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Strong cluster synchronization in complex semiconductor laser networks with time delay signature suppression

Liyue Zhang, Wei Pan, Lianshan Yan, Bin Luo, Xihua Zou, and Songsui Li

Open Access Open Access

Abstract

Cluster synchronization is a state where clusters of nodes inside the network exhibit isochronous synchronization. Here, we present a mechanism to realize the strong cluster synchronization in semiconductor laser (SL) networks with complex topology, where stable cluster synchronization is achieved with decreased correlation between dynamics of different clusters and time delay signature concealment. We elucidate that, with the removal of intra-coupling within clusters, the stability of cluster synchronization could be enhanced effectively, while the statistical correlation among dynamics of each cluster decreases. Moreover, it is demonstrated that the correlation between clusters can be further reduced with the introduction of dual-path injection and frequency detuning. The robustness of strong cluster synchronization on operation parameters is discussed systematically. Time delay signature in chaotic outputs of SL network is concealed simultaneously with heterogeneous inter-coupling among different clusters. Our results suggest a new approach to control the cluster synchronization in complex SL networks and may potentially lead to new network solutions for communication schemes and encryption key distribution.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Synchronization is a ubiquitous nonlinear phenomenon exhibited in complex networks with applications of a vast variety of natural and technology systems [1,2]. Complete isochronous synchronization in which all units in network evolve into the same dynamics has been studied extensively in previous studied [3,4]. However, many complex networks may display more ordinary and complicated synchronization patterns, such as cluster synchronization and chimera states [5–8]. In cluster synchronization, network self-organizes into subsets of units in which members within same clusters will synchronize to the uniform trajectory, while elements in different cluster exhibit diverse dynamics. And chimera state is a symmetry breaking phenomenon in which coherent and incoherent behavior coexist [9,10].

In general, cluster synchronization and chimera states are closely related patterns of partial synchronization, and chimera state can be considered as a division of network into clusters with synchronous one and incoherent one [11,12]. Different kinds of oscillator models have been adopted as fundamental units to investigate cluster synchronization and chimera states [12,13]. Particularly, systems of semiconductor lasers (SLs) are versatile and low-cost and have been extensively used as nonlinear elements to generate complex dynamical behaviors [14,15]. Especially, synchronization of SLs has attracted considerable attentions for its potential applications in secure communication [16–18], chaotic radar [19,20], random key distribution [21–23], photonic information processing [24,25] and neural networks [26–28].

Generally, the synchronization property and dynamics of two mutually coupled lasers have been extensively studied [29], but only recently pioneering works have begun to extend the mutually coupled SL system to a network scenarios, and to explore the relationship between synchronization characteristics and different network topologies. For example, zero-lag synchronization could be achieve between mutually coupled SLs when a relay element is added [30]. The bubbling effect and chaotic synchronization in star-type network with heterogeneous coupling delays and multiple optical injections are discussed systematically [31–33]. Experimental investigation shows that global synchronization and cluster synchronization can be observed in ring and fully-connected SL networks [34–36]. In recent years, the investigations of SL network synchronization are extending from small motifs to complex network scenarios in practical application perspective. With the introduction of complex network theory, it is shown that the different patterns of synchronization could be revealed by the symmetries of network topologies [11,12]. SLs in network are partitioned into different clusters based on the orbits of symmetry group. Members of same clusters will evolve into the same trajectories, and the dynamics of different clusters will not synchronize or be incoherent [37,38]. Additionally, these symmetry clusters in SL network are divided into distinct independently synchronizable groups, in which the stability of synchronization of individual clusters is decoupled from that in different groups [39]. However, the network realizations about the control of cluster synchronization in SL networks are still scarce [40,41]. Moreover, how to achieve effectively control and enhance the stability of cluster synchronization for SL networks with arbitrary complex topology is still an open question.

In this work, we develop a mechanism to enhance the stability of cluster synchronization in complex SL network, where the strong cluster synchronization is achieved with low correlation between the chaotic dynamics of different clusters. It is demonstrated that the stability of cluster synchronization is enhanced effectively with the removal of intra-coupling within clusters, parameter spaces for stable cluster synchronization with time averaged Root Mean Square synchronization error RMS$<0.01$ are effectively extended and the statistically correlation among dynamics of different cluster is weakened significantly either. Dual-path optical injection and frequency detuning is introduced to further decrease the values of cross correlation function between different clusters. Additionally, it is shown that the time delay signature in chaotic outputs of SL network could be suppressed simultaneously with the introduction of heterogeneous coupling delay between different clusters.

