# Wave Propagation in Periodic Metallic Structures with Equilateral Triangular Holes
## 3 Results
### Isotropic Behavior
Certainly, the propagation paths in \(x\) and \(y\) directions of the structure with double-ridged holes are different. Thus, this structure is expected to present a clear anisotropic behavior. However, at the low frequencies (below the hole cutoff at 60 GHz in the green region), both structures are quite isotropic. Only at frequencies above the hole cutoff frequency (red region), the shape of the hole has an impact on the anisotropy of the metasurface. An additional physical mechanism can also contribute to enhance the isotropy in the low-frequency region of the dispersion diagram. In Figures 9 and 10, we plot the magnitude of the \(C_{i}\) \"excitation\" coefficient in (1) for the first 14 modes of the triangular hole. The analysis is performed at different frequencies: 10, 20 and 40 GHz for the mirror-symmetric configuration, and 10, 20, 30 and 40 GHz for the glide-symmetric configuration (the mirror-symmetric configuration being in stopband at 30 GHz). Therefore, these figures decompose the total field at the hole surface into the modal basis discussed in Subsection 2.2 and Figure 2. In the insets, we also plot the total transverse electric field when propagation occurs along the \(x\) and \(y\) directions. We observe that, at all the  
Figure 8: Dispersion diagram of two mirror-symmetric structures with triangular (black lines) and double-ridged (red lines) holes. The right panel shows a detail of the double-ridged hole. The light line is displayed in gray. The cutoff frequency of the triangular and double-ridged waveguides is chosen to be the same, \(f_{c}=60\) GHz. Independently of the shape of the holes, the anisotropy of the structures is hardly noticeable below the cutoff frequency (green region). The geometrical parameters of the unit cells are: \(h_{\text{hole}}=1.5\) mm, \(g=0.2\) mm, \(d_{x}=d_{y}=4.5\) mm, \(h_{t}=2.88\) mm, \(a_{1}=b_{2}=1.46\) mm, and \(a_{2}=b_{1}=a_{1}/3\). considered frequencies, the total field on the hole has the same distribution independently on the propagation direction. At low frequencies, the most excited mode is #4, while at higher frequencies modes #1 (when propagation is along \(x\)) and #2 (when propagation is along \(y\)) become gradually more important. This trend happens for both symmetric configurations. We should note that mode #4 is the fundamental TM mode and modes #1 and #2 are the first TE modes (this is clear in Figure 2). This fact agrees well with the findings reported in [25], where it was discussed that the presence of a metallic surface close to a holey metasurface requires to consider both the fundamental TE and TM modes when computing the field on a hole. In the present work, the most relevant modes have a rather symmetric behavior with respect to the center of the triangle, which gives rise to the excitation of basically the same total field independently of the propagation direction. Figure 9: Normalized magnitude of the coefficients \(C_{i}\) in the mirror-symmetric configuration for wave propagation in \(x\) and \(y\) directions at different frequencies: (**a**) \(10\,\mathrm{GHz}\), (**b**) \(20\,\mathrm{GHz}\), (**c**) \(40\,\mathrm{GHz}\). The normalized magnitude of the tangential electric field, calculated according to Equation (1), is also displayed for both propagating directions. The geometrical parameters of the considered unit cells are: \(h_{hole}=$1.5\,\mathrm{mm}$\), \(g=$0.2\,\mathrm{mm}$\), \(h_{t}=$3\,\mathrm{mm}$\), and \(d_{x}=d_{y}=$4.5\,\mathrm{mm}$\). The dispersion diagram of the structure is displayed in the green curve of Figure 5a.