Open Access
Add to BibSonomy
Abstract
Soliton pulsation is one of the most fascinating phenomena in ultrafast fiber lasers, owing to its rich nonlinear dynamics and potential generation of high peak power pulse. However, it is still a challenge to efficiently search for pulsating soliton in fiber lasers because it requires a fine setting of laser cavity parameters. Here, we report the autosetting soliton pulsation in a passively mode-locked fiber laser. The parameters of electronic polarization controller are intelligently adjusted to search for pulsating soliton state by the improved depth-first search algorithm. Moreover, the intensity modulation depth of pulsating soliton could be flexibly controlled. These findings indicate that the intelligent control of a fiber laser is an effective way to explore on-demand soliton dynamics and is also beneficial to the optimization of ultrafast laser performance.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Passively mode-locked fiber lasers have attracted much attention due to their versatile applications ranging from fundamental science to industrial purposes [1–4], i.e., material processing, biomedicine, and nonlinear microscopy. The rapid developments of optical fiber fabrication and laser technologies lead to the great advances of ultrafast fiber laser performance. Thus, the fiber lasers are regarded as an alternative to the traditional solid-state ultrafast lasers [5]. In fact, in addition to an ultrashort pulse laser source, the ultrafast fiber laser is also a good platform for investigating the soliton nonlinear dynamics [6]. To date, different soliton dynamics have been observed in fiber lasers by properly adjusting the cavity parameters, such as soliton molecules [7–10], dissipative soliton resonance [11,12] and vector solitons [13–15]. As one of the most intriguing nonlinear dynamics in fiber lasers, the soliton pulsating behavior features the periodically oscillating localized structures when the soliton propagates in the laser cavity [16]. Indeed, based on the complex Ginzburg-Landau equation, a variety of pulsating structures have been revealed [17,18]. Apart from the fruitful dynamics of pulsating solitons, it was also found that the pulsating behavior can be used to generate ultrashort pulse with extremely high peak power for a given pump power lever [19]. Therefore, the investigation of soliton pulsation is meaningful to the fundamental physics and practical applications. In particular, by virtue of dispersive Fourier transformation (DFT) method, the real-time spectral dynamics of pulsating solitons could be identified, and thus, to further unveil the pulsating behavior [20–22].
On the other hand, in comparison to the ultrafast fiber lasers based on real saturable absorber, the polarization controller (PC) in fiber lasers mode-locked by nonlinear polarization rotation (NPR) offers one more degree of freedom to adjust the laser operation regime. Therefore, the mode-locked fiber lasers with NPR technique are generally employed to investigate different soliton dynamics. Recently, relying on the cavity feedback control and intelligent algorithms, the electronic polarization controller (EPC) enables the automatic mode locking and control of ultrafast fiber lasers [23–26]. Aiming at shortening the time of automatic mode locking [27–29] and tuning the mode locking status (i.e., pulse duration and spectral bandwidth) [24,28], versatile intelligent control algorithms have been proposed to drive the polarization searching through EPC in the laser cavity. Moreover, in combination with the real-time spectral analysis based on DFT, it is also able to realize the fast control of the mode-locked spectral bandwidth as well as pulse duration [30,31]. As for the pulsating solitons with desired parameters, the only way to achieve is precisely adjusting the cavity parameters, such as pump power and polarization states [32]. However, due to the unclear relationship between cavity parameters and output results, the generation of soliton pulsation is somewhat random and time consuming. Considering the significance of soliton pulsating behavior and systematical investigation of pulsating soliton dynamics, it would be interesting to develop the intelligent identification and control of pulsating solitons in ultrafast fiber lasers.
In this work, for the first time, we demonstrated the autosetting soliton pulsation in an NPR mode-locked fiber laser by improved depth-first search (DFS) algorithm. The automatic control of operation regime in the fiber laser was realized by intelligent polarization tuning through an EPC, which imitates the manually adjusting process of experienced experimenter. In the improved DFS algorithm, the search step for pulsating soliton regime could be dynamically adjusted according to the current matching degree. The pulsating solitons with different intensity modulation depths could be repeatedly achieved. We believe that the proposed intelligent control of the pulsating solitons would find potential applications in exploration of target soliton nonlinear dynamics and generation of high peak power pulses in fiber lasers.
