We report a study on a hybrid mode-locked fiber laser with two saturable absorbers: slow and fast, integrated in a single device. Amorphous antimony telluride (Sb2Te3) layer was deposited on side-polished fiber to form the slow saturable absorber due to the third order nonlinear susceptibility of Sb2Te3. Additionally, an unsymmetrical design of the device causes polarization-dependent losses and together with polarization controller allows to use a nonlinear polarization evolution to form the artificial fast saturable absorber. Sub-200 fs soliton pulses with 0.27 nJ of pulse energy were generated in the hybrid mode-locked Er-doped fiber laser. Differences in the dynamics of mode-locked laser are further investigated with the use of slow and fast saturable absorbers solely, and compared with the hybrid device. Joint operation of two saturable absorbers enhances the laser performance and stability. The conducted experiments allowed to define roles of each mechanism on the pulse shaping in the laser cavity.
© 2015 Optical Society of America
Since the first demonstrations of mode-locked fiber lasers based on carbon nanotubes (CNT) and afterwards on graphene [1–3], various nanomaterials have been investigated for saturable absorber (SA) application. At first, mainly carbon-based materials like graphene oxide and reduced graphene oxide were studied extensively . Further, topological insulators (TIs) emerged as a very promising class of saturable absorber materials, including bismuth selenide (Bi2Se3) , bismuth telluride (Bi2Te3)  and antimony telluride (Sb2Te3) , followed by transition metal sulfide semiconductors, such as molybdenum disulfide (MoS2)  or tungsten disulfide (WS2) . Very recently, a new material, black phosphorus, has appeared [10,11]. All of them have common features like broadband saturable absorption, straightforward fabrication and ease of integration into an all-fiber laser setup. Those qualities distinguish them from semiconductor-based SAs (SESAMs) and justify further research efforts.
Several schemes of integration of a low-dimensiona material into an all-fiber laser system have already been investigated . The most straightforward approach is to deposit a SA on fiber connector end facet, but it might result in relatively low damage threshold of the SA. Another attitude is to place it on the surface of a side-polished fiber [6,7,13–15]. Strong nonlinear response is possible through increasing the length of interaction and since only evanescent field is in the interaction with nonlinear material, the damage threshold is gradually increased . In this configuration the technology of material fabrication is less complicated, since single-layer or a few-layer structure is not required. It is possible to use micrometer thick, bulk structured material . Topological insulators are particularly interesting. They are characterized by the giant nonlinear refractive index , which is comparable to graphene, and the surface states of those materials consists of a single Dirac cone, while having insulating states in the bulk . However, the unsymmetrical design of a device causes (usually) undesirable polarization-dependent losses (PDL). Those might be avoided by shortening the length of the interaction  or the use of a tapered fiber . Nevertheless, it has never been pointed out that this kind of a device can serve as both slow saturable absorber (utilizing the saturable absorption property of a topological insulator in the ps time scale ) and as artificial, fast saturable absorber (on the basis of nonlinear polarization evolution (NPE), working as a polarizer together with a set of quarter- and half-wave plates). It has been noted before, that the use of two SAs might enhance the laser performance. Since there are some intrinsic differences between mode-locking with slow and fast saturable absorbers , hybrid scheme allows to take advantage of both [21–24]. It is possible to generate the shortest pulses with fast SAs (NPE, NOLM) , however self-starting is usually difficult to achieve. They are also the most sensitive to external factors. Mode-locking with slow saturable absorbers is characterized by better stability and is more easily achieved.
In hybrid mode-locking, a fast component is usually a dominant pulse shaping mechanism, which ensures efficient generation of ultrashort pulses . On the other hand, a slow SA facilitates the self-starting feature of a mode-locked laser and prevents multiple pulse operation, wave-breaking or continuous wave (CW) operation for higher pumping powers . This increases the available pumping range for stable, fundamental soliton operation, and hence the output power. It was shown theoretically that slow SA have significant impact on a mode-locking stability .
