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Abstract
In passively mode-locked fiber lasers (PMLFLs), the dissipative solitons (DSs) can self-organize to form complex structures through delicate interactions. However, it is still elusive to control these soliton structures by external influences. We here find that at a certain critical power, the location between two soliton molecules can be controlled by a slow modulated pump power. After applying the pump power with periodic fluctuation, two soliton molecules oscillate from the state of soliton molecular complex to stable distribution with maximum inter-molecular separation. During this process, the internal structure of each soliton molecule keeps steady. The slow gain depletion and recovery mechanism which plays a dominant role affects the motion of soliton molecules. These results could further expand the molecular analogy of spectroscopy and stimulate the development of optical information storage and processing.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In nonlinear optics, dissipative solitons (DSs) are localized structures in time and/or space dimensions, resulted from the composite balances between gain and loss, nonlinearity and dispersion/diffraction in resonant cavities [1]. As robust entities that can maintain long-term stability in a cavity, DSs have been proposed to use as an information carrier for optical buffer and signal processing [2,3]. In the passively mode-locked fiber lasers (PMLFLs), such particle-like structures can self-assemble into multi-soliton bound states, also termed soliton molecules (SMs) as the analogy with matter molecules is particularly striking in certain aspects [4–6]. Experimental and numerical studies have shown the internal motions of soliton molecules, akin to vibration, oscillation, synthesis and dissociation [7–12]. Soliton molecule can also function as a unit to interact with each other, showing various complex nonlinear dynamics processes. They can collide with each other resulting in dissociations or synthesis of new molecules, combine to form stable 2 + 2 soliton molecular complexes (SMC) or breather molecular complexes, bound states of soliton molecules, period doubling and harmonic mode locking of soliton molecules, and even “polymerize” into macromolecules and soliton crystals comprised up to thousands of optical soliton molecules [13–20].
The formation of these composite patterns is attributed to various nonlinear interaction mechanisms in PMLFLs. The direct soliton interaction is exerted on the neighboring solitons which locks their relative timing and phase, and certainly soliton profile dependent [21–24]. The slow gain depletion and recovery mechanism, that is the gain consumption of the former pulses results in the deceleration of the latter pulses, provides long-range repulsive forces for the soliton distribution [13,22,25,26]. The noise-mediated interaction (NMI) mechanism which is reminiscent of the Casimir-effect in quantum electrodynamics can be attractive at lager separation, and stable-state separation is related with the pedestal widths of solitons [27,28]. Another long-range soliton interaction occurs when the fields of dispersion wave of adjacent solitons are overlapped and whether the forces are attractive or repulsive depends on the phase between solitons and dispersive waves [22,23,29–31]. In addition, acoustic effect generated by electrostriction can induce attractive or repulsive interaction in fiber lasers, resulting in a new arrangement of solitons [32–35].
The motion of DSs in cavities can be manipulated externally. For the spatial DSs in semiconductor microcavity, they can be moved in the transverse plane by modulating holding beam [2]. In PMLFLs, it is found that an injected continuous wave (CW) can affect the structures and motions of solitons theoretically [36,37]. Either the soliton crystal, gas or liquid depends on the intensity and frequency of the CW, and harmonic passive mode locking will occur when the frequency of the CW coincides with the dispersion waves. Experiment work has discovered a resonance in the response of soliton molecules to external perturbation and achieved reversible all-optical switching between two bound states with different binding separations [38]. Note that the focus of these studies is to control the motion of particle-like solitons. Recently, the 2 + 2 SMC, exhibiting strong intra-molecular bonds and weak inter-molecular bonds, appears to extend the analogy between optical soliton bound states and matter molecules to molecular complexes which is universal in condensed-matter physics [16]. With that in mind, we naturally turn our attention to another problem, that is, whether it is possible to drive or control the motion of the soliton molecule as a unit and how it can be realized.
