Generation of various bound solitons in a carbon nanotube-based bidirectional mode-locked all-fiber ring laser

Bound solitons, as an interesting physical phenomenon operating at multi-pulse mode, are intensively investigated in unidirectional mode-locked lasers, both theoretically and experimentally. However, they have rarely been observed in bidirectional lasers, especially the bound solitons with anti-phase and in-phase have not even reported. Here, we demonstrate a systematic experimental observation of two-soliton bound states with phase differences of 0 (in-phase), $\pm \pi /2$, and $\pi$ (anti-phase). The laser is passively mode-locked by a carbon nanotube saturable absorber, with the unsaturated loss, saturation optical intensity, and modulation depth of 40 MW/cm², 52.3%, and 6.1%, respectively. Two synchronous single-soliton pulses in clockwise and counter-clockwise directions, with a repetition frequency of 63.92 MHz, were delivered simultaneously from the fiber laser. Besides, various two-soliton bound states with different pulse separations and phase differences were obtained when the pump power was up to 68  mW. Furthermore, not only the same, but also the different bound states were observed between the counter-propagation directions. The short- and long-term stability of the bound states were also investigated in laboratory environments. The results enrich the content of nonlinear dynamics of bound solitons and have potential applications in fiber optical communications.