# Revealing Optical Soliton Radiation and Encoding via Nonlinear Fourier Transform

Yi Zhou, Gai Zhou, Yuwen Qin, Songnian Fu, Alan Pak Tao Lau, Philippe

Grelu, Kenneth Wong

► To cite this version:

Yi Zhou, Gai Zhou, Yuwen Qin, Songnian Fu, Alan Pak Tao Lau, et al. Revealing Optical Soliton Radiation and Encoding via Nonlinear Fourier Transform. ACS photonics, 2024, 10.1021/acsphotonics.4c01623. http://dx.doi.org/10.1021/acsphotonics.4c01623

HAL Id: hal-04875146

https://hal.science/hal-04875146v1

Submitted on 20 Jan 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

# Revealing Optical Soliton Radiation and Encoding via Nonlinear Fourier Transform

Yi Zhou,\# Gai Zhou,\*\# Yuwen Qin, Songnian Fu, Alan Pak Tao Lau, Philippe Grelu, and Kenneth K. Y. Wong\*

<figure><img src="image_6.png" /></figure>

Cite This: https://doi.org/10.1021/acsphotonics.4c01623

<figure><img src="image_7.png" /></figure>

Read Online

## ACCESS

<figure><img src="image_8.png" /></figure>

Metrics & More

<figure><img src="image_9.png" /></figure>

Article Recommendations

<figure><img src="image_10.png" /></figure>

Supporting Information

**ABSTRACT:** Mode-locked lasers exhibit a rich diversity of nonlinear dynamics, often featuring the nontrivial coexistence of linear dispersive waves and coherent structures, especially in transient evolution involving multiple soliton pulses. The coexistence of solitons and an embedded dispersive wave background sets a challenge for characterizing and analyzing these transient dynamics. Here, we demonstrate the real-time full-field characterization of transient soliton dynamics in a mode-locked fiber laser using nonlinear Fourier transform (NFT) and high-bandwidth coherent homodyne detection, revealing new insights into the physics of optical soliton interactions within complex nonlinear systems. Such characterization includes the formation of multiple solitons amid wide relaxation oscillations, the switching of multiple solitons, and controlled soliton drifting with associated digital encoding. NFT proves its efficiency in separating and analyzing coherent structures among the dispersive wave radiation in fiber lasers. By implementation of the inverse NFT, the corresponding pure soliton distribution can be reconstructed. These findings shed new light on ultrafast transient dynamics in optics.

**KEYWORDS:** mode locking, nonlinear Fourier transform, adaptive trust region, soliton, ultrafast fiber lasers

<figure><img src="image_11.png" /></figure>

## INTRODUCTION

Ultrashort pulses generated from mode-locked lasers find major applications in material processing, optical communications, optical sensors, spectroscopy, and medicine.¹⁻⁵ Besides, mode-locked fiber lasers constitute an ideal test bed for investigating a wealth of dissipative soliton dynamics, which arise from the complex balance among nonlinearity, dispersion, gain, and loss.⁶,⁷ Recent studies, enabled by the development of advanced real-time characterization, have emphasized the investigation of the transient and nonequilibrium dynamics of ultrafast lasers. These involve the buildup of single and multiple dissipative soliton structures within the laser cavity,⁸⁻¹² the observation of pulsating dynamics such as oscillations of soliton molecules and complexes,¹³⁻¹⁷ and the chaotic dynamics of incoherent dissipative solitons such as soliton explosions and noise-like pulses.¹⁸⁻²¹ The investigation of these complex soliton dynamics is also motivated by the prospect of enhancing the palette of laser outputs to fit novel applications. Real-time characterization has been popularized in the ultrafast laser community owing to the generalization of high-bandwidth oscilloscopes having an extended buffer memory, coupled to the efficiency of ultrafast optical techniques. The latter include the time-stretch dispersive Fourier transform (TS-DFT) for the direct measurement of single-shot spectra at MHz repetition rates²²,²³ and time-lens spectroscopy that proved its efficiency in revealing short transient waveforms such as optical rogue waves within optical turbulence.²⁴,²⁵

Some of us proposed a combination of TS-DFT and time lens to achieve a multiscale characterization, called real-time optical spectro-temporal analyzer (ROSTA), for single-shot studies of laser dynamics.²⁶ Although the DFT and time-lens methods have successfully realized laser output monitoring, nonequilibrium dynamics involving a large number of optical solitons often featuring the nontrivial coexistence of embedded dispersive waves and coherent structures, the dynamics of solitons alone or of pure solitons in the laser cavity are still largely unexplored.

Considering the prerequisites to apply complex nonlinear characterization techniques, such as time lens, it is worthy to revisit linear characterization methods, combining their experimental simplicity with the assistance of advanced data processing techniques, in order to better retrieve multiple field components. Among advanced processing methods applicable

