Chaotic synchronization of a distant star-type laser network with multiple optical injections

Xue-Yan Xiong; Binglei Shi; Yanling Yang; Li Ge; Jia-Gui Wu

doi:10.1364/OE.403287

1. Introduction

Optical chaos has become a hot topic due to such excellent features as wide bandwidth, high complexity, and relatively easy implementation [1–4]. Novel applications of optical chaos have been found in secure communication [5], radar [6], physical random number generation (PRN) [7], time domain reflectometry [8], information processing [9] and even neural network [10–12]. Especially, optical chaos in semiconductor lasers (SLs) have been investigated extensively. Synchronization between chaotic SLs is a necessary condition to achieve secure communication and secure key distribution [1,2,5], and various coupling synchronization regimes have been proposed [5,13–23]. Moreover, cluster synchronization properties [24–27] in networks with coupled SLs have caught more and more attention. According to the experimental phenomenon of cluster synchronization in coupled laser networks in [25], synchronization is achieved between lasers in the same clusters while there is no synchronization between lasers from different clusters. Moreover, the influence of network topology structures on synchronization is still needed to be further investigated. Xu et al. theoretically investigated the cluster synchronization of mutually coupled vertical-cavity surface-emitting lasers in networks with symmetric topology [26]. The performance of cluster synchronization in semiconductor lasers networks of 12 different topologies is studied both analytically and numerically [27].

Taking realistic conditions into account, the star-type laser network topology structure [28–36] is extremely typical for more investigation. In a star-type laser network, side nodes (the star SLs nodes, S-SLN) within the same cluster are linked to a common central node (the hub SL node, H-SLN) in the other cluster. The chaotic synchronization between the nodes of a star-type laser network opens a route for a multi-directional communication scheme and advanced secure key distribution network [31–36]. For instance, Bourmpos et al. investigated the parameter sensitivity of the synchronization in a star network for S-SLNs asymmetrically coupled to an H-SLN [33], and found the critical intrinsic and operational variables for zero-lag synchronization between side nodes [34]. Xiang et al. studied theoretically the synchronization regime of a star laser network with heterogeneous time delays [35]. Argyris et al. reported an experimental study of the ultra-high quality chaotic synchronization between the S-SLNs in a star-type SL network [36]. These systems could be classed as single-injection star-type SL (SIST-SL) networks, where the light of the H-SLN is injected into the corresponding S-SLNs through a single-coupling link, and the S-SLNs are classed into a cluster while the H-SLN formed a unique cluster. All the above results are very valuable for guiding the implementation of an SL network in the real world. However, when considering practical circumstances, both a high synchronization among S-SLNs and a low correlation between the H-SLN and S-SLNs are desirable. The former offers the basis for secure-oriented applications among S-SLNs, while the latter could ensure the system’s security [21-22, 30 and 32]. For a distant SL network, the long-distance fiber link between the H-SLN and S-SLNs could offer a viable attack avenue for eavesdroppers when the output of the Hub node is highly similar to that of the S-SLNs [22,30]. Unfortunately, in most SIST-SL networks the correlation between the H-SLN and S-SLNs is significantly high, typically about 0.7 [20,21]. Based on these considerations, it is necessary to explore new ways of reducing the correlation between the H-SLN and S-SLNs.

In this work, we propose a distant multi-injection star-type SL (MIST-SL) network, with a novel multi-injection module (MIM) to obtain multiple optical injections from the H-SLN to the S-SLNs. We build both the experimental setup and a theoretical model to analyze the performance of the MIST-SL network in terms of the temporal waveforms, the spectral profiles, and the cross-correlation properties between nodes in the same cluster and nodes among different clusters. This MIST-SL network owns some advantages compared with previous schemes in [21,22,30], such as all-fiber construction, the long-distance malleability, and easy implementability. In this work, we first describe the configuration of the MIST-SL network and its required synchronization properties. Secondly, we show the experimental results for tests of cluster synchronization among nodes. Thirdly, we explore and discuss the theoretical model of the MIST-SL network. Finally, we draw a number of conclusions.

