Figure-nine fiber lasers can realize all-polarization-maintaining, self-started, highly stable mode-locked laser sources, and are very attractive for applications such as optical frequency combs, metrology, etc. In this work, we investigated a dispersion-managed, polarization-maintaining, Er-doped, ultrashort-pulse figure-nine fiber laser both experimentally and numerically. Stable, self-started, passive mode-locking operation was achieved in a wide net cavity dispersion region, covering the soliton, stretched pulse, and dissipative soliton mode-locking regimes. A 132 fs ultrashort pulse with spectral width of 46 nm was obtained in the stretched pulse mode-locking regime. The initial mode-locking process and dynamics inside the cavity, in addition to the fundamental characteristics of the output pulses, were examined via numerical analysis. Owing to the asymmetric configuration, the propagation behaviors were different between the two counter-propagation directions. It was found that a large breathing had already started before the passive mode-locking point in stretched pulse mode-locking operation. Intense overshoots were also observed at the beginning of passive mode-locking. Numerical results were almost in agreement with the experimental ones.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Ultrashort-pulse fiber lasers are compact, stable, and practical ultrashort pulse sources, and they play dominant roles in ultrashort-pulse laser applications . In those applications, environmental stability is a crucial aspect. Conventional single-mode fibers are generally used in fiber laser systems, but the polarization conditions of the propagating beam are sensitive to environmental perturbations, such as temperature variations, stress, and humidity.
By using polarization maintaining (PM) fibers, since the linear polarization state can be kept during propagation, environmentally stable ultrashort pulse fiber lasers can be realized. Nowadays, stable all-PM fiber lasers are desired, especially in the field of optical frequency combs.
So far, all-PM fiber lasers have been demonstrated using saturable absorption (SA) devices, including a semiconductor saturable absorber (SESA), single wall carbon nanotubes (SWNTs), graphene, etc [2–6]. So-called figure-eight fiber lasers, which use interactions between counter-propagating pulses in a nonlinear amplifying loop mirror (NALM), can be constructed with only PM fiber devices. An all-PM figure-eight fiber laser was reported in 2006 . However, it is difficult to achieve self-starting operation in a figure-eight fiber laser, and a starter such as an intensity modulator or additional saturable absorber is needed [7,8]. Recently, a specially designed all-PM fiber laser using nonlinear polarization evolution for mode-locking was reported . However, it needs careful design, and it is difficult to optimize the conditions.
In 2016, a figure-nine fiber laser consisting of an NALM including a reciprocal phase shifter was developed to realize a highly stable optical frequency comb. A figure-nine fiber laser can consist of only PM fiber devices . Thanks to the reciprocal phase shifter, which controls the phase difference between counter propagating beams, a figure-nine fiber laser can achieve self-start operation even if it consists of all-PM fiber devices. Following that report, figure-nine fiber lasers attracted a great deal of interest, and several studies on figure-nine fiber lasers have been reported [10–16].
Unlike small SA devices, an NALM is an optical fiber interferometer that includes an optical fiber amplifier. Moreover, since the dispersion map is different between propagation directions in an NALM, it is interesting to investigate the dynamics inside the cavity. Dispersion management is an important aspect, especially for optical frequency combs . Guo et al investigated the octave spanning supercontinuum generation and detection of carrier envelope offset frequency signal using dispersion managed, figure-nine Yb fiber laser . The dispersion was controlled using transmission type grating pair. So far, there have been no detailed reports of investigation of dispersion management in figure-nine fiber lasers.
In this work, we investigated the dispersion management of an all-PM figure-nine fiber laser. The characteristics of the dispersion managed figure-nine fiber laser were investigated both experimentally and numerically for a wide net cavity dispersion region. Soliton, stretched pulse, and dissipative soliton mode locking operations were achieved stably, and the characteristics were investigated for each regime. The initial process of passive mode-locking and the dynamics inside the cavity were examined by numerical analysis, in addition to the characteristics of the output pulses.
Figure 1 shows the configuration of the figure-nine fiber laser. The fiber laser consisted of all-polarization-maintaining fiber devices. As the gain device, we used 1.0 m of PM, Er-doped fiber with positive dispersion properties, whose second-order dispersion was β2 = + 17.2 ps2/km. In this work, since the obtained gain was larger than that at λ = 0.98 μm, a high-power LD operating at λ = 1.48 μm was used as the pump light source. The maximum power was 500 mW. A PM normal dispersion fiber was used for dispersion management of the NALM cavity, whose second-order dispersion was β2 = + 79.4 ps2/km. A reciprocal phase shifter consisting of Faraday rotators and wave plates was used to control the phase difference between the counter-propagating beams . A 50:50 coupler was used to construct the NALM. The output beam in the upper branch of the NALM was reflected at a 100% reflective mirror and went back to the coupler. A part of the beam was picked off at the polarization beam splitter (PBS) and used as the reflect port output. The output coupling ratio was adjusted using a half-wave plate. The other branch, which we call the output port, was used as the main output.
