## 2 Materials and Methods
\begin{bmatrix}\mathrm{E}_{0}\\ \mathrm{H}_{0}\end{bmatrix} \sum_{n=-\infty}^{\infty}\mathrm{J}_{n}(\mathrm{k}_{ \mathrm{c}0}\rho_{1})\;\mathrm{j}^{n}\;\mathrm{e}^{\mathrm{j}n}\;\mathrm{e}^{ \mathrm{j}n}(\varphi_{1}-\varphi^{\prime}). \tag{2}\]  
E\({}_{0}\) and H\({}_{0}\), in Equation (2), are the magnitudes of the incident electric and magnetic fields, whereas \(\rho_{1}\) and \(\varphi_{1}\) are the polar coordinates of \(\overline{\rho}\) in (O\({}_{1}\)), \(\mathrm{k}_{\mathrm{c}0}=\sqrt{\mathrm{k}_{0}^{2}-\beta^{2}}\) and \(\mathrm{J}_{n}(x)\) stands for the Bessel function of order \(n\) and argument \(x\)[51]. The \(\varphi\)-components of the incident field can be expressed as  
\[ \begin{bmatrix}\mathrm{E}_{\varphi}^{\mathrm{inc}}(\overline{\rho})\\ \mathrm{H}_{\varphi}^{\mathrm{inc}}(\overline{\rho})\end{bmatrix} = \begin{bmatrix} \mathrm{E}_{0}\\ \mathrm{H}_{0}\end{bmatrix} \sum_{n=-\infty}^{\infty}{}^{n}\overline{\overline{ \mathrm{G}}_{0}}(\rho_{1})\;\mathrm{j}^{n}\;\mathrm{e}^{\mathrm{j}n}(\varphi_ {1}-\varphi^{\prime}), \tag{3}\]  
where \({}^{n}\overline{\overline{\mathrm{G}}_{i}}(\rho)\) is the abbreviation for  
\[{}^{n}\overline{\overline{\mathrm{G}}_{i}}(\rho)= \begin{bmatrix}\frac{\beta n }{\mathrm{k}_{\mathrm{c}i}^{2}}\rho_{n}(\mathrm{k}_{\mathrm{c}i}\;\rho)&\frac {\mathrm{j}\omega\mu_{i}}{\mathrm{k}_{\mathrm{c}i}}\mathrm{j}_{n}^{\prime}( \mathrm{k}_{\mathrm{c}i}\;\rho)\\ -\frac{\mathrm{j}\omega\varepsilon_{i}+\sigma_{i}}{\mathrm{k}_{\mathrm{c}i}} \mathrm{j}_{n}^{\prime}(\mathrm{k}_{\mathrm{c}i}\;\rho)&\frac{\beta n}{\mathrm{ k}_{\mathrm{c}i}^{2}}\mathrm{J}_{n}(\mathrm{k}_{\mathrm{c}i}\;\rho)\end{bmatrix} , \tag{4}\]  
and \(\mathrm{j}_{n}^{\prime}(x)\) is the first derivative of the Bessel function of order n with respect to its argument, \(x\). The EM field \(\left[\overline{\mathrm{E}}^{\mathrm{sc}}(\overline{\rho}),\;\overline{ \mathrm{H}}^{\mathrm{sc}}(\overline{\rho})\right]\;\mathrm{e}^{-\mathrm{j} \beta z}=\;\left[\;\widetilde{\mathrm{E}}_{\mathrm{z}}^{\mathrm{sc}}(\overline {\rho})+\overline{\mathrm{E}}_{\mathrm{t}}^{\mathrm{sc}}(\overline{\rho})\;,\;\widetilde{\mathrm{z}}\;\mathrm{H}_{\mathrm{z}}^{\mathrm{sc}}(\overline{ \rho})+\overline{\mathrm{H}}_{\mathrm{t}}^{\mathrm{sc}}(\overline{\rho})\; \right]\;\mathrm{e}^{-\mathrm{j}\beta z}\) excited at any observation point \(\overline{\rho}\) of the cylindrical region (\(i\)), with \(i=0,\;1,\;2,\;3,\;\ldots,\;2L-1\), is referred to as the scattered field due to the existence of the \(L\) dielectric stratified cylinders of Figure 1. It should be noted that regions (\(i\)) with \(i=2,\;4,\;\ldots,\;2(s-1),\ldots,\;2(L-1)\) and \(i=3,\;5,\;\ldots,\;2(s-1)+1,\;\ldots,\;2(L-1)+1\) designate the cladding and the core regions for the \(s\)-th cylinder (\(s=2,\;3,\;\ldots,\;L\)), respectively. Regarding the local coordinate system of the \(q\)-th cylinder (O\({}_{q}\)), the longitudinal components \(\mathrm{E}_{z}^{\mathrm{sc}}\) and \(\mathrm{H}_{z}^{\mathrm{sc}}\) of the scattered field may be written in compact form as  
\[ \begin{bmatrix}\mathrm{E}_{z}^{\mathrm{sc}}(\overline{\rho})\\ \mathrm{H}_{z}^{\mathrm{sc}}(\overline{\rho})\end{bmatrix} =\sum\limits_{n=- \infty}^{\infty}\biggl{\{}(1-\delta_{i,0})\;\mathrm{J}_{n}(\mathrm{k  
cladding; this term is actually the first term in the sum of Equation (5) for \(i=1\). The EM field inside the core of each cylindrical inclusion is represented by the first term in the sum of Equation (5) for \(i=3,\ 5,\ \ldots,\ 2(L-1)+1\); the modal expansion contains only Bessel functions of the first kind since the EM field should be regular at the local origin \(\mathrm{O_{q}}\) (i.e., for \(\ \mathrm{\rho_{q}}\to 0\)). The expansion of the EM field inside the cylindrical layer of each inclusion should comprise both Bessel and Hankel functions since the (local) origin is not included in these regions.