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Abstract
Soliton molecule has the properties that similar to those of matter molecule, which brings great research value. The rich dynamics of soliton molecules depend on the complex interactions between pulses. As one of the important factors, gain dynamics has multi-dimensional effects on the interactions of soliton molecules. We here study the interaction process of soliton molecules under gain perturbation, and find the oscillation and collision behaviors of pulses within soliton molecules induced by gain perturbation, as well as the pulses recombination through interactions between soliton molecules. We believe that the energy change in the cavity and the energy transfer based on the continuous wave component play an important role in the evolution process of the dissociation of soliton molecules into three-pulses bunch and single pulse. These findings reveal the effects of gain on the motion of soliton molecules and provide a basis for exploring the control dynamics of soliton molecules.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Dissipative solitons (DSs) are ubiquitous local structures in non-equilibrium systems with particle-like properties. In passively mode-locked fiber lasers (PMLFLs), multiple solitons tend to coexist in the cavity under high pump power and self-organize to form soliton molecules (SMs) [1–3]. Due to their high similarity with matter molecules in some aspects, soliton molecules have always been a research hotspot in the field of dissipative soliton dynamics. On the one hand, soliton molecules have rich internal dynamics, such as the establishment of soliton molecules, vibrations or oscillations of relative phase and separation between pulses [4–9]. On the other hand, taking the soliton molecule as a structural unit, it also exhibits a variety of complex external dynamics. They can dissociate or synthesize new soliton molecules after collision with each other, and can also combine to form “2 + 2” soliton molecular complexes (SMCs), breathing soliton molecules, and even “polymerize” into solitons supramolecular or soliton crystals with thousands of soliton molecules [10–16]. Therefore, studying the characteristics of soliton molecules is of great significance to explore and excavate the analogy and connection between the micro domain and the macro world.
The formation these complex structures of soliton molecules is inseparable from the rich nonlinear interactions in PMLFL, such as pulse coherent overlap [1,17], acousto-optic effect [18–20], the slow gain depletion and recovery mechanism [21–23], dispersive waves [22,24–27], noise-mediated Casimir-like effects [28,29] and the echo mechanism [30]. Based on these interaction mechanisms, the transition of bound states has been studied [30,31]. Among them, both the acousto-optic effect, slow gain depletion and recovery mechanism and noise-mediated Casimir-like effects can provide a long-range repulsive force. Acousto-optic effect refers to the mechanism of soliton interaction based on the phonon generated by interaction between soliton and medium. Gain is another important factor in soliton interaction. On the one hand, when the former pulse pass through the gain medium, the gain will be consumed, and the recovery process of the gain takes a certain time which makes the deceleration of the latter pulses. Thus, the pulses are driven to be equally spaced in the laser cavity. On the other hand, the gain depletion caused by the pulse leads to the uneven distribution of the noise field, which leads to the interaction between the pulses. As a multi-pulse state, soliton molecule has more complex degrees of freedom than single pulses, so they will show more dimensional interaction processes under the influence of gain, which requires further observation and research.
In this work, based on PMLFL with nonlinear polarization rotation (NPR) technology, together with the dispersion Fourier transform (DFT) technique, we study the interaction process of soliton molecules with different initial inter-molecular separation under gain perturbation. It is found that the gain perturbation will lead to the oscillation and collision of the pulses constituting soliton molecules, and then they recombine through the inter-molecular interaction. In the evolution process of the splitting of “2 + 2” double SMs into “3 + 1” three-pulses bunch and single pulse, it is accompanied by the energy change in the cavity and the energy transfer based on the continuous wave (CW) component. For soliton molecules with different initial inter-molecular separation, their cleavage pathways are different. In the direct interaction of soliton molecules with an initial inter-molecular separation of picosecond magnitude, the soliton molecules undergo three processes successively: tight soliton molecular complex, intra-molecular oscillation dynamics, three-pulses bunch and single pulse. In the indirect interaction of soliton molecules with an initial inter-molecular separation of nanosecond magnitude, the soliton molecules have undergone three stages of intra-molecular periodic vibration dynamics, intra-molecular oscillation dynamics, three-pulses bunch and single pulse.