All of the above examples were based on the two separate devices. Herein, we present for the first time, hybrid mode-locking with the use of slow and fast SAs integrated in the single device. The hybrid device works as a slow saturable absorber and polarizer at the same time. Slow component is present due to the saturable absorption of Sb2Te3 layer deposited on the surface of the side-polished fiber, and fast component is realized due to the NPE (thanks to the polarizing property and polarization controller). Sub-200 fs soliton pulses with 0.27 nJ energy were stably generated at 33 MHz repetition rate in anomalous dispersion regime. In addition, we investigate the role of each mechanism on the pulse shaping in hybrid mode-locking by employing slow and fast SAs solely, and comparing with the hybrid device. In the role of the slow SA we use side-polished fiber with 1 mm long Sb2Te3 layer (with negligible PDL) and a fiber polarizer (to utilize NPE) as the fast SA.
2. Slow, fast and hybrid saturable absorbers characterization
Both slow and hybrid saturable absorbers were prepared by depositing Sb2Te3 on a side-polished fiber. Antimony telluride materials were synthesized by the direct fusion technique from pure elements (99.99 + %) in the quantities corresponding to the stoichiometric composition of Sb2Te3 compound. The material synthesis was performed in quartz ampoules closed in vacuum. The densification process was conducted under vacuum at uniaxial compressive pressure in the spark plasma sintering (SPS) apparatus .
Layers of Sb2Te3 were deposited on the side-polished fibers with pulsed magnetron sputtering technique (PVD) as described in detail in . The process of sputtering was conducted under power of Pe = 0.1 kW and at low current I = 0.05 A. The pressure of argon was kept stable at the level of 0.25 Pa. The deposition conditions were chosen to receive Sb2Te3 coatings of thin column crystalloid morphology, compact and not very porous. The distance between the core boundary and the polished surface is at the level of 1 μm. The side-polished fiber was covered with a mask in order to adjust the length of the deposition to 1 mm and 5 mm for slow and hybrid saturable absorbers, respectively. The exemplary image from scanning electron microscope (SEM) is presented in Fig. 1, showing the surface and the cross-section. The fracture of the layer has compact and continuous structure, with thickness of ~50 nm. Further characterization of the deposited layer, including Raman spectroscopy and XRD diffraction analysis, may be found in .
Nonlinear optical response of prepared devices was investigated in twin-detector all-fiber setup  with the use of 100-MHz 1-ps laser pump (Menlo Systems T-Light). Since these compounds are slow saturable absorbers , the measurement data are fitted with the following theoretical saturable absorber model  as a function of pulse fluence FP:
First, we investigate the device with 1 mm long layer of deposition. We observed that both linear transmission and characteristics of the saturable absorption property depend on the initial polarization state of light. The linear transmission changes from 40% to 74.4% and polarization dependent losses are at the level of 2.7 dB. Two exemplary nonlinear transmission curves are presented in Fig. 2(a) and 2(b). For input light experiencing higher transmission, modulation of 5.2% was observed from the level of 68% (Fig. 2(a)). The inverse slope, originating mainly from two-photon absorption, is visible. The saturation fluence is 110.3 μJ/cm2. For polarization experiencing lower linear transmission (49%, Fig. 2(b)), the modulation of 13% was measured (limited by the power of available pump source). Saturation fluence is 212.3 μJ/cm2. Despite that only a fraction of beam is interacting with the deposited Sb2Te3 layer, the mode-field diameter of a single mode-fiber (10.5 μm at 1550 nm) was assumed for fluence calculation in order to compare with performance of other SAs.
Subsequently, the saturable absorption property of hybrid device was investigated. Similarly to the previous case, we observed linear transmission and saturable absorption change as a function of incident polarization state of light. The linear transmission changes from 1.2% up to 62%. Corresponding polarization dependent losses are 17 dB. Two exemplary nonlinear transmission curves are presented in Fig. 2(c) and 2(d). For polarization experiencing lower absorption (Fig. 2(c)), 4.5% modulation depth was observed, slightly limited by inverse slope in saturable absorption, which additionally confirms the slow character of Sb2Te3 saturable absorption. Saturation fluence and non-saturable losses are at the level of 129 μJ/cm2 and 34.7%. Stronger modulation was observed for polarizations with higher absorption (Fig. 2(d)). The measured transmission modulation was 14.9%. Saturation fluence increases together with linear absorption and is equal to 228 μJ/cm2. We did not observe damage of any device during the measurements up to 2.6 GW/cm2 of peak power intensity.