2. System configuration

Figure 1 is the experimental setup of the MIST-SL network with a Hub SL node in one cluster (Cluster H, C_H) and n Star SL nodes within the other cluster (Cluster S, C_S). Numbers of 1550 nm InGaAsP/InP DFB-SLs and related optical components constitute the S-SLNs, and are linked to the H-SLN with tens of kilometers of single-mode fiber spools. In the H-SLN section, a chaotic signal is preliminarily generated in the SL via an external optical feedback. Next, the chaotic signal is transmitted through a multi-injection module (MIM) where the signal is divided into m parts. Each part of the signal passes through a non-isometric fiber branch, and the parts are combined again as a newly restructured signal at the MIM exit port. After that, the restructured signal is amplified by the EDFA and injected simultaneously into n S-SLNs through a 1: n optical coupler. Thus, the multiple optical injection from the H-SLN to S-SLNs is formed. It should be noted that the multi-injection here is different from that of previous star-type SL networks, where the optical signal from the H-SLN is directly injected into the S-SLNs in a single fiber link. In a MIST-SL network, the MIM is the key for the achieving a particularly low correlation among different clusters, and the high synchronization between the S-SLNs within the same cluster.

Fig. 1. Experimental setup of the MIST-SL network. H-SLN: Hub SL node. S-SLN: Star SL node. OC: optical circulator. PC: polarization controller. VA: variable attenuator. OI: optical isolator. MIM: multi-injection module. EDFA: erbium-doped fiber amplifier. PD: photodetector. OSA: optical spectrum analyzer. ESA: electronic spectrum analyzer. OPM: optical power meter.

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In the S-SLNs sections, all the SL chips have similar internal parameters and matched characteristics because they are selected from a common wafer. Series of variable attenuators (VAs) and polarization controllers (PCs) are used for adjusting the optical injection power (κ) and optical polarization of the H-SLN. Additionally, optical isolators (OIs) are used to prevent unwanted reflection disturbances. Moreover, a series of 10:90 optical couplers (OCs) are also inserted into the setup to sample light for monitoring and testing. Finally, test equipment is employed to measure the chaotic signals both optically and electronically. This equipment consists of an optical power meter (OPM), optical spectrum analyzer (Anritsu MS9740A, 0.02nm resolution), electronic spectrum analyzer (Keysight N9010A, 26.5GHz) and wideband digital oscilloscope (Keysight DSO91204A, 12GHz bandwidth). All SLs are driven by low noise SL controllers. In the experiment, m is set as 3 and n is 2 due to the limitations of laboratory equipment. The length of three fiber branches are set as about 1000 mm, 1409 mm and 1613 mm, respectively. The chip bias currents are fixed at 22 mA for the H-SLN, 11.38 mA for S-SLN1, and 11.32 mA for S-SLN2. The injection power κ from the H-SLN to each S-SLN is initially set at 170 µw. The chip temperatures are stabilized at 21.02°C for the H-SLN, 24.33°C for S-SLN1, and 24.68°C for S-SLN2. Under these circumstances, the laser free-running wavelengths are measured as 1553.483 nm for the H-SLN and 1553.514 nm for the S-SLNs. As a result, we obtain a positive frequency detuning Δf_H-S=3.85 GHz between the H-SLN and S-SLNs. These parameters are optimally chosen for chaotic synchronization among the SL nodes.

3. Experimental results

Figure 2 shows the experimentally recorded chaotic temporal waveforms and frequency spectra of lasers in the MIST-SL network with 10 kilometer fiber spools. In Figs. 2(a1)–2(a3), we see that the waveforms of the S-SLN1 and S-SLN2 in the C_S are highly similar to each other, whereas the waveform of the H-SLN in the C_H is quite different. Next, considering the spectra (both optical and electrical) in Figs. 2(b1)–2(b3), 2(c1)–2(c3), the spectra of the S-SLN1 and S-SLN2 are obviously similar, but the spectra of the H-SLN is different from that of the S-SLNs. For the low frequency part of RF spectra, we see that the signal in Fig. 2(c3) is significantly higher than in Figs. 2(c1) and 2(c2). This low frequency enhancement corresponds to the large temporal fluctuations in Fig. 2(a3), and results in the H-SLN temporal waveform being quite different from the S-SLN temporal waveforms. This indicates that there is significant synchronization between lasers within the same cluster and a low correlation among different clusters. Moreover, the bandwidths of S-SLN chaos signals can be further enhanced with careful optimization of laser operations parameters [20].

