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Abstract
Dissipative systems form various self-organized states owing to the abundant attractor structures. The study of the response of different self-organized states under collision perturbation is of great significance for understanding the dissipative nonlinear systems. The collision dynamics of single soliton and soliton molecules can not only assist the stability analysis of attractors, but also reveal the rich physical connotations of soliton interactions. Here, for the first time, the collision processes of single soliton and soliton molecules in different excited states are detected using the dispersive Fourier transform technology. The collision processes include the disintegration and rebuilding of soliton molecules as well as chaotic oscillating evolution, accompanied by the emergence of transition states such as triple binding state, soliton fusion and acceleration. According to whether the soliton molecule can return to its initial excited state, the collisions are classified as elastic and inelastic. The different interaction strength between solitons is an important condition for rebuilding stable soliton molecules. Numerical simulations show that the gain dynamics are the main physical origin of collisions. Our research will stimulate in-depth research on the interaction of self-organized states in nonlinear systems such as chemical molecules, and have potential applications in optical logic gates.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Soliton, a localized structure related to the balance between linear and nonlinear physical effects, has always been a crucial subject in nonlinear sciences such as fluids, plasma physics, and optics. In optics, the generalized concept of dissipative solitons is defined as a localized solution in space or time, resulting from the composite balance among dispersion, nonlinearity, gain, and loss [1]. As a dissipative system, passively mode-locked lasers (PMFLs) have always been an ideal platform for studying soliton dynamics, such as soliton explosions [2,3], soliton rains [4,5], soliton pulsations [6,7], and soliton molecules (SMs) [8,9]. With the recent development of real-time detection technology, especially the dispersive Fourier transform (DFT) technique, abundant detailed transient dynamics in mode-locked lasers have been observed [10,11]. Recently, this real-time measurement technique has been used to investigate many transient dynamics in mode-locked lasers, such as the buildup of soliton [12,13] and SMs [14,15], as well as the internal motion of soliton pairs [16,17], soliton triplets [18] or other soliton molecular complexes [19], etc.
Collision of solitons is a basic and important type of soliton interaction and has been widely studied in different systems [20–22]. In integrable systems, based on the nonlinear Schrödinger equation, two solitons collide elastically owing to the differences in phase and group velocities. It is worth noting that in dissipative systems, the collision of solitons has a more complex dynamic process. The interaction force between the solitons originates not only from the different phases and velocities, but is also related to the long-range force caused by the depletion and slow recovery of gain [23], as well as the medium-range force mediated by dispersive waves [24]. Moreover, it is also remarkable that soliton-bound states can be formed owing to this nonlinear interaction process [25]. SMs have been highlighted with remarkable properties in the analogy of matter molecules, not only because of their various stable or vibrating states, but also because of their different binding energies related to different excitation states [15]. In recent studies, soliton molecular complexes [19] and supramolecular complexes [26] have been discovered, showing similar behaviors between optical dissipative SMs and matter molecules. Moreover, the non-stationary dynamic process of matter particles has always been an important subject in particle physics, containing rich physical connotations such as the interaction of multi-particle systems. For example, ultra-cold atom collision is an important method for studying the non-stationary dynamics of particles. In a previous report, two important collision processes, that is, elastic and inelastic collisions, were discovered in the atom-molecule collisions with the same or different compositions [27,28]. Interestingly, the behavior of optical SMs in different excited states collision with a single soliton (SS) has been a fascinating basic collision problem in particle physics. This is because the collision between a SS and SM, has a richer dynamic process in analogy with matter particles. Because of the relatively better demonstration of attractor stability in dissipative systems, this type of collision has received extensive attention in previous works. Through theoretical predictions, it was found that SS and SM can lead to a variety of collision results [29], and most of these results have been demonstrated in the experiments [30–33]. However, the collision of solitons is a significantly short time-scale dynamic process. In the early traditional time-averaged measurements, owing to the lack of real-time detection techniques to obtain dynamic details, the collision dynamics between SS and SM can only be described in an insufficient frame.
