## 2 SIEs for Multilayered Dissipative Medium Structures  
A non-planar \(K\)-layered dissipative medium structure is illustrated in Figure 1. Let the frequency of excitation sources in homogeneous medium be \(\omega\). The electromagnetic constants at the \(i\)th-layer are denoted by conductivity \(\sigma_{i}\), permittivity \(\varepsilon_{i}\), and permeability \(\mu_{i}\), with complex wave number \(k_{i}\) and wave resistance \(\eta_{i}\). Let \(S_{i-1}\) and \(S_{i}\) denote the top and bottom surfaces of \(i\)th-layer medium of the region \(\mathrm{D}_{i}\), respectively, where \(i=0,1,2,\ldots,K\). In addition, there is no top surface on the half-space region \(\mathrm{D}_{0}\) and no bottom surface beneath the half-space region \(\mathrm{D}_{K}\). Figure 1: \(K\)-layered dissipative medium structure. Let \(\left(\mathbf{E}_{i}^{\text{source}},\mathbf{H}_{i}^{\text{source}}\right)\) denote the incident fields in region \(\text{D}_{i}\), which can be excited by electric dipole antennas or magnetic dipole antennas. According to the equivalent principle, the total fields \(\left(\mathbf{E}_{i},\mathbf{H}_{i}\right)\) in region \(\text{D}_{i}\) can be seen as a synergistic result of the incident fields \(\left(\mathbf{E}_{i}^{\text{source}},\mathbf{H}_{i}^{\text{source}}\right)\), fields \(\left(\mathbf{E}_{i,i-1}^{\text{IM}},\mathbf{H}_{i,i-1}^{\text{IM}}\right)\) excited by the surface currents on \(S_{i-1}\), and fields \(\left(\mathbf{E}_{i,i}^{\text{IM}},\mathbf{H}_{i,i}^{\text{IM}}\right)\) excited by the surface currents on \(S_{i}\). Hence, the total electric and magnetic fields at arbitrary point on surface \(S_{i-1}\) or \(S_{i}\) can be expressed as  
\[\mathbf{E}_{i}=\mathbf{E}_{i}^{\text{source}}+\mathbf{E}_{i,i-1}^{\text{IM}}+ \mathbf{E}_{i,i}^{\text{IM}}\;, \tag{1}\]  
\[\mathbf{H}_{i}=\mathbf{H}_{i}^{\text{source}}+\mathbf{H}_{i,i-1}^{\text{IM}}+ \mathbf{H}_{i,i}^{\text{IM}}\;. \tag{2}\]  
The electric and magnetic surface currents on surface \(S_{i}\) in region \(\text{D}_{i}\) are defined as \(\mathbf{J}_{i}=\mathbf{n}_{i}\times\mathbf{H}_{i}\) and \(\mathbf{M}_{i}=-\mathbf{n}_{i}\times\mathbf{E}_{i}\), where \(\mathbf{n}_{i}\) is the normal vector of \(S_{i}\) pointing to the inside of the region \(\text{D}_{i}\). Because electric and magnetic currents on each interface shall be continuous across the boundary, the electric and magnetic surface currents on \(S_{i-1}\) are \(\left(\mathbf{J}_{i-1}\;,\mathbf{M}_{i-1}\right)\) in region \(\text{D}_{i-1}\) and \(\left(-\mathbf{J}_{i-1}\;,\mathbf{-M}_{i-1}\right)\) in region \(\text{D}_{i}\). Based on the electric and magnetic field integral equations [17; 21; 22], the induced electric and magnetic fields \(\left(\mathbf{E}_{i,i-1}^{\text{IM}},\mathbf{H}_{i,i-1}^{\text{IM}}\right)\) and \(\left(\mathbf{E}_{i,i}^{\text{IM}},\mathbf{H}_{i,i}^{\text{IM}}\right)\) in region \(\text{D}_{i}\) excited by surface current \(\left(-\mathbf{J}_{i-1}\;,-\mathbf{M}_{i-1}\right)\) and \(\left(\mathbf{J}_{i}\;,\mathbf{M}_{i}\right)\) can be written as  
\[ \begin{array}{rcl}\mathbf{E}_{i,i-1}^{\text{IM}}&=&\mathbf{E}_{i}^{-I_{i-1} }+\mathbf{E}_{i}^{-M_{i-1}}=\eta_{i}L_{i}(-\mathbf{J}_{i-1})-K_{i}(-\mathbf{ M}_{i-1}),\\ \mathbf{E}_{i,i}^{\text{IM}}&=&\mathbf{E}_{i}^{I_{i}}+\mathbf{E}_{i}^{M_{i}}= \eta_{i}L_{i}(\mathbf{J}_{i})-K_{i}(\mathbf{M}_{i}),\\ \mathbf{H}_{i,i-1}^{\text{IM}}&=&\mathbf{H}_{i}^{-I_{i-1}}+\mathbf{H}_{i}^{-M_{ i-1}}=K_{i}(-\mathbf{J}_{i-1})+\frac{1}{\eta_{i}}L_{i}(-\mathbf{M}_{i-1}),\\ \mathbf{H}_{i,i}^{\text{IM}}&=&\mathbf{H}_{i}^{I_{i}}+\mathbf{H}_{i}^{M_{i}}=K _{i}(\mathbf{J}_{i})+\frac{1}{\eta_{i}}L_{i}(\mathbf{M}_{i}).\end{array}  \tag{3}\]  
The linear operator \(L_{i}\) and \(K_{i}\) can be defined as  
\[L_{i}(\mathbf{F})\;=\;-jk_{i}\int\limits_{S_{i}}\left[\mathbf{F}+\frac{1}{k_{ i}^{2}}\
abla(\
abla^{\prime}_{s}\cdot\mathbf{F})\right]G_{i}ds^{\prime}, \tag{4}\]  
\[K_{i}(\mathbf{F})\;=\;\int\limits_{S_{i}}\
abla G_{i}\times\mathbf{F}ds^{\prime}, \tag{5}\]  
where \(\
abla^{\prime}_{s}\) is the surface divergence of a vector field.