Fig. 2: Criterion of CWML.
a Gain dynamics in the dual-time scales. In the state of QSML, the gain is depleted and subsequently recovers between soliton crystals. The GHz solitons in the soliton crystal are bound through the GDR effect in the fast time scale. The quasi-single soliton (QSS) formed in the strongly-correlated (SC) length \({T}_{{sc}}\) is proposed to characterize the collective behavior of the soliton assembling. \({g}_{0}\), the gain depending on the slow time \(\tau\). \({g}_{1}\), the gain depending on the fast time \(T\). \({g}_{c}\), the average gain over the soliton crystal in the slow time scale. \({\tau }_{c}\), the period of soliton crystals. \({T}_{0}\) and \({T}_{1}\) are the start and end time of the soliton crystal, respectively. Ave., average. \({N}_{{sc}}\), the number of solitons in the SC length. \({T}_{R}\), the roundtrip time of the GHz solitons. b Criterion of CWML for stable mode-locking. In the left panel, the green dash and black solid curves represent the values of Y1 and Y2 as a function of the pulse energy, respectively, wherein the cross point (black dot) corresponds to the critical pulse energy for CWML, assuming \({q}_{0}=\bar{{g}_{0}}L\). The right panel shows the ratio of \({E}_{c}/{E}_{G}\) as a function of the pump power, wherein the black dash-dotted line indicates the threshold of CWML. c Experimental evolutions of a mode-locked fiber laser operating with a fundamental repetition rate of 4.6 GHz, i.e., evolving from rectangular-shape QSML (RSQSML) to CWML with increasing pump power
