## 2 Antenna Design Theory  
### Omnidirectional Circular Polarization Analysis  
The radiation slots of the omnidirectional circularly polarized (CP) antenna are illustrated in Figure 1. In the antenna design, radiation is achieved through a pair of X-shaped slots arranged in 180\({}^{\circ}\) rotational mirror symmetry around the central axis of the circular waveguide. These slots are oriented with an angular separation \(\alpha\) of 45\({}^{\circ}\), and their close proximity enables an approximation of 360\({}^{\circ}\) omnidirectional radiation coverage. According to the operating principles of slot antennas, the slots act as equivalent magnetic current sources when excited. The effective length of such slots is typically defined as the vertical projection relative to the feeding direction. Consequently, the slot length in this design is optimized to half the operating wavelength [15], ensuring impedance matching and efficient radiation. The mechanism of circular polarization generation is shown in the following diagram:  
Figure 1: Schematic diagram of electric field vector and parameter of X-shaped gap.  
Assuming that the total radiated electric field of the two gaps is \(\overset{\rightarrow}{E}\), the electric field radiated by the first gap is \(\overset{\rightarrow}{E_{1}}\), and the electric field radiated by the second gap is \(\overset{\rightarrow}{E_{2}}\), then the relationship between the radiated electric field of the two gaps and the total electric field is:  
\[\overset{\rightarrow}{E}=\overset{\rightarrow}{E_{1}}+\overset{\rightarrow} {E_{2}}=E_{x}\overset{\rightarrow}{a_{z}}+E_{y}\overset{\rightarrow}{a_{y}} \tag{1}\]  
Assuming \(E_{x}\) amplitude \(\hat{z}\) (\(\hat{z}>0\)), \(E_{y}\) amplitude \(\hat{y}\)(\(\hat{y}>0\)), and \(E_{y}\) is equal to the phase of \(E_{z}\), but because the center point of the two gaps is the same, the angle deviation of the physical direction \(\varphi\)(\(\varphi=90^{\circ}\)). Formula (1) can be written as:  
\[\overset{\rightarrow}{E}=\hat{z}\overset{\rightarrow}{a_{z}}e^{-j\kappa x} +\hat{y}\overset{\rightarrow}{a_{y}}e^{-j\kappa x}e^{-j\varphi} \tag{2}\]  
Its instantaneous expression is:  
\[\overset{\rightarrow}{E}(x,t)=\hat{z}\cos(\omega t-kx)\overset{\rightarrow} {a_{z}}+\hat{y}\cos(\omega t-kx+\varphi)\overset{\rightarrow}{a_{y}} \tag{3}\]  
Among:  
\[\left\{\begin{aligned} \overset{\rightarrow}{E_{z}}(x,t)=\hat{z} \cos(\omega t-kx)\\ \overset{\rightarrow}{E_{y}}(x,t)=\hat{y}\cos(\omega t-kx+\varphi) \end{aligned}\right. \tag{4}\]  
At \(x=0\), assume the amplitude of the radiated electric field of the two gaps \(\hat{z}=\hat{y}=A\):  
\[\left[\frac{\overset{\rightarrow}{E_{z}}(x,t)}{\hat{z}}\right]^{2}+\left[ \frac{\overset{\rightarrow}{E_{y}}(x,t)}{\hat{y}}\right]^{2}-\frac{2\overset {\rightarrow}{E_{z}}(x,t)\overset{\rightarrow}{E_{y}}(x,t)}{2\hat{y}}\cos \varphi=\sin^{2}\varphi \tag{5}\]  
The physical angles of the two slots are known as \(\varphi=90^{\circ}\):  
\[\left(\overset{\rightarrow}{E_{z}}(x,t)\right)^{2}+\left(\overset{ \rightarrow}{E_{y}}(x,t)\right)^{2}=A^{2} \tag{6}\]  
Benefiting from the symmetric configuration of the X-shaped slots, the balance between two orthogonal radiating components can be maintained by optimizing their slot positions and spacing \(\hat{z}=\hat{y}=A\), thereby generating circularly polarized (CP) waves. Furthermore, the X-shaped slot design enables dual-CP characteristics (i.e., simultaneous left-handed and right-handed circular polarization), providing identical transmit/receive performance for both polarization senses. This capability enhances adaptability to complex electromagnetic environments, such as multipath interference or polarization-mismatched scenarios in satellite communications.