In this work, by adjusting pump power, we realize the external modulation of the self-organization process from a four-pulses soliton bunch to the stable distribution of double soliton molecules in an ultrafast fiber laser. Two soliton molecules, formed by the four-pulses soliton bunch after increasing pump power, move away and close to each other under the drive of gain with periodic fluctuation until they reach the furthest distance. Such final distribution can only be broken when the pump power is reduced to the point where the soliton molecules dissociate and soliton bunch restores again. During the oscillation, the soliton molecules exhibit rigid body properties without structural changes. The periodicity of the pump power is realized by the built-in circuit of the pumped laser. With the help of numerical simulations, the effect of gain on the motion of soliton molecules is discussed. Our works further extend the analogy between optical soliton molecules and matter molecules for their robust structure and rich interactions. It opens a new way to the manipulation of large-scale optical-soliton-molecule compounds, suggesting applications in fast optical sampling and information transmission.
2. Experimental setup and fitting model
The experimental arrangement is a passively mode-locked fiber laser with nonlinear polarization rotation (NPR) technology, as sketched in Fig. 1(a). A home-made pump laser contains a 980 nm laser diode (LD) and a built-in circuit, which enables the output of periodic pump power via rapid modulations of current. 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 at 1550 nm. 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 12.6 m with the fundamental frequency of 16.89 MHz, and operates in the anomalous dispersion region with a net dispersion of -0.195 ps2.
Fig. 1. (a) Schematic diagram of the experimental setup. Pump laser: a laser diode (LD) together with a built-in circuit, WDM: wavelength division multiplexer, EDF: erbium-doped fiber, OC: optical coupler, PC: polarization controller, PD-ISO: polarization dependent isolator, DCF: dispersive compensation fiber, OSA: optical spectrum analyzer, PD: photodetector. (b) The model of the oscillatory self-organization dynamics. The red structure represents the soliton molecule 1 with the time-phase separation coordinate (τ1, α1), and the blue one represents the soliton molecule 2 with the coordinate (τ2, α2). (τ3, α3) describes the time and phase separations among the two soliton molecules. (c) The model of the final state of the dynamics. The intermediate disconnected “spring” together with some black points are used to describe that the two soliton molecules no longer oscillate towards each other with the maximum inter-molecular separation of τ3’, and remain stable under various interactions.
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, AQ6370D). Another branch is used for the measurement of real-time spectra by the time-stretch dispersive Fourier transformation (DFT) technology, which maps the pulse spectrum into a temporal waveform using chromatic dispersion [11,39]. The optical pulses are fed into a 1.05 km long dispersive compensation fiber (DCF) with the dispersion coefficient of −131.34 ps/(nm⋅km) at 1550 nm to stretch the pulses, then they are captured by a photodetector with a 45-GHz bandwidth (PD, Finisar XPDV2320R). The signal from the last branch (undispersed) is directly sent to another photodetector. Both of them are monitored by two serial connection high-speed oscilloscopes with a 59-GHz bandwidth and 200-GHz sampling rate (Tektronix DPO75902SX), thus yield the spectral evolution as well as the temporal evolution with a resolution of approximately 0.15 nm and 5 ps, respectively.
In our case, the laser generates soliton molecules comprising two solitons. Unlike a single pulse, the soliton molecule has two internal degrees of freedom corresponding to the temporal separation τ and the relative phase α, respectively. The oscillation model of the soliton molecules under gain modulation is provided in Fig. 1(b), we use the time-phase separation coordinates to refer to a set of variables (τ, α) of the soliton molecules. For each soliton molecule, (τ1, α1) represents the time and phase separation between the two solitons of soliton molecule 1 (SM1), and (τ2, α2) represents the separation of the other soliton-pair consisting the soliton molecule 2 (SM2). All of them are the intra-molecular freedom. It should be noted that τ1 and τ2 are considered to be equal in the present work, which are only stated to distinguish the two soliton molecules here. Besides, the coordinate of (τ3, α3) is utilized to describe the inter-molecular freedoms which are the separations between the two molecules. Since the forced oscillation of two soliton molecules is akin to the resonance of a spring oscillator with double oscillators, we regard the time separation between them as a “spring”, while the time separations within the soliton molecules are basically unchanged. Figure 1(c) shows the model of the final state of the dynamics. The soliton molecules no longer oscillate with the maximum inter-molecular separation of τ3’, which remains invariable under the influence of various interactions.