Once the optimum conditions of the phase shifter and the output ratio were found, self-started mode-locking could be achieved simply by increasing the power of the pump LD. Thanks to the all-PM configuration, this laser showed high long-term stability and excellent repeatability.
Figure 2 shows the observed optical spectra of the output pulses when the net cavity dispersion was −0.0137 ps2. The output coupling ratio was fixed at a small number less than 10%. A broad smooth pulse spectrum with a spectral width of 30 nm was obtained at the output port. A small peak at 1590 nm was considered as the Kelly sideband. The average power was 18 mW. For the reflect port, a spectrum with several dips was obtained. The average power was 89 μW, and the spectral width was 25 nm. The reason for the small average power at the reflect port was due to the very small output coupling ratio.
Figure 3 shows the autocorrelation trace and the pulse train of the output pulses. A second harmonic generation (SHG) type autocorrelator (Femtochrome Research FR-103XL), a fast pin photo diode (Thorlabs DET01CFC), and a wideband digital oscilloscope (Yokogawa DL9240L) were used for the measurement. A single-peak autocorrelation trace and a clear pulse train were observed stably, confirming that single-pulse mode-locking was achieved. The temporal width of the autocorrelation trace was 300 fs, and the corresponding pulse width was 197 fs under the assumption of a sech2 shape. A stable pulse train was observed, as shown in Fig. 3(b). For the reflect port, almost the same pulse train was observed. Owing to the low power, the autocorrelation trace was not able to be observed.
Figure 4 shows the observed RF spectra at the fundamental frequency and in a wide-range. The repetition frequency was 44.9 MHz, and the signal to noise ratio (SNR) was 65 dB for a resolution bandwidth (RBW) of 30 Hz. The RF spectra of higher order harmonics were observed stably in a wide frequency range, and stable passive mode-locking was confirmed.
Figure 5 shows the characteristics of the output pulses as a function of the net cavity dispersion, D. The magnitude of D was varied by changing the length of the PM-NDF. As the D was increased, the mode-locking condition was shifted from soliton to stretched pulse operation, and then to dissipative soliton mode-locking operation. Self-started passive mode-locking was achieved, and the mode-locking threshold was 200-250 mW at all dispersion conditions in this range. When the D was around zero, the spectral width was broadened, and the pulse width was shortened. In the large positive dispersion region, the spectral width was narrowed, and the temporal width was increased, up to a few picoseconds. The maximum power was increased as the D was increased. The maximum average power was limited by the single-pulse operation limit. The SNR was almost constant for all values of net cavity dispersion, D.
Figure 6 shows the observed pulse spectrum and autocorrelation trace at the output port when the net cavity dispersion was D = + 0.0036 ps2. In this case, stretched pulse mode-locking operation was achieved, and a wide optical spectrum with a spectral width of 46 nm was obtained. The temporal width of the autocorrelation trace was 204 fs, and the corresponding pulse width was 132 fs under the assumption of a sech2 pulse shape without any dispersion compensation. The pulse width of the transform-limited pulse was 80 fs. Stable long term operation was achieved more than 120 hours. The variations of output power, center wavelength, and spectral width were < ± 1%, < ± 0.04 nm, and < ± 0.3% for free running condition, respectively.
The characteristics of the output pulses for each dispersion regime are discussed in the following section.
3. Numerical analyses
Next, in order to investigate the pulse dynamics and mode-locking properties, we performed numerical analysis of the figure-nine fiber laser. We used the extended nonlinear Schrodinger equations, written as [17,18]19,20]. For the gain in the EDF, the saturation was considered as follows :
In this numerical analysis, the same configuration of the all-PM figure-nine fiber laser used in the experiment was assumed as the simulation model. The phase bias and output ratio were set as 60 degree and 10%, respectively. The mode-field diameter (MDF) and chromatic dispersion in each fiber were considered. Random noise was assumed as the initial condition. The gain was adjusted to achieve stable single-pulse mode-locking operation.
