4.9 円形テーパ導波管
導波菅のモード関数は，
\begin{align}
&\VEC{e}_n = \left\{
\begin{array}{ll}
\displaystyle{ \VEC{e}^{\TE}_n
= \VEC{a}_z \times \nabla _t \Psi ^{\TE}_n }
& \mbox{ (TE mode) } \\
\displaystyle{ \VEC{e}^{\TM}_n
= - \nabla _t \Psi ^{\TM}_n }
& \mbox{ (TM mode) } \\
\end{array} \right.
\\
&\VEC{h}_n = \left\{
\begin{array}{ll}
\displaystyle{ \VEC{h}^{\TE}_n
= - \nabla _t \Psi ^{\TE}_n }
& \mbox{ (TE mode) } \\
\displaystyle{ \VEC{h}^{\TM}_n
= - \VEC{a}_z \times \nabla _t \Psi ^{\TM}_n }
& \mbox{ (TM mode) } \\
\end{array} \right.
\end{align}
ただし，スカラ関数
$\Psi ^{\TE}_n $および
$\Psi ^{\TM}_n $は変数分離した$z$に依らないスカラー関数を示し，スカラヘルムホルツ方程式を満足する．ここでは，半径$a$の円形導波菅の場合を考え，TE$_{Mn}$モードのスカラ関数
$\Psi ^{\TE}_n$，
およびTM$_{Mn}$モードのスカラ関数
$\Psi ^{\TM}_n$は，
\begin{align}
&\Psi ^{\TE}_n = A_n^{\TE} J_M(k_{c,n}^{\TE} \rho) \Phi_M^{\TE}(\phi), \ \ \ \ \
k_{c,n}^{\TE} = \frac{\chi_n'}{a}, \ \ \ \ \
J_M'(\chi_n') = 0
\\
&\Psi ^{\TM}_n = A_n^{\TM} J_M(k_{c,n}^{\TM} \rho) \Phi_M^{\TM}(\phi), \ \ \ \ \
k_{c,n}^{\TM} = \frac{\chi_n}{a}, \ \ \ \ \
J_M(\chi_n) = 0
\end{align}
ただし，$\Phi_M^{\TE}(\phi)$，$\Phi_M^{\TM}(\phi)$は次式で表され，
TEモードとTMモードは，同じ正弦モード間，同じ余弦モード間で結合する
（両者をまとめて$\Phi_M(\phi)$とする）．
\begin{gather}
\Phi_M^{\TE}(\phi)=
\left\{ \begin{matrix} \sin (M \phi) \\ \cos (M \phi) \end{matrix} \right., \ \ \ \ \
\Phi_M^{\TM}(\phi)=
\left\{ \begin{matrix} -\cos (M \phi) \\ \sin (M \phi) \end{matrix} \right.
\end{gather}
伝送方程式の係数（TE-TE, TM-TM）
スカラ関数の微分について，
TE，TMモードをまとめて
$k_{c,l} = \frac{\bar{\chi}_l}{a}$，$k_{c,n} = \frac{\bar{\chi}_n}{a}$とおき，
\begin{eqnarray}
\frac{\partial \Psi_n}{\partial z}
&=& \frac{\partial}{\partial z} \Big( A_n J_M(k_{c,n} \rho) \Phi_M(\phi) \Big)
\nonumber \\
&=& A_n \frac{dk_{c,n}}{dz} \rho J_M'(k_{c,n}\rho) \Phi_M(\phi)
\nonumber \\
&=& A_n \left( - \frac{1}{a} \frac{da}{dz} k_{c,n} \right) \rho J_M'(k_{c,n}\rho) \Phi_M(\phi)
\label{eq:dPsindz}
\\
\left. \frac{\partial \Psi_n}{\partial z} \right|_{\rho=a}
&=& -A_n \frac{da}{dz} k_{c,n} J_M'(\bar{\chi}_n) \Phi_M(\phi)
\\
\frac{\partial \Psi_l}{\partial n}
&=& \frac{\partial}{\partial \rho} \Big( A_l J_M(k_{c,l} \rho) \Phi_M(\phi) \Big)
\nonumber \\
&=& A_l k_{c,l} J_M'(k_{c,l}\rho) \Phi_M(\phi)
\\
\left. \frac{\partial \Psi_l}{\partial n} \right|_{\rho=a}
&=& \frac{\partial}{\partial \rho} \Big( A_l J_M(k_{c,l} \rho) \Phi_M(\phi) \Big)
\nonumber \\
&=& A_l k_{c,l} J_M'(\bar{\chi}_l) \Phi_M(\phi)
\end{eqnarray}
さらに，
\begin{eqnarray}
&&\frac{\partial}{\partial n} \left( \frac{\partial \Psi_n}{\partial z} \right)
= \frac{\partial}{\partial \rho} \left\{