3. Results
3.1 Oscillatory self-organization dynamics between soliton molecules under gain modulation
We start at a pump power of P0 = 20.82 mW, where self-starting mode locking is accompanied with the generation of four pulses per cavity roundtrip (RT). Modulation fringes appear on the optical spectrum [Fig. 2(a)] measured by the OSA, and the pulse structures [Fig. 2(c)] show that the durations between neighboring pulses are 35 ps, 35 ps and 39 ps, respectively. As soon as the pump power is finely tuned to P0 = 21.03 mW, pulses self-assemble into a 2 + 2 SMC consisted of two soliton-pair molecules. As shown in Fig. 2(b), the average spectrum presents symmetric modulated fringes with a central dip, indicating out-of-phase bound solitons with a separation of 2 ps within each soliton pair. Figure 2(d) shows the temporal separation between soliton molecules 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.
The pump laser’s built-in circuit is programmed to output periodic pump power. The period of pump power is set as 8 s and average power is set as 21.03 mW where the SMC can exist in the cavity, and the modulation amplitude is set as 0.3 mW. As shown in Fig. 3(a), the pump power measured at room temperature exhibits a significant sine-like periodicity within 700 seconds. The temperature of the circuit and the experimental environment cause the slight drift of the pump fluctuation. This drift is a little bit obvious in comparison with the small amplitude of modulation. However, the pump power fluctuation is still periodic under our modulation, which can be proved by the Fourier transform of the pump power without direct current (DC) component, as shown in Fig. 3(c). The frequency of the measured pump fluctuation is 0.1314 Hz, corresponding to a 7.6 s average period. Such deviation with the set period is also considered to be caused by thermal effect of the current. The a, b and c points in Fig. 3(d) represent equilibrium positions of the pump power which are corresponding to the purple line in Fig. 3(a).
Fig. 3. (a) The pump power with a significant sine-like periodicity within 700 seconds. The purple dotted line represents the equilibrium positions in the whole sine-like oscillation process of gain. (b) Color plots made up of successive oscilloscope measurements at the cavity output show the temporal evolution of two soliton molecules over 170 s. The insert is the Fourier transform to the relative separation of two soliton molecules. (c) The Fourier transform to the measured output power of the pump laser. (d) A magnification of the red part in Fig. 3(a). (e) The recoverable-oscillation part magnified in the green circle. (f) The unrecoverable-oscillation part amplified in the black circle.
Overall dynamic process of soliton molecules is monitored by real-time spectral measurement. We perform a trigger operation on the high-speed oscilloscope and take the SM2 as the reference frame. The time domain trajectory and spectral evolution of the dynamics are recorded by taking the moment that generate the soliton molecular complex as the time origin (t0 = 0), and the whole process last nearly 170 s (see Visualization 1). In Fig. 3(b), we show a color plot of successive traces of the oscillation dynamics. The track of SM2 is plotted in blue line and set as zero point, so the SM1 represented in red line reflects the change of the temporal separation of the two soliton molecules. It’s clear that the separation experiences a periodic oscillation.
