Abstract

The isochronous cluster synchronization with time delay (TD) signature suppression in delay-coupled vertical-cavity surface-emitting laser (VCSEL) networks subject to variable-polarization optical injection (VPOI) is theoretically and numerically studied. Based on the inherent symmetries of network topology, parameter spaces for stable cluster synchronization are presented, and zero-lag synchronization are achieved for VCSELs in same clusters. Additionally, the TD signature reduction for the dynamics of VCSELs in the stable clusters are systematically discussed. It is shown that both moderate polarizer angle and frequency detuning between different clusters have strengthen the effect of TD signature suppression. Moreover, the isochronous cluster synchronization with TD signature concealment is also verified in another VPOI-VCSEL network with different topology, indicating the generality of proposed results. Our results shed a new light on the research of chaos synchronization and chaos-based secure communications in VCSEL networks.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Chaos synchronization in delay-coupled semiconductor lasers (SLs) has received considerable interests for its potential applications in secure communications [16], high speed random number generators (RNGs) [710], 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 [1723]. 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 [2023]. 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]:

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

$$A=\begin{pmatrix} 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\\$$
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]:

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

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

$$R_C= \frac{\textrm{max}(|C_T|)}{\left\langle |C_T (\Delta t)|\right\rangle}$$
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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37. N. Q. Li, W. Pan, S. Y. Xiang, L. S. Yan, B. Luo, and X. H. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photonics Technol. Lett. 24(23), 2187–2190 (2012). [CrossRef]  

38. Y. Liu, Y. Xie, Y. Ye, J. Zhang, S. Wang, Y. Liu, G. Pan, and J. Zhang, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017). [CrossRef]  

39. J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997). [CrossRef]  

40. B. D. MacArthur, R. J. Sánchez García, and J. W. Anderson, “Symmetry in complex networks,” Discret. Appl. Math. 156(18), 3525–3531 (2008). [CrossRef]  

41. L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, S. Y. Xiang, and N. Q. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photonics Technol. Lett. 24(19), 1693–1695 (2012). [CrossRef]  

References

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  4. P. Li, Q. Cai, J. G. Zhang, B. J. Xu, Y. M. Liu, A. Bogris, K. A. Shore, and Y. C. Wang, “Observation of flat chaos generation using an optical feedback multi-mode laser with a band-pass filter,” Opt. Express 27(13), 17859–17867 (2019).
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  7. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
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  21. Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, “Synchronization in small networks of time-delay coupled chaotic diode lasers,” Opt. Express 20(4), 4352–4359 (2012).
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  25. L. Y. Zhang, A. E. Motter, and T. Nishikawa, “Incoherence-mediated remote synchronization,” Phys. Rev. Lett. 118(17), 174102 (2017).
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  26. V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
    [Crossref]
  27. K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
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  28. D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32(20), 2960–2962 (2007).
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  29. L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
    [Crossref]
  30. R. M. Nguimdo, M. C. Soriano, and P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. 36(22), 4332–4334 (2011).
    [Crossref]
  31. J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009).
    [Crossref]
  32. S. Y. Xiang, W. Pan, A. J. Wen, N. Q. Li, L. Y. Zhang, L. Shang, and H. X. Zhang, “Conceal time delay signature of chaos in semiconductor lasers with dual-path injection,” IEEE Photonics Technol. Lett. 25(14), 1398–1401 (2013).
    [Crossref]
  33. S. S. Li and S. J. Chan, “Chaotic time-delay signature suppression in a semiconductor laser with frequency-detuned grating feedback,” IEEE J. Sel. Top. Quantum Electron. 21(6), 541–552 (2015).
    [Crossref]
  34. A. B. Wang, Y. B. Yang, B. J. Wang, B. B. Zhang, L. Li, and Y. C. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
    [Crossref]
  35. S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and H. N. Zhu, “Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback,” Opt. Commun. 284(24), 5758–5765 (2011).
    [Crossref]
  36. Y. Hong, “Experimental study of time-delay signature of chaos in mutually coupled vertical-cavity surface-emitting lasers subject to polarization optical injection,” Opt. Express 21(15), 17894–17903 (2013).
    [Crossref]
  37. N. Q. Li, W. Pan, S. Y. Xiang, L. S. Yan, B. Luo, and X. H. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photonics Technol. Lett. 24(23), 2187–2190 (2012).
    [Crossref]
  38. Y. Liu, Y. Xie, Y. Ye, J. Zhang, S. Wang, Y. Liu, G. Pan, and J. Zhang, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
    [Crossref]
  39. J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
    [Crossref]
  40. B. D. MacArthur, R. J. Sánchez García, and J. W. Anderson, “Symmetry in complex networks,” Discret. Appl. Math. 156(18), 3525–3531 (2008).
    [Crossref]
  41. L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, S. Y. Xiang, and N. Q. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photonics Technol. Lett. 24(19), 1693–1695 (2012).
    [Crossref]