2. Theory and model

As presented in Fig. 1, we consider a random network configuration composed of 11 mutually coupled SLs. Here, gray links denote the inter-coupling between SLs of different clusters and red dashed lines indicate the intra-coupling between SLs within same cluster. Adjacency matrix $A$ is introduced to describe the connectivity of SL network mathematically. $A_{m,n}=A_{n,m}=1$ if there are mutually coupled links between $\text {SL}_m$ and $\text {SL}_n$, and $A_{m,n}=A_{n,m}=0$ otherwise. Then the adjacency matrix $A$ is decoupled to coupling matrix $A_{in}$ and $A_{ex}$ as shown in Eq. (1) to interpret the internal (red dashed line) and external coupling (gray links) properties respectively, and $A=A_{in}+A_{ex}$.

$$\scalebox{0.67}{$\displaystyle A_{in}=\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} , ~~~ A_{ex}=\begin{pmatrix} 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \end{pmatrix}$}$$

 figure: Fig. 1.

Fig. 1. Schematic diagrams of SLs network with complex topology. (a) Symmetric networks composed of 11 SLs, red dashed lines are intra-coupling within clusters and gray lines represent inter-coupling among clusters; (b) Network topology with the removal of internal coupling within clusters; (c) Network topology with inter-coupling of dual-path from cluster $C_3=\{4, 6\}$ to the other clusters in SL network.

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Then, SLs labeled with same color are divided into same cluster based on the symmetry of network topology. The symmetries of a network are defined by the automorphism group Aut($A$) which is formed by all the permutation $R_g$ that hold the network topology invariant, i.e. $AR_g=R_gA$. Then the SLs which are permuted with each other by these symmetries compose the orbits of group Aut($A$), and the set of all the orbits define the partition of symmetry clusters in complex SL network, form candidates of cluster synchronization [11,12]. Based on the inherent symmetries of network topology, SLs in network of Fig. 1(a) are divided into five clusters as: $C_1=\{1, 8\}; C_2=\{2, 3, 7, 9\}; C_3=\{4, 6\}, C_4=\{5, 10\}, C_5=\{11\}$. As the adjacency matrix $A$ commutes with all the symmetry permutations, the dynamical equations of SLs within same cluster that mapped into each other will be same. So if they start from identical initial conditions, the dynamics of SLs will realize isochronous synchronization indefinitely. While different initial conditions among SLs will induce symmetry breaking and may impair the stability of cluster synchronization, and the purpose of our work is to enhance the stability of cluster synchronization.

The dynamical evolution of SLs network is governed by modified Lang-Kobayashi (L-K) equations with introduction of adjacency matrix to take into account mutually coupled term in network scenarios [42]:

