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Abstract
Extreme events (EEs) are rare and unpredictable, as have been observed in nature. Up to now, manipulating EEs has remained a challenge. Here, we experimentally observe the enhancement of EEs in a three cascade-coupled semiconductor laser system. Specifically, a continuous-wave optical injection semiconductor laser acts as the chaotic source with rare EEs, which is subsequently injected into a second laser for increasing the number of EEs. Interestingly, we find that the number and region size of EEs can be further enhanced by sequentially injecting into a third laser, i.e., a cascade-injection structure. Our experimental observations are in good agreement with the numerical results, which indicate that EEs can be significantly enhanced in wide injection parameter space due to the cascade-injection effect. Furthermore, our simulations show that the evoluation of the regions with enhanced EEs may be associated with the noise considered.
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1. Introduction
Hurricanes, tsunamis, earthquakes, market crashes, population extinctions, and other extreme events (EEs) are catching lots of researchers’ eyes because of their destructive effects in real-world scenarios [1–5]. The concept of EEs, also called rogue waves (RWs) in oceanography, was initially proposed to describe ultrahigh waves in the deep ocean [6]. However, the intrinsic scarcity and the evident technical difficulties in the implementation of experiments limit researchers’ understanding of the mechanism of triggering RWs. In 2007, Solli et al. proposed an analogy between ocean RWs and light fields in optical fibers and thus established a useful test platform to detect RWs in a short time [7]. Since then, a new field has been opened up and the studies of EEs have been booming. On top of all that, researchers turned to the rare EEs in other systems, such as spatially extended systems, plasmas, and lasers with modulation loss, chaotic semiconductor lasers [8–17]. Among them, the chaotic semiconductor laser has attracted wide attention for studying EEs due to its simplicity and ease of implementation.
The majority of early studies have been focused on exploring the formation scheme of EEs. For example, Bonatto et al. experimentally observed the rare, high-amplitude pulses in the continuous-wave optical injection semiconductor laser (cw-OISL), and its deviation from the Gaussian distribution, which can be recognized as EEs [18]. After that, they revealed that the occurrence of EEs in a crisis-like process can be predicted for a long time and those EEs can be either enhanced or suppressed by introducing noise in the same cw-OISL system [19]. In addition to the cw-OISL, EEs have also been demonstrated in time-delayed systems [20–23], coupled laser arrays [11,12], semiconductor lasers with a saturable absorber [9,10], and other systems [24–30]. In recent years, much effort has been devoted to controlling the EE generation. For instance, Perrone et al. numerically demonstrated that EEs can be suppressed via direct current modulation [31]. Jin et al. showed that step-up perturbations of the pump current can trigger extreme pulses on demand with more than 50% probability in the same system [32]. Our group experimentally demonstrated the enhancement/suppression of EEs in a semiconductor laser subjected to chaotic optical injection [33]. Very recently, Li et al. found the enhanced EEs through injecting the output of the cw-OISL into the energy redistribution module, origining from the temporal energy redistribution of the chaotic emission waveform [34]. The above studies provide meaningful insights into the control of EEs and attract more attention to the fields.
In this paper, we experimentally and numerically demonstrate that the number of EEs can be enhanced in the three cascade-coupled semiconductor laser system. Herein, we select a cw-OISL as the chaotic source, where the EE is only found in limited injection parameters regions. Then, the chaotic output of the cw-OISL with rare EEs is injected into a second laser. By observing its output, we find that the number of EEs is increased in the certain injection parameters. Finally, the output of the second laser with EEs is sent to a third laser. We see that the number and occurrence regions are significantly enhanced in the injection parameter space. Additionally, we discuss the effect of the noise and injected light field on triggering EEs in the third laser and reveal that the evolution of the enhanced regions in the injection parameter plane differs for high-frequency detuning and low-frequency detuning as the noise strength increases. Therefore, our results provide more insights into the evolution of EEs in cascade-coupled semiconductor lasers.
