1. Introduction

The term soliton was coined in 1965 to describe localized solutions of integrable nonlinear partial differential equations including the Korteveg de Vries and nonlinear Schrödinger equations. Such solutions are remarkable because they remain intact during interactions. Since then, solitons have been discovered and investigated in various branches of physics. In optics, the existence of solitons was first predicted and experimentally observed in optical fibers, which could be interpreted as a result of an interplay between dispersive and nonlinear Kerr effects [1,2]. A dramatic turning point occurred when it was found that solitary wave solutions did exist in a wide range of nonintegrable dissipative systems. In contrast to classical solitons, localized structures in non-conservative systems require continuous exchange of energy with external sources. New paradigm was soon introduced and self-organized solutions in nonlinear systems far from equilibrium were referred to as dissipative solitons (DSs) [3]. The basic equation describing the evolution of dissipative systems is the complex cubic-quintic Ginzburg-Landau equation (CQGLE), which takes into account the dispersion and nonlinearity as well as the gain and loss. The characteristics of DSs are predetermined by a given set of system parameters rather than by the initial conditions, which are markedly different from those of conservative solitons. Examples of DSs in optics include ultrashort pulses circulating in fiber lasers. Indeed, passively mode-locked fiber lasers not only allow the generation of well-shaped optical pulses, but also have been regarded as convenient test-beds for exploring complex soliton dynamics [4–13]. It is usually assumed that a mode-locked fiber laser operates in a period-1 state [14] when the output pulse repeats itself only after one roundtrip, i.e., the pulse monitored at a fixed point of the cavity possesses a stationary shape. In fact, the generation of stable DSs requires a careful adjustment on the laser parameters. More generally, the output pulse varies its temporal profile after each roundtrip and exhibits complicated nonlinear dynamics in the time domain. Soliton pulsations are among the most fascinating non-stationary phenomena manifested in dissipative optical systems [15]. In this regime, the pulse experiences periodic changes in its shape, amplitude, and width. This phenomenon was originally labeled as novel localized solutions of the CQGLE, and a number of numerical studies have subsequently been completed within this framework to analyze the influence of different system parameters on the pulsating behaviors [16–19]. In 2004, Soto-Crespo et al. reported the first experimental observation of soliton pulsations in a dispersion-managed mode-locked fiber laser [20]. Nevertheless, due to the poor frame rate of conventional diagnostic method, the occurrence of pulsating solitons was only identified by the periodic energy modulation of output pulse train on an oscilloscope.

In recent years, dispersive Fourier transform (DFT) has attracted tremendous interest in the field of ultrafast optics [21–28]. As an emerging spectroscopic technique, it can overcome the scan-speed limitation of traditional optical spectrum analyzers (OSA) and enable the single-shot spectral measurement of ultrashort optical pulses. This technique is implemented by stretching a pulse train in an optical fiber with sufficient dispersion accumulation, which effectively maps the spectrum of each ultrashort pulse into a time-domain waveform whose intensity mimics its spectrum [29]. By virtue of this powerful tool, plentiful experimental studies on soliton pulsations have been conducted [30–42]. In 2018, Du et al. reported the first observation of pulsating DSs in a normal-dispersion mode-locked fiber laser [31]. Later on, the feasibility of pulsating solitons with chaotic behaviors was confirmed by Wei et al. in a passively mode-locked fiber laser operating at anomalous-dispersion regime [32]. In 2020, a novel pulsating behavior was unveiled by Liu et al. in an ultrafast fiber laser. It features that the soliton undergoes periodic intensity variation but with invariable pulse energy during the pulsation process, which is referred to as invisible soliton pulsations [37]. In addition, Wang et al. revealed the phenomenon of asynchronous pulsations in the case of multi-soliton state, confirming that each soliton may evolve periodically in different ways [38]. Attempts have also been made to study pulsating vector solitons in polarization-insensitive fiber lasers [39–41]. In 2020, Luo et al. reported on the experimental observation of pulsating group-velocity-locked vector solitons in a nonlinear-multimode-interference based fiber laser [39]. Since then, some works have shown the existence of other pulsating vector solitons in mode-locked fiber lasers in the absence of polarization-discrimination devices [40,41]. Very recently, period-doubled solitons (PDSs) with additional pulse-energy modulations were presented by Chen et al. in a hybrid mode-locked fiber laser [42]. Since an in-line polarizer was adopted to implement the nonlinear-polarization-rotation technique, PDSs obtained in this case were scalar ones. Therefore, there is a strong motivation to explore the vector features of pulsating PDSs. Moreover, the aforementioned work does not present the whole complexity of possible pulsating behaviors. A natural question arises as to whether multiple-period intensity modulation can be involved in the pulsation process.

