Fluctuations in transverse laser patterns

We study the effects of white additive noise on a transverse pattern of the laser formed by the combination of three degenerate transverse modes. In the semiclassical description, the formation of this pattern corresponds to a process of spontaneous breaking of the cylindrical symmetry. The stationary probability distribution is calculated analytically within the adiabatic elimination of the atomic variables and the cubic approximation. Noise induces a slow random rotation of the pattern around the laser axis and a random distortion. The angular rotation is characterized by a diffusive evolution and restores completely the broken symmetry. We study the random motion of the four phase singularities of the pattern. We analyze the transverse process given by the decay of the unstable cylindrically symmetrical configuration and show that it is accompanied by anomalous fluctuations.