Open Access
Add to BibSonomyAbstract
Fiber-stretcher based phase-lock loop (PLL) is a mature technique in fiber mode-locked lasers for repetition-rate stabilization. However, undesired side effects may be induced if not properly handled, which is easily overlooked owing to the lack of single-shot spectral analyzers. Thanks to the ultrafast spectral analyzing capability of optical time-stretch, an intriguing spectral dynamics is observed in a repetition-rate-stabilized nonlinear polarization rotation (NPR) mode-locked laser. Under the dynamic state, the optical spectra of pulses undergo dramatic evolution in every round trip while the pulse energy is relatively constant. Indicated by estimated cross-spectral densities, such spectral dynamics results in noticeable degradation in optical spectral coherence. The physical origin of the round-trip evolved spectral dynamics is attributed to the local birefringence induced by the fiber stretcher. Therefore, the results are helpful for a proper use of fiber-stretcher based PLL in fiber lasers, particularly when a good spectral coherence is desired. Furthermore, our study has also provided a potentially useful optical source for applications where fast spectral modulation is desired.
© 2017 Optical Society of America
1. Introduction
Mode-locked fiber lasers have gained much attention for their ultrashort pulse width, high pulse energy and compactness [1–3]. Over the last two decades, mode-locked fiber lasers have been widely applied in micromachining [4,5], optical communication [6] and biological imaging [7–9]. Particularly, they have played important roles in the optical frequency metrology [10,11] and astronomical observations [12,13] by providing excellent optical frequency combs. For those applications, the stabilization of repetition rate is always necessary. Therein, a common approach is to stabilize the cavity length by using fiber-stretcher based phase-lock loop (PLL) [14,15]. By winding a section of fiber around a piezoelectric fiber stretcher, a fast tuning range of tens of microns over the cavity length can be obtained to instantaneously counteract the frequency drifting that arises from environmental perturbation, e.g., thermal and mechanical vibrations, and thus stabilizes the repetition rate of mode-lock lasers [16,17].
Although the fiber-stretcher based PLL has been widely adapted in stable mode-locked fiber lasers, undesired fast spectral dynamics can be induced under a sub-optimal configuration, which may be overlooked without single-shot spectral analysis. Here we report the observation of an intriguing spectral dynamics in a PLL-stabilized nonlinear polarization rotation (NPR) mode-locked fiber laser. Using optical time-stretch [18,19], round-trip resolved spectral evolution of the laser output is observed. With excessive stress in the fiber stretcher, it is observed that the pulse spectrum evolves in each round trip, which will hardly be captured by conventional optical spectrum analyzers (OSA). To explore the impact of such spectral dynamics, ultrafast spectral coherence measurement based on the Young’s type interferometry [20] is performed, and an obvious degradation of spectral coherence is observed. The physical origin can be attributed to the formation of polarization-rotation vector soliton (PRVS) [21–25] owing to large local birefringence and our hypothesis is confirmed by polarization-resolved temporal and spectral detections. This elucidates that the localized large birefringence induced by the fiber stretcher can convert a scalar soliton into vector one. Therefore, our findings could alert in cases when the fiber stretcher is utilized for repetition-rate stabilization.
2. Laser configuration and fiber-stretcher based PLL
The fiber laser used was a dispersion-managed NPR mode-locked laser with repetition rate stabilization by means of a fiber-stretcher based PLL, as shown in Fig. 1(a). The laser cavity consisted of 0.5m highly-doped erbium-doped fiber (EDF), 0.3m dispersion-compensating fiber (DCF), 1.5m standard single-mode fiber (SMF), a polarization controller (PC) and an optical integrated module (OIM). The OIM provided multiple functions, e.g., wavelength-division multiplexing, beam splitting (90/10, 10% output) and polarization-sensitive isolation in a single unit with a physical dimension of 0.55 × 5.5 cm and insertion loss of < 1.5 dB [26]. By fine-tuning the PC, the laser was stably mode-locked through NPR and the 6dB spectral bandwidth was as large as 75 nm [17] thanks to the careful dispersion management. The total cavity length was about 2.3 m which led to a repetition rate of 89.3 MHz.
