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Abstract
Due to the fascinating features, pulsating solitons attract much attention in the field of nonlinear soliton dynamics and ultrafast lasers. So far, most of the investigations on pulsating soliton are conducted in Er-doped fiber lasers. In this work, we reported the periodic transition between two evolving pulsating soliton states in an Yb-doped fiber laser. By using the real-time measurement techniques, the spectral and temporal characteristics of this transition state were investigated. Two evolving soliton pulsation states have similar evolution process, i.e., from pulsating towards quasi-stable mode-locked states. However, the details of the two processes are different, such as the pulse energy levels, pulsating modulation depths, duration of quasi-stable mode-locked states. The transition between two evolving soliton pulsation states could be attributed to the interaction of the polarizer and the varying polarization states of the pulse inside the laser cavity. The experimental results will contribute to the further understanding of soliton pulsating dynamics in dissipative optical systems.
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1. Introduction
Due to the dissipative properties of energy exchange with the external environment, the dissipative system is more complicated than Hamiltonian system. Dissipative solitons (DSs) are formed under the composite balance of dispersion, nonlinearity, gain and loss in such a dissipative system [1]. Thus, dissipative solitons can exhibit abundant fascinating nonlinear dynamics. As a typical dissipative system, ultrafast fiber laser constitutes an ideal platform for the exploration of nonlinear soliton dynamics. In addition to the stationary soliton states, the real-time measurement methods, such as dispersive Fourier transform (DFT) [2,3] and time-lens techniques [4,5], make the investigation of transient dissipative soliton dynamics possible. So far, various types of transient dissipative soliton phenomena have been studied, such as evolving soliton molecules [6–10], soliton explosion [11–15], soliton buildup [16–21], soliton pulsation [22–25].
Because of the distinctive localized structure, pulsating soliton has attracted much attention in field of nonlinear optics. The early research of pulsating soliton was mainly focused on the theoretical prediction, such as plain pulsating, creeping, and erupting solitons [26–29]. Since 2018, with the DFT technique, the spectro-temporal dynamics of pulsating soliton has been widely investigated experimentally [24,25,30–33]. It has been demonstrated that the variations of pulse amplitude and spectral bandwidth are the typical features of pulsating soliton. Besides that, more complex and fruitful experimental studies, such as double-periodic pulsating soliton [34], multi-soliton pulsations with synchronous and asynchronous behaviors [35], vector pulsating solitons [36], “invisible” soliton pulsation [37], also have been explored. The above reports on pulsating solitons were confined in the erbium-doped fiber lasers (EDFLs) with anomalous dispersion or net-normal dispersion. Few works on pulsating solitons are carried out in all-normal dispersion ytterbium-doped fiber lasers (YDFLs) [38]. Theoretically, since soliton pulsating is one of the intrinsic features of dissipative systems, there should be no significant difference of soliton pulsating dynamics between EDFLs and YDFLs. However, it’s still interesting to confirm it by studying the pulsating soliton in YDFLs. On the other hand, except for the stable soliton pulsations, transient dynamics of soliton pulsation also becomes a hot topic in recent years. Until now, various interesting researches have been reported. Q. Q. Huang et al. unveiled the transition process from stationary to soliton pulsation [39]. Z. W. Wei et al. showed the phenomenon of pulsating soliton with chaotic behavior [30]. And J. S. Peng et al. reported the breathing pulsating soliton explosion [40]. In addition, the buildup and dissociation dynamics of pulsating solitons have been studied [38,41]. However, these transient pulsation dynamics are all unidirectional and irreversible. Then a question would naturally arise as to whether the YDFLs could operate in a regime, where it could transit between two different pulsating states periodically.
In this work, we experimentally demonstrated the periodic transition dynamics between two evolving soliton pulsation states in an YDFL by utilizing the DFT technique. In both evolving pulsating states, the soliton starts from pulsation and becomes to quasi-stable mode-locked state. However, the two evolving pulsating regimes exhibit distinct characteristics, such as different pulse energy levels, energy modulation depths and duration of the quasi-stable soliton states. The interaction of the polarizer and the varying polarization states of the pulse inside the cavity results in the periodic transition between two evolving pulsating soliton states. The transition period could be changed by adjusting the cavity parameters, such as pump power and polarization state. These experimental results would deepen our understanding of the physical mechanism of pulsating solitons in ultrafast fiber lasers.
