## 1 Introduction
### Decoupling the Transmission Line Equations  
The prescribed set of Equation (8) constitutes a homogeneous set of ordinary differential equations of \(r\) and considering that the all vectors \([V_{M},~{}I_{M},~{}V_{E},~{}I_{E}]\) can be turned into exponential functions of \(r\) given by \(V_{M}=V_{M}e^{\beta r},~{}I_{M}=I_{M}e^{\beta r},~{}V_{E}=V_{E}e^{\beta r},~{}I _{E}=I_{E}e^{\beta r}\), where \(V_{M},~{}I_{M},~{}V_{E},~{}I_{E}\) are constants, i.e., not functions of \(r\). Thus, the system (8) can be transformed in an algebraic set of the following four equations  
\[\left\{ \begin{array}{c}\xi V_{M}=-\frac{\gamma^{2}}{jF}I_{M}-jMI_{E}\\ \xi I_{M}=-jFV_{M}\\ \xi V_{E}=-\frac{\gamma^{2}}{j\overline{n^{2}F}}I_{E}-jMI_{M}\\ \xi I_{E}=-jn^{2}FV_{E}\end{array} \right. \tag{9}\]  
Replacing \(I_{M}=-\frac{jF}{\xi}V_{M}\), \(I_{E}=-\frac{jn^{2}F}{\xi}V_{E}\), we obtain a set of two homogeneous equations  
\[\left\{ \begin{array}{c}\xi V_{M}=\frac{\gamma^{2}}{\xi}V_{M}-M\frac{n^{2}F}{ \xi}V_{E}\\ \xi V_{E}=\frac{\gamma^{2}}{\gamma^{2}}V_{E}-M\frac{\gamma}{\xi}V_{M}\end{array}  \right.~{}~{}\text{or}~{}\left\{ \begin{array}{c}\xi^{2}V_{M}=\gamma^{2}V_{M} -n^{2}MFV_{E}\\ \xi^{2}V_{E}=\gamma^{2}V_{E}-MF\end{array} \right.\]This then leads to the eigenvalue equations  
\[\left\{ \begin{array}{c}\left(\xi^{2}-\gamma^{2}\right)V_{M}+n^{2}MFV_{E}=0\\ MFV_{M}+\left(\xi^{2}-\gamma^{2}\right)V=0\end{array} \right.\]  
From the standard form of the eigenvalue problem we obtain through the determinant differential equations as follows  
\[\left(\xi^{2}-\gamma^{2}\right)^{2}-n^{2}M^{2}F^{2}=0,\text{ or }\xi^{2}=\gamma^{2} \pm nMF \tag{10}\]  
Hence the system has two eigenvalues and two mutually excluded or \"normal\" eigenvectors. The eigenvectors will be found by replacing \(\xi^{2}\) by its value. Thus, for \(\xi^{2}=\gamma^{2}-nMF\), \(nMFV_{E}-nMFV_{M}=0=>V_{M}=nV_{E}\) and the eigenvector is \(V_{S}=V_{M}+nV_{E}\) While for \(\xi^{2}=\gamma^{2}+nMF\), \(V_{M}=-nV_{E}\) and the eigenvector becomes \(V_{d}=V_{M}-nV_{E}\). Their respective \"current\" eigenvectors are related as \(\frac{I_{M}}{I_{E}}=\frac{V_{M}}{n^{2}V_{E}}=\frac{1}{n}\), thus \(I_{M}=\frac{I_{C}}{n}\) and \(I_{s}=I_{M}+\frac{I_{C}}{n}\), \(I_{d}=I_{M}-\frac{I_{C}}{n}\). Since the auxiliary \(M\) function has the sign of \(l\), the set \((V_{s},\ I_{s})\), for \(l=-l\) becomes equal to the set \((V_{d},\ I_{d})\). Thus, we can consider as a unique solution for the set \((V_{s},\ I_{s})\) and the integer '\(l\) 'varies \(-\infty\)\(to+\infty\), and of course \(\xi^{2}=\gamma^{2}-nMF\):  
\[\left\{ \begin{array}{c}\frac{\partial V_{s}}{\partial r}=-\frac{\xi^{2}}{jF} I_{s}\\ \frac{\partial I_{s}}{\partial r}=-jFI_{s}\end{array} \right. \tag{11}\]  
Furthermore, \(V_{s},\ I_{s}\) should be continuous functions at their boundaries although \(n(r)\) varies from layer to layer. This is achieved via the adjustment \(V_{s}=V_{M}+nV_{E}=2V_{M}\) and \(I_{s}=I_{M}+\frac{I_{E}}{n}=2I_{M}\), which are continuous functions of \(r\) by definition. \[\left\{ \begin{array}{c}\frac{\partial V_{M}}{\partial r}=-\frac{\xi^{2}}{jF }I_{M}\\ \frac{\partial I_{M}}{\partial r}=-jFI_{M}\end{array} \right. \tag{12}\]  
Another option for achieving continuity is to consider the functions \(V_{ss}=\frac{V_{M}}{n}+V_{E}\) and \(I_{ss}=nI_{M}+I_{E}\). In this case, \(V_{ss}=2V_{E}\) and \(I_{ss}=2I_{E}\) are also continuous. Thus  
\[\left\{ \begin{array}{c}\frac{\partial V_{E}}{\partial r}=-\frac{\xi^{2}}{jn^ {2}F}I_{E}\\ \frac{\partial I_{E}}{\partial r}=-jFn^{2}I_{E}\end{array} \right. \tag{13}\]  
Thus, the set of two coupled transmission lines (9) is equivalent to two independent transmission lines (12) and (13). The two waves represented by the equations of transmission lines (12) and (13), are geometrically normal because the first is related to the magnetic field and the second to the electric field that are geometrically normal for transmitted EM waves. This property is an inherent property of EM modes in optical fibers related to birefringence phenomena.