Open Access
Add to BibSonomy
Abstract
We have proposed and implemented for the first time to our knowledge a passive and all-optical pulse synchronization for polarization-maintaining fiber lasers. Specifically, the synchronization system was comprised of two independent Yb-doped and Er-doped mode-locked fiber lasers in a master-slave configuration. Master pulses were injected into the slave laser cavity consisting of a nonlinear amplifying loop mirror, which provided an effective fast intensity modulator due to the periodic introduction of nonreciprocal phase difference. As a result, robust and tight timing synchronization was achieved with a cavity mismatch tolerance of 800 µm and a relative timing jitter of 26 fs within 1-MHz bandwidth. In combination with the all-polarization-maintaining structure of fiber lasers, long-term stable operation was demonstrated over 12 hours without the need for temperature stabilization and vibration isolation. The implemented synchronous laser system could find immediate applications such as pump-probe microscopy, two-color spectroscopy and nonlinear frequency mixing.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Stable and precise synchronization of optical pulses plays an essential role in ultrafast science and technology. In particular, synchronized pulses with disparate colors are widely used in many applications, including time-resolved imaging and spectroscopy, nonlinear frequency conversion, pump-probe investigations, coherent pulse synthesis, and precise timing distribution [1–4]. Up to date, timing synchronization between two independent ultrafast lasers has been realized in various schemes. In pioneering works, active synchronization was demonstrated by using techniques of phase locked loops [5] and balanced optical cross correlator [6], which ultimately led to an ultralow residual jitter with attosecond precision [7–9]. To mitigate the stringent requirement of complicated feed-back system and high-speed electronics, passive synchronization was proposed and implemented based on the nonlinear optical cross-phase modulation (XPM) [10–12] in a shared laser cavity, where the high-bandwidth feedback was provided by the instantaneous nonlinear response.
Over the past years, the XPM-based passive synchronization has attracted increasing attention in fiber lasers due to enhanced nonlinear interaction within the confined and elongated transverse mode of the fiber waveguide [13, 14]. Compared with solid-state lasers, fiber lasers also offer the advantages of low cost, compactness, and higher efficiency. Moreover, a so-called master-slave configuration can readily be deployed to obtain the sufficient nonlinearity for self-synchronization [15–19], which enables to overcome limitations in the shared intra-cavity scheme, for instance unstable gain competition, poor spectral tunability, and mutual cross-talk. Recently, active-passive hybrid approach for fiber lasers was investigated by combining slow cavity feedback and fast passive self-locking, which resulted in a long-term stability and a low timing error [20].
However, in the aforementioned master-slave architecture, the slave fiber lasers were typically constructed with non-polarization-maintaining fibers, which is required to implement synchronous mode-locking based on nonlinear polarization rotation [21, 22]. In this case, the synchronization system inevitably suffers from running instability due to ambient perturbation on polarization states, such as temperature fluctuation, mechanical vibration and pressure change. Additionally, the initiation of synchronous operation usually needs careful tuning of polarization states in the slave laser cavity and polarization optimization on injected master pulses. Consequently, self-starting and long-term stable synchronization may be hard to retained in practice. To alleviate the disturbing effects, the fiber laser should be installed into a foamed polystyrene box [15] or be controlled with an active fashion [19], thus increasing the complexity of the whole system. Recently, broadband graphene was incorporated into a common branch of two fiber laser cavities to augment the synchronization stability both in passive [23,24] and active [25] fashions, although the initialized synchronization still needs intra-cavity polarization optimization.
Here, we proposed and implemented, for the first time to the best of our knowledge, an all-polarization-maintaining synchronization system for two fiber lasers in a master-slave layout. Specifically, the optical master pulses from an Yb-doped mode-locked fiber laser were fed into a nonlinear amplifying loop mirror in the slave Er-doped fiber laser cavity, which periodically introduced non-reciprocal phase shift difference by the nonlinear cross-phase modulation. As a result, synchronous mode-locking of the slave laser could be obtained with a large tolerance of cavity mismatch up to 800 µm. Additionally, the relative timing jitter was measured by cross-correlation technique, revealing a root-mean-square value of 26 fs in a Fourier frequency range from 0.1 Hz to 1 MHz. The robust and tight synchronization could be launched in a plug-and-play way, and maintained more than 12 hours with a fraction frequency error of 5 × 10−10. The self-starting, highly precise and long-term stable synchronization system could promote its use in broader practical applications.
