1.2 不連続部の散乱行列
散乱行列を次式で定義する．
\begin{gather}
\begin{pmatrix}
\VECi{b}_1 \\ \VECi{b}_2 \\
\end{pmatrix}
= \Big[ S \Big]
\begin{pmatrix}
\VECi{a}_1 \\ \VECi{a}_2 \\
\end{pmatrix}, \ \ \ \ \
\Big[ S \Big] =
\begin{pmatrix}
\big[ S_{11} \big] & \big[ S_{12} \big] \\
\big[ S_{21} \big] & \big[ S_{22} \big] \\
\end{pmatrix}
\end{gather}
ここで，$i=1,2$，$j=1,2$として
\begin{align}
&\big[ S_{i j} \big] =
\begin{pmatrix}
S_{ij,11} & S_{ij,12} & \cdots & S_{ij,1n} & \cdots \\
S_{ij,21} & S_{ij,22} & \cdots & S_{ij,2n} & \cdots \\
\vdots & \vdots & \ddots & \vdots & \\
S_{ij,m1} & S_{ij,m2} & \cdots & S_{ij,mn} & \cdots \\
\vdots & \vdots & & \vdots & \ddots \\
\end{pmatrix}
\\
&\VECi{b}_i =
\begin{pmatrix}
b^{(i)}_1 \\ b^{(i)}_2 \\ \vdots \\ b^{(i)}_m \\ \vdots
\end{pmatrix}, \ \ \ \ \
\VECi{a}_i =
\begin{pmatrix}
a^{(i)}_1 \\ a^{(i)}_2 \\ \vdots \\ a^{(i)}_m \\ \vdots
\end{pmatrix}
\end{align}
先に示した式
\begin{align}
&\VECi{a}_1 - \big[ \bar{P}_{21} \big]_T \VECi{a}_2
= -\VECi{b}_1 + \big[ \bar{P}_{21} \big]_T \VECi{b}_2
\\
&\big[ \bar{P}_{21} \big] \VECi{a}_1 + \big[ \bar{P}_{22} \big] \VECi{a}_2
= \big[ \bar{P}_{21} \big] \VECi{b}_1 + \big[ \bar{P}_{22} \big] \VECi{b}_2
\end{align}
を行列表示すると，
\begin{eqnarray}
\begin{pmatrix}
\big[ U \big] & -\big[ \bar{P}_{21} \big]_T \\
\big[ \bar{P}_{21} \big] & \big[ \bar{P}_{22} \big] \\
\end{pmatrix}
\begin{pmatrix}
\VECi{a}_1 \\ \VECi{a}_2 \\
\end{pmatrix}
&=&
\begin{pmatrix}
-\big[ U \big] & \big[ \bar{P}_{21} \big]_T \\
\big[ \bar{P}_{21} \big] & \big[ \bar{P}_{22} \big] \\
\end{pmatrix}
\begin{pmatrix}
\VECi{b}_1 \\ \VECi{b}_2 \\
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
-\big[ U \big] & \big[ \bar{P}_{21} \big]_T \\
\big[ \bar{P}_{21} \big] & \big[ \bar{P}_{22} \big] \\
\end{pmatrix}
\Big[ S \Big]
\begin{pmatrix}
\VECi{a}_1 \\ \VECi{a}_2 \\
\end{pmatrix}
\end{eqnarray}
よって，散乱行列$\Big[ S \Big]$は，
\begin{gather}
\Big[ S \Big] =
\begin{pmatrix}
-\big[ U \big] & \big[ \bar{P}_{21} \big]_T \\
\big[ \bar{P}_{21} \big] & \big[ \bar{P}_{22} \big] \\
\end{pmatrix}^{-1}
\begin{pmatrix}
\big[ U \big] & -\big[ \bar{P}_{21} \big]_T \\
\big[ \bar{P}_{21} \big] & \big[ \bar{P}_{22} \big] \\
\end{pmatrix}
\end{gather}
あるいは，次のように変形して計算すると，
\begin{eqnarray}
\VECi{b}_1
&=& \big[ \bar{P}_{21} \big]_T \big( \VECi{b}_2 +\VECi{a}_2 \big) - \VECi{a}_1
\nonumber \\
&=& \big[ \bar{P}_{21} \big]_T \Big\{ \big( \big[ S_{21} \big] \VECi{a}_1 + \big[ S_{22} \big]\VECi{a}_2 \big)
+\VECi{a}_2 \Big\} - \VECi{a}_1
\nonumber \\
&=& \Big\{ \big[ \bar{P}_{21} \big]_T \big[ S_{21} \big] - \big[ U \big] \Big\} \VECi{a}_1