Received: August 27, 2024

Revised: November 28, 2024

Accepted: December 3, 2024
<figure><img src="image_12.png" /><figcaption>**Figure 1.** Schematic illustration of the fiber laser experiment and NFT analysis. (a) Sketch of the experimental setup, including a mode-locked fiber laser and an external controller. The external CW light was coupled into the laser cavity to trigger pulse switching. The pump power is programmable by an AWG to achieve controllable multisoliton drifting. CW: wavelength-adjustable single-frequency laser, EOM: electro-optic modulator, AWG: arbitrary waveform generator, OIM: optical integrated module, PC: polarization controllers, WDM: wavelength division multiplexer, OC: output coupler, EDF: erbium-doped fiber, BPF: bandpass filter. (b) Principle of the real-time coherent homodyne detection configuration, the 90° hybrid generates in-phase (I) and quadrature (Q) copies of the laser signal which aids in the reconstruction of the single-sided complex-valued spectrum, as shown using the FFT domain representation. This real-time coherent homodyne detection configuration provides a high spectral resolution. The phase information on pulses is deterministically collected, and continuous measurement can be made without risking the loss of pulses due to the walk-off effect. (c) Optical spectrum of multiple pulses, and location of the injected CW light (green line) whose wavelength belongs to the gain bandwidth of multiple solitons. (d) Corresponding digital signal processing (DSP) flows of NFT analysis. (e) Full-field reconstructed mode-locked pulses in the time domain. (f) The discrete NF spectrum of multi-pulses corresponds to (e) after NFT is applied. The full-field time domain signal of four and six pulses corresponds to RT 20,000 in Figure 2 and RT 8200 in Figure 3, respectively.</figcaption></figure>

to ultrafast optics, the nonlinear Fourier transform (NFT) has recently attracted considerable attention.²⁷⁻³² From its inception, NFT is a powerful mathematical tool developed in the frame of the inverse scattering transform method to solve wave propagation in nonlinear integrable systems, which led to the celebrated soliton solutions of the nonlinear Schrödinger (NLS) equation.³³ NFT, applied to a temporal waveform, yields a nonlinear spectrum in the complex plane that is composed of a continuous part on the real axis featuring the dispersive (nonsoliton) components and a discrete set in the upper half complex plane, which corresponds to soliton components.²⁷ In the past few years, NFT has attracted significant attention for its potential in optical transmission systems using nonlinear optical fiber systems.²⁷⁻²⁹ The

applicative context of NFT was initially restricted to the situation of pulse propagation within an NLS medium. Nevertheless, it was shown that NFT could be applied to characterize the ultrafast dynamics of dissipative nonlinear systems under suitable normalization schemes, such as ultrafast lasers.³¹ For ultrafast fiber laser systems operating in the anomalous dispersion regime, there are large domains in the parameter space where the generated pulses, despite being intrinsically shaped by a dissipative soliton attractor, resemble NLS solitons, adopting a nearly hyperbolic secant waveform endowed with a slight frequency chirping.³⁴ As the nonlinear spectrum can be computed over successive cavity roundtrips, NFT can be viewed as a powerful way to decompose the output laser field between coherent soliton components and an
<figure><img src="image_13.png" /><figcaption>**Figure 2.** NFT data evolution obtained from the measurement of the four soliton formation within wide relaxation oscillation. (a,b) Evolution of the real-time laser output obtained from full-field measurement. (b) Corresponding to the zoom-in plot of (a). (c) Evolution of the imaginary and real parts of the NFT eigenvalues and temporal location distribution of individual solitons. (d) Instantaneous soliton temporal distribution obtained at the roundtrips marked by vertical dashed lines in (b). (e) Temporal location distribution of multiple soliton formation after NFT is applied corresponding to RT 10,000–20,000 in (a). (f) DEP and (g) phase evolution of multiple soliton formation with relaxation oscillation. Distribution of the NFT discrete eigenvalues of multiple soliton formation (depicted on a complex plane of nonlinear spectral parameter $\lambda$) in (h,i) corresponding to RT 10,500–11,000 and RT 19,500–20,000 in (a), respectively.</figcaption></figure>

embedded background of dispersive waves, therefore reducing the practical number of degrees of freedom to follow the laser dynamics accurately.

Moreover, there is major interest in following the energy exchanges between solitons and dispersive wave components. Soliton pulse reshaping generates dispersive waves, which can be resonantly reinforced over periodic perturbations such as successive cavity roundtrips. Reciprocally, a strong emission of dispersive waves can seed the formation of additional soliton pulses.¹¹ Therefore, even if the propagation medium significantly departs from the pure NLS system, NFT can be employed as a convenient nonlinear basis of analysis, highlighting key information about the nonequilibrium dynamical processes of optical solitons. In recent years, NFT has been effectively applied to characterize the simulated soliton dynamics in several nonlinear dissipative systems.³⁰⁻³²,³⁵ It is now highly desirable to test the ability of NFT to characterize versatile soliton dynamics in real-world

ultrafast laser experiments, where the presence of a large number of solitary modes can pose a challenge.

In this work, we demonstrate the successful full-field analysis of the transient soliton dynamics in an ultrafast fiber laser in the presence of embedded dispersive waves, as revealed through the NFT analysis. Our work encompasses the following dynamical situations: the formation of multiple solitons within wide relaxation oscillation, the switching of multiple solitons with reconfigurable temporal separation under external injection, the controllable drifting of optical solitons, and drifting-tailored digital encoding. The validity of our experimental results is confirmed by numerical simulations. Moreover, an original NFT scheme based on the ATR method is utilized to characterize optical soliton radiation, which can solve the large-sized discrete eigenvalues of complex pulses with low samples quickly and effectively. The eigenvalue probability distributions are used to classify different ultrafast laser regimes. Further, by using the inverse NFT (INFT), the
pure soliton distribution can be recovered, confirming that the filtered eigenvalues can characterize localized solitary modes with good accuracy. These results suggest that NFT can be used to identify localized solitons within a nonequilibrium evolution by excluding the embedded dispersive wave radiation influence, providing a unique picture of optical soliton interactions and assisting in the optimization of laser performance.