2. Experimental setup
The experimental setup of the PMLFL with NPR is shown in Fig. 1. A 980 nm semiconductor-pumped laser (PL) is connected to an external signal generator (SG) to realize the modulation of the pump power. The gain medium is a 1.5 m long erbium-doped fiber (EDF) whose group velocity dispersion of 40 ps2/km at 1550 nm. The remaining fibers in the cavity are all single-mode fiber (SMF) with a group velocity dispersion of -23 ps2/km@1550 nm. The pump light is coupled into the cavity via a 980 nm/1550 nm wavelength division multiplexer (WDM). The structure of a polarization dependent isolator (PD-ISO) sandwiched between two squeezed polarization controllers (PCs) acts as a saturable absorber to achieve mode locking. The total cavity length is 12.6 m with the fundamental frequency of 16.89 MHz. The net intracavity dispersion is -0.195 ps2, which means the laser operates in the anomalous dispersion region.
An optical coupler (OC) with a split ratio of 3:7 is used to output 70% of the light out of the resonator, which will be split into three paths through two more OCs. One branch is used to synchronously monitor the average spectrum of the pulse evolution via an optical spectrum analyzer (OSA, Yokogawa, AQ6370D). The DFT technique is applied in the other branch to obtain the real-time spectral evolution, which maps the pulse spectrum into a temporal waveform using chromatic dispersion. The optical pulse is stretched by a 1.05 km dispersive compensation fiber (DCF) with a dispersion coefficient of -131.34 ps/(km·nm) at 1550 nm, and then is fed into a photodetector (PD, Finisar XPDV2320R) with a bandwidth of 50 GHz and monitored in real time by two serial high-speed oscilloscopes (Tektronix DPO75902SX). Another optical pulse is not stretched, and is directly sent into the PD, so as to obtain the time domain evolution of the optical pulse. The bandwidth of the high-speed oscilloscope is 59 GHz, and the sampling rate can be up to 200 GS/s. The frequency domain resolution of the detection system is ∼0.1 nm, and the time domain resolution is 5 ps.
3. Results and discussions
3.1 Indirect interaction of soliton molecules
When the pump power is 20.82 mW, four-pulses bunch forms in the cavity, and the spectrum is shown in Fig. 2(a). Figure 2(c) shows the pulse structure, and the time separations of adjacent pulses are 35 ps, 35 ps and 39 ps, respectively. As soon as the pump power is finely tuned to 21.03 mW, the four-pulses bunch self-assemble into a “2 + 2” SMC. Figure 2(b) shows the spectrum of SMC, and the period of obvious modulation fringes is 4.02 nm, which caused by the interference between pulses constituting the soliton molecules. The center wavelength of SMC is 1562 nm, and the phase difference between adjacent pulses is π. According to the interference in Fig. 2(b), the intra-molecular separation is 2 ps, and inter-molecular separation is 35 ps.
Fig. 2. Optical spectra directly recorded by the OSA at the pump power of (a) 20.82 mW and (b) 21.03 mW. The pulse trains at the pump power of (c) 20.82 mW and (d) 21.03 mW.
By keeping the pump power and adjusting the PCs, the inter-molecular separation of the “2 + 2” SMC is gradually increased to the nanosecond level. By adjusting the signal generator, the pump laser output can be perturbed. The pump power is stabilized at 21.03 mW at the beginning. Then a sinusoidal signal with a modulation period of 8 s and a modulation amplitude of 0.3 mW is applied to the pump laser via the signal generator. The modulating signal is removed after one period and the pump power returns to the level of 21.03 mW. After that, a high-speed oscilloscope with a sampling rate of 100 Gs/s is used to monitor and record the evolution of the soliton molecules in real time. The time domain and spectrum of the soliton molecule in 10300 roundtrips (RTs) are shown in Figs. 3(a) and (b), respectively. The inter-molecular separation is 1.91 ns, and there is no interference between soliton molecules. Figures 3(c) and (d) show the first-order optical autocorrelation (AC). According to different characteristics, the evolution process of soliton molecules can be divided into the following three stages: intra-molecular periodic vibration, intra-molecular oscillation, three-pulses bunch and single pulse.
