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Abstract
The evolution of ultrafast laser technology hinges partially on the understanding of the soliton nonlinear dynamics. Recently, the concept of pure-quartic soliton (PQS) that arises from the balance of pure negative fourth-order dispersion (FOD) and nonlinearity was proposed to generate high peak power pulse. Herein, we investigate the generation of dissipative pure-quartic soliton (DPQS) in a fiber laser, which is balanced among the positive FOD, nonlinearity, gain and loss. The DPQS features the shape-preserving propagation despite the asymmetrical temporal profile at higher pulse energy. It is found that the asymmetrical temporal profile of DPQS is resulted from the mismatching of the phase shift profiles caused by self-phase modulation and FOD. Moreover, it is demonstrated that the DPQS possesses a higher energy-scaling ability compared to conventional dissipative soliton, owing to the nonlinear relationship between the pulse energy and pulse duration. These findings demonstrated that the employment of positive FOD could be a promising way for manipulation of optical pulse as well as the improvement of laser performance.
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1. Introduction
Ultrafast fiber lasers capable of generating short pulses have attracted much attention in the past decades owing to their important roles in many aspects of scientific and industrial fields, such as nonlinear imaging [1,2], micromachining [3], and optical communications [4]. Therefore, there is always a strong motivation to investigate the high-performance ultrafast fiber lasers. In fact, relying on the rapid development of fiber fabrication and laser technology, the performance of ultrafast fiber lasers has been improved greatly in recent years. In addition to being an ultrashort pulse source, the passively mode-locked fiber laser is actually a nonlinear optical system, which provides an excellent platform for investigating various soliton dynamics and nonlinear phenomena [5]. To date, by adjusting the cavity parameters such as nonlinearity, birefringence and dispersion, diverse soliton nonlinear dynamics have been observed in fiber lasers, i.e., vector soliton [6–8], dissipative soliton resonance [9–12], multi-soliton patterns [13–15]. Indeed, the discovery of the above-mentioned nonlinear phenomena not only deepens our understanding of soliton dynamics, but also is beneficial to improvement of ultrafast laser performance.
The dispersion engineering of an ultrafast fiber laser has been demonstrated as an effective strategy for exploring the soliton nonlinear dynamics and manipulating the temporal shape of optical solitons [16]. By engineering the cavity dispersion from anomalous to normal one, the conventional soliton [17,18], self-similar pulse [19,20] and dissipative soliton [5,21] can be obtained from the fiber lasers. In particular, the realization of dissipative soliton in the all-normal dispersion fiber laser promotes the output pulse energy to several tens of nanojoules because of its highly chirped characteristic [22,23]. Generally, the dispersion engineering of a fiber laser is only related to the second-order dispersion, namely group velocity dispersion (GVD). Meanwhile, the higher-order dispersion is generally deemed to be a detrimental effect to soliton propagation in fiber lasers [24,25]. However, it was demonstrated that the stable propagation of soliton-like pulse could be still obtained in the presence of negative fourth-order dispersion (FOD) and GVD in nonlinear optical systems, which is called as “quartic soliton” [26,27].
Recently, by virtue of the unique dispersion properties of photonic crystal waveguides, Blanco-Redondo et al. demonstrated that the soliton-like pulse could be formed through a balance between positive Kerr nonlinearity and purely negative FOD, which is termed as “pure-quartic soliton” (PQS) [28]. The concept of PQS stimulated the intensive efforts dedicated to the manipulation of soliton properties by introducing negative FOD [29–36]. The discovery of PQS enables the generation of higher peak power pulse with broader spectrum when comparing to that of conventional soliton if the pulse duration is short enough. Then the PQS was also verified in a passively mode-locked fiber laser by utilizing an intracavity pulse shaper as the dispersion engineering component [29]. The generation of PQS from an ultrafast fiber laser indicates that the mode-locked soliton with high peak power could be directly obtained from the fiber laser. Note that the formation of PQS was mostly dealing with the interaction of negative FOD and positive Kerr nonlinearity. Regarding the dissipative pure-quartic soliton (DPQS) relying on the balance among positive FOD, nonlinearity, gain and loss, it was found that the self-similar pulse could propagate stably in a fiber link with the distributed gain [37,38]. This result indicates that the DPQS could be potentially obtained in a mode-locked fiber laser. However, the investigation of DPQS in fiber lasers remains largely unexplored. Taking the fruitful dynamics of the PQSs into account, it would be interesting to explore the properties and dynamics of the DPQS in a mode-locked fiber laser dominated by the positive FOD.
