## 3 Applications and Numerical Results
### Propagation in Magnetized Plasma  
We first study wave propagation in anisotropic magnetized plasma in which the direction of propagation is along the \(z\)-axis and the external DC magnetic field is \(\mathbf{H}=H_{\text{DC}}\mathbf{z}\) as in [22] (chap. 8). In this case, the nonzero elements of the permittivity tensor, assuming the \(e^{\text{i}\omega t}\) time convention, are given by  
\[\varepsilon_{xx}(\omega) =\varepsilon_{yy}(\omega)=1-\frac{\left(1-\frac{\left|\
u\right\rangle }{\omega}\frac{\omega_{p}^{2}}{\omega^{2}}}{\left(1-\frac{\left|\
u\right\rangle }{\omega}\right)^{2}-\frac{\omega_{p}^{2}}{\omega^{2}}} \tag{16a}\] \[\varepsilon_{xy}(\omega) =-\varepsilon_{yx}(\omega)=-j\frac{\frac{\omega_{b}}{\omega} \frac{\omega_{p}^{2}}{\omega^{2}}}{\left(1-\frac{\left|\
u\right\rangle}{ \omega}\right)^{2}-\frac{\omega_{b}^{2}}{\omega^{2}}}\] (16b) \[\varepsilon_{zz}(\omega) =1-\frac{\omega_{p}^{2}}{\omega^{2}\left(1-\frac{\left|\
u\right\rangle }{\omega}\right)} \tag{16c}\]  
where \(\omega_{p}\) is the plasma frequency, \(\
u\) is the electron collision rate and \(\omega_{b}\) is the cyclotron frequency. In view of the tensor elements of Equation (16), the parameters of the CCPR model are derived through the simple algebraic manipulations shown in Table 2.  
We study the wave propagation of a Gaussian pulse normaly incident to a magnetized plasma slab of 9 mm in thickness, assuming that the static biasing magnetic is parallel to the direction of propagation. The plasma parameters are \(\omega_{p}=2\pi\times 50\times 10^{9}\) rad/s, \(\omega_{b}=3\times 10^{11}\) rad/s, and \(v_{c}=2\times 10^{10}\) Hz. The FDTD cell size is \(\Delta z=75\) um and the time step is chosen as \(\Delta t=Q\Delta z/c_{0}\), where \(Q=0.3\) is the Courant number and \(c_{0}\) is the velocity of light in free space. The Gaussian pulse is excited from the left side of the computational domain at the \(x\) component of the electric field. The domain is truncated by a 12 cell convolution perfectly matched layer (CPML) [23]. We calculate the reflection (\(\Gamma\)) and transmission (\(\Gamma\)) coefficients for right circularly polarized (RCP) and left circularized polarized (LCP), defined as  
\[T_{\text{RCP}}(\omega) =\frac{E_{xt}(\omega)}{E_{xt}(\omega)}+j\frac{E_{yt}(\omega)}{E_{ xi}(\omega)},\ \ \Gamma_{\text{RCP}}(\omega)=\frac{E_{xt}(\omega)}{E_{ xi}(\omega)}+j\frac{E_{yt}(\omega)}{E_{xi}(\omega)} \tag{17a}\] \[T_{\text{LCP}}(\omega) =\frac{E_{xt}(\omega)}{E_{xi}(\omega)}-j\frac{E_{yt}(\omega)}{E_{ xi}(\omega)},\ \ \Gamma_{\text{LCP}}(\omega)=\frac{E_{xt}(\omega)}{E_{ xi}(\omega)}-j\frac{E_{yt}(\omega)}{E_{xi}(\omega)} \tag{17b}\]  
where \(E_{xt}\) and \(E_{yt}\) are the \(x\) and \(y\) components of the transmitted electric field in the frequency domain, respectively, and \(E_{xt}\) and \(E_{yt}\) are likewise the reflected components. \(E_{xi}\) is the \(x\) component of the incident wave in the frequency domain. In Figure 1, we validate the proposed FDTD method using as a reference the semi-analytical solution calculated by the Berreman \(4\times 4\) method [24]. Excellent agreement is observed in a very broad spectral range (0-100 GHz).