Fig. 2. The temporal waveforms (the left column), optical spectra (the middle column), and RF spectra (the right column) of the S-SLN1 (the first row), the S-SLN2 (the second row) and the H-SLN (the third row). The time delay between temporal waveforms has been compensated, and the total length of temporal waveforms is 2000 ns. The red curves in the RF spectra are the background noise floor.

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In order to assess the correlation degree between two temporal waveforms, the well-known cross-correlation (CC) coefficient is introduced by the following equation [2,3]:

(1)$$C{C_{ij}}(\Delta \textrm{t}) = \frac{{\langle [{P_i}(t + \Delta t) - \langle {P_i}(t + \Delta t)\rangle ][{P_j}(t) - \langle {P_j}(t)\rangle ]\rangle }}{{{{\{ \langle {{[{P_i}(t + \Delta t) - \langle {P_i}(t + \Delta t)\rangle ]}^2}\rangle \langle {{[{P_j}(t) - \langle {P_j}(t)\rangle ]}^2}\rangle \} }^{1/2}}}}$$

where the subscripts i and j represent H-SLN, S-SLNs, respectively. Angle brackets represent time average, Δt is the time shift, P is the instance amplitude of temporal waveforms.

Figure 3 shows the correlation plots and calculated CC coefficient curves between arbitrary nodes of the MIST-SL network. From Figs. 3(a1)–3(a2), we can see that the max value of the CC coefficient (CC_max) reaches at 0.92, indicating good synchronization is achieved in the C_S which is composed of S-SLN1 and S-SLN2. In contrast, in Figs. 3(b1), 3(b2), 3(c1), and 3(c2), the correlations between the H-SLN and S-SLNs are much lower, and the CC_max is calculated as 0.17 for Fig. 3(b2) and as 0.16 for Fig. 3(c2). That is to say, the correlations between the C_H and the C_S stay at a low level. These CC_max values are much lower than that of the single-injection scheme, where the typical CC_max values between the H-SLN and S-SLNs are about 0.7 [20,21]. The main general attack scenario for this type of network is that an eavesdropper might extract a clue by comparatively analyzing the signals of the H-SLN and S-SLNs through intercepting fiber links [22,30]. However, the correlation between different clusters in the MIST-SL network is so low that no useful clue can be extracted. Thus, this attack scenario can be eliminated.

Fig. 3. The correlation plots (the upper row) and CC coefficient curves (the lower row) between arbitrary nodes of MIST-SL network. a1-a2: the correlation between the S-SLN1 and S-SLN2 in the C_S; b1-b2: the correlation between the S-SLN1 in the C_S and H-SLN in the C_H; c1-c2: the correlation between the S-SLN2 in the C_S and H-SLN in the C_H.

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Figure 4 presents the CC_max evolution of the MIST-SL system and the SIST-SL system under different injection powers κ and frequency detuning Δf_H-S between H-SLN and S-SLNs, to show the specific operation conditions for an SL network exhibiting the desired synchronous behavior. Firstly, for the MIST-SL situation, the CC_max values rise with the increase in injection power κ in Fig. 4(a1). The CC_max values between Star nodes (labeled by black squares) increase dramatically in the ‘I’ region (20µw<κ<150µw). The CC_max values between the Hub node and the Star nodes, on the other hand, grow slowly. Next, in the ‘II’ region (κ>150µw), CC_max between the Star nodes reaches levels over 0.9. Conversely, the CC_max values between the Hub node and the Star nodes retain a remarkably low level (less than 0.2). CC_max for different Δf_H-S values is given in Fig. 4(b1). In the whole investigated Δf_H-S range (from -20GHz to 20GHz), CC_max between the Star nodes reaches high levels in the “II” region (-12GHz<Δf_H-S<5GHz), while CC_max between the Hub node and the Star nodes always stay at a low level (less than 0.2). In comparison, Figs. 4(a2) and 4(b2) gives the evolution of CC_max between arbitrary nodes of the SIST-SL network, with similar operating conditions to those of the MIST-SL network. In Figs. 4(a2) and 4(b2), we see that CC_max between the Hub node and the Star nodes is over 0.6 in the “II” regions, indicating a strong correlation between the Hub node and Star nodes of the SIST-SL network. Therefore, the MIST-SL network can achieve good chaotic synchronization among the Star nodes within the same cluster and significantly low correlation between the Hub node in the C_H and Star nodes in the C_S for a relatively wide range of κ and Δf_H-S.

