## 2 Formulation of the Problem and Proposed Solution  
### Formulation of the Problem  
Figure 1 shows the geometry of the problem: a zero-thickness PEC disk of radius \(a\) buried at depth \(d\) in a homogeneous and isotropic lossy half-space, called medium 2, surrounded by a half-space filled with air, i.e., medium 1. The symbols \(\epsilon_{i}\) and \(\mu_{i}\) denote, respectively, the dielectric permittivity and the magnetic permeability of the \(i\)-th medium. A Cartesian coordinate system is introduced such that the \(xy\) plane and the \(z\) axis coincide, respectively, with the flat interface and the disk axis. An arbitrarily oriented Hertzian dipole located in air, of current density \(\underline{l}^{\text{inc}}(\underline{r})=\underline{l}_{0}\delta(\underline {r}-\underline{r}_{0})\), where \(\underline{r}=(x,y,z)\) and \(\underline{r}_{0}=(x_{0},y_{0},z_{0})\) are the vector positions of the observation point and the dipole, respectively, and \(\delta(\cdot)\) is the Dirac delta function, radiates an electromagnetic field, \(\left(\underline{E}^{\text{inc}}(\underline{r}),\underline{H}^{\text{inc}}( \underline{r})\right)\), thus exciting a surface current density on the disk, \(\underline{l}(x,y)\), that in turn generates a scattered field, \(\left(\underline{E}^{\text{sc}}(\underline{r}),\underline{H}^{\text{sc}}( \underline{r})\right)\). The induced surface current density is related to the jump across the disk of the tangential component of the scattered magnetic field, whereas the incident and scattered electric fields are related to each other by the boundary conditions on the disk surface, i.e.,  
\[\underline{l}(x,y)=\underline{z}\times\left(\underline{H}^{\text{sc}}\big{(}x,y,-d^{+}\big{)}-\underline{H}^{\text{sc}}\big{(}x,y,-d^{-}\big{)}\right), \tag{1a}\] \[\underline{z}\times\left(\underline{E}^{\text{inc}}(x,y,-d)+\underline{E}^{ \text{sc}}(x,y,-d)\right)=0,\] (1b) for \[x^{2}+y^{2}\leq a^{2}.\]  
Figure 1: Geometry of the problem. Hence, a boundary value problem for the Maxwell equation is established. By further imposing the Silver-Muller radiation condition and the power-boundedness condition, the uniqueness of the solution, if it exists, is guaranteed. Since both the incident and scattered electric fields can be represented by the convolution integral involving the dyadic Green's function of the problem at hand and the corresponding current density, the Silver-Muller radiation condition is automatically satisfied by the proper selection of Green's function. Moreover, because Green's function of a planar layered medium can be expressed in closed form only in the spectral domain [27; 28], the incident and scattered electric fields can be conveniently retrieved by means of the inverse Fourier transform of the corresponding spectral domain counterparts. In general, the electric field generated in medium \(i=1,2\) by a current density of the kind \(\overline{\int}_{j}(\underline{t})=\underline{\int}_{j}(x,y)\delta(z- \overline{z}_{j})\) located in medium \(j=1,2\) can be written as the superposition of a primary or free-space contribution, arising only for \(i=j\), and a secondary or scattered contribution, related to the layering of the medium [29], i.e.,  
\[\underline{E}_{i,j}(\underline{t})=\underline{E}_{i,j}^{(\mathrm{P})}( \underline{t})+\underline{E}_{i,j}^{(\mathrm{S})}(\underline{t}), \tag{2a}\] \[\underline{E}_{i,j}^{(\mathrm{P})}(\underline{t})= \begin{cases}\iint\limits_{- \infty}^{+\infty}\underline{\widetilde{G}}_{i,j}^{(\mathrm{P})}(u,v)\widetilde {\underline{\int}}_{j}(u,v)e^{-\mathrm{i}(ux+vy+R_{i}(u,v)|z-z_{j}|)}dudv&i=j \\ 0&i\
eq j\end{cases}