The optical soliton, as a self-localized structure in ultrafast resonators [1], is widely applied in plasma physics [2], optical systems [3], chemistry and biology [4], etc. Recently, by virtue of the physical versatility and potential applications, soliton molecules [5–10] have attracted much interest in ultrafast science, the concept of which is analogically transferred from matter particles [11,12]. Governed by the soliton interactions of repulsive and attractive forces, coexisting solitons could be temporally bound together, contributing to the formation of multisoliton structures [12–14]. Regarding the multisoliton complexes, molecular assemblies are generally characterized by two-dimensional degrees of temporal distributions and internal motions. The amounts of solitonic constituents and their binding patterns in the time domain are jointly interpreted as the isomeric assemblies, also an analogy to the concept of “soliton isomer” in previous works [15,16]. In parallel, intriguing internal motions within multisoliton complexes yield the isomeric dynamics with respect to views of sub-picosecond, even femtosecond, scales. Abundant isomeric assemblies and dynamics naturally make them candidates for solitonic manipulations [17,18], which brightens their application prospects, particularly in soliton encoding [19–21] and all-optical storage [22].

For the investigations of transient solitonic dynamics, the time-stretch dispersive Fourier transform (DFT) [23–27] technique enables access to consecutive shot-to-shot interferograms of soliton molecules through mapping spectral information onto temporal waveforms [28–30]. The DFT process makes it possible to retrieve versatile transient dynamics, such as soliton breathing [11,31,32], soliton collision [33], and soliton explosion [34]. However, for complicated multisoliton scenarios, their temporal features cannot be directly retrieved by the DFT technique, which requires another advanced detection technique to implement the accurate temporal characterization of multisoliton complexes. The time lens [35,36] is typically utilized to capture solitonic refined structures. Analogous to a space thin lens, the time lens can realize temporal magnification, settling the challenge that real waveforms of ultrashort pulses cannot be fully detected, because the narrow pulse durations and ps-level binding separations exceed the resolution range of photoelectric detection instruments. Recently, the time lens is usually utilized to temporally characterize soliton singlets [37,38], soliton molecules [36,39,40], and noise-like pulses [41]. However, concerning soliton molecules, previous works mainly focus on ones with simple structures containing a few solitonic constituents. The increased amounts of soliton constituents will dramatically add the complexity and need much more efficient ultrafast analysis methods. Thus, the combination of the DFT and the time lens becomes a popular scheme. One common interest is stimulating the on-going research for comprehensively unveiling the molecular dynamics and responding to the profound questions of matter-soliton analogy, as well as exploring the high-level multisoliton manipulation.

Multiple constituents within multisoliton complexes bring increasing degrees of freedom; thus temporal-spectral analysis is necessary for precisely probing their fine temporal distributions and internal motions. In this paper, we build an observation system based on the DFT and time-lens technique to investigate the complicated multisoliton structures. First, the real-time temporal distribution and optical spectra of typical soliton triplets are recorded, which verifies the high performance of the temporal-spectral characterization system. Based on this, we capture several kinds of soliton quartets with isomeric assemblies directly in the time domain, revealing their diversity of molecular configurations. Moreover, the intrinsic relation between relative phase evolution and energy exchange among soliton constituents is experimentally verified. Besides, the numerical simulation of dynamical multisoliton complexes provides sufficient theoretical support for unveiling the energy exchange within diverse internal motions. The above work deepens the long-lasting study of matter-soliton analogy, and unveils the internal mechanism of intriguing dynamics in complicated solitonic systems.

