Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation

Temporal and spatiotemporal soliton molecules in ultrafast fibre lasers

Ultrafast fibre lasers, characterized by ultrashort pulse duration and broad spectral bandwidth, have drawn significant attention due to their vast potential across a wide range of applications, from fundamental scientific to industrial processing and beyond. As dissipative nonlinear systems, ultrafast fibre lasers not only generate single solitons, but also exhibit various forms of spatiotemporal soliton bunching. Analogous to molecules composed of multiple atoms in chemistry, soliton molecules (SMs) - alias bound states - in ultrafast fibre lasers are a key concept for gaining a deeper understanding of nonlinear interaction and hold a promise for advancing high-capacity fibre-optic communications. SMs are particularly notable for their high degree of controllability, including their internal temporal separation, and relative phase differences, thereby suggesting new possibilities for manipulating multi-pulse systems. In this review, we provide a comprehensive overview of recent advancements in the studies of SMs with the multidimensional parameter space in ultrafast fibre lasers. Owing to the flexibility afforded by mode-locking techniques and dispersion management, various types of SMs - with diverse values of the soliton number, relative phase, pulse separation, carrier frequencies, and even modal dispersion - have been experimentally demonstrated. We also discuss other basic nonlinear optical phenomena observed in fibre lasers, including the formation, spatiotemporal pulsations, and interaction dynamics of SMs. Furthermore, we explore the multidimensional control of SMs through approaches such as gain modulation, polarization control, dispersion management, and photomechanical effects, along with their applications to optical data encoding. Finally, we discuss challenges and future development of multidimensional technologies for the manipulation of SMs.

Bound state of asymmetric pure-quartic solitons in motion

In this Letter, we investigate the binding mechanism and motion dynamics of the bound state consisting of two pure-quartic solitons (PQSs) with unequal intensities and find that their movement occurs as an entity under the Raman self-frequency shift. By calculating the forces that induce the relative motion between the unequal PQSs, we derive the balanced conditions for maintaining a near-constant separation and the constant phase profile between them. The predictions are validated by the numerical simulations. Our work provides insights into the intrinsic features of symmetry-broken localized structures in nonlinear-dispersive systems, stimulating interest in asymmetric multi-soliton states with intriguing dynamics.

Pure-high-even-order dispersion bound solitons complexes in ultra-fast fiber lasers

Temporal solitons have been the focus of much research due to their fascinating physical properties. These solitons can form bound states, which are fundamentally crucial modes in fiber laser and present striking analogies with their matter molecules counterparts, which means they have potential applications in large-capacity transmission and all-optical information storage. Although traditionally, second-order dispersion has been the dominant dispersion for conventional solitons, recent experimental and theoretical research has shown that pure-high-even-order dispersion (PHEOD) solitons with energy-width scaling can arise from the interaction of arbitrary negative-even-order dispersion and Kerr nonlinearity. Despite these advancements, research on the bound states of PHEOD solitons is currently non-existent. In this study, we obtained PHEOD bound solitons in a fiber laser using an intra-cavity spectral pulse shaper for high-order dispersion management. Specifically, we experimentally demonstrate the existence of PHEOD solitons and PHEOD bound solitons with pure-quartic, -sextic, -octic, and -decic dispersion. Numerical simulations corroborate these experimental observations. Furthermore, vibrating phase PHEOD bound soliton pairs, sliding phase PHEOD bound soliton pairs, and hybrid phase PHEOD bound tri-soliton are discovered and characterized. These results broaden the fundamental understanding of solitons and show the universality of multi-soliton patterns.

Buildup and synchronization regimes of a vector pure-quartic soliton molecule in a fiber laser cavity

Pure-quartic solitons (PQSs) have recently received increasing attention due to their energy-width scaling over the traditional soliton, which has expanded our understanding of soliton dynamics with high-order dispersion in nonlinear systems. Here, we numerically reveal the asynchronization and synchronization processes of the sub-pulse within the vector PQS molecule in a mode-locked fiber laser by solving the coupled Ginzburg-Landau equations. During the establishment of a vector PQS molecule, the repulsion, attraction, and finally stabilization processes have been observed. Specifically, sub-pulse disappearance, regeneration, and finally synchronization with the other pulses are also investigated. Our analysis of the pulse energy, time interval, and relative phase evolution dynamics with the round trip indicates that the asynchronization and synchronization within the vector PQS molecule associate tightly with the gain competition and the cross-phase modulation. Our findings provide insights into the internal mutual dynamics within the vector soliton molecule and offer guidance for the applications of PQS.

Synchronization, Desynchronization, and Intermediate Regime of Breathing Solitons and Soliton Molecules in a Laser Cavity

We report on the experimental and numerical observations of synchronization and desynchronization of bound states of multiple breathing solitons (breathing soliton molecules) in an ultrafast fiber laser. In the desynchronization regime, although the breather molecules as wholes are not synchronized to the cavity, the individual breathers within a molecule are synchronized to each other with a delay (lag synchronization). An intermediate regime between the synchronization and desynchronization phases is also observed, featuring self-modulation of the synchronized state. This regime may also occur in other systems displaying synchronization. Breathing soliton molecules in a laser cavity open new avenues for the study of nonlinear synchronization dynamics.

