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Abstract
Pulsating soliton in ultrafast fiber lasers has interesting non-stationary dynamics, which is one of the hot topics in field of nonlinear soliton. So far, most researchers only focused on the spectral and temporal characteristics of pulsating soliton. However, the vector features of pulsating soliton were rarely studied. In this work, we experimentally studied the pulsating vector solitons in an ultrafast fiber laser. Three categories of vector solitons with different polarization evolution characteristics could be obtained by adjusting the pump power and polarization controller, such as pulsating polarization-locked vector soliton (PLVS), pulsating polarization-rotation vector soliton (PRVS) and progressive pulsating PRVS. Interestingly, besides the basic polarization rotation with a period of 2 roundtrips, the polarization angle also evolves with progressive mode in the progressive pulsating PRVS state. The abundant results of pulsating vector solitons demonstrate that investigating vector features of nonlinear soliton dynamics is necessary and significant and would greatly enrich the research of soliton dynamics.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As a typical dissipative optical system, ultrafast fiber laser has been regarded as a perfect platform for investigating nonlinear soliton dynamics. By adjusting the cavity parameters, various kinds of fantastic soliton dynamics have been observed in ultrafast fiber lasers. Initially, the stable soliton states in the ultrafast fiber lasers have attracted primary attention of researchers, such as harmonic mode-locking [1,2], bound soliton [3,4], dissipative soliton resonance [5,6]. With the deepening of research and the development of measurement technology, the non-stationary soliton dynamics have become the hot topic in nonlinear soliton field, such as soliton pulsation [7–22], soliton explosion [23–26], and rogue waves [26–28]. These researches would be beneficial for enriching our understanding of nonlinear soliton dynamics.
As a distinctive non-stationary soliton dynamics in ultrafast fiber lasers, pulsating soliton has attracted extensive attention since it was first predicted theoretically [7]. Generally, in ultrafast fiber lasers due to the complex balances between gain, loss, dispersion and nonlinear effects [29], the pulse would evolve when circulating in the laser cavity with a period equal to the roundtrip time. Therefore, when the pulse is monitored at a fixed point of the cavity, it would be invariant with time, namely the stable mode-locking state. However, additional periodicity, an integer multiple of the round-trip time, could also be imposed on the pulse train by adjusting the cavity parameters, termed as pulsating soliton. The characteristic of pulsating soliton is that one or several parameters of the pulse, such as pulse profile, pulse width and peak power, etc., change periodically from one roundtrip to another [7,8]. The early investigation of pulsating soliton is mainly focused on its characteristics in the temporal domain. Theoretically, various evolution forms on the pulse profiles were predicted, such as plain pulsating, creeping, and exploding soliton [7,8,12,13]. Experimentally, since the pulsating soliton energy could be identified and measured by the oscilloscope and radio-frequency (RF) spectrum analyzer, many related works have been conducted [9–11,14]. As for the features of pulsating soliton in the spectral domain, it has just been unveiled in recent years by using the real-time diagnostic method of dispersive Fourier transform (DFT) technique [15–22]. These researches show that bandwidth breathing is the main spectral feature of pulsating soliton. Therefore, the features of pulsating solitons in both temporal and spectral domains have been investigated in depth in the ultrafast fiber lasers.
However, rare works on the vector features of pulsating solitons have been reported. Due to the manufacturing imperfections and externally applied random strain or bending, the single-mode fibers (SMFs) always support two orthogonal polarization modes [30,31]. It means that the vector feature is also one of the intrinsic properties of soliton in ultrafast fiber lasers composed of SMFs. Then investigating the vector features of pulsating solitons in ultrafast fiber lasers could further enrich and improve the research content of pulsating soliton. Very recently, Y. Luo et al. reported the pulsating group-velocity-locked vector soliton (GVLVS) in fiber lasers by using the DFT technique [21]. In fact, there are three categories of vector solitons in ultrafast fiber lasers, i.e. polarization-locked vector soliton (PLVS) [32,33], polarization rotation vector soliton (PRVS) [34,35], and GVLVS [36–38]. However, in Ref. [21] only the pulsating GVLVSs have been observed. Then a question would naturally arise as to whether the other categories of pulsating vector solitons could be achieved in ultrafast fiber lasers, such as pulsating PLVS and pulsating PRVS.
