As one of the most striking localized structures in dissipative systems, pulsating soliton has been widely studied in theory but rarely observed in experiments. Here, three typical types of soliton pulsations are experimentally demonstrated in an L-band normal-dispersion mode-locked fiber laser via the dispersive Fourier transform (DFT) technique. According to the distinctive features, they are classified as single-periodic pulsating soliton, double-periodic pulsating soliton and soliton explosion. These pulsations have common features such as energy oscillation, bandwidth breathing and temporal shift. However, the pulse is repeated every two oscillations for double-periodic pulsating soliton. When it comes to soliton explosion, because of the intermittent overdriven nonlinear effect induced by the extreme energy oscillation, the spectrum cracks into pieces at a periodic manner. To the best of our knowledge, it is the first time that both pure soliton pulsations and soliton explosion are observed experimentally in the same fiber laser. The results will enhance a more comprehensive understanding for the soliton pulsating phenomena.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Passively mode-locked fiber lasers have long been a fertile ground for investigations. For one thing, as an excellent candidate of the ultrashort pulse sources, mode-locked fiber lasers have various applications including in optical communication , spectroscopy , optical microscopy , biomedical science [4,5], etc. For another, due to the composite balance among dispersion, nonlinearity, gain and loss, there exists complex nonlinear optical phenomena in mode-locked fiber lasers, making them ideal playgrounds for testing various dissipative solitons and exploring unusual nonlinear dissipative dynamics . In fact, the generation of stable pulses is limited in a narrow range of the laser parameter space. More generally, the pulses are pulsating or chaotic and exhibit complex and interesting nonlinear behaviors due to the dissipative feature of a laser cavity . Therefore, revealing the transient and non-stationary properties of ultrafast lasers cannot only increase insights about the nonlinear dynamics, but also be of great significance to the development of compact, efficient and reliable laser systems [8–10].
Solitons, nonlinear dynamics and dissipative systems essentially constitute the three sources and component parts of the dissipative soliton concept . As part of the localized structures in nonlinear systems, soliton pulsations and explosions are used to describe the complex nonlinear optical phenomena . Generally, it is assumed that the laser operates in a stable regime when the pulse recovers exactly the same profile after each roundtrip. However, by carefully adjusting the system parameters of the fiber laser, the pulse suffers from instabilities and might acquire periodicities apart from the cavity roundtrip, because of which the pulsating soliton is formed . Soliton pulsations are fleeting transient dynamics with a timescale of microseconds or nanoseconds. Due to the lack of real-time experimental measurement instruments to capture the transient variations, the early investigations were focused on the theoretical analyses and numerical simulations [7,11–14]. It has been demonstrated that pulsating solitons do exist whether the net dispersion in the cavity is normal or anomalous [7,10]. To date, based on the complex Ginzburg-Landau equation (CGLE), a variety of pulsating solutions have been found theoretically, such as plain pulsating soliton, creeping soliton, erupting soliton (soliton explosion) , period doubling , extreme soliton pulsations , etc. All these solutions existing in different regions of the parameter space have a common feature: pulses vary in adjacent roundtrip but repeat periodically after multiple roundtrips. Notably, the pulsating solitons may become chaotic in the later stage of energy growth, where the spectra collapse abruptly and then recover gradually, that is, soliton explosions occur . Although the exploding behavior seems to be similar to chaos, it shows periodic characteristic, which falls in the criterion of the soliton pulsation . In other words, periodic soliton explosions can be called as soliton pulsations.
