## 1 Introduction
### Equivalent Circuits for Cylindrical Layers, Boundary Conditions, and Birefringence
For any given \(l\), using the resonance technique the \(\beta\) values of the two birefringence modes can be calculated. The Equation (16) is given as a MATLAB function in Appendix A. Let us consider for example a step-index fiber of \(n_{1}=1.54\), \(n_{2}=1.47\) the \(V_{M}\), \(V_{E}\), fundamental modes for \(V=3.3\), can be calculated and their \(\beta\)/\(k_{0}\) values are respectively 1.518934962534846 and 1.518340184686295, hence their birefringence is equal to 0.0004947 or 0.0391%. The \(\beta\)/\(k_{0}\) value for the equivalent mode V\({}_{\rm eq}\) was also calculated and was equal to 1.518638548412019 (that is approximately equal to the mean value of the previous \(\beta\)/\(k_{0}\) values), while the \(\beta\)/\(k_{0}\) value calculated classically by Bessel functions is equal to 1.518642063686336. These \(\beta\) values are very close differing only 0.0002315%. In the following Figure 4, the normalized birefringence of the step-index fibers for \(n_{1}=1.54\), \(n_{2}=1.47\), and of \(n_{1}=1.475\), \(n_{2}=1.47\) as functions of V are shown. We notice that for any V, the normalized birefringence is almost proportional to \(\Delta n^{2}=(n_{1}-n_{2})^{2}\), thus the birefringence of step-index fibers of very small \(\Delta n\) is negligible. For instance, for a value of V = 2.4, and \(\Delta n=1.54-1.47=0.07\), the birefringence is found to be 0.168 \(\times\) 0.0049 = 0.0008232 or \(\sim\)0.055% on the average \(\beta\), while for \(\Delta n=1.475-1.47=0.005\), the birefringence becomes 0.168 \(\times\) 0.000025 = 0.000042 or \(\sim\)0.0028% on the average \(\beta\). What is remarkable is that our method is sensitive and calculates it. Calculating \"Voltages\" \(V_{m}\), \(V_{e}\) and \"Currents\" \(I_{M}\), \(I_{E}\) and Resulting Fields  
For any given \(l\), using the resonance technique the \(\beta\) values of the two birefringence modes can be calculated. These \(\beta\) values are practically the same, thus we can consider them as equal or we can consider as the proper value of \(\beta\) the mean value of the two modes. Taking \(V_{M}=1\) at the center point  
Figure 4: Normalized birefringence of two step-index fibers with different refractive indexes as functions of their parameters V.  
of the fiber (\(r=0\)), the respective value of \(I_{M}\) at the same point can be calculated by the respective terminal impedance. Using the matrix relations between input-output for the equivalent successive T-circuits, the values of \(V_{M}\) and \(I_{M}\) at the rest thin cylindrical layers can be calculated. In fact, from the general theory of the telegrapher's equation we know that the inputs and outputs are associated via a transfer matrix as follows  
\[ \begin{array}{l}\left[V_{out}I_{out}\right]=\left(\begin{array}{cc}\cosh( \xi(r)\cdot\delta r)&Z(r)\cdot\sinh(\xi(r)\cdot\delta r)\\ \sinh(\xi(r)\cdot\delta r)/Z(r)&\cosh(\xi(r)\cdot\delta r)\end{array}\right) \left[\begin{array}{c}V_{in}\\ I_{in}\end{array}\right]\\ \approx\left(\begin{array}{cc}1&Z(r)\cdot(\xi(r)\cdot\delta r)\\ (\xi(r)\cdot\delta r)/Z(r)&1\end{array}\right)\left[\begin{array}{c}V_{in}\\ I_{in}\end{array}\right]\\ =\left(\begin{array}{cc}1&\xi^{2}(r)\cdot\delta r/jF(r)\\ jF(r)\cdot\delta r&1\end{array}\right)\left[\begin{array}{c}V_{in}\\ I_{in}\end{array}\right]\end{array}  \tag{17}\]  
In Equation (17), the characteristic impedance should be taken as \(Z(r)=\xi(r)/jF(r)\) to fit with the previous analysis. Using the relations \(nV_{E}=V_{M}\) and \(nI_{M}=I_{E}\) the respective values of their birefringence partners can also be calculated for every thin cylindrical layer \(r_{i}\). Finally, we obtain the actual fields via the relations  
\[ \begin{array}{l}\overline{H_{r}}=\frac{iI_{M}}{r},\ \overline{E_{r}}=\frac{I_{E}}{n^{2}r} \\ \overline{H_{\varphi}}=jIV_{M}\ /r-\frac{\beta}{F}I_{E}\\ \overline{E_{\varphi}}=IV_{E}/r+j\frac{\beta}{F}I_{M}\\ \overline{H_{z}}=\frac{l}{Fr}I_{E}+j\beta V_{M}\\ \overline{z}=-j\frac{l}{Fr}I_{M}+\beta V_{E}\end{array}