Abstract

We reported on the generation of versatile soliton molecules in a fiber laser mode-locked by a microfiber-based graphene saturable absorber (GSA). By virtue of the highly nonlinear effect of the microfiber-based GSA, the soliton molecules could be easily observed. In addition to regular soliton molecules, it is found that the “soliton atoms” in molecules could exhibit different characteristics and show ultra-narrow pulse separations, which was termed as ‘structural soliton molecule’. The pulse profiles of ‘structural soliton molecules’ were further reconstructed theoretically. The obtained results would give further insight towards understanding the dynamics of soliton molecules in fiber lasers.

© 2014 Optical Society of America

1. Introduction

Since temporal solitons in optical fibers were first observed by Mollenauer et al. in 1980 [1], they have been the fascinating subject of considerable theoretical and experimental studies in recent decades. When the interaction occurs among multiple temporal solitons, they could show some interesting multi-soliton behaviors, such as formation of soliton rain [24], soliton cluster [57], harmonically mode-locked solitons [810], and soliton molecules [1116]. Among them, soliton molecule is a type of stable multi-soliton structure, which has the capacity of restituting the equilibrium separation of solitons due to the Kerr mediated interaction that avoids the neighboring solitons locating too close or too far to each other. Since the soliton molecule formation is related to the soliton interactions that are able to bind the individual solitons stably, the soliton molecule would present more physical features than those of the single soliton. Therefore, investigations of soliton molecule in nonlinear systems could contribute to the further understanding of fundamental physics of solitons.

The passively mode-locked fiber lasers, which are actually nonlinear dissipative systems, are suitable for observing various soliton dynamics and nonlinear phenomena. Soliton molecule, as a type of soliton nonlinear phenomenon, could be frequently observed in fiber lasers depending on the proper cavity parameters [1116]. An effective approach to obtain soliton molecule is to increase the intracavity nonlinear effect or the pump power level, which makes the soliton split into multiple solitons and then bound together to form a soliton molecule. So far, much attention has been directed to the phase relationship, pulse separation and evolution of soliton molecules. It was found that the nonlinear effect inside the fiber laser cavity could play an important role in the evolution and dynamics of soliton molecules.

On the other hand, graphene, as a typical two-dimensional nanomaterial, has attracted much attention in the ultrafast laser community due to the excellent saturable absorption characteristics as well as the wide range of applications for ultrafast pulse generation in passively mode-locked fiber lasers [1724]. Apart from saturable absorption characteristics, graphene was also found to possess a giant nonlinear refractive index [25,26]. In particular, by increasing the interaction length between the graphene and propagation light, it has been demonstrated that the microfiber-based graphene saturable absorber (GSA) could be employed as highly nonlinear saturable absorption photonic device for pulse shaping in fiber lasers [27,28]. By virtue of highly nonlinear effect of microfiber-based GSA, the multi-pulse operation could be easily observed in fiber lasers [28], which could be potentially used to investigate the characteristics of soliton molecules. Moreover, obtaining soliton molecules in fiber lasers using mode-locked techniques such as nonlinear polarization rotation (NPR) and nonlinear amplifying loop mirror (NALM) [1116] requires high pump power or careful adjustments of PCs, which makes it somewhat inconvenient to investigate the dynamics and features of soliton molecules. Therefore, considering the excellent performance of microfiber-based GSA both in high nonlinear effect and saturable absorption, it would be interesting to know whether the soliton molecule generated from a fiber laser could exhibit somewhat different characteristics by using a microfiber-based GSA.

Here, we reported on the versatile soliton molecules generated from a passively mode-locked fiber laser by using a microfiber-based GSA. Taking advantage of high nonlinear effect of microfiber-based GSA, the soliton molecule could be easily formed in the fiber laser. The autocorrelation traces shows that the individual solitons (soliton atoms) in the molecules possessed different peak intensities and durations as well as ultra-narrow pulse separations, whose characteristics were distinguished from regular soliton molecules. Therefore, the obtained soliton molecules in this work were termed as ‘structural soliton molecules’. The profiles of ‘structural soliton molecules’ were further reconstructed theoretically. The obtained results further reveal the fundamental physics of soliton molecules, and demonstrate that the microfiber-based GSA is indeed a good candidate of highly nonlinear photonic device for investigating various soliton dynamics and nonlinear phenomena in fiber lasers.

2. Fabrication and characteristics of microfiber-based GSA

Firstly, the microfiber was fabricated by drawing the standard single mode fiber (SMF) with flame-brushing technique. Here, the diameter of microfiber including core and cladding was tapered to be about 8 μm. The graphene with a concentration of 0.075 mg/ml was dispersed in the Dimethylformamide (DMF) solution and ultrasonicated 30 mins for the uniformity. Then the graphene/DMF solution was dripped onto the microfiber and covered the waist region of the microfiber. The experimental setup for depositing the graphene onto the microfiber is the same as that in [8]. For controlling the deposition amount of graphene, the process was in situ observed by a microscope. When the graphene was deposited on the microfiber at a proper amount, we stopped the optical deposition and the residual DMF solution of the microfiber-based GSA was evaporated at room temperature. The as-prepared microfiber-based GSA was shown in Fig. 1(a). By injecting the visible light, the scattering evanescent field around the waist region was observed, as shown in Fig. 1(b). To investigate nonlinear optical characteristics of the fabricated microfiber-based GSA, the saturable absorption was measured by the power-dependent transmission technique, whose setup is the same as that we previously used [8]. The measured results are shown in Fig. 1(c). As can be seen here, the modulation depth is ~6.8% and the nonsaturable loss is ~36%, which clearly shows the saturable absorption characteristic of the microfiber-based GSA for passive mode-locking. It should be also to note that the parameters of the microfiber-based GSA such as modulation depth and non-saturable loss could be further improved by optimizing the deposited amount of graphene and the diameter of microfiber. Moreover, no evident polarization-dependent loss of the fabricated GSA could be observed.

 figure: Fig. 1

Fig. 1 (a) Microscopy image of microfiber-based GSA; (b) Scattering evanescent field of the microfiber through injecting the visible light; (c) Measured saturable absorption curve and the corresponding fitting curve.

