# Design of a Polarization-Insensitive and Wide-Angle Triple-Band Metamaterial Absorber
## 2 Modeling and Theoretical Analysis  
The schematic diagram of the structure designed in this paper is shown in Figure 1. Considering the electromagnetic properties and manufacturability, as well as the process of combining different materials, the top and bottom layers are made of copper with a thickness of H1 = 0.035 mm, and the middle dielectric layer is made of FR4 with a thickness of H = 1 mm. The conductivity of copper is 5.8107 S/m, the dielectric constant of FR4 is 4.3, and the loss tangent is 0.025. Different colors are used in Figure 1 to indicate the structural parameters of different layers. Dark blue A, orange B, and blue C correspond to the inner layer, middle layer, and outer layer structure, respectively. The bottom layer is completely covered by copper. The structural parameters after parameter optimization are shown in Table 1. As shown in Figure 2, the overall structure of the MMA is formed by the periodic combination of the unit structure shown in Figure 1. Since the sub-unit adopts a hexagonal shape, the setting of periodic boundary conditions is different from the conventional rectangular boundary condition setting, specifically set to the repetition period in the x direction, which is P \(\times\)\(\sqrt{3}/2\); furthermore, the other period direction forms a 60\({}^{\circ}\) angle with the x direction, and the repetition period is also P \(\times\)\(\sqrt{3}/2\). Strictly speaking, the boundary condition used here is a rectangular periodic boundary condition with an angle, i.e., a parallelogram boundary condition. The hexagonal unit cell extends infinitely in the two-dimensional space through the parallelogram boundary condition, thus forming a honeycomb structure. The schematic diagram of boundary conditions is shown in Figure 3.  
The simulation method used is the finite integral method, and the frequency domain solver used is shown in [44; 45; 46; 47; 48; 49; 50; 51]. The simulation frequency is set to 2-16 GHz. The incident direction of the plane wave is the negative direction of the Z axis, and the electric field vibration direction is the y direction. These simulation conditions are used as the standard in the following cases, where no explanation is given. For the structure simulated using the finite integral method [52; 53; 54; 55; 56; 57; 58; 59; 60], the reflectivity \(R(\omega)\) and transmittance \(T(\omega)\) after the incident wave acts on the structure can be calculated by the S parameters, \(R(\omega)=\left|S_{11}\right|^{2}\), \(T(\omega)=\left|S_{21}\right|^{2}\) (\(S_{11}\) is the input reflection coefficient,  
Figure 3: Schematic diagram of boundary conditions. Figure 2: Schematic diagram of periodic MMA. \(S_{21}\) is the transmission coefficient). According to [PERSON]'s thermal radiation law, the absorption rate can be calculated by the following formula:  
\[A(\omega)=1-R(\omega)-T(\omega) \tag{1}\]  
In the designed MMA, the copper covering the bottom layer effectively suppresses the transmission of electromagnetic waves, so the absorption rate can be further simplified to \(A(\omega)=1-R(\omega)\). The matching of the impedance of the MMA and the vacuum is often considered to be a manifestation of efficient absorption, where the impedance in the vacuum is 377 \(\Omega\). This indicates that the electromagnetic energy can enter the structure and be consumed, and there is no reflection or transmission [61; 62; 63; 64; 65; 66; 67; 68]. Since the thickness of the MMA (2H + H1 = 1.07 mm) is very small compared to the minimum wavelength (18.75 mm@16 GHz) of the incident wave, based on the equivalent medium theory [69], this MMA can be treated as a homogeneous medium, and its impedance can be described as follows:  
\[Z=\sqrt{\mu_{eff}/\varepsilon_{eff}} \tag{2}\]  
where \(\mu_{\text{eff}}\) and \(\varepsilon_{\text{eff}}\) represent the equivalent permeability and permittivity of MMA, respectively. At the same time, the refractive index and impedance of MMA can be calculated by reflection coefficient \(S_{11}\) and transmission coefficient \(S_{21}\).