2019 (4)

N. Jiang, A. K. Zhao, C. P. Xue, J. M. Tang, and K. Qiu, “Physical secure optical communication based on private chaotic spectral phase encryption/decryption,” Opt. Lett. 44(7), 1536–1539 (2019).
[Crossref]

P. Li, Q. Cai, J. G. Zhang, B. J. Xu, Y. M. Liu, A. Bogris, K. A. Shore, and Y. C. Wang, “Observation of flat chaos generation using an optical feedback multi-mode laser with a band-pass filter,” Opt. Express 27(13), 17859–17867 (2019).
[Crossref]

X. H. Zou, F. Zou, Z. Z. Cao, B. Lu, X. L. Yan, G. Yu, X. Deng, B. Luo, L. S. Yan, W. Pan, J. P. Yao, and A. M. J. Koonen, “A multifunctional photonic integrated circuit for diverse microwave signal generation, transmission, and processing,” Laser Photonics Rev. 13, 1800240 (2019).
[Crossref]

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Cluster synchronization of coupled semiconductor lasers network with complex topology,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–7 (2019).
[Crossref]

2018 (4)

2017 (6)

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

D. Z. Zhong, G. L. Xu, W. Luo, and Z. Z. Xiao, “Real-time multi-target ranging based on chaotic polarization laser radars in the drive-response VCSELs,” Opt. Express 25(18), 21684–21704 (2017).
[Crossref]

J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7(1), 42116 (2017).
[Crossref]

J. Shena, J. Hizanidis, P. Hövel, and G. Tsironis, “Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions,” Phys. Rev. E 96(3), 032215 (2017).
[Crossref]

L. Y. Zhang, A. E. Motter, and T. Nishikawa, “Incoherence-mediated remote synchronization,” Phys. Rev. Lett. 118(17), 174102 (2017).
[Crossref]

Y. Liu, Y. Xie, Y. Ye, J. Zhang, S. Wang, Y. Liu, G. Pan, and J. Zhang, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
[Crossref]

2015 (3)

S. S. Li and S. J. Chan, “Chaotic time-delay signature suppression in a semiconductor laser with frequency-detuned grating feedback,” IEEE J. Sel. Top. Quantum Electron. 21(6), 541–552 (2015).
[Crossref]

F. Böhm, A. Zakharova, E. Schöll, and K. Lüdge, “Amplitude-phase coupling drives chimera states in globally coupled laser networks,” Phys. Rev. E 91(4), 040901 (2015).
[Crossref]

C. P. Xue, N. Jiang, K. Qiu, and Y. X. Lv, “Key distribution based on synchronization in bandwidth-enhanced random bit generators with dynamic post-processing,” Opt. Express 23(11), 14510–14519 (2015).
[Crossref]

2014 (1)

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5(1), 4079 (2014).
[Crossref]

2013 (6)

M. Bourmpos, A. Argyris, and D. Syvridis, “Analysis of the bubbling effect in synchronized networks with semiconductor lasers,” IEEE Photonics Technol. Lett. 25(9), 817–820 (2013).
[Crossref]