To probe assemblies of multisoliton complexes, a real-time observation platform based on the DFT and time-lens technique is built, as shown in Fig. 1(a). The under-test multisoliton complexes are produced from a mode-locked fiber laser (MLFL), of which the mode-locking mechanism is the nonlinear polarization rotation (NPR). The pump energy is provided by a 980-nm semiconductor laser diode and injected into the laser resonator through a wavelength division multiplexer (WDM). The equivalent saturable absorber consists of a polarizer and two polarization controllers (PCs), and the isolator ensures the unidirectional transmission. The length of the laser resonator is about 6.47 m, consisting of a 1.5-m erbium-doped fiber (EDF, ${\beta }_2=+0.061{\mathrm{ps}}^2/\mathrm{m}$, EDF 80, OFS) as the gain medium and a 4.97-m single-mode fiber (SMF, ${\beta }_2=-0.022{\mathrm{ps}}^2/\mathrm{m}$). The laser output is partitioned into two branches and sent to the DFT and time-lens arms. In the DFT process, the real-time shot-to-shot interferograms of multisoliton complexes are mapped onto a temporal waveform through stretching in the 10-km SMF. A 43-GHz photodiode detector (PD) and a 20-GHz high-speed oscilloscope (OSC, 80 GS/s, Teledyne LeCroy) are utilized to detect the successive interferograms. Additionally, to evaluate the reliability of the spectral-temporal mapping, a commercial optical spectrum analyzer (OSA, Yokogawa AQ6370D) is utilized to record the time-averaged spectra.

For temporal observation, the asynchronous time lens based on the four-wave-mixing (FWM) effect is implemented to contribute direct insight into the finer molecular structure within multisoliton complexes. In the asynchronous time-lens system, another MLFL is constructed as the pump light in the FWM process, as shown in Fig. 1(b). For better interaction between the pump pulses and the under-test multisoliton complexes, it is necessary to guarantee that the repetition rates of the two are close enough. In order to realize it, we insert a tunable time delay line (TDL) to precisely regulate the repetition rate of the pump cavity. Figures 1(c) and 1(d) respectively demonstrate the temporal series of the signal and pump light detected by the OSC. As the repetition rates of two lasers are not locked, although all recorded as 31.51 ns, the OSC cannot distinguish the subtle distinctions between them. Besides, in the pump cavity, a segment of dispersion compensation fiber (DCF, 0.8 m) is inserted to adjust the net dispersion into the normal region, therefore generating pump pulses that can hold much high energy, of which the spectra are plotted in Fig. 1(e). Two bandpass filters (BPFs) respectively centered at 1590 nm and 1570 nm with the bandwidth of 13 nm guarantee that the multisoliton complexes and the pump light do not superimpose in the spectral domain, and two segments of SMF (250 m and 500 m) respectively serve as input dispersion and pump dispersion. When the signal and pump lights encounter in the highly nonlinear fiber (HNLF), the idle light centered around 1550 nm is efficiently generated in the FWM process. Ultimately, another BPF centered at 1550 nm with the bandwidth of 13 nm is harnessed to precisely extract the idle pulse, after which a 2.8-km DCF imposes a large output dispersion on the idle component, finally outputting the temporally magnified multisoliton complexes. Theoretically, the magnification $m$ rests with the ratio of the output dispersion to the input dispersion, which is about 28.75 in this system (see Appendix A), enough for temporal analysis.

The overall process is illustrated in Fig. 2. The DFT and time lens can image multisoliton complexes in two approaches, the former of which can acquire real-time spectra, via conducting fast Fourier transform (FFT) on which an autocorrelation (AC) trace containing molecular phases and binding separations can be obtained. Combined with the fine temporal structure acquired by temporal magnification, temporal-spectral characteristics of multisoliton complexes can be retrieved. Therefore, molecular assemblies of temporal distribution and internal motion can be completely unveiled.

The typical soliton triplets are prepared to estimate the usefulness and performance of the observation system, and the output power of the cavity in this scenario is 1.048 mW. Figure 3 depicts the observation results in detail. Preliminarily, the physical model of the soliton triplets is demonstrated in Fig. 3(a), in consideration of the temporal structure of which three solitons are termed as leading pulse (LP), middle pulse (MP), and trailing pulse (TP). Therefore, the essential degrees of freedom, i.e. the binding separations (${\tau }_{12}$, ${\tau }_{13}$, ${\tau }_{23}$) and relative phases (${\phi }_{12}$, ${\phi }_{13}$, ${\phi }_{23}$), can be clarified.

With respect to spectral observation, the interferogram of the soliton triplets obtained by the OSA is plotted in Fig. 3(b). Meanwhile, Fig. 3(c) demonstrates the consecutive recording of shot-to-shot interferograms with 2000 roundtrips, the close-up of which highlights the spectral details of soliton triplets. It can be concluded that the soliton triplets are in the stable operating condition. Via conducting FFT on the interferograms in Fig. 3(c), the first-order AC traces can be obtained and depicted in Figs. 3(d) and 3(e), from which the temporal separations and relative phases within the soliton triplets can be quantitatively retrieved. Obviously, it can be seen that the under-test soliton triplets are unequally spaced, ${\tau }_{12}$ and ${\tau }_{13}$ being relatively constant at $\sim 11.55\mathrm{ps}$ and $\sim 27.05\mathrm{ps}$ along the propagation.