Polarizing properties of devices were further analyzed in the experiment schematically presented in Fig. 3. Linearly polarized 1550 nm CW laser diode was used as a light source. Light was free-space coupled via polarization maintaining (PM) collimator. The first half-wave plate and polarization beam splitter (PBS) was used to ensure that only horizontally polarized light passes, which might be subsequently rotated with the other half-wave plate. The following fiber with light coupled through the collimator was kept as short and straight as possible, and without any strains to maintain the polarization state. The measurement results are presented in Fig. 4. Power of light incident to the hybrid component was set to 5 mW.
In the case of slow saturable absorber, short interaction length (1 mm) does not cause strong polarization dependent losses (polarization dependent losses are 2.7 dB). This allows to exclude the influence of NPE on the pulse shaping. The detailed characteristics are presented in Fig. 4. The transmission level changes from 40% for θ = 90° and 270° up to maximum value of 74.4% for polarization angles θ = 0° and 180°. Longer interaction length in the case of hybrid device (5 mm) caused significant increase of polarization-dependent losses, which are at the level of 17 dB i.e. that fabricated device works as a polarizer . The maximum transmission of 62% was observed at polarization angles θ = 0° and 180°, whereas the minimum value of 1.2% at θ = 90° and 270°.
Saturable absorption properties of slow and hybrid saturable absorbers are very similar. As proved by detailed simulations based on nonlinear Schrödinger equation, small differences in modulation depth have minor impact on spectral width and pulse duration in the anomalous cavity dispersion regime . As a consequence, this will allow for reliable comparison of slow and hybrid saturable absorbers: to define the influence of higher extinction ratio, while saturable absorption is maintained at the same level.
In the role of fast saturable absorber the nonlinear polarization evolution was used by employing the commercially available in-line fiber polarization beam splitter with extinction ratio of 26 dB.
In order to define the role of slow and fast saturable absorbers in hybrid mode-locking scheme, three saturable absorbers were subsequently spliced into the Er-doped fiber laser with ring-shaped cavity as schematically presented in Fig. 5. The experimental setup consists of 45-cm-long Er-doped fiber (nLight Liekki Er80 4/125, β2 = 0.0593 ps2/m) pumped via wavelength-division multiplexer (WDM) by 980 nm pump diode. Further, the laser contains an in-line polarization controller, a polarization-independent isolator and output coupler with 20% transmission. Total cavity length is 6.26 m, which corresponds to the net cavity dispersion of −0.1 ps2 (assuming β2 = −0.023 ps2/m for single-mode fiber). The length of the laser cavity in the experiment was maintained. The laser operation was analyzed and recorded using optical spectrum analyzer (Ando AQ-6315A), optical autocorrelator (APE PulseCheck), and radio-frequency (RF) spectrum analyzer (Agilent Technologies N9010A) connected to a fast photodiode (OptiLab PD-30), and oscilloscope (Agilent Technologies DSO-X 3034A).
Output spectra and autocorrelation functions for all three configurations are presented in Fig. 6. Essential parameters are compared in Table 1. In order to reliably compare output characteristics in all three configurations, the maximum pumping power enabling mode-locked operation without any instabilities was set as a standard for comparison. Since we observed the broadening of output spectra with increasing pump power in each case, this is also the point where the pulses had the maximum value of spectral bandwidth, and maximum output power.
The laser mode-locked by slow absorber solely is characterized by 8.7 nm wide full-width at half maximum (FWHM) spectrum centered at 1564.6 nm. We observed that by changing the setting of the polarization controller it is possible to change the spectral width of output spectrum of approximately 1 nm, which is associated with a change of modulation depth for different polarization states . The polarization controller was set to obtain the broadest spectral width. Transform-limited, 298 fs pulses are stably generated for pumping power in the range between 117 and 181 mW. The laser requires relatively high pumping power for mode-locking operation and this is associated with the fact that only the evanescent field is in the interaction with SA layer.
The use of fast saturable absorber allows for the generation of shorter, 225 fs transform-limited pulses with 11.5 nm FHWM bandwidth. The spectrum is centered at 1564.6 nm. The mode-locking operation was observed in the pumping range of 55-85 mW. For higher pumping power wave-breaking and CW-component were visible in the spectrum. It can be seen, that even at relatively low pumping power (which ensures stable mode-locking) the spectrum contains a very prominent side-band in the blue-wavelength region of the spectrum (around 10 dB higher than the peak in the central lobe of the soliton spectrum).