Fig. 4. The CC_max evolution under different injection powers κ (left column) and frequency detuning Δf_H-S (left column) between H-SLN and S-SLNs of the SL networks. The upper row is the MIST-SL case and the lower row is the SIST-SL case. The Δf_H-S is 3.85 GHz (a1, a2), the injection power is 170µw (b1, b2). The black squares denote the CC_max values between S-SLN1 and S-SLN2 in the C_S; the red diamonds denote the CC_max between the H-SLN in the C_H and S-SLN1 in the C_S; the blue triangles denote the CC_max between the H-SLN in the C_H and S-SLN2 in the C_S.

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Next, it is necessary to consider the influence of parameter mismatch since it is inevitable in reality. Figure 5 shows the effects of Δf_S-S mismatch between S-SLN1 and S-SLN2 on chaotic synchronization. Here, Δf_S-S is introduced by fine tuning the wavelength of S-SLN1 (through changing the temperature of the laser chip) and fixing the wavelength of S-SLN2. Then, the synchronization evolution could be investigated between S-SLN1, S-SLN2 and H-SLN. According to Figs. 5(a1), 5(b1), and 5(c1), CC_max between the Star nodes in the C_S is decreased sharply as the Δf_S-S mismatch increases. It demonstrates the high synchronization sensitivity of parameter mismatch in the clusters of the MIST-SL network. Figures 5(a1), 5(b1) and 5(c1) also show evidence that CC_max between nodes in different clusters preserves a low level (around 0.2) over the entire test Δf_S-S range. The blue triangles always keeps constant and a low level since the wavelengths of S-SLN2 and H-SLN are not change during the entire test Δf_S-S range. This feature promotes the security of chaos communication among the Star nodes within the same cluster [22,30]. The CC_max evolution in the SIST-SL network is very different from that of the MIST-SL network. As shown in Figs. 5(a2), 5(b2) and 5(c2), CC_max between the Hub node and Star nodes fluctuates significantly, and stays at a high level under specific Δf_S-S mismatch conditions, particularly for the positive Δf_S-S mismatch cases in Figs. 5(b2) and 5(c2).

Fig. 5. CC_max between arbitrary nodes as a function of Δf_S-S between S-SLN1 and S-SLN2 for different injection power levels. The injection power values are successively 50µw (a1), 100µw (b1), 140µw (c1), 100µw (a2), 150µw (b2), and 170µw (c2), respectively. The upper row is the MIST-SL case and the lower row is the SIST-SL case. The black squares denote the CC_max values between S-SLN1 and S-SLN2 in the C_S; the red diamonds denote CC_max between the H-SLN in the C_H and S-SLN1 in the C_S; the blue triangles denote CC_max between the H-SLN in the C_H and S-SLN2 in the C_S.

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In order to illustrate the malleability of the transmission distance, Fig. 6 shows the chaotic synchronization among nodes in the cluster C_H and C_S of MIST-SL network with 80 kilometer fiber spools. The temporal waveform of the H-SLN (Fig. 6(a3)) is distinct from that of the S-SLNs (Fig. 6(a1) and Fig. 6(a2)). Accordingly, the remarkably low CC_max (0.16 and 0.18) between H-SLN of the C_H and S-SLNs of the C_S is achieved. In contrast, the chaotic synchronization among the Star nodes within the same cluster is demonstrated by the high similarity between the temporal waveforms (Fig. 6(a1) and Fig. 6(a2)), the spindly correlation plot (Fig. 6(b1)) and the calculated CC_max≈0.81. The synchronization among the Star nodes in the C_S is distorted by the accumulated fiber dispersion and other nonlinearities [22,37]. Thanks to various advanced compensation devices, such as the dispersion compensating fiber (DCF), fiber Bragg grating (FBG), etc., the implementation of a long-distance MIST-SL network of over 100km is also very possible.