$$\begin{aligned} \dot{E_{m}}(t)=&\frac{(1+i\alpha)}{2}\left(G_{m}-\frac{1}{\tau_{p}}\right)E_{m}(t)\\ &+\sigma\sum_{n=1}^{D_{s}}A_{ex(m,n)}E_{n}(t-{\tau_{m,n}})exp-i(w_{n}{\tau_{m,n}}+\Delta w_nt)\\ &+\eta\sigma\sum_{n=1}^{D_{s}}A_{in(m,n)}E_{n}(t-{\tau_{m,n}})exp-i(w_{n}{\tau_{m,n}}+\Delta w_nt) \end{aligned}$$
$$\dot{N_{m}}(t)=\frac{p_{f}I_{th}}{q}-\frac{N_m(t)}{\tau_{e}}-G_{m}\|E_{m}(t)\|^2 \quad m, n =1,{\ldots}D_{s},$$
where $E_m$ indicates the slowly varying complex optical field, $N_m$ and $G_m(t)$ denote the carrier number and optical gain of $m$th SL in network, respectively. $\sigma \sum _{n=1}^{D_{s}}A_{ex(m,n)}E_{n}(t-{\tau _{m,n}})exp-i(w_{n}{\tau _{m,n}}+\Delta w_nt)$ represents the inter-coupling among SLs in different clusters with overall coupling strength $\sigma$, and $\eta \sigma \sum _{n=1}^{D_{s}}A_{in(m,n)}E_{n}(t-{\tau _{m,n}})exp-i(w_{n}{\tau _{m,n}}+\Delta w_nt)$ stands for intra-coupling between SLs within clusters with internal coupling strength $\eta \sigma$, the relationship between internal coupling strength and external coupling strength is characterized by coupling factor $\eta$, and $\tau _{m,n}$ is coupling delay from $\text {SL}_n$ to $\text {SL}_m$. The internal parameters of SL in network are set to be typical values as referenced in [38,39]: the carrier lifetime $\tau _e=2~\text {ns}$, photon lifetime $\tau _p=2~\text {ps}$, electron charge $q=1.6\times 10^{-19}~\text {C}$, the gain coefficient $G_m(t)=g[N_m(t)-N_0]/(1+\epsilon \Vert {E_m(t)} \Vert ^2)$, where transparency carrier density $N_0=1.5\times 10^8$, differential gain $g=1.5\times 10^{-8}~\text {ps}^{-1}$. Threshold current $I_{th}=14.7~\text {mA}$, $w_{m}$ is reference frequency corresponding to wavelength of free running SLs $\lambda _m=1550~\text {nm}$, $\Delta w_n=2\pi \Delta f_n$, and $\Delta f_n=f_m-f_n$ indicates the frequency detuning between $\text {SL}_m$ and $\text {SL}_n$. Unless otherwise stated, linewidth-enhancement factor $\alpha =5$ and gain saturation coefficient $\epsilon =5\times 10^{-7}$, coupling delay ${\tau _{m,n}}=2~\text {ns}$, frequency detuning $\Delta f_n=0~\text {Hz}$ and current factor $p_f=2.5$.

Based on the above modified L-K model, we start to investigate the mechanism to realize strong cluster synchronization. Essentially, internal coupling links between SLs within same clusters are removed to exclude the symmetry breaking effect induced by the different initial conditions of SLs and to enhance the stability of cluster synchronization. Therefore, all the calculations in this paper are conducted with random initial conditions of SLs in network. The corresponding network dynamics is illustrated in Fig. 2, and temporal trajectories for the intensity of SLs ($I_m(t)=|E_m(t)|^2$) in specific cluster is highlighted with colourful curves to be distinguished from dynamical evolution from SLs in different clusters. As shown in Figs. 2b(1)-b(4), separated curves displayed in Figs. 2a(1)-a(4) for all the four non-trivial clusters of SL network converge to overlapped trajectories, implying that stable isochronous synchronization within cluster is induced with the removal of internal coupling links. Additionally, the colourful trajectories of synchronized clusters is distinguishable from dynamical evolutions of other SLs in network (gray trajectories) which means that there is no synchronization between different clusters.

 figure: Fig. 2.

Fig. 2. Networks exhibiting strong cluster synchronization. a(1)-a(4) Optical intensity of SLs in different clusters as function of $t$ for networks in Fig. 1(a) with $\sigma =5\text {ns}^{-1}, \eta =1$; b(1)-b(4) Time evolution of SLs with the removal of internal coupling within clusters i.e. $\eta =0$, only a single colorful trajectory is clearly visible in this column because the curves overlap.

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The statistical coherence between SLs in network is evaluated by cross correlation, accounting for the possible coherence with time lag $\Delta t$. $\text {CCF}_{m,n}$ indicates the absolute value of Pearson correlation coefficient between the intensity of $\text {SL}_m~(I_m(t))$ and $\text {SL}_n~(I_n(t+\Delta t))$:

