## 6 Characterization of Surface Wave Modes
### Example of Surface-Wave Mode Characterization  
Take the dual-layer PEC surface shown in Figure 19a as an example. Its surface impedance of TM polarization at 10 GHz is analytically computed through the procedure listed above. The top layer is composed of periodic square apertures, which is exactly the same as the metascreen characterized in Section 5.3, and the extracted surface susceptibilities have already been shown in Figure 16. With \(\overline{\overline{X}}_{ee}\) and \(\overline{\overline{X}}_{mm}\) determined, expression of \(Z_{e}^{\text{TM}}\) is obtained using Equation (41), where \(\sin^{2}\theta\) is replaced as a function of \(Z_{\text{surf}}^{\text{TM}}\) through Equations (56) and (58). Then Equation (53) is solved using MATLAB, resultsof which are shown in Figure 19b with variable \(a\). Since the surface is lossless, its surface impedance is purely imaginary. The computed \(Z_{\text{surf}}^{\text{TM}}\) is compared with eigen-mode simulated results from ANSYS HFSS, and the computed results through the method in [39] are also shown in Figure 19b. It can be observed that the method proposed in this paper, which is based on surface susceptibilities, is more accurate than the one given in [39].  
Equation (53) can also be utilized to compute eigen-frequencies with given phase delay across one period, with which the dispersion diagram of surface-wave mode can be plotted. Still take TM polarization as the example. Suppose the phase delay across one period of unit cell along \(\hat{x}\) and \(\hat{y}\) are \(\phi_{x}\) and \(\phi_{y}\) respectively, then Equation (49) can be rewritten as  
\[Z_{\text{surf}}^{\text{TM}}=j\eta_{0}\sqrt{\frac{\phi_{x}^{2}+\phi_{y}^{2}}{k_{ 0}^{2}p^{2}}-1}, \tag{59}\]  
where \(\phi_{x}\) and \(\phi_{y}\) are variables in dispersion diagram and their relations represent the propagation direction. According to Equation (59), \(Z_{\text{surf}}^{\text{TM}}\) can be expressed as a function of \(k_{0}=2\pi f\), which makes all unknown quantities in Equation (53) related to \(f\). Consequently, \(f\) can be computed by solving Equation (53). By computing \(f\) for different groups of \((\phi_{x},\phi_{y})\) with this analytical method, the dispersion diagram can be solved.  
Figure 20a,b respectively show the computed dispersion diagrams with \(a=1.5\) mm and \(a=2.5\) mm for the dual-layer surface shown in Figure 19a, compared with simulated ones from eigen-mode solver. Figure 20c shows the relative error which is defined by  
\[\text{Relative Error}=\frac{|f_{\text{sim}}-f_{\text{com}}|}{f_{\text{sim}}}, \tag{60}\]  
where \(f_{\text{sim}}\) and \(f_{\text{sim}}\) are the simulated and computed eigen-frequencies respectively. It can be observed that the computed results agree well with eigen-mode simulations when \(f\) is small. As frequency increases, the wavelength \(\lambda\) becomes comparable to the fixed period \(p\), which makes the assumption of homogeneous effective invalid and results in quite large relative errors. For example, the period \(p=3\) mm is equal to one-tenth wavelength at \(f=10\) GHz but half wavelength at \(f=50\) GHz. It is also found that the case with \(a=2.5\) mm has larger error than the one with \(a=1.5\) mm, which should be due to the approximation of dipole interactions assumed in derivation of GSTCs [18].  
Figure 19: Dual-layer metasurface with periodic square apertures at top and ground plane at bottom: (**a**) configuration of surface with \(p=h=3\) mm (**b**) comparison between simulated and computed TM-polarized surface impedance of surface-wave mode at 10 GHz.