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Abstract
Dissipative solitons (DSs), due to the complex interplay among dispersion, nonlinear, gain and loss, illustrate abundant nonlinear dynamics behaviors. Especially, dispersion plays an important role in the research of DS dynamics in ultrafast fiber lasers. Previous studies have mainly focused on the effect of even-order dispersion, i.e., group velocity dispersion (GVD) and fourth-order dispersion. In fact, odd-order dispersions, such as third-order dispersion (TOD), also significantly influences the dynamics of DSs. However, due to the lack of dispersion engineering tools, few experimental researches in this domain have been reported. In this work, by employing a pulse shaper in ultrafast fiber laser, an in-depth exploration of the DS dynamics influenced by TOD was conducted. With the increase of TOD value, the stable single DS undergoes a splitting into two solitons and then enters explosion state, and ultimately evolves into a chaotic state. The laser operation state is correlated to dispersion profile, which could be controlled by TOD. Here, the positive dispersion at long-wavelength side will be gradually shifted to negative dispersion by increasing the TOD, where soliton effect will drive the transitions. These findings offer valuable insights into the nonlinear dynamics of ultrafast lasers and may also foster applications involving higher-order dispersion.
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1. Introduction
Optical solitons, serving as localized structures in nonlinear optical systems, have extensive applications in various fields, such as Bose-Einstein condensates [1], hydrodynamics [2], astronomy [3], and photonics [4]. Consequently, researchers have shown great interests in solitons over the past decades. In general, there are two primary categories of solitons: conservative [5,6] and dissipative solitons (DSs) [4,7,8]. Conservative solitons are characterized by minimal losses and the absence of external energy infusion. This results from an equilibrium finely tuned between nonlinearity and dispersion/diffraction. In contrast, DSs emerge from an intricate balance involving dispersion, nonlinearity, energy supply, and inherent losses within the physical system. Passively mode-locked fiber lasers have been confirmed as optimal platforms for studying nonlinear DSs dynamics. Within this context, numerous periodic or non-periodic processes are revealed, including soliton pulsations [9–11], soliton explosions [9,12–15], noise-like pulses [16–18] and multi-soliton states [19–22].
As mentioned, it is evident that dispersion plays an essential role in the research of DS dynamics. So far, considerable efforts have been dedicated to exploring the soliton nonlinear dynamics and improving laser performance through dispersion engineering. In 1993, Nelson et al. found that precise management of group velocity dispersion (GVD) in laser cavities could be beneficial for yielding high-energy pulses with highly linear chirp, which were termed as stretched pulses [23]. Later, it is found that the passive mode locking can be even achieved in an all-normal dispersion fiber laser, showing great potential in generating high-energy ultrashort pulses, namely DS [24,25]. One of the most interesting phenomena in fiber lasers, soliton explosions showed different characteristics when GVD was manipulated [26,27]. Moreover, pure-quartic solitons have been realized experimentally and numerically by introducing negative fourth-order dispersion to balance with positive Kerr nonlinearity [28]. In 2020, Runge et al. reported the first pure-quartic soliton generation in passively mode-locked fiber laser by engineering cavity dispersion with an intracavity pulse shaper [29]. The results indicate that pure-quartic solitons have potential on generating pulses with shorter duration, higher energy and flatter spectral envelope.
In addition to even-order dispersion, odd-order dispersion, such as third-order dispersion (TOD), has been demonstrated to greatly influence soliton dynamics. Initially, the effect of TOD on solitons was theoretically studied in fiber propagation. These studies revealed that TOD induces dispersive wave radiation, pulse-shape distortion and spectral asymmetry [30–33]. In more intricate systems, particularly in ultrafast lasers, researchers discovered that TOD also has great influence on soliton performance and dynamics [34–38]. V. L. Kalashnikov et al. reported that TOD could either stabilize or destabilize the pulse depending on the amount of GVD [37]. Subsequent studies demonstrated that TOD also has a crucial impact on bound soliton behavior [39,40], soliton explosion characteristics [41–43], and rogue wave generation [44,45]. Despite the intriguing DS dynamics under different TOD conditions, the lack of reliable TOD engineering tools has limited experimental investigations. So far, most researches have focused on numerical simulations, with only a few experimental results reported. In recent years, pulse shapers have emerged as powerful tools for intra-cavity dispersion control. They offer programmability and operational simplicity, allowing for dynamic manipulation of dispersion at arbitrary orders, including GVD, TOD, and higher orders [27,29,46]. Therefore, it would be both possible and interesting to experimentally investigate the effect of TOD on DS dynamics in ultrafast fiber lasers.
