## 2 Heat Equation with Dynamic Heat Capacity
### Analytical Solution of the Heat Equation with Dynamic Heat Capacity  
Let us consider the temperature distribution \(T(t,r)\) after a single laser pulse in a micron-sized focal zone. A relatively small change in the thermal conductivity of the glass matrix around the hot focal zone does not affect the dynamics of the temperature distribution in the hot focal zone. In fact, for the glass-forming materials under consideration, \(\lambda\) varies insignificantly with temperature and is about 1 W/m\({}^{2}\)K over a wide temperature range [64; 78; 79]. Thus, it is assumed that the thermal conductivity \(\lambda\) of the glass matrix does not depend on temperature. However, the change in dynamic heat capacity \(c_{dyn}(t)\) from \(c_{S}\) to \(c_{L}\) during heating is significant. As a first step, we consider the Debye relaxation law; see Equation (6). Thus, for spherical geometry, from Equation (1), we obtain Equation (7):  
\[\frac{\partial}{\partial t}T(t,r)-D_{0}\Delta T(t,r)=\frac{\Phi(t,r)}{\rho c_ {0}}+\varepsilon_{0}\frac{\partial}{\partial t}\int_{0}^{t}exp\bigg{(}-\frac{ t-\tau}{\tau_{0}}\bigg{)}T(t,r)d\tau, \tag{7}\]  
where \(D_{0}=\lambda/\rho c_{0}\) and the spherically symmetric heat source \(\Phi(t,r)=\Phi(r)F(t)\) is distributed in a spherical volume of radius \(r_{0}\). The decomposition of \(\Phi(t,r)\) into the product \(\Phi(r)F(t)\) is common for laser heating. Furthermore, after the end of the heating pulse, the temperature distribution \(T(t,r)\) does not depend on the shape of \(F(t)\); see below. In this article, we are interested in the dynamics of the temperature distribution \(T(t,r)\) after the heating pulse, when the shape of the heating pulse does not matter. Consider Equation (7) under the initial conditions \(\Phi(t,r)=0\) and \(T(t,r)=T_{in}\) for \(t\leq 0\), where \(T_{in}\) is the initial temperature in the glass matrix. Let \(T(t,r)\) be a bounded function and \(T(t,R_{0})=T_{in}\) for \(R_{0}\gg r_{0}\). Since we are considering a single heating pulse, it does not matter how large the parameter \(R_{0}\) is, as long as \(R_{0}\gg r_{0}\). Indeed, the temperature change is localized in a hot zone about a few micrometers in size; see below. The solution to this boundary value problem can be represented in the form of the following series (for details, see Appendix A):  
\[T(t,r)=T_{in}+\sum_{\alpha=1}\psi_{n}(t)\frac{sin(\pi nr/R_{0})}{r}, \tag{8}\]  
where the functions \(\psi_{n}(t)\) are determined by Equation (9). \[\psi_{n}^{\prime}(t)+\frac{\psi_{n}(t)}{\tau_{n}}=\frac{\Phi_{n}F(t)}{\rho c_ {0}}+\varepsilon_{0}\frac{\partial}{\partial t}\int_{0}^{t}exp\bigg{(}-\frac{ t-\tau}{\tau_{0}}\bigg{)}\psi_{n}^{\prime}(\tau)d\tau, \tag{9}\]  
where \(\Phi_{n}=\frac{2}{R_{0}}\int_{0}^{R_{0}}r\Phi(r)sin(\pi nr/R_{0})dr\) and \(\tau_{n}^{-1}=D_{0}(\pi n/R_{0})^{2}\). The functions \(\psi_{n}(t)\) satisfying Equation (9) are presented in Appendix A. The spatial distribution \(\Phi(r)\) of the heating power \(\Phi(t,r)\) can be arbitrary. For example, we consider a heat source uniformly distributed in a volume of radius \(r_{0}\) with volumetric density \(\Phi_{0}\) (in W/m\({}^{3}\)); then,  
\[\Phi_{n}=2R_{0}\Phi_{0}\frac{sin(\pi nr_{0}/R_{0})-(\pi nr_{0}/R_{0})\cdot cos (\pi nr_{0}/R_{0})}{(\pi n)^{2}}. \tag{10}\]In fact, the distribution of the intensity of light radiation in the spot of a focused laser beam, depending on the distance \(r\) from the center of the spot, can have a Gaussian bell-shaped character; see Figure 2. However, the intensity sufficient to induce nonlinear processes; for example, multiphoton absorption is located close to the center of the laser beam spot [5]. Thus, the model of a heat source uniformly distributed in a volume of radius \(r_{0}\) with a sharp drop at the periphery is more realistic than the bell-shaped Gaussian distribution. The shape of the change in heating power over time can be bell-shaped or more complex. However, this shape is not significant for the temperature distribution \(T(t,r)\) at \(t>\tau_{p}\); see below.