Disorder-mediated synchronization resonance in coupled semiconductor lasers

TL;DR

The paper addresses how fixed intrinsic frequency disorder in networks of delay-coupled semiconductor lasers hinders steady-state synchronization and cannot be treated by traditional master-stability theory. It analyzes a network of $M$ fully connected lasers using the Lang-Kobayashi equations and demonstrates the existence of an optimal weak coupling $κ^*$ that maximizes the steady-state synchronization measure $⟨S⟩$, with $κ^* ∝ 1/(M-1)$ and a total coupling cost that scales linearly with $M$. A theoretical framework recasts the delayed phase dynamics as a gradient flow on an effective thermodynamic potential $U(η)$, featuring a phase shift $φ=\tan^{-1}(α)+ω_0τ$ and a dressed coupling, which yields a critical coupling $𝕂_c$ that delineates near-synchronized steady states and the onset of chaos, thereby explaining the observed resonance. Importantly, nonzero amplitude–phase coupling $α$ is essential for the resonance, and the maximum synchronization level is largely independent of $M$, while the required coupling per link decreases as the network grows. These results offer a practical route to scalable, high-coherence, high-power emission in large laser arrays and point to a general mechanism by which moderate coupling can overcome static heterogeneity in nonlinear, delay systems.

Abstract

Disorder can profoundly influence synchronization in networks of nonlinear oscillators, sometimes enhancing coherence through external tuning. In semiconductor lasers, however, achieving high-quality steady-state synchronization is desired, while intrinsic and typically uncontrollable disorder poses a major challenge. Under fixed frequency disorder, we investigate homogeneous fully coupled external-cavity semiconductor lasers governed by the complex, time-delayed Lang-Kobayashi equations with experimentally relevant parameters and identify an optimal coupling strength that maximizes steady-state synchronization in the weak-coupling regime, which we term disorder-mediated synchronization resonance. This optimum appears for any fixed configuration of intrinsic frequency detuning and scales inversely with the number of lasers, leading to a linear scaling of the total coupling cost with the number of lasers. A theory based on an effective thermodynamic potential explains this disorder-mediated optimization, revealing a general mechanism by which moderate coupling can overcome static heterogeneity in nonlinear physical systems.

Figures (6)

• Figure 1: Illustration of a network of coupled semiconductor diode lasers. Experimentally, coupling is implemented using a spatial light modulator (SLM) placed at the back focal plane of a collimation lens, with the laser array located at the front focal plane. Lasers at different transverse positions are coupled through gratings of corresponding periods on the SLM, introducing a delay determined by the round-trip time $\tau$ between the array and the SLM. The intrinsic lasing frequencies of the individual lasers are randomly detuned due to unavoidable fabrication-induced heterogeneity.
• Figure 2: Optimized steady-state synchronization with fixed frequency disorder in a network of $M=24$ semiconductor diode lasers. (a) Maximization of synchronization measures $\langle S\rangle$ by an optimal coupling strength $\kappa^*$. (b) Average short-term final frequency detuning of the individual lasers versus $\kappa$, where the error bars represent the standard deviation calculated over a moving time window of size $3\tau$ with a step size of $0.3\tau$ for $t\in [50,100]\rm ns$. (c) Mean-square value of the normalized intensity fluctuations, $(I_i(t)-\langle I_{i}(t)\rangle)/\langle I_{i}(t)\rangle$, as a function of the coupling parameter $\kappa$, serving as a time-domain measure of the relative intensity noise (RIN) of individual lasers over a $50\,\rm ns$ time window. Different colors correspond to different lasers.
• Figure 3: Robustness and size scaling of optimized steady-state synchronization. The frequency disorders are independently sampled from a Gaussian distribution $\sigma_{\Delta} \mathcal{N}(0,1)$ with the standard deviation $\sigma_{\Delta}$. (a) Optimizing steady-state synchronization for different number $M$ of lasers for $\sigma_{\Delta} = 14\,\textnormal{rad/ns}$. (b) Statistical fluctuations of the optimized steady-state synchronization, where for each value of $M$, ten independent realizations of the frequency disorder for $\sigma_{\Delta} = 14\,\textnormal{rad/ns}$ are tested. The four curves in their respective statistical clouds are the averages and the shaded areas indicate the corresponding standard deviations. (c) Location of the optimized peak, $\kappa^*$, indicated by the averaged peak position over ten frequency disorder realizations. The resulting $\kappa^*$ values exhibit an inverse relationship with the system size: $\kappa^* \propto 1/(M-1)$, demonstrated for $M = [12,18,24,30,36,42,46,50,55,60,65,70,80,90,100]$. (d) Effect of frequency disorder strength as characterized by $\sigma_{\Delta}$ on the steady-state synchronization for $M=24$. For small values of $\sigma_{\Delta}$, the peak value of $\langle S\rangle$ can reach unity as in the corresponding disorder-free system.
• Figure 4: Schematic of local-minima selection, corresponding to the steady state of each laser. (a) Effective thermodynamic potential comprising a parabolic term, a shifted global minimum $-2\tau\Delta_i\eta_i(t)$, and a coupling-induced cosine component. The red rectangle marks the near-overlapping regime that yields nearly synchronized final frequencies. (b) Illustration of steady-state selection from the local minima of the effective potential landscape for different lasers at the critical effective coupling $\mathbb{K}_c$.
• Figure 5: Without frequency disorder, homogeneous all-to-all coupling yields complete synchronization, while nonzero amplitude–phase coupling $\alpha$ drives the system into chaotic regimes as the coupling increases. Panels (a,c,e) and (b,d,f) show, for $M=24$, the synchronization measure $\langle S \rangle$, a time-domain measure of RIN, and the local maxima of the normalized intensity for each laser $i$, as functions of $\kappa$ for $\alpha=5$ and $\alpha=0$. The time window is $[50,100]\,{\rm ns}$. Complete synchronization means that all $M=24$ lasers share the same RIN and the same local maxima of the normalized intensity.