## 3 Results and Discussion
For photon energies above a transition threshold energy \(E_{g}+E_{t}\), \(\varepsilon_{\text{i}}\) is expressed as the product of a variable band edge function \(G(E)\) and a Lorentz oscillator \(L(E)\):  
\[\varepsilon_{i}(E)=\frac{\left(E-E_{g}\right)^{2}}{\left(E-E_{g}\right)^{2}+E _{P}^{2}}L(E)~{}=~{}G(E)L(E),~{}~{}E>\left(E_{g}+E_{t}\right)\] (2a) where \[E_{P}\] is a weighting factor introduced to differentiate between Cody absorption and Lorentz absorption behavior. For energies below \[E_{g}+E_{t}\], the CL model introduces an exponential Urbach tail to account for absorption at energies below the optical bandgap:\[\varepsilon_{i}(E)=\frac{E_{t}G(E_{t})L(E_{t})}{E}\textit{exp}\left(\frac{E-E_{g}-E _{t}}{E_{u}}\right),\hskip 28.452756 ptE\leq(E_{g}+E_{t}) \tag{2b}\]  
where \(E_{u}\) is the Urbach width. Finally, the Cauchy-Urbach (CU) dispersion model [51] is used to describe the optical properties of the films in the NIR region (1000 nm to 1690 nm). In this model, the extinction coefficient, \(k\), is described using an exponential Urbach tail (pink line in Figure 4A)  
\[k(\lambda)=k_{0}\textit{exp}\left[\frac{E-E_{g}}{E_{u}}\right], \tag{3}\]  
where \(k_{0}\) is the absorption coefficient near the band edge. Additionally, \(n\) is described by:  
Figure 2:0  
where \(A\) and \(B\) are fitting parameters, and \(\lambda\) is the wavelength. By leveraging three complementary models, we have accurately determined the range of critical optical properties of our deposited SiC thin films, specifically the absorption coefficient (\(\alpha\)) in the telecom wavelength region (e.g., at 1550 nm). Mean square error (MSE) of the fitted models, along with the corresponding fitting parameters for each model, are provided in Tables S1 and S2. Figure 4B presents the \(n\) and \(k\) spectra of the AD and A3-annealed 650 \({}^{\circ}\)C and 700 \({}^{\circ}\)C depositions, respectively, based on the above-described fitting models. The \(n\) and \(k\) values at 500 nm and 1550 nm across all samples, processing conditions, and fitting models are summarized in Table 1, Tables S1 and S2. Across the visible region, both the TL and CL models yield consistent \(n\) values. For example, for 650 \({}^{\circ}\)C AD samples, the TL and CL models produced \(n\) values of 2.85 at 500 nm and 2.57 at 1550 nm (Figure 4(Bi)), with a measurement uncertainty of \(\pm\)0.01. Following the A3 annealing, the refractive index increased to 2.97 and 2.73 at 500 nm and 1550 nm, respectively. In contrast, the 700 \({}^{\circ}\)C AD films exhibited higher \(n\) (\(\sim\)2.94 at 500 nm and \(\sim\)2.67 at 1550 nm), which remained essentially unchanged after A3 annealing (Figure 4(Bii)). The insets of Figure 4(Bi,4 Bii) demonstrate that refractive index values obtained from the Cauchy model closely match those derived from the TL and CL models in the near-infrared (NIR) region for the corresponding samples. The refractive index, \(n\), at optical frequencies, can be described by the Lorenz-Lorenz (L-L) equation [52]:  
Figure 2:1  
where \(N\) is the concentration of dipoles (e.g., the number of atoms or molecules per m\({}^{3}\)), \(\alpha_{e}(\omega)\) is the electronic polarizability, and \(\varepsilon_{0}\) is the permittivity of vacuum. Above 10 THz, only the electronic polarization, with resonant frequencies usually in the order of \(\sim\)10\({}^{3}\) THz, survives and thus contributes to the dielectric constant. Equation (5) indicates that the concentration \(N\), representing the number of atoms per unit volume (film density), can significantly affect the index of refraction of the SiC thin films. This approach is practically applied in porous low-dielectric-constant (low-k) materials used in microelectronics, in which by reducing the film density, the dielectric constant (index of refraction) is decreased [53; 54].