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3. Laser performance and discussions

When the microfiber-based GSA was fabricated, it was incorporated into a fiber ring laser to achieve soliton mode-locking operation. The experimental setup of the passively mode-locked fiber laser with the microfiber-based GSA was shown in Fig. 2. Here, the fiber laser has a ring cavity with a length of ~77 m, corresponding to a fundamental repetition rate of ~2.67 MHz. A segment of ~10 m erbium-doped fiber (EDF) with a dispersion parameter of ~-17.3 ps/(nm·km) was used as the gain medium. The other fibers used to construct the fiber laser are all standard SMF. Two polarization controllers (PCs) were employed to adjust the polarization state of light. The unidirectional operation of the fiber laser was ensured by the polarization-independent isolator (PI-ISO). The laser output, which was taken by a 10% coupler, was recorded by an optical spectrum analyzer (Anritsu MS9710C) and an oscilloscope (LeCroy Wave Runner 104MXi, 1 GHz) with a photodetector (New Focus P818-BB-35F, 12.5GHz). Moreover, the profiles of soliton molecules were measured by a commercial autocorrelator.

 figure: Fig. 2

Fig. 2 Schematic of the passively mode-locked fiber laser with the microfiber-based GSA.

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In the experiment, the passive mode locking could be easily achieved by simply increasing the pump power to about ~85 mW. Since the microfiber-based GSA with highly nonlinear effect was incorporated, the fiber laser was inclined to emit multiple pulses [29]. Depending on the cavity parameter selections, the multi-pulse operation could be formed as harmonic mode locking, soliton rain and soliton molecules. Here, we only concentrated on the investigations of soliton molecules. Firstly, the regular soliton molecule was observed. The typical mode-locked spectrum of regular soliton molecule was shown in Fig. 3(a). As can be seen in Fig. 3(a), the central wavelength and the 3-dB spectral bandwidth are 1530.38 nm and 1.55 nm, respectively. Note that the typical modulation was not shown on the mode-locked spectrum, which could be caused by the independently or chaotically evolving phase among the “soliton atoms” [30]. Moreover, a cw component was presented on the mode-locked spectrum. However, it is not essential for the formation of soliton molecules in our experiments. The corresponding autocorrelation trace of soliton molecule was presented in Fig. 3 (b). 7 peaks with 1.63 ps separation were symmetrically shown on the autocorrelation trace and the intensities of the side peaks gradually decreased, which is the typical characteristic of regular soliton molecule. As we know, the autocorrelation trace could not directly reflect the pulse profile of soliton molecule. Therefore, in order to better characterize soliton molecule, Fig. 4 shows the plotted pulse profile of the soliton molecule as well as the corresponding autocorrelation trace with the same method as that of [31]. Here, the plotted autocorrelation trace is well consistent with the experimentally observed one. Thus, based on the calculated results of Fig. 4, it could be deduced that there were 4 pulses with the same characteristics (including the pulse widths and peak amplitudes) in the observed soliton molecule, which was categorized as a regular one.

 figure: Fig. 3

Fig. 3 Regular soliton molecule operation. (a) Spectrum; (b) Autocorrelation trace.

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 figure: Fig. 4

Fig. 4 Theoretically recovered regular soliton molecule. (a) Pulse profile; (b) Autocorrelation trace.

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Then the PCs were adjusted slightly. In this case, it could be seen that the profiles of autocorrelation traces varied and became distinctly different from the regular one, which we termed as ‘structural soliton molecules’. Figure 5 presents two types of structural soliton molecules and the theoretically reconstructed pulse profiles. As can be seen in Figs. 5(a) and 5(b), the mode-locked spectra of structural soliton molecules were similar to that of regular one shown in Fig. 3(a). Therefore, in the following we concentrated on the investigations of the structural soliton molecules in time domain. Figures 5(c)-5(h) present the observed two typical autocorrelation traces of structural soliton molecules and the corresponding recovered pulse profiles. Being different from the regular soliton molecule shown in Fig. 3(b), the side peaks of autocorrelation traces of structural soliton molecules are irregular, where they did not gradually decrease with the orders of the side peaks, as shown in Figs. 5(c) and 5(d). In addition, the interval among the solitons inside the molecule is about 388 fs and 329 fs, respectively, which is obviously narrower than that of regular one. In comparison of the autocorrelation traces between regular and structural soliton molecules, we believed that there exist fine structures inside the soliton molecules in our fiber laser. In order to better show the fine details of “atoms” inside the structural soliton molecules, we also plotted the profiles of the two types of structural soliton molecules according to the measured autocorrelation traces using the method reported in [31]. Figures 5(e) and 5(f) shows the recovered profiles of structural soliton molecules, and the corresponding calculated autocorrelation traces are presented in Figs. 5(g) and 5(h). As can be seen here, the calculated autocorrelation traces exhibit almost the same profiles as the measured ones, confirming that the “soliton atoms” of the observed structural soliton molecules are the same as those shown in Figs. 5(e) and 5(f). Obviously, the pulse profiles of the structural soliton molecules are different from the regular ones, which show the pulses (soliton atoms) with unequal intensities and durations.

 figure: Fig. 5

Fig. 5 Structural soliton molecules. (a) and (b) are mode-locked spectra; (c) and (d) represent the measured autocorrelation traces; (e) and (f) represent the theoretically plotted pulse profiles; (g) and (h) represent the recovered autocorrelation traces.