2. Experimental setup and principle
2.1 Experimental setup
Here, a dual-algorithm mechanism was employed to effectively obtain the pulsating solitons, as shown in Fig. 1(a). Firstly, the stable mode-locked regime is obtained through the evolutionary algorithm, which has been proved to be the fast way to search for mode locking state [27,28]. For the soliton pulsating states, it is demonstrated that the cavity generally locates around those of mode locking states. Therefore, after achieving the stable mode-locked state, we employed the improved DFS algorithm to intelligently locate the cavity parameters for the pulsating soliton states. The operation principle of the DFS algorithm is to generate a descendant of the most recently expanded node, and explore as far as possible along each branch before trying the next branch over [33]. However, different from the traditional DFS algorithm, the improved one introduces reward and punishment mechanisms to search for pulsating solitons more efficiently, as depicted in Fig. 1(b). Here, the improved DFS algorithm always takes the mode-locked point as the root node, and explores along different branch paths until the end of depth. Due to the introduction of reward and punishment mechanism, the search step for pulsating soliton regime could be dynamically adjusted according to the current matching degree. When the operation state is close to the target pulsating soliton state, the search step would be automatically shortened (reward). On the contrary, the search step would be extended (punishment). Thus, it could greatly improve the search efficiency.
Fig. 1. Flowchart for automatic search of mode locked state and pulsating soliton state. (a) Flowchart of dual-algorithm mechanism, where the first step is necessary for initialization; (b) Flowchart of the improved DFS algorithm, which introduces the reward and punishment mechanism on the basis of the traditional DFS algorithm.
The schematic of automatic mode-locked fiber laser is shown in Fig. 2(a). The fiber laser consists of a 9-m long erbium-doped fiber (EDF) with a dispersion parameter D≈-15ps/nm/km and a 2.1-m long single-mode fiber (SMF) with a dispersion parameter D=17 ps/nm/km, leading to the net-normal cavity dispersion. The polarization-dependent isolator (PD-ISO) ensures the unidirectional propagation of the fiber laser. An EPC (Phoenix Photonics, EPC-15-1-1-2) is used to adjust the polarization state of propagating light in the cavity, while the full polarization state control is realized through applying different DC voltages on it. Since the fiber laser is mode-locked by NPR technique, the operation regime could be flexibly controlled by adjusting the polarization state of the propagating light inside the laser cavity. The laser feedback system is composed of a high-speed ADC data acquisition card, an industrial personal computer (IPC) and a high-precision DAC module. The ADC acquisition card is a data sampling, sending and signal processing board equipped with a high-speed ADC. The ADC acquisition card is a data sampling, sending and signal processing board equipped with a high-speed ADC (5 GSps sampling rate and 10 bit sampling accuracy, 2 GHz bandwidth, maximum signal input voltage range ±250 mV), a Stratix5 high-configuration FPGA chip and a 2Gbyte DDR3 memory chip. The IPC, acting as a computing center, runs the evolutionary algorithm and improved DFS algorithm. The IPC receives data collected by the high-speed ADC acquisition card. After processing the data by the algorithm, IPC will output instructions as feedback signal to control the subsequent DAC (DAC8563 with 16bit resolution) function on EPC. The laser performance is also checked through a 20% fiber coupler which is simultaneously measured by an optical spectrum analyzer (OSA, Yokogawa, AQ6317C) and an oscilloscope (Tektronix, DSA-70804) with photodetector (Newport, 818-BB-35F, 12.5 GHz bandwidth, 1000-1650 nm operation waveband). For simplicity, these measurement devices outside the cavity are not illustrated in Fig. 2(a).
Fig. 2. (a) Schematic of the NPR-based mode-locked fiber laser for intelligently searching short-pulse regimes; (b)-(c) Acquisition schematics of the mode-locked regime and soliton pulsation regime. The black solid line represents the actual profile of the pulse train, and the blue frame is the separation limit of the acquired data processing. The black and red small circles represent the scattered sampling points of the high-speed ADC acquisition card and the pulse amplitude, respectively. The red and blue dashed lines represent the pulse average amplitude threshold (80% of the mode-locking pulse amplitude) and the pulse counting threshold (20% of the mode-locking pulse amplitude), respectively.
