Cluster synchronization in symmetric VCSELs networks with variable-polarization optical feedback

Mingfeng Xu; Wei Pan; Shuiying Xiang; Liyue Zhang

doi:10.1364/OE.26.010754

1. Introduction

As an excellent platform satisfied for different applications, the chaos synchronization in mutually coupled semiconductor lasers (SLs) has attracted considerable interests in the past decades [1–4]. On the other hand, the cluster synchronization in delay-coupled networks with chaotic units is an emergent phenomenon in nonlinearly complex dynamical systems. Recently, the research on such synchronization phenomenon has rapidly grown [5–8], more especially for networks with coupled SLs [9–19]. For instance, Röhm et al. discussed the existence of chimera states in a globally delay-coupled network of four lasers [9]. Bourmpos et al. explored the bubbling effect of dynamical behavior in a well-synchronized SLs network with star topology [10]. Xiang et al. numerically investigated the synchronization regime of star-type laser network with heterogeneous coupling delays [11]. Williams et al. experimentally studied the synchronization states in a ring of four identical lasers [15]. Argyris et al. experimentally investigated the synchrony of SLs in coupled networks [16]. Nixon et al. experimentally investigated the cluster synchronization in coupled laser networks, in which they found that lasers can be synchronized within same cluster and there is no synchronization amongst the clusters [17].

Compared to conventional edge-emitting semiconductor lasers, vertical-cavity surface-emitting lasers (VCSELs) exhibit several desirable characteristics, such as single longitudinal-mode operation, small threshold current, circular beam profile with narrow divergence, low cost, wafer-scale testing and easy large-scale integration into two-dimensional arrays [20,21]. These prominent advantages have stimulated a wide variety of applications, such as chaos-based secure communications [22,23], chaotic radar [24] and high speed random bit generators [25]. Moreover, the synchronization between coupled VCSELs has been investigated theoretically and experimentally. For instance, Gatare et. al. and Sciamanna et. al. theoretically discussed the polarization synchronization in unidirectionally coupled VCSELs [26, 27], Ozaki et. al. investigated the chaos synchronization in mutually coupled VCSELs with time delay [28], and Uchida et. al. and Hong et. al. experimentally demonstrated the synchronization in VCSELs [29,30].

However, the research on the synchronization of VCSELs in network scenarios is still lacked. Remarkably, Xia et. al. [31] discussed the broadband chaos synchronization in the simple star-type VCSELs network. Nevertheless, the synchronization regimes in a VCSELs network with more complex topology are still deserved for further investigation. In this paper, we theoretically investigated the symmetry-induced cluster synchronization in the VCSELs networks with different topologies. To obtain the optical feedback with variable rotated polarization, an intracavity rotating polarizer was introduced in the external cavity. The formation of cluster is discussed from the point view of the inherent topology of network, i.e. the symmetry. The dependences of cluster synchronization quality on the coupling strength, the current factor, the self-feedback strength and the polarizer angle are detailedly considered, respectively. More importantly, based on the topology of network, we proposed that the polarizer angle has profound impact on the formation of clusters in VCSELs networks.

2. Theoretical model

For simplicity, we define 0° polarization to be the x polarization (XP) mode of VCSELs and 90° polarization to be the y polarization mode. We also assume that the transmission axis of polarizer is oriented at a polarizer angle θ_pm with respect to the XP mode of VCSEL_m, which can be rotated between 0° and 90°. Adopting the parallel-polarization optical injection (PPOI), we can extend the spin-flip model (SFM) to take into account the mutually coupled VCSELs network with variable-polarization optical feedback (VPOF) as follows [32,33]:

\begin{array}{l} \dot{E_{m x}} & = (1 + i α) [(N_{m} - 1) E_{m x} + i n_{m} E_{m y}] - (γ_{a} + i γ_{p}) E_{m x} \\ + γ E_{m x} (t - τ_{f}) \cos^{2} (θ_{p m}) \exp (- i w_{m} τ_{f}) \\ + γ E_{m y} (t - τ_{f}) \cos (θ_{p m}) \sin (θ_{p m}) \exp (- i w_{m} τ_{f}) \\ + σ \sum_{n = 1}^{N_{s}} A_{m l} E_{n x} (t - τ_{r}) \exp (- i w_{n} τ_{r}) \end{array}

