## 4 Discussion and Comparison with Experiments
### LFC at very Low Average Laser Power  
It is also of interest to find such LFC modes that at extremely low average laser power and low pulse energy allow, nevertheless, obtaining a cutting speed which is although not very high, but still acceptable for practical applications.  
We search for such a mode in the low-frequency range PRR, in which, according to the ratio \(P=E_{\mathrm{f}}\), the average laser power in the material \(P\) is small due to a significantly lower repetition frequency \(f\) compared to the high-frequency range considered above. In this case, to obtain the maximum speed \(u\), we use according to (90) the maximum PRR in this range, i.e., as follows from (55), \(f=f_{2}\). As follows from Equation (91), a further increase in \(u\) is possible by reducing the step of following \(s_{1}\). To do this, according to (66), we use, as in the example above, a smaller filament radius \(r_{\mathrm{f}}\) (for example, as follows from (19), \(r_{\mathrm{f}}=0.244~{}\upmu\)m due to the use of a twice shorter wavelength \(\lambda=520\) nm). And we also use a smaller filament VED \(w\), for example, \(w=(0.9\)-\(1)2w_{dis}\), i.e., \(w=(115\)-\(128)~{}\mathrm{kJ/cm^{3}}\), instead of \(w=270~{}\mathrm{kJ/cm^{3}}\) considered above in the interpretation of the experiment with the LFC of tempered glass at the wavelength \(1040\) nm.  
To calculate particular parameters of such LFC, we follow, as above, the calculation algorithm shown in Figure 7. As input, we use the parameters of tempered glass indicated in the example above. A stage-by-stage (1, 2, \(\ldots\), 5) calculation of unknown irradiation parameters according to the scheme gives the following results:  
1. We take VED bounds as \(w_{1}=400~{}\mathrm{kJ/cm^{3}}\) and \(w_{2}=396\)-\(480~{}\mathrm{kJ/cm^{3}}\) according to calculations given in Table 1 for a very close value of \(r_{\mathrm{f}}=0.25~{}\upmu\)m.
2. The VED \(w\) is taken as \(w=1.8w_{dis}=115~{}\mathrm{kJ/cm^{3}}\). In this case the condition (12) \(w\leq w_{1}\) is evidently fulfilled, corresponding to a significant elimination of filament energy losses for thermal emission and thus providing for a driving force and an efficiency for LFC. We put \(\eta\approx 0.1\) for the considered low-frequency range, and also take \(C=C_{2}\approx 1.0~{}\mathrm{J/(g\times K)}\). The modification threshold according to (64) is then \(w_{mod}^{T}=1.22~{}\mathrm{kJ/cm^{3}}\).
3. The pulse energy in the material for the same filament length \(H=45~{}\upmu\)m as in Section 4.2 is from (17) \(E_{1}=1.21~{}\upmu\)J, and the step of filaments following is from (66) \(s_{1}=0.9s_{1 max}=4.74r_{\mathrm{f}}=1.16~{}\upmu\)m.
4. LFC throughput from (93) is \(\Omega=43~{}\upmu\mathrm{m^{2}/\upmu J}\), which is noticeably lower than the value \(\Omega\approx 59~{}\upmu\mathrm{m^{2}/\upmu J}\) obtained above in the high-frequency mode. From Table 3, for the limit \(f_{2}\), we find \(f_{2}=0.547\chi/{s_{1}}^{2}\), which gives \(f_{2}=162\)-\(203\) kHz for the above thermal diffusivity \(\chi\) of this material.
5. We take \(f=162\) kHz, i.e., \(f\leq f_{2}\). We then finally find the average laser power in the material and the relative velocity of the beam and target: \(P=196\) mW and \(u=0.19~{}\mathrm{m/s}\).  
As we can see, a proper choice of irradiation parameters allows LFC with an acceptable (though not very high) speed of smooth separation, reaching about \(0.2~{}\mathrm{m/s}\), when using a fairly small average laser power of about \(200\) mW and a very small pulse energy in the material of the order of \(1~{}\upmu\)J. For the LFC of tempered glass, when, as we saw above, it is required to simultaneously create two arrays of filaments in the thickness of the material, these figures will double, respectively, to about \(400\) mW and \(2~{}\upmu\)J.