Induced by the pump power with periodic fluctuation, the two soliton molecules begin to move with different group velocities in the cavity soon after the formation of the SMC, showing approaching and moving away from each other alternately. For the first 120 s of the laboratory time, the inter-molecular separations oscillate periodically between the order of 0.01 ns and 1 ns. The soliton molecules can always return to the state of the SMC after each oscillation period, which is defined as the recoverable-oscillation stage. Part of this stage is shown in Fig. 3(f). Then their separations oscillate from the order of 1 ns to 10 ns after 120 s, showing an increasing trend on the whole. This is the unrecoverable-oscillation stage, as shown in Fig. 3(e). Finally, they no longer move relatively to each other and the separation stabilizes at ∼19.3 ns. Figure 4 gives several typical single-shot DFT spectra of the dynamics, which change in step with the inter-molecular separation. The blue box corresponds to the recoverable-oscillation stage, while the red box represents the unrecoverable-oscillation stage. The maximal inter-molecular separation τ3’ is up to 33.88 ns among our experiments. It is worth noting that the state transitions between the soliton bunch and the soliton molecules are unidirectional. This virtually error-free regulating on the oscillatory self-organization dynamics of soliton molecules is also repeatable by reducing the pump power until a four-pulses bunch forms again.
Fig. 4. Typical single-shot spectra obtained with DFT in the evolution process of the oscillatory self-organization dynamics of soliton molecules. The black arrows indicate the change of the pump power, and the blue and red arrows indicate the direction of oscillation of the soliton molecules. The blue arrows represent distance and red arrows represent proximity.
Applying the Fourier transform, the frequency of the oscillation dynamics is 0.1365 Hz, as shown in the insert of Fig. 3(b), which is basically the same as the fluctuation frequency of pump power, directly verifying the modulation effect of pump power on the motion of soliton molecules. The gain of the laser cavity is directly affected by sine-like oscillation of pump power, providing effective repulsive force for soliton molecules that changes periodically along with the gain jitter due to the slow gain depletion and recovery mechanism. The repulsive force which plays a regulatory role, with other abundant interaction forces that either repulsive or attractive, affects the motions together. As shown in Fig. 3(e), in one oscillation period during the unrecoverable-oscillation stage, the group velocities of SM1 and SM2 are equal (v1 = v2) at points of A, B and C. At these moments, the repulsive force and the attraction force acting on the soliton molecules are balanced, corresponding to points a, b and c in Fig. 3(d). In the a-b stage, the gain first increases and then decreases, but its value is always higher than the equilibrium value. The repulsive force is greater than the attraction forces, so the two soliton molecules are away from each other, which shows that the motion of SM1 is slower than that of SM2 (v1 < v2) corresponding to stage A to B in Fig. 3(e). As the repulsive force changes synchronously with the gain, the resultant force which is repulsive first increases and then decreases, resulting in the relative speed between SM1 and SM2 decreases and then increases to the same. In the b-c stage corresponds to B-C stage in Fig. 3(e), the gain changes just opposite to the previous stage. The resultant force is attractive, making the soliton molecules get close to each other, where v1 > v2.
Gain fluctuation has the same effect on the motion of soliton molecules for the recoverable-oscillation stage. However, the difference is that the soliton molecules will undergo a mutation of speed during the departure process, corresponding to the point D in Fig. 3(f). Before the point D, the relative motion of the two soliton molecules is slow, and the inter-molecular separation changes by ∼73 ps within 3.1 s. After the point D, their relative motion is much faster as the separation changes by 2.325 ns in 2.7 s. We believe that this is the result of strong inter-molecular attraction generated by the interaction such as coherent overlap when the soliton molecules are very close to each other. With the increase of the distance between the soliton molecules, this strong attraction constraint gradually weakens and will be broken off at a certain interval (the point D), then the repulsive force becomes more obvious and the departure motion is accelerated. The soliton molecules undergo the two stages, gradually getting further apart until they reach somewhere the forces are balanced. That is, the final state of the oscillatory self-organization dynamics of soliton molecules, which is robust for it will only be broken when the pump power is reduced to support the pulse bunch.
