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Abstract
Optical soliton molecules exhibiting behaviors analogous to matter molecules have been the hotspot in the dissipative system for decades. Based on the dispersion Fourier transformation technique, the real-time spectral interferometry has become the popular method to reveal the internal dynamics of soliton molecules. The rising degrees of freedom in pace with the increased constitutes of soliton molecules yield more intriguing sights into the internal motions. Yet the soliton molecules with three or more pulses are rarely investigated owing to the exponentially growing complexity. Here, we present both experimental and theoretical studies on the soliton molecules containing three solitons. Different assemblies of the constitutes are categorized as different types of soliton triplet akin to the geometric isomer, including equally-spaced triplet and unequally-spaced triplet. Typical soliton triplets with different dynamics including regular internal motions, hybrid phase dynamics and complex dynamics involving separation evolution are experimentally analyzed and theoretically simulated. Specifically, the energy difference which remains elusive in experiments are uncovered through the simulation of diverse triplets with plentiful dynamics. Moreover, the multi-dimensional interaction space is proposed to visualize the internal motions in connection with the energy exchange, which play significant roles in the interplays among the solitons. Both the experimental and numerical simulations on the isomeric soliton triplets would release a larger number of degrees of freedom and motivate the potentially artificial configuration of soliton molecules for various ultrafast applications, such as all-optical buffering and multiple encoding for telecommunications.
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1. Introduction
Solitons, the self-organized localized structures through the balance between dispersion and nonlinearity, exist in lots of nonlinear systems such as hydrodynamics and optics [1,2]. In dissipative systems, the balance between gain and losses also plays a significance role in the generation of dissipative solitons [3,4], which are considered as versatile paradigm of ultrashort laser pulses with ultralow phase noise. In general, for mode-locked lasers, e.g., nonlinear fiber lasers, sufficiently-high pump power may lead to multi-soliton operations (soliton clusters, soliton crystals and so on) due to various nonlinear effects, such as peak-power clamping [5] and gain saturation [6]. More specifically, profited from the bounding effects via overlapping tails [7], dispersive waves [8] and etc., two or more dissipative solitons may achieve their local minima of the interaction potential, and subsequently assemble as the bound states of multiple pulses. The aforementioned collective and robust units, known as soliton molecules (SMs) [9,10], may propagate as a whole in laser systems. Besides the stationary SMs [11–13] on account of the strong trapping with well-defined separation, various dynamical SMs [14–17] have been predicted and observed in last two decades, namely the bouncing/vibrating intra-molecular motions [18] and the synchronized shaking inter-molecular momentum [19]. These interesting but complex nonlinear evolutions of SMs are mainly governed by the gain dynamics and nonlinearity, leading to abundant evolutions of internal separation and relative phase between the constitutes.
From the experimental point of view, the time scales of non-repetitive ultrafast soliton dynamics are far beyond the bandwidth limit of the measurement devices. While fortunately, these issues have been tackled down with the development of time stretch schemes such as time lens [20,21] and the dispersion Fourier transformation (DFT) technique [22–24]. For the later method, one constructs the real-time spectroscopy of the ultrafast pulses so as to gain the detailed physical insights of the transient dynamics of dissipative soliton including the buildup process [25–27], breathing soliton [28,29], explosion [30,31] and so on. Along with this line, in the past decade, diverse dynamics of two-soliton molecules have been theoretically predicted and experimentally observed, such as vibrating [32,33], shaking [34,35] and phase evolving soliton pairs (SPs) [36–38]. The primary reports on the phase evolution pave the new routine to investigate the internal motions of SPs, including the separation and relative phase retrieved from the interfering spectra. In addition, three-soliton molecules, i.e., the soliton triplets with varied dynamics are experimentally reported in our previous work [39], demonstrating more complicated internal motions among the pulses. Recently, soliton quadruplets consisting of four solitons are characterized as four families of isomers have been studied with considerably increased degrees of freedom (DOF) [40]. In these works, internal motions of SMs could be generally described by the evolution of relative phase and temporal separation, while the energy and location for each pulse still remain elusive. With three or more pulses in the SMs, the complexity of the dynamics would grow exponentially along with the increasing DOF. Therefore, the characteristics of each single pulse in SM are urgent to be uncovered which is rarely reported on the three-soliton molecules.
