## 4 Discussion and Comparison with Experiments
### LFC of Sapphire  
Cutting 300 \(\upmu\)m-thick sapphire by the laser with wavelength \(\lambda\) = 1.04 \(\upmu\)m has been performed experimentally [20]. As it can be read directly from the picture of the cut walls there with the scale shown, the step of filaments in the material is \(s_{1}\approx\) 4 \(\upmu\)m and the length of each filament is about \(H\approx\) 45 \(\upmu\)m. The level of the average power and the scan speed  
Figure 7: Schematic of the algorithm for calculating the main irradiation parameters \(E_{1}\), \(\tau\), \(s_{1}\) and \(f\) (highlighted in yellow), which are preferred for the LFC with a better smoothness of separation for the heat diffusion modification mechanism. For the alternative shock-wave mechanism, the smooth separation is not achieved, but this scheme, if formally to put in it \(\eta\) = 0 and \(f_{1}\) = \(f_{2}\), can be used in this case as well—to create a separation with filaments which are more straight and more uniform in diameter along the length. 5 consecutive stages are shown for calculation of unknowns \(E_{1}\), \(\tau\), \(s_{1}\) and \(f\). of the laser beam reported in [20] are respectively \(P<2\) W and \(u=30\) cm/s. The values of \(f\) and \(E_{1}\) are not reported in this paper. However, they can be evaluated based on the reported \(s_{1}\), \(P\) and \(u\). Indeed, the pulse repetition rate can be found from (90) as \(f=u\)/\(s_{1}\), which gives \(f=75\) kHz. Then according to the relation \(E_{1}=P/f\) for the indicated \(P\) and \(f\) we also find \(E_{1}<26.7\) uJ. To test the adequacy of the model, it is interesting to find out whether these LFC parameters \(E_{1}\), \(s_{1}\), \(f\), \(P\) and \(u\) can be explained based on the proposed model when using for calculations the model algorithm shown in Figure 7? For this \(w_{dis}\) for sapphire is taken as 78 kJ/cm\({}^{3}\)[57], also \(A_{L,P}=0.8\) and \(A_{P,H}=0.9\) as has been mentioned above. LFC of sapphire according to our model is not smooth, since the parameters of this crystalline material, as can be seen from Table 4, do not satisfy the necessary conditions (87) and (89) for smooth separation The predominant mechanism in this case is not a thermal mechanism, but rather a shock-wave mechanism of the impact zone formation around the filament. However, this is not favourable for obtaining smooth separation. Photos of the sidewalls of separation surfaces, which are given in the paper [20], confirm this conclusion about a non-smooth separation. For such a mechanism, we take \(\gamma_{sw}\) the same as \(\gamma_{tw}\), i.e., \(\gamma_{sw}\approx\gamma_{tw}=1/\mathrm{e}=0.37\). Also, we put that in the relation (14) \(K_{tw}\approx K_{sw}\), and hence, \(K_{sw}\approx 0.5\). We also use \(P_{c}\) (\(\lambda=1.04\) um) = 2.8 MW [60] and \(\Gamma=1\)-2. The filament radius is not reported in the considered experiment [20], but its calculation with Equation (19) at \(M^{2}=1.25\) for the given wavelength \(\lambda=1.04\) um, NA = 0.6 and \(q\approx 1\) gives an estimation \(r_{f}\approx 0.49\) um. According to the algorithm diagram in Figure 7, we use these parameters as input data. Further, on the stage 1, VED bounds are chosen as \(w_{1}=860\) kJ/cm\({}^{3}\) and \(w_{2}=1020\)-1230 kJ/cm\({}^{3}\) according to calculations given in Table 1 for a very close value of \(r_{f}=0.5\) um. The choice of the frequency range, low-frequency or high-frequency one, for calculations within the framework of the indicated dominant shock-wave modification mechanism, is not essential. In stage 2, the VED \(w\) is taken according to the condition (12) for the elimination of thermal emission losses as \(w<w_{1}=860\) kJ/cm\({}^{3}\), namely as \(w=460\) kJ/cm\({}^{3}\). For the shock-wave mechanism of modification, in this case, we put \(\eta=0\) and according to (46) use for this material the mechanical strength indicated in Table 2 \(\sigma_{c}=3.0\) GPa as \(w_{mod}^{sw}\), i.e., \(w_{mod}^{sw}=3.0\) kJ/cm\({}^{3}\).