Analysis and characterization of chaos generated by free-running and optically injected VCSELs

1. Introduction

Optical chaos has attracted widespread attention for its potential applications in secure optical communication [1,2], high-speed random bit generation [3–6], chaotic radar [7], compressive sensing [8] and reservoir computing [9]. The important chaotic sources can be readily generated by various types of semiconductor lasers (SLs), such as edge-emitting laser [10–13], vertical-cavity surface-emitting lasers (VCSELs) [14–17], semiconductor ring lasers [18], quantum well (QW) [19, 20] and dot (QD) lasers [21], quantum cascade lasers [22], and nano-lasers [23]. But, it is known that SLs belong to Class B type lasers [24] and output chaos because of the inclusion of more degrees of freedom, that is, under external forcing or modulation, including optical feedback [25, 26], current modulation [27], optoelectronic feedback [28, 29], mutual coupling [30], etc. Among them, external cavity semiconductor lasers (ECSLs) have attracted much attention due to their simplicity and the promising rich nonlinear dynamics [25]. However, the additional degree of freedom may impose restrictions in return. For example, an external cavity introduces the additional time-delay due to the finite propagation of light into the chaotic signals, which is known as ‘time-delay signature’ (TDS) in the literature [25] and may degrade the security of chaos-based communication, or prevent the generated random bit sequence from passing the standard randomness tests. Regarding this signature, various important schemes have been proposed and demonstrated theoretically and/or experimentally to suppress or even eliminate it [31–36]. For example, Wu et al have demonstrated experimentally and numerically that the TDS could be suppressed from the intensity chaos by adopting double optical feedback to SL [31]. Hong et al. have introduced a new concept to quantify the TDS in a three-cascade VCSELs system, and found that the TDS can be totally concealed over a wide frequency detuning region, while the resulting chaos has a higher bandwidth [32]. Xiang et al. have successfully removed the TDS from the chaotic output in ECSLs by using a complex dual chaos injection scheme [34]. Li et al. have proven that the TDS can be substantially suppressed both in phase and amplitude by means of chaos injection in conjunction with large values of the linewidth enhancement factor [35]. Jiang et al. have investigated that the TDS of chaotic signal can be suppressed by injecting the output of a chaotic general ECSL into an optical time lens module, where the bandwidth can be greatly enhanced [36]. Most of these schemes indeed provide promising solutions to the TDS concealment but at the expense of increasing the system complexity, which can be seen as a drawback for photonic integrated circuits (PICs) of chaos emitters. Thus, it is of prime importance to find alternative ways to generate optical chaos where the system structure is simple (i.e., no feedback or mutual coupling loop) and promising for future PICs. Fortunately, Virte et al. have found that deterministic polarization chaos can be generated in a free-running QD VCSEL (grown and described by Hopfer et al. [37]), which remains the first counter-example of a free-running laser generating chaos [38]. It is accepted that the nonlinear coupling between two elliptically polarized modes accounts for the chaos generation in a free-running VCSEL, which has also been reproduced and explained by using the well-known spin-flip model [39, 40]. Later on, they have proven the feasibility of the applications in random number generation and chaos synchronization [41, 42]. More recently, they have demonstrated experimentally similar polarization chaos can be obtained in off-the-shelf VCSELs, i.e., a commercially available quantum-well (QW) VCSEL in conjunction with the mechanically applied in-plane anisotropic strain [43]. Inspired by those important findings, we believe a comprehensive study on the characterization of chaotic dynamics in those simple VCSELs is called for, which can provide a better understanding of the influence of some key parameters on the chaos generation and pave the way to the wide spread use of solitary VCSELs for chaos-based applications as mentioned above.

In addition to TDS mentioned above, other properties of optical chaos have also been widely studied in the past few years, such as bandwidth [36], complexity [44], or the combination of bandwidth and complexity [26]. These properties are of prime importance to enhance the performance of chaos-based applications. For example, three-cascaded lasers with enhanced bandwidth and complexity have been used to generate physical random numbers, where the maximum generation rate can reach 1.2Tb/s [45]. Such a high bit rate indeed benefits from the enhanced properties of optical chaos. It should be noted that, even though the feasibility of chaos generation in a free-running VCSEL has been demonstrated, the bandwidth and complexity are rather low and thus cannot meet the demand of high bandwidth and security of modern communications. Thus, it would be of interest to study whether the properties of optical chaos generated by a free-running VCSEL can be greatly improved by using the some simple but effective approaches, for example, an optical injection scheme, which will be applied to our system and testified.

