Taking advantage of technology of spatio-temporal reconstruction and dispersive Fourier transform (DFT), we experimentally observed the buildup dynamics of dissipative soliton in an ultrafast fiber laser in the net-normal dispersion regime. The soliton buildup dynamics were analyzed in both the spectral and temporal domains. We firstly revealed that the appearing of the spectral sharp peaks with oscillation structures during the mode-locking transition is caused by the formation of structural dissipative soliton. The experimental results were explained by the numerical simulations. These findings would give some new insights into the dissipative soliton buildup dynamics in ultrafast fiber lasers.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Optical solitons, as localized waveforms arising from the balance of the nonlinear and dispersive interactions, have attracted much attention due to their important fundamental science and various applications. The passively mode-locked fiber lasers are regarded as the excellent tools for generating optical solitons as well as exploring soliton dynamics. So far, different pulse dynamics in ultrafast fiber lasers in different operation regimes have been reported. In an anomalous dispersion mode-locked fiber laser, the conventional soliton can be formed by balancing the group velocity dispersion (GVD) and the self-phase modulation (SPM) . It is well known that the energy achievable in conventional soliton laser systems is limited by the pulse breaking effect due to the soliton area theorem . To increase the soliton energy from a mode-locked fiber laser, the dispersion engineering of a laser cavity was proposed. It was demonstrated that, depending on the cavity dispersion, dissipative soliton can be formed in a fiber laser operating in the net-normal or all-normal dispersion regimes. Being different from the conventional soliton, the stable propagation of the dissipative soliton depends not only on the balance between the nonlinearity and dispersion but also on the gain and loss [3–7]. Therefore, from the viewpoint of theoretical prediction, the dissipative solitons would exhibit more complex dynamics comparing to the conventional solitons. On the other hand, the abundant nonlinear dynamics facilitate the various applications of dissipative solitons. Indeed, high performance pulse relying on the dissipative feature can be achieved from dissipative soliton fiber lasers [8–10], i.e., high-energy pulses based on wave-breaking-free effect. Therefore, unveiling the dissipative soliton dynamics in fiber lasers would be meaningful for both the fundamental physics and practical applications.
So far, in addition to the investigations aiming on higher pulse energy, many highlights striking emerging soliton dynamics have been reported in dissipative soliton fiber lasers, such as dissipative soliton molecules [11,12], dissipative soliton rain [13–15] and dissipative soliton resonance [16–19]. As a basic but important phenomenon of dissipative soliton, the soliton buildup dynamics could be used to understand the formation of a mode-locked pulse. Earlier studies on the starting dynamics of mode-locked laser systems were mostly focused on optimizing the self-starting performance by versatile approaches, for example, minimizing the pulse chirp , maximizing both the nonlinear mode variations and dynamic loss modulation , adjusting the carrier lifetime and the top reflector of the semiconductor saturable absorber . In addition, investigations on threshold power for the self-starting passive mode locking were also reported [23,24]. Recent reports have also investigated the self-starting time of harmonic mode-locking pulses , and the influence of nonlinearity and cavity dispersion on the pulse growth dynamics in ultrafast lasers . While the above mentioned efforts have mostly been made for the conventional solitons, less attention were paid to the buildup dynamics of the dissipative soliton. Recently, the research on the starting dynamics of dissipative solitons in a normal-dispersion femtosecond laser was reported by Li et al, and it was demonstrated that the large pulse energy was beneficial for the fast self-starting . However, the starting dynamics of dissipative soliton were only investigated in a large time scale, i.e., hundreds of microseconds, due to the limitation of diagnostic method. Furthermore, R. I. Woodward et al. also numerically simulated the radiation build-up dynamics of bright pulses in all-normal dispersion mode-locked fiber lasers. However, they mostly focused on the existence of slowly decaying dark solitons in the buildup process . Consequently, the transient dynamics of dissipative soliton buildup in normal dispersion fiber lasers have not been well understood yet. Thanks to the great advances in the detection technologies of ultrashort pulses, transient dynamics of solitons in fiber lasers can be experimentally measured in real-time manner. Recently, the dispersive Fourier-transform (DFT) method , a simple but very powerful real-time measurement technique, has been used to experimentally resolve ultrafast dynamics in laser systems. Based on the principle of the DFT technique in which the spectrum of each optical pulse is mapped into a temporal waveform using group-velocity dispersion, the transient spectral dynamics can be measured directly from a real-time oscilloscope. Indeed, the DFT method has recently employed to observe complex ultrafast nonlinear phenomena, such as optical rogue waves [30–32], soliton explosions [33,34], phase evolution of bound solitons [35–37] and transient vector dynamics of coherent/incoherent dissipative solitons [38,39] in ultrafast fiber lasers. Notably, the DFT technique was also applied to investigate the buildup dynamics of conventional soliton in the state-solid laser  or anomalous dispersion fiber laser [41,42]. Being different from the solid-state lasers, it was found that more nonlinear effect would be observed owing to the light confinement in a small mode area of optical fiber. In addition, as mentioned above the real-time buildup dynamics of dissipative solitons have not been fully revealed yet. Therefore, the exploration of the dissipative soliton buildup dynamics in ultrafast fiber lasers would be essential for further understanding the physical mechanism of dissipative soliton.