In this work, we experimentally observe the evolution of DS dynamics in an ultrafast fiber laser by dynamically adjusting TOD through a pulse shaper. The stable DS state could be obtained by setting suitable GVD parameters. With the increase of TOD values, the stable single DS splits into two DSs, then enters soliton explosion regime, and finally becomes chaotic state. The transitions among these states are related to dispersion changes at the long wavelength side induced by TOD. These findings provide a visual understanding of how TOD influences the DS dynamics and will contribute to exploring nonlinear soliton dynamics and advancing ultrafast laser technology.
2. Experimental setup
Figure 1 presents the experimental setup of the ultrafast fiber laser. The setup includes a 15 m erbium-doped fiber (EDF) with a dispersion parameter of -17.3 ps/km/nm, along with a 15 m standard single-mode fiber (SMF). The net cavity dispersion, in the absence of dispersion compensation, is approximately zero. A polarization controller (PC) is incorporated to manipulate the polarization states of the propagating light, and passive mode-locking is achieved using a commercial saturable absorber (Batop, SA-1550-35-2 ps). A polarization-independent optical integrated module (PI-OIM) is employed to fulfill the functionalities of a polarization-independent isolator, a wavelength division multiplexer, and a 10:90 output coupler. The laser output is concurrently monitored by an optical spectrum analyzer (OSA) and a high-speed real-time oscilloscope (Tektronix DSA-70804, 8 GHz) equipped with a photodetector (Newport 818-BB-35F, 12.5 GHz). Additionally, for real-time observation of spectral dynamics, a ∼15 km SMF (Corning SMF-28e) is inserted between the oscilloscope and the output port, enabling the dispersive Fourier transform (DFT) technique with the spectral resolution of 0.49 nm [47,48].
Here, the programmable pulse shaper (Finisar, Waveshaper 4000A) is integrated for flexible adjustment of GVD and TOD [29]. Based on the high-resolution liquid crystal on silicon (LCoS) technology, the pulse shaper could arbitrarily control the amplitude and phase characteristics of the incident light. The pulse shaper introduces an insertion loss of 3 dB and the group delay dispersion of the pulse shaper could tune from -127.5 to 127.5 ps2 at 1.55 µm. In this work, by adjusting the voltage on each pixel of LCoS chip to create phase masks with different frequency components, we can dynamically adjust each order of intracavity dispersion. A phase profile imposed on the propagating pulse can be described as:
3. Experimental results
The fiber laser achieves passive mode-locking at the pump power of 35 mW when the pulse shaper does not provide dispersion compensation. Nevertheless, no matter how the pump power varies, the laser consistently operates in a noise-like state. This is because the long SMF with anomalous dispersion in the laser system easily transforms dispersive waves or background noise into noise-like pulses [17]. Hence, to attain stable DS, we adjust the intra-cavity GVD through a pulse shaper. Note that the laser operates in stable single DS state when the GVD is set to the range of 7 to 8.2 ps2/km. By setting the GVD value at 7 ps2/km, the stable DS state is illustrated in Fig. 2. In Fig. 2(a), the spectrum reveals the distinctive rectangular spectral profile of the DSs, which is a typical characteristic of the ultrafast fiber laser with net-normal dispersion. Figure 2(b) presents pulse train with an interval of ∼145 ns. Figure 2(c) displays the autocorrelation trace, indicating the pulse width of ∼18.98 ps. Additionally, the radio-frequency (RF) spectrum, as shown in Fig. 2(d), centers at 6.898 MHz with a signal-to-noise ratio of 78 dB. The pulse train and shot-to-shot spectra over 500 roundtrips (RTs) with consistent intensity and profile are shown in Figs. 2(e) and (f), respectively. These results confirm the high stability of the DSs.
