## 3 Femtosecond Laser Direct Writing of Gradient Index Flat Lens
### Identification of Different Kinds of Permanent Modifications in GeS\({}_{2}\)-Based Glass  
The FLDW technique locally modifies the physicochemical properties of a transparent material initiated by nonlinear absorption of laser pulse energy. Depending on the writing parameters, different kinds of modifications can appear in such optical glasses. These permanent changes are commonly classified in the literature into different types [38; 39; 40]: Type I, Type II and Type III. Type I corresponds to a smooth and homogeneous variation of the refractive index. This type of variation appears at a low energy threshold and can be simply seen through contrast using optical microscopy in natural light. This permanent and isotropic variation of the refractive index is very interesting for imaging applications because it allows precise control of the spatial refractive index distribution, associated with a fine tunability of the index modulation amplitude. Type II modifications are characterized by the appearance of an anisotropy in the index variation, the orientation of which can be controlled by laser polarization. This can be easily revealed using polarized optical microscopy and the full waveplate technique. Indeed, in some glasses (mostly oxide ones), specific conditions of pulse duration, repetition rate and pulse energy can create some birefringent self-assembled quasiperiodic nanostructures. Subsequently, beyond certain laser intensity, Type III appears, characterized by the presence of voids resulting from micro-explosions. Finally, another regime of modification appears, especially whenusing a high-repetition-rate laser. It is often described inaccurately in the literature as a \"heat accumulation regime\", although it appears at low repetition rates as well when the deposited energy is quite high. This modification is not really a new type of modification because it is an isotropic variation of the refractive index like Type I. The difference is a significant increase in the size of the written area compared to the size of the laser spot due to heat diffusion. This regime is usually related to the pulse-to-pulse time interval (1/f, f being the repetition rate in Hz) being shorter than the heat diffusion time \(\tau_{\text{min}}\) needed to evacuate the heat from each pulse. \(\tau_{\text{min}}\) can be approximated using the following formula: \(\tau_{\text{min}}\sim w_{0}^{2}/(4D_{th})\), where \(w_{0}\) is the beam waist radius, \(D_{th}\) the thermal diffusion coefficient, expressed as \(D_{th}=\kappa/\left(\rho\text{-}C_{\text{p}}\right)\), where \(\kappa\) is the thermal conductivity, \(\rho\) the volumetric mass density and \(C_{\text{p}}\) the specific heat capacity. In this paper, we call this regime the \"spatial broadening regime\" [25]. Thus, discrimination between what has been called Type I and spatial broadening relies on visual measurements based on a partly subjective criterion. Usually, the size of Type I modifications only slightly depends on the pulse energy and it remains compatible with the estimated spot size. This contrasts with a spatial broadening regime, where the heat-affected zone increases with the pulse energy, resulting in tracks much bigger than the laser spot size. In the following, we define the transition between Type I modifications and a spatial broadening regime as being an increase of the object by about half of its \"initial value\". Figure 6 shows the processing windows related to different types of modifications obtained as a function of the scanning speed and the laser pulse energy in \(75(\text{GeS}_{2})-15(\text{In}_{2}\text{S}_{3})-10(\text{CsCl})\) glass. Two examples of typical laser track cross-sections taken in optical transmission are also given for both Type I and the spatial broadening regime. The faster the writing, the higher the energy needed to write a permanent optical change into a glass substrate.