3.2 Recoverable-oscillation stage
At the beginning of the recoverable-oscillation stage, a 2 + 2 SMC is just formed with an inter-molecular separation of 35 ps that is 17.5 times greater than the intra-molecular separation. Their real-time spectra overlap after being stretched by the dispersive fiber, and the motion of SM1 can be directly observed in the real-time spectrum which has distinct changes. Figure 5 shows a complete process of approaching and moving away from each other of this stage. The period from t1 to t5 corresponds to a mutually exclusive process. From t6 to t10, SM1 moves faster than SM2 and approaches to it. In Fig. 5(a), when t1 = 7.3 s, the two soliton molecules basically coincide in the time domain with a separation of tens of ps. The pulses that make up the two soliton molecules overlap directly and affect the inter-molecular separation and relative phase. Obvious interference fringes appear in the real-time spectrum at this point, as shown in Fig. 5(b), and the total energy of the soliton molecular complex is the largest. With the departure of two soliton molecules, their spectra overlap varies from two peaks to the single peak, and then completely separate with the disappear of interference fringes. At the time of t5 = 13.3 s, the SM1 moves away to the position with the maximum inter-molecular separation about 2 ns (103 times of intra-molecular separation). Next, the SM1 approaches SM2 from t6 = 13.6 s to t10 = 16.2 s. At the end of the stage, the time separation between the two molecules drops back to the order of ps. Interference fringes appear again in the real-time spectrum, and the total energy of the soliton molecular complex returns to the maximum.
Fig. 5. The evolutions of (a) time domain and (b) single-shot spectra obtained with DFT of soliton molecules at different times in the recoverable-oscillation stage (see Visualization 1).
We set the sampling rate of the high-speed oscilloscope as 200 GS/s, and then capture the SMC in the recoverable-oscillation stage. Figure 6(a) gives the real-time spectrum evolution of 10000 successive cavity roundtrips with clear and dense interference fringes. The Fourier-transform of each DFT single-shot spectrum is computed, which is equivalent to a first-order optical autocorrelation (AC). As shown in Fig. 6(b), the AC trajectory exhibits periodic oscillation between soliton molecules. Within a period (red dashed line), the soliton molecules experience the complete process of approaching and departure. Figure 6(c) illustrates the evolution of the relative separation τ3 (blue curve) and relative phase α3 (orange curve). The part in the green dashed box corresponds to that in the red dashed box in Fig. 6(b), which is persist for ∼1320 RTs. The relative separation τ3 changes periodicity with the number of RTs, showing that the two soliton molecules approach and move away from each other in cycles. During this evolution, the inter-molecular separation is maximum at the RT 8 which is 39.44 ps, while the minimum is 38.41 ps at the RT 8865. The relative phase α3 is continuously increasing as a function of roundtrip number and slides to a higher frequency, indicating 2 + 2 SMC has a sliding internal phase. But it is also affected by the periodic motions of the two soliton molecules as it exhibits obvious fluctuation that is consistent with the period of the motions. The inset gives the intra-molecular separation and phase τ1 (τ2, blue curve) is always 2 ps, and α1 (α2, orange curve) remains 0.35, indicating that the soliton molecules maintain their robustness under gain modulation. Figure 6(d) shows the two single-shot AC traces within the period highlighted in Fig. 6(b). The temporal separation is 38.55 ps at the RT1 3910 (blue curve), and 39.33 ps at the RT2 4680 (orange curve), which are the closest and the farthest positions during this period, respectively.
Fig. 6. Characterization of 2 + 2 SMC in the recoverable-oscillation stage of the dynamics. (a) The real-time spectral dynamics. (b) Autocorrelation traces calculated from the real-time spectra over 10000 roundtrips. (c) Evolution of the inter-molecular relative temporal separation (blue curve) and relative phase (orange curve). The insert gives evolution of the intra-molecular separation (blue curve) and relative phase (orange curve). (d) Two examples of single-shot autocorrelation trace corresponding to the RT1 3910 (bule curve) and RT2 4680 (orange curve).