Fig. 1 (a) Experimental setup of the repetition-rate stabilized mode-locked laser; inset: photo of home-made fiber stretcher. (b) RF spectrum of the laser output. (c) Zoom-in of the 12th harmonic RF tone without PLL stabilization. (d) Zoom-in of the 12th harmonic RF tone with PLL stabilization. PC, polarization controller; DCF, dispersion-compensating fiber; EDF, erbium-doped fiber; OIM, optical integrated module; PD, photodetector; BPF, band-pass filter; LPF, low pass filter; PID, proportional–integral–derivative controller.
Since the broadband fs fiber laser was sensitive to environmental disturbance, a fiber-stretcher based PLL was introduced to stabilize the laser repetition rate. As shown in the inset of Fig. 1(a), a section of SMF was wound around a fiber-stretcher consisting of two post holders whose separation was adjusted by a piezoelectric ceramic transducer actuated translation stage. To introduce active feedback, 10% of the laser output was launched into a photodetector, after which the 12th harmonic RF tone was selected by a band-pass filter (BPF). Then, a frequency reference at 1.07 GHz provided by a function generator (Agilent 8684A) was mixed with the 12th harmonic tone through a frequency mixer to generate a beating signal. After passing through a low-pass filter (LPF), the beating signal was launched into an analog PID controller (SRS SIM960), which generated the error signal. The following high-voltage (HV) driver amplified the error signal, and then applied it onto the fiber stretcher to introduce negative feedback for the cavity length stabilization. An oscilloscope monitored the error signal to confirm that the laser repetition rate was well stabilized. The RF spectrum of the laser output is shown in Fig. 1(b), which is consisted of multiple harmonic tones separated by 89.3 MHz. To verify the performance of fiber-stretcher based PLL, the 12th frequency tone was recorded repeatedly every 1 minute over ten minutes. As shown in Fig. 1(c), without switching on the PLL, the 12th tone drifted over 1 kHz, corresponding to an 80Hz drifting of the repetition rate. However, when the PLL was switched on, no measurable frequency drifting could be observed, as shown in Fig. 1(d), indicating a well stabilization of the laser repetition rate.
3. Real-time spectral-temporal observation of spectral dynamics
Despite the effectiveness of the fiber-stretcher based PLL in stabilizing repetition rate, particular attentions should be paid to the fiber deformation. When the fiber is deliberately over-stretched to compensate a large frequency drifting, an intriguing fast spectral dynamics was observed through the single-shot spectral analysis, which could be easily overlooked by conventional observing methods. In Fig. 2, the spectrally-dynamic states were characterized in both time and spectral domains. To provide a reference, the normal mode-locking state was characterized first, as presented in Figs. 2(a)-2(c). The time-domain waveform in Fig. 2(a) exhibits a periodic pulse train at a period of 11.2 ns. The corresponding peak intensity histogram over 2000 round trips is shown in the inset, which indicates a stable temporal intensity. The intensity distribution shows a standard Gaussian shape, and the full width at half maximum (FWHM) of Gaussian fitting is 0.018, i.e., a very small intensity fluctuation of about 2%. In addition to the time-domain characterization, the spectral-domain information was analyzed in a single-shot manner by optical time-stretch spectroscopy. The laser output was propagated through a spool of 4.7km SMF, which was then received by a 16GHz photodetector and recorded by a real-time oscilloscope. Owing to the dispersion of the SMF, different wavelength components traveled at different group velocities inside the fiber, and thus spectrum-to-time mapping was achieved at a mapping ratio of 82 ps/nm, corresponding to a spectral resolution of about 0.44 nm. In this way, the optical spectrum of every round trip was obtained from the temporal waveform, as shown in Fig. 2(b). Under the normal mode-locking state, the time-stretch waveform was consistent for every round trip, which represented a high spectral stability. This is clearly shown in Fig. 2(c), where four consecutive frames of optical spectrum indicated by the shadowed area in Fig. 2(b) are overlapped to compare with the optical spectrum obtained by a conventional OSA. Overall, the time-stretch waveform matches well with the OSA spectrum, which implies the effectiveness of optical time-stretch spectroscopy for real-time spectral observation. It should be pointed out that the negligible mismatches between the time-stretch spectroscopy and OSA can be attributed to the slightly wavelength-dependent loss and the higher-order dispersions in the SMF.
