## 3 Results  
The mode-matching formulation was implemented in MATLAB [49] and used to study the dispersion diagrams of the structures in Figure 2. The geometrical dimensions of the structure are \(r_{i}=2\,\mathrm{mm}\), \(r_{o}=4.7\,\mathrm{mm}\), \(r_{s}=4.7\,\mathrm{mm}\), \(\alpha=180^{\circ}\), and \(\delta=6\,\mathrm{mm}\). With these dimensions, the screen connects the inner and outer conductors of the coaxial lines, and as a result, the first mode has a cut-off frequency. This configuration is chosen so that the mode-matching formulation converges quickly: good convergence is achieved by only including higher order modes with \(n=1\). The convergence of the numerical result is evaluated by comparing the root-mean-squared deviation of the obtained samples of \(\beta\) for different numbers of coaxial line modes included in the computation. The convergence is deemed sufficient if the result deviates less than 1% when including an additional coaxial line mode. In the studied structures, 1 TEM, 17 TM, and 7 TE modes were found to ensure convergence. The dispersion diagrams of the analyzed twist-symmetric structures are plotted in Figure 4. The results obtained using the mode-matching formulation agree well with the ones computed using the Eigenmode Solver of CST. The curves in Figure 4 show that, with the definition of the twist operator in (1), the dispersion diagrams converge to that of the conventional periodic structure as the order of the symmetry increases. This observation is supported by the fact that the conventional periodic structure is equivalent to an \(\infty\)-fold twist-symmetric structure with the present definition of the twist operator. For alternative definitions of the twist operator (as the one used in [39]), this is not necessarily the case. A valuable insight that the mode-matching analysis directly provides comes from the analysis of the magnitudes of the different coaxial cable modes. This analysis gives us useful information about the field characteristics, as well as its most convenient way of being excited. Thus, the magnitudes of the coaxial cable mode coefficients for three values of the phase constant (\(\beta\)) are presented in Figure 5. The mode coefficients are normalized so that \(|\mathbf{a}|=1\). A first interesting observation in Figure 5 is that the magnitude of the TEM mode does not vary too much in the three cases shown in the figure. It can also be observed that the degrees of excitation of the TEM and the TE\({}_{11}\) modes are similar and greater than the one of the TM modes. However, the data in Figure 5a show that there is a significant variation of the magnitudes of the TM modes as the symmetry order (\(\xi\)) of the analyzed structures changes. This latter fact can be explained taking into account that the neighboring screens form lengthwise capacitors for the \(z\)-directed electric fields of these TM modes. The magnitudes of the TM modes are then expected to depend on the degree of overlap between consecutive metallic plates; namely, with the value of \(\xi\). A similar trend to that in Figure 4 is observed here, as the mode magnitudes in the twist-symmetric structures converge to those of the conventional periodic structure as \(\xi\) increases. Figure 5a also shows that the magnitude of the TE\({}_{11}\) increases as the symmetry order decreases. This is consistent with the fact that the cutoff frequency of the TE\({}_{11}\) mode is \(\sim\)15 GHz, and the dispersion curves in Figure 4 show that, for \(\beta=6\,^{\circ}/\)mm, the smaller \(\xi\) is, the closer the corresponding frequency appears to that cutoff frequency. In the cases shown in Figure 5b,c, since the corresponding modal solutions appear at frequencies above the TE\({}_{11}\) cutoff frequency, a significant decrease in the relative importance of the TM modes with respect to the TE\({}_{11}\) mode can be observed.