## Characteristics of Surface Waves Propagating on Grounded Anisotropic Slab  
### Infinitely Large Anisotropic Grounded Slab  
Figure 1 shows the geometry of the anisotropic slab backed by a perfect electric conductor. We assume that the grounded slab is infinite in both \(x\) and \(y\) directions, the surface waves propagate along the longitudinal \(+x\) direction with an \(e^{-j\bar{\mathbf{\mu}}_{x}x}\) propagation factor. The thickness of the slab is \(t\). The slab is considered as anisotropic as a general discussion, and the relative permittivity \(\overline{\bar{\mathbf{\epsilon}}}\) and permeability \(\overline{\overline{\mathbf{\mu}}}\) of the anisotropic slab are expressed as:  
\[\overline{\bar{\mathbf{\epsilon}}}=\left[ \begin{array}{ccc}\varepsilon_{x}&0&0 \\ 0&\varepsilon_{y}&0\\ 0&0&\varepsilon_{z}\end{array} \right],\,\overline{\bar{\mathbf{\mu}}}=\left[ \begin{array}{ccc}\mu_{x}&0&0\\ 0&\mu_{y}&0\\ 0&0&\mu_{z}\end{array} \right], \tag{1}\]  
where the diagonal elements represent the eigenvalues of \(\overline{\bar{\mathbf{\epsilon}}}\) and \(\overline{\bar{\mathbf{\mu}}}\), respectively, and their directions constitute the principle axes of the anisotropic slab. In order to analyze the characteristics of surface waves propagating along the grounded anisotropic dielectric slab, we separately consider the fields in anisotropic and air regions, then solve Maxwell's equations and apply boundary conditions across the interface. For the considered structure, the E-field and H-field along the \(x\) direction for both regions are expressed as:  
Region I  
\[E_{x}^{\mathrm{I}}=A_{\mathrm{I}}f_{e}(y)e^{-j\beta_{x}x}\sin(k_{z}z), \tag{2}\]  
\[H_{x}^{\mathrm{I}}=B_{\mathrm{I}}f_{h}(y)e^{-j\beta_{x}x}\cos(k_{z}z), \tag{3}\]  
Region II  
\[E_{x}^{\mathrm{II}}=A_{\mathrm{II}}f_{e}(y)e^{-j\beta_{x}x}e^{-k_{0z}z}, \tag{4}\]  
\[H_{x}^{\mathrm{II}}=B_{\mathrm{II}}f_{h}(y)e^{-j\beta_{x}x}e^{-k_{0z}z}, \tag{5}\]where \(A_{\rm I}\), \(A_{\rm II}\), \(B_{\rm I}\), and \(B_{\rm II}\) are the amplitude coefficients of the corresponding fields in each region. \(\beta_{x}\) is the phase constant of surface waves along the \(x\) direction. \(k_{z}\) is the wavenumber in the anisotropic medium in the \(z\) direction. The sine function of \(E_{x}^{\rm I}\) and cosine function of \(H_{x}^{\rm I}\) in the anisotropic medium are selected due to the vanishing electric field and maximum magnetic field on the perfect electric conductor (PEC) ground plane, respectively. Due to the fields of surface waves decaying exponentially away in air from the interface, the \(z\) component of fields in air region is expressed as \({\rm e}^{-k_{0}z}\), with \(k_{0}z\) real and positive. In addition, \(f_{e}(y)\) and \(f_{h}(y)\) represent the \(y\) components of E-field and H-field, respectively. For the infinitely large slab, we assume that the propagation constant in the \(y\) direction is \(R_{y}\), \(f_{e}(y)\) and \(f_{h}(y)\), then satisfy the following form:  
\[f_{e}(y)=f_{h}(y)=f(y)\ =\left\{ \begin{array}{c}e^{-R_{y}y}\ (y\geq 0)\\ e^{R_{y}y}\ (y\leq 0)\end{array} \right.. \tag{6}\]  
If \(R_{y}=0\), then \(\partial f(y)/\partial y=0\), which indicates that the field has no variation in the \(y\) direction. On the other hand, if \(R_{y}\
eq 0\), the field of surface wave is non-uniformly distributed along the \(y\) direction.