Fig. 2 Time- and spectral-domain characterization of the mode-locked laser under normal and dynamic states. (a) Time-domain waveform of the laser output. Inset: histogram of the intensity of pulse train over 2000 periods. (b) Time-stretch waveform of pulse trains showing a consistent optical spectrum in each round trip. (c) Comparing the average spectrum obtained by OSA with those captured by time-stretch spectroscopy over four periods indicated by the shadowed area in (b). (d)-(f): The same characterization for the case where the optical spectrum evolved at a period of two round-trips time. (g)-(i): The same characterization for the case where the optical spectrum evolved over multiple round trips.
In contrast, under a sub-optimal configuration (excessive stress at the fiber stretcher), an intriguing spectral dynamics was observed by the time-stretch spectroscopy, as shown in Figs. 2(d)-2(f). Under such condition, the time-domain waveform shown in Fig. 2(d) is almost the same as that of normal mode-locking state in Fig. 2(a), except for a slightly-increased intensity fluctuation of ~6%. However, the single-shot optical spectrum of every round trip is dramatically different from that of normal mode-locking case. As shown in Fig. 2(e), instead of a consistent spectrum in each round trip, the laser pulses exhibit a distinct spectrum in neighboring round trips, repeating every two round trips (referred to as spectral bifurcation in the following text). Again, four consecutive frames of time-stretch waveform are overlapped and compared with the OSA spectrum, as shown in Fig. 2(f). In this case, large mismatch can be observed since the OSA only measured an average spectrum over multiple round trips. However, since the pulse spectral evolution repeats every two round trips, the OSA falsely exhibits a stable spectrum owing to its speed limitation. Considering the time-domain waveform is also almost identical with the normal/stable case in Fig. 2(a), such a spectral dynamics can be easily overlooked under conventional observing methods. By tuning the intra-cavity PC or the pump power, more-complex spectrally-dynamic states were obtained. One such example is shown in Figs. 2(g)-2(i). The corresponding time-domain intensity waveform is shown in Fig. 2(g). In this case, the pulse train exhibits relatively larger fluctuation, and the FWHM of the corresponding Gaussian distribution is increased to 12%. In the spectral domain shown in Fig. 2(h), the optical spectrum varies in each round trip, even though not as significant as that in the spectral-bifurcation case in Fig. 2(e). The variation in spectra shapes during four consecutive round trips are more clearly observed in Fig. 2(i). On the other hand, the OSA still falsely manifested a stable optical spectrum in each scan, which indicated that although the spectrum varied from pulse to pulse, there was a stationary distribution of these spectra reached within a time (i.e. the spectral evolution period) much shorter than the scan time of the OSA.
4. Optical spectral coherence
There is no doubt that such spectral dynamics will impose significant impact on systems where broadband mode-locked sources are used. Under such states, even though the pulse energy is relatively constant between round trips, the spectrum shapes differ significantly, which can severely degrade the intensity stability of a system when only a certain region of the spectrum is utilized. In addition to the concern about the spectral intensity stability, the impact of such spectral dynamics on optical spectral coherence was also investigated. Here, the optical time-stretch was equipped with Young’s type interferometry to perform ultrafast characterization of the second-order spectral coherence, defined by the cross-spectral density function [20]. In brief, time-stretch interferometry was utilized to map spectral interferogram onto the time domain, which was then used to analyze the spectral coherence based on the visibility of the interference fringes. As shown in Fig. 3(a), the same spool of SMF was placed at the output port of a free-space Michelson interferometer consisting of a 50/50 coupler, two collimators and two mirrors. The optical path lengths of two arms were differed by a cavity round trip so that each pulse was interfered with its neighboring pulse.
Fig. 3 (a) Time-stretch interferometry used to measure the optical spectral coherence. (b)-(d) 500 consecutive single-shot interferograms overlapped together for both normal/stable and spectrally-dynamic cases. The red curves show the averages. (e)-(g) The 2D spectral interferograms generated from 500 single-shot 1D interferograms. (h)-(j) The estimated cross-spectral densities calculated from (e)-(g).
