Rapid progress in real-time spectroscopy uncovers the spatio-spectral scenarios of ultrashort pulses in dissipative systems. Varieties of transient soliton dynamics on different timescales have been revealed. Here, we report on an experimental observation of stationary and pulsating vector dissipative solitons in a nonlinear multimode interference (NL-MMI) based fiber laser with net normal dispersion. Polarization non-discrimination of the NL-MMI mode-locking facilitates the dissipative soliton trapping process. Two orthogonally polarized components are coupled together through oppositely shifting their central frequencies to form the group-velocity-locked vector dissipative solitons (GVLVDSs). Dispersive Fourier transform (DFT) based polarization resolved measurement enables insights into the transient polarization dynamics and the long-term evolution. Particularly, both stationary and pulsating GVLVDSs are obtained with appropriate parameter settings. It is found that the quasi-stationary pulsating manner is accompanied with recurrent spectral breathing and energy oscillation; the two orthogonally polarized components possess synchronous pulsating manners due to the cross-phase modulation induced trapping mechanism and the similar formation process. Additionally, chaotic pulsation is also captured in sense that the spectra cannot recover to their original profiles despite of the harmonic energy oscillation. All these findings can enhance our understanding towards soliton pulsation with the freedom of vectorial degree.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Passively mode-locked fiber lasers are of twofold significance for both extending the application scenarios and spreading the concept of dissipative solitons (DSs) [1,2]. Different from the conservative solitons formed by single balance of dispersion and nonlinearity in Hamiltonian systems, additional laser gain and loss jointly play roles in the formation and evolution processes of DSs . Governed by the Ginzburg-Landau equation (GLE), this composite balance brings about profound laser dynamics in not only the anomalous-dispersion regime but also the normal-dispersion regime [4,5]. In particular, spectral filtering effect gets involved in the formation of DSs with (net) normal dispersion, and the chirping characteristics allow for the delivery of ultrashort pulses with unprecedented energy . Since the pioneering reports on DSs  and gain-guided solitons  in 2006, the normal-dispersion mode-locking operation has become an intensive topic of research. Varieties of mode-locking techniques have been demonstrated for DS emission such as nonlinear polarization rotation (NPR) , semiconductor saturable absorber (SESA) , carbon nanotube (CNT)  and 2-D materials . Besides, nonlinear multimode interference paves a new way for passive mode-locking [13,14]. The nonlinear effect induced intensity modulation endows the geometry of single-mode fiber (SMF)-graded-index multimode fiber (GIMF)-SMF as an equivalent fast saturable absorber, which has been demonstrated for various pulses emission in different dispersion regimes [15–17]. The nonlinear multimode interference (NL-MMI) based passively mode-locked fiber laser also serves as an ideal platform for unveiling complex dynamics of ultrashort pulses.
From the theoretical aspect, localized solutions of GLE can be mainly categorized into three classes of stationary, pulsating and chaotic manners . The pulsating solitons are of equal importance to the stationary ones in terms of the common borders in the parameter space [19–21]. As a complex nonlinear phenomenon, soliton pulsation is generally verified by the periodic evolution of the spectral profile and pulse energy with respect to nanoseconds even microseconds timescales. Indeed, chaotic behaviors may be also involved in the pulsating structures, which indicates the evolution route to the chaos . Previous experimental studies of soliton pulsation are principally focused on the energy oscillation of pulses [23,24]. Recently, the time-stretch dispersive Fourier transform (TS-DFT) technique has been utilized for mapping the real-time spectral information into the time domain . Varieties of transient soliton dynamics on different timescales have been revealed, such as the birth of mode-locking , soliton explosion , formation and internal motions of soliton molecules  etc. This real-time spectroscopy technique also provides an optimal method to capture the transient evolution of the pulsating manners. Particularly, both pulsating solitons and pulsating DSs have been respectively studied in the anomalous/normal-dispersion regimes [27–30]. Consecutive recordings of the shot-to-shot spectra enable insight into the periodic spectral breathing even chaotic behaviors during soliton pulsation. These observations are meaningful for validating previous theoretical predictions and optimizing laser designing.
