## 4 Discussion and Comparison with Experiments
### LFC of Tempered Glass
Then the modification threshold when substituting these \(C_{2}\), \(\rho\) and \(\Delta T_{1}\) into (66) is \(w_{mod}^{T}\) = 0.655 kJ/cm\({}^{3}\). According to (17) for the indicated \(w\), \(r_{f}\), \(A_{L,P}\) and the filament length, as above, \(H\) = 45 um, the pulse energy in the material is \(E_{1}\) = 1.35 \(\upmu\)J. Using the same \(w_{dis}\), \(\gamma_{T}\), \(K_{tw}\) and \(A_{P,H}\), as above in this section, according to (66) we find the step as \(s_{1}\) = 0.9\(s_{max}\) =7.24 \(r_{f}\) =1.77 um. From (93) for such \(s_{1}\), \(E_{1}\) and \(H\), we find \(\Omega\)\(\approx\) 59 um\({}^{2}\)/\(\upmu\)J. For comparison, we recall that for the above-given example of the LFC of tempered glass at \(\lambda\) = 1.04 um, the LFC throughput was only \(\approx\) 23 um\({}^{2}\)/\(\upmu\)J, which is 2.6 times less. In this case, the calculation of the pulse duration according to (21), if each filament is created by only one pulse with the indicated energy \(E_{1}\) = 1.35 \(\upmu\)J at \(\Gamma\) = 1-2 and \(P_{c}\)(\(\lambda\)=520 nm) = 1.3 MW, gives \(\tau\) = 0.52-1.04 ps. But if the filament is created by a burst of pulses, for example, from \(N\) = 6 pulses, as in the example above, then the duration of a single pulse from the burst is \(\tau_{0}\)\(\approx\)\(\tau\)/\(N\), i.e., \(\tau_{0}\)\(\approx\) 86-172 fs. For the indicated above step \(s_{1}\) = 0.9\(s_{max}\) =1.77 um, which corresponds to the ratio (\(s_{1}\)/\(R\)) = 1.8, we find the limit\(f_{1}\) for this ratio according to Table 3 as \(f_{1}\) = 13.3 \(\chi\)/\(s_{1}\)\({}^{2}\). For the above-mentioned thermal diffusivity \(\chi\) of tempered glass and \(s_{1}\) = 1.77 um this relation gives \(f_{1}\) = 1.70-2.12 MHz. The PRR is chosen for certainty as \(f\) = 3\(f_{1}\), i.e., \(f\) = 3 \(\times\) 2.12= 6.36 MHz. For the average laser power in the material \(P\) = \(E_{1}\)\(f\) with indicated \(E_{1}\) and \(f\), we get \(P\) = 8.6 W. When creating in the material not one, but, as noted above, two arrays of filaments simultaneously, for example, by the multi-focusing method, the resulting pulse energy and average power are required, respectively, 2 times greater, i.e., \(E_{1}\) = 2.70 \(\upmu\)J and \(P\) = 17.2 W. The relative speed of the beam and material \(u\) = \(s_{1}\)\(f\) for indicated \(s_{1}\) and \(f\) is \(u\) = 11.3 m/s. As we can see, the indicated irradiation mode with almost half the \(w\), twice the shorter wavelength \(\lambda\) and PRR in the high-frequency range, in addition to maximising throughput for a smooth LFC of tempered glass, provides also for a very high cutting speed, 11.3 m/s,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Pulse** & **Pulse** & **Scan** & **Repetition** & **Average** & **Scanning** & **Interval** \\  & **Energy** & **Duration** & **Step** & **Rate** & **Power** & **Speed** & **Pulses in the** \\  & **\(E_{1}\), \(\upmu\)J** & \(\tau\), \(fs\) & \(s_{1}\), \(\upmu\)m & _f,_ \(kHz\) & _P,_ \(W\) & _u,_ \(m/s\) & **Burst,** \\ \hline Experiment [5,6] & - & \(<\)400 & - & - & \(<\)4 & \(<\)4 & 13 \\ \hline \multirow{2}{*}{Model} & \multirow{2}{*}{23} & \multirow{2}{*}{192–384} & \multirow{2}{*}{5.9} & 153–191 & 3.5–4.4 & 0.90–1.13 & \multirow{2}{*}{\(<\)13–17} \\ \cline{5  
which exceeds the speeds of previously known laser and non-laser glass cutting methods by tens and hundreds of times.