## 2 Antenna Structure
### Hybrid Coupler Design  
Various feeding networks are used for exciting the antennas to achieve dual circular polarization, such as sequential rotation feeding [25], hybrid couplers [26], coupled line [27], two ports with self-interference cancellation (SIC) circuits [28], \(\Gamma\) and fork-shaped strip lines [29], and so on. A four-port 3 dB hybrid coupler is utilized for feeding a circular patch to generate dual-circular polarization radiation (RHCP and LHCP). Two of its ports are considered to be input ports and the other two are output ports. Depending on which one of the input ports is excited, the direction of rotation (LHCP or RHCP) is determined. Figure 2 depicts the geometry of a 3 dB hybrid coupler. Using the even and odd modes analysis of this hybrid coupler [30], the resulting S-matrix based on the port arrangement shown in Figure 2 at the center frequency should be as follows:  
\[[S]=\frac{1}{\sqrt{2}} \begin{bmatrix}0&0&1&j\\ 0&0&j&1\\ 1&j&0&0\\ j&1&0&0\\ \end{bmatrix}  \tag{1}\]  
When choosing port 1 as an input port, the first row of the matrix is determined. The reflection is zero, and the transmission to port 2 is also zero. Therefore, port 2 is perfectly isolated from port 1. Port 3 and port 4 have an equal magnitude of insertion loss, which is equal to \(\frac{1}{\sqrt{2}}\) (or 3 dB). However, the phase difference between the ports is \(90^{\circ}\), which is indicated by the (\(j\)) operator in the matrix. Using the CST microwave studio suite, this coupler is designed at the center frequency of 3.7 GHz with the FR4 dielectric substrate and height (h) of 1.6 mm. The width of the coupler lines (\(W\)) can be calculated for each characteristic impedance (\(Z_{o}\)) as in (2) [30]:  
\[W=\left\{ \begin{array}{cc}\frac{8 hc^{4}}{e^{2}A-2}&for\frac{W}{h}\leq 2\\ \frac{2h}{\pi}[\text{B}-1-\ln(2\text{B}-1)+\frac{\epsilon_{\text{r}}-1}{2 \epsilon_{\text{r}}}\{\ln(\text{B}-1)+0.39-\frac{0.61}{\epsilon_{\text{r}}} \}]&\text{for}\ \text{W}/\text{h}\ >2\\ \end{array} \right. \tag{2}\]  
where \(\epsilon_{\text{r}}\) is the dielectric constant of the substrate, and  
\[\text{A}=\frac{Z_{o}}{60}\sqrt{\frac{\epsilon_{r}+1}{2}}+\frac{\epsilon_{r}- 1}{\epsilon_{r}+1}(0.23+\frac{0.11}{\epsilon_{r}});\ \text{B}=\frac{377\pi}{2Z_{o}\sqrt{\epsilon_{r}}}\]  
Figure 1: Structure of the proposed antenna: (**a**) front view, (**b**) hybrid coupler (sub.1), (**c**) the patch (sub.2); (**d**) side view. (Unit: mm). For \(Z_{o}=50\)\(\Omega\), \(W=3.1\) mm. For \(Z_{o}/\sqrt{2}=35\)\(\Omega\), \(W=5.4\) mm. The effective dielectric constant \(\varepsilon_{reff}\) of coupler lines is given by the following equation [30]:  
\[\varepsilon_{reff}=\frac{\varepsilon_{r}+1}{2}+\frac{\varepsilon_{r}-1}{2}(1+ \frac{12h}{w})^{-0.5} \tag{3}\]  
For \(Z_{o}=50\)\(\Omega\), \(\varepsilon_{reff}=3.27\). For \(Z_{o}/\sqrt{2}=35\)\(\Omega\), \(\varepsilon_{reff}=3.42\). The quarter wavelength of the coupler lines with respect to the center frequency \(f_{o}\) is formulated in (4) [24] as shown below:  
\[\frac{\lambda}{4}=\frac{\lambda_{o}}{4\sqrt{\varepsilon_{reff}}}=\frac{c}{4f_ {o}\sqrt{\varepsilon_{reff}}} \tag{4}\]  
For \(Z_{o}=50\)\(\Omega\), \(\frac{\lambda}{4}=11.2\) mm. For \(Z_{o}/\sqrt{2}=35\)\(\Omega\), \(\frac{\lambda}{4}=10.96\) mm. As a result of these parameters, the hybrid coupler is designed. Figure 3a illustrates the general structure of this hybrid coupler. The reflection coefficient (\(S_{11}\)) in Figure 3b shows a reflection coefficient (\(S_{11}\)) with a wide matching range of frequencies (3.18-5.07 GHz) with a value equal to \(-22.08\) dB at the center frequency of 3.7 GHz. The insertion loss of the coupler is relatively large due to the inherent 3 dB losses of the hybrid coupler and the losses of the FR4 dielectric substrate. Therefore, using lower loss dielectric materials for the substrate, such as Rogers (RO5880), can reduce these losses noticeably.