Auto-setting multi-soliton temporal spacing in a fiber laser by a hybrid GA-PSO algorithm

Multi-soliton operation in fiber lasers is a promising platform for the investigation of soliton interaction dynamics and high repetition-rate pulse. However, owing to the complex interaction process, precisely manipulating the temporal spacing of multiple solitons in a fiber laser is still challenging. Herein, we propose an automatic way to control the temporal spacing of multi-soliton operation in an ultrafast fiber laser by a hybrid genetic algorithm-particle swarm optimization (GA-PSO) algorithm. Relying on the intelligent adjustment of the electronic polarization controller (EPC), the on-demand temporal spacing of the double solitons can be effectively achieved. In particular, the harmonic mode locking with equal temporal spacing of double solitons is also obtained. Our approach provides a promising way to explore nonlinear soliton dynamics in optical systems and optimize the performance of ultrafast fiber lasers.

Linear coupling-related pulse splitting in fiber lasers

We demonstrate a unique pulse-splitting mechanism dominated by the linear coupling between two vector modes in a mode-locked fiber laser using polarization-maintaining fiber. As the linear coupling strength increases, the pulse experiences larger perturbations and manifests as stronger spectral sidebands. Correspondingly, the temporal pedestals possessing a higher intensity become untrapped and eventually evolve into a stable pulse. Such linear coupling-related pulse splitting is ubiquitous both in normal- and anomalous-dispersion regimes, fundamentally differing from that induced by the excessive nonlinear phase shift. Experimental observations fully sustain numerical results and provide a flexible approach to managing the number and energy of vector solitons.

Observation of bound solitons generated by a figure-9 fiber laser in the 1 µm band

The paper describes the observation of diverse bound-state patterns, including tightly bound states, loosely bound states, and composite bound states, in a figure-9 fiber laser. By performing dispersion management and using polarization-maintaining fibers with high gain coefficient, stable dispersion-managed solitons and bound solitons can be simultaneously generated. This work advances our understanding of complex soliton dynamics and presents a novel, to the best of our knowledge, approach for future applications of bound states. Additionally, the research involves integrated packaging, effectively enhancing overall work stability.

Investigations on diverse dynamics of soliton triplets in mode-locked fiber lasers

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.

Two-photon imaging of soliton dynamics

Optical solitary waves (solitons) that interact in a nonlinear system can bind and form a structure similar to a molecule. The rich dynamics of this process have created a demand for rapid spectral characterization to deepen the understanding of soliton physics with many practical implications. Here, we demonstrate stroboscopic, two-photon imaging of soliton molecules (SM) with completely unsynchronized lasers, where the wavelength and bandwidth constraints are considerably eased compared to conventional imaging techniques. Two-photon detection enables the probe and tested oscillator to operate at completely different wavelengths, which permits mature near-infrared laser technology to be leveraged for rapid SM studies of emerging long-wavelength laser sources. As a demonstration, using a 1550 nm probe laser we image the behavior of soliton singlets across the 1800-2100 nm range, and capture the rich dynamics of evolving multiatomic SM. This technique may prove to be an essential, easy-to-implement diagnostic tool for detecting the presence of loosely-bound SM, which often remain unnoticed due to instrumental resolution or bandwidth limitations.

Observation of the "invisible" pulsation of soliton molecules in a bidirectional ultrafast fiber laser

A novel optical soliton dynamics phenomenon, called "invisible" pulsation, has gradually attracted extensive interest in recent years, which can only be identified effectively with the help of real-time spectroscopy technique, i.e., dispersive Fourier transformation (DFT). In this paper, based on a new bidirectional passively mode-locked fiber laser (MLFL), the "invisible" pulsation dynamics of soliton molecules (SMs) is systematically studied. It is indicated that the spectral center intensity, pulse peak power and relative phase of SMs are periodically changed during the "invisible" pulsation, while the temporal separation inside the SMs is constant. The degree of spectral distortion is positively correlated with the pulse peak power, which verifies that self-phase modulation (SPM) is the inducement of spectral distortion. Finally, the universality of the SMs "invisible" pulsation is further experimentally verified. We believe our work is not only conducive to the development of compact and reliable bidirectional ultrafast light sources, but also of great significance to enrich the study of nonlinear dynamics.

Dichromatic "Breather Molecules" in a Mode-Locked Fiber Laser

Bound states of solitons ("molecules") occur in various settings, playing an important role in the operation of fiber lasers, optical emulation, encoding, and communications. Soliton interactions are generally related to breathing dynamics in nonlinear dissipative systems, and maintain potential applications in spectroscopy. In the present work, dichromatic breather molecules (DBMs) are created in a synchronized mode-locked fiber laser. Real-time delay-shifting interference spectra are measured to display the temporal evolution of the DBMs, that cannot be observed by means of the usual real-time spectroscopy. As a result, robust out-of-phase vibrations are found as a typical intrinsic mode of DBMs. The same bound states are produced numerically in the framework of a model combining equations for the population inversion in the mode-locked laser and cross-phase-modulation-coupled complex Ginzburg-Landau equations for amplitudes of the optical fields in the fiber segments of the laser cavity. The results demonstrate that the Q-switching instability induces the onset of breathing oscillations. The findings offer new possibilities for the design of various regimes of the operation of ultrafast lasers.

Soliton molecules and their scattering by a localized P T-symmetric potential in atomic gases

We propose a physical scheme to study the formation of optical soliton molecules (SMs), consisting of two solitons bound together with a π-phase difference, and the scattering of SMs by a localized parity-time (P T)-symmetric potential. In order to stabilize SMs, we apply an additional space-dependent magnetic field to introduce a harmonic trapping potential for the two solitons and balance the repulse interaction induced by the π-phase difference between them. On the other hand, a localized complex optical potential obeying P T symmetry can be created through an incoherent pumping and spatial modulation of the control laser field. We investigate the scattering of optical SMs by the localized P T-symmetric potential, which exhibits evident asymmetric behavior and can be actively controlled by changing the incident velocity of SMs. Moreover, the P T symmetry of the localized potential, together with the interaction between two solitons of the SM, can also have a significant effect on the SM scattering behavior. The results presented here may be useful for understanding the unique properties of SMs and have potential applications in optical information processing and transmission.