## EXPERIMENTS AND RESULTS

Optical solitons can be generated by the mode-locked fiber laser under the appropriate parameter settings and operating conditions. In this experiment, the real-time dynamics of multiple solitons were characterized in full field (amplitude and phase) by using coherent homodyne detection and NFT signal processing techniques. The corresponding experimental setup is shown in Figure 1a,b. The mode-locked multiple pulses are generated by the erbium gain-based ultrafast fiber laser with a nonlinear polarization rotation (NPR) mechanism. The details of the mode-locked fiber laser and the coherent homodyne detection can be found in Supporting Information S1. Figure 1e shows the full-field time domain retrieval of multiple laser pulses in a quasi-stationary state. The nonlinear spectrum is then calculated from such full-field information using the Zakharov-Shabat spectral approach. The details of the NFT processing and nonlinear spectrum calculation can be found in Supporting Information S8 and S9. Figure 1f displays the exemplary discrete nonlinear Fourier (NF) spectra for the multiple laser pulses shown in Figure 1e. Each point corresponds to an individual solitary mode, where the imaginary part of the eigenvalue Im(λ) is determined by the soliton amplitude, and the real part Re(λ) relates to the soliton center frequency and group velocity.

### Multiple Soliton Formation with Wide Relaxation Oscillation.
The formation process of multiple solitons from the background noise can be observed in Figure 2. It is initiated by switching on the pump power with a target value of 140 mW. Prominent relaxation oscillation can be observed in Figure 2a, which displays the temporal intensity evolution on a logarithmic scale. Whereas the four stable solitons are formed after relaxation oscillations, we note the absence of Q-switching and beating dynamics processes, in contrast to ref 11, therefore illustrating the multiplicity of soliton buildup dynamics. On the other hand, the early shaping of solitary modes within the relaxation oscillations, characterized by splitting, collision, and annihilation processes, bears similarity with optical turbulence.36 The appearance, proliferation, and clustering of solitary modes during relaxation oscillations lead to an increase in temporal complexity. Nevertheless, the transition leading to a large coherence can be clearly identified after RT 17,000 with the increase of the discrete-spectrum energy proportion (DEP, calculated by applying the Parseval principle, see Supporting Information S8), as shown in Figure 2f.

It is instructive to follow the history leading to the formation of an individual soliton, as in Figure 2b (corresponding to the dashed rectangle in Figure 2a). The corresponding eigenvalue evolution is shown in Figure 2c(i,ii). The evolution of Im(λ) indicates that initially four solitary modes arise from the background noise during the relaxation oscillation stage. Among them, three quasi-solitary modes (dashed lines in Figure 2) undergo annihilation one after another, in par with a transfer of energy to the survived solitary mode (red line in

Figure 2). The raised relaxation oscillation evolution of the survived solitary mode (red line) is resolved, markedly different from the relaxation evolution reported in ref 11. The group velocity difference among multiple solitary modes can be retrieved from the evolution of Re(λ), showing significant fluctuations during the relaxation oscillations (RT 13,000–18,000 in Figure 2). At the same time, the discrete spectrum features some eigenvalues with weaker imaginary parts and different real parts (black dots in Figure 2c), therefore representing pulse components having low energy, large temporal width, and narrow bandwidth: they tend to be quasi-CW backgrounds at different frequencies. We interpret such a collection as embedded dispersive wave radiation rather than solitary modes. Such dispersive waves arise during the involved multisoliton formation stage and disappear in the steady state. Therefore, these transient dispersive waves are different from the stationary resonant dispersive waves of solitons propagating in periodic media that generate temporal pulse pedestals and spectral sidebands.32 As the surviving solitary mode stabilizes from RT 19,000, the weak dispersive wave radiation annihilates [black dots in Figure 2c(i,ii)]. Several single-shot soliton temporal distributions suggest a major energy transfer and competition among multiple solitary modes during the formation process; see the variations of the intensities of solitary modes indicated by colored dots in Figure 2d. The evolutions of the intensities of the multiple solitary modes are consistent with the eigenvalues displayed in Figure 2. The distribution of the temporal locations of the multiple solitary modes is shown in Figure 2. The evolution of Im(λ), Re(λ), and the temporal location distribution for the other three survived solitons (Figure 2a), can be found in Supporting Information Figures S1 and S2.

To display more intuitively the intensity and group velocity evolution for the four survived solitons, we combine the eigenvalues involving these four solitons and those for two annihilated solitary modes in Figure 2e. The corresponding evolution of imaginary and real parts of eigenvalues can be found in Supporting Information Figure S3. A rise in the DEP in Figure 2f indicates that the participation of coherent structures gradually increases during the four-soliton formation, whereas the embedded dispersive wave radiation decreases. The phase evolution for multiple solitons is shown in Figure 2g. Here, the soliton phase is defined as the peak of the phase profile of each solitary mode. The phase of decaying solitary modes is also displayed in Figure 2g as colored dashed lines. The four surviving solitary modes (solid colored line) have a faster phase evolution compared to the decaying solitary modes. The solitary mode in purple line has the largest phase variation range, which is attributed to the fact that it reaches first a steady state.

