Traveling-wave states and secondary instabilities in optical parametric oscillators.

A complex order-parameter description of pattern formation in large aspect ratio optical parametric oscillators (OPOs) with flat end reflectors and uniform pumping is presented starting from the mean-field model of the OPO equations [G.-L. Oppo et al., Phys. Rev. A 49, 2028 (1994)]. It is shown that, in the nondegenerate case, the full OPO equations have an exact continuum family of traveling-wave (TW) solutions, which are preferred to standing-wave (SW) states found in the degenerate case. These solutions correspond to an off-axis emission for both signal and idler fields along two symmetric directions to satisfy momentum conservation in the parametric conversion process. Stability of TW versus SW solutions is investigated by deriving two coupled Newell-Whitehead-Segel equations describing the growth of SW or TW close to threshold. Analytical expressions for long-wavelength phase instabilities of the TW states above threshold are obtained from the coefficients of a Cross-Newell phase equation, and are shown to be the same for OPOs with high or low finesse for the pump field. By direct linear stability analysis of the TW solutions, it is also shown that the appearance of amplitude instabilities may reduce the region of stable TW states in the case of OPOs with a high finesse for the pump field. \textcopyright{} 1996 The American Physical Society.