We consider a model for a Kerr medium in a planar resonator, which takes into account the vectorial character of the radiation field. We analyze the spatial behavior of quantum fluctuations around a steady state, with a roll-pattern configuration in the beam cross section, using a Langevin treatment based on the Wigner representation. The spatial distribution of the quantum fluctuations around the roll pattern is dominated by the neutral ͑or Goldstone͒ mode, corresponding to rigid spatial displacements of the pattern. The spatial configuration of the field immediately outside the cavity input-output mirror depends on the time window over which fluctuations are averaged: only when the time window is on the order of the cavity lifetime the output field fluctuations are qualitatively similar to that of the intracavity field. The quantum correlations among the fields in play, as described by the full multimode model, turn out to be in good agreement with those predicted by a simple three-mode model.

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 fields, and situations of self-focusing and self-defocusing. For the case of self-defocusing and a linearly polarized driving field, there is a stripe pattern orthogonally polarized to the driving field. Such a pattern changes into a hexagonal pattern for an 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 hexagonal pattern ͑for linearly polarized driving͒ to a circularly polarized hexagonal pattern ͑for circularly polarized driving͒. Intermediate situations include a modified Hopf bifurcation at a finite wave number, leading to a time dependent pattern of deformed hexagons and a codimension 2 Turing-Hopf instability resulting in an elliptically polarized stationary hexagonal pattern. Our numerical observations of different spatiotemporal structures are described by appropriate model and amplitude equations.

We study the spatiotemporal dynamics, in one and two spatial dimensions, of two complex fields which are the two components of a vector field satisfying a vector form of the complex Ginzburg-Landau equation. We find synchronization and generalized synchronization of the spatiotemporally chaotic dynamics. The two kinds of synchronization can coexist simultaneously in different regions of the space, and they are mediated by localized structures. A quantitative characterization of the degree of synchronization is given in terms mutual information measures.