6.1 TE波とTM波の合成
任意の電磁界は，TE波とTM波の合成によって表すことができるので，まず，電界を次のように定義する．
\begin{eqnarray}
\VEC{E}
&=& \VEC{E}^{\TE} + \VEC{E}^{\TM}
\nonumber \\
&=& \left( E_\rho^{\TE} \VEC{a} _\rho + E_\phi^{\TE} \VEC{a} _\phi \right)
+ \left( E_\rho^{\TM} \VEC{a} _\rho + E_\phi^{\TM} \VEC{a} _\phi
+ E_z^{\TM} \VEC{a} _z \right)
\nonumber \\
&=& \left( E_\rho^{\TE} + E_\rho^{\TM} \right) \VEC{a} _\rho
+ \left( E_\phi^{\TE} + E_\phi^{\TM} \right) \VEC{a} _\phi
+ E_z^{\TM} \VEC{a} _z
\nonumber \\
&=& E_\rho \VEC{a} _\rho + E_\phi \VEC{a} _\phi + E_z \VEC{a} _z
\end{eqnarray}
同様にして，磁界は
\begin{eqnarray}
\VEC{H}
&=& \VEC{H}^{\TE} + \VEC{H}^{\TM}
\nonumber \\
&=& \left( H_\rho^{\TE} \VEC{a} _\rho + H_\phi^{\TE} \VEC{a} _\phi
+ H_z^{\TE} \VEC{a} _z \right)
+ \left( H_\rho^{\TM} \VEC{a} _\rho + H_\phi^{\TM} \VEC{a} _\phi \right)
\nonumber \\
&=& \left( H_\rho^{\TE} + H_\rho^{\TM} \right) \VEC{a} _\rho
+ \left( H_\phi^{\TE} + H_\phi^{\TM} \right) \VEC{a} _\phi
+ H_z^{\TE} \VEC{a} _z
\nonumber \\
&=& H_\rho \VEC{a} _\rho + H_\phi \VEC{a} _\phi + H_z \VEC{a} _z
\end{eqnarray}
これより，電界の$\rho$方向に沿う成分$E_\rho$，および$\phi$方向に沿う成分$E_\phi$は，次のようになる．
\begin{eqnarray}
E_\rho
&=& E_\rho^{\TE} + E_\rho^{\TM}
\nonumber \\
&=& \sum_m \left\{ \int_\gamma \frac{\gamma}{2}
\left( \bar{f}_m^{\TE} (\gamma) R_{E\rho}^{\TE}
+ \bar{f}_m^{\TM} (\gamma) R_{E\rho}^{\TM} \right)
e^{-jhz} d\gamma \right\}
\nonumber \\
&&\cdot \cos (m\phi + \alpha_m)
\label{eq:erho_tetm}
\\
E_\phi
&=& E_\phi^{\TE} + E_\phi^{\TM}
\nonumber \\
&=& \sum_m \left\{ \int_\gamma \frac{\gamma}{2}
\left( \bar{f}_m^{\TE} (\gamma) R_{E\phi}^{\TE}
+ \bar{f}_m^{\TM} (\gamma) R_{E\phi}^{\TM} \right)
e^{-jhz} d\gamma \right\}
\nonumber \\
&&\cdot \sin (m\phi + \alpha_m)
\label{eq:ephi_tetm}
\end{eqnarray}
ここで，
\begin{gather}
R_{E\rho} ^{\TE \brace \TM}
\equiv \pm {k \brace h} \{ J_{m-1}(\gamma \rho ) \pm J_{m+1}(\gamma \rho ) \}
\\
R_{E\phi} ^{\TE \brace \TM}
\equiv \mp {k \brace h} \{ J_{m-1}(\gamma \rho ) \mp J_{m+1}(\gamma \rho ) \}
\end{gather}
とおくと，
\begin{eqnarray}
&&\bar{f}_m^{\TE} R_{E{\rho \brace \phi}}^{\TE}
+ \bar{f}_m^{\TM} R_{E{\rho \brace \phi}}^{\TM}
\nonumber \\
&\equiv& f_m^{(+)}
