## Three-Layered MMs for Surface Plasmon Polariton Guiding  
Figure 6 illustrates a diagram of dielectric/graphene/dielectric multilayered MM. It should be mentioned that the MM under study is described by effective permittivity tensor as follows:  
\[\varepsilon=\varepsilon_{0}\left( \begin{array}{ccc}\varepsilon_{||}&0&0\\ 0&\varepsilon_{||}&0\\ 0&0&\varepsilon_{\perp}\end{array} \right) \tag{10}\]  
in the principal-axis coordinate system. The principal values of the tensor are obtained in [35; 36] by means of the theoretical approach provided in Section 2:  
\[\varepsilon_{||}=f_{g}\varepsilon_{g}+f_{1}\varepsilon_{1}+\big{(}1-f_{g}-f_{1 }\big{)}\varepsilon_{2} \tag{11}\]  
\[\varepsilon_{\perp}=\varepsilon_{g}\varepsilon_{1}\varepsilon_{2}/\big{(}f_{g }\varepsilon_{1}\varepsilon_{2}+f_{1}\varepsilon_{g}\varepsilon_{2}+\big{(}1- f_{g}-f_{1}\big{)}\varepsilon_{g}\varepsilon_{1}\big{)} \tag{12}\]  
where \(f_{g}=d_{g}/L\) and \(f_{1}=d_{1}/L\), \(L=d_{1}+d_{2}+d_{g}\) represent the graphene and first dielectric filling ratios, respectively, \(d_{g}\) is the thickness of the graphene layer, \(d_{1}\) is the thickness of the first dielectric sheet, \(d_{2}\) is the thickness of the second dielectric layer, \(\varepsilon_{1}\) is the permittivity of the first dielectric, and \(\varepsilon_{2}\) is the permittivity of the second dielectric. For the different cases presented in Figure 7, dispersion relations are obtained as follows  
\[\beta=k\sqrt{\frac{\varepsilon_{1}\varepsilon_{2}\varepsilon_{g}\big{(} \varepsilon_{1}-\varepsilon_{2}-\varepsilon_{1}f_{1}+\varepsilon_{2}f_{1}+ \varepsilon_{2}f_{g}-\varepsilon_{g}f_{g}\big{)}}{\varepsilon_{1}^{2} \varepsilon_{g}-\varepsilon_{2}^{2}\varepsilon_{g}-\varepsilon_{1}^{2} \varepsilon_{g}f_{1}+\varepsilon_{2}^{2}\varepsilon_{g}f_{1}+\varepsilon_{1}^ {2}\varepsilon_{2}f_{g}-\varepsilon_{1}^{2}\varepsilon_{g}f_{g}-\varepsilon _{2}\varepsilon_{In consequence, a surprising result is obtained: the thicknesses of the layers (Figure 7c,d) do not affect the dispersion of a (single) boundary mode, and it agrees with the dispersion of a conventional SPP at a metal-dielectric boundary. It can be concluded that the systems displayed in Figure 7c,d can support topological properties.  
Herein, we present dispersion diagrams calculated after carrying out the homogenization of two MMs. Thus, Figure 8 shows the dispersion branches of different cases of the multilayered MM. The dispersion branches of spoof SPPs at the MM interface with \(d_{\text{g}}=0.35\) nm are shown in Figure 8. It should be noted that the case displayed in Figure 8a (\(\varepsilon_{1}=18.8\)) corresponds to Equation (15). Thus, it can be concluded that the usage of the three-layered MM allows for the rich phenomenon, i.e., broadening of the frequency range of the SPP's existence by approximately 800 THz. The dispersion curve computed for the three-layered MM case has special features deserving of attention, besides the fact that the graph resembles behavior of a traditional metal SPP and multilayered MMs [21].  
For SPPs supported by the multilayered MM under study, the horizontal asymptote appears when \(\beta\) approaches infinity. Based on the dispersion equation [21], the corresponding condition is as follows:  
\[\varepsilon_{||}^{L}\varepsilon_{\perp}^{L}-\left(\varepsilon^{R}\right)^{2}=0 \tag{17}\]  
In other words, the position of horizontal asymptotes is determined by Equation (17). It is also concluded that \(\varepsilon_{||}^{L}\) must share the same sign with \(\varepsilon_{\perp}^{L}\). It is worthwhile noting that \(\varepsilon^{R}\) will drastically affect the curve type. Equation (17) holds only when both \(\varepsilon_{||}^{L}\) and \(\varepsilon_{\perp}^{L}\) are negative when \(\left(\varepsilon^{R}\right)^{2}\) is larger than \(\lim_{\omega\rightarrow\infty}\varepsilon_{||}\varepsilon_{\perp}\). The former scenario matches band 2 in Figure 9. Consequently, the dispersion branch possesses only one horizontal asymptote resembling the metal SPP case.