# Wave Propagation in Periodic Metallic Structures with Equilateral Triangular Holes
## 3 Results
### Applications  
In this subsection we compare the major properties of different isotropic metastructures possessing glide symmetry. For this study, three glide-symmetric structures with triangular, circular, and square holes are considered. The results are summarized in Tables 1 and 2. As a comparison criterion for Table 1, we consider an equilateral triangular hole whose side \(e=0.99d\), very close to the entire period, is identical to the side of a square hole as well as to the diameter of a circular hole. This is considered here as a \"maximum\" hole side. In order to calculate the frequency range in which the structures are regarded as isotropic and low-dispersive, we allow a maximum relative deviation of the refractive index of \(5\%\) with respect to its reference value at \(f\to 0\)  
Figure 11: Relative bandgap in the (**a**) mirror-symmetric and (**b**) glide-symmetric structures with equilateral triangular holes as a function of the period \(d\) and the triangle side \(e\). The geometrical parameters of the considered unit cells are: \(h_{\text{hole}}=1.5\,\text{mm}\), and \(g=0.2\,\text{mm}\). Figure 12: Relative bandgap in the (**a**) mirror-symmetric and (**b**) glide-symmetric structures with equilateral triangular holes as a function of the gap height \(g\) and the hole depth \(h_{\text{hole}}\). The geometrical parameters of the considered unit cells are: \(h_{t}=3.8\,\text{mm}\), and \(d=4.5\,\text{mm}\). and propagating direction 0\({}^{\rm o}\). The analysis of the \"isotropic\" degree of the structure is carried out by comparing the dispersion properties in the propagating directions of 0\({}^{\rm o}\) and 45\({}^{\rm o}\). For the maximum hole size, the triangular geometry offers the highest equivalent refractive index, as well as the least isotropic and low-dispersion bandwidth. Note that the circular hole is isotropic in a higher bandwidth compared to the other geometries; that is, it is less dispersive for different propagating directions. This could be associated to the rotational symmetry that possesses. Additionally, we calculate the maximum achievable bandgap in the three structures. In this calculation of the bandgap, the trends are different in the square, circular, and triangular holes. As previously discussed, the circular and square holes have an optimal value that maximizes the bandgap and beyond this value, the bandgap starts to close. As a consequence, in the case of maximum hole size, the bandgap within the first four propagating modes is very narrow in the case of circular and square holes (null in the case of the square geometry). However, in the case of the triangular hole, the bigger the hole is, the wider the bandgap. For this reason, the triangular geometry offers a wide bandgap when fixing the maximum hole size as a comparison criterion. In Table 2 we compare the optimal values of glide-symmetric structures with triangular, circular and square holes. We carry out a parameter sweep in order to find the hole sizes associated with the optimal major properties of the three structures. In order to compute the optimal bandwidths, a maximum deviation of the refractive index of 5% is allowed. However, we should remark that any other comparison criterion, as fixing the same refractive index in all the structures and looking for the optimal bandwidths, would have been equally valid. According to the results shown in Table 2, the square hole offers the highest equivalent refractive index (it is the densest structure) as well as the widest stopband. The triangular hole shows similar characteristics (high equivalent refractive index and wide bandgap), being less frequency-dispersive than the square hole. Furthermore, the circular hole is isotropic and low-dispersive in a larger bandwidth compared to the triangular and square geometries, as also pointed out in Table 1.  
The potential application of the different structures can readily be inferred from the properties observed in Tables 1 and 2.