Fig. 3. The indirect interaction process of soliton molecules: (a) time domain evolution; (b) spectral evolution; (c) autocorrelation trajectory of SM1; (d) autocorrelation trajectory of SM2; (e) evolution curve of cavity energy during indirect interactions of soliton molecules.
After applying gain perturbation, the adjacent pulses constituting the soliton molecules all start to vibrate periodically, which lasts from RT 1 to RT 3080. Figure 4(a) is the averaged spectrum at this stage, showing that there are two soliton molecules in the cavity at this time, and their central wavelengths are both 1562 nm. The red curve in the inset is AC of soliton molecule 1 (SM1) at 2000th RT, with an intra-molecular separation of 6.16 ps. The blue inset is the AC curve of soliton molecule 1 (SM2), and its intra-molecular separation is 5.04 ps. The soliton molecules are loose at this state. Figures 4(b) and (c) show the intra-molecular separation and relative phase evolution of two soliton molecules, respectively. The intra-molecular separation (blue curve) of SM1 has obvious periodic vibrational properties whose vibrational range is 5.875∼6.715 ps. The relative phases (orange curves) of adjacent pulses also vibrate periodically with the vibration of the interval, ranging from 2.32 to 5.17. The inset in Fig. 4(b) is an enlargement of the trajectory inside the red dashed box, indicating the intra-molecular separation and relative phase of the SM1 vibrate synchronously with a vibration period of 44 roundtrips. As shown in Fig. 4(c), the intra-molecular separation (blue curve) of SM2 also vibrates periodically, and the vibration range is 4.756∼5.875 ps. The maximum value of the intra-molecular separation of SM2 is equal to the minimum value of the intra-molecular separation of SM1, that is, SM2 is always tighter than SM1. In addition, although the relative phase (orange curve) of the pulses in the SM2 has undergone three large changes during the evolution, its overall has periodic vibration. The blue dashed box in the inset in Fig. 4(c) shows that the intra-molecular separation and relative phase of the SM2 are asynchronously vibrating with a difference in the vibrational period of π. The vibration period of the SM2 is also 44 roundtrips.
Fig. 4. The stage of intra-molecular periodic vibration: (a) averaged spectra of the two soliton molecules, and the insets are their ACs at RT2000; (b) the intra-molecular separation and relative phase evolution of SM1, the inset is an enlarged view of the trajectory within the red dashed box; (c) the intra-molecular separation and relative phase evolution of SM2, the inset is the enlarged view of the trajectory inside the blue dashed box.
From RT 3080, the amplitude of the internal vibration of the soliton molecules intensifies, and then they enter the stage of intra-molecular oscillation stage which continues until RT 5970. Figures 5(a) and (b) show the temporal and spectral evolution trajectories of this stage. In the early stage of the intra-molecular oscillation from RT 3080 to RT 5120, SM1 continues to vibrate periodically, and the period and range of the vibration remain unchanged. The change of the intra-molecular separation is given by the blue curve in Fig. 5(c). The pulses consisting SM2 have already started to oscillate violently, and its AC trajectory in this period can be observed from Fig. 3(d). Adjacent pulses of SM2 experience two collisions during the oscillation. The first collision occurs at RT 3550 which continues for 50 roundtrips. Then the pulses are accelerated away, and the intra-molecular separation of SM2 haves exceeded that of SM1 after RT 3963. At RT 4939, the intra-molecular pulses accelerate again, culminating in a second collision at RT 5120. The evolution of the intra-molecular separation of SM2 is shown by the orange curve in Fig. 5(c).
