## 1 Introduction
The energy of the electrons in the laser focus increases sharply during \(\tau_{laser}\) due to electron-photon interaction. The electrons then transfer their energy to the lattice through electron-phonon interaction. The thermalization process takes approximately tens or hundreds of picoseconds [6; 10; 38; 39]. Consequently, the duration of heating pulses \(\tau_{p}\) that heat the material can be on the order of tens to hundreds of picoseconds. Thus, the material can be locally heated above the glass transition temperature \(T_{g}\) and melting temperature \(T_{m}\). Heat then spreads through the material, and the hot focal zone is rapidly cooled below \(T_{g}\). In this article, we will consider laser pulses of moderate energy \(E_{p}\) (in the range 10-100 nJ), which is sufficient to locally heat the material well above the glass transition temperature \(T_{g}\) but not sufficient to destroy the material with the formation of voids. Thus, the intensity of the laser pulses considered in this article is below the material damage threshold, which, for example, is about \(10^{16}\) W/cm\({}^{2}\) for silica glass [40]. We focus on micrometer-scale local thermal perturbations caused by laser pulses near or below the threshold ionization intensity of a material. The threshold ionization intensity of most transparent solids ranges from \(10^{13}\) to \(10^{14}\) W/cm\({}^{2}\) for light with a wavelength of about 1 \(\upmu\)m [41; 42]. For example, the threshold intensity in silica glass and similar glass-forming materials is about \(10^{13}\) W/cm\({}^{2}\) for light pulses with a wavelength of about 1 \(\upmu\)m and a duration of 100 fs [41]. Thus, we will consider thermal processes in dielectric glasses below the threshold intensity of ionization of the material and in the time interval after electron-phonon thermalization, when there is no electron-hole plasma. Thus, we are interested in changes in the lattice temperature on a time scale outside the time interval when a multi-temperature model is usually considered [12; 13; 14]. In fact, we are interested in changes in lattice temperature that affect the structure of the material at moderate laser pulse energies and time \(t\) exceeding the electron-phonon thermalization time. The initial structure of glass-forming materials is not completely restored after the rapid heating-cooling cycle, which leads to a change in the local properties of the material in rapidly cooled areas. The local structure, specific volume, density, Rayleigh scattering loss, and refractive index of glass-forming materials significantly depend on the local cooling rate [19; 43; 44]; see Figure 1.  
Thus, we focus on the dynamics of the temperature distribution \(T(t,r)\) and study the local cooling rate \(-\mathcal{R}(t,r)\) near the focal region during laser-induced microstructuring. Measurements of the temperature distribution at the focus of the laser beam can be carried out using micro-Raman spectroscopy [45; 46; 47]. However, the time resolution of Raman spectroscopy (with a measuring pulse duration of about 10 ns and a repetition rate of 1 kHz) is not sufficient to detect ultrafast changes in the local temperature \(T(t,r)\). We found that the local cooling rate \(-\mathcal{R}(t,r)\) after the end of the heating pulse can reach more than \(10^{11}\) K/s in a thin layer around the focus of the laser beam. In fact, resolving such ultrafast temperature changes at the periphery of the hot focal zone requires a temporal and spatial resolution of at least about 0.1 ns and 10 nm, respectively, since the cooling rate \(-\mathcal{R}(t,r)\) reaches about 600 GK/s in a narrow layer around the hot zone at the laser focus (see below). Thus, the maximum change in the glass structure occurs at the periphery of the focal region (where the material undergoes ultrafast quenching).