2. Experimental setup
The schematic of the YDFL used in our experiment is shown in Fig. 1, which is based on nonlinear polarization rotation (NPR) mode-locking technique [42]. The fiber laser cavity consists of ∼0.8 m long YDF (Nufern SM-YSF-HI) with a dispersion parameter of -36 ps/nm/km, pumped by a 980 nm laser diode (LD) through a wavelength division multiplexer (WDM). The other fibers are ∼7.1 m HI-1060 fiber with a dispersion parameter of -38 ps/nm/km. Therefore, the total cavity length is 7.9 m and the net cavity dispersion is ∼0.177 ps2. A polarization controller (PC) is employed to adjust the polarization states of the propagating light in the cavity. In addition to ensuring unidirectional operation, the polarization-dependent isolator (PD-ISO) works with the PC to realize the mode-locking operation by NPR technique. A 5 nm band-pass filter (BPF) centered at 1064 nm is used to make the mode-locking operation more easily. A 30:70 optical coupler (OC) is used to output the laser. The average optical spectrum and the radio-frequency (RF) spectrum of the laser output are performed by using an optical spectrum analyzer (OSA), and a RF spectrum analyzer. In addition, a high-speed real-time oscilloscope with a photodetector is used to observe the spatial-temporal intensity evolution. To observe the real-time spectral dynamics evolution of the solitons with DFT technique [2], a ∼10 km single mode fiber is placed between the output port and oscilloscope. It is worth noting that, the shot-to-shot spectral resolution is estimated to be ∼0.25 nm for this DFT configuration [3].
Fig. 1. Schematic of the ultrafast fiber laser used in the experiment. WDM, wavelength division multiplexer; YDF, Yb-doped fiber; PC, polarization controller; PD-ISO, polarization-dependent isolator; BPF, band-pass filter; OC, optical coupler; SMF, single mode fiber; PD, photodetector; OSA, optical spectrum analyzer.
3. Experimental results
3.1 Stable mode-locking state
By increasing the pump power to the mode-locking threshold of ∼159 mW and properly tuning the PC, the stable mode-locking state could be easily observed. Figure 2 shows the output characteristics of the stable mode-locking operation. The mode-locked spectrum measured by OSA is presented in Fig. 2(a). The center wavelength is 1060.86 nm. As can be seen in Fig. 2(b), the RF spectrum has a peak at 26.1 MHz with the signal-to-noise ratio (SNR) of >50 dB. It indicates a stable mode-locking state. The autocorrelation trace with a pulse width of ∼2.31 ps is shown in Fig. 2(c). Figure 2(d) illustrates the pulse train over 100 roundtrips, which have the uniform intensity. To further demonstrate the stability of the mode-locked operation, the evolution of shot-to-shot spectra over 100 roundtrips measured by DFT technique is presented in Fig. 2(e). The corresponding normalized energy evolution is calculated by integrating the DFT spectra, as shown in the inset (yellow curve) of Fig. 2(e). Here, the profiles of the shot-to-shot spectra and energy intensities are almost indistinguishable from each other. Moreover, to confirm the accuracy of DFT, the average spectrum of 100 roundtrips of shot-to-shot spectra is also plotted with the green curve in Fig. 2(e). It is in good agreement with the linear spectrum measured by OSA in Fig. 2(a). All the results indicate that the fiber laser operates in the stable mode-locking regime.
Fig. 2. Stable mode-locking state. (a) Linear spectrum measured by OSA (Inset: log-scale spectrum); (b) RF spectrum with resolution bandwidth of 1 kHz; (c) autocorrelation trace; (d) pulse train; (e) shot-to-shot spectra measured by DFT (Inset: the corresponding energy evolution and average spectrum of 100 roundtrips).
3.2 Pulsating soliton state
When the pump power is increased to 173 mW, the pulsating soliton state can be obtained by finely adjusting the PC, which is shown in Fig. 3. Figure 3(a) shows the average spectrum of the pulsating soliton directly recorded by the OSA. There is only some slight difference between Fig. 2(a) and Fig. 3(a). The RF spectrum in Fig. 3(b) has several satellite peaks around the fundamental repetition frequency with a separation of 3.26 MHz, indicating that the laser operates in the pulsating soliton regime. In order to investigate the characteristics in temporal and spectral domains, the corresponding evolutions over 100 consecutive roundtrips are recorded and shown in Figs. 3(c) and 3(d), respectively. The temporal evolution of the soliton directly from the oscilloscope in Fig. 3(c) demonstrates that the pulse intensity varies periodically with 8 cavity roundtrips, which agrees well with the pulsating intensity modulation frequency of 3.26 MHz on RF spectrum. In Fig. 3(d), the real-time spectral evolution measured by DFT technique presents the spectral breathing with a period of ∼8 roundtrips. The corresponding energy evolution over 100 roundtrips also varies periodically. In addition, the average spectrum over 100 roundtrips is plotted in the inset of Fig. 3(d), which is consistent with the OSA spectrum. The typical characteristics, such as periodic spectral breathing, pulse intensity modulation, multi-peak RF spectrum, demonstrate that the laser operates in a stable pulsating soliton regime.
