# Analytically Regularized Evaluation of the Coupling of Planar Concentric Conducting Rings
## 2 Statement of the Problem  
Let us consider the geometry depicted in Figure 1: a set of \(Q\) coplanar concentric rings, with inner and outer radii \(r_{iq}\) and \(r_{qq}\), respectively, for \(q=1\ldots Q\). A plane wave impinges on the rings with an incidence angle \(\theta_{inc}\) with respect to \(z\) axis. Due to the symmetry the azimuthal angle of incidence is not essential and can be set to 0. Many different configurations are considered throughout the paper. All dimensions normalized to wavelength are summarized in Table 1. In particular, geometry #1 is composed of rings with a quite large width, very close to each other. In geometry #2, a regular lattice of four rings and slits is considered, whereas in geometry #3 the same rings as in geometry #2 are considered, but they are very close to each other. Finally, geometry #4 is again a regular lattice of rings and slits, with six narrow elements. Due to the revolution symmetry of the geometry, a cylindrical harmonics expansion can be adopted for currents and fields. Moreover, the radial and azimuthal components will be gathered in vector notation as  
\[\underline{E}^{(n)}(\rho,z)\equiv\underline{E}(\rho,z)= \begin{bmatrix}F_{r}( \rho,z)\\ -jF_{\varphi}(\rho,z)\end{bmatrix}  \tag{1}\]  
In (1) \(\underline{E}\) represents the \(n\)th harmonic of either the surface current density or the field. Note that since cylindrical harmonics are independent of each other, the analysis can be carried out harmonic by harmonic. Consequently the superscript \((n)\) is understood and will be omitted throughout the paper. The VHT of \(\underline{E}\) will also be introduced as  
\[\underline{E}(w,z)=\int_{0}^{\infty}\underline{H}_{n}(w\rho)\underline{E}( \rho,z)\rho d\rho \tag{2}\]  
where the kernel of VHT is defined as  
\[\underline{H}_{n}(w\rho)= \begin{bmatrix}J_{n}^{\prime}(w\rho)&\frac{n}{w\rho} J_{n}(w\rho)\\ \frac{n}{w\rho}J_{n}(w\rho)&J_{n}^{\prime}(w\rho)\end{bmatrix}  \tag{3}\]  
\(J_{n}(\cdot)\) being the Bessel function of the first kind and order \(n\) and the apex representing the derivative with respect to the argument. Some useful properties of the VHT can be found in [46] whereas its relationship with the Scalar Hankel Transform is quite evident and will be employed later on. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Label** & \(\mathbf{r_{i1}}\) & \(\mathbf{r_{o1}}\) & \(\mathbf{r_{i2}}\) & \(\mathbf{r_{o2}}\) & \(\mathbf{r_{i3}}\) & \(\mathbf{r_{o3}}\) & \(\mathbf{r_{i4}}\) & \(\mathbf{r_{o4}}\) & \(\mathbf{r_{i5}}\) & \(\mathbf{r_{i6}}\) & \(\mathbf{r_{o6}}\) \\ \hline Geom1 & 0.2 & 0.6 & 0.62 & 1.02 & 1.04 & 1.44 & - & - & - & - & - & - \\ Geom2 & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 & 1.6 & - & - & - & - \\ Geom3 & 0.2 & 0.4 & 0.42 & 0.62 & 0.64 & 0.84 & 0.86 & 1.06 & - & - & - & - \\ Geom4 & 0.2 & 0.25 & 0.3 & 0.35 & 0.4 & 0.45 & 0.5 & 0.55 & 0.6 & 0.65 & 0.7 & 0.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of configurations analyzed throughout the paper. All dimensions are normalized to the wavelength. Figure 1: Geometry of the problem: \(Q\) coplanar perfectly conducting concentric rings, with inner and outer radii \(r_{iq}\) and \(r_{oq}\), respectively, for \(q=1\ldots Q\). A plane wave impinges on the rings with an incidence angle \(\theta_{inc}\) with respect to \(z\) axis. Due to the symmetry the azimuthal angle of incidence is not essential. The incident electromagnetic field induces a surface current on the rings, which in turn generates a scattered field. Such scattered field can then be written in the spectral domain as in [47], the generalization to multiple rings being straightforward  
\[\underline{E}_{sc}(\rho,z)=j\frac{\zeta_{0}}{2}\sum_{q=1}^{Q}\int_{0}^{\infty} \underline{H}_{n}(w\rho)\underline{G}(w)\underline{I}_{q}(w)e^{-j|z|\sqrt{k_{0 }^{2}-w^{2}}}wdw \tag{4}\]  
where \(\zeta_{0}\) and \(k_{0}\) are the free space impedance and wavenumber, respectively.