In this work, the vector features of pulsating solitons were studied in an erbium-doped fiber (EDF) laser. Through adjusting the cavity parameters, the pulsating PLVS, pulsating PRVS and progressive pulsating PRVS were achieved. In the pulsating PLVS, the polarization angle keeps constant while the pulse intensity is pulsating periodically. In the pulsating PRVS with a polarization rotation period of 2 roundtrips, the polarization angle of the pulse varies within certain two values as its pulse intensity in odd (even) roundtrips is pulsating periodically. The pulsating PLVS and pulsating PRVS could be commonly understood as the general PLVS and PRVS with additional intensity modulation with a larger period. However, the progressive pulsating PRVS presents more complicated polarization evolution. While maintaining 2 roundtrips of polarization rotation, the polarization angle of each odd (even) pulse will shift from the last one. After a pulsation period, the polarization angle and pulse intensity of the total pulse recover to its original state. It means that the polarization evolves with a progressive method, so we termed this phenomenon as progressive pulsating PRVS. In all three states, the characteristics of output pulse in spectral and temporal domains are similar. However, there is a noticeable difference in the polarization evolution among these cases. It demonstrates that there might not be a direct relationship between the evolutions of pulse intensity and polarization state. These experimental results further unveil the vector features of pulsating solitons and enrich our understanding of the pulsating solitons in ultrafast fiber lasers.
2. Experimental setup
The schematic of the ultrafast fiber laser used in our experiment is shown in Fig. 1, which is a typical fiber laser mode-locked by a real saturable absorber (SA). A segment of 15 m EDF is used as the gain medium with a dispersion parameter of −17.3 ps/km/nm, pumped by a 976 nm laser diode through a wavelength division multiplexer (WDM). The other fibers are 10.47 m standard SMF with a 17 ps/km/nm dispersion parameter. Therefore, the net cavity dispersion is ∼0.105 ps2. Polarization controller 1 (PC1) is employed to adjust the polarization states of the propagating light in the cavity. A PVA-based carbon nanotube-SA (CNT-SA) is used to realize the mode-locking operation. The unidirectional operation is ensured by a polarization-independent isolator (PI-ISO). A 10:90 optical coupler (OC) is used to output the laser. Since there are no polarization sensitive components in the proposed fiber laser, it could be used to investigate the vector features of pulsating solitons. In order to resolve the two orthogonal polarization components of soliton, PC2 and a fiber-based polarization beam splitter (PBS) are placed after the output port. The laser output is simultaneously measured by an optical spectrum analyzer (OSA, Yokogawa, AQ6375B) and a high-speed real-time oscilloscope (Tektronix DSA70804B, 8 GHz) with a photodetector (Newport 818-BB-35F, 12.5 GHz).
3. Results and discussion
By increasing the pump power to 32 mW and appropriately adjusting the PC1, the mode-locking operation could be easily achieved owing to the saturable absorption of CNT-SA. Figure 2 summarizes the performance of the mode-locking soliton. In Fig. 2(a), the rectangular spectrum with steep edges on both sides demonstrates that the fiber laser operates in the dissipative soliton regime [29]. The spectrum is centered at 1561.53 nm with a 3 dB bandwidth of 11.37 nm. The corresponding pulse train and RF spectrum are provided in Figs. 2(b) and (c). The pulse interval of 123.15 ns corresponds to the RF spectral peak at the frequency of 8.12 MHz, the fundamental repetition rate. The signal-to-noise ratio of the RF spectrum is >50 dB, indicating the excellent stability of the mode-locking operation. Figure 2(d) shows the autocorrelation trace with a pulse width of ∼16.35 ps. All these results demonstrate the stable operation of the fiber laser.
Fig. 2. Stable dissipative soliton. (a) Spectrum; (b) Pulse-train; (c) RF spectrum; (d) Autocorrelation trace.
When the pump power is decreased to ∼30 mW slowly, the pulsating soliton state could be obtained. The results of pulsating soliton are illustrated in Fig. 3. Since the fiber laser deviated from the perfect mode-locking state, the spectrum was no longer rectangular. Due to the periodic spectral evolution of pulsating soliton and the average mode of the OSA measurements, the spectrum in this state has two arc edges, as shown in Fig. 3(a) [18]. In Fig. 3(b), it is evident that the pulse intensity varies with a period of ∼17.24 µs (140 roundtrips). In addition, the corresponding RF spectrum with 7 peaks is shown in Fig. 3(c). The frequency difference between the central peak and satellite peaks is ∼58 kHz, in good agreement with the pulsating period of 140 roundtrips. These results show the typical characteristics of pulsating soliton, indicating that the fiber laser operates in the pulsating soliton state.