+ \big[ \bar{P}_{21} \big]_T \Big\{ \big[ S_{22} \big] + \big[ U \big] \Big\} \VECi{a}_2
\nonumber \\
&\equiv& \big[ S_{11} \big] \VECi{a}_1 + \big[ S_{12} \big] \VECi{a}_2
\end{eqnarray}
また，
\begin{eqnarray}
\VECi{b}_2
&=& \big[ \bar{P}_{22} \big]^{-1}
\big[ \bar{P}_{21} \big] \big( \VECi{a}_1 -\VECi{b}_1 \big)+ \VECi{a}_2
\nonumber \\
&=& \big[ \bar{P}_{22} \big]^{-1} \big[ \bar{P}_{21} \big]
\Big\{ \VECi{a}_1 - \big( \big[ S_{11} \big] \VECi{a}_1 + \big[ S_{12} \big] \VECi{a}_2 \big) \Big\} + \VECi{a}_2
\nonumber \\
&=& \big[ \bar{P}_{22} \big]^{-1}
\big[ \bar{P}_{21} \big] \Big\{ \big[ U \big] - \big[ S_{22} \big] \Big\} \VECi{a}_1
+ \Big\{ \big[ U \big] - \big[ \bar{P}_{22} \big]^{-1}
\big[ \bar{P}_{21} \big] \big[ S_{12} \big] \Big\} \VECi{a}_2
\nonumber \\
&\equiv& \big[ S_{21} \big] \VECi{a}_1 + \big[ S_{22} \big] \VECi{a}_2
\end{eqnarray}
さらに，第1式を第2式に代入すると，
\begin{align}
&\big[ \bar{P}_{22} \big] \VECi{b}_2
= \big[ \bar{P}_{21} \big] \big( \VECi{a}_1 -
\big\{ \big[ \bar{P}_{21} \big]_T \big( \VECi{b}_2 +\VECi{a}_2 \big) - \VECi{a}_1 \big\}
\big) + \big[ \bar{P}_{22} \big] \VECi{a}_2
\nonumber \\
&\Big( \big[ \bar{P}_{22} \big]
+ \big[ \bar{P}_{21} \big] \big[ \bar{P}_{21} \big]_T \Big) \VECi{b}_2
= 2 \big[ \bar{P}_{21} \big] \VECi{a}_1
+ \Big( \big[ \bar{P}_{22} \big] - \big[ \bar{P}_{21} \big] \big[ \bar{P}_{21} \big]_T \Big)
\VECi{a}_2
\end{align}
よって，
\begin{eqnarray}
\VECi{b}_2
&=& \Big( \big[ \bar{P}_{22} \big]
+ \big[ \bar{P}_{21} \big] \big[ \bar{P}_{21} \big]_T \Big)^{-1}
\nonumber \\
&&\cdot \Big\{ 2 \big[ \bar{P}_{21} \big] \VECi{a}_1
+ \Big( \big[ \bar{P}_{22} \big] - \big[ \bar{P}_{21} \big] \big[ \bar{P}_{21} \big]_T \Big)
\VECi{a}_2 \Big\}
\nonumber \\
&\equiv& \big[ S_{21} \big] \VECi{a}_1 + \big[ S_{22} \big] \VECi{a}_2
\end{eqnarray}
したがって，
\begin{gather}
\big[ S_{21} \big]
= 2 \Big( \big[ \bar{P}_{22} \big] + \big[ \bar{P}_{21} \big] \big[ \bar{P}_{21} \big]_T \Big)^{-1} \big[ \bar{P}_{21} \big]
\end{gather}
これより，
\begin{gather}
\big[ S_{11} \big]
= \big[ \bar{P}_{21} \big]_T \big[ S_{21} \big] - \big[ U \big]
\end{gather}
逆に，第2式を第1式に代入して$\VECi{b}_2$を消去して整理すると次式が得られる，
\begin{gather}
\big[ S_{12} \big] = 2\Big( \big[ U \big] + \big[ \bar{P}_{21} \big]_T
\big[ \bar{P}_{22} \big]^{-1} \big[ \bar{P}_{21} \big] \Big)^{-1} \big[ \bar{P}_{21} \big]_T
\end{gather}
ただし，散乱行列の対称性より$\big[ S_{12} \big] $は次のように転置で求めることができる．
\begin{gather}
\big[ S_{12} \big] = \big[ S_{21} \big]_T
\end{gather}
さらに，
\begin{gather}
\big[ S_{22} \big]
= \big[ U \big] - \big[ \bar{P}_{22} \big]^{-1}
\big[ \bar{P}_{21} \big] \big[ S_{12} \big]
\end{gather}
