Polarization patterns in Kerr Media

We study spatiotemporal pattern formation associated with the polarization degree of freedom of the electric field amplitude in a mean field model describing a Kerr medium in a cavity with flat mirrors and driven by a coherent plane-wave field. We consider linearly as well as elliptically polarized driving field and situations of self-focusing and self-defocusing. For the case of self-defocusing and linearly polarized driving field there is a stripe pattern orthogonally polarized to the driving field. Such pattern changes into an hexagonal pattern for elliptically polarized driving field. 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 field ellipticity leads from a linearly polarized (for linearly polarized driving) to a circularly polarized hexagonal pattern (for circularly polarized driving). Intermediate situations include a modified Hopf bifurcation at finite wave numb..