Since no polarization sensitive components are involved in the proposed fiber laser, the vector features of pulsating soliton were investigated here. By using the PBS and PC2, the polarization resolved results of the pulsating vector soliton were recorded and illustrated in Fig. 4. The spectra of total output and two orthogonal components are displayed in Fig. 4(a). They have similar profiles and the same central wavelengths. Correspondingly, the pulse trains of total output and two components are shown in Figs. 4(c)-(e), respectively. It is obvious that the three pulse trains have synchronized evolution trend. The pulsating periods of all the pulse trains are ∼42.74 µs (347 roundtrips). In order to gain more insights, we provide the zoom-in pulse trains in Figs. 4(f)-(h). In each pulse train, the pulse intensity changes slowly and continuously with the fundamental repetition rate. In addition, we provide the peak intensity ratios of the corresponding pulses on the two axes in Fig. 4(b). They could represent the polarization angle evolution qualitatively based on the beam splitting method of PBS. It can be seen that the intensity ratio is almost invariable with time. All these results demonstrate that the fiber laser operates in the pulsating PLVS state. In this state, the polarization angle of pulse keeps constant while an intensity modulation with a period of 347 roundtrips is added on the pulse train.
Fig. 4. Pulsating PLVS. (a) Spectra of total output and two orthogonal components; (b) Peak intensity ratios of the corresponding pulses on the two axes; (c)-(e) Corresponding pulse trains; (f)-(h) Zoom-in pulse trains.
By carefully rotating PC1, the pulsating PRVS could also be achieved in our experiments. Figure 5(a) shows the spectra of total output and two orthogonal components of the pulsating PRVS. The spectral tops of the two components are slightly tilted. Correspondingly, the pulse trains of total output and two orthogonal components are presented in Figs. 5(c)-(e), respectively. The total pulse intensity varies with a period of ∼33.25 µs (270 roundtrips). Different from Figs. 4(d) and (e), there are two sets of pulse envelopes in each pulse train of the two polarization components in Figs. 5(d) and (e). One set of pulse envelopes has a completely higher intensity, while the other one has a completely lower intensity. However, every set of pulse envelops of the two components in Figs. 5(d) and (e) has the synchronized evolution trend with Fig. 5(c). For clarity, we also offer the zoom-in pulse trains in Figs. 5(f)-(h). It can be seen that pulse intensities along two polarization components alter between higher values and lower values every 2 roundtrips. It is the typical polarization rotation feature with 2 roundtrips. In addition, the pulse intensity in odd (even) roundtrips changes continuously and periodically. Then combining the intensity pulsation and polarization rotation features, it could be confirmed that the fiber laser operates in the pulsating PRVS state. Figure 5(b) presents the peak intensity ratios of the corresponding pulses on two axes. The peak intensity ratios of the odd-numbered pulses of two components with blue triangles are ∼2.12, while that of the even-numbered pulses of two components with red circles are ∼0.49. Therefore, we can conclude that in this state the polarization angle of pulse varies within the two specific values while the pulse intensity in odd (even) roundtrips pulsates periodically. It is just like an additional intensity modulation is imposed on the traditional PRVS with a period of 2 roundtrips, while the intensity modulation does not affect the polarization angle evolution of the pulse.
Fig. 5. Pulsating PRVS. (a) Spectra of total output and two orthogonal components; (b) Peak intensity ratios of the corresponding pulses on the two axes; (c)-(e) Corresponding pulse trains; (f)-(h) Zoom-in pulse trains.
In the experiments, when PC1 was adjusted slightly, we noticed an exceptional phenomenon, i.e., another type of pulsating PRVS with peculiar polarization evolution. In this pulsating soliton state, the total pulse intensity varies with a period of ∼8.13 µs (66 roundtrips), as shown in Fig. 6(a). As for the pulse trains of two components in Figs. 6(b) and (c), similar to Fig. 5, two sets of pulse envelops are also involved. However, their intensity evolutions are not synchronized but crossed, which is shown as two humps on the pulse trains. In addition, we can see that one set of pulse envelops on the horizontal (vertical) axis has a consistent evolution trend with the total pulse train. In contrast, the evolution trend of the other set has a time delay of ∼3.1 µs (25 roundtrips) with the total pulse train. Figures 6(d)-(f) present the corresponding zoom-in pulse trains, providing clearer features about this phenomenon. In Figs. 6(e) and (f), it can be seen that the intensity evolution in odd (even) roundtrips is continuous, but there is no apparent correlation between the intensities of adjacent pulses. It indicates that the soliton still possesses the polarization rotation feature with a period of 2 roundtrips, but it is different from the common polarization rotation characteristic. The peak intensity ratios of the corresponding pulses on the two axes in Fig. 6(g) indicate that the polarization angle of pulse in odd (even) roundtrips would shift from the last one periodically, i.e. a progressive method. These results demonstrate that besides the polarization rotation period of 2 roundtrips, there is another polarization evolution period in this state, which is equal to the total pulse intensity modulation period of ∼66 roundtrips. After one pulsation period, the polarization angle and pulse intensity recover to the original values. Due to the progressive polarization evolution method, we term this phenomenon as progressive pulsating PRVS.