A_n \left( - \frac{1}{a} \frac{da}{dz} k_{c,n} \right) \rho J_M'(k_{c,n}\rho) \Phi_M(\phi) \right\}
\nonumber \\
&=& -A_n \frac{1}{a} \frac{da}{dz} k_{c,n}
\Big\{ J_M'(k_{c,n}\rho) + \rho k_{c,n} J_M''(k_{c,n}\rho) \Big\} \Phi_M(\phi)
\end{eqnarray}
\begin{gather}
\left. \frac{\partial}{\partial n} \left( \frac{\partial \Psi_n}{\partial z} \right) \right|_{\rho=a}
= -A_n \frac{1}{a} \frac{da}{dz} k_{c,n}
\Big\{ J_M'(\bar{\chi}_n) + \bar{\chi}_n J_M''(\bar{\chi}_n) \Big\} \Phi_M(\phi)
\end{gather}
ここで，
\begin{gather}
J_M''(\bar{\chi}_n)
= \left( \frac{M^2}{\bar{\chi}_n^2} -1 \right) J_M(\bar{\chi}_n) - \frac{J_M'(\bar{\chi}_n)}{\bar{\chi}_n}
%= \frac{M^2-\bar{\chi}_n^2}{\bar{\chi}_n^2} J_M(\bar{\chi}_n)
\end{gather}
これより，
\begin{eqnarray}
\left. \frac{\partial}{\partial n} \left( \frac{\partial \Psi_n}{\partial z} \right) \right|_{\rho=a}
&=& -A_n \frac{da}{dz} \frac{\bar{\chi}_n}{a^2}
\frac{M^2-\bar{\chi}_n^2}{\bar{\chi}_n} J_M(\bar{\chi}_n) \Phi_M(\phi)
\nonumber \\
&=& -A_n \frac{da}{dz} \frac{M^2-\bar{\chi}_n^2}{a^2} J_M(\bar{\chi}_n) \Phi_M(\phi)
\end{eqnarray}
よって，異なるモードの場合（$n \ne l$），TEモードのとき，
\begin{eqnarray}
T_{I,ln}^{\TETE}
&=& \frac{(k_{c,l}^{\TE})^2}{(k_{c,l}^{\TE})^2 - (k_{c,n}^{\TE})^2}
\oint _C \Psi_l^{\TE} \frac{\partial}{\partial n} \left( \frac{\partial \Psi_n^{\TE}}{\partial z} \right) d\sigma
\nonumber \\
&=& \frac{(k_{c,l}^{\TE})^2}{(k_{c,l}^{\TE})^2 - (k_{c,n}^{\TE})^2}
A_l^{\TE} J_M(\chi_l') (-A_n^{\TE})
\nonumber \\
&&\cdot \frac{da}{dz} \frac{M^2-\chi_n^{\prime 2}}{a^2} J_M(\chi_n')
\int _0^{2\pi} \big( \Phi^{\TE} (\phi) \big)^2 a d\phi
\nonumber \\
&=& A_l^{\TE} A_n^{\TE} \frac{1}{a} \frac{da}{dz} (\chi_n^{\prime 2}-M^2)
\nonumber \\
&=& \frac{(k_{c,l}^{\TE})^2}{(k_{c,l}^{\TE})^2 - (k_{c,n}^{\TE})^2}
J_M(\chi_l') J_M(\chi_n') \frac{2\pi}{\epsilon_M}
\end{eqnarray}
ここで，絶対値をとらないTE$_{Ml}$，TE$_{Mn}$モードの正規化係数
$A_l^{\TE}$，$A_n^{\TE}$より，
\begin{eqnarray}
A_l^{\TE} A_n^{\TE}
&=& \sqrt{\frac{\epsilon _M}{\pi \big( \chi^{\prime 2}_l -M^2 \big) }}
\ \frac{1}{J_M (\chi '_l)} \cdot
\sqrt{\frac{\epsilon _M}{\pi \big( \chi^{\prime 2}_n -M^2 \big) }}
\ \frac{1}{J_M (\chi '_n)}
\nonumber \\
&=& \frac{\epsilon_M}{\pi} \frac{1}{\sqrt{\chi^{\prime 2}_l -M^2}}
\frac{1}{\sqrt{\chi^{\prime 2}_n -M^2}}
\frac{1}{J_M (\chi '_l)} \frac{1}{J_M (\chi '_n)}
\end{eqnarray}
よって，
\begin{eqnarray}
T_{I,ln}^{\TETE}
&=& \frac{1}{a} \frac{da}{dz} \frac{2(k_{c,l}^{\TE})^2}{(k_{c,l}^{\TE})^2 - (k_{c,n}^{\TE})^2}
\sqrt{\frac{\chi_n^{\prime 2}-M^2}{\chi_l^{\prime 2}-M^2}}
\nonumber \\