To the best of our knowledge, this is the first study to reveal several typical dynamics of the collision processes between SS and SM, where the time domain and real-time spectrum are measured using a high-speed oscilloscope and DFT technology. The real-time spectrum was analyzed, taking into consideration that the disintegration of SMs is a typical process in the collision dynamics between SSs and SMs. Different interaction strengths between solitons lead to different collision results. Through numerical simulations, we observed the drift speeds of pulses with different self-organized states under the influence of gain dynamics, and its relationship to the collision between SS and SM.
2. Experimental setup
Our experimental setup uses a fiber laser based on the NPR mode-locking technology as shown in Fig. 1. The gain medium is a segment of 1.5 m long erbium-doped fiber (EDF) whose group velocity dispersion is 40 ps2/km at 1550 nm. All the other fibers including device pigtails are standard single-mode fibers (SMFs) with a group velocity dispersion of −23 ps2/km. The polarization dependent isolator (PD-ISO) sandwiched by two polarization controllers (PCs) is served as an artificial saturable absorber to realize mode locking. The laser has a cavity length of 13.8 m with the fundamental frequency of 15.1 MHz, and operates in the anomalous dispersion region with a net dispersion of -0.223 ps2. We distributed a 1500 m dispersion-compensating fiber (DCF) with a dispersion parameter of -131.34 ps/nm/km to time stretch the real-time output of the laser. The laser output is split into three branches by two optical couplers (OCs). One branch is used to synchronously monitor the average spectra by an optical spectrum analyzer (OSA, Yokogawa, 70D). Another branch is used for the measurement of real-time spectra by the DFT. The signal from the last branch is directly sent to another photodetector. We used a 45 GHz bandwidth photodetector (PD, DiscoverySemi, DSC10H) to detect real-time output intensity, and recorded it using a high-speed oscilloscope with a 33 GHz bandwidth and a 100 GHz sampling rate (Tektronix DPO75902SX). Using these detection techniques, our time resolution and spectral resolution are 10 ps and 0.15 nm, respectively.
By increasing the pump power and properly adjusting the polarization controllers (PCs), a stationary SM mode-locking state can be achieved. Based on the peak-power clamping effect [34], increasing the pump power results in the formation of bound states of two or more identical pulses with a pulse interval of a few picoseconds. When increasing the pump power to a certain threshold value, the laser forms a stable mode-locked state, which consists of a SS and a SM. Moreover, the SM is either in a tight or loose bound state.
3. Results
3.1 Inelastic collision dynamics with SM transition
By slowly adjusting the pump power to 75.2mw, we obtain a situation where the loose SM transitions to a tight SM by colliding with a SS. As shown in Fig. 2(a), the collision dynamics can be divided into three processes: the approaching process, chaotic oscillating process, and repelling process. The time domain and spectral evolution of the pulses are illustrated in Figs. 2(b) and 2(c), respectively. As shown in Fig. 2(b), the relative speed of the colliding solitons undergoes an obvious change after the collision. The SS approaches the SM with a small relative speed, however, it exhibits a faster repulsion. Figure 2(c) illustrates the real-time spectra, and the interference structures of the SM indicate the change in the SM state before and after collision. The buffer can be realized by the continuous wave of the background during the collision, which will reduce the coherence between the dispersive wave and the solitons, resulting in chaos or shedding of the Kelly sidebands [35,36]. Figures 2(d) and 2(e) show the spectra of loose and tight SMs, respectively. The blue curves were measured by optical spectrum analyzer (OSA), and the red curves were measured by DFT. The real-time single-shot spectra agreed well with the time-averaged spectra. The center wavelengths of the two states are 1558.9 nm and 1559.5 nm.
Fig. 2. Inelastic collision between SS and SM: (a) The frame of SM transiting collision dynamics. (b) Real-ti me characterization of collision dynamics. (c) Real-time spectrum of collision dynamic. (d)–(e) Optical spectrum of loose and tight SMs (blue curve is measured by OSA, red curve is measured by DFT).