2. Experimental setup
Figure 1 shows the experimental setup for the synchronized fiber laser system in a polarization-maintaining (PM) master-slave structure, which is comprised of an Yb-doped fiber laser (YDFL) as the master laser and an Er-doped fiber laser (EDFL) as the slave one. The master fiber laser is based on an all-PM linear cavity, where two ends are terminated with a semiconductor saturable absorber mirror (SESAM) and a fiber Bragg grating (FBG) with a bandwidth of 0.2 nm. Stable passive mode-locking can simply be obtained by increasing the pump power around 40 mW. The resulting pulses are then tapped bidirectionally through a 2 × 2 optical coupler with a splitting ration of 5:95. Figure 2(a) gives the measured optical spectrum centered at 1063.8 nm. The double peaks shown in the spectrum are ascribed to the self-similar evolution of mode-locked pulses in a normal-dispersion fiber cavity [26]. The corresponding pulse duration of 19.5 ps can be deduced from the measured autocorrelation trace as shown in Fig. 2(c). Note that a piezoelectric transducer is glued with the intra-cavity fiber for actively stabilizing the cavity length, which will be elaborated later for the characterization of the synchronization system.
Fig. 1 Experimental setup for all-polarization-maintaining synchronization system. (a) Master Yb-doped mode-locked fiber laser. (b) Injection of master pulses into the slave Er-doped fiber laser cavity. The synchronous mode-locking is initiated by the additional non-reciprocal phase shift induced by the cross-phase modulation in the nonlinear amplifying loop mirror. (c) Schematic for the measurement of relative timing jitter. LD: Laser diode; SESAM: semiconductor saturable absorber mirror; WDM: wavelength division multiplexer; Yb: ytterbium-doped gain fiber; OC: output coupler; PZT: piezoelectric transducer; FBG: fiber Bragg grating; PS: phase shifter; Er: erbium-doped gain fiber; Col: collimator; PPLN: periodically-poled lithium niobate crystal; IF: interference filter; PD: photodiode; FFT: fast Fourier transform.
Fig. 2 Optical spectra of master (a) and slave (b) pulses. Autocorrelation traces for master (c) and slave (d) pulses.
The slave fiber laser consists of a nonlinear amplifying loop mirror (NALM), which serves as a fast artificial saturable absorber. The related mechanism relies on the intensity-dependent nonlinear phase shift difference between clockwise and anticlockwise directions in the Sagnac loop [27]. It is thus imperative to accumulate sufficient phase difference to initiate mode-locked operation. For this purpose, the Er-doped gain fiber is placed asymmetrically relative to the 2 × 2 balanced optical coupler. Additionally, a π/2 non-reciprocal phase shifter is used to further reduce the mode-locking threshold [28–30]. At one port of of the optical coupler, a 1-nm-bandwidth FBG is connected to effectively provide an end mirror and a spectral filter. Under a pump power above 280 mW, self-starting mode-locked pulses can be obtained with a pulse duration of 12.1 ps as shown in Fig. 2(d). Figure 2(b) shows the corresponding spectrum centered at 1569.8 nm, which exhibits a soliton-like shape with characteristic Kelly sidebands in a fiber cavity with anomalous net dispersion [31]. The cavity length can be manually tuned by translating one of the two collimators in the free space as shown in Fig. 1(b).
Now the two master-slave lasers are ready for synchronization. The master pulses at 1064 nm after an Yb-doped fiber amplifier (YDFA) are injected into the slave laser cavity via a wavelength division multiplexer (WDM). By carefully match the slave cavity length, synchronous mode-locked pulses can be observed as shown in Fig. 3(a). More specifically, the injected master pulses will induce additional phase shift in one direction of the NALM due to the cross-phase modulation, which provides a high-speed Kerr-based intensity modulation. Thanks to the all-PM design, the synchronization can be launched immediately with the light injection, without the need of any polarization state optimization. Figure 3(b) gives the measured relative power fluctuation after a linear polarizer, showing an excellent stability of 0.08% and 0.16% for master and slave laser outputs, respectively.
