Polarization patterns in Kerr Media

We study spatiotemporal pattern formation associated with the polarization degree of freedom of the electric eld amplitude in a mean eld model describing a Kerr medium in a cavity with at mirrors and driven by a coherent plane-wave eld. We consider linearly as well as elliptically polarized driving eld and situations of self-focusing and self-defocusing. For the case of self-defocusing and linearly polarized driving eld there is a stripe pattern orthogonally polarized to the driving eld. Such pattern changes into an hexagonal pattern for elliptically polarized driving eld. The range of driving intensities for which the pattern is formed shrinks to zero with increasing ellipticity. For the case of self-focusing, changing the driving eld ellipticity leads from a linearly polarized (for linearly polarized driving) to a circularly polarized hexagonal pattern (for circularly polarized driving). Intermediate situations include a modiied Hopf bifurcation at nite wave number leading to a time dependent pattern of deformed hexagons and a codimension two Turing-Hopf instability resulting in an elliptically polarized stationary hexagonal pattern. Our numerical observations of diierent spatiotemporal structures are described by appropriate model and amplitude equations.