## 4 Performance and Framework  
Some approximations made here are not always applicable and, in this section, we will establish the validity framework of the proposed algorithms. In particular, we should consider the two approximations \(\int_{0}^{\left(s+\frac{1}{2}\right)\Delta t}\delta(t-\tau)|\vec{E}(\vec{\tau}, \tau)|d\tau\simeq\int_{0}^{\left(s-\frac{1}{2}\right)\Delta t}\delta(t-\tau)| \vec{E}(\vec{\tau},\tau)|d\tau\) and \(|\vec{Q}_{a}^{s+\frac{1}{2}}|=|\vec{Q}_{a}^{s-\frac{1}{2}}|+\overline{\vec{ \xi}}((s-nt_{0}(k))\Delta t)|\vec{E}^{s-\frac{1}{2}}|^{2}\). Both expression are derived in [9] under the Born-Oppenheimer approximation [4] where the Kerr non-resonant virtual electronic transitions are in the order of 1 fs, so it can be considered an instantaneous event, which is modeled by the Dirac delta function. However, while the time stepping keeps beneath 1 fs, we could model the medium reaction by using a time narrow Gaussian function with a standard deviation of 1 fs. In our simulations, \(\Delta_{t}\sim 0.1\) fs, so we are adding in a fraction of \(\frac{1}{10}\) the standard deviation. In this particular situation, the approximation is assumed to be valid. We employ the full explicit form \(|\vec{Q}_{a}^{s+\frac{1}{2}}|=|\vec{Q}_{a}^{s-\frac{1}{2}}|+\overline{\vec{\xi} }((s-nt(k))\Delta_{t})|\vec{E}^{s-\frac{1}{2}}|^{2}\) in the place of right implicit form \(|\vec{Q}_{a}^{s+\frac{1}{2}}|=|\vec{Q}_{a}^{s-\frac{1}{2}}|+\overline{\vec{\xi }}((s-nt(k))\Delta_{t})|\frac{E^{\frac{s+\frac{1}{2}}}+E^{\frac{s-\frac{1}{2}} }}{2}|^{2}\) based on the same idea than before, \(\Delta_{t}\ll\tau_{1}\) and \(\Delta_{t}\ll\tau_{2}\) so \(e^{\frac{-(t+\Delta_{t})}{\tau_{1}}\sin\left(\frac{t+\Delta_{t}}{\tau_{2}} \right)}\simeq e^{\frac{-(t)}{\tau_{1}}\sin\left(\frac{t}{\tau_{2}}\right)}\). This expression is valid for all instant \(\forall t\). Hence, we fix the limit for this full explicit approach in the relation \(\Delta_{t}<1\) fs. The advantage of a fully explicit approach concerns the simulation time. Due to the updating connection in an implicit scheme, most of the algorithms should be re-updating to advance a time step. In classical FDTD, it is usually fixing the limits of the index in all nested loops. In a significant computational domain, for instance, 1.2 G cells, the updating demands 30 s. If the algorithm has to be repeated at least three times, the simulation should need around three times the the implicit scheme. Figure 3 shows the number of simulation status updates on the abscissa axis and on the ordinate axis the time needed in order to update these simulation states. Each step or update corresponds to executing the 14 algorithms shown by Figure 2. In addition to these algorithms that are necessary to perform a system status update, two numerical processes are employed. One exports simulation data or results (every 40 updates) and the other exports the complete simulation status (every 500 updates). The latter is done for two reasons. The first is to recover a state of a simulation in case of computer collapse (for example a failure in the computer electric power source). The second reason concerns the alteration of the computational domain (electromagnetic properties or/and size of the computational domain) without the need to restart the simulation from the first time stepping. In Appendix B the Table A1 lists the computational resources employed in the simulations. Under these conditions we reach a performance which can be summarized by the computational burdens and some performance parameters. Those relies on a computational domain size of 121.5 GBs in RAM, a maximum number of [PERSON]'s cell charged and computed of 1.2 Giga cells, a minimum number of [PERSON]'s cell computed of 13 Mega cells, a speed at minimum and maximum load of 6.5 Mcells/s and 38 Mcells/s respectability, a probe exportation time of 1.47 s and a computational domain backup time of 8.76 s. Therefore, Figure 3 depicts the time performance for a simulation carried out with dynamics nested loops.