## 4 Performance and Framework
Figure 6b shows the history of the power density at the critical threshold separating medium and high power regimes, along with a few steps of time. In this figure we can identify the interface air-fused silica due to the rings which record the history of a stationary wave formed after the interaction. We can observe a strong focusing process in the entrance of the fused silica block followed by an alternation of refocusing and the defocusing events induced and governed by the photoionization and Kerr effect competition, and we know that these are not diffraction artifacts because in the air, at the same laser intensity, we do not observe them. At this stage, we have to give a clear definition of the pulsed power density and the power history. The magnitude of pulsed power density is defined as \(\
abla\cdot\widetilde{S}(\vec{r},t)\), where \(\widetilde{S}(\vec{r},t)=\vec{E}(\vec{r},t)\wedge\vec{H}(\vec{r},t)\) is the Poynting vector [31]. Figure 6a sketches the pulsed power density. From this magnitude \(\
abla\cdot\widetilde{S}(\vec{r},t)\), we can calculate the delivered pulsed electromagnetic density of energy \(\int_{0}^{t^{\prime}}\
abla\cdot\widetilde{S}(\vec{r},t)dt\). Besides this, it is interesting to consider the range of values in Figure 6a. We can identify a negative power density that requires clarification. As the Poynting vector represents the flux of power through a given surface, the divergence of this magnitude represents the power density that comes into/out a particular point in a given region. In particular, in FDTD calculations, we are describing the power flux through the computational nodes that discretized the computational domain. In the density power accounting, the reference system is placed inside the cell that encloses the nodes. In this way, a negative power density, in a particular instance of time, in a local region, means a local sink of power \(\
abla\cdot\widetilde{S}(\vec{r},t)<0\) in that region, and vice versa, we have a local source of power as \(\
abla\cdot\widetilde{S}(\vec{r},t)>0\). These oscillations in the density power magnitude indicate a redistribution that is able to support the pulsed density power and the pulsed density energy traveling. Figure 5: Propagation of the pulsed beam given by a time sequence of instants which has been plotted before reaching the theoretical critical power, with an energy per pulse of 2.8 \(\upmu\)J allowing to reach an electron density of \(10^{27}\) m\({}^{-3}\), the beam splits, enclosing the charge and defocusing the beam. (**a**) Several instants of the pulsed square electromagnetic field when the power is beyond the critical power (High power regime). (**b**) Pulse propagation sequences during the travel inside the fused silica bulk for low power regime. (**c**) Sequence of frames that sketches the pulse along during its travel inside the fused silica bulk from the instance \(t_{1}=6290\Delta_{t}\) to \(t_{18}=9620\Delta_{t}\) (where \(\Delta_{t}=2.65765\ \times\ 10^{-16}\) s and Courant number 0.69). The difference between two time-steps is \(200\Delta_{t}\) in the sequence. The power of this sequence is in the threshold between low and high power regimes (critical power). Turning to the matter at hand, we define the history of the power density by applying the absolute value to the divergence of the Poynting vector \(|\
abla\cdot\vec{S}(\vec{r},t)|\). The bi-dimensional longitudinal cut of the history of the power density depicted in Figure 6b permits one to identify the air-fused silica interface. There, in that interface, we can appreciate the reflected beam and the strong focusing in the entrance due to the Kerr effect. We assume the same refractive index for both the axicon lens and the fused silica block in which the wave Bessel beam impinges. Although we assume nonlinear media, in the simulation results presented up to this point in the manuscript, we have considered that the fused silica is isotropic and homogeneous.