Fig. 5. The stage of intra-molecular oscillation: (a) time domain evolution; (b) spectral evolution; (c) the variation curve of the intra-molecular separation of two soliton molecules in the early stage
After the second intra-molecular collision of SM2 that starts from RT 5120, it enters the later stage of the intra-molecular oscillation. Figures 6(a) and (b) show the AC trajectories of SM1 and SM2 in this process that from RT 5000 to RT 5970. As shown in Fig. 6(a), the intra-molecular pulses of SM1 oscillate close from RT 5120, and collide at RT 5550. Then, its intra-molecular separation vibrates periodically between 1.6 and 2.2 ps, which lasts until RT 5790. Figure 6(b) shows that the adjacent pulses of SM2 are accelerated and approached, causing the pulses collide for the second time at RT 5120. And the satellite peak of SM2 disappears after the collision, illustrating that the collision causes the pulse annihilation. Figure 6(c) shows several typical spectra for expressing the indirect interaction process of soliton molecules more intuitively. The orange spectrum in Fig. 6(c) is the real-time spectrum of RT 5120. The spectrum of SM1 still has interference fringes, but that of SM2 have already disappeared which corresponds to the spectrum of a single pulse. It indicates that a pulse is annihilated after the collision, and the laser operates in the state of “2 + 1” soliton molecule and single-pulse. Combined with the spectral evolution in Fig. 5(b), we notice that the single pulse actually undergoes decay and re-establishment during RT 5120 to RT 5550. The purple spectrum in Fig. 6(c) shows that SM1 is still a loose soliton molecule at RT 5300, and the spectral intensity of the single pulse is almost completely attenuated. The corresponding pulse sequence at this time is shown as the purple curve in the inset, and the normalized intensity of the single pulse has decayed to 0.21. At RT 5600, the green spectrum shown in Fig. 6(c) shows that SM1 has evolved into a tight type, and the single pulse has reformed and the normalized intensity recover to 0.6 as shown in the green curve in the inset. The single pulse maintains stable evolution after RT 5550, while the SM1 enters a chaotic oscillation state. Furthermore, the Kelly sideband of SM1 enhances during the process of the decay and reformation of the single pulse. After the recovery of the single pulse and before the chaotic oscillation of the internal pulse of SM1, that is, during RT 5550 to RT 5790, the intensity of the Kelly sideband is further enhanced, as does the spectral substrate. Then, the intensities of the spectral substrate and Kelly sidebands are significantly weakened after the SM1 enters the chaotic oscillation. As shown in Fig. 6(d), the spectrum corresponding to RT 5600 has significantly higher spectral substrate and Kelly sideband intensities than those of RT 5300, while the spectrum corresponding to RT 5900 has lower spectral base and Kelly sideband intensities than RT 5300 and RT 5600. Actually, the reason for the enhancement of the spectral substrate is that the energy of the annihilated pulse after the collision is converted into a DC component. The global action of the DC component transfers the energy to the position of SM1 in the cavity, causing the internal chaotic oscillation of SM1. At the same time, new pulse is gradually formed and brings about a drastic change in the relative phase between the pulses, so the interference fringes on its spectrum are no longer symmetrically distributed, as shown in the light blue spectrum in Fig. 6(d).
Fig. 6. The late stage of intra-molecular oscillation: (a) AC trajectory of SM1; (b) AC trajectory of SM2; (c) typical spectra during the internal oscillation process; (d) single-shot spectra of RT 5600, RT 5300 and RT 5900.
The internal chaotic oscillation of SM1 continued to RT 5970 and a new pulse generates at the position of SM1, forming a three-pulses bunch together with SM1. After that, the laser operates in the state of “3 + 1” three-pulses bunch and single pulse. Figure 7(a) shows the AC trajectory of the three-pulses bunch. Adjacent pulses of the bunch move relatively under interaction. Among them, two adjacent pulses approach each other during RT 5970 to RT 7305, and collide at RT 7305. Subsequently, a second collision occurs at RT 7390, after which the pulse begins to accelerate away until RT 7460, and the time separation between them increases to 6.44 ps. The other pulse is affected by the collision and is relatively far away from the other two pulses from RT 7460. Figures 7(b) and (c) show the real-time spectra of RT 7460 and RT 10000, respectively, and Fig. 7(d) shows their ACs. The time separation between two adjacent pulses that collide is 6.16 ps and 5.59 ps in these two roundtrips, and the time separation between the pulse which moves far away and the adjacent pulse is increased from 8.11 ps to 25.18 ps.