\begin{array}{l} \dot{E_{m y}} & = (1 + i α) [(N_{m} - 1) E_{m y} + i n_{m} E_{m x}] + (γ_{a} + i γ_{p}) E_{m y} \\ + γ E_{m x} (t - τ_{f}) \cos (θ_{p m}) \sin (θ_{p m}) \exp (- i w_{m} τ_{f}) \\ + γ E_{m y} (t - τ_{f}) \sin^{2} \exp (- i w_{m} τ_{f}) \\ + σ \sum_{n = 1}^{N_{s}} A_{m l} E_{n y} (t - τ_{r}) \exp (- i w_{n} τ_{r}) \end{array}

\dot{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}^{*})]

\dot{n_{m}} = - γ_{s} n_{m} - γ_{N} [n_{m} (1 + {| E_{m x} |}^{2}) + {| E_{m y} |}^{2}) + i N_{m} (E_{m y} E_{m x}^{*} - E_{m x} E_{m y}^{*})]

where E_x and E_y are the slowly varying amplitudes 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 a N_s × N_s dimensional coupling matrix that describes the connectivity of the network, N_s is the size of network. If there exists a direct connection between node m and node l (m, l = 1, ..., N_s), A_ml = 1, otherwise A_ml = 0. σ is the overall coupling strength, and if A_ml equals to 1, node m and l will be mutually coupled with uniform coupling strength σ. γ is the self-feedback strength, μ is the current factor, α is the linewidth enhancement factor, τ_f = 1ns is the self-feedback delay and τ_r = 1ns is the coupling delay. In our simulations, a set of typical VCSEL parameters are considered, with decay rate of total carrier γ_N = 1ns⁻¹, spin-flip rate γ_s = 50ns⁻¹, linear birefringence γ_p = 30ns⁻¹, linear dichroism γ_a = 0.5ns⁻¹, central wavelength of VCSEL λ_m = 850nm with central frequency w_m = 2πc/λ_m and field decay rate k = 300ns⁻¹ [32].

Figure 1(a) presents a typically symmetric VCSELs network consisted of 7 mutually coupled VCSELs with VPOF, in which the nodes (VCSELs) with same color are classified as same cluster. This VCSELs network is composed of four clusters which is defined by the hidden symmetries of network, and could be written as (1), (2,3), (4,5) and (6, 7). A network could be described as a graph g = (V(g), E(g)) with vertex set V(g) and edge set E(g), and if there is an edge between two vertices, they are considered as adjacent. Actually, the symmetry of the network is a permutation of the vertices that can preserve the adjacency. And it can be mathematically represented as a set of permutation matrixes R_g, which can re-order the nodes of network in a way that preserve the dynamical equations unchanged (that is, each R_g commutes with A, i.e. R_g A = AR_g). These symmetries are manifest in the symmetries of the matrix that describes the network, referred to as the adjacency matrix [34]. The adjacency matrix of network plays an important role in modeling the dynamics of network, as it can provide the coupling between VCSELs in network.

Fig. 1 Schematic diagrams of VCSELs networks with different symmetric topologies. The nodes with the same color are defined in the same synchronized cluster.

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When all the symmetries of the network are identified, the nodes of the graph can be partitioned into clusters by distinguishing the orbits of the symmetry group. The orbits of the symmetry group are defined as the disjoint sets of nodes in all of the symmetry operations, and nodes in the same orbits are classified into same cluster. Therefore, nodes that are mapped into each other in the same cluster can be permuted by same symmetry and preserve the adjacency matrix unchanged, thus having the same equations of motion. For example, in the VCSELs network presented in Fig. 1(a), VCSEL 2 and VCSEL 3 can be permuted with each other and retain the adjacency of network unchanged, leading them to be in the same cluster and thus having the same dynamical equations. It means that as long as the two VCSELs are started with the same initial conditions, they will remain synchronized indefinitely. Otherwise, for random initial conditions, the stability of cluster synchronization depends on the reasonable selection of parameters of the VCSELs network. On the other hand, VCSEL1 cannot be permuted into any of the other VCSELs and therefore it can not synchronize with the others. In fact, this is the intimate relationship between the symmetry and dynamics in networks [34]. Moreover, the adjacency matrix A of the VCSELs network in Fig. 1(a) is presented as follows:

A = (\begin{matrix} 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix})

3. Numerical results and discussion

Figure 2 present the evolution and bifurcation of dynamics of total output I_T (I_T = |E_x|² + |E_y|²) of VCSELs for network in Fig. 1(a) to elaborate the stability of cluster synchronization with different values of polarizer angle. As shown in Fig. 2(a), all the clusters will split into trivial clusters of one node each for θ_p = 54°. And for the condition of θ_p = 72° [Fig. 2(b)], all four non-trivial clusters achieve stable synchronization as there are only four trajectories in this network. Obviously, the above contrast indicates that the polarizer angle of optical feedback plays an important role on the stability of cluster synchronization.

Fig. 2 The dynamics of VCSELs with different state, the panels correspond to different value of polarizer angle. (a) θ_p = 54°, (b) θ_p = 72° with σ = 120ns⁻¹, μ = 2, γ = 6ns⁻¹.

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To quantitatively evaluate the synchronization quality within same clusters, the time-averaged root-mean square (RMS) synchronization error is introduced, which is obtained by the calculation between the total outputs of VCSELs in same cluster over 50ns ≤ t ≤ 100ns with random initial conditions [8]. The RMS is given as:

RMS = \frac{\sum_{n = 1}^{N_{c}} \sqrt{(I_{Tn} (t) - {\hat{I}}_{T} (t)) / N_{e}}}{N_{c} {\hat{I}}_{T} (t)}

where N_c is the size of clusters, N_e is the length of outputs for VCSELs,

{\hat{I}}_{T} = \sum_{n = 1}^{N_{c}} I_{Tn} (t) / N_{c}

and the Î_T (t) is divided in Eq. 6 for normalization. Here, the threshold value of RMS is set to be 0.01, which means that we assume stable cluster synchronization can be achieved only when RMS < 0.01.

Moreover, we further introduce the cross-correlation function (CCF) to measure the linear relationship between the outputs between VCSELs in same clusters. For a delay-differential system, CCF is defined as follow [7,33]:

CCF = \frac{〈 [I_{Ti} (t) - 〈 I_{Ti} (t) 〉] \cdot [I_{Ti} (t + Δ (t)) - 〈 I_{Ti} (t + Δ (t)) 〉] 〉}{\sqrt{〈 {[I_{Ti} (t) - 〈 I_{Ti} (t) 〉]}^{2} 〉 \cdot 〈 {[I_{Tj} (t + Δ t)) - 〈 I_{Tj} (t + Δ t)) 〉]}^{2} 〉}}

where 〈·〉 denotes time average, Δt ∈ [−10ns, 10ns] denotes the lag time. For a given value Δt, the CCF measures the tendency of cloud of points (I_Ti(t), I_Tj (t)) to be aligned along a straight line and thus measures a linear relationship between I_Ti and I_Tj. Here, the threshold value of CCF for stable cluster synchronization is set to be 0.95.

In order to explore the effects of polarizer angle on the stability of cluster synchronization, Fig. 3 presents the dependences of RMS and CCF on the polarizer angle θ_p of optical feedback with different current factor μ. It is worth mentioning that for μ = 2, stable cluster synchronization can be easily obtained (RMS < 0.01 and CCF > 0.95) with appropriate choice of polarizer angle, while with the increment of current factor, there will no stable cluster synchronization in VCSELs network, which means that the overall trend of RMS will be modified by the current factor. Figure 4 presents the time series of cluster (2, 3) and the spectra power of VCSEL 2 with different values of μ. It is shown that, with the increment of μ from 2 to 3, the output bandwidth of VCSEL2 will increase from 17.3 GHz to 27.3 GHz. And as discussed in previous works by Argyris et. al, the higher frequency components of SLs are much more difficult to achieve synchrony [16,35]. Moreover, it can be seen from Fig. 3 that, the stability of cluster synchronization is quite sensitive to the polarizer angle θ_p, and small values of RMS (high values of CCF) exist in the domain of polarizer angle for VCSELs with YP mode optical feedback (θ_p is closed to 90°), leading the clusters to achieve synchrony.