S. Y. Xiang, W. Pan, A. J. Wen, N. Q. Li, L. Y. Zhang, L. Shang, and H. X. Zhang, “Conceal time delay signature of chaos in semiconductor lasers with dual-path injection,” IEEE Photonics Technol. Lett. 25(14), 1398–1401 (2013).
[Crossref]

N. Q. Li, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and S. Y. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quantum Electron. 19(4), 0600109 (2013).
[Crossref]

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

A. B. Wang, Y. B. Yang, B. J. Wang, B. B. Zhang, L. Li, and Y. C. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
[Crossref]

Y. Hong, “Experimental study of time-delay signature of chaos in mutually coupled vertical-cavity surface-emitting lasers subject to polarization optical injection,” Opt. Express 21(15), 17894–17903 (2013).
[Crossref]

2012 (3)

N. Q. Li, W. Pan, S. Y. Xiang, L. S. Yan, B. Luo, and X. H. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photonics Technol. Lett. 24(23), 2187–2190 (2012).
[Crossref]

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, S. Y. Xiang, and N. Q. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photonics Technol. Lett. 24(19), 1693–1695 (2012).
[Crossref]

Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, “Synchronization in small networks of time-delay coupled chaotic diode lasers,” Opt. Express 20(4), 4352–4359 (2012).
[Crossref]

2011 (2)

R. M. Nguimdo, M. C. Soriano, and P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. 36(22), 4332–4334 (2011).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and H. N. Zhu, “Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback,” Opt. Commun. 284(24), 5758–5765 (2011).
[Crossref]

2010 (1)

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

2009 (3)

J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009).
[Crossref]

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
[Crossref]

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79(2), 026210 (2009).
[Crossref]

2008 (3)

R. Vicente, I. Fischer, and C. R. Mirasso, “Synchronization properties of three delay-coupled semiconductor lasers,” Phys. Rev. E 78(6), 066202 (2008).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

B. D. MacArthur, R. J. Sánchez García, and J. W. Anderson, “Symmetry in complex networks,” Discret. Appl. Math. 156(18), 3525–3531 (2008).
[Crossref]

2007 (1)

2005 (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazz-Lodi, P. Colet, I. Fischer, J. Garcia Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

2004 (1)

F. Y. Lin and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004).
[Crossref]

2003 (1)

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[Crossref]

1997 (1)

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[Crossref]

Abraham, N. B.

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[Crossref]

Amano, K.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Anderson, J. W.

B. D. MacArthur, R. J. Sánchez García, and J. W. Anderson, “Symmetry in complex networks,” Discret. Appl. Math. 156(18), 3525–3531 (2008).
[Crossref]

Annovazz-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazz-Lodi, P. Colet, I. Fischer, J. Garcia Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Argyris, A.

M. Bourmpos, A. Argyris, and D. Syvridis, “Analysis of the bubbling effect in synchronized networks with semiconductor lasers,” IEEE Photonics Technol. Lett. 25(9), 817–820 (2013).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazz-Lodi, P. Colet, I. Fischer, J. Garcia Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Aviad, Y.

Bai, W. L.

Bogris, A.

Böhm, F.

F. Böhm, A. Zakharova, E. Schöll, and K. Lüdge, “Amplitude-phase coupling drives chimera states in globally coupled laser networks,” Phys. Rev. E 91(4), 040901 (2015).
[Crossref]

Bourmpos, M.

M. Bourmpos, A. Argyris, and D. Syvridis, “Analysis of the bubbling effect in synchronized networks with semiconductor lasers,” IEEE Photonics Technol. Lett. 25(9), 817–820 (2013).
[Crossref]

Brunner, D.

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

Cai, Q.

Cao, Z. Z.

X. H. Zou, F. Zou, Z. Z. Cao, B. Lu, X. L. Yan, G. Yu, X. Deng, B. Luo, L. S. Yan, W. Pan, J. P. Yao, and A. M. J. Koonen, “A multifunctional photonic integrated circuit for diverse microwave signal generation, transmission, and processing,” Laser Photonics Rev. 13, 1800240 (2019).
[Crossref]

Chan, S. J.

