3.2 軸対称導波管のモード関数の内積（TE-TE，TM-TM）
2つのモードともTEモード，あるいはTMモードの場合，モード関数の内積は，次のようにスカラー関数によって求めることができる．
\begin{gather}
\iint_S \VEC{e} \cdot \widehat{\VEC{e}} \ dS
= \iint_S \VEC{h} \cdot \widehat{\VEC{h}} \ dS
= \iint_S \big( \nabla_t \Psi \big) \cdot \big( \nabla_t \widehat{\Psi} \big) dS
\end{gather}
グリーンの第一定理
\begin{gather}
\iint _S \left( \Phi \nabla _t^2 \Psi + \nabla _t \Phi \cdot \nabla _t \Psi \right) dS
= \oint _C \Phi \frac{\partial \Psi}{\partial n} d\sigma
\end{gather}
および
\begin{gather}
\nabla _t^2 \Psi_{mn} + k_{c,mn}^2 \Psi_{mn} = 0, \ \ \ \ \
\nabla _t^2 \widehat{\Psi}_{m'n'} + \widehat{k}_{c,m'n'}^2 \widehat{\Psi}_{m'n'} = 0
\end{gather}
より，
\begin{eqnarray}
I_{mn,\widehat{m'n'}}
&\equiv& \iint_S \big( \nabla_t \Psi_{mn} \big) \cdot \big( \nabla_t \widehat{\Psi}_{m'n'} \big) dS
\nonumber \\
&=& \widehat{k}_{c,m'n'}^2 \iint_S \Psi_{mn} \widehat{\Psi}_{m'n'} dS
+ \oint _C \Psi_{mn} \frac{\partial \widehat{\Psi}_{m'n'}}{\partial n} d\sigma \label{eq:Iihj1}
\end{eqnarray}
ただし，$\partial/\partial n$は積分範囲$S$からの外向き法線方向の微分，
$d\sigma $は$S$の周囲にとった周回積分路$C$に沿う線要素を示す．逆に，$\Psi_{mn}$と$\widehat{\Psi}_{m'n'}$を交換して，
\begin{gather}
I_{mn,\widehat{m'n'}}
= k_{c,mn}^2 \iint_S \widehat{\Psi}_{m'n'} \Psi_{mn} dS
+ \oint _C \widehat{\Psi}_{m'n'} \frac{\partial \Psi_{mn}}{\partial n} d\sigma
\end{gather}
グリーンの第一定理から得られた2つの式を用いて面積分の項を消去すると，
\begin{align}
&\big( k_{c,mn}^2 - \widehat{k}_{c,m'n'}^2 \big) I_{mn,\widehat{m'n'}}
\nonumber \\
&= k_{c,mn}^2 \oint _C \Psi_{mn} \frac{\partial \widehat{\Psi}_{m'n'}}{\partial n} d\sigma
- \widehat{k}_{c,m'n'}^2 \oint _C \widehat{\Psi}_{m'n'} \frac{\partial \Psi_{mn}}{\partial n} d\sigma
\end{align}
これより，$k_{c,mn} \neq \widehat{k}_{c,mn'}$のとき，
\begin{align}
&I_{mn,\widehat{m'n'}}
\nonumber \\
&= \frac{1}{k_{c,mn}^2 - \widehat{k}_{c,m'n'}^2}
\left( k_{c,mn}^2 \oint _C \Psi_{mn} \frac{\partial \widehat{\Psi}_{m'n'}}{\partial n} d\sigma
- \widehat{k}_{c,m'n'}^2 \oint _C \widehat{\Psi}_{m'n'} \frac{\partial \Psi_{mn}}{\partial n} d\sigma
\right)
\nonumber
\end{align}
積分範囲（周回積分路C）が$z$軸（導波路断面に直交）に関して回転対称な場合，断面上の法線方向成分$n$は$\rho$方向であり，その微分は次のようになる．
\begin{eqnarray}
