## 3 Results
### Modeling
#### 3.1.1 Optical Modeling
We can note that above 6 mm, \(L_{eff}\) starts to draw away  
\[\overset{\rightarrow}{J_{S1}}=\frac{\left(Z_{S}\overset{\rightarrow}{E_{T1}} -Z_{T}\overset{\rightarrow}{E_{T2}}\right)}{Z_{S}^{2}-Z_{T}^{2}},\overset{ \rightarrow}{J_{S2}}=\frac{\left(Z_{S}\overset{\rightarrow}{E_{T2}}-Z_{T} \overset{\rightarrow}{E_{T1}}\right)}{Z_{S}^{2}-Z_{T}^{2}} \tag{1}\]0 Propagation constant was plotted as function of SiO\({}_{2}\) and SiN thickness for (**a**) 3 SiO\({}_{2}\) interlayers and (**b**) 4 SiO\({}_{2}\) interlayers. (**c**) the propagation constant is given as a function of the number of interlayers. The layer thickness was fixed at 125 nm for SiN layers and at 90 nm for SiO\({}_{2}\) layers. from the linear behavior. That value for \(L_{\textit{eff}}\) is similar to that shown by [PERSON] et al. [30] (\(L_{\textit{eff}}\) = 6 mm) for Si WGs. In their study, Si WGs had a dimension of 106 nm \(\times\) 475 nm with a length equal to 14.9 mm and guided a fundamental TE mode at the 1500 nm wavelength. That result indicates that 6 mm is a good WG length for our purposes. Once we established the number of layers, we calculated the range of WG widths that offer less loss, as is shown in \[\overset{\rightarrow}{J_{S1}}=\frac{\left(Z_{S}\overset{\rightarrow}{E_{T1}} -Z_{T}\overset{\rightarrow}{E_{T2}}\right)}{Z_{S}^{2}-Z_{T}^{2}},\overset{ \rightarrow}{J_{S2}}=\frac{\left(Z_{S}\overset{\rightarrow}{E_{T2}}-Z_{T} \overset{\rightarrow}{E_{T1}}\right)}{Z_{S}^{2}-Z_{T}^{2}} \tag{1}\]1 In Figure 8b, losses are plotted as a function of wavelength and WG width. In particular, for WG widths below 6 \(\upmu\)m, a reduction in the range of guided wavelengths is obtained. As is shown in Figures 5a and 8a, for WGs fabricated with three SiO\({}_{2}\) interlayers, the cut-off wavelength is about 3.1 \(\upmu\)m. This particular can be solved for WG widths wider than 6 \(\upmu\)m (Figure 8b) to obtain a wider range of guided wavelengths (from 300 nm to 3.1 \(\upmu\)m). In order to perform an experimental test of the waveguiding behavior by using laser sources emitting in the visible and telecom wavelengths, we requested of the manufacturer five WGs with widths ranging between 7 and 10 \(\upmu\)m.  
For the 9.4 \(\upmu\)m-width WG, we report in Figure 9 the simulation (left) of fundamental TE/TM modes for 1550 nm and 2640 nm. \[\overset{\rightarrow}{J_{S1}}=\frac{\left(Z_{S}\overset{\rightarrow}{E_{T1}} -Z_{T}\overset{\rightarrow}{E_{T2}}\right)}{Z_{S}^{2}-Z_{T}^{2}},\overset{ \rightarrow}{J_{S2}}=\frac{\left(Z_{S}\overset{\rightarrow}{E_{T2}}-Z_{T} \overset{\rightarrow}{E_{T1}}\right)}{Z_{S}^{2}-Z_{T}^{2}} \tag{1}\]2 (**a**) Effective refractive index and (**b**) loss are plotted as a function of WG width and wavelength for 3 SiO\({}_{2}\) interlayers. The layer thickness was 125 nm for SiN layers and 90 nm for SiO\({}_{2}\) layers. \[\overset{\rightarrow}{J_{S1}}=\frac{\left(Z_{S}\overset{\rightarrow}{E_{T1}} -Z_{T}\overset{\rightarrow}{E_{T2}}\right)}{Z_{S}^{2}-Z_{T}^{2}},\overset{ \rightarrow}{J_{S2}}=\frac{\left(Z_{S}\overset{\rightarrow}{E_{T2}}-Z_{T} \overset{\rightarrow}{E_{T1}}\right)}{Z_{S}^{2}-Z_{T}^{2}} \tag{1}\]3 (**a**) Free-carrier absorption (FCA) effective length (\(L_{\textit{FCA,eff}}\)) vs. WG length (L) for different numbers of SiO\({}_{2}\) interlayers. The linear behavior is also plotted in the graph (red dot) and (**b**) shows \(L_{\textit{FCA,eff}}\) for 3 SiO\({}_{2}\) interlayers below 10 mm. A thorough WG characterization was performed by using the four lasers previously described. The inset of Figure 2 shows the WG behavior in the visible range, at 635 nm. All the light observed in the picture is solely due to a scattering process along the WG. Figure 9 (right) shows the NIR and MIR beam profiles guided through the WG, as recorded by the InSb camera. Such images, in agreement with the simulations (Figure 9 left), are a demonstration of single-mode waveguiding in the IR.