## 2 Theoretical Background
### Hollow Waveguide of Arbitrary Cross Section  
Consider a uniform waveguide which has perfectly electrically conducting (PEC) walls, has an axis aligned with the \(z\)-axis, and has a cross-sectional contour \(L_{0}\) in the \(x\)-\(y\) plane which is smooth and of arbitrary shape. The cut-off wave numbers \(k_{c}\) of the modes are determined to be those wavenumbers \(k\) where non-trivial solutions of the homogeneous Dirichlet BVP for the Helmholtz equation  
\[\Delta U(q)+k^{2}U(q)=0\]  
occur in the interior domain D with boundary \(L_{0}\) satisfying the boundary condition  
\[U(q)\rfloor_{q\in L_{0}}=0.\]  
Here, \(U=E_{z}\) denotes the \(z\)-component of the electric field in the guide (see Figure 1) corresponding to the \"cut-off\" values of generally complex wavenumbers \(k\), which are eigenvalues of the formulated Dirichlet BVP. (Note that the spectrum of the Dirichlet boundary eigenvalue problem considered with respect to the spectral parameter \(k\) in a domain bounded by a closed piecewise smooth contour is a set of isolated positive eigenvalues accumulating at infinity. However, the development of efficient techniques for calculating the spectrum, which is actually an objective of the present study, remains an urgent task.) We seek the solution in the form of a single-layer potential  
\[U(q)=\int_{L_{0}}G_{2}(k|p_{0}-q|)Z_{0}(p_{0})dl_{p_{0}} \tag{1}\]  
where the unknown density function \(Z_{0}(p_{0})\) at each \(p_{0}\) on \(L_{0}\) is directly associated with the normal component of the electric field  
\[Z_{0}(p_{0})\,\sim\frac{\partial E_{z}}{\partial n_{p_{0}}}. \tag{2}\]  
Here, the free-space Green's function \(G_{2}\) is related to the Hankel function of the first kind and order \(0\),  
\[G_{2}(k|p_{0}-q|)=-\frac{i}{4}H_{0}^{(1)}(k|p_{0}-q|), \tag{3}\]  
depending on the distance \(|p_{0}-q|\) between observation point \(q\) and point \(p_{0}\) on the contour \(L_{0}\). The choice of a first-kind Hankel function is consistent with an assumed time-harmonic variation of the electromagnetic field of the form \(exp(-i\omega t)\) where \(\omega=kc,c\) denoting the speed of light. This time factor is suppressed throughout. Enforcement of the boundary condition on the contour \(L_{0}\),  
\[U(q)=0\,,q\in L_{0}, \tag{4}\]  
leads to the homogeneous surface integral equation  
\[\int_{L_{0}}G_{2}(k|p_{0}-q|)Z_{0}(p_{0})dl_{p_{0}}=0,\;q\in L_{0}. \tag{5}\]  
Now parameterize the points \(q(\theta_{0})=(x_{0}(\theta_{0}),\;y_{0}(\theta_{0}))\) on the contour \(L_{0}\) by smooth periodic functions \(x_{0},\;y_{0}\) so that \(x_{0}(-\pi)=x_{0}(\pi),\;y_{0}(-\pi)=y_{0}(\pi)\). With this parameterization, the differential of arc length is  
\[l_{0}(\tau_{0})=\sqrt{\left(x_{0}^{{}^{\prime}}(\tau_{0})\right)^{2}+\left(y_ {0}^{{}^{\prime}}(\tau_{0})\right)^{2}}, \tag{6}\]  
and we may reformulate the electric surface integral Equation (5) in the form  
\[G(k)z_{0}=0, \tag{7}\]  
where the integral operator-valued function (OVF) \(G(k)\) of the (in general, complex) spectral parameter \(k\) is defined by  
\[G(k)z_{0}(\tau_{0})\equiv-\frac{i}{4}\int_{-\pi}^{\pi}H_{0}^{(1)}(kR_{00}( \theta_{0},\tau_{0}))z_{0}(\tau_{0})d\tau_{0},\]  
with the kernel \(H_{0}^{(1)}(kR_{00}(\theta_{0},\tau_{0}))\), the function \(z_{0}(\tau_{0})=l_{0}(\tau_{0})Z_{0}(\tau_{0})\) and a distance function  
\[U(q)\rfloor_{q\in L_{0}}=0.\]0  
The homogeneous Equation (7) is the problem of the characteristic numbers (CNs) for the OVF \(G(k)\). Although the CNs for the _closed_ structures studied in this paper are real, the CNs for _open_ structures have a non-zero imaginary part. \[U(q)\rfloor_{q\in L_{0}}=0.\]1 Hollow waveguide of arbitrary cross section. Let us now briefly summarise the relevant results of [1] concerning the spectral theory of OVFs.