Fig. 6. The experimental results for the MIST-SL network with 80 kilometer fiber spools. a1-a3: the recorded temporal waveforms of S-SLN1, S-SLN2 and H-SLN, respectively. b1-a3: the correlation plots between S-SLN1 and S-SLN2 in the C_S, between S-SLN1 in the C_S and H-SLN in the C_H, and between S-SLN2 in the C_S and H-SLN in the C_H, respectively.

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4. Theoretical model and related discussion

The SL dynamics can be described by the famous Lang-Kobayashi (L-K) equations [38]. The model of the MIST-SL network is then established by modifying the L-K equations with multi-optical injection terms. For the H-SLN of the C_H:

(2)$$\begin{array}{ll} \frac{{d{E_{HL}}(t )}}{{dt}} &= \frac{1}{2}({1 + i{\alpha_{HL}}} )\left( {{G_{HL}} - \frac{1}{{{\tau_P}}}} \right){E_{HL}}(t )+ {K_{HL}}{E_{HL}}({t - {\tau_{HL}}} )exp ({ - i2\pi {f_{HL}}{\tau_{HL}}} )\\ &\mathop {}\limits_{}^{} + \sqrt {2\beta {N_{HL}}} {\chi _{HL}} \end{array} . $$

(3)$$\frac{{dNHL(t )}}{{dt}} = \frac{I}{e} - \frac{{NHL(t )}}{{{\tau _e}}} - GHL{|{EHL(t )} |^2}.$$

(4)$${E_{H - SLN}} = 1/\sqrt m \times \left\{ \begin{array}{l} {}^{}{}^{}{E_{HL}}(t - {\tau_1}\textrm{ - }{\tau_0})\exp ( - i2\pi {f_{HL}}({\tau_1} + {\tau_0}))\\ + {E_{HL}}(t - {\tau_2}\textrm{ - }{\tau_0})\exp ( - i2\pi {f_{HL}}({\tau_2} + {\tau_0}))\\ + {E_{HL}}(t - {\tau_3}\textrm{ - }{\tau_0})\exp ( - i2\pi {f_{HL}}({\tau_3} + {\tau_0}))\\ \cdots \\ + {E_{HL}}(t - {\tau_m}\textrm{ - }{\tau_0})\exp ( - i2\pi {f_{HL}}({\tau_m} + {\tau_0})) \end{array} \right\}.$$

For the S-SLNs of the C_S:

(5)$$\begin{array}{ll} \frac{{d{E_{SLj}}(t)}}{{dt}} &= \frac{1}{2}(1 + i{\alpha _{SLj}})({G_{SLj}} - \frac{1}{{{\tau _p}}}){E_{SLj}}(t) + {K_{H - }}_S{E_{H - SLN}}exp ({i2\pi \Delta {f_{H - S}}t} )\\ & \mathop {}\limits_{}^{} + \sqrt {2\beta {N_{SLj}}} {\chi _{SLj}} \end{array} . $$

(6)$$\frac{{d{N_{^{SLj}}}(t)}}{{dt}} = \frac{I}{q} - \frac{{{N_{SLj}}(t)}}{{{\tau _n}}} - {G_{SLj}}{|{{E_{SLj}}(t)} |^2}.$$