\left( R_{E{\rho \brace \phi}}^{\TE} + R_{E{\rho \brace \phi}}^{\TM} \right)
+ f_m^{(-)}
\left( R_{E{\rho \brace \phi}}^{\TE} - R_{E{\rho \brace \phi}}^{\TM} \right)
\nonumber \\
&\equiv& f_m^{(+)} R_{E{\rho \brace \phi}}^{(+)}
+ f_m^{(-)} R_{E{\rho \brace \phi}}^{(-)}
\nonumber \\
&=& \left( f_m^{(+)} + f_m^{(-)} \right) R_{E{\rho \brace \phi}}^{\TE}
+ \left( f_m^{(+)} - f_m^{(-)} \right) R_{E{\rho \brace \phi}}^{\TM}
\end{eqnarray}
ただし，$\bar{f}_m^{\TE} $，$\bar{f}_m^{\TM}$は，TE波，TM波の円筒波スペクトラムを示し，新たに，スペクトラム$ f_m^{(+)}$，$ f_m^{(-)}$を定義している．
\begin{gather}
\bar{f}_m^{\TE} = f_m^{(+)} + f_m^{(-)}
\\
\bar{f}_m^{\TM} = f_m^{(+)} - f_m^{(-)}
\end{gather}
よって，
\begin{gather}
f_m^{(+)} = \bar{f}_m^{\TE} + \bar{f}_m^{\TM}
\\
f_m^{(-)} = \bar{f}_m^{\TE} - \bar{f}_m^{\TM}
\end{gather}
ここで，
\begin{gather}
R_{E{\rho \brace \phi}}^{(+)}
= R_{E{\rho \brace \phi}}^{\TE} + R_{E{\rho \brace \phi}}^{\TM}
\\
R_{E{\rho \brace \phi}}^{(-)}
= R_{E{\rho \brace \phi}}^{\TE} - R_{E{\rho \brace \phi}}^{\TM}
\end{gather}
これより，
\begin{eqnarray}
R_{E\rho}^{(+)}
&& R_{E\rho}^{\TE} + R_{E\rho}^{\TM}
\nonumber \\
&=& k \{ J_{m-1}(\gamma \rho ) + J_{m+1}(\gamma \rho ) \}
- h \{ J_{m-1}(\gamma \rho ) - J_{m+1}(\gamma \rho ) \}
\nonumber \\
&=& (k + h) J_{m+1}(\gamma \rho ) + (k - h) J_{m-1}(\gamma \rho )
\\
R_{E\rho}^{(-)}
&=& R_{E\rho}^{\TE} - R_{E\rho}^{\TM}
\nonumber \\
&=& k \{ J_{m-1}(\gamma \rho ) + J_{m+1}(\gamma \rho ) \}
+ h \{ J_{m-1}(\gamma \rho ) - J_{m+1}(\gamma \rho ) \}
\nonumber \\
&=& (k - h) J_{m+1}(\gamma \rho ) + (k + h) J_{m-1}(\gamma \rho )
\\
R_{E\phi}^{(+)}
&=& R_{E\phi}^{\TE} + R_{E\phi}^{\TM}
\nonumber \\
&=& - k \{ J_{m-1}(\gamma \rho ) - J_{m+1}(\gamma \rho ) \}
+ h \{ J_{m-1}(\gamma \rho ) + J_{m+1}(\gamma \rho ) \}
\nonumber \\
&=& (k + h) J_{m+1}(\gamma \rho ) - (k - h) J_{m-1}(\gamma \rho )
\\
R_{E\phi}^{(-)}
&=& R_{E\phi}^{\TE} - R_{E\phi}^{\TM}
\nonumber \\
&=& - k \{ J_{m-1}(\gamma \rho ) - J_{m+1}(\gamma \rho ) \}
- h \{ J_{m-1}(\gamma \rho ) + J_{m+1}(\gamma \rho ) \}
\nonumber \\
&=& (k - h) J_{m+1}(\gamma \rho ) - (k + h) J_{m-1}(\gamma \rho )
\end{eqnarray}
また，磁界の$\rho$方向に沿う成分$H_\rho$，および$\phi$方向に沿う成分$H_\phi$は，
\begin{eqnarray}
H_\rho
