## 3 Numerical Analysis  
### Comparison of Different Methods  
In this section, three different methods are employed to compare the propagation characteristics of surface waves on an anisotropic grounded dielectric slab. In the first method, it is assumed that the slab is infinitely large in both \(x\) and \(y\) directions, and there is no field variation along the \(y\) direction. This method has been utilized in most open literature, but it can only provide approximate results for real engineering applications, in which finite-width slabs are usually employed, and the field distributions along the \(y\) direction are non-uniform. The propagation characteristics are summarized in Equations (A1)-(A6) in Appendix A. The second method is based on [PERSON]'s method [22]. This method considers the field variations along the \(y\) direction of the finite-width slab. However, the analysis assumes that \(H_{z}=0\) and \(H_{y}=0\) for quasi-TM and quasi-TE modes, respectively. Based on this approximate method, we derived the closed-form expressions for propagation characteristics of quasi-TM and quasi-TE modes for the considered structure, which can be obtained using Equations (A8)-(A12) and (A14)-(A18), respectively, as summarized in Appendix B. The third one is based on our derivations by using Equations (19)-(21) and (36)-(38). In our method, hybrid surface wave modes on grounded dielectric slab with finite width are analyzed, which addresses the main concern and provides a more accurate solution for practical applications. As an example, we use an anisotropic PTFE substrate in [21] with \(\varepsilon_{x}=2.95\), \(\varepsilon_{y}=2.89\), \(\varepsilon_{z}=2.45\), and \(\mu_{x}=\mu_{y}=\mu_{z}=1\) to analyze the propagation constants of surface waves, and the results at\(f=16\) GHz are shown in Figure 3a. In addition, the corresponding results of isotropic slab with \(\varepsilon_{x}=\varepsilon_{y}=\varepsilon_{z}=2.45\) and \(\mu_{x}=\mu_{y}=\mu_{z}=1\) are also shown in Figure 3b for comparison. In our calculation, we are looking for solutions that surface waves propagate along the \(x\) direction and decay in the \(z\) direction away from the interface, so that \(\beta_{x}\) and \(k_{0z}\) should be positive real numbers. For the first method, the slab width is assumed to be infinite with \(R_{y}=0\), while for the other two methods, the slab width is set as \(2a=37.5\) mm, with \(R_{y}\
eq 0\). Figure 3a,b both indicate that the results of finite-width slab obtained by [PERSON]'s method and our method are very close. However, it is interesting to note that for a fixed slab thickness, the propagation constant \(\beta_{x}\) obtained from the finite-width slab (\(R_{y}\
eq 0\)) is smaller than that from the infinite one (\(R_{y}=0\)). In other words, in order to excite the same order of surface waves with an equal \(\beta_{k}\), the finite-width slab should be thicker than the infinite one. In addition, it is seen that as the thicknesses of finite- and infinite-width slabs decease, the finite one can reach the condition \(\beta_{k}/k_{0}=1\) with a larger \(t\). Under this condition, the surface waves will transform into leaky waves. Furthermore, comparing the case of anisotropic with isotropic slabs, it is seen that the anisotropic property of the slab has significant effects on both quasi-TM and quasi-TE modes, which will be discussed in detail in the next section. As shown in Figure 4a, the values of \(\beta_{x}/k_{0}\) as a function of \(t/\lambda_{0}\) with different slab widths are compared. In this study, our method based on Equations (19)-(21) and (36)-(38) is employed for the case of finite-width slab, while for the case of \(a=\infty\), the first method is utilized. It is seen that when the slab width increases, the values of \(\beta_{x}/k_{0}\) increase, and they are closer to the case of \(a=\infty\) with \(R_{y}=0\). This phenomenon also validates our derivations in a way.