Bound-state-like Q-switched mode-locking in erbium-doped fiber lasers

In this paper, we explore the vibrational behavior of bound-state envelopes on the $Q$-switched mode-locking (QML) spectrum and propose the concept of bound-state-like. We demonstrate an erbium-doped fiber laser using a saturable absorber and nonlinear polarization rotation to generate bound-state-like QML. Numerical simulations based on the coupled Ginzburg-Landau equations solved by the fourth-order Runge-Kutta method support the experimental results. The bound-state-like QML evolution patterns with gain- or polarization-state variations in both experiments and simulations. Our results deepen the understanding of QML pulses in mode-locked fiber lasers and provide a foundation for studying mode-locking nonlinear evolutionary paths and $Q$-switched behavior.