Disorder-induced localization of excitability in an array of coupled lasers

We report on the localization of excitability induced by disorder in an array of coupled semiconductor lasers with a saturable absorber. Through numerical simulations we show that the exponential localization of excitable waves occurs if a certain critical amount of randomness is present in the coupling coefficients among the lasers. The results presented in this Rapid Communication demonstrate that disorder can induce localization in lattices of excitable nonlinear oscillators, and can be of interest in the study of photonics-based random networks, neuromorphic systems, and, by analogy, in biology, in particular, in the investigation of the collective dynamics of neuronal cell populations.

Figure 1

The spatiotemporal dynamics corresponding to the diffusive regime: The excitability wave emanates from the center of the array where the intensity noise is added. Field intensity is plotted vs laser ($x$ axis) and time ($y$ axis) in (a), while in (b) the corresponding phase evolution is shown, demonstrating a substantial locking among the lasers during the firing events. Parameters used are $D=0.1$ and $K_{i,i\pm 1}=K_0=0.1\forall i$.

Figure 2

In the presence of disorder, the excitable behavior of the laser array is localized as the spatiotemporal dynamics depicted in (a) shows. The average intensity across the array averaged over the “firing events” of 150 multiple realizations of disorder in the coupled lasers' system is fitted by an exponential function [dashed red line in (b)]. Parameters used are $r=0.4$, $D=0.1$, and $K_0=0.5$; $n=150$ coupled lasers have been considered. We excluded from the fit the central laser as well as its three nearest neighbors on each side.

Figure 3

The phase transition from a diffusive to localized regime is illustrated by plotting the average localization exponent $\langle \alpha \rangle$ and relative standard deviation vs the randomness strength $r$, for laser chains having different numbers of elements $n$ (see legend). The amount of randomness necessary to achieve localization decreases by increasing the lattice size. Each point has been calculated averaging over five values of $\alpha$ obtained from 150 different realizations of disorder with the same strength $r$. The remaining parameters used are the same as in Fig. 2.

Figure 4

The average localization exponent dependence on the noise strength $D$ for $n=150$ coupled lasers. Each point, and relative standard deviation, is the result of an average of over 20 different values of $\langle \alpha \rangle$, each one obtained through 150 realizations of the disorder and with the same value of $D$. Parameters used are $K_0=0.5$ and $r=0.4$.