## 2 Materials and Methods  
### Theory and Design Strategy  
Figure 1 demonstrates single and glide-symmetric double CPPW structures. As mentioned in [36], if \(h_{2}=h_{3}\) and \(m=p/2\), the unit cell has glide symmetry, as demonstrated in Figure 1. Full-wave electromagnetic simulators are capable of modeling these structures to derive their dispersion characteristics. On the other hand, the longitudinal transmission line model for these geometries is presented using the circuit model described in [36], as shown in Figure 2. The circuit approach developed by [PERSON], with detailed information provided in Appendix A and in [49], was utilized for the circuit structure pertinent to this problem. This methodology combines the circuit model from Figure 2 with the necessary parameters from Appendix A, resulting in terminals for single and double corrugations  
with a short circuit configuration. The phase constant of single Floquet mode supported by the unit cell of the periodic structure under consideration can be determined using the following equation [36]:  
\[\cos\left(\beta_{x}p\right)=\frac{A(f)+D(f)}{2}=A(f). \tag{1}\]  
The parameters \(\beta_{x}\), \(p\), \(A(f)\), and \(D(f)\) in Equation (1) represent the Floquet phase constant associated with the periodic structure with respect to the \(x\) direction, the unit cell's period, and the matrix elements \(A\) and \(D\), which is a function of frequency and of the \(ABCD\) matrix referring to the unit cell, respectively. This study will investigate the scattering matrix representation of the unit cell in both the input and output reference planes. We will determine the scattering matrix for each structure in the circuit model inside the unit cell, as illustrated in Figure 2. As shown in Figure 1, the next step is to stack the scattering matrices of these structures on top of each other to get the full set of S-parameters for the corrugated waveguides' periodic structure. Figure 2 demonstrates the representations of subcircuits. The scattering matrices of the block structures can be seen to cascade from the first block (S1) to the last block (S5), ending with the scattering matrix of the whole structure. S matrices of all models given in Figure 2a,b are explained in the Appendix A. The phase constant for the structure's dominant mode can be found using the eigenvalue equation given below [5] after obtaining the unit cell's S-parameters:  
\[\cos\left(\beta_{x}p\right)=\frac{1+S_{11}^{2}+S_{21}^{2}}{2S_{21}}. \tag{2}\]  
The two parameters \(S_{11}\) and \(S_{21}\) in Equation (2) indicate the scattering parameter components of the unit cell. Full-wave simulators can accomplish multimodal excitation for the unit cell. We can utilize the generalized scattering matrix elements from full-wave simulators to determine the passband and stopband regions of the periodic structure within the following equation [5]:  
\[ \begin{bmatrix}\mathbf{I}&-\mathbf{S_{11}}\\ \mathbf{0}&-\mathbf{S_{12}}\end{bmatrix}\begin{bmatrix}\mathbf{b_{1}}\\ \mathbf{a_{1}}\end{bmatrix} +\lambda \begin{bmatrix}-\mathbf{S_{12}}&\mathbf{0} \\ -\mathbf{S_{22}}&\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{b_{1}}\\ \mathbf{a_{1}}\end{bmatrix} =0. \tag{3}\]  
The AFGSM method can be used for finding band edge frequencies in a symmetric unit cell structure in the case of multimode excitation, and it is described in the literature with the equation below [5]:  
\[X_{\pm}^{full}=2\,\text{Im}(S_{n,n}\pm S_{n,n+P})-\sum_{k=M+1}^{P}|S_{n,k}\pm S _{n,k+P}|^{2}=0. \tag{4}\]  
There are a total of \(P\) modes, with \(M\) and \(P-M\) being the number of propagating and non-propagating modes in the waveguide used in Equation (4). \(S_{i,j}\) is the generalized scattering matrix element, and \(n\) is the input port that corresponds to the dominant propagating waveguide mode.