## 4 Performance and Framework
\begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{5}{c}{**Parameters**} \\ \hline
**Name** & **Symbol** & **Value** & **Unit** & **Ref.** \\ \hline Gap energy & \(E_{g}\) & 9 & eV & [28] \\ \hline Third-order elec-tric susceptibility & \(\chi_{0}^{(3)}\) & \(10^{-22}\) & \(\frac{\text{m}^{2}}{\text{V}^{2}}\) & [28] \\ \hline Raman sinusoidal time & \(\tau_{1}\) & 12.2 & fs & [29] \\ \hline Raman response fraction & \(1-\alpha\) & 0.18 & - & [29] \\ \hline Electron recombi-nation time & \(\tau_{r}\) & 150 & fs & [10] \\ \hline Mobility & \(\mu_{fe^{-}}\) & \(1.13\,\times\,10^{-22}\) & \(\frac{\text{m}^{2}}{\text{V}^{2}}\) & [21] \\ \hline Density of valence electrons & \(N_{0}\) & \(10^{22}\) & cm\({}^{-3}\) & [10] \\ \hline Electron mass reduced particle & \(m^{*}_{\text{vc}}\) & 0.5 & m\({}_{0}\) & [10] \\ \hline Work function & \(W_{f}\) & 5 & eV & [28] \\ \hline Avalanche ioniza-tion coefficient & \(\Theta\) & \(10^{-3}\) & \(\frac{\text{m}^{2}}{\text{W}^{2}}\) & [28] \\ \hline Raman decay time & \(\tau_{2}\) & 32 & fs & [29] \\ \hline Kerr response fraction & \(\alpha\) & 0.82 & - & [29] \\ \hline Laser characteristic time for electrons production & \(\tau_{d}\) & 3 & ps & [10] \\ \hline Impact ionization efficiency & \(\eta\) & 0.01 & - & [28] \\ \hline Fitting parameter for the spatial expansion of the valence wave function & \(\Omega\) & 0.8 & - & [10] \\ \hline Metal & a & \(19.2+j2.7\) & \(\text{PHz}\) & \\ parameters & b & \(1.3-j3.9\) & \(\text{PHz}\) & [15] \\ \hline Axicon refractive index & \(n_{a}\) & 1.45 & - & [30] \\ \hline \end{tabular}
\end{table} 
Table 3: Media main parameters. * Low power regime: \(E_{0}=10^{4}\frac{\mathrm{V}}{\mathrm{m}}\) and \(I=2.46\ \times\ 10^{14}\frac{\mathrm{W}}{\mathrm{m}^{2}}\). * Critical power regime: \(E_{0}=3.5\ \times\ 10^{5}\frac{\mathrm{V}}{\mathrm{m}}\) and \(I=8.61\ \times\ 10^{16}\frac{\mathrm{W}}{\mathrm{m}^{2}}\). * High power regime: \(E_{0}=10^{6}\frac{\mathrm{V}}{\mathrm{m}}\) and \(I=2.45\ \times\ 10^{18}\frac{\mathrm{W}}{\mathrm{m}^{2}}\). The power intensity \(I=Max(\widetilde{I}(\tau,t))\) is calculated as the maximum power that flows through a section of \(\sigma_{w}^{2}\), being \(\sigma_{w}\) the pulsed laser transverse waist, \(\sigma_{\ell}\) the pulsed length in the propagation direction and \(z_{p}\) the central pulse location at the calculation time. Hence, \(\widetilde{I}(\vec{r},t)=\iint\limits_{-\sigma_{w}}^{\sigma_{w}}dxdy\int \limits_{z_{p}-\sigma_{\ell}}^{z_{p}+\sigma_{\ell}}dz\hat{z}\cdot\frac{ \widetilde{S}(\vec{r},t)}{\sigma_{w}^{2}}\). To validate the algorithm, which includes nonlinear effects during light propagation, we have investigated the features of the propagation regime depending on the laser intensity. Figure 5a,b redraw the distribution evolution of the pulsed square electric field module for simulation performed for different laser intensity sources. Figure 5a plots the iso-slices that depicts \(|\vec{E}|^{2}\) inside the fused silica bulk for high laser power in the system. From top to bottom, the time evolution sequence reveals the defocusing process of the Bessel beam, which ends by forming some filamentation structures in the light distribution. At the time \(t=5720,\Delta_{t}=1.52\) ps, the pulsed square electric field module begins to penetrate the interface air-fused silica. All successive time plots correspond to a propagation inside the fused silica block. Figure 5b illustrates, for the same time moments, but a higher initial power source, the square electric field module during the propagation inside SiO\({}_{2}\) material. We complement the picture with a simulation carried out in the medium power regime. From the instant of time, \(t_{1}=1.67\) ps to the instant \(t_{18}=2.56\) ps, Figure 5c illustrates the antagonistic trade-off between the focusing and the defocusing process experienced by the Bessel beam along with its travel through the bulk fused silica target.