The first experimental evidence of pulsating solitons was observed in a dispersion-managed mode-locked fiber laser where the short and long period pulsations could appear under different laser parameters . However, due to the insufficient of diagnostic method, the dynamics was only investigated by the periodic changes of the pulse energy in the oscilloscope traces. Recently, the transient spectral dynamics of solitons can be investigated experimentally, benefiting from the dispersive Fourier transform (DFT) method . Taking advantage of group velocity dispersion, the temporal waveform of the pulse is stretched and mapped into spectrum, thus the real-time spectral evolution is obtained . Therefore, various kinds of dissipative soliton dynamics have been observed in ultrafast fiber lasers through DFT method, such as soliton buildup dynamics [16–18], soliton rain , soliton molecule [20–22] and dissipative soliton resonance . This powerful technique has also enabled fruitful experimental studies on soliton pulsations and soliton explosions [8,9,14,24–29]. In 2017, the real-time evolutions of soliton self-organization and pulsation were observed in an anomalous-dispersion mode-locked fiber laser . The authors demonstrated that the periodic radiation dispersion waves are in sync with the pulsation. Later, for the first time, the pulsating solitons with the spectrum breathing and oscillating structures in a mode-locked fiber laser at normal-dispersion regime were detected . Soon after, in an anomalous-dispersion cavity, Wei et al. provided an experimental observation of the pulsating soliton with chaotic behavior . As for the experimental verification of soliton explosions, more works have been reported, including successive soliton explosions , duration-tunable soliton explosions , mutually ignited soliton explosions , solitons explosions induced by soliton collision , etc. However, according to the theoretical predictions, there should be a great variety of undiscovered soliton pulsations existing in such vast regions in the parameter space [7,10]. The reports mentioned above are far from covering all the intriguing dynamics in nonlinear optical systems. Therefore, detailed experimental analyses about soliton pulsations, especially the transitions between different pulsating states, are still desiderata.
In this work, we report on the experimental observations of three typical types of soliton pulsations in an L-band normal-dispersion mode-locked fiber laser by utilizing the DFT technique. Specifically, we record the real-time spectra and the temporal evolutions of the pulsations and classify them into single-periodic pulsating soliton, double-periodic pulsating soliton and soliton explosion according to the distinctive features. The pulsating state transitions are achieved by adjusting the pump power and polarization state. These pulsations have common features such as energy oscillation, bandwidth breathing and temporal shift. However, the pulse repeats itself after every two periods of oscillation for double-periodic pulsating soliton. When it comes to soliton explosion, the spectrum cracks into pieces during the late stage of spectral broadening. As far as we know, it is the first time that both pure soliton pulsations and soliton explosion are observed respectively under different parameter settings of a single fiber laser. These results provide novel insights into understanding the complicated nonlinear dynamics in the unstable mode-locking condition.
2. Experimental setup
As a test-bed system, we build a typical normal-dispersion mode-locked fiber laser in L-band using nonlinear polarization rotation (NPR) technology, as shown in Fig. 1. As the gain medium, a 9.5 m long normal-dispersion erbium-doped fiber (EDF, Fibercore, I-25(980/125)) is forward pumped by a 976 nm laser diode (LD) through a fused wavelength division multiplexer (WDM). The signal absorption coefficients of the EDF are 23.9 dB/m at 979 nm and 41.14 dB/m at 1530 nm, respectively. Two polarization controllers (PCs) and a polarization dependent isolator (PD-ISO) are used as an artificial saturable absorber for mode locking, and the PD-ISO is also employed to ensure unidirectional operation. Owing to the intrinsic birefringence of the fiber, the NPR technology can also act as an artificial birefringent filter . The laser output is directed through the 79% port of an optical coupler (OC). The group velocity dispersion of the EDF is 40 ps2/km at 1550 nm. All the other fibers including device pigtails (~8.3 m in total) are standard single-mode fibers (SMFs) with a group velocity dispersion of −23 ps2/km at 1550 nm. The net dispersion of the cavity is 0.19 ps2.
As shown in Fig. 1, the output of the laser is split into two branches by another OC, the left branch is for the average spectrum monitoring by an optical spectrum analyzer (OSA, Yokogawa, AQ6370D), the right one is for the temporal and spectral evolutions monitoring. The temporal evolution is obtained when the signal is directly captured (undispersed) by a photodetector with a 45-GHz bandwidth (PD, DiscoverySemi, DSC10H) and sent to a real-time oscilloscope with a 33-GHz bandwidth (Tektronix DPO75902SX). The real-time spectra of the laser pulse are measured by DFT technology, where the signal is fed into a 1.5 km long dispersive compensation fiber (DCF, YOFC DM1010-D, −131.34 ps/(nm⋅km) @ 1545 nm) to stretch the pulses and thus yield the spectral evolution. As a result, the real-time spectral resolution is estimated to be 0.152 nm in our case.
