# Analysis and Design of Tunable THz 1-D Leaky-Wave Antennas Based on Nematic Liquid Crystals
## 2 Theoretical Analysis
### Tunable Leaky Waveguide Filled with NLC  
From the radiating viewpoint, the structure considered thus far is a conventional 1-D PRS-based LWA. Therefore, the beam will scan with frequency \(f\) through the approximate formula [55]:  
\[\theta_{0}\simeq\sin^{-1}\sqrt{\varepsilon_{\rm r}-\left(\frac{c}{2 hf}\right) ^{2}} \tag{5}\]  
where \(c\) is the speed of light in vacuum, \(\varepsilon_{\rm r}\) the real part of the relative permittivity of the dielectric material filling the waveguide, and \(h\) the height of the waveguide in the vertical direction--here, the \(z\)-axis. However, our goal is to have beamscanning at a fixed frequency around 1 THz. For this purpose, we now fill the waveguide with an NLC material (mixture w-1825, see [38] and the references therein) as shown in Figure 1, third schematic from the left. As is known [57], the dielectric properties of NLC change when an external low-driving voltage is applied across the cell (see, e.g., Ref. [38; 58] for details). A rigorous electromagnetic analysis would require modeling the NLC with a full permittivity tensor [59]. Under this condition, the problem can no longer be decomposed into TE and TM modes, and dyadic Green's functions [60] or coupled transmission-line equations [61] are required. Nevertheless, as discussed in [61], when the ratio of anisotropy defined as \(\delta\varepsilon:=2\Delta\varepsilon/(\varepsilon_{\rm e}+\varepsilon_{ \rm o})\) (with \(\varepsilon_{\rm o}(\varepsilon_{\rm e})\) as the ordinary (extraordinary) relative permittivity value of the NLC and \(\Delta\varepsilon=|\varepsilon_{\rm e}-\varepsilon_{\rm o}|\) as its dielectric anisotropy) is less than 50% (as in the case of the mixture w-1825 [35]), the TE-TM coupling can safely be neglected. Hereafter, TE and TM modes are referred with respect to the \(xz\)-plane; thus, the TE and TM modes are described by \(\{E_{y},H_{x},H_{z}\}\) and \(\{H_{y},E_{x},E_{z}\}\), respectively. This assumption allows for greatly simplifying the problem. In particular, we model the NLC as a _lossy_ uniaxial crystal with a diagonal relative permittivity tensor with components given by \(\varepsilon_{xx}=\varepsilon_{\rm e}\) and \(\varepsilon_{yy}=\varepsilon_{zz}=\varepsilon_{\rm o}\) when no bias (off state) is applied and \(\varepsilon_{xx}=\varepsilon_{yy}=\varepsilon_{\rm o}\) and \(\varepsilon_{zz}=\varepsilon_{\rm e}\) at the high-voltage limit (on state) when the NLC optical axis is aligned with the \(z\)-axis [58]. The NLC orientation at the off-state is determined by properly placing alignment layers, e.g., polyamide films rubbed along the \(x\)-axis--a standard practice in LC technology. The thickness of such layers is in the order of tens of nm; hence, their effect at the THz frequency of operation can be neglected. The relative permittivity values of the considered NLC material are \(\varepsilon_{\rm o}=2.42(1-j0.08)\) and \(\varepsilon_{\rm e}=3.76(1-j0.05)\) (see, e.g., Ref. [38; 58]). We consider the case where the NLC voltage bias is applied along the \(z\)-axis, leading to a rotation of the NLC optical axis in the \(xz\)-plane. Hence, the orientation of the NLC optical axis can be described by a tilt angle \(\delta\), measured from the \(x\)-axis, and the NLC relative permittivity tensor is given by  
\[\tilde{\varepsilon}= \begin{bmatrix}\varepsilon_{\rm o}+\Delta\varepsilon\cos^ {2}\delta&0&\Delta\varepsilon\cos\delta\sin\delta\\ 0&\varepsilon_{\rm o}&0\\ \Delta\varepsilon\cos\delta\sin\delta&0&\varepsilon_{\rm o}+\Delta\varepsilon \sin^{2}\delta\end{bmatrix} . \tag{6}\]  
Although the dynamics of the NLC as a function of the driving voltage are nonlinear (see, e.g., Ref. [58]), this aspect is secondary in the current context: our goal is to determine the tunable range of the antenna (ultimately bound by the 'off' and 'on' states) and to optimize its radiation performance.