  Three-dimensional (3D) dissipative soliton existed in spatiotemporal
mode-locked (STML) multimode fiber laser has been demonstrated to be a
promising formalism for generating high-energy femtosecond pulses, which
unfortunately exhibit diverse spatiotemporal dynamics that have not been fully
understood. Completely modeling the STML multimode fiber lasers can shed new
light on the underlying physics of the spatiotemporal dynamics and thus better
manipulate the generation of high-quality energic femtosecond pulses, which
however is still largely unmet. To this end, here we theoretically investigate
a dynamic multimode gain model of the STML multimode fiber laser by exploring
the multimode rate equation (MMRE) in the framework of generalized multimode
nonlinear Schr\"odinger equation. Using this dynamic multimode gain model, the
attractor dissection theory is revisited to understand the dominant effects
that determine the modal composition of 3D dissipative soliton. Specifically,
by varying the numerical aperture of the multimode gain fiber (MMGF), different
gain dynamics that correspond to distinct types of gain attractors are
observed. As a result, two distinguishing STML operation regimes, respectively
governed by the multimode gain effect and spatiotemporal saturable absorption,
are identified. In the latter regime, especially, 3D dissipative solitons
present bistability that there exist bifurcated solutions with two different
linearly polarized (LP) mode compositions. To verify the theoretical findings,
the experimental implementation shows that the state of STML can be switched
between different LP modes, and confirms the presence of bistability.
Particularly, the 3D-soliton shaping mechanism that is governed by the
multimode gain effect is testified for the first time, to the best of our
knowledge.
