# Magnetooptics in Cylindrical Structures
## 3 Helmholtz Equations  
The Helmholtz wave equations in anisotropic media follow from Equation (1) and take the form [31]  
\[\
abla\times\left[\tilde{\mu}^{(i)-1}\left(\
abla\times\mathbf{E}^{( i)}\right)\right]=\omega^{2}\tilde{\epsilon}^{(i)}\mathbf{E}^{(i)}, \tag{9a}\] \[\
abla\times\left[\tilde{\epsilon}^{(i)-1}\left(\
abla\times \mathbf{H}^{(i)}\right)\right]=\omega^{2}\tilde{\mu}^{(i)}\mathbf{H}^{(i)}. \tag{9b}\]  
With restrictions to the terms of zero and first order in \(\mu_{1}^{(i)}\ll\mu_{0}^{(i)}\) and \(\epsilon_{1}^{(i)}\ll\varepsilon_{0}\) (including \(\mu_{z}^{(i)}\approx\mu_{0}^{(i)}\) and \(\epsilon_{z}^{(i)}\approx\varepsilon_{0}\)), the substitutions from Equation (4) provide  
\[\left(\
abla^{2}+\omega^{2}\mu_{0}\varepsilon_{0}\right)\left( \mathrm{e}^{-\mathrm{j}\varphi}\delta_{+}E_{+}+\mathrm{e}^{\mathrm{j}\varphi} \delta_{-}E_{-}+E_{z}\hat{z}\right)\] \[-\omega\left(\mu_{0}^{(i)}\epsilon_{0}^{(i)}\right)^{1/2}\left( \frac{\epsilon_{1}^{(i)}}{\epsilon_{0}^{(i)}}+\frac{\mu_{1}^{(i)}}{\mu_{0}^{ (i)}}\right)\left(\frac{\mu_{0}^{(i)}}{\epsilon_{0}^{(i)}}\right)^{1/2}\frac{ \partial}{\partial z}\left(\delta_{+}\mathrm{e}^{-\mathrm{j}\varphi}H_{+}+ \hat{\phi}_{-}\mathrm{e}^{\mathrm{j}\varphi}H_{-}+\hat{z}H_{z}\right)=0,\] (10a) and \[\left(\
abla^{2}+\omega^{2}\mu_{0}\varepsilon_{0}\right)\left( \mathrm{e}^{-\mathrm{j}\varphi}\delta_{+}H_{+}+\mathrm{e}^{\mathrm{j}\varphi} \delta_{-}H_{-}+H_{z}\hat{z}\right)\] \[+\omega\left(\mu_{0}^{(i)}\epsilon_{0}^{(i)}\right)^{1/2}\left( \frac{\epsilon_{1}^{(i)}}{\epsilon_{0}^{(i)}}+\frac{\mu_{1}^{(i)}}{\mu_{0}^{ (i)}}\right)\left(\frac{\epsilon_{0}^{(i)}}{\mu_{0}^{(i)}}\right)^{1/2}\frac{ \partial}{\partial z}\left(\delta_{+}\mathrm{e}^{-\mathrm{j}\varphi}E_{+}+ \delta_{-}\mathrm{e}^{\mathrm{j}\varphi}E_{-}+\hat{z}E_{z}\right)=0. \tag{10b}\]  
Here \(\
abla^{2}\) denotes the Laplacian  
\[\
abla^{2}=\frac{\partial^{2}}{\partial\varrho^{2}}+\frac{1}{\varrho}\frac{ \partial}{\partial\varrho}+\frac{1}{\varrho^{2}}\frac{\partial^{2}}{\partial \varphi^{2}}+\frac{\partial^{2}}{\partial z^{2}}.\]  
The Helmholtz partial differential Equation (10) can be separated assuming the solutions  
\[\mathbf{E}_{\
u}^{(i)}\left(\varrho,\varphi,z\right) = \left[\mathrm{e}^{-\mathrm{j}\varphi}\delta_{+}E_{\
u+}^{(i)} \left(\varrho\right)+\mathrm{e}^{\mathrm{j}\varphi}\delta_{-}E_{\
u-}^{(i)} \left(\varrho\right)+E_{\
u,z}^{(i)}\left(\varrho\right)\hat{z}\right]\mathrm{ e}^{\mathrm{j}\left(\
u\varphi-\beta z\right)}, \tag{11a}\] \[\mathbf{H}_{\
u}^{(i)}\left(\varrho,\varphi,z\right) = \left[\mathrm{e}^{-\mathrm{j}\varphi}\delta_{+}H_{\
u+}^{(i)} \left(\varrho\right)+\mathrm{e}^{\mathrm{j}\varphi}\delta_{-}H_{\
u-}^{(i)} \left(\varrho\right)+H_{\
u,z}^{(i)}\left(\varrho\right)\hat{z}\right]\mathrm{ e}^{\mathrm{j}\left(\
u\varphi-\beta z\right)}, \tag{11b}\]  
where \(\
u\) is an integer. Then \(\frac{\partial}{\partial z}\rightarrow-\mathrm{j}\beta\), \(\frac{\partial}{\partial\varphi}\rightarrow\mathrm{j}\
u\). The Helmholtz equation for the electric field becomes  
\[\delta_{+}\mathrm{e}^{-\mathrm{j}\varphi}\left\{\left[\left( \frac{\mathrm{d}^{2}}{\mathrm{d}\varrho^{2}}+\frac{1}{\varrho}\frac{\mathrm{d }}{\mathrm{d}\varrho}\right)+\left(\omega^{2}\epsilon_{0}^{(i)}\mu_{0}^{(i)}- \beta^{2}\right)-\frac{\left(\
u-1\right)^{2}}{\varrho^{2}}\right]E_{\
u+}^{(i) }+\mathrm{j}\beta\omega\frac{\epsilon_{1}^{(i)}\mu_{0}^{(i)}+\mu_{1}^{(i)} \epsilon_{0}^{(i)}}{\epsilon_{0}^{(i)}}H_{\
u+}^{(i)}\right\}\] \[\
abla\times\left[\tilde{\epsilon}^{(i)-1}\left(\
abla\times \mathbf{H}^{(i)}\right)\right]=\omega^{2}\tilde{\mu}^{(i)}\mathbf{H}^{(i)}. \tag{9b}\]0 \[\
abla\times\left[\tilde{\epsilon}^{(i)-1}\left(\
abla\times \mathbf{H}^{(i)}\right)\right]=\omega^{2}\tilde{\mu}^{(i)}\mathbf{H}^{(i)}. \tag{9b}\]1The corresponding Helmholtz equation for magnetic field is related to Equation (12a) by the duality transformation, i.e., \(\mathbf{E}\rightarrow\pm\mathbf{H}\), \(\mathbf{H}\rightarrow\mp\mathbf{E}\), and \(\tilde{\epsilon}\leftrightarrow\tilde{\mu}\).