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Generally, the pulse separations inside the soliton molecules are always several or tens of picoseconds [1215]. However, in our experiment the pulse interval of the structural soliton molecule is measured to be ~350 fs, which is much narrower than those of the regular soliton molecules. We think that the stable propagation of the structural soliton molecules with ultra-narrow separation could be attributed to the excellent saturable absorption ability of microfiber-based GSA. As we know, apart from the saturable absorption effect of the GSA, the highly nonlinear effect also exists in the microfiber-based GSA. Under the condition of highly nonlinear effect, the mode-locked soliton could split some excessive energy close to itself in time domain to balance the accumulated cavity nonlinearity. It should be noted that, compared with artificial saturable absorbers such as NPR and NALM, inserting a real SA (i.e., GSA) into the cavity would be more efficiently to obtain mode-locking state. Therefore, when the pulse was broken into some optical waves with narrow separations, the microfiber-based GSA facilitates the quick reshaping of the splitting waves to be stable solitons even if the intensities of the pulses are low. Then the reshaped structural solitons propagated together as one unit and thus the structural soliton molecules were formed in the fiber laser. In addition, although the pulses inside the soliton molecules might experience some variations when propagating in the laser cavity, the output status at the point of output coupler could be fixed. Therefore, the statically theoretical analyses [31] are still suitable for analyzing the pulse profiles of the observed structural soliton molecules.

4. Conclusion

In summary, we have demonstrated the generation of the structural soliton molecules in a fiber laser by employing a microfiber-based GSA. Taking advantage of the highly nonlinear effect generated by the microfiber-based GSA, both the regular and structural soliton molecules could be easily observed. By carefully analyzing the autocorrelation traces, the pulse profiles of structural soliton molecules were further identified. It is found that the “atoms” inside the structural soliton molecules could possess different characteristics such as different pulse durations and peak intensities. Moreover, the separations of “soliton atoms” inside the structural soliton molecules are much narrower than those of regular ones. The observed results would further enhance the understanding of fundamental physics of soliton molecules, and demonstrate that the microfiber-based GSA is indeed an excellent highly-nonlinear photonic device for investigating versatile soliton nonlinear dynamics in fiber lasers.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 61378036, 61307058, 11304101, 11474108, 11074078), the PhD Start-up Fund of Natural Science Foundation of Guangdong Province, China (Grant No. S2013040016320), and the Scientific and Technological Innovation Project of Higher Education Institute, Guangdong, China (Grant No. 2013KJCX0051). Z.-C. Luo acknowledges the financial support from Zhujiang New-star Plan of Science & Technology in Guangzhou City (Grant No. 2014J2200008).

References and links

1. L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980). [CrossRef]  

2. S. Chouli and P. Grelu, “Rains of solitons in a fiber laser,” Opt. Express 17(14), 11776–11781 (2009). [CrossRef]   [PubMed]  

3. Y. F. Song, L. Li, H. Zhang, Y. Shen, D. Y. Tang, and K. P. Loh, “Vector multi-soliton operation and interaction in a graphene mode-locked fiber laser,” Opt. Express 21(8), 10010–10018 (2013). [CrossRef]   [PubMed]  

4. Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014). [CrossRef]  

5. F. Amrani, M. Salhi, P. Grelu, H. Leblond, and F. Sanchez, “Universal soliton pattern formations in passively mode-locked fiber lasers,” Opt. Lett. 36(9), 1545–1547 (2011). [CrossRef]   [PubMed]  

6. F. Amrani, M. Salhi, H. Leblond, A. Haboucha, and F. Sanchez, “Intricate solitons state in passively mode-locked fiber lasers,” Opt. Express 19(14), 13134–13139 (2011). [CrossRef]   [PubMed]  

7. 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]   [PubMed]  

8. Z. C. Luo, M. Liu, H. Liu, X. W. Zheng, A. P. Luo, C. J. Zhao, H. Zhang, S. C. Wen, and W. C. Xu, “2 GHz passively harmonic mode-locked fiber laser by a microfiber-based Topological Insulator saturable absorber,” Opt. Lett. 38(24), 5212–5215 (2013). [CrossRef]   [PubMed]  

9. Y. C. Meng, S. M. Zhang, X. L. Li, H. F. Li, J. Du, and Y. P. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012). [CrossRef]   [PubMed]  

10. C. Lecaplain and P. Grelu, “Multi-gigahertz repetition-rate-selectable passive harmonic mode locking of a fiber laser,” Opt. Express 21(9), 10897–10902 (2013). [CrossRef]   [PubMed]  

11. P. Grelu and J. M. Soto-Crespo, “Temporal soliton “molecules” in mode-locked lasers: collisions, pulsations, and vibrations,” Lect. Notes Phys. 751, 137–173 (2008).

