# Magnetooptics in Cylindrical Structures
## 4 A Simple Cylindrically Layered Structure
In Equation (20), the arguments of the Bessel functions for the core become \[\kappa_{\pm}^{(2)2} = \left(\frac{2\pi}{\lambda_{\mathrm{vac}}}\right)^{2}\left(\kappa _{m\pm}^{(2)}\kappa_{e\pm}^{(2)}-N^{2}\right)\equiv-\gamma_{\pm}^{2}. \tag{19b}\]7 \[\kappa_{\pm}^{(2)2} = \left(\frac{2\pi}{\lambda_{\mathrm{vac}}}\right)^{2}\left(\kappa _{m\pm}^{(2)}\kappa_{e\pm}^{(2)}-N^{2}\right)\equiv-\gamma_{\pm}^{2}. \tag{19b}\]8  
and the arguments of modified Bessel functions for the cladding become  
\[\kappa_{\pm}^{(2)2} = \left(\frac{2\pi}{\lambda_{\mathrm{vac}}}\right)^{2}\left(\kappa _{m\pm}^{(2)}\kappa_{e\pm}^{(2)}-N^{2}\right)\equiv-\gamma_{\pm}^{2}. \tag{19b}\]9  
Figure 1:0  
Here \(\kappa_{e\pm}^{(i)}=\kappa_{e0}^{(i)}\pm\kappa_{e1}^{(i)}\), \(\kappa_{m\pm}^{(i)}=\kappa_{m0}^{(i)}\pm\kappa_{m1}^{(i)}\), \(i=1,2\), denote the CP relative permittivities and relative permeabilities in the core (\(i=1\)) and in the cladding (\(i=2\)). The evaluation of Equation (20) confirms that there are no first order effects of \(\varepsilon_{1}^{(1)},\mu_{1}^{(1)},\varepsilon_{1}^{(2)}\), or \(\mu_{1}^{(2)}\) on \(N\) for \(\
u=0\), i.e., for TE and TM modes, as expected [14; 35]. Solutions to the eigenvalue equation, Equation (20), will now be illustrated on a circular cylindrical dielectric waveguide operating at the wavelength \(\lambda_{\rm vac}\) =1.550 \(\upmu\)m with the core made from yttrium iron garnet, Y\({}_{3}\)Fe\({}_{5}\)O\({}_{12}\) (YIG), and the cladding made from gallium substituted YIG, Y\({}_{3}\)Fe\({}_{5-x}\)Ga\({}_{x}\)O\({}_{12}\) (GaYIG). The material parameters as functions of the diamagnetic substitution are collected in Figure 1:1 Their choice was inspired by the situations in pure and gallium substituted yttrium iron garnets [36; 37]. Table 2 provides the summary of parameters in the isotropic waveguide (\(\kappa_{e1}^{(i)}=\kappa_{m1}^{(i)}=0\)). The waveguide consists of a core of refractive index \(n_{1}=2.200\) and a cladding of refractive index \(n_{2}\left(x\right)<n_{1}\) monotonously decreasing with the diamagnetic substitution, \(x\). The choice of \(V-\)number, i.e., \(V=\left(2\pi/\lambda_{\rm vac}\right)\Delta a=\left(2\pi/\lambda_{\rm vac} \right)\left(n_{1}^{2}-n_{2}^{2}\right)^{1/2}a\approx 2.400<2.405\) for all \(x\) corresponds to monomode regime. This requires a corresponding adjustment of \(a\), the core radius. Figure 2 shows the dependence of \(a\) and \(\Delta\) on \(x\). Table 2 further contains the solution to the isotropic eigenvalue equation, Equation (21), \(N_{0}\), and the penetration depth in the cladding, \(\delta=\lambda_{\rm vac}/\left(2\pi\right)\left(N_{0}^{2}-n_{2}^{2}\right)^{- 1/2}\), a parameter useful in the evaluation of waveguide cross section. Figure 3 shows the effect of \(x\) on the effective guide index, \(N_{0}\), and the refractive index in the cladding, \(n_{2}\). The eigenvalues \(N_{\pm}\propto\beta_{\pm}\) distinguish the solutions to Equation (20). Here \(N_{+}\) denotes the solution for \(\
u=+1\) (\(\kappa_{e1}^{(i)}>0\) and \(\kappa_{m1}^{(i)}>0\), \(i=1,2\)), \(N_{-}\) denotes that for \(\
u=-1\) (\(\kappa_{e1}^{(i)}>0\) and \(\kappa_{m1}^{(i)}>0\)). The magnetization reversal results in the exchange of \(N_{+}\) and \(N_{-}\). For example, at \(\
u=+1\) for \(-\kappa_{e1}^{(i)}>0\) and \(-\kappa_{m1}^{(i)}>0\) (corresponding to the magnetization reversal, \(M\rightarrow-M\)), the solution is \(N_{-}\). The eigenvalue \(N_{0}\) in the isotropic waveguide (\(\kappa_{e1}^{(i)}=0\) and \(\kappa_{m1}^{(i)}=0\), corresponding to \(M=0\)) takes the same value for \(\
u=\pm 1\). The forward (\(\beta_{\pm}>0\)) propagation was assumed with \(N_{\pm}>0\). The eigenvalues \(-N_{\pm}>0\) of Equation (20) represent the solutions for the reversed propagation (\(-\beta_{\pm}>0\)). The propagation reversal does not change \(|N_{\pm}|\) the absolute values of \(N_{\pm}\), as required for the nonreciprocal propagation.