# A Low-Profile Dual-Polarized Transmitarray with Enhanced Gain and Beam Steering at Ku Band
## 2 UC Design  
The top and side view of the proposed UC is illustrated in Figure 1, the green color representing the substrate and the yellow color the metal. Three identical layers constitute the UC optimized for the phase compensation of the feed incident beam. A symmetric geometry is chosen to allow the dual polarization operation. The metal layer consists of a metal ring, with an outer radius R\({}_{1}\) and inner radius R\({}_{2}\), and an inner circle of radius R\({}_{3}\). In the design, the ranges for searching for the optimized dimensions are listed in Table 1. They are defined after the preliminary simulations. Eight UCs are designed by changing the values of R\({}_{1}\), R\({}_{2}\), and R\({}_{3}\) to achieve 3-bit phase with low transmission loss. The specific dimensions of the eight UCs are listed in Table 2 with the corresponding transmission magnitude and phase. The substrate used is Rogers RT/duroid 5880 (\(\epsilon_{r}\) = 2.2, tan\(\delta\) = 0.0009). The thickness of the substrate is t = 1.575 mm (0.063\(\lambda_{0}\)). The metal thickness is 35 \(\upmu\)m. The periodicity of the UC is P = 8.65 mm (0.346\(\lambda_{0}\)), \(\lambda_{0}\) = 25 mm being the central wavelength at the frequency f\({}_{0}\) = 12 GHz. The electric field of an incident wave on the UC with arbitrary linear polarization can be expressed by Equation (1) [44]:  
\[\overline{\mathrm{E}}_{\mathrm{inc}}=\mathrm{E}_{\mathrm{x}}\overline{\mathrm{x}} +\mathrm{E}_{\mathrm{y}}\overline{\mathrm{y}} \tag{1}\]  
where \(\mathrm{E}_{\mathrm{x}}\) and \(\mathrm{E}_{\mathrm{y}}\) are the incident electric field components along the orthogonal linear polarizations. Each UC can be modeled as a 2 \(\times\) 2 transmission matrix relating to input and output fields found in literature [44]:  
\[ \begin{pmatrix}\overline{\mathrm{E}}_{\mathrm{x,out}}\\ \overline{\mathrm{E}}_{\mathrm{y,out}}\end{pmatrix} = \begin{pmatrix}\mathrm{T}_ {\mathrm{x}}&\mathrm{T}_{\mathrm{xy}}\\ \mathrm{T}_{\mathrm{yx}}&\mathrm{T}_{\mathrm{yy}}\end{pmatrix}\begin{pmatrix} \overline{\mathrm{E}}_{\mathrm{x,in}}\\ \overline{\mathrm{E}}_{\mathrm{y,in}}\end{pmatrix}  \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value (mm)** \\ \hline P & 8.65 \\ \(\mathrm{R}_{1}\) & 3.8–4.325 \\ \(\mathrm{R}_{2}\) & 3.7–4.225 \\ \(\mathrm{R}_{3}\) & 2.55–4.11 \\ \(\mathrm{T}\) & 1.575 \\ \hline \end{tabular}
\end{table} 
Table 1: Range for geometrical parameter optimization. Figure 1: (**a**) Top and (**b**) side views of proposed monolithic UC, consisting of three identical metal layers. Yellow part represents metal, and green part is substrate. \begin{table}
\begin{tabular}{c c c c c c} \hline
**UC No.** & **R\({}_{1}\) (mm)** & **R\({}_{2}\) (mm)** & **R\({}_{3}\) (mm)** & \(\mathrm{I}\,\mathrm{S}_{2,1}\) (dB)** & \(\angle\)**S\({}_{2,1}\) (deg)** \\ \hline
1 & 4.325 & 4.225 & 4.11 & \(-\)0.86 & 46 \\
2 & 4.325 & 4.225 & 4.07 & \(-\)0.58 & 89 \\
3 & 4.325 & 4.225 & 3.96 & \(-\)0.92 & 136 \\
4 & 4.20 & 4.10 & 3.83 & \(-\)0.20 & 179.3 \\
5 & 4.25 & 4.15 & 3.60 & \(-\)0.71 & 225 \\
6 & 4.325 & 4.20 & 3.07 & \(-\)1.23 & 270.6 \\
7 & 4.20 & 4.10 & 2.63 & \(-\)0.17 & 314 \\
8 & 3.80 & 3.70 & 2.55 & \(-\)1.16 & 359.6 \\ \hline \end{tabular}
\end{table} 
Table 2: Geometrical dimensions for 3-bit phases and corresponding magnitudes. where T\({}_{\rm xx}\), T\({}_{\rm yy}\) and T\({}_{\rm xy}\), T\({}_{\rm yx}\) are the co-polar and cross-polar transmission coefficients, respectively. If the off-diagonal terms T\({}_{\rm xy}\), T\({}_{\rm yx}\) are null terms, then dual-polarized operation can be expected. Thus, for dual-polarized operation, the transmission matrix T is [44]:  
\[\rm T= \begin{pmatrix}|T_{\rm xx}|e^{i\angle T_{\rm xx}}&0\\ 0&|T_{\rm yy}|e^{i\angle T_{\rm yy}}\end{pmatrix}  \tag{3}\]  
where \(\angle T_{\rm xx}\) and \(\angle T_{\rm yy}\) are the phase shifts applied to x- and y-polarized waves respectively.