12. M. Grapinet and P. Grelu, “Vibrating soliton pairs in a mode-locked laser cavity,” Opt. Lett. 31(14), 2115–2117 (2006). [CrossRef]   [PubMed]  

13. X. M. Liu, “Dynamic evolution of temporal dissipative-soliton molecules in large normal path-averaged dispersion fiber lasers,” Phys. Rev. A 82(6), 063834 (2010). [CrossRef]  

14. A. Zaviyalov, P. Grelu, and F. Lederer, “Impact of slow gain dynamics on soliton molecules in mode-locked fiber lasers,” Opt. Lett. 37(2), 175–177 (2012). [CrossRef]   [PubMed]  

15. X. H. Li, Y. S. Wang, W. Zhang, and W. Zhao, “Experimental observation of soliton molecule evolution in Yb-doped passively mode-locked fiber lasers,” Laser Phys. Lett. 11(7), 075103 (2014). [CrossRef]  

16. X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012). [CrossRef]  

17. Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009). [CrossRef]  

18. H. Zhang, Q. L. Bao, D. Y. Tang, L. M. Zhao, and K. P. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009). [CrossRef]  

19. H. Zhang, D. Y. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010). [CrossRef]  

20. Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010). [CrossRef]   [PubMed]  

21. Z. Q. Luo, M. Zhou, J. Weng, G. M. Huang, H. Y. Xu, C. C. Ye, and Z. P. Cai, “Graphene-based passively Q-switched dual-wavelength erbium-doped fiber laser,” Opt. Lett. 35(21), 3709–3711 (2010). [CrossRef]   [PubMed]  

22. A. Martinez, K. Fuse, B. Xu, and S. Yamashita, “Optical deposition of graphene and carbon nanotubes in a fiber ferrule for passive mode-locked lasing,” Opt. Express 18(22), 23054–23061 (2010). [CrossRef]   [PubMed]  

23. H. Kim, J. Cho, S. Y. Jang, and Y. W. Song, “Deformation-immunized optical deposition of graphene for ultrafast pulsed lasers,” Appl. Phys. Lett. 98(2), 021104 (2011). [CrossRef]  

24. Y. Cui and X. Liu, “Graphene and nanotube mode-locked fiber laser emitting dissipative and conventional solitons,” Opt. Express 21(16), 18969–18974 (2013). [CrossRef]   [PubMed]  

25. H. Zhang, S. Virally, Q. L. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012). [CrossRef]   [PubMed]  

26. G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014). [CrossRef]  

27. N. Zhao, M. Liu, H. Liu, X. W. Zheng, Q. Y. Ning, A. P. Luo, Z. C. Luo, and W. C. Xu, “Dual-wavelength rectangular pulse Yb-doped fiber laser using a microfiber-based graphene saturable absorber,” Opt. Express 22(9), 10906–10913 (2014). [CrossRef]   [PubMed]  

28. P. F. Zhu, Z. B. Lin, Q. Y. Ning, Z. R. Cai, X. B. Xing, J. Liu, W. C. Chen, Z. C. Luo, A. P. Luo, and W. C. Xu, “Passive harmonic mode-locking in a fiber laser by using a microfiber-based graphene saturable absorber,” Laser Phys. Lett. 10(10), 105107 (2013). [CrossRef]  

29. D. Anderson, M. Desaix, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave breaking in nonlinear-optical fibers,” J. Opt. Soc. Am. B 9(8), 1358–1361 (1992). [CrossRef]  

30. 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]  

31. A. Luo, H. Liu, N. Zhao, X. Zheng, M. Liu, R. Tang, Z. C. Luo, and W. C. Xu, “Observation of three bound states from a topological insulator mode-locked soliton fiber laser,” IEEE Photon. J. 6(4), 1501508 (2014). [CrossRef]  

