## 4 Discussion and Comparison with Experiments
### LFC of Tempered Glass
Moreover, to obtain the smoothness of the separation as a whole, and not only between filaments of one array, it is important that at least one of these arrays of filaments covers both the hardened layer itself and also enters the area of the non-hardened remaining bulk material [14]. With the width of the set of filaments \(H\) = 45 \(\upmu\)m considered here, which exceeds the thickness of the hardened surface layer of 20 \(\upmu\)m, this is possible. Therefore, when creating not one, but simultaneously two these arrays of filaments in the material, for example, using the multi-focusing method, the required total pulse energy and average power are, respectively, twice as much as those indicated above, i.e., \(E_{1}\) = 23 \(\upmu\)J and \(P\) = 3.5-4.4 W. This estimate closely corresponds to the average power reported by the authors of the experiment for the separation of this plate: \(P\) < 4 W. Besides, for the higher PRR \(f\) = 700 kHz above mentioned, when \(u\) = 4.1 m/s, for the same \(E_{1}\) = 23 \(\upmu\)J, we find for the average laser power \(P\) = \(E_{1}\)\(f\) the value 16.1 W, which is also close to the value \(>\) 16 W mentioned in Refs. [5,6]. If each filament is created by a burst of pulses, then for the burst heating to be effective, its duration \(\tau_{burst}\) should not exceed the time \(t_{T1}\) for heat confinement in the filament with the radius \(r_{f}\) mentioned above in (4), i.e., \(\tau_{burst}\) \(<\)\(r_{f}^{2}\)/(4\(\chi\)) [39]. For the \(r_{f}\) and \(\chi\) indicated here, we obtain then an estimate: \(\tau_{burst}\) \(<\) 120-150 ns. Given that for \(N\) pulses in the burst, the distance between them in time is estimated by the obvious ratio as \(\Delta t\) = \(\tau_{burst}\)/(\(N\) - 1), for the number of pulses in the burst \(N\) = 6, we then find \(\Delta t\) \(<\) 24-30 ns, and for \(N\) = 10, we find respectively \(\Delta t\) \(<\) 13.3-16.7 ns. Both of these estimates of \(\Delta t\) are in a good agreement with the intervals between pulses \(\Delta t\) = 13 ns in the burst used for this laser source in the discussed experiment. The calculated parameters are summarised in Table 6. As can be seen, they are in a good agreement with the experimental values \(\tau\), \(P\), \(u\) and \(\Delta t\). For comparison, we calculate for this material the throughput of the LFC \(\Omega\) for the most favourable conditions of maximising \(\Omega\) indicated above in Section 3.4. To do this, as mentioned above, we also apply a twice shorter wavelength \(\lambda\) = 0.52 um and assume that according to (19), the filament radius at the same NA = 0.6 becomes \(r_{f}\) = 0.244 um, i.e., also decreases by half compared to the previously used value \(r_{f}\) = 0.49 um for \(\lambda\)=1.04 um. We use VED bounds \(w_{1}\) = 400 kJ/cm\({}^{3}\) and \(w_{2}\) = 396-480 kJ/cm\({}^{3}\) according to calculations given in Table 1 for a very close value of \(r_{f}\) = 0.25 um. We use the value of \(w\) based on the condition indicated above in Section 3.4\(w\) = 2\(w_{dis}\), i.e., \(w\) = 128 kJ/cm\({}^{3}\). In this case, as can be seen, the condition (12) \(w\)\(\leq\)\(w_{1}\) is fulfilled, under which, according to our model, there are no significant energy losses from the filament to thermal emission, providing thus for the driving force and efficiency of the LFC process. Also, following the condition (iii) in Section 3.4, we choose \(f\) in the high-frequency range (54), for example, as \(f\) = 3\(f_{1}\). For so high\(f\), \(f\) \(>\)\(f_{1}\), we put according to (61) \(\eta\)\(\approx\) 1. Such \(\eta\), as follows from (60), corresponds to a single pulse heating at the edge of the modification zone by the value \(\Delta T_{1}\) = (\(T_{strain}\) - \(T_{i}\))/2 = 275 \({}^{\circ}\)C, i.e., from \(T_{i}\) = 20 \({}^{\circ}\)C to 295 \({}^{\circ}\)C. For such a temperature range, we take the value \(C_{2}\)\(\approx\) 0.98 J/(g\(\times\)K) as the \"low-temperature\" heat capacity \(C\) = \(C_{2}\).