The combination of two saturable absorbers, slow and fast, in a single device allows for the generation of the shortest pulses with the broadest spectrum. While the shortening of the pulse duration compared to the NPE is modest, the impact on available pumping range, output power and stability is significant (see Table 1). The laser pumping range, which ensures stable mode-locking, was significantly increased in comparison to pure NPE (210 mW vs. 85 mW). The output power and pulse energy are also very high when compared to commonly reported values with the use nanomaterial-based saturable absorbers [2–6,8,9,14]. The spectrum shape is similar to the fast-saturable-absorber case, which indicate that NPE is dominant pulse shaping mechanism. However, the prominent Kelly’s sideband, which was present in the pure NPE spectrum is now significantly suppressed. The addition of slow saturable absorber stabilize the pulse generation and prevents pulse breaking for higher pumping powers, which is nicely rendered to broad pumping range and high output power. This is in a good agreement with theoretical considerations  and experiments conducted previously with the use of slow and fast SESAMs . Here, the mode-locking operation was possible for two distinctive states of polarization. The lack of possibility of continuously tuning the spectral width is associated with the fact, that a simple fiber PC, which was used in the experiment, does not provide the possibility to smoothly change the angle of polarization. Hybrid mode-locked laser was observed to be the most stable and much more resistant to external factors.
Of course the same pulse duration might be achieved by using a real saturable absorber with higher modulation depth. However, a real SAs always have some non-saturable losses, which are usually proportional to the modulation depth. Higher modulation depth is connected with higher losses, which limits the output power. Hence, a hybrid mode-locking scheme allows for comparable performance (also in terms of stability), as in the case of real saturable absorber with high modulation depth, but with introducing lower non-saturable losses.
4. Hybrid mode-locked laser: detailed characterization
The hybrid mode-locked laser starts to operate when pumped to 120 mW, once mode-locked the laser was working for powers in the range from 69 mW up to 210 mW. For higher pumping power modulations become visible in the spectrum. The optical spectrum recorded for the highest pumping power is presented in Fig. 6(e). The spectrum has typical shape for soliton pulses with symmetrical Kelly’s sidebands . The spectrum is centered at 1568.8 nm and has FWHM of 13.3 nm.
The measured pulse duration is 195 fs as obtained after deconvolution from autocorrelation (AC) function showed in Fig. 7 assuming hyperbolic secant shape. The produced pulses are nearly transform-limited with time-bandwidth product (TBP) of 0.316,which is another typical feature of solitary pulses. The inset of Fig. 7 presents the AC function in 5 ps span, proving single pulse operation. No pre- or post-pulses were visible in the measurement.
The RF spectrum is presented in Fig. 8(a). The fundamental repetition rate is 33.07 MHz with high signal to noise ratio of 73 dB. The stable mode-locking operation was indicated by steady RF spectrum with broad spectrum of harmonics measured up to 3 GHz (inset). The output pulse train recorded by the photodiode and the oscilloscope is depicted in Fig. 8 (b). The pulses are equally spaced by 30.2 ns, which corresponds to fundamental repetition rate and 6.26 m long cavity. Output pulses are characterized by very good amplitude stability. The average output power is 9.0 mW, which conforms to 0.27 nJ pulse energy and 1.22 kW peak power. The soliton order N may be determined by the following equation :26].
As a conclusion, we have reported the study on the hybrid mode-locked Er-doped fiber laser. The laser was mode-locked by two saturable absorbers: slow (real) and fast (artificial), integrated in the one device. The layer of Sb2Te3 deposited on the side-polished fiber works as a saturable absorber and as a polarizer with 17 dB extinction ratio. In addition, we studied the impact of each saturable absorber on the hybrid mode-locking. Nonlinear polarization evolution was observed to be the major pulse shaping mechanism, responsible for spectrum shape and ultrashort pulse duration. On the other hand, slow saturable absorber significantly improves the laser stability, preventing from wave-breaking, multiple pulse operation and CW-operation for higher pumping powers. The joint operation of those two mechanisms resulted in stable ultrafast operation with broad range of available pumping powers and high output power. The hybrid mode-locking scheme was observed to be the most efficient. Sub-200 fs pulses with 13.3 nm of FWHM bandwidth were stably generated with 33 MHz repetition rate and output power of 9.0 mW.
