Open Access
Add to BibSonomy
Abstract
In this study, we conduct a numerical evaluation of the impact of the recovery time of a saturable absorber (SA) on the output performance of an Yb-doped fiber laser operating in the dissipative soliton regime. Particularly, we evaluate the output pulse characteristics, such as the pulse width, spectral bandwidth, pulse peak power, and pulse energy depending on the change in recovery time. Applying a too-slow SA recovery time above a certain critical value is shown to make the output pulse unstable and broken. Furthermore, we demonstrate that there is an optimum recovery time range for stable dissipative soliton pulse generation, depending on the cavity dispersion and modulation depth of the SA. Further, we perform an additional numerical simulation of the pulse compression to demonstrate the relationship between the output dechirped pulse width and SA recovery time. The optimum approach for the generation of the shortest dechirped pulses in the dissipative soliton regime will be to construct a fiber laser cavity with a small normal cavity group velocity dispersion and use an SA with an appropriate recovery time.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High-energy pulsed lasers have attracted significant interest in scientific and industrial applications. Particularly, owing to their advantages such as compactness, easy alignment, and good beam quality, fiber lasers have been considered attractive light sources for the generation of pulsed outputs [1]. To produce pulsed outputs, mode-locking has been the commonly used technique, which involves the phase locking of multiple longitudinal cavity modes. Generally, mode-locked fiber lasers can be achieved by a balance of nonlinearity and group velocity dispersion (GVD). When a fiber laser cavity is in an anomalous cavity GVD, conventional soliton pulses are generated as a result of the balanced of GVD and self-phase-modulation (SPM) effects. However, obtaining high-energy pulses in the case of conventional soliton mode-locking is very difficult, because the output pulses would break up owing to the imbalance between the GVD and SPM effects when the pulse energy is increased. Using a fiber laser cavity with a normal GVD, linearly chirped pulses with high energy can be generated. The chirped output pulses can be readily compressed to femtosecond pulses through a dispersive medium. The dissipative soliton mode-locking technique is used to generate linearly chirped pulses. The dissipative soliton pulses are generated through a balance among gain, loss, dispersion, nonlinearity, spectral filtering, and saturable absorption within a laser cavity.
Until now, there have been numerous works on dissipative solitons. In 2006, Chong et al. first demonstrated the concept of dissipative soliton using an all-normal-dispersion (ANDi) fiber cavity at 1 μm [2]. Subsequently, an ytterbium (Yb)-doped fiber laser with pulse energy above 20 nJ was demonstrated using the same principle by Chong et al. in 2007 [3]. Kieu et al. demonstrated 80 fs dechirped pulses with a 31 nJ pulse energy from a dissipative soliton fiber laser [4]. Renninger et al. showed the existence of an area theorem specified for dissipative solitons and explained that energy quantization causes multiple pulses [5]. Cheng et al. reported a dissipative soliton generation scheme based on nonlinear polarization rotation (NPR) in an ANDi fiber laser [6]. Apart from the aforementioned instances, many studies on the output properties of dissipative soliton fiber lasers have been conducted [7–11]. Furthermore, quite a few numerical and analytical studies regarding the influence of the cavity parameters on pulse-shaping dynamics in dissipative soliton fiber laser have been conducted [12–15]. Ouyang et al. numerically studied the pulse-shaping dynamics by varying four parameters such as the position and bandwidth of the filter and the location and output ratio of the output coupler [12]. Bao et al. conducted a numerical simulation on influence of saturable absorber (SA) parameters such as the modulation depth and saturation power on the output pulse characteristics in dissipative soliton fiber lasers [13]. Zhang et al. reported the effects of different cavity conditions such as the cavity length, filter bandwidth, nonlinear phase shifts, and initial fields on the output pulses of a dissipative soliton fiber laser [14]. Recently, Chi et al. reported a nonlinearity optimization method in a dissipative soliton mode-locked fiber laser to boost the pulse peak power [15].