References

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  1. L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
    [Crossref]
  2. S. Chouli and P. Grelu, “Rains of solitons in a fiber laser,” Opt. Express 17(14), 11776–11781 (2009).
    [Crossref] [PubMed]
  3. Y. F. Song, L. Li, H. Zhang, Y. Shen, D. Y. Tang, and K. P. Loh, “Vector multi-soliton operation and interaction in a graphene mode-locked fiber laser,” Opt. Express 21(8), 10010–10018 (2013).
    [Crossref] [PubMed]
  4. Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
    [Crossref]
  5. F. Amrani, M. Salhi, P. Grelu, H. Leblond, and F. Sanchez, “Universal soliton pattern formations in passively mode-locked fiber lasers,” Opt. Lett. 36(9), 1545–1547 (2011).
    [Crossref] [PubMed]
  6. F. Amrani, M. Salhi, H. Leblond, A. Haboucha, and F. Sanchez, “Intricate solitons state in passively mode-locked fiber lasers,” Opt. Express 19(14), 13134–13139 (2011).
    [Crossref] [PubMed]
  7. 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] [PubMed]
  8. Z. C. Luo, M. Liu, H. Liu, X. W. Zheng, A. P. Luo, C. J. Zhao, H. Zhang, S. C. Wen, and W. C. Xu, “2 GHz passively harmonic mode-locked fiber laser by a microfiber-based Topological Insulator saturable absorber,” Opt. Lett. 38(24), 5212–5215 (2013).
    [Crossref] [PubMed]
  9. Y. C. Meng, S. M. Zhang, X. L. Li, H. F. Li, J. Du, and Y. P. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
    [Crossref] [PubMed]
  10. C. Lecaplain and P. Grelu, “Multi-gigahertz repetition-rate-selectable passive harmonic mode locking of a fiber laser,” Opt. Express 21(9), 10897–10902 (2013).
    [Crossref] [PubMed]
  11. P. Grelu and J. M. Soto-Crespo, “Temporal soliton “molecules” in mode-locked lasers: collisions, pulsations, and vibrations,” Lect. Notes Phys. 751, 137–173 (2008).
  12. M. Grapinet and P. Grelu, “Vibrating soliton pairs in a mode-locked laser cavity,” Opt. Lett. 31(14), 2115–2117 (2006).
    [Crossref] [PubMed]
  13. X. M. Liu, “Dynamic evolution of temporal dissipative-soliton molecules in large normal path-averaged dispersion fiber lasers,” Phys. Rev. A 82(6), 063834 (2010).
    [Crossref]
  14. A. Zaviyalov, P. Grelu, and F. Lederer, “Impact of slow gain dynamics on soliton molecules in mode-locked fiber lasers,” Opt. Lett. 37(2), 175–177 (2012).
    [Crossref] [PubMed]
  15. X. H. Li, Y. S. Wang, W. Zhang, and W. Zhao, “Experimental observation of soliton molecule evolution in Yb-doped passively mode-locked fiber lasers,” Laser Phys. Lett. 11(7), 075103 (2014).
    [Crossref]
  16. X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
    [Crossref]
  17. Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
    [Crossref]
  18. H. Zhang, Q. L. Bao, D. Y. Tang, L. M. Zhao, and K. P. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
    [Crossref]
  19. H. Zhang, D. Y. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
    [Crossref]
  20. Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
    [Crossref] [PubMed]
  21. Z. Q. Luo, M. Zhou, J. Weng, G. M. Huang, H. Y. Xu, C. C. Ye, and Z. P. Cai, “Graphene-based passively Q-switched dual-wavelength erbium-doped fiber laser,” Opt. Lett. 35(21), 3709–3711 (2010).
    [Crossref] [PubMed]
  22. A. Martinez, K. Fuse, B. Xu, and S. Yamashita, “Optical deposition of graphene and carbon nanotubes in a fiber ferrule for passive mode-locked lasing,” Opt. Express 18(22), 23054–23061 (2010).
    [Crossref] [PubMed]
  23. H. Kim, J. Cho, S. Y. Jang, and Y. W. Song, “Deformation-immunized optical deposition of graphene for ultrafast pulsed lasers,” Appl. Phys. Lett. 98(2), 021104 (2011).
    [Crossref]
  24. Y. Cui and X. Liu, “Graphene and nanotube mode-locked fiber laser emitting dissipative and conventional solitons,” Opt. Express 21(16), 18969–18974 (2013).
    [Crossref] [PubMed]
  25. H. Zhang, S. Virally, Q. L. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
    [Crossref] [PubMed]
  26. G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
    [Crossref]
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    [Crossref] [PubMed]
  28. P. F. Zhu, Z. B. Lin, Q. Y. Ning, Z. R. Cai, X. B. Xing, J. Liu, W. C. Chen, Z. C. Luo, A. P. Luo, and W. C. Xu, “Passive harmonic mode-locking in a fiber laser by using a microfiber-based graphene saturable absorber,” Laser Phys. Lett. 10(10), 105107 (2013).
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  31. A. Luo, H. Liu, N. Zhao, X. Zheng, M. Liu, R. Tang, Z. C. Luo, and W. C. Xu, “Observation of three bound states from a topological insulator mode-locked soliton fiber laser,” IEEE Photon. J. 6(4), 1501508 (2014).
    [Crossref]

2014 (6)

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

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] [PubMed]

X. H. Li, Y. S. Wang, W. Zhang, and W. Zhao, “Experimental observation of soliton molecule evolution in Yb-doped passively mode-locked fiber lasers,” Laser Phys. Lett. 11(7), 075103 (2014).
[Crossref]

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

N. Zhao, M. Liu, H. Liu, X. W. Zheng, Q. Y. Ning, A. P. Luo, Z. C. Luo, and W. C. Xu, “Dual-wavelength rectangular pulse Yb-doped fiber laser using a microfiber-based graphene saturable absorber,” Opt. Express 22(9), 10906–10913 (2014).
[Crossref] [PubMed]

A. Luo, H. Liu, N. Zhao, X. Zheng, M. Liu, R. Tang, Z. C. Luo, and W. C. Xu, “Observation of three bound states from a topological insulator mode-locked soliton fiber laser,” IEEE Photon. J. 6(4), 1501508 (2014).
[Crossref]

2013 (5)

2012 (4)

2011 (3)

2010 (5)

H. Zhang, D. Y. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Z. Q. Luo, M. Zhou, J. Weng, G. M. Huang, H. Y. Xu, C. C. Ye, and Z. P. Cai, “Graphene-based passively Q-switched dual-wavelength erbium-doped fiber laser,” Opt. Lett. 35(21), 3709–3711 (2010).
[Crossref] [PubMed]

A. Martinez, K. Fuse, B. Xu, and S. Yamashita, “Optical deposition of graphene and carbon nanotubes in a fiber ferrule for passive mode-locked lasing,” Opt. Express 18(22), 23054–23061 (2010).
[Crossref] [PubMed]

X. M. Liu, “Dynamic evolution of temporal dissipative-soliton molecules in large normal path-averaged dispersion fiber lasers,” Phys. Rev. A 82(6), 063834 (2010).
[Crossref]

2009 (4)

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]

Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

H. Zhang, Q. L. Bao, D. Y. Tang, L. M. Zhao, and K. P. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

S. Chouli and P. Grelu, “Rains of solitons in a fiber laser,” Opt. Express 17(14), 11776–11781 (2009).
[Crossref] [PubMed]

2006 (1)

1992 (1)

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
[Crossref]

Amrani, F.

Anderson, D.

Bao, Q.

H. Zhang, D. Y. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

Bao, Q. L.

H. Zhang, S. Virally, Q. L. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
[Crossref] [PubMed]

H. Zhang, Q. L. Bao, D. Y. Tang, L. M. Zhao, and K. P. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Basko, D. M.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Blau, W. J.

