1. Introduction

Traditional optical solitons originate from the balance between negative second-order dispersion (β₂) and self-phase modulation (SPM) effect in fiber lasers, and saturable absorber also plays an important role in passively mode-locked fiber lasers [1–4]. The energy (E) is proportional to the first power of inverse pulse duration (τ):$E = {{2|{\beta _2}|} / {\gamma \tau }},$ which sets limits for energy ascension [5,6]. In 2016, Redondo et al. found pure-quartic solitons (PQSs) that originate from the balance of negative β₄ and SPM effect in photonic crystal waveguide [7,8]. The energy of PQSs (E_PQS) is proportional to the third power of inverse τ: ${E_{PQS}} = {{2.87|{\beta _4}|} / {\gamma {\tau ^3}}}$(energy-width scaling law), indicating that PQS fiber lasers have great potential in generating high-energy ultrashort soliton pulses [9]. The research on PQSs in fiber and fiber lasers begins in 2018, Redondo et al. designed the micro-structured fiber to obtain PQSs, laying a foundation for the research of PQSs fiber lasers [10]. Later, they achieved dispersion-managed PQSs fiber laser [11,12]. The nonlinear Schrödinger equation (NLSE) and its extended form are the basic equations for describing pulse transmission in fibers and the ideal tool for modeling fiber lasers [13–15]. Redondo et al. proved that the combination of β₂ and β₄ can improve the performance of fiber lasers modeled by NLSEs which provides theoretical support to construct PQSs fiber lasers [16,17].

The relationship between soliton dynamics and laser structural parameters can be revealed through numerical simulation. Simulation will provide the theoretical basis to analyze soliton dynamics in experiments [18,19]. Pulsating solitons and erupting solitons are closely related to period-doubling bifurcations and are effective paths to chaos in fiber lasers, so they have attracted extensive attention. Pulsating conventional solitons [20,21] and erupting conventional solitons [22,23] exist in fiber lasers model by NLSEs. Average dispersion and small-signal gain can affect pulsating dynamics [24] and filters can affect erupting dynamics [25–27]. Dispersive Fourier transform technique that appeared in recent years has proved and revealed the unique feature of pulsating solitons and erupting solitons in experiments [28]. For PQS dynamics, only the vector characteristics [29] and the oscillation characteristics [30] have been studied recently. In 2022, Luo et al. first discovered pulsating PQSs caused by energy redistribution in the NLSEs-based mode-locked fiber laser [31]. It can be seen that the research on PQS dynamics is not sufficient and needs to be carried out urgently.

In this context, we numerically investigate the pulsating PQSs, creeping PQS molecules, and erupting PQSs in passively mode-locked fiber lasers. We investigate the influence of saturation power, small-signal gain, and output coupler on PQS dynamics. The results show that the fiber laser (without an intra-cavity filter) can obtain stationary PQSs, pulsating PQSs, and creeping PQS molecules by changing the above three parameters. At the same time, considering the influence of spectral filter effect and high-order dispersion on erupting solitons, the introduction of an intra-cavity narrow bandwidth filter makes fiber laser output stationary PQSs, pulsating PQSs, and erupting PQSs. This paper can provide a physical mechanism for the unique dynamics of PQSs.

2. Numerical method

Figure 1 is the configuration of the PQSs fiber laser, which contains a pump source, wavelength division multiplexer (WDM), Er-doped fiber (EDF), isolator, saturable absorber (SA), pulse shaper, single-mode fiber (SMF), and an output coupler (OC). The intra-cavity filter in red dotted line is introduced in the research of section 3.4 to explore its impact on PQS dynamics. Pulse transmission is simulated by NLSE:

 

Fig. 1. Configuration of PQSs fiber laser.

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U is the slowly varying complex envelope of pulses. γ, ${\beta _m} = \frac{{{d^m}\beta }}{{d{\omega ^m}}}{|_{\omega = {\omega _0}}}$ represents the nonlinear coefficient and m-order dispersion coefficient of fibers. T is the time coordinate in the reference frame that moves with pulses at group velocity. g is EDF gain given by:

The pulse shaper is constructed based on a spatial light modulator in experiments to realize intra-cavity high-order dispersion management. That is, compensating for intra-cavity β₂ and third-order dispersion (β₃) while providing large negative β₄. It is modeled by multiplying the electric field by phase in frequency domain in simulation [9]. For the pulse shaper in this simulation, β₂, β₃, and β₄ are taken as 19.40 ps²/km, -0.12 ps³/km, and -80.0022 ps⁴/km to fully compensate for intra-cavity β₂, β₃ to nearly zero and provide sufficient β₄ to obtain PQSs. Other fixed parameter values in this simulation are as follows: γ = 0.0016 W⁻¹m⁻¹, β₂ = 21.58 ps²/km, β₃ = 0.12 ps³/km, β₄= -0.0022 ps⁴/km, Ω_g = 50 nm and L_EDF = 1 m for EDF; γ = 0.0013 W⁻¹m⁻¹, β₂ = -21.4 ps²/km, β₃ = 0.12 ps³/km, β₄ = -0.0022 ps⁴/km and L_SMF = 20.4 m for SMF; q₀ = 0.7. The time resolution of this simulation is 50 fs and the center wavelength is set near 1550 nm determined by EDF.