In this work, we numerically investigate the generation and properties of DPQS in a passively mode-locked fiber laser. The cavity dispersion is designed to be dominated by the positive FOD. With the increasing pulse energy, the DPQS shows the asymmetrical temporal profile owing to the mismatching of the self-phase modulation (SPM)- and FOD-induced phase shift profiles. In addition, we demonstrated that the nonlinear relationship between the pulse energy and duration will lead to a higher energy scaling ability of DPQS than that of conventional dissipative soliton.
2. Numerical simulations and discussions
The numerical simulation model is based on the schematic of the typical ultrafast fiber laser with the distributed positive FOD fiber, as depicted in Fig. 1. To analyze the properties of the DPQS in a passively mode-locked fiber laser, we use the Ginzburg-Landau equation to simulate the generation and propagation of the DPQS. For the purpose of matching the practical laser setup, the action of individual component on the propagating soliton is considered, and the lumped cavity model that treats each element in the laser cavity separately is employed. The soliton evolution is governed by the following equation that includes the FOD:
Fig. 1. Schematic of the DPQS fiber laser for simulations. OC: output coupler; SA: saturable absorber.
Here, A is the slowly varying amplitude of the optical soliton, α is the attenuation coefficient, β2 is the second-order dispersion, β4 is the FOD, γ is the effective nonlinear coefficient, g is the gain coefficient, and Ωg is the gain bandwidth. For the single-mode fiber, the g is 0. For the EDF, the gain coefficient g is expressed as the following saturated gain model:
where g0 is the small-signal gain coefficient, Esat is the gain saturation energy, |A(t)|2 is the instantaneous pulse power. Here, the adjustment of Esat is corresponding to the variation of the pump power in the practical situation. Then the SA is modelled by a power-dependent transfer function: where q0 is the modulation depth of the SA, |A(t)|2 is temporal power profile of the pulse, and Psat is the saturation power for SA. The SA provides the nonlinear loss that triggers the mode-locking mechanism. The Ginzburg-Landau equation is numerically solved with the standard split-step Fourier method. In the simulations, the q0 and Psat are set to be 0.5 and 100W, respectively. Moreover, the γ and Ωg are 0.003W-1m-1 and 49 nm, respectively. To investigate the properties of the DPQS relying on pure positive FOD, the β2 is set to be 0.By adjusting the gain saturation energy (Esat), the stable DPQS could be observed with a weak Gaussian pulse as the initial seed. Figure 2 presents the spectral and temporal profiles of DPQS which is obtained with the Esat of 10 pJ and the FOD of 10 ps4/km. In this case, the spectrum of DPQS exhibits the rectangular-like profile, which is a typical feature of dissipative soliton, as shown in Fig. 2(a). Nevertheless, note that the temporal profile of the DPQS suffers a symmetrical pedestal comparing to conventional dissipative soliton, it is because the FOD results in higher stretching velocities on two pulse wings than the central part of the optical soliton [37,38]. Moreover, the frequency chirp of the DPQS is almost linear across the central part while undergoes changes at both wings. In order to show the stability of the generated DPQS, we plotted the spectral and temporal evolutions for 1000 roundtrips in Figs. 2(c) and 2(d), respectively. It can be seen that the spectral and temporal profiles of the DPQS are both nearly indistinguishable from each other with the cavity roundtrips, indicating that the DPQS fiber laser is operating in a stable regime.
Fig. 2. Mode-locked DPQS with FOD of 10 ps4/km and Esat of 10 pJ. (a) Spectrum with log and linear coordinates; (b) pulse and chirp; evolution of (c) spectrum and (d) pulse with 1000 roundtrips.
