## 3 Algorithms
\begin{table}
\begin{tabular}{l l} \hline Step 0: & Update 6D-Discrete Incident Electric Field Array \\ \hline Step 1: & Total field / Scattering field algorithm on \(E\) \\ \hline Step 2: & \(|\mathcal{G}_{a}^{s+\frac{1}{2}}|=|\mathcal{G}_{a}^{s-\frac{1}{2}}|+\overline{ \mathcal{F}}\left(s\Lambda_{t}-\frac{(k-k_{0})\Lambda_{t}}{|L_{E_{0}}}\right)|E ^{s-\frac{1}{2}}|^{2}\) \\ \hline Step 3: & \(E^{s+\frac{1}{2}}=E^{s-\frac{1}{2}}+\mathcal{C}_{B}^{s}\left(\
abla\wedge H^{ 2}-g_{C}\overline{f}_{Metal}^{s-\frac{1}{2}}\right)\) \\ \hline Step 4: & \(\overline{f}_{Metal}^{s+\frac{1}{2}}=g_{C}\overline{f}_{Metal}^{s-\frac{1}{2}}+ g_{b}\left(\overline{E}^{s+\frac{1}{2}}-\overline{E}^{s-\frac{1}{2}}\right)\) \\ \hline Step 5: & \(|\overline{A}^{s+\frac{1}{2}}|=|\overline{A}^{s+\frac{1}{2}}|+|\overline{E}^{s +\frac{1}{2}}|\Lambda_{t}\) \\ \hline Step 6: & \(\overline{1}_{ct}^{\frac{s+1}{2}}=\overline{T}_{ct}^{\frac{s+1}{2}}+\frac{ \Lambda_{t}\sqrt{E_{t}}|\overline{E}^{s+\frac{1}{2}}|e^{s\Lambda_{t}(s+\frac{ 1}{2})(t_{E}+g_{c})}}{(E_{t}+g_{c})\sqrt{2\pi n_{t}}\left(1-\frac{1}{2}|\Lambda ^{s+\frac{1}{2}}|\right)}\sqrt{g_{ct}}=0,...,W_{f}\) \\ \hline Step 7: & \(n_{pt-c}^{s+\frac{1}{2}}=N_{0}\sum\limits_{\delta=0}^{W_{f}}\left|\overline{T}_ {ct}^{\frac{s+1}{2}}\right|^{2}\) \\ \hline Step 8: & \(n_{fe}^{s+1}=\frac{2-\Lambda_{t}\left(f_{e}^{s+\frac{1}{2}}-f_{e}^{s+\frac{1} {2}}-f_{e}^{s+\frac{1}{2}}\right)}{2+\Delta_{t}\left(f_{e}^{s+\frac{1}{2}}-f_{ e}^{s+\frac{1}{2}}-f_{e}^{s+\frac{1}{2}}\right)}n_{fe}^{s}+\frac{2f_{e}^{s+ \frac{1}{2}}\Lambda_{t}}{2+\Delta_{t}\left(f_{e}^{s+\frac{1}{2}}-f_{e}^{s+\frac {1}{2}}-f_{e}^{s+\frac{1}{2}}\right)}n_{ph-e}^{s+\frac{1}{2}}\) \\ \hline Step 9: & CPML on Magnetic Field \\ \hline Step 10: & Update 6D-Discrete Incident Magnetic Field Array \\ \hline Step 11: & Total field / Scattering field algorithm on \(H\) \\ \hline Step 12: & \(\overline{H}^{s+1}=\overline{H}^{\frac{s}{2}}-\frac{\Lambda_{t}}{\overline{R}} \
abla\wedge\overline{E}^{s+\frac{1}{2}}\) \\ \hline Step 13: & CPML on Magnetic Field \\ \hline Step 14: & \(\overline{E}^{s-\frac{1}{2}}=\overline{E}^{s+\frac{1}{2}}\), \(\overline{f}_{Metal}^{s-\frac{1}{2}}=\overline{f}_{Metal}^{s+\frac{1}{2}}\) \\ \(n_{fe^{-}}^{s}=n_{fe^{+}}^{s+1};\overline{H}^{s}=H^{s+1};|\overline{G}_{a}^{s- \frac{1}{2}}|=|\overline{G}_{a}^{s+\frac{1}{2}}|\) \\ \hline \end{tabular}
\end{table} 
Table 1: The list of the new explicit algorithms added to the classical FDTD scheme in order to deal with the nonlinear effects. The current list of algorithms is depicted in the form of a flow chart and it is complemented by the global integration view shown in Figure 2. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Stability Conditions**} \\ \hline All cases & \(\Delta_{t}\leq\frac{n_{i}}{\varepsilon_{0}}\sqrt{\frac{1}{\Delta_{t}^{2}}+\frac{ 1}{\Delta_{y}^{2}}+\frac{1}{\Delta_{t}^{2}}}\) \\ \hline Metal & \(\mathcal{R}[a]\geq 0\) & \(\mathcal{R}[b]\geq 0\) \\ \hline Kerr & \(|\widetilde{E}_{0}|<\sqrt{\frac{\varepsilon_{0}}{D_{f}\chi_{0}^{(3)}}}Min \Big{[}\frac{1}{\sqrt{a}},\frac{1}{\sqrt{1-a}}\Big{]}\) \\ Raman & \\ \hline Photoionization & \(\sigma_{fe^{-}}\geq 0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of stability conditions. Note that \(Min[]\) selects the minimum value in a list. Figure 2: Global flow chart of the proposed GVADE-FDTD algorithm.