## 6 Characterization of Surface Wave Modes
### Characterization Parameters of Surface-Wave Mode  
For space-wave mode, the scattering properties of EM surfaces are mainly described by reflection and transmission coefficients. In contrast, surface-wave mode is an eigen propagation mode of surfaces, which is usually characterized by the transverse wave vector or the surface impedance. The transverse wave vector \(\vec{k}_{t}\) is defined as  
\[\vec{k}_{t}=\hat{x}k_{x}+\hat{y}k_{y}=\hat{k}_{t}|\vec{k}_{t}|, \tag{42}\]  
which stands for the propagation characteristics of surface wave along surfaces. Since surface-wave is a slow-wave mode, \(|\vec{k}_{t}|\) is usually larger than \(k_{0}\). Compared with the space-wave mode (Equation (12)), the wave vector of the surface-wave mode can be expressed as  
\[\vec{k}=\hat{x}k_{x}+\hat{y}k_{y}-\hat{z}j\gamma_{z}, \tag{43}\]  
since EM fields exponentially decay along \(\hat{z}\) direction. \(\gamma_{z}\) is the complex propagation constant along \(\hat{z}\) direction and it is a real number for surface-wave mode. According to Equations (42) and (43), the relations between \(\vec{k}_{t}\) and \(\gamma_{z}\) can be expressed as  
\[|\vec{k}_{t}|^{2}=k_{0}^{2}+\gamma_{z}^{2}. \tag{44}\]  
The surface impedance \(\overline{\overline{\overline{Z}}}_{\text{surf}}\) is defined as the ratio of tangential electric fields to tangential magnetic fields along one side of EM surfaces, for example the field components in region \(z>0\). The relations between \(\vec{E}\), \(\vec{H}\) and \(\overline{\overline{Z}}_{\text{surf}}\) are usually expressed as  
\[\left.\vec{E}_{\text{tan}}\right|_{z=0^{+}}=\overline{\overline{Z}}_{\text{ surf}}\cdot(\hat{z}\times\vec{H}|_{z=0^{+}}). \tag{45}\]  
Figure 17: Comparison between simulated and computed transmission coefficients with \(\theta\) = 45\({}^{o}\) of the zero-thickness PEC metascreen shown in Figure 15a: (**a**) magnitude (**b**) phase. Since field components in region \(z<0\) are not considered, Equation (45) is sometimes called opaque impedance boundary conditions, while the ones based on sheet impedances introduced in Section 3 are denominated as transparent IBCs [42]. Similar to sheet impedances, surface impedance is a two-dimensional tensor for general cases and can be simplified as a scalar with the condition of isotropy. Suppose that the surface wave propagates towards \(+\hat{x}\) direction (\(k_{y}=0\)) along a periodic isotropic surface. Then the \(\hat{z}\) components of EM fields can be expressed as  
\[E_{z} =E_{0}e^{-i|\vec{k}_{t}|x-\gamma_{z}z} \tag{46a}\] \[H_{z} =H_{0}e^{-i|\vec{k}_{t}|x-\gamma_{z}z}. \tag{46b}\]  
Based on the relations between transverse and longitudinal components of EM fields [43], field components along \(\hat{x}\) and \(\hat{y}\) directions can be respectively expressed as  
\[E_{x} =j\gamma_{z}E_{0}e^{-i|\vec{k}_{t}|x-\gamma_{z}z}/|\vec{k}_{t}| \tag{47a}\] \[E_{y} =k_{0}\eta_{0}H_{0}e^{-i|\vec{k}_{t}|x-\gamma_{z}z}/|\vec{k}_{t}|\] (47b) \[H_{x} =j\gamma_{z}H_{0}e^{-i|\vec{k}_{t}|x-\gamma_{z}z}/|\vec{k}_{t}|\] (47c) \[\vec{k}=\hat{x}k_{x}+\hat{y}k_{y}-\hat{z}j\gamma_{z}, \tag{43}\]0  
For TE polarization, \(E_{x}=E_{z}=H_{y}=0\). According to Equation (45), the surface impedance \(Z_{\text{surf}}^{\text{TE}}\) can be expressed as  
\[\vec{k}=\hat{x}k_{x}+\hat{y}k_{y}-\hat{z}j\gamma_{z}, \tag{43}\]1  
Similarly, for TM polarization,  
\[\vec{k}=\hat{x}k_{x}+\hat{y}k_{y}-\hat{z}j\gamma_{z}, \tag{43}\]2  
According to Equations (48) and (49), \(Z_{\text{surf}}\) is capacitive for TE polarization and inductive for TM polarization. The traditional method to determine \(\vec{k}_{t}\) or \(Z_{\text{surf}}\) is based on eigen-mode solver of commercial simulation softwares [44]. The input variables to eigen-mode solvers are the phase delay \(\phi\) across one period of unit cell along the propagation direction of surface wave, from which the value of \(|\vec{k}_{t}|\) can be determined. With given surface structures and \(\phi\), eigen-mode solvers will compute the corresponding eigen-frequency.