## 4 Discussion and Comparison with Experiments
### LFC of Sapphire
In this case, when \(s_{1}/R\) = 1.8, according to Table 2 the frequency limits are \(f_{1}\) = 13.3\(\chi\)/\(s_{1}\)\({}^{2}\) and \(f_{2}\) = 0.547\(\chi\)/\(s_{1}\)\({}^{2}\). For the thermal diffusivity of sapphire at high temperatures \(\chi\) = (1.5-2.4) \(\times\) 10\({}^{-2}\) cm\({}^{2}\)/s [57] and the above step \(s_{1}\), we find: \(f_{1}\) = 1.2-1.9 MHz and \(f_{2}\) = 49-78 kHz. From this, it can be seen that the repetition rate \(f\) = 75 kHz extracted above from the experimental data is close to the \(f_{2}\) limit of the low-frequency range (53), i.e., \(f\approx f_{2}\). The results of calculations in comparison with the experimental data are summarised in Table 5. It can be seen that the calculated \(E_{1}\), \(s_{1}\), \(P\) and \(u\) are in good agreement with the values obtained from the experiment when \(f\) is chosen as 75 kHz, i.e., close to the calculated limit \(f_{2}\) of the low-frequency range. However, the model also predicts, in this case, another, high-frequency range (54), emphf \(\geq\) f\({}_{1}\), which has not been studied in the experiment [20]. If we choose \(f\) in this range, for example, as \(f\) = 2.5 MHz (i.e.,\(f\geq f_{1}\) =1.2-1.9 MHz), then according to the expression \(P\) = \(E_{1}\)/, using the above value \(E_{1}\) = 19.5 \(\upmu\)J, we find a much higher required average laser beam power \(P\approx\) 49 W than \(P\) = 1.5 W estimated above in the low-frequency range. The scanning speed of the beam relative to the material according to (90) at such \(f\) and \(s_{1}\) is very large: \(u\) = \(s_{\rm f}\)/ = 10.3 m/s. If we choose \(f\) even more, for example, \(f\) = 6 MHz, then with the same \(s_{1}\) we find \(u\) = \(s_{\rm f}\)/ = 24.6 m/s and \(P\) =117 W.  
These particular numbers show that the high-frequency range \(f\geq f_{1}\) provides for high, in particular, for a sapphire--very high, cutting speeds of 25 m/s and more, which by tens and hundreds of times exceed cutting speeds of glass, sapphire or other transparent materials with known mechanical methods and by tens times exceeds the characteristic rate of separation with the known methods of thermo-cleaving by irradiation with continuous-wave lasers. However, for such high-speed LFC modes, a relatively high average laser power of tens of W is required. At the same time, it can be seen that in the low-frequency mode, the laser separation requires a much lower average power of the laser beam (for example, as we estimated above and as confirmed by the experiment, for sapphire--at the level of 1.5 W), at which, nevertheless, the processing separation speed is provided, \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **Pulse** & **Pulse** & **Scanning** & **Repetition** & **Average** & **Scanning** \\  & **Energy** & **Duration** & **Step** & **Rate Limit** & **Rate** & **Power** & **Speed** \\  & **\(E_{1}\), \(\upmu\)J** & \(\tau\), **ps** & \(s_{1}\), \(\upmu\)m** & \(f_{2}\), kHz** & \(f\), kHz** & \(P\), **W** & **\(u\), m/s** \\ \hline Experiment [20] & \textless{}26.7 & - & 4 & - & 75 & \textless{}2 & 0.30 \\ \hline Model & 19.5 & 3.5–7.0 & 4.1 & \(f_{2}\) = 49–78 & 75 & 1.5 (\(f\) = 75 kHz) & 0.31 \\ \hline \end{tabular}
\end{table} 
Table 5: Comparison of experimental [20] and theoretical values of the \(E_{1}\), \(s_{1}\), \(P\) and \(u\) parameters for the LFC sapphire. although not a very high, but acceptable for practical applications, specifically--about 0.3 m/s. Due to the correct choice of the repetition frequency for this material, if not a smooth separation, but, as already mentioned above, increased uniformity and straightness of the filaments is obtained.