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Abstract
We report on the manipulation of extreme events (EEs) in a slave spin-polarized vertical-cavity surface-emitting laser (spin-VCSEL) subject to chaotic optical injection from a master spin-VCSEL. The master laser is free-running but yielding a chaotic regime with obvious EEs, while the slave laser originally (i.e., without external injection) operates in either continuous-wave (CW), period-one (P1), period-two (P2), or a chaotic state. We systematically investigate the influence of injection parameters, i.e., injection strength and frequency detuning, on the characteristics of EEs. We find that injection parameters can regularly trigger, enhance, or suppress the relative number of EEs in the slave spin-VCSEL, where the large ranges of enhanced vectorial EEs and average intensity of both vectorial and scalar EEs can be achieved with suitable parameter conditions. Moreover, with the help of two-dimensional correlation maps, we confirm that the probability of occurrence of EEs in the slave spin-VCSEL is associated with the injection locking regions, outside which enhanced relative number of EEs regions can be obtained and expanded with augmenting the complexity of the initial dynamic state of the slave spin-VCSEL.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Rogue waves (RWs), characterized by high impact, abnormal magnitude, unpredictability, and deviation from Gaussian statistics, have been receiving a great deal of attention in systems of natural, technical, and societal settings [1]. The first observation of RWs was on the ocean, which caused numerous shipwrecks and loss of human lives [2]. Nevertheless, it is difficult to thoroughly investigate the RW phenomena in a natural environment owing to the unpredictable and intrinsic rareness properties of RWs. Remarkably, in laser systems, the external pumping and gain media are analogous to the wind on seawater. More importantly, the nonlinear Schrödinger equation used to describe pulse propagation in optical fibers is the same equation that oceanographers utilize to describe water waves. Therefore, many investigations have focused on optical rogue waves (ORWs), i.e., extreme events (EEs), in laser systems. In 2007, Solli et al. first reported the appearance of ORWs in the process of supercontinuum generation in optical fibers [3]. The pioneering research has resulted in the flourish of the studies of EEs in optical systems, such as a glass fiber [4], random media [5], and a tapered graded-index nonlinear fiber [6]. In addition to the optical systems mentioned above, much effort has been devoted to studying EEs generated in semiconductor lasers. On the one hand, the study of the behavior of EEs can serve as a powerful tool to understand the underlying physics in different semiconductor laser situations. On the other hand, semiconductor lasers displaying various dynamical states including chaos are excellent and controllable platforms to study and predict extreme fluctuations under controlled conditions. EEs generated from semiconductor lasers can be used in practical applications for imaging and sensing. For example, controlling the occurrence of pulse with ultra-high intensity can reduce signal-to-noise ratio and thus improve the image quality. For sensing, a small change in a parameter can be easily detected because a pulse with high peak is generated at the same time. Therefore, EEs have been widely presented in extensive different types of configurations based on semiconductor lasers [1,7–22].
One category is focused on understanding the mechanisms of triggering the EEs. For example, Bonatto et al. experimentally and theoretically investigated the giant intensity pulses identified as RWs in an optically injected semiconductor laser and they provided a framework for understanding the observed ORWs as the result of a deterministic nonlinear process [7]. Afterwards, they found that these ORWs could be predicted over a long expected period of time, which were generated by a crisis-like process and were enhanced or suppressed through employing the noise [10]. In addition, Mercier et al. numerically demonstrated the appearance of EEs deviating the intensity statistics to Gaussian-shape statistics in a laser diode with phase-conjugate feedback. They confirmed that EEs emerged from a sequence of bifurcations on self-pulsating solutions, i.e., the external cavity modes [13]. Another category is concentrated on the formation and control management of EEs. For example, Dal Bosco et al. reported the production of EEs in a diode laser subject to phase-conjugate feedback. They demonstrated the effect of increasing the feedback strength on the properties of EEs, such as a transition to non-Gaussian statistics of the pulse intensity, an increased number of EEs, and a deviation from the log-Poisson statistics of the time between EEs [8]. Perrone et al. numerically studied the EEs in an optically injected semiconductor laser via direct current modulation. The results showed that under the scenario of an adequate range of frequency and amplitude parameters, the EEs could be completely suppressed by the modulation, which has motivated many experimental and theoretical investigations in other natural systems [12]. Most of them indeed offer promising platforms to research EEs, but they are all based on conventional semiconductor lasers, which generate chaos only when external perturbation exists. An intensive and comprehensive study of the appearance of EEs in a new type of semiconductor lasers, i.e., spin-polarized vertical-cavity surface-emitting lasers (spin-VCSELs) is still not carried out.