The collective dynamics of multiple soliton formation with pronounced relaxation oscillation can be classified by a statistical interpretation of the eigenvalue distributions. Figure 2h,i shows the two-dimensional probability distribution of the eigenvalues on the complex plane for the different evolution states during the buildup. These distributions were obtained from the eigenvalue ensemble computed from 500 consecutive pulse measurements. The pronounced instability exhibited during the soliton relaxation oscillation is shown in Figure 2h. The surviving four solitary modes (colored solid dots) and annihilated solitary modes (colored cycles) span over wide areas, which is attributed to the rapidly changing positions of the individual eigenvalues. The embedded dispersive wave
<figure><img src="image_14.png" /><figcaption>**Figure 3.** Full-field and NFT data evolution obtained from the measurement of multiple soliton switching. (a,b,f) Real-time dynamics of laser output evolution obtained from full-field measurements. (b,f) Corresponding to the zoom-in plot of (a). White arrow in (a) corresponds to the injection components to trigger multiple soliton switching. (c,g) Evolution of imaginary and real parts of eigenvalues and temporal location distribution of individual solitons. (d,h) Instantaneous soliton temporal distribution obtained at the roundtrips marked by vertical dashed lines in (b,f), respectively. (e,i) Pure soliton temporal distribution by INFT corresponding to (d,h), respectively.</figcaption></figure>

radiation corresponding to the low-lying eigenvalues (black dots) can also be observed in the distribution. The asymptotic quasi-stationary state of the four surviving solitons can be seen from the clustering of the eigenvalue distribution in Figure 2i. Thus, the eigenvalue statistical distributions can be used to define the stability characteristics of ultrafast lasers and to reveal different dynamic evolution features.

In the present case, four solitons formed from background noise with a significant amplitude of relaxation oscillations. This specific evolution also illustrates the high sensitivity to initial conditions, even under a narrow optical spectrum bandwidth. The NFT analysis could retrieve the intense competition and energy transfer, leading to four quasi-stationary solitary modes. The relaxation oscillation possesses

the same oscillation period and random mode phase difference as the multiple spikes (see Figure S3i). The detailed buildup process of each solitary mode could be clearly resolved according to the eigenvalue evolution by removing the embedded CW background. The finally survived solitary mode with the highest intensity (purple and yellow lines in Figures 2e and S3i) does not correspond to the initial highest intensity solitary mode (blue and green lines), demonstrating the uncertainty of final soliton formation under the intense competition of multiple solitary modes.

**Dynamic Switching of Optical Solitons.** The switching of multiple solitons with reconfigurable temporal separation can be triggered by cross-gain saturation, which we induce by injecting a CW light whose wavelength belongs to the gain
<figure><img src="image_15.png" /><figcaption>**Figure 4.** Comparison of the waveform recovered by INFT and the exact one. The discrete spectra for INFT are, respectively, computed by the (a) software library FNFT and (b) new NFT scheme based on the ATR method. (c) Numerical simulation of the temporal intensity evolution for multiple soliton switching. The black arrow corresponds to the injection event.</figcaption></figure>

bandwidth. Increasing the pump power to 172 mW, a stable soliton bunch involving six solitons is generated. The initial stability of the soliton bunch is apparent from RT 7000–7300 in Figure 3a. To trigger the soliton switching, we launched a CW at a wavelength of 1550.3 nm (green line in Figure 1c) and an average power of ~0.175 mW into the laser cavity. The dramatic temporal switching behavior of multiple solitons can be observed in Figure 3a, with an enhanced noise background resolved from the full-field characterization in Figure 3b (and RT 7330 in Figure 3d). First, the intensities of the six solitons swiftly decrease to a level close to the background noise. Meanwhile, the solitons radiate strong dispersive waves that increase the embedded dispersive wave background, as confirmed by the evolution of Im(λ), see in particular the region indicated by the green arrow in Figure 3. One solitary mode (green line) attenuates earlier than the other modes (Figure 3). During their transient decay, the six solitary modes maintain almost the same group velocity, as indicated by the evolution of Re(λ) (Figure 3). The intense energy transfer of multiple solitary modes can be found in several single-shot soliton temporal distributions (Figure 3d), which is consistent with the evolution of Im(λ) (Figure 3). The corresponding temporal location distributions of multiple solitons are shown in Figure 3.

Subsequently, a transient stage where 12 solitary modes are regenerated from the background noise and then decay is shown in Figure 3f (corresponding to regime 2 in Figure 3a). This is corroborated by the evolution of Im(λ) shown in Figure 3, which indicates the increase and decay of the intensities of multiple solitary modes. The obvious intensity difference, asynchronous regeneration, and decay of the multiple solitary modes could be resolved. The evolution of Re(λ) indicates that the multiple solitary modes have close group velocities when they reach their highest intensities (RT 7500 to 7560 in Figure 3), while these velocities are notably different during the regeneration and decay stage. The corresponding distributions of the single-shot temporal

intensity and temporal location are shown in Figure 3h,g(iii), respectively. By carefully selecting the eigenvalues corresponding to the solitary wave modes and performing the INFT, we remove the contribution of the embedded dispersive wave background and reconstruct pure soliton temporal distributions, as illustrated in the single-shot reconstructions in Figure 3e,i. Such distributions clearly disclose the impact of dynamic switching. Compared to the mixture of solitary modes and an embedded dispersive wave background (Figure 3d,h), the largely different intensity distributions and temporal separation of reconstructed multiple solitary modes by INFT further demonstrate how the dispersive wave background indeed buffers solitons and affects the stability of coexisting solitons.

In the final stage, 10 solitary modes rebuild from the background and undergo collisions and annihilations, subsequently followed by the asymptotically stable evolution of six surviving solitons. Although, in the end, the number of solitons is recovered, their temporal distribution is different from that before switching state. Such dynamic switching of multiple solitons with reconfigurable temporal separation could find potential applications in ultrafast spectroscopy, information encoding, and computation.

