## 2 Structural Design and Principles  
Herein, we propose an ultra-thin and ultra-wideband linear polarizer, and the unit cell is shown in Figure 1a. It consists of three layers. The top layer comprises the double semicircular gap patches. The intermediate layer is FR-4 with a dielectric constant of 4.3 and loss tangent of 0.025. The bottom layer is copper with a conductivity of 5.8 \(\times\) 10\({}^{7}\) S/m and a thickness of 0.035 mm. The detailed structural parameters are as follows: \(p\) = 6 mm, \(a\) = \(b\) = \(c\) = 1 mm and \(t\) = 1.5 mm.  
In order to clearly understand the principle of polarization conversion, we used CST Microwave Studio for the simulation analysis. The principle of the polarization converter is shown in Figure 1b. The \(u\)-\(v\) axis is obtained by rotating the \(x\)-\(y\) axis 45\({}^{\circ}\) counterclockwise. Since the structures have symmetry along the \(u\)-\(v\) axis, they have the same polarization conversion capability regardless of whether the electric field of the incident wave is in the \(x\) or \(y\) direction. Assume that the incident and reflected electric fields are along the \(x\) and \(y\) axes, respectively. \(E_{ip}\) (\(E_{iu}\)) and \(E_{rv}\) (\(E_{ru}\)) represent the electric field components of the incident and reflected waves on the \(v\) (\(u\)) axis, respectively. Because of the anisotropy of the structure, if \(r_{uu}\) = \(r_{vv}\), \(\Delta\varphi\) = \(\varphi_{uu}\) - \(\varphi_{vv}\) = \(\pi\), then the electric field along the \(x\) axis will be reversed, and the synthetic reflection of \(E_{rv}\) and \(E_{ru}\) will propagate along the \(y\) axis. Herein, \(r_{uu}\)(\(r_{vv}\)) and \(\varphi_{uu}\)(\(\varphi_{vv}\)) represent the amplitude and phase along the \(u\), \(v\) axes, respectively. The incident \(x\)-polarized wave is decomposed to the \(u\), \(v\) axes, and the simulated amplitude and phase difference are shown in Figure 2. Obviously, the amplitudes of \(r_{uu}\) and \(r_{vv}\) are almost coincident and the phase difference is about 180\({}^{\circ}\) \(\pm\) 37\({}^{\circ}\) at 14.8-28.0 GHz, and the  
Figure 1: (**a**) Schematic diagram of unit structure parameters. (**b**) Diagram of the working principle of polarization conversion.  
phase difference at 15.45 GHz, 20.82 GHz and 26.94 GHz is equal to 180\({}^{\circ}\), which means that it has an efficient conversion capability in the ultra-wideband range.  
We also give the equivalent circuit models based on transmission line theory to analyze the behavior of the proposed structure, as depicted in Figure 3. The top metal structure can be equivalent to \(R_{1}\), \(L_{1}\) and \(C_{1}\) in series. The spacing between the double semicircular gap patches exhibits capacitance and can be equated to \(C_{d}\). The length of h transmission line represents the dielectric substrate. \(Z_{0}\) and \(Z_{1}\) are the equivalent characteristic impedance of the free space and dielectric substrate, respectively. The equivalent admittance and impedance of the top structures are as follows:  
\[Y_{re}=\frac{1}{R_{1}+j\omega L_{1}+\frac{1}{j\omega\overline{C_{1}}}}+j\omega C _{d}, \tag{1}\]  
\[Z_{re}=\frac{1}{Y_{re}}\]  
The equivalent impedance of the dielectric substrate is as follows:  
\[Z_{s}=jZ_{0}\tan(\beta h), \tag{2}\]  
\(\beta\), \(h\) are the phase constant and thickness of the dielectric substrate, respectively.  
The equivalent impedance of the polarizer is expressed as follows:  
\[Z=\frac{Z_{re}\cdot Z_{s}}{Z_{re}+Z_{s}}, \tag{3}\]  
The equivalent impedance of the polarizer can be determined by tuning the structure period, thickness, double semicircular gap patch parameters, etc.  
Figure 3: The equivalent circuit model.  
Figure 2: The amplitude and phase difference of \(u\) and \(v\) axes.