Fig. 3. Stable pulsating soliton state. (a) Spectrum measured by OSA; (b) RF spectrum with resolution bandwidth of 1 kHz; (c) pulse train; (d) shot-to-shot spectra measured by DFT (Inset: energy evolution and average spectrum).
3.3 Transition state between two evolving soliton pulsations
By further increasing the pump power to 183 mW and finely tuning the PC, the laser works in a novel transition state between two evolving soliton states, as plotted in Fig. 4. Different from the smooth spectra of the stable soliton and stable pulsating soliton states in Figs. 2(a) and 3(a), there are lots of unstable burrs on the spectrum measured by OSA, as shown in Fig. 4(a). It means that some complex transient dynamics happened in this state. By using the DFT technique, the real-time evolution of this state was studied. Figure 4(b) shows the evolution of shot-to-shot spectra over 8000 roundtrips. Here, we can clearly see that the soliton transits between two main stages periodically. The corresponding energy evolution with white curve in Fig. 4(b) and the corresponding pulse train in Fig. 4(d) also verify the periodic evolution.
Fig. 4. Periodic transition state. (a) Spectrum measured by OSA; (b) shot-to-shot spectra of 8000 roundtrips measured by DFT and the corresponding energy evolution; (c) average spectra of two quasi-stable states (1650-1700th and 3450-3500th, respectively); (d) pulse train.
In order to investigate the dynamics in detail, we extract more than one whole transition process (1 to 3500 roundtrips) of Fig. 4(b) to Fig. 5. It mainly consists of two processes from pulsating to quasi-stable mode-locked states with different pulse energy levels. Six representative regions are shown in Figs. 5(b)-(g), respectively. As demonstrated in Fig. 5(b), the spectrum and pulse energy during 50 to 200 roundtrips are almost constant. It implies that the fiber laser initially operates in a quasi-stable state. Then, the soliton energy starts to increase abruptly from 285th roundtrip until the appearance of pulsation at 300th roundtrip. Correspondingly, the spectrum broadens slightly during the process of energy increase, as shown in Fig. 5(c). At the beginning of the pulsation, the modulation depth is ∼9% and the period is ∼3 roundtrips. The spectral breathing could also be seen in Fig. 5(c). After thousands of roundtrips, the pulsation becomes less and less obvious progressively. The energy modulation depth is almost 0 and no spectral evolution could be identified from ∼1590th roundtrip. The laser enters into quasi-stable mode-locked state. Note that the pulse energy decreases a little (from 1 to ∼0.85) during the pulsation process. After ∼127 roundtrips of quasi-stable soliton state, the pulse energy begins to decrease at 1717th roundtrip suddenly, as shown in Fig. 5(e). When the pulse energy drops from 0.85 to ∼0.58, the pulsation starts again. Different from the soliton pulsation in Fig. 5(c), here the modulation depth could reach ∼19% at first. Then, the modulation depth and spectral breathing also become smaller and smaller over roundtrips, finally to ∼0 at 3000th roundtrip, as seen in Figs. 5(f) and 5(g). At the same time, the modulation period is ∼12 roundtrips at the beginning (1738th roundtrip), then it decreases to 9 roundtrips until no energy modulation could be recognized. The soliton goes to quasi-stable mode-locked state again, which would last for 500 roundtrips, much longer than that in first evolving pulsation process. The results indicate that the laser operates in a state, where it transits between two evolving pulsating soliton states periodically. In each evolving pulsating soliton state, it starts from pulsation and finally enters into quasi-stable soliton state. Although the main evolution trends are same in the two evolving pulsation states, the detail evolutions are different, such as the pulse energy level, modulation depth and pulsation period, duration of quasi-stable mode-locked state, and so on. Figure 5(h) shows the corresponding temporal evolution of 1-3500th roundtrips. The white dashed line in Fig. 5(h) is the horizontal reference line. Interestingly, we observed some slightly temporal shift of the pulse train during the whole transition process, especially during ∼1700th-2500th roundtrips. It could be contributed to the change in spectrum during the process, which could be identified from the average spectra of these two quasi-stable states (1650-1700th and 3450-3500th, respectively) in Fig. 4(c).
