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Abstract
Pulsating soliton (PS) as a local structure of nonlinear systems has induced substantial interest in nonlinear photonics and ultrafast lasers. However, the interaction mechanism between PSs has not been fully studied. Here, the vital role of the asymmetric dispersive wave (DW) in PSs interaction is investigated for the first time. Based on the complex Ginzburg-Landau equation (CQGLE), we find that the asynchronous pulsating soliton molecule (PSM) composed of strong PSs and weak PSs will produce frequency shift due to the asymmetric DW, and the state of the PS can be transferred through the DW during the collision between PSs and PSM. Moreover, we firstly characterize the PS with asymmetric DW in experiment, and observe the drift of PSM, which agree with our simulation that the asymmetric DW can cause the frequency shift of PSMs. Our results provide new insights into the multi soliton interaction of nonlinear systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Dissipative solitons (DSs) are important local structures for studying dissipative nonlinear systems, which are formed by the complex balance among dispersion, nonlinearity, gain and loss [1]. With the development of passively mode-locked fiber lasers (PMLFLs) and real-time detection based on dispersive Fourier transform technology (DFT) and time-lens technology, the various fascinating dynamics of DSs can be fully characterized [2]. The exciting study such as rogue wave [3], soliton explosion [4,5], internal motion of soliton molecules [6,7], the buildup of soliton [8], soliton collision [9], and soliton molecule [10] strongly improve the content of nonlinear science. These rich dynamic processes contain complex physical mechanisms, and put forward more requirements for the study of DSs theory. So further reveal the interaction mechanisms of DSs behind these complex processes from mathematical analysis or numerical simulations is urgent.
When the soliton is disturbed periodically in the laser, it will maintain the state of the soliton by radiating the resonant DW, satisfying the phase matching condition represented as Kelly sideband in the spectrum. DW has a pivotal role in the interaction between waves in nonlinear fiber lasers [11]. Recent theories and experiments have proved that DWs can provide medium-range interaction forces in the process of soliton interaction. Solitons can exchange weakly dissipative DWs, and the interaction can lead to the formation of bound states caused by harmonic synchronization [12]. At the same time, in dissipative systems, solitons can form new solitons by radiating DWs in the process of gain saturation [13]. It can be seen that the dissipative system can regulate the soliton interaction and system state by radiating DWs.
Dissipative nonlinear systems have abundant soliton self-organized states due to the attractor structure, and the pulsating soliton (PS) is related to the limited cycle. As a periodic local structure in a dissipative nonlinear system, PS is an important means of system energy regulation. PSs solutions with various evolutionary processes, such as plain, erupting, and creeping, have been numerically demonstrated based on the complex Ginzburg-Landau equation (CQGLE) [14,15]. Using DFT technology, the dynamics of PS has also been extensively studied, including single-period pulsation and double-period PS [16], pulsating soliton molecule (PSM) [17], asynchronous pulsation [18], snake PS [19], PSs with central wavelength oscillation [20], subharmonic entrainment of PSs [21,22]. In these studies, the self-organization structure formed by soliton in different pulsating states and the interaction between them are lacking. The questions about PSs’ interactions are fascinating and have profound implications for the expansion of PS dynamics.
In this research, we investigate the effect of asymmetric DW in PSs interaction for the first time. We investigate the asynchronous PSM composed of strong PSs and weak PSs in simulation. We find the frequency shift of PSM due to asymmetric DW, and during the collision between PS and PSM the state of PS can be transferred through DW. Additionally, we firstly observe PS with asymmetric DW and the drift of PSM based on experiment. What’s more, the simulations are consistent with experiments that asymmetric DW can cause the frequency shift of PSMs.