Although several experimental and theoretical studies on the characteristics of dissipative soliton fiber lasers have been conducted, several issues still exist that should be properly understood or addressed. Among them is technically interesting topic that focuses on the impact of the recovery time of an SA on the output pulse features in a dissipative soliton fiber laser incorporating an SA. Although high-quality dissipative soliton pulses can be readily produced from an NPR-based fiber laser cavity, the incorporation of an SA into a fiber laser cavity to generate dissipative soliton pulses becomes prevalent owing to advantages such as the ease of self-starting and vibrational stability. Our group previously conducted numerical evaluations on the influence of key parameters of an SA on the output pulse stability in the conventional soliton mode-locking of a fiber laser [16,17]. These evaluations focused on the impact of the modulation depth and recovery time of an SA on a dispersion-managed soliton fiber laser. In addition to our evaluation, several research groups have studied the impact of the SA parameters including the recovery time on the fiber laser performance; however, their studies were also limited to conventional soliton mode-locking [18–22].
Herein, we present our theoretical study results on the impact of the recovery time of an SA on the dissipative soliton pulses in an ytterbium-doped fiber (YDF) laser. To the best of the authors’ knowledge, no reports on the influence of the SA recovery time on the dissipative soliton pulse formation in a fiber laser have been presented previously. Particularly, the effects of the SA recovery time on the output pulse features, such as the pulse width, spectral bandwidth, pulse peak power, and pulse energy, are numerically evaluated in this work. The use of a too-slow SA recovery time above a certain critical value is demonstrated to make the output pulse unstable. Further, an optimum recovery time range for stable dissipative soliton pulse generation is observed, depending on the cavity dispersion and modulation depth of the SA. Furthermore, we conducted an additional simulation for the dechirped output pulses under various recovery times. We demonstrate that the dechirped pulses under an appropriate SA recovery time exhibit a shorter temporal width compared to the those under a faster recovery time.
2. Numerical modeling
Figure 1 shows a schematic of a dissipative soliton Yb-doped fiber laser for this simulation. The laser resonator was assumed to consist of a 1-m length YDF, a short length of the single mode fiber (SMF), a filter, a 20:80 output coupler, and an SA. The component parameters for the simulation are listed in Table 1.
Table 1. Parameters used in the simulation
To study the dissipative soliton pulse dynamics in this fiber laser configuration, a numerical calculation based on the extended nonlinear Schrödinger equation (Eq. (1)) [23] and the split-step Fourier method [23] were conducted.
3. Simulation results
Our numerical simulation was conducted in a normal cavity dispersion regime by varying the SA recovery time. First, we analyzed the optimum recovery time range required for an SA to generate stable output pulses for various cavity GVD values. Figure 2(a) shows the temporal pulse shape change with an increase in the recovery time under the condition of a 0.209 ps2 cavity GVD and 70% modulation depth. The pulse width was observed to increase with the recovery time up to ∼1.59 ps from 0.1 ps. Remarkably, the output pulses became unstable when the recovery time of the SA crossed over ∼1.59 ps. The corresponding optical output spectra of the output pulses are shown in Fig. 2(b). The 3-dB bandwidth was observed to increase with the recovery time. The spectra exhibited steep edges, which are a typical feature of dissipative solitons. Figure 2(c) and Fig. 2(d) show the temporal shape and spectral shape of the output pulses at an SA recovery time of 1.6 ps. It is obvious that the output pulse was broken under this condition. Stable dissipative soliton pulses could not be obtained at a recovery time larger than 1.59 ps.
Fig. 2. Output pulse change with an increase in SA recovery time under a cavity GVD of 0.209 ps2 and modulation depth of 70%. (a) Temporal shape and (b) spectral shape. (c) Temporal shape and (d) spectral shape of the output pulse at an SA recovery time of 1.6 ps under a cavity GVD of 0.209 ps2 and modulation depth of 70%.
