## 2 Modeling Electromagnetic Fields in Nonlinear Media
\[\
abla\wedge\vec{E}(\vec{r},t) = -\mu\frac{\partial\vec{H}(\vec{r},t)}{\partial t}\]2 \[\
abla\wedge\vec{E}(\vec{r},t) = -\mu\frac{\partial\vec{H}(\vec{r},t)}{\partial t}\]3  
In the carrier generation rate, the term associated with the photo-ionized electron density \(n_{ph-e^{-}}(\vec{r},t)\) produced in the conduction band is a function of \(|\vec{E}(\vec{r},t)|\). From Equation (5) in [10] we know that \(\frac{\partial}{\partial t}\bigg{[}n_{ph-e^{-}}(\vec{r},t)-N_{0}\sum\limits_ {c}|T_{cv}(t)|^{2}\bigg{]}=0\) and hence \(n_{ph-e^{-}}(\vec{r},t)-N_{0}\sum\limits_{c}|T_{cv}(t)|^{2}=cte\). If this constant \(cte=0\), then \(n_{ph-e^{-}}(\vec{r},t)=0\) when \(\vec{E}(\vec{r},t)=\bar{0}\). Combining this result with Equations (3) and (4) in [10], we calculate \(n_{ph-e^{-}}(\vec{r},t)\) by Equation (15). \[\
abla\wedge\vec{E}(\vec{r},t) = -\mu\frac{\partial\vec{H}(\vec{r},t)}{\partial t}\]4  
Finally the functions related to the generation by ionization \(f_{d}(\vec{r},t)\), impact \(f_{\eta}(\vec{r},t)\) and avalanche \(f_{a}(\vec{r},t)\), as well as the function related to the recombination of the free-charge \(f_{r}(\vec{r},t)\), are given by the Equations (16)-(19), respectively. \[f_{d}(\vec{r},t) = \begin{cases}\frac{1}{\tau_{d}}&\text{if }|\vec{E}(\vec{r},t)|\geq \frac{E_{max}}{20}\\ \frac{20|\vec{E}(\vec{r},t)|}{\tau_{d}E_{max}}&\text{if }|\vec{E}(\vec{r},t)|< \frac{E_{max}}{20}\end{cases}  \tag{16}\] \[f_{\eta}(\vec{r},t) = \begin{cases}\eta&\text{if }|\vec{E}(\vec{r},t)|\geq\frac{E_{max}}{20}\\ \frac{20|\vec{E}(\vec{r},t)|}{E_{max}}&\text{if }|\vec{E}(\vec{r},t)|< \frac{E_{max}}{20}\end{cases}  \tag{17}\]\[f_{a}(\vec{r},t) = \begin{cases}\frac{\Theta|\vec{E}(\vec{r},t)|^{2}}{\zeta}&\text{ Harmonic field}\\ \Theta\Big{(}\vec{E}(\vec{r},t)\wedge\vec{H}(\vec{r},t)\Big{)}\cdot\hat{z}&\text{ No harmonic field}\end{cases}  \tag{18}\] \[f_{r}(\vec{r},t) = \begin{cases}\frac{1}{\tau_{r}}&\text{if }|\vec{E}(\vec{r},t)|\geq\frac{E_{max}}{20}\\ \frac{20|\vec{E}(\vec{r},t)|}{\tau_{Emax}}&\text{if }|\vec{E}(\vec{r},t)|<\frac{E_{max}}{20} \end{cases}  \tag{19}\]  
In Equations (16)-(19), we find that the factor of maximum electric field reduction, which is suggested twenty times by [10], is taken into consideration. Nevertheless, when the photoionization process is considered in a truncated region where the magnitude of \(|\vec{E}(\vec{r},t)|\) is from the beginning higher than the threshold \(\frac{E_{max}}{20}\) reference value, these logical conditions can be suppressed in the nested loops, and this increases the algorithm performance. In other words, this simplification is possible when we assume the description with nonlinear effects in a particular region of the computational domain. So, the electromagnetic field that arrives at that region has a magnitude higher than the threshold \(\frac{E_{max}}{20}\). In Equation (18), we assume that \(\frac{|\vec{E}(\vec{r},t)|^{2}}{\zeta}\) is the fluence with \(\zeta\) being the medium impedance. However, this is a particular case only valid when the electromagnetic field is a harmonic function in time. In general, and assuming a propagation along the \(z\)-axis, the power per surface is given by the Poynting vector.