2. Numerical results
PSs are characterized by periodic energy oscillation and directly related to the gain dynamics of fiber lasers. Recently, the nonequilibrium dynamics with PSs as the elementary constituents have been observed, especially the collision dynamics of PSs [23]. The experiments put forward a demand for exploring the interaction mechanism between PSs. Here, the complex PSs dynamics in PMLFLs is investigated based on CQGLE with the terms of the gain saturation [24]:
The dynamics of two initial hyperbolic secant pulses are shown in Fig. 1 under the system parameter${\; }{\textrm{g}_0} = 2.58,{\; }\; r = 2$, ${\; }\varepsilon = 0.58$, ${\; }\mu ={-} 0.12$, ${\; }\nu ={-} 0.1$, ${\; }\beta = 0.05$, ${I_s} = 0.5$, where the intervals between strong PS and weak PS in Fig. 1(a) and Fig. 1(c) are 35 ps and 7.5 ps, respectively. In Fig. 1(a), the two PSs have different pulsation degrees, in which the left soliton characterizes with stronger oscillations. It can be seen from Fig. 1(a) that the two PSs do not move relative to the reference frame. Figure 1(b) shows the energy evolution of the two PSs from RT600 to RT660, and they have the same energy oscillation period (4 RTs) but the opposite oscillation phase. It means that they coexist in the form of asynchronous pulsation and follow different periodic solutions, which has been investigated in L-band PMLFL [18]. The pulsation dynamics change as shown in Fig. 1(c) when the interval between the two PSs is reduced. What’s more, both the two PSs show periodic creeping of different degrees due to the interaction, and the left PS evolves with stronger oscillations. As demonstrated in Fig. 1(d), the two PSs compose as PSM and pulsate asynchronously in an opposite-phase energy oscillation manner with 4 RTs, which reflects the robustness of the two PSs. What stands out in Fig. 1(c) is that the PSM has a positive drift velocity in addition to the peristaltic evolutions of the two PSs, indicating that the PSM has strong internal actions.
Fig. 1. Asynchronous PSs with different intervals. (a) Distant asynchronous PSs. (b) Energy evolution of distant asynchronous PSs. (c) asynchronous PSM. (d) Energy evolution of asynchronous PSM.
From Fig. 2 we can see that the drift velocity decreases exponentially with the increase of the two PSs interval, which means the interaction intensity of the two PSs is positively correlated with the overlap of the two PSs. The two PSs merge into a stable PS when the interval is less than 5 ps. What is interesting about the data is that the PSM is guided by the strong PS and drifts towards it. It indicates that there is an interaction mechanism inside the PSM.
In order to fully characterize the interaction of PSs, we use the cross-correlation frequency resolved optical gating (X-FROG):
Fig. 3. X-FROG: Asynchronous PSs with distant intervals at (a) RT648 and (b) RT650; Asynchronous PSM at (c) RT3164 and (d) RT3166.
To better prove the interaction of DWs and PSs, we keep the system parameters and change the initial conditions of the pulse. The dynamics of PSM with negative drift velocities are obtained as shown in Fig. 4(a). The trailing strong PS causes coherent fusion between the PSM and the low-frequency DW as shown in Fig. 4(b), causing red-shift and negative drift velocity of the PSM. What stands out in Fig. 4 is that the different pulsating degree of the two PSs significantly affects the drift of the PSM. Due to the asymmetry of the weak PS’s DW and the strong PS’s DW, the overall frequency shift of the PSM is caused and related to the coherent overlap, which consistent with the conclusion shown in Fig. 2.
Based on the frequency shift of asynchronous PSMs, we build the collision dynamics between PSs and asynchronous PSMs to explore more details of the interaction mechanism. The system parameters are set as ${{g}_0} = 2.58,\; {r} = 2,{\; }\varepsilon = 0.58,\;{\mu} ={-} 0.12,\; {\nu } ={-} 0.1,{\; }\beta = 0.05,{\; \; }{I_s} = 0.732$. It can be seen from Fig. 5(a) that the PS and the PSM approach to each other at a relatively constant speed and finally collide. The trailing PS is transformed into strong PS, meanwhile, the strong PS of PSM is transformed into weak PS, which indicates the state exchange during the collision dynamic. Afterwards, the two trailing PSs are assembled into an asynchronous PSM and gradually separated from the leading PS. Figure 5(c) shows the collision dynamics when the system parameter is set as ${{g}_0} = 2.58,\; {r} = 2,\;{\varepsilon } = 0.58, {\mu} ={-} 0.12,\; {\nu } ={-} 0.1,{\; }\beta = 0.05,{\; }{{I}_s} = 0.73$. Unlike elastic collision, three PSs finally form a PSM, which indicates that the strong PS and the weak PS on the trailing edge do not exchange states during the collision dynamic.