&=& H_\rho^{\TE} + H_\rho^{\TM}
\nonumber \\
&=& Y_w \sum_m \left\{ \int_\gamma \frac{\gamma}{2}
\left( \bar{f}_m^{\TE} (\gamma) R_{H\rho}^{\TE}
+ \bar{f}_m^{\TM} (\gamma) R_{H\rho}^{\TM} \right)
e^{-jhz} d\gamma \right\}
\nonumber \\
&&\cdot \sin (m\phi + \alpha_m)
\nonumber \\
&\equiv& Y_w \sum_m \left\{ \int_\gamma \frac{\gamma}{2}
\left( f_m^{(+)} (\gamma) R_{H\rho}^{(+)}
+ f_m^{(-)} (\gamma) R_{H\rho}^{(-)} \right)
e^{-jhz} d\gamma \right\}
\nonumber \\
&&\cdot \sin (m\phi + \alpha_m)
\end{eqnarray}
ここで，
\begin{gather}
R_{H\rho} ^{\TE \brace \TM}
\equiv \pm {h \brace k} \{ J_{m-1}(\gamma \rho ) \mp J_{m+1}(\gamma \rho ) \}
\end{gather}
これより，
\begin{eqnarray}
R_{H\rho}^{(+)}
&=& R_{H\rho}^{\TE} + R_{H\rho}^{\TM}
\nonumber \\
&=& h \{ J_{m-1}(\gamma \rho ) - J_{m+1}(\gamma \rho ) \}
- k \{ J_{m-1}(\gamma \rho ) + J_{m+1}(\gamma \rho ) \}
\nonumber \\
&=& -(k + h) J_{m+1}(\gamma \rho ) - (k - h) J_{m-1}(\gamma \rho )
\nonumber \\
&=& -R_{E\rho}^{(+)}
\\
R_{H\rho}^{(-)}
&=& R_{H\rho}^{\TE} - R_{H\rho}^{\TM}
\nonumber \\
&=& h \{ J_{m-1}(\gamma \rho ) - J_{m+1}(\gamma \rho ) \}
+ k \{ J_{m-1}(\gamma \rho ) + J_{m+1}(\gamma \rho ) \}
\nonumber \\
&=& (k - h) J_{m+1}(\gamma \rho ) + (k + h) J_{m-1}(\gamma \rho )
\nonumber \\
&=& R_{E\rho}^{(-)}
\end{eqnarray}
また，
\begin{eqnarray}
H_\phi
&=& H_\phi^{\TE} + H_\phi^{\TM}
\nonumber \\
&=& Y_w \sum_m \left\{ \int_\gamma \frac{\gamma}{2}
\left( \bar{f}_m^{\TE} (\gamma) R_{H\phi}^{\TE}
+ \bar{f}_m^{\TM} (\gamma) R_{H\phi}^{\TM} \right)
e^{-jhz} d\gamma \right\}
\nonumber \\
&&\cdot \cos (m\phi + \alpha_m)
\nonumber \\
&\equiv& Y_w \sum_m \left\{ \int_\gamma \frac{\gamma}{2}
\left( f_m^{(+)} (\gamma) R_{H\phi}^{(+)}
+ f_m^{(-)} (\gamma) R_{H\phi}^{(-)} \right)
e^{-jhz} d\gamma \right\}
\nonumber \\
&&\cdot \cos (m\phi + \alpha_m)
\end{eqnarray}
ここで，
\begin{gather}
R_{H\phi} ^{\TE \brace \TM}
\equiv \pm {h \brace k} \{ J_{m-1}(\gamma \rho ) \pm J_{m+1}(\gamma \rho ) \}
\end{gather}
これより，
\begin{eqnarray}
R_{H\phi}^{(+)}
&=& R_{H\phi}^{\TE} + R_{H\phi}^{\TM}
\nonumber \\
&=& h \{ J_{m-1}(\gamma \rho ) + J_{m+1}(\gamma \rho ) \}
- k \{ J_{m-1}(\gamma \rho ) - J_{m+1}(\gamma \rho ) \}
\nonumber \\