2. Experimental setup
Figure 1 displays the experimental setup for three cascade-coupled semiconductor lasers. In this implementation, three commercial distributed feedback (DFB) semiconductor lasers (Wuhan 69 Inc) are used. Three lasers with a threshold current of approximately 12 mA are driven and controlled by a current source and a thermoelectric controller (ILX Lightwave LDC-3724B). Here, a master-slave architecture is built to generate the chaotic signal, which was widely used to observe EEs in previous works [18,19,31–35]. A narrow linewidth tunable laser (TL, Newkey Photonics NLC13) is employed as the master laser and its output is injected into the slave laser 1 (SL1) through the polarization controller (PC1), variable attenuators (VA1), and optical circulator (CIR1). The current and temperature of SL1 are fixed at 27 mA and 25 ℃, which leads to a lasing wavelength of 1551.170 nm. The frequency detuning $\Delta {f_1} = {f_{TL}} - {f_{SL1}}$ between the TL and SL1 can be adjusted by changing the lasing wavelength of TL. The output power P1 of TL is 11 μW (see point A). The output of SL1 is sent successively to PC2, VA2, CIR2, and SL2. The current of SL2 is also fixed at 27 mA but its temperature is adjusted in real-time to form the frequency detuning $\Delta {f_2} = {f_{SL1}} - {f_{SL2}}$ between the SL1 and SL2. The injection power P2 is measured from point B. Likewise, the output of SL2 is injected into SL3 through the PC3, VA3, and CIR3. The current of SL3 is also fixed at 27 mA but its temperature is varied to achieve the frequency detuning $\Delta {f_3} = {f_{SL2}} - {f_{SL3}}$ between the SL2 and SL3. The injection power P3 is measured from point C. Finally, the output of SL3 is transmitted into a photodetector (Fiber-Photonics, bandwidth: 31 GHz) for photoelectric transformation. The data is collected by a real-time oscilloscope (LeCroy WaveMaster820Zi-B, sampling rate of 80 GS/s, bandwidth 20 GHz).
Fig. 1. Schematic diagram of the experimental setup. TL: tunable laser; SL1: slave laser 1; SL2: slave laser 2; SL3: slave laser 3; PC: polarization controller; VA: variable attenuator; CIR: optical circulator; PD: photodetector; OSC: oscilloscope.
3. Theoretical model
To explore the evolution of EEs, we also simulate the theoretical model for three cascade-coupled semiconductor lasers based on the modified Lang-Kobayashi equations [36], which can be described as [18,19,37]:
To characterize the correlation between two lasers, we use the cross-correlation coefficient (CC), which is computed as follows [39–41]:
To quantitatively distinguish EEs from all events, we employ two common criteria. One threshold $T1$ is defined as the average height of pulses, $\left\langle H \right\rangle $, plus 6 times the standard deviation of the distribution of pulse height, $\sigma $[18,19,23,31,32,35]. An event is identified as an EE if its height exceeds the threshold. Another criterion is the abnormality index (AI). The AI of event n can be defined as $A{I_n} = H{f_n}/{H_{1/3}}$, where $H{f_n}$ is the difference between the peak height of the event and the mean height of all events in the time series and ${H_{1/3}}$ is the average value of the first third of the highest values of $H{f_n}$. Any event that yields an abnormality index greater than 2 is considered extreme [20–22,29,42,43]. Such a threshold is called $T2$ in the following.
Besides, we calculate the probability density function (PDF, which describes the probability of different pulse intensities)) of ${I_{\max }}$ to estimate the data extremeness. The Weibull distribution can be fitted to evidence the deviation from a negative exponential (k = 1) or a Rayleigh (k = 2) distribution, which is computed as follows [11]:
4. Results and discussion
4.1 Experimental results
In this section, we display the experimental observations for chaotic signals of three lasers in the window of $10\textrm{ }\mu \textrm{s}$ (we record the experimental data for $10\textrm{ }\mu \textrm{s}$), and the corresponding time series and PDFs of the peak intensity are shown in Fig. 2. First, we can see from Fig. 2(a) that the EE is hardly found using T1. By changing the injection parameters, the SL1 still works in a chaotic state, but we cannot see obvious EEs. However, we can observe the EEs based on T2, which sets a lower threshold. This result indicates that the generation of EEs usually needs careful selection of the laser injection parameters [43]. For comparison, Fig. 2(b) shows the output of SL2. One can see the chaotic waveform in Fig. 2(a1) is shifted into a waveform with a randomly generated giant pulse in Fig. 2(b1) and its amplitude is beyond the threshold T1. The long-tail PDF in Fig. 2(b2) also confirms the appearance of EEs in the output of SL2. Quantitatively, the number of EEs is counted to 818 using T1. Similar phenomena were also found in our previous work [33]. Further, the time series sampled from the SL3 is plotted in Fig. 2(c1), where the occurrence of EEs (the pulses exceed the T1) becomes more frequent and the T1 value becomes smaller in Fig. 2(c3), which causes the number of EEs significantly increases from 818 to 1721. This result indicates that the EEs can be dramatically enhanced in three cascade-coupled semiconductor laser systems. We speculate that the physical mechanism is the optical injection effect, i.e., the SL1with rare EEs is injected to the SL2, where new dynamics including high pulsing fluctuations can be generated outside of the injection-locking region by properly choosing the injection parameters. This is also the case for injecting the SL2 to SL3.
