## 3 Results
### Modeling
#### 3.1.1 Optical Modeling
However, n\({}_{\text{eff}}\) for TM\({}_{01}\) becomes constant for WG widths above 10 \(\upmu\)m. On the other hand, mode losses were calculated, as we show  
Figure 3: n\({}_{\text{eff}}\) vs. SiO2 thickness at 1550 nm. Four SiN layer thicknesses: (**a**) 75 nm, (**b**) 100 nm, (**c**) 125 nm, and (**d**) 150 nm. in Figure 4b. As we can observe, for a reduced lateral dimension (below 3 \(\upmu\)m) of the WG cross-section, light confinement is not possible along the WG (Figure 4a), and, therefore, the mode losses are significantly incremented as a result of the scattering losses increase (Figure 4b). In this context, we have simulated the cut-off wavelength for different interlayer numbers, as shown in Figure 5a. Furthermore, in Figure 5b, the mode loss and the effective index at cut-off wavelengths are plotted. A dramatic increase in mode loss is observed with zero SiO\({}_{2}\) interlayers. When the number of interlayers is incremented, a similar mode loss (close to \(-5\) dB/cm) may be obtained for three or more interlayers. These results demonstrate that a WG fabricated with three SiO\({}_{2}\) layers performs well for single mode guiding radiation at wavelengths from 1.5 \(\upmu\)m to about 3 \(\upmu\)m, which represents an ideal working range for our purposes. The effective length (\(L_{\textit{eff}}\)) was simulated in order to work in the regime where linear loss still dominates over non-linear loss. In that regime, TPA and TPA-induced free-carrier absorption (FCA) were not produced. The second process is because free-carriers that are excited to the conduction band could further absorb photons and be excited to higher  
Figure 4: (**a**) n\({}_{\text{eff}}\) vs. WG width for TM00 and TM01 is plotted. Inside the plot, different simulated profiles of the fundamental TM modes for a WG width of 9 \(\upmu\)m at 1550 nm are shown. (**b**) Mode loss calculated for that range of WG widths is shown. The inset shows a zoom for the range of WG widths between 5 \(\upmu\)m and 20 \(\upmu\)m. These values were obtained for a multilayer WG with 3 SiO2 layers and 4 SiN layers. Figure 5: (**a**) n\({}_{\text{eff}}\) vs. wavelength for different numbers of SiO\({}_{2}\) interlayers. The cut-off wavelength is also plotted in the graph (yellow curve) and in (**b**) n\({}_{\text{eff}}\) at the cut-off wavelength and mode losses vs. wavelength are plotted. energy states. These carriers are distributed along the length of the device over the FCA effective length, \(L_{FCA,eff}\):  
\[L_{FCA,eff}=\frac{(1-exp(-2\alpha_{lin}L))}{2\alpha_{lin}}. \tag{6}\]  
The factor of two in the denominator is due to the dependence of free-carrier density on the square of the optical intensity. As shown in Figures 3 and 4, an increment of n\({}_{\text{eff}}\) occurs when we increment the number of SiO\({}_{2}\) interlayers. In order to understand the minimal number of SiO\({}_{2}\) interlayers to obtain a wide range of guided wavelengths, the propagation constant was calculated as a function of SiN and SiO\({}_{2}\) thickness for different numbers of layers (Figure 6). The propagation constant (also known as transmission constant) is the measure of the change in amplitude and phase per unit distance of the field vector. Calculations performed show an increment in the propagation constant given by the layer thickness increment (Figure 6a,b), whilst a reduction in the propagation constant is produced by incrementing the number of layers constituting the WG (Figure 6c). Figure 7a shows the effective length (\(L_{eff}\)) calculated for the cut-off refractive index for different numbers of interlayers. We can observe that \(L_{eff}\) differs from the linear behavior, (i.e., the FCA losses are higher). In particular, in Figure 7b, \(L_{eff}\) for three SiO\({}_{2}\) interlayers is plotted. \(L_{eff}\) of the device is a function of the physical length (L) and the linear loss (\(\alpha_{\text{lin}}\)). In that regime, the linear loss still dominates over the non-linear loss and, thus, we can calculate the optimal WG length.