Fig. 3 Pulse trains (a) and power stability (b) for synchronized master and slave pulses. σ indicates the relative fluctuation.
3. Results and discussion
Next we turn to characterize the implemented synchronization system. Figure 4(a) shows the cavity detuning diagram for repetition-rate changes when the slave laser operates at a pump power of 354 mW above the mode-locking threshold. The tolerance of the cavity-length mismatch is found to be about 460 µm, which corresponds to 635-Hz variation of free spectral range at a center repetition rate of 20.3 MHz. The achieved tolerance is longer than the conventional method for the passive synchronization [18]. In the scheme based on the nonlinear-polarization-rotation effect, the variation of the cavity length would inevitably alter the polarization evolution of pulses inside the laser resonator, which might prohibit the synchronous mode locking. In contrast, the all-polarization-maintaining structure used in our scheme favors resistance to the ambient perturbation, which would improve the tolerance and robustness of achieved synchronization. In the experiment, the group velocity mismatch between the master pulse at 1064 nm and slave pulse at 1570 nm is about 1.7 ps/m [20], corresponding to a walk-off length of 11.5 m for the duration of the master pulse. The long interaction length for the XPM is favorable for enhancing the tolerance of the passive locking. For master pulses with a peak power about 100 W, the induced nonlinear phase shift is 0.13 m−1 assuming a nonlinear coefficient of 1.3 W−1 km−1 of the optical fiber. The additionally introduced difference of the nonlinear phase shift between two circulating directions in the NALM of the slave cavity would play an essential role in the realization of the synchronous mode-locking.
Fig. 4 (a) Measured repetition rate of the slave laser as a function of its cavity-length variation. Red solid dots and blue open circles indicate synchronized and non-synchronized regimes, respectively. The intermediate Q-switched mode-locked regime is represented with red open circles. The dashed line is given by a theoretical mode in the absence of synchronization. (b) Optical spectrum of the slave laser for various cavity-length detuning. The dashed lines indicate a small wavelength shift about 0.2 nm. The inset shows the close-up of the spectral peaks.
When the cavity-length change is away from the locking range, the repetition rate will jump to the value defined by a free-running laser, thus showing a typical linear dependence on the cavity detuning. Notably, an intermediate Q-switched mode-locked state is observed in the negative detuning regime, which indicates that the injected pulses impose a significant influence on the free-running slave laser. It can also be noticed that the negative capture range is slightly larger than the one in the positive regime. The asymmetric tolerance of cavity-length detuning has been observed in many other schemes [10,13,23]. The underlying mechanism is generally complicated depending the cavity configuration and the pulse characteristics. Numerical solutions of the involved coupled nonlinear SchrÃűdinger equations could help to explore exotic properties of the relevant locking mechanism [22].
Figure 4(b) shows the output spectrum of the synchronized slave laser for various cavity-length detuning. A spectral redshift of 0.02 nm within the capture range was observed as increasing the cavity length, which is expected for an anomalous-dispersion laser cavity. It is the XPM-induced wavelength shift that compensates the cavity-length variations owing to the self-adaption of the pulse group velocity [10]. In general, the amount of the spectral shift should depend on the accumulated dispersion in the laser cavity and the strength of nonlinear XPM effect. In our case, the resulting wavelength shift is relatively smaller than that obtained in previous master-slave schemes [14, 15, 20]. This feature of stable spectral positioning during the synchronization operation is favorable for applications requiring precise working wavelengths, such as coherent anti-Stokes Raman spectroscopy.
In the following, we will further investigate two main factors that affect the locking range of pulse synchronization. In our experiment, we found that the locking range could be significantly increased by running the slave laser below the mode-locking threshold. In particular, the maximum tolerance of the cavity detuning reaches 800 µm at an optimum pump power of 234 mW as shown in Fig. 5(a). In this case, the required phase difference in NALM for mode locking is compensated by the Kerr-based phase shift by injected pulses. The other important factor to improve the capture range is the master-injecting pulse energy. As presented in Fig. 5(b), the range of the cavity-length tolerance increases linearly with augmenting the pulse energy. The fitted slopes are found to be 188 µm/nJ and 312 µm/nJ for two specific cases above and below the mode-locking threshold, respectively. Note that in the sub-threshold case, the synchronous mode-locking can only be obtained with sufficiently large injection pulse energy as expected.