Fig. 2. Time series and PDFs of the peak intensity for (a) SL1, when the injection power ${P_1} = 11.15{\ \mathrm{\mu} \mathrm{W}},$ $\Delta {f_1} = \textrm{ 8 GHz}$, (b) SL2 when normalized injection power ${P_2} = 0.046,\Delta {f_2} ={-} 37\textrm{ GHz}$, and (c) SL3, when normalized injection power ${P_3} = 0.063,\Delta {f_3} ={-} 36\textrm{ GHz}$. The red and green dashed lines represent the threshold $T1$ and $T2$ for detecting EEs, respectively. The magenta curves denote the Weibull distributions from the peak intensity.
4.2 Numerical results
To gain more insights into the evolution of EEs, we also numerically investigate the output of three lasers in this section. First, we discuss the output of SL1 in injection parameter regions (${P_{inj}},\Delta {f_1}$), as shown in Fig. 3(a), where the time window for counting the number of EEs is set at 10 μs. Moreover, the 0-1 test for chaos is also performed to identify the chaotic region, and the result is shown in Fig. 3(b) [44]. From these figures, we can find that EEs only occur in specific injection parameters rather than all chaotic regions. This result is consistent with previous conclusions [18,19,31,35]. In Fig. 4(a), we depict the evolution of EEs as a function of the frequency detuning, where the injection strength is fixed at ${P_{inj}} = \textrm{60 n}{\textrm{s}^{ - 2}}$. One can see the EE is only triggered in a narrow window of the frequency detuning. For comparison, we select two injection parameters (${P_{inj}} = \textrm{60 n}{\textrm{s}^{ - 2}},\Delta {f_1} = 0.22\textrm{ GHz}$) and (${P_{inj}} = \textrm{60 n}{\textrm{s}^{ - 2}}, \Delta {f_1} = 0.35\textrm{ GHz}$). For the former, the EEs can be observed based on T1 as shown in Fig. 4(b). For the latter, EEs are not observed under either criterion.
Fig. 3. (a) Maps of the number of EEs generated from SL1 in the (${P_{inj}},\Delta {f_1}$) plane and (b) the corresponding 0-1 test for chaos.
Fig. 4. (a) The number of EEs generated from SL1 (the pulses exceed the threshold $T1$) as a function of $\Delta {f_1}$ in SL1. (b),(c) Typical time series of SL1. (b) $\Delta {f_1} = 0.22\textrm{ GHz}$, (c) $\Delta {f_1} = 0.35\textrm{ GHz}$, where ${P_{inj}} =$ $\textrm{60 n}{\textrm{s}^{ - 2}}$. The red and green dashed lines represent the threshold $T1$ and $T2$ for detecting EEs, respectively.
Next, we pay attention to the output of SL2 with the chaotic injection from SL1, where two different cases are considered, i.e., the chaotic injection signals with EEs [see Fig. 4(b)] or without EEs [see Fig. 4(c)]. As illustrated in Fig. 5(a), numerous pulses surpass the red threshold $T1$, indicating a clear and notable enhancement of EEs, with the number of EEs reaching 992. The PDF distinctly deviates from a Gaussian distribution. We conduct a fitting of the PDFs using the Weibull distribution, revealing a shape parameter value of 1.103. Similarly, in the scenario where there are no EEs in SL1, 524 pulses are observed exceeding the red threshold. In this case, the fitted shape parameter of the PDF in Fig. 5(b) is determined to be 1.783. From these results, we find that the generation of EEs can be enhanced by chaotic optical injection. A similar conclusion has been found in a DFB laser subject to chaotic optical injection from another DFB laser with optical feedback [33].