It is well known that the dissipative soliton in the fiber laser is stabilized by balancing the dispersion, Kerr nonlinearity, gain and loss [5]. Generally, the conventional dissipative soliton will show multi-soliton dynamics [13–15] or wave breaking phenomenon [39] with the higher cavity nonlinearity. Therefore, it would be interesting to check what happens to the DPQS if the nonlinear effect is increased. To this end, Esat was further adjusted to 100 pJ. In this case, a notable phenomenon can be observed on the spectral and temporal profiles of the DPQS. It can be seen that the DPQS evolves into an asymmetrical temporal profile. In addition, the frequency chirp of the DPQS is nonlinear across the temporal profile, as plotted in Fig. 3(b). The corresponding DPQS spectrum evolves from the rectangular-like profile to the asymmetrical “M” shape, as presented in Fig. 3(a), which is actually dependent on the intracavity soliton energy. From the simulation results, despite the asymmetrically spectral and temporal profiles of the DPQS, the DPQS can still propagate stably in the fiber laser, as shown in Figs. 3(c) and 3(d). It should be noted that the DPQS will break up under the condition of 10 ps4/km FOD if the Esat is increased to over 743 pJ. Here, no multi-soliton operation could be observed with the increasing Esat. The wave breaking effect of the DPQS is similar to the conventional dissipative soliton, which is owing to the overdriven nonlinear effect experienced by the DPQS in the laser cavity. Note that we have also obtained the spectral and temporal profiles of mode-locked soliton similar to those of positive-FOD supported self-similar pulse evolution [38] in our fiber laser with proper parameter settings. However, in this case the self-similar evolution of the mode-locked pulse is just an intermediate state when the operation of the fiber laser is evolving into the stable DPQS state. And the stable DPQS shows the asymmetrical temporal profile instead of the parabolic-like profile of self-similar pulse. Maybe more precise parameters should be set to obtain the stable self-similar pulse evolution in our fiber laser.
Fig. 3. DPQS operation with FOD of 10 ps4/km and Esat of 100 pJ. (a) Mode-locked spectrum; (b) pulse and chirp; (c) spectral evolution; (d) temporal evolution.
It has been demonstrated that the spectral bandwidth and temporal profile of dissipative solitons are related to the cavity dispersion. Thus, to further reveal the characteristics of the DPQS, the spectral and temporal evolutions of DPQS were investigated with different values of cavity FOD in the proposed fiber laser. The results are shown in Fig. 4. From Fig. 4(a), we can see that the spectrum of the DPQS broadens with the decreasing FOD value, which is also a typical trend for soliton fiber lasers [16]. However, the spectrum of DPQS maintains the asymmetrical “M” shape during the broadening process, indicating that the asymmetrical “M” shape is the typical feature of DPQS fiber lasers, as plotted in Fig. 4(b). In addition, the temporal profiles of DPQS with different cavity FOD values still exhibit the asymmetrical characteristic despite of the different durations, as shown in Fig. 4(b). In order to investigate the DPQS fiber laser in more details, we plotted the pulse duration and the 3-dB spectral bandwidth versus Esat in Figs. 4(c) and 4(d). It can be observed that the pulse duration and spectral bandwidth increase with the increasing Esat for different cavity FOD values, whose trend is similar to those of conventional dissipative solitons in fiber lasers [40].
Fig. 4. DPQS operation with different FOD coefficients, β4 = 10 ps4/km (red), 40 ps4/km (blue), 70 ps4/km (green), 100 ps4/km (black). (a) Optical spectra; (b) pulse profiles; (c) and (d) spectral width and pulse duration over a range of Esat coefficients.
The energy scaling ability of the soliton fiber lasers is crucial for practical applications. Therefore, it would be meaningful to identify the energy scaling ability of the DPQS fiber laser. Figure 5(a) shows the pulse energy evolution with the increasing Esat under the condition of different FOD values. Moreover, the highest achievable energy is about 3.2 nJ for FOD of 100 ps4/km. In fact, the pulse energy was limited by the overdriven nonlinear effect in the laser cavity, which is partially determined by the pulse peak power. As we know, the pulse peak power is dependent on the pulse energy and pulse duration. Then we plotted the energy-duration scaling of the proposed DPQS fiber laser in Fig. 5(b). It should be noted that the pulse energy and pulse duration for conventional dissipative soliton possess a linear relationship [40]. However, the pulse energy of DPQS is demonstrated to be proportional to the cubic of the pulse duration by numerical fitting, indicating that the pulse energy scaling ability of DPQS is larger than conventional dissipative soliton fiber lasers for longer pulse duration. Here, we should stress that the cubic relationship between the pulse energy and pulse duration is not a universal law for DPQS. As we know, the achievable pulse energy in a fiber laser is dependent on several parameters such as the fiber dispersion and modulation depth of SA. Thus, when the modulation depth of SA is varied, the energy-width scaling property will also change correspondingly. However, the higher energy scaling ability of DPQS than that of conventional dissipative soliton can be found in a large range of cavity parameters according to our simulation results.