\frac{\partial \Psi_{mn}}{\partial n}
&=& \frac{\partial}{\partial \rho} \Big\{ A_{mn} f_{mn} (k_{c,mn} \rho) \Phi(m\phi) \Big\}
\nonumber \\
&=& A_{mn} \frac{df_{mn}}{d\rho} \Phi(m\phi)
\nonumber \\
&=& A_{mn} k_{c,mn}
\Big\{ B_{mn}^N J'_m \big( k_{c,mn} \rho \big)
- B_{mn}^J N'_m \big( k_{c,mn} \rho \big) \Big\} \Phi(m\phi)
\end{eqnarray}
ここで，
\begin{align}
&f'_{mn} \equiv B_{mn}^N J'_m \big( k_{c,mn} \rho \big)
- B_{mn}^J N'_m \big( k_{c,mn} \rho \big)
\\
&g'_{m'n'} \equiv \widehat{B}_{m'n'}^N J'_{m'} \big( \widehat{k}_{c,m'n'} \rho \big)
- \widehat{B}_{m'n'}^J N'_{m'} \big( \widehat{k}_{c,m'n'} \rho \big)
\end{align}
これより，
\begin{align}
&\frac{\partial \Psi_{mn}}{\partial n}
= A_{mn} k_{c,mn} f'_{mn} \Phi(m\phi)
\\
&\frac{\partial \widehat{\Psi}_{m'n'}}{\partial n}
= \widehat{A}_{m'n'} \widehat{k}_{c,m'n'} g'_{m'n'} \Phi(m' \phi)
\end{align}
ただし，
\begin{gather}
J_m'(\xi)= \frac{dJ_m(\xi)}{d\xi}, \ \ \ \ \
N_m'(\xi)= \frac{dN_m(\xi)}{d\xi}
\end{gather}
よって，周回積分は，
\begin{eqnarray}
&&\oint _C \Psi_{mn} \frac{\partial \widehat{\Psi}_{m'n'}}{\partial n} d\sigma
\nonumber \\
&=& A_{mn} \widehat{A}_{m'n'} \oint _C f_{mn} (k_{c,mn} \rho) \Phi (m\phi)
\frac{dg_{m'n'} (\widehat{k}_{c,m'n'} \rho)}{d\rho} \Phi(m' \phi) \rho d\phi
\nonumber \\
&=& A_{mn} \widehat{A}_{m'n'}
\left[ \rho f_{mn} (k_{c,mn} \rho) \frac{dg_{m'n'}(\widehat{k}_{c,m'n'} \rho)}{d\rho}
\right]_{\rho_1}^{\rho_2}
\int_0^{2\pi} \Phi_m (\phi) \Phi_{m'} (\phi) d\phi
\end{eqnarray}
このとき，両者ともTEモード，および両者ともTMモードの場合，各々，
\begin{eqnarray}
\int_0^{2\pi} \Phi_m^{\TE} (\phi) \Phi_{m'}^{\TE} (\phi) d\phi
&=& \int_0^{2\pi} \begin{matrix} \sin (m \phi) \sin (m' \phi) \\
\cos (m \phi) \cos (m' \phi) \end{matrix} d\phi
\nonumber \\
&=& \delta _{mm'} \frac{2\pi}{\epsilon _m}
\\
\int_0^{2\pi} \Phi_m^{\TM} (\phi) \Phi_{m'}^{\TM} (\phi) d\phi
&=& \int_0^{2\pi} \begin{matrix} (-\cos (m \phi)) (-\cos (m' \phi)) \\
\sin (m \phi) \sin (m' \phi)\end{matrix} d\phi
\nonumber \\
&=& \delta _{mm'} \frac{2\pi}{\epsilon _m}
\end{eqnarray}
ここで，
\begin{gather}
\epsilon _m = \left\{
\begin {array}{ll}
1 & (m=0) \\
2 & (m=1,2, \cdots )
\end{array} \right.