In this paper, we first study nonlinear dynamics of a free-running VCSEL, where two widely used measures including the 0-1 test for chaos and permutation entropy ($\text{PE}$) are introduced to analyze and characterize the effects of the key parameters on the chaotic region. Further, we construct a master-slave VCSEL configuration and focus on the bandwidth and unpredictability of the salve VCSEL as well as their relation to chaos synchronization between master and slave VCSELs. Finally, a three VCSEL system is suggested, which allows for high-quality chaos synchronization between enhanced chaos with extremely wide bandwidth and high complexity.

2. Theory

Our simulations are based on the simulating the well-known spin-flip model. The rate equations for a free-running VCSEL, i.e., the master laser in a master-slave configuration are written as [39, 42]

Since we attempt to enhance the properties of chaotic dynamics generated by a free-running VCSEL through optical injection, a master-slave scheme will be used. Here the rate equations of the slave VCSEL extended from the basic spin-flip model read [42]

Polarization dynamics of VCSEL in different systems have been studied in many works [16, 17, 46–48], in this study, we analyze and characterize VCSEL dynamics by using the widely used 0-1 test for chaos, which has been detailed in [49], and the powerful $\text{PE}$, which is derived from the information theory, as a complexity measurement. The computation of $\text{PE}$ has been given in several previous works; one can refer to [50] and references therein for more details. Here, we would like to emphasize that large $\text{PE}$ indicates high unpredictability degree, which is a highly desired property to ensure security for chaos-based communication systems. One can have $0\le \text{PE}\le 1$, with$\text{PE}=0$corresponding to a regular, predictable dynamic, and$\text{PE}=1$ to a fully random, unpredictable one.

In our simulations, the embedding parameters$D_x=5$(embedding dimension) and${\tau }_e=1$(embedding delay) are specified as in [50]. Each time series is obtained after sampling process, where the sampling period is${\Omega }_s=10\text{ps}$. To meet the statistical significance of the results, each time series is then divided into several disjointed sections with$T=5000$points, and a statistical average over different windows is performed to compute$H$.

In addition, we will consider the correlation between chaotic time series. By following [51], we can define the cross-correlation coefficient as

3. Results

In this section, we present the numerical simulation results by integrating the rate Eqs. (1)-(6) with the forth-order Runge-Kutta algorithm. The results are mainly represented by using high-resolution two-dimensional maps of 0-1 test for chaos and $\text{PE}$computed from intensity time series. The effects of important parameters including the pump rate $\mu$, the linear birefringence rate ${\gamma }_p$, the spin-flip relaxation rate ${\gamma }_s$, and the linewidth enhancement factor $\alpha$ on chaotic dynamics will be explored in some details. It should be noted that since the results for RCP and LCP are almost identical, only those for RCP are presented, unless otherwise stated.

3.1 Effects of the parameters on the chaos region of a free-running VCSEL

In [39], Virte et al. have shown the stability and instability of a free-running VCSEL by using combined methods of direct numerical simulations and a continuation method. Here we show more details about dynamics of free-running VCSELs. Figure 1 shows the time series and the corresponding RF spectra as the pump rate $\mu$increases, where the parameters are $\alpha \text{=}3$, ${\gamma }_p={25}_{}{\text{ns}}^{-1}$, and ${\gamma }_s={20}_{}{\text{ns}}^{-1}$. From the time-series plots it is clear that a solitary VCSEL can be destabilized through a Hopf bifurcation and other further complex bifurcations by increasing the pump rate. Under proper conditions, it can operate in various dynamical regimes, including steady state, periodicity, quasiperiodicity and complex dynamics including chaos. The corresponding spectra clearly verify these dynamics. Our results agree well with those in [39], which make us become more confident about our following studies.

 

Fig. 1 (a) Intensity time series and (b) the corresponding spectrum of a free-running VCSEL, where$\alpha \text{=}3$, ${\gamma }_p={25}_{}{\text{ns}}^{-1}$, and${\gamma }_s={20}_{}{\text{ns}}^{-1}$.