S. S. Li and S. J. Chan, “Chaotic time-delay signature suppression in a semiconductor laser with frequency-detuned grating feedback,” IEEE J. Sel. Top. Quantum Electron. 21(6), 541–552 (2015).
[Crossref]

Chen, W.

Citrin, D. S.

Colet, P.

R. M. Nguimdo, M. C. Soriano, and P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. 36(22), 4332–4334 (2011).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazz-Lodi, P. Colet, I. Fischer, J. Garcia Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Cuenot, J. B.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[Crossref]

Davis, P.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
[Crossref]

Deng, X.

X. H. Zou, F. Zou, Z. Z. Cao, B. Lu, X. L. Yan, G. Yu, X. Deng, B. Luo, L. S. Yan, W. Pan, J. P. Yao, and A. M. J. Koonen, “A multifunctional photonic integrated circuit for diverse microwave signal generation, transmission, and processing,” Laser Photonics Rev. 13, 1800240 (2019).
[Crossref]

Dubrova, E.

P. Li, Y. Guo, Y. Q. Guo, Y. L. Fan, X. M. Guo, X. L. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. C. Wang, and A. B. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

Escalona-Morán, M. A.

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

Fan, Y.

Fan, Y. L.

P. Li, Y. Guo, Y. Q. Guo, Y. L. Fan, X. M. Guo, X. L. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. C. Wang, and A. B. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

Fischer, I.

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

R. Vicente, I. Fischer, and C. R. Mirasso, “Synchronization properties of three delay-coupled semiconductor lasers,” Phys. Rev. E 78(6), 066202 (2008).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazz-Lodi, P. Colet, I. Fischer, J. Garcia Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Garcia Ojalvo, J.

A. Argyris, D. Syvridis, L. Larger, V. Annovazz-Lodi, P. Colet, I. Fischer, J. Garcia Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Goedgebuer, J. P.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[Crossref]

Guo, X.

Guo, X. M.

P. Li, Y. Guo, Y. Q. Guo, Y. L. Fan, X. M. Guo, X. L. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. C. Wang, and A. B. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

Guo, X. X.

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Guo, Y.

Guo, Y. Q.

P. Li, Y. Guo, Y. Q. Guo, Y. L. Fan, X. M. Guo, X. L. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. C. Wang, and A. B. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

Hagerstrom, A. M.

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5(1), 4079 (2014).
[Crossref]

Hao, Y.

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Hicke, K.

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

Hirano, K.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Hizanidis, J.

J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7(1), 42116 (2017).
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J. Shena, J. Hizanidis, P. Hövel, and G. Tsironis, “Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions,” Phys. Rev. E 96(3), 032215 (2017).
[Crossref]

Hong, Y.

Hövel, P.

J. Shena, J. Hizanidis, P. Hövel, and G. Tsironis, “Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions,” Phys. Rev. E 96(3), 032215 (2017).
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Inoue, M.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Jiang, N.

Kanter, I.

Koonen, A. M. J.

X. H. Zou, F. Zou, Z. Z. Cao, B. Lu, X. L. Yan, G. Yu, X. Deng, B. Luo, L. S. Yan, W. Pan, J. P. Yao, and A. M. J. Koonen, “A multifunctional photonic integrated circuit for diverse microwave signal generation, transmission, and processing,” Laser Photonics Rev. 13, 1800240 (2019).
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Kovanis, V.

J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7(1), 42116 (2017).
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Kurashige, T.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Larger, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazz-Lodi, P. Colet, I. Fischer, J. Garcia Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[Crossref]

Levy, P.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[Crossref]

Li, J. F.

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Li, K.

Li, L.

Li, N. Q.

