# Low-Profile High-Efficiency Transmitarray Antenna for Beamforming Applications
## 2 Design of 2D Partial H-Plane Waveguide Slot Array Antenna  
The compact partial H-plane waveguide in Figure 1 was proposed in [18]. The propagation constant (\(\beta_{y}\)) of a longitudinal direction and cut-off frequency (\(f_{c}\)) can be calculated by the following [18]:  
\[\beta_{y}=\sqrt{\beta_{0}^{2}-\beta_{z1}^{2}-\left(\frac{2m\pi}{w}\right)^{2} }(m=0,\ 1,\ 2,\ \cdots) \tag{1}\]  
\[f_{c}=\frac{c}{2\pi}\sqrt{\beta_{z1}^{2}+\left(\frac{2m\pi}{w}\right)^{2}}(m= 0,\ 1,\ 2,\ \cdots) \tag{2}\]where \(c\) and \(w\) are the velocity of light and the width of the partial H-plane waveguide, respectively. \(\beta_{0}\) and \(\beta_{z1}\) are the wave number in the air and the wave number of the \(z\)-direction in the region 1, respectively. Figure 2a shows the structure of the 1 \(\times\) 10 partial H-plane waveguide slot array antenna. The slot length is nearly \(\lambda_{0}/2\), and its width is assumed to be small. To be excited in phase for all the slots, the array with slots spaced \(\lambda_{\mathrm{g}}/2\) apart and with alternating slots on the opposite side of the center line is employed [22; 23]. The longitudinal shunt slot does not radiate when the slot is located at the centered line because the transverse current and the longitudinal magnetic field are zero in the dominant mode. The longitudinal magnetic field symmetrically increases with offset distance from the centered line of the top plane of a partial H-plane waveguide. The slot offset is determined to be at a maximum value of 4.25 mm considering a fabrication process so that maximum excitation level at each slot can be achieved. The extended ground is designed to place the unit cell of an FSS on the slot symmetrically. The unit cell of an FSS is described in detail in Section 3. The dimensions of the designed 1 \(\times\) 10 partial H-plane waveguide slot array antenna operating at 12 GHz are listed in Table 1. In summary, the slot length (\(l\)) and the slot periodicity (\(p\)) are nearly \(\lambda_{0}/2\) and \(\lambda_{\mathrm{g}}/2\), respectively, and are optimized for in-phase excitation at all slots. The slot width is 0.35 mm in this paper, so that can be assumed to be small. Also, when the slot offset (\(x\)) increases, the aperture efficiency is improved. Figure 2b shows the structure of the 8 \(\times\) 10 partial H-plane waveguide slot array antenna with power divider operating at 12 GHz. To maintain the electric and magnetic field distribution in the divided partial H-plane waveguide, the vane of the input port of the power divider is split and connected to the output ports. The 8 \(\times\) 10 partial H-plane waveguide slot array antenna is designed using a seven two-way power divider, as shown in Figure 2b. To excite the dominant mode of the designed array antenna, the coaxial transition as a feeding structure is used considering a fabrication. A coaxial probe from the narrow sidewall is inserted into the rectangular intaglio in the metal vane, and it is located at about a quarter wavelength long distance from the end metal wall. After optimizing the feeding structure, we find the dimension of a rectangular intaglio in the metal vane. The width and height of the rectangular intaglio are 3.4 mm and 2.26 mm, respectively. We confirm that all transmission is achieved, and the dominant mode is excited at partial H-plane waveguide at 12 GHz. The full-wave simulated reflection coefficient and realized gain of the designed array antenna are presented in Figure 3a and b, respectively. The simulated reflection coefficient is \(-\)23.67 dB at the operation frequency, and the \(-\)10 dB fractional bandwidth is 0.93%. The simulated maximum gain of the designed slot array antenna is 25.3 dBi at 12 GHz. As the width of the partial H-plane waveguide is inherently smaller than that of the conventional waveguide, the partial H-plane waveguide slot array antenna can have more radiating slots than the conventional waveguide slot array antenna in the same design area. Therefore,  
Figure 1: Structure of the partial H-plane waveguide.