## 1 Introduction  
Cutting up glass and other optically transparent materials in a desired shape and size is widely used in building construction and architecture, medicine, the automotive industry, mobile phone and tablet manufacture, electronics and many other industries. The development of new high-throughput, energy-efficient and environmentally friendly cutting technologies is important for large industrial-scale production. In comparison with traditional mechanical cutting laser-assisted cutting is promising owing to the prospects of obtaining a higher quality of the surface finish, the possibility of achieving tens or hundreds of times higher speed and environmental friendliness of the overall process. The use of lasers makes it possible to easily cut materials such as tempered and non-tempered glass, hard and superhard transparent materials, for which traditional mechanical methods are inefficient, requiring too much time and labour. Various methods of laser cutting of transparent materials are known:  
- The exposure to continuous-wave (cw) lasers, usually CO\({}_{2}\), Nd: YAG, or CO lasers (see, e.g., review [1] and references cited therein, also papers [2; 3]), where the laser energy is absorbed on the surface or in the volume of the material. This induces thermal stresses that cause the initiation and propagation of material cleaving both into the material depth and also along the laser path on the surface. The separation of parts occurs without applying any auxiliary mechanical stress in the irradiated region, and walls of the cut are smooth and practically do not contain any defects and chips;  
- Another method is exposure to ultrashort pulses, usually of femtosecond (fs) or picosecond (ps) pulse-periodic lasers when scanning the focused laser beam creates one or several arrays of thin extended filamentary plasma objects (shortly called below as filaments) inside transparent or semi-transparent materials (see the schematic in Figure 1). After such an exposure the material is split along the surface of the arrangement of these arrays when an external mechanical tensile stress is applied there (see, e.g., [4; 5; 6]). These elongated filaments can be created by the well-known nonlinear optical phenomenon of self-focusing of a high-intensity laser beam (see, for example, [7; 8], as well as reviews [9; 10] and references therein). A repeatable traversing of the beam through the material is used to create several arrays in the depth direction and thus to elongate filaments [4; 8] (see Figure 1). The method of \"multi-focusing\", where the initial beam is split employing a special optical scheme into two or more beams having each of those focused at a certain depth inside the material [4] also enables creating several arrays of filaments differing in the depth of their location (as in Figure 1), but with a single pass of the beam. Other methods, such as diffraction optics, Bessel non-diffracting beams created with axicons, or intentionally astigmatic lenses (see, e.g., [11; 12; 13; 14; 15; 16; 17; 18]), as well as the filament creation by a burst of pulses rather than by a single pulse [17; 19], have also opened up a way where the total length of filaments, continuous or discontinuous, may be increased even by a single pass of the beam, thus allowing one to split materials of a greater thickness (up to few millimetres). The described methods of laser-induced filamentation cutting (LFC) of transparent materials can be classified as \"green technologies\", because such methods use cleaving and hence give a \"zero-width\" cut. As a result, there is practically no release of the removed material in the form of polluting debris, vapours or droplets. Besides, LFC, if it would be capable of providing high-energy efficiency, a high throughput, and/or high quality of cut sidewalls, could also be important from an economical point of view. As it is well known, just the world market for mobile phones, tablets and wearable electronics  
Figure 1: Irradiation scheme explaining the main parameters of the model.