\label{eq:epsilon_m}
\end{gather}
ただし，$m=0$ のときの$\sin m \phi$は除く．
TE，TMをまとめると，
\begin{gather}
\int_0^{2\pi} \Phi_m (\phi) \Phi_{m'} (\phi) d\phi = \delta _{mm'} \frac{2\pi}{\epsilon _m}
\end{gather}
よって，
\begin{gather}
\oint _C \Psi_{mn} \frac{\partial \widehat{\Psi}_{m'n'}}{\partial n} d\sigma
= A_{mn} \widehat{A}_{mn'}
\widehat{k}_{c,mn'} \Big[ \rho f_{mn} g_{mn'}' \Big]_{\rho_1}^{\rho_2}
\delta _{mm'} \frac{2\pi}{\epsilon _m}
\end{gather}
同様にして，
\begin{gather}
\oint _C \widehat{\Psi}_{m'n'} \frac{\partial \Psi_{mn}}{\partial n} d\sigma
= A_{mn} \widehat{A}_{mn'}
k_{c,mn} \Big[ \rho g_{mn'} f_{mn}' \Big]_{\rho_1}^{\rho_2}
\delta _{mm'} \frac{2\pi}{\epsilon _m}
\end{gather}
これより，$k_{c,mn} \neq \widehat{k}_{c,mn'}$のとき，
\begin{align}
&I_{mn,\widehat{m'n'}}
\nonumber \\
&= A_{mn} \widehat{A}_{mn'}
\frac{k_{c,mn} \widehat{k}_{c,mn'} }{k_{c,mn}^2 - \widehat{k}_{c,mn'}^2}
\Big[ k_{c,mn} \rho f_{mn} g_{mn'}' - \widehat{k}_{c,mn'} \rho f'_{mn} g_{mn'}
\Big]_{\rho_1}^{\rho_2}
\delta _{mm'} \frac{2\pi}{\epsilon _m}
\nonumber
\end{align}
両者とも，同一の導波路のモード関数であれば，$k_{c,mn} \neq k_{c,mn'}$のとき，
\begin{align}
&I_{mn,m'n'}
\nonumber \\
&= A_{mn} A_{mn'}
\frac{k_{c,mn} k_{c,mn'} }{k_{c,mn}^2 - k_{c,mn'}^2}
\Big[ k_{c,mn} \rho f_{mn} f_{mn'}' - k_{c,mn'} \rho f'_{mn} f_{mn'}
\Big]_{\rho_1}^{\rho_2} \delta _{mm'} \frac{2\pi}{\epsilon _m}
\nonumber
\end{align}
積分範囲を導波路断面と一致させると，
$C$上で$\Psi_{mn}, \Psi_{mn'}=0$，あるいは
$\displaystyle{\frac{\partial \Psi_{mn}}{\partial n},
\frac{\partial \Psi_{mn'}}{\partial n}=0}$ のとき，上式はゼロ，つまり
$k_{c,mn} \neq k_{c,mn'}$
のとき，$I_{mn,m'n'}=0$
が成り立ち，異なるモードが直交していることもわかる．
逆に，$I_{mn,\widehat{m'n'}}$を消去すると，
\begin{align}
&\big( k_{c,mn}^2 - \widehat{k}_{c,m'n'}^2 \big) \iint_S \Psi_{mn} \widehat{\Psi}_{m'n'} dS
\nonumber \\
&= \oint _C \left( \Psi_{mn} \frac{\partial \widehat{\Psi}_{m'n'}}{\partial n}
- \widehat{\Psi}_{m'n'} \frac{\partial \Psi_{mn}}{\partial n} \right) d\sigma
\end{align}
$k_{c,mn} \neq k_{c,m'n'}$のとき，
\begin{align}
&\iint_S \Psi_{mn} \widehat{\Psi}_{m'n'} dS
\nonumber \\
&= \frac{1}{k_{c,mn}^2 - \widehat{k}_{c,m'n'}^2}
\oint _C \left( \Psi_{mn} \frac{\partial \widehat{\Psi}_{m'n'}}{\partial n}
- \widehat{\Psi}_{m'n'} \frac{\partial \Psi_{mn}}{\partial n} \right) d\sigma
\end{align}
上式右辺は周回積分より，
\begin{gather}
A_{mn} \widehat{A}_{mn'}
\frac{1}{k_{c,mn}^2 - \widehat{k}_{c,mn'}^2}
\Big[ \widehat{k}_{c,mn'} \rho f_{mn} g_{mn'}' - k_{c,mn} \rho f'_{mn} g_{mn'}
\Big]_{\rho_1}^{\rho_2}
\delta _{mm'} \frac{2\pi}{\epsilon _m}
\end{gather}
一方，左辺は面積分より，