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Figure 2 shows the typical one-parameter bifurcations of a free-running VCSEL as the pump current is increased, where the parameters are the same as those in Fig. 1. Both the results for RCP and LCP are shown to confirm that they are identical and thus one of them is needed for illustrative purposes. In the bifurcation diagram, the maximum and minimum of intensity time series are shown. As can be seen from this figure, a free-running VCSEL allows for a diversity of dynamics via a period-doubling route to chaos, which is similar to the one found in a solitary spin VCSEL [20]. This is expected since the physical mechanism of instability for both of them is the nonlinear coupling between two elliptically polarized modes.

 

Fig. 2 One-parameter bifurcation diagrams of a free-running VCSEL with increasing pump current: (a) RCP and (b) LCP. The parameters are the same as those in Fig. 1.

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To obtain a global view of the dynamics and the effects of key parameters mentioned above, we show high-resolution two-dimensional maps of the 0-1 test for chaos and $\text{PE}$ side by side. Let us first consider the effects of ${\gamma }_p$ (x-axis) and $\mu$ (y-axis). These two parameters can be seen as controllable parameters since one can easily change the pump current $\mu$ in the experiments while the birefringence ${\gamma }_p$can be controlled through the applied in-plane anisotropic strain [43]. The results for the 0-1 test for chaos and $\text{PE}$ in the (${\gamma }_p$,$\mu$) plane are shown in Fig. 3, where several values of are considered. In the case of the 0-1 test for chaos, chaos is highlighted by red, while in the case of $\text{PE}$, blue represents steady-state ($\text{PE}=0$) and yellow stands for high values of $\text{PE}$, viz., $\text{PE}>0.9$. Several phenomena can be discovered from this figure. First, given a proper${\gamma }_p$, increasing $\mu$is helpful to obtain chaos, which is consistent with the results in Figs. 1 and 2. Second, in all cases considered, the chaotic region resembles a well-defined ‘tongue ’-like shape and the tongue tip is located at the bottom left corner of the parameter space. As ${\gamma }_s$is decreased, the tongue becomes larger, which means that the chaotic region grows in size, and in the meantime, the tongue tip is moved towards smaller values of ${\gamma }_p$and$\mu$. Actually, the fact that the smaller the spin-flip relaxation rate ${\gamma }_s$, the larger region of chaos can be obtained coincides with the finding in spin VCSELs [20]. Third, as ${\gamma }_s$reaches a critical value, roughly${\gamma }_s=25{\text{ns}}^{\text{-1}}$, the chaotic region saturates. Finally, the results for the 0-1 test for chaos and $\text{PE}$ agree well with each other. That is, chaotic region always corresponds to the region with values of $\text{PE}$ close to 1. In addition, $\text{PE}$ provides more information about dynamics; for example, the region for $\text{PE}=0$ means steady state, while other values between 0 and 1 represent periodicity and quasiperodicity other than chaos.

 

Fig. 3 (a)The two-dimensional map of the 0-1 test for chaos and (b) $\text{PE}$ of the free-running VCSEL in the (${\gamma }_p$,$\mu$) plane as ${\gamma }_s$is varied, where$\alpha \text{=}3$. (a1, b1)${\gamma }_s={100}_{}{\text{ns}}^{-1}$, (a2, b2)${\gamma }_s={50}_{}{\text{ns}}^{-1}$, (a3, b3)${\gamma }_s={25}_{}{\text{ns}}^{-1}$, and (a4, b4)${\gamma }_s=5_{}{\text{ns}}^{-1}$ .

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Figure 4 shows the results of the 0-1 test for chaos and $\text{PE}$ in the (${\gamma }_s$,$\mu$) plane for several values of ${\gamma }_p$. For a large value of ${\gamma }_p$, chaos is generated for a VCSEL pumped extremely above the threshold [see Figs. 4(a1) and 4(b1)]. As ${\gamma }_p$decreases, the chaotic region expands; see Figs. 4(a2, a3) and 4(b2, b3). However, as ${\gamma }_p$is further decreased, chaos is found in a very small area [see Figs. 4(a4) and 4(b4)]. All of the information can be obtained either from the 0-1 test for chaos or from the $\text{PE}$.