As is known, a spin-VCSEL is a new type of spin laser and a specific example of spintronic devices that evinces various superiorities, including low threshold, ultra-fast polarization dynamics, and flexible control of the lasing output [23–25]. Especially, a spin-VCSEL can yield polarization chaos with no need for any external perturbation [26], which has huge potential in multitudinous fields, such as secure communication, secure key distribution, and lidar ranging [27–29]. Benefiting from these merits, it is believed that spin-VCSELs can be applied in a wide range of fields, such as the investigation of EEs. Interestingly, VCSELs are known to present multi-mode emission, which could potentially lead to the appearance of vectorial EEs. In 2017, Uy et al. reported the occurrence of vectorial and scalar EEs in the polarization dynamics of VCSELs with optical feedback, which separately correspond to the emission of a high-power pulse in both linear polarizations (LPs) simultaneously and in single LP [17]. Inspired by this research, we preliminarily demonstrated vectorial and scalar EEs in the chaotic dynamics of a free-running spin-VCSEL [22], which might provide a novel platform for the study of EEs with a simple structure and open up new research fields into spin-VCSELs. Although our previous work showed that the excitation of EEs including vectorial and scalar EEs can be facilitated by adjusting pump power and pump ellipticity, an interesting question has not been addressed: can these EEs be regularly triggered, enhanced, or suppressed by injecting into another solitary spin-VCSEL under different initial dynamical states?
In this paper, we introduce chaotic optical injection to realize the controlment of EEs in the slave spin-VCSEL. Herein, the master spin-VCSEL exhibits in chaotic regimes with EEs and the slave spin-VCSEL operates in various different dynamical regimes, including continuous-wave (CW), period-one (P1), period-two (P2), and chaotic states, which can be divided into four configurations, i.e., a chaotic spin-VCSEL unidirectionally injecting to a CW spin-VCSEL (configuration 1), a P1 spin-VCSEL (configuration 2), a P2 spin-VCSEL (configuration 3), and a chaotic spin-VCSEL (configuration 4). We focus on the influences of the injection strength and frequency detuning on the relative number of EEs, the relative number of vectorial EEs, and the average intensity of vectorial EEs and scalar EEs. Our results show that injection parameters display an important influence on the evolution of EEs including vectorial and scalar EEs for four distinct configurations, where the enhancement and suppression of the relative number of EEs (all EEs or vectorial EEs) can be regularly controlled. Compared with the master spin-VCSEL, the average intensity of both vectorial and scalar EEs in the slave spin-VCSEL can be significantly enhanced under suitable parameter conditions. More specifically, the two-dimension maps of maximum cross-correlation coefficient (CC) and the relative number of EEs in the injection parameter space are investigated, from which we can conclude that the variation of EEs in the slave spin-VCSEL is related to the injection locking regions.