Beyond the scalar model of passively mode-locked fiber lasers, the vectorial nature develops the framework towards polarization dynamics of vector solitons under polarization non-discriminated mode-locking mechanisms . Governed by the coupled Ginzburg-Landau equation (CGLE), two orthogonally polarized components trap each other to form the multi-soliton structure through cross-phase modulation (XPM) . Hence the vector solitons are characterized by richer behaviors than their scalar counterparts. Apart from soliton trapping in the anomalous-dispersion regime, referring to polarization locked vector solitons , polarization rotating solitons  and group-velocity-locked vector solitons (GVLVSs) , DS trapping  has also been demonstrated in the normal-dispersion regime with different net birefringence. In contrast to the GVLVSs with two sets of Kelly sidebands, gradient edges of the rectangle spectrum are interpreted as the indicator of the group-velocity-locked vector dissipative solitons (GVLVDSs) . This spectral feature derives from the spectral superposition of the two polarized components with obvious central wavelength difference. The NL-MMI based equivalent saturable absorber is essentially a polarization non-discriminated mode-locking technique. Hereby, the NL-MMI based ultrafast lasing platform can also release the vectorial degree for ultrashort pulses. The investigations on vector solitons will enhance the understanding towards the rich polarization dynamics of this multi-soliton structure, as well as extend the potential applications in nano-optics and high-capacity fiber optic communications.
Indeed, the mergence of pulsating manners and vectorial nature will shed a new light on the transient ultrafast dynamics of multi-soliton complexes in polarization directions. In this paper, we experimentally investigate the stationary and pulsating vector DSs in a NL-MMI based fiber laser in normal-dispersion regime. Polarization non-discrimination of this equivalent saturable absorber facilitates the vectorial evolution from DSs to GVLVDSs. By utilizing the DFT based polarization resolved measurement, both the transient polarization dynamics and long-term evolution are recorded for the stationary and pulsating GVLVDSs. Additionally, the pulsating manners of the two orthogonally polarized components are also unfolded for the first time. This investigation will enrich the framework towards pulsation dynamics with the freedom of vectorial degree in dissipative systems.
2. Experiment setup
The schematic diagram of the all-fiber laser is depicted in Fig. 1. A segment of erbium-doped fiber (EDF, OFS EDF-80) is utilized to provide the optical gain, which is pumped by two 980-nm laser diodes (LDs) through a wavelength-division-multiplexer (WDM) and a WDM/TAP/ISO hybrid module. This hybrid module is also utilized to output the lasing pulses at the 10% reflection port and ensure the unidirectional operation. In order to introduce the NL-MMI based mode-locking mechanism, an SMF-GIMF-SMF geometry (shown in inset 1) based SA is inserted inside the ring-cavity oscillator. Two polarization controllers (PCs) are implemented at both sides to adjust the polarization states. The GIMF shown in inset 1 is ∼50 cm with anomalous dispersion of -0.028 ps2/m (@1550 nm). The EDF is ∼4.30 m with normal dispersion of + 0.061 ps2/m (@1550 nm). All the pigtails are SMF with overall length of 7.05 m and anomalous dispersion of -0.022 ps2/m (@1550 nm). The total cavity length is ∼ 11.85 m and the net dispersion is managed to + 0.093 ps2. Thus the fiber laser approaches into the (net) normal-dispersion regime. An optical spectrum analyzer (OSA YOKOGAWA, AQ6370C) is used to record the time-averaged optical spectra. A 33-GHz bandwidth oscilloscope (Agilent Technologies, DSA-X 93304Q) together with a fast photodiode (PD, bandwidth 12 GHz) is utilized to capture the pulse trains. Additionally, DFT based polarization resolved measurement paves access to the transient polarization dynamics of GVLVDSs as illustrated in inset 2. A fiber-based polarization beam splitter (PBS) is connected to the output port of the laser cavity to resolve the vector pulses. A 10 km SMF with anomalous dispersion of -230 ps2 is utilized to stretch the pulses, and the shot-to-shot polarization resolved spectra unveil the real scenarios of GVLVDSs.
3. Results and discussion
By increasing the pump power to 614.09 mW and appropriately adjusting the PCs, the stable self-started mode-locking is achieved in the normal-dispersion regime and the output power is 15.05 mW. The typical spectrum is presented in Fig. 2(a). The rectangular spectral shape indicates the emission of DSs. The increased in-band absorption introduced by the 4.3-m EDF imposes on the redshift of the central wavelength towards 1600 nm. To verify the stability of the NL-MMI based mode-locking operation, time-averaged DS spectra are recorded every minute for 250 mins. The 2D contour plot is illustrated in Fig. 2(b), where no obvious change is observed on the rectangle spectra. Through reducing the pump power to 240.81 mW, the fiber laser approaches the fundamental mode-locking regime due to the pump hysteresis effect with the output power of 4.21 mW. The corresponding oscilloscope traces are recorded in Fig. 2(c). The pulse interval is ∼57.67 ns, which agrees with the cavity length of the fiber laser. The uniform pulse intensity declares that no pulsating manners are observed in terms of the temporal view. In addition, the NL-MMI mode-locking is of polarization non-discrimination. Hereby, the emitted pulses are essentially vector DSs composed of two orthogonally polarized components. Different from the GVLVSs with a distinct indicator of the Kelly sidebands, the gradient spectral edges or some spectral spikes can help us to recognize the vector DSs in the normal-dispersion regime. Thus it can be seen that this NL-MMI based passively mode-locked fiber laser can serve as an ideal platform for investigating the vectorial nature of DSs.
