## 1 Introduction
As is evident in the schematic, the variation is analogous to the arc length of the circular sector cut by the ellipse for each discretization step. Figure 1: Example of the alternating character of the local index value from a thin shell radial discretization. Additionally, photonic fibers made of silica with a set of small air holes around their centers will have many cylindrical layers of varying refractive indexes along \(\theta\). In Figure 2, we show an example of a PCF with a hexagonal lattice of 6 rows of air holes. Again, radial discretization with thin, circular cylindrical layers results in alternating cuts through air holes and silica of a PCF as depicted in the schematic. The main aim in the present work is to develop the original RTL method so as to be able to include all such UOF cases via an appropriate transformation to mathematically equivalent COF cases. In the following sections, we first introduce the exact mathematical formulation of the discretization process for [PERSON]'s equations in the context of the RTL method in the following section. We also show how to perform an algebraic decoupling of the resulting transmission line equations and the transfer function method for computing the actual fields from the eigenvalues obtained. In Section 3, the same tools are applied in four more difficult cases of UOF with either, asymmetry or eccentricity of the central core. We also perform the same analysis for elliptic and rectangular cores as well as for the PCF case. 2 Mathematical Equivalence of Homogeneous Circular Cylindrical Layers to Electric Transmission Lines  
The basis for the application of the previously introduced RTL method is the radial discretization of all cylindrical fibres via a separation into a succession of thin cylindrical layers, each one with its own constant refractive index \(n\). These layers can be made to extend outside of the cladding in order to take into consideration the effect of the surrounding air (\(n=1\)). Each thin cylindrical layer could have thickness \(\delta r\) proportional to each average radius \(r\) which means that given discrete steps as \(\delta r=r_{2}-r_{1}\) with \(r=\frac{r_{1}+r_{2}}{2}\) one has  
\[\frac{r_{2}-r_{1}}{r_{2}+r_{1}}=\frac{c}{2}=>\left\{ \begin{array}{l}\frac{1 +\xi}{1-\frac{\xi}{2}}r_{1}=r_{2}\;(out)\\ \\ \frac{1-\xi}{1+\frac{\xi}{2}}r_{2}=r_{1}(in)\end{array} \right. \tag{1}\]  
Figure 2: Schematic depiction of the alternating step index resulting from radial discretization in a standard photonic crystal fiber (PCF). For any such circular cylindrical layer Maxwell equations (for a constant wavelength, i.e., constant frequency \"\(\omega\)\") can be written in their standard form as  
\[\left\{ \begin{array}{c}\
abla X\overset{\rightarrow}{E}=-j\omega\mu_{0} \overset{\rightarrow}{H}\\ \
abla X\overset{\rightarrow}{H}=j\omega\varepsilon_{0}n^{2}(r)\overset{ \rightarrow}{E}\end{array} \right. \tag{2}\]  
Taking into consideration the relations \(\omega\mu_{0}=k_{0}z_{0}\) and \(\omega\varepsilon_{0}=\frac{k_{0}}{z_{0}}\) where \(k_{0}=\frac{\omega}{c},\ z_{0}=120\pi\) and replacing \(z_{0}\overset{\rightarrow}{H}\) with \(\overset{\rightarrow}{H}\) in order \(\overset{\rightarrow}{E}\) and \(\overset{\rightarrow}{H}\) to have the same units \(\left(\mathrm{V/m}\right)\), Maxwell equations become then  
\[\left\{ \begin{array}{c}\
abla X\overset{\rightarrow}{E}=-jk_{0}\overset{ \rightarrow}{H}\\ \
abla X\overset{\rightarrow}{H}=jk_{0}n^{2}(r)\overset{\rightarrow}{E} \end{array} \right. \tag{3}\]  
In circular cylindrical geometry of coordinates (\(r\), \(\varphi\), \(z\)) the following set of three partial differential equations can be derived by the first vector Maxwell equation as  
\[\left\{