 

Fig. 4 (a) The two-dimensional map of the 0-1 test for chaos and (b)$\text{PE}$of the free-running VCSEL in the (${\gamma }_s$,$\mu$) plane as ${\gamma }_p$is varied, where$\alpha \text{=}3$, (a1, b1)${\gamma }_p={100}_{}{\text{ns}}^{-1}$, (a2, b2)${\gamma }_p={50}_{}{\text{ns}}^{-1}$, (a3, b3)${\gamma }_p={25}_{}{\text{ns}}^{-1}$, and (a4, b4)${\gamma }_p=5_{}{\text{ns}}^{-1}$

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We now are interested to uncover changes in the dynamics of a free-running VCSEL in the (${\gamma }_s$,${\gamma }_p$) plane as the pump current is varied. As an example, Fig. 5 shows the results for the 0-1 test for chaos and $\text{PE}$, where $\alpha \text{=}3$and $\mu$is increased from 2 to 10. It can be clearly seen that chaotic dynamics is extremely sensitive to the variation of$\mu$. For a small value of$\mu$, chaos is mainly obtained for smaller values of ${\gamma }_s$and${\gamma }_p$. As $\mu$increases, chaos can be seen in a larger range of ${\gamma }_p$and the whole range of ${\gamma }_s$, indicating chaotic region expands in size. However, as$\mu$ is larger than a critical value, the portion and location of chaotic region remain almost unchanged as $\mu$ is further increased. This also indicates that one can reduce the independence of chaos on ${\gamma }_s$and${\gamma }_p$by always pumping VCSEL at large values of the injection current, which is very useful for experimental investigations. Again, the results for both measures, i.e., the 0-1 test for chaos and $\text{PE}$ are in good agreement with each other.

 

Fig. 5 (a)The two-dimensional map of the 0-1 test for chaos and (b) $\text{PE}$ of the free-running VCSEL with varying ${\gamma }_s$(x-axis) and ${\gamma }_p$(y-axis), where$\alpha =3$, (a1, b1): $\mu \text{=}2$, (a2, b3)$\mu \text{=5}$, (a3,b3)$\mu \text{=8}$,(a4,b4) $\mu \text{=10}$

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It is well known that in conventional lasers, the linewidth enhancement factor $\alpha$, which quantify the coupling between amplitude and phase of the electric field, plays an important role in the dynamics [35]. It is of interest to study if this parameter plays a similar role in the free-running VCSEL dynamics. In Fig. 6, we present the results for the 0-1 test for chaos and $\text{PE}$in the (${\gamma }_s$,${\gamma }_p$) plane for a fixed value of pump current but several different values of $\alpha$. Comparison among these subfigures clearly indicates that increasing $\alpha$is beneficial to the enhancement of chaos region. In other words, it is always easier to obtain chaos in a free-running VCSEL with larger values of$\alpha$, which provides necessary instructions for laser design and fabrications for chaos-based applications.

 

Fig. 6 (a1-c1, a2-c2) The two-dimensional map of the 0-1 test for chaos and (a3-c3, a4-b4) $\text{PE}$of the free-running VCSEL with varying ${\gamma }_s$(x-axis) and ${\gamma }_p$(y-axis). Left: $\mu \text{=}5$; right: $\mu \text{=8}$. First row: $\alpha \text{=}1$, second: $\alpha \text{=2}$, third: $\alpha \text{=6}$.

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3.2 Enhanced chaotic dynamics in an optically injected VCSEL and their synchronization issue

As has been discussed above, although a free-running VCSEL allows for chaos generation in a wide range of parameters, some drawbacks still exist, such as low bandwidth and complexity. In this section, we first construct a master-slave configuration based on two free-running VCSELs and discuss bandwidth and complexity of the slave VCSEL, as well as synchronization between two VCSELs. Then, a high-quality chaos synchronization scheme for synchronizing enhanced dynamics generated by optically injected VCSELs is proposed and demonstrated numerically.

