# Magnetooptics in Cylindrical Structures
## 2 Maxwell Equations
\right) \tag{2b}\]  
For the electric and magnetic wave field vectors in TCP representation, where \(E_{\pm}^{(i)}=E_{\pm}^{(i)}\left(\varrho,\varphi,z\right)\) and \(H_{\pm}^{(i)}=H_{\pm}^{(i)}\left(\varrho,\varphi,z\right)\), \(E_{z}^{(i)}=E_{z}^{(i)}\left(\varrho,\varphi,z\right)\) and \(H_{z}^{(i)}=H_{z}^{(i)}\left(\varrho,\varphi,z\right)\),\[\mathbf{E}^{(i)}\left(\varrho,\varphi,z,t\right) = \left(\mathrm{e}^{-\mathrm{j}\varphi}\hat{\varrho}_{+}E_{+}^{(i)}+ \mathrm{e}^{\mathrm{j}\varphi}\hat{\varrho}_{-}E_{-}^{(i)}+E_{z}^{(i)}\hat{z} \right)\exp\left(\mathrm{j}\omega t\right), \tag{3a}\] \[\mathbf{H}^{(i)}\left(\varrho,\varphi,z,t\right) = \left(\mathrm{e}^{-\mathrm{j}\varphi}\hat{\varrho}_{+}H_{+}^{(i) }+\mathrm{e}^{\mathrm{j}\varphi}\hat{\varrho}_{-}H_{-}^{(i)}+H_{z}^{(i)}\hat{ z}\right)\exp\left(\mathrm{j}\omega t\right), \tag{3b}\]  
the Maxwell curl equations in the transverse (\(\hat{\varrho}_{\pm}\)) and axial (\(\hat{z}\)) components are given by  
\[\hat{\varrho}_{+}\mathrm{e}^{-\mathrm{j}\varphi}\left[-\frac{ \partial E_{+}^{(i)}}{\partial z}+2^{-1/2}\left(\frac{\partial}{\partial \varrho}-\mathrm{j}\frac{1}{\partial\frac{\partial}{\partial\varphi}}\right) E_{z}^{(i)}\right]+\hat{\varrho}_{-}\mathrm{e}^{\mathrm{j}\varphi}\left[\frac{ \partial E_{-}^{(i)}}{\partial z}-2^{-1/2}\left(\frac{\partial}{\partial \varrho}+\mathrm{j}\frac{1}{\partial}\frac{\partial}{\partial\varphi}\right) E_{z}^{(i)}\right] \tag{4a}\] \[+ 2^{-1/2}\mathrm{z}\left[\frac{1}{\varrho}E_{+}^{(i)}+\left( \frac{\partial}{\partial\varrho}+\mathrm{j}\frac{1}{\varrho}\frac{\partial}{ \partial\varphi}\right)E_{+}^{(i)}-\frac{1}{\varrho}E_{-}^{(i)}-\left(\frac{ \partial}{\partial\varrho}-\mathrm{j}\frac{1}{\varrho}\frac{\partial}{ \partial\varphi}\right)E_{-}^{(i)}\right]\] \[= -\omega\left(\mathrm{e}^{-\mathrm{j}\varphi}\hat{\varrho}_{+} \mu_{+}^{(i)}H_{+}^{(i)}+\mathrm{e}^{\mathrm{j}\varphi}\hat{\varrho}_{-}\mu_{ -}^{(i)}H_{-}^{(i)}+\mu_{z}^{(i)}H_{z}^{(i)}\hat{z}\right),\] and \[\hat{\varrho}_{+}\mathrm{e}^{-\mathrm{j}\varphi}\left[-\frac{ \partial H_{+}^{(i)}}{\partial z}+2^{-1/2}\left(\frac{\partial}{\partial \varrho}-\mathrm{j}\frac{1}{\varrho}\frac{\partial}{\partial\varphi}\right) H_{z}^{(i)}\right]+\hat{\varrho}_{-}\mathrm{e}^{\mathrm{j}\varphi}\left[\frac{ \partial H_{-}^{(i)}}{\partial z}-2^{-1/2}\left(\frac{\partial}{\partial \varrho}+\mathrm{j}\frac{1}{\varrho}\frac{\partial}{\partial\varphi}\right) H_{z}^{(i)}\right]\] \[+ 2^{-1/2}\mathrm{z}\left[\frac{1}{\varrho}H_{+}^{(i)}+\left( \frac{\partial}{\partial\varrho}+\mathrm{j}\frac{1}{\varrho}\frac{\partial}{ \partial\varphi}\right)H_{+}^{(i)}-\frac{1}{\varrho}H_{-}^{(i)}-\left(\frac{ \partial}{\partial\varrho}-\mathrm{j}\frac{1}{\varrho}\frac{\partial}{ \partial\varphi}\right)H_{-}^{(i)}\right]\] \[= \omega\left(\mathrm{e}^{-\mathrm{j}\varphi}\hat{\varrho}_{+} \varepsilon_{+}^{(i)}E_{+}^{(i)}+\mathrm{e}^{\mathrm{j}\varphi}\hat{\varrho}_{- }\varepsilon_{-}^{(i)}E_{-}^{(i)}+\varepsilon_{z}^{(i)}E_{z}^{(i)}\hat{z} \right),\]  
where \(\varepsilon_{\pm}^{(i)}=\varepsilon_{0}^{(i)}\pm\varepsilon_{1}^{(i)}\), and \(\mu_{\pm}^{(i)}=\mu_{0}^{(i)}\pm\mu_{1}^{(i)}\). The TCP unit vectors, \(\hat{\varrho}_{\pm}\), are related to the Cartesian unit vectors \(\hat{x}\) and \(\hat{y}\) and to the circular cylindrical unit vectors \(\hat{\varrho}\) and \(\hat{\varphi}\) according to  
\[\hat{\varrho}_{\pm}=2^{-1/2}\left(\hat{\varrho}\pm\mathrm{j}\hat{\varphi} \right)\mathrm{e}^{\pm\mathrm{j}\varphi}=2^{-1/2}\left(\hat{x}\pm\mathrm{j} \hat{y}\right). \tag{5}\]  
From now, the factor \(\exp\left(\mathrm{j}\omega t\right)\) will be dropped out.