Figure 6 shows the results for the number of EEs in SL2 as a function of the injection strength ${k_{inj1}}$, where $\Delta {f_2} \in [ - 30,20]\textrm{ GHz}$. In the case of Fig. 6(a), where there are many EEs triggering in SL1, windows for the enhanced EEs (over the gray line) can be easily observed. For this case of $\Delta {f_2} = {-}30\,\textrm{GHz}$, the enhancement effect on EEs is dramatically larger than that in the other two cases. Moreover, the case of Fig. 6(b), where there are no EEs in SL1, can still generate a large number of EEs in SL2 while the other two cases demonstrate a limited capability to trigger EEs. Therefore, we infer that the negative detuning parameter ($\Delta {f_2} < 0\textrm{ GHz}$) has a larger and more obvious enhancement effect on EEs in SL2.
Fig. 5. Typical chaotic time series and PDFs of the peak intensity in SL2 with chaotic optical injection from SL1, where ${k_{inj1}} = 50\textrm{ n}{\textrm{s}^{\textrm{ - 1}}},\Delta {f_2} ={-} 30\textrm{ GHz}$. (a1),(a2) $\Delta {f_1} = 0.22\textrm{ GHz}$, (b1),(b2) $\Delta {f_1} = 0.35\textrm{ GHz}$. The red and green dashed lines represent the threshold $T1$ and $T2$ for detecting EEs, respectively. The magenta curves denote the Weibull distributions from the peak intensity.
Fig. 6. The number of EEs versus the injection strength ${k_{inj1}}$ in SL2, where (a) $\Delta {f_1} = 0.22\textrm{ GHz}$ and (b) $\Delta {f_1} = 0.35\textrm{ GHz}$. Gray lines represent the number of EEs generated in SL1.
To gain a global view of the effects of injection parameters, we plot two-dimensional maps of the number of EEs in SL2, defined by $T1$, and corresponding CC between SL1 and SL2 in the (${k_{inj1}},\Delta {f_2}$) plane with SL1 operating in two sets of detuning parameters, as displayed in Fig. 7. In Figs. 7(b1-b2), the $C{C_{SL1,SL2}}$ in the yellow region is greater than 0.7, indicating that there is a high correlation between the output of SL1 and SL2, which is caused by the injection-locking effect. In Fig. 7(a1), the number of EEs injected by SL1 corresponds to the blue region, and the remaining red and yellow regions indicate the enhancement of EEs in SL2. Compared with SL1, SL2 has a larger parameter region that triggers EEs. In particular, EEs in SL2 are significantly enhanced under negative detuning parameters in the low correlation region, which confirms our previous speculation. We also find the triggering and enhancement phenomenon of EEs in SL2 when the output of SL1 is chaotic but with no EEs occurring, as shown in Fig. 7(a2). As a result, we reveal that chaotic light injection can trigger or enhance EEs, regardless of the presence or absence of EEs in SL1.
Then, the output of SL2 is injected into SL3 to discuss the influence of further chaotic light injection (i.e., by injecting the chaotic output of SL2 into SL3) on EEs. Figure 8 displays typical time series and PDFs of SL3 when the output of SL2 with two sets of parameters in Fig. 4 is injected into SL3. Notably, the time series becomes more complex, and more pulses exceed the red threshold $T1$. The calculated number of EEs in SL3 is 1610 and 804, whereas the number of EEs in SL2 is 992 and 524, respectively. Besides, we also apply Weibull distribution for PDF fitting, and the values of ${k_{a,b}}$ are 1.040 and 1.931. These analyses indicate a further heightened triggering of EEs in SL3 through the introduction of the secondary chaotic injection.