&=& \frac{1}{a} \frac{da}{dz} \frac{2\chi_l^{\prime 2}}{\chi_l^{\prime 2} - \chi_n^{\prime 2}}
\sqrt{\frac{\chi_n^{\prime 2}-M^2}{\chi_l^{\prime 2}-M^2}}
\nonumber \\
&=& -\frac{1}{a} \frac{da}{dz} \frac{2\chi_l^{\prime 2}}{\chi_n^{\prime 2} - \chi_l^{\prime 2}}
\sqrt{\frac{\chi_n^{\prime 2}-M^2}{\chi_l^{\prime 2}-M^2}}
\end{eqnarray}
また，TMモードのとき，
$\Psi_l=0$，$\Psi_n=0$ (on C)ゆえ，
\begin{eqnarray}
T_{I,ln}^{\TMTM}
&=& \frac{-(k_{c,n}^{\TM})^2}{(k_{c,l}^{\TM})^2 - (k_{c,n}^{\TM})^2}
\oint _C \frac{\partial \Psi_n^{\TM}}{\partial z} \frac{\partial \Psi_l^{\TM}}{\partial n} d\sigma
\nonumber \\
&=& \frac{-(k_{c,n}^{\TM})^2}{(k_{c,l}^{\TM})^2 - (k_{c,n}^{\TM})^2}
(-A_n^{\TM})
\nonumber \\
&&\cdot \frac{da}{dz} k_{c,n}^{\TM} J_M'(\bar{\chi}_n)
A_l^{\TM} k_{c,l}^{\TM} J_M'(\bar{\chi}_l)
\int _0^{2\pi} \big( \Phi^{\TE} (\phi) \big)^2 a d\phi
\nonumber \\
&=& A_n^{\TM} A_l^{\TM} \frac{da}{dz}
\frac{(k_{c,n}^{\TM})^2}{(k_{c,l}^{\TM})^2 - (k_{c,n}^{\TM})^2}
\frac{\chi_n \chi_l}{a^2}
J_M'(\chi_n) J_M'(\chi_l) a \frac{2\pi}{\epsilon_M}
\end{eqnarray}
ここで，絶対値をとらないTM$_{Ml}$，TM$_{Mn}$モードの正規化係数
$A_l^{\TM}$，$A_n^{\TM}$より，
\begin{eqnarray}
A_l^{\TM} A_n^{\TM}
&=& \sqrt{\frac{\epsilon_M}{\pi}} \frac{1}{\chi_l J_M'(\chi_l)} \cdot
\sqrt{\frac{\epsilon_M}{\pi}} \frac{1}{\chi_n J_M'(\chi_n)}
\nonumber \\
&=& \frac{\epsilon_M}{\pi} \frac{1}{\chi_l \chi_n}
\frac{1}{J_M' (\chi_l)} \frac{1}{J_M' (\chi_n)}
\end{eqnarray}
よって，
\begin{eqnarray}
T_{I,ln}^{\TMTM}
&=& \frac{1}{a} \frac{da}{dz}
\frac{2(k_{c,n}^{\TM})^2}{(k_{c,l}^{\TM})^2 - (k_{c,n}^{\TM})^2}
\nonumber \\
&=& \frac{1}{a} \frac{da}{dz} \frac{2\chi_n^2}{\chi_l^2 - \chi_n^2}
\nonumber \\
&=& -\frac{1}{a} \frac{da}{dz} \frac{2\chi_n^2}{\chi_n^2 - \chi_l^2}
\end{eqnarray}
同じモードの場合（$n=l$），
円形導波菅モードをTEモードとTMモードをまとめて，
\begin{gather}
\Psi_n = A_n J_M(k_{c,n} \rho) \Phi_M (\phi)
\label{eq:Psin}
\end{gather}
とすると，
\begin{gather}
&\nabla_t \Psi_n
= A_n \left\{ \frac{\partial J_M(k_{c,n} \rho)}{\partial \rho} \Phi_M (\phi) \VEC{a}_\rho
+ J_M(k_{c,n} \rho) \frac{\partial \Phi_M(\phi)}{\rho \partial \phi} \VEC{a}_\phi \right\}
\end{gather}
\begin{align}
&(\nabla_t \Psi_n) \cdot (\nabla_t \Psi_n)
\nonumber \\
&= A_n^2 \left\{ \left( \frac{\partial J_M(k_{c,n} \rho)}{\partial \rho} \Phi_M (\phi) \right)^2
+ \left( J_M(k_{c,n} \rho) \frac{\partial \Phi_M(\phi)}{\rho \partial \phi}\right)^2 \right\}
\nonumber \\
&= A_n^2 \left\{ J_M^{\prime 2}(k_{c,n} \rho) k_{c,n}^2 \Phi^2_M (\phi)
+ J_M^2(k_{c,n} \rho) \frac{1}{\rho^2} \left( \frac{\partial \Phi_M(\phi)}{\partial \phi} \right)^2
\right\}
\end{align}
\begin{align}