$${\text{CCF}_{m,n}}= \left\vert \frac{\Big\langle\left[I_m(t)-\left\langle{I_m(t)}\right\rangle\right]\cdot{\left[I_n(t+{\bigtriangleup} t)-\left\langle{I_n(t+{\bigtriangleup} t)}\right\rangle\right]}\Big\rangle}{\sqrt{\left\langle\left[I_m(t)-\left\langle{I_{m}(t)}\right\rangle\right]^2\right\rangle\cdot{\left\langle\left[I_{n}(t+{\bigtriangleup} t)-\left\langle{I_{n}(t+{\bigtriangleup} t)}\right\rangle\right]^2\right\rangle}}}\right\vert_\text{max}$$
where $I_{m}=|E_m(t)|^2$ is the intensity of SLs in network, $\langle {\cdot } \rangle$ denotes time average, $\bigtriangleup t \in [-5, 5]~\text {ns}$ denotes lag time. Fig. 3(a) displayed the pairwise correlation $\text {CCF}_{m,n}$ between all the SLs in network of Fig. 1(a). It can be seen that there is a high correlation between all the clusters, but the values of $\text {CCF}_{m,n}$ for SLs within same cluster are still less than 0.75 as there is no stable synchronization within clusters. Then internal coupling is deleted within clusters, the values of $\text {CCF}_{m,n}$ between SLs in same cluster increase to 1 as presented in Fig. 3(b), indicating isochronous synchronization is achieved successfully by our mechanism. Moreover, coherence among different clusters are significantly decreased either. In strong chimera, synchronous part of network can be sustained by the incoherent injection from the rest of network, and chimera states can be seen as the coexistence of synchronized cluster and incoherent cluster [10,43]. Compared with strong chimera, we term this proposed mechanism with enhanced cluster synchronization stability and low coherence among clusters as strong cluster synchronization.

 figure: Fig. 3.

Fig. 3. Correlation between SL in network. (a) Pairwise cross correlation $\text {CCF}_{m,n}$ between SLs in network Fig. 1(a) with $\sigma =5\text {ns}^{-1}, \eta =1$; (b) Pairwise cross correlation $\text {CCF}_{m,n}$ between SLs with the removal of internal coupling within clusters with $\sigma =5\text {ns}^{-1}, \eta =0$.

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In order to quantify the robustness of strong cluster synchronization on the operation parameters of SL network quantitatively and obtain the parameter spaces for stable cluster synchronization, we calculate the time averaged root-mean square (RMS) synchronization error for the chaotic outputs of SLs in all the four non-trivial clusters [37]. We have calculated the values of RMS with random initial conditions for many cases with different parameters. It is found that, the evolution trajectories of SLs within same cluster are all overlapped when the values of RMS is less than 0.01. And in contrast, when there is no synchrony between the chaotic dynamics of SLs, the values of RMS is higher than 0.01. Thus, for the purpose of comparison and convenience, a threshold value RMS=0.01 is introduced in our study, and stable synchronization is defined to be achieved in cluster $C_n$ if $\text {RMS}<0.01$. Fig. 4 plots the evolution maps of RMS in the parameter spaces of $\alpha \times \epsilon$ and $p_f\times \sigma$ for networks in Figs. 1(a) and (b), respectively. As shown in Figs. 4b(1)-b(4) and d(1)-d(4), high quality of isochronous synchronization for clusters $C_1-C_4$ ($\text {RMS}<0.01$) can be achieved in a widespread parameter spaces of $\alpha \times \epsilon$ and $p_f\times \sigma$ with the removal of internal coupling links within clusters, indicating the robustness of proposed scheme on the operation parameters of SL network. Moreover, parameter spaces for stable cluster synchronization are expanded effectively compared with that in Fig. 4a(1)-a(4) and c(1)-c(4), which means the synchronizability of clusters is facilitated with the removal of intra-coupling. While, how does the synchronizablity of clusters depend on intra-coupling? In order to provide an answer to the above question, the distribution of RMS as function of coupling factor $\eta$ with different coupling strength $\sigma$ is presented. The initial conditions of SLs are chosen randomly over all the calculation. As shown in Fig. 5, the values of RMS for each cluster in network are all monotonically decreasing function of coupling factor $\eta$, indicating that stability of cluster synchronization is impaired with the increment of internal coupling strength. Previous work in network theory proved that compared with the situation with highly correlated injection, stable synchronization can be sustained and facilitated by the injection with low correlation from network, this mechanism can also be regarded as a analog of noise induced synchronization [43]. The pairwise cross correlation between SLs calculated in Fig. 3 shows that dynamics of SLs within same clusters are highly correlated. Moreover, as shown in Fig. 4 that cluster synchronization in SLs network can be improved significantly with the removal of internal coupling, i.e. $\eta =0$, which means that there is no coupling with highly correlated dynamics in network. Since the chaotic dynamics of SLs within same cluster are high correlated, increasing values of $\eta$ will give rise to strong correlated injection from the internal coupling within clusters, and leading to the dynamics of SLs lose synchrony. The curves in Fig. 5 with different markers indicate the calculation of RMS with different external coupling strength $\sigma$. And it is found that the parameter spaces of $\eta$ for $\text {RMS}<0.01$ with external coupling strength $\sigma =5\text {ns}^{-1}$ will shrink significantly compared to the scenario with external coupling strength $\sigma =3\text {ns}^{-1}$. Since the complexity of chaotic dynamics of SLs will increase with $\sigma$, and thus enhance the degree of difficulty to overcome the symmetry breaking that induced by different initial conditions of SLs.