In this paper, we report the real-time observation of pulsating period-doubled vector solitons in a polarization-insensitive mode-locked fiber laser by utilizing the DFT technique. PDSs with various pulsating behaviors could be observed in the laser through proper adjustment of an intra-cavity polarization controller (PC). The single-periodic PDS pulsation was achieved, which has a pulsating period ranging from tens to hundreds of cavity roundtrips. More complicated PDS pulsation with three combined intensity-modulation periods (i.e., double-periodic PDS pulsation) was formed. The vector features of these pulsating PDSs were analyzed based on polarization-resolved single-shot spectra.

2. Experimental setup

The upper part of Fig. 1 shows the schematic of our proposed mode-locked fiber laser based on single-wall carbon nanotubes (SWNTs). It has a ring cavity configuration. The laser contained a 0.9-m erbium-doped fiber (EDF, Liekki Er80-8/125) and a 11.7-m single-mode fiber (SMF) with dispersion parameters of -20 ps²/km and −23 ps²/km, respectively. The total length of the cavity was around 12.6 m and the net-cavity dispersion was estimated to be −0.3 ps². The gain medium (i.e., EDF) was pumped by a 500-mW laser diode (LD) via a 980/1550-nm wavelength division multiplexer (WDM). A 10:90 output coupler (OC) was utilized to tap 10% of the laser power for measurement, while an in-line PC was engaged to alter the net-cavity birefringence. In addition, a polarization-insensitive isolator (ISO) was applied to force the unidirectional propagation of the lasing pulse. The as-prepared SWNTs saturable absorber was fused inside the cavity to trigger the mode-locking operation. An optical band-pass filter was located straight after the ISO, which has a 3-dB bandwidth of ∼1.2-nm. The output pulses were simultaneously monitored by a commercial OSA (Yokogawa AQ6370C), a 500-MHz digital oscilloscope (Rigol DS4054) together with a 1.2-GHz photodetector (Thorlabs DET01CFC), and a 50-GHz radio frequency (RF) spectrum analyzer (Rohde & Schwarz FSU50). As illustrated in the lower part of Fig. 1, a fiber-based polarization beam splitter (PBS) was connected to the laser output port, which was utilized to separate the two orthogonal polarization components of vector solitons. Besides, an external PC was placed between the OC and PBS to compensate the linear birefringence introduced by the pigtails of these devices. In order to realize polarization-resolved real-time spectral measurements, the two outputs of the PBS subsequently counter-propagated through a spool of 24-km dispersion-compensating fiber (DCF). This mapped the pulse spectra for the two polarized modes into the temporal domain, which allowed us to detect them via a high-speed electronic system comprising a 33-GHz real-time oscilloscope (100-GSa/s sampling rate, Tektronix DPO73304SX) and two identical 15-GHz photodetectors (HP 11982A). The dispersion coefficient of the DCF is ∼200 ps²/km. As a result, the spectral resolution is about 0.06 nm for our DFT configuration.

 

Fig. 1. Schematic of our proposed mode-locked fiber laser and polarization-resolved real-time spectral measurement system. PD1 and PD2, two identical photodetectors; OSC, high-speed electric oscilloscope.

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3. Results and discussion

At the pump power of 71 mW, the fiber laser operated at period-1 state and stationary solitons were generated in the cavity. Soliton pulsations could also be produced by adjusting the intra-cavity PC and detailed experimental results have been covered in our previous work [40]. The aim of this paper focuses on PDSs with additional pulse-energy modulations, as well as their vector features. Further increasing the pump power to 97 mW, a phenomenon of period-doubling bifurcations could occur by adjusting the intra-cavity PC, and the corresponding laser performance is presented in Fig. 2. Figure 2(a) illustrates the oscilloscope trace of output pulse train. The pulse-to-pulse separation is 60.6 ns, which matches well with the cavity length of 12.6 m. However, the intensity of soliton pulse is no longer uniform but alternates between two values, exhibiting a so-called period-2 state [43,44]. As shown in the blue curve of Fig. 2(b), the optical spectrum recorded with the OSA is centered at 1572.3 nm and the spectral bandwidth is about 0.7 nm. The central wavelength of output pulses was pre-fixed by the optical band-pass filter in the experiment. Figure 2(c) demonstrates the corresponding RF spectrum, which shows that the pulse repetition rate is 16.49 MHz. It should be noted that the repetition rate is determined by the cavity length rather than by the pump power, excluding the likelihood of passive Q-switching operation in the fiber laser. Moreover, there are two symmetric sidebands with a frequency offset of 8.24 MHz from the central peak. The new frequency component locates exactly at the half of the cavity repetition rate while its amplitude is intense enough, undoubtedly verifying the appearance of period-doubling phenomenon. The transient spectral evolution with consecutive cavity roundtrips is shown in Fig. 2(d), which manifests typical characteristics of PDSs [42]. Figures 2(e) and 2(f) depict the real-time spectra of output pulses versus the odd and even roundtrips before entering the PBS, respectively. It can be seen that each spectrum in the odd (even) roundtrips is nearly indistinguishable, and no evident spectral breathing behavior occurs. The average of 100 roundtrip-to-roundtrip spectra (red curve) shown in Fig. 2(b) is consistent with the steady-state spectrum measured by the OSA (blue curve), confirming the validity of our real-time spectral measurement system. Figure 2(g) displays the evolution of pulse energy for this case, which is achieved by piecewise integration of the power spectral density in Fig. 2(d). One can see that the output pulse energy takes one of the two alternative values every second roundtrips. All of these results indicate that the fiber laser operates in a pure period-2 state.

