## 2 Design and Modeling  
As shown in Figure 1a, the yellow part is a metal film and its thickness is \(h_{1}\) after uniform slicing; the gray part is a transparent substrate and its thickness is \(h_{2}\) after uniform slicing. Among those which, according to the structural design formula summarized in the literature [15], can achieve continuous adjustable transmission spectrum in the mid infrared band, the unit structure period \(P_{x}=P_{y}=4\) um is adopted in this article, and the metal film is etched with circle holes arranged periodically with radius \(R=1.5\) um. According to different stacking modes, the following models are given: as shown in Figure 1b, multi-layer single transparent substrate is stacked; as shown in Figure 1c, multilayer single metal films are stacked; and as shown in Figure 1d, it is a single-layer transparent substrate stacked with a single-layer metal film. The simulation part in this paper adopts CST microwave studio for simulation, and the specific experimental environment parameters are set as follows: based on Floquet theorem, periodic boundary conditions are used, so only one periodic element structure of the research object needs to be considered in the actual CST Studio calculation. This disposal can not only effectively reduce the calculation time, but also attain the electromagnetic field distribution of the element structure. Therefore, the \(X\) and \(Y\) boundaries of the element structure are set as periodic boundary conditions, and the \(Z\) boundary is set as open boundary conditions. In addition, free electron gas Drude model [13] is used to describe metal materials. Moreover, the calculation part of the article adopts the two port transmission line theory for analysis, as follows:  
The microwave network with any number of ports can be characterized by \(Z\), \(Y\) and \(S\)[14]. In fact, many microwave networks are composed of two or more two port networks in cascade. Under this premise, it is convenient to define each two port network with a \(2\times 2\) transmission matrix or \(ABCD\) matrix.  
The \(ABCD\) matrix of two port network can be defined by the following formula:  
\[V_{1}=AV_{2}+Bl_{2} \tag{1}\]  
\[I_{1}=CV_{2}+Dl_{2} \tag{2}\]  
Figure 1: Unit structure diagram and structure diagram under different cascading conditions. (**a**) Multiple layers of different materials; (**b**) Single transparent material; (**c**) Single metallic materials; (**d**) Single-layer metal materials and transparent materials.  
Or in matrix form:  
\[\begin{bmatrix}V_{1}\\ I_{1}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}V_{2}\\ I_{2}\end{bmatrix} \tag{3}\]  
Cascading of multiple two port networks can be expressed as:  
\[\begin{bmatrix}V_{1}\\ I_{1}\end{bmatrix}=\begin{bmatrix}A_{1}&B_{1}\\ C_{1}&D_{1}\end{bmatrix}\cdots\begin{bmatrix}A_{n}&B_{n}\\ C_{n}&D_{n}\end{bmatrix}\begin{bmatrix}V_{n+1}\\ I_{n+1}\end{bmatrix} \tag{4}\]  
At the same time, the parameters of the two port network are converted into:  
\[A=\frac{(1+S_{11})(1-S_{22})+S_{12}S_{21}}{2S_{21}} \tag{5}\]  
\[B=Z_{0}\frac{(1+S_{11})(1+S_{22})-S_{12}S_{21}}{2S_{21}} \tag{6}\]  
\[C=\frac{1}{Z_{0}}\frac{(1-S_{11})(1-S_{22})-S_{12}S_{21}}{2S_{21}} \tag{7}\]  
\[D=\frac{(1-S_{11})(1+S_{22})+S_{12}S_{21}}{2S_{21}} \tag{8}\]  
Through measurement or theoretical analysis, the description of these effects can be obtained and expressed as a two port 'black box'. The nature of the transition section can be represented by the network parameters (\(Z\), \(Y\), \(S\) or \(ABCD\)) of the two port network. For the transparent substrate or membrane structure with large size (relative to the response band) in practical applications, it often occurs that too many grid divisions lead to excessive program load when conducting structural simulation.