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Bonaccorso, F.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Cai, Z. P.

Cai, Z. R.

P. F. Zhu, Z. B. Lin, Q. Y. Ning, Z. R. Cai, X. B. Xing, J. Liu, W. C. Chen, Z. C. Luo, A. P. Luo, and W. C. Xu, “Passive harmonic mode-locking in a fiber laser by using a microfiber-based graphene saturable absorber,” Laser Phys. Lett. 10(10), 105107 (2013).
[Crossref]

Chen, S. Q.

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Chen, W. C.

P. F. Zhu, Z. B. Lin, Q. Y. Ning, Z. R. Cai, X. B. Xing, J. Liu, W. C. Chen, Z. C. Luo, A. P. Luo, and W. C. Xu, “Passive harmonic mode-locking in a fiber laser by using a microfiber-based graphene saturable absorber,” Laser Phys. Lett. 10(10), 105107 (2013).
[Crossref]

Chen, Y.

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Cheng, X.

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Cheng, Y.

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Cho, J.

H. Kim, J. Cho, S. Y. Jang, and Y. W. Song, “Deformation-immunized optical deposition of graphene for ultrafast pulsed lasers,” Appl. Phys. Lett. 98(2), 021104 (2011).
[Crossref]

Chouli, S.

Coghlan, D.

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Cui, Y.

Desaix, M.

Dong, N. N.

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Du, J.

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Y. C. Meng, S. M. Zhang, X. L. Li, H. F. Li, J. Du, and Y. P. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
[Crossref] [PubMed]

Egorov, O.

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]

Ferrari, A. C.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Fuse, K.

Gao, C.

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Godbout, N.

Gordon, J. G.

L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
[Crossref]

Grapinet, M.

Grelu, P.

Haboucha, A.

Hao, Y. P.

Hasan, T.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Hu, X.

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Huang, G. M.

Huang, X. G.

Iliew, R.

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]

Jang, S. Y.

H. Kim, J. Cho, S. Y. Jang, and Y. W. Song, “Deformation-immunized optical deposition of graphene for ultrafast pulsed lasers,” Appl. Phys. Lett. 98(2), 021104 (2011).
[Crossref]

Jiang, G. B.

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Kim, H.

H. Kim, J. Cho, S. Y. Jang, and Y. W. Song, “Deformation-immunized optical deposition of graphene for ultrafast pulsed lasers,” Appl. Phys. Lett. 98(2), 021104 (2011).
[Crossref]

Knize, R. J.

H. Zhang, D. Y. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

Kockaert, P.

Leblond, H.

Lecaplain, C.

Lederer, F.

A. Zaviyalov, P. Grelu, and F. Lederer, “Impact of slow gain dynamics on soliton molecules in mode-locked fiber lasers,” Opt. Lett. 37(2), 175–177 (2012).
[Crossref] [PubMed]

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]

Li, C.

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Li, H. F.

Li, L.

Li, X.

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Li, X. H.

X. H. Li, Y. S. Wang, W. Zhang, and W. Zhao, “Experimental observation of soliton molecule evolution in Yb-doped passively mode-locked fiber lasers,” Laser Phys. Lett. 11(7), 075103 (2014).
[Crossref]

Li, X. L.

Li, Y.

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Lin, Z. B.

P. F. Zhu, Z. B. Lin, Q. Y. Ning, Z. R. Cai, X. B. Xing, J. Liu, W. C. Chen, Z. C. Luo, A. P. Luo, and W. C. Xu, “Passive harmonic mode-locking in a fiber laser by using a microfiber-based graphene saturable absorber,” Laser Phys. Lett. 10(10), 105107 (2013).
[Crossref]

Lisak, M.

Liu, H.

Liu, J.

P. F. Zhu, Z. B. Lin, Q. Y. Ning, Z. R. Cai, X. B. Xing, J. Liu, W. C. Chen, Z. C. Luo, A. P. Luo, and W. C. Xu, “Passive harmonic mode-locking in a fiber laser by using a microfiber-based graphene saturable absorber,” Laser Phys. Lett. 10(10), 105107 (2013).
[Crossref]

Liu, M.

Liu, X.

Liu, X. M.

X. M. Liu, “Dynamic evolution of temporal dissipative-soliton molecules in large normal path-averaged dispersion fiber lasers,” Phys. Rev. A 82(6), 063834 (2010).
[Crossref]

Loh, K. P.

Y. F. Song, L. Li, H. Zhang, Y. Shen, D. Y. Tang, and K. P. Loh, “Vector multi-soliton operation and interaction in a graphene mode-locked fiber laser,” Opt. Express 21(8), 10010–10018 (2013).
[Crossref] [PubMed]

H. Zhang, D. Y. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

H. Zhang, Q. L. Bao, D. Y. Tang, L. M. Zhao, and K. P. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Luo, A.

A. Luo, H. Liu, N. Zhao, X. Zheng, M. Liu, R. Tang, Z. C. Luo, and W. C. Xu, “Observation of three bound states from a topological insulator mode-locked soliton fiber laser,” IEEE Photon. J. 6(4), 1501508 (2014).
[Crossref]

Luo, A. P.

Luo, Z. C.

Luo, Z. Q.

Martinez, A.

Massar, S.

Meng, Y. C.

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
[Crossref]

Ni, Z.

Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Ning, Q. Y.

Ping, L. K.

Popa, D.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Privitera, G.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Quiroga-Teixeiro, M. L.

Salhi, M.

Sanchez, F.

Shen, D.

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Shen, Y.

Shen, Z.

Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Song, Y. F.

Song, Y. W.

