## 4 Results  
During mathematical simulation on the basis of Equations (1)-(10), the temperature field in the optical fiber and duralumin plate was calculated, and the boundaries of temperature fronts corresponding to the regions where various phase states of quartz glass can occur, namely liquid, gaseous and plasma, were determined. The melting temperature of quartz (optical fiber) corresponds to 1440 K, vaporization in the fiber corresponds to 2706 K, and plasma formation starts at 5000 K.  
Figure 7 shows an example of the calculated temperature field for radiation with wavelength \(\lambda_{0}\) = 1550 nm and power \(P\) = 1 W in a time interval of 1 s after the introduction \begin{table}
\begin{tabular}{l l} \hline \hline
**Description** & **Value** \\ \hline Boltzmann’s constant & \(1.38\times 10^{-23}\) J/K \\ Core refractive index & \(1.4617\) \\ Charge of electron & \(1.6\times 10^{-19}\) C \\ Concentration & \(1.72\times 10^{15}\) m\({}^{-3}\) \\ Input power & \(1.10\times\) W \\ Formation energy GeE\({}^{\prime}\) & \(4.0054\times 10^{-19}\) J \\ Light velocity & \(3\times 10^{8}\) m/s \\ Core radius & \(4.1\times 10^{-6}\) m \\ Core layer thickness & \(4.1\times 10^{-6}\) m \\ Magnetic constant & \(1.2566\times 10^{-6}\) H/m \\ Drift mobility of electrons & \(5\times 10^{-3}\) m\({}^{2}\)/(V\(\cdot\)s) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of constant parameters of mathematical model. of radiation. The time interval was chosen from the experimental data [13]. As seen in Figure 7, the temperature front of the plasma formation comes out in the region of the fiber shell but does not reach the protective-strengthening coating. Experiments show that the size of the plasma spark moving along the fiber and the defects resulting from this motion do not exceed the size of the fiber core [18]. This circumstance suggests that optical breakdown, as a process of the plasma spark motion,  
Figure 7: Temperature field distribution in the axisymmetric setting (symmetry axis r = 0) in 1 s after introducing radiation into the fiber: (**a**) general view; (**b**) near the heating area. begins at the moment when the temperature front of the plasma formation reaches the core-shell \(S_{12}\) boundary. Within the framework of the constructed model, it is possible to estimate the time of occurrence of the optical breakdown \(t_{ab}\) for different wavelengths and powers. Plots of the optical breakdown time dependences on the input optical power for four wavelengths are shown in Figure 8.  
As seen in Figure 8, the time required for the initiation of optical breakdown in the fiber decreases with increasing power according to the law close to the exponential one. At high radiation powers (\(P_{max}=10\) W), the plasma spark initiation time lies in a narrow time range from 0.09 us for \(\lambda_{0}=1080\) nm to 0.18 us for \(\lambda_{0}=2050\) nm. At low radiation powers, the initiation time increases with increasing wavelength. As a result of the calculations for different wavelengths, the minimum \(P_{min}\) power values necessary to initiate optical breakdown were estimated. The results of the calculations are shown in Table 4, which shows that the minimum power \(P_{min}\) required to initiate optical breakdown increases with increasing wavelength \(\lambda\). At low powers, there is a significant difference in the initiation times (see Figure 8). At minimum powers, the plasma boundary reaches the border of the fiber core in 1 ms and does not move further. Figure 9 shows the temperature field in the fiber for radiation with wavelength \(\lambda_{0}=1550\) nm and power \(P_{min}=400\) mW. The temperature front of plasma formation does not cross the boundary \(S_{12}\) core-shell of the fiber. \begin{table}
\begin{tabular}{c c c c c} \hline \(\lambda\), nm & 1080 & 1310 & 1550 & 2050 \\ \hline \(P_{min}\), mW & 270 & 335 & 400 & 630 \\ \(T_{max}\), K & 14,685 & 14,671 & 14,630 & 14,477 \\ \hline \end{tabular}
\end{table} 
Table 4: Minimum power required to initiate optical breakdown.