## 2 Background: Formulation of the Problem and Matrix Equation  
Figure 1 shows the geometry of the problem in hand: a perfectly electrically conducting (PEC) polygonal cross-section strip in a homogeneous and isotropic half-space (medium 2) of relative dielectric permittivity \(\varepsilon_{\tau 2}\) placed on a grounded planar layer (medium 1) of relative dielectric permittivity \(\varepsilon_{\tau 1}>\varepsilon_{\tau 2}\) and thickness \(d\). Thus, \(k_{l}=k_{0}\sqrt{\varepsilon_{\tau l}}\) with \(l\in\{\,1,2\}\) denotes the wavenumber in medium \(l\), where \(k_{0}=\omega\sqrt{\varepsilon_{0}l_{0}}=2\pi/\lambda_{0}\) is the free-space wavenumber in vacuum, \(\omega\) is the angular frequency, \(\varepsilon_{0}\) and \(\mu_{0}\) are the dielectric permittivity and the magnetic permeability of vacuum, respectively, and \(\lambda_{0}\) is the wavelength in vacuum.  
Let us introduce a Cartesian coordinate system \((x,y,z)\) with the origin on the ground plane, the \(y\) axis parallel to the strip axis and the \(z\) axis orthogonal to the interface between the involved media. Moreover, let us number the \(L\) sides of the strip clockwise and introduce a local Cartesian coordinate system \((x_{i},y,z_{i})\) on the \(i\)-th side, of length \(2a_{i}\), with the origin at the center of the side itself in the position \((\overline{x}_{i},\overline{z}_{i})\) with respect to the main coordinate system, such that the \(z_{i}\) axis is orthogonal to the \(i\)-th side and oriented in the outward direction, and the angle of the \(x_{i}\) axis with respect to the \(x\) axis is denoted by \(\varphi_{i}\).  
The analysis of the propagation needs to assume a behavior of the fields along the \(y\) axis of the kind \(e^{-j\
u y}\), where \(v=v_{R}-jv_{I}\) is the mode propagation constant. According to the procedure detailed in Ref. [16], the field can be written as the superposition of the fields generated by the surface current densities on each side of the considered strip. Moreover, remembering that the edge behavior of the components of the surface current density on the \(i\)-th side is [44]  
\[J_{iy}(x_{i}),\frac{\partial J_{ix_{i}}(x_{i})}{\partial x_{i}}\overset{x_{i \rightarrow\pm a_{i}}}{\sim}\left(1\mp\frac{x_{i}}{a_{i}}\right)^{t_{i}^{\pm} -1} \tag{1}\]  
with \(t_{i}^{\pm}\geq 1/2\), and that the transverse component of the surface current density is continuous even on the wedges, i.e., \(C_{i}=J_{qx_{i}}\big{(}a_{q}\big{)}=J_{ix_{i}}(-a_{i})\), where \(q\) and \(i\) denote two consecutive sides, the following spectral domain integral representation for the longitudinal and the transverse components of the electric field in the upper half-space can be established [16]:  
\[E_{s}(x,y,z)=E_{s}^{t}(x,y,z)+E_{s}^{y}(x,y,z) \tag{2}\]  
with \(s\in\{x_{j},y\}\), and the superscripts \(t\) and \(y\) denote the functional dependence on the transverse current and the longitudinal current, respectively, where  
\[E_{s}^{r}(x,y,z)=j\frac{e^{-j\
u y}}{2\omega\varepsilon_{0}\varepsilon_{r2}} \Big{[}\overline{E}_{s,P}^{r}(x,y,z)+\overline{E}_{s,S}^{r}(x,y,z)\Big{]} \tag{3}\]  
with \(r\in\{t,y\}\), and \(P\) and \(S\) denote the primary (free-space) contribution and the scattered contribution (due to the dielectric slab and the ground plane), respectively,  
\[\begin{split}\overline{E}_{x_{j},P}^{t}(x,y,z)=\overset{L}{i=1} \Bigg{\{}\frac{C_{i}}{2\pi i}\underset{C}{\int}\Big{\{}\frac{F_{ij}(u,x,z)}{e^ {-jnu_{q}}}-\frac{F_{ij}(u,x,z)}{e^{jnu_{i}}}\Big{\}}du+\\ +\underset{C}{\int}\;U_{i,j}(u,x,z)\overline{J}_{ix_{i}}(u)V_{i}( u,x,z)du\Bigg{\}}\end{split} \tag{4a}\] \[\begin{split}\overline{E}_{x_{j},P}^{y}(x,y,z)=\overset{L}{i=1} \underset{C}{\int}\;vT_{i,j}(u,x,z)\overline{J}_{iy}(u)V_{i}(u,x,z)du,\end{split} \tag{4b}\]  
Figure 1: Geometry of the problem.