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Phys. Rev. E, 58, 3843-3853 (1998).
The problem of two-dimensional (2D), transverse, noise-sustained pattern formation is theoretically and numerically studied, in the case of an optical parametric oscillator for negative signal detuning. This gives a complete analysis of a 2D, convective, pattern forming system which is also relevant to more general 2D physical systems. For the optical parametric oscillator the transversal walk-off due to the nonlinear crystal birefringence, exploited to phase-match the frequency down-conversion process, turns the instability to convective up to a certain threshold. In this regime noise-sustained patterns can be observed. These structures are a macroscopic manifestation of amplified microscopic noise which, in the context of optics, can be of quantum nature. Directly observable properties of the near- and far-field as well as statistical properties of the spectral intensity help to distinguish noise- from dynamics-sustained structures. Moreover, the analysis indicates that the walk-off term breaks the rotational symmetry of the 2D model. This causes a preferential selection of the stripe orientation, which would be otherwise random, being the modulus of the wave-vector the only restricted value. At the convective threshold an entire set of spatial modes becomes unstable; whereas the threshold of absolute instability depends on the relative orientation of the mode. Beyond the threshold for absolute instability this causes the co-existence, in the linear regime of evolution, of modes that are absolutely unstable and others that are only convectively unstable. The numerical solutions of the dynamical equations of the system under study confirm the analytical predictions for the value of the instability thresholds and the kind of pattern selected. Moreover, they allow to investigate the nonlinear regime showing qualitatively the co-existence of modes with different type of instability and giving a quantitative characterization of the transition from noise-sustained to dynamics-sustained structures.
PACS numbers: 46.65.Sf, 42.50.-p, 42.65.Yj
Pere Colet e-mail: [email protected] http://www.imedea.uib.es/~pere/