## 1 Introduction
### Equivalent Circuits for Cylindrical Layers, Boundary Conditions, and Birefringence  
Taking into consideration the transmission line theory, it can be proved that each layer of infinitesimal thickness \(\delta r\) is equivalent to a T-circuit as the one shown in Figure 3  
\[\left\{ \begin{array}{c}Z_{B}=\frac{\xi}{jF}\text{tanh}\Big{[}\frac{(\xi \delta r)}{2}\Big{]}\\ Z_{p}=\frac{\xi}{jF\text{sinh}(\xi\delta r)}\end{array} \right.\]For \(\xi\delta r\ll 1\) the impedances can be approximated by the equivalent relations  
\[\left\{ \begin{array}{c}Z_{B}=\frac{\xi^{2}\left(\frac{\xi r}{T}\right)}{jF}\\ Z_{p}=\frac{1}{jF\delta r}\end{array} \right. \tag{14}\]  
If \(\xi^{2}>0\), both \(Z_{B},~{}Z_{p}\) are \"capacitive\" reactances, for \(\xi^{2}<0\) however \(Z_{B}\) becomes \"inductive\" reactance. For \((V_{E},~{}I_{E})\) the approximate respective impedances of the T-circuit are given as  
\[\left\{ \begin{array}{c}Z^{\prime}{}_{B}=\frac{\xi^{2}\left(\frac{\xi r}{T} \right)}{j^{2}F}\\ Z^{\prime}{}_{p}=\frac{1}{j^{2}F\delta r}\end{array} \right. \tag{15}\]  
As previously stated, the functions \((V_{M},~{}I_{M})\) of each layer are continuous at the cylindrical boundaries of the layer, thus if we divide the fiber (including a sufficient number of air layers) in successive thin layers and replace them by their equivalent T-circuits, an overall lossless transmission line is formed with only reactive elements. For given \"\(l\)\", the \"\(\beta\)\" values that lead to the resonance of the overall transmission line are the eigenvalues of the whole optical fiber. When a transmission line is in resonance, at any arbitrary point \(r_{0}\) of the line, the sum of reactive impedances arising from the successive T-circuits on the left and right sides of \(r_{0}\) should be equal to zero, thus the equation giving the eigenvalues of the transmission line is the following:  
\[\left\{\dot{Z}_{L,r_{0}}+\dot{Z}_{R,r_{0}}=0\right. \tag{16}\]  
Equation (16) provides the eigenvalues \"\(\beta\)\" for a given \"\(l\)\", where \(\dot{Z}_{L,r_{0}},~{}\dot{Z}_{R,r_{0}}\) are the overall reactive impedances of successive T-circuits on the left and right of \(r_{0}\), using Equations (14) or (15). The value of \(r_{0}\) is usually given by the core radius. For the same \"\(l\)\" the Equations (14) and (15) give usually slightly different values of '\(\beta\)'. This phenomenon is called \"Birefringence\". For circular step index fibers, the birefringence is negligible; however, for elliptic fibers and fibers of any other non-circular cores, the birefringence phenomenon could be not negligible. In order to calculate the overall reactive impedances on the left and right of \(r_{0}\) we should find the impedances for \(r\to 0\) and for \(r\to\infty\). As we proceed to \(0\) or to \(\infty\), the remaining piece of transmission line becomes \"homogeneous\", i.e., its overall reactive impedance is equal to its characteristic impedance given by \(Z=\frac{\xi}{jF}~{}\left(\sigma\frac{\xi}{jn^{2}F}\right)\). Then we must have  
\[r\to\infty:~{}F\to\beta^{2}r,~{}MF\to 0,~{}\xi\to\sqrt{\beta^{2}-n^{2}}~{} \to~{}Z_{r\to\infty}=0\] \[r\to 0:~{}F\to\frac{\xi^{2}}{r},~{}\xi\to\frac{l}{r}~{}\to~{}Z_{r\to 0 }=\frac{1}{j|l|}~{}\left(\sigma\frac{1}{jn^{2}|l|}\right)\]  
Figure 3: The equivalent quadrupole for each cylindrical sector. For \(l=0\)\(Z_{r\to 0}=\infty\) (open circuit at the center of the equivalent transmission line) It is useful to notice that there is an equivalence between our formulation and the classic formulation modes of optical fibers. In particular, for \(l=0\), the modes (\(V_{M}\),\(I_{M}\)) are the TM modes, while the modes (\(V_{E}\),\(I_{E}\)) are the TE modes. For \(l>0\), the modes (\(V_{M}\),\(I_{M}\)) are the HE modes, while the modes (\(V_{E}\),\(I_{E}\)) are their HE birefringence modes. For \(l<\)0 the modes (\(V_{M}\),\(I_{M}\)) are the EH modes, while the modes (\(V_{E}\),\(I_{E}\)) are their EH birefringence modes.