where E is the slowly varying optical field, and N represents the carrier number. The subscripts HL and SLj stand for the Hub laser and for the jth Star laser (j=1, 2, …, n) in the MIST-SL network. n is the number of Star nodes in C_S. The multi-optical injection term E_H-SLN is expressed as Eq. (4) to characterize the MIM in the experimental setup. The subscript m is the number of injection branches. The τ₁, τ₂, τ₃ and τ_m are the time delays of each branch. τ₀ is the fiber time delay from MIM to the Star lasers (S-SLNs). Next, in Eq. (5), the E_H-SLN inject simultaneously into the S-SLNs. K_H-_S and Δf_H-_S represent the injection strength and frequency detuning from the H-SLN to S-SLNs. For the convenience of calculation, the internal parameters of the SL lasers are assumed to be identical. τ₀ sets as 0ns since the output of MIM is simultaneously injected into to S-SLNs. Then, we could focus on the effect of MIM on the chaotic synchronization. Next, the parameter values are: the linewidth enhancement factor α=3, the photon lifetime τ_p=2 ps, the carrier lifetime τ_n=2 ns, the electronic charge q=1.602×10⁻¹⁹ C, the normalized Gaussian white noise χ, the spontaneous emission rate β=1.1×10³ s⁻¹, the gain coefficient G = g(N(t)-N₀)/(1+s|E(t)|²), where the differential gain coefficient s=1.5×10⁻⁸ ps⁻¹, the transparent carrier number N₀=1.5×10⁸, and the gain saturation factor ρ=1×10⁻⁷. The time delay of each branch is τ₁=4.9 ns, τ₂=6.9 ns, τ₃=7.9 ns. It is imaginable that the correlation between the H-SLN and S-SLNs should not be offset but be strengthened, if the time delays of fiber branches have integer multiple relationship with the external cavity feedback delay τ_HL. Therefore, in principle, the non-convention relationship between times delays in our setup is necessary. The feedback time delay is τ_HL=5.5 ns, the feedback strength is K_HL=2.5 ns⁻¹, and the pump current is I=29 mA. For simplicity, the free running wavelength of the lasers is set as λ=1550 nm (with a corresponding central frequency f_HL₌1.94×10¹⁴Hz), and the value of K_H-_S is variable and is set to the optimal value. The software Matlab and fourth-order Runge-Kutta algorithms are employed to numerically simulate the dynamics of the SL nodes and the performance of the MIST-SL network.

To illustrate the characteristic of larger MIST-SL network, we theoretically simulate the synchronization when n is set as 2 (the left part) and 5 (the right part) in Fig. 7. One can see that the temporal waveforms of S-SLN1 and S-SLN2 (Figs. 7(a1) and 7(a2)) are nearly identical while the temporal waveforms of H-SLN (Fig. 7(a3)) are very different from that of S-SLNs. As a result, the CC_max between S-SLN1 and S-SLN2 (Fig. 7(b1)) is almost equal to 1, and the CC_max between H-SLN and S-SLNs are about 0.13 (Figs. 7(b2) and 7(b3)). The simulation results are confirmed with the experiment observations in Fig. 2 and Fig. 6. Moreover, for the n = 5 case (Figs. 7(c)–7(d)), the CC_max between laser nodes show similar property with that of n = 2 case. Based on the simulation and experiment, it could be reasonable to suppose that the MIST-SL network has similar performance for larger scale networks.

Fig. 7. The theoretical simulation of the MIST-SL network when the number of S-SLNs is set as 2 (the left part) and 5 (the right part). a1-a3: the temporal waveforms of S-SLN1, S-SLN2 and H-SLN, respectively. b1-b3: the correlation plots between S-SLN1 and S-SLN2 in the C_S, between SLN1 in the C_S and H-SLN in the C_H, and between S-SLN2 in the C_S and H-SLN in the C_H, respectively. c1-c6: the temporal waveforms of S-SLN1, S-SLN2, S-SLN3, S-SLN4, S-SLN5 and H-SLN, respectively. d1-d6: the correlation plots between S-SLN1 and S-SLN2, between S-SLN2 and S-SLN3, between S-SLN3 and S-SLN4, between S-SLN4 and S-SLN5, between S-SLN5 and S-SLN1, between SLN5 and H-SLN, respectively.

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To analyze the general correlation feature among nodes of different clusters in a star-type SL network, Fig. 8 shows the simulated CC_max evolution maps in parameter spaces of K_H-S and Δf_H-S. For the SIST-SL network case, as shown in Fig. 8(a1), the high synchronization between Star nodes in the C_S (CC_max >0.99) region covers the right half of the parameter space. Additionally, in Figs. 8(b1) and 8(c1), the correlation between the Hub node and Star nodes in different clusters increases rapidly with increasing K_H-S, and leads to a high correlation (CC_max >0.5) region marked by the white dash lines. We find that the high correlation regions in Figs. 8(b1) and 8(c1) almost coincide with the high synchronization region in Fig. 8(a1). Regrettably, it reveals that the appropriate K_H-S and Δf_H-S range that satisfies the high synchronization between the Star nodes in the C_S and the low correlation between the Hub node of C_H and the Star nodes of C_S in the SIST-SL network is very narrow. In comparison, for the MIST-SL network case, the high synchronization region almost covers the right half of the parameter space in Fig. 8(a2). Interestingly, in Figs. 8(b2) and 8(c2), the high correlation (CC_max >0.5) region disappears from the whole investigated parameter space. It means that the whole right half parameter space has the appropriate parameter range for the requirements of low correlation between the Hub node of C_H and the Star nodes of C_S in the MIST-SL network, along with the high synchronization quality between Star nodes of C_S.