E_sat, P_sat, g₀, and output ratio of OC (R_out) are set to 300 pJ, 200 W, 200 m^-1, and 20% to acquire enough gain to obtain stationary PQSs under different β₄. Fig. 2(a) shows the typical oscillation tail of PQSs. Fig. 2(b) exhibits that the change in spectrum intensity is less pronounced but becomes broader and flatter. Fig. 2(c) and (d) presents the PQS time profile evolution and PQS spectrum evolution to demonstrate its stability. All the above results are salient features of stationary PQSs. Typically, conventional solitons exhibit pulsating dynamics or erupting dynamics with modulated system parameters. Therefore, it would be interesting to examine what happens to PQSs if regulate system parameters. To this end, we comprehensively analyze the effects of system parameters on PQS dynamics.

 

Fig. 2. Logarithmic scale (a) time profile and (b) spectrum of stationary PQSs obtained with different β₄ of pulse shaper, inset in (a): time profile. Evolution of (c) time profile and (d) spectrum when β₄ in pulse shaper is set to -80.0022 ps⁴/km, inset in (d): energy.

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3. Influence of parameters on pure-quartic soliton dynamics

3.1 Influence of saturation power

P_sat is the product of saturable flux and effective mode field area of laser acting on SA, which is correlated with material parameters and optics design of SA. Taken E_sat, g₀, and R_out as fixed values of 300 pJ, 900 m^-1, and 20%, respectively. P_sat is linearly increased from 10 W in steps of 10 W to study the impact of SA on PQSs dynamic without an intra-cavity filter. The fiber laser outputs stationary PQSs as P_sat ≤ 600 W and PQS exhibits oscillation in the time-domain (Fig. 3(a)) with the breathing of spectrum (Fig. 3(b)) when P_sat increases to 670 W. Furthermore, energy also changes periodically. In other words, pulsating PQSs appear. Figure 3(c) shows the spectrum evolution within 20 RTs and proves that pulsating period is about 5 RTs. Figure 3(d)-(i) provide single-shot spectrums and single-shot time profiles within one pulsating period to further analyze the cause of pulsating PQSs. Fig. 3(d)-(h) present the intensity exchange between central and sidebands on both sides during pulsating, and spectrums return to the original state after 5 RTs. The intensity of the oscillatory tail in the corresponding logarithmic scale time profiles in Fig. 3(i) fluctuates and shifts back to its original position after 5 RTs. The energy exchange between the main part and the oscillating tail of PQSs results in its pulsating behavior [31].

 

Fig. 3. Evolution of (a) time profile and (b) spectrum when P_sat= 670 W, inset in (b): energy. (c) Extracted spectrum evolution of (b). Single-shot (d)-(h) spectrums and (i) time profiles (logarithmic scale) at different RTs in one pulsating period.