Note that with just a single high-speed photodetector, we cannot conduct simultaneous measurements of the spectral and temporal evolutions of the output pulses. However, the pulsating soliton manifests itself as a quasiperiodic process , the spectral evolution can still correspond to the temporal evolution.
The stationary soliton state could be easily achieved by increasing the pump power and properly adjusting the PCs, a detailed description can be found in our previous work . The subject of this paper is the pulsating soliton, which can periodically change its shape, amplitude and bandwidth, and recover the original state after multiple roundtrips rather than one. Based on the CGLE, various complicated pulsating behaviors of solitons and their existence regions in the parameter space have been found [7,10]. In our experiments, we adjust the pump power and PCs to get different pulsating states. Note that not all dynamics predicted numerically can be easily observed experimentally. So far, we discover three types of pulsating soliton with complicated behaviors, as described below.
3.1. Single-periodic pulsating soliton
When the pumping power is 174.9 mW, the pulsating soliton with single period can be obtained by finely adjusting the PCs. Corresponding output performances are shown in Fig. 2. To confirm the validation of DFT, the spectrum directly recorded by the OSA and the one by averaging the DFT data are compared and plotted in Fig. 2(a). Different from the extremely steep spectral edges for stationary solitons , the spectra of pulsating solitons show arcuate edges. In addition, there is a weak continuous wave (CW) emission near the center wavelength, which exists in the form of noise floor in the temporal domain. The spatio-temporal and spatio-spectral dynamics of 7006 consecutive roundtrips are recorded by the real-time oscilloscope, as shown in Figs. 2(c) and 2(d), respectively. Figure 2(e) provides the pulse energy evolution by summing the intensity of all sampling points of a roundtrip. As can be seen in Figs. 2(c)–2(e), the pulse periodically changes its energy, chirp and spectral composition, and oscillates back and forth relative to its average position in the temporal domain, indicating that we are dealing with pulsating soliton. The pulsating period is about 808 roundtrips. One of the unusual features of the pulsating soliton in our case is that it exhibits a recurrent slow drop of the spectral bandwidth followed by a sharp drop, and finally, a dramatic growth. Therefore, the average spectrum demonstrates both arcuate edges. Comparing the Figs. 2(d) and 2(e), we can see that the growth of energy is in sync with the growth of the spectral bandwidth. For the pulse evolution in the time domain, we notice obvious amplitude pulsation. In addition, there is a periodical temporal shift, showing similar feature to the “double creeping” soliton in . Here, we attribute it to the change of group velocity caused by the variety of spectral composition . The radio-frequency (RF) spectrum in Fig. 2(b) shows the fundamental cavity repetition rate of 11.2103 MHz and the main six satellite peaks induced by the intensity modulation of pulsations. The 138.8 kHz frequency difference between peaks is consistent well with the pulsating period of ∼808 roundtrips demonstrated in Figs. 2(c)–2(e). However, due to the insufficient detection resolution, it is hard to observe all the features of the pulsating soliton experimentally such as duration and profile.