2. Theoretical model
In our simulation, the spin-flip model (SFM) is employed to describe the nonlinear dynamics of the free-running spin-VCSEL, i.e., the master spin-VCSEL in a master-slave configuration, which is given in Eqs. (1–3) [22,30,31]. The associated rate equations for the slave spin-VCSEL subject to optical injection can be expressed by Eqs. (4–6) [32–34]. The details of these equations are given in the following:
To characterize the correlation between two lasers, we consider the CC, which is calculated as follows [28,32,37]:
To distinguish EEs from all other events, we employ one common criterion abnormality index ($AI$), which has been widely used in previous investigations [13,17,22]. The $AI$ of event n can be defined as:
where $H{f_n}$ represents the difference between the peak height of the event n and the mean height of all events in the time series and ${H_{1/3}}$ stands for the average value of the first third of the highest values of $H{f_n}$. When $AI \ge 2$, an event corresponding to the height of the peak intensity can be considered as an EE. Moreover, the relative number of events referring to the proportion of all the peaks in the total intensity ${I_{tot}}$ or the modal intensities ${I_{X, Y}}$, the relative number of EEs representing the proportion of EEs in all peaks in the total intensity ${I_{tot}}$ or the modal intensities ${I_{X, Y}}$, the relative number of vectorial EEs corresponding to the proportion of vectorial EEs among all EEs, and the average intensity of vectorial EEs and scalar EEs are analyzed to evaluate the characteristics of EEs generated in the slave spin-VCSEL [17,22].3. Results and discussion
3.1 Dynamic characteristics of the master and slave spin-VCSELs
Figure 1 shows the dynamic characteristics of the total intensity ${I_{tot}}$ and the modal intensities ${I_{X, Y}}$ of the free-running master and slave spin-VCSELs in detail through time series, where ${I_{tot}} = {I_X} + {I_Y}$. As shown in Figs. 1(a1)–(a3), for ${\eta ^M} = 2.96$, ${P^M} ={-} 0.22$, the dynamic behavior of the free-running master spin-VCSEL presents a chaotic state, where random-like intensity pulses are displayed. We would like to emphasize that EEs can be detected under the selected conditions, which are the same as those given in Fig. 2 in Ref. [17]. Under different values of ${\eta ^S}$ and ${P^S}$, the free-running slave spin-VCSEL can operate in various dynamical regimes, including CW, P1, P2, and chaotic states. The time sequences with a constant intensity, two intensity extrema, four intensity extrema, and even more extrema can be observed from Figs. 1(b)–(e), respectively. In the following analyses, we are interested in whether the occurrence of EEs generated by the master spin-VCSEL can lead to the appearance, enhancement, or suppression of EEs in the slave spin-VCSEL operating in different initial states as shown in Figs. 1(b)–(e). Therefore, for comparison purposes, four configurations, i.e., a chaotic spin-VCSEL unidirectionally injecting to a CW spin-VCSEL (configuration 1), a P1 spin-VCSEL (configuration 2), a P2 spin-VCSEL (configuration 3), and a chaotic spin-VCSEL (configuration 4), are comprehensively investigated.
Fig. 1. Time series of the total intensity ${I_{tot}}$ and the modal intensities ${I_{X, Y}}$ in different dynamic states of (a) the master spin-VCSEL and (b-e) the slave spin-VCSEL: (a) ${\eta ^M} = 2.96$, ${P^M} ={-} 0.22$, (b) ${\eta ^S} = 2.96$, ${P^S} ={-} 0.8$, (c) ${\eta ^S} = 2.96$, ${P^S} ={-} 0.6$, (d) ${\eta ^S} = 3.34$, ${P^S} ={-} 0.21$, (e) ${\eta ^S} = 3$, ${P^S} ={-} 0.08$, which corresponds to chaos, CW, P1, P2, and chaos states, respectively.
Fig. 2. (a1-d1, a3-d3) Chaotic time series and (a2-d2, a4-d4) evolution of the distribution of the relative number of events as a function of peak intensity in ${I_{tot}}$ of the slave spin-VCSEL with chaotic optical injection, where the free-running slave spin-VCSEL operates in (a) CW, (b) P1, (c) P2, and (d) chaos, respectively. (a1-d1, a2-d2) $\mathrm{\Delta }v ={-} 20\;\textrm{GHz}$, ${k_{inj}} = 10\;\textrm{n}{\textrm{s}^{ - 1}}$, (a3-d3, a4-d4) $\mathrm{\Delta }v ={-} 20\;\textrm{GHz}$, ${k_{inj}} = 60\;\textrm{n}{\textrm{s}^{ - 1}}$. The red lines correspond to the threshold of $AI = 2$.
