## 2 Design Theory and Methodology  
### Design and Structures  
As shown in Figure 1, a chessboard metasurface typically consists of alternating arrangements of the AMC1 block and AMC2 block, resembling a checkerboard pattern, with each block containing n \(\times\) n unit cells. When a plane wave is incident upon this metasurface, the two types of AMC blocks exhibit different phase and amplitude responses to the incident wave:  
\[E_{s_{1}}=A_{1}E_{i_{1}}e^{i\varphi_{1}} \tag{1}\]  
\[E_{s_{2}}=A_{2}E_{i_{2}}e^{i\varphi_{2}} \tag{2}\]  
where \(E_{s_{1}}\) and \(E_{s_{2}}\), respectively, denote the electric fields of incident and reflected waves for AMC1 and AMC2 blocks, while \(\varphi_{\text{n}}\) and \(A_{\text{n}}\) represent the reflection phase and amplitude for the incident wave by both AMC types. By utilizing the definition of RCS, it is known that compared to a PEC with the same size (with a reflectivity of 1), the RCS reduction in dB can be calculated as follows [25]:  
\[\text{{RCS } reduction}=10\log_{10}\frac{\underset{\underset{\tau\to \infty}}{lim}\left[4\pi r^{2}\frac{\left|E_{s}\right|^{2}}{\underset{\underset{ \tau\to\infty}}{lim}\left[4\pi r^{2}(1)^{2}\right]}\right]}{\underset{\underset {\tau\to\infty}}{lim}\left[4\pi r^{2}(1)^{2}\right]} \tag{3}\]  
Considering the equal areas of both types of AMC, it is reasonable to assume that \(E_{i_{1}}=E_{i_{2}}\). Additionally, due to the presence of the ground layer, it is evident that \(A_{n}=1\). By synthesizing the formulas mentioned above, the RCS reduction can be represented using \(\varphi_{1}\) and \(\varphi_{2}\). To achieve a reduction greater than \(-10\) dB, the following condition must be satisfied:  
\[20\log_{10}\left(\frac{e^{i\varphi_{1}}+e^{i\varphi_{2}}}{2}\right)<-10\ dB \tag{4}\]  
Solving the above equation, it becomes apparent that a phase difference ranging from \(143^{\circ}\) to \(217^{\circ}\) can result in an RCS reduction exceeding \(10\) dB. For simplicity and to ensure  
Figure 1: AMC chessboard metasurface configuration.  
the effectiveness of the RCS reduction in practical applications, an effective phase difference of 180\({}^{\circ}\)\(\pm\) 30\({}^{\circ}\) is adopted.  
Due to the complex correlation between the phase response and geometric structure, it is challenging to define these relationships precisely. To achieve diverse reflect phase, we manipulate the geometrical structure of the metasurface using ten parameters.  
During the parameterization process of AMC, a square canvas with dimensions of w \(\times\) w, determined by the unit cell width, w, is initially prepared. On this canvas, two rectangular conductive elements, with predefined dimensions, center coordinates, and rotation angles, are positioned as depicted in Figure 2a. These rectangles are then symmetrically duplicated along the canvas's diagonal, resulting in four identical rectangles. Subsequent symmetrical duplication along both the horizontal and vertical axes, with the canvas center as the reference point, yields a unit cell composed of 16 rectangular elements, as shown in Figure 2b, and the unit cells are constructed using an FR4 dielectric substrate, which possesses a thickness of 3 mm and a relative permittivity of 4.4 [26]. Governed by a set of 10 parameters, this methodology enables the sculpting of the unit cell's geometry. Strategic operations such as symmetrical duplication facilitate the creation of unit cells with various configurations, thereby enabling diverse phase responses. Moreover, the unit cells generated by this method exhibit rotational symmetry (remaining identical upon a 90\({}^{\circ}\) rotation), thereby offering polarization insensitivity.