## 4 Fabrication and Characterization of a Fresnel GRIN Plate
Using the same optical material (our GeS\({}_{2}\)-based chalcogenide) and an identical variation of index \(\Delta\)n, we achieved the same optical path difference (OPD). The phase difference then depended on the wavelength as follows:  
\[\Delta\varphi(\lambda_{\text{IR}})=\frac{2\pi}{\lambda_{\text{IR}}}\cdot \text{OPD}=\Delta\varphi(\lambda_{0})\cdot\frac{\lambda_{0}}{\lambda_{\text{IR}}} \tag{6}\]  
where e.g., \(\lambda_{\text{IR}}=3.3\) um is the desired wavelength in the infrared range and \(\lambda_{0}=0.55\) um is the chosen wavelength in the visible range. We neglected the material dispersion here, assuming near conservation of the material dispersion when written by fs-laser (to be investigated in the future). Finally, we chose the diameter of this Fresnel lens to have a similar spatial resolution (about 3 um) in the visible range to the Fresnel lens designed in the mid-infrared. To characterize this Fresnel lens, we started by measuring the different focal points due to the different diffraction orders of this lens. For a first-order Fresnel lens with a gradient index, the focal length f\({}_{\text{m}}\) associated with the diffraction order m is [34]  
\[\text{f}_{\text{m}}(\lambda)=\frac{\text{r}_{1}^{2}}{4\text{m}\lambda} \tag{7}\]  
where r\({}_{1}\) the radius of the first Fresnel zone of the lens (in our case about 330 um). The measurement of the different focal lengths of the plate was done with a microscope and spectral filter with a spectral band centered on 550 nm. At this wavelength, the focal length of the first order of diffraction (the most important for a first-order Fresnel lens) was f\({}_{1}(\lambda)=7\) mm. Figure 10 shows an image of the field diaphragm of the microscope by the Fresnel lens positioned at about 7 mm from the lens. Figure 10: (**a**) The phase profile designed at a central wavelength of 550 nm. (**b**) Microscopic photograph of the graded index Fresnel lens written using the femtosecond laser direct writing technique (\(\times\)4 objective). (**c**) Image of the field diaphragm of the microscope by the Fresnel lens at the first order (\(\times\)10 objective, 550 nm filter). We wanted to measure the variation of the focal length f as a function of the different orders of diffraction of the laser written lens and also the spectral dependence of the focal shift \(\Delta\)f/f. For these measurements, we used an Olympus BX51 visible microscope to which we added different visible filters with a spectral bandwidth of 10 nm (Thorlabs FKB-VIS-10). Figure 11 shows a comparison of the focal shift against the wavelength. We note that the image focal point was brighter for the first order of diffraction as we had designed our Fresnel lens to order 1. We measured three successive orders of diffraction with respective focal lengths of f, f/2, and f/3. Based on Figure 11, the measurements of the defocus as a function of wavelength seemed to meet our expectations concerning the first order of diffraction. We note that the depth of field of the focal point was several tens of micrometers so it was difficult to accurately find the best focus. At this step, the losses associated with the diffraction efficiency of the Fresnel lens (typically between 40% and 70% at \(\lambda_{0}\)) and the losses of brightness to illuminate the object due to the addition of spectral filters did not allow us to measure the defocus of each diffraction order. Only the main focal lengths of these orders (f/2 and f/3) were measured and corresponded well to the expected value compared to order 1. A bench dedicated to the measurements of focal lengths and defocus of graded index lenses in the infrared is under preparation and will allow a more detailed analysis of the optical characteristics of these GRIN Fresnel lenses in the future. In addition, adding more layers (N) to the discretization will result in much higher diffraction efficiency in agreement with the theoretical considerations.