## 5 Tunable Hyperbolic MM Made of TCO-Dielectric Multilayered MMs  
The proposed MM is displayed in Figure 11. Herein, \(d_{TCO}\) and \(d_{d}\) are the widths of the TCO and dielectric sheets, correspondingly. A TCO/PbS MM is considered, aiming to study the dispersive properties of SPPs. It is assumed that \(\varepsilon_{d}=18.8\) for PbS sheets, and \(\varepsilon_{TCO}\) is computed by means of the parameters outlined in [38]. For the instance of the anisotropic material, the EM wave dispersions can be sketched for MM/dielectric and the MM/TCO instances. Let us concentrate on a semi-infinite AZO (Aluminum Zinc Oxide)/PbS MM example while tackling the numerical calculations. Moreover, a homogenization of the MM matching the MM/dielectric and MM/TCO boundaries is considered. Sketching the branches of the permittivities (\(\varepsilon_{\perp}\) and \(\varepsilon_{||}\)) and the dielectric constant (\(\varepsilon_{R}\)) allows us to consider the optical properties. Moreover, we consider different frequency ranges, distinguishing them based on the properties of and relation among \(\varepsilon_{\perp}\), \(\varepsilon_{||}\), and \(\varepsilon_{R}\)[39]. (1) \(\varepsilon_{\perp}>\varepsilon_{R}>\varepsilon_{||}\) and \(\varepsilon_{||}<0\) (cyan);  
(2) \(\varepsilon_{\perp}>\varepsilon_{R}>\varepsilon_{||}\) (green);  
(3) \(\varepsilon_{||}>\varepsilon_{R}>\varepsilon_{\perp}\) and \(\varepsilon_{\perp}<0\) (gray);  
(4) \(\varepsilon_{R}>\varepsilon_{||}>\varepsilon_{\perp}\) and \(\varepsilon_{\perp}<0\) (magenta) and  
(5) \(\varepsilon_{\perp}<\varepsilon_{||}<0\) (orange). It is worthwhile mentioning that the effective parameters will dramatically influence the SPPs, as outlined in [39]. The TCO filling ratio,\(f_{TCO}\), is used here as the main mechanism controlling the effective optical parameters. Figures 12 and 13 display characteristic dispersion branches, considering different TCO filling factors. Different branch colors are applied, aiming to distinguish various frequency ranges. One can distinguish three SPPs fitting three different ranges in the case of \(f_{TCO}=0.3\) and MM/dielectric boundary. Moreover, two kinds of SPPs are observed in the instance of the MM/TCO boundary. For the MM/TCO case, the upper waves are in the gray region. The \(f_{TCO}=0.5\), \(0.7\), \(0.9\) cases are associated with three exceptional cases. In the case of \(f_{TCO}=0.5\), two types of SPPs appear in two color ranges, i.e., cyan and gray for the MM/dielectric and MM/TCO boundaries. The upper short branches in the case of \(f_{TCO}=0.7\), lying in the gray range, are presented in Figures 12 and 13. The other SPPs belong to the orange range. The supplementary SSPs belonging to the magenta range are observed with respect to a MM/dielectric boundary. A dark-magenta color displays an extension approach for the magenta region in the case of \(\varepsilon_{R}=2.25\). One may observe two kinds of SPPs belonging to the orange and cyan ranges for the MM/TCO instance and three types of SPPs lying in the cyan, orange, and gray ranges for the MM/dielectric case with a filling factor \(f_{TCO}=0.9\)  
Figure 11: Interface of multilayered MM made of periodically changing transparent-conducting oxide (TCO) and dielectric layers. The MM/dielectric (\(\varepsilon_{R}=1\) or \(\varepsilon_{R}=2.25\)) and MM/TCO boundaries are investigated. It should be underlined that one SPP occurs in the cyan range for different TCO filling factors in the MM/dielectric case. The former is outlined in Figure 12. It is of importance that the case \(\varepsilon_{R}=\varepsilon_{ITO}\) opens the gateway for possible applications, as the SPP always occurs in the cyan range for different filling factors (Figure 13). Following the essential condition for the presence of the Dyakonov-like SPP (DSW) [35], the SPP in the green resembles the behavior of a DSW. Thus, it is important to address DSW [40], or the Dyakonov defined in [35].