## 2 Design and Modeling
### Two-Port Transmission Line Theory  
For microwave networks with arbitrary ports, we can characterize them using the \(Z\), \(Y\), and \(S\) parameters [24]. However, it is important to note that many microwave networks actually comprises multiple two-port networks. In this case, a \(2\times 2\) transmission matrix (also known as an ABCD matrix) can be used to define the situation of each dual-port network. The details are as follows:  
\[V_{1}=AV_{2}+Bl_{2}, \tag{7}\]  
\[I_{1}=CV_{2}+Dl_{2}, \tag{8}\]  
Or in matrix form:  
\[ \begin{bmatrix}V_{1}\\ I_{1}\end{bmatrix} = \begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}V_{2}\\ I_{2}\end{bmatrix} , \tag{9}\]  
The cascading of a network can be represented as:  
\[ \begin{bmatrix}V_{1}\\ I_{1}\end{bmatrix} = \begin{bmatrix}A_{1}&B_{1}\\ C_{1}&D_{1}\end{bmatrix} \cdots \begin{bmatrix}A_{n}&B_{n}\\ C_{n}&D_{n}\end{bmatrix}\begin{bmatrix}V_{n+1}\\ I_{n+1}\end{bmatrix} , \tag{10}\]  
Furthermore, convert the parameters of the network to:  
\[A=\frac{(1+S_{11})(1-S_{22})+S_{12}S_{21}}{2S_{21}}, \tag{11}\]  
\[B=Z_{0}\frac{(1+S_{11})(1+S_{22})-S_{12}S_{21}}{2S_{21}}, \tag{12}\]  
\[C=\frac{1}{Z_{0}}\frac{(1-S_{11})(1-S_{22})-S_{12}S_{21}}{2S_{21}}, \tag{13}\]\[D=\frac{(1-S_{11})(1+S_{22})+S_{12}S_{21}}{2S_{21}}, \tag{14}\]  
As shown above, \(V_{1}\) is the total voltage flowing into the port, and \(V_{2}\) is the total voltage flowing out of the port; \(I_{1}\) is the total current flowing into the port, and \(I_{2}\) is the total current flowing out of the port; \(S_{11}\) is the input reflection coefficient passing through the port network, \(S_{22}\) is the output transmission coefficient passing through the port network, \(S_{12}\) is the feedback coefficient passing through the port network, and \(S_{21}\) is the forward transmission coefficient passing through the port network; and \(Z_{0}\) is the characteristic impedance. Through measurement or theoretical analysis, a description of the 'black box' effects on these dual ports can be obtained. The characteristic of the transition section can be expressed through the network parameters (\(Z\), \(Y\), \(S\), or ABCD) of the two-port network. Generally speaking, in practical applications, simulating the model structure of thick transparent substrates or thin films (relative to the response band) may result in an excessive program load due to the need for excessive grid partitioning. However, the transmission line cascade theory calculation is a useful method for obtaining the characteristic parameters of a single thin layer material through calculation. By cascading these parameters according to the requirements, this method can effectively solve the problems mentioned above. The transmission spectrum of the cascade calculation and direct simulation under different conditions are compared and analyzed as shown in Figure 2. For a single sapphire transparent substrate in Figure 2a, the cascade calculation results of different thicknesses are in good agreement with the direct simulation results. Obviously, the transmission spectrum shows periodic fluctuations. This phenomenon can be explained by the etalon effect [25]:  
\[t=-\frac{(1-R)^{2}e^{-ab}}{1+R^{2}e^{-2 ab}-2 Re^{-ab}\text{cos}\,\theta} \tag{15}\]  
Herein, \(a\) and \(b\) are the absorption coefficient and the thickness of the sample, respectively. \(R\) is the reflectivity of the one-sided reflection; \(\theta=4\pi nb/\lambda_{0}\); \(\lambda_{0}\) is the wavelength of the radiation in the vacuum, and \(n\) is the refractive index. As the wavelength increases, \(\theta\) changes, leading to periodic changes in cos\(\theta\), ultimately leading to periodic oscillations in transmittance. In Figure 2c, we used a Fourier-transform infrared spectrometer to measure the infrared transmittance of the sapphire substrate in the 2.5-6 \(\upmu\)m range.