## 2 A gradient Index Plate in a Multispectral Camera
### GRIN Fresnel Lenses  
In order to reduce the index variation and preserve the small thickness of this component, the classical GRIN lens can be substituted with a Fresnel GRIN lens. In a classical Fresnel, the continuous surface of the lens is cut into a series of discontinuous rings [33; 34]. Similarly, a GRIN Fresnel lens works on the same principle as the classical version: the continuous radial variation of the refractive index is cut into a series of discontinuous rings. The Fresnel versions of a spherical lens and a GRIN lens are shown in Figure 2.  
A GRIN lens works similarly to a conventional lens, and the radial index variation corresponds to the radial variation of the sag surface for a conventional lens. The surface sag (\(\mathrm{z_{Fresnel}}\)) of a lens is defined by its curvature and its conicity. The surface sag of a Fresnel lens is described in Equation (3) [35], and it corresponds to the sag surface of a conventional lens that is divided into a series of rings. \[\left\{ \begin{matrix}\mathrm{z_{Fresnel}}(\mathrm{r})&=&\frac{\mathrm{cr}^{2 }}{1+\sqrt{1+k}\mathrm{c}^{2}\mathrm{r}^{2}}\;\mathrm{mod}\;\mathrm{H_{Fresnel }}\\ &&\mathrm{H_{Fresnel}}&=&\frac{\lambda_{0}}{\mathrm{n}_{0}-1}\end{matrix} \right. \tag{3}\]  
where \(\mathrm{c}\) is the curvature, \(\mathrm{k}\) is the conic constant, \(\mathrm{mod}\) is the modulo operation, \(\lambda_{0}\) is the central wavelength and \(\mathrm{H_{Fresnel}}\) is the etching depth of a first-order Fresnel lens. In the case of a GRIN Fresnel lens, the surface sag is replaced by an index profile distribution (\(\mathrm{n_{Fresnel}}\)) and so \(\mathrm{H_{Fresnel}}\) corresponds to the total index variation \(\Delta\mathrm{n_{Fresnel}}\):  
\[\left\{ \begin{matrix}\mathrm{n_{Fresnel}}(\mathrm{r})&=&\frac{\mathrm{n_{r}} \mathrm{r}^{2}}{1+\sqrt{1-(1+k)\mathrm{n_{r}}^{2}\mathrm{r}^{2}}}\;\mathrm{ mod}\;\Delta\mathrm{n_{Fresnel}}\\ &&\Delta\mathrm{n_{Fresnel}}&=&\frac{\lambda_{0}}{\mathrm{t}}\end{matrix} \right. \tag{4}\]  
where \(\mathrm{t}\) is the thickness of the GRIN plate and \(\mathrm{n_{r}}\) is the coefficient of index variation (as in Equation (1)). The Zemax optical design software does not offer predefined Fresnel GRIN surfaces in its surface library. However, the user can specify a specific surface by using the \"User Define Surface\" function. Thus, we created a GRIN Fresnel surface for which the refractive index profile is defined by Equation (4). This surface was then implemented in the Zemax software and we defined the base index of the material (\(75(\mathrm{GeS_{2}})-15(\mathrm{In_{2}S_{3}})-10(\mathrm{CsCl})\)) at \(\mathrm{n}=2.08\). The GRIN Fresnel lens was then optimized to meet the specifications of the camera with a spectral bandwidth of \([3.2;3.4]\)\(\mathrm{\SIUnitSymbolMicro m}\). We obtained a GRIN lens with a diameter of 3.8 mm that was composed of 38 rings, with the outer ring having a width of approximately 24.8 \(\mathrm{\SIUnitSymbolMicro m}\). This 100 \(\mathrm{\SIUnitSymbolMicro m}\) thick GRIN Fresnel lens had a refractive index variation coefficient of \(\mathrm{n_{r}}\sim-0.67\) and a conic constant of \(\mathrm{k}\sim-1.9\). Only one channel is modeled and represented in Figure 3 because each channel was the same, with a small adaptation of the total index variation as a function of the central  
Figure 2: From a classical lens to a Fresnel lens and from a GRIN lens to a GRIN Fresnel lens. wavelength of the filter. Figure 3 shows a good conservation of the modulation transfer function (MTF) over the whole field, indicating a good contrast restitution. Although the contrasts of the monochromatic MTF (\(\lambda=3.3\) um) were lower than the diffraction limit, the monochromatic MTF showed good conservation of the contrasts up to the Nyquist frequency (greater than 0.3). The image quality of the optical channel was thus considered good. However, the polychromatic MTF (\(\lambda\in[3.2;\,3.4]\) um) dropped the contrasts at Nyquist frequencies below 0.1 due to the huge chromaticity of the first-order Fresnel lens.