Further calculations were performed to obtain various output pulse parameters such as the temporal width, 3-dB bandwidth, time-bandwidth product (TBP), pulse peak power, and pulse energy, and the results are shown in Fig. 3. A slight increase in the output pulse width from 4.77 ps to 4.88 ps was observed as the recovery time was varied from 0.1 ps to 1.59 ps, while the spectral bandwidth increased from 5.94 nm to 6.79 nm. The TBP values, which were estimated from the calculated pulse width and 3-dB bandwidth, were observed to increase from 7.96 to 9.3, as shown in Fig. 3(b). An interesting finding was that both the pulse peak power and pulse energy increased with increasing recovery time, as shown in Fig. 3(c). For the calculated recovery time range of 0.1–1.59 ps, the pulse peak power was estimated as 78.79–99.58 W, whereas the pulse energy was from 393.07 to 513.87 pJ. These results imply that the use of an SA with a very short recovery time would not be suitable for obtaining large spectral bandwidth dissipative soliton pulses. Moreover, an SA with a very short recovery time is not desirable for generating high energy dissipative soliton pulses. The values of these four pulse parameters initially increased with the recovery time and then saturated. Depending on the SA recovery time, we can consider three different SA operation regimes for dissipative soliton formation in a fiber laser, as shown in Table 2. Mode detailed explanations on the underpinned pulse formation and breaking mechanism in each regime is presented in Section 4.
Fig. 3. Output pulse characteristic variations vs. SA recovery time under a condition of a cavity GVD of 0.209 ps2 and modulation depth of 70%. (a) Temporal width and 3-dB bandwidth. (b) TBP. (c) Pulse peak power, and energy.
Table 2. Three SA operation regimes depending on the SA recovery time under a condition of a cavity GVD of 0.209 ps2 and modulation depth of 70%
In our previous study on the impact of the recovery time on the mode-locking performance of a dispersion-managed fiber laser [17], we reported that the SA recovery time have significantly influences the output pulse characteristics in the case of a normal cavity GVD. According to the summarized results in Fig. 3, the SA recovery time has a non-negligible impact on dissipative soliton generation in a Yb-doped fiber laser.
Further, to evaluate the impact of the cavity GVD on the maximum recovery time for stable dissipative soliton generation, we conducted a numerical simulation by changing the cavity GVD into 0.4 ps2 while maintaining the modulation depth at 70%. This simulation was also performed for various recovery times. Figure 4(a) illustrates the temporal shape change of the output pulses, and its corresponding spectrum is shown in Fig. 4(b). In the case of a cavity GVD of 0.4 ps2, both the temporal and spectral variation tendencies are similar those of a cavity GVD of 0.209 ps2. The maximum recovery time for stable dissipative soliton generation substantially increased to 8.79 ps, which is ∼5.3 times that the in the case of a cavity GVD of 0.209 ps2. Figure 4(c) and Fig. 4(d) show the temporal shape and spectral shape of the output pulse atan SA recovery time of 9 ps. In this condition, the output pulse was observed to be unstable and broken.
Fig. 4. Output pulse change with an increase of SA recovery time under a cavity GVD of 0.4 ps2 and modulation depth of 70%. (a) Temporal shape and (b) Spectral shape. (c) Temporal shape and (d) spectral shape of the output pulse at an SA recovery time of 9 ps under a cavity GVD of 0.4 ps2 and modulation depth of 70%.
Figure 5 illustrates the temporal width, 3-dB bandwidth, TBP, pulse peak power, and pulse energy of the output pulses for the case of a 70% modulation depth and cavity GVD of 0.4 ps2. All the pulse parameters mentioned above were observed to increase with the recovery time, i.e., from 0.1 to 8.79 ps; specifically, the pulse width and 3-dB bandwidth increased from 9.98 ps to 10.65 ps and 4.21 nm to 5.48 nm, respectively. The TBP increased from 11.82 to 16.42. The pulse peak power increased from 41.20 W to 66.39 W, whereas the pulse energy increased from 428.99 pJ to 769.44 pJ. It was reaffirmed that the use of an SA with a very short recovery time is not desirable for obtaining dissipative soliton pulses with a large spectral bandwidth and large pulse energy.