In this work, we report both the experimental observations and numerical simulations on the dissipative soliton buildup dynamics in an erbium-doped fiber (EDF) laser operating in the net-normal dispersion regime. Due to the pulse shaping from the noise background to mode-locking operation, the transient structural dissipative soliton  were formed towards the stable mode locking, which results in the formation of the transient structural spectral patterns, i.e., sharp spectral peaks with oscillation structures at the edges of mode-locked spectrum. The obtained results enable us to have a better understanding for the dissipative soliton buildup dynamics in ultrafast fiber lasers, which will be beneficial for both the nonlinear optics and ultrafast laser communities.
2. Experimental setup and results
The experimental setup of the fiber laser for investigating the dissipative soliton buildup dynamics is shown in Fig. 1. It consists of a wavelength-division-multiplexed (WDM) coupler, a 10-m-long EDF, a polarization-dependent isolator (PD-ISO), two sets of polarization controllers (PCs), a carbon nanotube (CNT) saturable absorber (SA), a coupler with 10% output. A 10 m EDF with a dispersion parameter of −44.5 was used as the gain medium. Other fibers in cavity are 12 m standard single-mode fiber (SMF). Therefore, the net cavity dispersion is ~0.3 ps2. In order to initiate (or stop) the self-starting mode locking operation, we placed a mechanical chopper between the pump laser and the WDM. To obtain the dissipative soliton easily, the fiber laser was mode-locked by the hybrid mode-locking technique, which was realized with a CNT-based SA [44,45] and nonlinear polarization rotation (NPR). Two PCs were used to adjust the polarization state of the circulating light. A PD-ISO provides the unidirectional operation and the polarization selectivity. By combining the PCs and PD-ISO, the laser mode-locking parameters can be tuned, such as central wavelength and spectral bandwidth [46,47]. Another 10:90 couplers was used to measure the soliton buildup dynamics both in temporal and frequency domains with the DFT measurement technique simultaneously. Optical signal from one port was detected by a photodetector (Newport 818-BB-35F, 12.5 GHz) and sent to a real-time oscilloscope (Tektronix DSA-70804, 8 GHz), while the other was linking to a dispersive element (~14 km long SMF) to perform DFT real-time spectral measurement [29,48]. Therefore, the real-time spectral resolution by DFT is ~0.43 nm for our setup.
In the following, the dissipative soliton buildup dynamics in the net-normal dispersion fiber laser will be investigated. The self-started mode-locking could be easily achieved by increasing the pump power to 12.7 mW as the cavity parameters were properly adjusted. The laser performance of the dissipative soliton is displayed in Fig. 2. The typical spectrum of rectangular profile for dissipative soliton was obtained with a center wavelength of 1562 nm [4,5]. The pulse-train operated at a fundamental repetition rate of 9.27 MHz and the pulse duration is 16.21 ps. Moreover, the radio-frequency (RF) spectrum shows a signal-to-noise ratio of >50 dB, indicating that the dissipative soliton fiber laser operates in a relatively stable mode.
In order to investigate the dissipative soliton buildup process in the ultrafast fiber laser, then we used the mechanical chopper to periodically chop the 980 nm pumping laser when the pump power was a little higher than the mode-locking threshold. Therefore, the transition from cw to mode-locked operation could be easily obtained. By virtue of DFT, the spatio-spectral dynamics of the dissipative soliton mode-locking process for 2700 consecutive roundtrips could be recorded by the real-time oscilloscope, as shown in Fig. 3(a). Similar to the case of conventional soliton buildup dynamics [41,42], the spectral broadening process could be also observed. However, it is worth noting that the highest structured spectral spikes spread along both edges for dissipative soliton during the soliton buildup (also see the Visualization 1). Moreover, it can be seen that there is a spectral bandwidth oscillation on the real-time spectral dynamics in Fig. 3(a), which was caused by the incomplete balance among the dispersion, nonlinearity, gain and loss before achieving the stable dissipative soliton mode locking. However, the oscillation degree of the spectral bandwidth could be tuned or even eliminated by carefully adjusting the pump power level. As for the spatio-temporal dynamics of the dissipative soliton build up process, it can be seen from the Fig. 3(b) where the pulse evolved with an increasing trend in the peak intensity, then decreased to a certain value and finally become stable mode locking state in fiber laser. In addition, Fig. 3(c) also provides the pulse energy evolution in dissipative soliton buildup regime. Here, it should be also noted that the roundtrip 1 in Fig. 3 is not corresponding to the moment the chopper opened, because the measurement window of oscilloscope was adjusted to 400 μs to resolve the spectral and temporal dynamics with enough sampling points.
