## Characteristics of Surface Waves Propagating on Grounded Anisotropic Slab
### Finite-Width Anisotropic Grounded Slab  
Figure 2 shows the grounded anisotropic slab with a finite width \(2a\) and thickness \(t\). By applying the boundary conditions at the interfaces \(y=\pm a\), the propagation constant in the \(y\) direction \(R_{y}\) can then be determined. For analysis, the cross-section of the grounded slab is sub-divided into six regions, as shown in Figure 2b. Region I is the anisotropic slab, while the others are air regions. Since the fields in two corner Regions V and VI are very small, which are assumed to be negligible [22], we only take into account Regions I, II, III, and IV for analysis. The E-field and H-field along the \(x\) direction for Regions I and II have the same expressions as Equations (2)-(5). However, due to the discontinuity at \(y=\pm a\), reflected waves exist along the \(y\) direction. Therefore, the \(y\) components \(f_{e}(y)\) and \(f_{h}(y)\) of the E-Field and H-field are expressed as:  
\[f_{e}(y)=e^{-R_{y}y}+r_{1}e^{R_{y}y}, \tag{22}\]  
\[f_{h}(y)=e^{-R_{y}y}+r_{2}e^{R_{y}y}, \tag{23}\]  
where \(r_{1}\) and \(r_{2}\) represent the magnitude ratios of \(e^{R_{y}y}\) and \(e^{-R_{y}y}\) for \(f_{e}(y)\) and \(f_{h}(y)\), respectively. The fields in both Regions I and II in \(y\) and \(z\) directions can be obtained by substituting Equations (22) and (23) into Equations (7)-(14). In Regions III and IV, the fields are decaying away from the interfaces \(y=\pm a\) along the \(y\) direction. Therefore, the \(y\) components of fields can be expressed as \(\mathrm{e}^{-R_{0y}y}\) and \(\mathrm{e}^{R_{0y}y}\), respectively. Based on the continuity conditions of the tangent fields at the interface, the E-field and H-field along the \(x\) direction for Regions III and IV are given as follows:  
Region III  
\[E_{x}^{\mathrm{III}}=A_{\mathrm{III}}\mathrm{e}^{-R_{0y}y}e^{-j\beta_{x}x}\sin( k_{z}z), \tag{24}\]  
\[H_{x}^{\mathrm{III}}=B_{\mathrm{III}}\mathrm{e}^{-R_{0y}y}e^{-j\beta_{x}x}\cos (k_{z}z), \tag{25}\]  
Region IV  
\[E_{x}^{\mathrm{IV}}=A_{\mathrm{IV}}\mathrm{e}^{R_{0y}y}e^{-j\beta_{x}x}\sin( k_{z}z), \tag{26}\]  
\[H_{x}^{\mathrm{IV}}=B_{\mathrm{IV}}\mathrm{e}^{R_{0y}y}e^{-j\beta_{x}x}\cos( k_{z}z), \tag{27}\]  
where \(A_{\mathrm{III}}\), \(B_{\mathrm{III}}\), \(A_{\mathrm{IV}}\), and \(B_{\mathrm{IV}}\) are corresponding amplitude coefficients. The E-field and H-field in the \(y\) and \(z\) directions can then be found as:  
Region III  
\[E_{y}^{\mathrm{III}}=\frac{e^{-j\beta_{x}x}\mathrm{e}^{-R_{0y}y}\sin(k_{z}z)} {j\big{(}k_{0}^{2}-\beta_{x}^{2}\big{)}}\big{[}-\beta_{x}R_{0y}A_{\mathrm{III}} -\omega\mu_{0}k_{z}B_{\mathrm{III}}\big{]}, \tag{28}\]  
\[H_{y}^{\mathrm{III}}=\frac{e^{-j\beta_{x}x}\mathrm{e}^{-R_{0y}y}\cos(k_{z}z)} {j\big{(}k_{0}^{2}-\beta_{x}^{2}\big{)}}\big{[}-\beta_{x}R_{0y}B_{\mathrm{III}} -\omega e_{0}k_{z}A_{\mathrm{III}}\big{]}, \tag{29}\]  
Figure 2: Surface waves on the interface of air region and grounded anisotropic slab with finite width: (**a**) Perspective view; and, (**b**) Cross-section view. \[E_{z}^{\text{III}}=\frac{e^{-i\beta_{x}x}\text{e}^{-R_{0y}y}\cos(k_{z}z)}{j(k_{0}^{2 }-\beta_{x}^{2})}[\beta_{x}k_{z}A_{\text{III}}+\omega\mu_{0}R_{0y}B_{\text{III}}], \tag{30}\]  
\[f_{h}(y)=e^{-R_{y}y}+r_{2}e^{R_{y}y}, \tag{23}\]0  
Region IV  
\[f_{h}(y)=e^{-R_{y}y}+r_{2}e^{R_{y}y}, \tag{23}\]1  
\[f_{h}(y)=e^{-R_{y}y}+r_{2}e^{R_{y}y}, \tag{23}\]2  
\[f_{h}(y)=e^{-R_{y}y}+r_{2}e^{R_{y}y}, \tag{23}\]3  
\[f_{h}(y)=e^{-R_{y}y}+r_{2}e^{R_{y}y}, \tag{23}\]4  
The boundary condition across the interface \(z=t\) provides the same result as Equation (19), while the field continuities at the interfaces \(y=\pm a\) give  
\[