# 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
This effect is also appreciable from the dispersion curves depicted in Figure 2(a),b, where it is seen that the normalized phase (in solid lines) and attenuation constant (in dashed lines) dispersion curves cross each other (resonant condition for having broadside beams [63, 64]) in the range of 0.85-0.95 THz. In Figure 2(b), the same results of Figure 2(a) are obtained for the lossless version of the optimized structure in order to obtain \(\hat{\beta}_{\text{rad}}\) and \(\hat{\alpha}_{\text{rad}}\), which are essential to compute the radiation efficiency (see Equation (4)) and, in turn, the gain. Figure 2: (a) A schematic 3-D view of a 1-D unidirectional LWA based on PRS and NLC, (b) its cross section along the \(yz\)-plane, and (c) the related transverse equivalent network. The biasing scheme is shown only in (b). In Figure 3a, we also report the fixed frequencies that we selected to evaluate the scan performance through NLC biasing with black, violet, and magenta dotted vertical lines. The rationale behind the choice of these frequencies is as follows. The lowest frequency (the dotted vertical black line in Figure 3a,b) corresponds to the frequency for which, at the 'on' state (see the red-colored dispersion curves), the TM leaky mode is at the leaky cutoff (viz., \(\hat{\beta}=\hat{\alpha}\)) and, thus, radiates efficiently close to broadside (we recall that, for \(\hat{\beta}<\hat{\alpha}\), the gain of 1-D unidirectional LWAs abruptly decreases). The highest frequency (the dotted vertical magenta line in Figure 3a,b) corresponds to the frequency for which, at the 'off' state (see the blue-colored dispersion curves), the TM leaky mode radiates at endfire (viz., \(\hat{\beta}=1\)). The intermediate frequency (the dotted vertical violet line in Figure 3a,b) is simply a middle-of-the-way choice that represents, in our opinion, a sub-optimal condition at which it is convenient to have the antenna work. In particular, in Figure 3c, the beamwidth and the beam angle are reported as a function of the NLC tilt angle. The results for the beam angle are in agreement with the formula for the scan rate (viz., Equation (7)), which predicts a greater tunable range for angles either close to the broadside or endfire. However, the results for the beamwidth show that, as the scan rate increases, the beamwidth dynamic range also increases, which is not a desirable property for beamscanning applications. Figure 3d completes the picture, showing the gain as a function of the average NLC tilt angle, and revealing that there is only a slight improvement in working close to the endfire (magenta curve) with respect to the intermediate condition (violet curve), which instead ensures a quasi-consistent beamwidth through scanning. For these reasons, in the  
Figure 3: **(a,b)** Dispersion curves \(\hat{\beta}\) (solid lines) and \(\hat{\alpha}\) (dashed lines) vs. \(f\) for NLC states continuously varying from the ‘off’ (blue) to ‘on’ state (red). The results in **(a,b)** are for the lossy and lossless cases, respectively. Dotted vertical colored lines highlight the frequency at which the antenna performance shown in **(c,d)** are evaluated. **(c)** The beam angle \(\theta_{0}\) (in solid lines) and half-power beamwidth \(\Delta\theta\) (in dashed lines), and **(d)** gain \(G\) as the optical axis \(\delta\) tilts from the horizontal \(z\)-axis \(\delta=0^{\circ}\) to the vertical \(x\)-axis (\(\delta=90^{\circ}\)) for the frequencies shown in **(a)**. following, we only report results for the fixed frequency of \(f=1.049\) THz (the violet curves in Figure 3c,d), as this choice represents an optimal trade-off.