Although the temporal separations can be indirectly retrieved through the DFT-based real-time spectral observation, this is only applicable for soliton molecules with few constituents. It lacks access to temporal information of soliton molecules, which particularly cannot distinguish the exact temporal distributions in multisoliton scenarios. Here, the time lens is utilized to make up for this deficiency, unveiling detailed temporal characteristics of the soliton triplets. As demonstrated in Fig. 3(f), the temporal waveforms are captured with 2000 roundtrips, which verify the asymmetrical molecular configuration. Furthermore, the temporal reconstruction equips us with a quantitative insight, particularly into pulse intensity and molecular separations, which are limned in Fig. 3(g). The red curve represents the normalized total intensity of the soliton triplets, from which it can be known that pulse energy does not fluctuate drastically, just within the marked region, certificating that the under-test soliton triplets keep a relatively steady state. Note that the efficiency of FWM is affected by the polarization state and external environmental perturbations; the energy fluctuation of the reconstructed soliton triplets may result from this. The bottom blue curves represent ${\tau }_{12}$ and ${\tau }_{13}$, which are respectively constant at $\sim 11.54\mathrm{ps}$ and $\sim 27.05\mathrm{ps}$. From the above quantitative retrieval, the molecular separations recorded by the time lens match well with the results retrieved from AC traces in the DFT technique. The above results manifest the superior performance of the temporal-spectral analysis, laying the foundation for subsequent observations of multisoliton complexes.

Akin to matter particles, soliton isomers are characterized by processing the same number of solitons yet with different temporal assemblies. They originate from the self-organization of optical solitons in dissipative systems, thus particularly existing in multisoliton complexes owing to their multiple constituents. It is worth noting that the property of temporal magnification of the time lens can help to obtain the internal structure directly and visually from the temporal perspective, making it the natural candidate for the observation of soliton isomers.

In our work, via altering the status of mode-locking, soliton quartets with diverse molecular assemblies can be formed. Utilizing the time lens, diversified intermolecular structures are visualized, among which four representative assemblies are parallelly delineated in Fig. 4. In the first place, we captured the almost equally spaced soliton quartets, the temporal waveform of which is demonstrated in Fig. 4(a). The shot-to-shot interferograms and AC traces are respectively plotted in Figs. 4(b) and 4(c). As one can imagine that the permutation of four solitons in the time domain can lead to multiple temporal structures, therefore apart from the equally spaced soliton quartets, unequally spaced ones are also common in the laser cavity. Here follow several other isomeric configurations of soliton quartets captured in the experiment. Following their temporal assemblies, we severally term them as (2 + 2), (3 + 1), and (2 + 1 + 1) soliton quartets. The captured temporal waveform of (2 + 2) soliton quartet is demonstrated in Fig. 4(d). It is equivalently regarded as the combination of two tightly bound soliton pairs. Additionally, the consecutive interferograms and the corresponding AC traces are jointly plotted in Figs. 4(e) and 4(f). For the other two scenarios in Figs. 4(g)–4(l), the (3 + 1) assembly is featured by one soliton triplet and one single soliton, while the (2 + 1 + 1) assembly is more likely formed by one soliton pair and two soliton singlets.

The above isomerism of soliton quartets demonstrates the versatility of temporal assemblies. Note that the temporal interval of constituents is the key degree of freedom in multisoliton complexes, which is analogous to the bond length between matter particles, determining the extent to how they interact with each other [42]. Except for the equally spaced scenarios, unequally spaced molecular complexes are also prevalent, which is facilitated by the fact that solitonic long-range and short-range interactions are well matched. It is obvious that it is difficult to distinguish the molecular assemblies of these soliton quartets just through the real-time interferograms and AC traces, but the time lens can solve this problem through directly recording temporal assemblies. Therefore, the time lens is applicable to probe the isomeric properties of multisoliton complexes, and some more complicated observation results are shown in Appendix B. Besides, from the applicational standpoint, the isomerism of multisoliton complexes is suitable for the pulse-counting-based optical encoding, and the time lens could make optical encoding become more visualized.