All-fiber saturable absorber based on slightly tapered graded index multimode fiber for conventional and bound soliton generation

A slightly tapered graded index multimode fiber (GIMF) is demonstrated as the saturable absorber (SA) for a conventional soliton and tightly bound soliton generation. The SA device is simply a GIMF spliced with single-mode fiber (SMF) at its two ends. The waist diameter of the slightly tapered fiber is ∼23.8µm, located at one of the splicing points of SMF and GIMF. The mode-locked output of conventional solitons and tightly bound solitons can be obtained in a ring fiber laser based on such a SA. The central wavelength of the output conventional solitons is 1563.5 nm, and the pulse width is as short as 473 fs at the fundamental frequency of 19.53 MHz. The central wavelength of the tightly bound solitons is 1563.5 nm, with a pulse width of 636 fs and a separation of 1.89 ps. The device is featured with a simple and robust structure and stable operation capability and can effectively overcome the difficulty in requirement of precise control of the GIMF length, which make it attractive in ultrafast lasers and optical fiber communication systems.

Novel optical soliton molecules formed in a fiber laser with near-zero net cavity dispersion

Soliton molecules (SMs) are stable bound states between solitons. SMs in fiber lasers are intensively investigated and embody analogies with matter molecules. Recent experimental studies on SMs formed by bright solitons, including soliton-pair, soliton-triplet or even soliton-quartet molecules, are intensive. However, study on soliton-binding states between bright and dark solitons is limited. In this work, the formation of such novel SMs in a fiber laser with near-zero group velocity dispersion (ZGVD) is reported. Physically, these SMs are formed because of the incoherent cross-phase modulation of light and constitute a new form of SMs that are conceptually analog to the multi-atom molecules in chemistry. Our research results could assist the understanding of the dynamics of large SM complexes. These findings may also motivate potential applications in large-capacity transmission and all-optical information storage.

Multi-Pulse Bound Soliton Fiber Laser Based on MoTe2 Saturable Absorber

Bound solitons have become a hot topic in the field of nonlinear optics due to their potential applications in optical communication, information processing and radar systems. However, the trapping of the cascaded bound soliton is still a major challenge up to now. Here, we propose and experimentally demonstrate a multi-pulse bound soliton fiber laser based on MoTe₂ saturable absorber. In the experiment, MoTe₂ nanosheets were synthesized by chemical vapor deposition and transferred to the fiber taper by optical deposition. Then, by inserting the MoTe₂ saturable absorber into a ring cavity laser, the two-pulse, three-pulse and four-pulse bound solitons can be stably generated by properly adjusting the pump strength and polarization state. These cascaded bound solitons are expected to be applied to all-optical communication and bring new ideas to the study of soliton lasers.

Real-Time Access to Collisions between a Two-Soliton Molecule and a Soliton Singlet in an Ultrafast Fiber Laser

Optical solitons in ultrafast fiber lasers, as a result of dual balances between dispersion and nonlinearity as well as gain and loss, enable various soliton interactions. Soliton collisions are among the most intriguing soliton interactions, which fuel the understanding for particle-like properties of solitons. Here, we experimentally investigate the transient dynamics of collisions between a two-soliton molecule and a soliton singlet in a mode-locked fiber laser. By means of the dispersive Fourier transform technique, the evolving spectral interferograms of different collision scenarios are measured in real time. In particular, the “quasi-elastic” collision is observed, which shows that the soliton-molecule state remains unaltered after the collision and the group-velocity difference between the soliton molecule and the singlet is changed. It is directly demonstrated that a bond exchange occurs between the colliding solitons. By tuning the intra-cavity polarization controller, the dynamic processes of other collision outcomes, including the annihilation of a soliton in the soliton molecule as well as the formation of a stable unequally spaced soliton triplet, are also revealed. Our work facilitates a deeper understanding of soliton collision dynamics in ultrafast fiber lasers.

Superlocalization Reveals Long-Range Synchronization of Vibrating Soliton Molecules

We implement a superlocalization method in the time domain that allows the observation of the external motion of soliton molecules in a fiber ring cavity laser with unprecedented accuracy. In particular, we demonstrate the synchronization of two oscillating soliton molecules separated by several nanoseconds, with intermolecular oscillations following the same pattern as the intramolecular motion of the individual molecules. These experimental findings indicate an interplay between the different interaction mechanisms that coexist inside the laser cavity, despite their very different characteristic ranges, timescales, strengths, and physical origins.

Elastic and inelastic collision dynamics between soliton molecules and a single soliton

Dissipative systems form various self-organized states owing to the abundant attractor structures. The study of the response of different self-organized states under collision perturbation is of great significance for understanding the dissipative nonlinear systems. The collision dynamics of single soliton and soliton molecules can not only assist the stability analysis of attractors, but also reveal the rich physical connotations of soliton interactions. Here, for the first time, the collision processes of single soliton and soliton molecules in different excited states are detected using the dispersive Fourier transform technology. The collision processes include the disintegration and rebuilding of soliton molecules as well as chaotic oscillating evolution, accompanied by the emergence of transition states such as triple binding state, soliton fusion and acceleration. According to whether the soliton molecule can return to its initial excited state, the collisions are classified as elastic and inelastic. The different interaction strength between solitons is an important condition for rebuilding stable soliton molecules. Numerical simulations show that the gain dynamics are the main physical origin of collisions. Our research will stimulate in-depth research on the interaction of self-organized states in nonlinear systems such as chemical molecules, and have potential applications in optical logic gates.

