# Dual-Band Shared-Aperture Antenna with Pattern and Polarization Diversity
## 2 Antenna Design and Analysis
### Vivaldi Antenna Design
The linear polarized electric field of the Vivaldi antenna enters the circular polarization converter. Therefore, the electric field is decomposed into two orthogonal electric fields along the \(u\) and \(v\) directions, which are indicated as \(E_{u}\) and \(E_{v}\). Because the different structural composition of the polarizer along the \(u\) and \(v\) directions leads to different equivalent dielectric constants in the two paths, the two electric fields in the \(u\) and \(v\) directions produce a phase difference of 90\({}^{\circ}\). The equivalent dielectric constant in the direction of the medium polarization converter of the \(u\) direction can be written as [30]  
\[\varepsilon_{u}=\varepsilon_{1}W_{2}+\varepsilon_{2}W_{3} \tag{2}\]  
where \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are the relative permittivity of the air and the medium, respectively. The equivalent dielectric constant in the \(v\) direction can be written as [30]  
\[\varepsilon_{v}=\frac{\varepsilon_{1}\varepsilon_{2}}{\varepsilon_{1}W_{2}+ \varepsilon_{2}W_{3}} \tag{3}\]  
Therefore, the phase difference between them can be written as  
\[\alpha=\Delta\beta\cdot H_{2}=(\beta_{u}-\beta_{v})\cdot H_{2}=\frac{2\pi f \cdot H_{2}}{c}(\sqrt{\varepsilon_{u}}-\sqrt{\varepsilon_{v}}) \tag{4}\]  
where \(\beta_{u}\) and \(\beta_{v}\) are the phase constants in the \(u\) and \(v\) directions, respectively. \(H_{2}\) represents the height of the medium polarization converter. A circular polarization wave can be generated when two orthogonal linear polarization waves are of the same magnitude and have a phase difference of \(\pm 90^{\circ}\). According to the above equation, the circular polarization radiation can be realized by adjusting the medium slabs and air widths to control the phase difference of the electric field in the \(u\) and \(v\) directions. The medium polarization converter designed in this paper can not only convert the Vivaldi antenna from linear polarization to circular polarization but also improve the radiation gain of the Vivaldi antenna. As Figure 9b shows, when the electromagnetic wave of the Vivaldi antenna is transmitted to free space along the \(+z\)-axis direction through a medium polarization converter, the medium does not change the propagation path of the  
Figure 9: Medium polarization converter structure and electromagnetic wave passage path. (**a**) Top view, and (**b**) side view. wave. However, when the wave from the bottom is oblique into the medium polarization converter, the wave, after two refractions, will converge in an axial direction according to [PERSON]'s law, which will improve the gain. Figure 10a shows the high-frequency reflection coefficient comparison of the Vivaldi antenna in the hollow cone with and without the polarization converter. It can be seen that the addition of the medium polarization converter causes little fluctuation in the reflection coefficient of the Vivaldi antenna, but the working bandwidth does not change significantly. The IMBW of the antenna is 21.6-29.6 GHz (8 GHz, 31.2%). The gain of the Vivaldi antenna with and without a polarization converter is shown in Figure 10b. Before incorporating the medium polarization converter, the Vivaldi antenna can achieve a maximum gain of 17 dBi at 29 GHz, and the gain within the bandwidth is relatively stable. Adding the medium polarization converter helps to improve the gain consistently across the entire working bandwidth. With the addition of the polarization converter, the axial ratio is less than 3 dB within 21.2-29.9 GHz, which proves that the proposed antenna has good circular polarization radiation ability in the high-frequency band. The peak gain of the proposed antenna is 18.9 dBi, and the gain fluctuation in the whole bandwidth is relatively flat, which is less than 3 dBi within the axial ratio bandwidth (ARBW) of 21.2-29.9 GHz. The gain within the ARBW is higher than 16 dBi. The efficiency across the 2.09-11.61 GHz (9.52 GHz, 138.9%) and 21.6-29.6 GHz (8 GHz, 31%) are higher than 89.4% and 84%, respectively, as shown in Figure 10c.