## 6 Characterization of Surface Wave Modes
### Equivalent Circuit Model and Transverse Resonance Method  
An alternative method to determine \(\vec{k}_{t}\) or \(Z_{\text{surf}}\) of surface-wave mode is introduced below. Different from the method based on eigen-mode solvers, the value of \(Z_{\text{surf}}\) can be analytically computed by solving the transverse resonance equation established through equivalent circuit model. The equivalent circuit model of a dual-layer surface with ground plane at bottom is shown in Figure 18, where the zero-thickness PEC top layer is represented as sheet impedance \(Z_{e}\) and the ground plane is modelled as a shorted load. For simplicity, the substrate between top layer and ground plane is selected as air, which has a thickness of \(h\) and is modelled as a transmission line with characteristic impedance equals \(Z_{0}\). \(Z_{0}\) can be expressed as  
\[Z_{0}^{\text{TE}}=\eta_{0}/\cos\theta \tag{50}\]  
for TE polarization and  
\[Z_{0}^{\text{TM}}=\eta_{0}\cos\theta \tag{51}\]  
for TM polarization respectively. For the space-wave mode, \(\theta\) is the incident angle related to direction of wave vector \(\vec{k}\). However, as discussed in Section 2, \(\theta\) of surface-wave mode is an unknown complex number, as a special case of space-wave mode. Based on Equation (43), \(\cos\theta\) can be expressed as  
\[\cos\theta=\gamma_{z}/(jk_{0}), \tag{52}\]  
while the value of \(\gamma_{z}\) is also undetermined. According to the equivalent circuit model, \(Z_{\text{surf}}\) represents the impedance looking down from the reference plane (dashed line) located along the top face of the given surface, while the impedance looking up from reference plane is \(Z_{0}\). Based on the knowledge of microwave engineering [10], \(Z_{\text{surf}}\) can be expressed as  
\[\frac{1}{Z_{\text{surf}}}=\frac{1}{Z_{e}}+\frac{1}{Z_{0}\tanh(\gamma_{z}h)}. \tag{53}\]  
In order to compute \(Z_{\text{surf}}\), it is required to determine the values of \(Z_{e}\), \(Z_{0}\) and \(\gamma_{z}\) first. The transverse resonance technique [45] demonstrates that, for the existence of surface-wave mode, it has to satisfy the condition  
\[\frac{1}{Z_{\text{up}}(z)}+\frac{1}{Z_{\text{down}}(z)}=0, \tag{54}\]  
where \(Z_{\text{up}}(z)\) and \(Z_{\text{down}}(z)\) represent the impedances looking in opposite directions from any observation plane along the transmission line. Equation (54) is also called transverse resonance equation. As discussed before, along the reference plane selected in Figure 18, \(Z_{\text{up}}(z)=Z_{0}\) and \(Z_{\text{down}}(z)=Z_{\text{surf}}\). Consequently,  
\[Z_{0}=-Z_{\text{surf}}. \tag{55}\]  
Take TM polarization as an example, which is the dominant mode of surface wave propagating along surfaces with ground plane, \(\gamma_{z}\) can be expressed as  
\[\gamma_{z}=-jk_{0}Z_{\text{surf}}^{\text{TM}}/\eta_{0}, \tag{56}\]  
based on Equations (51) and (52). According to Equations (55) and (56), both \(Z_{0}\) and \(\gamma_{z}\) in Equation (53) can be expressed as functions of \(Z_{\text{surf}}\). Thus, if \(Z_{e}\) is also determined or related to \(Z_{\text{surf}}\), the value of \(Z_{\text{surf}}\) can be computed from Equation (54). Figure 18: Equivalent circuit model of a dual-layer surface with ground plane at bottom. In [39], an extraction method of \(Z_{e}\) based on simulations of space-wave mode is utilized to solve Equation (53) for surface-wave mode characterization. One set of unit cell simulation for the given dual-layer surfaces with normal incidence (\(\theta=0^{\circ}\)) is required. With simulated reflection coefficients \(\Gamma\), the corresponding surface impedance \(Z_{\text{surf}}\) of space-wave mode can be computed through  
\[Z_{\text{surf}}=Z_{0}\frac{1+\Gamma}{1-\Gamma}. \tag{57}\]  
Once \(Z_{\text{surf}}\) has been extracted, the sheet impedance \(Z_{e}\) can be obtained from the relations defined by Equation (53), where \(\gamma_{z}\) should be modified as \(jk_{0}\) due to normal incidence of space-wave mode.