## 2 Design of the FSS
### Design Evolution and Configuration  
The design flow chart is shown as in Figure 1. The design process can be divided into four steps. The first step is the selection of the initial structural model. We choose the square gap ring element as the initial model (see Figure 1a). In the second step, in order to achieve miniaturization, we convolute the square stripe in the unit cell (see Figure 1b). To further achieve miniaturization, we add branches to the outermost square ring patch for interdigital processing (see Figure 1c). Finally, we carry out specific parameter optimization.  
The proposed FSS unit cell is shown in Figure 2. This compact convoluted and interdigitated arrangement effectively increases the corresponding equivalent inductance. By adjusting the distance of the slot, the corresponding equivalent capacitance can be increased. Therefore, by using convolution and interdigitation technologies, the internal coupling is well controlled, the phenomenon of grating lobes is avoided, and the space efficiency and miniaturization are improved to the maximum extent. In addition, this centrosymmetric structure improves the stability of FSS under different polarizations and incident angles and ensures the practicability of FSS.  
The proposed FSS with \(2\times 2\) array elements is shown in Figure 3, in which a metal layer is sandwiched by two dielectric layers. The array is arranged in a square grid. In the figure, \(D_{x}\) and \(D_{y}\) are the periods of the proposed FSS in the \(x\)-axis and \(y\)-axis directions, and their values can be equal or unequal, but the values must meet the FSS requirements for array spacing to avoid or delay the appearance of grating lobes. To make the structure highly symmetrical and enhance the angular and polarization stability of the new structure, \(D_{x}=D_{y}\) is taken here. \(W_{1}\) and \(W_{2}\) are the widths of the inner and outer branches of the square ring aperture. Also, \(W_{1}=W_{2}\) is taken here. \(G\) is the width of the aperture. To be able to insert more branches, the value of \(G\) should be as small as possible. However, the value must not be too small, otherwise, the electromagnetic wave will no longer propagate along with the curved aperture.  
Figure 1: Design flowchart: (**a**) initial structure; (**b**) convoluted square stripes; (**c**) added branches to be convoluted and interdigitated stripe.  
Figure 2: Diagrammatic sketch of the proposed unit: (**a**) top view; (**b**) side view.  
The model consists of a double-layer dielectric material and a single-layer metal surface, where the metal layer is located in the center of the dielectric layer. The green part is the dielectric substrate. The orange part is the metal, and its thickness can be ignored in the simulation process. The material of the top and the bottom dielectric layer is Rogers RT 5880 (tm) with \(\varepsilon_{\mathrm{r}}=2.2\), \(\tan\delta=0.0009\), and \(H=0.8\)mm. The proposed FSS has a miniaturized size of only 3.94 mm \(\times\) 3.94 mm (0.09\(\lambda\)\(\times\) 0.09\(\lambda\), where \(\lambda\) is the free space wavelength at 7.5 GHz). Through analysis and optimization by HFSS software, the specific structural parameters are listed in Table 1.