To gain insight into the transient dynamics of dissipative soliton in the fiber laser, we provided four typical spectra during the dissipative soliton buildup corresponding to different roundtrips in Fig. 4. From Fig. 4, it can be seen that the spectral bandwidth becomes larger and larger when evolving into the stable mode-locking state. Meanwhile, the spectra also exhibit the structured profiles, where the highest interference peaks generally locate at the two edges of the evolving spectrum. Specifically, two sharp spectral peaks with oscillation structures around them could be seen on both edges of the pulse spectrum near the stable mode locking operation, as shown in Fig. 4(c). Following this, after a certain number of roundtrips, the bandwidth of the spectrum still broadens slowly until reaching the stable mode-locking state. In this case, a rectangular shape spectrum which is the typical characteristic of the dissipative soliton in fiber laser could be seen [4,5], indicating that the fiber laser operated in a stable mode-locking state, as presented in Fig. 4(d). Note that the real-time spectrum measured by the DFT is not as smooth as that by OSA shown in Fig. 2(a). It is because the measurement mode of OSA is an average one.
3. Simulation results
3.1 Laser setup and numerical model
The schematic of the fiber ring laser cavity for simulations is depicted in Fig. 5, which is conceptual but general enough for us to investigate the soliton buildup dynamics of passively mode-locked lasers. The ring cavity consists of the active EDF, SMF as the passive fiber, a SA, and an output coupler (OC).
Numerical simulations are based on an extended nonlinear Schrodinger equation , which is solved with a standard symmetric split-step algorithm for both segments of the active and passive fibers. The modeling includes the physical terms such as the self-phase modulation, the group velocity dispersion of fiber, and saturated gain with a finite bandwidth:
Here, is the slowly varying amplitude of the pulse envelope, the variable and represent the propagation distance and the time, respectively. and are denoted as the fiber dispersion and the cubic refractive nonlinearity of the fiber, respectively. The bandwidth of the gain spectrum is , describes the gain function of the EDF which could be given by:
Where is the small-signal gain, and is the pulse energy, is the gain saturation energy relying on the pump power. The SA is modelled by a simplified transfer function :
Here, is the modulation depth, is the instantaneous pulse power, and is the saturation power for SA.
The initial condition used in our numerical simulation is an arbitrary weak pulse. When the lightwave evolves in the cavity, after a cavity roundtrip, the result of calculation of pulse is then used as a new initial signal in the next roundtrip calculation.
3.2 Simulation results
Then the dissipative soliton buildup dynamics in the net-normal dispersion cavity was simulated by the above-mentioned model. The detailed parameter settings can be found in Table 1. The simulation grid consists of points in time domain and 2216 nodes in space at the cavity round-trip. The temporal resolution of the simulation is 7.3 fs. Here, the overview of the dissipative soliton buildup dynamics was presented firstly by plotting the light propagation in the laser cavity over 2700 roundtrips both in spectral and temporal domains from the simulation results in Fig. 6. In Fig. 6(a), it can be seen that the laser operates in the quasi-cw state firstly, and then the spectral bandwidth broadens as the lightwave propagates in the laser cavity. Note that the sharp-peaked edges were also created during the spectral broadening process, which was in accordance with the experimental results. Figure 6(b) illustrates the pulse evolution in 2700 roundtrips. We can see that the peak intensity of the pulse increased firstly, then decreased sharply until the stable mode locking operation was realized. Then we plotted the pulse energy evolution by integrating the pulse profiles in Fig. 6(c). The pulse energy evolution is still consistent with the experimental results.
