## 2 Materials and Methods
\begin{bmatrix}\int_{M}(\mathrm{k}_{\text{c}\text{l}}\alpha_{q}) \,\overline{\mathbb{I}}\\ \overline{\mathrm{M}}\overline{\overline{\mathrm{G}}}_{1}^{\text{H}}(\alpha_{ q})\end{bmatrix} \, \begin{bmatrix}\left(\begin{matrix}\dot{a}_{M}^{q}\\ \dot{b}_{M}^{q}\end{matrix}\right)\end{bmatrix} \, \begin{bmatrix}\left(\begin{matrix} \dot{a}_{M}^{q}\\ \dot{b}_{M}^{q}\end{matrix}\right)\end{bmatrix} \, \begin{bmatrix}\left(\begin{matrix} \dot{a}_{M}^{q}\\ \dot{b}_{M}^{q}\end{matrix}\right)\end{bmatrix} \] (11) \[+\,\sum\limits_{ \begin{subarray}{c}s=2\\ s\
eq p\end{subarray} }^{L}\mathrm{H}_{n-M}^{(2)}(\mathrm{k}_{\text{c}\text{l}} \mathrm{D}_{sp})\,\,\mathrm{e}^{\mathrm{i}(n-M)\varphi_{sp}} \begin{bmatrix} \int_{M}(\mathrm{k}_{\text{c}\text{l}}\alpha_{q})\,\overline{\mathbb{I}}\\ \overline{\mathrm{M}}\overline{\overline{\mathrm{G}}}_{1}^{\text{J}}(\alpha_{ q})\end{bmatrix} \, \begin{bmatrix}\left(\begin{matrix}\dot{a}_{M}^{2(s-1)}\\ \dot{a}_{n}^{2(s-1)}\end{matrix}\right)\\ \end{bmatrix} \, \begin{bmatrix}\left(\begin{matrix}\dot{a}_{M}^{2(s-1)}\\ \dot{a}_{n}^{2(s-1)}\end{matrix}\right)\end{bmatrix} \] \[=\delta_{q,1}\,\mathrm{j}^{M}\,\mathrm{e}^{-\mathrm{i}M\varphi_{ }} \begin{bmatrix}\int_{M}(\mathrm{k}_{\text{c}\text{l}}\alpha_{q})\,\overline{ \mathbb{I}}\\ \overline{\mathrm{M}}\overline{\overline{\mathrm{G}}}_{0}^{\text{J}}(\alpha_{ 1})\end{bmatrix} \,\begin{bmatrix}\left( \begin{matrix}\dot{a}_{0}\\ \dot{a}_{0}^{q}\end{matrix} \right)\,\overline{\mathbb{I}}\\ \endHankel function [51], the z- and \(\varphi\)-components of the electromagnetic far-field are given by the analytical expressions of Equation (12)  
\[ \begin{bmatrix}\mathrm{E}_{\mathrm{z}}^{\mathrm{sc}}(\overline{\rho})\\ \mathrm{H}_{\mathrm{z}}^{\mathrm{sc}}(\overline{\rho})\end{bmatrix} =\mathrm{e}^{- \mathrm{j}\ (\mathrm{k}_{\mathrm{0}}\ \rho_{1}-\pi/4)}\sqrt{\frac{2}{\pi\mathrm{k}_{\mathrm{0}}\rho_{1}}}\sum_{n=- \infty}^{\infty}\mathrm{e}^{\mathrm{j}\mu(\sigma_{1}+\pi/2)}\ \begin{bmatrix}c_{n}^{1}\\ d_{n}^{1}\end{bmatrix}  \tag{12a}\] \[ \begin{bmatrix}\mathrm{E}_{\varphi}^{\mathrm{sc}}(\overline{\rho})\\ \mathrm{H}_{\varphi}^{\mathrm{sc}}(\overline{\rho})\end{bmatrix} = \begin{bmatrix} 0&Z_{0}(\sin\theta^{\prime})^{-1}\\ -(Z_{0}\sin\theta^{\prime})^{-1}&0\end{bmatrix} \ \begin{bmatrix} \mathrm{E}_{\mathrm{z}}^{\mathrm{sc}}(\overline{\rho})\\ \mathrm{H}_{\mathrm{z}}^{\mathrm{sc}}(\overline{\rho})\end{bmatrix} , \tag{12b}\]  
where \(\mathrm{k}_{\mathrm{0}}=\sqrt{\mathrm{k}_{\mathrm{0}}^{2}-\beta^{2}}=\mathrm{k }_{\mathrm{0}}\sin\theta^{\prime}\) and \(Z_{0}=\sqrt{\mu_{0}/\epsilon_{0}}\). The radial component of the Poynting vector is given by:  
\[\mathrm{S}_{\rho}^{\mathrm{sc}}(\overline{\rho})=\frac{1}{2Z_{0}\sin\theta^{ \prime}}\ \Bigl{[}\bigl{|}\mathrm{E}_{\mathrm{z}}^{\mathrm{sc}}(\overline{\rho}) \bigr{|}^{2}+Z_{0}^{2}\ \bigl{|}\mathrm{H}_{\mathrm{z}}^{\mathrm{sc}}(\overline{\rho}) \bigr{|}^{2}\Bigr{]}. \tag{13}\]  
The total radial scattered power, \(\mathrm{P}_{\rho}^{\mathrm{sc}}\), may be found by integrating \(\mathrm{S}_{\rho}^{\mathrm{sc}}(\overline{\rho})\) on the surface of a cylinder of infinite radius, \(\alpha_{1}\to\infty\), and finite length, dz. Consequently, the total scattering cross section, \(\sigma_{t}\), of the structure is obtained by [50]  
\[\sigma_{t}=\frac{\mathrm{P}_{\rho}^{\mathrm{sc}}}{\bigl{|}\mathrm{S}^{\mathrm{ inc}}(\overline{\rho})\bigr{|}\ \mathrm{dz}}\ \Rightarrow\ \sigma_{t}=\frac{1}{\mathrm{k}_{\mathrm{0}}}\ \Bigl{[}\frac{2}{\sin\theta^{ \prime}}\Bigr{]}^{2}\ \frac{1}{\mathrm{E}_{0}^{2}+Z_{0}^{2}\mathrm{H}_{0}^{2}}\sum_{n=- \infty}^{\infty}\biggl{(}\Bigl{|}\ c_{n}^{1}\Bigr{|}^{2}+\Bigl{|}Z_{0}\ d_{n}^{1 }\Bigr{|}^{2}\biggr{)}, \tag{14}\]  
where \(\mathrm{S}^{\mathrm{inc}}(\overline{\rho})\) is the Poynting vector of the incident plane wave.