*3.1. Refractive Index*

Variable angle spectroscopic ellipsometry (VASE) was used to measure the refractive index of the coatings and we limited ourselves to coatings made of the thinnest grade of nanoparticles where the final material does significantly scatter the light (empirical observation). VASE provides ellipsometric raw data (here *ψ* and Δ angles) that need to be parametrized in order to extract the complex refractive index *n*¯ = *n* + *ik* [26].

The optimisation approach requires knowledge of the refractive index *n*¯ *NP* of CeO2 nanoparticles, their packing density *φc* inside the coating (1 − *φc* being the void fraction or porosity), and the thickness of the coating *h*. We then compute the ellipsometric angles of the coating with a refractive index calculated via the Bruggeman effective-medium approximation [27] of a flat nano-porous film with no roughness and deposited of a semi-infinite silicon substrate with an oxidized top surface, namely a 2 nm thick silica layer.

To guide the input parameters, we used the coating thickness measured by mechanical profilometry (*h* = 18 ± 3 nm), the packing density *φc* = 0.64 of a random close packing structure, and the refractive index *n*¯ *NP* based on literature data [28] and parametrized according to [29]. The optimisation is then performed through the Horiba software and eventually yields *φc* = 0.61, *h* = 15 nm, a model of *n*¯ *NP* (not shown), and ultimately, a very good agreemen<sup>t</sup> with ellipsometric raw data (Figure 4 left). The refractive index of CeO2 nanoparticles is comparable ye<sup>t</sup> smaller in magnitude than bulk cerium oxide, which could be attributed to a synthetic route that is likely to lead to a material different from bulk CeO2.

We thus obtain a satisfactory measurement of the refractive index of the nanoporous coating of CeO2 on a specific range of photon energy (1 − 5 eV, i.e., *λ* = 1250 − 250 nm, Figure 4): *n*¯ shows a significant absorption of light in the UV-visible regime up to *λ* ≈ 500 nm where the material becomes a loss-less dielectric with *n* ≈ 1.75, *k* ≈ 0. We assume it remains true even up to the mid-IR range, for instance *λ* = 4 μm, which we will also use to characterize the transmittance of the coating in this regime.

Also, we assume that this measurement obtained on a specific coating is intrinsic and thus holds, whatever the thickness of a homogeneous coating made out of the same grade of particles.

**Figure 4.** (**Left**): Examples of ellipsometric angles *ψ* and Δ 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 CeO2 coating obtained out of these ellipsometric measurements.

#### *3.2. Reflection Features on Silicon Substrates*

Light reflection is due to a refractive index mismatch at an interface between two different media. The reflectance *R* = *IR*/*<sup>I</sup>*0 is the ratio between the reflected intensity *IR* over the incident intensity of light *I*0 and can be calculated using the Fresnel equation at normal incidence as follows: *R* = [(*<sup>n</sup>*1 − *<sup>n</sup>*3)(*<sup>n</sup>*1 + *n*3)]<sup>2</sup> where the *ni*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 (*<sup>n</sup>*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 *λ* is obtained at a thickness *h* = *<sup>λ</sup>*/(4*n*2) when *n*2 = (*<sup>n</sup>*1*<sup>n</sup>*3) 1/2. More precisely, such a result is obtained through the

calculation of the multi-layer reflection coefficient *R* = |*r*|<sup>2</sup> 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]:

$$r = \frac{r\_{12} + r\_{23} \exp\left(2i\beta\right)}{1 + r\_{12}r\_{23} \exp\left(2i\beta\right)},\tag{1}$$

with *β* = <sup>2</sup>*πn*2*h*/*<sup>λ</sup>* the phase shift in the coating of thickness *h*, and *r*12 = (*<sup>n</sup>*1 − *<sup>n</sup>*2)/(*<sup>n</sup>*1 + *n*2) and *r*23 = (*<sup>n</sup>*2 − *<sup>n</sup>*3)/(*<sup>n</sup>*2 + *n*3) the amplitude Fresnel coefficients at interfaces between media 1 and 2, and 2 and 3 respectively. Notice that the *ni*s may admit complex values. Here, medium 1 is air (*<sup>n</sup>*1 ≡ 1), medium 2 is the nanoporous CeO2 coating (see Figure 4 right for *n*2), and layer 3 is the silicon substrate (*<sup>n</sup>*3 found in Reference [32]).

**Figure 5.** Intensity reflection coefficient *R* against wavelength for four different coating thickness (values given in red); in red the measurement, in black the calculation according to Equation (1) with no free parameter, in dashed blue the reflection coefficient of a bare silicon substrate. The value of *R* denoted by the arrow is the best extinction calculated from Equation (1) in the visible to near-infra-red (IR) range.

