## 3 Results and Discussions
### The Multiple Hybrid Cascade Calculation of the Transparent Substrate and Metal Film  
Through the analysis and discussion of a single transparent substrate and a single metal film material, the cascade calculation results between the same material and the direct simulation results can achieve a basically consistent result.  
As shown in Figure 6 above, when the transparent substrate material with a thickness of 0.1 um is combined with the circle hole metal gold film etched with a radius of 1.5 um, the structure diagram is as shown in Figure 1d, which is different from the single material cascade. When the metal film material is 0.1 um thick gold, the thickness of transparent substrate material is changed. The results show that when the thickness of transparent substrate is 0.1 um, the distribution of optical response \(S\) parameters in the mid infrared band is basically consistent. The difference is that, as shown in Figure 6a, the amplitude of cascade calculation is about 0.2 less than the simulation result, that is, the transmittance decreases by about 0.15. When the thickness of transparent substrate is 1 um, the results are as shown in Figure 6b. The simulation results are fitted with 8 th degree polynomial, and the peak position distribution and amplitude size of \(S\) parameter are basically consistent. In other words, the thicker the transparent substrate is, the higher the coincidence degree is in the case of cascaded calculation fitting.  
For the cascading calculation error of thin transparent substrate, the equivalent circuit model method can be used for further analysis. According to the literature [18; 19; 20], the surface impedance of metal materials can be calculated by:  
\[Z_{s1}=\frac{D}{\sigma}-j\frac{\pi}{\omega\varepsilon_{0}(\varepsilon_{r}+1)D \ \mathrm{ln}\left\{\csc(\pi g/2D)\right\}} \tag{11}\]  
\[Z_{s2}=\frac{1}{\sigma}=\frac{1+j\omega\tau}{\sigma_{02}} \tag{12}\]  
\[Z_{d}=\mathrm{j}Z_{c}\tan(\beta l)=\frac{1}{j\omega C_{d}} \tag{13}\]  
where \(g\), \(l\) and \(D\) are the geometric parameters and \(\varepsilon_{r}\) is the relative permittivity of the transparent media. \(\sigma_{02}\) is the conductivity related to metal materials. \(C_{d}\) is the effective capacitance. It can be seen from Formulas (11)-(13) that when the metal film is deposited on the surface of the transparent substrate, coupling occurs to a certain extent, and the coupling strength is related to the thickness of the substrate. Therefore, it is also explained that when only transparent substrates are used for cascading calculation, the  
Figure 5: Cascade response when only containing metallic film (Au) on 120–750 THz band, wherein the black curve is the case with a thickness of 0.1 μm, that is, the cascade benchmark.  
calculation results are highly consistent with the results of direct simulation. When metal materials are involved in the cascade calculation, the degree of coincidence will be correspondingly reduced. At the same time, when the frequency of the response band increases, that is, the wavelength decreases, more electrons can be excited in the hole to participate in the resonance mechanism, thus generating more surface plasma. It will further increase the effective refractive index \(n_{\mathit{eff}}\), leading to changes in the resonant peak [21; 22; 23]. In other words, when the response frequency increases, the accuracy degree of cascade calculation will decrease.