## 5 Conclusions  
In this paper, an analytical model of laser filamentation cutting (LFC) of transparent materials is proposed, which predicts the process parameters for achieving high cutting speeds (of several m/s), as well as a mode important for practical applications--the so-called \"smooth\" laser cutting, when the surface of the sidewalls of the cut has a better smoothness and practically does not contain roughness with periods of less than a few microns. As a result, laser-cut plates have an increased resistance to damage and cracking due to mechanical impacts during further use, which is important in applications, for example, as mobile phone displays, car windshields, etc. Such parameters quantified in the model are the radiation parameters--the step of filaments following in the material \(s_{1}\), the pulse repetition frequency \(f\) of the laser (in addition to the other two necessary parameters analytically calculated in our previous work--the pulse energy \(E_{1}\) in the material and the pulse duration \(\tau\)). As the model shows, \(E_{1}\), \(\tau\), \(s_{1}\) and \(f\) are a function of the radius \(r_{f}\) and the length \(H\) of the filament, the volumetric energy density \(w\) of the filament plasma, the bounds \(w_{1}\) and \(w_{2}\), which determine the ranges of \(w\) for an elimination of the energy losses of the filament to thermal radiation, as well as the threshold \(w_{mad}\) for the modification critical for obtaining separation of the material. According to the model, obtaining a cut with a better smoothness is determined by several necessary conditions, the implementation of each of them requires a certain choice of radiation and material parameters. These necessary conditions are:  
* Obtaining high energy efficiency of the separation process by eliminating the energy losses of the filament due to thermal radiation. To do this, the pulse energy \(E_{1}\) in the material according to the model is chosen in such a way that the plasma of the filament is pumped to the VED \(w\), which satisfies the condition \(w\leq w_{1}\), or \(w\geq w_{2}\). The limits of \(w_{1}\) and \(w_{2}\), for which usually \(w_{1}\) < \(w_{2}\), are determined in the model by the thermal and optical properties of the material and are numerically of the order of several hundred kJ/cm\({}^{3}\);
* Ensuring increased uniformity of the diameter of each emerging filament along its length. For this purpose, the pulse duration \(\tau\) is matched with the pulse energy \(E_{1}\) so that the power of a single laser pulse \(P_{1}\approx E_{1}/\tau\) is approximately equal to the critical power \(P_{c}\) for self-focusing, i.e., \(P_{1}\) = \(\Gamma P_{c}\), where \(\Gamma\) = 1-2;
* Creation in the volume of the material of a continuous modified layer from the filament to the filament with the value of the volumetric energy density released in it (in [J/cm\({}^{3}\)]) not lower than the minimum (or threshold) value \(w_{mad}\), necessary to obtain a separation. For this purpose, the step \(s_{1}\) of filaments following in the material is selected in such a way that the zones of modification critical for separation from neighbour filaments join or even overlap with each other; Formation of straight filaments of a regular shape in each of their arrays by minimising their influence on each other, i.e., minimising the heating and temperature gradient from the previous filament at the location and at the time of occurrence of the next, new filament. To do this, the PRR \(f\) of laser pulses should be selected in one of two ranges found in the model, conventionally called as high-frequency range,\(f\geq f_{1}\), and low-frequency range,\(f\leq f_{2}\), where the limits \(f_{1}\) and \(f_{2}\) differ by several tens of times with\(f_{1}>f_{2}\); Formation of the impact zone from the filament without destruction in the form of scattered cracking in the solid material as a result of laser irradiation.