## 3 Applications and Numerical Results
### Terahertz Wave Propagation through a Nematic Liquid Crystal Cell  
For NLC materials, the relative permittivity tensor is given by  
\[\vec{\bar{\varepsilon}}(\omega)=\left(\begin{array}{ccc}\varepsilon_{o}+ \Delta\varepsilon\cos^{2}\theta\cos^{2}\phi&\Delta\varepsilon\cos^{2}\theta \sin\phi\cos\phi&\Delta\varepsilon\sin\theta\cos\theta\cos\phi\\ \Delta\varepsilon\cos^{2}\theta\sin\phi\cos\phi&\varepsilon_{o}+\Delta \varepsilon\cos^{2}\theta\sin^{2}\phi&\Delta\varepsilon\sin\theta\cos\theta \sin\phi\\ \Delta\varepsilon\sin\theta\cos\theta\cos\phi&\Delta\varepsilon\sin\theta\cos \theta\sin\phi&\varepsilon_{o}+\Delta\varepsilon\sin^{2}\theta\end{array} \right), \tag{18}\]  
\begin{table}
\begin{tabular}{c c} \hline
**Tensor Element** & **Parameters: \(\sigma\), \((c_{p},a_{p})\)** \\ \hline \(\varepsilon_{xx}\) & \(\varepsilon_{0}\frac{\omega_{p}^{2}v}{v^{2}+\omega_{b}^{2}}\cdot\left(-0.5 \omega_{p}^{2}\frac{v+j\omega_{b}}{v^{2}+\omega_{b}^{2}},-v+j\omega_{b}\right)\) \\ \(\varepsilon_{xy}\) & \(-\varepsilon_{0}\frac{\omega_{p}^{2}\omega_{b}}{v^{2}+\omega_{b}^{2}}\cdot \left(-0.5\omega_{p}^{2}\frac{-\omega_{b}+jv}{v^{2}+\omega_{b}^{2}},-v+j\omega_ {b}\right)\) \\ \(\varepsilon_{zz}\) & \(\varepsilon_{0}\omega_{p}^{2}/v\), \((-0.5\omega_{p}^{2}/v^{2},-v)\) \\ \hline \end{tabular}
\end{table}
Table 2: CCPR parameters for magnetized plasma.  
Figure 1: The magnitude and phase of the transmission/reflection coefficients for RCP/LCP waves propagating through magnetized plasma calculated by the proposed FDTD method and the Berreman \(4\times 4\) matrix method.  
where \(\theta\) and \(\phi\) are the tilt and twist angles of the spatially varying LC molecular orientation, respectively, and \(\varepsilon_{o}\) and \(\varepsilon_{e}\) are the frequency-dependent ordinary and extraordinary dielectric permittivities, with \(\Delta\varepsilon=\varepsilon_{e}-\varepsilon_{o}\). The tilt angle is defined as the angle between the nematic director, a unit vector parallel to the NLC optical axis, and the \(x\)-\(y\) plane, whereas the twist angle is measured between the \(x\) axis and the projection of the director on the \(x\)-\(y\) plane. The ordinary and extraordinary dielectric permittivities are described as the sum of CCR terms according to Equation (4).  
We demonstrate the efficiency of the proposed FDTD formulation for the case of THz wave propagation through an LC cell. The NLC material is 5 CB, whose material dispersion has been described via the modified Lorentz model [25], and the equivalent CCR parameters are obtained using Table 1. The real and imaginary parts of the extraordinary and ordinary relative permittivity (\(\varepsilon_{o,e}=e^{\prime}-je^{\prime\prime}\)) in the considered spectral range are shown in Figure 2a. A layer of 0.5 mm is embedded in a substrate material with a refractive index of 1.5.  
An \(x\) polarized plane wave impinges on the slab, and the transmittance after the slab is calculated. The FDTD cell size is chosen to be equal to 0.5 \(\upmu\)m, and the computational domain is terminated by a 12-cell CPML. The FDTD time step is set to \(\Delta t=0.3\Delta z/c_{0}\) and the simulation is run for 60,000 time steps. A direct comparison of the transmittance of the structure obtained by the proposed FDTD scheme and the reference matrix technique is provided in Figure 2b for indicative values of the \(\theta\) and \(\phi\) angles. Excellent agreement is observed.