Fig. 7. The stage of three- pulses bunch and single pulse: (a) AC trajectories; (b) single-shot spectra of RT 7460; (c) single-shot spectra of RT 10000; (d) ACs of RT 7460 and RT 10000.
Since the gain perturbation affects the energy of the resonator to some extent, the energy of the cavity during the indirect interactions of soliton molecules is shown in Fig. 3(e). At different stages of the indirect interactions of the soliton molecules, the energy in the cavity also changes. During the stage of intra-molecular periodic vibration and the early stage of the intra-molecular oscillation, the laser operates in the state of “2 + 2” SMC, and the energy in the cavity is basically stable. At RT 5120, the pulses that make up SM2 collide, resulting in the annihilation of one of the pulses and the decay of the other one, and the corresponding energy in the cavity decreases rapidly. At the same time, the part of the energy of the annihilating pulse begins to convert into the energy of the dispersive wave of SM1, so its Kelly sideband intensity increases. With the recovery of the single pulse, the energy in the cavity also increases slowly. Until the single pulse reforms at RT 5550, the energy in the cavity returns to the level before the collision. Due to the formation of the DC component, the energy in the cavity continues to increase. At RT 5790, the intra-molecular pulses of SM1 begins to chaotically oscillate, accompanied by the generation of new pulses, so the energy in the cavity decreases rapidly. The gain is depleted after experiencing the violent oscillation of energy, and after a period of recovery, the energy in the cavity increases slowly. Finally, it reaches stability and the laser continues to operate in the stage of “3 + 1” three pulses bunch and single pulse.
3.2 Direct interactions of soliton molecules
By adjusting the PCs and pump power, the state of the laser cavity is restored to the tight soliton molecular complex. On this basis, we also use the signal generator to apply a short-term sinusoidal signal to the pump laser, so that the pump power is disturbed and the direct interaction between soliton molecules occur. Figure 8(a), (b) and (c) show the time-domain, the real-time spectra and AC trace of this dynamics, respectively. It can be divided into three stages: tight soliton molecular complex, intra-molecular oscillation dynamic, three-pulses bunch and single pulse.
Fig. 8. The direct interaction process of soliton molecules: (a) time domain evolution; (b) spectral evolution; (c) autocorrelation trajectory; (d) evolution curve of cavity energy during direct interactions of soliton molecules.
The stage of tight soliton molecular complex corresponds to the RT 1 to RT 1900 in Fig. 8. Figure 9(a) shows one of the single-shot spectra, and the fine modulation fringes on it are the result of interference between soliton molecules. The ACs in Fig. 9(b) shows that these two soliton molecules constituting the SMC have the same structure, and the intra-molecular separations are 2.04 ps and 2.13 ps, respectively. Figure 9(c) shows that the intra-molecular time separations fluctuate around 2 ps during the evolution, indicating that the two soliton molecules remain tight in this stage, and the intra-molecular relative phases show a step-like mutation after the gain perturbation. Figure 9(d) shows the evolution of the inter-molecular separation and relative phase. The inter-molecular separation represented generally tends to decrease with obvious fluctuations, indicating that the two soliton molecules oscillate simultaneously during the relatively close process. Moreover, the SMC has a sliding internal phase.
Fig. 9. The stage of tight soliton molecular complex: (a) the single-shot spectrum of RT 1000; (b) the ACs of RT 1 and RT 1900; (c) the evolutions of the intra-molecular separation and relative phase; (d) the evolutions of the inter-molecular separation and relative phase.
RT 1901 to RT 5275 in Fig. 8 is the intra-molecular oscillation state. The evolution of the ACs at this stage is shown in Fig. 10(a), which exhibit the soliton molecules undergo three states in turn: starting, chaotic and accelerating oscillation. In order to describe the evolution more intuitively, Fig. 10(a) gives the dynamic models of the starting and accelerating oscillation phases. Both the red and blue balls represent pulses, and constitute two soliton molecules, respectively. The red soliton molecule is denoted as SM1, and the blue one is denoted as SM2, which are both tight soliton molecules.