2.2 Discrimination of the mode-locked regime
The signals detected by the high-speed ADC data acquisition card are discrete voltage points, as the circles shown in Figs. 2(b) and 2(c), while the black line represents the actual signal waveform. Here, the collected data is uniformly divided according to the cavity roundtrip time, as indicated by the blue border. It should be noted that in the mode locking and soliton pulsation discrimination diagrams shown in Figs. 2(b)-(c), the values of the sampling point higher than the pulse count thresholds 2 and 4 are recognized as mode-locked pulses, otherwise they are categorized into noise [29]. For a preset pump power in a fiber laser, generally the achievable average amplitude of the single-pulse mode locking regime could be predicted. Then the average pulse amplitude (denoted as ${A_{average}}$) could be given by:
Here, ${A_i}$ represents the maximum value inside each blue border, which corresponds to the pulse amplitude. N is the number of splits for the collected data in the algorithm. The average pulse amplitude is an important basis for identifying the mode-locking state in the fiber laser. However, it is not enough to precisely distinguish the mode-locked regime only with the average pulse amplitude. Because other types of mode-locking regimes such as the noise-like pulse and pulsating soliton could also satisfy the criterion. In this case, firstly an expected average pulse amplitude, i.e., threshold 1 in Fig. 2(b), should be preset according to the pump power level. When the average pulse amplitude is larger than the preset threshold 1, the pulse uniformity and the repetition rate should be considered to check whether the fiber laser operates in stable single-pulse regime. The pulse repetition rate could be obtained by counting the number of the mode-locked pulses with amplitudes larger than threshold 2 in each blue border. If the single-pulse regime is confirmed, the amplitude jitter of the mode-locked pulse train should be further checked, which is defined as: Here, $Max({{A_1},{A_2},\ldots ,{A_N}} )$ represents the maximum value among the collected data, while $Min({{A_1},{A_2},\ldots ,{A_N}} )$ represents the minimum value. The pulse amplitude jitter indicates the stability of the mode-locking state to some extent. The smaller the pulse amplitude jitter is, the more likely it is to be the stable mode-locking regime. To track the stable single-pulse mode locked state, we define a composite fitness function related to average pulse amplitude, pulse count and amplitude jitter:2.3 Discrimination of the pulsating soliton regime
After obtaining the stable single-pulse mode locking operation, generally the pulsating soliton can be observed by further adjusting the soliton peak power. In fact, one pronounced characteristic of the pulsating soliton is that the amplitude of the output pulse periodically changes with time, as shown in in Fig. 2(c). Therefore, the discrimination of pulsating soliton regime can be preliminarily realized through the pulse amplitude jitter. If the amplitude jitter is detected in the collected data, we need to further check that the amplitude jitter is periodic or not. Provided that the pulse amplitude jitter is periodic, it can be confirmed that the soliton pulsation is obtained in the fiber laser. Moreover, the intensity modulation depth is also an important parameter for pulsating solitons. Here, we define the modulation depth as the ratio of the peak-to-valley value to their sum, which satisfies:
3. Results and discussion
3.1 Autosetting mode-locked soliton with the evolutionary algorithm
In order to investigate the passive mode locking by evolutionary algorithm, firstly the pump power was adjusted to 29.1 mW, above the mode-locked threshold. In the experiments, the initial polarization state was set to a random state. When applying the composite fitness function-based evolutionary algorithm to the laser, the autosetting of single-pulse mode-locking could be achieved. The evolution curves are shown in Fig. 3, which correspond to the maximum fitness value (black squared points) and the average fitness value (red round points), respectively. As mentioned above, the fitness value is defined jointly by the average pulse amplitude, pulse count and amplitude jitter. It can be seen that with the increasing generation number, the evolution of the maximum and average fitness values are as expected. The maximum fitness value increases rapidly and then flattens out. There is a small decline in the first few generations, such as the third and sixth generations. Compared to the maximum fitness value, the average one increases slower. However, it is still able to converge to the optimum fitness value for stable mode-locking.
