## 3 Numerical Analysis
### Field Distributions of Surface Waves  
The field distributions of surface waves propagating on a finite-width grounded dielectric slab are examined in this section. For instance, we use an anisotropic slab with \(\varepsilon_{x}\) = 2.95, \(\varepsilon_{y}\) = 2.89, \(\varepsilon_{z}\) = 2.45, and \(\mu_{x}\) = \(\mu_{y}\) = \(\mu_{z}\) = 1. The slab width is = 400 mm. The height of the slab is \(t\) = 5 mm (0.27\(\lambda_{0}\) at calculated frequency \(f\) = 16 GHz). By substituting the values \(\beta_{x}\), \(R_{y}\), \(R_{oy}\), \(k_{z}\), \(k_{oz}\), and \(r_{1}\) obtained from Equations (19)-(21) and (36)-(38) into Equations (7)-(14) and (24)-(35), the field distributions in all of the regions can be obtained. Based on Figure 3, it is known that when the slab thickness is 0.27\(\lambda_{0}\), quasi-TM\({}_{0}\), and quasi-TE\({}_{1}\) surface wave modes are excited. Figure 6 shows the distributions of E- and H-field components of the quasi-TM\({}_{0}\) mode on the y-z cut-plane at \(x\) = 0 mm in Regions I and II. It should be mentioned that the magnitude of E-field component along the \(x\) direction \(\mid E_{x}\mid\) is normalized, and the magnitudes of other components are then obtained accordingly. Based on the field distributions of the quasi-TM\({}_{0}\) mode, the following phenomena are observed: (1) Most fields aretrapped inside the slab and bound onto the interface at \(z=5\) mm. When it is away from the interface, the field decay rapidly; (2) Due to the finite width of the slab, the fields non-uniformly distribute along the \(y\) direction; (3) Figure 6a shows that for the E-field components, \(\left.\right|E_{x}\left.\right|\) and \(\left.\right|E_{z}\left.\right|\) are much larger than \(\left.\right|E_{y}\left.\right|\); Figure 6b indicates that for the H-field components, \(\left.\right|H_{x}\left.\right|\) and \(\left.\right|H_{z}\left.\right|\) are very small compared to \(\left.\right|H_{y}\left.\right|\); (4) On the ground plane at \(t=0\), the tangential E-fields \(\left.\right|E_{x}\left.\right|\) and \(\left.\right|E_{y}\left.\right|\) are zero, while the normal component \(\left.\right|E_{z}\left.\right|\) is strong. On the other hand, for the H-fields, \(\left.\right|H_{x}\left.\right|\) and \(\left.\right|H_{y}\left.\right|\) are maximum, while \(\left.\right|H_{z}\left.\right|\) is zero when \(t=0\).  
In addition, the field distributions of quasi-TE\({}_{1}\) mode on the \(y\)-\(z\) cut-plane at \(x=0\) mm are plotted in Figure 7. Several similar phenomena are observed when compared to Figure 6. However, contrary to the quasi-TM\({}_{0}\) mode, it is seen from Figure 7a that the E-field components of quasi-TE\({}_{1}\) mode \(\left.\right|E_{x}\left.\right|\) and \(\left.\right|E_{z}\left.\right|\) are small as compared to \(\left.\right|E_{y}\left.\right|\). While for the H-field distributions in Figure 7b, \(\left.\right|H_{x}\left.\right|\) and \(\left.\right|H_{z}\left.\right|\) are much larger than \(\left.\right|H_{y}\left.\right|\). These reasonable observations further prove the reliability of our proposed method. It is also expected that the proposed method is very promising for analyzing other higher-order surface wave modes for a thicker slab.