To further illustrate the pulse evolution, Fig. 3 shows eight typical spectra within a pulsating period. Similar to the numerical simulation results in , spectral bandwidth breathing and amplitude pulsation can be observed (see Visualization 1), which are caused by the incomplete balance among the dispersion, nonlinearity, gain and loss within one roundtrip. As shown in Figs. 3(a)–3(e), the multiple peaks arise at the edges of the spectra, then diffuse along both edges during the spectrum widening interval, and finally decay rapidly before the bandwidth reaches the wide extreme point, showing almost the same process as the dissipative soliton buildup dynamics in the normal-dispersion fiber laser . Subsequently, as shown in Figs. 3(e)–3(h), the spectrum shrinks. We recall that, owing to the pulse shaping mechanism in the normal dispersion regime, it is typical that the spectrum of high energy pulses evolve gradually to the M-shape when the mode-locked fiber lasers operate at high nonlinearity [16,31]. C. Lecaplain et al. proposed a generic explanation for the formation of these M-shaped spectra that the large fringes appearing at the edges of the spectrum are caused by discontinuities in the spectral phase . Generally, spectral phase jumps appear as soon as the gain saturation becomes large enough. According to the real-time spectral evolution in Fig. 4(a), the multiple peaks have roughly fixed frequencies, indicating that the four-wave mixing is playing an important role during the evolution . In addition, the spectral broadening effect is attributed to the increase of the self-phase modulation effect with the pulse energy increasing. The subsequent drop can be explained as spectral filter effect, because broadened spectrum are subjected to more loss induced by the gain filter and the intrinsic birefringent filter .
The change of the pump power influences the intracavity energy and nonlinear effect, which significantly affect the pulsation state . Once soliton pulsating state is achieved with a given setting of PCs, the changes of the pulsation dynamics can be observed by adjusting the pump power. The spectral evolutions and the corresponding temporal evolutions with increasing pump power are shown in Figs. 4 and 5, respectively. As shown in Figs. 4(a) and 4(b), the energy oscillation decreases with the increase of the pump power; meanwhile, the spectral bandwidth variation interval becomes smaller; further, the pulsation period declines. Similar transition has been observed by changing the quintic nonlinear gain coefficient in numerical simulations . When the pump power is increased to 211.8 mW (Fig. 4(c)), the pulsation disappears, instead, a strong CW noise component induced by large instabilities emerges. As shown in the inset of Fig. 4(c), the corresponding energy is scattered throughout the cavity cycle in the form of noise floor. Further increasing the pumping power, large instabilities are prevented by sharing the intracavity energy between two pulses, the mode-locked fiber laser steps into the double-soliton pulsation regime, as shown in Figs. 4(d)–4(f). Same as single soliton pulsation case, the oscillations of energy and spectral bandwidth get smaller and the pulsation period declines. At higher power levels, solitons pulsation with more pulses can be observed. It is worth mentioning that the average spectrum of multi-soliton pulsation presents a relatively flat top without CW component. Conversely, when the pump power is decreased under 174.9 mW, the growing soliton energy modulation leads to the complete disruption of the mode-locking regime, the laser enters the CW regime of operation.
Pulses in stationary multi-soliton mode-locked regime generally possess same characteristics due to the effect of soliton energy quantization. However, for multi-soliton pulsation, the enhancement or attenuation of any signal will affect the gain of all other signals . In order to get more information of multi-soliton pulsation, the time intervals between adjacent solitons are carefully tuned by adjusting the PCs to avoid the overlap of their real-time spectra. Figures 6(a) and 6(b) show the spectral evolution and the corresponding energy evolution of triple-soliton pulsation at the pumping power of 261.8 mW, respectively. In a cavity roundtrip, we define the leading soliton as soliton 1, the median one as soliton 2, and the trailing one is soliton 3. A higher resolution energy evolution is plotted in Fig. 6(c), the insets show two representative real-time spectra at the 763th and 800th roundtrips. Obviously, due to the different evolution rates of the pulsating process, the evolutions of the three solitons in the process of energy rise are asynchronized while the paces of energy decline are almost the same.
3.2. Double-periodic pulsating soliton
Mode-locked lasers generally present an important hysteresis with respect to the pumping power, which is applied in our experiment of generating double-periodic pulsations. We start from the single-periodic pulsation with intense energy oscillations, as shown in Fig. 2. Subsequently, we diminish the pump power slowly and tune the PCs finely, meanwhile, monitor the pulse evolution by real-time spectral measurement. Finally, we obtain a more complex pulsating dynamics at 173.3 mW, namely, double-periodic pulsating soliton. The corresponding output performances are shown in Fig. 7. Figure 7(a) presents the OSA-measured spectrum (black curve) and the average of 7006 consecutive single-shot spectra (red curve). To further illustrate the pulse evolution, Fig. 7(b) shows the real-time spectral evolution in three-dimensional (3D) format. Figures 7(c)–7(e) show the temporal evolution, the spectral evolution and the corresponding energy evolution within 7006 roundtrips, respectively. It can be observed that the double-periodic pulsation has almost all the features of the single-periodic pulsation described above, such as energy oscillation, bandwidth breathing, temporal shift, etc. The difference is that the pulse repeats itself every two oscillation intervals. In this sense, we call it a double-periodic pulsating soliton. The adjacent oscillations with unequal period exhibit different extrema of both energy and spectral bandwidth, and the larger modulation corresponds to the longer period. In addition, as shown in Figs. 7(b) and 7(e), the multiple peaks at the edges of the spectra are stronger during the oscillation with larger energy modulation.