In the same way, Fig. 7 offers four representative theoretical spectra and the corresponding pulse profiles at the roundtrips of 104th, 163th, 222nd and 1700th so as to be in contrast with the experimental results and investigate the details during the soliton buildup dynamics. In the frequency domain, from the upper row of Fig. 7, it was found that the spectral interference patterns appeared on the evolving spectra and the interference depth decreased with the propagation roundtrips, and then disappeared when realizing the stable mode locking operation. In addition, the spectral peaks with oscillation structures at both edges of evolving spectra could be observed before the stable mode locking operation in the simulation results, which is in agreement with the experimental results. In the time domain, the pulse profiles evolving with the cavity roundtrips were shown in the lower row of Fig. 7. Due to the dispersion, the pulse duration broadened with the increase of roundtrips until reaching the limitation of the effective gain bandwidth and then became stable. Therefore, the SA and the effective gain bandwidth in the laser cavity would cut off the wings of the pulse and spectrum , as illustrated in Fig. 7(f). Moreover, it can be seen that there are also two small humps located at two edges of the evolving soliton in Fig. 7(g). All these features make the dissipative soliton as a structural one in the laser cavity. From the simulation results, it is confirmed that the spectral pattern appears on the pulse spectrum when the evolving soliton becomes a structural one . Thus, it can be concluded that the transient spectral characteristics are dependent on evolution features in the time domain due to the pulse shaping effect towards stable mode locking.
From both the experimental and numerical simulation results, we could find the highest structured spectral spikes spread along both edges for dissipative soliton during the buildup process. And no spectral intensity oscillations were found during the dissipative soliton buildup. However, the spectral evolution characteristics of the conventional soliton in the anomalous dispersion regime can be concluded as that the highest spectral components with strong spectral oscillation behavior are concentrated in the central part of mode-locked soliton during the buildup process [41,42]. The difference of the spectral evolutions in the anomalous and normal dispersion regimes could be attributed to the different pulse shaping mechanisms for two types of solitons. In addition, the high amplitude waves could be observed during the buildup process of dissipative soliton, i.e., Fig. 3(b), which is also different from the case of conventional soliton buildup. On the other hand, in this work the soliton buildup dynamics in EDF lasers with net-normal dispersion were demonstrated experimentally and theoretically. However, in order to fully investigate the soliton buildup dynamics, the fiber lasers at other wavebands or with all-normal dispersion regime should be studied in the future to explore whether there exist some other different characteristics in the dissipative soliton buildup dynamics. Moreover, although the dissipative soliton buildup dynamics in time domain have not be clearly revealed in our experiment, the problem can be solved by using the time lens measurement [51–53] that can be employed to provide a temporal magnification of the evolving pulses. Then the pulse profile in the picosecond or sub-picosecond regime can be measured directly from a real-time oscilloscope. Nevertheless, since the general spectral evolution tendency of both the experiments and simulations are almost the same, the simulation results of the temporal pulse evolution can still provide a detailed insight into the real-time soliton buildup dynamics in the time domain.
In summary, we have experimentally and theoretically investigated the dissipative soliton buildup dynamics in an ultrafast fiber laser in the net-normal dispersion regime. Owing to the pulse shaping mechanism in the normal dispersion regime, the spectral interference pattern with highest intensities at both edges of the mode-locked spectrum could be observed during the dissipative soliton buildup process. And the spectral evolution dynamics could be explained by the transient structured soliton formation during the pulse shaping from the noise background. The obtained results would pave the way for further investigation of the complex soliton dynamics in nonlinear optical systems, which would be also helpful for the communities dealing with the dissipative solitons and fiber lasers.
National Natural Science Foundation of China (NSFC) (11304101, 11474108, 61307058, 61378036); Guangdong Natural Science Funds for Distinguished Young Scholar (2014A030306019); Program for the Outstanding Innovative Young Talents of Guangdong Province (2014TQ01X220); Program for Outstanding Young Teachers in Guangdong Higher Education Institutes (YQ2015051); Science and Technology Project of Guangdong (2016B090925004).
