## 3 Temperature Distribution \(T(t,r)\) in Glasses under Local Thermal Perturbations
### Dependence of \(T(t,r)\) on the Size of the Heating Zone
For example, consider the cooling curves calculated at the periphery of the hot zone for borosilicate (\(T_{g}=900\) K [62]) and sodium-time-silicate glasses (\(T_{g}=840\) K [56]); see Figures 10b and 14, respectively. It should be noted that the effect associated with the time dispersion of the dynamic heat capacity is significant well above the glass transition temperature \(T_{g}\). Indeed, in the case of \(r_{0}=1\) um, the modification process occurs within approximately 80 ns at \(r_{1}=0.95\) um and \(\tau_{0}\geq 1\) us; see Figure 14a. However, without taking into account the time dispersion of the dynamic heat capacity (at \(\tau_{0}=0\)), this process occurs in less than 20 ns; see Figure 14a. In the case of \(r_{0}=2\) um, the modification process occurs during a longer period of about 300 ns at \(r_{1}=1.98\) um and \(\tau_{0}>1\) us; see Figure 10b. However, without taking into account the time dispersion of the dynamic heat capacity (at \(\tau_{0}=0\)), this process occurs in less than 50 ns; see Figure 10b. To summarize, we can conclude that the maximum cooling rate exists in the outer regions of the heating zone in the shell of about \((0.97\pm 0.02)r_{0}\). The cooling rate reaches several hundred GK/s in the first nanosecond after the laser pulse. Then, the cooling rate decreases. However, even 100 ns after the laser pulse, \(-\mathcal{R}(t,r)\) in the outer shell of the hot zone still exceeds 1 GK/s. It is noteworthy that in the center of the heating zone, the cooling rate is three orders of magnitude less than \(-\mathcal{R}(t,r)\) at the periphery. Since regions quenched at different cooling rates have different physical properties, strong gradients  
Figure 14: Time dependence of \(T(t,r_{1})\) at the periphery of the hot zone at \(r_{1}=0.95\) μm (**a**) and \(r_{1}=0.98\) μm (**b**) at the same parameters as in Figure 3 (\(\tau_{0}=0\), 0.1 μs, 1 μs, 10 μs, and 100 μs—squares, circles, triangles up, triangles down, and diamonds). Inset (**a**) shows the time dependence of \(T(t,0)\) in the center of heating zone (at \(\tau_{0}=10\) μs). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Time** & **Distance** & **Coiling Rate** \\ \(\mathbf{t_{1}}\) & \(\mathbf{r_{max}}\) & \(-\mathcal{R}(t_{1}r_{max})\) \\
**ns** & \(\mathbf{\mu}\)**m** & **GK/s** \\ \hline
0.5 & 0.985 & 410 \\
1 & 0.98 & 170 \\
5 & 0.94 & 31 \\
10 & 0.92 & 16 \\
50 & 0.80 & 4 \\
100 & 0.70 & 2.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Cooling rate \(-\mathcal{R}(t,r)\) for sodium-time-silicate glass at \(r_{0}=1\) μm. of physical properties should exist in the material predominantly at a distance of about \(0.9r_{0}\) from the center of the heating zone. This effect is significantly enhanced by the time dispersion of the dynamic heat capacity. In conclusion, let us briefly discuss laser microprocessing in the cumulative heating mode. Typically, cumulative heating is achieved at a laser pulse repetition rate of hundreds of kHz or more. Usually, this mode operates at frequencies from 200 kHz to 10 MHz [3]. Indeed, the temperature \(T(t,0)\) in the center of the hot zone relaxes in a time of the order of \(r_{0}^{2}/D_{0}\), which for the glasses under consideration is about 4 us at \(r_{0}=1\)um; see inset in Figure 14a. Thus, in this case, the cumulative effect can be achieved at a frequency of about 250 kHz or more. Note that the cumulative effect leads to stable average heating of the material in the heating zone. Thus, the effect of laser pulse repetition can be approximately simulated by shifting the local temperature distribution \(T(t,r)\) upward by a certain temperature difference \(\Delta T\), which increases with the increasing pulse repetition frequency. It is clear that with increasing \(\Delta T\), the modification process will occur over a longer time interval--see Figure 14--where \(T(t,r_{1})\) should be shifted upward by the amount \(\Delta T\).