Dynamics and stability of periodic spatial patterns in the optical parametric oscillator

The stability and dynamics of a new class of periodic solutions is investigated when a degenerate optical parametric oscillator (OPO) system is forced by an external pumping field with a periodic spatial profile. Both sinusoidal behavior as well as localized hyperbolic (front and pulse) behavior are considered in this model. The stability and bifurcation behaviors of these transverse structures are studied numerically.; Periodic solutions are shown to be stabilized by the nonlinear parametric interaction between the pump and signal fields and diffraction, attenuation, and periodic external pumping. Specifically, sinusoidal solutions result in robust and stable con figurations while well-separated and more localized field structures often undergo bifurcation to new steady-state solutions. Numerical simulations and studies of the solutions are provided.; The systematic study of the linear stability and bifurcation structure of solitary wave, front, and periodic wave-train solutions which arise in the OPO are examined near the onset of instability for the plane wave solution. The dynamics describe the onset of spatial patterns beyond the equilibrium state. A complete analytic characterization is given of pulse and front solutions. Supporting computational evidence is given for the stability structure of such solutions. Periodic wave-train solutions are found of the Jacobi elliptic form and are explored numerically.