## 4 Orthogonality and Discussions  
When the metal of an MIM waveguide is a PEC, field profiles of entire modes (including a transverse electromagnetic mode) can be expressed easily, and the sinusoidal modal variations are found in the slit of the waveguide, while field profiles cannot be expressed in the PEC region. The orthogonal property is completely satisfied, because inner products among the sinusoidal harmonics in the silt are perfectly orthogonal. (In this work, only cosine functions are considered, because only a normally incident wave is dealt with.) The \(n\)-th modal electromagnetic fields of the parallel plate waveguide in Figure 4a can be expressed as follows:  
\[H_{y}^{n}\ =\ cos(y_{n}x)e^{-k_{n}z}, \tag{5}\] \begin{table}
\begin{tabular}{c c c} \hline
**Permititivities of Metal** & **2g = 4 nm** & **2g = 8 nm** \\ \hline \(\varepsilon_{r}=-14.88\) & 4.3822 & 2.9284 \\ \(\varepsilon_{r}=-14.88-j0.13\) & \(4.3820-j0.0194\) & \(2.9283-j0.0093\) \\ \(\varepsilon_{r}=-14.88-j0.26\) & \(4.3815-j0.0388\) & \(2.9281-j0.0185\) \\ \(\varepsilon_{r}=-14.88-j0.39\) & \(4.3805-j0.0582\) & \(2.9277-j0.0277\) \\ \hline \end{tabular}
\end{table} 
Table 2: Guide wavelengths (\(\lambda_{g}\)) with respect to the Ag permittivity and modified permittivities, when the gap width \(2g\) of 4 and 8 nm are used. (\(\varepsilon_{r}=-14.88-j0.39\) at \(\lambda_{0}=582.08\) nm [28]). \[E_{x}^{n}\ =\ -\frac{1}{j\omega e}\frac{\partial H_{y}^{n}}{\partial z}\, \tag{6}\]  
\[\gamma_{n}\ =\ \frac{2n\pi}{2g}\ \ \ \text{and}\ \ \ k_{n}\ =\ \sqrt{\gamma_{n}^{2}-\omega^{2}\mu \varepsilon},n\ =\ 0,1,2,\cdots. \tag{7}\]  
From the geometry in Figure 4a, the field profiles of the first three even modes are described in Figure 4b. Note that the scales are different for the slit (white) and metal (light orange) regions. By applying higher-order modes and the first three modes into the MMT, the transmittances are solved. The orthogonal properties among the modes are calculated and confirmed from the following equations [30]:  
\[\frac{1}{\eta_{0}}\frac{k_{0}}{k_{n}}\frac{1}{2g}\int_{-g}^{g}\!\!\left[E_{x}^ {n}(H_{y}^{m})^{*}\right]\!dx\ =\ \delta_{nm}, \tag{8}\]  
\[\delta_{nm}\ =\ \left\{ \begin{array}{l}1,n\ =\ m\\ 0,n\
eq m\end{array} \right.. \tag{9}\]  
Figure 4: Geometries of different metal–insulator–metal (MIM) waveguides for gap width \(2g=8\) nm, when a transverse magnetic (TM) polarized electromagnetic wave is normally incident: (**a**) a PEC as the conducting material in the MIM waveguide and (**b**) the corresponding field profiles of the guided modes (transverse electromagnetic (TEM), TM\({}_{2}\), and TM\({}_{4}\) modes); (**c**) a practical metal as the conducting material in the MIM waveguide, such as silver and the corresponding field profiles for a point spectrum (PS) (**d**) and a discretized continuous spectrum (DCS) (**e**). Note that different scales are shown within the periods of (**b**,**d**). From Equations (8) and (9), the calculated result is 1 when the modes are the same, and 0 when the two modes are different. Therefore, the orthogonal property is completely satisfied, when the metal of the MIM waveguide is a PEC. When the metal has a finite conductivity, the geometry of the single slit can be depicted by Figure 4c with the blue-colored metallic region and the corresponding field profiles shown in Figure 4d,e. The equations for the field profiles are represented as [9]:  
\[H_{y}^{n} = \left\{ \begin{array}{l}\frac{cosh(x_{i,n}x)}{cosh(x_{i,n}S)}e^{- k_{i,n}z},0<x<g\\ e^{-k_{m,n}(x-S)}e^{-k_{z,n}z},g<x<\infty\end{array} \right.. \tag{10}\]  
By inserting \(\varepsilon_{r,m}\) into Equations (1) and (2), the \(n\)-th transverse propagation constants (\(\kappa_{i,n}\) and \(\kappa_{m,n}\)) and the \(n\)-th propagation constant (\(k_{z,n}\)) are easily obtained. The field profiles can then be obtained from Equation (10) and are described in Figure 4d,e, which stand for the field profiles of a point spectrum (PS) and a discretized continuous spectrum (DCS) [9], respectively.