## 2 Problem Formulation
### Enforcement of Symmetry and Periodicity  
Due to the higher-symmetric in the structures, the generalized Floquet theorem [9] is used to analyze them. Therefore, only one sub-unit cell needs to be considered. The longitudinal and transversal cross-sectional views of the sub-unit cell are shown in Figure 3.  
It should be noted that, for the conventional periodic structure, the sub-unit cell is also the unit cell. The amplitudes of the coaxial line modes at Sides 1 and 2 of the sub-unit cell are referred to as \(a_{v}^{\pm}\) and \(b_{v}^{\pm}\), respectively. The subscript \(v\) is a combination of the angular and radial mode indices of a coaxial transmission line (\(m\) and \(n\) in the modal field expressions in Appendix A), and the superscript indicates the direction of propagation.  
Figure 3: Cross-sectional views of the sub-unit cell in the: (**a**) longitudinal plane and (**b**) transversal plane at the position of the screen.  
Figure 2: Analyzed periodically loaded coaxial transmission lines: (**a**) 2-, (**b**) 3-, and (**c**) 4-fold twist-symmetric structures and (**d**) conventional periodic structure.  
The Bloch modes supported by the structures are found through an eigenvalue problem that relates the coaxial line modes at the two sides of the sub-unit cell. The eigenvalue problem is constructed using the transfer matrix of the sub-unit cell, \(\underline{\mathbf{T}}\), which relates the modes as:  
\[\begin{bmatrix}\mathbf{b}^{+}\\ \mathbf{b}^{-}\end{bmatrix}=\underline{\mathbf{T}}\begin{bmatrix}\mathbf{a}^{+ }\\ \mathbf{a}^{-}\end{bmatrix}\;. \tag{2}\]  
The matrix \(\underline{\mathbf{T}}\) can be decomposed into a transfer matrix for the screen, \(\underline{\mathbf{T}}_{s}\), and two transfer matrices for the coaxial line between two screens, \(\underline{\mathbf{P}}_{s/2}\), as:  
\[\underline{\mathbf{T}}=\underline{\mathbf{P}}_{s/2}\,\underline{\mathbf{T}}_{ s}\,\underline{\mathbf{P}}_{s/2}=\begin{bmatrix}\underline{\mathbf{P}}_{s/2}^{+ }&\underline{\mathbf{0}}\\ \underline{\mathbf{0}}&\underline{\mathbf{P}}_{s/2}^{-}\end{bmatrix}\; \underline{\mathbf{T}}_{s}\begin{bmatrix}\underline{\mathbf{P}}_{s/2}^{+}& \underline{\mathbf{0}}\\ \underline{\mathbf{0}}&\underline{\mathbf{P}}_{s/2}^{-}\end{bmatrix} \tag{3}\]  
where \(\underline{\mathbf{P}}_{s/2}^{\pm}\) are diagonal matrices with the entries \(\mathrm{e}^{\mp\mathrm{i}k_{\mathrm{i}}\pm\delta/2}\). It is noted here and explicitly shown in Section 2.3 that \(\underline{\mathbf{T}}_{s}\) is a block partitioned matrix with two row partitions and two column partitions. By applying the generalized Floquet boundary conditions to (2), we get:  
\[\underline{\mathbf{P}}_{s/2}\,\underline{\mathbf{T}}_{s}\,\underline{\mathbf{ P}}_{s/2}\begin{bmatrix}\mathbf{a}^{+}\\ \mathbf{a}^{-}\end{bmatrix}=\mathrm{e}^{-\mathrm{i}\beta\xi}\underline{ \mathbf{Q}}\begin{bmatrix}\mathbf{a}^{+}\\ \mathbf{a}^{-}\end{bmatrix} \tag{4}\]  
where \(\beta\) is the sought phase constant and \(\underline{\mathbf{Q}}\) is a block diagonal matrix with entries \(\underline{\mathbf{q}}_{v}\) given in [46]:  
\[\underline{\mathbf{q}}_{v}=\begin{cases}1&\text{for TEMor TM/TE with $m=0$}\\ \begin{bmatrix}\cos(2\pi m/\xi)&\sin(2\pi m/\xi)\\ -\sin(2\pi m/\xi)&\cos(2\pi m/\xi)\end{bmatrix}&\text{for TM/TE with $m\
eq 0$}\;.\end{cases} \tag{5}\]  
It should be noted that it is matrix \(\underline{\mathbf{Q}}\) that introduces the twist symmetry in the formulation and that higher order modes with an angular variation are required to observe the effect of twist symmetry. Furthermore, for the TM and TE modes with \(m=0\), \(\underline{\mathbf{q}}_{v}\) is a scalar, and for \(\xi=1\), \(\underline{\mathbf{Q}}\) reduces to the identity matrix. The characteristic equation of the structure under study is found by solving the eigenvalue problem (4); namely, by searching for the values of \(\beta\) that satisfy:  
\[\det[\underline{\mathbf{P}}_{s/2}\,\underline{\mathbf{T}}_{s}\,\underline{ \mathbf{P}}_{s/2}-\mathrm{e}^{-\mathrm{i}\beta\xi}\underline{\mathbf{Q}}]=0\;. \tag{6}\]