# Analysis and Design of Tunable THz 1-D Leaky-Wave Antennas Based on Nematic Liquid Crystals
## 2 Theoretical Analysis
### Open Leaky Waveguide  
We now replace the top PEC plate with a PRS (see Figure 1, the second from the left), which enforces a surface impedance boundary condition of the type \((\underline{H}^{+}-\underline{H}^{-})\times\underline{z}_{0}=\underline{E}/ \,\mathrm{Z}_{\mathrm{s}}\), where the \({}^{+(-)}\) superscripts refer to the field immediately above (below) the PRS, \(\underline{z}_{0}\) is the unit vector along the \(z\)-axis, and \(\mathrm{Z}_{\mathrm{s}}\) is the surface impedance of an isotropic lossless metasurface, which is thus described by a single scalar reactance value \(X_{\mathrm{s}}=\mathrm{Im}[Z_{\mathrm{s}}]\). We also assume the PRS to be highly reflective (i.e., \(X_{\mathrm{s}}\ll\eta_{0}\), with \(\eta_{0}\simeq 120\pi\,\Omega\) as the vacuum impedance) so as to slightly perturb the modal configuration of the \(\mathrm{TM}_{10}\) mode.  
However, the partially open nature of the problem will now allow the propagation wavenumber to be generally complex [44; 48]. This perturbed version of the \(\mathrm{TM}_{10}\) will be referred to as the TM _leaky_ mode when \(\mathrm{Im}[k_{z}]>0\): it will represent the dominant contribution to radiation when \(\beta>\alpha\) and \(0<\alpha\ll k_{0}\)[49; 50], where \(\beta=\mathrm{Re}[k_{x}]\) and \(\alpha=-\mathrm{Im}[k_{x}]\) are the phase and attenuation constants of the leaky mode, and \(k_{0}\) is the vacuum wavenumber. Since the modal field of the TM leaky mode is constant with respect to the \(y\)-axis, we will study its propagation properties with the aid of a transverse equivalent network (TEN) [51], as will be discussed in more detail in Section 3.  
As is well-known [1], one-dimensional (1-D) PRS-based LWAs, such as those considered here, radiate half of a conical beam. The beam angle \(\theta_{0}\) (measured from the vertical \(z\)-axis) and \(-3\) dB beamwidth \(\Delta\theta\) are determined by the normalized phase \(\hat{\beta}:=\beta/k_{0}\) and attenuation \(\hat{\alpha}:=\alpha/k_{0}\) constants through the following approximate relations:  
\[\theta_{0} \simeq\arcsin\hat{\beta} \tag{1}\] \[\Delta\theta \simeq 2\hat{\alpha}/\cos\theta_{0}. \tag{2}\]  
The directivity \(D\) is instead given by [52]:  
\[D\simeq 1.2/\hat{\alpha}, \tag{3}\]  
and it coincides with the gain \(G\) for a lossless structure. When material losses have to be accounted for (as in the case discussed in the next Section 2.3), it is \(G=\eta_{\mathrm{rad}}D\), where the radiation efficiency \(\eta_{\mathrm{rad}}\) can be estimated as [53]:  
\[\eta_{\mathrm{rad}}\simeq\frac{\hat{\beta}_{\mathrm{rad}}\hat{\alpha}_{\mathrm{ rad}}}{\hat{\beta}\hat{\alpha}} \tag{4}\]  
Figure 1: From left to right, an air-filled PEC/PMC waveguide, which evolves first in an open leaky waveguide where the PEC top plate is replaced by a PRS, is then entirely filled with an NLC to allow for dynamic beamsteering and, finally, is only partially filled with an NLC material to compensate for losses.  
where \(\hat{\beta}_{\rm rad}\) and \(\hat{\mathbb{x}}_{\rm rad}\) are the phase and attenuation constants only due to radiation and can thus be obtained by evaluating the propagation complex wavenumber of the _lossless_ structure, i.e., by assuming no material losses.  
These formulas only hold for infinite structures; however, more accurate formulas exist for their truncated version [52; 54; 55; 56]. For the cases analyzed here, it will be seen that no substantial differences are expected with respect to the infinite case; thus, Equations (1)-(4) are accurate enough.