Fig. 5. Energy-width scaling properties of the DPQS for different FOD values, β4 = 10 ps4/km (red), 40 ps4/km (blue), 70 ps4/km (green), 100 ps4/km (black). (a) Pulse energy versus pulse width; (b) energy1/3 versus pulse width
By utilizing the interaction between positive FOD and Kerr nonlinearity, we numerically investigated the DPQS formation in a passively mode locked fiber laser. In fact, recently it is indicated that the self-similar pulse can propagate stably in an optical fiber link with gain and positive FOD. The symmetrical “M” shape of the pulse spectrum and triangle-like temporal profile can be observed [37,38]. As we know, the stable propagation of the dissipative soliton in a fiber laser is dependent on the balance among the gain, loss, dispersion and Kerr nonlinearity [5]. In this case, when the pulse energy is increased with the increasing Esat, the profile mismatching of the phase shifts induced by the Kerr nonlinearity and FOD become more evident, which leads to a special chirp characteristic. Therefore, when the DPQS evolves into an equilibrium state, the asymmetrical feature in the temporal and spectral domains can be observed simultaneously. In our simulations, the asymmetrical spectral and temporal profiles can be observed for different FOD values, which is different from those of traditional dissipative solitons. Thus, we think that this might be a unique characteristic of DPQS in fiber lasers. Moreover, the relationship between the DPQS energy and duration is demonstrated to be a nonlinear one, which is different from linear one of conventional dissipative solitons. Here, we should note that the DPQS shows a long tail on the temporal profile, which can be considered as energy storage ability of DPQS. In this way, the relationship between the DPQS energy and duration will show a nonlinear one, and thus, the energy scaling ability of DPQS is higher than that of conventional dissipative solitons. Note that the FOD is designed to be distributed along the laser cavity. However, in practice the FOD is generally compensated in a lumped way by a spectral-pulse shaper. In fact, we have also simulated the DPQS fiber laser with similar parameters in a lumped way of FOD compensation. And we still observed the similar properties of DPQS in the fiber laser, for example, asymmetrical temporal and spectral profiles. As for the experimental observation of DPQSs in fiber lasers, the commercially available waveshaper or the spatial light modulator can be introduced into the laser cavity to manage the dispersion, which has been reported for the investigation of PQS in the anomalous dispersion fiber laser [29]. Finally, only the DPQS erbium-doped fiber laser operating at 1.55 µm waveband was numerically simulated. However, we believed that the DPQS can be also obtained from other fiber lasers operating at different wavebands (i.e., Yb-doped fiber laser), if the appropriate instrument for dispersion engineering is available.
3. Conclusion
In conclusion, we have investigated the properties of DPQSs in a passively mode-locked fiber laser dominated by positive FOD. The formation of DPQS in the fiber laser is relying on the balance among the positive FOD, Kerr nonlinearity, gain and loss. With the increasing pulse energy, the temporal profile of the mode-locked DPQS evolves from symmetrical to asymmetrical ones while maintaining the shape-preserving propagation. Moreover, it is found that the DPQS has a higher pulse energy scaling ability when comparing to that of conventional dissipative soliton, owing to the nonlinear energy-width relationship. Our findings will give new insights into the PQS nonlinear dynamics in dissipative optical systems, and also be beneficial to design a fiber laser system that supports high-energy pulse output.
Funding
National Natural Science Foundation of China (11874018, 11974006, 61805084, 61875058, 62175069); Key-Area Research and Development Program of Guangdong Province (2018B090904003, 2020B090922006); Science and Technology Program of Guangzhou (2019050001); Basic and Applied Basic Research Foundation of Guangdong Province (2019A1515010879, 2021A1515012315).
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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