We performed the characterization of intensity reflectance *R* under normal incidence of light in the range *λ* = 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 λ* 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 expectingly increase with the thickness of the coating. Owing to the absence of a fitting parameter, the agreemen<sup>t</sup> 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 *λ* = 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.

#### *3.3. Enhanced Transmission on Silicon Substrates*

Along with anti-reflection behaviour comes a possible enhanced transmission provided by the ARC. Now, we work with a bare substrate which has two polished sides and which is thick enough (350 μm) to exhibit an incoherent behaviour [30], so that we can neglect the interferences. We nevertheless take into account the multiple reflections for calculating the total optical path from which the total transmittance follows:

$$T = \frac{(1 - R\_{ACS})(1 - R\_{AS})}{1 - R\_{SCA}R\_{AS}} \exp\left(-aL\right),\tag{2}$$

with *R* is the reflectance at the different interfaces where subscripts *A*, *C*, and *S* stand for air, coating, and substrate respectively, *α* = 4 *<sup>π</sup>k*3/*<sup>λ</sup>* is the extinction coefficient in the substrate with *k*3 as the imaginary part of the refractive index of the substrate of thickness *L*. *RACS* and *RSAC* are calculated following the procedure of Equation (1); in the case of a bare substrate, replace *RACS* and *RSCA* with *RAS*.

We measured the transmission *T* of a two-sided polished substrate coated with a micro-structured pattern using a spectro-imaging set-up (described in Section 4.4 and in [33,34]). Owing to the imaging capability of the device, it is possible to select the measurement place in a heterogeneous sample such as ours, see Figure 6A with a spatial resolution of about 25 μm in this specific case, but which depends on the magnification. We thus performed a reference measurement on the bare substrate, the result of which is shown in Figure 6B (blue symbols) and which agrees well with the calculated Equation (2) and tabulated values [32] (blue line). We notice a residual peak at *λ* = 4.2 μm which we attribute to CO2 absorption, and which we have difficulty in systemically removing. When the measurement was performed and averaged on a specific part of the pattern—namely the bright lines of Figure 6A—we observed a significant increase of transmittance, see the red symbols of Figure 6B, and which is parametrized with Equation (2) where the thickness *h* of the coating is left as a free parameter in the range 300–600 nm. It is clear that the experimental transmission is properly framed by the calculated *T* although the best agreemen<sup>t</sup> is found for *h* = 400 nm which does not perfectly match the actual thickness *h* = 260 nm. The slight discrepancy could be due either to the fact that the sample is actually slightly slanted and not perfectly perpendicular to the incident beam, or possibly also that the pixel size of the imaging set-up ( ≈25 μm) is not very small compared to the width of the pattern ≈100 μm), which could induce some 'blurring' in the measured transmittance. Yet, it definitely corroborates that the anti-reflection coating is accompanied with an enhanced transmission; such a coating could thus be used to enhance the silicon transmission when this material is used as an optical component for mid-IR imaging purposes for instance, and is particularly efficient at some specific wavelengths but is obviously wavelength dependent, see Section 5.

**Figure 6.** (**A**) Imaging the intensity transmission *T* at *λ* = 4 μm for a micro-structured pattern. (**B**) Spectral transmission *T* in the mid-IR range. Symbols are the local average of the wavelength dependent transmittance for the bare silicon substrate (blue) and for the pattern (red, average along the bright lines). The solid blue line is the theoretical transmission of an incoherent silicon substrate; the solid red line corresponds to a calculated *T* with a coating thickness *h* = 400 nm, whereas the pale red zone shows the parametrization of *T* with *h* ranging from 300 to 600 nm.

#### **4. Materials And Methods**

#### *4.1. Dispersions of* CeO2*Nanoparticles*

Fairly monodisperse cerium oxide (CeO2) nanoparticle dispersions were synthesized and kindly provided by Solvay with two grades: a small diameter (4 ± 1 nm) at volume fraction *φ*0 = 1.2 × <sup>10</sup>−2, pH = 1.6, and a large diameter (40 ± 10 nm) at volume fraction *φ*0 = 5.6 × <sup>10</sup>−2, pH = 4.2, see Figure 7. These acidic dispersions are formulated in water with acetic acid in order to ensure the long-term colloidal stability of the particles (month/years at 4 ◦C) with no additional additives such as surfactants. Transmitted-electron microscopy (TEM EOL JEM 2200FS FEG HR 200 kV) observations reveal the polyhedral shape of the particles, along with a significant size range which was otherwise also estimated using dynamic light scattering (values given above). The dispersions are used as such for the assembly of the nanoparticles. Importantly, we also obtained the very same results concerning coatings, using a commercial dispersion available at Sigma-Aldrich (catalog number 289744-500g).

**Figure 7.** Transmitted-electron microscopy. (TEM) of CeO2 nanoparticles deposited on the TEM grid, with diameters in the range of 4 ± 1 nm (**left**) and 40 ± 10 nm (**right**).
