## 2 Transmission Cross-Sections (TCSs) through a Subwavelength Slit for a Lossless Conducting Plate  
Figure 1 shows the geometry of an MIM waveguide, including a gap width of \(2g\) and a metal plate thickness of \(d\). In this section, \(2g\) is fixed as \(0.03\lambda_{0}\), while \(d\) is a variable and the structure is constant  
along the \(y\)-axis. An electromagnetic plane wave with parallel polarization propagates along the \(z\)-axis. In the metallic region, the permittivity of the metal \(\varepsilon_{r}\) is generally negative and will be varied in this work to investigate resonant transmitances. We consider only even modes, because even modes are sufficient for describing normal incidence in the MIM waveguide. The dispersion equation is the most fundamental equation in the MIM waveguide and obtained as follows [10]:  
\[\tanh(\kappa_{i,m}g)=-(\kappa_{m,n}\times\varepsilon_{r,j})/(\kappa_{i,n} \times\varepsilon_{r,m}), \tag{1}\]  
\[\text{where }(k_{z,n})^{2}=(\kappa_{m,n})^{2}+\omega^{2}\mu\varepsilon_{r,m}=( \kappa_{i,n})^{2}+\omega^{2}\mu\varepsilon_{r,j}, \tag{2}\]  
where \(\varepsilon_{r,m}\) and \(\varepsilon_{r,i}\) are permittivities in the metal and the insulator, respectively, \(\kappa_{m,n}\) and \(\kappa_{i,n}\) are \(n\)th transverse propagation constants in the metal and the insulator, respectively, and \(k_{z,n}\) is the \(n\)th propagation constant along the \(z\)-axis. The subscripts \(n\) in \(\kappa_{m,n}\), \(\kappa_{i,n}\), and \(k_{z,n}\) commonly indicate the \(n\)th modes, which are coupled to one another in Equation (2). The roots \(\{\kappa_{m,n}\}\)s of the dispersion equation are found using [PERSON]'s method [10] while varying the geometrical parameter of \(2g\) and the relative permittivity of \(\varepsilon_{r,m}\) for the metal. The TCS transmitted into the right free space is defined as:  
\[\text{TCS}[m]\ =\ \frac{P_{t}}{W_{i}}\ =\ \frac{1}{2W_{i}}\int_{S}Re[\overrightarrow{ E}\times\overrightarrow{H^{*}}]\cdot\Big{(}\overrightarrow{a}_{z}ds\Big{)}, \tag{3}\]  
where \(P_{t}\) and \(W_{i}\) are transmitted power (W) and incident power density (W/m), respectively. As another quantity, the transmittance into the right free space is also widely used [6; 10] and defined as:  
\[\tau[\text{dimensionless}]\ =\ \frac{1}{2g}\frac{P_{t}}{W_{i}}\ =\ \frac{1}{2g}\cdot\frac{1}{2W_{i}}\int_{S}Re[ \overrightarrow{E}\times\overrightarrow{H^{*}}]\cdot\Big{(}\overrightarrow{ a}_{z}ds\Big{)}. \tag{4}\]  
From Equation (3), the TCSs were obtained by the MMT and plotted in Figure 2a-d as a function of the permittivities of real metals. Figure 2a shows the TCS through a PEC slit versus the thickness of  
Figure 1: Geometry of a single metallic slit. a metallic plate of \(d\). The maximum periodic TCSs are observed at a plate thickness of \((0.486+0.5n)\lambda_{0}\), where \(n\) is an integer. Figure 2b-d shows the TCSs in terms of thickness of the metallic plate when the permittivities of a metal \(\varepsilon_{r,m}\) are \(-2000\), \(-100\), and \(-5\), respectively. The maximum periodic TCSs in Figure 2b-d are also observed at specific thicknesses of the metallic plate. The value of the maximum TCS for the PEC case in Figure 2a is found to be about \(0.3183\), which is the same as a known theoretical value [27]. By contrast, the maximum TCSs in Figure 2b-d do not reach the theoretical TCS value, because the incident wave from Figure 1 is slightly absorbed into the metallic plate. This could be understood by the fact that a lower metallic permittivity causes greater incident wave absorption into the metallic region [9]. From Equation (4), the transmittance \(\tau\) is obtained as described in the inset of Figure 2a. While the pattern is the same as the TCS, numerical values and transmitting units are different, as represented in Equation (4). Guide wavelengths are helpful to understand resonant transmission characteristics and can be solved from Equations (1) and (2).