Self-Frequency Shift in Transmission of Asymmetric Pulse in Optical Medium

Linear and nonlinear effects often induce a pulse self-frequency shift as it propagates along with an optical medium. Here, we theoretically investigate the transmission dynamics of asymmetric pulses propagating along with an optical medium in the temporal and spectral domains. Due to the asymmetric nonlinear phase-shift effect in the optical medium, the peak wavelength of asymmetric pulses exhibits a redshift or a blueshift in the spectral domain, while it slows down or speeds up in the temporal domain. Our results show that the peak wavelength shift initiated by a temporal or spectral asymmetric pulse depends not only on the pulse intensity, but also on the initial pulse chirp and dispersion of optical medium. We find that the peak wavelength shift of the asymmetric pulse increases with the pulse intensity and the initial pulse chirp, together with the spectrum width. The temporal and frequency shifts of the asymmetric pulses are found to be sensitive to the asymmetry ratio as well. These excellent properties may lead to the realization of a self-frequency shift-based tunable light source by launching asymmetric pulses into an optical medium.

Internal dynamics in bound states of unequal solitons

The bound states (BSs) of solitons are found to have intriguing internal dynamics in ultrafast lasers. Here, we explore the binding mechanism and internal motions of asymmetric bound state (ABS) solitons constituted by unequal solitons at short-range with their tails directly overlapped. Experiments and simulations show that the periodic energy flux between two solitons, mediated by their overlapped tails, gives rise to a balanced separation and energy distribution across the ABS. The motion mechanisms of strong and weak solitons are discussed in detail. This work provides insights into the binding mechanism and internal dynamics of BSs.

Bound-State Soliton and Noise-Like Pulse Generation in a Thulium-Doped Fiber Laser Based on a Nonlinear Optical Loop Mirror

We demonstrate a thulium-doped, mode-locked, all-fiber laser capable of operating in two generation regimes: dispersion-managed soliton and noise-like pulse (NLP). Employing a nonlinear optical loop mirror as an artificial saturable absorber, the oscillator generated optical pulses with a fundamental pulse repetition frequency of ~15.795 MHz. The total net dispersion of the laser cavity had a slightly anomalous group delay dispersion value of −0.016 ps2. After appropriate adjustment of a polarization controller, bound states of a dispersion-managed soliton composed of three pulses with fixed soliton separations were also observed. NLP generation, tunable over 35 nm from 1943.5 to 1978 nm, was also presented in the same laser setup. To our knowledge, this is the first report of the generation of tunable NLPs in a mode-locked thulium-doped fiber laser based on a nonlinear loop mirror saturable absorber.

Interactions of the solitons in periodic driven-dissipative systems supporting quasibound states in the continuum

The paper is devoted to the dynamics of dissipative gap solitons in the periodically corrugated optical waveguides whose spectrum of linear excitations contains a mode that can be referred to as a quasi-bound state in the continuum. These systems can support a large variety of stable bright and dark dissipative solitons that can interact with each other and with the inhomogeneities of the pump. One of the focus points of this work is the influence of slow variations of the pump on the behavior of the solitons. It is shown that for the fixed sets of parameters the effect of pump inhomogeneities on the solitons is not the same for the solitons of different kinds. The second main goal of the paper is systematic study of the interaction between the solitons of the same or different kinds. It is demonstrated that various scenarios of intersoliton interactions can occur: The solitons can repulse each other or get attracted. In the latter case, the solitons can annihilate, fuse in a single soliton, or form a new bound state depending on the kinds of the interacting solitons and on the system parameters.

Periodic attraction and repulsion within the tight-bound π-phase soliton molecule

In the over-pumped dissipative system, the single pulse is prone to split into multi-soliton modes, among which the soliton molecule (SM) comprising two pulses has attracted much interest recently. In this Letter, the tight-bound SM with the π-phase-difference, a soliton pair predicted to be unstable observed in fiber lasers, is found to have oscillating separation with excellent stability. For the first time, to the best of our knowledge, we reveal the mechanism of the π-phase SM to circumvent the irreversible repulsion and the role of dispersive waves on the SM. During the periodic propagation, the destructive interference between solitons produces the repulsion while the dispersive waves give rise to the attractive force, leading to the dynamic oscillating behavior of the SM. The numerical simulation reproduces the experimental observation and offers panoramic insights into the nonlinear interactions between multiple components in dissipative systems.

Quantum limited timing jitter of soliton molecules in a mode-locked fiber laser

Soliton molecules in mode-locked lasers are expected to be ideal self-organization patterns, which warrant stability and robustness against perturbations. However, recent ultra-high resolution optical cross-correlation measurements uncover an intra-molecular timing jitter, even in stationary soliton molecules. In this work, we found that the intra-molecular timing jitter has a quantum origin. Numerical simulation indicates that amplified spontaneous emission (ASE) noise induces a random quantum diffusion for soliton pulse timing, which cannot be compensated by soliton binding mechanism. By suppressing indirectly coupled timing jitter at close-to-zero cavity dispersion, a record-low 350 as rms intra-soliton-molecular jittering is obtained from an Er-fiber laser in experiment. This work provides insight into the fundamental limits for the instability of multi-soliton patterns.