Previous numerical results [7,10] demonstrated that the pulsating state could transform from single-periodic pulsation to double-periodic pulsation when one or several system parameters were changed. The maximal pulse energy during the pulsating period split into two and took one of the two alternative values every other oscillation. In 2004, the first experimental observation of double-periodic soliton pulsation was demonstrated in a dispersion-managed mode-locked fiber ring laser and verified by the periodic changes of the pulse energy in the oscilloscope traces . However, the measured pulse trains were far from adequate in describing the specific behaviors of the soliton pulsation because of the insufficient resolution and response speed of the oscilloscope. Herein, for the first time, the remarkable features of double-periodic soliton pulsation are characterized in terms of temporal evolution and spectral evolution. The transformation of pulsation state is attributed to the modification of the nonlinear transmission function in the cavity caused by adjusting the pumping power and polarization state. This is qualitatively consistent with the period doubling bifurcation phenomenon caused by the change of system parameters in the theoretical predictions . Further, we demonstrate experimentally that the minimum energy also bifurcates in the case of double-periodic pulsation. In addition, the period-doubling phenomena for multi-soliton are also observed at higher power levels taking advantage of the hysteresis. One of these cases is shown in Fig. 8, the mode-locked fiber laser operates in double-soliton double-periodic pulsation regime when the pump power is 210.7 mW. In a word, our experiments will enhance a more comprehensive understanding of the double-periodic pulsating soliton.
3.3. Soliton explosion
By finely tuning the PCs, we observe that the pulsating soliton suffers erupting instabilities and the operation mode of the fiber laser switches to soliton explosion. Figure 9 summarizes the performances of the soliton explosion at the pump power of 168.2 mW. Figure 9(a) shows the OSA-measured spectrum (black curve) and the average of 7006 consecutive single-shot spectra (red curve). The real-time evolution of the optical spectrum is reflected in Figs. 9(b) and 9(c). Figure 9(d) provides the pulse energy evolution corresponding to Fig. 9(c). We can see that the erupting soliton evolution starts from a relatively stable mode-locked state. After a while, the spectrum experiences a sharp drop of the bandwidth followed by a dramatic growth, and very soon, cracks into pieces like an explosion. These completely chaotic, but well-localized spectral structures restore the original relatively stable mode-locked profile after about 100-roundtrip time, presenting analogies with the “cooling process” in real explosions.
To illustrate the explosion behavior of the pulses in more detail, the typical spectra over an explosion period are presented in Fig. 10 (see Visualization 2). As can been seen, the exploding evolution seems similar to the pure pulsating process, the pulse is also characterized as a spectral bandwidth breathing process. However, the amplitude of the pulse energy oscillation is much larger than that of pure pulsation. Correspondingly, the spectral multiple peaks caused by spectral phase disturbance are stronger. In the later stage of energy growth, the overdriven nonlinear effect leads to an abrupt spectral collapse process , as shown in Figs. 10(e)–10(g). Subsequently, a new soliton emerges from the fragments of the burst with the dissipation of energy. Due to the energy oscillation, the intracavity nonlinear effect level oscillates in the interval between the stationary and chaotic regimes, resulting in pulse breaking at a periodic manner. The soliton explosion is a transition state between the pure pulsation and the chaos in our case.