 figure: Fig. 4.

Fig. 4. Parameter landscape of RMS for chaotic outputs of all the four non-trivial clusters in SL network. a(1)-a(4), c(1)-c(4) Evolution maps of RMS in parameter space of $\alpha \times \epsilon$ with coupling strength $\sigma =\eta \sigma =5\text {ns}^{-1}$, current factor $p_f=2.5$ and $p_f\times \sigma$ with linewidth enhancement factor $\alpha =5$ and gain saturation coefficient $\epsilon =5\times 10^{-7}$ for original SL network in Fig. 1(a). b(1)-b(4), d(1)-d(4) Prameter landscape of RMS for network in Fig. 1(b) with the removal of internal coupling within clusters, i.e.$\eta =0$.

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 figure: Fig. 5.

Fig. 5. Influence of internal coupling strength on the synchronizability of clusters. The valus of RMS as function of $\eta$ for different values of $\sigma$, (a)-(d) $C_1, C_2, C_3, C_4$.

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Moreover, a restricted mutual visibility for different group of users is greatly requisite for the practical application of SLs network. In strong cluster synchronization, we demonstrate that the coherence between dynamics of specific clusters and the others in network can be attenuate with the introduction of frequency detuning and dual-path optical injection. Figures 6(a)-(b) calculate the pairwise cross correlation between all the SLs in network in Figs. 1(a) and (b) with the frequency detuning between cluster $C_3$ and the other clusters, i.e. $f_{C_3}-f_{C_n}=10$GHz. It can be found that the coherence between clusters is decreased effectively with the introduction of frequency detuning compared that of Fig. 3. Furthermore, dual path optical injection is introduced into the coupling links between cluster $C_3=\{4, 6\}$ and all the other clusters as shown in Fig. 1(c). The coupling delays from SLs in cluster $C_3$ to the other SLs in netwrok are set to be $\tau _{C_3}(1)=2$ns and $\tau _{C_3}(2)=3.5$ns, and coupling delays from SLs of all the other clusters to cluster $C_{3}$ equal to $\tau _{C_3}(1)$ and $\tau _{C_3}(2)$. Coupling strength of each path is set to be $0.5\sigma$ to preserve the overall coupling strength uniform. As additional dynamical dimension is introduced with dual-path optical injection, the coherence between cluster $C_3$ and the other clusters is further reduced as presented in Figs. 6(c)-(d).

 figure: Fig. 6.

Fig. 6. Pairwise cross correlation $\text {CCF}_{m,n}$ for all the SLs in network. (a)-(b) The values of $\text {CCF}_{m,n}$ for SLs in network of Figs. 1(a) and (b) with frequency detuning between cluster $C_3$ and the other clusters $\Delta f=10$GHz ($n=1, 2, 4$). (c) The values of $\text {CCF}_{m,n}$ for SLs in network of Fig. 1(c) with internal coupling. (d) The values of $\text {CCF}_{m,n}$ for SLs in network of Fig. 1(c).