&(\nabla_t \Psi_n) \cdot (\nabla_t \Psi_n) \Big|_{\rho=a}
\nonumber \\
&= A_n^2 \left\{ J_M^{\prime 2}(\chi_n) k_{c,n}^2 \Phi^2_M (\phi)
+ J_M^2(\chi_n) \frac{1}{a^2} \left( \frac{\partial \Phi_M(\phi)}{\partial \phi} \right)^2
\right\}
\end{align}
軸対称ゆえ，$\vartheta$を定数として$\frac{da}{dz} = \tan \vartheta $．よって，
\begin{gather}
\iint_S \frac{\partial \VEC{e}_n}{\partial z} \cdot \VEC{e}_n dS
= - \frac{1}{2} \frac{da}{dz} \int_\sigma (\nabla_t \Psi_n) \cdot (\nabla_t \Psi_n) \ d\sigma
\end{gather}
周回積分は，半径$a$の円形導波管の管壁に沿う積分経路ゆえ，
\begin{eqnarray}
&&\int_\sigma (\nabla_t \Psi_n) \cdot (\nabla_t \Psi_n) d\sigma
\nonumber \\
&=& \int_0^{2\pi} (\nabla_t \Psi_n) \cdot (\nabla_t \Psi_n) \rho \Big|_{\rho = a} d\phi
\nonumber \\
&=& A_n^2 a \left\{ J_M^{\prime 2}(\chi_n) k_{c,n}^2 \int_0^{2\pi} \Phi^2_M (\phi) d\phi \right.
\nonumber \\
&&\left. + J_M^2(\chi_n) \frac{1}{a^2} \int_0^{2\pi} \left( \frac{\partial \Phi_M(\phi)}{\partial \phi} \right)^2 d\phi \right\}
\nonumber \\
&=& A_n^2 a \left\{ J_M^{\prime 2}(\chi_n) k_{c,n}^2 \frac{2\pi}{\epsilon_M}
+ J_M^2(\chi_n) \frac{1}{a^2} M^2 \frac{2\pi}{\epsilon_M} \right\}
\nonumber \\
&=& A_n^2 \frac{2\pi}{\epsilon_M} \frac{1}{a}
\left\{ \chi_n^2 J_M^{\prime 2}(\chi_n) + M^2 J_M^2(\chi_n) \right\}
\end{eqnarray}
ここで，
\begin{align}
&\int_0^{2\pi} \Phi^2_M (\phi) d\phi = \frac{2\pi}{\epsilon_M}
\\
&\int_0^{2\pi} \left( \frac{\partial \Phi_M(\phi)}{\partial \phi} \right)^2 d\phi = M^2 \frac{2\pi}{\epsilon_M}
\end{align}
両者ともTEモードのとき，$J_M'(\chi_n') =0$ より，
\begin{eqnarray}
\oint_\sigma (\nabla_t \Psi_n^{\TE}) \cdot (\nabla_t \Psi_n^{\TE}) d\sigma
&=& (A_n^{\TE})^2 \frac{2\pi}{\epsilon_M} \frac{1}{a} \cdot M^2 J_M^2(\chi_n')
\nonumber \\
&=& \frac{\epsilon _M}{\pi \big( \chi^{\prime 2}_n -M^2 \big) } \frac{1}{J_M^2 (\chi '_n)}
\frac{2\pi}{\epsilon_M} \frac{1}{a} \cdot M^2 J_M^2(\chi_n')
\nonumber \\
&=& \frac{1}{a} \frac{2M^2}{\chi^{\prime 2}_n -M^2}
\end{eqnarray}
よって，
\begin{eqnarray}
T_{I,nn}^{\TETE}
&=& \iint_S \frac{\partial \VEC{e}_n^{\TE}}{\partial z} \cdot \VEC{e}_n^{\TE} dS
\nonumber \\
&=& - \frac{1}{2} \frac{da}{dz} \oint_\sigma (\nabla_t \Psi_n^{\TE}) \cdot (\nabla_t \Psi_n^{\TE}) \ d\sigma
\nonumber \\
&=& - \frac{1}{a} \frac{da}{dz} \frac{M^2}{\chi^{\prime 2}_n -M^2}
\end{eqnarray}
また，両者ともTMモードのとき，$J_M(\chi_n) =0$より，
\begin{eqnarray}
\oint_\sigma (\nabla_t \Psi_n^{\TM}) \cdot (\nabla_t \Psi_n^{\TM}) d\sigma
&=& (A_n^{\TM})^2 \frac{2\pi}{\epsilon_M} \frac{1}{a} \cdot \chi_n^2 J_M^{\prime 2}(\chi_n)
\nonumber \\
&=& \frac{\epsilon_M}{\pi} \frac{1}{\chi_n^2 J_M^{\prime 2}(\chi_n)}