Fig. 3. Convergence of the evolutionary algorithm with the maximum (squared black points) and average (red round points) fitness defined by the pulse amplitude, pulse count and pulse amplitude jitter.
As an example, Fig. 4 illustrates the laser performance when the maximum fitness value evolved after 20 generations. Figure 4(a) shows the mode-locked pulse train. It can be seen that the intensity of the pulse train is uniform and the pulse amplitude jitter is small. The radio frequency (RF) spectrum of the passive mode locking was measured, as depicted in Fig. 4(b). The RF peak locates at the fundamental repetition rate of 18.65 MHz with a signal-to-noise ratio (SNR) of >50 dB, showing that the fiber laser operates in the stable single-pulse regime. The measured autocorrelation trace indicates that the pulse duration is 8.94 ps, if the Gaussian profile is assumed, as plotted in Fig. 4(c). Figure 4(d) demonstrates that the mode-locked pulse centers at 1556.69 nm. Note that the mode-locked spectrum features the rectangular profile covering a spectral range from 1544.88 nm to 1568.50 nm, which is a typical characteristic of net-normal dispersion fiber lasers [34]. These findings indicate that the proposed composite fitness function-based evolutionary algorithm could be employed to efficiently generate stable single-pulse mode locking in the fiber laser.
Fig. 4. Experimental results of the fundamental mode-locked regime. (a) Pulse train; (b) RF spectrum; (c) Autocorrelation trace; (d) Spectrum.
3.2 Autosetting soliton pulsation with the improved DSF algorithm
Generally, the pulsating soliton can be obtained if the soliton average power is properly adjusted after the single-pulse mode locking, because the origin of soliton pulsating behavior is from the imbalance between nonlinear effect and dispersion. In addition, it is demonstrated that tuning polarization state of mode-locked pulse leads to the adjustment of cavity loss (soliton power) in NPR-based fiber laser [16]. Therefore, when the stable mode-locked operation was achieved, the current drive voltages of the EPC and pump power level were used as the initial conditions for searching pulsating soliton regime. Then we reset the target pulse state to the pulsating soliton regime and ran the improved DFS algorithm. In this algorithm, it firstly explores the polarization setting of any one of the 6 directions from the stable mode-locked point, which corresponds to the increase/decrease of the drive voltages of the EPC. Therefore, the soliton average power could be dynamically adjusted with the variable drive voltages of the EPC, which is beneficial to observe pulsating soliton. After searching process with improved DSF algorithm, the soliton intensity changes periodically from one round trip to another, while keeping staying in the single-pulse regime. This phenomenon indicates the pulsating soliton was generated from the fiber laser. It should be noted that there is a range of pulsation modulation depth in each fiber laser and it’s ∼0-0.4 in our case. Then, a series of target spacings for the modulation depth are preset as 0-0.1, 0.1-0.2, 0.2-0.3, 0.3-0.4, respectively. Otherwise, the algorithm will fail to find the target soliton pulsation states. When the soliton pulsation state with the modulation depth in target spacing is found, the searching process will stop and monitoring mode will start. These results are shown in Fig. 5. As can be seen here, the modulation depth of the pulsating soliton is adjusted from 0.042 to 0.363 according to the preset target pulsating soliton status. Here, it should be addressed that the small modulation depth, such as 0.042 in Fig. 5(a), and the preset fitness value (90% of the ideal one) are not contradictory. On one hand, the fitness value is influenced not only by amplitude jitter, but also average amplitude and pulse count. On the other hand, the modulation depth of 0.042 is achieved after the confirmation of soliton pulsation state with periodic amplitude jitter. In addition, note that the pulsating periods are variable with different modulation depths, which is a general behavior of pulsating with fixed pump power. Figure 6 shows two typical shot-to-shot spectra, corresponding pulsating soliton states of Figs. 5(a) and (d). The pulsating characteristics could be obviously seen here. Then the spectral properties of the pulsating soliton were investigated. It is observed that the bandwidth of the mode-locked spectrum is increased with increasing modulation depth, as shown in Fig. 7. In particular, the two sharp edges of the rectangular mode-locked spectrum gradually bend, which is resulted from the averaged effect of the spectral bandwidth breathing dynamics [35]. These results demonstrate that the soliton pulsation state with specific modulation depth could be efficiently located by the improved DFS algorithm.
