## 3 Temperature Distribution \(T(t,r)\) in Glasses under Local Thermal Perturbations
### Influence of the Shape and Duration of Heating Pulses on \(T(t,r)\)  
Thus, for model calculations, we use a heating pulse with a power density \(\Phi_{0}\)\(sin(\pi t/\tau_{p})\), acting during the time interval \(\tau_{p}\) in a spherical region with a radius \(r_{0}=1\)\(\upmu\)m. Let us com \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Substance**} & **Density in Solid State** & \begin{tabular}{c} **Specific Heat** \\ **Capacity in the** **Solid** **State** \\ **Solid State** \\ \end{tabular} & \begin{tabular}{c} **Volumetric Heat** \\ **Capacity in the** **Solid** **State** \\ **Solid State** \\ \end{tabular} & \begin{tabular}{c} **Volumetric Heat** \\ **Capacity in Solid** **State** \\ **Static** \\ **(**\(\varepsilon_{0}\)**)** \\ \end{tabular} & \begin{tabular}{c} **Dynamic Heat** \\ **Capacity Parameter** \\ **(**\(\varepsilon_{0}\)**)** \\ \end{tabular} & \begin{tabular}{c} **Thermal** \\ **Conductivity** \\ **\(\lambda\)** \\ **\(\lambda\)** \\ **W/mK** \\ \end{tabular} \\ \hline \
\begin{tabular}{c} Sodium-line- \\ silicate glass \\ \end{tabular} & 2.8 & 1.14 & 3.2\(\cdot\)10\({}^{6}\) & 4.4\(\cdot\)10\({}^{6}\) & 0.273 & 1 \\ \hline
\begin{tabular}{c} Borosilicate glass \\ \end{tabular} & 2.23 & 1.1 & 2.45\(\cdot\)10\({}^{6}\) & 3.8\(\cdot\)10\({}^{6}\) & 0.355 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermal parameters of silicate glasses used for model calculations. Figure 2: Schematic diagram of the distribution of the intensity of light radiation in a spot of a focused laser beam depending on the distance \(r\) from the center of the spot (dashed line) and the intensity distribution sufficient to induce nonlinear processes such as multiphoton absorption in a spot with radius \(r_{0}\) (solid line). pare the results for heating pulses with \(F(t)\) with half-sinusoidal and rectangular shapes; see Figures 3 and 4. Let the initial temperature \(T_{in}=300\) K, \(\tau_{p}=300\) ps, \(E_{p}=15\) nJ, and the volume of the spherical heating zone \(V_{0}=\frac{4\pi}{3}r_{0}^{3}\). Thus, \(\Phi_{0}\) is equal to \(\frac{\pi}{2}E_{p}/\tau_{p}V_{0}\) and \(E_{p}/\tau_{p}V_{0}\) for half-sinusoidal and rectangular heating pulses, respectively. In this case, the energy of these pulses is the same and equal to \(E_{p}\). The temperature distribution \(T(t,r)\) can be calculated according to Equations (8) and (9) at various \(\tau_{0}\). For example, for sodium-time-silicate glass (see Table 1), \(T(t,r)\) at \(\tau_{0}=0\), \(0.1\)\(\mu\)s, and \(10\)\(\mu\)s is represented in Figures 3 and 4. The temperature distribution \(T(t,r)\) is calculated at \(R_{0}\gg r_{0}\) (for example, at \(R_{0}=30\)\(\mu\)m). It follows from direct calculations that the result does not depend on \(R_{0}\), at least on the time scale \(t<{R_{0}}^{2}/4D_{0}\), where \({R_{0}}^{2}/4D_{0}\) for the glasses under consideration is about 3-4 ms. However, we are interested in changes in the temperature distribution \(T(t,r)\) on a time scale of microseconds and less when the temperature changes are localized within a few micrometers; see Figure 3.  
The influence of the time dispersion of the dynamic heat capacity \(c_{dyn}(t)\) is significant; see Figures 3 and 4. The difference between equilibrium (\(\tau_{0}=0\)) and non-equilibrium  
Figure 4: Time dependence of \(T(t,0)\) in the center of the hot zone during (**a**) and after (**b**) the heating pulses of half-sinusoidal shape (filled symbols) and rectangular shape (open symbols) at the same parameters as in Figure 3 (\(\tau_{0}=0\), \(0.1\)\(\mu\)s, and \(10\)\(\mu\)s—squares, circles, and triangles, respectively). Inset (**a**) shows pulse power as a function of time.