\begin{eqnarray}
&&\iint_{S_C} \Psi_{mn} \widehat{\Psi}_{m'n'} dS
\nonumber \\
&=& A_{mn} \widehat{A}_{m'n'}
\iint_{S_C} f_{mn} (k_{c,mn} \rho) \Phi_m(\phi)
g_{m'n'} (\widehat{k}_{c,m'n'} \rho) \Phi_{m'} (\phi) dS
\nonumber \\
&=& A_{mn} \widehat{A}_{m'n'} \int_{\rho_1}^{\rho_2} f_{mn} g_{m'n'} \rho d\rho
\int_0^{2\pi} \Phi_m \Phi_{m'} d\phi
\nonumber \\
&=& A_{mn} \widehat{A}_{mn'} \delta _{mm'} \frac{2\pi}{\epsilon _m}
\int_{\rho_1}^{\rho_2} f_{mn} g_{mn'} \rho d\rho
\end{eqnarray}
半径方向$\rho$の積分は，ベッセル関数の不定積分公式
\begin{gather}
\int zuv \ dz = \frac{z ( uv'-u'v ) }{\alpha^2- \beta^2}
\end{gather}
より，$k_{c,mn} \neq k_{c,mn'}$のとき，
\begin{align}
&\int_{\rho_1}^{\rho_2} f_{mn} g_{mn'} \rho d\rho
\nonumber \\
&= \left[ \frac{\rho}{k_{c,mn}^2 - \widehat{k}_{c,mn'}^2}
\left\{ \widehat{k}_{c,mn'} f_{mn} g_{mn'}'
- k_{c,mn} f_{mn}' g_{mn'} \right\}
\right]_{\rho_1}^{\rho_2}
\end{align}
したがって，次のように周回積分の結果と一致することが確認できる．
\begin{align}
&\iint_{S_C} \Psi_{mn} \widehat{\Psi}_{m'n'} dS
\nonumber \\
&= \delta _{mm'} \frac{2\pi}{\epsilon _m}
\frac{A_{mn} \widehat{A}_{mn'}}{k_{c,mn}^2 - \widehat{k}_{c,mn'}^2}
\Big[ \widehat{k}_{c,mn'} \rho f_{mn} g_{mn'}' - k_{c,mn} \rho f'_{mn} g_{mn'}
\Big]_{\rho_1}^{\rho_2}
\end{align}
次に，$k_{c,mn} = k_{c,mn'}(\neq 0)$のとき，ベッセル関数の不定積分公式
\begin{gather}
\int zu_1 u_2 dz
= \frac{1}{2} \left\{ \frac{z^2}{\alpha^2} u_1' u_2'
+ \left( z^2 - \frac{\nu ^2}{\alpha ^2} \right) u_1 u_2 \right\}
\end{gather}
より，
\begin{gather}
\int_{\rho_1}^{\rho_2} f_{mn} g_{mn'} \rho d\rho
= \frac{1}{2} \left[ \rho^2 f_{mn}' g_{mn}'
+ \left( \rho^2 - \frac{m^2}{k_{c,mn}^2} \right) f_{mn} g_{mn'} \right]_{\rho_1}^{\rho_2}
\end{gather}
これより，
\begin{align}
&\iint_{S_C} \Psi_{mn} \widehat{\Psi}_{m'n'} dS
\nonumber \\
&= A_{mn} \widehat{A}_{mn'} \delta _{mm'} \frac{2\pi}{\epsilon _m}
\frac{1}{2} \left[ \rho^2 f_{mn}' g_{mn}'
+ \left( \rho^2 - \frac{m^2}{k_{c,mn}^2} \right) f_{mn} g_{mn'} \right]_{\rho_1}^{\rho_2}
\end{align}
$I_{mn,\widehat{m'n'}}$は，$k_{c,mn} = \widehat{k}_{c,m'n'}$のとき，
\begin{eqnarray}
I_{mn,\widehat{m'n'}}
&=& k_{c,mn}^2 \iint_S \Psi_{mn} \widehat{\Psi}_{m'n'} dS
+ \oint _C \frac{\partial \Psi_{mn}}{\partial n} \widehat{\Psi}_{m'n'} d\sigma
\nonumber \\
&=& k_{c,mn}^2 \iint_S \Psi_{mn} \widehat{\Psi}_{m'n'} dS
+ \oint _C \Psi_{mn} \frac{\partial \widehat{\Psi}_{m'n'}}{\partial n} d\sigma
\end{eqnarray}
ここで，
\begin{gather}
\oint _C \Psi_{mn} \frac{\partial \widehat{\Psi}_{m'n'}}{\partial n} d\sigma
= \oint _C \widehat{\Psi}_{m'n'} \frac{\partial \Psi_{mn}}{\partial n} d\sigma \ \ \ \ \