Widely stretchable soliton crystals in a passively mode-locked fiber laser

We present the first direct demonstration of a new type of stable and extremely elastic soliton crystals, the bond length and bond strength of which can be individually controlled in a wide range. The stretching and compressing can be repeated many times, conserving the overall structure by incorporating a highly asymmetric tunable Mach-Zehnder interferometer into a specially designed passively mode-locked fiber laser. The temporal structure and dynamics of the generated soliton crystals were measured using an asynchronous optical sampling system with picosecond resolution. We demonstrated that a stable and robust soliton crystal can be formed by two types of primitive structures: single dissipative solitons and (or) pairs each consisting of a dissipative soliton and a pulse with a lower amplitude. Continuous stretching and compression of the soliton crystal by an extraordinarily high factor of more than 30 has been demonstrated, the smallest recorded separation between the pulses being as low as 5 ps, corresponding to an effective repetition frequency of 200 GHz. Collective pulse dynamics, including soliton crystal cracking and transformation of crystals comprising high/low-amplitude pulse pairs to the crystals of similar pulses, has been observed experimentally.

Synthesis and dissociation of soliton molecules in parallel optical-soliton reactors

Mode-locked lasers have been widely used to explore interactions between optical solitons, including bound-soliton states that may be regarded as "photonic molecules". Conventional mode-locked lasers normally, however, host at most only a few solitons, which means that stochastic behaviours involving large numbers of solitons cannot easily be studied under controlled experimental conditions. Here we report the use of an optoacoustically mode-locked fibre laser to create hundreds of temporal traps or "reactors" in parallel, within each of which multiple solitons can be isolated and controlled both globally and individually using all-optical methods. We achieve on-demand synthesis and dissociation of soliton molecules within these reactors, in this way unfolding a novel panorama of diverse dynamics in which the statistics of multi-soliton interactions can be studied. The results are of crucial importance in understanding dynamical soliton interactions and may motivate potential applications for all-optical control of ultrafast light fields in optical resonators.

Orbital-angular-momentum-resolved diagnostics for tracking internal phase evolution in multi-bound solitons

The generation of multi-bound solitons is a fascinating subject of investigation in many conservative and dissipative systems, such as photonics, fluid mechanics, Bose-Einstein condensates, and so on. In this study, we demonstrate the successful extraction of phase dynamics between solitons in bound multiple solitons with up to seven constituents in a mode-locked Er laser system. By mapping the internal phase motions of multi-bound solitons to the spatial phase movement of cylindrical vector beams using orbital angular momentum (OAM)-based diagnostics, different categories of internal pulsations are revealed. We show that bound state of four solitons exhibits linear drifting relative phase evolution dynamics; while for bound multiple solitons with constituents from five to seven pulses, stationary relative phase dynamics are observed. These findings highlight the possibility of the OAM-based method access to the internal motion of multi-soliton molecules with more freedom of degrees and fuel the analogy with research on chemistry molecule complex.

Molybdenum Carbide Buried in D-Shaped Fibers as a Novel Saturable Absorber Device for Ultrafast Photonics Applications

Study of nonlinear laser-matter interactions in 2D materials has promoted development of photonics applications. As a typical MXene material, molybdenum carbide (Mo₂C) has attracted much attention because of its graphene-like structure. Here, a type of D-shaped fiber (DF)-buried Mo₂C saturable absorber (SA) fabricated by magnetron-sputtering deposition (MSD) and sol-gel technique is reported. The Mo₂C material was buried between the bottom DF and the upper amorphous silica fabricated by sol-gel technology. Therefore, the DF-based SA effectively solves the problem of material shedding and aging, thus improving the stability and damage threshold of the fiber laser. Application of the SA in erbium-doped fiber laser and stable passive Q-switched operation with a maximum pulse energy of 430.47 nJ is realized. By adjusting the polarization state and pump power, high-power mode-locked pulses are generated with a pulse duration and output power of 199 fs and 54.13 mW, respectively. Further, bound-state soliton pulses are obtained with a pulse width of 312 fs and soliton interval of 1.26 ps for the first time based on MXene materials. Moreover, by application of the SA in ytterbium-doped fiber lasers, a stable dissipative soliton mode-locked pulse is obtained with a pulse width of 23 ps. These results indicate that the DF-based buried Mo₂C as a novel SA provides a reliable method for all-fiber and multifunctional high-power ultrafast laser.

Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains

We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al., Phys. Rev. E 77, 036219 (2008) and Phys. Rev. E 95, 012203 (2017) for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.

Applications of Few-Layer Nb2C MXene: Narrow-Band Photodetectors and Femtosecond Mode-Locked Fiber Lasers

Although the physicochemical properties of niobium carbide (Nb₂C) have been widely investigated, their exploration in the field of photoelectronics is still at the infancy stage with many potential applications that remain to be exploited. Hence, it is demonstrated here that few-layer Nb₂C MXene can serve as an excellent building block for both photoelectrochemical-type photodetectors (PDs) and mode-lockers. We show that the photoresponse performance can be readily adjusted by external conditions and that Nb₂C NSs exhibit a great potential for narrow-band PDs. The demonstrated mechanism was further confirmed by work functions predicted by density functional theory calculations. In addition, as an optical switch for passively mode-locked fiber lasers, ultrastable pulses can be demonstrated in the telecommunication and mid-infrared regions for Nb₂C MXene, and as high as the 69th harmonic order with 411 MHz at the center wavelength of 1882 nm can be achieved. These intriguing results indicate that few-layer Nb₂C nanosheets can be used as building blocks for various photoelectronic devices, further broadening the application prospects of two-dimensional MXenes.