&=& (k + h) J_{m+1}(\gamma \rho ) - (k - h) J_{m-1}(\gamma \rho )
\nonumber \\
&=& R_{E\phi}^{(+)}
\\
R_{H\phi}^{(-)}
&=& R_{H\phi}^{\TE} - R_{H\phi}^{\TM}
\nonumber \\
&=& h \{ J_{m-1}(\gamma \rho ) + J_{m+1}(\gamma \rho ) \}
+ k \{ J_{m-1}(\gamma \rho ) - J_{m+1}(\gamma \rho ) \}
\nonumber \\
&=& -(k - h) J_{m+1}(\gamma \rho ) + (k + h) J_{m-1}(\gamma \rho )
\nonumber \\
&=& -R_{E\phi}^{(-)}
\end{eqnarray}
$z$成分については，
\begin{eqnarray}
E_z
&=& E_z^{\TM}
= -j \sum _m \left\{ \int_\gamma \bar{f}_m^{\TM} (\gamma) \gamma^2
J_m ( \gamma \rho ) e^{-jhz} d\gamma \right\} \cos (m\phi + \alpha_m)
\nonumber \\
&=& -j \sum _m \left\{ \int_\gamma
\left( f_m^{(+)} (\gamma) - f_m^{(-)} (\gamma)\right) \gamma^2
J_m ( \gamma \rho ) e^{-jhz} d\gamma \right\} \cos (m\phi + \alpha_m)
\nonumber \\
&=& \sum_m \left( E_{z m}^{(+)} + E_{z m}^{(-)} \right)
\cos (m\phi + \alpha_m)
\nonumber
\end{eqnarray}
ここで，
\begin{gather}
E_{z m} ^{(\pm )}
= \mp j \int _\gamma f_m^{(\pm )} (\gamma )
e^{-jhz} J_m (\gamma \rho ) \gamma ^2 d \gamma
\end{gather}
また，
\begin{eqnarray}
H_z &=& H_z^{\TE}
= j Y_w \sum _m \left\{ \int_\gamma \bar{f}_m^{\TE} (\gamma) \gamma^2
J_m ( \gamma \rho ) e^{-jhz} d\gamma \right\} \sin (m\phi + \alpha_m)
\nonumber \\
&=& j Y_w \sum _m \left\{ \int_\gamma
\left( f_m^{(+)} (\gamma) + f_m^{(-)} (\gamma)\right) \gamma^2
J_m ( \gamma \rho ) e^{-jhz} d\gamma \right\} \sin (m\phi + \alpha_m)
\nonumber \\
&\equiv& \sum_m \left( H_{z m}^{(+)} + H_{z m}^{(-)} \right)
\sin (m\phi + \alpha_m)
\nonumber
\end{eqnarray}
ここで，
\begin{gather}
H_{z m} ^{(\pm )}
= j Y_w \int _\gamma f_m^{(\pm )} (\gamma )
e^{-jhz} J_m (\gamma \rho ) \gamma ^2 d \gamma
= \mp Y_w E_{z m} ^{(\pm )}
\end{gather}
これより，
\begin{gather}
E_\rho
= \sum_m \left( E_{\rho m}^{(+)} + E_{\rho m}^{(-)} \right)
\cos (m\phi + \alpha_m)
\equiv \sum_m \left( \hat{E}_{\rho m,c}^{(+)} + \hat{E}_{\rho m,c}^{(-)} \right)
\\
E_\phi
= \sum_m \left( E_{\phi m}^{(+)} + E_{\phi m}^{(-)} \right)
\sin (m\phi + \alpha_m)
\equiv \sum_m \left( \hat{E}_{\phi m,c}^{(+)} + \hat{E}_{\phi m,c}^{(-)} \right)
\\
E_z
= \sum_m \left( E_{z m}^{(+)} + E_{z m}^{(-)} \right)
\cos (m\phi + \alpha_m)