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From another viewpoint, as a crucial parameter, the time delay signature which possess detectable features will lead to unavoidable security issue and be detrimental to the applications of optical chaos. In strong cluster synchronization, we employee heterogeneous inter-coupling delays among clusters to suppress the time-delay signature in the chaotic dynamics of SLs in network. Time dependent auto-correlation function (ACF) is adopted as a quantifier to evaluate time delay signature [37].

$$C_T= \frac{\Big\langle\left[I_m(t)-\left\langle{I_m(t)}\right\rangle\right]\cdot{\left[I_m(t+{\bigtriangleup} t)-\left\langle{I_m(t+{\bigtriangleup} t)}\right\rangle\right]}\Big\rangle}{\sqrt{\left\langle\left[I_m(t)-\left\langle{I_{m}(t)}\right\rangle\right]^2\right\rangle\cdot{\left\langle\left[I_{m}(t+{\bigtriangleup} t)-\left\langle{I_{m}(t+{\bigtriangleup} t)}\right\rangle\right]^2\right\rangle}}}$$
For purpose of comparison, the time-dependent ACF of network in Fig. 1(b) is presented in Figs. 7a(1)-a(4). It can be seen that, an obvious peak is extracted at ${\Delta t}=2$ns, which indicates successfully identification of time delay signature. On the other hand, with the introduction of heterogeneous inter-coupling delay, the information of time delay signature in ACF is inhibited significantly as shown in Figs. 7b(1)-b(4). Furthermore, we calculate the peak to mean ratio $R_C$ in the vicinity of time delay to evaluate the time delay signature quantitatively, and $R_C=max(|C_T|)/\langle |C_T| \rangle$. As heterogeneous delays will weaken the residual periodicity induced by external cavity, it can be clearly found in Fig. 7(c) that, the time delay signature suppression can be improved significantly with the introduction of heterogeneous inter-coupling delay in SL network. Moreover, the isochronous synchronization of SLs within same clusters could still be preserved simultaneously, and dynamical trajectories of SLs within same clusters are overlapped as shown in Figs. 7d(1)-d(4).

 figure: Fig. 7.

Fig. 7. Time delay signature suppression in complex network. a(1)-a(4) The values of ACF in network Fig. 1(b) with homogeneous inter-coupling delay ${\tau _{m,n}}=2$ns; b(1)-b(4) The values of ACF in network Fig. 1(b) with heterogeneous inter-coupling delays $\tau _{C_1}=0.85$ns, $\tau _{C_2}=1.25$ns, $\tau _{C_3}=1.85$ns, $\tau _{C_4}=2.35$ns, $\tau _{C_5}=0.75$ns, where $\tau _{C_n}$ is the coupling delays from SLs of cluster $C_n$ to the other SLs in network; (c) The value of peak signal to mean ratio $R_C$ for different $\text {SL}_n$ with homogeneous and heterogeneous inter-coupling delays, d(1)-d(4) Dynamical evolutions ($I_m(t)=|E_m(t)|^2$) of SLs in different clusters.

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Furthermore, the generality of our results is validated in a SLs network with the structure of Nepal power grid (Fig. 8(a)) [44]. Based on the inherent symmetry of network topology, SLs in network is divided into five clusters with three untrivial clusters $C_1=\{1,2,3,4,5\}$, $C_2=\{6,10,11\}$ and $C_3=\{9,12,13,14,15\}$. It is shown that, the synchronizability of SLs network could still be improved with the removal of internal coupling between SLs within clusters in SLs network with different structure. Separated dynamical trajectories of SLs within same clusters in Figs. 8b(1)-b(3) will converge to overlapped curves as shown in Figs. 8c(1)-c(3), indicating the achievement of stable cluster synchronization. Moreover, calculation of cluster synchronization with the removal of internal coupling is also conducted with coupling delay $\tau _{m,n}=20$ns and $\tau _{m,n}=100$ns to investigate the scalability and feasibility of strong cluster synchronization, stable synchronization of three untrivial clusters could still be preserved, and it needs to be noted that the evolving time for dynamics of SLs converge to synchronous manifold will increase with the coupling delays as presented in Figs. 8d(1)-d(3) and Figs. 8e(1)-e(3). While, the time sale of synchronization is still on the nanosecond and has little effect on the feasibility of implementation.

 figure: Fig. 8.

Fig. 8. Strong cluster synchronization in SLs network with structure of Nepal power grid. (a) Schematic diagrams of Nepal power grid network structure. b(1)-b(3) Dynamical trajectories of SLs in network with internal coupling, i.e. $\eta =1$ and $\sigma =10\text {ns}^{-1}$, $\tau _{m,n}=2$ns; c(1)-c(3) Dynamical trajectories of SLs in network with the removal of internal coupling, i.e. $\eta =0$ and $\sigma =10\text {ns}^{-1}$, $\tau _{m,n}=2$ns; cluster synchronization of cluster $C_1$, $C_2$ and $C_3$ with $\tau _{m,n}=20$ns d(1)-d(3) and $\tau _{m,n}=100$ns e(1)-e(3).