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The interaction between solitons is closely associated with the shape of interacting solitons. For solitons produced by the balance between SPM and combinatorial dispersion, oscillatory tails can create a potential barrier and prevent the combination of two adjacent solitons [32]. PQSs have periodic oscillation tails and more complex structures. PQSs split into PQS molecules with a further increase of P_sat. The interactions between PQSs will present interesting and complicated behavior [33]. Figure 4(g)-(i) display the field autocorrelation trace through the Fourier transform of Fig. 4(d)-(f), which could reflect the complex interaction between solitons more clearly [19,25,28,34]. There will have 2N-1 peaks in field autocorrelation traces as the number of solitons is N. Fig. 4(g)-(i) further prove that we have observed two PQSs. At the same time, Fig. 4(g)-(i) also present periodic mutual repulsion and attraction between two PQSs which make PQS molecules “creeping” in the time domain (Fig. 4(a)-(c)) and periodically “breathing” during spectrum evolution (Fig. 4(d)-(f)), evoking pulse energy varies periodically. The creeping (breathing) period is about 29 RTs, 31 RTs, and 19 RTs, respectively. Figure 4(j)-(l) and Fig. 4(m)-(o) prove that PQS molecules will have a small time shift during creeping and return to the original position at the beginning of a new creeping period under four different values of P_sat. The small time shift leads to a slight change in intensity and interval of spectrum fringes, but P_sat does not cause a change in the distance between two PQSs. The oscillation of PQS molecules relates to the enhancement of dispersion waves in simulation. Creeping PQS molecules’ single-shot spectrum has similar multi-sidebands float as single pulsating PQSs [32], which indicates that the formation of creeping PQS molecules has a bearing on intra-cavity large negative β₄. Creeping PQS molecules will evolve into stationary single PQSs as present in Fig. 4(b), (c), they will also evolve into stationary single PQSs when taken RTs number as 5000 and other parameters are the same in Fig. 4(a), indicating that creeping is a transitional state. Although the creeping characteristics display in Fig. 4(b), (c) are similar, the two stationary single PQS appears at different time delay positions. Such differences can be contributed to the nonlinear phase shift accumulated after each RT. And the intensity-dependent nonlinear phase shift could affect the shape of spectrums profiles. Figure 4(k),(l),(n), and (o) present creeping PQSs molecules having different intensities, resulting in different nonlinear phase shift accumulated during one creeping period so that new PQSs after creeping appear at different time delay positions.

 

Fig. 4. Evolution of (a)-(c) time profile, (d)-(f) spectrum, and (g)-(i) field autocorrelation traces when P_sat = 740 W, 760 W, 770 W, inset in (d)-(f): energy. Single-shot (j)-(l) spectrums (logarithmic scale) and (m)-(o) time profiles for creeping PQS molecules at different RTs in one creeping period.

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3.2 Influence of small-signal gain

The change of pump power in experiments corresponds to altering g₀ in simulation, which is an essential element in fiber laser systems. And it will change intra-cavity soliton characteristics. Ω_g, E_sat, P_sat, and R_out are taken as 50 nm, 300 pJ, 200 W, and 20% in this section. The influence of pump power on PQS dynamics is investigated by linearly increasing g₀ from 1 m^-1 with a step of 0.1 m^-1. Intra-cavity pulse shaper makes fiber laser outputs stationary PQSs as g₀ ≤ 46.8 m^-1, however, splitting occurs when g₀ is increased to ∼ 1.1 m^-1 in fiber lasers without a pulse shaper. These results mean PQSs can endure large nonlinear phase shifts without splitting and have large pulse energy [30]. Pulsating PQSs and stationary PQSs appear alternately (similar to reference [31]) when g₀ is in the range of 46.9 m^-1-750 m^-1, which will not repeat here. PQSs begin to split when g₀ is further increased to 751 m^-1, and creeping PQS molecules occur as g₀ = 776 m^-1. The field autocorrelation traces in Fig. 5(g)-(i) present periodic mutual repulsion and attraction between two PQSs resulting in PQS molecules periodically creeping and periodically breathing, that is, creeping PQS molecules. For creeping PQS molecules shown in Fig. 5(a)-(c), the creeping period remains at about 20 RTs, 20 RTs, and 21 RTs, and the distance between two PQSs increases with the increase of g₀. Spectrum evolution of creeping PQS molecules in Fig. 5(d)-(f) reveals that the larger creeping distance makes the spectrum breathing more obvious, that is, exhibits more obvious pulsating behavior. It should be noted that when the RTs number in Fig. 5(a)-(c) is taken as 5000, creeping PQS molecules will eventually evolve into stationary single PQSs which proves that creeping is a transitional state.

 

Fig. 5. Evolution of (a)-(c) time profile, (d)-(f) spectrum and (g)-(i) field autocorrelation traces when g₀ = 777 m^-1, 841 m^-1, 895 m^-1, inset in (d)-(f): energy. Single-shot (j)-(l) spectrums (logarithmic scale) and (m)-(o) time profiles for creeping PQS molecules at different RTs in one creeping period.

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Figure 5(j)-(o) present single-shot spectrums and single-shot time profiles in one creeping period to further investigate factors affecting the properties of creeping PQS molecules. PQSs molecules show a small time shift during creeping and return to the original state at the start of a new creeping period under three different values. Its spectrum intensity and fringe spacing remain unchanged with the dramatic change in sideband intensity, which is similar to the research in reference [14]. However, the variation of distance between fringes on creeping PQSs molecules spectrum is more pronounced with more obvious high-intensity and low-intensity sidebands. For traditional soliton molecules generated by the balance of β₂ and SPM effect, the distance between fringes is driven by the phase difference between solitons. Spectrum Kelly sidebands are related to the phase matching between dispersive waves and solitons. Multi-sidebands in the PQSs spectrum, which can be attributed to the intra-cavity large negative β₄ [7], facilitate the energy exchange of spectrum center versus multi-sidebands. In other words, the appearance of creeping PQS molecules is related to intra-cavity large β₄.