Fig. 5 (a) Cavity mismatch tolerance versus pump power of the slave laser. The dashed line indicates the mode-locking threshold in the free-running condition. (b) Cavity mismatch tolerance versus injection pulse energy from the master laser for above- and sub-threshold cases. The solid lines are given by linear fits.
In order to demonstrate the long-term stability, we have recorded the repetition rates of two synchronized fiber lasers by using digital frequency counters (Tektronix FCA3103). Figure 6(a) gives the result for an acquisition time as long as 12 hours, which shows a frequency discrepancy of 0.01 Hz in standard deviation. The corresponding frequency error relative to repetition rate is calculated to be 5 × 10−10. On the other hand, the overall frequency difference for individual fiber laser is about 45 Hz, which is mainly due to the indoor-temperature fluctuation of 1.5 °C. Considering a maximum capture range of 800 µm, our synchronization system could in principle resist a temperature variation of 37 °C. Assisted with the PM mode-locking mechanism, the presented system could potentially operate outdoors beyond the laboratory environment.
Fig. 6 (a) Repetition-rate changes for master and slave pulses within 12 hours (top panel) and their discrepancy (bottom panel). The Δfr indicates the frequency discrepancy compared to the average value of the measured repetition rate for the master laser. (b) Repetition-rate fluctuations of master laser with and without active cavity-length locking (top panel). Repetition-rate fluctuations of slave laser with and without synchronization (bottom panel).
We then further investigate the frequency transfer by the passive all-optical synchronization. For this purpose, we actively lock the repetition rate of the master laser by using a PZT for cavity-length stabilization. The standard-deviation fluctuations of repetition rate are 0.5 mHz and 0.6 Hz with and without locking, respectively, as shown in Fig. 6(b). With the help of synchronization, the initial 2.0-Hz fluctuation of repetition rate for the slave laser is substantially reduced to 0.5 mHz as well, which indicates a high-precision frequency transfer between the two synchronized fiber lasers.
Finally, to quantitatively evaluate the short-term stability of passive synchronization, the timing jitter between the two-color pulses is measured by using the optical cross-correlation technique. As shown in Fig. 1(c), the two synchronized pulses are spatially combined by a WDM. One of the two beams can be delayed by a linear translation stage (Thorlabs LTS300/M) to control the temporal overlap. The mixed beam is then focused into a 25-mm periodically-poled lithium niobate (PPLN) crystal for sum-frequency generation (SFG). The SFG signal at 634 nm is steered through a spectral filer and detected by a gain-variable photodiode (Thorlabs PDA100A2). The output electronic signal passes a 1-MHz low-pass filter for subsequent analysis. Figure 7(a) shows the cross-correlation trace measured by remotely scanning the translational stage. The cross-correlation bandwidth τ3 is measured to be 22.9 ps, which is in agreement with the value calculated by the formula , where τ1,2 denote the durations of the two synchronized pulses. Therefore, the timing jitter should be negligible in our synchronization system.
Fig. 7 (a) Cross-correlation trace between the two synchronized pulses. The green dot at the half amplitude indicates the position to measure the timing jitter. (b) The voltage fluctuations within a 1-MHz bandwidth at the maximum and middle points at the correlation trace. (c) Timing jitter power spectral density (left) and the integrated timing jitter in Fourier domain (right).
