## 2 Modeling Electromagnetic Fields in Nonlinear Media
Finally, to model the metal, we consider a complex electrical permittivity \(\widetilde{e}(\omega)=\frac{a}{j\omega+b}\) where \(a\) and \(b\) are complex numbers that fit experimental data [15]. To model the dispersive media, we assume an electric permittivity, which is frequency-dependent as the calculations are devoted to ultrafast laser propagation, with bandwidth-limited pulses that exhibit wavelength dispersion. In ref. [2], the general vector auxiliary differential equation technique allows considering nonlinearities or dispersive characteristics of a medium. They are introduced into the Ampere-Maxwell equation by means of current densities. These currents correspond intrinsically to the temporal variation of the polarization vectors. Thereby, the following Equations (8)-(11) can be derived to correlate the currents \(\vec{J}_{Metal}(\vec{r},t)=\frac{\partial P_{Metal}(\vec{r},t)}{\partial t}\), \(\vec{J}_{Kerr}(\vec{r},t)=\frac{\partial\vec{P}_{Kerr}(\vec{r},t)}{\partial t }\), \(\vec{J}_{Raman}(\vec{r},t)=\frac{\partial\vec{P}_{Raman}(\vec{r},t)}{\partial t}\), \(\vec{J}_{fe^{-}}(\vec{r},t)=\frac{\partial\vec{P}_{fe^{-}}(\vec{r},t)}{\partial t}\) with the electric field:  
\[\vec{J}_{Metal}(\vec{r},t) = \frac{ae_{0}}{b}\frac{\vec{E}(\vec{r},t)}{\partial t}-\frac{1}{b} \frac{\vec{J}_{Metal}(\vec{r},t)}{\partial t} \tag{8}\] \[\vec{J}_{Kerr}(\vec{r},t) = \alpha e_{0}\chi_{0}^{(3)}\frac{\partial}{\partial t}\bigg{[}\vec {E}(\vec{r},t)\int_{0}^{t}\delta(t-\tau)|\vec{E}(\vec{r},\tau)|^{2}d\tau\bigg{]}\] (9) \[\vec{J}_{Raman}(\vec{r},t) = \Psi\frac{\partial}{\partial t}\bigg{[}\vec{E}(\vec{r},t)\int_{0 }^{t}e^{\frac{-(t-\tau)}{\tau_{2}}}\sin(\frac{t-\tau}{\tau_{1}})U(t-\tau)| \vec{E}(\vec{r},\tau)|^{2}d\tau\bigg{]}\] (10) \[\
abla\wedge\vec{E}(\vec{r},t) = -\mu\frac{\partial\vec{H}(\vec{r},t)}{\partial t}\]0We define a free-charge density of current as an Ohmic current, where the conductivity \(\sigma_{fe^{-}}(\vec{r},t)=q\mu_{fe^{-}}n_{fe^{-}}(\vec{r},t)\) is given by the following frequency dependent expression:  
\[\
abla\wedge\vec{E}(\vec{r},t) = -\mu\frac{\partial\vec{H}(\vec{r},t)}{\partial t}\]1  
where \(q\) is the electron charge, \(\mu_{fe^{-}}\) the free carriers of charge mobility, \(n_{fe^{-}}(\vec{r},t)\) the free-charge density and \(n_{ph-e^{-}}(\vec{r},t)\) the photo-ionized electron density. This definition for the density of current Equation (11) is similar to Equation (9) in [10]. In Equation (13), we employ free-charge which is governed by Equation (3), or generation Equation (13) and recombination Equation (14) rates. In [10], the authors explain that electron density \(n_{ph-e^{-}}(\vec{r},t)\) produced in the conduction band only describes the interaction of an electron with the laser electric field and accounts for the lattice periodicity, but does not account for possible collisions with phonons, ions, or other electrons. Due to these collisions, the coherence between the excited electron to the CB and its parent ion is destroyed and may introduce deviations from the above-predicted evolution of the electron density in the CB with respect to time. Indeed, without collisions, the produced electron density was shown to oscillate with time as a conduction electron may go back and forth to the VB through the action of the electromagnetic field [10; 16]. Obviously, in an oscillation, the energy stays constant [17]. Hence, the authors of [10] defined a free state as a state where the electrons can be heated by inverse bremsstrahlung (IB) and are decorrelated from the parent ion. The density of charge associated with this free state is called here density of free-charge \(n_{fe^{-}}(\vec{r},t)\), and its evolution along the time is modeled by Equation (3). Nevertheless, the \(\mu_{fe^{-}}\), the free carriers of charge mobility, then we employ experimental data, which allow the determination of this parameter. The method is explained in detail in Section 5.