Fig. 10. The stage of intra-molecular oscillation: (a) the evolution of ACs of this stage; (b) ACs of intra-molecular pulses during the starting oscillation phase; (c) ACs of intra-molecular pulses during the chaotic oscillation phase; (d) the ACs of RT 2350 and RT 4880.
After a period of stable evolution of SMC, gain perturbation makes the adjacent pulses of the SM1 first start to oscillate, while the intra-molecular separation of the SM2 remains unchanged. The intra-molecular AC evolution during this phase is shown in Fig. 10(b), showing the two soliton molecules have different evolution processes. At the end of the starting oscillation state, the internal pulse of SM2 is “ignited” under the influence of SM1, thereby also starting to oscillate internally. At this point, it enters the chaotic oscillation state. The evolution of AC trajectory is shown in Fig. 10(c). The internal oscillations of the two soliton molecules are random and alternate, while the soliton molecules are close to each other with inter-molecular separation is reduced from 48.94 ps to 44.05 ps. Subsequently, the intra-molecular oscillation enters the acceleration state that starts from the RT 3010 and ends at RT 5275. In the whole acceleration oscillation phase, SM2 experiences two accelerations causing by the collision of intra-molecular pulses, while SM1 just oscillates without acceleration. At the same time, the inter-molecular separation decreases as a whole, indicating the soliton molecules are close to each other. Figure 10(d) shows the single AC during the starting and accelerating oscillation states. The intra-molecular separations of the two soliton molecules at this time are 2.14 ps and 5.19 ps at RT 2350, and SM1 has already started to oscillate, while the SM2 has not yet started. At RT 4880of the accelerating oscillation state, the intra-molecular separations are 5.19 ps and 9.77 ps, respectively. The intra-molecular separation of SM1 is almost unchanged, but SM2 is already transformed into a loose type at this time.
RT 5275 to RT 10300 in Fig. 8 are the three-pulses bunch and single pulse state. The AC traces in Fig. 11(a) shows that the pulses have different evolution processes under different interactions. The first process is the annihilation and recombination of pulses. In the late stage of the intra-molecular oscillation, the pulses constituting SM1 collide, and its structure dissociates after the third collision. Then, one of the pulses gradually approaches SM2. and form bunch of three-pulses, as shown in the region I of Fig. 11(a). At RT 6134, two of the pulses constituting the three-pulses bunch annihilate after colliding with each other, after which only two pulses exist and form a loose soliton molecule under weak interaction. Figure 11(d) shows the evolution in the spectral domain, and the dashed box in the figure is the evolution process of the Kelly sidebands. With the annihilation of pulses, the intensities of Kelly sidebands are significantly enhanced, as is the spectral substrate intensity. Figure 11(e) shows the single-shot spectra before and during the annihilation of the pulses, corresponding to RT 6000 and RT 6410, respectively. Figure 11(f) shows a magnification of the local spectra in the black dashed box of Fig. 11(e). At RT 6000, the intensity of the Kelly sideband is lower than 0.1, while its intensity increases to 0.32 at RT 6410. Bifurcations also appear in the Kelly sidebands, which may be due to the simultaneous phase-matching conditions of dispersive waves of two adjacent wavelengths. At the same time, the intensity of the spectral substrate also increased by an order of magnitude. Therefore, we believe that the energy radiated by pulse annihilation is transformed into a continuous DC component in the cavity after soliton collision. With the evolution to RT 6660, two pulses regenerate at the location of the double-pulses bunch and beyond. The regeneration process of the pulse is shown in the dashed box in Fig. 11(c). The laser operates again in the state of “3 + 1” three-pulses bunch and single pulse, which corresponds to the region III in Fig. 11(a). As shown in the yellow curves in Figs. 11(e) and (f), the intensities of the Kelly sidebands and the spectral substrate at RT 9500 both decrease compared with the RT 6410, and are basically the same as the one before pulses annihilate at RT 6000. Therefore, we believe that the generation of new pulses is the related to the DC component in the cavity.