Fig. 5. Collision dynamics of PS and PSM. (a) Elastic collision. (b) X-FROG of elastic collision at RT3880. (c) Inelastic collision. (d) X-FROG of inelastic collision at RT4180.
We calculate the X-FROG of elastic collision dynamics as shown in Fig. 5(a). The coherent overlaps between the three PSs are inconsistent, which can be seen from Fig. 5(b) that the middle PS and the trailing PS produce stronger coherent overlaps (zoom A and zoom B). The energy resonance transfer is occurred in the process of chaotic collision. Both the middle PS and the trailing PS have strong DW, and the energy transfer is provided by the coherently overlaps of it. The average DW frequencies of the leading and trailing PSs are -53.1 THz and 158.4 THz, respectively. Therefore, the frequency shift obtained through the coherent overlap of the middle PS and the other two PSs is negative, which indicates that the drift direction of the middle soliton is consistent with trailing soliton. The coherent overlap enhances the energy transfer of the DW, and finally the trailing PS inherit the strong pulsation of the middle PS, resulting in the drift of the new PSM. Figure 5(c) shows the X-FROG of the three PSs during the inelastic collision. As illustrated in Fig. 5(c), unlike the elastic collision, the coherent overlap between the middle PS and other two PSs are basically same (zoom C and zoom D). The strong DW radiated by the middle PS make interaction with the other two PSs, so these three PSs form a stable PSM. Comparing the two dynamics in Fig. 5, it can be found that the asymmetric DW leads to inconsistent bonding strengths between PSs, thus forming elastic and inelastic collisions.
3. Experimental setup and results
The schematic configuration of the fiber laser in our experiment is shown in Fig. 6. The ring cavity comprises 1 m er-doped fiber (EDF, the group velocity dispersion is 12 ps2/km at 1550 nm) and 6.7 m single-mode fiber (SMF, the group velocity dispersion is -23 ps2/km at 1550 nm). Two PCs (PC1 and PC2) together with a polarization-dependent isolator (PD-ISO) realize the nonlinear-polarization-rotation (NPR) mode locking. The cavity is positively pumped by a 980 nm laser diode through a 980/1550 wavelength-division multiplexer (WDM). OC1 and OC2 output the light in cavity and divide it into two channels. After two 45 GHz bandwidth photodetector (PD, DiscoverySemi, DSC10H), One of them is stretched through a 1.5 km dispersion compensation fiber to measure the real-time spectrum, and the other is measured in time-domain. The high-speed oscilloscope has a 33 GHz bandwidth and a 100 GHz sampling rate (Tektronix DPO75902SX). The repetition rate of the laser is 27.6 MHz and the dispersion of the cavity is -0.142 ps2.
Fig. 6. Scheme of the passive mode-locked Er-doped fiber laser based on NPR. WDM, wavelength division multiplexer; EDF, Er-doped fiber; PC, polarization controller; PD-ISO, polarization-dependent isolator; OC, optical coupler; DCF, dispersion compensation fiber.
The PS state is achieved with appropriate adjustment on the PC at a pump power of 240 mW. The average spectrum is depicted in Fig. 7(a), which shows that PS has broadened Kelly sidebands. What’s more, The Kelly sideband is asymmetrically distributed and short wave has higher intensity than that of long wave. As shown in Fig. 7(b), the intracavity power supports the operation of three PSs. Figure 7(d) shows the pulse peak evolution of strong PS and weak PS. The two PSs have the same change period, and the peak extremum is 0.48 and 0.44, respectively. One possible implication of the peak extremum difference is that gain is consumed by solitons in the time front. As shown in enlarged view of Fig. 7(b), both weak PS and strong PS have asymmetric tails, which is related to the asymmetric DW. The time domain characteristics in Fig. 7(c) more clearly show the asymmetric DW. Strong DW with faster speed is in the time front and corresponds to shorter wavelength under abnormal dispersion, which is consistent with the characteristics of Kelly sideband in Fig. 7(a). The real-time spectrum in Fig. 7(e) also indicates that the PS has both synchronous and asynchronous DW, which have been discussed in [25] in simulation. The Kelly sideband at long wave has a periodic drift synchronized with the pulsating period, which corresponds to the characteristics in Fig. 7(a). What’s more, Kelly sideband at short wave can be divided into two parts, one part is periodic, and the other part has a fixed central wavelength and higher intensity, which is consistent with Fig. 7(b).