Recent advances in real-time spectrum measurement of soliton dynamics by dispersive Fourier transformation

Mode-locking lasers have not only produced huge economic benefits in industrial fields and scientific research, but also provided an excellent platform to study diverse soliton phenomena. However, the real-time characterization of the ultrafast soliton dynamics remains challenging for traditional electronic instruments due to their relatively low response bandwidth and slow scan rate. Consequently, it is urgent for researchers to directly observe these ultrafast evolution processes, rather than just indirectly understand them from numerical simulations or averaged measurement data. Fortunately, dispersive Fourier transformation (DFT) provides a powerful real-time measurement technique to overcome the speed limitations of traditional electronic measurement devices by mapping the frequency spectrum onto the temporal waveform. In this review, the operation principle of DFT is discussed and the recent progress in characterizing the ultrafast transient soliton dynamics of mode-locking lasers is summarized, including soliton explosions, soliton molecules, noise-like pulses, rogue waves, and mode-locking buildup processes.

Bound states of light bullets in passively mode-locked semiconductor lasers

In this paper, we analyze the dynamics and formation mechanisms of bound states (BSs) of light bullets in the output of a laser coupled to a distant saturable absorber. First, we approximate the full three-dimensional set of Haus master equations by a reduced equation governing the dynamics of the transverse profile. This effective theory allows us to perform a detailed multiparameter bifurcation study and to identify the different mechanisms of instability of BSs. In addition, our analysis reveals a non-intuitive dependence of the stability region as a function of the linewidth enhancement factors and the field diffusion. Our results are confirmed by direct numerical simulations of the full system.

Heteronuclear soliton molecules in optical microresonators

Optical soliton molecules are bound states of solitons that arise from the balance between attractive and repulsive effects. Having been observed in systems ranging from optical fibres to mode-locked lasers, they provide insights into the fundamental interactions between solitons and the underlying dynamics of the nonlinear systems. Here, we enter the multistability regime of a Kerr microresonator to generate superpositions of distinct soliton states that are pumped at the same optical resonance, and report the discovery of heteronuclear dissipative Kerr soliton molecules. Ultrafast electrooptical sampling reveals the tightly short-range bound nature of such soliton molecules, despite comprising cavity solitons of dissimilar amplitudes, durations and carrier frequencies. Besides the significance they hold in resolving soliton dynamics in complex nonlinear systems, such heteronuclear soliton molecules yield coherent frequency combs whose unusual mode structure may find applications in metrology and spectroscopy.

Solitonic Fixed Point Attractors in the Complex Ginzburg–Landau Equation for Associative Memories

It was recently shown that the nonlinear Schrodinger equation with a simplified dissipative perturbation features a zero-velocity solitonic solution of non-zero amplitude which can be used in analogy to attractors of Hopfield’s associative memory. In this work, we consider a more complex dissipative perturbation adding the effect of two-photon absorption and the quintic gain/loss effects that yields the complex Ginzburg–Landau equation (CGLE). We construct a perturbation theory for the CGLE with a small dissipative perturbation, define the behavior of the solitonic solutions with parameters of the system and compare the solution with numerical simulations of the CGLE. We show, in a similar way to the nonlinear Schrodinger equation with a simplified dissipation term, a zero-velocity solitonic solution of non-zero amplitude appears as an attractor for the CGLE. In this case, the amplitude and velocity of the solitonic fixed point attractor does not depend on the quintic gain/loss effects. Furthermore, the effect of two-photon absorption leads to an increase in the strength of the solitonic fixed point attractor.

Formation of optical supramolecular structures in a fibre laser by tailoring long-range soliton interactions

Self-assembly of fundamental elements through weak, long-range interactions plays a central role in both supramolecular DNA assembly and bottom-up synthesis of nanostructures. Optical solitons, analogous in many ways to particles, arise from the balance between nonlinearity and dispersion and have been studied in numerous optical systems. Although both short- and long-range interactions between optical solitons have attracted extensive interest for decades, stable soliton supramolecules, with multiple aspects of complexity and flexibility, have thus far escaped experimental observation due to the absence of techniques for enhancing and controlling the long-range inter-soliton forces. Here we report that long-range soliton interactions originating from optoacoustic effects and dispersive-wave radiations can be precisely tailored in a fibre laser cavity, enabling self-assembly of large numbers of optical solitons into highly-ordered supramolecular structures. We demonstrate several features of such optical structures, highlighting their potential applications in optical information storage and ultrafast laser-field manipulation.

Breathing dissipative solitons in mode-locked fiber lasers

Dissipative solitons are self-localized coherent structures arising from the balance between energy supply and dissipation. Besides stationary dissipative solitons, there are dynamical ones exhibiting oscillatory behavior, known as breathing dissipative solitons. Substantial interest in breathing dissipative solitons is driven by both their fundamental importance in nonlinear science and their practical applications, such as in spectroscopy. Yet, the observation of breathers has been mainly restricted to microresonator platforms. Here, we generate breathers in a mode-locked fiber laser. They exist in the laser cavity under the pump threshold of stationary mode locking. Using fast detection, we are able to observe the temporal and spectral evolutions of the breathers in real time. Breathing soliton molecules are also observed. Breathers introduce a new regime of mode locking into ultrafast lasers. Our findings may contribute to the design of advanced laser sources and open up new possibilities of generating breathers in various dissipative systems.

