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.

Figure 1

(Color online) Free energy $f\left(\zeta \right)-f(\zeta =0)$ as a function of magnetization $\zeta$ at different temperatures. From high to low: $T=0.910$, $T_o=0.717$, $T=0.616$, $T_c=0.548$, and $T=0.504$. $T_o$ marks the appearance of the unstable minimum, and $T_c$ is the transition temperature at which the solution with $\zeta >0$ becomes thermodynamically stable.

Figure 2

(Color online) (a) Magnetization $\zeta$ and (b) energy $e$ as a function of temperature. Full lines correspond to the thermodynamically stable solution of Eq. (18), whereas dashed lines are the unstable solutions. At $T_c=0.548$, a first-order thermodynamic phase transition takes place, while at $T_o=0.717$ unstable solutions appear.

Figure 3

(Color online) The entropy of saddles $\sigma$ as a function of potential energy $e$.

Figure 4

(Color online) Time dependence of the self-correlation function $F\left(t\right)$ at different temperatures. From left to right: $T=0.1,0.5,0.6,1.0$ $(T_c=0.548)$. Time is in normalized units.

Figure 5

(Color online) (a) Relaxation time ${\tau }_c$ of the self-correlation function $F_c\left(t\right)$ as a function of $T$. Full line corresponds to stable state (paramagnetic above $T_c$ and ferromagnetic below it). Dashed lines refer to unstable solutions. (b) Relaxation time ${\tau }_c^G$ of the collective correlation function $G_c\left(t\right)$ as a function of $T$. Full and dashed lines are the same as in (a).