Optimal sparse networks for synchronization of semiconductor lasers

TL;DR

This work tackles the challenge of achieving robust synchronization in large arrays of semiconductor lasers that exhibit intrinsic frequency detuning. By combining the Lang-Kobayashi model with time-delayed coupling and an island-based genetic algorithm, the authors show that optimally sparse coupling topologies can outperform fully connected networks in terms of global phase and frequency locking. A thermodynamic-potential framework maps the delayed-phase dynamics to an energy landscape, revealing that sparsity concentrates coupling on laser pairs with large detuning and leads to hub structures that stabilize the synchronized state; the optimal connectivity scales as $\chi^*\propto 1/(M-1)$. The findings offer a scalable, cost-effective route to robust, steady-state synchronization in disordered laser arrays, with broad implications for large-scale photonic networks and coherent beam generation.

Abstract

The inevitable random frequency differences among semiconductor lasers present an obstacle to achieving their collective coherence, but previous worked showed that fully (all-to-all) coupled networks can still be synchronized even in the weakly coupling regime. An outstanding question is whether sparsely coupled network structures exist that lead to strong synchronization. This paper gives an affirmative answer: optimal sparse coupling configurations can be found which enables near-complete synchronization. Quite surprisingly, with respect to synchronization, certain sparse networks can outperform fully coupled networks, when the weights of coupling are placed dominantly on the laser pairs with large frequency differences. The counterintuitive phenomenon can be explained by a thermodynamic potential theory that maps the time-delay-induced phase dynamics to an energy landscape. These findings suggest a scalable and cost-effective approach to achieving robust, steady-state synchronization of semiconductor lasers in the presence of disorder and noise.

Figures (16)

• Figure 1: Schematic diagram of laser coupling architectures. (a) Programmable laser coupling via re-programmable diffraction pattern of the SLM. (b–d) Candidate coupling structures for synchronization under frequency disorder: all-to-all, nearest-neighbor, and selectively sparse configurations, where both the horizontal and vertical axes denote the laser No., and a colored (blank) block at $(i,j)$ indicates that lasers $i$ and $j$ are coupled (uncoupled).
• Figure 2: Optimal synchronization state and coupling configuration for $M=24$ lasers. (a) Connectivity resonance in sparse networks with frequency disorders: shown are $\langle S \rangle$ values from 1000 random sparse networks for different $\chi$ values, with the orange curve being the average. The red star marks the high value $\langle S \rangle = 0.96$ in an optimized sparse network for $\chi^*=0.4$ found by a genetic algorithm. The blue curve represents the benchmark corresponding to the homogeneous all-to-all coupling, defined by $K_{ij}=\kappa(1-\delta_{ij})$. (b) Spatiotemporal evolution of the phase function $\cos\Omega_i(t)$ of the laser field and (c) time series of the normalized total field intensity $\tilde{I}_{\rm tot}$ for the optimal sparse network. (d) Scatter plot of the final frequencies $f_{\rm final}$ of the synchronized lasers versus their initial mean $\langle f_{\rm initial}\rangle$ for 100 frequency-disorder realizations, corresponding to 100 data points, respectively. The final frequencies are quantized in intervals of $N/\tau$. Points below the dashed line indicate that the final frequency is statistically lower than the initial mean frequency. Parameter values are: pump rate $J_0 = 4\gamma_{n}(N_{0}+\gamma/g) \approx 3.67\times 10^8\,\textnormal{ns}^{-1}$ and $\sigma_{\Delta}=14\,\textnormal{rad/ns}$.
• Figure 3: Scaling behavior associated with connectivity resonance. (a) For a fixed frequency disorder with random seed $\texttt{rng(1)}$ [as in Fig. \ref{['fig:GA_time_series']}(a)], $1000$ coupling configurations are randomly generated at each connectivity $\chi$, and the corresponding $\langle S\rangle$ values are shown. (b) For each $\chi$ value in (a), the configuration yielding the highest $\langle S\rangle$ is used to compute the final frequencies of all lasers. These frequencies are linearly fitted within moving windows of size $\tau$ with a step size of $0.1\tau$ and the error bars indicate the resulting standard deviations. (c) Using the same random seed $\texttt{rng(1)}$, the scaling behavior of the optimal connectivity $\chi^{*}$ is examined across different laser numbers $M=[12, 18, 24, 30, 36, 42, 46, 50, 55, 60, 65, 70, 80, 90, 100]$.
• Figure 4: Emergence of hub structure in the optimal sparse laser network. (a) Histogram of the percentage of coupled links as a function of the frequency differences $\Delta_{i}-\Delta_{j}$ ($i,j = 1,\ldots, M$) for optimal coupling $\kappa = \kappa^{*}$ in homogeneous all-to-all coupled networks. The continuous curve illustrates the shape of the distribution. (b) Same legend as in (a) but for the optimal sparse laser networks found by the genetic algorithm. (c) Heat maps of the mean $\langle K_{ij}\rangle$ of the coupling matrix elements $K_{ij}$ for the optimal sparse networks, averaged over $100$ frequency disorder realizations. (d) Histogram of the average number of connections per laser, defined as $\langle\xi_{i}\rangle=\sum_j\langle K_{ij}/\kappa^{f}\rangle$ and calculated from the matrix elements in (c), for the optimal sparse networks. The horizontal dashed line indicates the baseline given by the mean value, $\mu_{\langle\xi\rangle}=\sum_i \langle\xi_i\rangle/M$, plus one standard deviation, $\sigma_{\langle\xi\rangle}$. Hub lasers are defined as those with $\langle\xi_{i}\rangle$ exceeding this baseline. Also shown on the right-hand side is the absolute value of the initial frequency detuning per laser averaged over $100$ frequency disorder realizations, with standard deviations indicated by the error bars.
• Figure 5: Understanding synchronization in sparse networks. (a,b) Schematic illustrations of steady-state selection from the potential landscape for different lasers under homogeneous all-to-all coupling and sparse coupling, respectively. The cross intersections of Eq. (\ref{['eq: ECM_criteria_1']}) are marked by the red dots in panels (a) and (b). These intersections fall within the light-shaded region that satisfies Eq. (\ref{['eq: ECM_criteria_2']}). Together, they define the steady-state solutions $\{\eta^{*}_{12}\}$ and $\{\eta^{*}_{24}\}$. These solutions correspond to two representative effective detuning, $\mathbb{F}_{12}$ and $\mathbb{F}_{24}$ associated with the $12$-th and $24$-th lasers, respectively. The green and purple oscillatory curves represent stronger and weaker effective coupling strengths. The inset in (b) shows the effective thermodynamic potential in \ref{['eq:thermodynamic_potential']}, consisting of a parabolic term and an oscillatory sinusoidal component. A smaller effective coupling $\mathbb{K}^{\rm sparse}_i$ produces a smoother potential with fewer steady-state solutions.