Mode-locking transitions in nanostructured weakly disordered lasers

We report on a statistical approach to mode-locking transitions of nanostructured laser cavities characterized by an enhanced density of states. We show that the equations for the interacting modes can be mapped onto a statistical model exhibiting a first-order thermodynamic transition, with the average mode energy playing the role of inverse temperature. The transition corresponds to a phase locking of modes. Extended modes lead to a mean-field-like model, while in the presence of localized modes, as due to a small disorder, the model has short-range interactions. We show that simple scaling arguments lead to observable differences between transitions involving extended modes and those involving localized modes. We link the thermodynamic transition to a topological singularity of the phase space, as previously reported for similar models. Finally, we solve the dynamics of the model, predicting a jump in the relaxation time of the coherence functions at the transition.