Fig. 5. Output pulse characteristic variations vs. SA recovery time under a cavity GVD of 0.4 ps2 and modulation depth of 70%. (a) Temporal width and 3-dB bandwidth. (b) TBP. (c) Pulse peak power, and energy.
We then evaluated the change in the maximum SA recovery time for stable dissipative soliton generation depending on the cavity GVD. For this simulation, five different cavity GVD values (0.151, 0.209, 0.266, 0.324, and 0.4 ps2) were considered, and the modulation depth of the SA was fixed at 70%. Four characteristics of the output pulses, i.e., the temporal width, 3-dB bandwidth, pulse peak power, and pulse energy, were calculated, as shown in Fig. 6. As the cavity GVD increased, the pulse width broadened, and the 3-dB bandwidth narrowed. Additionally, the pulse peak power decreased as the pulse energy increased. These changes in the pulse characteristics depending on the cavity GVD are well known. One noticeable result in this simulation is that the recovery time range for stable dissipative soliton generation increased as the cavity GVD was enlarged. In other words, the maximum recovery time to ensure stable dissipative pulse generation is influenced by the magnitude of the net normal cavity GVD, and it increases as the cavity GVD increases. The recovery time range for stable dissipative soliton generation depending on the cavity GVD is summarized in Table 3.
Fig. 6. (a) Pulse width, (b) 3-dB bandwidth, (c) pulse peak power, and (d) pulse energy of the output pulses as a function of the recovery time for different cavity GVDs of 0.151, 0.209, 0.266, 0.324, 0.4 ps2 with a modulation depth of 70%.
Table 3. Recovery time range for stable dissipative soliton generation depending on cavity GVD at a fixed modulation depth of 70%
Subsequently, the modulation depth dependence of the recovery time range for stable dissipative soliton generation was evaluated under a fixed cavity GVD of 0.151 ps2. The results with six modulation depth values of 25, 30, 40, 50, 60, and 70% are illustrated in Fig. 7 in terms of the pulse width, 3-dB bandwidth, pulse peak power, and pulse energy. As the modulation depth increased, the pulse width decreased, the corresponding pulse spectral width increased, pulse peak power increased, and pulse energy increased. However, the recovery time range for stable dissipative soliton generation was observed to decrease as the modulation depth increased. The recovery time range for stable dissipative soliton generation depending on the modulation depth of the SA is summarized in Table 4.
Fig. 7. (a) Pulse width, (b) 3-dB bandwidth, (c) pulse peak power, and (d) pulse energy of the output pulses as a function of the recovery time for different modulation depths of 25, 30, 40, 50, 60, 70% with a cavity GVD of 0.151 ps2.
Table 4. Recovery time range for stable dissipative soliton generation depending on modulation depth of an SA at a fixed cavity GVD of 0.151 ps2.
Further, we calculated the maximum SA recovery time for stable dissipative soliton generation as a function of the cavity GVD for various modulation depth values. The calculated results are presented in Fig. 8, in which several remarkable features can be identified regarding the relationship between the recovery time, cavity GVD, and modulation depth. As the cavity GVD value approaches zero, the maximum recovery time for stable dissipative soliton generation decreases. The maximum recovery time near the zero cavity GVD was very small irrespective of the modulation depth. This indicates that an SA with an ultrafast recovery time must be used in this operation regime. However, in the regime with high cavity GVD, stable dissipative soliton pulses can be obtained using an SA with a relatively slow recovery time; an SA with a much slower recovery time can be used if it has a smaller modulation depth. However, the use of an SA with a slow recovery time in the regime with large cavity GVD inevitably induces a wider temporal width and a narrower spectral width of the output dissipative soliton pulses.