 

Fig. 2. Pure period-2 state. (a) Pulse train, (b) Optical spectrum measured by the OSA and the average of 100 transient spectra, (c) The corresponding RF spectrum, (d) 100 roundtrip-to-roundtrip spectra, (e)-() Real-time spectra of output pulses versus the odd and even roundtrips before entering the PBS, and (g) Pulse-energy evolution with cavity roundtrips.

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By fine adjusting the intra-cavity PC while fixing the pump power (i.e., 97 mW), pulsating PDSs could be observed in the laser, as presented in Fig. 3. Owing to the limitation of scan rate, the OSA-recorded spectrum is invariant with time, which is akin to that shown in Fig. 2(b). By virtue of the DFT technique, the single-shot spectra of output pulses over the consecutive odd and even roundtrips before entering the PBS are mapped in Figs. 3(a) and 3(b), respectively. One can see that the two sets of shot-to-shot spectra actually exhibit the same evolution characteristics. The single-shot spectrum in the odd (even) roundtrips oscillates periodically, which confirms the occurrence of soliton pulsation with a period being equal to around 50 roundtrips. Furthermore, the single-shot spectral profile evolves symmetrically in the process of pulsation. Figure 3(c) depicts the corresponding evolution of pulse energy. In contrast to the case in Fig. 2(f) where the pulse energy alternates between the two values, a diversity of energy values can be obtained here instead of two fixed amounts. After each roundtrip the pulse energy jumps from a high (low) value to a low (high) one as it should be for a period-2 state. In addition, the pulse energy in the odd (even) roundtrips changes periodically and the same pattern is repeated every ∼50 successive roundtrips. Thus, such pulsating behavior is referred to as single-periodic PDS pulsation.

 

Fig. 3. PDSs with a short-period pulsation. (a)-(b) Real-time spectra of output pulses over the odd and even roundtrips before entering the PBS, (c) The evolution of pulse energy with consecutive cavity roundtrips, (d)-(e) Polarization-resolved transient spectra in the odd roundtrips, (f)-(g) Polarization-resolved transient spectra in the even roundtrips, (h) Pulse-energy evolution of vertical-axis polarized mode with cavity roundtrips, and (i) Pulse-energy evolution of horizontal-axis polarized mode with cavity roundtrips.

