## 3 Unconventional Fibers
### Application to Elliptic Core Fibers  
The method was applied in the calculation of fundamental modes of a fiber of elliptic core of \(a\) and \(b\) major and minor semi-axis, respectively, with refractive index \(n_{1}=1.54\), and a cladding value of \(n_{2}=1.47\) (Figure 5) for various wavelengths (defined by various V factor values \(V=2\pi b\sqrt{n_{1}^{2}-n_{2}^{2}}\)) and four ellipticity ratios \(a/b=1.1,1.3,1.5, and 2.0. Results are presented in tabulated format (Tables 1-4) compared with previous results calculated with Mathieu functions together with differences and relative differences showing a deviation which goes as only \(0.01/0.123\%\) on the average. Note the results in Tables 1-4 are not normalised the same way as in Figure 4, they are simply the odd modes, \(\mathrm{b}_{11}=\beta_{\mathrm{o}}/k_{\mathrm{o}}\). We can see that for small ellipticities especially, the results compare very well for all V values quite well with the Mathieu Functions results. As the ellipticity becomes very large as in Table 4, the results begin to differ. However, the accuracy of the Mathieu Functions used in this case is not known precisely, so the trend is correct and the actual difference could be debated. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**a/b = 1.1** & **Mathieu** & **RTL** & **Differences** & **Relative Diff. (0/00)** \\ \hline
**V** & **b11, No** & **b11, No** & & \\ \hline
1.5 & 1.487454917000000 & 1.48753837672558 & \(-0.00008345972558\) & 0.056109079 \\
1.7 & 1.49324500000000 & 1.49459896158188 & \(-0.00020598615188\) & 0.137945315 \\
1.9 & 1.4984577000000 & 1.49872279558264 & \(-0.00026509558264\) & 0.17691229 \\
2.1 & 1.50302975000000 & 1.503132165850878 & \(-0.00028271585078\) & 0.188097309 \\
2.3 & 1.506994250000000 & 1.50727053366314 & \(-0.00027628366314\) & 0.183334252 \\
2.5 & 1.51041850000000 & 1.51067588097959 & \(-0.00025738097959\) & 0.170403752 \\
2.7 & 1.513376830000000 & 1.51360949624267 & \(-0.00023266624267\) & 0.153739794 \\
2.9 & 1.515936970000000 & 1.51614747492828 & \(-0.00020780492828\) & 0.13708019 \\
3.1 & 1.518160870000000 & 1.51834484072766 & \(-0.00018397072766\) & 0.12117996 \\
3.3 & 1.520101500000000 & 1.52026286667918 & \(-0.000161186679180\) & 0.106036787 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of ellipse for the fundamental modes. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**a/b = 1.3** & **Mathieu** & **RTL** & **Differences** & **Relative Diff. (0/00)** \\ \hline
**V** & **b11, No** & **b11, No** & & \\ \hline
1.5 & 1.491188512000000 & 1.491027765079550 & 0.000160746920450 & 0.107797853 \\
1.7 & 1.49702899000000 & 1.49689763751180 & 0.000131352488200 & 0.087742114 \\
1.9 & 1.502119714000000 & 1.501986592174880 & 0.000133121825120 & 0.088622647 \\
2.1 & 1.5064715230000000 & 1.506335918376860 & 0.000135604623140 & 0.090014727 \\
2.3 & 1.510205927000000 & 1.51039049600890 & 0.000166877399110 & 0.110499764 \\
2.5 & 1.5134235000000000 & 1.51319602523000 & 0.000274974770000 & 0.150319773 \\
2.7 & 1.5161701202200000 & 1.518978403413190 & 0.00027278586810 & 0.179783432 \\
2.9 & 1.5185134800000000 & 1.51828023017018870 & 0.000293162981130 & 0.193059189 \\
3.1 & 1.520539300000000 & 1.520228501721160 & 0.000310798278840 & 0.204400030 \\
3.3 & 1.52222981900000000 & 1.521974104375010 & 0.000324085624990 & 0.212892341 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of ellipse for the fundamental modes. Figure 5: Elliptic fiber with three indicative elliptic thin layers. Inside the ellipse \(r<b\) (\(n=n_{1}\)), outside the ellipse \(r>a\) (\(n=n_{2}\)) and partly outside \(b<r<a\) (\(n_{1}>n>n_{2}\)). The steady component of the refractive index for the calculations for each radius \(r\) is defined as \(n_{1}\) for \(r<b\), \(n_{2}\) for \(r>a\), and as \(\left(n_{1}\varphi_{1}+n_{2}\varphi_{2}\right)/\pi\) when \(b<r<b\), where \(\varphi_{1}\), \(\varphi_{2}\) are the arcs of the circle of radius \(r\), inside and outside the ellipse in the upper semi ellipse.