Isochronous cluster synchronization in delay-coupled VCSEL networks subjected to variable-polarization optical injection with time delay signature suppression

Liyue Zhang; Wei Pan; Lianshan Yan; Bin Luo; Xihua Zou; Mingfeng Xu

doi:10.1364/OE.27.033369

1. Introduction

Chaos synchronization in delay-coupled semiconductor lasers (SLs) has received considerable interests for its potential applications in secure communications [1–6], high speed random number generators (RNGs) [7–10], generation of neuron-like dynamics [11], chaotic radar [12,13] and reservoir computing [14], etc. Different from the most of previous works that focused only on the synchronization patterns in simple scenarios with two or three SLs [15,16], there have some pioneering works extending the investigations to network realization of SLs and diverse synchronization patterns have been reported in complex SL networks [17–23]. More recently, cluster synchronization in SL networks, a new synchronization pattern in which SLs are divided into disjoint subsets based on the inherent symmetry of network topology, has obtained gradually intrests [20–23]. In cluster synchronization, SLs within same cluster can achieve isochronous synchronization and dynamics among different clusters are asynchronous [24,25]. However, the isochronous cluster synchronization regime in the vertical cavity surface-emitting laser (VCSEL) networks subjected to variable-polarization optical injection (VPOI) has never been reported and still deserve for further study.

From another viewpoint, the chaotic dynamics of delay-coupled SLs possess the detectable and recurrent features, which correspond to the optical round-trip time in the external cavity. As a result, this time delay (TD) signature leads to unavoidably serious security issues in the practical applications of chaos synchronization, as the chaotic dynamics of SLs correlated to themselves with a delay time. For example, the security of chaotic communication systems will degrade seriously if an eavesdropper can retrieve the TD signature [26]. Moreover, the TD signature induces recurrent information and thus resultantly reduce the randomness of RNGs [27] and it can also compromise the accuracy in chaotic ranging and chaotic radar. Unfortunately, it has been shown that the TD signature can be directly extracted by statistical analysis of the intensity time-series of lasers [28,29], and, what was even worse, the TD signature concealed successfully in the time domain also can be retrieved from the phase of laser emission [30]. Recently, more and more strategies have been proposed to suppress TD signature in two different ways. On the one hand, the TD signature can be deducted by modifying the structure of systems, such as the SLs with double optical feedback [31], dual-path optical injection [32], fiber Bragg grating feedback [33], and incoherent delayed self-interference of laser emission [34]. On the other hand, it can also be concealed by taking advantages of the interactions between different polarization modes of VCSELs [35,36].

Nevertheless, the studies on the TD signature reduction in network scenarios are still sorely lacked and most of related works are constrained to three lasers [37,38], greatly limiting the scope of practical applications of chaos synchronization. Here, we both theoretically and numerically investigate the isochronous cluster synchronization in complex VCSEL networks subject to VPOI with TD signature suppression. The parameter spaces for stable cluster synchronization are numerically studied, and then the TD signature reduction in the VPOI-VCSEL networks are proposed and discussed systematically. Moreover, the generality of our results are validated in a different topology of VPOI-VCSEL network.

2. Theoretical model

For the theoretical model, spin-flip model (SFM) is adopted and extended to the network scenarios by taking into account the delay-coupled VCSELs with VPOI as follows [35,39]:

(1)$$\begin{aligned}\dot{E_{mx}}=&k(1+i\alpha)\left[(N_m-1)E_{mx}+in_mE_{my}\right]-(\gamma_a+i\gamma_p)E_{mx}\\ &+\sigma\sum_{l=1}^{D_{s}}A_{ml}E_{lx}(t-\tau_{in})cos^2(\theta_{pl})e^{{-}i(w_{l}\tau_{in}+\Delta wt)}\\ &+\sigma\sum_{l=1}^{D_{s}}A_{ml}E_{ly}(t-\tau_{in})cos(\theta_{pl})sin(\theta_{pl})e^{{-}i(w_{l}\tau_{in}+\Delta wt)} \end{aligned}$$