The time-stretch interferograms of all three cases were captured and shown in Figs. 3(b)-3(d). 500 single-shot interferograms of each case were captured and overlapped in gray color. Assume the 1D single-shot interferograms, i.e., those shown in Figs. 3(b)-3(d), are expressed as X(λ), the 2D interferograms can be generated according to
where 〈〉 denotes the ensemble average. As shown in Fig. 3(e), the 2D interferogram of the stable mode-locked case exhibits a well-defined square-grid pattern with high contrast, which is resulted from the high visibility of the interference fringes across the spectrum. On the other hand, Figs. 3(f) and 3(g) show sheared grids or even diagonal strips, respectively for the spectral bifurcation and the more-complex cases, which are resulted from the self-correlation of the low-coherence wavelength components [20]. Figures 3(h)-3(j) show the estimated cross-spectral density (CSD) functions recovered from the visibilities of 2D spectral interferograms. As expected, a high degree of coherence correlation across the spectrum can be observed for the normal/stable case in Fig. 3(h), which is manifested as the large square area with CSD close to 1.0. This indicates that different spectral components are highly correlated with each other. In contrast, for both the spectral bifurcation and the more-complex cases, i.e., Figs. 3(i) and 3(j), respectively, the estimated CSD functions exhibit an obvious degradation in coherence: the maximum value of the CSD functions have been reduced to around 0.8 and the high correlation areas have significantly shrunk. The coherence degradation can be attributed to the large spectral amplitude fluctuation from frame to frame. Therefore, such spectral dynamics potentially induced by a fiber-stretcher based PLL in a mode-locked laser deserves more attention, particularly for the applications requiring broadband spectral coherence.5. Polarization-rotating nature of the spectral dynamics
The physical origin of the spectral dynamics is attributed to be the formation of polarization-rotation vector soliton (PRVS). Admittedly, such suspicion is counter-intuitive, since the formation of vector soliton is not favorable in NPR mode-locked lasers owing to the existenceof polarization-sensitive isolation inside the cavity. To verify the mechanism of such spectral dynamics, polarization-resolved characterization in both time and spectral domain was conducted. First, the pulse trains of the two spectrally-dynamic cases were passed through a polarization beam splitter (PBS, 1 × 2), and the output from one of the output ports were received by a PD and recorded by a real-time oscilloscope. The polarization state before the linear polarizer was adjusted by a PC to maximize the intensity modulation depth on the output pulse train after the polarizer. For the spectral bifurcation case shown in Fig. 4(a), the pulse intensity shows a large but regular fluctuation at a period of two round-trip times. This implies that the output pulses from the laser have different polarization orientations at different round trips. The corresponding RF spectrum is shown in Fig. 4(c). It is clear that comparing to Fig. 1(b), new frequency components with slightly lower intensities appear in between the harmonics of fundamental repetition rate. The spacing between the new frequency components and the original harmonic tones is exactly half of the laser repetition rate, i.e. 44.6 MHz. Those features are consistent with that of PRVSs [21–25]. To further confirm the PRVS generation, polarization-resolved spectral detection was performed by replacing the aforementioned PD and real-time oscilloscope with the OSA, and the signals from both ports of the PBS were recorded, as shown in Figs. 4(e) and 4(f), respectively. The PC was fine tuned to maximize the spectral intensity on the horizontal polarization axis, which is shown in the red curve of Fig. 4(e). On the other hand, the corresponding optical spectrum on the vertical polarization axis is shown in blue curve. Obviously, the spectra from two polarization axis have distinct shapes, which further proves the generation of vector solitons.
Fig. 4 Verification of polarization-rotating vector soliton state for spectral bifurcation (left column) and more complex cases (right column). (a), (b) Time domain pulse train after passing through a polarization beam splitter (PBS). (c), (d) RF spectrum. (e), (f) Polarization-resolved spectrum.
Same characterization was performed for the case of more-complex spectral dynamics, as shown on the right column of Fig. 4. Consistent with the spectral evolution shown in Fig. 2(h), the pulse intensity after the PBS also evolves from pulse to pulse. The corresponding RF spectrum in Fig. 4(d) exhibits much denser frequency components at a tone spacing of around 4.5 MHz. Therefore, the more complexed spectrally-dynamic state corresponds to a PRVS state with lower rotating frequency (i.e. slower rotating speed). The polarization-resolved spectra in Fig. 4(f) further validate the vector nature of those spectral dynamics.
