## 3 Model
### Strategy for Obtaining Better Smoothness of Cut Walls
#### 3.2.1 Laser Pulse Energy
That is if filaments follow each other having spacing several times (about 3 to 6) greater than their diameter when usually high \(w\) are required for LFC (see below), \(A_{L\cdot p}\) is taken close to a maximal saturated experimental value about 0.80 \(\pm\) 0.05 [46]. But in other cases, when a pulse produces about an order of magnitude lower level of VED, a smaller value, \(A_{L\cdot p}\) = 0.3-0.4 [46] is taken as a better value for the model. This happens if LFC uses filaments that overlap partially, or for a single pulse from a burst of several very close in time pulses when the filament is produced with the burst of pulses. From Equations (1) and (16) a simple relationship follows for the laser pulse energy inside the material as a function of \(w\), as well as the length \(H\) and the radius \(r_{f}\) of the filament:  
\[E_{1}=\pi r_{f}^{2}Hw/A_{L-p}, \tag{17}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multicolumn{3}{c}{\(w_{1}\) **(k/cm3)**} \\ \cline{2-5}  & \(r_{f}\) = **0.25 \(\upmu\)m** & \(r_{f}\) = 0.5 \(\upmu\)m & \(r_{f}\) = 1.5 \(\upmu\)m & **\(w_{2}\) (k/cm3)** \\ \hline Corning Gorilla Glass 5 & 400 & 336 & 248 & 396–480 \\ Eagle XG@Display glass & 396 & 322 & 244 & 460–540 \\ Quartz glass & 356 & 296 & 226 & 640–750 \\ Sapphire & 1070 & 860 & 620 & 1020–1230 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: VED bounds \(w_{1}\) and \(w_{2}\) calculated from (13,15) for \(\beta_{max}=0.12\) and several materials at different filament radii \(r_{f}\) for \(K_{tw}\approx 0.5\). where \(w\) satisfies to the condition (12). On the other hand, if \(r_{f}\), \(H\) and \(E_{1}\) are known from experiment, the VED can be estimated from (17) as:  
\[w=A_{L-P}E_{1}/\Big{(}\pi r_{f}^{2}H\Big{)}. \tag{18}\]  
For the correct comparison of model estimations with experimental results, it is worth noticing that Gaussian radius at the \((1/\text{e}^{2})\)-level for VED profile, \(r_{f}\)\((1/\text{e}^{2})\), relates to the \(r_{f}\)\((1/\text{e})\) radius of our model as \(r_{f}\)\((1/\text{e})=(1/\sqrt{2})r_{f}\)\((1/\text{e}^{2})\). We also assume that for self-focusing \(r_{f}\)\((1/\text{e})\) can be given in terms of the waist radius \(r_{0}\) for the classical focusing of the Gaussian beam in a transparent material, i.e., \(r_{f}=qr_{0}\). Here a coefficient \(q\) can be attributed to a plasma defocusing and beam propagation effects and in general, may depend both on the VED and pulse duration, \(q=q(w,\tau)\). But still within indicated below a range of interest of \(w\) one can put \(q\approx 1\). As for the parameter \(r_{0}\)\((1/\text{e})\), it is given by the ratio \(r_{0}=(\lambda M^{2})/(\sqrt{2}\tau\text{NA})\)[32]. Thus, one can put for the filament formed by self-focusing:  
\[r_{f}=q(w,\tau)\frac{\lambda M^{2}}{\sqrt{2}\tau\text{NA}}\;. \tag{19}\]  
An important consequence of this ratio for further modelling is that, as \(q\approx 1\), the filament radius \(r_{f}\), like \(r_{0}\), decreases with shortening the laser wavelength \(\lambda\) and with an increase of the numerical aperture of beam focusing NA. To confirm that \(q\) is very close to 1, one can estimate the diffusion length of electrons \(L_{e}\approx 2(D_{e}\tau)^{1/2}\) excited by an intense laser for the pulse duration \(\tau\). Here \(D_{e}\) is the electron diffusion coefficient, which for crystalline materials has a characteristic value of the order of 1 cm\({}^{2}\)/s, and for amorphous materials--less than this value by about an order of magnitude or more. For a considered below characteristic value for LFC \(\tau\) = 0.2-0.4 ps and \(D_{e}=1\) cm\({}^{2}\)/s, one get \(L_{e}\) = (0.9-1.3)\(10^{-2}\)\(\upmu\)m. Then the inequality \(L_{e}<<r_{0}\) is fulfilled even for smallest filament radii considered in this paper, around \(r_{0}\approx 0.24\)\(\upmu\)m (which is given by the above formula for \(\lambda\) = 0.52 \(\upmu\)m) and for the largest value \(D_{e}=1\) cm\({}^{2}\)/s.