# Wave Propagation in Periodic Metallic Structures with Equilateral Triangular Holes
## 1 Introduction  
Holey metasurfaces are obtained by etching small holes on a metallic plate in a periodic or locally periodic arrangement [1]. The simplicity of the fabrication process together with the advantageous wave-propagation properties obtained has boosted the interest in this class of structures. In fact, a single metallic sheet etched with holes can support plasmon-like waves (the so-called spoof-plasmons) even in the absence of dielectric or losses [2], in close similarity to the propagation in corrugated surfaces [3, 4]. This phenomenon can be explained by considering that the presence of the periodic texture of holes gives rise to an equivalent impedance of the surface that actually allows for the propagation of bounded waves [3, 4]. An alternative explanation in terms of equivalent plasma-like dielectric has also beenpresented [2]. The absence of material substrates in these fully metallic structures is quite beneficial because of the consequent reduction of losses. It leads to potential applications in microwave and photonics technology, not only for circuit devices [5] but also for antenna applications where the bounded waves are converted to leaky waves, responsible for highly directive radiation features [6]. In a first stage, only a single surface was considered in the analysis of holey metasurfaces, with no close scatterers interacting with that surface. However, in the last few years, a number of works have shown that a second metasurface, placed in the proximity of the first one and parallel to it, can drastically change the propagation properties of the complete structure. The wavenumber and the phase velocity of the resulting guided modes heavily depend on the mutual transverse shift between the metasurfaces. Interesting configurations are the mirror- and the glide-symmetric ones. In the former, the metasurfaces are simply mirrored one into the other one while, in the latter, they are off-shifted by half a period along the periodicity directions (see Figure 1). In this last configuration, modes propagating between the surfaces are low-dispersive over a large frequency band [7; 8]. Furthermore, their stopbands benefit of stronger field attenuation and of a larger bandwidth with respect to the mirror-symmetric configuration [9; 10]. Finally, glide-symmetric unit cells are able to produce higher levels of anisotropy [11], and higher magnetic responses [12]. In passband operation, the modal phase constant (and then the equivalent refractive index of the mode) can easily be tuned by varying the geometry of the periodic lattice, leading to the design of miniaturized structures [13; 14] and synthesis of low-loss ultra-wideband graded-index lenses. For example, Luneburg lenses have been reported to provide beam scanning in a wide angular region without pattern deterioration [7; 11; 15]. In stopband operation, these surfaces can efficiently isolate microwave circuits and enhance the performance of gap waveguides [16; 17; 18; 19; 20]. Apart from the previous control on the the behaviour of holey structures by acting on the geometric parameters of the holes and the lattice, it has also been shown that different shapes of the hole can render additional convenient features. For instance, square holes provide ultra-large non dispersive isotropic propagation while rectangular ones allow for a certain degree of anisotropy in the waveguide [21]. Circular and elliptical holes were also studied due to their simpler fabrication [22; 23; 24]. In the present work, we study metasurfaces made of holes with equilateral-triangle shape and show that, despite the triangle not being invariant under rotations of \(90^{\circ}\), an unexpected high degree of isotropy is achieved, especially along the two orthogonal periodicity directions. We compare the two extreme configurations of mirror- and glide-symmetric metasurfaces. On one side, we observe that the isotropy is conserved in both cases, which points out that this feature is more related to the hole shape than to the metasurface mutual arrangement.