## 3 Transmittance through a Subwavelength Slit on a Real Metallic Plate  
In Section 2, transmittance results were obtained, when a metal was a PEC or lossless conductor having a finite conductivity, and transmittances for lossy metals are discussed in this section. Figure 1 is again adopted for the geometry, but the conductor is changed into a realistic metal characterized by a finite conductivity with loss, which depends on the function of THz frequencies. The relative permittivity of a real metal has not only a real part, but also an imaginary part considering metallic loss. The metal we deal with is silver (Ag), and its relative permittivity is \(-14.88-j0.39\) when \(\lambda_{0}=582.08\) nm [28]. The complex numbered \(\varepsilon_{r,m}\) can be inserted into the dispersion equation known as Equation (1). The corresponding transverse propagation constants (\(\kappa_{m,n}\) and \(\kappa_{i,n}\)) and propagation constant (\(k_{z,n}\)), which are all complex numbers, can then be easily solved. The reflection and transmission coefficients at each boundary can be subsequently calculated by the MMT, obtaining the overall transmission results [9; 10; 11; 12]. The transmittances are represented by Figure 3a,b, when the gap widths \(2g\) become 4 and 8 nm, respectively. The transmittances are investigated by gradually reducing, and finally eliminating, the imaginary part of \(\varepsilon_{r,m}\), even if the actual relative permittivity at \(\lambda_{0}=582.08\) nm is \(14.88-j0.39\)[28]. Figure 3a shows the transmittances through a subwavelength slit versus the thickness of a metallic plate when the gap width \(2g\) is 4 nm (\(\sim\)0.0069\(\lambda_{0}\)). With no imaginary part in \(\varepsilon_{r,m}\), periodically obtained peak transmittances are maintained although the thickness of the plate increases, because there is no metallic loss, producing the same result trend shown in Figure 2. By contrast, resonant peak transmittances are gradually reduced as the imaginary part of \(\varepsilon_{r,m}\) increases. This can be understood that increasing the plate thickness makes electromagnetic waves to pass through the metal over a longer distance, causing more power to be lost due to metallic loss. As the imaginary part of the relative permittivity increases up to \(-j0.39\), attenuation phenomena can be observed clearly. Figure 3b shows the transmittances by varying the relative permittivities, when the gap width \(2g\) becomes 8 nm (0.0137\(\lambda_{0}\)). The resonant plate thicknesses are maintained despite increased imaginary parts of the permittivities, and the peak transmittances are gradually reduced as the imaginary part \begin{table}
\begin{tabular}{c c c} \hline \hline
**Permitivities in Metal** & **Guide Wavelength (\(\Lambda_{C}\))** & **Maximum TCS Interval** \\ \hline PEC & \(1.0\lambda_{0}\) & \(0.5\lambda_{0}\) \\ \(\varepsilon_{r}=-2000\) & \(1.1124\lambda_{0}\) & \(0.4495\lambda_{0}\) \\ \(\varepsilon_{r}=-100\) & \(1.4406\lambda_{0}\) & \(0.3471\lambda_{0}\) \\ \(\varepsilon_{r}=-5\) & \(3.0307\lambda_{0}\) & \(0.1650\lambda_{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Guide wavelengths with respect to metallic permittivities and computed maximum TCS intervals. Figure 3: Transmittance as a function of Ag plate thickness: (**a**) \(2g=4\) nm; (**b**) \(2g=8\) nm. The blue asterisks are remarked from [29], when \(\varepsilon_{r,m}=-14.88-j0.39\). of \(\varepsilon_{r,m}\) increases. However, the interval, at which resonant peaks occur, is increased compared to the interval in Figure 3a. This can be understood using Table 2, which shows the guide wavelengths as functions of the gap width \(2g\) and the relative permittivity of Ag. From each value, the real part of the guide wavelength represents the interval of the periodic transmission peaks, and the imaginary part stands for the attenuation of their peaks.