## 1 Introduction
\begin{array}{c}\frac{1}{r}\frac{\partial E_{z}}{\partial\varphi}- \frac{\partial E_{\varphi}}{\partial z}=-jk_{0}H_{r}\\ \frac{\partial E_{r}}{\partial z}-\frac{\partial E_{z}}{\partial r}=-jk_{0}H_{ \varphi}\\ \frac{1}{r}\frac{\partial\left(rE_{\varphi}\right)}{\partial r}-\frac{1}{r} \frac{\partial E_{r}}{\partial\varphi}=-jk_{0}H_{z}\end{array} \right. \tag{4}\]  
Applying a Fourier Transform along \"z\" and \"\(\varphi\)\" with wave numbers \"\(\beta\)\" and \"\(l\)\", where \(l\) is integer (because along \"\(\varphi\)\" we have Fourier series of period \(2\pi\)), the set (4) becomes:  
\[\left\{ \begin{array}{c}\frac{il}{r}\overline{E_{z}}-j\beta\overline{E_{ \varphi}}=-jk_{0}\overline{H_{r}}\\ j\beta\overline{E_{r}}-\frac{\partial\overline{E_{r}}}{\partial\pi}=-jk_{0} \overline{H_{\varphi}}\\ \frac{1}{r}\frac{\partial\left(r\overline{E_{\varphi}}\right)}{\partial r}- \frac{il}{r}\overline{E_{r}}=-jk_{0}\overline{H_{z}}\end{array} \right. \tag{5}\]  
In (5) we use new variables \(\overline{E_{r}},\ \overline{E_{\varphi}},\ \overline{E_{z}},\ \overline{H_{r}},\ \overline{H_{ \varphi}},\ \overline{H_{z}}\) to denote the Fourier Transforms of the respective electromagnetic field components. Furthermore, replacing \(\beta\) and \(r\) by their reduced variables according to the following relations:  
\[\left\{ \begin{array}{c}\frac{\beta}{k_{0}}=>\beta\\ rk_{0}=>r\end{array} \right.\]  
Then (5) takes the form  
\[\left\{ \begin{array}{c}\frac{il}{r}\overline{E_{z}}-j\beta\overline{E_{ \varphi}}=-j\overline{H_{r}}\\ j\beta\overline{E_{r}}-\frac{\partial\overline{E_{z}}}{\partial r}=-j\overline{H _{\varphi}}\\ \frac{1}{r}\frac{\partial\left(r\overline{E_{\varphi}}\right)}{\partial r}- \frac{il}{r}\overline{E_{r}}=-j\overline{H_{z}}\end{array} \right. \tag{6}\]  
Following a similar approach, the second Maxwell vector Equation (3) can be written in the form  
\[\left\{ \begin{array}{c}\frac{il}{r}\overline{H_{z}}-j\beta\overline{H_{ \varphi}}=jn^{2}(r)\overline{E_{r}}\\ j\beta\overline{H_{r}}-\frac{\partial\overline{H_{z}}}{\partial r}=jn^{2}(r) \overline{E_{\varphi}}\\ \frac{1}{r}\frac{\partial\left(r\overline{E_{\varphi}}\right)}{\partial r}- \frac{il}{r}\overline{H_{r}}=jn^{2}(r)\overline{E_{z}}\end{array} \right. \tag{7}\]  
Furthermore, following a cumbersome analysis, it is possible to prove that the system of Equations (6) and (7) can be transformed in a set of four differential Equation (8), relating the equivalent \"voltage\" and \"current\" functions \(V_{M},\ I_{M},\ V_{E},\ I_{E}\) defined as follows:\[V_{M}=\frac{i\overline{H_{\varphi}}+\beta\overline{r_{z}}}{jF}\] \[I_{M}=\frac{r\overline{H_{\varphi}}}{j}=\frac{\beta\overline{r_{ \varphi}}-i\overline{z}}{j}\] \[V_{E}=\frac{i\overline{r_{z}}+i\overline{F}\overline{E_{z}}}{F}\] \[I_{E}=n^{2}r\overline{E_{r}}=l\overline{H_{z}}-\beta r\overline {H_{\varphi}}\]  
where we use the notation \(F=\frac{(\beta r)^{2}+\beta^{2}}{r}\)  
\[\left\{ \begin{array}{c}\frac{\partial V_{M}}{\partial r}=-\frac{\gamma^{2}} {jF}I_{M}-jMI_{E}\\ \frac{\partial I_{M}}{\partial r}=-jFV_{M}\\ \frac{\partial V_{E}}{\partial r}=-\frac{\gamma^{2}}{j\overline{n^{2}F}}I_{E} -jMI_{M}\\ \frac{\partial I_{E}}{\partial r}=-jn^{2}FV_{E}\end{array} \right. \tag{8}\]  
In (8) we introduced the total propagation factor \(\gamma^{2}=\frac{l^{2}}{r^{2}}+\beta^{2}-n^{2}\) and the auxiliary function \(M=\frac{2l\beta}{\left[(\beta r)^{2}+l^{2}\right]F}\). At this point it is noticed that \(V_{M},~{}I_{M},~{}V_{E},~{}I_{E}\) are continuous functions at the boundaries because the tangential components of electric and magnetic fields \(\stackrel{{\rightarrow}}{{H_{\varphi}}}\stackrel{{ \rightarrow}}{{H_{z}}}\) and \(\stackrel{{\rightarrow}}{{E_{\varphi}}}\stackrel{{ \rightarrow}}{{E_{z}}}\) on the cylindrical surface are continuous functions passing the boundaries' of the cylindrical layer.