\frac{2\pi}{\epsilon_M} \frac{1}{a} \cdot \chi_n^2 J_M^{\prime 2}(\chi_n)
\nonumber \\
&=& \frac{2}{a}
\end{eqnarray}
よって，
\begin{eqnarray}
T_{I,nn}^{\TMTM}
&=& \iint_S \frac{\partial \VEC{e}_n^{\TM}}{\partial z} \cdot \VEC{e}_n^{\TM} dS
\nonumber \\
&=& - \frac{1}{2} \frac{da}{dz} \oint_\sigma (\nabla_t \Psi_n^{\TM}) \cdot (\nabla_t \Psi_n^{\TM}) \ d\sigma
\nonumber \\
&=& - \frac{1}{a} \frac{da}{dz}
\end{eqnarray}
比較のため，面積分表示の式でも求め，同様の結果が得られることを確認する．まず，
$a=a(z)$として，
\begin{gather}
\frac{\partial k_{c,n}}{\partial z}
= \frac{\partial a}{\partial z} \frac{\partial k_{c,n}}{\partial a}
= \frac{\partial a}{\partial z} \frac{\partial}{\partial a} \left( \frac{\chi_n}{a} \right)
= -\frac{da}{dz} \frac{\chi_n}{a^2}
= -\frac{1}{a} \frac{da}{dz} k_{c,n}
\end{gather}
\begin{gather}
2 k_{c,n} \frac{\partial k_{c,n}}{\partial z} \iint _S \Psi_l \Psi_n dS
= 2 k_{c,n} \left( -\frac{1}{a} \frac{da}{dz} k_{c,n} \right) \frac{\delta_{ln}}{k_{c,n}^2}
= -\frac{2}{a} \frac{da}{dz} \delta_{ln}
\end{gather}
$n=l$ のとき，
\begin{eqnarray}
T_{I,nn}^{\TETE}
&=& (k_{c,n}^{\TE})^2 \iint _S \frac{\partial \Psi_n^{\TE}}{\partial z} \Psi_n^{\TE} dS
\\
T_{I,nn}^{\TMTM}
&=& (k_{c,n}^{\TM})^2 \iint _S \frac{\partial \Psi_n^{\TM}}{\partial z} \Psi_n^{\TM} dS
- \frac{2}{a} \frac{da}{dz}
\end{eqnarray}
ここで，式\eqref{eq:Psin}，式\eqref{eq:dPsindz}を再記して，
\begin{align}
&\Psi_n = A_n J_M(k_{c,n} \rho) \Phi_M (\phi)
\\
&\frac{\partial \Psi_n}{\partial z}
= A_n \left( - \frac{1}{a} \frac{da}{dz} k_{c,n} \right) \rho J_M'(k_{c,n}\rho) \Phi_M(\phi)
\end{align}
面積分について，両者をまとめると，
\begin{gather}
\iint _S \frac{\partial \Psi_n}{\partial z} \Psi_n dS
= -\frac{1}{a} \frac{da}{dz} A_n^2 k_{c,n}
\int_0^a J_M'(k_{c,n}\rho) J_M(k_{c,n}\rho) \rho^2 d\rho \cdot
\frac{2\pi}{\epsilon_M}
\end{gather}
ベッセル関数の積分については，まず，
\begin{gather}
\frac{d}{d\rho} \left\{ \rho^2 J_M^2(k_{c,n} \rho) \right\}
= 2\rho J_M^2(k_{c,n} \rho) + \rho^2 \cdot 2 J_M(k_{c,n} \rho) J_M'(k_{c,n} \rho) k_{c,n}
\end{gather}
不定積分して，
\begin{eqnarray}
&&\int \frac{d}{d\rho} \left\{ \rho^2 J_M^2(k_{c,n} \rho) \right\} d\rho
\nonumber \\
&=& \rho^2 J_M^2(k_{c,n} \rho)
\nonumber \\
&=& 2 \int \rho J_M^2(k_{c,n} \rho) d\rho
+ 2 k_{c,n} \int \rho^2 J_M(k_{c,n} \rho) J_M'(k_{c,n} \rho) d\rho
\end{eqnarray}
よって，
\begin{gather}
k_{c,n} \int \rho^2 J_M(k_{c,n} \rho) J_M'(k_{c,n} \rho) d\rho
= \frac{1}{2} \rho^2 J_M^2(k_{c,n} \rho)
- \int \rho J_M^2(k_{c,n} \rho) d\rho
\end{gather}
右辺の第2項は不定積分公式
\begin{gather}
\int \rho J_M^2(k_{c,n} \rho) d\rho
= \frac{1}{2} \left\{ \rho^2 J_M^{\prime 2} (k_{c,n} \rho)