S. Y. Xiang, W. Pan, A. J. Wen, N. Q. Li, L. Y. Zhang, L. Shang, and H. X. Zhang, “Conceal time delay signature of chaos in semiconductor lasers with dual-path injection,” IEEE Photonics Technol. Lett. 25(14), 1398–1401 (2013).
[Crossref]

N. Q. Li, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and S. Y. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quantum Electron. 19(4), 0600109 (2013).
[Crossref]

N. Q. Li, W. Pan, S. Y. Xiang, L. S. Yan, B. Luo, and X. H. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photonics Technol. Lett. 24(23), 2187–2190 (2012).
[Crossref]

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, S. Y. Xiang, and N. Q. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photonics Technol. Lett. 24(19), 1693–1695 (2012).
[Crossref]

Li, P.

Li, P. X.

Li, S. S.

S. S. Li and S. J. Chan, “Chaotic time-delay signature suppression in a semiconductor laser with frequency-detuned grating feedback,” IEEE J. Sel. Top. Quantum Electron. 21(6), 541–552 (2015).
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Lin, F. Y.

F. Y. Lin and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004).
[Crossref]

Liu, J. M.

F. Y. Lin and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004).
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Liu, X.

Liu, X. L.

P. Li, Y. Guo, Y. Q. Guo, Y. L. Fan, X. M. Guo, X. L. Liu, K. A. Shore, E. Dubrova, B. Xu, Y. C. Wang, and A. B. Wang, “Self-balanced real-time photonic scheme for ultrafast random number generation,” APL Photonics 3(6), 061301 (2018).
[Crossref]

Liu, Y.

Y. Liu, Y. Xie, Y. Ye, J. Zhang, S. Wang, Y. Liu, G. Pan, and J. Zhang, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
[Crossref]

Y. Liu, Y. Xie, Y. Ye, J. Zhang, S. Wang, Y. Liu, G. Pan, and J. Zhang, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
[Crossref]

Liu, Y. M.

Locquet, A.

Lu, B.

X. H. Zou, F. Zou, Z. Z. Cao, B. Lu, X. L. Yan, G. Yu, X. Deng, B. Luo, L. S. Yan, W. Pan, J. P. Yao, and A. M. J. Koonen, “A multifunctional photonic integrated circuit for diverse microwave signal generation, transmission, and processing,” Laser Photonics Rev. 13, 1800240 (2019).
[Crossref]

X. H. Zou, W. L. Bai, W. Chen, P. X. Li, B. Lu, G. Yu, W. Pan, B. Luo, L. S. Yan, and L. Y. Shao, “Microwave photonics for featured applications in high-speed railways: Communications, detection, and sensing,” J. Lightwave Technol. 36(19), 4337–4346 (2018).
[Crossref]

Lüdge, K.

F. Böhm, A. Zakharova, E. Schöll, and K. Lüdge, “Amplitude-phase coupling drives chimera states in globally coupled laser networks,” Phys. Rev. E 91(4), 040901 (2015).
[Crossref]

Luo, B.

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Cluster synchronization of coupled semiconductor lasers network with complex topology,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–7 (2019).
[Crossref]

X. H. Zou, F. Zou, Z. Z. Cao, B. Lu, X. L. Yan, G. Yu, X. Deng, B. Luo, L. S. Yan, W. Pan, J. P. Yao, and A. M. J. Koonen, “A multifunctional photonic integrated circuit for diverse microwave signal generation, transmission, and processing,” Laser Photonics Rev. 13, 1800240 (2019).
[Crossref]

X. H. Zou, W. L. Bai, W. Chen, P. X. Li, B. Lu, G. Yu, W. Pan, B. Luo, L. S. Yan, and L. Y. Shao, “Microwave photonics for featured applications in high-speed railways: Communications, detection, and sensing,” J. Lightwave Technol. 36(19), 4337–4346 (2018).
[Crossref]

N. Q. Li, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and S. Y. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quantum Electron. 19(4), 0600109 (2013).
[Crossref]

N. Q. Li, W. Pan, S. Y. Xiang, L. S. Yan, B. Luo, and X. H. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photonics Technol. Lett. 24(23), 2187–2190 (2012).
[Crossref]

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, S. Y. Xiang, and N. Q. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photonics Technol. Lett. 24(19), 1693–1695 (2012).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and H. N. Zhu, “Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback,” Opt. Commun. 284(24), 5758–5765 (2011).
[Crossref]

Luo, W.

