## 3 Model
### Speed and Average Power of the Beam  
The speed \(u\) of the relative movement of the laser beam and the material (and, correspondingly, the processing speed) is defined by the known formula:  
\[u=s_{1}f, \tag{90}\]  
where the choice of \(s_{1}\) and \(f\) is defined above depending on material and radiation parameters. For example, for the thermal modification mechanism, if the condition (33) of a short duration \(t_{E}\) of heat release is fulfilled, we obtain the following estimate for \(f\approx f_{1}\), or \(f\approx f_{2}\):  
\[u\approx a\chi/s_{1}, \tag{91}\]  
where \(a=a_{1}\) or \(a=a_{2}\) (and \(a_{1}>>a_{2}\)) correspond according to (53) to the frequency limits\(f_{1}\) and \(f_{2}\), respectively.  
It is seen from (89) that a high speed \(u\) (and processing speed, respectively) can be obtained by using the high-frequency range, where \(a\) is higher. Also, \(u\) increases with a decrease in the step of the filaments following \(s_{1}\), which according to (66) is achieved by reducing \(w\), as well as by reducing the filament radius \(r_{f}\), if, as follows from the discussion of the Formula (19) above, a shorter wavelength \(\lambda\) is used. But at the same time, according to the ratios (56,57), where \(f_{1,2}\propto 1/{s_{1}}^{2}\), a higher pulse repetition rate \(f\) is also required. Besides, as follows from (89), \(u\) increases for materials with a higher thermal diffusivity \(\chi\).  
The necessary average power \(P\) of the beam is found in terms of \(E_{1}\) and \(f\) obtained above, according to the obvious relation \(P=E_{1}f\). For example, for the thermal mechanism of the impact zone formation, when the condition (33) of short duration of heat dissipation is fulfilled, at \(f\approx f_{1}\) or at \(f\approx f_{2}\), with taking into account the foregoing ratio \(f_{1,2}=a\chi/{s_{1}}^{2}\)  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **Quartz Glass** & \
\begin{tabular}{c} **Corning** \\ **EAGLE XGTM** \\ \end{tabular} &
\begin{tabular}{c} **Corning** \\ **Gorilla Glass 5** \\ \end{tabular} & **Sapphire** \\ \hline \multirow{2}{*}{**Amorphous material**} & \(\frac{a_{T}Y(T_{ststunin}-T_{t})}{\sigma_{c}}\) & 0.11–0.14 & 0.28 & 0.14–0.19 & - \\ \cline{2-6}  & \(\frac{C_{2}\rho(T_{stunin}-T_{t})}{(1+\gamma)\sigma_{c}}\) & 1.5–2.0 & 1.6 & 0.32–0.42 & - \\ \hline \multirow{2}{*}{**Crystalline material**} & \(\frac{a_{T}Y(T_{stunin}-T_{t})}{\sigma_{c}}\) & - & - & - & 1.3–2.2 \\ \cline{2-6}  & \(\frac{C_{2}\rho(T_{stun}-T_{t})}{(1+\gamma)\sigma_{c}}\) & - & - & - & 1.5–1.9 \\ \hline \end{tabular}
\end{table}
Table 4: Verification of criteria (86,88) for amorphous materials and (87,89) for crystalline materials, the simultaneous fulfilment of which is preferable for obtaining a separation with better smoothness. The parameters \(\alpha_{T}\), \(Y\), \(T_{ststunin}\), \(T_{melt}\), \(\sigma_{c}\), \(C_{2}\) and \(\rho_{2}\), required for calculations, are taken from Table 2. Initial temperature is \(T_{i}=20\)\({}^{\circ}\)C, heat preservation factor is \(\eta=0.76\).  
where \(a=a_{1}\) for \(f\approx f_{1}\) and \(a=a_{2}\) for \(f\approx f_{2}\), expressions (17) for \(E_{1}\) and (66) for \(s_{1 max}\) and with choosing \(s_{1}\) as \(s_{1}\)= 0.9\(s_{1 max}\) (when according to Table 3\(a=a_{1}=13.3\) or \(a=a_{2}=0.547\)), we obtain the following estimate of \(P\):  
\[P\cong\frac{0.309\pi}{\gamma_{tw}\ K_{tw}A_{P-H}\ A_{L-P}}\frac{w}{(w-w_{dis})} aH\chi w_{mod}^{\Gamma}. \tag{92}\]  
It can be seen that \(P\) decreases many times for the low-frequency range, where \(a\) is less than in the high-frequency range, as well as for filaments with shorter length \(H\), for materials with a lower thermal diffusivity \(\chi\) and having a smaller threshold \(w_{mod}^{\Gamma}\). An increase in the conversion coefficients \(A_{L-P}\), \(A_{P-H}\) and \(K_{tw}\) also leads to \(P\) decrease. It is also seen that \(P\) does not depend on \(\gamma_{f}\), i.e., on the wavelength, in contrast to the discussed above estimate (91) for \(u\).