References and links
2. L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B 65(2), 277–294 (1997). [CrossRef]
3. N. Akhmediev and A. Ankiewics, Dissipative Solitons (Springer, 2005).
5. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008). [CrossRef]
6. A. Cabasse, B. Ortaç, G. Martel, A. Hideur, and J. Limpert, “Dissipative solitons in a passively mode-locked Er-doped fiber with strong normal dispersion,” Opt. Express 16(23), 19322–19329 (2008). [CrossRef] [PubMed]
7. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012). [CrossRef]
9. F. W. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1–2), 58–73 (2008). [CrossRef]
10. W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18(1), 389–398 (2012). [CrossRef] [PubMed]
11. Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, “Relative phase locking of pulses in a passively mode-locked fiber laser,” J. Opt. Soc. Am. B 20(5), 863 (2003). [CrossRef]
12. 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]
14. S. Chouli and P. Grelu, “Soliton rains in a fiber laser: An experimental study,” Phys. Rev. A 81(6), 063829 (2010). [CrossRef]
15. A. Niang, F. Amrani, M. Salhi, P. Grelu, and F. Sanchez, “Rains of solitons in a figure-of-eight passively modelocked fiber laser,” Appl. Phys. B 116(3), 771–775 (2014). [CrossRef]
16. W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78(2), 023830 (2008). [CrossRef]
18. X. Liu, “Pulse evolution without wave breaking in a strongly dissipative-dispersive laser system,” Phys. Rev. A 81(5), 053819 (2010). [CrossRef]
19. 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] [PubMed]
24. F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond Solid-State Lasers,” IEEE J. Quantum Electron. 28(10), 2097–2122 (1992). [CrossRef]
25. C. Bao and C. Yang, “Harmonic mode-locking in a Tm-doped fiber laser: Characterization of its timing jitter and ultralong starting dynamics,” Opt. Commun. 356, 463–467 (2015). [CrossRef]
26. M. Popov and O. Gat, “Pulse growth dynamics in laser mode locking,” Phys. Rev. A 97(1), 011801 (2018). [CrossRef]
29. K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013). [CrossRef]
30. C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108(23), 233901 (2012). [CrossRef] [PubMed]
33. A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36 (2015). [CrossRef]
34. M. Liu, A. P. Luo, Y. R. Yan, S. Hu, Y. C. Liu, H. Cui, Z. C. Luo, and W. C. Xu, “Successive soliton explosions in an ultrafast fiber laser,” Opt. Lett. 41(6), 1181–1184 (2016). [CrossRef] [PubMed]
35. 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] [PubMed]
36. Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (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] [PubMed]
39. K. Krupa, K. Nithyanandan, and P. Grelu, “Vector dynamics of incoherent dissipative optical solitons,” Optica 4(10), 1239–1243 (2017). [CrossRef]
40. G. Herink, B. Jalali, C. Ropers, and D. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016). [CrossRef]
41. X. Wei, C. Zhang, B. Li, and K. K. Y. Wong, “Observing the spectral dynamics of a mode-locked laser with ultrafast parametric spectro-temporal analyzer.” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2015), paper STh3L.4. [CrossRef]
42. X. Wei, B. Li, Y. Yu, C. Zhang, K. K. Tsia, and K. K. Y. Wong, “Unveiling multi-scale laser dynamics through time-stretch and time-lens spectroscopies,” Opt. Express 25(30), 29098–29120 (2017). [CrossRef]
44. S. Yamashita, “A tutorial on nonlinear photonic applications of carbon nanotube and graphene,” J. Lightwave Technol. 30(4), 427–447 (2012). [CrossRef]
45. F. Wang, A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari, “Wideband-tuneable, nanotube mode-locked, fibre laser,” Nat. Nanotechnol. 3(12), 738–742 (2008). [CrossRef] [PubMed]
47. W. S. Man, H. Y. Tam, M. S. Demokan, P. K. A. Wai, and D. Y. Tang, “Mechanism of intrinsic wavelength tuning and sideband asymmetry in a passively mode-locked soliton fiber ring laser,” J. Opt. Soc. Am. B 17(1), 28–33 (2000). [CrossRef]
48. Y. C. Tong, L. Y. Chang, and H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33(11), 983–985 (1997). [CrossRef]
49. A. Cabasse, B. Ortaç, G. Martel, A. Hideur, and J. Limpert, “Dissipative solitons in a passively mode-locked Er-doped fiber with strong normal dispersion,” Opt. Express 16(23), 19322–19329 (2008). [CrossRef] [PubMed]
50. I. A. Yarutkina, O. V. Shtyrina, M. P. Fedoruk, and S. K. Turitsyn, “Numerical modeling of fiber lasers with long and ultra-long ring cavity,” Opt. Express 21(10), 12942–12950 (2013). [CrossRef] [PubMed]
51. B. Li, S.-W. Huang, Y. Li, C. W. Wong, and K. K. Y. Wong, “Panoramic-reconstruction temporal imaging for seamless measurements of slowly-evolved femtosecond pulse dynamics,” Nat. Commun. 8(1), 61 (2017). [CrossRef] [PubMed]
52. P. Suret, R. E. Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016). [CrossRef] [PubMed]
53. M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016). [CrossRef] [PubMed]