3.1 Soliton mode-locking regime
First, we examined the net anomalous cavity dispersion condition, where the length of the NDF was 0 cm and the net cavity dispersion was −0.032 ps2. In this case, self-started, stable soliton mode-locking was observed.
Figure 7 shows the initial process of passive mode-locking in this condition. The pulse width and peak power of the most intense pulse components are shown. The general behaviors were similar to those observed in the analysis of a fiber ring laser with an SA device . During the round trip, the intense short-pulse component survived and was amplified in the NALM. Above a certain threshold, the amplified pulse suffered the soliton compression effect. Then, the temporal width was decreased, and the peak power suddenly started increasing, in this case after about 300 rounds. Since the generated intense pulse dominated the gain of the EDF, the other noise components were decreased, and finally, a stable ultrashort single pulse was generated. A large overshoot was observed, especially for the peak power. This phenomenon was not observed in the ring cavity type fiber laser with a saturable absorber . It was considered that this overshoot was caused by the NALM, which is a nonlinear loop mirror including a fiber amplifier. The magnitude of this overshoot was reduced as the gain was decreased.
Figure 8 shows the characteristics of the output pulses. The experimentally observed optical spectrum at the output port is also shown for comparison. A sech2-shaped spectrum with Kelly sidebands was obtained at the output port. This spectral shape showed good agreement with the experimentally observed one. It is interesting to note that the sideband components were well suppressed at the reflect port. This shaping effect is caused by interference between counter-propagating pulses in the NALM. For the temporal shape, an almost chirp-free sech2-shaped pulse was obtained at the output port. The pulse width was about 150 fs FWHM in the simulation, which was slightly narrower than that observed in the experiment. It was considered that the experimentally observed pulse suffered the effect of chromatic dispersion of the connection fibers.
The magnitudes of the Kelly sidebands were decreased as the gain was decreased. A clean sech2-shaped spectrum without Kelly sidebands was achieved by optimizing the gain. The spectral width was decreased and the temporal width was increased as the gain was decreased.
Figure 9 shows the dynamics of pulse propagation inside the cavity for the clockwise (cw) and counter clockwise (ccw) directions. There are both normal and anomalous dispersion regions, and the temporal width changed as a function of the location inside the cavity. Due to the asymmetric configuration, there are obvious differences between the cw and ccw directions, except for the common reflect port. There was a local minimum in the positive-dispersion EDF. The shortest pulse was achieved around the fiber coupler in this case. Similar behaviors were observed between the cw and ccw directions. The average peak power in the ccw direction was larger than that in the cw direction.
Figure 10 shows the variation of pulse width and peak power of the most intense propagation pulse inside the cavity for the initial process of mode-locking. The cw direction was considered in this case. The peak power built up at the mode-locking point. The small temporal breathing started after the mode-locking point. Almost the same results were obtained for the ccw direction.
3.2 Stretched pulse mode-locking regime
Next, we examined the condition when NDF was 45 cm. In this case, the net cavity dispersion, D, was + 0.003 ps2, and stretched pulse mode-locking operation was achieved.
Figure 11 shows the initial process of passive mode-locking. During the round trip, the intense pulse gradually survived due to the properties of the NALM. Initially, the pulse width was broadened, and above a certain threshold, it started narrowing. Then the peak power rapidly increased (at around 100 rounds in this case), and stable passive mode-locking was achieved after a few overshoot oscillations. Similar to the soliton mode-locking regime, the magnitude of overshoot was decreased as the gain was decreased.
Figure 12 shows optical spectra of the output pulse. The wide pulse spectra with a width of 40 nm were generated. At the output of the fiber coupler, an ultrashort pulse of ~100 fs was obtained. The numerical results showed similar properties to the experimental ones.
Figure 13 shows the dynamics of the pulse propagation inside the cavity. The propagating pulse showed two minima at both positive and negative dispersion parts during one round trip. This is a typical characteristic of stretched pulse mode-locking. The pulse width varied from 100 up to 800 fs and the breathing ratio was about 8 for the cw direction. For the ccw direction, the breathing ratio was ~6. The location of the local minima and the propagation behavior were different between the cw and ccw directions due to the difference of the dispersion map. The shortest pulse was obtained around the coupler after interference. The average peak power in the ccw direction was larger than that in cw one.