(2)$$\begin{aligned} \dot{E_{my}}=&k(1+i\alpha)\left[(N_m-1)E_{my}-in_mE_{mx}\right]+(\gamma_a+i\gamma_p)E_{my}\\ &+\sigma\sum_{l=1}^{D_{s}}A_{ml}E_{lx}(t-\tau_{in})cos(\theta_{pl})sin(\theta_{pl})e^{{-}i(w_{l}\tau_{in}+\Delta wt)}\\ &+\sigma\sum_{l=1}^{D_{s}}A_{ml}E_{ly}(t-\tau_{in})sin^2(\theta_{pl})e^{{-}i(w_{l}\tau_{in}+\Delta wt)} \end{aligned}$$

(3)$$ \dot{N_{m}}=\gamma_N[\mu-N_m(1+|E_{mx}|^2+|E_{my}|^2)+in_m(E_{mx}E_{my}^*-E_{my}E_{mx}^*)]$$

(4)$$ \dot{n_{m}}=-\gamma_sn_m-\gamma_N[n_m(|E_{mx}|^2+|E_{my}|^2)+iN_m(E_{my}E_{mx}^*-E_{mx}E_{my}^*)] $$

where $E_x$ and $E_y$ denote the linear polarizations of the XP and YP components. $N$ is the total carrier inversion between conduction and valence bands, while $n$ accounts for the difference between carrier inversions with opposite spins. $A$ is the adjacency matrix that illustrates the topology of VCSEL network, $A_{ml}=1$ if $\textrm {VCSEL}_m$ is directly coupled to $\textrm {VCSEL}_l$, and $A_{ml}=0$ otherwise. $D_s$ represents the network size, and $D_s$=9 for the networks in Fig. 1. $\sigma$ is the uniform coupling strength among VCSELs, $\mu$ is the normalized current factor ($\mu =1~\textrm {corresponds to threshold current}$), $\alpha$ is the linewidth enhancement factor, $\tau _{in}=1.25\textrm {ns}$ is the coupling delay, $\theta _p$ is the variable polarizer angle (with respect to XP), and $w_m=2\pi c/\lambda _m$ is the central frequency of VCSEL with central wavelength $\lambda _m=850\textrm {nm}$. $\Delta w=2\pi \Delta f$ and $\Delta f=f_m-f_l$ represents the frequency detuning between VCSELs. The other typical VCSEL parameters include field decay rate $k=300\textrm {ns}^{-1}$, total carrier decay rate $\gamma _N=1\textrm {ns}^{-1}$, linear dichroism $\gamma _a=1\textrm {ns}^{-1}$, linear birefringence $\gamma _p=30\textrm {ns}^{-1}$, and spin-flip rate $\gamma _s=50\textrm {ns}^{-1}$ [35].

Fig. 1. Schematic diagrams of delay-coupled VPOI-VCSEL networks with two different network topologies.

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Figure 1 presents two different delay-coupled VPOI-VCSEL networks, and the same-colored VCSELs are classified into the same cluster or synchronizable sub-cluster. Mathematically, the VCSEL network can be described as a graph $g=(V(g),E(g))$, where $V(g)$ is the vertex set and $E(g)$ is the set of edges, and two vertices are adjacent if there is an edge between them. Then we can represent the network as an adjacency matrix, and the symmetry of network is a permutation of the vertices and make the adjacency matrix unchanged [40]. As VCSELs in same clusters are mapped into each other in the symmetry operations, we can separate the VCSELs in network into different clusters after all the permutations of the network vertices. As mentioned before, the permutation of VCSELs in same clusters will preserve the adjacency matrix of network unchanged, and thus make them having the same dynamical equations. Therefore, if VCSELs in same cluster start with same initial conditions, isochronous synchronization can be achieved indefinitely. Otherwise, the stability of isochronous cluster synchronization will depends on the parameters choice of VCSEL network for random initial conditions [24]. Moreover, the adjacency matrix A of the VCSEL network in Fig. 1(a) is presented as follows:

