## 4 Discussion and Comparison with Experiments
### LFC of Sapphire
In stage 3, we calculate the laser pulse energy in the material \(E_{1}\), which is not reported directly in the considered experiment [20]. For the above input parameters \(A_{L,P}\), \(r_{f}\) and \(H\), as well as the \(w\) chosen on the previous stage, Equation (17) gives then \(E_{1}=19.5\) uJ. Also, according to (46), we calculate the modification radius \(R_{SW}\) for the shock-wave mechanism--for the above values of the input parameters \(\gamma_{sw}\), \(K_{sw}\), \(A_{P,H}\), \(r_{f}\), \(w_{dis}\) and the VED \(w\) chosen on the previous stage, this gives \(R_{SW}=2.25\) um. If smooth separation were possible in this case, then calculating the maximum step of filaments following \(s_{1 max}\) to obtain it would give according to (22) \(s_{1 max}=2R_{SW}=4.5\) um. The choice of the step value as \(s_{1}=0.9s_{1 max}=1.8R_{SW}\), when the impact zones from individual filaments overlap with each other, would lead to an estimate \(s_{1}=4.1\) um. In stage 4, we find the pulse duration \(\tau\) for the case when the filament is formed by only one pulse. In Equation (21) we use the value \(E_{1}\approx 19.5\) uJ found on the previous stage, and also use the input parameters indicated above: \(P_{c}\)(\(\lambda=1.04\) um) = 2.8 MW [60] and \(\Gamma=1\)-2. This gives \(\tau=3.5\)-7.0 ps. For the LFC throughput using (93) for the found above \(s_{1}=4.1\) um, \(H\approx 45\) um and \(E_{1}\approx 19.5\) uJ, we get: \(\Omega\approx 9.5\) um\({}^{2}\)/uJ. In stage 5 for the PRR\(f=75\) kHz mentioned above and extracted from experimental values and also for the calculated \(E_{1}\approx 19.5\) uJ we find the average laser power in the material as \(P=E_{1}f=1.5\) W. Also, for the above \(s_{1}\) and the same \(f\), we find from (90): \(u=s_{1}f=31\) cm/s. This estimate is in good agreement with the scan speed \(u=30\) cm/s reported in the experiment under discussion. In the absence of a smooth separation in LFC for the given material, when it is only a matter of mechanical weakening this material in the irradiation zone regardless of the quality of the mechanical separation obtained after irradiation (i.e., not necessarily smooth), it seems that lower pulse energies and average beam powers can be chosen, but probably not more than about 1.5-2 times lower, compared to the \(E_{1}\) and \(P\) indicated here above and calculated as for smooth separation. Also, the parameter \(\Gamma\), which describes the multiplicity of exceeding for the power of a single pulse \(P_{1}\) over the critical power for self-focusing \(P_{\rm c}\) and is necessary for estimating the pulse duration according to the Formula (21), can be 2-3 times lower or higher, i.e., \(\Gamma\) instead of 1-2 can be of the order of \(\Gamma\approx\) 0.25-5. If still the indicated above, calculated \(E_{1}\) and \(P\) are used, then the step of filaments following can also be chosen more than the found value \(s_{1}\) = 4.1 \(\upmu\)m up to several tens of per cent. For the given model calculations, the shock-wave mechanism of the impact zone formation around each filament has been considered as the dominant. However, an alternative thermo-deformation mechanism gives a practically comparable contribution, as this mechanism has a very close modification threshold: according to (68) \(w_{mod}^{TD}\approx\) 2.4-3.6 kJ/cm\({}^{3}\), while \(w_{mod}^{SW}\approx\) 2.4-3.0 kJ/cm\({}^{3}\). Therefore, the radial transfer of energy from the filament to the surrounding material due to thermal diffusion, which is characteristic of this alternative mechanism, is also important for obtaining separation. In this relation, when choosing the pulse repetition frequency \(f\) to obtain more regular and more straight filaments, it seems preferable, as in the case of the thermal mechanism, to use frequencies in the two found frequency ranges (54) and (55): \(f\geq f_{1}\) and \(f\leq f_{2}\), or according to (78) \(f\geq f^{*}\).