Fig. 5. Details of the transition state. (a) Shot-to-shot spectra of 3500 roundtrips and corresponding energy evolution; (b)-(g) 6 representative regions; (h) pulse train.
4. Discussions
In our experiments, by increasing the pump power and adjusting the PC, stable mode-locking state and stable pulsating soliton state could be achieved in the YDFL. When finely adjusting the cavity parameters, a transition state between two progressively evolving pulsating soliton states could be observed. There are two evolving pulsating soliton stages during each transition process. The first stage of high pulse energy lasts ∼1432 roundtrips. Soliton pulsation starts with modulation depth of 9% and period of 3 roundtrips, then gradually becomes to quasi-stable mode-locked state. Suddenly, the soliton will enter into the second stage, which lasts ∼1784 roundtrips. In this stage, the beginning soliton pulsation has larger modulation depth of 19% and period of 12 roundtrips. Similar with the first stage, the soliton pulsation will become weaker and weaker, and finally become quasi-stable. Here, the quasi-stable soliton state lasts for longer time of ∼500 roundtrips. The two stages will happen alternatively with period of ∼3216 roundtrips. In addition, when the cavity parameters were fixed, the periodic transition state could last for hours. The laser could restore the transition state after restarting by just increasing pump power, but the period of the transition varies from time to time. Note that when the pump power was further increased beyond 183 mW, the laser will operate in multi-soliton state.
The proposed fiber laser is mode locked by the NPR technique. The PD-ISO not only ensures the unidirectional operation of the laser cavity, but also serves as a polarizer. It will modulate the pulse intensity depending on the polarization state of the pulse at the point of polarizer. When the pump power and PC are set properly, the polarization state of the pulse at the point of polarizer will be constant after 1 or several roundtrips. Then the fiber laser will operate at stable soliton or stable pulsating soliton states. The polarization states of the pulse in these cases are at the balanced points (stable solitons) or balanced regions (stable pulsating solitons). However, if the polarization state of the pulse at the point of polarizer deviates from the balanced points and regions, the laser will be in a transition state. In our work, the polarization state is far away from the balanced point at first, then the NPR technique tries to modulate the pulse to stable states in a progressive way. It can be identified by the gradual evolution from pulsating to quasi-stable state. In addition, the gradually changing energy during each stage could also confirm the progressive variation of the polarization state. At the end of each stage, the polarization state of pulse is very close to the balanced points, corresponding to the quasi-stable state. However, it could not reach the fully stable state with these parameter settings. With the continuous variation of the polarization state, the pulse will jump into another completely different region. The NPR effect will try to make the pulse stable again. Then, similar process from soliton pulsation to quasi-stable soliton will happen. But the evolving details are different because of the different polarization evolving regions. The laser transits between the two stages periodically. The average spectra of these two quasi-stable states shown in Fig. 4(c) are slightly different, which also confirms that the laser operates in two different states. The sudden energy changes at the edge of each stage could be contributed to the jumps between two polarization regions, which lead to the different loss of the cavity. It should be noted that different types of pulsating solitons and stable mode-locking solitons could be achieved by adjusting the cavity parameters in fiber lasers with almost all mode-locking techniques, which indicates that it’s also possible to see this state in other fiber lasers. Therefore, in our opinion, the periodic transition state is a critical state, which might have no relation to the mode-locking techniques, but mostly depends on the cavity parameters, such as cavity dispersion, pump power and polarization states.
5. Conclusion
In conclusion, we experimentally observed stable soliton, stable pulsating soliton states in an YDFL. With proper cavity parameters, the periodic transition dynamics between two evolving soliton pulsation states could be achieved. Due to the progressively evolving polarization state of the pulse at the point of polarizer, the laser will operate at the varying pulsating soliton state. With the cavity settings in this work, the laser could not reach fully stable soliton states. It will transit between two evolving pulsating states periodically, where different variation details could be observed. However, the main evolution trends from pulsating to quasi-stable soliton states are similar. All these findings provide a new insight to the investigation of pulsating soliton and dissipative soliton in ultrafast fiber lasers.
Funding
Key-Area Research and Development Program of Guangdong Province (2020B090922006); National Natural Science Foundation of China (11904339, 11974006, 12274149, 61875058, 62175069); Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515012315); Guangzhou Science and Technology Plan Project (202201010202).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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