Transverse patterns in nondegenerate intracavity second-harmonic generation

Pattern formation is theoretically investigated in the process of second-harmonic generation in an externally driven plane-plane optical cavity which uses phase matching of type II. It is shown that in this system a symmetry-breaking polarization instability at the fundamental frequency can lead to the spontaneous appearance of self-organized states, which can be periodic or quasiperiodic in space. A weakly nonlinear analysis in the vicinity of the bifurcation point where these patterns emerge is developed by deriving both a complex order parameter equation of the Swift-Hohenberg type close to resonance, and a set of amplitude equations for the neutral modes far from resonance. The amplitude equations reveal the existence of multistability and pattern coexistence, and show that, besides the most common periodic patterns such as rolls, squares, and hexagons, quasipatterns with an arbitrary orientational order can be selected by the nonlinearity in this system. Numerical simulations of the original field equations are presented and the main predictions of the weakly nonlinear analysis are compared with the numerical results.