## 3 Two-Layered MMs for Surface Plasmon Polariton Guiding  
Figure 2 is a schematic view of a hyperbolic multilayered MM consisting of periodically changing graphene and dielectric layers. The effective-medium theory [21] can be applied if the width of any sheet is much smaller than the wavelength of radiation. This approach allows us to describe the optical response in such a structure. Herein, we present the dispersion branches calculated after carrying out the homogenization of two MMs. The dispersion pattern for the case \(\varepsilon_{1}=\varepsilon_{3}\), \(\varepsilon_{2}=\varepsilon_{4}\) and \(d_{1}\
eq d_{2}\
eq d_{3}\
eq d_{4}\) is presented in Figure 3. Figure 3 demonstrates that the SPPs branch at the interface separating MMs with \(d_{1}\) = 0.35 nm, \(d_{2}\) = 10 nm, \(d_{3}\) = 0.25 nm, and \(d_{4}\) = 10.1 nm. Moreover, the dispersion diagrams of SPPs, characterized by the effective properties outlined in [21], are displayed in Figure 3.  
The tunability feature of the frequency ranges of SPPs is the main outcome of changing the Fermi energy. It is worthwhile noting that the decrease of the Fermi energy \(\mu\) causes movement of the dispersion diagrams to lower frequency ranges. On the contrary, the increase of the Fermi energy \(\mu\) will stimulate their movement to higher frequencies. As illustrated in Figure 3, the smallest asymptotic frequency is achievable in the case of the smallest Fermi energy. The presented controllability feature provides possibilities to engineer the SPPs by the Fermi energy of the graphene layers. Consequently, special phase-matching tools, such as a grating or prism coupling [22], near-field excitation [23], and end-fire coupling [24], are needed for their excitation via three-dimensional beams [3]. Because of the graphene being treated as lossless, \(\beta\) is complex resulting in a finite propagation length, presented in Figure 3b. The next case to deal with, i.e. \(\varepsilon_{1}=\varepsilon_{3}\), \(\varepsilon_{2}\
eq\varepsilon_{4}\) and \(d_{1}\
eq d_{2}\
eq d_{3}\
eq d_{4}\), is studied by sketching four modes in Figure 4 at the interface of the MMs with \(\varepsilon_{4}\) = 2.25. As it follows from Figure 4, the case \(\mu=0.5\) eV is characterized by the smallest asymptotic frequency. It is worthwhile noting, that \(Re(\beta)\
eq 0\) allows for the quasi-bound [21], leaky region of the dispersion equation. Figure 2: A schematic diagram of the boundary surface separating two different infinite multilayered MMs designed by a stack of periodically changing graphene and dielectric layers. Here, dielectric and graphene sheets are denoted by indexes “1, 2”, correspondingly. Figure 4: The dispersion of surface plasmonic polaritons (SPPs) (**a**); propagation lengths (**b**) and absorption (**c**) at different Fermi energies, where the number of graphene layers \(N=1\), \(d_{d}=10\) nm, \(\varepsilon_{1}=\varepsilon_{3}\), \(\varepsilon_{2}\
eq\varepsilon_{4}\), and \(d_{1}\
eq d_{2}\
eq d_{3}\
eq d_{4}\), and the blue line in (**a**) denotes the free space light line. Figure 3: The dispersion of SPPs (**a**); propagation lengths (**b**) and absorption (**c**) at different Fermi energy, where number of graphene layers, i.e. \(N=1\), \(d_{\text{d}}=10\) nm, \(\varepsilon_{1}=\varepsilon_{3}\), \(\varepsilon_{2}=\varepsilon_{4}\) and \(d_{1}\
eq d_{2}\
eq d_{3}\
eq d_{4}\) and the blue line in (**a**) denotes the free space light line. Following the previous instances, the transverse magnetic (TM) mode dispersion plots are used to deal with the case of \(\varepsilon_{1}=\varepsilon_{3}\), \(\varepsilon_{2}\
eq\varepsilon_{4}\) and \(d_{1}=d_{3}\), \(d_{2}=d_{4}\). Four various modes are presented in Figure 5. The outcome of tuning the fill ratios is examined as displayed in Figure 5. Firstly, it is worth discussing how the dielectric thickness \(d_{\rm d}\) impacts the dispersion branch (see Figure 5a). It is found that an increase of \(d_{\rm d}\) causes a shift of the upper limit to higher frequencies.