(5)$$A=\begin{pmatrix} 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\\$$

Based on the inherent symmetries of network topology, the network in Fig. 1(a) can be divided into three untrivial clusters that contain more than one VCSEL, which are defined as cluster I ($\textrm {VCSEL}_2$ and $\textrm {VCSEL}_3$), cluster II ($\textrm {VCSEL}_4$ , $\textrm {VCSEL}_5$, $\textrm {VCSEL}_6$, and $\textrm {VCSEL}_7$), and cluster III ($\textrm {VCSEL}_8$ and $\textrm {VCSEL}_9$). It is worth noting that there exists partial synchronization in cluster II for a wide range of parameter space. Therefore, we can further divide the cluster II into two sub-clusters, i. e. cluster II$_a$ ($\textrm {VCSEL}_4$ and $\textrm {VCSEL}_5$) and cluster II$_b$ ($\textrm {VCSEL}_6$ and $\textrm {VCSEL}_7$) for convenience.

3. Numerical results and discussions

The root-mean square (RMS) synchronization error is adopted to evaluate the synchronization quality of clusters in network. The values of RMS are obtained by the calculation between the intensity time series of VCSELs ($I_T=|E_x|^2+|E_y|^2$) in same cluster with random initial conditions as follows [20,24]:

(6)$$RMS=\frac{\sum_{m=1}^{D_{c}}\sqrt{\left \langle{I_{Tm}(t)-\hat{I}_{T}(t)}\right \rangle}}{D_{c}\hat{I}_{T}(t)}$$

where $D_c$ is the dimention of the clusters, $\left \langle \cdot \right \rangle$ denotes the time average, and $\hat {I}_{T}(t)=\sum _{m=1}^{D_{c}}I_{Tm}(t)/D_{c}$. The threshold value of RMS for stable isochronous cluster synchronization is set to be 0.01, which means that stable isochronous cluster synchronization is assumed to be achieved for $\textrm {RMS}\;<\;0.01$.

To explore the parameter spaces for stable isochronous cluster synchronization, Fig. 2 presents the values of RMS for different clusters in network as function of coupling strength $\sigma$, current factor $\mu$, polarizer angle $\theta _{p}$ for optical injection and linewidth enhancement factor $\alpha$ which is the internal parameter of VCSELs. It is shown that the stable isochronous cluster synchronization can be obtained in a wide range of parameters space, which validates that the topology of network plays an important role on the synchronization scheme of VPOI-VCSEL networks. Furthermore, there is an intra-cluster deviation for cluster II. With the modulation of parameters, the cluster splits into two sub-clusters, i. e. cluster II$_a$ ($\textrm {VCSEL}_4$ and $\textrm {VCSEL}_5$) and cluster II$_b$ ($\textrm {VCSEL}_6$ and $\textrm {VCSEL}_7$). As shown in Figs. 2a(2)–2a(4) and Figs. 2b(2)–2b(4), the parameter spaces of stable cluster synchronization for cluster II$_a$ and cluster II$_b$ are much wider than that for cluster II.

Fig. 2. a(1)-a(5) The values of RMS as functions of coupling strength $\sigma$ and current factor $\mu$ for different clusters in network with $\theta _{p}= 50^\circ$ and $\alpha =3$. b(1)-b(5) The values of RMS as function of linewidth enhancement factor $\alpha$ and polarizer angle $\theta _{p}$ with $\sigma =35\textrm {ns}^{-1}$ and $\mu =2.5$.

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Figure 3 presents the dynamical evolution and bifurcation for the intensity of VCSELs in cluster II. When linewidth enhancement factor $\alpha =1$, all the four VCSELs of cluster II synchronize isochronously as shown in Fig. 3a(1). For $\alpha =2$ (Fig. 3a(2)), cluster II has split into two smaller sub-clusters, each of which includes two VCSELs. And when $\alpha =3$ (Fig. 3a(3)), these four VCSELs lose synchrony eventually. Moreover, Figs. 3b(1)–3b(4) illustrate the intra-cluster deviation as function of network parameters systematically, which clearly show that there is a bifurcation of cluster II with the modulation of parameters in network. Meanwhile, cluster II$_a$ and cluster II$_b$ can still achieve stable isochronous synchronization in a wide range of parameter spaces after the intro-cluster bifurcation.

