# Analysis and Design of Tunable THz 1-D Leaky-Wave Antennas Based on Nematic Liquid Crystals
## 3 Optimization and Design
### Design of the PRS  
The sheet impedance of the PRS used in the optimized structure described in Section 3.1 is \(Z_{\mathrm{s}}=j60\,\Omega\). Homogenized metasurfaces similar to those used for the synthesis of the AMC walls can also be used to realize such a PRS.  
However, as opposed to the synthesis of the AMC, where the thickness of the substrate is instrumental to realize the zero-phase condition at the desired frequency, it is here of little importance, since we only require a metasurface that behaves as a PRS. This is fully characterized by a single scalar purely imaginary value of \(Z_{\mathrm{s}}\). Moreover, the need to use the PRS as an electrode for the application of the signal driving the liquid crystal demands that the PRS is connected as it acts as one of the electrodes. Additionally, this requires a high filling factor as the LC materials under the gaps suffer from lower reorientation as driven only by the fringe field.  
Hence, patch and strip-grating unit cells do not represent optimal choices for our PRS as there are either nonconnected topologies or intrinsically low filling-factor geometries. Conversely, fishnet-like unit cells, such as those studied in [25, 58], possess both the above-mentioned features of having a connected topology and a relatively high filling factor. However, since we deal with a 1-D LWA with PMC walls, the unit cell will be periodic only along the propagation \(x\)-axis. This is a relevant difference with respect to the fishnet  
Figure 4: (**a**) Phase of the reflection coefficient vs. \(f\) (solid black line) of the patch unit cell for realizing the PMC. The vertical dotted line highlights that the phase of the reflection coefficient is zero at the desired frequency of 1.049 THz. (**b**) Sheet reactance vs. \(f\) (solid black line) of the transverse slot for realizing the PRS. The parameters are in the text.  
like unit cell studied in [25; 58], which is instead 2-D periodic. In particular, we note that the original 2-D periodic fishnet-like unit cell is a superposition of a longitudinal slot and a transverse slot; in our case study, the TM leaky mode will only couple with the transverse slot, and the PMC symmetry along the \(xz\)-planes will cancel the contribution of the longitudinal slot.  
Indeed, the longitudinal slot behaves as an \(x\)-oriented magnetic dipole, which is parallel to the PMC walls, and thus it does not radiate (the dual case of an electric dipole parallel to a PEC wall [67]). Therefore, the PRS will be realized considering only a transverse slot unit cell in a 2-D periodic environment, with the contribution of the longitudinal slot being negligible for the reasons above.  
The sheet impedance is then extracted by means of conventional Floquet-Bloch full-wave simulations of such a unit cell for different choices of design parameters. After a parametric analysis, we obtained the desired sheet impedance value (see Figure 4b) for the following design parameters: the period \(p_{2}=75\,\upmu\)m, the slot length \(s=67.5\,\upmu\)m, and the slot width \(w=7\,\upmu\)m. In the simulation of the unit-cell, a substrate of Zeonor of thickness \(50\,\upmu\)m is considered so that the extracted \(Z_{\text{s}}\) considers the inhomogeneous character of the materials at the interface.  
The frequency-dispersive character of \(Z_{\text{s}}\) is then used in the implementation of the surface impedance boundary condition in the CST [68] full-wave simulation of the entire structure. Although approximate, this approach allows for a great reduction of computational burden while maintaining excellent accuracy as demonstrated in previous studies [25; 55].