Fig. 1. Schematic diagram of dissipative soliton formation in fiber mode-locked lasers. (a) Dissipative soliton generation requires double balance between dispersion and nonlinearity as well as gain and loss; (b) the formation of conventional soliton relies on the balance between anomalous dispersion and self-phase modulation (SPM); (c) the formation of dispersion-less soliton relies on the balance between spectral filtering and self-phase modulation; (d) the dispersion in the process of formation of spatiotemporal dissipative soliton includes chromatic dispersion and intermode dispersion (steady propagation of spatiotemporal dissipative solitons in multi-mode fiber lasers needs delicate balance between gain, nonlinear loss, linear loss, nonlinearity and dispersion)

Fig. 2. Schematic diagram of second-order dispersive soliton formation in fiber mode-locked lasers. (a) Self-phase modulation (SPM) and anomalous dispersion can be balanced in the cavity, resulting in stable solitons; (b) both normal and anomalous dispersion exist, and the pulse undergoes periodic broadening and compression by the combined effect of dispersion and nonlinearity, forming near-zero dispersion solitons; (c) in an all-normal-dispersion laser cavity, both normal dispersion and nonlinearity result in pulse broadening, and pulse width broadening is minimized by the addition of a spectral filter to reduce the pulse broadening, thus ensuring the self-consistency of the pulse transmission; (d) optical spectra and pulse trains of anomalous dispersive soliton (left), near-zero dispersive soliton (middle), and all-normal dispersive soliton (right)

Fig. 3. Higher even-order dispersive solitons. (a) Schematic diagram of fourth-order dispersion soliton formation in fiber lasers. The anomalous fourth-order dispersion leads to a temporal broadening of the pulse but no change the spectral shape, while the self-phase modulation leads to a broadening of the spectrum without disturbing the pulse in time domain. The two effects together ensure that fourth-order dispersion solitons are produced ^[66]. (b) Spectrum (left) and pulse train (right) of a fourth-order dispersive soliton with an exponentially decaying periodic oscillatory tail near the center of the pulse, and the center of the pulse exhibits an approximately Gaussian shape^[67]. (c) Spectra of high-even-order dispersive solitons and temopral characterization in linear (left) and logarithmic (right) coordinates, where blue, red, green, and black curves denote the time domain and spectra of numerically computed sixth-, eighth-, tenth-, and sixteenth-order dispersive solitons, respectively^[42]

Fig. 4. Dispersion-less solitons^[74]. (a) Mechanism of the dispersion-less solitons. The loss due to spectral filtering decreases as the filtering order increases. When the spectral filtering order tends to infinity, the spectral filtering effect and self-phase modulation reach equilibrium, and the parametric gain and uniform loss reach equilibrium. (b) Temporal pulse train (left) and optical spectrum (right) of a dispersion-less soliton

Fig. 5. Spatiotemporal dissipative solitons in multimode fiber lasers. (a) Mechanism of spatiotemporal dissipative solitons. The formation of spatiotemporal dissipative solitions relies on the intracavity effects that balance intermodal dispersion, such as nonlinear and dissipative effects. (b) Refractive index distributions and light propagation trajectories in graded-index fiber (top) and step-index fiber (bottom). (c) Schematic diagrams of pulse reactions for three mechanisms as intermode dispersion increases: Pulses for different spatial modes can bind together through Kerr nonlinearity under weak intermode dispersion (top). Pulses slightly walk off under intermediate intermode dispersion. Spatiotemporal saturable aborser can reset the walk-off by saturable absorbing weak modes and redistributing energy among modes (middle). And under large intermodal dispersion, spatial coupling can attenuate the weak mode pulses and redistribute energy from the mother-mode to the child-modes and align the modal pulse timing (bottom)

Fig. 6. Spatiotemporal dissipative solitons with large intermode dispersion. (a)‒(c) Beam profile, pulse trance, and optical spectrum of the output pulse in the multi-transverse-mode spatiotemporal mode-locking state; (d)‒(g) beam profiles (far-field and near-field), pulse trains, and optical spectra of near-single-transverse-mode (LP11 mode) locking; (h) simulation results of the evolution of modal walk-off in the cavity

Fig. 7. All step-index fiber spatiotemporal mode-locked laser. (a) Pulse trains and radio-frequency spectra (i), beam profiles (ii), optical spectra and spectral filtering results (iii) of spatiotemporal mode-locking. (b) Evidence of picosecond of large intermode pulse walk off. (i) Optical spectra of the output pulses with different fiber lengths, all showing some modulation. (ii) As a result of the modulation, additional peaks appear in the Fourier transform of the spectra, and the positions of the peaks generally follow the net intermodal dispersion, indicated by the solid lines. (iii) For the 3.9 m long fiber laser, mode-locking without strong spectral modulation can be observed by adjusting the spatial coupling. (iv) Output beam profile for the state in panel (iii) and white curves are horizontal and vertical slices, which are Gaussian-like. (v) Second-order autocorrelation trace of the input pulse and of the pulse sampled at the four positions shown in panel (iv). Since the mode lock contains negligible higher order modes and intermodal walks, the trace is smooth. (c) Optical spectrum, beam profile, first-order autocorrelation curve, and second-order autocorrelation curve of mode-locking output pulses obtained from simulation. (d) Intracavity change of the mode-resolved pulse center position, and the right panel is an illustration of the mother-child coupling picture