More rigorous deduction of timing jitter can be obtained by the Fourier analysis of the cross-correlation signal at its half maximum. In this position, the temporal variation linearly translates into the intensity fluctuation. The related slope is about 112 mV/ps in the experiment. We record the temporal evolution of the SFG signal at the half-intensity level as shown in Fig. 7(b). Consequently, the 4.8-mV variation without correction of detector noise and intrinsic intensity fluctuation indicates that the resultant timing jitter has an upper bound of 42 ± 9 fs. The uncertainty is due to the limited vertical resolution of the oscilloscope. Figure 7(c) shows the power spectral density from 0.1 Hz to 1 MHz in the Fourier domain, which is measured by a signal source analyzer (AnaPico APPH6040). The integrated root-mean-square timing jitter of 26 fs is obtained, which reveals tight synchronization enabled by our novel master-slave scheme. In principle, the Yb- and Er-doped fiber lasers could have an intrinsic jitter at the level of 100 as [32, 33], which would contribute negligible portion to the achieved relative timing jitter between the synchronized pulses. The achieved timing jitter is comparable to the results obtained in previously reported schemes with similar pulse durations for two synchronized fiber lasers [18, 23, 24]. In principle, shorter pulse durations could lead to a smaller relative timing jitter in the context of same synchronization mechanism. Recently, laser pulses with durations of 13 fs and 51 fs have enabled a timing jitter as low as 0.75 fs in a detection bandwidth of 350 kHz [19]. Due to the fast response of nonlinearity, the passive locking based on the cross-phase modulation effect would be effective to suppress high-frequency noises. Another limiting factor for obtaining ultra-low timing jitter is the residual low-frequency noise. There have been studies showing that sub-femtosecond timing jitter can be obtained by using a hybrid synchronization technique including the low-bandwidth active locking and the fast-response passive locking [20].
4. Conclusions
In summary, we have presented an all-polarization-maintaining synchronized fiber laser system, which was comprised of two independent Yb-doped and Er-doped mode-locked fiber lasers in the master-slave configuration. The robust self-starting and long-term stable operation was made possible by leveraging the nonlinear amplifying loop mirror as the high-speed intensity modulator and fast artificial saturable absorber for synchronous mode-locking in the full polarization-maintaining structure. The tolerance of the cavity-length mismatch was about 800 µm, which could be further improved by increasing the injected pulse energy. The large capture range enabled a passive locking of the relative repetition rate over 12 hours, notably with a frequency error of 5×10−10. The high-precision frequency transfer between the synchronized lasers was also verified by combing active locking of the master laser and passive synchronization of the slave laser. The measured timing jitter was 26 fs in a Fourier frequency range from 0.1 Hz to 1 MHz. The timing jitter can be reduced with the help of shortening the synchronized pulse durations by cavity dispersion management, or/and resorting to low-bandwidth active jitter cancellation to remove the slow timing drifts [34, 35]. Benefiting from the full polarization-maintaining architecture, the presented passive all-optical synchronization system would be attractive in applications beyond laboratorial operations, such as remote clock stabilization and distribution [3]. Additionally, recent studies have demonstrated the possibility for the passive synchronization technique to transfer the offset frequency of a mode-locked laser besides the repetition frequency [36, 37]. This observation may not only arouse interests for fundamental investigation of the involved nonlinear coupling mechanism, but also promote applications in duplicating stability and accuracy among frequency combs with alleviative requirement of sophisticated active feedback.
