5.3 TE波
TE波の電界の$z$成分は$E_z^{\TE} = 0$，磁界の$z$成分$H_z^{\TE} $は，
\begin{eqnarray}
H_z^{\TE}
&=& \frac{1}{j\omega \epsilon} \left( \frac{\partial ^2 }{\partial z ^2 } + k^2 \right) \psi
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TE} (\gamma)
\frac{1}{j\omega \mu} \left( \frac{\partial ^2 }{\partial z ^2 } + k^2 \right) \psi _{\gamma, m, h} d\gamma
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TE} (\gamma)
\frac{\gamma^2}{j\omega \mu} \psi _{\gamma, m, h} d\gamma
\end{eqnarray}
ただし，$f_m^{\TE} (\gamma)$はTE波の円筒波スペクトラムを示す．また，$z$軸に直交する横断面内電界成分は，
\begin{eqnarray}
E_{\rho }^{\TE}
&=& -\frac{1}{\rho } \frac{\partial \psi }{\partial \phi }
= \sum _m \int_\gamma f_m^{\TE} (\gamma) \left( -\frac{1}{\rho }
\frac{\partial \psi _{\gamma, m, h}}{\partial \phi } \right) d\gamma
\\
E_{\phi }^{\TE}
&=& \frac{\partial \psi }{\partial \rho }
= \sum _m \int_\gamma f_m^{\TE} (\gamma)
\frac{\partial \psi _{\gamma, m, h}}{\partial \rho} d\gamma
\end{eqnarray}
TE波の横断面内磁界成分は，
\begin{eqnarray}
H_{\rho }^{\TE}
&=& \frac{1}{j\omega \mu} \frac{\partial ^2 \psi }{\partial \rho \partial z}
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TE} (\gamma) \frac{1}{j\omega \mu}
\frac{\partial ^2 \psi _{\gamma, m, h}}{\partial \rho \partial z} d\gamma
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TE} (\gamma) \frac{-jh}{j\omega \mu}
\frac{\partial \psi _{\gamma, m, h}}{\partial \rho} d\gamma
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TE} (\gamma) Y^{\TE}
\left( -\frac{\partial \psi _{\gamma, m, h}}{\partial \rho} \right) d\gamma
\\
H_{\phi }^{\TE}
&=& \frac {1}{j\omega \mu} \frac{1}{\rho }
\frac{\partial ^2 \psi }{\partial \phi \partial z}
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TE} (\gamma) \frac{1}{j\omega \mu} \frac{1}{\rho}
\frac{\partial ^2 \psi _{\gamma, m, h}}{\partial \phi \partial z} d\gamma
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TE} (\gamma) \frac{-jh}{j\omega \mu} \frac{1}{\rho}
\frac{\partial \psi _{\gamma, m, h}}{\partial \phi} d\gamma
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TE} (\gamma) Y^{\TE} \left( -\frac{1}{\rho}
\frac{\partial \psi _{\gamma, m, h}}{\partial \phi} \right) d\gamma
\end{eqnarray}
ここで，
\begin{align}
&Y^{\TE} \equiv \frac{h}{\omega \mu}
= \frac{k}{\omega \mu} \cdot \frac{h}{k}
= Y_w \frac{h}{k}
\left( \equiv \frac{1}{Z^{\TE}} \right)
\\
&Y_w = \frac{k}{\omega \mu} = \sqrt{\frac{\epsilon}{\mu}}
= \frac{1}{Z_w}
\end{align}
ベッセル関数の関係式
\begin{gather}
\frac{d J_m (\gamma \rho )}{d \rho }
= \frac {\gamma }{2} \{ J_{m-1}(\gamma \rho )-J_{m+1}(\gamma \rho ) \} \\
J_{m}(\gamma \rho )
= \frac{\gamma \rho }{2m} \{ J_{m-1}(\gamma \rho ) + J_{m+1}(\gamma \rho ) \}
\end{gather}
を用いて微分等を行うと，
\begin{eqnarray}
\frac{\partial \psi _{\gamma, m, h}}{\partial \rho}
&=& \frac{\partial J_m ( \gamma \rho )}{\partial \rho}
\cdot \begin{matrix} \sin \\ \cos \end{matrix} \ m \phi \cdot e^{-jhz}
\nonumber \\
&=& \frac {\gamma }{2} \{ J_{m-1}(\gamma \rho )-J_{m+1}(\gamma \rho ) \}
\cdot \begin{matrix} \sin \\ \cos \end{matrix} \ m \phi \cdot e^{-jhz}
\\
\frac{1}{\rho } \frac{\partial \psi _{\gamma, m, h}}{\partial \phi}
&=& \frac{1}{\rho } J_m(\gamma \rho )
\cdot m \ \begin{matrix} \cos \\ -\sin \end{matrix} \ m \phi \cdot e^{-jhz}
\nonumber \\
&=& \frac{1}{\rho }
\left[ \frac {\gamma \rho}{2m} \{ J_{m-1}(\gamma \rho )+J_{m+1}(\gamma \rho ) \} \right]
