Vector fractional-order solitons in birefringent optical fibers

Yet another important family of temporal optical solitons, the fractional-order optical solitons (FOSs), can be generated by the interplay between fractional-order dispersion and self-phase modulation due to Kerr nonlinearity in a fiber laser cavity, which provides a promising avenue for exploring unique soliton dynamics in fractional-order nonlinear systems. We theoretically investigate the stationary properties of vector fractional-order optical solitons (VFOSs) in a fractional-order dispersion birefringent optical fiber, and then the dynamics of VFOSs in a fractional-dispersion-dominant mode-locked fiber laser. Investigating the formation dynamics of the FOSs, we find that the vector soliton tails are decaying nonexponentially, and the frequency chirps of two orthogonal polarization components complement each other. Due to the varied fractional phase profiles, the temporal pulse, spectral profile, central frequency offset, time-bandwidth product, and energy-width scaling are closely associated with the L\'evy index \ensuremath{\alpha} $(1<\ensuremath{\alpha}<2)$. The spectral profiles of the mode-locked VFOSs evolve from the fish-fin structure with no sidebands $(\ensuremath{\alpha}=1.1)$ to the hyperbolic-secant shape with noticeable sidebands $(\ensuremath{\alpha}=1.7)$, and a spectral-shifting feature is observed during the VFOS formation in the cavity, generation of asymmetric sidebands, and complex dispersion wave radiation. We discuss in detail the polarization dynamics of the VFOSs for $\ensuremath{\alpha}=1.1$ and 1.7. Our findings will offer significant insight into both the fundamental working principle and the fascinating nonlinear dynamics of the fractional-order optical soliton laser.