Effective low-dimensional dynamics of a mean-field coupled network of slow-fast spiking lasers

Low-dimensional dynamics of large networks is the focus of many theoretical works, but controlled laboratory experiments are comparatively very few. Here, we discuss experimental observations on a mean-field coupled network of hundreds of semiconductor lasers, which collectively display effectively low-dimensional mixed mode oscillations and chaotic spiking typical of slow-fast systems. We demonstrate that such a reduced dimensionality originates from the slow-fast nature of the system and of the existence of a critical manifold of the network where most of the dynamics takes place. Experimental measurement of the bifurcation parameter for different network sizes corroborates the theory.

Figure 1

Experimental setup and typical dynamics. (a) The light emitted by a VCSEL array is converted to an electrical signal which, after filtering and nonlinear transformation, is reinjected into the single current source driving all lasers. (b) Single-laser time traces showing periodic oscillations, chaotic bursting, and spiking (pumping currents 195.0, 196.5, and 187.0 mA). (c) Mean-field time trace of 451 lasers (182.1 mA), showing MMOs. (d) Top panel: total intensity of all 451 lasers, middle and bottom panels: intensity of two different lasers (pumping is 189.9 mA).

Figure 2

Dynamics of small networks. Different subpopulations examples [(a), (b) and (c)] show different dynamics (left). Each uncoupled population has a different laser threshold distribution (right), and the black dashed line shows $\langle I_{th}\rangle$ for each population [(a) 185.63 mA, (b) 182.88 mA, (c) 180.81 mA]. Red dash-dotted line: current value used in all measurements (193.50 mA).

Figure 3

Numerical simulation of a single laser modeled by Eq. (1) with $N=1$. The black and the green curve are the two branches of the critical manifold (unstable when dashed). The green and black points are attracting on the manifold they belong to. The magenta point is the minimum of the parabolic part of the critical manifold, where its stability changes. The red point is the (unstable) intersection of the two critical manifolds: $\alpha =2,k=0.7,\gamma =4.0{10}^{-3},\varepsilon ={10}^{-4},A=1,{\delta }_0=1.25$. (a) 3D phase space, and (b) projection in the $(x,w)$ plane shows the semiconductor laser relaxation oscillations.

Figure 4

Numerical simulations of ${10}^4$ coupled lasers modeled by (1). Only three laser dynamics are plotted (dashed and dotted lines). The mean dynamics is plotted in continuous black. The parameters $\frac{{\delta }_i-〈\delta 〉}{〈\delta 〉}$ are independent Gaussian variables of zero mean and standard deviation $1\times {10}^{-3}$ with $〈\delta 〉=1.2045$, $k=0.7$, $A=1$, $\alpha =2$, $\gamma =4\times {10}^{-3}$, $\epsilon ={10}^{-4}$. Black dot: intersection between the two slow manifold branches. The parabola ${\mathrm{\Sigma }}_y$ and the straight line ${\mathrm{\Sigma }}_x$ constitute the critical manifold calculated for a single laser with parameter $\mathrm{\Delta }$.

Figure 5

Bifurcation diagrams of three networks of different sizes (D:19, $\langle I_{th}\rangle =185.32\text{mA}$; E:$251\langle I_{th}\rangle =183.70\text{mA}$; F:451). Right: bifurcation curves for the uncoupled elements, and the dashed line indicates $\langle I_{th}\rangle$.

Figure 6

Upper bifurcation parameter value depending on average laser threshold value for increasing sample size. Smaller samples (lighter blue) are distributed along a straight line. When the sample grows (darker blue) the bifurcation parameter converges to a well-defined value.

Figure 7

Distribution of laser threshold currents. The vertical lines are the average and standard deviation of the distribution.

Figure 8

Far field image showing incoherent superposition of the 451 laser intensities.