Fig. 2. Stable single DS regime. (a) Spectrum; (b) Pulse train; (c) Autocorrelation trace; (d) RF spectrum; (e) Temporal evolution and (f) Spectral evolution over 500 RTs. Inset: average spectrum (yellow curve).
In order to investigate the impact of TOD on DS dynamics in ultrafast fiber laser, we adjust the intra-cavity TOD using the pulse shaper while the other cavity parameters are fixed. The laser operates in a stable single DS state if TOD value is set to below -0.8 ps3/km, and transits to the dual-DSs regime when TOD is set to the value between -0.8 to -1.2 ps3/km. At the TOD of -1.2 ps3/km, the dual-DS state is shown in Fig. 3. From Fig. 3(b), we observed a slant at the top of the rectangular spectrum, which is caused by the blue shift in spectral domain induced by the negative TOD [31]. As evidence of the blue shift, the statistical central positions of spectra in Figs. 2(a) and 3(b) are 1557.97 and 1556.95 nm, respectively [18]. Compared to Fig. 2(a), despite the spectral blue shift, new longer wavelength components emerge in Fig. 3(b). As depicted in Fig. 3(a), the dispersion profile of pulse shaper is characterized by an upward-opening symmetrical parabola with the introduction of only GVD (black curve). Following the incorporation of TOD of -1.2 ps3/km (red curve), the slope of the dispersion profile at long wavelengths decreases. This indicates a gradual shift of long wavelengths from normal dispersion towards zero dispersion. It is worth noting that in Fig. 3(a), only the dispersion profile provided by the pulse shaper is presented, without considering the other fibers and components in the cavity. It is enough to qualitatively demonstrate the trend of laser’s dispersion profile with the addition of TOD. The appearance of new longer wavelength components and dispersion shift would introduce soliton effects and results in the soliton splitting [33]. In addition, the bandwidth is a little narrower than that in Fig. 2(a), which could be attributed to the decreased pulse energy after splitting and reduced self-phase modulation. The pulse trains are illustrated in Fig. 3(c), and the temporal interval between the two solitons is close to 11 ns. As tested for several times, the laser could evolve into the dual-DSs regime with separations of ∼11 ns once the TOD was set to -1.2 ps3/km. Figures 3(d), (e) and (f) display the temporal and spectral evolution of the two DSs over 500 RTs, respectively. The averaged DFT spectra, depicted by the red curve in Figs 3(e) and 3(f), demonstrate agreement with the spectrum recorded by the OSA.
Fig. 3. Dual-DSs regime with the TOD of -1.2 ps3/km. (a) Dispersion profile of pulse shaper; (b) Spectrum; (c) Pulse train; (d) Temporal evolution over 500 RTs; Spectral evolution of (e) soliton 1 and (f) soliton 2 over 500 RTs. Inset: average spectrum (red curve).
Subsequently, the TOD is further increased to -1.6 ps3/km, the laser operates in single soliton state again, but not stable. Figure 4(b) showcases the pulse evolution over 600 RTs. The intermittent appearance of high-intensity pulses suggests the potential occurrence of soliton explosion. Additionally, the unstable spectral spikes on the corresponding spectrum in Fig. 4(a) are common characteristics associated with soliton explosion. To further verify it, we utilize DFT technique to measure real-time spectra, as illustrated in Fig. 4(c). Here, the quasi-stable single pulse experiences explosion instability from time to time, indicating the fact that soliton explosion occurs in the experiments. Note that there are tails on the trailing edge of pulse train in Fig. 4(b) when soliton explosion happens. It could be attributed to the appearance of ultra-high intensity pulse during the exploding process. In fact, whether the soliton explosion appears on the leading or trailing edge of pulse depends on the sign of TOD. For negative TOD in this work, the explosions occur in the pulse leading edge [42]. However, due to the bandwidth limitation of photodiode and oscilloscope, the details of soliton explosion in temporal domain could not be identified in the experiments. With the increase of TOD, an inflection point in the dispersion profile occurs at the wavelength of 1568.3 nm (indicated by the blue circle in Fig. 3(a)). It shifts the trend of dispersion at long wavelength (>1568.3 nm), aligning it similar with that of negative GVD region. These results mean that more energy is transferred into the anomalous dispersion region, implying that the soliton has higher energy and nonlinearity. The broadened spectrum serves as confirmation. The further increased nonlinearity would break the equilibrium of dual-DSs, leading the laser into a soliton explosion state. However, it’s not high enough for dual-soliton explosion. One soliton might dissipate when the other soliton exploded because of the gain competition. Therefore, the laser operated in single soliton explosion state with TOD of -1.6 ps3/km. Note that the longer side of the spectrum was cut by the pulse shaper with operational range of 1526.0-1568.7 nm.
