# Wave Propagation in Periodic Metallic Structures with Equilateral Triangular Holes
## 3 Results
### Dispersion Diagram  
In order to simplify the analysis of the structure, a reference unit cell is defined with the following geometrical parameters: Period \(d_{x}=d_{y}=d=4.5\,\mathrm{mm}\), triangle height \(h_{t}=3\,\mathrm{mm}\), hole depth \(h_{\mathrm{hole}}=1.5\,\mathrm{mm}\), gap height \(g=0.5\,\mathrm{mm}\), and relative permittivities \(\varepsilon_{r}=\varepsilon_{\mathrm{hole}}=1\). Figure 3 shows the dispersion diagrams of the mirror- and glide-symmetric unit cells. The interval \(\Gamma\)-X refers to wave propagation along the \(x\)-direction and \(\Gamma\)-Y to wave propagation along the \(y\)-direction. As shown in the figure, an excellent agreement is found between the results computed with the proposed mode-matching method (solid lines) and \(CST\) (dashed lines). The dispersion diagrams associated with the glide-symmetric structures are here plotted with red lines. In Figure 3, it can be appreciated that the stop band between the first and second modes disappears, as expected for this symmetric configuration. Furthermore, the frequency dispersion of propagating modes is considerably reduced compared to the mirror-symmetric structure (blue lines). Surprisingly, both configurations with mirror and glide symmetries present a very high isotropic behavior, despite the triangle is asymmetric in \(x\) and \(y\) directions with associated wave propagation paths clearly different. Specifically, the glide-symmetric structure exhibits a isotropic behavior in a wider range of frequencies. This issue will be treated in more detail in the next subsection. Figure 4 shows the effects on the dispersion diagram of varying the size of the triangular hole (\(h_{t}\)) in the mirror- and glide-symmetric configurations. As it can be noticed, both structures become denser (higher equivalent refractive index) as the triangle is larger, since its interaction with the propagating wave is greater. Additionally, the bandgap regions also become wider as the size of the hole increases. In Figure 5, the influence of the gap height \(g\) on the dispersion diagram is analyzed. It is observed that the equivalent refractive index is inversely proportional to the gap height; namely, the structure is denser for smaller \(g\). Additionally, the glide-symmetric configuration is denser compared to the mirror-symmetric one (i.e., the green curves at the \(\Gamma\) point show that the second mode propagates at lower frequencies in glide-symmetric cells). This fact was already exploited in [24] to increase the compression factor of a wideband [PERSON] fish-eye lens made of glide-symmetric elliptical holes. Figure 3: Dispersion diagram of the glide-symmetric (red lines) and mirror-symmetric (blue lines) unit cells. Dashed lines represent the results extracted from \(CST\) and solid lines the results extracted from the mode-matching. The light line is displayed in gray. Finally, Figure 6 shows the influence of the hole depth \(h_{\text{hole}}\) on the dispersion diagram. In this case, the equivalent refractive index is directly proportional to \(h_{\text{hole}}\); that is, the structure is denser as the hole is deeper. Similarly to the findings in [21; 23; 24], once the depth is increased beyond a threshold value, \(h_{\text{hole}}\) has no longer influence on the structure. This can be explained by the fact that the modes inside the hole are evanescent (operation frequency below the cutoff frequency of the triangular hole) and, therefore, they exponentially attenuate inside the hole without \"reaching\" the ground at the bottom of the hole, so that the position of this bottom ground has an almost negligible influence on the metasurface behavior. Figure 4: Dispersion diagram of the (**a**) mirror-symmetric and (**b**) glide-symmetric unit cells when varying the height of the triangle. Dashed lines represent the results extracted from \(CST\) and solid lines the results extracted from the mode-matching. The light line is displayed in gray.