In the case of a complex evolution of multisoliton involving a large number of eigenvalues in the discrete spectrum, e.g., 15 and 12 solitary modes during multiple soliton formation with relaxation oscillation and dynamic switching (Figures 2a and 3h), the standard fast NFT (FNFT) method based on the polynomial approach<sup>30,32,37</sup> has some difficulty in accurately retrieving these eigenvalues. For illustration, we used the conventional software library FNFT,<sup>38</sup> with default parameters, to compute the discrete spectrum of the single-shot waveform shown in Figure 3h and then recover the waveform with all computed eigenvalues and spectral coefficients. The comparison between the recovered and exact waveforms is shown in Figure 4a. The intensity profile of the recovered waveform is quite different from the exact one, which is attributed to the search errors of the discrete eigenvalues. To compute the
<figure><img src="image_16.png" /><figcaption>**Figure 5.** Controllable soliton drifting and exemplary multiletter encoding. Real-time spatial–temporal evolution of four soliton drifting in (a) three different pump driver voltage levels in a period. (b) Four different voltage levels in a period. (c) Successive recording of the encoded streams. The pump power is set at 140 mW and modulated by an electronic signal as embedded in the inset. Each frame is composed of four time slots and represents a single letter. (d) Encoded streams are decoded as {1020110231111}, representing a word {HIKIU}. The four drifting-defined regimes are with different colors. (e) Visualization of each letter. (f) Evolution of imaginary and real parts of eigenvalues and temporal distribution of the four solitons. (g) DEP evolution of the four drifting solitons.</figcaption></figure>

discrete spectrum accurately, we use the new NFT scheme based on the ATR method.³⁹ For the spectral coefficients, we use the bidirectional algorithm with the cutting-point criterion generalized from ref 40. With this new NFT scheme, the recovered waveform shown in Figure 4b is consistent with the exact one. All discrete spectra calculated from the experimental data presented in this work are computed by the new ATR–NFT scheme. To evaluate the accuracy of the eigenvalue search methods in the case of multiple soliton radiation, we quantify the retrieval errors based on generated multi-eigenvalue solitons which are similar to the radiation shown in Figure 3f and compare the accuracy of eigenvalue searching by using FNFT and ATR methods. This comparison shows that the computation accuracy of the ATR (adaptive trust-region) method is improved by 3 orders of magnitude compared to that of the FNFT (see Supporting Information Figure S4). Moreover, by suitable choice of eigenvalues and INFT to exclude the embedded dispersive wave background [black dots during Im(λ) and Re(λ) evolution] as indicated in Figure 1d, we can reconstruct the pure multiple solitary mode formation with relaxation oscillation and dynamic switching shown in Figure S5. The DEP and phase evolution of multiple soliton switching can be found in Supporting Information Figures S6 and S7.

To provide additional insight into the multisoliton switching in the mode-locked fiber laser under injection of an external

CW light, we perform numerical simulations based on the lumped propagation model, where each cavity component is modeled by a separate equation (see *Supporting Information S1*). The simulated temporal intensity evolution of six soliton switching (Figure 4c) agrees qualitatively well with the experimental result (Figure 3a) under an external CW light coupled into the laser cavity. The initial six solitons first decay owing to the gain cross saturation, then rebuild and split into 12 solitons (indicated by the dashed lines in Figure 4c). Next, the 12 solitary modes swiftly attenuate and rebuild to form 10 new solitary modes. Some solitary modes annihilate; finally, only six solitary modes survive with a different temporal separation compared to the initial one.

**Controllable Soliton Drifting and Encoding under Pump Modulation.** Whereas the CW laser injection highlighted the possibility of reshuffling the temporal distribution of a multisoliton bunch, we here test and analyze the possibility of controlling the group velocity of the whole multisoliton bunch. Here, the synchronous drifting of intra-cavity solitons can be simply and flexibly controlled by programming the pump power. To demonstrate this, we first apply a periodic electric modulation to the pump laser consisting of three different voltage levels. The corresponding temporal drifting of four solitons is observed in Figure 5a. Figure 5b illustrates the control of four different drifting speeds under four distinct applied voltage levels in a modulation
period. As a higher voltage level corresponds to a higher pump power, the four solitons are shown to accelerate when the pump power decreases. The physical reason is that the gain reduction induces a soliton frequency shift toward the shorter wavelength, entailing an increase of the group velocity in the situation of a laser cavity having a net anomalous dispersion. The corresponding numerical simulation of the synchronous multisoliton drifting can be found in Supporting Information Figure S8.

Since the deterministic manipulation of multisoliton patterns attracts extensive interest in ultrafast optics,⁴¹ we subsequently implement the programmable tailoring of the multiple soliton drifting for digital encoding. Based on the four different drifting speeds shown in Figure 5b, here, we introduce four drifting-defined regimes to represent the logical states 0, 1, 2, and 3 of a new quaternary encoding format with high distinguishability (drifting speeds of −95, 0, 103, and 202 fs/RT for “0”, “1”, “2”, and “3”, respectively). To exemplify this drifting-tailored digital encoding, we design an electronic modulation signal to stimulate continuous transition among these four drifting-defined regimes. Successive recording of the encoded streams corresponding to four soliton temporal evolution is shown in Figure 5c. In our test, the 1 bit time slot is set as 100 µs (encoding speed of 10 kHz). An exemplary “HKU” is encoded into real-time streams and decoded by the different drifting speeds (Figure 5c,d). For visualization, the multiletter streams are shown in Figure 5e. The intensity [Figure 5f(i)] and group velocity variation [Figure 5f(ii)] during four tons drifting can be resolved in the evolution of imaginary and real parts of eigenvalues. The corresponding temporal distribution and DEP evolution of the four drifting solitons are shown in Figure 5f(iii),g, respectively. The recording of {1020|1023|1111} allows the construction of encoding string {H|K|U} with high fidelity. This multiletter test verifies the availability of the drifting-tailored digital encoding and the potential for a higher capacity in all-optical storage. No obvious hysteresis process can be observed during different time slots, demonstrating that a higher encoding speed is achievable. To further verify the availability of this drifting-tailored encoding, another exemplary encoded string can be found in Supporting Information Figures S9 and S10. The global control illustrated here through external injection or programmable pump modulation (Figures 3 and 5) provides enhanced flexibility upon multiple soliton manipulation, which would be potentially useful in applications like soliton communications and soliton-based optical computations.