Figure 14 shows the variation of the temporal width and peak power of the propagating pulse inside the cavity for the initial process of passive mode-locking. It is interesting to note that the large temporal breathing had already started before the passive mode-locking was achieved. In addition, the largest breathing was observed just before the passive mode-locking point for the ccw direction shown in Fig. 14(b). The pulse width varied from 100 fs up to 1.8 ps, and the breathing ratio was up to 18 for this largest breathing point. After this point, the shortest pulse was generated and then stable passive mode-locking was achieved after a few overshoot oscillations.
For the cw direction, as shown in Fig. 14(a), the breathing started before the passive mode-locking point as well. The magnitude of the largest breathing was almost the same as that of the steady state, and a large breathing peak like that in Fig. 14(b) was not observed. It was considered that the large breathing peak in the ccw direction was caused by the wide pulse spectra generated by a larger nonlinear phase shift in the ccw direction.
In order to discuss the physical mechanisms involved, we conducted a numerical simulation for a ring cavity with a similar structure. A transmission-type SA device, such as SWNTs, was assumed as the mode locker . Figure 15 shows the variation of the pulse width and peak power of the propagating pulse inside the cavity. As shown in Fig. 15, a similar peak of breathing arose just before the ML point. Breathing before the ML point was also observed but its magnitude was small before the peak of breathing. An overshoot was not observed even when the gain was large. For the cw direction, a breathing peak before the ML point was not observed. It was considered that the breathing peak before the ML point was caused by the cavity configuration in the ccw direction.
Figure 16 shows the enlarged variation of pulse width and distribution of second order dispersion for the breathing peak point in Fig. 14. In Fig. 16(b), there is only one minimum point of pulse width during one round trip. The pulse width was the shortest just before the EDF. It started broadening in normal dispersive EDF, and took maximum at the output of NDF, and then the pulse width was monotonically decreased in the anomalous dispersive SMF. In this case, the chirping property was always positive (up-chirping). Since the spectral width had been broadened in this propagation length, the large temporal breathing was observed.
At the steady state of mode-locking shown in Fig. 13(b), there were two local minima, and the chirping property alternated between positive and negative during one round trip. So we can see that the pulse dynamics inside the cavity was changed at the mode-locking point. The similar behavior was also observed for SWNT fiber laser shown in Fig. 15.
In Fig. 16(a) showing the enlarged pulse dynamics in the cw direction, there were two local minima of the pulse width. As the result, the maximum value of the pulse width was smaller than that of ccw direction. It is interesting to note that the pints of local minima in Fig. 13(a) were shifted from those in Fig. 16(a).
3.3 Dissipative soliton mode-locking regime
Finally, we examined the case when NDF was 60 cm. In this case, D was + 0.04 ps2, and dissipative soliton mode-locking operation was achieved.
Figure 17 shows the initial process of passive mode-locking. The passive mode-locking was achieved with a relatively small number of round trips. Generally, a spectral filter is used to achieve dissipative soliton mode-locking. In this work, however, we did not use any spectral filter, but dissipative soliton mode-locking was achieved nevertheless. It was considered that the effect of gain filtering works as a spectral filter of this fiber laser. A weak overshoot was observed at the beginning of passive mode-locking. Its magnitude was much smaller than those observed in the soliton and stretched pulse mode-locking regimes. The magnitude of the overshoot increased as the gain was increased.
Figure 18 shows the optical spectra and temporal shapes of output pulses. Optical spectra with steep edges at both sides were obtained. For the temporal shape, a Gaussian-like picosecond pulse with linear chirping properties was obtained. These are typical characteristics of dissipative soliton mode-locking [20,22,23]. Thanks to the relatively broad temporal width,a high-energy single pulse can be generated in this regime. The magnitude of the intensity fluctuation was very small. Similar properties to the experimental results were achieved in the numerical analysis.
Figure 19 shows the dynamics of pulse propagation inside the cavity for this dissipative soliton mode-locking condition. The pulse width became shortest just before the normal-dispersion fibers, and there was only one local minimum. These are typical properties of dissipative soliton mode-locking. Similar to the previous results, the average peak power was larger for the ccw direction than that for the cw one.
Figure 20 shows the variation of the temporal width and peak power of a propagating pulse inside the cavity for the initial process of passive mode-locking. The cw direction was considered in this case. The small breathing started before passive mode-locking was achieved. The magnitude of the breathing gradually increased along the propagation length, and it took the largest value after the passive mode-locking. Almost the same behaviors were obtained for the ccw direction.
In the dispersion-managed figure-nine fiber laser, stable passive mode-locking was achieved for a wide net dispersion region. This is the advantage of the figure-nine fiber laser using an NALM, which showed a fast response and large modulation depth. The large modulation depth is required to achieve stable stretched pulse mode-locking with wide pulse spectra [24,25].
