# Magnetooptics in Cylindrical Structures
## 4 A Simple Cylindrically Layered Structure
The eigenvalue difference, \((N_{+}-N_{-})\propto(\beta_{+}-\beta_{-})\), a measure of nonreciprocity, as a function of \(x\) was computed for two cases, i.e., for a magnetic core and a nonmagnetic cladding, denoted as and for the magnetic core and magnetic cladding, denoted as \(\Delta N\). A considerable enhancement of \((N_{+}-N_{-})\) is predicted in the latter case. The trends are shown in Figure 1:2  
In the waveguide with the nonmagnetic cladding, the distance \(d=\lambda_{\rm vac}/\left[4\left(N_{+}-N_{-}\right)\right]\) required for the azimuth rotation on the axis \(\varrho=0\) by \(\pi/4\) remains practically independent of \(x\), \(d\approx 198\)\(\upmu\)m. On the other hand, in the waveguide with magnetic core and magnetic cladding, the required \(d\) is reduced to \(d\approx 140\)\(\upmu\)m for \(x=0.2\). The results are collected in Figure 1:3  
In practice, it would be desirable to reduce Ga content and consequently \(\Delta\). This would enable the monomode regime at a higher core radius and at a higher Curie temperature in the cladding. As indicated in Figure 4, this would also increase the magnetooptic contribution from the cladding and improve the weak guiding regime with reduced axial field components. The characteristics were evaluated for the core made of YIG. The Ce\({}^{3+}\) substitution for Y\({}^{3+}\) in Ce\({}_{y}\)Y\({}_{3-y}\)Fe\({}_{5}\)O\({}_{12}\) iron garnets can significantly improve magnetooptic activity at the communication wavelength \(1.55\)\(\upmu\)m with respect to that in YIG [38]. Figure 1:4 The effective guide index, \(N_{0}\) and the refractive index, \(n_{2}\), of the cladding in a monomode isotropic dielectric cylindrical waveguide as a function of Ga content, \(x\), in \(\mathrm{Y_{3}Fe_{5-x}Ga_{x}O_{12}}\), forming the cladding. Figure 1:5 The core radius, \(a\), and the parameter, \(\Delta=\sqrt{n_{1}^{2}-n_{2}^{2}}\) in a monomode dielectric cylindrical waveguide as a function of Ga content, \(x\), in yttrium iron garnet, \(\mathrm{Y_{3}Fe_{5-x}Ga_{x}O_{12}}\), in the cladding. The symbols \(n_{1}\) and \(n_{2}\) denote the indices of refraction in the core and in the cladding, respectively. The \(V-\)number was fixed at \(V=2.400\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(x\) & **0** & **0.2** & **0.4** & **0.6** & **0.8** & **1.0** & **1.2** & **1.4** \\ \hline \(n\) & 2.200 & 2.188 & 2.176 & 2.164 & 2.152 & 2.140 & 2.128 & 2.116 \\ \(\overline{M}\) & 1 & 0.806 & 0.617 & 0.450 & 0.289 & 0.167 & 0.083 & \(-\)0.022 \\ \(\kappa_{\epsilon 1}\times 10^{3}\) & 2 & 1.611 & 1.233 & 0.90 & 0.578 & 0.333 & 0.167 & \(-\)0.044 \\ \(\kappa_{m1}\times 10^{3}\) & 1 & 0.806 & 0.617 & 0.450 & 0.289 & 0.167 & 0.083 & \(-\)0.022 \\ \hline \end{tabular}
\end{table} 
Table 1: Effect of diamagnetic substitution, \(x\), in yttrium iron garnet, Y\({}_{3}\)Fe\({}_{5-x}\)Ga\({}_{x}\)O\({}_{12}\), on the real index of refraction, \(n\), relative magnetization, \(\overline{M}\), and the off-diagonal relative permittivity and permeability tensor elements, \(\kappa_{\epsilon 1}\) and \(\kappa_{m1}\). Figure 4: Difference in effective guide indices, \(N_{+}-N_{-}\), for the \(\pm\) transverse circular polarizations in a monomode dielectric circular cylindrical waveguide with the axial magnetization as a function of Ga content, \(x\), in yttrium iron garnet, Y\({}_{3}\)Fe\({}_{5-x}\)Ga\({}_{x}\)O\({}_{12}\), in the cladding. \(N_{+}-N_{-}\) is plotted for two cases: (1) For the case where both the core and the cladding are characterized by the permittivity and permeability tensors (magnetic cladding). (2) For the case where the core remains characterized by the permittivity and permeability tensors while the isotropic cladding is characterized by scalar permittivity and permeability (nonmagnetic cladding).