## 3 Model
### Strategy for Obtaining Better Smoothness of Cut Walls
#### 3.2.1 Laser Pulse Energy
Thus the dominant part of filament energy is effectively absorbed in submicron vicinity of the filament, without escaping from the micron-size affected zone. \(\sigma\) = 5.67 \(\times\) 10\({}^{-12}\) W/(cm\({}^{2}\ \times\) K\({}^{4}\)) is Stefan constant. \(D\) is a dimensionless parameter, \(D\approx 2.5\div 3\). \(b\approx 2900\)\(\upmu\cdot\)K is a so-called Wien's displacement constant. \(\lambda_{ln}\) is a high-absorption spectrum boundary in the short-wavelength range, defined by a condition that for \(\lambda\leq\lambda_{ln}\) the material absorption coefficient \(\alpha(\lambda)\geq 10^{5}\ \text{cm}^{-1}\). Calculated from (15) values of \(T_{1}\) and \(T_{2}\), for \(\beta_{max}\) conditionally taken for estimations as \(\beta_{max}\) = 0.12, are equal to several tens of thousands of degrees. For estimations we put that \(K_{sw}\approx K_{tw}\). In combination with (14), it gives \(K_{tw}\approx K_{sw}\approx 0.5\). VED bounds \(w_{1}\) and \(w_{2}\) calculated from (13) for such \(K_{tw}\) and with the use of data on \(w_{dis}\) from [52] are given in Table 1. As seen, \(w_{1}\) and \(w_{2}\) are equal to few hundred kJ/cm\({}^{3}\) depending on the material. There is also a \"window of unfavourable VED\" \(w\), defined by an opposite to (12) inequality: \(w_{1}<w<w_{2}\). For this range of \(w\), the relative part of thermal emission losses \(\beta\) becomes very significant and even the dominant and can exceed the allowed limiting level \(\beta_{max}\) by many times (conditionally \(\beta_{max}\) is taken for estimations in Table 1 as \(\beta_{max}=0.12\)). According to (15), a decrease in the filament radius \(r_{f}=r_{1/e}\) in a range of practical interest from about 1.5 \(\upmu\)m to 0.25 \(\upmu\)m provides an increase of \(T_{1}\). It makes the window of \"unfavourable\" \(w\) to shrink and even to disappear for some materials, in particular, as Table 1 shows, for sapphire and Corning Gorilla Glass 5. If the filament is formed due to the nonlinear process of self-focusing, the laser beam propagates in its immediate vicinity, as is known [4; 21], in a \"non-Gaussian\" way--mainly along curvilinear trajectories in a so-called \"energy reservoir\" surrounding the filament. In other words, an effective waveguide results from the refractive index gradient induced by the laser heating and symmetrically distributed across the filament axis. The laser energy drains from such a waveguide into the filament laterally, from different sides, due to multiple reflections and absorptions of the filament for lateral rays coming from the waveguide. In this way, a quite long and homogeneous for most of its length filament is created. The waveguide exists in that region along the length where the filament itself exists--a region of space, both in the filament itself and around it--where concentrated heating of the material and related phenomena such as temperature gradient are most significant. In this lateral energy supply only a part (\(A_{L\cdot p}<1\)) of laser pulse energy inside the material \(E_{1}\) transforms into the plasma energy \(E_{p}\), that is:  
\[E_{p}=A_{L\cdot p}E_{1}. \tag{16}\]  
The remaining part, (\(1-A_{L\cdot p}\)), of \(E_{1}\) traverses the material through such waveguide without being converted to the filament plasma energy. The lateral energy supply to the filament also takes place when it is formed by Bessel beam, i.e., by laser focusing with a conical lens--axicon [11]. The efficient conversion of the incident laser pulse energy inside the material \(E_{1}\) into the plasma energy of the filament \(E_{p}\) corresponds to the maximisation of the filament absorptivity \(A_{L\cdot p}\) in (16). As mentioned in Section 2 above, the absorptivity \(A_{L\cdot p}\) is taken for the model based on the proposed semi-empirical approach, i.e., from the available experimental data, where due to its non-linear character \(A_{L\cdot p}\) demonstrates a dependence on the VED \(w\) (see, e.g., [46]).