# Enhancement of the Purcell Effect by the Wire Metamaterials Formed by the Hexagonal Unit Cells
## 3 Results  
In the following, we will demonstrate how the proposed structure can be used to control the properties of the metamaterial, mode patterns of a surface plasmon, and value of the Purcell factor. It is worthwhile examining the resonance of the perpendicular permittivity. The frequency dispersion of \(\varepsilon_{\perp}\) is shown in Figures 2 and 3. Due to the low level of losses, the real part of \(\varepsilon_{\perp}\) changed the sign of the resonance. It is worthwhile noting that the regime \(\varepsilon_{\perp}=0\) facilitated the second topological transition, observable for \(d=20\), \(30\), and \(40\) nm but not for the case \(d=10\) nm. It should be mentioned that the resonance of the Purcell factor corresponded to the topological transition [23]. We also took into account the parallel component of the effective permittivity, due to the reason that the Purcell factor of a highly anisotropic metamaterial is defined by the factor \(\varepsilon_{\parallel}/\varepsilon_{\perp}\). Figures 2 and 3 show \(\varepsilon_{\parallel}\) versus the frequency. The surface plasmon polariton dispersion, calculated according to Equation (6), is shown in Figures 4 and 5. For comparison, the mode patterns of conventional surface waves propagating at the interface, separating the metal and dielectric, are also shown. It is of significant importance to mention that the plasmon associated with an ordinary wave was proven to be more effective in reducing the peak frequency of the local density of states (LDOS). It is obvious that by varying the filling factor, one can tune the peak frequency to any wavelength. Figure 3: (**a**) Dependences of the real parts of the perpendicular and parallel permittivity components, upon frequency of the silver nanowire metamaterial. (**b**) Dependence of the imaginary part of the perpendicular permittivity component, upon frequency. \(d\) = 30 nm. Figure 2: (**a**) Dependences of the real parts of the perpendicular and parallel permittivity components, upon frequency of the silver nanowire metamaterial. (**b**) Dependence of the imaginary part of the perpendicular permittivity component upon frequency. \(S\) = 70 nm. It can be observed, in Figures 4 and 5, that if \(\rho\to 1\), Re(\(\beta\)) (Figures 4a and 5a) and Im(\(\beta\)) (Figures 4b and 5b) are approaching the dispersion of the conventional surface plasmon polaritons, propagating at the metal/dielectric boundary. It is worthwhile mentioning that a comparison of the result with a rigorous calculation (the full wave method) is presented in Figure 4a for the case of \(d=10\) nm. The results we obtained from the full wave method are demonstrated with the blue markers. It should be mentioned that the agreement between the results we obtained using different methods was very good. In doing so, the dispersion curves were approaching the conventional surface plasmon polaritons curve, in the case of the increase in \(d\). The same tendency can be observed in Figure 5a, in the case of the decrease in \(S\). The determined estimates of the Purcell factor are presented in Figure 6. It can be observed in Figure 6a that the Purcell factor was increased by increasing the diameter of the nanowires. Moreover, it can be clearly observed in Figure 6b that the Purcell factor was increased by decreasing the distance between the nanowire. In doing so, it can be clearly observed, by displaying the Purcell factor versus the frequency in Figure 6, that the maxima of the Purcell factor were dramatically affected by the geometry of the composite. It is worthwhile noting that it might be found with the Purcell factor  
Figure 4: Mode patterns of the surface waves at the interface of metamaterial and dielectric (\(\epsilon_{\text{d}}=11.8\)). \(S=70\) nm. Dependencies of the real (**a**) and imaginary (**b**) parts of the propagation constant, upon frequency. Figure 5: Mode patterns of the surface waves at the interface of metamaterial and dielectric (\(\epsilon_{\text{d}}=11.8\)). \(d=30\) nm.