# Wave Propagation in Periodic Metallic Structures with Equilateral Triangular Holes
## 3 Results
### Isotropic Behavior  
In this subsection, we give a physical insight on the unusual wave propagation isotropy observed in periodic structures with equilateral triangular holes. Firstly, we extend the analysis of wave propagation in \(x\) and \(y\) directions to all spatial directions (both \(k_{x}d/\pi\) and \(k_{y}d/\pi\) varying in the range \([-1,1]\)). Figure 7 shows isofrequency color-map plots for the mirror- and glide-symmetric configurations. For the sake of conciseness, only the first propagating mode is considered in the color maps. Additionally, white-dashed circumferences have been included to easily compare the isotropy condition in both structures. As the isofrequency plots illustrate, both mirror- and glide-symmetric structures are quite isotropic for frequencies below \(20\,\mathrm{GHz}\) (\(h_{t}/\lambda_{0}<1/5\)). Note that in the plots, radial color symmetry indicates perfect isotropy. However, as previously discussed in Section 3.1, the glide-symmetric configuration keeps the isotropy condition in a wider range of frequencies. The mirror-symmetric structure starts to show its anisotropic behavior near the stopband region (above \(25\,\mathrm{GHz}\)), where the frequency dispersion is more evident. While less pronounced, the anisotropic behavior of the structure also becomes apparent at these frequencies in the glide-symmetric configuration, specially for wave propagation along the \(45^{\circ}\) direction (\(k_{x}=k_{y}\)). Figure 6: Dispersion diagram of the (**a**) mirror-symmetric and (**b**) glide-symmetric unit cells when varying the hole depth. 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. The geometrical parameters of the unit cells are: \(g=0.5\,\mathrm{mm}\), \(h_{t}=3\,\mathrm{mm}\), and \(d_{x}=d_{y}=4.5\,\mathrm{mm}\). Figure 7: Isofrequency plot for the (**a**) mirror-symmetric and (**b**) glide-symmetric configurations. To compare the isotropy level, perfect circles are marked in white. The geometrical parameters of the unit cells are: \(g=0.2\,\mathrm{mm}\), \(h_{t}=3\,\mathrm{mm}\), \(h_{\mathrm{hole}}=1.5\,\mathrm{mm}\), and \(d_{x}=d_{y}=4.5\,\mathrm{mm}\). The isotropic behavior of the structures can be related to the electrical size of the equilateral triangular hole and the cutoff frequency of the equivalent triangular waveguide. If the modal functions in the equilateral triangular waveguide are sorted according to their cutoff frequencies, the cutoff frequency of the first mode (labeled as #1 in Figure 2) is much higher compared to other canonical waveguides, such as the square or circular waveguides of the same lateral size [26; 30]. This means that, in comparison with the same geometric size, the \"effective\" electrical size of the triangular hole is much smaller compared to other holes used in isotropic periodic holey structures. To illustrate this fact, let us consider an equilateral triangular waveguide whose side \(e\) is identical to the side of a square waveguide as well as to the diameter of a circular waveguide. The cutoff frequency of the first modal function in the triangular hole (\(m=1\), \(n=0\)) filled with air is \((2c/3e)/(c/2e)=1.333\) times higher than the cutoff of the fundamental mode of a square hole, and \((2c/3e)/(1.8412c/\pi e)=1.138\) times higher than the cutoff of the fundamental mode of a circular hole. To further test this hypothesis, we compare in Figure 8 the dispersion features of our triangular structure with a benchmark holey metasurface whose holes have a double-ridged shape. The geometry of the holes are chosen so that their fundamental modes have the same cutoff frequency (60 GHz). Under this assumption, the size of the triangular hole is found to be much larger than the double-ridged hole. The same will be true for square and circular geometries, being the triangular hole one of the canonical geometries with higher cutoff frequency.