Lv, Y. X.

MacArthur, B. D.

B. D. MacArthur, R. J. Sánchez García, and J. W. Anderson, “Symmetry in complex networks,” Discret. Appl. Math. 156(18), 3525–3531 (2008).
[Crossref]

Mihara, T.

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79(2), 026210 (2009).
[Crossref]

Mirasso, C. R.

K. Hicke, M. A. Escalona-Morán, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1501610 (2013).
[Crossref]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

R. Vicente, I. Fischer, and C. R. Mirasso, “Synchronization properties of three delay-coupled semiconductor lasers,” Phys. Rev. E 78(6), 066202 (2008).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazz-Lodi, P. Colet, I. Fischer, J. Garcia Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

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L. Y. Zhang, A. E. Motter, and T. Nishikawa, “Incoherence-mediated remote synchronization,” Phys. Rev. Lett. 118(17), 174102 (2017).
[Crossref]

Murphy, T. E.

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5(1), 4079 (2014).
[Crossref]

Naito, S.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Nguimdo, R. M.

Nishikawa, T.

L. Y. Zhang, A. E. Motter, and T. Nishikawa, “Incoherence-mediated remote synchronization,” Phys. Rev. Lett. 118(17), 174102 (2017).
[Crossref]

Oowada, I.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Ozaki, M.

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79(2), 026210 (2009).
[Crossref]

Pan, G.

Y. Liu, Y. Xie, Y. Ye, J. Zhang, S. Wang, Y. Liu, G. Pan, and J. Zhang, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
[Crossref]

Pan, W.

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Cluster synchronization of coupled semiconductor lasers network with complex topology,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–7 (2019).
[Crossref]

X. H. Zou, F. Zou, Z. Z. Cao, B. Lu, X. L. Yan, G. Yu, X. Deng, B. Luo, L. S. Yan, W. Pan, J. P. Yao, and A. M. J. Koonen, “A multifunctional photonic integrated circuit for diverse microwave signal generation, transmission, and processing,” Laser Photonics Rev. 13, 1800240 (2019).
[Crossref]

X. H. Zou, W. L. Bai, W. Chen, P. X. Li, B. Lu, G. Yu, W. Pan, B. Luo, L. S. Yan, and L. Y. Shao, “Microwave photonics for featured applications in high-speed railways: Communications, detection, and sensing,” J. Lightwave Technol. 36(19), 4337–4346 (2018).
[Crossref]

M. F. Xu, W. Pan, S. Y. Xiang, and L. Y. Zhang, “Cluster synchronization in symmetric VCSELs networks with variable-polarization optical feedback,” Opt. Express 26(8), 10754–10761 (2018).
[Crossref]

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

N. Q. Li, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and S. Y. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quantum Electron. 19(4), 0600109 (2013).
[Crossref]

S. Y. Xiang, W. Pan, A. J. Wen, N. Q. Li, L. Y. Zhang, L. Shang, and H. X. Zhang, “Conceal time delay signature of chaos in semiconductor lasers with dual-path injection,” IEEE Photonics Technol. Lett. 25(14), 1398–1401 (2013).
[Crossref]

N. Q. Li, W. Pan, S. Y. Xiang, L. S. Yan, B. Luo, and X. H. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photonics Technol. Lett. 24(23), 2187–2190 (2012).
[Crossref]

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, S. Y. Xiang, and N. Q. Li, “Conceal time-delay signature of mutually coupled vertical-cavity surface-emitting lasers by variable polarization optical injection,” IEEE Photonics Technol. Lett. 24(19), 1693–1695 (2012).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and H. N. Zhu, “Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback,” Opt. Commun. 284(24), 5758–5765 (2011).
[Crossref]

Panajotov, K.

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79(2), 026210 (2009).
[Crossref]

Pecora, L. M.

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5(1), 4079 (2014).
[Crossref]

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazz-Lodi, P. Colet, I. Fischer, J. Garcia Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
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Prati, F.

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
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Qiu, K.

Regalado, J. M.

