## 2 Planar Ultra-Wideband FSS Design
\begin{array}{l}\varepsilon_{zx}=\ \iint\limits_{\Omega}\varepsilon( \vec{z})A_{1z}^{x}d\Omega\\ \varepsilon_{zy}=\ \iint\limits_{\Omega}\varepsilon(\vec{z})A_{1z}^{y}d\Omega\\ \varepsilon_{zz}=\ \iint\limits_{\Omega}\varepsilon(\vec{z})(1+A_{1z}^{z})d \Omega\\ \end{array} \right.. \tag{16}\]  
It is shown from Equations (1) to (16) that the calculated effective dielectric constant of the homogenization field is composed of two parts: (1) The dielectric constant of the background material and the dopant material, while in this paper they are FR4 and perforated air through holes, respectively; (2) vector field correction term \(\mathbf{A}(x,\xi)\), which is related to the frequency. References [16; 17] utilized the gradient of the scalar potential function as the correction term in the small scale. However, when this correction term is used in the metamaterial studies, the equivalent electromagnetic parameters are only influenced by quasi-static field, the contribution of the rotational field is not involved. In terms of this issue, this paper introduces vector field correction term into consideration, thus, both divergence field and rotational field are involved. Furthermore, in comparison with the traditional Maxwell-Garnett theory, which homogenizes the dielectric constant to a constant value, this multiscale homogenization method can reveal the dielectric constant variation related with frequency. Based on the above theory, the electromagnetic parameters of the periodic units can be calculated. The equivalent relative dielectric constant of the PM FSS structure in Figure 1 is shown in Figure 2a. This curve states that the medium has an effective dielectric constant which changes in the 2.68 to 2.72 range within 6 to 25 GHz. The change in value is the effect of the vector field correction term related to the frequency. The variation is small because of the thin FSS unit thickness and the relatively simple background. The slight dielectric constant variation has little impact on the radome manufacture, even the process specification of distinguished corporation may have a 0.05 relative dielectric constant variation [27]. The FSS unit is simulated using the Finite Element Method (FEM) method in the infinite periodic structure, as shown in Figure 2b. The simulated transmission coefficients result of the transverse electric (TE) incident wave through the planar FSS surfaces are shown in Figure 3, where the equivalent medium means an equivalent 2.70 dielectric constant medium in the figure, and the transmission phase means the phase variation of the transmitted wave from the FSS unit upper surface to the lower surface. The two curves are the results of the equivalent 2.70 dielectric constant medium and the PM FSS, respectively, and the results of the 30\({}^{\circ}\) incident wave are also shown. From the comparison of the results, the two FSS with different media have very close transmission coefficients within the  
Figure 2: (**a**) The equivalent relative dielectric constant of the PM; (**b**) the schematic diagram of the FSS unit in infinite periodic boundary simulation. operating frequency range, especially the transmission coefficients phase. As far as the normalized transmission coefficients is concerned, transmission coefficients in the 2 to 8 GHz range are almost the same because the FSS is not in its working band, small dielectric constant variations can hardly change the transmission coefficients. In the 8 to 17 GHz range, the equivalent 2.70 dielectric constant medium has better transmission coefficient values because this FSS structure is intended for an ideal 2.70 dielectric constant, while the PM FSS dielectric constant is not equal to it, thus reducing the performance. And in the 17 to 25 GHz range, the wavelength of the incident wave gets much closer to the unit dimension and perforation size, thus more portions of the incident wave are transmitted through the FSS.