Traditional polarization resolved measurement can display the time-averaged spectra of the two orthogonally polarized components. Increasing the pump power to 462.52 mW, the typical polarization resolved spectra are shown in Fig. 3(a) and the output power is around 9.5 mW. Due to the absence of the Kelly sidebands, the two spectral spikes can help us to resolve the total spectrum onto the vertical and horizontal axes by adjusting the PC outside the laser cavity. In particular, the two orthogonally polarized components possess different central wavelengths and spectral shapes. Their spectra are not entirely overlapped. The 3-dB bandwidths of the vertical and horizontal spectra are 5.86 nm and 7.92 nm, respectively. The total spectrum is the superposition of these two spectra, thus inducing the gradient spectral edges or the aforementioned spectral spikes. Accordingly, the corresponding pulse trains are of uniform pulse intensity as depicted in Fig. 3(b)–3(d) and the two orthogonally polarized components own the different output power of 1.80 mW and 5.20 mW. To gain insight into the real-time spectral information, DFT based polarization resolved measurement is also utilized for the GVLVDSs. Figures 3(e)–3(g) present the 2D contour plots of the shot-to-shot spectra before and after the polarization resolved measurements with 1735 consecutive roundtrips. The corresponding single-shot spectra of the first roundtrip are illustrated as the insets in Fig. 3(e)-(g), which are highly in accordance with the polarization resolved time-averaged spectra, as depicted in Fig. 3(a). It is found that no obvious variations of the spectral profiles are observed for the total and the polarization resolved shot-to-shot spectra; the intensities are uniform along roundtrips. Hence, the two orthogonally polarized components are stably trapped together to form GVLVDSs with polarization locking state. Meanwhile, the formed vectorial complexes are stationary GVLVDSs without pulsating manners in terms of the mainly invariable spectral profile.
Decreasing the pump power to 383.47 mW and properly adjusting the PCs, pulsating vector DSs are also observed in this fiber laser with the output power of 8.42 mW. Figure 4(a) presents the time-averaged spectra before and after the polarization resolved measurement. The total spectrum and the polarization resolved spectra are all characterized by the similar features to the stationary ones. Obvious central wavelength shift of the two orthogonally polarized components manifests the generation of GVLVDSs. By using the polarization resolved measurement, two orthogonally polarized components are obtained. The corresponding output powers are measured as 1.3 mW and 4.2 mW. Furthermore, the DFT process is introduced to illustrate the real-time pulsating manners. It is noted that, in these multiple-pulse states, all pulses are of similar pulsating manners and we chose one pulse as a representative to illustrate the soliton dynamics. Accordingly, the 2D contour plots of the shot-to-shot total spectra with 1735 consecutive roundtrips are respectively displayed in Fig. 4(b)–4(d). It is found that the spectra of the GVLVDSs pulsate periodically along roundtrips. The energy and bandwidth variations versus to roundtrips of the total spectra are extracted in Fig. 4(e) and 4(f). Both the spectral profile and pulse energy oscillate with a period of ∼280 roundtrips, corresponding to 16.15 µs. The oscillations are almost harmonic with small changing amounts. This pulsating manner is regarded as a quasi-stationary state considering that the pulse can recover its original spectrum after specific roundtrips. Particularly, the recurrent broadening of the spectral profile principally derives from the enhanced self-phase modulation (SPM) induced by the increasing pulse energy. Inversely, the intra-cavity filtering effect will clamp the spectral profile to the minimal bandwidth. Figures 4(g)–4(i) respectively present the single-shot spectra of the GVLVDSs and two polarized components at roundtrip 730, 800, and 870. The spectral oscillations are restricted in a small range. This slightly pulsation state is just nearby the bifurcation point. By finely adjusting the settings, the pulsating GVLVDSs may approach into the stationary states or other severer pulsating states with large diverging values of the energy and spectral profile. Additionally, the 2D contour plots of the polarization resolved shot-to-shot spectra indicate that the two orthogonally polarized components synchronously inherit the pulsating manners. They are characterized by both oscillating energy and breathing spectral profile with the identical period of ∼280 roundtrips.