3.2 Effects of the injection parameters on the characteristics of EEs
To begin with, we study the effects of different injection conditions on the chaotic time series and the relative number distribution of events of peak intensity in ${I_{tot}}$ generated from the slave spin-VCSEL in different initial dynamic states. Some representative examples are presented in Fig. 2, in which the first and second columns include the results of EEs generated from the slave spin-VCSEL with the selected frequency detuning of $\Delta v = -20\;\textrm{GHz}$ and injection strength of ${k_{inj}} = 10\;\textrm{n}{\textrm{s}^{ - 1}}$, and the third and fourth columns show the results of a larger injection strength of ${k_{inj}} = 60\;\textrm{n}{\textrm{s}^{ - 1}}$. Under the scenarios of configurations 1 and 2, the weak fluctuations around the mean value are found in Figs. 2(a1) and 2(b1) for ${k_{inj}} = 10\;\textrm{n}{\textrm{s}^{ - 1}}$, whereas more spiking pulses much higher than the mean value appear, i.e., more EEs, take place in Figs. 2(a3) and 2(b3) for ${k_{inj}} = 60\;\textrm{n}{\textrm{s}^{ - 1}}$. Correspondingly, one can see almost symmetrical distributions of peak intensities in ${I_{tot}}$ of the slave spin-VCSEL as shown in Figs. 2(a2) and 2(b2), which further confirms the rare EEs for ${k_{inj}} = 10\;\textrm{n}{\textrm{s}^{ - 1}}$; when the injection strength is increased to $60\;\textrm{n}{\textrm{s}^{ - 1}}$, the distributions in Figs. 2(a4) and 2(b4) deviate significantly from the Gaussian distribution and exhibit distinct long tails, differing from those in Figs. 2 (a2) and 2(b2) and indicating the appearance of more EEs. Nevertheless, in the cases of configurations 3 and 4 [see Figs. 2(c) and 2(d)], rich EEs are always obtained for the two considered injection strengths, which means that increasing ${k_{inj}}$ has no obvious influence on the characteristics of EEs under the situation considered. Therefore, it is of interest to know the characteristics of EEs as the injection strength ${k_{inj}}$ is varied continuously in a wide range for the four configurations proposed. Moreover, the distribution of the time separation of two sequential EEs is also an important feature of EEs, which has been widely reported in optical feedback systems and sole laser system [8,13,17,22]. Herein, we plot the typical distributions of the time separation in our proposed chaotic optical injection system in Fig. 3, i.e., the distribution of the waiting time ${\omega _n} = \textrm{log}({{t_{n + 1}}/{t_n}} )$ between two successive EEs appearing at times ${t_n}$ and ${t_{n + 1}}$. It can be seen that for four configurations mentioned above, the statistical properties of the time separation between continuous EEs follow two distinct log-Poisson laws with different slopes.
Fig. 3. Statistical distributions of the waiting time in a logarithmic scale between consecutive EEs in ${I_{tot}}$ for four different configurations: (a) configuration 1, (b) configuration 2, (c) configuration 3, and (d) configuration 4, where $\mathrm{\Delta }v ={-} 20\;\textrm{GHz}$, ${k_{inj}} = 60\;\textrm{n}{\textrm{s}^{ - 1}}$. The dashed red lines are fitting curves.
In the following, we further investigate the evolution of the relative number of EEs generated in the slave spin-VCSEL when the injection strength ${k_{inj}}$ is used as a control parameter for three values of the frequency detuning $\Delta v$. For comparison purposes, the relative numbers of EEs in the master spin-VCSEL are shown in the gray dashed lines of Figs. 4, 5, and 6. The case of $\Delta v ={-} 20\;\textrm{GHz}$ is shown in Fig. 4, from which we can discover several phenomena. First, in all configurations considered, as ${k_{inj}}$ is varied, the evolution of the relative numbers of EEs in ${I_{tot}}$, ${I_X}$, and ${I_Y}$ show a common feature that the relative number of EEs reaches a maximal value when the injection strength increases to a certain value. In addition, in the selected range of ${k_{inj}}$, compared with the master spin-VCSEL, the relative number of EEs in the slave spin-VCSEL can be suppressed in some regions of ${k_{inj}}$ and be enhanced in the other regions. In other words, the injection strength ${k_{inj}}$ can act as a control parameter for enhancing or suppressing the generation of EEs in the slave spin-VCSEL. Second, when the initial state complexity of the slave spin-VCSEL is enhanced, the relative number of EEs can reach a maximum value at a smaller ${k_{inj}}$, and the corresponding maximum value gradually decreases. Third, it can be found that the influence of ${k_{inj}}$ on the relative number of EEs gradually weakens with the increase of the initial state complexity of the slave spin-VCSEL. For example, for configurations 1 and 2, a significant enhancement or suppression region can be discovered in Figs. 4(a1)–(a3) and 4(b1)–(b3), whereas the effect of ${k_{inj}}$ diminishes significantly in the cases of configurations 3 and 4 as shown in Figs. 4(c1)–(c3) and 4(d1)–(d3).