Fig. 8. Calculated maximum recovery time of a saturable absorber as a function of the cavity GVD with different modulation depths, which is required to generate stable dissipative soliton pulses.
Finally, a question remains regarding the dependence of the output dechirped pulse width on the SA recovery time. This question can be easily answered by using the calculated 3-dB bandwidth of the output dissipative soliton pulses. However, to demonstrate the relationship between the output dechirped pulse width and the SA recovery time, we performed an additional numerical simulation on the pulse compression. In this simulation, the chirp was compensated for using a grating compressor. For this simulation, we assumed the use of a free-space grating with a groove frequency of 1250 lines/mm and a grating period of 800 nm. Figure 9 shows an output dissipative soliton pulse and its dechirped pulse for both recovery time cases of 0.1 ps and 8.79 ps. The modulation depth and cavity GVD were 70% and 0.4 ps2, respectively. In the case of the 0.1 ps recovery time, the pulse duration of the dissipative solitons was ∼9.98 ps, whereas the dechirped pulse width was ∼737 fs. In the case of the 8.79 ps recovery time, the pulse duration of the dissipative solitons was ∼10.65 ps, whereas the dechirped pulse width was ∼596 fs.
Fig. 9. Temporal shape and dechirped pulse shape for recovery time of (a) 0.1 ps and (b) 8.79 ps with modulation depth of 70% at a cavity GVD of 0.4 ps2. Blue line: frequency chirp.
Figure 10(a) shows the change in the dechirped pulse duration as a function of the SA recovery time under a modulation depth of 70% and a cavity GVD of 0.4 ps2. When the SA recovery time was larger than 1 ps, no significant change in the dechirped pulse duration was observed. Additionally, we evaluated the effect of the cavity GVD change on the dechirped pulse duration, assuming that the SA has a maximum recovery time, which is allowed for stable dissipative soliton generation (see Table 3). The results are presented in Fig. 10(b). The shorter dechirped pulses could be obtained as the cavity GVD was reduced. These results indicate that the optimum approach for generating the shortest dechirped pulses in the dissipative soliton regime would be to construct a fiber laser cavity with a small net normal cavity GVD and use an SA with a relatively slow recovery time.
Fig. 10. (a) Change of dechirped durations as a function of the recovery time with modulation depth of 70% at a cavity GVD of 0.4 ps2. (b) Change of dechirped durations as a function of the cavity GVD with modulation depth of 70% and maximum recovery times at each cavity GVD.
4. Pulse formation and breaking mechanism
As aforementioned, we can consider three different SA operation regimes for dissipative soliton formation depending on the SA recovery time. The pulse formation and breaking mechanism in each regime can be explained as following.
4.1 Too-slow recovery time regime (unstable pulse)
As shown in the following Fig. 11(a), the trailing edge of the input pulses cannot be efficiently carved by the saturable absorption effect due to a too slow recovery time, even if the leading edge experiences sufficient saturable absorption. This means that asymmetric output pulses with a wide temporal width are produced in this operation regime [17]. It is thus difficult to impose a sufficient nonlinear phase shift-induced linear chirp on the pulses in the SMF due to a low peak power of the carved pulses within the laser cavity in Fig. 1. Therefore, stable dissipative soliton pulses cannot be formed above a certain critical value of the SA recovery time due to the broken balance among the parameters of spectral filtering, Kerr nonlinearity, GVD, and cavity gain. Note that the composite balance among spectral filtering, Kerr nonlinearity, GVD, and cavity gain is essential to forming dissipative soliton [10].
Fig. 11. Pulse formation mechanisms in three different SA operation regimes: (a) too-slow recovery time regime, (b) appropriate recovery time regime, and (c) fast recovery time regime.