Fig. 11. The stage of three-pulses bunch and single pulse: (a) AC trajectories and (b) dynamics models; the phase of pulses annihilation and recombination: (c) time domain evolution and (d) spectral evolution; (e) single-shot spectra of RT 6000, RT 6410 and RT 9500; (f) Zoom in on the details of the black dashed box in Fig. 6(e)
Figure 8(d) shows the energy evolution of the cavity during the direct interactions of soliton molecules. In the stage of tight SMC, the energy is stable due to the stable structure of the two tight soliton molecules under the strong interactions. As the intra-molecular pulses begin to oscillate, they interact with each other and complex evolution in the energy within the cavity occur. The collision of adjacent pulses always accompanied by a dramatic increase in the energy, which is then dissipated as the evolution of soliton structure. In the late stage of the accelerating oscillation, the energy reaches a minimum. At this time, one of the soliton molecules changes from a tight type to a loose type, and the intra-molecular spacing of the other soliton molecule is also increased compared with the stage of SMC. With the dissociation and recombination of the structure of the soliton molecules, three-pulses bunch is formed. The effects of coherent overlap diminish with the increase of the separations between pulses, which cause the intracavity energy increases. From RT 6134 to RT 6600, two pulses of the three-pulses bunch annihilate after collision. The energy is radiated in the cavity, which converted into a continuous DC component and strength the dispersive wave, so the energy in the cavity is greatly increased in a short period of time. The global action of the DC component diverts the pulse energy, allowing it to regenerate at a new location, so the intracavity energy is again reduced to the level before the pulses annihilate.
Although the stages experienced by soliton molecules in direct or indirect interactions are different, the result of gain perturbation is to change the operating state of the laser from “2 + 2” soliton molecules to “3 + 1” three-pulses bunch and single pulse. From the perspective of nonlinear dynamics, these two states actually correspond to different attractors in the system. The gain disturbance breaks the stability of the attractor of the system corresponding to the state of “2 + 2” soliton molecules. With the generation of the DC component, the system gradually converges to the attractor of the state of the three-pulses bunch and the single soliton. The differences in the stages that the soliton molecules undergo during the two interactions, the energy evolution are also different, that is, there is a large gap between the initial and final states of energy in the indirect interactions of the soliton molecules, while it eventually almost returns to the initial level in the direct interactions of the soliton molecules.
4. Conclusion
In summary, we have demonstrated the oscillation and collision behavior of pulses within soliton molecules, as well as the phenomenon of pulse recombination through interactions between soliton molecules. First, the laser output a “2 + 2” soliton molecular complex composed of two loose soliton molecules, and the initial inter-molecular separation is on the order of nanoseconds. After applying gain perturbation, the evolution of the two soliton molecules goes through the stages of the intra-molecular periodic vibration, intra-molecular oscillation, three-pulses bunch and single pulse in sequence. Afterwards, by adjusting the polarization controller, the initial inter-molecular separation is decreased to the picosecond level. The gain perturbation is also applied, and the evolution of the soliton molecules sequentially experience tight soliton molecular complex, intra-molecular oscillation, three-pulses bunch and single pulse. Although the evolution processes of the soliton molecules with different initial states are different, the common point is that the splitting and pulse recombination of the soliton molecules are accompanied by the energy change in the cavity and the energy transfer based on the DC component. We believe our findings will provide an important idea to study and harness multi-soliton interactions.
Funding
National Key Research and Development Program of China (Grant No. 2018YFB0504400); National Natural Science Foundation of China (Grant No. 12274238, Grant No. 61835006, Grant No.62205159); Natural Science Foundation of Tianjin City (Grant No. 19JCZDJC31200); Special Project for Cooperation in Basic Research of Beijing, Tianjin and Hebei (Grant No. 21JCZXJC00010).
Disclosures
The authors declare no conflicts of interest.
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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