Fig. 4. DS explosion regime. (a) Spectrum; (b) Temporal evolution and (c) Spectral evolution over 600 RTs. Inset: energy evolution (red curve).
As we further increase the TOD to -2.2 ps3/km, dispersion at longer wavelength enters a more negative region, as shown by the green curve in Fig. 3(a). The heightened nonlinearity results in the laser transitioning into a chaotic state. Figure 5(a) illustrates the further broadened spectrum, with an inset in Fig. 5(a) showing the autocorrelation trace. The autocorrelation trace reveals a coherent narrow peak atop a broad pedestal, providing clear evidence of noise-like pulses. Figures 5(c) and (d) depict temporal and spectral evolution over 600 RTs, respectively. Note that there is obvious temporal shift in pulse train of Fig. 5(c), which could be attributed to the dramatic variation of statistical central position of the spectrum during the chaotic state. Figure 5(b) displays the RF spectrum, wherein the pedestals arise from inherent stochastic intensity fluctuations. Note that when the TOD is set beyond -2.2 ps3/km, the laser remains in the chaotic state.
Fig. 5. Chaotic regime. (a) Spectrum, Inset: Autocorrelation trace; (b) RF spectrum; (c) Temporal evolution and (d) Spectral evolution over 600 RTs.
4. Discussion
In our experiments, the stable DS state could be achieved by adding GVD of 7 ps2/km. Then, we utilize the pulse shaper for intra-cavity dispersion engineering to investigate the impact of TOD on the dynamics of DSs. After introducing TOD of -1.2 ps3/km, the spectrum undergoes a blue shift due to the negative TOD. The dispersion at the long wavelength side shifts from positive to near zero region, initiating the soliton effect and subsequently leading to soliton splitting. In the temporal domain, the single DS splits into two DSs with the same characteristics. Then, further increasing the TOD results in more wavelength components entering the negative dispersion region. So, more energy will be transferred into the anomalous dispersion region, indicating that the soliton energy and nonlinearity become higher. It will disrupt the equilibrium of the dual-DSs state and soliton explosion appears. In this state, the spectrum exhibits a broader bandwidth with unstable spikes. Finally, when the TOD was increased to -2.2 ps3/km, the laser entered the chaotic state, which could be attributed to the further increased energy in anomalous dispersion regime. Here, the asymmetry in the spectrum caused by TOD is evident through the whole process. It should be noted that at different GVD values from 7 to 8.2 ps2/km, the laser could experience the similar evolution trend by changing the TOD. However, the transition values of TOD are a little different.
5. Conclusion
In conclusion, we explore the impact of TOD on the dynamics of DSs in an ultrafast fiber laser. From a spectral perspective, the DSs exhibit a pronounced spectral tilt in correlation with the increasing intra-cavity TOD. This phenomenon is caused by the frequency shift induced by the radiation of dispersion waves under negative TOD. In the temporal domain, the stable single DS splits into two DSs, and further to explosion and chaotic regimes, all within the framework of dynamically increasing TOD. This is closely associated with the dispersion profile shift, which will be influenced by TOD. These results not only unveil the crucial role of TOD in ultrafast lasers but also present a new avenue for controlled and optimized performance of ultrafast lasers through the manipulation of TOD.
Funding
Key-Area Research and Development Program of Guangdong Province (2020B090922006, 2023B0909010002); National Natural Science Foundation of China (11904339, 11974006, 12274149, 62175069, 62375091); Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515012315, 2022A1515011760); Guangzhou Science and Technology Plan Project (202201010202).
Disclosures
The authors declare that there are no conflicts of interest to this article.
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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