5.4 TM波
TM波の磁界の$z$成分は$H_z^{\TM} = 0$，電界の$z$成分$E_z^{\TM} $は，
\begin{eqnarray}
E_z^{\TM}
&=& \frac{1}{j\omega \epsilon} \left( \frac{\partial ^2 }{\partial z ^2 } + k^2 \right) \psi
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TM} (\gamma)
\frac{1}{j\omega \epsilon} \left( \frac{\partial ^2 }{\partial z ^2 } + k^2 \right) \psi _{\gamma, m, h} d\gamma
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TM} (\gamma)
\frac{\gamma^2}{j\omega \epsilon} \psi _{\gamma, m, h} d\gamma
\end{eqnarray}
ただし，$f_m^{\TM} (\gamma)$はTM波の円筒波スペクトラムを示す．また，$z$軸に直交する横断面内磁界成分は，
\begin{eqnarray}
H_{\rho }^{\TM}
&=& \frac{1}{\rho } \frac{\partial \psi }{\partial \phi }
= \sum _m \int_\gamma f_m^{\TM} (\gamma) \frac{1}{\rho }
\frac{\partial \psi _{\gamma, m, h}}{\partial \phi } d\gamma
\\
H_{\phi }^{\TM}
&=& - \frac{\partial \psi }{\partial \rho }
= \sum _m \int_\gamma f_m^{\TM} (\gamma)
\left( -\frac{\partial \psi _{\gamma, m, h}}{\partial \rho} \right) d\gamma
\end{eqnarray}
TM波の横断面内電界成分は，
\begin{eqnarray}
E_{\rho }^{\TM}
&=& \frac{1}{j\omega \epsilon} \frac{\partial ^2 \psi }{\partial \rho \partial z}
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TM} (\gamma) \frac{1}{j\omega \epsilon}
\frac{\partial ^2 \psi _{\gamma, m, h}}{\partial \rho \partial z} d\gamma
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TM} (\gamma) \frac{-jh}{j\omega \epsilon}
\frac{\partial \psi _{\gamma, m, h}}{\partial \rho} d\gamma
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TM} (\gamma) Z^{\TM}
\left( -\frac{\partial \psi _{\gamma, m, h}}{\partial \rho} \right) d\gamma
\\
E_{\phi }^{\TM}
&=& \frac {1}{j\omega \epsilon} \frac{1}{\rho } \frac{\partial ^2 \psi }{\partial \phi \partial z}
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TM} (\gamma)
\frac{1}{j\omega \epsilon} \frac{1}{\rho}
\frac{\partial ^2 \psi _{\gamma, m, h}}{\partial \phi \partial z} d\gamma
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TM} (\gamma)
\frac{-jh}{j\omega \epsilon} \frac{1}{\rho}
\frac{\partial \psi _{\gamma, m, h}}{\partial \phi} d\gamma
\nonumber \\
&=& \sum _m \int_\gamma f_m^{\TM} (\gamma)
Z^{\TM} \left( -\frac{1}{\rho}
\frac{\partial \psi _{\gamma, m, h}}{\partial \phi} \right) d\gamma
\end{eqnarray}
ここで，
\begin{align}
&Z^{\TM} \equiv \frac{h}{\omega \epsilon}
= \frac{k}{\omega \epsilon} \cdot \frac{h}{k}
= Z_w \frac{h}{k}
\left( \equiv \frac{1}{Y^{\TM}} \right)
\\
&Z_w = \frac{k}{\omega \epsilon} = \sqrt{\frac{\mu}{\epsilon}}
= \frac{1}{Y_w}
\end{align}
これより，TM（no $H_z$）波の各成分は，
\begin{eqnarray}
E_{\rho }^{\TM}
&=& -Z_w \sum _m \int_\gamma f_m^{\TM} (\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
\\
E_{\phi }^{\TM}
&=& -Z_w \sum _m \int_\gamma f_m^{\TM} (\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
\\
E_z^{^{\TM}}
&=& \frac{1}{j\omega \epsilon} \sum _m \int_\gamma f_m^{^{\TM}} (\gamma) \gamma^2
J_m ( \gamma \rho ) \ \begin{matrix} \sin \\ \cos \end{matrix} \ m \phi
\cdot e^{-jhz} d\gamma
\\
H_{\rho }^{\TM}
&=& \sum _m \int_\gamma f_m^{\TM} (\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
\\
H_{\phi }^{\TM}
&=& -\sum _m \int_\gamma f_m^{\TM} (\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
\end{eqnarray}
上側と下側を入れ換えて，
\begin{eqnarray}
E_{\rho }^{\TM}
&=& -Z_w \sum _m \int_\gamma f_m^{\TM} (\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
\\
E_{\phi }^{\TM}
&=& -Z_w \sum _m \int_\gamma f_m^{\TM} (\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
\\
E_z^{^{\TM}}
&=& \frac{1}{j\omega \epsilon} \sum _m \int_\gamma f_m^{^{\TM}} (\gamma) \gamma^2
J_m ( \gamma \rho ) \ \begin{matrix} \cos \\ \sin \end{matrix} \ m \phi
\cdot e^{-jhz} d\gamma
\\
H_{\rho }^{\TM}
&=& \sum _m \int_\gamma f_m^{\TM} (\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_{\phi }^{\TM}
&=& -\sum _m \int_\gamma f_m^{\TM} (\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
\end{eqnarray}
また，
\begin{align}
&\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}
\\
&\bar{f}_m^{\TM} \equiv \frac{Z_w}{k} f_m^{\TM}
\end{align}
とおくと，
\begin{align}
&-\sin (m\phi + \alpha_m) =
\begin{matrix} -\sin \\ \cos \end{matrix} \ m \phi
\ \ \ \ \
\begin{matrix} (\alpha_m =0) \\ (\alpha_m = -\pi/2) \end{matrix}
\\
&f_m^{\TM} = \frac{k}{Z_w} \bar{f}_m^{\TM}
= Y_w \bar{f}_m^{\TM} k
\end{align}
これより，
\begin{eqnarray}
E_{\rho }^{\TM}
&=& -\sum _m \int_\gamma \bar{f}_m^{\TM} (\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
\\
E_{\phi }^{\TM}
&=& \sum _m \int_\gamma \bar{f}_m^{\TM} (\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
\\
E_z^{\TM}
&=& -j \sum _m \int_\gamma \bar{f}_m^{\TM} (\gamma) \gamma^2
J_m ( \gamma \rho ) \cos (m\phi + \alpha_m) e^{-jhz} d\gamma
\\
H_{\rho }^{\TM}
&=& -Y_w \sum _m \int_\gamma \bar{f}_m^{\TM} (\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_{\phi }^{\TM}
&=& -Y_w \sum _m \int_\gamma \bar{f}_m^{\TM} (\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
\end{eqnarray}