Fig. 3 RMS and CCF as the function of polarizer angle and current factor for different clusters. (a), (d) cluster (2, 3), (b), (e) cluster (4, 5), (c), (f) cluster (6, 7) with coupling strength σ = 120ns⁻¹ and self-feedback strength γ = 6ns⁻¹.

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Fig. 4 The time series of cluster (2, 3) with different state and the power spectra of VCSEL 2, the panels correspond to different value of current factor. (a), (d) μ = 2, (b), (e) μ = 2.5 and (c), (f) μ = 3 with σ = 120ns⁻¹, θ_p = 63°, γ = 6ns⁻¹.

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The above results demonstrate that both the current factor μ and polarizer angle θ_p have significant effects on the stability of cluster synchronization. On the other hand, it is also essential to investigated the influences of parameters of VCSELs network on the stability of cluster synchronization, and to explore the optimal operation parameters that can achieve stable cluster synchronization. Figures 5(a)–5(c) present the maps of RMS in the parameter space of coupling strength σ and self-feedback strength γ for all three clusters, and Figs. 5(d)–5(f) show the RMS as function of polarizer angle θ_p and current factor μ. It is found that stable cluster synchronization (RMS < 0.01) can be achieved for a wide range of parameter space. More importantly, it is worth indicating that the operation range of cluster (2, 3) is obviously wider than the other clusters, which means that even when all the other clusters lose their synchrony, cluster (2, 3) can still remain synchronized.

Fig. 5 Two dimensional maps of RMS in the parameter space of coupling strength σ and self-feedback strength γ with θ_p = 63°, μ = 2.5. (a) cluster (2, 3), (b) cluster (4, 5), (c) cluster (6, 7); The RMS as function of polarizer angle θ_p and current factor μ with σ = 120ns⁻¹ and γ = 6ns⁻¹. (d) cluster (2, 3), (e) cluster (4, 5), (f) cluster (6, 7).

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Furthermore, we proposed that based on the symmetries of network topology, the polarizer angle does not merely affect the stability of cluster synchronization, but also has a significant influence on the formation of clusters. Figure 1(b) shows a symmetric network with four VCSELs, where the polarizer angle of optical feedback of these VCSELs are identical and thus all the four VCSELs belong to the same cluster. And the adjacency matrix for the network in Fig. 1(b) and 1(c) is presented as follows:

A = (\begin{matrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{matrix})

As shown in Fig. 6(a), all the VCSELs acheive stable cluster synchronization and there are only one trajectory. However, as shown in Fig. 6(b), the VCSEls network will be separated to two clusters with two lasers when we adjust the polarizer angle of optical feedback for VCSEL1 and VCSEL 2 from 72° to 63° [presented in Fig. 1(c)], in which stable cluster synchronization can still be achieved as there are two synchronized trajectories. This effect opens a new route to chaos-based secure communication networks built upon VCSELs as the selected users can be chosen by adopting the different polarizer angle.

Fig. 6 The dynamics of VCSELs with different values of polarizer angle. (a) θ_p1 = θ_p2 = θ_p3 = θ_p4 = 72°, (b) θ_p1 = θ_p2 = 63° and θ_p3 = θ_p4 = 72°, with σ = 120ns⁻¹, μ = 2, γ = 12.5ns⁻¹.

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4. Conclusion

In conclusion, the symmetry-induced cluster synchronization in VCSELs networks with variable-polarization optical feedback in complex topology are systematically investigated, in which the symmetry of network topology plays an important role in the formation of cluster. The stability of synchronization under different key parameters of VCSLs are also discussed. More importantly, it is presented that the polarizer angle of optical feedback has significant effect not only on the stability of cluster synchronization, but also on the formation of clusters. Our work suggest the applicability of cluster synchronization in different delay-coupled SLs networks.

Funding

National Natural Science Foundation of China (NSFC) (61775185, 61274042, 61674119, 61306061, 61405167); National 863 Program of China (2015AA016903).

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Cluster synchronization in symmetric VCSELs networks with variable-polarization optical feedback

Abstract

1. Introduction

2. Theoretical model

3. Numerical results and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (8)

Optics Express