## 6 Characterization of Surface Wave Modes
### Equivalent Circuit Model and Transverse Resonance Method
By substituting the determined value of \(Z_{e}\) into Equation (53) for surface-wave mode, there is only one unknown \(Z_{\text{surf}}^{\text{TM}}\) in this equation, which can be solved by numerical tools like MATLAB. Based on the method derived in [39], it is assumed that sheet impedance \(Z_{e}\) is not related to the incident angle \(\theta\), so its value extracted from the space-wave simulation with \(\theta=0^{\circ}\) is directly utilized for characterization of surface-wave mode. However, as discussed in previous sections, \(Z_{e}\) is \(\theta\)-dependent in many cases, thus its value of surface-wave mode can be quite different from the one of space-wave mode with \(\theta=0^{\circ}\). Still take TM polarization as the example. If the top-layer surface is zero-thickness PEC metafilm, based on Equation (30) given in Section 3, \(Z_{e}^{\text{TM}}\) is independent of \(\theta\), and method of [39] still works. But if the top layer is metascreen, \(Z_{e}^{\text{TM}}\) is related to \(\sin^{2}\theta\) according to Equation (41) derived in Section 5. In this case, the assumption of \(\theta\)-independent \(Z_{e}\) is invalid, leading to the inaccuracy of computed results. Thus, the method that directly extracts \(Z_{e}\) from simulations of space-wave mode is not a general solution for different kinds of surfaces. Here it is proposed to determine \(Z_{e}\) from surface susceptibilities \(\overline{\overline{X}}_{ee}\) and \(\overline{\overline{X}}_{mm}\), based on relations derived in previous sections. For example, \(Z_{e}^{\text{TM}}\) of metascreens can be readily determined through Equation (41), once \(\chi_{mm}^{yy}\), \(\chi_{ee}^{zz}\) and \(\sin^{2}\theta\) are determined. Although the value of \(\theta\) for surface-wave mode is unknown, the \(\sin^{2}\theta\) term required can be expressed as  
\[\sin^{2}\theta=\frac{k_{0}^{2}+\gamma_{z}^{2}}{k_{0}^{2}}, \tag{58}\]  
according to Equation (44). Thus, \(\sin^{2}\theta\) can be written as a function of \(\gamma_{z}\), and further replaced by \(Z_{\text{surf}}\) through Equation (56). Consequently, there is still one unknown \(Z_{\text{surf}}\) in Equation (53), which can be numerically solved. With the method based on surface susceptibilities, more accurate results can be obtained rather than the one based on method given in [39], since \(Z_{e}\) is considered as \(\theta\)-dependent. The procedure to compute surface-wave \(Z_{\text{surf}}\) of dual-layer impenetrable surfaces based on surface susceptibilities, equivalent circuit model and transverse resonance technique is listed below:  
1. carry out limited sets of space-wave simulations of the top-layer surface and extract its surface susceptibilities from simulated scattering coefficients;
2. determine the expression of \(Z_{e}\) based on \(\overline{\overline{X}}_{ee}\) and \(\overline{\overline{X}}_{mm}\) through related formulas derived in previous sections;
3. substitute \(Z_{e}\) into Equation (53) and solve the equation of \(Z_{\text{surf}}\) with numerical tools. One example computing surface impedance of a dual-layer metasurface, which is composed of one layer of zero-thickness metascreen at top with ground plane at bottom, will be given in Section 6.3.