Generalized complex Swift-Hohenberg equation for optical parametric oscillators

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Abstract
A generalized complex Swift-Hohenberg equation including diffraction and nonlinear resonance terms is derived for spatially extended nondegenerate optical parametric oscillators (OPOs) with flat end mirrors. For vanishing pump detuning this equation becomes the complex Swift-Hohenberg (SH) equation valid also for lasers. Nevertheless the similarities between OPOs and lasers are limited, since the diffractive character of OPOs is lost when the diffraction coefficients of signal and idler fields are equal. This manifests, e.g., in the absence of advection by traveling waves (TWs), a clear difference with lasers. When pump detuning is nonzero a nonlinear resonance develops, as it occurs in degenerate OPOs. This nonlinear resonance is essential in order to properly describe the TWs that OPOs support, and describes the bistability between TWs. This leads to the appearance of localized structures, which we also report here.

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