Fig. 5. Pulsating solitons with different modulation depths. (a) D=0.042; (b) D=0.153; (c) D=0.268; (d) D=0.363.
Fig. 6. Shot-to-shot spectra of pulsating soliton states of Figs. 5(a) and (d) after DFT process. (a) D=0.042; (b) D = 0.363.
3.3 Discussion
In the experiments, the dual-algorithm mechanism was employed to achieve the stable single-pulse and pulsating soliton operations. That is, firstly the automatic mode locking of the fiber laser was enabled by evolutionary algorithm. Then considering the characteristics of the pulsating solitons, the improved DFS algorithm was applied to obtain the pulsating soliton. The spectral bandwidth of pulsating soliton will vary with the modulation depth. The variation of spectral bandwidth is attributed to the spectral (temporal) breathing dynamics. Recently, the DFT technique has been widely used to characterize the real-time spectral dynamics of mode-locked soliton, enabling the single-shot spectral measurement by an oscilloscope [36–40]. Therefore, if the modulation depth of the pulsating soliton could be mapped into the spectral breathing ratio, the automatic searching of soliton pulsation with target modulation depth could be also obtained by establishing the fitness function based on real-time spectral characteristics. In addition, the pulsating period changes when approaching the pulsating soliton states with different modulation depths, which could be attributed to sole polarization state adjustment with the fixed pump power. In fact, both of the polarization state and pump power have influence on the pulsating period and modulation depth of pulsating solitons [18]. Therefore, it is expected that the coordinated tuning of pump power and polarization state by the improved DFS algorithm could keep the pulsating period unchanged when applying different modulation depths on the pulsating soliton. In our experiment, the autosetting of soliton pulsation was investigated in ultrafast fiber laser with net-normal dispersion regime, where the single soliton pulsation could be achieved more easily than that in the anomalous dispersion one. In fact, the improved DFS algorithm could be applied to various pulsating soliton types. Therefore, it is expected that the autosetting of other types of pulsating solitons, such as dual soliton pulsation and double-period pulsating soliton can be explored. In these cases, more parameters should be involved in the algorithm to set the corresponding target pulsating soliton states. Finally, only the automatic searching of the soliton pulsating operation was considered in this work. Note that many soliton nonlinear phenomena existed in passively mode-locked fiber lasers, such as multi-soliton patterns [7–10], vector solitons [13,14] and dissipative soliton resonance [11,12]. Thus, future efforts could be also directed to the autosetting of other soliton nonlinear dynamics, which will pave the way for the investigation of soliton nonlinear behavior in a more efficient manner.
4. Conclusion
In summary, we have investigated the automatic mode locking and soliton pulsation in a fiber laser by evolutionary algorithm and improved DFS algorithm, respectively. The cavity feedback is controlled intelligently by the EPC. Particularly, the autosetting soliton pulsation with target modulation depth could be realized by applying the composite fitness function. These findings indicate that the desirable nonlinear soliton dynamics could be efficiently obtained by intelligent control of a fiber laser, which will prove to be meaningful to the various communities interested in soliton nonlinear dynamics and fiber laser technologies.
Funding
Key-Area Research and Development Program of Guangdong Province (2018B090904003, 2020B090922006); National Natural Science Foundation of China (11874018, 11974006, 61805084, 61875058); Guangzhou Science and Technology Program key projects (2019050001); Basic and Applied Basic Research Foundation of Guangdong Province (2019A1515010879, 2021A1515012315); Open Fund of State Key Laboratory of Advanced Optical Communication Systems and Networks (2020GZKF010); Shenzhen-Hong Kong Cooperation Zone for Technology and Innovation (HZQB-KCZYB-2020082).
Disclosures
The authors declare that there are no conflicts of interest to this article.
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.