6. Discussion
Even though it is now confirmed that the physical origin of the spectral dynamics is the formation of PRVS, questions might be aroused on how it can be supported in a NPR mode-locked laser and why the spectrum of vector solitons in our configuration varies in each round trip. The fundamental mechanism that answers these questions can be attributed to the conversion between scalar and vector solitons in a NPR mode-locked laser. It has been proved that scalar and vector soliton can co-exist and convert between each other in a NPR mode-locked laser by introducing localized birefringence inside the cavity [27]. In our case, the fiber deformation owing to excessive stress in the fiber stretcher is responsible for the localized birefringence. Such scalar-vector conversion is also responsible for the round-trip evolving spectrum in PRVS. Owing to the linear polarizer inside the laser cavity, the two orthogonal soliton components will couple and interfere with each other at a different relative phase when the vector soliton is converted to a scalar one by passing through the polarizer, thus resulting in a different spectrum in every round trip. Therefore, it is believed that such phenomenon is unique to PRVS formed under NPR mode-locking scheme, which has never been reported before in literatures on conventional PRVS.
Therefore, in applications where high spectral intensity stability and spectral coherence are required, our findings alert that cares should be taken when fiber stretchers are utilized in NPR mode-locked fiber lasers for stabilization. To avoid such dynamic state, it is suggested that the fiber stretcher should have a large bending radius, and that the stress on the fiber stretcher should be minimized to avoid the onset of vector soliton. At the same time, the unique feature of PRVS in NPR mode-locked laser which is revealed for the first time using time-stretch spectroscopy in this work, will promote the further exploration in the vector soliton. Last but not least, the PRVS formed in NPR mode-locked lasers may even find some applications where fast and pseudo-random spectral intensity modulation is desired, e.g., the optical compressed sensing using broadband source [28,29]
7. Conclusions
In conclusion, for the first time to the best of our knowledge, a fast spectral dynamics has been observed in a PLL-stabilized NPR mode-locked fiber laser using single-shot spectral analysis. Under the spectrally-dynamic state, the optical spectra of the pulses exhibit dramatic variation in each round trip even though the pulse energy is relatively consistent. Such spectral dynamics reduces spectral stability and second-order spectral coherence. The results thus draw the attentions to the appropriate application of fiber-stretcher based PLL in fiber mode-locked lasers, which facilitates achieving the ultimate performance of fs mode-locking. In addition, our findings contribute to the study of vector soliton and the dynamic-spectrum mode-locked laser, which is useful for fast pseudo-random spectral intensity modulation.
Funding
Research Grants Council of the Hong Kong Special Administrative Region, China (Project Nos. HKU 17205215, HKU 17208414, and CityU T42-103/16-N) and National Natural Science Foundation of China (N_HKU712/16); Innovation and Technology Fund (GHP/050/14GD); and University Development Fund of HKU.
References
1. M. E. Fermann, M. Hofer, F. Haberl, and S. P. Craig-Ryan, “Femtosecond fibre laser,” Electron. Lett. 26(20), 1737–1738 (1990). [CrossRef]
2. I. N. Duling, “All-fiber ring soliton laser mode locked with a nonlinear mirror,” Opt. Lett. 16(8), 539–541 (1991). [CrossRef] [PubMed]
3. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18(13), 1080–1082 (1993). [CrossRef] [PubMed]
4. L. Shah, A. Arai, S. Eaton, and P. Herman, “Waveguide writing in fused silica with a femtosecond fiber laser at 522 nm and 1 MHz repetition rate,” Opt. Express 13(6), 1999–2006 (2005). [CrossRef] [PubMed]
5. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008). [CrossRef]