+ \left( \rho^2 - \frac{M^2}{k_{c,n}^2} \right) J_M^2 (k_{c,n} \rho) \right\}
\end{gather}
より，
\begin{align}
&k_{c,n} \int \rho^2 J_M(k_{c,n} \rho) J_M'(k_{c,n} \rho) d\rho
\nonumber \\
&= -\frac{1}{2} \left\{ \rho^2 J_M^{\prime 2} (k_{c,n} \rho)
- \frac{M^2}{k_{c,n}^2} J_M^2 (k_{c,n} \rho) \right\}
\end{align}
よって，新たに次の不定積分公式が得られる．
\begin{align}
&\int J_M(k_{c,n} \rho) J_M'(k_{c,n} \rho) \rho^2 d\rho
\nonumber \\
&= \frac{1}{2 k_{c,n}^3} \Big\{ -(k_{c,n} \rho)^2 J_M^{\prime 2} (k_{c,n} \rho)
+ M^2 J_M^2 (k_{c,n} \rho) \Big\}
\end{align}
半径$a$の円形導波管に対して定積分して，
\begin{align}
&\int_0^a J_M(k_{c,n} \rho) J_M'(k_{c,n} \rho) \rho^2 d\rho
\nonumber \\
&= \frac{1}{2 k_{c,n}^3} \left[ -\chi_n^2 J_M^{\prime 2} (\chi_n)
+ M^2 \Big\{ J_M^2 (\chi_n) - J_M^2 (0) \Big\} \right]
\end{align}
両者ともTE$_{Mn}$モードのとき（$M \ne 0$），
\begin{gather}
\int J_M(k_{c,n}^{\TE} \rho) J_M'(k_{c,n}^{\TE} \rho) \rho^2 d\rho
= \frac{M^2}{2 (k_{c,n}^{\TE})^3} J_M^2 (\chi_n')
\end{gather}
また，両者ともTM$_{Mn}$モードのとき，
\begin{eqnarray}
\int J_M(k_{c,n}^{\TM} \rho) J_M'(k_{c,n}^{\TM} \rho) \rho^2 d\rho
&=& -\frac{\chi_n^2}{2 (k_{c,n}^{\TM})^3} J_M^{\prime 2} (\chi_n)
\nonumber \\
&=& -\frac{a^2}{2 k_{c,n}^{\TM}} J_M^{\prime 2} (\chi_n)
\end{eqnarray}
よって，
\begin{eqnarray}
T_{I,nn}^{\TETE}
&=& (k_{c,n}^{\TE})^2 \iint _S \frac{\partial \Psi_n^{\TE}}{\partial z} \Psi_n^{\TE} dS
\nonumber \\
&=& (k_{c,n}^{\TE})^2 \left( -\frac{1}{a} \frac{da}{dz} (A_n^{\TE})^2 k_{c,n}^{\TE} \right)
\frac{M^2}{2 (k_{c,n}^{\TE})^3} J_M^2 (\chi_n')
\frac{2\pi}{\epsilon_M}
\nonumber \\
&=& -(A_n^{\TE})^2 \frac{1}{a} \frac{da}{dz} M^2 J_M^2 (\chi_n') \frac{\pi}{\epsilon_M}
\nonumber \\
&=& -\frac{\epsilon _M}{\pi \big( \chi^{\prime 2}_n -M^2 \big) } \frac{1}{J_M^2 (\chi '_n)}
\frac{1}{a} \frac{da}{dz} M^2 J_M^2 (\chi_n') \frac{\pi}{\epsilon_M}
\nonumber \\
&=& - \frac{1}{a} \frac{da}{dz} \frac{M^2}{\chi^{\prime 2}_n -M^2}
\end{eqnarray}
線積分表示の式より求めた結果と一致する．
また，
\begin{eqnarray}
T_{I,nn}^{\TMTM}
&=& (k_{c,n}^{\TM})^2 \iint _S \frac{\partial \Psi_n^{\TM}}{\partial z} \Psi_n^{\TM} dS
- \frac{2}{a} \frac{da}{dz}
\nonumber \\
&=& (k_{c,n}^{\TM})^2 \left( -\frac{1}{a} \frac{da}{dz} (A_n^{\TM})^2 k_{c,n}^{\TM} \right)
\left( -\frac{\chi_n^2}{2 (k_{c,n}^{\TM})^3} J_M^{\prime 2} (\chi_n) \right)
\frac{2\pi}{\epsilon_M}
\nonumber \\
&&- \frac{2}{a} \frac{da}{dz}
\nonumber \\
&=& (A_n^{\TM})^2 \frac{1}{a} \frac{da}{dz} \chi_n^2 J_M^{\prime 2} (\chi_n) \frac{\pi}{\epsilon_M}
- \frac{2}{a} \frac{da}{dz}
\nonumber \\
&=& \frac{\epsilon_M}{\pi} \frac{1}{\chi_n^2 J_M^{\prime 2}(\chi_n)}
