# Circularly Polarized Vivaldi Antennas Integrated with Septum-like Polarizer
## 2 Materials and Methods
### CP Vivaldi Antenna with Ideal Dual Ports  
To verify the design approach of the proposed CP Vivaldi antenna using a pulse train-shaped polarizer, a dual port design is considered as shown in Figure 2. The proposed CP antenna consists of two orthogonal Vivaldi antennas in the \(xz\) and \(yz\) planes, with the antenna in the \(xz\) plane containing a pulse train-shaped polarizer whose width is widened in a geometric sequence. As the phase difference between the dual ports is set to \(0^{\circ}\) or \(180^{\circ}\), the CP Vivaldi antenna operates in a right-handed circular polarization (RHCP) or left-handed circular polarization (LHCP) mode, respectively. The impedance of the dual ports and feeding lines is set to 100 \(\Omega\) considering the connection with the feeding network in the next step. In addition, the dielectric slots exist for orthogonal physical assembly. For broadband impedance matching, the feeding structure of the open-ended tapered micro-strip lines (or feeding radial stubs) and circular cavity are adopted, the same as for conventional CP Vivaldi antennas. Two orthogonal Vivaldi antennas are implemented as exponential slot edges for broadband impedance matching, and the shape of the slot edges is determined from the following formula:  
\[x=\text{SSP}+\frac{\text{SW}_{m}}{2}\times e^{\varphi_{m}z} \tag{1}\]  
where SSP, SW\({}_{m}\), and \(\varphi_{m}\) represent the starting point, width, and tapering angle of the slot. \(m=1\) and \(m=2\) represent the cases for the \(xz\) and \(yz\) planes, respectively. Also, RD is set to be 10 mm. The detailed structure of the CP Vivaldi in the \(xz\) and \(yz\) planes is shown in Figure 3. The pulse train-shaped polarizer is designed with increasing pulse spacing (PD\({}_{n}\)) and width (PW\({}_{n}\)) for broadband impedance matching, where \(n\) represents the stage index of the pulse train. The total number of stages in the pulse trains is set to be nine for the wide axial ratio (AR) bandwidth. As the pulse train-shaped polarizer is added, the impedance mismatch occurs for the Vivaldi antenna in the \(xz\) plane, causing an imbalance in the radiation intensity of the two orthogonal antennas. Thus, by optimizing the slot width (SW\({}_{m}\)) and the shape of the exponential slot edge in the \(xz\) and \(yz\) planes, the radiation intensity of the two orthogonal polarizations can be balanced. To reduce the  
Figure 1: Comparison of a typical septum polarizer for conventional waveguides and a proposed polarizer for Vivaldi antennas.  
number of optimization cases, the \(\text{PD}_{n}\) and \(\text{PW}_{n}\) of each stage are defined by the following geometric sequence models:  
\[\text{PD}_{n+1}=\text{PD}_{n}+\text{PD}_{\text{add}} \tag{2}\]  
\[\text{PW}_{n+1}=\text{PW}_{n}+\text{PW}_{\text{add}} \tag{3}\]  
where the \(\text{PD}_{1}\) and \(\text{PW}_{1}\) correspond to the initial pulse distance and width, and \(\text{PD}_{\text{add}}\) and \(\text{PW}_{\text{add}}\) correspond to the common radio of the geometric sequence. Finally, ten variables of PS, \(\text{PD}_{1}\), \(\text{PW}_{1}\), \(\text{PD}_{\text{add}}\), \(\text{PW}_{\text{add}}\), SSP, \(\text{SW}_{1}\), \(\text{SW}_{2}\), \(\varphi_{1}\), and \(\varphi_{2}\) are specified that mainly determine the performance of the proposed CP Vivaldi antenna. However, the performance optimization faces the challenge of being time-consuming due to the multiple variables.  
Figure 3: Detailed view of the proposed CP Vivaldi antenna with a pulse train-shaped polarizer. (**a**) Perspective view, (**b**) \(xz\) plane, and (**c**) \(yz\) plane. The slot edges increase exponentially, and the spacing and width of the pulse train-shaped polarizer widen in a geometric sequence for broadband AR.  
Figure 2: Proposed CP Vivaldi antenna with ideal two ports. (**a**) Perspective view, (**b**) top and (**c**) bottom view of the Vivaldi antenna in the \(xz\) plane, (**d**) top and (**e**) bottom view of the Vivaldi antenna in the \(yz\) plane.