Furthermore, we present the results depicting the number of EEs in SL3 as a function of the injection strength ${k_{inj2}}$ in Fig. 9, where $\Delta {f_3} \in [ - 35,15]\textrm{ GHz}$. For the case of $\Delta {f_3} = 0\textrm{ GHz}$ and $\Delta {f_3} = 15\textrm{ GHz}$, the number of EEs increases first and then decreases with the increase of injection strength ${k_{inj2}}$, and finally trends to stationary values, while for the case of $\Delta {f_3} ={-} 35\textrm{ GHz}$, the number of EEs exhibits an initial increase, followed by a decrease, then a subsequent rise, and ultimately concludes with a decrease. Regarding the enhancement of EEs, there are no windows for the enhanced EEs in the case of 15 GHz, while windows can be observed for the other two cases. For the case of $\Delta {f_3} = 0\textrm{ GHz}$, a small range of weak enhancement appears, which suggests a weak enhancement effect on EEs in SL3. For the case of $\Delta {f_3} ={-} 35\textrm{ GHz}$, two wider ranges of the injection strength ${k_{inj2}}$ for the enhancement and a more significant enhancement effect (more EEs are triggered) than the case of $\Delta {f_3} = 0\textrm{ GHz}$ are shown. It is noteworthy that the curves exhibit a similar trend when there are no EEs in SL1. Based on the above results, we consider that there would be a more diverse enhancement phenomenon of EEs under negative frequency detuning parameters ($\Delta {f_3} < 0\textrm{ GHz}$).
In order to analyze the further enhancement phenomenon in SL3 more comprehensively, we plot high-resolution two-dimensional maps of the number of EEs in SL3 and corresponding $C{C_{SL2,SL3}}$ in the (${k_{inj2}},\Delta {f_3}$) plane. In Fig. 10(a1), the number of EEs injected by SL2 corresponds to the green region, the blue region indicates the suppression of EEs, and the red region represents the enhancement. In Fig. 10(a1), two regions of enhancement in the plane are mainly found, one is located outside the yellow region in Fig. 10(b1) (low correlation region with $C{C_{SL2,SL3}}$ lower than 0.7), and the other is located inside the yellow region (high correlation region with $C{C_{SL2,SL3}}$ ranging from 0.7 to 0.95). Interestingly, we find that the enhancement mainly appears under negative frequency detuning parameters. In the low correlation region, the threshold decreases due to the lower average output power of SL3 under partial negative detuning parameters, thus there are more oscillating pulses counted as EEs in the same time interval. In the high correlation region, we also discover the enhancement, which, we believe, is mainly attributed to the strong interaction of light fields between SL2 and SL3. Since the injection strength is larger, EEs may also be triggered in SL3 under specific frequency detuning when the dynamics of EEs in SL2 are injected into SL3. Moreover, the complex chaotic dynamics of SL3 can generate additional EEs, thus leading to further enhancement. Likewise, the conclusion is true when there are no EEs triggered in SL1, as shown in Figs. 10(a2-b2). Based on the above discussion, we can conclude that the secondary chaotic injection can indeed further enhance the generation of EEs whether in the low correlation region or the high correlation region. We speculate that the occurrence of EEs is caused by the generated new nonlinear dynamics due to the interaction between the chaotic injection light field and the SL3. Under the scenario of cascade chaotic injection, the nonlinear dynamics became more complex/pulsing and thus may cause more EEs under certain injection parameters.
Finally, we investigate the role played by the noise on the further enhancement behavior of EEs in SL3 of the cascade-coupled laser system, inspired by some previous works [45,46]. Figure 11 illustrates the output of SL3 with different values of the noise strength. In Figs. 11(a1-c1), we find that the pulse intensity gradually decreases with the increase of the noise strength, and the long-tail feature in the PDF diagram is diminished, the fitted values of ${k_{a,b,c}}$ are computed as 1.083, 1.297 and 1.236. This means that the interplay between the noise and injected light field can regulate and control the triggering of EEs. To further study such regulation phenomenon, we display maps of the number of EEs in the (${k_{inj2}},\Delta {f_3}$) plane under several noise strength values in Fig. 12. It is noticed that the red region corresponding to the enhancement with high $\Delta {f_3}$ (0 GHz∼40 GHz, -20 GHz∼-40 GHz) is shrinking with stronger noise. However, the case of low $\Delta {f_3}$ is a little complex. With moderate noise strength, the red-enhanced region shrinks while it expands when considering stronger noise.
