## 3 Green's Function Formulation of the Problem  
We consider the two-dimensional electromagnetic scattering problem for a human arm model composed of concentric cylindrical layers of skin, fat, muscle, cortical bone and bone marrow. Figure 1 shows the cross-sectional reference geometry of the cylindrical structure, which is located in the transverse direction. The wavenumber of the background medium is \(k_{b}\). The stratified model is irradiated by a monochromatic electromagnetic wave having the frequency \(\omega\) excited by an infinitely long line source parallel to the structure. The Green's function \(G_{b}(x,y)\) of the structure satisfies the equation  
\[\
abla^{2}G_{b}(x,y)+k_{b}^{2}G_{b}(x,y)=-\delta(x-y) \tag{2}\]  
and has the solution  
\[G_{b}(x,y)=\frac{i}{4}\,H_{0}^{(1)}(k_{b}|x-y|), \tag{3}\]  
which is simply the field of the line source located at an arbitrary point \(y\in\mathbb{R}^{2}\) outside the structure. Here, \(H_{0}^{(1)}\) denotes the zeroth order Hankel function of the first kind and can be replaced by using the addition theorem for Hankel functions as follows [25]:  
\[H_{0}^{(1)}(k_{b}|x-y|)=\frac{i}{4} \begin{cases}\sum\limits_{n=- \infty}^{\infty}H_{n}^{(1)}(k_{b}\rho)\,J_{n}(k_{b}\rho^{\prime})\,e^{in(\phi -\phi^{\prime})},&\text{if }\rho>\rho^{\prime}\\ \sum\limits_{n=-\infty}^{\infty}H_{n}^{(1)}(k_{b}\rho^{\prime})\,J_{n}(k_{b} \rho)\,e^{in(\phi-\phi^{\prime})},&\text{if }0<\rho<\rho^{\prime}\end{cases}  \tag{4}\]  
where the vectors \(\vec{\rho}=(\rho,\phi)\) and \(\vec{\rho}_{s}=(\rho^{\prime},\phi^{\prime})\) denote the observation and the source points, respectively. The line source is assumed to be located outside the cylindrical structure. The \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Tissue** & \(\epsilon_{\infty}\) & \(\Delta\epsilon_{1}\) & \(\Delta\epsilon_{2}\) & \(\Delta\epsilon_{3}\) & \(\tau_{1}\times 10^{-9}\) & \(\tau_{2}\times 10^{-9}\) & \(\tau_{3}\times 10^{-9}\) \\ \hline
**Marrow** & 4.162 & 1.321 & 0.4162 & 26.08 & 0.02899 & 0.501 & 10.14 \\
**Bone** & 6.915 & 5.487 & 1.653 & 94.31 & 0.03378 & 0.4537 & 13.19 \\
**Muscle** & 31.45 & 23.27 & 6.746 & 3101 & 0.02212 & 0.5729 & 39.26 \\
**Fat** & 4.113 & 1.341 & 0.2853 & 119.3 & 0.02728 & 0.4887 & 29.28 \\
**Dry skin** & 27.44 & 12.73 & 17.1 & 417.9 & 0.0366 & 0.5684 & 7.581 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Debye parameters of the arm tissues for the frequency range from 100 MHz to 2 GHz [23; 24]. zeroth order Hankel function appearing in this expression can adequately be replaced by using the addition theorem for Hankel functions as follows:  
\[H_{0}^{(1)}(k_{1}|x-y|)=\sum_{n=-\infty}^{\infty}H_{n}^{(1)}(k_{1}\rho_{>})\,J_{n }(k_{1}\rho_{<})\,e^{in(\phi-\phi^{\prime})} \tag{5}\]  
where \(\rho_{>}=max(\rho,\,\rho^{{}^{\prime}})\) and \(\rho_{<}=min(\rho,\,\rho^{{}^{\prime}})\)[25]. The line source situated at \(\rho=\rho_{0},\,\phi=\phi_{0}\) has the current density  
\[J(\rho,\phi)=\frac{I_{0}}{\rho_{0}}\,\delta(\rho-\rho_{0})\,\delta(\phi-\phi_{ 0}), \tag{6}\]  
where \(I_{0}\) denotes the current strength. In the TM case, the incident electric field generated by the line source has only a \(x_{3}\)-component, given by  
\[\vec{E}^{i}(x)=u^{i}(x)\,\vec{\varepsilon}_{3},\quad\vec{x}=(x_{1},x_{2})\in \mathbb{R}^{2} \tag{7}\]  
with  
\[u^{i}(x)=-\frac{k_{b}\,I_{0}}{4\,\omega\,\epsilon_{0}}\,H_{0}^{(1)}(k_{b}| \vec{\rho}-\vec{\rho}_{s}|) \tag{8}\]  
where \(H_{0}^{(1)}\) denotes the Hankel function of the first kind with order zero and \(k_{b}\) is the wavenumber of the background medium, and the vectors \(\vec{\rho}=(\rho,\phi)\) and \(\vec{\rho}_{s}=(\rho_{0},\phi_{0})\) denote the observation and the source points, respectively. In what follows is the line source assumed to be located outside the cylindrical structure.