## ■ CONCLUSIONS

The prospect of utilizing the dynamical regimes of ultrafast lasers that go beyond stationary single-pulse mode locking represents a challenging vast potential for innovative applications, such as optical information processing and computing. In particular, a great challenge lies in performing an effective real-time full-field characterization of the complex ultrafast laser dynamics, which can mix strongly interacting multiple solitons and embedded dispersive wave radiation. Although recent advances in understanding and categorizing such complex laser dynamics were made through theory and numerical modeling by NFT,³²⁻³⁵ comparing the theoretical and numerical results carefully with the experiment is essential. Here, we demonstrated that the NFT scheme based on the ATR method could solve large-sized discrete eigenvalues of complex pulses with low samples quickly and effectively. This

represents an improvement of 3 orders of magnitude in the computation accuracy compared with previous work.³⁰,³²,⁴² As an additional important prospect, such ATR-NFT scheme can be applied to characterize the dynamics of soliton crystal in Kerr microresonators, as well as multisoliton in the topological temporally mode-locked laser.⁴³⁻⁴⁵ Our results reveal the complex multiple-soliton interactions and the transient emergence and decay of multiple coherent structures. Indeed, the analysis of laser radiation by NFT offers a natural framework for the description of localized coherent features embedded in dispersive waves.

To summarize the present work, we studied laser radiation involving a large number of optical solitons (12–15 solitary modes) during transient evolution by combining the ATR-NFT and coherent homodyne detection techniques. We investigated the multisoliton formation amid wide relaxation oscillations and confirmed the large complexity of such buildup, where the intense competition among multiple solitary modes leads to collisions and annihilations, resulting in patterns that varied from one buildup event to another. We also implemented control of multisoliton bunches, with the dynamic switching leading to reconfigurable temporal separation of the multisoliton pattern under external injection, and the efficient control of multisoliton drifting, leading to a successful test of drifting-tailored digital encoding. The eigenvalues of the multisoliton pattern were retrieved by using NFT, whose real and imaginary parts correspond to the frequency and amplitude of pulses, respectively. Soliton components can be isolated from the embedded dispersive wave background according to the eigenvalue distributions, which also demonstrate that the dispersive wave background is an energy buffer that affects the dynamics of coexisting multiple solitons. Further, the eigenvalue probability distribution was used to classify different lasing regimes according to the energy ratio of coherent structures. With the help of INFT, the pure soliton distribution can be recovered with a high accuracy. These findings shed new insights into the ultrafast dynamics of optical solitons, and we anticipate the wide application of a similar soliton manipulation approach through external injection or programmable pump power in the optimization of laser performance. Finally, we point out that the combination of NFT with the real-time full-field characterization technique provides a comprehensive frame of characterization and analysis, which helps understand better the complex dynamics of nonlinear systems mixing coherent structures and dispersive waves and motivates the investigation of broader innovative applications.

## ■ ASSOCIATED CONTENT

### ● Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsphotonics.4c01623.

Overview of the methods for experiments and simulations; error comparison of eigenvalue search by use of FNFT and ATR methods; pure soliton temporal evolution by INFT; DEP and phase evolution of optical soliton switching; numerical simulation of optical soliton drifting; more exemplary multiletter tests with a time slot of 100 µs; NFT and nonlinear spectrum computation; and trace methodology of discrete eigenvalues (PDF)
## AUTHOR INFORMATION

### Corresponding Authors

**Gai Zhou** – Advanced Institute of Photonics Technology, School of Information Engineering, and Guangdong Provincial Key Laboratory of Photonics Information Technology, Guangdong University of Technology, Guangzhou 510000, China; Photonics Research Institute, Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong 999077, China; Email: gaizer3085@gdut.edu.cn

**Kenneth K. Y. Wong** – Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong 999077, China; Advanced Biomedical Instrumentation Centre, Hong Kong 999077, China; orcid.org/0000-0003-3668-3539; Email: kywong@eee.hku.hk

### Authors

**Yi Zhou** – Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong 999077, China; orcid.org/0000-0002-8701-2834

**Yuwen Qin** – Advanced Institute of Photonics Technology, School of Information Engineering, and Guangdong Provincial Key Laboratory of Photonics Information Technology, Guangdong University of Technology, Guangzhou 510000, China

**Songnian Fu** – Advanced Institute of Photonics Technology, School of Information Engineering, and Guangdong Provincial Key Laboratory of Photonics Information Technology, Guangdong University of Technology, Guangzhou 510000, China

**Alan Pak Tao Lau** – Photonics Research Institute, Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong 999077, China

**Philippe Grelu** – Laboratoire ICB UMR 6303 CNRS, Université de Bourgogne, Dijon 21000, France

Complete contact information is available at:
https://pubs.acs.org/10.1021/acsphotonics.4c01623

### Author Contributions

\#Y.Z. and G.Z. contributed equally to this work. K.K.Y.W. and A.P.T.L. initiated the project. Y.Z. and G.Z. performed the experiments and simulations. A.P.T.L. and K.K.Y.W. supervised and guided the project. Y.Z., G.Z., Y.Q., S.F., A.P.T.L., P.G., and K.K.Y.W. discussed the results and contributed to the writing of the paper.