m \ \begin{matrix} \cos \\ -\sin \end{matrix} \ m \phi \cdot e^{-jhz}
\nonumber \\
&=& \frac {\gamma }{2} \{ J_{m-1}(\gamma \rho )+J_{m+1}(\gamma \rho ) \}
\cdot \begin{matrix} \cos \\ -\sin \end{matrix} \ m \phi \cdot e^{-jhz}
\end{eqnarray}
これより，TE（no $E_z$）波の各成分は，
\begin{eqnarray}
E_{\rho }^{\TE}
&=& \mp \sum _m \int_\gamma f_m^{\TE} (\gamma)
\frac {\gamma }{2} \{ J_{m-1}(\gamma \rho )+J_{m+1}(\gamma \rho ) \}
\nonumber \\
&&\cdot \begin{matrix} \cos \\ \sin \end{matrix} \ m \phi \cdot e^{-jhz}
d\gamma
\\
E_{\phi }^{\TE}
&=& \pm \sum _m \int_\gamma f_m^{\TE} (\gamma)
\frac {\gamma }{2} \{ J_{m-1}(\gamma \rho )-J_{m+1}(\gamma \rho ) \}
\nonumber \\
&&\cdot \begin{matrix} \sin \\ -\cos \end{matrix} \ m \phi \cdot e^{-jhz}
d\gamma
\\
H_{\rho }^{\TE}
&=& \mp Y_w \sum _m \int_\gamma f_m^{\TE} (\gamma) \frac{h}{k}
\frac{\gamma }{2} \{ J_{m-1}(\gamma \rho )-J_{m+1}(\gamma \rho ) \}
\nonumber \\
&&\cdot \begin{matrix} \sin \\ -\cos \end{matrix} \ m \phi \cdot e^{-jhz}
d\gamma
\\
H_{\phi }^{\TE}
&=& \mp Y_w \sum _m \int_\gamma f_m^{\TE} (\gamma) \frac{h}{k}
\frac{\gamma }{2} \{ J_{m-1}(\gamma \rho )+J_{m+1}(\gamma \rho ) \}
\nonumber \\
&&\cdot \begin{matrix} \cos \\ \sin \end{matrix} \ m \phi \cdot e^{-jhz}
d\gamma
\\
H_z^{\TE}
&=& \pm \frac{1}{j\omega \mu} \sum _m \int_\gamma f_m^{\TE} (\gamma) \gamma^2
J_m ( \gamma \rho ) \ \begin{matrix} \sin \\ -\cos \end{matrix} \ m \phi
\cdot e^{-jhz} d\gamma
\end{eqnarray}
ここで，
\begin{eqnarray}
\cos \left( m\phi -\frac{\pi}{2} \right)
&=& \cos m \phi \cos \frac{\pi}{2} + \sin m \phi \sin \frac{\pi}{2}
\nonumber \\
&=& \sin m\phi
\end{eqnarray}
より，
\begin{gather}
\cos (m\phi + \alpha_m) \equiv
\begin{matrix} \cos \\ \sin \end{matrix} \ m \phi
\ \ \ \ \
\begin{matrix} (\alpha_m = 0) \\ (\alpha_m = -\pi/2) \end{matrix}
\end{gather}
ここで，
\begin{gather}
\bar{f}_m^{\TE} \equiv \mp \frac{1}{k} f_m^{\TE}
\end{gather}
とおく．また，
\begin{eqnarray}
\sin \left( m\phi -\frac{\pi}{2} \right)
&=& \sin m \phi \cos \frac{\pi}{2} + \cos m \phi \sin \frac{\pi}{2}
\nonumber \\
&=& -\cos m\phi
\end{eqnarray}
ゆえ，
\begin{gather}
\sin (m\phi + \alpha_m) =
\begin{matrix} \sin \\ -\cos \end{matrix} \ m \phi
\ \ \ \ \
\begin{matrix} (\alpha_m = 0) \\ (\alpha_m = -\pi/2) \end{matrix}
\end{gather}
また，
\begin{gather}
Y_w f_m^{\TE} \frac{h}{k} = \mp Y_w \bar{f}_m^{\TE} h
\end{gather}
これより，
\begin{eqnarray}
E_{\rho }^{\TE}
&=& \sum _m \int_\gamma \bar{f}_m^{\TE} (\gamma) k
\frac {\gamma }{2} \{ J_{m-1}(\gamma \rho )+J_{m+1}(\gamma \rho ) \}
\nonumber \\
&&\cdot \cos (m\phi + \alpha_m) e^{-jhz} d\gamma
\\
E_{\phi }^{\TE}
&=& -\sum _m \int_\gamma \bar{f}_m^{\TE} (\gamma) k
\frac {\gamma }{2} \{ J_{m-1}(\gamma \rho )-J_{m+1}(\gamma \rho ) \}
\nonumber \\
&&\cdot \sin (m\phi + \alpha_m) e^{-jhz} d\gamma
\\
H_{\rho }^{\TE}
&=& Y_w \sum _m \int_\gamma \bar{f}_m^{\TE} (\gamma) h
\frac{\gamma }{2} \{ J_{m-1}(\gamma \rho )-J_{m+1}(\gamma \rho ) \}
\nonumber \\
&&\cdot \sin (m\phi + \alpha_m) e^{-jhz} d\gamma
\\
H_{\phi }^{\TE}
&=& Y_w \sum _m \int_\gamma \bar{f}_m^{\TE} (\gamma) h
\frac{\gamma }{2} \{ J_{m-1}(\gamma \rho )+J_{m+1}(\gamma \rho ) \}
\nonumber \\
&&\cdot \cos (m\phi + \alpha_m) e^{-jhz} d\gamma
\\
H_z^{\TE}
&=& j Y_w \sum _m \int_\gamma \bar{f}_m^{\TE} (\gamma) \gamma^2
J_m ( \gamma \rho ) \sin (m\phi + \alpha_m) e^{-jhz} d\gamma
\end{eqnarray}