metrology : Fabry-Perot
(190.0190) Nonlinear optics : Nonlinear optics
(190.3270) Nonlinear optics : Kerr effect
(270.3100) Quantum optics : Instabilities and chaos
M. Yu, C. J. McKinstrie, and Govind P. Agrawal, "Temporal modulation instabilities of counterpropagating waves in a finite dispersive Kerr medium. I. Theoretical model and analysis," J. Opt. Soc. Am. B 15, 607-616 (1998)