In this paper, we have not detected the temporal evolution of the soliton explosion, because the corresponding results are included in the past work . Yu et al. reported a similar soliton explosion and provided the temporal evolution, which is complementary with our experiments.
In addition, because of the resolution and limited memory of our oscilloscope, only 7006 consecutive roundtrips of the mode-locked spectra could be recorded. In this case, less than two explosions could be recorded each time. Actually, the explosions process repeats successively, although the distance between “eruptions” fluctuates slightly.
The pulsating behavior of solitons in dissipative systems has long been a striking phenomenon. Although it has been theoretically demonstrated that pulsating solutions exist in a huge range of the laser parameter space [7,10,13], indeed, these dynamics need strict and unquantifiable cavity parameter condition in experiments. It is difficult to characterize the evolution of the pulse shape, for the pulses generated by mode-locked fiber lasers generally has extremely short durations. As a result, we focus on the energy and spectral evolutions and commit to building a mode-locked fiber laser with high energy and broad bandwidth. Further, the approximately rectangular spectra in normal dispersion regime make the pulsating states more recognizable. Therefore, special cavity designs with normal dispersion and weak nonlinear effect are employed .
In this paper, we report the experimental observations of three typical types of soliton pulsations. Recall that there is a direct relation between the physical parameters of the mode-locked fiber laser and the coefficients of the CGLE . To get a thorough understanding of the soliton pulsations, several theoretical works should be referred together [7,10,12,13]. Based on the CGLE, various kinds of pulsating solutions have been predicted theoretically, each of them exists in a particular region of the parameter space. By adjusting one or more parameters continuously and crossing the borders of these regions, the transitions between the three major classes of solutions (stationary, pulsating, and chaotic solutions) are predicted . However, in experiment, some parameter regions cannot be reached for the difficulty to adjust the specific parameters as precisely and continuously as in numerical calculation . Therefore, not all pulsating dynamics predicted numerically can be easily observed in experiment.
In our work, the changes of pump power and polarization settings correspond to the adjustments of the coefficients in CGLE to some extent . Herein, we qualitatively discuss the influencing factors of pulsating state. As depicted in Fig. 4, once single-periodic pulsating state is achieved at a given polarization setting, adjusting the pump power affects the pulsation intensity but does not change the type of pulsation. There is a similar evolution in numerical simulations when changing the quintic nonlinear gain coefficient . At slightly lower power levels, the transformations from single-periodic pulsation to double-periodic pulsation or soliton explosion can be achieved by finely tuning the PCs. But more generally, the laser enters the CW or noise-like operating regimes. In other words, we demonstrate that the route from pulsations to chaos can be through period-doubling bifurcation, periodic chaos, or an abrupt transition, which agree qualitatively well with the simulation results [7,10]. Further, we note that the three obvious differences among the pulsations are the intensity of the multiple peaks, the modulation depth of the energy and spectral bandwidth. They indicate that the soliton pulsations are sensitive to the dissipative processes such as the nonlinear effect and spectral filtering.
In summary, by adjusting the pumping power and polarization state, we have experimentally observed three typical types of soliton pulsations in a normal-dispersion mode-locked fiber laser via the DFT technique: single-periodic pulsating soliton, double-periodic pulsating soliton and soliton explosion. Despite the common features such as energy oscillation, bandwidth breathing and temporal shift, each kind of these pulsations has its distinctive features. The pulse is repeated every two oscillation periods for double-periodic pulsating soliton. When it comes to soliton explosion, because of the intermittent overdriven nonlinear effect induced by the extreme energy oscillation, the spectrum cracks into pieces at a periodic manner. Surely, the limited examples presented in this work do not represent the whole complexity of pulsating dynamics. However, we believe that these results will contribute to a more comprehensive understanding of the soliton pulsating phenomena, and expect that this work can stimulate further research both in experimental investigations and theoretical analyses.
National Key Research and Development Program of China (2018YFB0504401, 2018YFB0703501); National Natural Science Foundation of China (NSFC) (61835006, 11674177, 61775107); Tianjin Natural Science Foundation (16JCZDJC31000); 111 Project (B16027).
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