Variational approach to study soliton dynamics in a passive fiber loop resonator with coherently driven phase-modulated external field

We report a detailed semianalytical treatment to investigate the dynamics of a single cavity soliton (CS) and two copropagating CSs separately in a Kerr mediated passive optical fiber resonator which is driven by a phase-modulated pump. The perturbation is dealt with by introducing Rayleigh's dissipation function in the framework of a variational principle that results in a set of coupled ordinary differential equations describing the evolution of individual soliton parameters. We further derive closed-form expressions for quick estimation of the temporal trajectory, drift velocity, and the phase shift accumulated by the CS due to the externally modulated pump. We also extend the variational approach to solve a two-soliton interaction problem in the absence as well as in the presence of the externally modulated field. In the absence of a phase-modulated field, the two copropagating solitons can attract, repulse, or propagate independently depending on their initial delay. The final state of interaction can be predicted through a second-order differential equation which is derived by the variational method. While in the presence of the phase-modulated field, the two-soliton interaction can result in annihilation, merging, breathing, or a two-soliton state depending on the detuning frequency and the pump power. Variational treatment analytically predicts these states and portrays the related dynamics that agrees with the full numerical simulation carried out by solving the normalized Lugiato-Lefever equation. The results obtained through this variational approach will enrich the understanding of complex pulse dynamics under a phase-modulated driving field in passive dissipative systems.

Real-time characterization of spectral instabilities in a mode-locked fibre laser exhibiting soliton-similariton dynamics

The study of dissipative solitons in mode-locked lasers reveals a rich landscape of interaction dynamics resulting from the interplay of nonlinearity, dispersion and dissipation. Here, we characterize a range of instabilities in a dissipative soliton fibre laser in a regime where both conventional soliton and similariton propagation play significant roles in the intracavity pulse shaping. Specifically, we use the Dispersive Fourier Transform technique to perform real-time spectral measurements of buildup dynamics from noise to the generation of stable single pulses, phase evolution dynamics of bound state “similariton molecules”, and several examples of intermittent instability and explosion dynamics. These results show that the instabilities previously seen in other classes of passively mode-locked fibre lasers are also observed in the presence of strong nonlinear attraction of similariton evolution in an optical fibre amplifier.

Real-time access to the coexistence of soliton singlets and molecules in an all-fiber laser

Recent progress in time-stretch spectroscopy accelerates the exploration of ultrafast dynamics in mode-locked lasers through mapping the spectral information into time domain. Here, we report on real-time access to the coexistence of soliton singlets and molecules in an all-fiber laser mode-locked by a 45° tilted fiber grating. By virtue of the dispersive Fourier transform process, spectral information of the pulse trains under multi-pulse states can be resolved. It is identified that soliton singlets and soliton molecules coexist in one cavity roundtrip with different assembling forms. In addition, consecutive recordings of the shot-to-shot spectra further enable insight into the transient dynamics of soliton molecules. Particularly, varieties of internal motions, including the diverging/oscillating phase evolution and the temporal separation vibration, are validated loosely bound intra-soliton molecules. All of these findings unveil the scenarios of multi-soliton phenomena in fiber lasers, as well as highlight the significance of pulse interaction towards both scientific research and practical applications.

Self-organized structures of bound states in a 2 μm dispersion-managed mode-locked fiber laser

We report on the experimental generation of various self-organized structures of bound states in a near zero-dispersion mode-locked fiber laser. When the pump power is fixed at 492 mW, appropriately adjusting polarization controllers, the switching of the cavity feedback results in the evolution from the single pulse to the dispersion-managed soliton (i.e., stretched-pulse) pair. With the increase of pump power, bound states composed of more than two pulses can also be observed. Our results of the self-organized structures might enlarge the data-carrying capacity of current fiber-optical communication systems and benefit the investigation of nonlinear dynamics of bound states in fiber lasers at 2 μm.

Bound states of different pulses based on third-order dispersion

Bound states of double pulses with different wavelengths are observed in a Tm-doped passively mode-locked fiber laser with near-zero net cavity group-velocity dispersion and strong third-order dispersion. The double pulses in the bound states exhibit different pulse durations and peak powers. Simulations show that the two pulses experience different compressing and broadening processes during the intra-cavity evolution. This demonstration reveals the existence of a new form of bound states and enriches the nonlinear dynamics of multi-pulse mode locking.

Various soliton molecules in fiber systems

Generation and propagation of various soliton molecules (SMs) in fiber systems are reviewed. SMs can survive either in fibers or fiber lasers. A dispersion-managed (DM) fiber link is the only platform for SM demonstration, while various fiber lasers can support different SMs. The fundamental unit of SMs can be conventional solitons generated in an anomalous dispersion regime, stretched pulses in DM fiber lasers, parabolic pulses, or gain-guided solitons in a normal dispersion regime. SMs with typically close soliton separation are presented. In addition, we demonstrate a new kind of SM with nanosecond soliton separation. The narrow spectral filtering is required for the generation of SMs with such long-distance interaction.

Breather Wave Molecules

We present both a theoretical description and experimental observation of the nonlinear mutual interactions between a pair of copropagative breathers in the framework of the focusing one-dimensional nonlinear Schrödinger equation. As a general case, we show that the resulting bound state of breathers exhibits moleculelike behavior with quasiperiodic oscillatory dynamics (i.e., internal coherent interactions and pulsations), while for commensurate conditions the molecule oscillations become exactly periodic. Our theoretical model is confirmed by an experimental observation of shaped moleculelike breather light waves propagating in a nearly conservative optical fiber system. Our work sheds new light on the existence of localized wave structures and recurrence dynamics beyond the multisoliton complexes.