H. Kim, J. Cho, S. Y. Jang, and Y. W. Song, “Deformation-immunized optical deposition of graphene for ultrafast pulsed lasers,” Appl. Phys. Lett. 98(2), 021104 (2011).
[Crossref]

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
[Crossref]

Sun, Z.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Tang, D. Y.

Y. F. Song, L. Li, H. Zhang, Y. Shen, D. Y. Tang, and K. P. Loh, “Vector multi-soliton operation and interaction in a graphene mode-locked fiber laser,” Opt. Express 21(8), 10010–10018 (2013).
[Crossref] [PubMed]

H. Zhang, D. Y. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

H. Zhang, Q. L. Bao, D. Y. Tang, L. M. Zhao, and K. P. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Tang, P. H.

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Tang, R.

A. Luo, H. Liu, N. Zhao, X. Zheng, M. Liu, R. Tang, Z. C. Luo, and W. C. Xu, “Observation of three bound states from a topological insulator mode-locked soliton fiber laser,” IEEE Photon. J. 6(4), 1501508 (2014).
[Crossref]

Torrisi, F.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Umran, F. A.

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Virally, S.

Wang, F.

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Wang, G. Z.

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Wang, H.

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Wang, J.

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Wang, X.

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Wang, Y.

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Wang, Y. S.

X. H. Li, Y. S. Wang, W. Zhang, and W. Zhao, “Experimental observation of soliton molecule evolution in Yb-doped passively mode-locked fiber lasers,” Laser Phys. Lett. 11(7), 075103 (2014).
[Crossref]

Wen, S. C.

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Z. C. Luo, M. Liu, H. Liu, X. W. Zheng, A. P. Luo, C. J. Zhao, H. Zhang, S. C. Wen, and W. C. Xu, “2 GHz passively harmonic mode-locked fiber laser by a microfiber-based Topological Insulator saturable absorber,” Opt. Lett. 38(24), 5212–5215 (2013).
[Crossref] [PubMed]

Weng, J.

Wu, M.

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Xing, X. B.

P. F. Zhu, Z. B. Lin, Q. Y. Ning, Z. R. Cai, X. B. Xing, J. Liu, W. C. Chen, Z. C. Luo, A. P. Luo, and W. C. Xu, “Passive harmonic mode-locking in a fiber laser by using a microfiber-based graphene saturable absorber,” Laser Phys. Lett. 10(10), 105107 (2013).
[Crossref]

Xu, B.

Xu, H. Y.

Xu, S. H.

Xu, W. C.

Yamashita, S.

Yan, Y.

Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Yang, Z.

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Yang, Z. M.

Ye, C. C.

Yu, W.

Zaviyalov, A.

Zavyalov, A.

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]

Zhang, H.

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Y. F. Song, L. Li, H. Zhang, Y. Shen, D. Y. Tang, and K. P. Loh, “Vector multi-soliton operation and interaction in a graphene mode-locked fiber laser,” Opt. Express 21(8), 10010–10018 (2013).
[Crossref] [PubMed]

Z. C. Luo, M. Liu, H. Liu, X. W. Zheng, A. P. Luo, C. J. Zhao, H. Zhang, S. C. Wen, and W. C. Xu, “2 GHz passively harmonic mode-locked fiber laser by a microfiber-based Topological Insulator saturable absorber,” Opt. Lett. 38(24), 5212–5215 (2013).
[Crossref] [PubMed]

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X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

H. Zhang, D. Y. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
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H. Zhang, Q. L. Bao, D. Y. Tang, L. M. Zhao, and K. P. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
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G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Zhang, S. F.

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Zhang, S. M.

Zhang, W.

X. H. Li, Y. S. Wang, W. Zhang, and W. Zhao, “Experimental observation of soliton molecule evolution in Yb-doped passively mode-locked fiber lasers,” Laser Phys. Lett. 11(7), 075103 (2014).
[Crossref]

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Zhao, C. J.

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Z. C. Luo, M. Liu, H. Liu, X. W. Zheng, A. P. Luo, C. J. Zhao, H. Zhang, S. C. Wen, and W. C. Xu, “2 GHz passively harmonic mode-locked fiber laser by a microfiber-based Topological Insulator saturable absorber,” Opt. Lett. 38(24), 5212–5215 (2013).
[Crossref] [PubMed]

Zhao, L.

H. Zhang, D. Y. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

Zhao, L. M.

H. Zhang, Q. L. Bao, D. Y. Tang, L. M. Zhao, and K. P. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

Zhao, N.

N. Zhao, M. Liu, H. Liu, X. W. Zheng, Q. Y. Ning, A. P. Luo, Z. C. Luo, and W. C. Xu, “Dual-wavelength rectangular pulse Yb-doped fiber laser using a microfiber-based graphene saturable absorber,” Opt. Express 22(9), 10906–10913 (2014).
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A. Luo, H. Liu, N. Zhao, X. Zheng, M. Liu, R. Tang, Z. C. Luo, and W. C. Xu, “Observation of three bound states from a topological insulator mode-locked soliton fiber laser,” IEEE Photon. J. 6(4), 1501508 (2014).
[Crossref]

Zhao, W.

X. H. Li, Y. S. Wang, W. Zhang, and W. Zhao, “Experimental observation of soliton molecule evolution in Yb-doped passively mode-locked fiber lasers,” Laser Phys. Lett. 11(7), 075103 (2014).
[Crossref]

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
[Crossref]

Zheng, X.

A. Luo, H. Liu, N. Zhao, X. Zheng, M. Liu, R. Tang, Z. C. Luo, and W. C. Xu, “Observation of three bound states from a topological insulator mode-locked soliton fiber laser,” IEEE Photon. J. 6(4), 1501508 (2014).
[Crossref]

Zheng, X. W.

