Three-dimensional self-organized patterns in the field profile of a ring cavity resonator

We consider the paraxial model for a nonlinear resonator filled with a saturable absorber, beyond the mean field limit. We develop a general treatment to study the modulational instabilities which give rise to pattern formation in the propagation direction z and in the transverse plane (x, y). For appropriate parametric domains we observe the first example of system self-organization in space and time proceeding from competition of linear effects with nonlinear medium properties and resonator feedback. In particular, we could observe either 3D global structures or structures which are localized in the transverse plane, showing different lengths in the longitudinal direction and endlessly travelling in the resonator.