Optical soliton molecular complexes in a passively mode-locked fibre laser

Ultrashort optical pulses propagating in a dissipative nonlinear system can interact and bind stably, forming optical soliton molecules. Soliton molecules in ultrafast lasers are under intense research focus and present striking analogies with their matter molecules counterparts. The recent development of real-time spectral measurements allows probing the internal dynamics of an optical soliton molecule, mapping the dynamics of the pulses’ relative separations and phases that constitute the relevant internal degrees of freedom of the molecule. The soliton-pair molecule, which consists of two strongly bound optical solitons, has been the most studied multi-soliton structure. We here demonstrate that two soliton-pair molecules can bind subsequently to form a stable molecular complex and highlight the important differences between the intra-molecular and inter-molecular bonds. The dynamics of the experimentally observed soliton molecular complexes are discussed with the help of fitting models and numerical simulations, showing the universality of these multi-soliton optical patterns.

It has recently been shown that optical solitons can form stably bound states, so-called soliton molecules. Here, Wang et al. demonstrate stable soliton molecule complexes and explore the different bonds represented by the inter- and intra-molecular coupling.

Generation of pulse-width controllable dissipative solitons and bound solitons by using an all fiber saturable absorber

A passively mode-locked fiber laser with controllable pulse width is demonstrated by use of an all-fiber saturable absorber based on a hybrid no-core fiber (NCF)-graded index multimode fiber (GIMF) structure incorporated into an Er-doped fiber ring cavity. Such a hybrid NCF-GIMF structure has a tunable intracavity filtering effect. As a result, the mode-locking operation is achieved with controllable pulse width and spectral bandwidth in the normal dispersion regime by only stretching the fiber device. Soliton pulses with the pulse width of 7.7-23 ps are generated, and bound solitons with variable widths are also experimentally demonstrated. The results obtained reveal the versatility and flexibility of the NCF-GIMF structured device in controlling the pulse dynamics for different experimental requirements.

Mutually ignited soliton explosions in a fiber laser

Soliton explosions are among the most intriguing nonlinear dynamics in dissipative systems, manifesting themselves as a self-recovered localized structure when suffering explosive instabilities. Herein, we report on the investigation of soliton explosions in an ultrafast fiber laser operating in the multi-soliton regime. It is demonstrated that explosion of one soliton could be induced by another one through the soliton interactions mediated by the transient gain response of an erbium-doped fiber. We denote this phenomenon as "mutually ignited soliton explosions" when referring to the multi-soliton regime. The results provide the first investigation of soliton explosions in the multi-soliton regime and, therefore, will enhance a more comprehensive understanding of the soliton exploding phenomenon.

Time-stretch probing of ultra-fast soliton dynamics related to Q-switched instabilities in mode-locked fiber laser

Ultra-fast soliton dynamics is one of the most attractive phenomena in the mode-locked fiber laser. However, the formation and breakup of solitons are difficult to observe, due to the transient nature of the process. Using the time-stretch technique, we are able to trace the real-time evolution of the soliton bound state formation and mode-locking build-up. Q-switched instabilities exist in both booting processes. Moreover, we find that the evolving patterns of soliton bound states are highly dependent on their initial conditions. Here, two types of soliton pairs are observed in the cavity and their typical forming dynamics are recorded and analyzed. Our findings uncover a diverse set of soliton dynamics in a mode-locked fiber laser and thus promote our understandings about complex dynamics in nonlinear optical systems. These results also provide a valuable reference for further theoretical studies.

Stationary and oscillatory bound states of dissipative solitons created by third-order dispersion

We consider the model of fiber-laser cavities near the zero-dispersion point, based on the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity and third-order dispersion (TOD) term. It is known that this model supports stable dissipative solitons. We demonstrate that the same model gives rise to several specific families of robust bound states of solitons. There are both stationary and dynamical bound states, with constant or oscillating separation between the bound solitons. Stationary states are multistable, corresponding to different values of the separation. Following the increase of the TOD coefficient, the stationary bound state with the smallest separation gives rise to the oscillatory one through the Hopf bifurcation. Further growth of TOD leads to a bifurcation transforming the oscillatory bound state into a chaotically oscillating one. Families of multistable three- and four-soliton complexes are found too, the ones with the smallest separation between the solitons again ending by the transition to oscillatory states through the Hopf bifurcation.

Observation of Wavelength Tuning and Bound States in Fiber Lasers

We report an experimental observation of wavelength tuning and bound states in fiber lasers. A Mach-Zehnder interferometer (MZI) is adopted as an intra-cavity tunable filter to realize large-scale wavelength tuning and bandwidth controlling. By finely manipulating the MZI and intra-cavity polarization state, continuous wavelength-tunable operation from 1550.7 nm to 1580.8 nm is achieved. Meanwhile, the spectral bandwidth varying from 1.85 nm to 3.41 nm is also controlled by broadening the free spectrum range (FSR) of the MZI. Additionally, with modest polarization adjustment, both tightly and loosely bound states are experimentally observed, which can be validated by the numerical simulations. The results indicate that the proposed fiber laser is attractive for telecommunication systems, on account that the tuning feature can be applied to wavelength-division multiplexer (WDM) and the various soliton bound states could contribute to the high-level modulation format.