By finely tuning the PCs under the pump power of 280.27 mW, another mode-locking state is obtained with the output power of 3.42 mW. Figure 5(a) depicts the time-averaged total spectrum and polarization resolved spectra characterized by the spectral features of GVLVDSs. The short-wavelength edges become gradient, implying the presence of a severer pulsating manner. Noted that the long-wavelength edges of the spectra remain steep. It should be attributed to the spectral clamping effect induced by the limited gain bandwidth beyond 1600 nm. All these spectra are sharply cut off near the gain edge of the EDF. Corresponding long-term pulses trains without DFT are presented in Fig. 5(b). Recurrent evolution of the pulse intensities is clearly observed, which validates the pulsating manners of the GVLVDSs and their two orthogonally polarized components. Two polarization resolved components show the output power of 0.78 mW and 2.41 mW. The pulsation period is identical, while the oscillating amounts of the pulse intensities are quite different. After DFT, it is found that the real-time spectral profiles are no longer uniform and randomly vary along roundtrips. We average the total and polarization resolved shot-to-shot spectra as illustrated in Fig. 5(c)–5(e). They all agree well with the time-averaged spectra recorded by the OSA. In order to get insight into the transient evolution of the spectral profiles. 3D contour plots of the shot-to-shot spectra are respectively displayed in Fig. 5(f)–5(h). In particular, both recurrent pulsating manners and stochastic evolution of the real-time spectra are captured. The twofold spectral evolutions are seemingly independent. One derives from the periodic DS pulsation; the other one is introduced by the spectral instability of the DSs. The insets show the extracted energy oscillation versus roundtrips. Although the spectra of the pulsating GVLVDSs and the two polarized components cannot recover to their original profiles, they are still characterized by harmonic energy oscillations with the identical period.
To our best knowledge, it is the first time to investigate the pulsating manners of DSs in the vectorial model of fiber lasers. Previous works demonstrated that pulsating manners of DSs can transform the steep spectral edges into gradient ones, which is interpreted as a distinct indicator of the pulsating DSs . However, it seems inconspicuous for the observations depicted in Fig. 4. The time-averaged spectrum of the pulsating GVLVDSs still possesses steep edges. We deem that this phenomenon should be ascribed to the small oscillating amount of the spectral profiles. The slight spectral differences of the pulsating pulses will not notably reshape the edges in the time-averaged spectra. Furthermore, recurrent oscillation in the shot-to-shot spectra becomes more complicated for the two orthogonally polarized components. In our case, polarization locking state of the GVLVDSs excludes the polarization evolution induced spectral oscillation of these two components. Especially, polarization rotation of the GVLVDSs can also bring about similar shot-to-shot spectral evolution with respect to the pulsating manners. The observed spectral oscillation may be the superposition of both the pulsating manners and the polarization rotation. Moreover, the synchronization of the pulsating manners is related to the soliton trapping and formation mechanisms. XPM can balance the polarization-mode dispersion. For the polarization locked GVLVDSs, the fixed parameter settings dominate the similar pulsating manners in spite of different oscillating amounts for the pulse energy and the spectral profile. Hereby, the mergence of pulsating manners and vectorial nature spread an intriguing scenario towards the transient ultrafast dynamics of multi-soliton complexes.
In summary, we investigate the mergence of pulsating manners and vectorial nature of DSs in a NL-MMI based fiber laser with net normal dispersion. The geometry of SMF-GIMF-SMF is utilized to initialize the mode-locking, as well as facilitate the vectorial evolution from DSs to GVLVDSs. DFT based polarization resolved measurement enables the access to the real-time scenarios of the polarization dynamics and long-term evolution. Particularly, both stationary and pulsating GVLVDSs are obtained with appropriate parameter settings. The quasi-stationary pulsating manner is found nearby the bifurcation point with recurrent spectral breathing and energy oscillation. It is found that the two orthogonally polarized components are of synchronous pulsating manners due to the XPM induced trapping mechanism and the similar formation process. Additionally, pulsating manners with chaotic spectral evolution are also captured in sense that the spectra cannot recover to their original profiles despite of the harmonic energy oscillation. All these findings can enrich the research framework towards pulsating manners of DSs with the freedom of vectorial degree.
Ministry of Education - Singapore (MOE2019-T1-001-111); National Natural Science Foundation of China (61775067, 61775072); Science Fund for Creative Research Groups of the Nature Science Foundation of Hubei (2018CFA004); Major Projects of Technical Innovation of Hubei (2018AAA040); Fundamental Research Funds for the Central Universities (HUST2017KFXKJC002); Science and Technology Program of Shenzhen, China (JCYJ20160531194407693).
The authors declare no conflicts of interest.
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