## Circular Polarizer and Circularly Polarized Patch Antenna Design
### Design of Circular Polarizer
It can be attributed to the structure difference along the two orthogonal tilted directions of the DSSRs. Since we can derive the radiation field from the surface current on the radiator, the total far-field \(E_{Z\mathrm{T}}\) radiating from the polarizer as it integrates into antennas can be expressed as follows [28]:  
\[E_{Z\mathrm{T}}=k_{\#1}E_{\#1}+k_{\#2}E_{\#2}, \tag{1}\]  
where \(E_{\#1}\) and \(E_{\#2}\) are the electric field components relative to mode #1 and #2, respectively, and \(k_{\#1}\) and \(k_{\#2}\) are corresponding weight coefficients. The calculation of \(k_{\#1}\) and \(k_{\#2}\) can be carried out by applying weight functions (for example, one possible matrix representation is \(\left[ \begin{array}{cc}1&1\\ -1&1\end{array} \right]\), which takes into account the interaction between \(E_{\#1}\) and \(E_{\#2}\)), by multiplication and integration on both sides of Equation (1). Furthermore, the AR of the radiated EM field can be calculated by using \(k_{\#1}\) and \(k_{\#2}\) in terms of the following equation [28; 29]:  
\[AR=\sqrt{\frac{\left|k_{\#1}\right|^{2}\cos^{2}r+\left|k_{\#1}\right|\left|k_ {\#2}\right|\sin 2r\cos\Delta\varphi+\left|k_{\#2}\right|^{2}\sin^{2}r}{\left|k_{ \#1}\right|^{2}\sin^{2}r-\left|k_{\#1}\right|\left|k_{\#2}\right|\sin 2r\cos \Delta\varphi+\left|k_{\#2}\right|^{2}\cos^{2}r}}. \tag{2}\]  
In this way, for the polarizer we found that \(k_{\#1}=k_{\#2}\), \(\left| \begin{array}{cc}E_{\#1}&\approx&\left|\begin{array}{cc}E_{\#2}\\ \end{array}\right|\end{array} \right.\), and \(\angle E_{\#2}-\angle E_{\#1}=90^{\circ}\) (\(\angle E_{\#2}=45^{\circ}\), \(\angle E_{\#1}=-45^{\circ}\), which are referring to the \(+\)-axis). Thus, a RHCP polarizer is achieved. In contrast, if we properly change the polarizer and control the parameters such that \(\angle E_{\#2}-\angle E_{\#1}=-90^{\circ}\) (\(\angle E_{\#2}=-45^{\circ}\), \(\angle E_{\#1}=45^{\circ}\)), the LHCP characteristics can be achieved. The offset distance of the bottom split can be used to tune the phase difference between \(E_{\#1}\) and \(E_{\#2}\). Further investigation and validation to the physics of the circular polarizer were carried out in terms of the waveguide model, as shown in Figure 4a. As mentioned previously, the polarizer is  
Figure 3: Surface current distribution on the DSSRs of the polarizer at its modes #1 and #2, corresponding eigenfrequencies \(f_{\#1}=2.5189\) GHz and \(f_{\#2}=2.5190\) GHz, respectively, in terms of the eigenmode model. illuminated by LP waves below, the bottom side of the waveguide model is set to wave port 1, and the polarizer is suspended above. Also, a perfect electric conductor (PEC) and a perfect magnetic conductor (PMC) boundary are assigned to the model's opposite sides lying on the x- and y-axis, respectively. Furthermore, wave port 2 is set up on the top side of the model. Thus, as an incident LP plane wave transmitting from wave port 1 and passing through the polarizer, a CP wave would be expected to be found at wave port 2 as the operation implied by circular polarizers. Based on such a model, it was found that revising the offset of the dual bottom splits that go along the opposite direction to each other results in different sensing circular polarizations. Also, with a proper equal offset distance assigned to the dual bottom splits, such as 3.5 mm (along the direction as the bottom splits shown in Figure 4a), a considerable CP (RHCP) property can be achieved. In fact, we can fix the offset distance in the \(3.5\pm 0.5\) mm range, where an acceptable CP and impedance matching performance can be maintained. While revising the dual bottom splits at such an equal distance, opposite sensing CP (LHCP) characteristics can be obtained. With appropriate separations between the polarizer surface and the model's upper and lower sides maintained, and a carefully optimized substrate size of the polarizer, the transmission coefficient \(\mid\)S\({}_{21}\mid\) between the wave port 1 and 2 was calculated as shown in Figure 4b.