## 3 Results  
### Modeling  
#### 3.1.1 Optical Modeling  
In order to guide a wide range of wavelengths from visible to IR, a SiN multilayer WG structure was designed to guide those and host non-linear optical processes between a TEM-polarized pump and signal modes with NIR-MIR wavelengths. The basic WG structure that was studied is shown in Figure 1. The refractive indexes of SiO2, SiN, and air were found in the relevant literature [55; 56]. We changed the WG dimensions and number of layers to optimize the waveguiding behavior in a wide emission spectrum range, as well as to host the non-linear optical process involving NIR and MIR wavelengths. The cross-section dimension of the surrounding air was selected according to WG dimensions; the distance between the WG edge and the simulation boundary was set to 2.5 \(\upmu\)m. In addition, a 4-\(\upmu\)m SiO2 substrate was modeled. That thickness was chosen considering the SOI wafer used for fabrication (the thickness of the SOI wafer before processing was 1 \(\upmu\)m for the device layer, 4 \(\upmu\)m for the SiO2 layer, and 650 \(\upmu\)m for the substrate). When working with interlayers, it is convenient to define a transition boundary condition, because a discontinuity in the tangential electric field is clearly present [57]. Transition boundary conditions are characterized by:  
\[\overset{\rightarrow}{J_{S1}}=\frac{\left(Z_{S}\overset{\rightarrow}{E_{T1}} -Z_{T}\overset{\rightarrow}{E_{T2}}\right)}{Z_{S}^{2}-Z_{T}^{2}},\overset{ \rightarrow}{J_{S2}}=\frac{\left(Z_{S}\overset{\rightarrow}{E_{T2}}-Z_{T} \overset{\rightarrow}{E_{T1}}\right)}{Z_{S}^{2}-Z_{T}^{2}} \tag{1}\]  
\[Z_{S}=\frac{-j\omega\mu}{k}\frac{1}{tan(kd)},Z_{T}=\frac{-j\omega\mu}{k}\frac{ 1}{sin(kd)},\;k=\omega\sqrt{(\varepsilon+(\sigma(j\omega)))\mu} \tag{2}\]  
where \(\mu\) (relative permeability), \(\varepsilon\) (relative permittivity), and \(\sigma\) (electrical conductivity) are all taken from SiN. \(E_{T1,2}\) are the transversal components of the electric field, and \(k\) is the wavenumber. The transition boundary thickness (d) is computed as 1 A. A tetrahedral grid with a maximum-element size below \(\lambda/20\)[58] was used to mesh the computational model. Thereafter, the electric field distribution was calculated assuming [PERSON]'s equations:  
\[\overset{\rightarrow}{\
abla}\times\left(\overset{\rightarrow}{\
abla}\times \overset{\rightarrow}{E}\right)-k_{0}^{2}\varepsilon_{r}\overset{\rightarrow} {E}=0 \tag{3}\]  
where \(E\) is the electric field and \(k_{0}\) is the free-space wavenumber. Resolving this equation, we can obtain the effective refractive index given by:  
\[n=\sqrt{\varepsilon_{r}\mu_{r}} \tag{4}\]  
\[\varepsilon_{r}=\varepsilon_{r}+\widetilde{\varepsilon}_{r}=\underline{n}^{2 }=(n+i.\kappa)^{2}. \tag{5}\]  
Due to the tolerance and robustness of the fabrication method, and based on previous papers [45], a working range of values was defined for the different WG dimensions (i.e., SiN and SiO\({}_{2}\) thickness). Therefore, WG widths and layer thicknesses were calculated, respectively, within the 0.5-20 \(\upmu\)m and 50-500 nm size intervals. A summary of the simulated model behavior is reported in Figure 3, where the effective refractive index (n\({}_{\text{eff}}\)) of the guided fundamental single mode is plotted vs. the SiO\({}_{2}\) interlayer thickness. The SiO\({}_{2}\) thickness values below 50 nm were strongly affected by mesh size and computation errors. From the initial results, we observed that the waveguiding effectiveness dramatically increased for a SiO\({}_{2}\) thickness between 50 and 150 nm, when the WG is fabricated with more than three SiO\({}_{2}\) interlayers. Actually, a simulation of the model behavior for different WG widths was carried out. In Figure 4, a comparative study for the TM\({}_{00}\) and TM\({}_{01}\) modes at 1550 nm is plotted for a WG with three SiO\({}_{2}\) interlayers. In Figure 4a, a quasi-constant n\({}_{\text{eff}}\) value for TM\({}_{00}\) with WG widths above 4 \(\upmu\)m can be observed.