In this paper, we demonstrated the intriguing observation of the soliton triplet dynamics in a near-zero-dispersion fiber laser, which is fully interpreted by the experimental analysis and numerical simulations based on the generalized nonlinear Schrödinger equation (GNLSE). In this work, stationary equally-spaced triplets (EST) and unequally-spaced triplets (UST) with either stationary dynamic and evolving dynamics are recorded by the DFT based real-time spectrometer and analyzed through the first-order autocorrelation. In addition, numerical simulations are conducted to reproduce the experimental results and further predict more diversified soliton triplets analogous to the geometric isomer. Beyond the regular dynamic, hybrid intramolecular dynamics and evolving separations are investigated. It is worth noting that energy difference is calculated to interpret the phase dynamics associated with the energy-exchange, whose characterizations are extremely challenging in experiments. To this end, for visualizing the internal motions with multiple DOF, we have implemented the analysis of multi-dimensional interaction space to demonstrate the evolution of separation, relative phase and energy difference. In this work, the experimental and numerical results on the isomeric soliton triplets enrich the framework of intramolecular dynamics through releasing more DOF, opening the routine for the synthesis of large SMs and artificial manipulation of evolving SMs.
2. Experimental results
We conducted the experimental investigations in a nonlinear-polarization-rotation based mode-locked fiber laser, where the shot-to-shot measurements of the soliton triplet dynamics are realized with the DFT spectrometer system. With this platform, we have observed variety configurations of isometric soliton triplets.
2.1 Experimental setup
As schematically presented in Fig. 1, a 0.8-m-long erbium-doped fiber (EDF 80, OFS, + 0.061 ps2/m at 1530 nm) is pumped by a 980 nm laser diode (LD) via a hybrid module containing a wavelength division multiplexer (WDM) and an isolator, and the later guarantees the unidirectional laser propagation. In this fiber laser architecture, the total cavity length is around 3.7 m including 2.9-m long single-mode fiber (SMF, −0.022 ps2/m), resulting in a near-zero-dispersion laser cavity (net dispersion ∼ −0.015 ps2). Here, the mode-locking is enabled with a polarizer sandwiched by two polarization controllers - a structure equivalent to saturable absorbers (SA) [11,27,28]. The generated ultrafast pulses are at first extracted from the cavity with a 10:90 optical coupler (OC). Then, their real-time spectra are measured by a DFT-based spectrometer built with a 10-km SMF, a photodiode detector (PD, 43 GHz bandwidth) and a high-speed oscilloscope (OSC, 20 GHz bandwidth, 80 GSa/s, Lecroy), as shown in Fig. 1(b). With this method, the recorded shot-to-shot interferograms conceal the internal dynamics of the SMs, which can be retrieved through the first-order autocorrelation traces [35].
Fig. 1. (a) Schematic setup; (b) Real-time spectrometer system; (c) Graphical representation of the internal motions of soliton triplets. (d) Conceptual illustrations for 4 typical configurations of isomeric soliton triplets.
2.2 Observations of four representative configurations of isometric soliton triplets
As long as the pulses are considered as distinct entities which are sufficiently separated from each other and have the same intensity profile, we provide a conceptual illustration of a soliton triplet constituted by lead pulse (LP), center pulse (CP) and tail pulse (TP). As schematically shown in Fig. 1(c), the internal separation between solitons LP and CP, CP and TP, LP and TP are designated as the notation Δτ12, Δτ23 and Δτ13, while Δφ12, Δφ23 and Δφ13 denotes the corresponding relative phases. To gain more physical insight into the dynamics in terms of the soliton energy, we use ΔE12, ΔE23 and ΔE13 to represent the energy difference between the solitons. Besides the EST (Δτ12 = Δτ23) with fixed temporal separation, UST with fixed temporal separation are also common cases to obtain. Due to the weaker interaction between the solitons with larger separation in the UST, the internal motions are tended to be different with that of closer two, resulting in the hybrid phase evolving soliton triplet. Moreover, the internal separations among pulses can also change along the propagation, which is described as “1 + 1 + 1” type UST, as conceptually pictured in Fig. 1(d).