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3. Conclusion

In conclusion, we propose a mechanism that can realize the strong cluster synchronization in SLs network with complex topology, where stable cluster synchronization is achieved with suppressed correlation between dynamics of different clusters and time delay signature concealment. It is demonstrated that the stability of cluster synchronization can be improved with the removal of internal coupling within clusters, and correlation among different clusters in network could be decreased at the same time. The robustness of strong cluster synchronization on the operation parameters of network is validated with the calculation of RMS in the parameter spaces of $\alpha \times \epsilon$ and $p_f\times \sigma$. Moreover, the correlation between different clusters can be further weakened with the introduction of frequency detuning and dual-path optical injection. Additionally, it is demonstrated that time delay signature can also be concealed effectively in strong cluster synchronization with heterogeneous inter-coupling delay. And the generality of our results is validated in a SLs network with the structure of Nepal power grid. Our results suggest a new control mechanism to optimize the synchronizability of complex SL network and may lead to new schemes in potential optical chaos based applications.

Funding

National Key Research and Development Program of China (2021YFB2801901); National Natural Science Foundation of China (62104203); Sichuan Province Science and Technology Support Program (2022YFG0026); Fundamental Research Funds for the Central Universities (2682022CX024).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Data availability

No data were generated or analyzed in the presented research.

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Figures (8)
Fig. 1.
Fig. 1. Schematic diagrams of SLs network with complex topology. (a) Symmetric networks composed of 11 SLs, red dashed lines are intra-coupling within clusters and gray lines represent inter-coupling among clusters; (b) Network topology with the removal of internal coupling within clusters; (c) Network topology with inter-coupling of dual-path from cluster $C_3=\{4, 6\}$ to the other clusters in SL network.

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Fig. 2.
Fig. 2. Networks exhibiting strong cluster synchronization. a(1)-a(4) Optical intensity of SLs in different clusters as function of $t$ for networks in Fig. 1(a) with $\sigma =5\text {ns}^{-1}, \eta =1$; b(1)-b(4) Time evolution of SLs with the removal of internal coupling within clusters i.e. $\eta =0$, only a single colorful trajectory is clearly visible in this column because the curves overlap.

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Fig. 3.
Fig. 3. Correlation between SL in network. (a) Pairwise cross correlation $\text {CCF}_{m,n}$ between SLs in network Fig. 1(a) with $\sigma =5\text {ns}^{-1}, \eta =1$; (b) Pairwise cross correlation $\text {CCF}_{m,n}$ between SLs with the removal of internal coupling within clusters with $\sigma =5\text {ns}^{-1}, \eta =0$.

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Fig. 4.
Fig. 4. Parameter landscape of RMS for chaotic outputs of all the four non-trivial clusters in SL network. a(1)-a(4), c(1)-c(4) Evolution maps of RMS in parameter space of $\alpha \times \epsilon$ with coupling strength $\sigma =\eta \sigma =5\text {ns}^{-1}$, current factor $p_f=2.5$ and $p_f\times \sigma$ with linewidth enhancement factor $\alpha =5$ and gain saturation coefficient $\epsilon =5\times 10^{-7}$ for original SL network in Fig. 1(a). b(1)-b(4), d(1)-d(4) Prameter landscape of RMS for network in Fig. 1(b) with the removal of internal coupling within clusters, i.e.$\eta =0$.

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Fig. 5.
Fig. 5. Influence of internal coupling strength on the synchronizability of clusters. The valus of RMS as function of $\eta$ for different values of $\sigma$, (a)-(d) $C_1, C_2, C_3, C_4$.

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Fig. 6.
Fig. 6. Pairwise cross correlation $\text {CCF}_{m,n}$ for all the SLs in network. (a)-(b) The values of $\text {CCF}_{m,n}$ for SLs in network of Figs. 1(a) and (b) with frequency detuning between cluster $C_3$ and the other clusters $\Delta f=10$GHz ($n=1, 2, 4$). (c) The values of $\text {CCF}_{m,n}$ for SLs in network of Fig. 1(c) with internal coupling. (d) The values of $\text {CCF}_{m,n}$ for SLs in network of Fig. 1(c).