Fig. 6. Progressive pulsating PRVS. (a)-(c) Pulse trains of total output and two orthogonal components, respectively; (d)-(f) Zoom-in pulse trains;(g) Peak intensity ratios of the corresponding pulses on the two axes.
In this work, we investigated the vector features of three types of pulsating solitons. In these cases, all the total outputs possess almost the same characteristics of pulsating solitons, such as the similar spectral profile and pulse intensity modulation. However, there is a big difference among the three cases on the polarization evolution. In the pulsating PLVS and pulsating PRVS states, the polarization angle evolutions of the pulse are similar to the traditional PLVS and PRVS. The difference lies in the additional intensity pulsations. In the progressive pulsating PRVS state, besides the intensity modulation, the polarization angle of pulse evolves progressively while maintaining the polarization rotation period of 2 roundtrips. It is totally different from the above pulsating PRVS. Note that in the progressive pulsating PRVS state, the time delay between the evolutions in even-numbered and odd-numbered pulse trains could be adjusted by rotating PC2. Then, under particular circumstances, a similar phenomenon as Fig. 5 also will occur. However, they are essentially different. On the other hand, since the pulse intensity of pulsating soliton changes continuously, it is an ideal platform to study the relationship between the pulse intensity and polarization state. These results show that the change in polarization state and pulse intensity can be synchronized, such as pulsating PLVS and pulsating PRVS, or unsynchronized, such as progressive pulsating PRVS. Therefore, the evolution of pulse polarization state and intensity may be independent of each other. Then, it would be significant to supplement the missing investigation on vector features of various soliton dynamics.
4. Conclusion
In conclusion, we experimentally observed three kinds of pulsating solitons with different polarization characteristics in an ultrafast fiber laser, namely, pulsating PLVS, pulsating PRVS, and progressive pulsating PRVS. In the pulsating PLVS and pulsating PRVS states, apart from the pulse intensity pulsation, the polarization angles would keep constant or change between two certain values, just like the traditional PLVS and PRVS. Especially, the progressive pulsating PRVS illustrates the progressive polarization angle evolution together with the polarization rotation with period of 2 roundtrips. It demonstrates that the polarization state evolution of pulse is independent of the intensity variation. All these findings provide a new perspective for the investigation of the vector features of soliton dynamics.
Funding
Key-Area Research and Development Program of Guangdong Province (2018B090904003, 2020B090922006); National Natural Science Foundation of China (11874018, 11974006, 61805084, 61875058); Science and Technology Program of Guangzhou (2019050001); Guangdong Basic and Applied Basic Research Foundation (2019A1515010879); Open Fund of the Guangdong Provincial Key Laboratory of Fiber Laser Materials and Applied Techniques (South China University of Technology, 2019-2); Open Fund of State Key Lab of Advanced Communication Systems and Networks, Shanghai Jiao Tong University (2020GZKF010).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. A. Grudinin and S. Gray, “Passive harmonic mode locking in soliton fiber lasers,” J. Opt. Soc. Am. B 14(1), 144–154 (1997). [CrossRef]
2. S. Zhou, D. G. Ouzounov, and F. W. Wise, “Passive harmonic mode-locking of a soliton Yb fiber laser at repetition rates to 1.5 GHz,” Opt. Lett. 31(8), 1041–1043 (2006). [CrossRef]
3. D. Tang, W. Man, H. Tam, and P. Drummond, “Observation of bound states of solitons in a passively mode-locked fiber laser,” Phys. Rev. A 64(3), 033814 (2001). [CrossRef]
4. C. Mou, S. V. Sergeyev, A. G. Rozhin, and S. K. Turitsyn, “Bound state vector solitons with locked and precessing states of polarization,” Opt. Express 21(22), 26868–26875 (2013). [CrossRef]
5. W. Chang, A. Ankiewicz, J. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78(2), 023830 (2008). [CrossRef]