5. Conclusion
In summary, we have experimentally and numerically investigated the triggering of EEs in three cascade-coupled semiconductor lasers. Note that a small number of EEs are observed in the output of the chaotic source (SL1). When the chaotic signal is cascade injected into the SL3, the number and generation region can be further enhanced compared with SL2. Besides, we have also discussed the effect of the noise strength on the enhancement behavior of EEs and found that the enhanced regions corresponding to high and low $\Delta {f_3}$ evolve differently in the injection parameter plane with the increase of the noise strength. Different from previous works in EEs [18–23,31–35,42], we have focused on the the manipulation of EEs by optical injection and further enhancement by a cascade injection structure. Such a mechanism may be suitable for any optical injection systems. Therefore, our results are important for gaining further insights into the generation and controlling of EEs, as well as provide some useful theoretical guidance for optimizing chaos-based applications where EEs should be avoided, such as the generation of high-quality random numbers [44].
Fig. 7. (a1),(a2) Maps of the number of EEs generated from SL2 in the (${k_{inj1}},\Delta {f_2}$) plane and (b1),(b2) corresponding CC between the SL2 and SL1. (a) $\Delta {f_1} = 0.22\textrm{ GHz}$, (b) $\Delta {f_1} = 0.35\textrm{ GHz}$.
Fig. 8. Typical time series and PDFs of the peak intensity in SL3 with chaotic optical injection from SL2, where ${k_{inj1}} = 50\,\textrm{n}{\textrm{s}^{\textrm{-1}}},\,\Delta {f_2} ={-}30\,\textrm{GHz},\,{k_{inj2}} = 55\,\textrm{n}{\textrm{s}^{\textrm{-1}}},\,\Delta {f_3} ={-}35\,\textrm{GHz}$. (a1),(a2) $\Delta {f_1} = 0.22\textrm{ GHz}$, (b1),(b2) $\Delta {f_1} = 0.35\textrm{ GHz}$. The red and green dashed lines represent the threshold $T1$ and $T2$ for detecting EEs, respectively. The magenta curves denote the Weibull distributions from the peak intensity.
Fig. 9. The number of EEs versus the injection strength ${k_{inj2}}$ in SL3, where (a) $\Delta {f_1} = 0.22\textrm{ GHz}$ and (b) $\Delta {f_1} = 0.35\textrm{ GHz}$. Gray lines represent the number of EEs generated in SL2.
Fig. 10. (a1),(a2) Maps of the number of EEs generated from SL3 in the (${k_{inj2}},\Delta {f_3}$) plane and (b1),(b2) corresponding CC between the SL2 and SL3. (a) $\Delta {f_1} = 0.22\textrm{ GHz}$, (b) $\Delta {f_1} = 0.35\textrm{ GHz}$.
Fig. 11. Typical time series and PDFs of the peak intensity in SL3 with chaotic optical injection from SL2, and the noise strength is (a) $D = {10^{ - 3}}\textrm{ n}{\textrm{s}^{ - 1}}$, (b) $D = {10^{ - 2}}\textrm{ n}{\textrm{s}^{ - 1}}$, and (c) $D = {10^{ - 1}}\textrm{ n}{\textrm{s}^{ - 1}}$. The red and green dashed lines represent the threshold $T1$ and $T2$ for detecting EEs, respectively. The magenta curves denote the Weibull distributions from the peak intensity. The other parameters are the same as those in Fig. 8(a).
Fig. 12. Maps of the number of EEs generated from SL3 in the (${k_{inj2}},\Delta {f_3}$) plane, where (a) $D = {10^{ - 3}}\textrm{ n}{\textrm{s}^{ - 1}}$, (b) $D = {10^{ - 2}}\textrm{ n}{\textrm{s}^{ - 1}}$, and (c) $D = {10^{ - 1}}\textrm{ n}{\textrm{s}^{ - 1}}$. The other parameters are the same as those in Fig. 8(a).
Funding
National Natural Science Foundation of China (62001317, 62004135, 62111530301, 62171305); Innovative and Entrepreneurial Talent Program of Jiangsu Province (JSSCRC2021527); The Postdoctoral Fellowship Program of CPSF (GZC20231883).
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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