Fig. 4. Relative number of EEs as a function of ${k_{inj}}$ in (a1-d1) the total intensity ${I_{tot}}$, (a2-d2) the X LP intensity ${I_X}$, and (a3-d3) the Y LP intensity ${I_Y}$ with $\mathrm{\Delta }v ={-} 20\;\textrm{GHz}$, where the free-running slave spin-VCSEL operates in (a) CW, (b) P1, (c) P2, and (d) chaos, respectively. The gray dashed lines represent the relative number of EEs in each polarization mode of the master spin-VCSEL.
Fig. 5. Relative number of EEs as a function of ${k_{inj}}$ in (a1-d1) the total intensity ${I_{tot}}$, (a2-d2) the X LP intensity ${I_X}$, and (a3-d3) the Y LP intensity ${I_Y}$ with $\mathrm{\Delta }v = 0\;\textrm{GHz}$, where the free-running slave spin-VCSEL operates in (a) CW, (b) P1, (c) P2, and (d) chaos, respectively. The gray dashed lines represent the relative number of EEs in each polarization mode of the master spin-VCSEL.
Fig. 6. Relative number of EEs as a function of ${k_{inj}}$ in (a1-d1) the total intensity ${I_{tot}}$, (a2-d2) the X LP intensity ${I_X}$, and (a3-d3) the Y LP intensity ${I_Y}$ with $\mathrm{\Delta }v = 30\;\textrm{GHz}$, where the free-running slave spin-VCSEL operates in (a) CW, (b) P1, (c) P2, and (d) chaos, respectively. The gray dashed lines represent the relative number of EEs in each polarization mode of the master spin-VCSEL.
Figure 5 shows the situation of $\Delta v = 0\;\textrm{GHz}$. Similarly, for the four configurations, the regions of enhanced and suppressed relative number of EEs can be controlled by adjusting ${k_{inj}}$. However, compared with the case of negative detuning, e.g., $\Delta v ={-} 20\;\textrm{GHz}$, the relative number of EEs in the slave spin-VCSEL does not increase or decrease significantly, which is basically close to that in the master spin-VCSEL. This indicates that the frequency detuning also has an impact on the relative number of EEs to some extent, and the selection of $\Delta v = 0\;\textrm{GHz}$ may lead to a weak effect of ${k_{inj}}$ on the variation of the relative number of EEs. Moreover, it is also worth noting that the closer the complexity of the initial state of the slave spin-VCSEL is, the more similar the evolution of the relative number of EEs is. For instance, the relative numbers of EEs in the cases of configurations 1 and 2 show very similar developing trends, which can also be confirmed under the scenarios of configurations 3 and 4.
Considering the frequency detuning of 30 GHz, Fig. 6 shows the calculated relative number of EEs of the slave spin-VCSEL in the proposed four configurations. In every case considered here, similar phenomena are observed in the total intensity ${I_{tot}}$, the X LP intensity ${I_X}$, and the Y LP intensity ${I_Y}$. Nevertheless, for the four different configurations, the evolution of the relative number of EEs shows different or even completely opposite trends, i.e., it is gradually descending with increasing ${k_{inj}}$ as shown in Fig. 6(a), while it is first growing in a small range of ${k_{inj}}$ and then saturating as presented in Figs. 6(c) and 6(d). Remarkably, the variation trends for configurations 3 and 4 are almost identical, illustrating that enhancing the complexity of the initial state of the spin-VCSEL has almost no effect on the relative number of EEs under the given injection parameters, for the considered frequency detuning. In addition, we can discover that when the complexity of the initial state of the slave spin-VCSEL is increased, the region of the enhanced relative number of EEs expands gradually, even can be realized over the whole parameter range of ${k_{inj}}$, as shown in Figs. 6(c2)–(c3) and 6(d2)–(d3). The above results show that, the two injection parameters, i.e., the frequency detuning and injection strength, have an important influence on the relative number of EEs for four distinct configurations, where the evolution of the relative number of EEs can be controlled by manipulating them regularly.
