# Dispersion Management and Pulse Characterization of Graphene-Based Soliton Mode-Locked Fiber Lasers
## 1 Introduction  
Research on ultrafast fiber lasers has been extensively conducted in efforts to reach the optimized design and performance at the highest possible level, which can open up wider possibilities of their applications [1; 2; 3]. Among the mechanisms used to generate ultrafast fiber lasers, the saturable absorption effect in passive mode-locking has attracted significant attention due to its advantage of low cost, high stability, high robustness, and good beam quality [4; 5; 6; 7]. Contrarily, the active mode-locking technique, which utilizes optical modulator and external electrical devices, has limited flexibility in design. The saturable absorption mechanism in passive mode-locking can be further classified into either natural or artificial saturable absorbers (SAs). Artificial SAs, which are enabled by nonlinear effects occurring in optical fibers, can be driven by several approaches, such as nonlinear polarization rotation [8; 9], nonlinear amplifying loop mirror [10; 11; 12], and nonlinear optical loop-mirror [1]. They can offer automatic self-start-up of mode-locked operation even without manual triggering conditions, as required by most of their natural analogs [1]. Furthermore, their properties of a high damage threshold and easy implementation in a fiber laser cavity are also attractive factors. However, there is still a considerable challenge in achieving the shortest possible pulse duration by using this technique, and their vulnerability towards environmental changes also still remains a problem. Concurrently,passive mode-locking by natural saturable absorbers has emerged as the preferred strategy to generate ultrashort pulse, due to their excellent optical properties and high compatibility with fiber laser systems. Recently, various kinds of materials from the group of topological insulators [13; 14; 15], transition metal dichalcogenides [16; 17; 18; 19], and transition metal oxides [20; 21] have been widely explored and demonstrated as new potential saturable absorbers. MXenes, black phosphorus, metal-organic frameworks, bismuthene, and antimonene are among the 2D materials that have attracted considerable research interest in their application as saturable absorbers due to their unique optical properties [22; 23; 24; 25]. It is important to point out that the rise and advancement of these materials has been initiated following the success of graphene, the pioneer of all 2D materials as a versatile saturable absorber, which is still remarkable to date [26; 27; 28; 29]. The superiority of graphene for saturable absorber applications, which is highlighted in the recent review by [PERSON] et al. [30], also deserves to be mentioned. It is undeniable that graphene possesses a short recovery time, zero bandgap energy, ultra-broadband absorption, gapless linear dispersion of Dirac electrons, high modulation depth, and wide absorption wavelength bandwidth, which are desirable for femtosecond pulse generation [30]. In general, femtosecond pulse lasers realized by soliton mode-locking are very useful for absolute distance measurement [31], optical sensing [32], and coherent communication [33] due to their high laser stability and high pulse quality with a low chirp. The net anomalous dispersion from the soliton-based mode-locked regime is an advantage in balancing the mutual effect of dispersion and non-linear Kerr effect imposed on the pulse propagation [34]. In expanding the capability of soliton mode-locked fiber lasers, several attempts have been made to produce tunable pulse width, repetition rate, or spectral bandwidth of the laser output [35; 36], which are normally achieved by introducing the filtering effect. However, the dependence of the aforementioned pulse characteristics on cavity dispersion has not been thoroughly determined. Thus, analysis of the spectral and temporal characteristics of the output pulses based on the net cavity dispersion is still necessary.