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Figures 3(d) and 3(e) depict the polarization-resolved transient spectral evolution of pulsating PDSs in the odd roundtrips after passing through the PBS. It can be seen that the intensities of both the polarized modes change periodically with a single period of around 50 roundtrips. Surprisingly, the intensity-modulation behaviors of the two orthogonal polarization modes in the odd roundtrips are asynchronous, i.e., when one component has a maximal spectral intensity, the other always gets a minimal one. The real-time spectra of the two polarized components in the even roundtrips are displayed in Figs. 3(f) and 3(g), manifesting the same evolution characteristics as shown in Figs. 3(d) and 3(e). Such asynchronous phenomenon can be ascribed to the combination of the intensity modulation and polarization rotation of vector solitons [40]. To gain more insight, we plotted the roundtrip-to-roundtrip pulse-energy evolution of the two polarized components, as illustrated in Figs. 3(h) and 3(i). It can be seen that when the pulse energy of one polarization component alters from a high (low) value to a low (high) one after each roundtrip, the other component shows an opposite change in its pulse energy. This is an indication that we are dealing with polarization-rotation state locking to two times of the cavity roundtrip. It should be emphasized that this phenomenon is not caused by the intensity modulation of PDSs with a fixed orientation of polarization ellipse. On the other hand, it is found that the pulse energy of each polarized mode in the odd and even roundtrips separately oscillate with an anti-phase evolution trend, as shown in Figs. 3(h) and 3(i). These results imply that in addition to a polarization-ellipse rotation with a period being equal to two cavity roundtrips, there exists another slower polarization modulation. After every second roundtrips, the orientation of polarization ellipse is shifted by a small amount from the last one in this case. Furthermore, it was found that the polarization-evolution period is identical to that of the pulsation. The polarization-ellipse orientation and pulse energy of the pulsating PDSs return back to its own original value every ∼50 cavity roundtrips. Very recently, Li et al. experimentally demonstrated a novel type of pulsating vector soliton in an ultrafast fiber laser [41]. The polarization-ellipse orientation evolves progressively while maintaining a polarization-rotation period of two cavity roundtrips. It should be mentioned that they treated with the cases where vector solitons operate at period-1 state rather than period-2 state. In our present work, we first identify progressive polarization-evolution features of pulsating PDSs, to the best of our knowledge. It is shown that two periods (i.e., period-2 and ∼50 roundtrips) appear in combination during this single-periodic PDS pulsation. In addition, the underlying spatio-spectral dynamics of pulsating PDSs are unveiled by utilizing the DFT technique.

PDSs with a long-period pulse-energy modulation could also be observed in the laser. The shot-to-shot spectra of the two orthogonally polarized components over the odd and even roundtrips are demonstrated in Figs. 4(a)–4(d), which illustrates a longer period of pulsation compared with that shown in Figs. 3(d)–3(g). Moreover, the intensity-modulation behaviors of the two orthogonal polarization modes in the odd (even) roundtrips are still asynchronous, unambiguously confirming the rotation of polarization ellipse. The polarization state here recovers to its original one every ∼220 roundtrips, matching well with the pulsation period. Again, Figs. 4(e) and 4(f) present the corresponding pulse-energy evolution of the two polarization modes with successive cavity roundtrips. In contrast to Figs. 3(h) and 3(i) where the pulse energy undergoes a nearly sinusoidal modulation, the long-periodic pulsation here features a quick growth followed by a slow decrease in each oscillation period. However, these two cases all share the same evolution characteristics, as is evident from the experimental results. It indicates that the polarization state of single-periodic pulsating PDSs evolves with two combined modulation periods, which could be an intrinsic feature of pulsating period-doubled vector solitons in dissipative optical systems.

 

Fig. 4. PDSs with a long-period pulsation. (a)-(b) Polarization-resolved transient spectra in the odd roundtrips, (c)-(d) Polarization-resolved transient spectra in the even roundtrips, (e) Corresponding pulse-energy evolution of vertical-axis polarized mode with cavity roundtrips, and (f) Corresponding pulse-energy evolution of horizontal-axis polarized mode with cavity roundtrips.

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By further adjustment of the PC, a more complex pulsating pattern could be obtained in the fiber laser. As shown in Fig. 5(a), the single-shot spectrum of the output pulse in the odd roundtrips before entering the PBS oscillates with a short period, which is accompanied by a long-period intensity modulation. Figure 5(b) demonstrates the corresponding shot-to-shot spectra in the even roundtrips, illustrating the same pulsating behavior. The roundtrip-to-roundtrip evolution of pulse energy is mapped in Fig. 5(c). After each roundtrip the pulse energy alters from a high (low) value to a low (high) one as it should be for a period-2 state. Moreover, two energy-modulation periods (i.e., period-∼36 and ∼532 roundtrips) are involved in this process. Thus, such pulsating behavior is referred to as double-periodic PDS pulsation. Figures 5(d) and 5(e) present the polarization-resolved transient spectral evolution of the output pulses in the odd roundtrips after passing through the PBS. It can be seen that the intensities of both the polarized modes oscillate with a combination of short and long pulsating periods. Remarkably, the intensity-modulation behaviors of the two orthogonal polarization components in the odd roundtrips are still asynchronous. The polarization-resolved real-time spectra in the even roundtrips are mapped in Figs. 5(f) and 5(g), exhibiting the same evolution characteristics as shown in Figs. 5(d) and 5(e). Obviously, the anti-phase intensity modulation between the two polarization modes proves that an additional slower polarization modulation of pulsating period-doubled vector solitons exists indeed. For better clarity, we plotted the roundtrip-to-roundtrip pulse-energy evolution of the two polarization modes, as illustrated in Figs. 5(h) and 5(i). It is shown that the pulse energy of the polarization component along the vertical axis jumps from a high (low) value to a low (high) one after each roundtrip while the horizontal-axis polarization mode has an opposite variation in its pulse energy. This is an indication that the fast polarization-ellipse rotation is locked to twice of the cavity roundtrip time. In addition, the pulse energy of each polarization mode in the odd and even roundtrips separately oscillate with an anti-phase double-periodic energy-modulation behavior, as shown in Figs. 5(h) and 5(i). This is the first experimental observation of pulsating period-doubled vector solitons with three combined polarization-modulation periods (i.e., period-2, ∼36 and ∼532 roundtrips) involved in pulsating dynamics.