Fig. 8. The simulated evolution maps of CC_max in the parameter space of K_H-S and Δf_H-S. The upper row is the SIST-SL case and the lower row is the MIST-SL case. The left column: the CC_max maps between S-SLN1 and S-SLN2 in the C_S; the middle column: the CC_max maps between the H-SLN in the C_H and S-SLN1 in the C_S; the last column: the CC_max maps between the H-SLN in the C_H and S-SLN2 in the C_S. The white dash line marks the boundary with condition CC_max=0.99, while the black dash line marks the boundary with condition CC_max=0.5.

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Moreover, the effect of MIM branch number on the correlation among clusters is investigated in Fig. 9, where the number of the MIM branch varies from 1 to 6. As shown in Fig. 9(a), CC_max between Star nodes in C_S suffer rare affection with the increasing number of the MIM branch. For a different MIM branch case, CC_max gets very high (CC_max>0.99) once K_H-S>45ns⁻¹. Next, in Figs. 9(b) and 9(c), CC_max monotonically decreases with the increase of MIM branch number under the same K_H-S condition. The more branches are used in the MIM, the lower the CC_max received. Therefore, multi-injection is necessary for the simultaneous realization of low correlation between different clusters and high synchronization within the same cluster. There are several ways to further improve the system synchronization performance, such as manufacturing lasers with better intrinsic parameter consistency, better long-distance fiber pairs, and choosing consistent active devices in system etc.

Fig. 9. Simulated CC_max evolution between arbitrary lasers of the MIST-SL network with different MIM branch numbers as m=1, 2, 3, 4, 5, 6 respectively, a: the CC_max between S-SLN1 and S-SLN2 in the C_S; b: CC_max between H-SLN in the C_H and S-SLN1 in the C_S; c: CC_max between H-SLN in the C_H and S-SLN2 in the C_S.

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5. Conclusions

In this paper, we experimentally and theoretically investigate the chaotic synchronization performance of a MIST-SL network, where a novel multi-injection module is introduced and lasers are linked with long distance fiber. Significant low correlation (corresponding CC_max<0.2) is successfully achieved between the H-SLN and S-SLNs in different clusters. We extensively investigate the effect of injection power and frequency detuning, as well as the frequency mismatch between laser nodes, and determine the specific operating conditions for desired synchronous outputs. Moreover, in the 80 kilometer distant MIST-SL network, high synchronization between S-SLNs in the same cluster and dramatically low correlation (CC_max≈0.18) between the H-SLN and S-SLNs in different clusters are also achieved, showing the distance malleability of MIST-SL network. Additionally, the theoretical simulations reveal the broad operation parameter range for the correlation suppression the Hub node and Star nodes among different clusters, and the effectiveness of the multi-injection module. From a broader perspective, the MIM could be transplanted into other topological types of laser networks, such as mesh, ring, or other hybrid types, and offers promise for various high applications, such as giant neurons networks, long-distance chaos communication networks or secure key distribution networks.

Funding

National Natural Science Foundation of China (11474233, 61875168); China Postdoctoral Science Foundation (2017M612885); Chongqing Postdoctoral Science Foundation Special Funded project (Xm2017008); Fundamental Research Funds for the Central Universities (XDJK2019B059).

Disclosures

The authors declare no conflicts of interest.

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Chaotic synchronization of a distant star-type laser network with multiple optical injections

Abstract

1. Introduction

2. System configuration

3. Experimental results

4. Theoretical model and related discussion

5. Conclusions

Funding

Disclosures

References

Cited By

Figures (9)

Equations (6)

Optics Express