## 4 Orthogonality and Discussions
The PS and DCS correspond to the guided and radiation modes in the dielectric slab waveguide, respectively. The PS can describe the sinusoidal electromagnetic field patterns mainly in the \"insulator\", while the DCS depicts the sinusoidal field patterns mainly in the \"metal\" region [9; 10]. The right-hand side of Equation (8), for a metal having a finite conductivity, is not solved exactly as 1 or 0, which indicates the orthogonality is not perfectly satisfied, but the modes are applied into the MMT and the corresponding transmittances are solved. The field profiles of the PS and the DCS are described in Figure 4d,e, respectively, where the gap width \(2g\) is 4 nm and \(\varepsilon_{r,m}\) is \(-14.88\). Figure 4d shows the field profiles for the first mode of the PS and may correspond to the first field profile of Figure 4b. The field shape in the white insulator region is almost maintained; however, the field shape in the sky-blue metal region is different. The field at the boundary between the insulator and the metal penetrates into the metal region, because the metal is no longer a PEC. Figure 4e describes the field profiles of the first, second, and third modes of the DCS. The field profiles represent sinusoidal harmonics in the metal region, and various field patterns in the metal region can be described by superposition of the modes in the DCS. The numbers of the PS and the DCS used in the MMT formulation are 1 and 50, respectively. Only one mode of the PS is applied to solve the transmittance, because the slit is too narrow compared to the wavelength. That being said, a sufficient number of modes of the DCS should be applied, given the wideness of the metal region compared to that of the slit region [12]. The sufficient number of DCS modes is obtained as follows: We start to apply 5 trial DCS modes and investigate the reflection and transmission coefficients. Computing the coefficients by increasing the number of the DCS modes to 5 are repeated, and then the coefficients are arranged with respect to the number of DCS modes. Once the convergence is found, the updated number of the DCS modes are obtained. From the viewpoint of orthogonality, the PS and the DCS are clearly independent, because the field patterns of both are formed in different regions; therefore, the result of Equations (8) and (9) between the two approaches 0. However, the orthogonal properties among 50 DCS modes require investigation, since the metal is not a PEC. Figure 5 indicates in Figure 5a the results of Equations (8) and (9) of the first 10 DCS modes when \(\varepsilon_{r,m}\) of the MIM waveguide is \(-14.88\). Each cell represents the result of Equations (8) and (9) between two modes. What we need to pay attention to Equations (8) and (9) for the MIM waveguide is the interval of integration. The interval should cover the entire waveguide junction including the metal region as well as the gap region of \(2g\), while the interval of integration \(2g\) in Equations (8) and (9) is still valid in the case of the PEC waveguide. The diagonal components approach approximately 1 when the DCS modes used in Equations (8) and (9) are identical, while the off-diagonal results go to approximately 0 when the modes differ. For the first 10 modes of the DCS, the values from Equations (8) and (9) are within [1.0082, 1.0111] for the same modes, whereas the values are within [\(4.878\times 10^{-5}\), 0.00340] for different modes. These computed results are still effective up to the next 40 DCS modes. The orthogonality changes slightly but remains almost constant despite the inclusion of an imaginary part in the permittivity of the metal. The field profiles of the PS and the DCS in Figure 4d,e change slightly, although \(\varepsilon_{r,m}\) becomes \(-14.88-j0.39\). Figure 5b shows the orthogonal results of Equations (8) and (9) among the first 10 DCS modes, when \(\varepsilon_{r,m}\) is \(-14.88-j0.39\). The solved results are very similar to Figure 5a, even though the imaginary part of the permittivity is included.