## 3 Dispersion Relations, Group Velocities and Frequency Range of Backward Surface Waves
Figure 2: Dispersion relations and corresponding group velocities of the TM-polarized surface states supported by a homogenous dielectric medium and an MMDM with the filling factor being set as \(f=0.7\) in the UDL case. Relative permittivities \(\varepsilon_{i}=1\) and \(\varepsilon_{d}=18.8\), refer to the partner dielectric medium (air) and dielectric layers (PbS), respectively. (**a**) Dispersion relations with different unit-cell thicknesses. (**b**) Group velocities with different unit-cell thicknesses. (**c**) The frequency range and the relevant upper and lower threshold frequencies of the anomalous dispersions as a function of the unit-cell thickness. The dispersion relations of the backward SWs can also be modulated by changing the filling factor \(f\) in the UDL case. In Figure 3, we presented the dispersion relations, relevant group velocities and anomalous dispersion frequency range of the TM-polarized surface states with different filling factors when the unit-cell thickness was set as \(L=0.3\lambda_{p}\). For \(f\) above 0.87, the dispersion curve unfolds normal dispersive when \(\beta\) is in the vicinity of the light line. With the increasing \(\beta\), once the frequency reaches \(\Omega_{\text{up}}\), the dispersion curve becomes anomalous dispersive until it is cut off at \(\Omega_{\text{S}}\) with related \(\beta\) achieving infinity. When \(f\) is within the range of 0.64-0.87, the lower threshold frequency \(\Omega_{\text{low}}\) of the anomalous dispersion is smaller than \(\Omega_{\text{S}}\). Yet, the whole dispersion curve always maintains normal dispersive with \(f\) lower than 0.64. In general, regardless of the unit-cell thickness, the filling factor should surpass 0.5 in order to support anomalous dispersion relations. Figure 3: Dispersion relations and corresponding group velocities of the TM-polarized surface states supported by a homogenous dielectric medium and an MMDM with the unit-cell thickness being set as \(L=0.3\lambda_{p}\) in UDL case. The material parameters are set as the same values as those in Figure 2. (**a**) Dispersion relations with different filling factors. (**b**) Group velocities with different filling factors. (**c**) The frequency range and the relevant upper and lower threshold frequencies of the anomalous dispersions as a function of the filling factor. For the EMA method, the dispersion relations are independent of the order of the dielectric and metallic layers, as the effective permittivity tensors of the MMDM in both the UDL and UML cases remain the same. However, different orders of dielectric and metallic layers may lead to different surface plasmon modes, which will cause further impact on the dispersions of the surface states. Therefore, the order of the metallic and dielectric layers may be a key factor for the existence of backward SWs. In Figure 4, we showed the dispersion relations of TM-polarized surface state in the UML case while keeping other conditions unchanged. Compared to the UDL case, there is a blue shift of the dispersion curves calculated by the transfer-matrix method from one of the EMA method. Under such a circumstance, though still being affected by both the unit-cell thickness and filling factor, the dispersion relations maintain normal dispersive. Moreover, in the limiting case with an extremely large unit-cell thickness, the dispersion curve tends to coincide with that of the SPPs excited at a single interface of a semi-infinite metal and air, due to the absence of coupling effect. In Figure 5, we displayed the profiles of the tangential components \(H_{t}\) of the magnetic field and the time-averaged Poynting vector \(S_{t}\) according to the propagation constants and related frequencies acquired at Points A and B in Figure 2a and Point C in Figure 4a. For the UDL case, the field is confined at the boundary between the dielectric and metallic layers of the first bilayer unit, which is adjacent to the partner medium, as shown in Figure 5a,c.