## 2 Formulation of the Problem  
Figure 1 shows an infinite zero-thickness resistive plane of resistivity \(R_{e}\) with a circular hole of radius \(a\) filled by a zero-thickness resistive disk of resistivity \(R_{i}\
eq R_{e}\). A cylindrical coordinate system \((\rho,\phi,z)\), with the \(z\)-axis orthogonal to the structure and the origin located at the center of the disk, and a spherical coordinate system \((r,\theta,\phi)\), such that \(\rho=rs_{\theta}\) and \(z=rc_{\theta}\), where \(s_{t}=\sin t\) and \(c_{t}=\cos t\), are sketched. A plane wave, propagating in the half-space \(z>0\) of electric field \(\underline{E}^{\text{inc}}(\underline{r})=\underline{E}_{0}e^{-\underline{i} \cdot\underline{r}}\) and magnetic field \(\underline{H}^{\text{inc}}(\underline{r})=\underline{H}_{0}e^{-\underline{i} \cdot\underline{r}}\), where \(\underline{r}\) identifies the observation point and \(\underline{E}_{0}\) and \(\underline{H}_{0}\) are constant  
vectors, such that \(\underline{H}_{0}=\frac{1}{\omega\mu_{0}}\underline{k}\times\underline{E}_{0}\); \(\underline{k}=-k_{0}\big{(}s_{0}c_{\phi_{0}-\phi}\hat{\rho}+s_{\phi_{0}}s_{\phi_ {0}-\phi}\hat{\phi}+c_{\phi_{0}}\hat{\rho}\big{)}\) is the wave vector; \(k_{0}=\omega\sqrt{\varepsilon_{0}\mu_{0}}=2\pi/\lambda\) is the free space wavenumber; \(\omega\) is the angular frequency; \(\varepsilon_{0}\) and \(\mu_{0}\) are the dielectric permittivity and the magnetic permeability of the free space; \(\lambda\) is the free space wavelength; and the angles \(\theta=\theta_{0}\) and \(\phi=\theta_{0}\) identify the incidence direction, impinges onto the structure generating a scattered field, i.e., \(\big{(}\underline{E}^{\text{sc}}(\underline{r}),\underline{H}^{\text{sc}}( \underline{r})\big{)}\). Such a field in the upper/lower half-space can be interpreted as the superposition of the field reflected/transmitted by the homogeneous resistive plane, \(\Big{(}\underline{E}^{\text{ref/tr}}(\underline{r}),\underline{H}^{\text{ ref/tr}}(\underline{r})\Big{)}\), and the field diffracted due to the discontinuity in the material, \(\Big{(}\underline{E}^{\text{difftr}}(\underline{r}),\underline{H}^{\text{difftr }}(\underline{r})\Big{)}\). Of course, the total field, \((\underline{E}(\underline{r}),\underline{H}(\underline{r}))\), is given by the superposition of the incidence field and the scattered field. The following two-sided generalized boundary conditions have to be verified by the fields [47]:  
\[\hat{z}\times\big{(}\underline{E}\big{(}\rho,\phi,0^{+}\big{)}-\underline{E} \big{(}\rho,\phi,0^{-}\big{)}\big{)}=\underline{0}, \tag{1a}\] \[\frac{1}{2}\hat{z}\times\big{(}\underline{E}\big{(}\rho,\phi,0^{+}\big{)}+ \underline{E}\big{(}\rho,\phi,0^{-}\big{)}\big{)}\times\hat{z}=R_{i}\underline{ I}_{i}(\rho,\phi)+R_{e}\underline{I}_{e}(\rho,\phi),\] (1b) for \[\rho\geq 0\] and \[0\leq\phi<2\pi\], where \[\underline{I}_{i}(\rho,\phi)= \begin{cases}\hat{z}\times(\underline{H}(\rho, \phi,0^{+})-\underline{H}(\rho,\phi,0^{-}))&\rho<a\\ \underline{0}&\rho>a\end{cases} ,\] (2a) \[\underline{I}_{e}(\rho,\phi)= \begin{cases}\underline{0}&\rho<a\\ \hat{z}\times(\underline{H}(\rho,\phi,0^{+})-\underline{H}(\rho,\phi,0^{-}))& \rho>a\end{cases}  \tag{2b}\]  
define the effective surface current densities on the disk and the holed plane, respectively. Supposing \(\mathfrak{R}\{R_{e}\}>0\), conditions (1), together with the power boundedness condition and the Silver-Muller radiation condition on the diffracted field, guarantee the unique solvability of the boundary value problem for the Maxwell equations at hand [53]. Such a problem can be equivalently formulated in terms of surface integral equations for the effective surface current densities whose kernels involve the free-space dyadic Green's function [54].