# High-Quality, InN-Based, Saturable Absorbers for Ultrafast Laser Development
## 2 Material Characterization
(**c**) Evolution of the nonlinear transmittance versus the incident pulse fluence at 1560 nm. Solid lines are fitting curves to Equation (2). Figure 2b shows the transmission increase concerning the linear transmission for the maximum light intensity (40 kW), and as a function of the sample position (relative to the focus \(z=0\)), fitted by [38; 39]. \[T=1-\frac{1}{2\sqrt{2}}\frac{\alpha_{2}I_{0}L_{eff}}{1+(z/z_{R})^{2}}, \tag{1}\]  
where \(T\) is the transmission, \(I_{0}\) is the intensity of the incident pulse; \(L_{\text{eff}}\) is the effective length of the InN layer; \(z\) is the position of the sample, with respect to the focal point; and \(z_{R}\) the Rayleigh distance. The extracted nonlinear absorption coefficients are \(\alpha_{2}=-3.5\times 10^{3}\), \(-13.5\times 10^{3}\), \(-3\times 10^{4}\), and \(-2.7\times 10^{4}\) cm/GW for samples S0, S0\({}^{\prime}\), S1, and S2, respectively. A study of the nonlinear transmission at the focal point and as a function of the variable impinging fluence is shown in Figure 2c, together with a fit to the following equation [14]:  
\[T=T_{\text{ns}}\frac{\text{ln}\Big{(}1+T_{\text{lin}}/T_{\text{ns}}\Big{(}e^{F /F_{\text{sat}}-1}\Big{)}\Big{)}}{F/F_{\text{sat}}}, \tag{2}\]  
where \(T_{\text{lin}}\) and \(T_{\text{ns}}\) correspond to the saturable and non-saturable transmittance coefficients respectively; \(F\) is the fluence, estimated from the spatial characteristics of the incident beam (beam radius and Rayleigh distance); and \(F_{\text{sat}}\) represents the saturated fluence. The parameters extracted from the fits are listed in Table 1. The modulation depth (MD), calculated as \(T_{\text{ns}}-T_{\text{lin}}\), was 23.2% for S1 and 17.3% for S2, while S0 and S0\({}^{\prime}\) have modulation depths of 16.7% and 30.6%, respectively. A higher saturable absorption response was obtained for samples S1 and S2 compared to reference samples S0 and S0\({}^{\prime}\), as shown in Figure 2b. The maximum nonlinear transmission change \(\Delta T\), calculated as the ratio between the non-saturable and linear transmission coefficients, was obtained for S1 (241%) and S2 (715%), which means at least a two-fold improvement for S0 and S0\({}^{\prime}\). Note that the reduced transmission change in S1 with respect to S2 is due to the smaller InN layer thickness, and thus to a higher \(T_{\text{lin}}\) coefficient (16.4% for S1 and 2.8% for S2). As depicted in Figure 2c, the samples have a saturable absorption behavior at high input fluences. This difference can be explained by the reduction of the bandgap attained in samples S1 and S2 compared to previous results. In order to achieve the optical bleaching of samples S1 and S2 (maximum transmission), a higher excitation fluence is required. This implies that the damage threshold of samples S1 and S2 may be higher than for reference samples S0 and S0\({}^{\prime}\), and hence there is a wider operation energy region. As it is shown in Figure 2c, these materials tolerate high-incident fluence. In our experimental setup, the damage threshold value could not be measured, due to the peak power limitation of the laser cavity. Our group has recently reported the highest value of incident energy fluence achievable for this type of passive mode-locked lasers (1 TW/cm\({}^{2}\)) without performing any apparent damage to the bulk InN saturable absorber [40], denoting the viability of these materials for high-power, ultrafast applications. This suggests promising results as saturable absorbers in ultrafast lasers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & \(F_{\text{sat}}\) **(\(u\))/cm\({}^{2}\)** & \(T_{\text{lin}}\) **(\%)** & \(T_{\text{ns}}\) **(\%)** & \(\Delta T\) **(\%)\({}^{1}\)** \\ \hline S0 & 357.2 & 27.8 & 44.5 & 160 \\ S0\({}^{\prime}\) & 383.8 & 12.7 & 43.3 & 340 \\ S1 & 756.6 & 16.4 & 39.6 & 241 \\ S2 & 831.5 & 2.81 & 20.1 & 715 \\ \hline \hline \end{tabular}  
\({}^{1}\)\(\Delta T\) has been calculated as \(100\times T_{\text{ns}}/T_{\text{lin}}\).  
\end{table}