In our previous investigation [22], vectorial EEs appearing simultaneously in each linear polarization component is a very interesting and sound discovery in the spin-VCSEL. Herein, we would like to further study the evolution of vectorial EEs in the spin-VCSEL for the four configurations. In Fig. 7, we plot the relative number of vectorial EEs in the total intensity ${I_{tot}}$ as ${k_{inj}}$ is varied, which evolves in a similar way to that of all EEs in ${I_{tot}}$. From the example of the selected frequency detuning of $\Delta v ={-} 20\;\textrm{GHz}$ as shown in Figs. 7(a1)–(d1), we can see that the relative numbers of vectorial EEs in the cases of four configurations show analogous evolution tendencies: first growing and then gradually decreasing with increasing the injection strength. Moreover, with the increase of the initial state complexity of the slave spin-VCSEL, the relative number of vectorial EEs can reach a maximum value at a smaller injection strength. This interesting feature can also be identified in Figs. 4(a1)–(d1). For comparison purposes, the relative number of vectorial EEs in the master spin-VCSEL is also presented in the gray dashed lines in Fig. 7. It can be found that the relative number of vectorial EEs is enhanced almost throughout the considered range of ${k_{inj}}$. For the case of $\Delta v = 0\;\textrm{GHz}$, the relative number of vectorial EEs grows rapidly in the range of the weak injection strength and then declines for the strong injection strength. As shown in Figs. 7(a2)–(d2), the window for the enhanced vectorial EEs shrinks in size when the initial state of the slave spin-VCSEL becomes more complex. As for the situation of $\Delta v = 30\;\textrm{GHz}$, different from the cases of $\Delta v ={-} 20\;\textrm{GHz}$ and $\Delta v = 0\;\textrm{GHz}$, for the four different configurations, the relative numbers of vectorial EEs exhibit inconsistent evolvement regulations: gradually saturating and then rapidly decreasing in Fig. 7(a3), first greatly increasing and then gradually reducing after reaching its maximum in Fig. 7(b3), and basically remaining unchanged in Figs. 7(c3) and 7(d3). Likewise, the relative number of vectorial EEs is almost enhanced in the whole parameter range. From these results, we can conclude that for the four configurations, injection parameters show different influences on the variations of the relative number of vectorial EEs. Under suitable parameter conditions, a large range of enhanced vectorial EEs can be obtained in all four cases, especially in configurations 1 and 2.
Fig. 7. Relative number of vectorial EEs as a function of ${k_{inj}}$ in the total intensity ${I_{tot}}$, where (a1-d1) $\mathrm{\Delta }v ={-} 20\;\textrm{GHz}$, (a2-d2) $\mathrm{\Delta }v = 0\;\textrm{GHz}$, and (a3-d3) $\mathrm{\Delta }v = 30\;\textrm{GHz}$. The free-running slave spin-VCSEL operates in (a) CW, (b) P1, (c) P2, and (d) chaos, respectively. The gray dashed lines represent the relative number of vectorial EEs in total polarization mode of the master spin-VCSEL.
We are also interested in revealing the effect of injection parameters on average intensity of vectorial EEs and scalar EEs. We monitor this property in Fig. 8 by plotting average intensity of vectorial EEs and scalar EEs detected in the total intensity ${I_{tot}}$ as a function of ${k_{inj}}$ when $\Delta v ={-} 20\;\textrm{GHz}$, $\Delta v = 0\;\textrm{GHz}$, and $\Delta v = 30\;\textrm{GHz}.$ We notice that the average intensities of vectorial EEs are consistently more intense for the entire range of ${k_{inj}}$ through comparing with the scalar counterparts, except the case where vectorial EEs are completely suppressed to zero. In addition, by comparison to the master spin-VCSEL, the average intensity of both vectorial and scalar EEs in slave spin-VCSEL can be significantly enhanced when the injection parameters are set properly.
Fig. 8. Average intensity of vectorial EEs (diamonds) and scalar EEs (crosses) as a function of ${k_{inj}}$ in the total intensity ${I_{tot}}$, where (a1-d1) $\mathrm{\Delta }v ={-} 20\;\textrm{GHz}$, (a2-d2) $\mathrm{\Delta }v = 0\;\textrm{GHz}$, and (a3-d3) $\mathrm{\Delta }v = 30\;\textrm{GHz}$. The free-running slave spin-VCSEL operates in (a) CW, (b) P1, (c) P2, and (d) chaos, respectively. The black dashed lines and gray dashed lines represent average intensity of vectorial EEs and scalar EEs in the total polarization mode of the master spin-VCSEL, respectively.