3.3 Influence of an output coupler

OC is also an indispensable device in fiber laser systems for determining the intra-cavity soliton intensity. R_out refers to the proportion of output to extra-cavity in this simulation. For example, R_out = 51% means that the proportion of output to extra-cavity is 51% and the proportion of intra-cavity circulation is 49%. Ω_g, E_sat, P_sat, and g₀ are taken as 50 nm, 300 pJ, 200 W, and 200 m^-1 in this section. R_out is linearly reduced from 90% to 10% in steps of 1% to investigate the influence of OC on PQS dynamics. The change of R_out will also cause pulsating PQSs or creeping PQS molecules. Figure 6 presents the time profile evolution, spectrum evolution, field autocorrelation traces, and energy change of creeping PQS molecules when R_out is taken as 60%, 31%, and 30%, respectively. Creeping PQS molecules in Fig. 6 show two different creeping periods during evolution and it may be due to the different degrees of periodic repulsion and attraction between two PQSs during the early creeping period and later evolution as present in the field autocorrelation traces of Fig. 6(d)-(f). Figure 7(a)-(d) provide single-shot spectrums and single-shot time profiles of one creeping period for two different creeping periods in Fig. 6(b)(i) for further analysis. The creeping period is 12 RTs at the initial creeping stage, later there is an obvious interaction between two PQSs after about 250 RTs, which alters the creeping period to 8 RTs. Two PQSs have different intensities under two different periods, but there still will be a certain time shift during creeping. Figure 7(a) and (b) indicate the intensity exchanges of its spectrum center and sidebands in one creeping period. Such intensity exchange makes the appearance of creeping PQS molecules, which is also the cause of conventional creeping soliton molecules [14]. PQSs spectrum has low-intensity multi-sidebands under the influence of β₄ [9]. PQS molecules spectrum has similar multi-sidebands with intensity changes occurring during creeping in Fig. 7(a) and (b). Variations in single-shot spectrums suggest that the generation of creeping PQS molecules is related to the intensity change of multi-sidebands, which can also be further attributed to large intra-cavity negative β₄.

 

Fig. 6. Evolution of (a)-(c) time profile, (d)-(f) field autocorrelation and (g)-(i) spectrum when R_out = 60%, 31%, 30%, inset in (g)-(i): energy.

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Fig. 7. Single-shot (a)(b) spectrums (logarithmic sca09le) and (c)(d) time profiles for creeping PQS molecules at different RTs in one different creeping period in Fig. 6(b)(h).

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3.4 Influence of intra-cavity filter

Spectral filter effects play a key role in intra-cavity pulse evolution [21,22,25–27,35]. Intra-cavity filters will induce multi-pulsing instability [25,26] or result in pulsating solitons [21] and erupting solitons [22]. It is necessary to explore the influence of intra-cavity filters on PQS dynamics. A Lorentzian filter with a transfer function $F(\omega ) = 1/[1 + {(\omega /\varDelta \omega )^2}]$ has been added to the fiber cavity to investigate the effect of filter bandwidth and its presence on PQS dynamics. ω is the instantaneous angular frequency and Δω is the bandwidth. Figure 8 presents different states of PQSs under different intra-cavity filter bandwidths and g₀. E_sat, P_sat, and R_out are taken as 300 pJ, 200 W, and 20% to compare the results with Section 3.2. It can be found that when intra-cavity filter bandwidth is in the range of 30 nm-50 nm, PQSs are not easy to split as altered g₀. When intra-cavity filter bandwidth is in the range of 15 nm-30 nm, PQSs are more likely to split or pulsate. As intra-cavity filter bandwidth is in the range of 5 nm-15 nm, PQSs will split and form irregular evolution when g₀ is less than 1 m^-1. Such results present that an intra-cavity filter with a bandwidth of 5 nm-15 nm has the greatest influence on PQS dynamics. Therefore, we set the intra-cavity filter bandwidth as 10 nm which is the intermediate value of such range to further study the effects of E_sat, P_sat, and R_out on PQS dynamics with an intra-cavity narrowband filter. The results reveal that the existence of filter results in erupting PQSs. PQSs, pulsating PQSs, and erupting PQSs appear with the modulated E_sat, P_sat, and R_out, which means the three parameters have qualitatively consistent effects on PQS dynamics. Considering that g₀ in simulation corresponds to pump power in experiments, which is the most common variable, this section analyzes erupting PQSs as alter g₀ as an example.

