# A Low-Profile Dual-Polarized Transmitarray with Enhanced Gain and Beam Steering at Ku Band
## 4 The TA Antenna  
In this work, a focal length-to-diameter ratio (F/D) of 0.6 was chosen, and a maximum D size of 200 mm was considered. Based on this requirement, a suitable feed antenna was selected: in particular, a substrate integrated waveguide (SIW)-based 2 \(\times\) 2 array antenna [46] was used, as shown in Figure 8. This feed antenna array consists of two Rogers RT/duroid 5880 substrates (\(\varepsilon_{\mathrm{r}}\) = 2.2, tan\(\delta\) = 0.0009). The thickness of the first substrate is T\({}_{1}\) = 0.381 mm, and for the second is T\({}_{2}\) = 1.575 mm. The length and width of the substrates are L = 35 mm, W = 35 mm, respectively. The diameters of the vias connecting both grounds and the diame  
Figure 7: Transmission magnitude 1S\({}_{2,1}\) | and transmission phase \(\angle\)S\({}_{2,1}\) of UC\({}_{6}\) as a function of frequency f, obtained by CST and ECM simulation. (**a**) Transmission magnitude; (**b**) transmission phase. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline C\({}_{0}\) & 0.323 pF \\ C\({}_{\mathrm{g}}\) & 0.042 pF \\ C\({}_{1}\) & 0.019 pF \\ L\({}_{\mathrm{r}}\) & 3.727 nH \\ L\({}_{\mathrm{c}}\) & 1.729 nH \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of circuit components of UC\({}_{6}\) ECM. ter of the circular slot in the first ground are D\({}_{1}\) = 0.62 mm and D\({}_{2}\) = 15.05 mm, respectively. The patch width, length are P\({}_{1}\) = 7.7 mm, P\({}_{2}\) = 7.7 mm, and the dimensions regarding lateral and corner slot in the patch are P\({}_{3}\) = 2.75 mm, P\({}_{4}\) = 1.44 mm, and P\({}_{5}\) = 2.18 mm, respectively shown in Figure 8. Specifically, the 2 \(\times\) 2 array in [46] is chosen because it has a \(-\)10 dB beamwidth of approximately 66\({}^{\circ}\), which aligns with the chosen F/D ratio and the target D size. This feed antenna supports a bandwidth of BW = 2.5 GHz (10-12.5 GHz) with an impedance bandwidth of BW\({}_{z}\) = 20.83%, having a peak gain G\({}_{\rm f0}\) = 12.93 dB and G\({}_{\rm f1}\) = 12.83 dB at f\({}_{0}\) = 12 GHz and f\({}_{1}\) = 12.4 GHz, respectively [46]. A 22 \(\times\) 22 sized TA is designed and implemented using the proposed UC. As a result of this design approach, the width of the TA is D = 22 \(\times\) 8.65 mm = 190.3 mm and the focal distance is F = 0.6 \(\times\) D = 114.18 mm. The transmission phase of the UC of the TA needs to be adjusted to steer the beam in the desired direction. The phase adjustment of the UC compensates the spatial phase delay, which is introduced due to the distance, d\({}_{\rm p}\), between the feed antenna and the corresponding pth UC (see also Figure 9a). For optimal beam forming in the targeted direction, (\(\theta_{0}\), \(\varphi_{0}\)), the required phase shift across the TA is computed as follows. The phase of the transmitted electric field for the pth UC is [47]:  
\[\psi(x_{\rm P},y_{\rm P})=-k_{0}x_{\rm P}\sin\theta_{0}\cos\varphi_{0}-k_{0}y_{ \rm P}\sin\theta_{0}\sin\varphi_{0} \tag{4}\]  
where \((x_{\rm P},y_{\rm P})\) is the coordinate of the pth UC and k\({}_{0}\) is the free space wavenumbers.