## 2 Theoretical Background
### Waveguide with an Infinitesimally Thin Strip Placed Inside at Varying Locations  
In this section, we consider how the cut-off wavenumbers of the isolated waveguide are perturbed by the introduction of an infinitesimally thin PEC strip; the semi-width of the strip will be denoted \(w\), and its centre is the geometric centre of the rectangular waveguide, which is also taken as the origin of the Cartesian coordinates (\(x_{0}=y_{0}=0\)). The strip is parallel to the wider wall of the waveguide; its semi-width varies in the interval \(0.001\leq w\leq 0.25\). The evolution of the cut-off wavenumbers \(\gamma_{m,n}\), as the semi-width \(w\) of the strip increases, is shown in Figure 9.  
Figure 8: Rectangular waveguide with sides \(a\) and \(b,\ (b>a)\). 3 types of metallic insertions are considered: (**a**) an infinitely thin strip; (**b**) a thick strip with rounded corners; (**c**) an inner conductor of elliptic cross section. The location of the centre of symmetry of the inner conductor is denoted \((x_{0},y_{0})\), and the angle \(\alpha\) specifies its orientation. It can be seen that the cut-off wavenumbers \(\gamma_{m,n}\) of the \(TM_{2,n}(n=1,2,3)\) modes are nearly independent of the strip width, indeed nearly independent of the strip's presence or absence. These modes have an even number \(m=2\) of oscillations between the wider walls of the waveguide; the distribution of the modal electromagnetic field for modes does not cross the locations of the intensity maxima, and, hence, the strip does not sensibly perturb this distribution. On the other hand, the cut-off wavenumbers \(\gamma_{m,n}\) of the \(TM_{1,n}(n=1,\ 2,\ 3,\ 4)\) modes are noticeably affected, increasing in value as \(w\) increases. We now consider how the cut-off wavenumbers of the isolated waveguide are perturbed when the centre of the strip is held fixed at \((x_{0},y_{0})=(0,0)\), is rotated anticlockwise by an angle \(\alpha\) of \(30^{\circ}\) and \(90^{\circ}\). The results are displayed in Figure 10a,b, respectively. In contrast to the case \(\alpha=0^{\circ}\), the cut-off wavenumbers of _all_ modes are increasingly perturbed as the semi-width increases. Common to the cut-off wavenumbers computed at different orientation angles \(\alpha\) (\(0^{\circ}<\alpha<90^{\circ}\)) is the splitting of the \(TM_{2,2}\) and \(TM_{1,4}\) modes; in the empty waveguide, these modes are degenerate but as the semi-width of strip embedded in rectangular waveguide increases, the computed cut-off wavenumbers of these two modes diverge significantly. When the strip lies in the vertical position (\(\alpha=90^{\circ}\)) exactly, it is not obvious whether the \(TM_{2,2}\) and \(TM_{1,4}\) modes split or not. To the degree of resolution shown in Figure 10b), the plots of the cut-off wavenumbers \(\gamma_{2,2}(w)\) and \(\gamma_{1,A}(w)\), as functions of the semi-width \(w\), do not reveal any signs of mode splitting. However, the _MAR_ enables us to reveal the true situation by means of high-precision calculation of the spectral dependence \(\mathit{cond}\ A_{N}(ka)\) in the neighbourhood of the cut-off wavenumbers \(\gamma_{2,2}(0)=\gamma_{1,A}(0)=7.02481473\) of the \(TM_{2,2}\) and \(TM_{1,4}\) modes existing in the hollow rectangular waveguide. When the strip lies in the vertical position (\(\alpha=90^{\circ}\)), it is parallel to the smaller sides of the rectangle. In this orientation, the plots \(\mathit{cond}\ A_{N}(ka)\) presented in Figure 11 demonstrate the evolution of the cut-off wavenumbers \(\gamma_{2,2}(w)\) and \(\gamma_{1,A}(w)\) as the strip semi-width \(w\) increases. Our extensive calculations establish that modes remain degenerate up to a critical semi-width \(w_{0}\) lying between \(0.25\) and \(0.3\). When \(w=0.25\), Figure 11a shows that the resonance value of the function \(\mathit{cond}\ A_{256}(ka)\) occurs at a single point \(ka=7.02480\).