## 2 Formulation of the Problem
However, according to the procedure proposed in [48], for problems with azimuthal symmetry, all the field components can be expanded in Fourier series and, by invoking the VHT, the \(n\)-th azimuthal harmonic of the scattered electric field can be written in the spectral domain as follows:  
\[\underline{E}_{t}^{\text{sc}(n)}(\rho,z)= \begin{pmatrix}\underline{E}_{\rho}^ {\text{sc}(n)}(\rho,z)\\ -\text{j}\underline{E}_{\phi}^{\text{sc}(n)}(\rho,z)\end{pmatrix} =\int\limits_ {0}^{+\infty}\underline{\mathbf{H}}^{(n)}(wp)\underline{\widetilde{\mathbf{G} }}(w)\Big{(}\underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)+\underline{\tilde{ \mathbf{J}}}_{e}^{(n)}(w)\Big{)}e^{-\text{j}\sqrt{\underline{k}_{0}^{2}-w^{2} }|z|}wdw, \tag{3a}\]  
Figure 1: Geometry of the problem. \[E_{z}^{\text{sc}(n)}(\rho,z)=-\text{j}\frac{\text{sgn}(z)}{2\omega\epsilon_{0}} \int\limits_{0}^{+\infty}J_{n}(w\rho)\Big{(}\dot{I}_{i,C}^{(n)}(w)+\dot{I}_{e,C}^ {(n)}(w)\Big{)}e^{-\text{j}\sqrt{k_{0}^{2}-w^{2}}|z|}w^{2}dw,\] (3b) where \[\sqrt{k_{0}^{2}-w^{2}}=-\text{j}\sqrt{w^{2}-k_{0}^{2}}.\] \[\underline{\mathbf{H}}^{(n)}(w\rho)= \begin{pmatrix}J_{n}^{\prime}(w\rho)& \frac{nJ_{n}(w\rho)}{w\rho}\\ \frac{nJ_{n}(w\rho)}{w\rho}&J_{n}^{\prime}(w\rho)\end{pmatrix}  \tag{4}\]  
is the kernel of the VHT of order \(n\) (VHT\({}_{n}\)), and \(J_{v}(\cdot)\) and \(J_{v}^{\prime}(\cdot)\) are, respectively, the Bessel function of the first kind and order \(\
u\) and its first derivative with respect to the argument [55],  
\[\underline{\widetilde{\mathbf{G}}}(w)= \begin{pmatrix}\widetilde{\text{G}}_{ \text{C}}(w)&0\\ 0&\widetilde{\text{G}}_{\text{D}}(w)\end{pmatrix} = \begin{pmatrix}-\frac{\sqrt {k_{0}^{2}-w^{2}}}{2\omega\epsilon_{0}}&0\\ 0&-\frac{\omega\eta_{0}}{2\sqrt{k_{0}^{2}-w^{2}}}\end{pmatrix}  \tag{5}\]  
is related to the spectral domain counterpart of the free-space dyadic Green's function [48], and  
\[\underline{\tilde{\mathbf{J}}}_{s}^{(n)}(w)= \begin{pmatrix}\dot{I}_{s,C}^{(n)} (w)\\ -\dot{I}_{s,\text{D}}^{(n)}(w)\end{pmatrix} =\int\limits_{0}^{+\infty}\underline{ \mathbf{H}}^{(n)}(w\rho)\underline{\mathbf{J}}_{s}^{(n)}(\rho)\rho d\rho \tag{6}\]  
with \(s=i,e\), is the VHT\({}_{n}\) of \(\underline{\tilde{\mathbf{J}}}_{s}^{(n)}(\rho)\), i.e., the \(n\)-th azimuthal harmonic of \(\underline{\tilde{\mathbf{J}}}_{s}(\rho,\phi)\). Hence, the surface integral equations can be replaced by an infinite set of independent coupled dual integral equations in the VHT domain:  
\[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \Big{[}\Big{(}\underline{\widetilde{\mathbf{G}}}(w)-R_{\underline{1}} \underline{\tilde{\mathbf{J}}}\Big{)}\underline{\tilde{\mathbf{J}}}_{i}^{(n)} (w)+\underline{\widetilde{\mathbf{G}}}(w)\underline{\tilde{\mathbf{J}}}_{e}^ {(n)}(w)\Big{]}wdw=-\underline{\mathbf{E}}_{t}^{\text{inc}(n)}(\rho,0)\text{ for }\rho<a,\] (7a) \[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)wdw=\underline{0}\text{ for }\rho>a,\] (7b) \[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \left[\underline{\widetilde{\mathbf{G}}}(w)\underline{\tilde{\mathbf{J}}}_{i}^ {(n)}(w)+\Big{(}\underline{\widetilde{\mathbf{G}}}(w)-R_{\underline{1}} \underline{\tilde{\mathbf{J}}}_{e}^{(n)}(w)\Big{]}wdw=-\underline{\mathbf{E}}_ {t}^{\text{inc}(n)}(\rho,0)\text{ for }\rho>a,\] (7c) \[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \underline{\tilde{\mathbf{J}}}_{e}^{(n)}(w)wdw=\underline{0}\text{ for }\rho<a,\] (7d) with \[n\in\mathbb{Z}\], where \[\underline{\mathbf{E}}_{t}^{\text{inc}(n)}(\rho,0)=-\text{j}^{n+1}e^{-\text{ j}n\phi_{0}}\underline{\mathbf{H}}^{(n)}\big{(}k_{0}\delta_{\theta_{0}}\rho\big{)} \underline{\mathbf{E}}_{0t}\] (8) is the \(n\)-th azimuthal harmonic of the transverse component of the incident electric field.