In the experiment, by appropriately adjusting the pump power and the intracavity polarizations, the mode-locked laser operation has been established and variety configurations of soliton triplets are observed. In this framework, we have obtained stationary EST and UST with either stationary dynamic or hybrid phase dynamics, which are systematically demonstrated by the consecutive spectra, first-order autocorrelation (AC) trace and interaction space. As shown in Fig. 2, the stationary EST depicted at the first row is verified by the equally-spaced straight spectral fringes and five AC peaks, which are pictured in Fig. 2(a) and (b) respectively. The corresponding interaction space in Fig. 2(c) shows uniformly distributed points at nearly fixed positions, indicating the unchanged separations and relative phases between the pulses. As for the stationary UST demonstrated at the second row, the spectral fringes and the corresponding AC peak traces are straight but unequally-spaced compared to the stationary UST. Seven straight peaks in Fig. 2(e) manifest the fixed pulse distribution of the soliton triplets. And the corresponding interaction space in Fig. 2(f) shows the same distribution as the EST, indicating the stationary pulse intensity evolution without noticeable energy exchange.
Fig. 2. Experimental results of stationary EST (first row), stationary UST (second row), UST with hybrid phase dynamic (third row) and “1 + 1 + 1” type UST (last row). (a), (d), (g) and (j) spectral interferograms; (b), (e), (h) and (k) first-order autocorrelation traces; (c), (f), (i) and (l) interaction spaces.
Apart from the stationary dynamics, USTs with evolving dynamics are relatively less challenging to obtain as the pulse energy would flow among the constitutes. In this case, the third row in Fig. 2 demonstrates the UST with hybrid phase evolution, where the consecutive spectra in Fig. 2(g) showing stepping fringes, thus implying the stepping phase evolution between two closer pulses. Again, the corresponding first-order AC trace in Fig. 2(h) shows the similar distribution of seven straight AC peaks with the stationary UST, indicating the unequally-spaced three solitons with fixed separations. Nevertheless, the interaction space shown in Fig. 2(i) characterizes the tardy evolution of stepping phase between two closer pulses and oscillating phase between two farther pulses. Moreover, UST with varying separations and hybrid phase evolution is obtained, as demonstrated at the last row. The consecutive spectra in Fig. 2(j) shows intermittently flip towards longer wavelength, different to the linear spectral fringes observed above. And the evolution traces in first-order AC in Fig. 2(k) oscillate obviously, implying the varying separation between either closer and farther pulses. The interaction space in Fig. 2(l) characterizes the sliding phase between two closer pulses and oscillating phase between two farther pulses, indicating the “1 + 1 + 1” type UST.
Compared with our previous work in Ref. [39], the experimental results described above demonstrate more intriguing phase evolutions, which extensively reveal novel regimes of internal-motion dynamics for soliton triplets. It is also worth mentioning that, either in our previous [39] or current work, analysis based on DFT interferograms and AC traces can only provide us limited information about mutual dynamics between pulses, e.g. relative phases and internal motions. However, more details about position evolutions and energy fluctuations for each pulse remain unknown. From this point of view, it is therefore essential to perform systematic numerical simulations for better understanding of diversified gain-dynamic induced intramolecular complexities of soliton triplets.