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[Crossref]

Reidler, I.

Rhodes, W. T.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[Crossref]

Rontani, D.

Rosenbluh, M.

Rosso, O. A.

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

Roy, R.

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5(1), 4079 (2014).
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San Miguel, M.

J. M. Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[Crossref]

Sánchez García, R. J.

B. D. MacArthur, R. J. Sánchez García, and J. W. Anderson, “Symmetry in complex networks,” Discret. Appl. Math. 156(18), 3525–3531 (2008).
[Crossref]

Schöll, E.

F. Böhm, A. Zakharova, E. Schöll, and K. Lüdge, “Amplitude-phase coupling drives chimera states in globally coupled laser networks,” Phys. Rev. E 91(4), 040901 (2015).
[Crossref]

Sciamanna, M.

M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. E 79(2), 026210 (2009).
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D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32(20), 2960–2962 (2007).
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Shang, L.

S. Y. Xiang, W. Pan, A. J. Wen, N. Q. Li, L. Y. Zhang, L. Shang, and H. X. Zhang, “Conceal time delay signature of chaos in semiconductor lasers with dual-path injection,” IEEE Photonics Technol. Lett. 25(14), 1398–1401 (2013).
[Crossref]

Shao, L. Y.

Shena, J.

J. Shena, J. Hizanidis, V. Kovanis, and G. P. Tsironis, “Turbulent chimeras in large semiconductor laser arrays,” Sci. Rep. 7(1), 42116 (2017).
[Crossref]

J. Shena, J. Hizanidis, P. Hövel, and G. Tsironis, “Multiclustered chimeras in large semiconductor laser arrays with nonlocal interactions,” Phys. Rev. E 96(3), 032215 (2017).
[Crossref]

Shiki, M.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, and S. Yoshimori, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
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Shore, K. A.

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S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and H. N. Zhu, “Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback,” Opt. Commun. 284(24), 5758–5765 (2011).
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[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. Schematic diagrams of delay-coupled VPOI-VCSEL networks with two different network topologies.
Fig. 2.
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 $\mu =2.5$.
Fig. 3.
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 $\theta _p$.
Fig. 4.
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 $\theta _{p}= 50^\circ$.
Fig. 5.
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 $\Delta f=35\textrm {GHz}$.
Fig. 6.
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 $\theta _p$.
Fig. 7.
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 $\theta _p=50^\circ$.

Equations (8)

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E m x ˙ = k ( 1 + i α ) [ ( N m 1 ) E m x + i n m E m y ] ( γ a + i γ p ) E m x + σ l = 1 D s A m l E l x ( t τ i n ) c o s 2 ( θ p l ) e i ( w l τ i n + Δ w t ) + σ l = 1 D s A m l E l y ( t τ i n ) c o s ( θ p l ) s i n ( θ p l ) e i ( w l τ i n + Δ w t )
E m y ˙ = k ( 1 + i α ) [ ( N m 1 ) E m y i n m E m x ] + ( γ a + i γ p ) E m y + σ l = 1 D s A m l E l x ( t τ i n ) c o s ( θ p l ) s i n ( θ p l ) e i ( w l τ i n + Δ w t ) + σ l = 1 D s A m l E l y ( t τ i n ) s i n 2 ( θ p l ) e i ( w l τ i n + Δ w t )
N m ˙ = γ N [ μ N m ( 1 + | E m x | 2 + | E m y | 2 ) + i n m ( E m x E m y E m y E m x ) ]
n m ˙ = γ s n m γ N [ n m ( | E m x | 2 + | E m y | 2 ) + i N m ( E m y E m x E m x E m y ) ]
A = ( 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 )
R M S = m = 1 D c I T m ( t ) I ^ T ( t ) D c I ^ T ( t )
C T = [ I T ( t ) I T ( t ) ] [ I T ( t + Δ t ) I T ( t + Δ t ) ] [ I T ( t ) I T ( t ) ] 2 [ I T ( t + Δ t ) I T ( t + Δ t ) ] 2
R C = max ( | C T | ) | C T ( Δ t ) |

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