The filed autocorrelation trace helps to obtain more details on the collision dynamics. Figure 3(a) shows the filed autocorrelation trace obtained by performing fast Fourier transformations of the real-time spectra. The distance between the central peak and the satellite peaks in the filed autocorrelation trace refers to the temporal separation between two of the three solitons. As shown in Fig. 3(a), a stable loose SM with 5.3 ps temporal separation collides with a SS. The velocity was 0.019 ps per round-trip. At approximately RT10123 (point 1 in Fig. 3(a)), when the SS separates from the left soliton of the SM (LSM) of approximately 60 ps, the SM disintegrates under the perturbation of a SS. Then, the SS undergoes chaotic oscillation with the broken molecule. At approximately RT15121 (point 2 in Fig. 3(a)), the broken molecule rebuilds to be a tight molecule with a 3.2 ps temporal separation. Figure 3(b) shows the separation and relative phases between the solitons that form the molecule. We note that the relative phase oscillates simultaneously with the interval and changes from 0.10π (0.3 rad) to 1.14π (3.5 rad). This means that the separation and relative phase of the loose SMs both have poor stability. Figure 3(c) shows the energy evolution of collision dynamics. With the increasing energy of a SS, the total energy of the three solitons increases simultaneously. At approximately RT10123 to RT15121, the bond energy of the molecule changes with the chaotic oscillation, leading to energy oscillation of the SM until the tight SM builds up. Before the disintegration of the SM, the energy was fixed at approximately 0.97. With the interval oscillation, the energy also oscillates and finally becomes constant at approximately 0.86. This means that the SM completes the state transition during the collision, from a weakly bound state with a low binding energy to a more stable tightly bound state. In this process, part of the binding energy of the SM may be transferred to the energy of the SS.
Fig. 3. Details of inelastic collision between SS and SM: (a) Filed autocorrelation trace of the collision between SS and loose SM and loose soliton transitions. (b) The relative phase evolution of loose SM. (c) Energy evolution of collision dynamics (blue curve for total energy of three solitons, red curve for SM).
3.2 Elastic collision
The elastic collision between a SS and a SM can be achieved by further improving the pump level to 89.7mw. Figure 4(a) shows the schematic of the main collision process. Similar to the inelastic collision, the collision dynamics can be divided into three processes: the approaching process, chaotic oscillating process, and repelling process. By contrast, the states of the SM remain the same before and after collision, while the relative speed between a SS and SM does not change. Figures 4(b) and 4(d) show the collision dynamics in spectra and time evolution, respectively. As shown in Fig. 4(d), the weaker pulse is a SS and the brighter pulse represents a SM. The SS and SMs meet and separate at a constant relative speed. As shown in Fig. 4(c), the spectra of the molecule remain the same before and after collision, which means that the structure of the soliton molecule is rebuilt after collision. This is a typical elastic collision. From approximately RT7000 to RT8000, the evolution of interference fringes on the spectra in Fig. 4(b) indicates that the three solitons undergo a complicated interaction process. Similarly, the decrease of the coherence between soliton and dispersive wave leads to the shedding of Kelly sidebands.
Fig. 4. Elastic collision between SS and SM: (a) The frame of elastic collision dynamics. (b) Real-time spectrum of collision dynamics. (c) Optical spectrum before and after the collision. (d) Real-time characterization of the collision dynamics.
Real-time spectra measured by DFT provide sufficient resolution to build up the dynamics in the complicated collision region. Figure 5(a) shows the filed autocorrelation trace of the entire process. Four satellite peaks can be clearly seen in the autocorrelation trajectory, which is the result of the interference of the three solitons at unequal intervals. Figures 5(b) and 5(c) show a close-up figure of the collision region. The complicated collision region can be divided into four parts. First, when the separation between SS and the right soliton of the SM (RSM) is close to the inner soliton separation of the SM, the SM disintegrates. Then, the soliton system with three solitons oscillates near the equilibrium distance of approximately 2 ps. After going through an accelerated separation process, the system expands to another transient state only lasting for approximately 80 roundtrips, that is, the triple binding state with an interval of 3.2 ps. As the system shrinks, the unstable state eventually disappears. In part III, the three solitons repeat the oscillation dynamics in process I at a balanced distance of approximately 2 ps for approximately 150 roundtrips. At RT 7585, the oscillation disappears, the soliton repels violently, and one of the solitons accelerates away from the other two solitons. The main process in the fourth stage is the formation of a new SM. The distance between the escaped SS and the remaining two solitons is much larger than the distance between the two solitons that formed the new SM. Therefore, the interaction force between the SS and the other two solitons is much smaller than the internal force between these two solitons. This different strength of interaction provides the conditions for the two solitons to form a new SM. Finally, the stable SM forms, which have the same separation and relative phase as those before the collision.
