## 2 Theoretical Background
### Waveguide with an Infinitesimally Thin Strip Placed Inside at Varying Locations
The first row of the table shows the cut-off wavenumbers \(\gamma_{m,n}^{(0)}\) for the infinitesimally thin strip; subsequent pair of rows show the perturbed cut-off wavenumbers \(\gamma_{m,n}=\gamma_{m,n}^{(0)}+\delta_{m,n}\) (upper row) and the relative perturbation \(\Delta_{m,n}=\delta_{m,n}/\gamma_{m,n}^{(0)}\) (lower row) when the thin strip is replaced by an elliptic strip of increasing thickness. Note that the perturbations \(\delta_{m,n}\) (and relative perturbations \(\Delta_{m,n}\)) are always positive. Table 9 presents a similar comparison when the elliptic strips are replaced by rounded rectangular strips of width \(2w\) and thickness \(2d\); the rounding radius is \(\varrho=0.25d\); the relative thickness \(d/w\) at values 0.01, 0.02 and 0.05 are examined. Again, note that the perturbations \(\delta_{m,n}\) (and relative perturbations \(\Delta_{m,n}\)) are always positive. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multicolumn{1}{c}{\(w=0.25\)} & \multicolumn{2}{c}{\(w=0.3\)} & \multicolumn{2}{c}{\(w=0.4\)} & \multicolumn{2}{c}{\(w=0.48\)} \\ \hline \(condA_{256}\) & \(\gamma_{m,n}\) & \(condA_{256}\) & \(\gamma_{m,n}\) & \(condA_{256}\) & \(\gamma_{m,n}\) & \(condA_{256}\) & \(\gamma_{m,n}\) \\ \hline
7.02480 & 7.024814 & 6.98625 & 6.986267 & 7.02256 & 7.022578 & 7.02480 & 7.024814 \\ \hline  & & 6.99220 & 6.992208 & 7.02266 & 7.022682 & & \\ \hline  & & 7.02480 & 7.024814 & 7.02480 & 7.024814 & & \\ \hline \end{tabular}
\end{table} 
Table 7: Fine effects of mode splitting. The calculated cut-off wavenumbers \(\gamma_{m,n}\) for the strips of varying width examined in Figure 11a–d. The columns labelled \(cond\ A_{256}\) record the values \(ka\) where the local maxima of \(cond\ A_{256}(ka)\) occur; the adjacent entries are the calculated cut-off wavenumbers \(\gamma_{m,n}\). Figure 12: Geometry of the strip-like inner conductors: on the left, an elliptic cylinder (\(a^{\prime}/b^{\prime}\ll 1\)); on the right, a thick strip with rounded ends (\(d/w\ll 1\)). The magnitude of the perturbation to the cut-off wavenumbers in a hollow waveguide by (one or more inner conductors basically depends on two factors: conductor location inside the waveguide and its size. It is convenient to measure the conductor size by the ratio \(\eta_{S}\) of the cross-sectional area \(S_{c}\) of the conductor and the cross-sectional area \(S_{w}\) of the hollow waveguide, i.e., \(\eta_{S}=S_{c}/S_{w}\). The shifts in the cut-off wavenumbers can be directly related to the values \(\eta_{S}\). The rectangular waveguide with rounded off corners has cross-sectional area \(S_{w}=ab-(4-\pi)\rho^{2}\), where \(\rho\) is the rounding off radius. Similarly, the cross-sectional area \(S_{tc}\) of the conductor of non-zero thickness \(d\) and semi-width \(w\) is \(S_{tc}=2 wd-(4-\pi)\rho^{2}\). The cross-sectional area \(S_{cs}\) of the elliptic strip is \(S_{cs}=\pi a^{\prime}b^{\prime}=\frac{1}{2}wd\). Even prior to calculation, it is crystal clear that the shifts \(\delta_{m,n}\) will be positive because insertion of an inner conductor into the waveguide, whether infinitesimally thin or of non-zero thickness, reduces the \"working\" cross-sectional area of the waveguide, thus leading to positive shifts in the spectrum. Table 8 shows that as the elliptic strip thickens, the increase in the shifts \(\Delta_{m,n}\) is proportional to the increase in cross-sectional area \(S_{es}\). In all cases presented in Table 8, the semi-width \(w\) of the elliptic strip is fixed (at 0.25), so the cross-sectional area \(S_{es}\) is proportional