Fig. 7. PS with asymmetrical DW. (a) Spectrum. (b) Temporal evolution. (c) Temporal profile of strong PS at RT696 and RT721. (d) Energy evolution of strong PS and weak PS. (e) Spectral evolution of strong PS. Typical spectra at (f) RT470, (g) RT488 and (h) RT500.
The PSM state is achieved with appropriate adjustment on the PC. Figure 8(a) reveals that two trailing PSs assemble as PSM, and the time interval is 60 ps, which is far greater than the pulse width, and indicates the long interaction length of PSs [12]. Figure 8(d) shows the pulse peak evolution of strong PS and weak PS. The two PSs have the same change period of 43 RT, and the peak extremum is 0.48 and 0.42, respectively. The gain consumption in Fig. 8 is more than that in Fig. 7, therefore, the peak extremum of weak PS in Fig. 8 is lower than that in Fig. 7. As PSM has faster drift speed and shorter average wavelength, PSM moves to PS in Fig. 8(a). It can be found from enlarged view of Fig. 8(a) that PSs in PSM still have asymmetrical DW. In addition, Fig. 8(b-c) shows that DWs of the weak PS and the strong PS have coherent overlap, which leads to the blue shift of the PSM, and it shows a faster drift under abnormal dispersion.
Fig. 8. Asynchronous PSM. (a) Temporal evolution. Temporal profile at (b) RT1983 and (c) RT14998. (d) Energy evolution of strong PS and weak PS.
PS and PSM collide due to their different drift velocities, which leads to chaotic pulsation, as shown in Fig. 9. It can be seen from Fig. 9(a) that the soliton quantity changes complicatedly with the process of soliton annihilation and regeneration. Figure 9(b-d) shows three typical temporal states of the chaotic pulsation dynamics. The graphs show that new solitons generate on the side of strong DW. The DW at time-front has higher intensity and is asymmetrically with the PS, and it is easier to obtain gain, so it is more conductive to generate new PS. What’s more, the newly generated PS inherit the asynchrony.
Fig. 9. Chaotic pulsation. (a) Temporal evolution. Temporal profile at (b) RT5012, (c) RT5123 and (d) RT5197.
DW promotes the complex PS interaction and is the medium of soliton medium range interaction. Based on the collision dynamics obtained from simulation and experiment, it is shown that asymmetric DW can lead to the drift of asynchronous PSM. The parameter space of the laser pulsating state is narrow, while the elastic collision and inelastic collision process of the PSs have a narrower parameter space as a more special pulsating dynamics, and the interval between the PSs is uncontrollable, which limits the in-depth exploration of the current experiment. Intelligent mode-locking has been applied to realize the pulsating mode-locked state [26,27]. In the future work, we hope to optimize the laser and find the parameter space corresponding to the interaction of PSs through the intelligent mode-locking technology based on reinforcement learning, and explore the interaction physical law of PSs.
4. Conclusion
In conclusion, this research set out to investigate the interaction mechanism of PSs based on asymmetric DWs in simulations and experiments. Based on CQGLE, we find that strong PS and weak PS are organized into a stable asynchronous PSM due to the overlap between them, and the asymmetric DW in PSM results in the frequency shift of PSM. What’s more, the elastic and inelastic collision between PS and PSM are related to the exchange of DW, which can exchange pulsating states. In experiment, we completely characterize the PSs with asymmetric DW, and find that the PSM consisting of strong PS and weak PS has frequency shift, which supports the simulation results. Last but not least, the collision between PSM and PS leads to chaos, and new solitons can be generated by the asynchronous DW. Our results not only provide a new perspective on the interaction of PSs in dissipative systems, but also pave the way for the physical understanding of complex nonlinear systems.
Funding
National Key Research and Development Program of China (2018YFB0504400); National Natural Science Foundation of China (12274238, 61835006, 62205159); Natural Science Foundation of Tianjin City (19JCZDJC31200); Special Project for Cooperation in Basic Research of Beijing, Tianjin and Hebei (21JCZXJC00010); Tianjin Research Innovation Project for Postgraduate Students (2021YJSB083).
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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