## 2 A gradient Index Plate in a Multispectral Camera
### GRIN Fresnel Lenses
We noted a drop in the MTF at low frequencies due to the different defocused diffraction orders which created a parasitic background that lowered the quality of the contrast. To significantly increase the polychromatic MTF, the solution was to design a thirty-order Fresnel lens. However, manufacturing such a high-order Fresnel lens is difficult, so a first step was to work on manufacturing a first-order Fresnel lens and to test it under monochromatic illumination. The fs-laser direct writing technique discretizes the profile into several index levels, so we needed to determine a compromise between the imaging quality of the Fresnel plate and the spatial resolution of the writing technique that discretized the variation of the index of refraction. For a first-order Fresnel lens as studied herein, we can define the diffraction efficiency as a function of the index versus wavelength variation (Equation (5)) [36]. \[ \begin{cases}\eta_{m,\lambda}^{N}\,=\,\sin\mathrm{c}^{2}[\alpha(\lambda)- \mathrm{m}]\sin\mathrm{c}^{2}\left[\frac{\lambda_{0}}{\lambda N}\right]\\ \alpha(\lambda)=\mathrm{p}\frac{\lambda_{0}}{\lambda}\frac{\mathrm{n}(r\lambda )}{\mathrm{n}(r\lambda_{0})}\end{cases}  \tag{5}\]  
where \(\mathrm{p}\) is the Fresnel order, \(\mathrm{m}\) is the considered diffraction order, \(\lambda_{0}\) is the reference wavelength, \(\mathrm{N}\) is the number of discretization levels and \(\mathrm{n}(r,\lambda)\) is the refractive index at the radial position \(r\) and the wavelength \(\lambda\). Figure 3: **(a)** Zemax 2D layout of a GRIN Fresnel lens with (from left to right) the pupil, the GeS\({}_{2}\) sample with a 100 μm thick GRIN Fresnel lens in the center of the sample and the image surface. Below, two Modulation Transfer Functions (MTF) given for three different half-fields of view (\(0^{\circ}\), \(5^{\circ}\) and \(10^{\circ}\)). **(b)** The MTF is given for the central wavelength (\(\lambda=3.3\) μm). **(c)** The MTF is given for three different wavelengths of the spectral bandwidth: 3.2 μm, 3.3 μm and 3.4 μm. Figure 4 shows a good diffraction efficiency (up to 90% for the [3.2; 3.4] um spectral bandwidth) for a first-order Fresnel lens discretized into eight index levels. Sixteen levels of discretization might be better, but would require rings thinner than 1 um, which cannot be achieved with our laser direct writing technique. Indeed, with our setup, the laser waist (i.e., the writing spatial resolution) in GeS\({}_{2}\) is about 1.1 um. We modeled a GRIN Fresnel lens in our GeS\({}_{2}\)-based material, which is presented in Figure 5. For 100 um thick laser writing, the total gradient index was about \(3.3\times 10^{-2}\), which was achievable with our laser writing technique. With the numerical aperture specification of the imaging system, the last ring needed was about 24.8 um thick, so we needed a lateral spatial resolution of about 3.1 um to be able to write this last ring in such a way that we could discretize it into eight levels. Indeed, at this step, we needed to discretize this phase profile because our fs-laser direct writing software can only work at constant power, whereas ideally it would be preferable to have it continuously variable along the trajectory. Figure 4: Graphic representation of the diffraction efficiency as a function of the wavelength for the designed first-order Fresnel lens at wavelength 3.3 μm. Only the diffraction efficiency of the first order of diffraction is shown. The different colors represent the different discretization levels. The dashed line represents the central wavelength of this design. Figure 5: Refractive index profile of the GRIN Fresnel lens made in the chalcogenide glass \(75(\text{GeS}_{2})-15(\text{In}_{2}\text{S}_{3})-10(\text{CsCl})\): blazed on the top and discretized on the bottom.