3. Numerical simulations
To gain better insight into the experimental observations, we numerically simulated various laser operation states of our fiber laser. The modeled laser cavity is similar as the schematic setup shown in Fig. 1 and we have successfully reproduced the experimental results by varying the laser gain. Here, a lumped propagation model [1541,42] is used to describe each component of the cavity by a separate equation, and the pulse propagation in the fiber system is modeled by a generalized nonlinear Schrödinger equation (GNLSE) in the following form:
3.1 EST with regular dynamics
To start, we report the stationary EST at the upper row in Fig. 3. The internal separations between two pulses maintain unchanged during the propagation in the cavity as shown in Fig. 3(a). The consecutive spectra in Fig. 3(b) and AC trace in Fig. 3(c) are the same as that of the experimental stationary EST depicted in Fig. 2(a) and (b). The retrieved relative phases pictured in Fig. 3(d) show unchanged Δφ12 and Δφ13, indicating the stationary evolution of the soliton triplet which agree well with the experimental results in Fig. 2(a-c). In addition, the lower row in Fig. 3 demonstrates the EST with oscillation phase evolution. Different from the stationary EST, the consecutive spectra shown in Fig. 3 (f) exhibit equally-spaced but oscillating fringes, while the equally-spaced five peaks in the AC trace in Fig. 3(g) verifies the equally-spaced pulses. And the synchronous oscillating relative phases of Δφ12 and Δφ13 in Fig. 3(h) conform the oscillating phase evolution of EST.
Fig. 3. Simulation results of stationary EST (upper row) and oscillation phase EST (lower row): (a) and (e) temporal distributions; (b) and (f) spectral interferograms; (c) and (g) AC traces; (d) and (h) relative phases.
3.2 UST with regular dynamics
Besides the common dynamical EST can be obtained in our model, the UST with regular phase evolution can also be achieved as demonstrated in Fig. 4. Both stationary UST and oscillating phase UST numerically studied and resolved are displaced at the upper row and the lower row, respectively. The pulses depicted in Fig. 4(a) and (e) show the assemble of two closer constitutes and a distant one, but the separations are fixed as the seven straight peaks shown in both AC traces in Fig. 4(b) and (f). It is worth noting that two AC traces exhibit similar peak distribution while the corresponding pulse distributions are different, which was mentioned in our previous work [37] that the exact pulse distribution are unable to be calculated. The spectra of the stationary UST show straight fringes in Fig. 4(b), corresponding to the unchanged relative phases retrieved and pictured in Fig. 4(d) which agree well with the experimental result. Yet the UST with oscillating phase is verified by the oscillation of the spectra fringes in Fig. 4(g), as well as the relative phases demonstrated in Fig. 4(h) respectively.
Fig. 4. Simulation results of stationary UST (upper row) and oscillating phase UST (lower row): (a) and (e) temporal distributions; (b) and (f) spectral interferograms; (c) and (g) AC traces; (d) and (h) relative phases.
For better comprehension, energy difference between pulses is introduced to quantitatively analyze the internal motions. In the case of the UST with regular dynamics, the energy differences are calculated and depicted in Fig. 5(a) and (c). The energy difference of stationary UST can be regarded as nearly unchanged by ignoring the fast jitter/vibration due to the limited resolution of the model. As for the oscillating phase UST, the energy differences show apparent oscillation back and forth around zero. Here, ΔE12 and ΔE23 experience opposite oscillation corresponding to the opposite oscillating phases shown in Fig. 4(h), indicating the energy exchange guided phase evolution. In order to clearly illustrate the internal dynamics associated with energy difference, we perform a new concept of interaction space to describe the multi-dimensional evolution, as shown in Fig. 5(b) and (d). Where the x, y, z axis are τcos(φ), τsin(φ) and ΔE, respectively. The projection on xy plane is the interaction plane usually used to describe the internal motions, and the green dotted circles represent the plane z = 0. The interaction points in Fig. 5(b) with concentrated distribution near the z = 0 plane represent the stationary triplet, while that in Fig. 5(d) distributed both up and down around the z = 0 plane represent the oscillating phase triplet. This multi-dimensional interaction space can reveal the relation between internal motions and energy difference.
Fig. 5. Stationary and oscillating phase UST with energy differences (a, c) and multi-dimensional interaction spaces (b, d).
3.3 UST with hybrid phase dynamics
Beyond the same phase evolution among pulses, different phase evolutions could also occur in a soliton triplet, especially the UST due to the different interaction induced by the different separation and pulse difference. Figure 6 demonstrates the UST with fixed separations but different phase evolution termed as hybrid phase dynamics. It's worth noting that the fixed separations of the triplets are left out.
