## 5 Discussion
It turns out that ultrafast cooling of the material occurs at a rate of more than \(10^{11}\) K/s in a thin shell in the region from \(0.95\tau_{0}\) to \(0.99\tau_{0}\). However, in the center of the heating zone, the cooling rate is three orders of magnitude lower than at the periphery. Thus, the local structure of glass after laser modification in the center of the hot zone should differ significantly from the structure at the periphery in a thin shell \(\Delta r\) of about \(0.05\) um, which is consistent with experiments [19; 33; 35; 39; 45; 46; 47; 48; 49; 50; 51; 52; 85]. In fact, the experimentally observed thin shell at the periphery of the laser beam focus is visually about \(\Delta r/r_{0}\approx 10\%\); i.e., \(\Delta r\) is about \(0.5\) um at \(r_{0}\approx 5\) um [19; 33; 35]. Of course, the visual size \(\Delta r\) is smoothed out due to the diffraction limit of optical images of laser-induced microbeads. Indeed, for such laser-induced microbeads, the shell size \(\Delta r\) is about \(0.05\) um in high-resolution scanning electron microscopy images [49; 85]. In addition, the shell size \(\Delta r\) is smoothed due to the high-frequency repetition of laser pulses during laser processing in the cumulative heating mode. The rate \(-\mathcal{R}\approx 10^{11}\) K/s of ultrafast cooling of the material in a thin shell at the periphery of the heating zone is sufficient to freeze the liquid structure corresponding to a temperature \(T_{f}\) that is significantly higher than the initial temperature \(T_{in}\). In fact, the difference \(\Delta T=T_{f}-T_{in}\) can be on the order of several hundred K. Indeed, the material at the periphery of the heating zone is locally quenched-in within a time interval \(\Delta t\) of about \(10\)-\(100\) ns; see Figure 14. In this case, the liquid structure of the material should remain frozen, since the distance \(\sqrt{\Delta tD_{SD}}\) of self-diffusion of glass molecules does not exceed the order of several nanometers with the self-diffusion coefficient \(D_{SD}\) of the order of \(10^{-9}\) m\({}^{2}\)/s [64]. Suppose the material is locally heated and then rapidly cooled to the initial temperature of \(T_{in}\) of the matrix material. Thus, if a rapidly cooled material has a locally frozen density \(\rho\left(T_{f}\right)\) corresponding to a certain temperature \(T_{f}=T_{in}+\Delta T\), then the local density change \(\Delta\rho/\rho\) is about \(\alpha_{V}\Delta T\), where \(\alpha_{V}\) is the volumetric thermal expansion coefficient. For example, for borosilicate glasses, \(\Delta\rho/\rho\) is about \(0.2\%\) at \(\Delta T=200\) K, \(\alpha_{V}=3\alpha\), and linear thermal expansion \(\alpha=3.3\cdot 10^{-6}\) 1/K [60]. Then, the local stress can reach the order of \(K_{B}\alpha_{V}\Delta T\) (about \(65\) MPa) with a bulk modulus \(K_{B}=33\) GPa [60; 61]. For sodium-lime-silicate glass, \(\Delta\rho/\rho\) is about \(0.5\%\) at \(\Delta T=200\) K, \(\alpha_{V}=3\alpha\), and linear thermal expansion \(\alpha=7.7\cdot 10^{-6}\) 1/K [59]. Then, the local stress can reach about \(K_{B}\alpha_{V}\Delta T\approx 250\) MPa with a bulk modulus \(K_{B}=55\) GPa [55; 58]. This local stress can cause a change in the local birefringence on the order of several percent [86]. A similar change in the local refractive index, proportional to \(\Delta\rho/\rho\), can be caused by a change in the local density [13]. This estimate is consistent with femtosecond laser anisotropic nanostructuring in transparent materials for optical data storage [3]. Of course, the stress caused by the laser pulse then relaxes over time to some residual value. This residual stress may be the subject of a separate study. In this work, we focus on the unique capability of femtosecond laser microstructuring deep inside transparent glasses. This possibility can be used for optical data storage. In fact, changes in the refractive index with changes in local density \(\Delta\rho/\rho\) and changes in the birefringence due to local stress are sufficient for data storage [13].