\cdot \frac{1}{a} \frac{da}{dz} \chi_n^2 J_M^2 (\chi_n) \frac{\pi}{\epsilon_M}
- \frac{2}{a} \frac{da}{dz}
\nonumber \\
&=& \frac{1}{a} \frac{da}{dz} - \frac{2}{a} \frac{da}{dz}
\nonumber \\
&=& -\frac{1}{a} \frac{da}{dz}
\end{eqnarray}
これについても，線積分表示の式より求めた結果と一致する．
伝送方程式の係数（TE-TM, TM-TE）
TMモードの電界モード関数の微分とTEモードの場合，先に示したように
$\VEC{e}_l^{\TM} \to \frac{\partial \VEC{e}_l^{\TM}}{\partial z}$とすると，
$\Psi_l^{\TM} \to \frac{\partial \Psi_l^{\TM}}{\partial z}$より
（$ J_M'(\chi_n') =0$，$J_M(\chi_l) =0$），
\begin{eqnarray}
%&&\iint_S \VEC{e}_n^{\TE} \cdot \frac{\partial \VEC{e}_l^{\TM}}{\partial z} \ dS
%\nonumber \\
%&=& -\iint_S \left( \nabla_t \Psi_n^{\TE} \times \nabla_t \frac{\partial \Psi_l^{\TM}}{\partial z} \right) \cdot \VEC{a}_z dS
%\nonumber \\
%&=& -\oint_C \Psi_n^{\TE} \frac{\partial}{\partial \sigma}
%\left( \frac{\partial \Psi_l^{\TM}}{\partial z} \right) d\sigma
%\nonumber \\
\oint_C \frac{\partial \Psi_n^{\TE}}{\partial \sigma}
\frac{\partial \Psi_l^{\TM}}{\partial z} d\sigma
= A_n^{\TE} J_M(\chi_n') A_l^{\TM} \frac{da}{dz} k_{c,l} J_M'(\chi_l)
\int_0^{2\pi} \frac{d \Phi_M^{\TE}(\phi)}{ad\phi} \Phi_M^{\TM}(\phi) a d\phi
\end{eqnarray}
ここで，
\begin{eqnarray}
\int_0^{2\pi} \frac{d \Phi_M^{\TE}(\phi)}{d\phi} \Phi_M^{\TM}(\phi) d\phi
&=& \int_0^{2\pi} M \begin{matrix} \cos (M \phi) \\ -\sin (M \phi) \end{matrix}
\begin{matrix} (-\cos (M \phi)) \\ \sin (M \phi) \end{matrix}
d\phi
\nonumber \\
&=& -\frac{2M \pi}{\epsilon_M}
\end{eqnarray}
TE$_{Mn}$モードの正規化係数$A_n^{\TE}$，
TM$_{Ml}$モードの正規化係数$A_l^{\TM}$
\begin{eqnarray}
A_n^{\TE}
&=& \sqrt{\frac{\epsilon _M}{\pi \big( \chi^{\prime 2}_n -M^2 \big) }} \ \frac{1}{J_M (\chi '_n)}
\\
A_l^{\TM}
&=& \sqrt{\frac{\epsilon_M}{\pi}} \frac{1}{\chi_l J_M'(\chi_l)}
\end{eqnarray}
より，
\begin{gather}
A_n^{\TE} A_l^{\TM}
= \frac{\epsilon_M}{\pi} \frac{1}{\sqrt{ \chi^{\prime 2}_n -M^2}}
\frac{1}{J_M (\chi '_n)} \frac{1}{\chi_l J_M'(\chi_l)}
\end{gather}
これより，
\begin{eqnarray}
%\iint_S \VEC{e}_n^{\TE} \cdot \frac{\partial \VEC{e}_l^{\TM}}{\partial z} \ dS
-T_{V,ln}^{\TMTE}
= T_{I,nl}^{\TETM}
&=& \frac{\epsilon_M}{\pi} \frac{1}{\sqrt{ \chi^{\prime 2}_n -M^2}}
\frac{1}{a} \frac{da}{dz} \left( -\frac{2M\pi}{\epsilon_M} \right)
\nonumber \\
&=& -\frac{1}{a} \frac{da}{dz} \frac{2M}{\sqrt{ \chi^{\prime 2}_n -M^2}}
\end{eqnarray}
伝送方程式
以上をまとめると，
\begin{align}
&T_{I,nl}^{\TETM} = -T_{V,ln}^{\TMTE}
= -\frac{1}{a} \frac{da}{dz} \frac{2M}{\sqrt{ \chi^{\prime 2}_n -M^2}}
\\
&T_{I,nn}^{\TETE} = -T_{V,nn}^{\TETE}
= - \frac{1}{a} \frac{da}{dz} \frac{M^2}{\chi^{\prime 2}_n -M^2}