 

Fig. 8. State distributions under different filter bandwidths and g₀.

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Ω_g, E_sat, P_sat, and R_out are taken as 50 nm, 100 pJ, 70 W, and 20%, respectively. The PQSs first pulsate, then erupt as g₀ = 3.9 m^-1, Fig. 9(a),(b), and (e) show the spectrum evolution, time profile evolution, and energy evolution of such alteration. By analyzing the time profile evolution (Fig. 9(c)), spectrum evolution (Fig. 9(d)), energy change (Fig. 9(f)), single-shot time profile (Fig. 9(h)), and single-shot spectrum (Fig. 9(j)) of pulsating PQSs. It can be found that compared with the pulsating without an intra-cavity filter, the oscillation in the time domain and the “breathing” of spectrums are more obvious with an intra-cavity filter. Nevertheless, the narrow intra-cavity filter makes the multi-sidebands in single-shot spectrums disappear, indicating that pulsating under this condition is closely connected with tail oscillation. Erupting traditional solitons is characterized by continuous or periodic time shift, spectrum fragmentation, and energy improvement, which some researchers characterize as a kind of pulsating [36,37], but more researchers classify it as soliton explosion [23,38–41]. Time profile evolution in Fig. 9(a) shows the continuous multiple time shift of PQSs, that is, the typical characteristics of erupting. Single-shot time profiles of time shifts in Fig. 9(i) reveal that the erupting direction of PQSs is not fixed, but it will cause an increase in pulse intensity and energy (Fig. 9(e)). Fragmented single-shot spectrums given in Fig. 9(g) further prove the erupting PQSs after pulsating PQSs. High-order dispersion and high-order nonlinearity have important influences on erupting traditional solitons [39], and proper pairwise conjugation of these high-order effects can eliminate erupting [38,40]. Therefore, erupting PQSs can be attributed to large intra-cavity negative β₄ in this simulation.

 

Fig. 9. Evolution of (a) time profile and (b) spectrum when Ω_g, E_sat, P_sat, R_out, g₀ are taken as 50 nm, 100 pJ, 70 W, 20%, and 3.9 m^-1. Extracted (c) time profile evolution and (d) spectrum evolution of (a)(b). (e)(f) Energy evolution of 1-800 RTs and 1-300 RTs. Single-shot (g)(h) spectrums and (i)(j) time profiles at different RT of erupting and one pulsating period.

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4. Discussion

PQSs, pulsating PQSs, and creeping PQS molecules appear when altering the saturation power of SA, small-signal gain, and the splitting ratio of an output coupler. The appearance of pulsating PQSs can be attributed not only to its oscillation tail of time profile [32] but also to the intensity exchange between multi-sidebands and the center of soliton spectrums by analyzing PQSs single-shot spectrums and PQSs single-shot time profiles in one pulsating period or one creeping period. The nonlinear phase shift accumulated (intensity-dependent) after solitons run one RT will affect the shape of spectrums or time profiles. Interaction between solitons is closely associated with the shape of interacting solitons. For solitons produced by the balance between SPM and combinatorial dispersion, oscillation tails can create a potential barrier and prevent the combination of two adjacent solitons [32]. PQSs have periodic oscillation tails and more complex structures. The interactions between PQSs will present interesting and complicated behavior [33] including creeping PQS molecules. Moreover, intra-cavity filters will affect soliton transmission in passively mode-locked fiber lasers. The presence of an intra-cavity filter makes PQSs show erupting dynamics differ from those without an intra-cavity filter. That is, PQSs appear stationary, pulsating to erupting in turn with the modulated parameters, which is associated with the change of soliton spectrum induced by a narrow bandwidth filter. Erupting PQSs can also be attributed to large intra-cavity β₄ considering the influence of high-order dispersion and high-order nonlinearity on erupting traditional solitons besides filter [38,40]. These findings provide new insights into PQS dynamics in ultrafast fiber lasers.

5. Conclusion

In conclusion, we have numerically investigated the pulsating PQSs, creeping PQS molecules, and erupting PQSs in passively mode-locked fiber lasers. Saturation power, small-signal gain, and splitting ratio of an output coupler will all affect the PQS dynamics. PQSs, pulsating PQSs, and creeping PQS molecules appear without an intra-cavity filter as alter the above parameters. The presence of an intra-cavity filter makes PQSs show erupting dynamics, that is, PQSs appear stationary, pulsating, and erupting with the modulated parameters, which is associated with the change of soliton spectrum induced by a narrow bandwidth filter. The high-energy creeping PQS molecules in this simulation have application value for high-capacity and long-distance communication. These results deepen our understanding of PQS dynamics.