4.2 Appropriate recovery time regime (stable dissipative soliton)
As shown in the following Fig. 11(b), symmetric short pulses are produced due to the efficient pulse carving by the saturable absorption effect for both the leading and trailing edges of the pulses. The nonlinear phase shift-induced linear chirp, which is imposed on the pulses in the SMF, is sufficient and the amplitude modulation by spectral filtering is fully conducted. The composite balance among spectral filtering, Kerr nonlinearity, GVD, and cavity gain can be completely obtained for stable dissipative soliton formation. The composite balance allows for almost similar characteristics of the output pulses in terms of temporal width and spectral bandwidth within this recovery time range.
4.3 Fast recovery time regime (stable dissipative soliton)
As shown in the following Fig. 11(c), symmetric short pulses are still produced due to the saturable absorption effect for both the leading and trailing edges of the pulses. However, the pulse carving capability of the SA is less efficient in this regime than in the appropriate recovery time regime, due to a too fast recovery time. The temporal width of the carved pulses in this regime is broader than in the appropriate recovery time regime. The nonlinear phase shift-induced linear chirp, which is imposed on the pulses in the SMF, is limited and the amplitude modulation by spectral filtering is thus restricted. Therefore, the output dissipative soliton pulses exhibit a small temporal width together with a limited spectral bandwidth even though stable dissipative soliton operation is obtained.
We believe that the recovery time of 1.59 ps is the critical value that separates the slow recovery time and appropriate recovery time regime for a particular fiber laser cavity of Fig. 1 under a condition of a cavity GVD of 0.209 ps2 and modulation depth of 70%.
5. Conclusion
In conclusion, we numerically evaluated the influence of the SA recovery time on the dissipative soliton formation in a Yb-doped fiber laser. The SA recovery time was found to have a non-negligible influence on the output pulse characteristics, such as the pulse width, spectral bandwidth, pulse peak power, and pulse energy. We demonstrated that there is an optimum recovery time range for stable dissipative soliton pulse generation, and the range varies depending on the cavity dispersion and modulation depth of the SA. One important finding, which was made from this study is that the optimum approach for generating the shortest dechirped pulses in the dissipative soliton regime would be to establish a fiber laser cavity with a small net normal cavity GVD and use an SA with a relatively slow recovery time.
We believe that the results of this study could provide a useful guide for choosing an appropriate SA for the implementation of an efficient dissipative soliton fiber laser. Furthermore, the results would help optimize the performance of a dissipative soliton fiber laser.
Funding
University of Seoul (Basic Study and Interdisciplinary R&D Fund (2021), For Ju Han Lee); National Research Foundation of Korea (2021R1A2C1004988, For Jinho Lee).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. M. Fermann and I. Hartl, “Ultrafast fiber laser technology,” IEEE J. Sel. Top. Quantum Electron. 15(1), 191–206 (2009). [CrossRef]
2. A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14(21), 10095–10100 (2006). [CrossRef]
3. A. Chong, W. H. Renninger, and F. W. Wise, “All-normal-dispersion femtosecond fiber laser with pulse energy above 20 nJ,” Opt. Lett. 32(16), 2408–2410 (2007). [CrossRef]
4. K. Kieu, W. H. Renninger, A. Chong, and F. W. Wise, “Sub-100 fs pulses at watt-level powers from a dissipative-soliton fiber laser,” Opt. Lett. 34(5), 593–595 (2009). [CrossRef]