Fig. 5. Filed autocorrelation trace of elastic collision between SS and SM: (a) Filed autocorrelation trace of collision dynamics. (b) Close-ups of the collision region A. (c) Close-ups of collision region B.
Figure 6 shows more details regarding the data of the collision dynamics. As shown in Fig. 6(a), during the collision dynamics, the energy changes synchronized with the state of system and are mainly related to the separation between solitons. As the states of the SMs before and after the collision are the same, which is a typical feature of elastic collision. To study the stability of SM under the perturbation of a SS, Fig. 6(b) illustrates the separation and the relative phase of SM during the SM disintegration. The phase stability disintegrates earlier than that of the separation. Note that the phase of SM is more sensitive than the separation inside SM, unlike the loose SM. During the collision process II, the appearance of two satellite peaks means that three solitons self-organize to the triple bound state with equal separation. The three-soliton state can be considered as a combination of two soliton molecular systems, where the interaction of the two systems is transmitted through the solitons they share. Figure 6(c) shows the typical autocorrelations. The changes in the intensity of the first-order autocorrelation satellite peaks indicate that the phases of the three solitons are changing asynchronously, which shows that in this stage, the interval between solitons is stable relative to the relative phase between solitons. These two soliton pair systems mainly experience phase dynamics. We noticed that in Fig. 5(b), the state transition experienced two strong repulsive processes. The soliton acceleration occurs in RT7315 and RT7587 [points Acceleration 1 and Acceleration 2 in Fig. 5(c)]. Figure 6(d) shows the autocorrelation at the corresponding roundtrips. Considering the continuity of the autocorrelation trajectory evolution and only one satellite peak, it shows that there are two solitons in the cavity, where soliton fusion occurs. According to the law of soliton energy, soliton fusion is unstable under this pump energy, so soliton will be regenerated. The new soliton repels due to the energy effect [37]. The distance between the two solitons in RT7587 is smaller than that in RT7315, which leads to a stronger repulsion process in the system, resulting in the difference in the interaction between the three solitons.
Fig. 6. Details of elastic collision between SS and SM: (a) Energy evolution of the collision. (b) The separation (blue curve) and relative phase between SM (red curve) from RT6200 to RT7300. (c) Filed autocorrelation trace at RT7368 (blue curve), RT7410 (red curve), and RT7421 (yellow curve). (d) Filed autocorrelation trace at RT7315 (blue curve) and RT7587 (red curve).
3.3 Inelastic collision dynamics with SM broken
To analyze the dissociation of SMs in the general collision process, we changed the soliton organization state by adjusting the pump power to 79.3mw. The situation in which the SM disintegrates by the perturbation of a SS is shown in Fig. 7. Figure 7(a) shows the real-time evolution in the time domain. Resembling the situations discussed above, the SS approaches the SM at a certain relative speed. The SS and SMs remain stable when they are separated. Figure 6(b) shows the spectral evolution of collision dynamics. The stable spectral structure of the soliton changes to a chaotic interference spectral structure after collision.
Fig. 7. Inelastic collision between SS and SM: (a) Time domain of the collision between the SS and compact SM. (b) The real-time spectra of the collision between the SS and compact SM measured via TS-DFT.
