## 2 Methods and Materials  
The model considers elongated filaments either created inside the material by a Gaussian beam at its self-focusing point or by an extended in length non-diffracting Bessel beam free of self-focusing, as shown in Figures 1-3. In either case, these objects are designated hereinafter as \"filaments\", since their effect on the sample separation is determined, as will be shown below, by their characteristic radius and the volumetric energy density (VED) they acquire, regardless of whether they are due to a self-focusing or Bessel beam. The laser action on the sample surface mentioned in [32; 34; 35] is not addressed in the paper. Each filament is created by a single pulse. Figure 3: Schematic of producing the extended-length focusing inside the transparent material employing axicons. Figure 2: The follow-up to the Figure 1, where other parameters of the model are explained. An individual filament for the sake of simplicity is considered as a cylinder of initial radius \(r_{f}\) (at the 1/e-level) and a length \(H>>r_{f}\), with a volumetric energy density \(w\) uniform along the length, \(E_{1}\) is the laser pulse energy inside the material, \(E_{p}\) is the energy of the filament plasma, \(Q\) is the heat released in the surrounding filament material as a result of relaxation of the filament plasma. The main parameters considered in the modelling of a standard scheme of LFC (see, e.g., [4; 8; 17; 39]) are presented in Figures 1 and 2. In this scenario, the focused beam of a pulse-periodic ps-, or sub-ps laser is scanned over the material and creates one or several arrays of plasma filaments, with each of these arrays being different from others by its depth inside the sample. To describe properly the laser heating of the filament a parameter such as a VED of the filament, \(w\), is used instead of laser pulse fluence or intensity. This feature \(w\), measured in J/cm\({}^{3}\), is directly related to the filament temperature. Since many of the processes involved (glass softening, cracking, dissociation, ionisation et al.) have a temperature-activated nature, such an approach enables one to estimate most simply the temperature of the filament in terms of the VED and to tackle the mechanisms and the nature of laser-induced phenomena behind the thus produced laser cleavage. Also, a rectangular temporal shape of the pulse is assumed as a model simplification. Typical experimental photos of sidewalls of the cuts (see, e.g., [4; 8; 14; 17; 20]) suggest that it is a good approximation for modelling to consider an individual filament as a cylinder with a height \(H\) (see Figures 1 and 2). Another simplifying approximation is that the filament VED is considered as uniform along the cylinder length, but with initially a Gaussian distribution over the radius, with \(r_{f}\) being the filament radius at the 1/e-level of VED and \(r_{f}<<H\). Then a simple integration over the space gives a simple and transparent expression for the filament plasma energy:  
\[E_{p}=\Big{(}\pi r_{f}^{2}H\Big{)}w. \tag{1}\]  
The parameter \(\pi r_{f}^{2}H\) here corresponds to an effective volume of the mentioned cylindrical filament, and we is a maximal VED value on filament axis. A similar expression \(E_{p}=\pi r_{\textit{eff}}{}^{2}Hw_{max}\) could also be written for a non-Gaussian transverse distribution, if it has an effective concentration around the filament axis rather than being widely distributed across space, in particular, for filaments produced with Bessel beams (\(w_{max}\) here is a maximal value of VED in this non-Gaussian distribution, and \(r_{\textit{eff}}\) is its effective radius, which is not so obvious as the mentioned (1/e) Gaussian radius. It can be calculated as \(r_{\textit{eff}}^{2}=E_{p}/\left(\pi Hw_{max}\right)=\left[\int_{0}^{\infty}w (r)2\pi rdr\right]/\left(\pi w_{max}\right)\). However, it requires information on the \(w(r)\) dependence, which is not always known.