# Wave Propagation in Periodic Metallic Structures with Equilateral Triangular Holes
## 2 Formulation
### Mode Matching  
The modal expansion of the tangential fields in the surface of the holes in both configurations in Figure 1 can be expressed as [21]  
\[\mathbf{E}_{t}^{\text{WG}}(x,y,z=-g/2) =\sum_{i=1}^{N}r_{i}^{-}C_{i}\mathbf{\Phi_{i}}(x,y) \tag{1}\] \[\mathbf{H}_{t}^{\text{WG}}(x,y,z=-g/2) =\sum_{i=1}^{N}r_{i}^{+}Y_{i}C_{i}\left[\hat{\mathbf{z}}\times \mathbf{\Phi_{i}}(x,y)\right] \tag{2}\]  
where \(C_{i}\) is the coefficient of the \(i\)-th mode, \(\mathbf{\Phi_{i}}(x,y)=\phi_{x,i}\hat{\mathbf{x}}+\phi_{y,i}\hat{\mathbf{y}}\) is the modal vector function that represents the \(i\)-th mode inside the hole, and \(Y_{i}\) is the wave admittance of the \(i\)-th mode. The coefficients \(r_{i}^{\pm}=1\pm\mathrm{e}^{-2\mathrm{i}x\mathrm{i}h_{\text{hole}}}\) take into account the short circuit at the end of the hole, with \(k_{zi}=\sqrt{\varepsilon_{\text{hole}}k_{0}^{2}-k_{ti}^{2}}\) being the wavenumber in the \(z\)-direction of the triangular waveguide, \(h_{\text{hole}}\) the hole depth, and \(k_{ti}\) the transverse wavenumber, which here corresponds to the cutoff wavenumber (\(k_{c,mn}\)) of the triangular waveguide [26]. The tangential fields in the gap region can be described as the following Floquet-series expansion [23]:  
\[\mathbf{E}_{t}^{\text{gap}} =\frac{1}{d_{x}d_{y}}\sum_{p=-\infty}^{\infty}\sum_{s=-\infty}^{ \infty}\mathrm{e}^{-\mathrm{j}(k_{x,p}x+k_{y,y}y)}\,\mathbf{\tilde{e}}_{t,ps} ^{\text{gap}}(z) \tag{3}\] \[\mathbf{H}_{t}^{\text{gap}} =\frac{1}{d_{x}d_{y}}\sum_{p=-\infty}^{\infty}\sum_{s=-\infty}^{ \infty}\mathrm{e}^{-\mathrm{j}(k_{x,p}x+k_{y,y}y)}\,\mathbf{\tilde{h}}_{t,ps} ^{\text{gap}}(z) \tag{4}\]  
where \(k_{x,p}=k_{x,0}+2\pi p/d_{x}\), \(k_{y,s}=k_{y,0}+2\pi s/d_{y}\), \(p\) and \(s\) are the integer values specifying the order of the Floquet harmonics, \(d_{x}\) is the period in \(x\) direction, \(d_{y}\) is the period in \(y\) direction, and \(\mathbf{\tilde{e}}_{t,ps}^{\text{gap}}(z)\) and \(\mathbf{\tilde{h}}_{t,ps}^{\text{gap}}(z)\) are the field amplitudes of each transversal electric and magnetic Floquet harmonic, respectively. Both field-amplitude terms are expressed as a sum of sine and cosine functions [22,23]. After applying the corresponding boundary conditions (continuity of electric and magnetic tangential fields at \(z=-g/2\) and \(z=g/2\) for the glide case, and at \(z=-g/2\) and \(z=0\) for the non-glide case), the following linear system of equations is obtained [22,23]:  
\[\sum_{i=1}^{N}C_{i}\,\alpha_{ri}=0\qquad\qquad(r=1,\dots,N) \tag{5}\]  
where \(N\) is the number of modal functions considered for the triangular hole and  
\[\alpha_{ri}=\mathrm{j}k_{0}\eta_{0}d_{x}d_{y}Y_{i}I_{ri}+\frac{r_{i}^{-}}{r_{ i}^{+}}\sum_{p=-\infty}^{\infty}\sum_{s=-\infty}^{\infty}\widetilde{f}_{ps}(k_{x, ps})\beta_{ri}(k_{x,p},k_{y,s},k_{z,ps})\;. \tag{6}\]  
By setting the determinant of \(\alpha_{ri}\) to zero, the dispersion equation can be obtained. In Equation (6), \(I_{ri}\) is the inner product of the \(r\)-th and \(i\)-th modal functions. Under the assumption of perfect conductors, the modes are orthogonal to each other and, therefore, \(I_{ri}\) can be reduced to  
\[I_{ri}=\int_{S_{\text{hole}}}\mathbf{\Phi}_{r}(x,y)\cdot\mathbf{\Phi}_{i}(x,y) \,dS= \begin{cases}\int_{-\epsilon/2}^{\epsilon/2}\int_{0}^{ht}\mathbf{\Phi}_{ r}(x,y)\cdot\mathbf{\Phi}_{i}(x,y)\,\mathrm{d}x\mathrm{d}y\;,&r=i\\ 0\;,&r\
eq i\end{cases}  \tag{7}\]where the integration limits are defined by triangle side \(e\) and the triangle height \(h_{t}\). Both terms are related in an equilateral triangle according to \(h_{t}=e\sqrt{3}/2\).