Figure 8(a) shows the filed autocorrelation trace of the collision dynamics. The separation and relative phase of the SM are 3.2 ps and 1.14π (3.5 rad), respectively. The constant relative velocity was 0.067 ps per roundtrip. At approximately RT14143 (point 1 in Fig. 8(a)), the SM disintegrates and the RSM attracts a SS. It is noteworthy that the separation of the SM is unstable when the distance between the SS and RSM is 20 ps. The stability of the SM in this situation is much stronger than that of the loose SM, but weaker than that in the second situation. As shown in Fig. 8(b) in detail, SS and RSM attract and when the separation between a SS and RSM is equal to the separation of the broken SM, the relative movement of a SS and SM becomes slow, and three solitons experience the same oscillation dynamics as the second collision situation, which has been discussed previously. During the collision dynamics, the solitons also separate quickly owing to the exclusion. The strong repulsion of solitons due to soliton fusion and regeneration can also occur. Finally, the three solitons jitter slightly over time, chaotically and asynchronously. It is worth noting that unlike the elastic collision situation, after the acceleration process, the interval between the three solitons is almost equal and greater than 20 ps, that is, the interaction strength between the solitons is weak and without difference. This is an important reason why SMs cannot be rebuilt. Figure 8(c) shows the energy evolution of the collision dynamics. The energy of the system increases because the bound energy of the solitons is lower after collision. To better characterize the collision dynamics, the phase diagrams of SS-RSM and SMs are shown in Figs. 8(d) and 8(e), respectively. As shown in Fig. 8(d), the SS-RSM dynamics experiences a spiral in the phase diagram, which means that the relative phase between SS and RSM rotates with a constant velocity. After the SM disintegrating [black curve in Fig. 8(d), the relative phase between SS and RSM is -0.67π], the evolution of the relative phase between SS and RSM experiences a turning point and becomes slow. Similarly, the disintegration dynamics of SM are shown in Fig. 8(e), where the red curve illustrates that the relative phase of the SM experiences oscillation before the separation instability.
Fig. 8. Details of inelastic collision between SS and SM: (a) Filed autocorrelation trace calculated from the spectra, exhibiting the separation among three solitons during the entire collision region. (b) Close-ups of the SM disintegration region. (c) Energy evolution of collision dynamic. (d) Dynamic phase diagram of disintegration of bound state under SS perturbation from RT13750 to RT 14229 (From red to black). (e) Dynamic phase diagram of SS and LSM from RT13750 to RT14229. (From red to black).
4. Analysis and discussion
In the experiment, we obtain the collision dynamics of SS and SMs with different configurations, including elastic collision and inelastic collision. The change of soliton molecular configuration during collision and the different group velocity between SS and SMs are directly related to the properties of SM and the physical effects in the cavity. Soliton molecules act as the attractors in a nonlinear dissipative system, and they are the result of the equilibrium of the interaction between solitons. In previous theoretical work, the basic interaction of two solitons in a dissipative system has been extensively studied [25,38,39]. Generally, the dynamics of passively mode-locked fiber lasers can be qualitatively described by the complex Ginzburg Landau equation (CQGLE), which is the averaging model [40]:
The stable state of a binary SM occurs at the extremum of its potential function. Ignoring the gain dynamics effects in Eq. (1), the interaction of two dissipative solitons may be regarded as equations of motion for a mechanical system with two degrees of freedom in the presence of friction. Through the variational method, the normalized potential energy of the binary system can be represented as [41] :
where s and ψ are the relative separation and phase between the solitons in the molecule, respectively. Here $b = \frac{{ - ({r - {\textrm{g}_0}} )+ \beta {\eta ^2}}}{{{\eta ^2}}}$ and $\eta $ is the amplitude of the soliton. In this model, the SM can be regarded as a binary dynamic system, whose stable state appears at the potential minima. We set the parameters as $r = 2,\; \beta = 0.35,\; {\textrm{g}_0} = 2.6,\; \eta = 0.5$, and the normalized potential energy as a function of the separation and relative phases is shown in Fig. 9. There is a set of local minima in the potential energy function when ψ is 0 or π, which means that the SM has multi-stable states, similar to material molecules. Considering the potential energy distribution in Fig. 8, the inelastic collision with SM transition in experiment can be regarded as the transition of a SM breaking through the potential barrier from point A to point B.In experiment, the interaction between single soliton and soliton molecule is related to the pump level, and this means that the collision dynamics is related to the gain factor. To confirm and better interpret these complex collision dynamics, we considered a simulation of the averaging model based on Eq. (1) with the gain dynamics. When we set the parameters as$D = 1,\mathrm{ }r = 2,\mathrm{ }\varepsilon = 0.58,\mathrm{ }\beta = 0.35,\mathrm{ }\mu ={-} 0.12,\mathrm{ }\nu = 0,\mathrm{ }{I_s} = 0.28,\mathrm{ }{\mathrm{g}_0} = 2.6,\mathrm{ }\Gamma = 0.001$, the inelastic collision situation is shown in Fig. 10(a). The two initial pulses on the right form an oscillating soliton pair. It is known that solitons drift under the influence of the fast gain response coefficient Γ. The soliton pair moves faster than the SS, which means that different soliton organization states will have different drift speeds under a fast gain response. As in the experiment with a SS colliding with a tight SM, when the three solitons are equal spaced, the soliton pair dissociates and then the SS and LSM form a new SM. Considering the changes in the molecular structure of the soliton before and after the collision, it can be deduced that their drift speeds are also different, demonstrating that the gain relaxation has different effects on different soliton self-organization states. Comparing the SM state before and after the collision, it can be found that the soliton pair after the collision is tighter and has weaker oscillations, leading to different drift speeds of the two SMs, indicating that this collision is an inelastic collision. The elastic collision can be obtained when the parameters are as $D = 1,\; r = 2,\; \varepsilon = 0.58,\; \beta = 0.5,\; \mu ={-} 0.12,\; \nu = 0,\; {I_s} = 0.32,\; {\textrm{g}_0} = 2.5,\; \Gamma = 0.002$, shown in Fig. 10(b). During the collision, the three solitons formed a short-term soliton triplet state. Finally, the SS and LSM formed a new soliton pair. The new soliton pair is separated from a SS owing to the difference in the drift speed. It should be noted that the state of the soliton pairs before and after the collision is consistent, which shows that this is an elastic collision.
Fig. 10. Inelastic and elastic collision between SS and SM based on CQGLE: (a) Time domain evolution of inelastic collision. (b) Time domain evolution of elastic collision.
Although the average model cannot compare the experimental results quantitatively, our results show that SSs and different soliton molecular configurations will obtain different group velocity responses under the influence of gain dynamics. On the other hand, the inelastic and elastic collision under different gain parameters, provide insights into the gain dynamics effect for the experiment. In fact, it is also of great significance to discuss the collision of SMs with SSs of different relative velocities, which has been analyzed in previous theoretical work [42,43]. Note that, the collision threshold of SMs in different excited states is an important parameter to understand the stability of attractor structure and the interaction of multiple solitons in dissipative systems. By the way, our numerical simulation points out the influence of gain dynamics on the drift speed of soliton, indicating that controllable collision dynamics can be achieved by regulating the pump source. Interestingly, the regulation of soliton dynamics through the regulation of the pump source has been reported, including the regulation of the internal dynamics of SMs [44] and the interaction of SM pairs [45]. In future work, we will conduct more in-depth research on the properties of SMs through controllable collision dynamics.
5. Conclusion
We reported the first direct observation of collision dynamics between a SS and a SM through real-time, single-shot spectral measurements supported by the DFT method. We point out that SMs in different excited states have different responses to the collision perturbation of a SS. The dissociation of the bound state is a typical process of the SS-SM collision dynamics. Moreover, the tighter SM can withstand greater perturbation of a SS. In the chaotic oscillation process, we observed the relationship between soliton fusion and soliton acceleration. The different separation between solitons caused by soliton acceleration means different interaction strengths between solitons, which is an important condition for the rebuilding of stable SMs. Finally, through numerical simulations, we show that the gain dynamics process causes different soliton self-organized states to have different group velocities, which is the main reason for the collision of SMs and SSs in dissipative systems. The investigation of these intriguing collision processes facilitates an understanding of the particle nature of solitons. In our investigations, when the soliton interacts in the dissipative system, the processes involve the three-body interaction of the soliton, as well as the complexity of the energy changes. This is a significant supplement to previous work on soliton collisions. This work will facilitate a wide range of studies on collision dynamics between different self-organized states, including pulsation solitons and spatiotemporal solitons. Similarly, the collision dynamics of optical SMs will inspire in-depth research on the interaction of self-organized structures in dissipative nonlinear systems such as chemical molecules, binary star systems, and condensed matter physics. More interestingly, the control of the initial conditions of collisions to realize the transformation of soliton molecular states has potential applications in optical logic gates.
Funding
National Key Research and Development Program of China (2018YFB0504400); National Natural Science Foundation of China (11674177, 61640408, 61775107); Natural Science Foundation of Tianjin City (19JCZDJC31200).
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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