# Analysis and Design of Tunable THz 1-D Leaky-Wave Antennas Based on Nematic Liquid Crystals
## 4 Results  
According to the analysis in the previous Section 3.1, a 3-D model of the optimized 1-D unidirectional NLC-based LWA of length \(L=1.5\) mm (which is about 5\(\lambda_{0}\) at 1 THz) was designed in CST Microwave Studio and simulated with the time-domain solver. The results of Sections 3.2 and 3.3 were exploited to reduce the complexity of the problem; thus, the PRS was implemented with a surface impedance boundary condition, whereas the AMC walls were implemented as PMC boundary conditions. The lateral distance between the two PMC is \(d=75\,\upmu\)m. The antenna was excited with a waveguide port operating in its first-order TM mode, and another waveguide port was placed at the end of the antenna to simulate the presence of an absorber. The NLC mixture w-1825 is defined as a uniaxial crystal with diagonal components as in Equation (6). Full-wave simulations of the structure were performed at the fixed frequency of 1.049 THz for a tilt angle that switched from \(0^{\circ}\) to \(90^{\circ}\) at steps of \(15^{\circ}\). Full-wave results are shown for both the radiation efficiency and peak gain in Figure 5a,b, respectively, where they are reported as black circles and compared with the analytical results (solid black lines), showing satisfactory agreement. Analytical results were obtained using Equations (3) and (4) for the efficiency and directivity, respectively, with the numerical values of \(\hat{\beta}\) and \(\alpha\) obtained from the dispersion analysis as described in Section 3.1. In order to have a more complete picture of the radiation performance of the proposed structure, we also report the normalized radiation patterns on the elevation plane for four different bias states (corresponding to \(\delta=0^{\circ}\), \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\)) obtained with CST and compared with the expected analytical leaky-wave patterns. The latter were computed using the numerical values of \(\hat{\beta}\) and \(\hat{\alpha}\) into the following formula [54]:  
\[\hat{p}(\theta)=\frac{1+\sin^{2}t(\theta)/\sinh^{2}a}{1+t(\theta)^{2}/a^{2}} \tag{9}\]  
where \(t(\theta)=b-l\sin\theta\) with \(l=k_{0}L/2\), \(b=\hat{\beta}l\), and \(a=\hat{\alpha}l\) are normalized variables. The comparison is shown in Figure 5c), where a good agreement between full-wave (colored dots) and analytical (colored solid lines) results is observed for the last three bias states; the analytical results for the first state appear to underestimate the effective beam angle, which is most likely due to edge effects, since the beam peak is not far from the endfire. A 3-D view of the radiation pattern obtained from a full-wave simulation of the last bias state is shown as an inset in Figure 5c. Full-wave 3-D radiation patterns for all bias states were obtained but are not reported for the sake of brevity. Indeed, as can be inferred from Figure 5, the radiation patterns  
Figure 5: (**a**) Radiation efficiency \(\eta_{\text{rad}}\), and (**b**) gain (in dB) vs. NLC tilt angle \(\delta\) for \(f=1.049\) THz. The CST results are shown as black circles, and the numerical results are shown as solid black lines. (**c**) Normalized radiation patterns at \(f=1.049\) THz for four different LC states—namely, \(\delta=0^{\circ}\), \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) in black, light blue, green, and yellow, respectively. The CST results are shown as colored circles, and the analytical results are shown as solid colored lines. The inset of (**c**) shows the normalized 3-D radiation pattern for the last NLC state for which the beam peak points at approximately \(45^{\circ}\) (as expected from Figure 3c). The intensity of the normalized 3-D radiation pattern goes from \(-20\) dB (green) to \(0\) dB (red). are similar to one another apart from the different beam angles. The gain peak and the beamwidth, which are the most relevant antenna figures of merit, are slightly affected by the variation of the tilt angle as predicted by the analysis of Figure 3c,d (see the violet curve).