\equiv \sum_m \left( \hat{E}_{z m,c}^{(+)} + \hat{E}_{z m,c}^{(-)} \right)
\end{gather}
ここで，
\begin{eqnarray}
E_{\rho m} ^{(\pm )} &=& \frac{1}{2} \int _0^{\infty } f_m^{(\pm )} (\gamma )
\Big\{ (k \pm h) J_{m+1}(\gamma \rho ) + (k \mp h) J_{m-1}(\gamma \rho ) \Big\}
\nonumber \\
&&\cdot e^{-jhz} \gamma d \gamma
\label{eq:e-rho-m} \\
E_{\phi m} ^{(\pm )} &=& \frac{1}{2} \int _0^{\infty } f_m^{(\pm )} (\gamma )
\Big\{ (k \pm h) J_{m+1}(\gamma \rho ) - (k \mp h) J_{m-1}(\gamma \rho ) \Big\}
\nonumber \\
&&\cdot e^{-jhz} \gamma d \gamma
\label{eq:e-phi-m} \\
E_{z m} ^{(\pm )} &=& \mp j \int _0^{\infty } f_m^{(\pm )} (\gamma )
e^{-jhz} J_m (\gamma \rho ) \gamma ^2 d \gamma
\label{eq:e-z-m}
\end{eqnarray}
また，
\begin{eqnarray}
H_\rho
&=& \sum_m \left( H_{\rho m}^{(+)} + H_{\rho m}^{(-)} \right)
\sin (m\phi + \alpha_m)
\equiv \sum_m \left( \hat{H}_{\rho m,c}^{(+)} + \hat{H}_{\rho m,c}^{(-)} \right)
\nonumber \\
&=& Y_w \sum_m \left( -E_{\rho m}^{(+)} + E_{\rho m}^{(-)} \right)
\sin (m\phi + \alpha_m)
\\
H_\phi
&=& \sum_m \left( H_{\phi m}^{(+)} + H_{\phi m}^{(-)} \right)
\cos (m\phi + \alpha_m)
\equiv \sum_m \left( \hat{H}_{\phi m,c}^{(+)} + \hat{H}_{\phi m,c}^{(-)} \right)
\nonumber \\
&=& Y_w \sum_m \left( E_{\phi m}^{(+)} - E_{\phi m}^{(-)} \right)
\cos (m\phi + \alpha_m)
\\
H_z
&=& \sum_m \left( H_{z m}^{(+)} + H_{z m}^{(-)} \right)
\sin (m\phi + \alpha_m)
\equiv \sum_m \left( \hat{H}_{z m,c}^{(+)} + \hat{H}_{z m,c}^{(-)} \right)
\nonumber \\
&=& Y_w \sum_m \left( -E_{z m}^{(+)} + E_{z m}^{(-)} \right)
\sin (m\phi + \alpha_m)
\end{eqnarray}
ここで，
\begin{gather}
\hat{E}_{\rho m,c}^{(\pm)}
= E_{\rho m}^{(\pm )} \cos (m \phi + \alpha _{m})
\\
\hat{E}_{\phi m,c}^{(\pm)}
= E_{\phi m}^{(\pm )} \sin (m \phi + \alpha _{m})
\\
\hat{E}_{z m,c}^{(\pm)}
= E_{z m}^{(\pm )} \cos (m \phi + \alpha _{m})
\end{gather}
また，
\begin{gather}
\hat{H}_{\rho m,c}^{(\pm)}
= \mp \sqrt{\frac{\epsilon }{\mu }}
E_{\rho m}^{(\pm )} \sin (m \phi + \alpha _{m})
\\
\hat{H}_{\phi m,c}^{(\pm)}
= \pm \sqrt{\frac{\epsilon }{\mu }}
E_{\phi m}^{(\pm )} \cos (m \phi + \alpha _{m})
\\
\hat{H}_{z m,c}^{(\pm)}
= \mp \sqrt{\frac{\epsilon }{\mu }}
E_{z m}^{(\pm )} \sin (m \phi + \alpha _{m})
\end{gather}