3.3 Unrecoverable-oscillation stage
We can see the evolutions in time domain and frequency domain of the unrecoverable-oscillation stage from Fig. 7(a) and 7(b), respectively. At the time of t11 = 127.2 s, the inter-molecular separation is about 2.5 ns and the real-time spectrum has already separated completely. The group velocity v1 of SM1 is less than v2 of SM2, so the SM1 is moving to the left with respect to the SM2, showing they are far away from each other. The SM1 first slows down and then accelerates as the temporal separation increases until it reaches the position with a relative separation of nearly 8 ns corresponding to t12 = 131.1 s. Later, the SM1 moves to the right because v1 is larger than v2. Its group velocity is decreasing to the same speed of SM2 while it is approaching the SM2 tardily, but it can't be back to where it was. After approaching for a distance ∼2 ns, the SM1 moves slower than SM2 again with t13 = 134.5 s, and then the above processes repeat. As we can see that at different moments, the distances of moving away from each other is always greater than that of moving close to each other, so it shows a trend of departure. Up to t20 = 160.2 s, the two soliton molecules are moving in sync and stabilize in this state, remaining relatively static with an invariant relative temporal separation τ3’ about 19.3 ns that is also the maximum inter-molecular separation of this periodic oscillation dynamics. It can be seen from Fig. 7(b) that spectral evolution is synchronized with the evolution in time domain.
Fig. 7. The evolutions of (a) time domain and (b) single-shot spectra obtained with DFT of soliton molecules at different times in the unrecoverable-oscillation stage (see Visualization 1).
4. Discussion
The effect of soliton interactions distinctly demonstrates the particle-like nature of solitons, which promotes to external manipulation of the soliton motions [36–38]. Based on the important differences between the intra-molecular and inter-molecular bonds in a stable SMC, we have proposed a method to control the soliton molecules. Although the slide- and oscillating-phase dynamics exist, the intra-molecular bonds manifest low sensitivity to external perturbations, while the inter-molecular bonds are more sensitive to external perturbations [16]. Thus, we take the soliton molecule as the function object under the condition that just focus on its separations and ignore the changes in phase. The gain is changed by controlling the current of built-in circuit settled inside the pump laser and drive the soliton molecules move with each other periodically under the slow gain depletion and recovery mechanism. The gain medium used in our experiment is erbium-doped fiber, whose relaxation oscillation time is about 10 ms. In order to eliminate the effect of relaxation oscillation, we set a relative long modulation period. It is worth noting that the laser will output a four-pulses soliton bunch when the pump power is 20.08 mW and five pulses at 22.53 mW. Therefore, in order to strictly ensure that the laser always operates in the state of two soliton molecules during the modulation process, we set the modulation amplitude to 0.3 mW. The oscillatory process of soliton molecules starts at the state of 2 + 2 SMC and end with the sable distribution with the maximum inter-molecular separation, involving multiple interactions during different stages.
The periodically varying force provided by the gain fluctuation results in the repulsion of soliton molecules all the time. In recoverable-oscillation state, the attractive forces generated by the overlap of the pulse envelope, such as coherent overlap interaction and dispersion waves, make the soliton molecules move close to each other. Those strong attractions involved in the recoverable-oscillation stage are comparable to the oscillatory repulsive forces and the soliton molecules can return to the state of SMC after each period of oscillation. With the modulation of the gain, the inter-molecular interval gradually oscillates from a few picoseconds to nanoseconds. It worth noting that the strength of attractive forces between solitons decreases exponentially with increasing spacing [40], while the repulsive force is in the same order of magnitude as the previous stage since it is mainly controlled by gain. At this point, the periodic equilibrium points along the cavity formed by electrostriction effect could contribute to the reciprocating oscillations in the process of moving away. In the cavity composed of SMF, the distance between two adjacent equilibrium points induced by sound waves typically on the order of a few ns [9], which is basically consistent with the experimental observation in Fig. 3(e). Such long-range optomechanical interaction is weaker than the short-range attractions, so that soliton molecules can hardly return to the state of SMC as the approaching process is shorter than the distance process, that is, the unrecoverable-oscillation stage. Each period of oscillation begins at the end of the previous period, so it appears the trend of moving away constantly. In addition, we have considered the important effect of CW on the motion of solitons [22,23,27–31,36,37]. But in the soliton molecule oscillation process, there is no CW component in the spectra, so we think the CW has no obvious effect on the oscillatory motion of the soliton molecule.