Zhou, M.

Zhu, P. F.

P. F. Zhu, Z. B. Lin, Q. Y. Ning, Z. R. Cai, X. B. Xing, J. Liu, W. C. Chen, Z. C. Luo, A. P. Luo, and W. C. Xu, “Passive harmonic mode-locking in a fiber laser by using a microfiber-based graphene saturable absorber,” Laser Phys. Lett. 10(10), 105107 (2013).
[Crossref]

ACS Nano (1)

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano 4(2), 803–810 (2010).
[Crossref] [PubMed]

Adv. Funct. Mater. (1)

Q. L. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. Shen, K. P. Loh, and D. Y. Tang, “Atomic layer graphene as saturable absorber for ultrafast pulsed laser,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Appl. Phys. Lett. (4)

H. Zhang, Q. L. Bao, D. Y. Tang, L. M. Zhao, and K. P. Loh, “Large energy soliton erbium-doped fiber laser with a graphene-polymer composite mode locker,” Appl. Phys. Lett. 95(14), 141103 (2009).
[Crossref]

H. Zhang, D. Y. Tang, R. J. Knize, L. Zhao, Q. Bao, and K. P. Loh, “Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser,” Appl. Phys. Lett. 96(11), 111112 (2010).
[Crossref]

H. Kim, J. Cho, S. Y. Jang, and Y. W. Song, “Deformation-immunized optical deposition of graphene for ultrafast pulsed lasers,” Appl. Phys. Lett. 98(2), 021104 (2011).
[Crossref]

G. Z. Wang, S. F. Zhang, F. A. Umran, X. Cheng, N. N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

IEEE Photon. J. (1)

A. Luo, H. Liu, N. Zhao, X. Zheng, M. Liu, R. Tang, Z. C. Luo, and W. C. Xu, “Observation of three bound states from a topological insulator mode-locked soliton fiber laser,” IEEE Photon. J. 6(4), 1501508 (2014).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Phys. Lett. (3)

P. F. Zhu, Z. B. Lin, Q. Y. Ning, Z. R. Cai, X. B. Xing, J. Liu, W. C. Chen, Z. C. Luo, A. P. Luo, and W. C. Xu, “Passive harmonic mode-locking in a fiber laser by using a microfiber-based graphene saturable absorber,” Laser Phys. Lett. 10(10), 105107 (2013).
[Crossref]

X. H. Li, Y. S. Wang, W. Zhang, and W. Zhao, “Experimental observation of soliton molecule evolution in Yb-doped passively mode-locked fiber lasers,” Laser Phys. Lett. 11(7), 075103 (2014).
[Crossref]

Y. Chen, M. Wu, P. H. Tang, S. Q. Chen, J. Du, G. B. Jiang, Y. Li, C. J. Zhao, H. Zhang, and S. C. Wen, “The formation of various multi-soliton patterns and noise-like pulse in a fiber laser passively mode locked by a topological insulator based saturable absorber,” Laser Phys. Lett. 11(5), 055101 (2014).
[Crossref]

Opt. Commun. (1)

X. Li, Y. Wang, W. Zhao, W. Zhang, X. Hu, C. Gao, H. Zhang, Z. Yang, H. Wang, X. Wang, C. Li, and D. Shen, “Numerical investigation of soliton molecules with variable separation in passively mode-locked fiber lasers,” Opt. Commun. 285(6), 1356–1361 (2012).
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Opt. Express (9)

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Y. F. Song, L. Li, H. Zhang, Y. Shen, D. Y. Tang, and K. P. Loh, “Vector multi-soliton operation and interaction in a graphene mode-locked fiber laser,” Opt. Express 21(8), 10010–10018 (2013).
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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).
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Y. C. Meng, S. M. Zhang, X. L. Li, H. F. Li, J. Du, and Y. P. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
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C. Lecaplain and P. Grelu, “Multi-gigahertz repetition-rate-selectable passive harmonic mode locking of a fiber laser,” Opt. Express 21(9), 10897–10902 (2013).
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N. Zhao, M. Liu, H. Liu, X. W. Zheng, Q. Y. Ning, A. P. Luo, Z. C. Luo, and W. C. Xu, “Dual-wavelength rectangular pulse Yb-doped fiber laser using a microfiber-based graphene saturable absorber,” Opt. Express 22(9), 10906–10913 (2014).
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X. M. Liu, “Dynamic evolution of temporal dissipative-soliton molecules in large normal path-averaged dispersion fiber lasers,” Phys. Rev. A 82(6), 063834 (2010).
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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).
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Phys. Rev. Lett. (1)

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Figures (5)

Fig. 1
Fig. 1 (a) Microscopy image of microfiber-based GSA; (b) Scattering evanescent field of the microfiber through injecting the visible light; (c) Measured saturable absorption curve and the corresponding fitting curve.
Fig. 2
Fig. 2 Schematic of the passively mode-locked fiber laser with the microfiber-based GSA.
Fig. 3
Fig. 3 Regular soliton molecule operation. (a) Spectrum; (b) Autocorrelation trace.
Fig. 4
Fig. 4 Theoretically recovered regular soliton molecule. (a) Pulse profile; (b) Autocorrelation trace.
Fig. 5
Fig. 5 Structural soliton molecules. (a) and (b) are mode-locked spectra; (c) and (d) represent the measured autocorrelation traces; (e) and (f) represent the theoretically plotted pulse profiles; (g) and (h) represent the recovered autocorrelation traces.

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