Funding
Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning; National Natural Science Foundation of China (11727812); Science and Technology Innovation Program of Basic Science Foundation of Shanghai (18JC1412000).
Acknowledgments
The authors thank Dr. Junsong Peng for insightful discussions.
References
1. M. C. Fischer, J. W. Wilson, F. E. Robles, and W. S. Warren, “Invited Review Article: Pump-probe microscopy,” Rev. Sci. Instrum. 87, 031101 (2016). [CrossRef]
2. J. Kim and F. X. Kärtner, “Attosecond-precision ultrafast photonics,” Laser & Photon. Rev. \4, 432 (2010). [CrossRef]
3. S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78, 021101 (2007). [CrossRef] [PubMed]
4. R. K. Shelton, L.-S. Ma, H. C. Kapteyn, M. M. Murnane, J. L. Hall, and J. Ye, “Phase-Coherent Optical Pulse Synthesis from Separate Femtosecond Lasers,” Science 293, 1286 (2001). [CrossRef] [PubMed]
5. R. K. Shelton, S. M. Foreman, L.-S. Ma, J. L. Hall, H. C. Kapteyn, M. M. Murnane, M. Notcutt, and J. Ye, “Subfemtosecond timing jitter between two independent, actively synchronized, mode-locked lasers,” Opt. Lett. 27, 312–314 (2002). [CrossRef]
6. T. Miura, H Nagaoka, K. Takasago, K. Kobayashi, A. Endo, K. Torizuka, M. Washio, and F. Kannari, “Active synchronization of two mode-locked lasers with optical cross correlation,” Appl. Phys. B 75, 19 (2002). [CrossRef]
7. T. R. Schibli, J. Kim, O. Kuzucu, J. T. Gopinath, S. N. Tandon, G. S. Petrich, L. A. Kolodziejski, J. G. Fujimoto, E. P. Ippen, and F. X. Kaertner, “Attosecond active synchronization of passively mode-locked lasers by balanced cross correlation,” Opt. Lett. 28, 947–949 (2003). [CrossRef] [PubMed]
8. M. Xin, K. Şafak, M. Y. Peng, A. Kalaydzhyan, W.-T. Wang, O. D. Mücke, and F. X. Kärtner, “Attosecond precision multi-kilometer laser-microwave network,” Light: Science & Appl. 6, e16187 (2017). [CrossRef]
9. H. Tian, Y. Song, J. Yu, H. Shi, and M. Hu, “Optical-optical synchronization between two independent femtosecond Yb-fiber lasers with 10−20 instability in 105s,” IEEE Photon. J. 9, 1 (2017). [CrossRef]
10. C. Furst, A. Leitenstorfer, and A. Laubereau, “Mechanism for self-synchronization of femtosecond pulses in a two-color Ti:sapphire laser,” IEEE J. Sel. Top. Quan. Electron. 2, 473 (1996). [CrossRef]
11. Z. Wei, Y. Kaboyashi, and K. Torizuka, “Passive synchronization between femtosecond Ti:sapphire and Cr:forsterite lasers,” Appl. Phys. B 74, s171 (2002). [CrossRef]
12. C. J. Zhu, J. F. He, and S. C. Wang, “Generation of synchronized femtosecond and picosecond pulses in a dual-wavelength femtosecond Ti:sapphire laser,” Opt. Lett. 30, 561–563 (2005). [CrossRef] [PubMed]
13. M. Rusu, R. Herda, and O. G. Okhotnikov, “Passively synchronized erbium (1550-nm) and ytterbium (1040-nm) mode-locked fiber lasers sharing a cavity,” Opt. Lett. 29, 2246–2248 (2004). [CrossRef] [PubMed]
14. M. Rusu, R. Herda, and O. G. Okhotnikov, “Passively synchronized two-color mode-locked fiber system based on master-slave lasers geometry,” Opt. Express 12, 4719–4724 (2004). [CrossRef] [PubMed]
15. D. Yoshitomi, Y. Kobayashi, M. Kakehata, H. Takada, K. Torizuka, T. Onuma, H. Yokoi, T. Sekiguchi, and S. Nakamura, “Ultralow-jitter passive timing stabilization of a mode-locked Er-doped fiber laser by injection of an optical pulse train,” Opt. Lett. 31, 3243–3245 (2006). [CrossRef] [PubMed]
16. C. Zhou, Y. Cai, L. Ren, P. Li, S. Cao, L. Chen, and M. Zhang, “Passive synchronization of femtosecond Er- and Yb-fiber lasers by injection locking,” Appl. Phys. B 97, 445 (2009). [CrossRef]
17. Li Yao, Gu Xiaorong, Yan Ming, E Wu, and Heping Zeng, “Square nanosecond mode-locked Er-fiber laser synchronized to a picosecond Yb-fiber laser,” Opt. Express 17, 4526–4532 (2009). [CrossRef]