The work presented in this paper was supported by the National Science Centre (NCN, Poland) under the project “Topological insulators as a new class of saturable absorbers for fiber lasers” (decision no. DEC-2014/13/B/ST7/01699) and by the Faculty of Electronics, Wroclaw University of Technology (grant no. B50168). This scientific work has also been partially financed as a research postdoctoral project (decision no. DEC-2014/12/S/ST8/00582) from the resources assigned for science by National Science Centre (NCN, Poland) and by AGH University of Science and Technology (Statutory grant no. 220.127.116.118). We acknowledge Maciej Kowalczyk (Laser & Fiber Electronics Group, Wroclaw University of Technology) for helpful discussions.
References and links
1. A. Martinez and Z. Sun, “Nanotube and graphene saturable absorbers for fibre lasers,” Nat. Photonics 7(11), 842–845 (2013). [CrossRef]
2. Q. L. Bao, H. Zhang, Y. Wang, Z. H. Ni, Z. X. 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]
3. T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube–polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), 3874–3899 (2009). [CrossRef]
4. J. Boguslawski, J. Sotor, G. Sobon, R. Kozinski, K. Librant, M. Aksienionek, L. Lipinska, and K. M. Abramski, “Graphene oxide paper as a saturable absorber for Er- and Tm-doped fiber lasers,” Photonics Res. 3(4), 119–124 (2015). [CrossRef]
5. C. Zhao, Y. Zou, Y. Chen, Z. Wang, S. Lu, H. Zhang, S. Wen, and D. Tang, “Wavelength-tunable picosecond soliton fiber laser with Topological Insulator: Bi2Se3 as a mode locker,” Opt. Express 20(25), 27888–27895 (2012). [CrossRef] [PubMed]
6. J. Lee, J. Koo, Y. M. Jhon, and J. H. Lee, “A femtosecond pulse erbium fiber laser incorporating a saturable absorber based on bulk-structured Bi2Te3 topological insulator,” Opt. Express 22(5), 6165–6173 (2014). [CrossRef] [PubMed]
7. J. Boguslawski, G. Sobon, R. Zybala, and J. Sotor, “Dissipative soliton generation in Er-doped fiber laser mode-locked by Sb2Te3 topological insulator,” Opt. Lett. 40(12), 2786–2789 (2015). [CrossRef] [PubMed]
8. H. Liu, A.-P. Luo, F.-Z. Wang, R. Tang, M. Liu, Z.-C. Luo, W.-C. Xu, C.-J. Zhao, and H. Zhang, “Femtosecond pulse erbium-doped fiber laser by a few-layer MoS2 saturable absorber,” Opt. Lett. 39(15), 4591–4594 (2014). [CrossRef] [PubMed]
9. P. Yan, A. Liu, Y. Chen, J. Wang, S. Ruan, H. Chen, and J. Ding, “Passively mode-locked fiber laser by a cell-type WS2 nanosheets saturable absorber,” Sci. Rep. 5, 12587 (2015). [CrossRef] [PubMed]
10. S. B. Lu, L. L. Miao, Z. N. Guo, X. Qi, C. J. Zhao, H. Zhang, S. C. Wen, D. Y. Tang, and D. Y. Fan, “Broadband nonlinear optical response in multi-layer black phosphorus: an emerging infrared and mid-infrared optical material,” Opt. Express 23(9), 11183–11194 (2015). [CrossRef] [PubMed]
11. J. Sotor, G. Sobon, M. Kowalczyk, W. Macherzynski, P. Paletko, and K. M. Abramski, “Ultrafast thulium-doped fiber laser mode locked with black phosphorus,” Opt. Lett. 40(16), 3885–3888 (2015). [CrossRef] [PubMed]
12. G. Sobon, “Mode-locking of fiber lasers using novel two-dimensional nanomaterials: graphene and topological insulators [Invited],” Photonics Res. 3(2), A56–A63 (2015). [CrossRef]
13. J. Koo, J. Lee, C. Chi, and J. H. Lee, “Passively Q-switched 1.56 μm all-fiberized laser based on evanescent field interaction with bulk-structured bismuth telluride topological insulator,” J. Opt. Soc. Am. B 31(9), 2157–2162 (2014). [CrossRef]