Fig. 3. The dynamical evolution of VCSELs in cluster II for different values of linewidth enhancement factor $\alpha$ with $\sigma =35\textrm {ns}^{-1}$, $\mu =2.5$ and $\theta _{p}= 50^\circ$, for $\alpha =1$ (a(1)), $\alpha =2$ (a(2)), and $\alpha =3$ (a(3)). b(1)-b(4) RMS values of cluster II, II$_a$ and II$_b$ (intra-cluster deviation) as function of $\alpha , \sigma , \mu$ and $\theta _p$.

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On the basis of synchronization in cluster II$_a$ and cluster II$_b$, we begin to demonstrate the simultaneous TD signature suppression of such VPOI-VCSEL netwok in the following parts. To estimate TD signatures both in the intensity and phase series of VCSELs in cluster II$_a$ and cluster II$_b$, time-dependent auto-correlation function (ACF) is induced and defined as follows [28,30]:

(7)$$C_T= \frac{\Big\langle\left[I_{T}(t)-\left\langle{I_{T}(t)}\right\rangle\right]\cdot{\left[I_{T}(t+{\Delta} t)-\left\langle{I_{T}(t+{\Delta} t)}\right\rangle\right]}\Big\rangle}{\sqrt{\left\langle\left[I_{T}(t)-\left\langle{I_{T}(t)}\right\rangle\right]^2\right\rangle\cdot{\left\langle\left[I_{T}(t+{\Delta} t)-\left\langle{I_{T}(t+{\Delta} t)}\right\rangle\right]^2\right\rangle}}}$$

where $\langle {\cdot } \rangle$ denotes time average, ${\Delta} t \in [-5, 5]~\textrm {ns}$ denotes the lag time, and the time series used for calculation are selected within $t=[40,450]~\textrm {ns}$, which is long enough to keep transients extinct. $I_{T}$ denotes the total outputs of VCSELs and will be changed to the phase series to calculate $C_T (\varphi )$. For a given value of $\Delta t$, the ACF measures a linear relationship between $I_{T} (t)$ and $I_{T} (t+{\Delta} t)$.

The dynamical evolution and TD signature identification (both in intensity and phase domain) of VCSELs in cluster II$_a$ and cluster II$_b$ with different values of frequency detuning $\Delta f$ are presented in Fig. 4. Here, $\Delta f$ denotes the frequency detuning between VCSELs within cluster I and the other clusters. It can be seen from Fig. 4b(2) and Fig. 4c(2) that, there exist peak values at $\Delta t$=2.5 ns, which corresponds to the optical round-trip time in the external cavity, i. e. two times of the coupling delay $2\times \tau _{in}$=2.5 ns with $\Delta f$=0 GHz. Moreover, TD signatures are significantly suppressed both in intensity and phase by inducing frequency detuning.

Fig. 4. Intensity time series of cluster II$_a$ and II$_b$ (a(1)-a(3)); ACF of intensity (b(1)-b(3)) and phase series (c(1)-c(3)) for different frequency detuning $\Delta f$; RMS, $R_c$ and $R_C (\varphi )$ as function of frequency detuning $\Delta f$ (d(1)-d(3)), with $\alpha =2$, $\sigma =20\textrm {ns}^{-1}$, $\mu =2.5$ and $\theta _{p}= 50^\circ$.

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Furthermore, in order to evaluate the TD signature quantitatively, the quantity peak signal to mean ratio are introduced [35,41]:

(8)$$R_C= \frac{\textrm{max}(|C_T|)}{\left\langle |C_T (\Delta t)|\right\rangle}$$

where max($|C_T|$) is the maximum value of $|C_T|$ in the vicinity of TD signature $\tau$, and $\left \langle |C_T (\Delta t)|\right \rangle$ denotes the mean value. The $R_C$ is calculated in the vicinity of optical round trip $\tau$=2.5 ns and the maximum value of $C_T$ at $\Delta t$=0 is excluded. Figs. 4d(1)–4d(3) present the RMS, $R_C$, and $R_C (\varphi )$ as a function of frequency detuning, respectively. The results clearly show that, the synchronization of VCSELs in cluster II$_a$ and II$_b$ are still preserved with the existence of frequency detuning, while TD signature reduction can be remarkably improved with the introduction of frequency detuning between clusters.

