# Analysis and Design of Tunable THz 1-D Leaky-Wave Antennas Based on Nematic Liquid Crystals
## 3 Optimization and Design  
### Optimization of the Ideal Structure  
According to the assumptions made in Section 2, we develop here a TEN model (see Figure 2) for the structure described in Section 2.4. This TEN can be studied through the transverse resonance technique [51], which provides the relevant dispersion equation  
\[\tilde{Y}_{0}+\tilde{Y}_{\rm{s}}+\tilde{Y}_{\rm{TL}}(\tilde{Y}_{\rm{sp}}, \tilde{Y}_{\rm{NLC}})=0 \tag{8}\]  
where the 'bar' (\(\cdot\)) denotes normalization to \(1/\eta_{0}\) (with \(\eta_{0}\simeq 120\pi\,\Omega\) as the vacuum impedance), whereas \(\tilde{Y}_{0}\), \(\tilde{Y}_{\rm{sp}}\), and \(\tilde{Y}_{\rm{NLC}}\), represent the normalized wave admittances of air, spacer, and NLC media, respectively. \(\tilde{Y}_{\rm{s}}\) is the normalized surface admittance of the PRS, and \(\tilde{Y}_{\rm{TL}}\) is the normalized input admittance of the three-section transmission-line terminated with a short circuit representing the ground plane. The expression of \(\tilde{Y}_{\rm{TL}}\) is a function of both \(\tilde{Y}_{\rm{sp}}\) and \(\tilde{Y}_{\rm{NLC}}\) and is not reported here for the sake of brevity. For TM polarization, the characteristic normalized admittances \(\tilde{Y}_{\rm{c}}\) of isotropic non-magnetic media, such as the spacer and air, are, \(\tilde{Y}_{\rm{c}}=\varepsilon_{\rm{r}}/\sqrt{\varepsilon_{\rm{r}}-\tilde{k} _{z}^{2}}\), with \(\varepsilon_{\rm{r}}\) as the relative permittivity of the medium, whereas, for anisotropic uniaxial media, such as the NLC, we have \(\hat{Y}_{\text{NLC}}=\varepsilon_{zz}/\sqrt{\varepsilon_{zz}-(\varepsilon_{zz}/ \varepsilon_{xx})\hat{k}_{z}^{2}}\) (see, e.g., Ref. [38]). According to the previous description of the NLC tensor, it becomes clear that the roots of this equation change as the tilt angle of the NLC optical axis changes. Root-finding methods, such as the Pade algorithm [62] can be employed to obtain the wavenumber dispersion of the higher order TM leaky mode. It is worthwhile recalling that the domain of the function at the left-hand sides of Equation (8) is a two-sheeted Riemann surface. The proper \(\text{Im}[k_{x}]<0\) and improper \(\text{Im}[k_{x}]>0\) sheets are connected through Sommerfeld branch cuts \(\text{Im}[k_{x}]=0\); leaky modes are found on the improper sheet [48]. In our analysis, we are interested in evaluating the wavenumber dispersion of the TM leaky mode as a function of both the frequency and tilt angle and for different choices of the sheet impedance value \(Z_{\text{s}}\) and the NLC filling factor defined as \(\text{FF}=h_{\text{NLC}}/h\). We stress that the analytical techniques that are typically adopted to maximize the gain in 1-D unidirectional LWAs [52, 55] cannot be applied here because of the anisotropy of the NLC layer and its non-negligible losses. In addition, the results from [52, 55] refer to typical frequency-scanning 1-D unidirectional LWAs. Here, the tunable feature enabled by the NLC cell requires the antenna designer to find the parameters that may enhance the maximum angular range at a fixed frequency--not only the gain. The search for optimal values of FF and \(Z_{\text{s}}\), thus, requires a parametric numerical investigation, whose final results are here reported (a qualitative analysis is instead discussed in [40]). Specifically, we found that the choice of \(\text{FF}=0.2\) and \(Z_{\text{s}}=j60\,\Omega\) represents an optimal trade-off among the directivity, radiation efficiency, and tunability. The numerical results of this optimized structure are reported in the frequency range of 0.6-1.2 THz for different NLC states in Figure 2(a) (the colors shade from blue to red as the NLC state switches from 'off' to 'on'), for a cavity height of \(h=96\,\upmu\)m, which corresponds approximately to a resonant cavity at 1 THz when the NLC is in its off state (note that \(h\simeq\lambda_{0}/(2\sqrt{\text{Re}[\varepsilon_{o}]})\) and that the NLC in its off state and the Zeonor are almost index-matched).