Interpretable disorder-promoted synchronization and coherence in coupled laser networks

TL;DR

The paper shows that deliberate parameter heterogeneity in delayed-coupled laser networks can stabilize frequency-synchronized states that are unstable in homogeneous arrays, enabling coherent scaling to large networks. Using the Lang-Kobayashi model derived from Maxwell-Bloch equations, it demonstrates that intermediate levels of disorder in parameters such as the linewidth enhancement factor $\alpha$, detuning $\omega$, and coupling $\kappa$ promote a stable state $E_j(t)=r_j^* e^{i(\Omega t+\delta_j^*)}$ with $\delta_j^*\approx0$ and a sharp spectrum around the minimum linewidth mode $\Omega_{ML}$. This disorder-promoted synchronization persists across irregular networks and scales to networks with up to $\mathcal{O}(10^3)$ lasers, providing a principled strategy to improve coherence for high-power photonics. The results contrast with non-delayed or gain-models lacking phase-amplitude coupling, highlighting the essential influence of time delay, nonlinear gain, and network topology on synchronization behavior.

Abstract

Coupled lasers offer a promising approach to scaling the power output of photonic devices for applications demanding high frequency precision and beam coherence. However, maintaining coherence among lasers remains a fundamental challenge due to desynchronizing instabilities arising from time delay in the optical coupling. Here, we depart from the conventional notion that disorder is detrimental to synchronization and instead propose an interpretable mechanism through which heterogeneity in the laser parameters can be harnessed to promote synchronization. Our approach allows stabilization of pre-specified synchronous states that, while abundant, are often unstable in systems of identical lasers. The results show that stable synchronization enabling coherence can be frequently achieved by introducing intermediate levels of random mismatches in any of several laser constructive parameters. Our results establish a principled framework for enhancing coherence in large laser networks, offering a robust strategy for power scaling in photonic systems.

Figures (20)

• Figure 1: Coupled-laser systems. (a) Schematic of a laser array coupled through a spherical mirror. The light paths are depicted for the blue laser, showing that the coupling strength decays with inter-laser distance due to mirror diffraction. (b) Decaying network topology. In both panels, the line thickness represents the coupling strength, and the shades of gray indicate the heterogeneity in laser parameters.
• Figure 2: Synchronization dynamics and coherence in coupled disordered lasers. (a) Spectrum of stationary solutions (amplitude $r^*$ and frequency $\Omega$ pairs) for $\tau = 1$ and $0.15$ ns, showing the multistability of the LK model. Stable and unstable modes are marked in blue and red, respectively. (b) Time series of the electric fields as the lasers transition from a homogeneous configuration for $t\in[0,400]$ ns (where $\omega_j=0$, $\forall j$) to a heterogeneous configuration for $t\in[400,600]$ ns (where $\omega_j$ follows a normal distribution). The small-amplitude signals indicate the imaginary component of the individual fields $E_j$, while the high-amplitude signal represents the combined field $E=\sum_jE_j$. For the homogeneous configuration, the lasers converge to a limit cycle with zero phase shift but distorted waveforms, whereas the heterogeneous configuration outputs sinusoidal waves with constant, but negligible, phase shifts among lasers. The frequency and amplitude dynamics are included in Fig. S7 supplemental_mat. (c) Power spectrum of the combined steady-state field for the homogeneous (red) and heterogeneous (blue) configurations. The simulations are on the 10-laser decaying network for $\kappa=0.54\, \rm{ns}^{-1}$ and $\tau=0.15$ ns.
• Figure 3: Synchronization stability for varying disorder levels in laser arrays. (a) Lyapunov exponent $\lambda_{\rm max}$ versus disorder level $\sigma_p$ in coupling strength (left) and frequency detuning (right) for three network topologies with 10 lasers: all-to-all (blue), decaying (green), and ring (orange). The lines indicate the median across 1,000 realizations of parameter disorder, while shaded areas indicate the first and third quartiles. (b) Fraction of realizations that stabilize the frequency-synchronized state. (c) Standard deviation of the power spectrum of the combined steady-state field $E(t)$, averaged over random initial conditions. The standard deviation measures the frequency bandwidth of $E(t)$, as illustrated in Fig. \ref{['fig:dynamics']}(c). (d) Average size of the basin of attraction. See SM supplemental_mat, Sec. SIII, for numerical analysis.
• Figure 4: Scalability analysis for large laser arrays. Violin plots showing the impact of increasing array sizes on synchronizability. (a) Width of the interval of $\sigma_p$ for which at least 50% of all disorder realizations lead to a frequency-synchronized system. (b) Area under the curve (AUC) representing the fraction of synchronized systems as a function of $\sigma_p$. These two measures are based on the curves displayed in Fig. \ref{['fig:results_random']}(b). In both panels, the distributions include 100 realizations of disorder applied to $\alpha$, $\omega$, and $\kappa$ in a decaying network. See SM supplemental_mat, Sec. SIII, for simulation details.
• Figure S1: Multistability of the LK model for a homogeneous laser array. Stationary solutions (amplitude $r^*$ and frequency shifts $\Omega$) for increasing time delay $\tau$ (top to bottom panels) and coupling strength $\kappa$ (left to right panels) for a decaying network of 10 lasers. Stable and unstable modes are marked in blue and red, respectively. Note that the number of stationary solutions increases as a function of $\tau$ and $\kappa$.