The periodic oscillation of soliton molecules under gain control can be understood as the process of constantly escaping and being trapped by one-dimensional finite potential wells, which refers to a potential well with zero potential inside the well and a finite value of V0 outside the well in the field of quantum mechanics. These potential wells are arranged in sequence. The initial state of the dynamics, a soliton molecule complex consisted with two soliton-pair solitons, is located in a weak well. The potential energy of the two soliton molecules increases gradually under the disturbance of various interactions with gain as the dominant factor, and then they are easy to break through the well wall. Then they fall into the next finite potential well with their potential energy gradually decreases and returns to the state of soliton molecule complex again. Until finally, the soliton molecules are trapped in a strong finite potential well. The “strong” and “weak” are used to describe the potential of the well wall. At this time, the potential energy generated by gain fluctuation and other interactions cannot support the soliton molecules to escape from the strong potential well. Further increasing the gain variation, that is, providing more energy to the soliton molecules, can make them break through the barrier.
To further verify the periodic modulation effect of gain on the oscillatory self-organization dynamics of soliton molecules, we simulate the evolution based on the complex Ginzburg-Landau equation (CQGLE) with gain dynamics terms, which can model the propagation of dissipative solitons in the FMLFLs [41–43]:
We consider a constant phase difference (α1 = π/2, α2 = π/4) and separation (τ1 = τ2 = 5 ps) between the two solitons within each soliton-pair molecule, and set the inter-molecular separation and phase as α3 = π/4 and τ3 = 12 ps respectively. A set of calculations are performed starting from the initial condition and fixed value of parameters at D = 1, r = 2, ε = 0.58, β = 0.5, µ = −0.12, υ = 0, Is = 0.16, g0 = 2.88. The roundtrip time of soliton molecules is set to 268 ns. We add a sinusoidal variation to the parameter of Γ to make the gain fluctuate with a period of 100 round trip time, which can simulate the sine-like fluctuation of the pump power in the experiment. The simulation result is shown in Fig. 8. Soliton molecules oscillates and constantly move away from the furthest stable state after thousands of roundtrips evolution and keep relatively static, which are consistent to our experimental observations to some extent.
Fig. 8. The numerical simulation of temporal intensity profile evolution under a sinusoidal variation gain.
At present, the solutions to some soliton molecular dynamics have been theoretically demonstrated in the huge range of laser parameter space [16,44–46]. In our wok, the oscillatory period of soliton molecules in the laser cavity under the gain modulation is as long as 7.6 s, which is very long in the photonic scales. Due to the limitation of computational capacity and speed, it’s hard for us to focus on the whole dynamics through numerical simulation. In other words, at least 28 million roundtrips are needed to simulate the motion of soliton molecules in one oscillation period, in order to find the parameter space of the forced periodic oscillation dynamics of soliton molecules accurately. Hence, we just explore the influence of parameters related to gain on the motion of soliton molecules for the time being.
5. Conclusion
In summary, we have demonstrated the resonance modulation of the gain on the level of soliton molecules, and experimentally realized the transition of different pulse states by applying varying pump power excitation. A four-pulses soliton bunch formed a soliton molecular complex that was sensitive to gain changes when pump power increased. Induced by the pump power of periodic fluctuation, the soliton molecules oscillated away from each other under various interactions, while kept their robustness at the same time. The inter-molecular separation oscillated from the order of 0.01 ns to 10 ns on long time scale of ∼170 s. The final distribution was stable where the inter-molecular separation was maximum until the gain in the cavity was reduced to support the soliton bunch state again. The numerical simulations were discussed to find the relationship between the oscillation of soliton molecules and gain fluctuation. However, it shall be further studied to find out a clear correlation between the gain parameters and the behavior of soliton molecules. We believe our findings will provide a new 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 Nos. 11674177, 61640408, 61775107); Natural Science Foundation of Tianjin City (Grant No. 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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