### Funding

This work was supported by the National Natural Science Foundation of China (62205068), the Research Grants Council of the Hong Kong Special Administrative Region of China (HKU 17212824, HKU 17210522, HKU C7074–21G, HKU R7003-21, and HKU 17205321), and the Innovation and Technology Commission of the Hong Kong SAR Government (MHP/073/20, MHP/057/21, and Health@InnoHK program).

### Notes

The authors declare no competing financial interest.

## REFERENCES

(1) Keller, U. Recent developments in compact ultrafast lasers. Nature 2003, 424, 831–838.

(2) Pang, M.; He, W.; Jiang, X.; Russell, P. St. J. All-optical bit storage in a fibre laser by optomechanically bound states of solitons. Nat. Photonics 2016, 10, 454–458.

(3) Klein, A.; Masri, G.; Duadi, H.; Sulimany, K.; Lib, O.; Steinberg, H.; Kolpakov, S. A.; Fridman, M. Ultrafast rogue wave patterns in fiber lasers. Optica 2018, 5, 774–778.

(4) Oktem, B.; Ülgüdür, C.; Ömer Ilday, F. Soliton-similariton fibre laser. Nat. Photonics 2010, 4, 307–311.

(5) Haus, H. A.; Wong, W. S. Solitons in optical communications. Rev. Mod. Phys. 1996, 68, 423–444.

(6) Grelu, P.; Akhmediev, N. Dissipative solitons for mode-locked lasers. Nat. Photonics 2012, 6, 84–92.

(7) Dudley, J. M.; Dias, F.; Erkintalo, M.; Genty, G. Instabilities, breathers and rogue waves in optics. Nat. Photonics 2014, 8, 755–764.

(8) Herink, G.; Jalali, B.; Ropers, C.; Solli, D. R. Resolving the buildup of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate. Nat. Photonics 2016, 10, 321–326.

(9) Peng, J.; Sorokina, M.; Sugavanam, S.; Tarasov, N.; Churkin, D. V.; Turitsyn, S. K.; Zeng, H. Real-time observation of dissipative soliton formation in nonlinear polarization rotation mode-locked fibre lasers. Commun. Phys. 2018, 1, 20.

(10) Liu, X.; Yao, X.; Cui, Y. Real-Time Observation of the Buildup of Soliton Molecules. Phys. Rev. Lett. 2018, 121, 023905.

(11) Liu, X.; Pang, M. Revealing the buildup dynamics of harmonic mode-locking states in ultrafast lasers. Laser Photonics Rev. 2019, 13, 1800333.

(12) Zhou, Y.; Ren, Y.-X.; Shi, J.; Mao, H.; Wong, K. K. Y. Buildup and dissociation dynamics of dissipative optical soliton molecules. Optica 2020, 7, 965–972.

(13) Hamdi, S.; Coillet, A.; Grelu, P. Real-time characterization of optical soliton molecule dynamics in an ultrafast thulium fiber laser. Opt. Lett. 2018, 43, 4965–4968.

(14) Wang, Z. Q.; Nithyanandan, K.; Coillet, A.; Tchofo-Dinda, P.; Grelu, P. Optical soliton molecular complexes in a passively mode-locked fibre laser. Nat. Commun. 2019, 10, 830.

(15) Peng, J.; Zhao, Z.; Boscolo, S.; Finot, C.; Sugavanam, S.; Churkin, D. V.; Zeng, H. Breather molecular complexes in a passively mode-locked fiber laser. Laser Photonics Rev. 2021, 15, 2000132.

(16) Zhou, Y.; Ren, Y.-X.; Shi, J.; Wong, K. K. Y. Breathing dissipative soliton molecule switching in a bidirectional mode-locked fiber laser. Adv. Photonics Res. 2022, 3, 2100318.

(17) Zhou, Y.; Shi, J.; Ren, Y.; Wong, K. K. Y. Reconfigurable dynamics of optical soliton molecular complexes in an ultrafast thulium fiber laser. Commun. Phys. 2022, 5, 302–311.

(18) Runge, A. F. J.; Broderick, N. G. R.; Erkintalo, M. Observation of soliton explosions in a passively mode-locked fiber laser. Optica 2015, 2, 36–39.

(19) Krupa, K.; Nithyanandan, K.; Grelu, P. Vector dynamics of incoherent dissipative optical solitons. Optica 2017, 4, 1239–1244.

(20) Zhou, Y.; Ren, Y.-X.; Shi, J.; Wong, K. K. Y. Breathing dissipative soliton explosions in a bidirectional ultrafast fiber laser. Photon. Res. 2020, 8, 1566–1572.

(21) Wang, Z. Q.; Nithyanandan, K.; Coillet, A.; Tchofo-Dinda, P.; Grelu, P. Buildup of incoherent dissipative solitons in ultrafast fiber lasers. Phys. Rev. Res. 2020, 2, 013101.

(22) Desbois, J.; Gires, F.; Turnois, P. A new approach to picosecond laser pulse analysis shaping and coding. IEEE J. Quantum Electron. 1973, 9, 213–218.

(23) Goda, K.; Tsia, K.; Jalali, B. Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena. Nature 2009, 458, 1145–1149.

(24) Foster, M. A.; Salem, R.; Geraghty, D. F.; Turner-Foster, A. C.; Lipson, M.; Gaeta, A. L. Silicon-chip-based ultrafast optical oscilloscope. Nature 2008, 456, 81–84.

