## 2 Formulation
 \tag{5}\]  
where  
\[\tilde{Q}_{pxx}=\frac{\varepsilon_{0}\varepsilon_{p,xx}}{j\omega-a_{p,xx}} \tilde{E}_{x},\ \tilde{Q}_{pxx}^{\prime}=\frac{\varepsilon_{0}\varepsilon_{p,xx}^{ \ast}}{j\omega-a_{p,xx}^{\ast}}\tilde{E}_{x} \tag{6}\]  
are auxiliary variables. The corresponding variables \(\tilde{Q}_{pxy}\), \(\tilde{Q}_{pxy}^{\prime}\), \(\tilde{Q}_{pxz}\) and \(\tilde{Q}_{pxz}^{\prime}\) are obtained in a similar manner. Next, we convert Equation (6) into the time domain:  
\[\frac{dQ_{pxx}}{dt}-a_{pxx}Q_{pxx}=c_{pxx}\varepsilon_{0}E_{x}, \tag{7a}\] \[\frac{dQ_{pxx}^{\prime}}{dt}-a_{pxx}^{\ast}Q_{pxx}^{\prime}=c_{pxx}^{\ast} \varepsilon_{0}E_{x}. \tag{7b}\]  
Since the electric field \(E_{x}\) is a real quantity in the time domain, we conclude from Equations (7a) and (7b) that \(Q_{pxx}^{\prime}=Q_{pxx}^{\ast}\), and as a result, only the variable \(Q_{pxx}\) is needed for the simulation. Similar conclusions can be obtained for the other variables. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Expression** & **Parameters of the CCPR Model: \(\sigma\), \(\langle\varepsilon_{p},a_{p}\rangle\)** \\ \hline [PERSON] & \(\frac{\omega_{p}^{2}}{j\omega(j\omega+v)}\) & \(\varepsilon_{0}\omega_{p}^{2}/\
u\), \((-0.5\omega_{p}^{2}/\
u\), \(-v)\) \\ \hline [PERSON] & \(\frac{\Delta\varepsilon}{1+j\omega\tau}\) & \(0\), \((0.5\Delta\varepsilon/\tau\), \(-1/\tau)\) \\ [PERSON] & \(\frac{\Delta\varepsilon\omega_{p}^{2}}{\omega_{p}^{2}+j\omega\delta+(j\omega)^ {2}}\) & \(0\), \(\left(\frac{j\Delta\varepsilon\omega_{p}^{2}}{2\sqrt{\omega_{p}^{2}-\delta^{2}} },-\delta-j\sqrt{\omega_{p}^{2}-\delta^{2}}\right)\) \\ \hline Critical Points & \(A_{p}\Omega_{p}\left(\frac{e^{j\phi_{p}}}{\Omega_{p}+\omega-j\Gamma_{p}}\right.\) & \(0\), \((A_{p}\Omega_{p}(\sin\phi_{p}-j\cos\phi_{p}),-\Gamma_{p}-\Omega_{p}j)\) \\  & \(\left.+\frac{e^{-j\phi_{p}}}{\Omega_{p}-\omega+j\Gamma_{p}}\right)\) \\ \hline Sellmeier & \(\frac{B\lambda^{2}}{\lambda^{2}-C},\lambda=\frac{\omega}{2\pi c}\) & \(0\), \(\left(-\frac{1}{2}jB\frac{2\pi c}{\sqrt{c}},j\frac{2\pi c}{\sqrt{c}}\right)\) \\ \hline Modified Lorentz & \(\frac{a_{0}+ja_{1}\omega}{b_{0}+j\omega b_{1}+(j\omega)^{2}}\) & \(0\), \(\left(c_{p1}=\frac{a_{0}+a_{p1}}{a_{p1}-a_{p2}},a_{p1}=0.5\Bigl{(}-b_{1}+\sqrt{ b_{1}^{2}-4b_{0}}\Bigr{)}\), \\  & \(c_{p2}=a_{1}-c_{p1},a_{p2}=0.5\Bigl{(}-b_{1}+\sqrt{b_{1}^{2}-4b_{0}}\Bigr{)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: CCPR parameters for the description of various known dispersion models. Subsequently, we convert Equation (5) into the time domain:  
\[ \begin{split}&(\
abla\times\mathbf{H})_{x}=\\ &\varepsilon_{0}\varepsilon_{\infty,xx}\frac{dE_{x}}{dt}+\sigma_{ xx}^{\varepsilon}E_{x}+\sum_{p=1}^{M_{xx}}\frac{d}{dt}\Big{(}Q_{pxx}+Q_{pxx}^{*} \Big{)}\\ &+\varepsilon_{0}\varepsilon_{\infty,yy}\frac{dE_{y}}{dt}+\sigma_ {xy}^{\varepsilon}E_{y}+\sum_{p=1}^{M_{xy}}\frac{d}{dt}\Big{(}Q_{pxy}+Q_{pxy}^ {*}\Big{)}\\ &+\varepsilon_{0}\varepsilon_{\infty,xz}\frac{dE_{z}}{dt}+\sigma_ {xz}^{\varepsilon}E_{z}+\sum_{p=1}^{M_{xz}}\frac{d}{dt}\Big{(}Q_{pxz}+Q_{pxz}^ {*}\Big{)}\end{split}  \tag{8}\]  
and using the formula \(z+z^{*}=2\Re(z)\), with \(\Re(\cdot)\) denoting the real part, we obtain  
\[ \begin{split}&(\
abla\times\mathbf{H})_{x}=\\ &\varepsilon_{0}\varepsilon_{\infty,xz}\frac{dE_{x}}{dt}+\sigma_ {xx}^{\varepsilon}E_{x}+2\sum_{p=1}^{M_{xx}}\Re\Big{(}\frac{dQ_{pxx}}{dt} \Big{)}\\ &+\varepsilon_{0}\varepsilon_{\infty,xy}\frac{dE_{y}}{dt}+\sigma_ {xy}^{\varepsilon}E_{y}+2\sum_{p=1}^{M_{xy}}\Re\Big{(}\frac{dQ_{pxy}}{dt} \Big{)}\\ &+\varepsilon_{0}\varepsilon_{\infty,xz}\frac{dE_{z}}{dt}+\sigma_ {xz}^{\varepsilon}E_{z}+2\sum_{p=1}^{M_{xz}}\Re\Big{(}\frac{dQ_{pxz}}{dt} \Big{)}\end{split}