First, the bandwidth of the slave VCSEL is studied. The parameters for the master VCSEL modeled by Eqs. (1)-(3) are chosen such that the laser operates in a chaotic regime. The corresponding parameters are $\mu \text{=}2{,}_{}\alpha \text{=}3{,}_{}{\gamma }_p={25}_{}{\text{ns}}^{-1}$and ${\gamma }_s={20}_{}{\text{ns}}^{-1}$, and kept constant in what follows. For simplicity, we consider identical parameter values for the master and slave lasers, but the frequency detuning between them will also be introduced for the purpose of enhancing the output dynamics of the slave laser. It is expected that under proper conditions, chaos injection will gives rise to chaos operation in the slave VCSEL, which is modeled by Eqs. (4)-(6), in the wide parameter space (In fact, the slave laser also operates in a chaotic regime in the absence of injection for the considered parameters.). For this reason, the 0-1 test will not be applied to the slave VCSEL. Two examples of chaos bandwidth of the VCSEL are shown in Fig. 7, where only the results for RCP are shown since RCP and LCP exhibit identical dynamical behavior. Here the bandwidth is defined as the range between DC and the frequency that contains 80% of the spectral power [52]. In Fig. 7 (a) we fix the injection ratio $k_{in}$ but vary the frequency detuning $\Delta f$ in a wide range, i.e., $k_{in}={60}_{}{\text{ns}}^{\text{-}1}$. It is interesting to observe that one can increase the bandwidth by considering large values of frequency detuning, especially for a large positive detuning. This produces a concave shape, with its bottom corresponding to low bandwidth range equal to that of the master free running VCSEL. This is attributed to the injection-locking effect as observed in other conventional optically injected lasers [53]. The asymmetry of the curve shape is caused by the relatively large value of linewidth enhancement factor, i.e., $\alpha \text{=}3$, which accounts for the coupling between amplitude and phase of the electric field. Figure 7 (b) shows the variation of chaos bandwidth of the slave VCSEL as a function of the injection ratio $k_{in}$for a fixed frequency detuning $\Delta f\text{=}{30}_{}\text{GHz}$. It is clearly seen that for the considered positive detuning, as the injection ratio increases, the bandwidth is improved gradually and almost saturates at very large values of $k_{in}$.

 

Fig. 7 The bandwidth of the slave VCSEL as a function of (a) frequency detuning $\Delta f$and (b) injection strength $k_{in}$, where (a) $k_{in}=60{\text{ns}}^{\text{-1}}$, and (b) $\Delta f\text{=}{30}_{}\text{GHz}$. Other parameters are$\mu \text{=}2{,}_{}\alpha \text{=}3{,}_{}{\gamma }_p={25}_{}{\text{ns}}^{-1}$and${\gamma }_s={20}_{}{\text{ns}}^{-1}$.

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Figure 8 shows a two-dimensional map of the bandwidth of the slave VCSEL computed from the RCP. The horizontal axis corresponds to frequency detuning$\Delta f$, while the vertical axis to the injection ratio $k_{in}$. The V-shape blue area indicates the injection –locking region, where the master and slave VCSELs exhibit identical bandwidth. Outside this region, chaos bandwidth enhancement is observed and its asymmetric property is seen, where positive detuning is preferred for the enhancement of chaos bandwidth, in accordance with the observation in other optical injection systems [52]. This means that the injection scheme used in this study is highly effective for enhancing the limited bandwidth of a free-running VCSEL. In addition, these results further confirm those in Fig. 7.

 

Fig. 8 Two-dimensional map of the bandwidth of the slave VCSEL shown in the plane of ($\Delta f$,$k_{in}$). Other parameters are $\mu \text{=}2{,}_{}\alpha \text{=}3{,}_{}{\gamma }_p={25}_{}{\text{ns}}^{-1}$and${\gamma }_s={20}_{}{\text{ns}}^{-1}$.

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Next, we will study the unpredictability property of optical chaos generated by the slave VCSEL. To this end, we adopt $\text{PE}$, whose parameter setting is given in Section 2, to quantify chaos complexity. Figure 9 shows the $\text{PE}$ variation against either frequency detuning $\Delta f$ or injection strength $k_{in}$. The parameter setting is the same as that in Fig. 7. A comparison between Fig. 7 and Fig. 9 shows that one can expect the simultaneous enhancement of bandwidth and complexity under proper circumstances. We further present the two-dimensional $\text{PE}$ map in Fig. 10. As expected, higher $\text{PE}$values are obtained outside the injection-locking region, especially for positive detuning values. This trend coincides well with bandwidth variation shown in Fig. 8. This indicates that it is possible to obtain enhanced chaotic dynamics with broadband bandwidth >30 GHz and high complexity ($\text{PE}\text{~}1$) in wide regions of the parameter space of interest.