In the numerical analysis, the order of mode-locking threshold was soliton mode-locking (ML) > dissipative soliton ML > stretched pulse ML for the condition of phase bias = 60 degree and output ratio = 10%. As the magnitude of threshold was increased, the required round number for passive mode-locking was increased.
As we discussed above, the temporal breathing was observed for the stretched pulse mode locking condition before the stable passive mode-locking was achieved. In particular, an intense peak of temporal breathing was observed in the ccw direction. In addition, the intense overshoots were observed at the beginning of the passive mode-locking. It was considered that these overshoots were generated as the results of over-amplification in the transient process to the steady state of mode-locking. Although the characteristics of this behavior depend on the configuration of laser cavity, it was considered that these are unique features of a fiber laser with an NALM, which is a nonlinear loop mirror with a saturable amplifier.
The characteristics of figure-nine fiber laser depend on the parameters in the cavity. In the numerical analysis, when the transmission ratio to coupling ratio was 60: 40, the wider and clean pulse spectrum with spectral width of 46 nm was obtained in the stretched pulse mode-locking condition (D = + 0.04 ps2). The roles of the reflect port and the output port can be reversed by changing the phase difference in the reciprocal phase shifter in the numerical analysis. Passive mode-locking was achieved at almost the same phase difference point. Among three mode-locking regimes, the range of phase differences for passive mode-locking was the widest for dissipative soliton mode-locking. It is expected that the laser performance can be improved more by the optimization of laser configuration and parameters in the cavity.
We investigated the characteristics and dynamics of a dispersion-managed, all-polarization-maintaining, passively mode-locked, Er-doped figure-nine fiber laser both experimentally and numerically. Stable, self-started, passive mode-locking operation was achieved in a wide net cavity dispersion region. Soliton, stretched pulse, and dissipative soliton mode-locking operations were achieved with high stability, both experimentally and numerically. A 132 fs ultrashort pulse with spectral width of 46 nm was obtained in the stretched pulse mode-locking condition.
In numerical analysis, the initial process of passive mode-locking, the characteristics of the output pulse, and the dynamics inside the cavity were investigated for soliton, stretched pulse, and dissipative soliton mode locking regimes. The pulse propagation behaviors were different between the cw and ccw directions due to the asymmetry of the nonlinear amplifying loop mirror (NALM). Intense overshoots were observed at the beginning of passive mode-locking for soliton and stretched pulse mode-locking. Large breathing was also observed before the passive mode-locking point for stretched pulse mode-locking. These are unique features of fiber lasers using NALMs. The numerical results were almost in agreement with the experimental ones.
1. M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultarfast Lasers, Technology and Applications (Marcel Dekker, 2003).
2. N. Nishizawa, Y. Seno, K. Sumimura, Y. Sakakibara, E. Itoga, H. Kataura, and K. Itoh, “All-polarization-maintaining Er-doped ultrashort-pulse fiber laser using carbon nanotube saturable absorber,” Opt. Express 16(13), 9429–9435 (2008). [CrossRef] [PubMed]
3. L. C. Sinclair, I. Coddington, W. C. Swann, G. B. Rieker, A. Hati, K. Iwakuni, and N. R. Newbury, “Operation of an optically coherent frequency comb outside the metrology lab,” Opt. Express 22(6), 6996–7006 (2014). [CrossRef] [PubMed]
4. N. Kuse, C.-C. Lee, J. Jiang, C. Mohr, T. R. Schibli, and M. E. Fermann, “Ultra-low noise all polarization-maintaining Er fiber-based optical frequency combs facilitated with a graphene modulator,” Opt. Express 23(19), 24342–24350 (2015). [CrossRef] [PubMed]
5. Y. Feng, X. Xu, X. Hu, Y. Liu, Y. Wang, W. Zhang, Z. Yang, L. Duan, W. Zhao, and Z. Cheng, “Environmental-adaptability analysis of an all polarization-maintaining fiber-based optical frequency comb,” Opt. Express 23(13), 17549–17559 (2015). [CrossRef] [PubMed]
6. J. Sotor, J. Bogusławski, T. Martynkien, P. Mergo, A. Krajewska, A. Przewłoka, W. StrupiŃski, and G. SoboŃ, “All-polarization-maintaining, stretched-pulse Tm-doped fiber laser, mode-locked by a graphene saturable absorber,” Opt. Lett. 42(8), 1592–1595 (2017). [CrossRef] [PubMed]
8. B. Xu, A. Martinez, S. Y. Set, C. S. Goh, and S. Yamashita, “Polarization maintaining, nanotube-based mode-locked lasing from figure of eight fiber laser,” IEEE Photonics Technol. Lett. 26(2), 180–182 (2014). [CrossRef]
9. J. Szczepanek, T. M. Kardaś, C. Radzewicz, and Y. Stepanenko, “Ultrafast laser mode-locked using nonlinear polarization evolution in polarization maintaining fibers,” Opt. Lett. 42(3), 575–578 (2017). [CrossRef] [PubMed]
10. N. Kuse, J. Jiang, C.-C. Lee, T. R. Schibli, and M. E. Fermann, “All polarization-maintaining Er fiber-based optical frequency combs with nonlinear amplifying loop mirror,” Opt. Express 24(3), 3095–3102 (2016). [CrossRef] [PubMed]
11. Y. Ozeki and T. Fukazu, “A wavelength-tunable, polarization-maintaining picosecond figure-nine fiber laser,” CLEO, 2016.