6. X. Wu, D. Tang, H. Zhang, and L. Zhao, “Dissipative soliton resonance in an all-normal-dispersion erbium-doped fiber laser,” Opt. Express 17(7), 5580–5584 (2009). [CrossRef]
7. J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. 85(14), 2937–2940 (2000). [CrossRef]
8. 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]
9. J. M. Soto-Crespo, M. Grapinet, P. 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]
10. L. Zhao, D. Tang, F. Lin, and B. Zhao, “Observation of period-doubling bifurcations in a femtosecond fiber soliton laser with dispersion management cavity,” Opt. Express 12(19), 4573–4578 (2004). [CrossRef]
11. B. Zhao, D. Tang, L. Zhao, P. Shum, and H. Tam, “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]
12. W. Chang, A. Ankiewicz, N. Akhmediev, and J. Soto-Crespo, “Creeping solitons in dissipative systems and their bifurcations,” Phys. Rev. E 76(1), 016607 (2007). [CrossRef]
13. 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]
14. Y. Song, Z. Liang, H. Zhang, Q. Zhang, L. Zhao, D. Shen, and D. Tang, “Period-doubling and quadrupling bifurcation of vector soliton bunches in a graphene mode locked fiber laser,” IEEE Photonics J. 9(5), 1–8 (2017). [CrossRef]
15. 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]
16. 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]
17. 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]
18. X. Wang, Y.-G. Liu, Z. Wang, Y. Yue, J. He, B. Mao, R. He, and J. Hu, “Transient behaviors of pure soliton pulsations and soliton explosion in an L-band normal-dispersion mode-locked fiber laser,” Opt. Express 27(13), 17729–17742 (2019). [CrossRef]
19. 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]
20. 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]
21. Y. Luo, Y. Xiang, P. P. Shum, Y. Liu, R. Xia, W. Ni, H. Q. Lam, Q. Sun, and X. Tang, “Stationary and pulsating vector dissipative solitons in nonlinear multimode interference based fiber lasers,” Opt. Express 28(3), 4216–4224 (2020). [CrossRef]
22. J. Chen, X. Zhao, T. Li, J. Yang, J. Liu, and Z. Zheng, “Generation and observation of ultrafast spectro-temporal dynamics of different pulsating solitons from a fiber laser,” Opt. Express 28(9), 14127–14133 (2020). [CrossRef]
23. S. T. Cundiff, J. M. Soto-Crespo, and N. Akhmediev, “Experimental evidence for soliton explosions,” Phys. Rev. Lett. 88(7), 073903 (2002). [CrossRef]
24. A. F. Runge, N. G. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36–39 (2015). [CrossRef]
25. 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]
26. K. Krupa, K. Nithyanandan, and P. Grelu, “Vector dynamics of incoherent dissipative optical solitons,” Optica 4(10), 1239–1243 (2017). [CrossRef]
27. C. Lecaplain, P. Grelu, J. 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]
28. Z. Liu, S. Zhang, and F. W. Wise, “Rogue waves in a normal-dispersion fiber laser,” Opt. Lett. 40(7), 1366–1369 (2015). [CrossRef]
29. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012). [CrossRef]
30. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. 12(8), 614–616 (1987). [CrossRef]
31. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes,” J. Opt. Soc. Am. B 5(2), 392–402 (1988). [CrossRef]
32. S. T. Cundiff, B. Collings, N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. Knox, “Observation of polarization-locked vector solitons in an optical fiber,” Phys. Rev. Lett. 82(20), 3988–3991 (1999). [CrossRef]
33. D. Tang, H. Zhang, L. Zhao, and X. Wu, “Observation of high-order polarization-locked vector solitons in a fiber laser,” Phys. Rev. Lett. 101(15), 153904 (2008). [CrossRef]
34. V. Afanasjev, “Soliton polarization rotation in fiber lasers,” Opt. Lett. 20(3), 270–272 (1995). [CrossRef]
35. Y. F. Song, H. Zhang, and D. Y. Tang, “Polarization rotation vector solitons in a graphene mode-locked fiber laser,” Opt. Express 20(24), 27283–27289 (2012). [CrossRef]
36. M. Islam, C. Poole, and J. Gordon, “Soliton trapping in birefringent optical fibers,” Opt. Lett. 14(18), 1011–1013 (1989). [CrossRef]
37. L. Zhao, D. Tang, H. Zhang, X. Wu, and N. Xiang, “Soliton trapping in fiber lasers,” Opt. Express 16(13), 9528–9533 (2008). [CrossRef]
38. D. Mao, X. Liu, and H. Lu, “Observation of pulse trapping in a near-zero dispersion regime,” Opt. Lett. 37(13), 2619–2621 (2012). [CrossRef]
