## 2 Few-Layer Transition Metal Dichalcogenides
### Nonlinear Optical Properties  
#### 2.3.1 Saturable Absorption  
Effective candidate materials for SAs possess a strong intensity-dependent loss, which can be described by a two-level model:  
\[\alpha(I)=\frac{\alpha_{\rm sat}}{1+I/I_{\rm sat}}+\alpha_{\rm non-sat} \tag{1}\]  
where \(\alpha_{\rm sat}\) is the modulation depth (saturable loss), \(\alpha_{\rm non-sat}\) is the non-saturable loss and \(I_{\rm sat}\) is the saturation intensity (required intensity to reduce the absorption by 0.5\(\alpha_{\rm sat}\)). Many studies have measured the nonlinear absorption of few-layer TMD materials (e.g., using Z-scan and balanced twin-detector setups) [30]. Saturable absorption in TMDs was first revealed  
Figure 3: Fabrication of few-layer MoS\({}_{2}\) flakes by liquid-phase exfoliation (LPE), after [32]: (**a**) liquid dispersion of flakes; (**b**) AFM image; (**c**) distribution of flake heights from AFM. by [PERSON] _et al._, using few-layer MoS\({}_{2}\) LPE dispersions containing a majority of 6-7 layer flakes: under intense 800 nm excitation, the sample transmission increased by 75%, showing a stronger nonlinear response than graphene [47]. The intraband relaxation time was also shown to be \(\sim\)30 fs, confirming that few-layer MoS\({}_{2}\) can behave as an ultrafast saturable absorber [47]. These liquid dispersions were impractical for applications in fibre lasers, but inspired further nonlinear studies on fibre-compatible integration schemes. Few-layer TMD-polymer composites have received particular attention. Using MoS\({}_{2}\), such devices have been fabricated and characterized at \(\sim\)1 \(\upmu\)m [32, 44], \(\sim\)1.5 \(\upmu\)m [34, 35, 36, 37, 39, 40, 41, 42, 45, 48] and \(\sim\)2 \(\upmu\)m [49] (corresponding to the gain bands for the principal active ions used in fibre lasers: ytterbium (Yb), erbium (Er) and thulium (Tm)), and a wide range of SA properties have been reported [30]. Saturable loss values have been reported from \(\sim\)1% [40] to \(\sim\)11% [37] with saturation intensities varying from \(\sim\)1 MW/cm\({}^{2}\)-130 MW/cm\({}^{2}\)[42]. A persistent concern, however, is the high non-saturable losses exhibited by TMD SAs to date (often >10% [30]). While the high gain of fibre lasers can often tolerate such loss, it is desirable to reduce this value to improve device efficiency and the operation performance of pulsed fibre lasers [1]. Strong saturable absorption has also been observed in other MoS\({}_{2}\)-based devices, such as MoS\({}_{2}\)-coated fibre tips, where modulation depths up to 35% and saturation intensities of 0.35 MW/cm\({}^{2}\) have been reported, although this was accompanied by very high non-saturable losses of \(\sim\)35% [34, 35]. A small, but growing number of studies have produced and characterized SAs using other TMD materials, including WS\({}_{2}\)[42, 50, 51, 52, 53, 54], MoSe\({}_{2}\)[42, 55, 56] and WSe\({}_{2}\)[42]; all have reported similar optical properties to MoS\({}_{2}\), including values of modulation depth and saturation intensity. Figure 4 presents the optical characterization of a polymer composite including 6-7 MoSe\({}_{2}\) flakes. The linear absorption profile (Figure 4a) reveals the characteristic excitonic features (labelled A and B [22]), which are a well-known feature of TMDs and can be used to identify such materials. The absorption is relatively flat, but finite, throughout the near-infrared (the origin of which is discussed later). Z-scan experiments were used to measure the intensity-dependent absorption (Figure 4b), which is well-fitted by Equation (1) to reveal the device parameters: 4.7% modulation depth, 3.4 MW/cm\({}^{2}\) saturation intensity and 6.5% non-saturable loss. Figure 4: Characterization of MoSe\({}_{2}\)-PVAcomposite, after [55]: (**a**) linear absorption; (**b**) Z-scan measurement, fitted with a two-level saturable absorber (SA) model (solid red line) to determine device performance parameters. Labels A and B refer to excitonic transitions, as described in Section 2.1.