To obtain a global view of the effects of the injection strength and the frequency detuning on the properties of EEs in the slave spin-VCSEL, we plot two-dimensional maps of the relative number of EEs in the $({{k_{inj}},\; \Delta v} )$ plane as shown in Fig. 9, where the blue-green region indicates that the relative number of EEs in the slave spin-VCSEL is similar to that in the master spin-VCSEL (the relative numbers of EEs in total polarization mode, X LP mode, and Y LP mode of the master spin-VCSEL are 2.71%, 2.23%, and 2.66%, respectively), and the dark red region corresponds to the relative number of EEs enhanced. Considering that the occurrence of EEs in the slave lasers may be associated with the correlations between the master and slave lasers, we also display a global view of the maximum CC in the parameter space of the injection parameters. As shown in Fig. 10, where the maximum CC of the total LP, X LP, and Y LP computed from two spin-VCSEL are included in the first column, second column, and third column, respectively. It is clear that a high correlation can be only seen in the injection locking region, which is marked in dark red. From the comparison of four configurations, we can discover that the more complex the initial dynamic state of the slave laser is, the larger the injection locking region is. Remarkably, within the injection locking region, the relative number of EEs in the slave spin-VCSEL is close to that in the master spin-VCSEL. Nevertheless, outside the injection locking region, there exists significantly enhanced relative number of EEs regions, which expand with the enhancement of the complexity of the initial dynamic state of the slave spin-VCSEL.
Fig. 9. Maps of the relative number of EEs in the $({{k_{inj}}, \Delta v} )$ plane. (a1-d1) The total intensity ${I_{tot}}$, (a2-d2) the X LP intensity ${I_X}$, and (a3-d3) the Y LP intensity ${I_Y}$, (a) CW, (b) P1, (c) P2, and (d) chaos.
Fig. 10. Maps of the maximum CC between the master spin-VCSEL and slave spin-VCSEL in the $({{k_{inj}}, \Delta v} )$ plane. (a1-d1) The total intensity ${I_{tot}}$, (a2-d2) the X LP intensity ${I_X}$, and (a3-d3) the Y LP intensity ${I_Y}$, (a) CW, (b) P1, (c) P2, and (d) chaos.
4. Conclusion
In conclusion, we have numerically investigated the generation of EEs based on a spin-VCSEL subject to chaotic optical injection. Herein, we consider four different configurations, i.e., a chaotic spin-VCSEL unidirectionally injecting to a CW spin-VCSEL (configuration 1), a P1 spin-VCSEL (configuration 2), a P2 spin-VCSEL (configuration 3), and a chaotic spin-VCSEL (configuration 4). There are some important features that can be identified. Firstly, injection parameters, i.e., the frequency detuning and injection intensity, have different effects on the relative number of EEs in the total polarization, X LP, and Y LP modes of the slave laser, where regular regulation of the EEs can be achieved. Secondly, the variation of the relative number of vectorial EEs in the total polarization mode is consistent with that of all EEs. In addition, a large range of enhanced vectorial EEs can be obtained from all four cases by flexibly adjusting the injection parameters, especially for configuration 1. Thirdly, the comparison of the average intensities of vectorial EEs and scalar EEs shows that the former are consistently more intense than the latter, except for the case where vectorial EEs are completely suppressed to zero. Moreover, the average intensity of both vectorial and scalar EEs in the slave spin-VCSEL can be significantly enhanced as the injection parameters are set suitably. Finally, as the initial dynamical state of the slave spin-VCSEL becomes increasingly complex, the region of high correlation between the master laser and the slave laser will become larger, outside which there exists enhanced relative number of EEs regions expanding with increasing the complexity of the initial dynamic state of the slave spin-VCSEL. These results are important for providing an intensive and comprehensive study of the appearance of EEs in spin-VCSELs and can be extended to other injection systems for controlling the likelihood of EEs.
Funding
National Natural Science Foundation of China (62001317, 62004135, 62111530301, 62171305); Natural Science Research of Jiangsu Higher Education Institutions of China (20KJA416001, 20KJB510011); Natural Science Foundation of Jiangsu Province (BK20200855).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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