18. K. Huang, X. Gu, H. Pan, E. Wu, and H. Zeng, “Synchronized fiber lasers for efficient coincidence single-photon frequency upconversion,” IEEE J. Sel. Topics Quantum Electron. 18, 562 (2012). [CrossRef]
19. D. Yoshitomi and K. Torizuka, “Long-term stable passive synchronization between two-color mode-locked lasers with the aid of temperature stabilization,” Opt. Express 22, 4091–4097 (2014). [CrossRef] [PubMed]
20. B. Tsai, S. Wu, C. Hu, W. Hsiang, and Y. Lai, “Subfemtosecond hybrid synchronization between ultrafast Yb and Er fiber laser systems by controlling the relative injection timing,” Opt. Lett. 38, 3456–3459 (2013). [CrossRef] [PubMed]
21. X. Gu, K. Huang, Y. Li, H. Pan, E Wu, and H. Zeng, “Temporal and spectral control of single-photon frequency upconversion for pulsed radiation,” Appl. Phys. Lett. 96, 131111 (2010). [CrossRef]
22. S. Wu, W. Hsiang, and Y. Lai, “Laser dynamics and relative timing jitter analysis of passively synchronized Er- and Yb-doped mode-locked fiber lasers,” J. Opt. Soc. Am. B 31, 1508–1515 (2014). [CrossRef]
23. J. Sotor, G. Sobon, J. Tarka, I. Pasternak, A. Krajewska, W. Strupinski, and K. M. Abramski, “Passive synchronization of erbium and thulium doped fiber mode-locked lasers enhanced by common graphene saturable absorber,” Opt. Express 22, 5536–5543 (2014). [CrossRef] [PubMed]
24. M. Zhang, E. J. R. Kelleher, A. S. Pozharov, E. D. Obraztsova, S. V. Popov, and J. R. Taylor, “Passive synchronization of all-fiber lasers through a common saturable absorber,” Opt. Lett. 36, 3984–3986 (2011). [CrossRef] [PubMed]
25. D. Li, H. Xue, Y. Wang, M. Qi, W. Kim, C. Li, J. Riikonen, Z. Ren, J. Bai, H. Lipsanen, and Z. Sun, “Active synchronization and modulation of fiber lasers with a graphene electro-optic modulator,” Opt. Lett. 43, 3497–3500 (2018). [CrossRef] [PubMed]
26. A. Chong, L. G. Wright, and F. W. Wise, “Ultrafast fiber lasers based on self-similar pulse evolution: a review of current progress,” Rep. Prog. Phys. 78, 113901 (2015). [CrossRef] [PubMed]
27. M. E. Fermann, F. Haberl, M. Hofer, and H. Hochreiter, “Nonlinear amplifying loop mirror,” Opt. Lett. 15, 752–754 (1990). [CrossRef] [PubMed]
28. T. Jiang, Y. Cui, P. Lu, C. Li, A. Wang, and Z. Zhang, “All PM Fiber Laser Mode Locked With a Compact Phase Biased Amplifier Loop Mirror,” IEEE Photon. Tech. Lett. 28, 1786 (2016). [CrossRef]
29. F. Chen, Q. Hao, and H. Zeng, “Optimization of an NALM Mode-Locked All-PM Er:Fiber Laser System,” IEEE Photon. Tech. Lett. 29, 2119 (2017). [CrossRef]
30. W. Hänsel, 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 123, 41 (2017). [CrossRef]
31. S. M. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806 (1992). [CrossRef]
32. Y. Song, C. Kim, K. Jung, H. Kim, and J. Kim, “Timing jitter optimization of mode-locked Yb-fiber lasers toward the attosecond regime,” Opt. Express 19, 14518–14525 (2011). [CrossRef] [PubMed]
33. T. K. Kim, Y. Song, K. Jung, C. Kim, H. Kim, C. H. Nam, and J. Kim, “Sub-100-as timing jitter optical pulse trains from mode-locked Er-fiber lasers,” Opt. Lett. 36, 4443–4445 (2011). [CrossRef] [PubMed]
34. D. Yoshitomi, Y. Kobayashi, H. Takada, M. Kakehata, and K. Torizuka, “100-attosecond timing jitter between two-color mode-locked lasers by active-passive hybrid synchronization,” Opt. Lett. 30, 1408–1410 (2005). [CrossRef] [PubMed]
35. D. D. Hudson, S. M. Foreman, S. T. Cundiff, and J. Ye, “Synchronization of mode-locked femtosecond lasers through a fiber link,” Opt. Lett. 31, 1951–1953 (2006). [CrossRef] [PubMed]
36. M. Betz, F. Sotier, F. Tauser, S. Trumm, A. Laubereau, and A. Leitenstorfer, “All-optical phase locking of two femtosecond Ti:sapphire lasers: a passive coupling mechanism beyond the slowly varying amplitude approximation,” Opt. Lett. 29, 629–631 (2004). [CrossRef] [PubMed]
37. N. Kuse, A. Ozawa, Y. Nomura, I. Ito, and Y. Kobayashi, “Injection locking of Yb-fiber based optical frequency comb,” Opt. Express 20, 10509–10518 (2012). [CrossRef] [PubMed]
