## 3 Results and Discussions
### The Cascade Calculation of the Single Material Type  
Herein, the substrate containing only a single transparent substrate is sliced. The structure is shown in Figure 1b, and the substrate material is sapphire. As shown in Figure 2a, the thickness \(h_{2}\) of the substrate is 0.5-3 um respectively, with a step of 0.5 um. Through the full wave simulation of CST microwave studio in the 60-100 THz, the transmission curve of the solid line part in Figure 2a is obtained. It is worth noting that the transmission curve shows periodic fluctuations due to the etalon effect [16], which can be characterized by the following formula:  
\[\text{t}=-\frac{(1-R)^{2}e^{-ab}}{1+R^{2}e^{-2 ab}-2 Re^{-ab}\text{cos}\,\theta} \tag{9}\]  
where \(a\) is the absorption coefficient; \(b\) is the thickness of the sample; \(R\) is the reflectivity of single-sided reflection; \(\theta=4\pi n\text{/}\lambda_{0}\); the \(\lambda_{0}\) is the wavelength of radiation in vacuum and \(n\) is the refractive index. At the same time, its \(S\) parameter can be obtained through simulation analysis, and the transmittance curve as shown by the dotted line in the figure can be obtained by substituting Formulas (5)-(8). It can be seen from the comparison in the figure that the results obtained by full wave simulation through the CST microwave studio are completely consistent with the results obtained by simulating the transparent substrate with a thickness of 0.1 um firstly, and then substituting the \(S\) parameters into the transmission line theoretical formula for multiple cascades. Herein, the slices with a thickness of 0.1 um are used as the cascade benchmark, and the calculation of substrate materials with different thicknesses is achieved by controlling the number of times of cascades. In order to further verify the results, different kinds of materials are compared. The results are shown in Figure 2b for SiO\({}_{2}\) and Figure 2c for Si. The simulation results are completely consistent with the theoretical calculation results. Owing to the fact that the transmittance of the metal film with a certain thickness in the visible and infrared bands is substandard, the method of etching the micro/nano structure on the film surface is used to enhance its optical transmission performance. The structure is shown in Figure 1c. Among them, the radius \(R\) of circle hole is 1.5 um. As shown in the solid line part of the curve in Figure 3, the thickness h1 of the metal film is 0.3-1 um, respectively, and the step size is 0.2 um. The full wave simulation was also carried out in the 60-100 THz band through CST microwave studio to obtain its transmittance response. As shown in the literature [17], when the incident light is perpendicular to the metal film, the position of the resonant transmission peak \(\lambda_{\text{max}}\) excited by SPPs mode in the square array with period \(P\) can be approximately calculated with the following formula:  
\[\lambda_{\text{max}}=\frac{p}{\sqrt{i^{2}+j^{2}}}\sqrt{\frac{\epsilon_{d} \epsilon_{m}}{\epsilon_{d}+\epsilon_{m}}} \tag{10}\]  
where \(i\) and \(j\) are the scattering orders along the X axis and Y axis, respectively, both of which are integers; \(\epsilon_{d}\) and \(\epsilon_{m}\) are the relative permittivity of the dielectric and metal materials, respectively. According to Formula (10), for square array, when \(\epsilon_{d}\) and \(\epsilon_{m}\) are determined, the array period \(P\) determines the position of the transmission peak excited by the SPPs mode. Further, the \(S\) parameter of the metal film with a thickness of 0.1 um is obtained by simulating. Then, multiple cascades are carried out by substituting Formulas (5)-(8) to obtain the transmittance distribution of metal films with different thicknesses, as shown by the dotted line in the figure. Under different condition of metal (gold, silver and aluminum) materials, the results of direct simulation and multiple cascade calculation are completely consistent with those of the case that only a single transparent substrate material is included.