 

Fig. 9 $\text{PE}$ of the slave VCSEL as a function of (a) frequency detuning $\Delta f$and (b) injection strength $k_{in}$, where (a) $k_{in}={60}_{}{\text{ns}}^{-1}$ and (b) $\Delta f\text{=}{30}_{}\text{GHz}$. Other parameters are $\mu \text{=}2{,}_{}\alpha \text{=}3{,}_{}{\gamma }_p={25}_{}{\text{ns}}^{-1}$and${\gamma }_s={20}_{}{\text{ns}}^{-1}$.

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Fig. 10 Two-dimensional $\text{PE}$map computed from the slave VCSEL shown in the plane of ($\Delta f$,$k_{in}$). Other parameters are $\mu \text{=}2{,}_{}\alpha \text{=}3{,}_{}{\gamma }_p={25}_{}{\text{ns}}^{-1}$and${\gamma }_s={20}_{}{\text{ns}}^{-1}$.

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We further consider the correlation between the master and slave VCSELs. The reason is that we want to obtain high-quality chaos synchronization between optimized chaotic signals, i.e., exhibiting higher bandwidth and complexity. The cross-correlation coefficient given in Eq. (8) is used to quantify the correlation. The result is shown in Fig. 11. It is clear that high-quality chaos synchronization is seen only in the injection—locking region, where the correlation is extremely high ($C>0.95$). However, in this region, the bandwidth and complexity of the slave output are low due to the injection-locking effect. Outside this region, the slave output can be enhanced due to the interaction between the injection light and the electric field of the slave VCSEL. For these reasons, the master-slave configuration is not the desirable system for chaos-based communications which require chaos possessing high complexity and broadband bandwidth.

 

Fig. 11 The calculated cross-correlation coefficient shown in the plane of ($\Delta f$,$k_{in}$). Other parameters are $\mu \text{=}2{,}_{}\alpha \text{=}3{,}_{}{\gamma }_p={25}_{}{\text{ns}}^{-1}$and${\gamma }_s={20}_{}{\text{ns}}^{-1}$.

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Finally, we propose a novel chaotic system that allows for perfectly synchronizing enhanced chaotic signals generated by optically injected VCSELs. Specifically, it consists of three VCSELs, i.e., one free-running master VCSEL and two free-running slave VCSELs; both slaves are subject to unidirectional injection from the same master, while there is no direct link between the two slaves. In fact, such a synchronization and communication scheme has been applied to other lasers [54–56]. Figure 12 shows a typical example of high-quality chaos synchronization between the two slave lasers. The results for intensity time series, RF spectrum, and time-shift cross-correlation coefficient clearly indicate that the slave laser output is enhanced as expected and can be highly correlated under proper conditions, while their correlation to the master laser is rather low. The next step is to construct chaos-based communication based on the current synchronization scheme, which offers enhanced security and high-speed message exchange (either unidirectional or bidirectional) [56]. However, the corresponding study is beyond the scope of the current article and will be carried out elsewhere.

 

Fig. 12 (a) Intensity time traces and (b) RF spectrum of three VCSELs, (c) the correlation between any two lasers of the prosed system, where $\Delta f\text{=}{30}_{}\text{GHz}$ and $k_{in}={60}_{}{\text{ns}}^{-1}$. Other parameters are $\mu \text{=}2{,}_{}\alpha \text{=}3{,}_{}{\gamma }_p={25}_{}{\text{ns}}^{-1}$and ${\gamma }_s={20}_{}{\text{ns}}^{-1}$.

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4. Conclusion

In summary, we have studied the dynamics of free-running and optically injected VCSELs, with the help of the 0-1 test for chaos and $\text{PE}$. The effects of several key parameters on chaotic dynamics of VCSELs are clarified based on high-resolution two-dimensional color maps of these two measures. We have also considered the master-slave configuration, where the chaotic output of the slave VCSEL can be greatly enhanced under proper conditions. Finally, we have proposed a three-laser scheme to synchronize the bandwidth- enhanced and complexity-improved chaos generated by the slave VCSELs. Additionally, no time-delay signature is expected since neither feedback nor mutual coupling is introduced. Therefore, these findings are of interest for chaos-based applications, such as secure communications and random number generation.