Soliton isomerism is shown not only in versatile temporal configurations but also in abundant internal motions. Here we engage intensive insight into gain-governed dynamics in soliton isomers, unveiling the intramolecular energy exchange within the (3 + 1) soliton quartets. The conceptual process of the energy exchange in the evolving (3 + 1) soliton quartets is illustrated in Fig. 5(a). Real-time interferograms are depicted in Fig. 5(b), recording 12,000 consecutive roundtrips. According to the assembly form, we naturally take the trailing pulse (the rightmost one) as the reference, and the quantitative analysis of the relative phases between the front three pulses and the referential pulse is severally conducted. From the phase trajectories depicted in Fig. 5(c), it can be found that they all decrease on the whole. As the phase velocity depends on the pulse intensity due to the Kerr effect, it can be concluded that the trailing pulse is weaker than the front three pulses. Besides, the relative phases of the leading and the second pulse to the trailing pulse exhibit very similar evolutionary trends with a stepping decrement, relying on the intensity differences among the three. Meanwhile, the retrieved relative phase between the third and the trailing solitons characterizes an upward and downward fluctuation, superimposed on a decreasing trend. These different phase evolutionary curves unveil the imbalance of the binding interaction inside the soliton quartets.

Theoretically, the internal dynamics of dissipative soliton molecules are mainly dominated by intramolecular energy exchange [30], and the time lens is utilized to obtain the amplitude of every constituent of the (3 + 1) soliton quartets. The internal solitonic structure of the evolving scenario is recorded by the time lens, shown in Fig. 5(d), indicating that there is indeed the energy exchange that occurs to the soliton quartets. Obviously, the intensity of the front three pulses is higher than that of the trailing one, therefore verifying the aforementioned conclusion. Figure 5(e) exhibits the normalized intensity of every soliton monomer along the propagation, four curves of which from the bottom to the top characterize the energy fluctuations of four pulses. It can be witnessed that energy exchange corresponds well to the evolution of the relative phases. The energy exchange between pulses alters the intensity of each pulse, which affects their phase velocity due to the Kerr effect, thereby varying the trend of the relative phase. Here, several certain frames within roundtrips 4620–5720 are selected as the analysis, of which temporal profiles are parallelly listed in Fig. 5(f). Energy exchange within the evolving soliton quartets can be clearly witnessed. In this selected range, the intensity of the second pulse first decreases and then increases, while the energies of the first and the third pulses all show the opposite trend. When it comes to the relative phases, the turning points correspond to the periodic flipping of the pulse intensity relation among these three constituents. Other time windows where energy exchanges exist also demonstrate this regularity, showing the exact relation between the relative phase evolutions and the pulse intensities, as well as validating the dominance of pulse energy on internal motions of multisoliton complexes.

In this scenario, the temporal observation results via the time lens suggest that energy exchange mainly exists within the former three pulses. In the front soliton triplets, the energy trend of the middle pulse is opposed to that of its immediate neighbors. When adding up the intensity of the front soliton triplets, it characterizes the feature of relative stability (bottom blue curve). It reflects the relative closure of energy exchange in the tightly bound triplets. It can be speculated that closer separations of the former soliton triplets provide convenience for energy transferring, whereas the longer separation obstacles the energy exchange between the former soliton triplets and the trailing pulses. All these analyses demonstrate the impressive capability of the time lens in unveiling the accurate temporal distributions and the energy exchange of dynamical multisoliton complexes, which particularly provides a new approach in further investigations on multisoliton complexes.