5. W. H. Renninger, A. Chong, and F. W. Wise, “Area theorem and energy quantization for dissipative optical solitons,” J. Opt. Soc. Am. B 27(10), 1978–1982 (2010). [CrossRef]
6. Z. Cheng, H. Li, and P. Wang, “Simulation of generation of dissipative soliton, dissipative soliton resonance and noise-like pulse in Yb-doped mode-locked fiber lasers,” Opt. Express 23(5), 5972–5981 (2015). [CrossRef]
7. A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25(2), 140–148 (2008). [CrossRef]
8. J. Zhou, P. Yan, D. Wang, and M. Gong, “Experimental and theoretical study of the weak-modulation all-normal-dispersion mode-locked fiber lasers,” J. Opt. Soc. Am. B 27(8), 1589–1593 (2010). [CrossRef]
9. D. S. Kharenko, O. V. Shtyrina, I. A. Yarutkina, E. V. Podivilov, M. P. Fedoruk, and S. A. Babin, “Highly chirped dissipative solitons as a one-parameter family of stable solutions of the cubic-quintic Ginzburg-Landau equation,” J. Opt. Soc. Am. B 28(10), 2314–2319 (2011). [CrossRef]
10. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nature Photon 6(2), 84–92 (2012). [CrossRef]
11. W. Lin, S. Wang, S. Xu, Z.-C. Luo, and Z. Yang, “Analytical identification of soliton dynamics in normal-dispersion passively mode-locked fiber lasers: from dissipative soliton to dissipative soliton resonance,” Opt. Express 23(11), 14860–14875 (2015). [CrossRef]
12. C. Ouyang, L. Chai, H. Zhao, M. Hu, Y. Song, Y. Li, and C. Wang, “Pulse-shaping dynamics controlled by four structural parameters in an all-normal-dispersion mode-locked fiber laser,” J. Opt. Soc. Am. B 26(10), 1875–1881 (2009). [CrossRef]
13. C. Bao, X. Xiao, and C. Yang, “Pulse shaping effects of fast saturable absorbers in dissipative soliton fiber lasers,” Opt. Commun. 308, 20–25 (2013). [CrossRef]
14. H. Zhang, S. Zhang, X. Li, and M. Han, “Optimal design of higher energy dissipative-soliton fiber lasers,” Opt. Commun. 335, 212–217 (2015). [CrossRef]
15. H. Chi, B. Liu, Y. Song, M. Hu, L. Chai, W. Shen, X. Liu, and C. Wang, “Nonlinearity optimization of dissipative-soliton fiber laser for generation of pulses with 350 kW peak power,” High Power Laser Sci. Eng. 6, e27 (2018). [CrossRef]
16. J. Jeon, J. Lee, and J. H. Lee, “Numerical study on the minimum modulation depth of a saturable absorber for stable fiber laser mode locking,” J. Opt. Soc. Am. B 32(1), 31–37 (2015). [CrossRef]
17. J. Lee, S. Kwon, and J. H. Lee, “Numerical investigation of the impact of the saturable absorber recovery time on the mode-locking performance of fiber lasers,” J. Lightwave Technol. 38(15), 4124–4132 (2020). [CrossRef]
18. F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers–What’s the difference?” IEEE J. Sel. Top. Quantum Electron. 4(2), 159–168 (1998). [CrossRef]
19. R. Herda and O. G. Okhotnikov, “Dispersion compensation-free fiber laser mode-locked and stabilized by high-contrast saturable absorber mirror,” IEEE J. Quantum Electron. 40(7), 893–899 (2004). [CrossRef]
20. S. K. Truitsyn, B. G. Bale, and M. P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012). [CrossRef]
21. L. Jin, C. Zhang, N. Hayashi, S. Y. Set, and S. Yamashita, “The Impact of saturable absorber recovery time in hybrid mode-locked fiber laser design,” in Conference on Lasers and Electro-Optics, Technical Digest (Optical Society of America, 2019), Paper JW2A.94.
22. T.-H. Chen, Y.-H. Lin, C.-H. Cheng, C.-T. Tsai, Y.-H. Lin, L.-T. Kao, Y.-C. Chi, Z. Luo, and G.-R. Lin, “Incorporating MoS2 saturable absorption with nonlinear polarization rotation for stabilized mode-locking fibre lasers,” Laser Phys. Lett. 15(7), 075102 (2018). [CrossRef]
23. G. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, 2008).
24. O. Shtyrina, M. Fedoruk, S. Turitsyn, R. Herda, and O. Okhotnikov, “Evolution and stability of pulse regimes in SESAM-mode-locked femtosecond fiber lasers,” J. Opt. Soc. Am. B 26(2), 346–352 (2009). [CrossRef]
