## 3 Optical Features of \(\mathrm{CeO_{2}}\) Nanoporous Coatings
### Reflection Features on Silicon Substrates  
Light reflection is due to a refractive index mismatch at an interface between two different media. The reflectance \(R=I_{R}/I_{0}\) is the ratio between the reflected intensity \(I_{R}\) over the incident intensity of light \(I_{0}\) and can be calculated using the Fresnel equation at normal incidence as follows: \(R=[(n_{1}-n_{3})(n_{1}+n_{3})]^{2}\) where the \(n_{i}\)s stand for the refractive indices of the two media, and which also holds when the refractive indices are complex values. As an example, silicon in air (\(n_{1}=1\), \(n_{3}>3\)) displays a reflectance of around \(R>40\%\) in the visible to near-infrared range, see the dashed blue line in Figure 5.  
An anti-reflective coating (ARC) with an adequate refractive index and thickness can lower this light reflection at an interface between two different media. The simplest ARC with a refractive index \(n_{2}\) leading to \(R=0\) at normal incidence and at a single wavelength of incident radiation \(\lambda\) is obtained at a thickness \(h=\lambda/(4n_{2})\) when \(n_{2}=(n_{1}n_{3})^{1/2}\). More precisely, such a result is obtained through the  
Figure 4: **(Left)**: Examples of ellipsometric angles \(\psi\) and \(\Delta\) collected for eight angles over a given range of energies (symbols) and best parametrization (solid lines). **(Right)**: Complex refractive index (\(n=n+ik\)) of a CeO\({}_{2}\) coating obtained out of these ellipsometric measurements.  
calculation of the multi-layer reflection coefficient \(R=|r|^{2}\) based on \(r\), the amplitude Fresnel coefficient of a three-layer stack where layer 1 stands for air (refractive index \(n_{1}=1\), semi-infinite), layer 2 stands for the coating, and layer 3 is the semi-infinite substrate [30; 31]:  
\[r=\frac{r_{12}+r_{23}\exp{(2i\beta)}}{1+r_{12}r_{23}\exp{(2i\beta)}}, \tag{1}\]  
with \(\beta=2\pi n_{2}h/\lambda\) the phase shift in the coating of thickness \(h\), and \(r_{12}=(n_{1}-n_{2})/(n_{1}+n_{2})\) and \(r_{23}=(n_{2}-n_{3})/(n_{2}+n_{3})\) the amplitude Fresnel coefficients at interfaces between media 1 and 2, and 2 and 3 respectively. Notice that the \(n_{i}\)s may admit complex values. Here, medium 1 is air (\(n_{1}\equiv 1\)), medium 2 is the nanoporous CeO\({}_{2}\) coating (see Figure 4 right for \(n_{2}\)), and layer 3 is the silicon substrate (\(n_{3}\) found in Reference [32]).  
We performed the characterization of intensity reflectance \(R\) under normal incidence of light in the range \(\lambda=250-1300\) nm with bench-top, fairly basic equipment; the main limitation of our set-up comes from the illumination and collection of optical fibres which do not work exactly at normal incidence but accommodate some angular opening, which is somewhat detrimental to the fine measurement of \(R\). Figure 5 shows measurements of \(R\)_vs_\(\lambda\) when the coating is deposited on the polished side of a silicon wafer. Here, we show the results for coatings with four different thickness (red curves) along with the reference reflectance of the bare silicon substrate (blue dashed curve) and the calculated reflectance, black curve with Equation (1).  
In all cases, the reflectance is lowered by the presence of the coating (Figure 5), and for the three thickest coatings, it even exhibits a vanishing reflectance at several wavelengths, which expecingly increase with the thickness of the coating. Owing to the absence of a fitting parameter, the agreement between the experiments and the calculation is satisfactory, especially in the visible to near-IR range. The slight discrepancy that shows for the thickest coating could be due to the fact that we are not working at a perfect normal incidence.  
The ARC is particularly efficient for \(h=157\) nm where \(R<1\%\) in a significant range of wavelengths in the near-IR range \(\lambda=1000-1300\) nm, Figure 5. In Section 5, we give a systematic mapping of \(R\) highlighting the regions where such a coating performs the best in terms of thickness and wavelength.