Search

The formation of multi-petals and multi-spirals is analysed with the help of a generalized dispersion relation, taking advantage of the invariance of the rotation frequency of a pattern. It results in an eigenvalue problem, which allows us to predict the rotation frequency and the radial curvature of the pattern, in good agreement with exact calculations: A stationary pattern is predicted to look like petals, while a uniform rotating pattern is predicted to have some curvature, leading to a spiral shape.

We present numerical simulations of a simplified model for the ring cavity device in the case of round-trip time equal to the atomic radiative lifetime. Real pattern effects are of primordial importance, especially in small systems where the boundary conditions seem the key for the pattern selection. For a plane-wave input, our system displays not only hexagons, as in the mean-field model, but also stationary rolls, squares and octagons with a weakly chaotic structure. All these structures are also obtained in the case of bell-shaped input profile with large aspect ratio; they even appear regular and wonderful, as if boundary effects stabilize them.