12. T. Honda, S. Y. Set, and S. Yamashita, “Effects of non-reciprocal phase bias in Figure-8/9 fiber lasers,” CLEO2017.
13. Y. Li, N. Kuse, A. Rolland, Y. Stepanenko, C. Radzewicz, and M. E. Fermann, “Low noise, self-referenced all polarization maintaining Ytterbium fiber laser frequency comb,” Opt. Express 25(15), 18017–18023 (2017). [CrossRef] [PubMed]
14. W. Hansel, H. Hoogland, M. Giunta, S. Schmid, T. Steinmetz, R. Doubek, P. Mayer, S. Dobner, C. Cleff, M. Fischer, and R. Holzwarth, “All polarization-maintaining fiber laser architecture for robust femtosecond pulse generation,” Appl. Phys. B 41(1), 123 (2017). [CrossRef]
15. Y. Ozeki, T. Asai, J. Shou, and H. Yoshimi, “Multicolor stimulated Raman scattering microscopy with fast wavelength-tunable Yb fiber laser,” IEEE J. Select. Topics in Quantum Electron. 25, 7100211 (2019).
16. D. Kim, D. Kwon, B. Lee, and J. Kim, “Polarization-maintaining nonlinear-amplifying-loop-mirror mode-locked fiber laser based on a 3 × 3 coupler,” Opt. Lett. 44(5), 1068–1071 (2019). [CrossRef] [PubMed]
17. L. Nugent-Glandorf, T. A. Johnson, Y. Kobayashi, and S. A. Diddams, “Impact of dispersion on amplitude and frequency noise in a Yb-fiber laser comb,” Opt. Lett. 36(9), 1578–1580 (2011). [CrossRef] [PubMed]
18. Z. Guo, Q. Hao, S. Yang, T. Liu, H. Hu, and H. Zeng, “Octave-spanning supercontinuum generation from an NALM mode-locked Yb-fiber laser system,” IEEE Photonics J. 9(1), 1600507 (2017). [CrossRef]
19. G. P. Agrawal, Nonlinear Fiber Optics, 5th edition (Academic, 2013).
20. N. Nishizawa, L. Jin, H. Kataura, and Y. Sakakibara, “Dynamics of a dispersion-managed passively mode-locked Er-doped fiber laser using single wall carbon nanotubes,” Photonics 2(3), 808–824 (2015). [CrossRef]
21. T. Schreiber, B. Ortaç, J. Limpert, and A. Tünnermann, “On the study of pulse evolution in ultra-short pulse mode-locked fiber lasers by numerical simulations,” Opt. Express 15(13), 8252–8262 (2007). [CrossRef] [PubMed]
22. F. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1-2), 58–73 (2008). [CrossRef]
23. N. Nishizawa, Y. Nozaki, E. Itoga, H. Kataura, and Y. Sakakibara, “Dispersion-managed, high-power, Er-doped ultrashort-pulse fiber laser using carbon-nanotube polyimide film,” Opt. Express 19(22), 21874–21879 (2011). [CrossRef] [PubMed]
24. H. H. Liu and K. K. Chow, “Enhanced stability of dispersion-managed mode-locked fiber lasers with near-zero net cavity dispersion by high-contrast saturable absorbers,” Opt. Lett. 39(1), 150–153 (2014). [CrossRef] [PubMed]
25. 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 (2015). [CrossRef]