References
1. J. Koch, E. Fadeeva, M. Engelbrecht, C. Ruffert, H. H. Gatzen, A. Ostendorf, and B. N. Chichkov, “Maskless nonlinear lithography with femtosecond laser pulses,” Appl. Phys. A 82(1), 23–26 (2006). [CrossRef]
2. B. Öktem, I. Pavlov, S. Ilday, H. Kalaycıoğlu, A. Rybak, S. Yavaş, M. Erdoğan, and F. Ö. Ilday, “Nonlinear laser lithography for indefinitely large-area nanostructuring with femtosecond pulses,” Nat. Photonics 7(11), 897–901 (2013). [CrossRef]
3. N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, and C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013). [CrossRef]
4. C. Kong, C. Pilger, H. Hachmeister, X. Wei, T. H. Cheung, C. S. W. Lai, N. P. Lee, K. K. Tsia, K. K. Y. Wong, and T. Huser, “High-contrast, fast chemical imaging by coherent Raman scattering using a self-synchronized two-colour fibre laser,” Light: Sci. Appl. 9(1), 25 (2020). [CrossRef]
5. F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1-2), 58–73 (2008). [CrossRef]
6. Ph. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012). [CrossRef]
7. Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, “Phase-locked soliton pairs in a stretched-pulse fiber laser,” Opt. Lett. 27(11), 966–968 (2002). [CrossRef]
8. M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett. 95(14), 143902 (2005). [CrossRef]
9. A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative soliton molecules with independently evolving or flipping phases in mode- locked fiber lasers,” Phys. Rev. A 80(4), 043829 (2009). [CrossRef]
10. B. Ortaç, A. Zaviyalov, C. K. Nielsen, O. Egorov, R. Iliew, J. Limpert, F. Lederer, and A. Tünnermann, “Observation of soliton molecules with independently evolving phase in a mode-locked fiber laser,” Opt. Lett. 35(10), 1578–1580 (2010). [CrossRef]
11. W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78(2), 023830 (2008). [CrossRef]
12. Z. C. Luo, W. J. Cao, Z. B. Lin, Z. R. Cai, A. P. Luo, and W. C. Xu, “Pulse dynamics of dissipative soliton resonance with large duration-tuning range in a fiber ring laser,” Opt. Lett. 37(22), 4777–4779 (2012). [CrossRef]
13. S. T. Cundiff, B. C. Collings, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Observation of polarization-locked vector solitons in an optical fiber,” Phys. Rev. Lett. 82(20), 3988–3991 (1999). [CrossRef]
14. L. M. Zhao, D. Y. Tang, H. Zhang, X. Wu, and N. Xiang, “Soliton trapping in fiber lasers,” Opt. Express 16(13), 9528–9533 (2008). [CrossRef]
15. Q. Y. Ning, H. Liu, X. W. Zheng, W. Yu, A. P. Luo, X. G. Huang, Z. C. Luo, W. C. Xu, S. H. Xu, and Z. M. Yang, “Vector nature of multi-soliton patterns in a passively mode-locked figure-eight fiber laser,” Opt. Express 22(10), 11900–11911 (2014). [CrossRef]
16. B. Zhao, D. Y. Tang, L. M. Zhao, and P. Shum, “Pulse-train nonuniformity in a fiber soliton ring laser mode-locked by using the nonlinear polarization rotation technique,” Phys. Rev. A 69(4), 043808 (2004). [CrossRef]
17. N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63(5), 056602 (2001). [CrossRef]
18. J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70(6), 066612 (2004). [CrossRef]
19. W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme soliton pulsations in dissipative systems,” Phys. Rev. E 92(2), 022926 (2015). [CrossRef]
20. Z. W. Wei, M. Liu, S. X. Ming, A. P. Luo, W. C. Xu, and Z. C. Luo, “Pulsating soliton with chaotic behavior in a fiber laser,” Opt. Lett. 43(24), 5965–5968 (2018). [CrossRef]