Fig. 6. Simulation results of UST with hybrid phase dynamics. (a) and (e) temporal distributions; (b) and (f) spectral interferograms; (c) and (g) energy differences; (d) and (h) interaction spaces; (i) and (j) multi-dimension interaction spaces
The upper row demonstrates the UST involving stationary dynamic and sliding phase evolution. The successive spectra in Fig. 6(b) show distinct drifting fringes to shorter wavelength, indicating the sliding phase evolution between two closer pulses CP and TP. The corresponding energy difference ΔE23 is depicted as blue line in Fig. 6(c), which shows periodic variation between −0.18% and −0.02%. Comparing to the oscillation phase evolution, the consistent energy difference less than 0 would lead to the continuous decrease of the relative phase Δφ23, namely sliding phase evolution shown by the helical trajectory in Fig. 6(d). The other calculated energy difference ΔE12 in Fig. 6(c) is similar with that of the stationary dynamic, verified by the almost straight trajectory in the interaction space.
The UST with hybrid phase dynamics consisting of stepping phase evolution and sliding phase dynamic is demonstrated at the lower row. The spectral interferogram in Fig. 6(f) show evident stepping fringes towards shorter wavelength, implying the stepping phase evolution between two closer constitutes of the triplet. The corresponding energy difference depicted in Fig. 7(g) shows the similar periodic variation but between −0.12% and 0.01%, signifying that the pulse energies of two constitutes will reach equal periodically during the evolution. Consequently, the relative phase would stop sliding when the energy difference equal to 0, resulting in the small step of the phase versus roundtrip as marked on the trajectory in Fig. 6(h). The energy differences between LP and CP show slight variation between −0.04% and 0.02%, corresponding to the slow slide of the relative phase.
Fig. 7. Simulation results of “1 + 1 + 1” type UST with complex dynamics. (a), (e) and (i) temporal distributions; (b), (f) and (j) spectral interferograms; (c), (g) and (k) energy differences; (d), (h) and (l) interaction spaces.
Additionally, the multi-dimensional interaction spaces are depicted in Fig. 6(i) and Fig. 6(j) to describe the relation between phase evolution and energy difference. With respect to the sliding phase evolution, the interaction points are distributed under z = 0 plane and the projection will overlap the whole circle τ=Δτ23. Different from the oscillating phase guided by the energy exchange between two pulses, the persistent sign of the energy difference could be regarded as energy flow from one to another. Therefore, the relative phase exhibits long-term slide rather than back and forth owing to the retentive difference in the carrier envelope phase speed. Also, the multi-dimension interaction space pictured in Fig. 6(j) demonstrates that the fast phase slide is associated with the sparser interaction points related to the larger energy difference, and vice versa. And the stepping phase appears exactly where the interaction points are distributed on the z = 0 plane.
3.4 “1 + 1 + 1” type UST with complex dynamics
Finally, when three pulses of the soliton triplet are close to each other, the attraction and repulsion would be evidently stronger pronounced among the constitutes, resulting in the drastic interplays containing variations of relative phase and separation. The triplet with two independent separation evolutions can be defined as “1 + 1 + 1” type UST as shown in Fig. 7. Noting that two retrieved internal separations are depicted together with the energy differences.
The upper row in Fig. 7 shows “1 + 1 + 1” type UST with oscillating separations and hybrid phase evolution. The successive spectra in Fig. 7(b) show flipping fringes towards shorter wavelength, which is resulted from the oscillating separations shown by the green and orange line depicted in Fig. 7(c). The retrieved Δτ12 and Δτ23 experience opposite oscillation with the same amplitude of 0.12 ps, testifying the opposite moving directions of the pulses LP and TP relative to CP. The calculated energy differences depicted in Fig. 7(c) imply the standard sliding phase between LP and CP, oscillating phase between CP and TP, which are verified by the trajectories in Fig. 7(d). This simulation agrees well with the experimental results of “1 + 1 + 1” type UST. The multi-dimensional interaction space in Fig. 8(a) demonstrates the standard distribution of the interaction points with respect to the sliding phase and oscillating phase evolution, respectively.
