## 2 Amorphous Silicon Waveguides
### Linear and Nonlinear Optical Characterization
\[P_{T}=e^{-\alpha_{0}L}\frac{\text{Im}\Big{(}\sqrt{2|\text{Im}(\gamma)|L_{\text {eff}}P_{0}}+\sqrt{1+2|\text{Im}(\gamma)|L_{\text{eff}}P_{0}}\Big{)}}{\sqrt{(2| \text{Im}(\gamma)|L_{\text{eff}}P_{0})(1+2|\text{Im}(\gamma)|L_{\text{eff}}P_{ 0})}}P_{0} \tag{3}\]  
using the waveguide loss \(\alpha_{0}\) (estimated to be 4.7 dB/cm) and the pulse peak power that can be extracted from the experimental data, Im\(\{\gamma\}\) was evaluated by fitting the experimental curve, yielding a value of 15.5 \(\pm\) 2 (Wm)\({}^{-1}\). In order to compare the performance of a-Si waveguides to other nonlinear material systems, it is useful to employ a figure of merit (FOM) that accounts both for the Kerr response and the nonlinear losses. This is achieved by the following FOM [42]:  
\[\text{FOM}=\frac{1}{4\pi}\frac{\text{Re}\{\gamma\}}{\text{Im}\{\gamma\}} \tag{4}\]  
which yields a value of 4.7 for the a-Si waveguides used in this study. This is one order of magnitude higher than typical values quoted for c-Si nonlinear waveguides [3]. The FCA coefficient (\(\alpha_{\text{FC}}\)) and the carrier dynamics were assessed by performing two different experiments as described in details in [28]. The pulse propagation in the presence of TPA can be described by the following equation [42]:  
\[\frac{d}{dz}I(z,t)=-[\alpha_{0}+\alpha_{\text{FC}}N_{\text{FC}}(t)]I(z,t)- \beta I^{2}(z,t) \tag{5}\]  
where \(N_{\text{FC}}\) is the number of free carriers per unit volume and \(\beta\) the TPA coefficient. The first experiment, aimed at the evaluation of free carrier recombination time, employed a pump-probe scheme as described in [28]. Results of this experiment are reported in Figure 3 (left panel). It can be noted that when the peak power of the pump pulses was relatively low (\(P_{0}\) < 10 dBm in the waveguide), the transmission power of the probe was almost constant in time, while for \(P_{0}\) > 10 dBm depletion of the probe beam was observed in the present of a pump pulse. Figure 3: (**Left**) Free carrier recovering time experimental trace (\(P_{0}\) = 10 W); and (**Right**) example of the transmission curve used to estimate \(\alpha_{\text{FC}}\) (\(L\) = 2.26 cm); black circles are the experimental data, the red trace is the fitting function and the purple line represents the theoretical curve without TPA (two-photon absorption) and FCA (Free Carrier Absorption) effects [28]. This can be attributed to the abrupt rise of \(N_{\text{FC}}\) induced by TPA and can be described through the following equation:  
\[\text{Re}\{\gamma\}=\frac{\sqrt{P_{i}(L)/P_{s}(L)}}{\text{n}P_{p}(0)L_{\text{eff }}} \tag{1}\]0  
after the initial power drop, caused by the TPA-excited free-carriers, a two-phase recovery process can be observed: a very fast process responsible for the initial recovery of about 10% of transmission, and a second (slower) process yielding a relatively long power-recovery tail. Experimental data can be fitted by the superposition of two exponential functions with time constants \(\tau_{1}<50\) ps (fast recovery) and \(\tau_{2}=1.87\) ns. In order to evaluate the additional loss induced by free carriers, a second experiment was carried out using a single CW beam. In this case, the beam intensity evolution along the waveguide can be written as [42]:  
\[\text{Re}\{\gamma\}=\frac{\sqrt{P_{i}(L)/P_{s}(L)}}{\text{n}P_{p}(0)L_{\text{eff }}} \tag{1}\]1  
\(\alpha_{0}\), \(\beta\) and \(\tau\) parameters were previously determined. The value of \(\alpha_{\text{FC}}\) can then be determined by experimentally measuring the transmission curve of a waveguide and fitting the data to Equation (7). This is shown in Figure 3 (right), which indicates a value \(\alpha_{\text{FC}}=0.8\times 10^{-17}\) cm\({}^{2}\). This is only slightly lower than that measured in c-Si waveguides (\(\alpha_{\text{FC}}=1.1\times 10^{-17}\) cm\({}^{2}\)) [42].