6. G. P. Agrawal, Fiber-Optic Communiation Systems, 3rd ed. (Wiley, 2002).
7. C. Xu and F. W. Wise, “Recent advances in fiber lasers for nonlinear microscopy,” Nat. Photonics 7(11), 875–882 (2013). [CrossRef] [PubMed]
8. N. G. Horton, K. Wang, D. Kobat, C. G. Clark, F. W. Wise, C. B. Schaffer, and C. Xu, “In vivo three-photon microscopy of subcortical structures within an intact mouse brain,” Nat. Photonics 7(3), 205–209 (2013). [CrossRef] [PubMed]
9. C. W. Freudiger, W. Yang, G. R. Holtom, N. Peyghambarian, X. S. Xie, and K. Q. Kieu, “Stimulated Raman scattering microscopy with a robust fiber laser source,” Nat. Photonics 8(2), 153–159 (2014). [CrossRef] [PubMed]
10. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002). [CrossRef] [PubMed]
11. A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature 482(7383), 68–71 (2012). [CrossRef] [PubMed]
12. T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012). [CrossRef] [PubMed]
13. T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321(5894), 1335–1337 (2008). [CrossRef] [PubMed]
14. B. R. Washburn, S. A. Diddams, N. R. Newbury, J. W. Nicholson, M. F. Yan, and C. G. Jørgensen, “Phase-locked, erbium-fiber-laser-based frequency comb in the near infrared,” Opt. Lett. 29(3), 250–252 (2004). [CrossRef] [PubMed]
15. J. Rauschenberger, T. Fortier, D. Jones, J. Ye, and S. Cundiff, “Control of the frequency comb from a modelocked Erbium-doped fiber laser,” Opt. Express 10(24), 1404–1410 (2002). [CrossRef] [PubMed]
16. T. Walbaum, M. Löser, P. Gross, and C. Fallnich, “Mechanisms in passive synchronization of erbium fiber lasers,” Appl. Phys. B 102(4), 743–750 (2011). [CrossRef]
17. B. Li, X. Wei, S. Tan, J. Kang, and K. K. Y. Wong, “Compact and stable temporally magnified tomography using a phase-locked broadband source,” Opt. Lett. 41(7), 1562–1565 (2016). [CrossRef] [PubMed]
18. F. Coppinger, A. S. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 47(7), 1309–1314 (1999). [CrossRef]
19. K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013). [CrossRef]
20. Y. Xu, X. Wei, Z. Ren, K. K. Y. Wong, and K. K. Tsia, “Ultrafast measurements of optical spectral coherence by single-shot time-stretch interferometry,” Sci. Rep. 6(1), 27937 (2016). [CrossRef] [PubMed]
21. S. Cundiff, B. Collings, and W. Knox, “Polarization locking in an isotropic, modelocked soliton Er/Yb fiber laser,” Opt. Express 1(1), 12–21 (1997). [CrossRef] [PubMed]
22. B. C. Collings, S. T. Cundiff, N. N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Polarization-locked temporal vector solitons in a fiber laser: experiment,” J. Opt. Soc. Am. B 17(3), 354–365 (2000). [CrossRef]
23. L. M. Zhao, D. Y. Tang, X. Wu, H. Zhang, and H. Y. Tam, “Coexistence of polarization-locked and polarization-rotating vector solitons in a fiber laser with SESAM,” Opt. Lett. 34(20), 3059–3061 (2009). [CrossRef] [PubMed]
24. S. V. Sergeyev, C. Mou, A. Rozhin, and S. K. Turitsyn, “Vector solitons with locked and precessing states of polarization,” Opt. Express 20(24), 27434–27440 (2012). [CrossRef] [PubMed]
25. M. Liu, A. P. Luo, Z. C. Luo, and W. C. Xu, “Dynamic trapping of a polarization rotation vector soliton in a fiber laser,” Opt. Lett. 42(2), 330–333 (2017). [CrossRef] [PubMed]
26. X. Wei, S. Xu, H. Huang, M. Peng, and Z. Yang, “Compact all-fiber ring femtosecond laser with high fundamental repetition rate,” Opt. Express 20(22), 24607–24613 (2012). [CrossRef] [PubMed]
27. Z. Wu, D. Liu, S. Fu, L. Li, M. Tang, and L. Zhao, “Scalar-vector soliton fiber laser mode-locked by nonlinear polarization rotation,” Opt. Express 24(16), 18764–18771 (2016). [CrossRef] [PubMed]
28. B. T. Bosworth and M. A. Foster, “High-speed ultrawideband photonically enabled compressed sensing of sparse radio frequency signals,” Opt. Lett. 38(22), 4892–4895 (2013). [CrossRef] [PubMed]
29. B. T. Bosworth, J. R. Stroud, D. N. Tran, T. D. Tran, S. Chin, and M. A. Foster, “High-speed flow microscopy using compressed sensing with ultrafast laser pulses,” Opt. Express 23(8), 10521–10532 (2015). [CrossRef] [PubMed]
