Formation and evolution dynamics of soliton molecules in spatiotemporal mode-locked fiber laser

Spatiotemporal mode-locked (STML) fiber lasers have garnered significant interest in recent years for their unique ability to generate high-energy ultrashort pulses while simultaneously supporting complex transverse field structures. In particular, the formation of soliton molecules opens new avenues for ultrafast signal processing, high-bit-rate optical communications, and precision metrology. In this work, we presented a mode-resolved investigation into the birth, temporal evolution, and nonlinear dynamics of multiple soliton molecules in an all-fiber STML laser. Additionally, our mode-resolved analysis revealed that the transverse modes evolved synchronously during soliton-molecule formation while maintaining a relatively stable modal energy distribution. We further observed that both pulse spacing and relative phase gradually converged to constant values after a transient stage, indicating the formation of stable soliton molecules sustained by the balance among dispersion, nonlinearity, and gain-loss dynamics. Our study not only advances fundamental understanding of multi-soliton physics in complex laser cavities but also provides a numerical basis for engineering tailored spatiotemporal soliton molecules with controllable molecular order, spacing, and stability.