## 1 Introduction  
Glass-forming materials are often used to create optical elements and components for optical integrated circuits. In the last two decades, the technology for the femtosecond laser microfabrication of various optical structures inside glass-forming materials has been intensively developed. Due to the extremely high peak intensities at the focus of femtosecond laser pulses, nonlinear processes such as multiphoton absorption can lead to significant energy absorption even within transparent materials such as glass. Thus, the femtosecond laser-induced microstructuring of glass-forming materials opens up wide possibilities for various applications. Laser heating can cause irreversible local phenomena at the beam focus, such as phase transitions [1], the formation of bubbles or microfluidic channels [2; 3; 4; 5], changes in chemical composition or refractive index [6; 7] such as the diffusion and aggregation of silver ions [6], or photo-oxidation [7]. Spatially selective laser-induced crystallization or structure changes in glasses allows the direct recording of channel waveguides [8; 9; 10; 11; 12; 13; 14]. In fact, laser-written waveguides (with improved mode structures of guided light) can be fabricated through local laser heating inside a glass matrix [8; 9]. Thus, optical integrated circuits can be fabricated using a laser beam that induces local structural changes in a fglass [15; 16; 17; 18]. Photopolymerization and photodamage by highly focused laser pulses can be used in microchemistry and stereolithography [6; 19]. Moreover, the femtosecond laser structuring of glass-forming materials can be used for the optical long-term storage of information [3; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30], photonics [18], the fabrication of phase gratings [31], nanogratings [32], and quantum dots that can be used in various devices [33; 34; 35]. Notably, optical storage based on glass-forming materials has the potential to replace magnetic storage in the quest to provide high-speed, high-capacity, low-power, low-cost, highly secure, and long-term data storage [25; 26; 27; 28; 29; 30]. However, for the development of these technologies, theoretical models of thermal processes occurring during spatially selective laser-induced structuring of glass-forming materials are required. For example, understanding the dynamics of the laser crystallization process of amorphous Ge films is important for transistor technology, photovoltaic devices, particle detectors, and photodetectors [36]. In fact, the laser crystallization method makes it possible to control the local temperature inside the material and avoid random nucleation [36]. The ability to form and control the dynamics of changes in the temperature distribution \(T(t,r)\) inside the material is very important for the opportunity to control the morphology of the laser-written structure inside the glass matrix [32; 22; 37]. Thus, a deep understanding of the thermal processes inside glass-forming materials under fast laser thermal perturbations is required. In this paper, we focus on local temperature changes, especially the local cooling rate \(-\mathcal{R}(t,r)\) in the focal region during laser-writing processes in a glass matrix (where \(\mathcal{R}(t,r)\) is the rate of temperature change). Femtosecond laser-induced microstructuring is a promising tool because the structures induced by femtosecond laser pulses can be even smaller than the optical diffraction limit of the beam focusing since multiphoton absorption is most effective in the central part of the beam's focus. Typically, in laser-induced microstructuring, the laser pulse duration \(\tau_{laser}\) is on the order of 100 fs, and the radiation wavelength is about 1 \(\upmu\)m. When a transparent material is irradiated with a powerful femtosecond laser pulse focused into a micro-sized focusing zone, multiphoton absorption of the laser radiation energy via electrons of the irradiated material occurs.