We further investigate the influence of polarizer angle on the TD signature suppression. The dynamical evolution and calculation of $C_T$ and $C_T (\varphi )$ of VCSELs in cluster II$_a$ and II$_b$ are presented in Fig. 5 with three distinct choices of $\theta _p$, which obviously indicate that the TD signature can be successfully concealed both in the intensity and phase series with moderate optical injection polarizer angle. Furthermore, the values of RMS, $R_C$ and $R_C (\varphi )$ as function of $\theta _p$ are further calculated in Figs. 6a(1)–6a(3), respectively. It is shown that, the TD signature is quite sensitive to polarizer angle and the values of $R_C$ achieve its minimum value at intermediate polarizer angles. The low values of $R_C$ for intermediate polarizer angle can be explained by the evolution of polarization states of VCSEL in network. To provide physical insight into the suppression of TD signature at critical values of $\theta _p$, Fig. 6(b) shows the Poincar$\acute {\textrm {e}}$ sphere for the dynamics of $\textrm {VCSEL}_4$ with $\theta _p=2^\circ$ and $\theta _p=50^\circ$. It can be seen that, the dynamics of XP mode are in dominant for $\theta _p=2^\circ$ as most of the points reside in the X-axis. However, for the situation of $\theta _p=50^\circ$, there is no dominant polarization mode and points spread all over the sphere. Hence, the interaction between the two polarization modes contributes to the suppression of TD signature.

Fig. 5. Intensity time series of cluster II$_a$ and II$_b$ (a(1)-a(3)); ACF of intensity (b(1)-b(3)) and phase series (c(1)-c(3)) for different polarizer angles $\theta _p$ with $\sigma =20\textrm {ns}^{-1}, \alpha =2, \mu =2.5$ and $\Delta f=35\textrm {GHz}$.

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Fig. 6. a(1)-a(3) The RMS, $R_C$ and $R_C (\varphi )$ of VCSELs in cluster II$_a$ and II$_b$ as function of optical injection polarizer angle $\theta _p$ with $\sigma =20\textrm {ns}^{-1}, \alpha =2, \mu =2.5$ and $\Delta f=35\textrm {GHz}$. b(1)-b(2) Evolution of polarization states described by the Poincar$\acute {\textrm {e}}$ sphere for delay-coupled VCSEL network with different polarizer angles $\theta _p$.

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Finally, the generality of our proposed results are validated in another delay-coupled VPOI-VCSEL network with different topology (Fig. 1(b)). Figure 7 show the dynamical evolution of the cluster containing $\textrm {VCSEL}_{6, 7, 8, 9}$ and the corresponding ACF calculated from intensity and phase series, respectively. It is demonstrated that, again, the isochronous synchronization of two sub-clusters and TD signature suppression are simultaneously implemented, which indicates that our result are applicable to different network topologies.

Fig. 7. Dynamical evolution (a) and calculation of ACF in intensity (b) and phase (c) series for VCSELs in network of Fig. 1(b), with $\sigma =20\textrm {ns}^{-1}, \alpha =2, \mu =2.5, \Delta f=35\textrm {GHz}$, and $\theta _p=50^\circ$.

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4. Conclusion

In conclusion, the isochronous cluster synchronization and TD signature suppression are achieved simultaneously in delay-coupled VCSEL networks subjected to VPOI. The influence of network parameters on the stability of cluster synchronization and TD signature reduction are investigated systematically. The generality of proposed results are validated in different topologies of VCSEL network. Our results offer a new insight on the chaos-based applications in VCSEL networks.

Funding

National Natural Science Foundation of China (61775185); Sichuan Province Science and Technology Support Program (2018HH0002, 2019JDJQ0022); the "111" Plan (B18045).

Disclosures

The authors declare no conflicts of interest.

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Isochronous cluster synchronization in delay-coupled VCSEL networks subjected to variable-polarization optical injection with time delay signature suppression

Abstract

1. Introduction

2. Theoretical model

3. Numerical results and discussions

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (7)

Equations (8)

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