## Three-Layered MMs for Surface Plasmon Polariton Guiding
The dispersion branch under such constraints bears a close resemblance to a metal and multilayered MM SPP [21] (see Figure 8). Similar to \(\omega_{\text{sp}}\) for metal, the upper bound for SPP is marked by the location of the lowest horizontal asymptote. \(\omega_{\text{upper}}\) denotes the upper bound frequency, aiming to separate the upper bound of multilayered SPPs and that of metal SPPs (i.e., \(\omega_{\text{sp}}\)). One can easily prove that it is possible to excite multilayered SPPs at any frequency lower than \(\omega_{\text{upper}}\) in band 2. Figure 7: Schematic of graphene-dielectric multilayered MM. The figures correspond to (**a**) Equation (13), (**b**) Equation (14), (**c**) Equation (15), (**d**) Equation (16). The dispersion curve (Figure 8) is obtained following the hypothesis of the EM field decaying exponentially in the \(z\) direction. It should be noted that not all modes with a certain frequency below \(\omega_{\text{upper}}\) is SPP in this study. The former takes place even though the dispersion curve displayed in Figure 10 resembles the behavior of conventional metal SPP. One should be assured that both \(k_{z}^{L}\) and \(k_{z}^{R}\) are imaginary when aiming to obtain the multilayered SPPs. It can be concluded, based on Figure 10, that the two cut-off points, \(\omega_{\text{lower}}\) and \(\omega_{\text{upper}}\) cut the frequency range into three regions. One may define \(\omega_{\text{upper}}\) as the frequency of the lowest horizontal asymptote in the dispersion branch. It is worthwhile noting that \(\omega_{\text{lower}}\), which is lower than \(\omega_{\text{upper}}\), is the frequency where \(k_{z}^{L}\) and \(k_{z}^{R}\) are converted from real to imaginary. The imaginary part of \(k_{z}^{L}\) in a multilayered MM is negligible, while the real part exists as indicated in Figure 10, in region 1. The former proves the possibilities of the EM wave propagating into the material, resulting in a little damping. Region 1 is ended by frequency \(\omega_{\text{lower}}\). The same frequency begins region 2. \(k_{z}^{L}\) possesses a negligible real part and relatively large imaginary part in region 2. The former describes the evanescence of the field is in this region. The imaginary part of \(k_{z}^{L}\) disappears and the real part of \(k_{z}^{L}\) occurs if \(\omega_{\text{upper}}\) is lower than the frequency (in region 3). The wavevector component \(k_{z}^{R}\) in the dielectric is shown in the right figure in Figure 10. In the same way, the wave can propagate into the dielectric in areas 1 and 3, however, it is bounded to the boundary in area 2. The SPPs can be controlled by tuning the Fermi energy. The former is in line with the dependence of \(\varepsilon_{\perp}\) on the Fermi energy [21]. It is worthwhile noting that the decreases in the Fermi energy \(\mu\) will cause a shift of the dispersion curves to higher wavelengths or lower frequencies. On the contrary, they will be shifted to higher frequencies because of the increases in the Fermi energy \(\mu\). The smallest asymptotic frequency is attained in the case of the smallest Fermi energy, as indicated in Figure 9. The former tunability property enables the SPP to be modified by the Fermi energy of the graphene layers. Figure 8: The impact of (**a**) \(\epsilon_{1}\), (**b**) Fermi energy \(\mu\), (**c**) graphene filling ratio,\(f_{\text{g}}\), and (**d**) dielectric filling ratio\(f_{1}\) on the dispersion diagrams of the SPPs. \(f_{\text{g}}=0.1\),\(f_{\text{a}}=0.5\) in (**a**). \(f_{\text{g}}=0.1\),\(f_{\text{a}}=0.1\) in (**b**). \(\mu=1.5\) eV in (**a**,**c**,**d**). Considering the next case, i.e., \(\varepsilon_{1}\
eq\) const, in Figure 8a, three modes are represented at the interface of MM with \(\varepsilon_{2}=18.8\). As displayed in Figure 8a, the smallest asymptotic frequency matches the case \(\varepsilon_{1}=18.8\). Also, \(Re(\beta)\
eq 0\) allows for the presence of the quasi-bound [37], leaky range of the dispersion equation.