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.

Solitons can form bound states that are frequently referred to as soliton molecules as they exhibit molecule‐like dynamics. The build‐up phase of the optical soliton molecule remains elusive. Here, by means of a time‐stretch technique that enables real‐time access to the spectral and temporal dynamics, rich nonlinear processes involved in the build‐up of soliton molecules are revealed in an ultrafast fibre laser. Specifically, the formation of closely‐ and well‐separated bound solitons are resolved. In both cases, the build‐up phases consist of three nonlinear stages including mode locking, soliton splitting, and soliton interactions. For closely‐separated bound solitons, soliton interactions display wide diversities in repeated measurements, including soliton attraction, repelling, collision, vibration, and annihilation. For well‐separated bound solitons, repulsive interactions dominate the soliton interactions. Numerical simulations corroborate these experimental observations. Furthermore, a conceptually different soliton molecule, the intermittent‐vibration soliton molecule, is discovered and characterized. It is the intermediate state between the vibrational and stationary soliton molecules. The author's findings could assist in the understanding of the build‐up phase of localized structures in different dissipative systems.

Formation of coherent structures and patterns from unstable uniform state or noise is a fundamental physical phenomenon that occurs in various areas of science ranging from biology to astrophysics. Understanding of the underlying mechanisms of such processes can both improve our general interdisciplinary knowledge about complex nonlinear systems and lead to new practical engineering techniques. Modern optics with its high precision measurements offers excellent test-beds for studying complex nonlinear dynamics, though capturing transient rapid formation of optical solitons is technically challenging. Here we unveil the build-up of dissipative soliton in mode-locked fibre lasers using dispersive Fourier transform to measure spectral dynamics and employing autocorrelation analysis to investigate temporal evolution. Numerical simulations corroborate experimental observations, and indicate an underlying universality in the pulse formation. Statistical analysis identifies correlations and dependencies during the build-up phase. Our study may open up possibilities for real-time observation of various nonlinear structures in photonic systems.

Modern high-power lasers exhibit a rich diversity of nonlinear dynamics, often featuring nontrivial co-existence of linear dispersive waves and coherent structures. While the classical Fourier method adequately describes extended dispersive waves, the analysis of time-localised and/or non-stationary signals call for more nuanced approaches. Yet, mathematical methods that can be used for simultaneous characterisation of localized and extended fields are not yet well developed. Here, we demonstrate how the Nonlinear Fourier transform (NFT) based on the Zakharov-Shabat spectral problem can be applied as a signal processing tool for representation and analysis of coherent structures embedded into dispersive radiation. We use full-field, real-time experimental measurements of mode-locked pulses to compute the nonlinear pulse spectra. For the classification of lasing regimes, we present the concept of eigenvalue probability distributions. We present two field normalisation approaches, and show the NFT can yield an effective model of the laser radiation under appropriate signal normalisation conditions.

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Breathing solitons are nonlinear waves in which the energy concentrates in a localized and oscillatory fashion. Similarly to stationary solitons, breathers in dissipative systems can form stable bound states displaying molecule-like dynamics, which are frequently called breather molecules. So far, the experimental observation of optical breather molecules and the real-time detection of their dynamics are limited to diatomic molecules, that is, bound states of only two breathers. In this work, the observation of different types of breather complexes in a mode-locked fiber laser: multibreather molecules, and molecular complexes originating from the binding of two breather-pair molecules or a breather pair molecule and a single breather is reported. The intermolecular temporal separation of the molecular complexes attains several hundreds of picoseconds, which is more than an order of magnitude larger than that of their stationary soliton counterparts and is a signature of long-range interactions. Numerical simulations of the laser model support the experimental findings. Moreover, nonequilibrium dynamics of breathing solitons are also observed, including breather collisions and annihilation. This work opens the possibility of studying the dynamics of many-body systems in which breathers are the elementary constituents.

The chestnut-like structure mesporous silicon sphere@C@void@ nitrogen-doped carbon (MSN@C@void@N-C) composite is designed and prepared successfully by introducing an internal carbon layer as a shell layer on the surface of a mesporous silicon core and then using nickel oxide as template to obtain a cavity between a carbon-coated mesporous silicon core and an external nitrogen-doped carbon layer. The influences of the double carbon layer and cavity on the morphology and electrochemical properties of the composite are systematically investigated by scanning electron microscope (SEM), transmission electron microscope (TEM), X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS), and galvanostatic charge–discharge tests. The results show that the coordination of the double carbon layer and the middle cavity can not only protect the silicon core from electrolyte corrosion but also improve the electron transmission rate of silicon-based materials as well as provide the space to accommodate the volume expansion of silicon without destruction of the electrode structure. It has been found that the MSN@C@void@N-C composite exhibits excellent electrochemical performance. The first discharge specific capacity is 2499 mAh g–1 and still maintains a discharge specific capacity of 1372 mAh g–1 with a capacity retention rate of 54.9% after 150 cycles. Therefore, the reasonable designs of the structure and morphology for Si/C composites are of great significance for improving the electrochemical performance of silicon-based materials, and this work provides a helpful exploration for development of the next-generation high-energy density lithium-ion batteries.

Soliton explosion refers to a striking nonlinear dynamics in dissipative systems. In this state, a dissipative soliton collapses but returns back to its original state afterwards. Yet, the origin of such exotic soliton dynamics remains elusive. Here it is revealed that soliton collision can induce soliton explosions in a mode-locked fibre laser, benefiting from synchronous measurements of the spatio-temporal intensity evolution and the real-time spectra evolution using dispersive Fourier transform. Up to seven nonlinear regimes are observed successively in the laser by increasing the pump power only, including single-pulse mode locking, standard soliton explosions, noise-like mode locking, stable double pulsing, soliton collision induced explosions, soliton molecules, and double-pulse noise-like mode locking. These experimental findings are conducive to understand complex soliton dynamics in many nonlinear dissipative systems.