14. N. H. Park, H. Jeong, S. Y. Choi, M. H. Kim, F. Rotermund, and D. I. Yeom, “Monolayer graphene saturable absorbers with strongly enhanced evanescent-field interaction for ultrafast fiber laser mode-locking,” Opt. Express 23(15), 19806–19812 (2015). [CrossRef] [PubMed]
15. H. Jeong, S. Y. Choi, F. Rotermund, Y. H. Cha, D. Y. Jeong, and D. I. Yeom, “All-fiber mode-locked laser oscillator with pulse energy of 34 nJ using a single-walled carbon nanotube saturable absorber,” Opt. Express 22(19), 22667–22672 (2014). [CrossRef] [PubMed]
16. J. Liu, S. Liu, and J. Wei, “Origin of the giant optical nonlinearity of Sb2Te3 phase change materials,” Appl. Phys. Lett. 97(26), 261903 (2010). [CrossRef]
17. H. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, “Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface,” Nat. Phys. 5(6), 438–442 (2009). [CrossRef]
18. Y. Meng, G. Semaan, M. Salhi, A. Niang, K. Guesmi, Z. C. Luo, and F. Sanchez, “High power L-band mode-locked fiber laser based on topological insulator saturable absorber,” Opt. Express 23(18), 23053–23058 (2015). [CrossRef] [PubMed]
19. N. Kumar, B. A. Ruzicka, N. P. Butch, P. Syers, K. Kirshenbaum, J. Paglione, and H. Zhao, “Spatially resolved femtosecond pump-probe study of topological insulator Bi2Se3,” Phys. Rev. B 83(23), 235306 (2011). [CrossRef]
20. F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorber – what’s the difference,” IEEE J. Sel. Top. Quantum Electron. 4(2), 159–168 (1998). [CrossRef]
21. M. Guina, N. Xiang, A. Vainionpää, O. G. Okhotnikov, T. Sajavaara, and J. Keinonen, “Self-starting stretched-pulse fiber laser mode locked and stabilized with slow and fast semiconductor saturable absorbers,” Opt. Lett. 26(22), 1809–1811 (2001). [CrossRef] [PubMed]
23. B. Xu, A. Martinez, S. Y. Set, C. S. Goh, and S. Yamashita, “A net normal dispersion all-fiber laser using a hybrid mode-locking mechanism,” Laser Phys. Lett. 11(2), 025101 (2014). [CrossRef]
24. M. A. Chernysheva, A. A. Krylov, C. Mou, R. N. Arif, A. G. Rozhin, M. H. Rummelli, S. K. Turitsyn, and E. M. Dianov, “Higher-order soliton generation in hybrid mode-locked thulium-doped fiber ring laser,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1100908 (2014). [CrossRef]
25. J. Szczepanek, T. M. Kardaś, M. Michalska, C. Radzewicz, and Y. Stepanenko, “Simple all-PM-fiber laser mode-locked with a nonlinear loop mirror,” Opt. Lett. 40(15), 3500–3503 (2015). [CrossRef] [PubMed]
26. J. C. Chen, H. A. Haus, and E. P. Ippen, “Stability of lasers mode locked by two saturable absorbers,” IEEE J. Quantum Electron. 29(4), 1228–1232 (1993). [CrossRef]
27. R. Zybala and K. T. Wojciechowski, “Anisotropy analysis of thermoelectric properties of Bi2Te2.9Se0.1 prepared by SPS method,” in 9th European Conference On Thermoelectrics AIP Conference Proceedings (ECT, 2012), pp. 393–396. [CrossRef]
28. R. Grange, M. Haiml, R. Paschotta, G. J. Spuhler, L. Krainer, M. Golling, O. Ostinelli, and U. Keller, “New regime of inverse saturable absorption for self-stabilizing passively mode-locked lasers,” Appl. Phys. B 80(2), 151–158 (2005). [CrossRef]
29. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011). [CrossRef]
30. 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]
31. H. Jeong, S. Y. Choi, F. Rotermund, and D. I. Yeom, “Pulse width shaping of passively mode-locked soliton fiber laser via polarization control in carbon nanotube saturable absorber,” Opt. Express 21(22), 27011–27016 (2013). [CrossRef] [PubMed]
32. S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28(8), 806–807 (1992). [CrossRef]
33. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).