To better understand the isomeric dynamics in the dissipative solitonic system, we further conduct the numerical simulation. Referring to the laser cavity shown in Fig. 1(a), similar parameters are set in our modeled laser cavity. A lumped propagation model is utilized to describe each component of the cavity by a separate equation, and the pulse propagation in the ring cavity is modeled by a generalized nonlinear Schrödinger equation (GNLSE) in the following form: $$\frac{\partial \psi }{\partial z}+\frac{i}{2}{\beta }_2\frac{{\partial }^2\psi }{\partial t^2}-\frac{i}{6}{\beta }_3\frac{{\partial }^3\psi }{\partial t^3}-(\frac{g}{2}+\frac{{\partial }^2}{{\mathrm{\Omega }}^2\partial t^2})\psi =-\frac{\alpha }{2}\psi +i\gamma {|\psi |}^2\psi ,$$(1)where $\psi$ is the complex amplitude of the optical pulses, $z$ is the propagation distance, and $t$ is the propagation time in the cavity. $\gamma$ is the Kerr nonlinear coefficient. $\mathrm{\Omega }$ is the full width at half maximum of the gain fiber. Additionally, $g$ and $\alpha$ are the gain and attenuation coefficients. For the passive fiber (SMF), $g=0$, while for the active fiber (EDF), $g$ can be expressed as $$g=g_0\exp (-{\int }_0^{T_R}{|\psi |}^2\mathrm{d}t/E_S),$$(2)where $g_0$ is the small signal gain, $E_S$ describes the saturation energy of the EDF, and $T_R$ is the integration time. In our simulation, through altering the small signal gain, isomeric dynamics in the experimental results are reproduced. In the first place, abundant temporal assembly forms of soliton quartets are successfully obtained. With the appropriate parameter settings, we obtain (2 + 1 + 1), (1 + 2 + 1), (1 + 1 + 2), (3 + 1) soliton quartets, which are plotted in Fig. 6. They demonstrate quite different temporal characteristics just as what we observe in our experiments. It manifests the self-organization property of solitonic constituents within the laser cavity.

Further, to shed insight into internal motions in soliton isomers, we exemplify two dynamical processes of (3 + 1) soliton quartets and (1 + 2 + 1) soliton quartets in our simulation. For the evolving state of (3 + 1) soliton quartets, the simulated consecutive interferograms with 200 roundtrips are depicted in Fig. 7(a). The relative phases between the solitonic constituents are retrieved, shown in Fig. 7(b). Moreover, we provide the intensity curves of every pulse in the evolution process, plotted in Fig. 7(c). It should be noticed that there remains energy exchange between soliton constituents. As the intensity of the first pulse is all along higher than that of the fourth pulse with almost the identical intensity difference, therefore their relative phase ${\phi }_{14}$ keeps decreasing in a fixed velocity. Within the range of roundtrips 30 to 50, energy exchange occurs to the (3 + 1) soliton quartets, mainly between the second pulse and the third pulse, leading to the result that the intensity of the second pulse is higher than that of the fourth pulse while the third pulse is weaker than the fourth pulse. The relative phases ${\phi }_{24}$ and ${\phi }_{34}$ accordingly decrease and increase, respectively, whereafter, the four constituents are accompanied with persistent energy differences and slight periodic energy fluctuations. This numerical simulation validates that the energy exchange between adjacent two constituents is important for phase-dominated soliton dynamics.

The similar phenomenon occurs to another isomeric soliton quartet with (1 + 2 + 1) assembly in the simulation, which is depicted in Figs. 7(d)–7(f). In this scenario, energy exchange also alters the velocity of the phase sliding. To sum up, the above simulation records the process of energy exchange within two isomeric soliton quartets. It characterizes the correspondence between soliton energy and molecular phases, which powerfully manifests the dominance of pulse energy on internal motions, unveiling the intriguing isomeric dynamics in multisoliton complexes.

In conclusion, the isomeric dynamics of multisoliton complexes are investigated in depth with the assistance of a time-lens and DFT technique. Utilizing the magnification property of the time lens, the isomeric assemblies of soliton quartets are probed, intuitionally showing the diversity of molecular configurations. In parallel, we observe the fine energy exchange that occurs to the evolving (3 + 1) soliton quartets, which validates that the relative phase evolutions of soliton quartets are mainly dominated by the pulse energy. As a complement, the numerical simulation of the generation of soliton isomers is also manifested in the framework of the GNLSE, and similar isomeric dynamics are also successfully attained. The simulation results agree well with our experimental observation, verifying the gain-governed dynamics in mode-locked fiber lasers. The above work emphasizes the analogy between optical solitons and matter particles, and opens up the real-time characterization of multisoliton complexes. It would particularly assist in the artificial manipulations of ultrashort pulses, thus occupying a vital position in the realms of pulse-counting-based encoding and all-optical storage.