 

Fig. 5. PDSs with double-periodic pulsation. (a)-(b) Real-time spectra of output pulses over the odd and even roundtrips before entering the PBS, (c) The evolution of pulse energy with consecutive cavity roundtrips, (d)-(e) Polarization-resolved transient spectra in the odd roundtrips, (f)-(g) Polarization-resolved transient spectra in the even roundtrips, (h) Pulse-energy evolution of vertical-axis polarized mode with cavity roundtrips, and (i) Pulse-energy evolution of horizontal-axis polarized mode with cavity roundtrips.

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In our previous work [40], pulsating polarization-rotation-locked and quasi-polarization-locked vector solitons were demonstrated with the help of the polarization-resolved DFT spectral characterization. We also obtained double-periodic pulsation in the cavity. It was confirmed that pulsating solitons can also host a variety of complex polarization-evolution behaviors, generalizing previous observations made in the case of stationary vector solitons. Here, however, we treat with the cases where the vector solitons operate at period-2 state instead of period-1 state, which was realized by further increasing the pump power and adjusting the intra-cavity PC. Both single- and double-periodic PDS pulsations were achieved in the cavity. The experimental results indicate that these pulsating PDSs can be categorized as the polarization-rotation-locked vector solitons. However, pulsating quasi-polarization-locked PDSs were not obtained in our current laser cavity.

The period-doubling bifurcation is an intrinsic feature of passively mode-locked fiber lasers and its appearance is independent of the specific pulse profile [43,44]. The origin of this phenomenon may be attributed to the strong nonlinear interaction of mode-locked pulses with the cavity components [14]. Therefore, it is expected that when the pulse peak power becomes intense by increasing the pump power, the laser output could experience periodic intensity fluctuations. Specifically, the intensity of mode-locked pulse returns back to its original value after every two cavity roundtrips, exhibiting period-doubling bifurcation. Period-2 states with additional pulse-energy modulations have been theoretically predicted within the framework of the CQGLE [20]. It was shown that the appearance of new periodicity in the pulsation process is related to series of bifurcations while any number of pulsating periods can be involved with an appropriate selection of the cavity parameters. In our experiments, we observed PDSs with a variety of pulsating behaviors (single- and double-periodic ones) by tuning the PC, illustrating that the pulsating period depends strongly on the choice of system parameters. On the other hand, polarization-insensitive mode-locked fiber lasers potentially admit vector solitons. With suitable net-cavity birefringence values, coherent coupling between the two polarization components takes place. In the case of pulsating PDS operation regime, the polarization ellipse rotates with a period being equal to twice of cavity roundtrip time. Moreover, the intensity-modulation behaviors of the two orthogonal polarization modes in the odd (even) roundtrips are always asynchronous, confirming the additional slower polarization modulation. Consequently, the polarization-ellipse rotation of pulsating PDSs is modulated with a combination of two or more periods, which corresponds to that of the pulse-energy modulation. It should be noted that a limited number of examples shown in this work do not represent all possible cases. Considering a wealth of complex nonlinear dynamics hosting in dissipative systems, we speculate that other polarization evolution behaviors could also be observed under certain conditions. The underlying mechanism may be related to the variation of laser cavity parameters, which could alter the coherent coupling of the two orthogonal polarized modes. We are sure that our present work would facilitate a further study on this issue, as it is interesting from both theoretical and experimental points of view.

4. Conclusion

We have experimentally demonstrated different types of pulsating PDSs in a SWNT mode-locked fiber laser. Pulsating periods could vary from tens to hundreds of cavity roundtrips during single-periodic PDS pulsation, and three periods of intensity modulation were involved in the process of double-periodic PDS pulsation. The polarization-resolved real-time spectra of these pulsating period-doubled vector solitons has been recorded by virtue of the DFT method. It has been shown that the evolution of polarization state is accompanied by additional slower modulation periods in addition to the period of two cavity roundtrips. Our results deepen the understanding of multiple-period pulsating vector solitons in mode-locked fiber lasers.