21. Z. Wang, Z. Wang, Y. Liu, R. He, J. Zhao, G. Wang, and G. Yang, “Self-organized compound pattern and pulsation of dissipative solitons in a passively mode-locked fiber laser,” Opt. Lett. 43(3), 478–481 (2018). [CrossRef]
22. Y. Du, Z. Xu, and X. Shu, “Spatio-spectral dynamics of the pulsating dissipative solitons in a normal-dispersion fiber laser,” Opt. Lett. 43(15), 3602–3605 (2018). [CrossRef]
23. T. Hellwig, T. Walbaum, P. Groß, and C. Fallnich, “Automated characterization and alignment of passively mode-locked fiber lasers based on nonlinear polarization rotation,” Appl. Phys. B 101(3), 565–570 (2010). [CrossRef]
24. X. Shen, W. Li, M. Yan, and H. Zeng, “Electronic control of nonlinear-polarization-rotation mode locking in Yb-doped fiber lasers,” Opt. Lett. 37(16), 3426 (2012). [CrossRef]
25. D. Radnatarov, S. Khripunov, S. Kobtsev, A. Ivanenko, and S. Kukarin, “Automatic electronic-controlled mode locking self-start in fibre lasers with non-linear polarisation evolution,” Opt. Express 21(18), 20626–20631 (2013). [CrossRef]
26. S. L. Brunton, X. Fu, and J. N. Kutz, “Extremum-seeking control of a mode-locked laser,” IEEE J. Quantum Electron. 49(10), 852–861 (2013). [CrossRef]
27. U. Andral, R. Si Fodil, F. Amrani, F. Billard, E. Hertz, and P. Grelu, “Fiber laser mode locked through an evolutionary algorithm,” Optica 2(4), 275–278 (2015). [CrossRef]
28. R. I. Woodward and E. J. R. Kelleher, “Towards ‘smart lasers’: self-optimisation of an ultrafast pulse source using a genetic algorithm,” Sci. Rep. 6(1), 37616 (2016). [CrossRef]
29. G. Q. Pu, L. L. Yi, L. Zhang, and W. S. Hu, “Intelligent programmable mode-locked fiber laser with a human-like algorithm,” Optica 6(3), 362–369 (2019). [CrossRef]
30. G. Pu, L. Yi, L. Zhang, C. Luo, Z. Li, and W. Hu, “Intelligent control of mode-locked femtosecond pulses by time-stretch-assisted real-time spectral analysis,” Light: Sci. Appl. 9(1), 13 (2020). [CrossRef]
31. J. Girardot, F. Billard, A. Coillet, É. Hertz, and Ph. Grelu, “Autosetting mode-locked laser using an evolutionary algorithm and time-stretch spectral characterization,” IEEE J. Sel. Top. Quantum Electron. 26(5), 1–8 (2020). [CrossRef]
32. R. J. Deissler and H. R. Brand, “Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. Lett. 72(4), 478–481 (1994). [CrossRef]
33. R. E. Korf, “Depth-first iterative-deepening: An optimal admissible tree search,” Artificial Intelligence 27(1), 97–109 (1985). [CrossRef]
34. A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14(21), 10095–10100 (2006). [CrossRef]
35. H. J. Chen, Y. J. Tan, J. G. Long, W. C. Chen, W. Y. Hong, H. Cui, A. P. Luo, Z. C. Luo, and W. C. Xu, “Dynamical diversity of pulsating solitons in a fiber laser,” Opt. Express 27(20), 28507–28522 (2019). [CrossRef]
36. G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017). [CrossRef]
37. K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118(24), 243901 (2017). [CrossRef]
38. X. M. Liu, X. Yao, and Y. D. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018). [CrossRef]
39. M. Liu, Z. W. Wei, H. Li, T. J. Li, A. P. Luo, W. C. Xu, and Z. C. Luo, “Visualizing the “invisible” soliton pulsation in an ultrafast laser,” Laser Photonics Rev. 14(4), 1900317 (2020). [CrossRef]
40. Z. W. He, Y. Q. Du, C. Zeng, B. Q. Jiang, D. Mao, Z. P. Sun, and J. L. Zhao, “Soliton metamorphosis dynamics in ultrafast fiber lasers,” Phys. Rev. A 103(5), 053516 (2021). [CrossRef]
