## 2 Planar Ultra-Wideband FSS Design  
FSS usually uses a dielectric plate as the structural support of the planar unit. Study results show that high performance FSS relied on not only excellent structural design, but also its high-performing medium. Reference [6] proposed a grid square structure FSS, which can reach a broad bandwidth, but many medium parameters were not mentioned. Reference [8] used a high cost dielectric medium plate with a relative dielectric constant of 2.65 and 2 mm thickness to accomplish the FSS in reference [6], but merits of low profile, flat passband, and stable performance for different polarizations and various incident angles within 30\({}^{\circ}\) are achieved. These works are remarkable, but it will be highly valued in practical applications if low-cost materials can be promoted to a superior performance and realize wideband FSS. The PM proposed in this paper is a composite medium structure and can meet the requirement of wideband. Furthermore, the total weight and fabricating cost of huge radome can be significantly decreased by air through holes. The frequently-used material with a relative dielectric constant of 4.4 is adopted in this article. Using the through holes structure, a 2.70 equivalent dielectric constant can be obtained and an FSS structure is shown in Figure 1. The structure is partially optimizedbase on references [6; 8], both the processing difficulties and FSS performance have been taken into consideration for the geometric dimensions. There are four perforations through each FSS periodic unit medium, which aim to reduce the equivalent dielectric constant of the dielectric plate and decrease the weight, red cylinders in Figure 1 are the air through holes. The top and bottom layers are the same metallic patches, shown as the blue patches in Figure 1. The middle layer is a square metallic grid shown as the yellow part. The unit is a \(4\times 4\) mm\({}^{2}\) square with the following values: \(h=2\) mm, \(w=2\) mm, \(a=0.5\) mm, \(b=0.5\) mm, \(d=1\) mm. With the blueprint of the PM structure and FSS frame, a detailed investigation of the dielectric constant of the PM FSS must be performed. The multiscale homogenization method is utilized in this paper for the numeric analysis of the PM dielectric constant. The solutions of the multiscale problems are obtained by solving the multiscale equations respectively [16; 17]. When applying the Maxwell equations to the composite structure multiscale problems solving, there are:  
\[\left\{ \begin{array}{rl}j\omega\epsilon\mathbf{E}&=&\
abla\times\mathbf{H}\\ j\omega\mu\mathbf{H}&=&-\
abla\times\mathbf{E}\end{array} \right., \tag{1}\]  
and the electric field vector in passive region follows:  
\[\
abla\times\frac{1}{\mu}\
abla\times\mathbf{E}-\omega^{2}\epsilon\mathbf{E}\ =\ 0. \tag{2}\]  
In multiscale problems, \(\xi\) represents a microscale variable, and \(x\) represents a large-scale variable. The electric field of multiscale problem can be resolved into the average field and microscale field:  
\[\mathbf{E}(x,\xi)\ =\ \mathbf{E}(x)+\mathbf{A}(x,\xi), \tag{3}\]  
where \(\mathbf{A}(x,\xi)\) is the correction term in small scale and is an arbitrary vector field which can be applied to situations of both rotational field and divergence field. Substituting Equation (3) into Equation (2):  
\[\
abla\times\
abla\times[\mathbf{E}(x)+\mathbf{A}(x,\xi)]-\mathbf{k}^{2}\varepsilon_{r}[ \mathbf{E}(x)+\mathbf{A}(x,\xi)]\ =\ 0, \tag{4}\]  
Figure 1: Perforated medium frequency selective surface (PM FSS) structure: (**a**) 3D semitransparent perspective view; (**b**) semitransparent side view; (**c**) plan view of the top and bottom layers and (**d**) plan view of the middle layer. \[\left\{ \begin{array}{l}\varepsilon_{yx}=\ \iint\limits_{\Omega}\varepsilon( \vec{x})A_{1y}^{x}d\Omega\\ \varepsilon_{yy}=\ \iint\limits_{\Omega}\varepsilon(\vec{z})(1+A_{1y}^{y})d\Omega\\ \varepsilon_{yz}=\ \iint\limits_{\Omega}\varepsilon(\vec{z})A_{1y}^{z}d\Omega\\ \end{array} \right., \tag{15}\]  
\[\left\{