Formation and evolution of roll patterns in optical parametric oscillators

We study pattern formation in a degenerate optical parametric oscillator (OPO) by including diffractive effects due to the propagation in the optical cavity. The solution of zero amplitude for the OPO signal is found to be unstable to periodically modulated perturbations whenever the detuning parameter of the signal field is negative. A comparison between analytical solutions and numerical simulations for the roll structure shows excellent agreement for a wide range of input amplitudes. The threshold for the appearance of rolls is lower than the usual OPO threshold obtained in the plane-wave limit, especially for large detunings. When the input power is increased, roll patterns lose stability, leading to filamentation and optical turbulence.