(25) Suret, P.; Koussaifi, R. E.; Tikan, A.; Evain, C.; Randoux, S.; Szwaj, C.; Bielawski, S. Single-shot observation of optical rogue waves in integrable turbulence using time microscopy. Nat. Commun. 2016, 7, 13136.
(26) Wei, X.; Li, B.; Yu, Y.; Zhang, C.; Tsia, K. K.; Wong, K. K. Y. Unveiling multi-scale laser dynamics through time-stretch and time-lens spectroscopies. *Opt. Express* **2017**, *25*, 29098–29120.

(27) Turitsyn, S. K.; Prilepsky, J. E.; Le, S. T.; Wahls, S.; Frumin, L. L.; Kamalian, M.; Derevyanko, A. Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives. *Optica* **2017**, *4*, 307–322.

(28) Zhang, W. Q.; Gui, T.; Zhang, Q.; Lu, C.; Monro, T. M.; Chan, T. H.; Lau, A. P. T.; Shahraam Afshar, V. Correlated eigenvalues of multi-soliton optical communications. *Sci. Rep.* **2019**, *9*, 6399–6417.

(29) Zhou, G.; Sun, L.; Lu, C.; Lau, A. P. T. Multi-symbol digital signal processing techniques for discrete eigenvalue transmissions based on nonlinear Fourier transform. *J. Light. Technol.* **2021**, *39*, 5459–5467.

(30) Sugavanam, S.; Kopae, M. K.; Peng, J.; Prilepsky, J. E.; Turitsyn, S. K. Analysis of laser radiation using the nonlinear Fourier transform. *Nat. Commun.* **2019**, *10*, 5663.

(31) Chekhovskoy, I. S.; Shtyrina, O. V.; Fedoruk, M. P.; Medvedev, S. B.; Turitsyn, S. K. Nonlinear Fourier transform for analysis of coherent structures in dissipative systems. *Phys. Rev. Lett.* **2019**, *122*, 153901.

(32) Wang, Y.; Fu, S.; Kong, J.; Komarov, A.; Klimczak, M.; Buczynski, R.; Tang, X.; Tang, M.; Qin, Y.; Zhao, L. Nonlinear Fourier transform enabled eigenvalue spectrum investigation for fiber laser radiation. *Photon. Res.* **2021**, *9*, 1531–1539.

(33) Shabat, A.; Zakharov, V. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. *Sov. Phys. JETP* **1972**, *34*, 62.

(34) Grelu, P. Solitary waves in ultrafast fiber lasers: From solitons to dissipative solitons. *Opt. Commun.* **2024**, *552*, 130035.

(35) Turitsyn, S. K.; Chekhovskoy, I. S.; Fedoruk, M. P. Nonlinear Fourier transform for analysis of optical spectral combs. *Phys. Rev. E* **2021**, *103*, L020202.

(36) Turitsyna, E. G.; Smirnov, S. V.; Sugavanam, S.; Tarasov, N.; Shu, X.; Babin, S. A.; Podivilov, E. V.; Churkin, D. V.; Falkovich, G.; Turitsyn, S. K. The laminar–turbulent transition in a fibre laser. *Nat. Photonics* **2013**, *7*, 783–786.

(37) Wahls, S.; Poor, H. V. Fast numerical nonlinear Fourier transforms. *IEEE Trans. Inf. Theory* **2015**, *61*, 6957–6974.

(38) Wahls, S.; Chimmalgi, S.; Prins, P. J. FNFT: A software library for computing nonlinear Fourier transforms. *J. Open Source Software* **2018**, *3*, 597.

(39) He, J.; Jianping, L.; Qin, Y.; Lin, N.; Yu, X.; He, Y.; Xu, O.; Peng, D.; Xiang, M.; Zhou, G.; Fu, S. Adaptive trust-region-based algorithm for the discrete eigenvalue evaluation of the direct nonlinear Fourier transform. *Opt. Lett.* **2022**, *47*, 4195–4198.

(40) Prins, P. J.; Wahls, S. Soliton phase shift calculation for the Korteweg–de Vries equation. *IEEE Access* **2019**, *7*, 122914–122930.

(41) Liu, Y.; Huang, S.; Li, Z.; Liu, H.; Sun, Y.; Xia, R.; Yan, L.; Luo, Y.; Liu, H.; Xu, G.; Sun, Q.; Tang, X.; Shum, P. P. Phase-tailored assembly and encoding of dissipative soliton molecules. *Light Sci. Appl.* **2023**, *12*, 123.

(42) Ryczkowski, P.; Närhi, M.; Billet, C.; Merolla, J.-M.; Genty, G.; Dudley, J. M. Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser. *Nat. Photonics* **2018**, *12*, 221–227.

(43) Cole, D. C.; Lamb, E. S.; Del'Haye, P.; Diddams, S. A.; Papp, S. B. Soliton crystals in Kerr resonators. *Nat. Photonics* **2017**, *11*, 671–676.

(44) Karpov, M.; Pfeiffer, M. H. P.; Guo, H.; Weng, W.; Liu, J.; Kippenberg, T. J. Dynamics of soliton crystals in optical micro-resonators. *Nat. Phys.* **2019**, *15*, 1071–1077.

(45) Leefmans, C. R.; Parto, M.; Williams, J.; Li, G. H. Y.; Dutt, A.; Nori, F.; Marandi, A.; Marandi, A. Topological temporally mode-locked laser. *Nat. Phys.* **2024**, *20*, 852–858.