(k_{c,mn} = \widehat{k}_{c,m'n'})
\label{eq:ki_eq_kj}
\end{gather}
これより，
\begin{eqnarray}
I_{mn,\widehat{m'n'}}
&=& k_{c,mn}^2 \iint_S \Psi_{mn} \widehat{\Psi}_{m'n'} dS
+ \frac{1}{2} \oint _C \frac{\partial}{\partial n} \Big( \Psi_{mn} \widehat{\Psi}_{m'n'} \Big) d\sigma
\nonumber \\
&=& A_{mn} \widehat{A}_{m'n'} \delta _{mm'} \frac{\pi}{\epsilon _m}
\left[ k_{c,mn}^2 \rho^2 f_{mn}' g_{mn'}'
+ \Big( k_{c,mn}^2 \rho^2 - m^2 \right) f_{mn} g_{mn'}
\nonumber \\
&&+ k_{c,mn} \rho \Big( f_{mn} g_{mn'}' + f'_{mn} g_{mn'} \Big)
\Big]_{\rho_1}^{\rho_2}
\nonumber
\end{eqnarray}
ただし，
\begin{gather}
\Big[ \rho f_{mn} g_{mn'}' \Big]_{\rho_1}^{\rho_2}
= \Big[ \rho g_{mn'} f_{mn}' \Big]_{\rho_1}^{\rho_2}
\end{gather}
実際に計算すると，
\begin{eqnarray}
&&\Big[ \rho
\big\{ B_{mn}^N J_m (k_{c,mn} \rho ) - B_{mn}^J N_m (k_{c,mn} \rho ) \big\}
\nonumber \\
&& \cdot
\big\{ \widehat{B}_{mn'}^N J'_m (k_{c,mn'} \rho ) - \widehat{B}_{mn'}^J N'_m (k_{c,mn} \rho ) \big\}
\Big]_{\rho_1}^{\rho_2}
\nonumber \\
&&- \Big[ \rho
\big\{ \widehat{B}_{mn}^N J_m (k_{c,mn} \rho ) - \widehat{B}_{mn}^J N_m (k_{c,mn} \rho ) \big\}
\nonumber \\
&& \cdot
\big\{ B_{mn'}^N J'_m (k_{c,mn'} \rho ) - B_{mn'}^J N'_m (k_{c,mn} \rho ) \big\}
\Big]_{\rho_1}^{\rho_2}
\nonumber \\
&=& \Big[ \rho \big( -B_{mn}^N \widehat{B}_{mn'}^J J_m N'_m - B_{mn}^J \widehat{B}_{mn'}^N N_m J'_m
\big) \Big]_{\rho_1}^{\rho_2}
\nonumber \\
&&- \Big[ \rho \big( - \widehat{B}_{mn'}^N B_{mn}^J J_m N'_m -\widehat{B}_{mn'}^JB_{mn}^N N_m J'_m
\big) \Big]_{\rho_1}^{\rho_2}
\nonumber \\
&=& \big( B_{mn}^J \widehat{B}_{mn'}^N - B_{mn}^N \widehat{B}_{mn'}^J \big)
\nonumber \\
&&\cdot \Big[ \rho
\big\{ J_m(k_{c,mn} \rho) N'_m(k_{c,mn}) - J_m'(k_{c,mn}) N_m(k_{c,mn}) \big\}
\Big]_{\rho_1}^{\rho_2}
\nonumber \\
&=& \big( B_{mn}^J \widehat{B}_{mn'}^N - B_{mn}^N \widehat{B}_{mn'}^J \big)
\left[ \rho
\frac{2}{\pi k_{c,mn} \rho} \right]_{\rho_1}^{\rho_2}
\nonumber \\
&=& 0
\end{eqnarray}
両者とも，同一の導波路のモード関数であれば，
\begin{align}
&I_{mn,m'n'}
= A_{mn} A_{mn'} \delta _{mm'} \frac{\pi}{\epsilon _m}
\Big[ k_{c,mn}^2 \rho^2 f_{mn}' f_{mn'}'
\nonumber \\
&+ \Big( k_{c,mn}^2 \rho^2 - m^2 \Big) f_{mn} f_{mn'}
+ k_{c,mn} \rho \Big( f_{mn} f_{mn'}' + f'_{mn} f_{mn'} \Big)
\Big]_{\rho_1}^{\rho_2}
\end{align}
特に，同一モードの場合，
\begin{eqnarray}
I_{mn,mn}
&=& A_{mn}^2 \frac{\pi}{\epsilon _m}
\Big[ k_{c,mn}^2 \rho^2 f_{mn}^{\prime 2}
\nonumber \\
&&+ \Big( k_{c,mn}^2 \rho^2 - m^2 \Big) f_{mn}^2
+ 2k_{c,mn} \rho f_{mn} f_{mn}' \Big]_{\rho_1}^{\rho_2}
\end{eqnarray}