\\
&T_{I,nn}^{\TMTM} = -T_{V,nn}^{\TMTM}
= - \frac{1}{a} \frac{da}{dz}
\\
&T_{I,ln}^{\TETE} = -T_{V,nl}^{\TETE}
= -\frac{1}{a} \frac{da}{dz} \frac{2\chi_l^{\prime 2}}{\chi_n^{\prime 2} - \chi_l^{\prime 2}}
\sqrt{\frac{\chi_n^{\prime 2}-M^2}{\chi_l^{\prime 2}-M^2}} \ \ \ (n \ne l)
\\
&T_{I,ln}^{\TMTM} = -T_{V,nl}^{\TMTM}
= -\frac{1}{a} \frac{da}{dz} \frac{2\chi_n^2}{\chi_n^2 - \chi_l^2} \ \ \ (n \ne l)
\end{align}
$n$ と $l$ を入れ換えて，
\begin{align}
&T_{I,ln}^{\TETM} = -T_{V,nl}^{\TMTE}
= -\frac{1}{a} \frac{da}{dz} \frac{2M}{\sqrt{ \chi^{\prime 2}_l -M^2}}
\\
&T_{I,nl}^{\TETE} = -T_{V,ln}^{\TETE}
= -\frac{1}{a} \frac{da}{dz} \frac{2\chi_n^{\prime 2}}{\chi_l^{\prime 2} - \chi_n^{\prime 2}}
\sqrt{\frac{\chi_l^{\prime 2}-M^2}{\chi_n^{\prime 2}-M^2}} \ \ \ (l \ne n)
\\
&T_{I,nl}^{\TMTM} = -T_{V,ln}^{\TMTM}
= -\frac{1}{a} \frac{da}{dz} \frac{2\chi_l^2}{\chi_l^2 - \chi_n^2} \ \ \ (l \ne n)
\end{align}
展開モード数をTE，TMについて$N_{\TE}$，$N_{\TM}$とすると，
\begin{align}
&\frac{dV^{\TM}_l}{dz}
= - Z^{\TM}_l I^{\TM}_l
- \sum _n^{N_{\TM}} T^{\TMTM}_{V,ln} V^{\TM}_n
- \sum _n^{N_{\TE}} T^{\TMTE}_{V,ln} V^{\TE}_n
\nonumber \\
&(l=1,2, \cdots, N_{\TM})
\\
&\frac{dI^{\TM}_l}{dz}
= - Y V^{\TM}_l
+ \sum _n^{N_{\TM}} T^{\TMTM}_{V,nl} I^{\TM}_n \ \ \ (l=1,2, \cdots, N_{\TM})
\\
&\frac{dV^{\TE}_l}{dz}
= - Z I^{\TE}_l
- \sum _n^{N_{\TE}} T^{\TETE}_{V,ln} V^{\TE}_n \ \ \ (l=1,2, \cdots, N_{\TE})
\\
&\frac{dI^{\TE}_l}{dz}
= - Y^{\TE}_l V_l^{\TE}
+ \sum _n^{N_{\TE}} T^{\TETE}_{V,nl} I^{\TE}_n
+ T_{V,-l}^{\TMTE} \sum _n^{N_{\TM}} I^{\TM}_n
\nonumber \\
&(l=1,2, \cdots, N_{\TE})
\end{align}
ここで，
\begin{align}
&T_{V,ln}^{\TMTM}
= \frac{1}{a} \frac{da}{dz} \frac{2\chi_l^2}{\chi_l^2 - \chi_n^2} \ \ \ (l \ne n)
\\
&T_{V,ll}^{\TMTM} = \frac{1}{a} \frac{da}{dz}
\\
&T_{V,ln}^{\TMTE} = \frac{1}{a} \frac{da}{dz} \frac{2M}{\sqrt{ \chi^{\prime 2}_n -M^2}}
\\
&T_{V,nl}^{\TMTM}
= \frac{1}{a} \frac{da}{dz} \frac{2\chi_n^2}{\chi_n^2 - \chi_l^2} \ \ \ (n \ne l)
\\
&T_{V,ln}^{\TETE}
= \frac{1}{a} \frac{da}{dz} \frac{2\chi_n^{\prime 2}}{\chi_l^{\prime 2} - \chi_n^{\prime 2}}
\sqrt{\frac{\chi_l^{\prime 2}-M^2}{\chi_n^{\prime 2}-M^2}} \ \ \ (l \ne n)
\\
&T_{V,ll}^{\TETE}
= \frac{1}{a} \frac{da}{dz} \frac{M^2}{\chi^{\prime 2}_l -M^2}
\\
&T_{V,nl}^{\TETE}
= \frac{1}{a} \frac{da}{dz} \frac{2\chi_l^{\prime 2}}{\chi_n^{\prime 2} - \chi_l^{\prime 2}}
\sqrt{\frac{\chi_n^{\prime 2}-M^2}{\chi_l^{\prime 2}-M^2}} \ \ \ (n \ne l)
\\
&T_{V,nl}^{\TMTE}
= \frac{1}{a} \frac{da}{dz} \frac{2M}{\sqrt{ \chi^{\prime 2}_l -M^2}}
\equiv T_{V,-l}^{\TMTE}
\end{align}