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Fig. 7.
Fig. 7. Time delay signature suppression in complex network. a(1)-a(4) The values of ACF in network Fig. 1(b) with homogeneous inter-coupling delay ${\tau _{m,n}}=2$ns; b(1)-b(4) The values of ACF in network Fig. 1(b) with heterogeneous inter-coupling delays $\tau _{C_1}=0.85$ns, $\tau _{C_2}=1.25$ns, $\tau _{C_3}=1.85$ns, $\tau _{C_4}=2.35$ns, $\tau _{C_5}=0.75$ns, where $\tau _{C_n}$ is the coupling delays from SLs of cluster $C_n$ to the other SLs in network; (c) The value of peak signal to mean ratio $R_C$ for different $\text {SL}_n$ with homogeneous and heterogeneous inter-coupling delays, d(1)-d(4) Dynamical evolutions ($I_m(t)=|E_m(t)|^2$) of SLs in different clusters.

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Fig. 8.
Fig. 8. Strong cluster synchronization in SLs network with structure of Nepal power grid. (a) Schematic diagrams of Nepal power grid network structure. b(1)-b(3) Dynamical trajectories of SLs in network with internal coupling, i.e. $\eta =1$ and $\sigma =10\text {ns}^{-1}$, $\tau _{m,n}=2$ns; c(1)-c(3) Dynamical trajectories of SLs in network with the removal of internal coupling, i.e. $\eta =0$ and $\sigma =10\text {ns}^{-1}$, $\tau _{m,n}=2$ns; cluster synchronization of cluster $C_1$, $C_2$ and $C_3$ with $\tau _{m,n}=20$ns d(1)-d(3) and $\tau _{m,n}=100$ns e(1)-e(3).

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Equations (5)

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$$\scalebox{0.67}{$\displaystyle A_{in}=\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} , ~~~ A_{ex}=\begin{pmatrix} 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \end{pmatrix}$}$$
$$\begin{aligned} \dot{E_{m}}(t)=&\frac{(1+i\alpha)}{2}\left(G_{m}-\frac{1}{\tau_{p}}\right)E_{m}(t)\\ &+\sigma\sum_{n=1}^{D_{s}}A_{ex(m,n)}E_{n}(t-{\tau_{m,n}})exp-i(w_{n}{\tau_{m,n}}+\Delta w_nt)\\ &+\eta\sigma\sum_{n=1}^{D_{s}}A_{in(m,n)}E_{n}(t-{\tau_{m,n}})exp-i(w_{n}{\tau_{m,n}}+\Delta w_nt) \end{aligned}$$
$$\dot{N_{m}}(t)=\frac{p_{f}I_{th}}{q}-\frac{N_m(t)}{\tau_{e}}-G_{m}\|E_{m}(t)\|^2 \quad m, n =1,{\ldots}D_{s},$$
$${\text{CCF}_{m,n}}= \left\vert \frac{\Big\langle\left[I_m(t)-\left\langle{I_m(t)}\right\rangle\right]\cdot{\left[I_n(t+{\bigtriangleup} t)-\left\langle{I_n(t+{\bigtriangleup} t)}\right\rangle\right]}\Big\rangle}{\sqrt{\left\langle\left[I_m(t)-\left\langle{I_{m}(t)}\right\rangle\right]^2\right\rangle\cdot{\left\langle\left[I_{n}(t+{\bigtriangleup} t)-\left\langle{I_{n}(t+{\bigtriangleup} t)}\right\rangle\right]^2\right\rangle}}}\right\vert_\text{max}$$
$$C_T= \frac{\Big\langle\left[I_m(t)-\left\langle{I_m(t)}\right\rangle\right]\cdot{\left[I_m(t+{\bigtriangleup} t)-\left\langle{I_m(t+{\bigtriangleup} t)}\right\rangle\right]}\Big\rangle}{\sqrt{\left\langle\left[I_m(t)-\left\langle{I_{m}(t)}\right\rangle\right]^2\right\rangle\cdot{\left\langle\left[I_{m}(t+{\bigtriangleup} t)-\left\langle{I_{m}(t+{\bigtriangleup} t)}\right\rangle\right]^2\right\rangle}}}$$
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