## 1 Introduction
Each filament is created by a single laser pulse of a focused beam from a pulse-periodic fs, or ps laser, \(\lambda\) is its wavelength, \(\tau\) is pulse duration, \(f\) is pulse repetition rate (PRR), \(u\) is the velocity of beam scanning across the material. The focusing system and the laser beam are described by the standard parameters NA and \(M^{2}\). \(L\) is the thickness of the irradiated sample, \(s_{1}\) is the distance between neighbouring filaments inside the material, \(H\) is the length of the single filament, and at the same time, the width of one array of filaments, \(d\) is the vertical distance between adjacent arrays of filaments. The beam scans the material in the \(xy\)-plane, the \(z\)-axis is directed into the material depth and coincides with the direction of the filament axis. comprises the production of more than 1 billion displays and protective coatings annually for them from thin plates (about 0.1 to 0.70 mm thick) of chemically-tempered, non-tempered glass and sapphire. Besides this, there are also many other applications in the automotive industry, instrumentation, construction, creation of art objects, and other industries, where transparent plates with a relatively great thickness of about 2 to 10 mm or more are required, and the materials used there can include not only glasses, but also other various transparent or semitransparent materials, including superhard and refractory ones, difficult to handle by traditional mechanical processing. Examples are sapphire, diamond, transparent conductors and semiconductors, SiC, GaN, ZnSe, ZnS, CaF\({}_{2}\), KCl, KBr, etc. Despite the aforementioned advantages, a better and deeper level of understanding of the LFC process is still required to loosen constraints acting individually or in various combinations. These are: (i) a low separation rate of only a few mm/s or cm/s in some cases [4; 13]; (ii) a lack of smoothness of resulting sidewalls of the cut [4; 17; 20]; and (iii) the need to use a high average power laser beam for separation--about a few hundreds of Watts-- in particular for materials with thicknesses greater than several millimetres. The solution to avoid these shortcomings might be found based on an analytical model of the complex phenomenon of LFC with a pulse-periodic beam scanning over the sample with a high pulse repetition rate of up to several megahertz. As evidenced by a number of reviews [1; 9; 10; 21; 22; 23], the numerous experimental and theoretical papers quoted therein, as well from the number of other references [24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35], a great number of authors in the filamentation physics field have for several decades considered various aspects of individual filament formation, but have not modelled the efficiency, speed and surface quality of cut walls for material separation processes induced with a large array of filaments. This filamentation physics area should be distinguished from the studied here specific field of \"LFC physics\", which demands consideration of much higher levels of radiation pumping of filaments used for obtaining a material separation, with correspondingly much higher filament temperatures (as will be shown below, in some cases of the order of \(10^{5}\) K). Besides, although the effects of the repetition rate on the heating of a transparent material by pulsed-periodic radiation of an ultrafast laser have been considered earlier, they were studied not for extended filaments, but almost exclusively for point-like focusing, used for laser direct writing waveguides in a transparent material [36; 37]. The effect of a stationary and also a moving pulsed-periodic beam has been studied, but at a beam displacement step much smaller than the laser spot diameter [38].