Fig. 8. Corresponding multi-dimensional interaction spaces of “1 + 1 + 1” type UST with complex dynamics.
The middle row demonstrates the “1 + 1 + 1” type UST with both attraction and repulsion between pulses as shown in Fig. 7(e), verified by the retrieved separations which are displayed by the continuously decreasing green line and slightly increasing orange line in Fig. 7(g). Therefore, the successive spectra in Fig. 7(f) show twisty winding, flipping and shifting fringes, which is resulted from the oscillation along with the varying separations. The calculated energy difference ΔE12 implies the standard sliding phase evolution, manifested by the helical trajectory in Fig. 7(h). Nevertheless, ΔE23 are mainly less than 0 but larger than 0 at some roundtrips as the blue line shown in Fig. 7(g), leading to the sliding phase evolution containing turning phase marked out by the turning back points on the helical trajectory in Fig. 8(b). Besides the standard interaction points distribution of sliding phase evolution, the turning phase appears when the interaction points are just above the z = 0 plane. Noted that the projections no longer trace the fixed circle, corresponding to the internal separation variations.
The bottom row demonstrates the “1 + 1 + 1” type UST with chaotic separation variation. The pulses in Fig. 7(i) exhibit both attraction and repulsion between CP and TP, as the green line shows in Fig. 7(k). The retrieved separation Δτ12 slightly vibrates with the amplitude of 0.16 ps and Δτ23 varies from 0.8 ps to 1.24 ps, leading to the scrambled shifting fringes in the spectral interferogram in Fig. 7(j). The calculated energy difference ΔE12 less than 0 implies the standard sliding phase evolution, demonstrated by the helical trajectory in Fig. 7(l). While chaotically varying ΔE23 gives rise to the phase evolution consisting of both sliding and oscillating phase, shown by the chaotic trajectory containing helical and oscillating evolution. In the multi-dimensional interaction spaces shown in Fig. 8(c), the interaction points between LP and CP exhibit the standard distribution of sliding phase evolution with slight offtrack trajectory on the projection. However, the distribution of interaction points between CP and TP exhibits the coexist of sliding phase evolution and oscillating phase evolution at different roundtrips. And the corresponding projection demonstrates both inward and outward deviation of the trajectory, verifying the complicated variation of the internal separation. With the help of the multi-dimensional interaction space, we can intuitively analyze the internal motions associated with the pulse energies.
4. Conclusions
Compared to the SP, the DOF of soliton triplets with increased constitutes yield more intriguing sights into the internal motions. Assisted with the DFT based real-time spectral interferometry, typical soliton triplets with either stationary dynamic and evolving dynamics are experimentally obtained. Compared to the simplex dynamics of soliton triplets demonstrated in our previous work [39], more configurations of diversified soliton triplets are illustrated with hybrid intramolecular dynamics consisting of different internal motions. In addition, numerical simulations not only reproduce the experimental results, but also predict soliton triplets with both regular dynamics and hybrid dynamics, as well as the evolving separations. In order to explain the relation between internal motions and pulse energies, a model of mode-locked fiber laser emitting soliton triplets is established based on the GNLSE. Numerical results not only agree well with the experimental results, but also demonstrate more variety of soliton triplet with plentiful dynamics, including EST with stationary dynamic and oscillation phase evolution, UST with stationary dynamic and hybrid phase dynamics. Moreover, “1 + 1 + 1” type USTs with complex dynamics are investigated with the evolving internal separations, reaching further analogy between the matter molecules and soliton molecules. Associated with the energy difference, the internal motions containing separation and relative phase are fully analyzed which are visually demonstrated by the proposed multi-dimensional interaction spaces. These investigations on the isomeric soliton triplets release numerous DOF with diversified dynamics and stimulate the comprehension about the complexities of many-body interactions in the framework of matter waves, gravitational waves and Bose-Einstein condensates.
Funding
China Postdoctoral Science Foundation (2022M711243); National Natural Science Foundation of China (62275097).
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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