積分範囲$S_0$が導波管 #2の断面と同じ場合，$[\bar{P}_{22} \big] = \big[ U \big]$となり，不連続部でSelf-Reactionが連続となる．
同様にして，すでに求めた式
\begin{align}
&\big[ \bar{P}_{12} \big]_T \VECi{a}_1 - \VECi{a}_2
= -\big[ \bar{P}_{12} \big]_T \VECi{b}_1 + \VECi{b}_2
\\
&\big[ \bar{P}_{11} \big] \VECi{a}_1 + \big[ \bar{P}_{12} \big] \VECi{a}_2
= \big[ \bar{P}_{11} \big] \VECi{b}_1 + \big[ \bar{P}_{12} \big] \VECi{b}_2
\end{align}
を行列表示して，
\begin{eqnarray}
\begin{pmatrix}
\big[ \bar{P}_{12} \big]_T & -\big[ U \big] \\
\big[ \bar{P}_{11} \big] & \big[ \bar{P}_{12} \big] \\
\end{pmatrix}
\begin{pmatrix}
\VECi{a}_1 \\ \VECi{a}_2 \\
\end{pmatrix}
&=&
\begin{pmatrix}
-\big[ \bar{P}_{12} \big]_T & \big[ U \big] \\
\big[ \bar{P}_{11} \big] & \big[ \bar{P}_{12} \big] \\
\end{pmatrix}
\begin{pmatrix}
\VECi{b}_1 \\ \VECi{b}_2 \\
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
-\big[ \bar{P}_{12} \big]_T & \big[ U \big] \\
\big[ \bar{P}_{11} \big] & \big[ \bar{P}_{12} \big] \\
\end{pmatrix}
\Big[ S \Big]
\begin{pmatrix}
\VECi{a}_1 \\ \VECi{a}_2 \\
\end{pmatrix}
\end{eqnarray}
よって，散乱行列$\Big[ S \Big]$は，
\begin{gather}
\Big[ S \Big] =
\begin{pmatrix}
-\big[ \bar{P}_{12} \big]_T & \big[ U \big] \\
\big[ \bar{P}_{11} \big] & \big[ \bar{P}_{12} \big] \\
\end{pmatrix}^{-1}
\begin{pmatrix}
\big[ \bar{P}_{12} \big]_T & -\big[ U \big] \\
\big[ \bar{P}_{11} \big] & \big[ \bar{P}_{12} \big] \\
\end{pmatrix}
\end{gather}
あるいは，次のように変形して計算すると，
\begin{eqnarray}
\VECi{b}_2
&=& \big[ \bar{P}_{12} \big]_T \big( \VECi{a}_1 +\VECi{b}_1 \big) - \VECi{a}_2
\nonumber \\
&=& \big[ \bar{P}_{12} \big]_T \Big\{ \VECi{a}_1 + \big( \big[ S_{11} \big] \VECi{a}_1 + \big[ S_{12} \big] \VECi{a}_2 \big) \Big\} - \VECi{a}_2
\nonumber \\
&=& \big[ \bar{P}_{12} \big]_T \Big\{ \big[ U \big] + \big[ S_{11} \big] \Big\} \VECi{a}_1
+ \Big\{ \big[ \bar{P}_{12} \big]_T \big[ S_{12} \big] - \big[ U \big] \Big\} \VECi{a}_2
\nonumber \\
&\equiv& \big[ S_{21} \big] \VECi{a}_1 + \big[ S_{22} \big] \VECi{a}_2
\end{eqnarray}
また，
\begin{eqnarray}
\VECi{b}_1
&=& -\big[ \bar{P}_{11} \big]^{-1}
\big[ \bar{P}_{12} \big] \big( \VECi{b}_2 -\VECi{a}_2 \big)+ \VECi{a}_1
\nonumber \\
&=& -\big[ \bar{P}_{11} \big]^{-1} \big[ \bar{P}_{12} \big]
\Big\{ \big( \big[ S_{21} \big] \VECi{a}_1 + \big[ S_{22} \big] \VECi{a}_2 \big)
- \VECi{a}_2 \Big\} + \VECi{a}_1
\nonumber \\
&=& \Big\{ \big[ U \big] - \big[ \bar{P}_{11} \big]^{-1}
\big[ \bar{P}_{12} \big] \big[ S_{21} \big] \Big\} \VECi{a}_1
\nonumber \\
&&+ \big[ \bar{P}_{11} \big]^{-1}
\big[ \bar{P}_{12} \big] \Big\{ \big[ U \big] - \big[ S_{22} \big] \Big\} \VECi{a}_2
\nonumber \\
&\equiv& \big[ S_{11} \big] \VECi{a}_1 + \big[ S_{12} \big] \VECi{a}_2
\end{eqnarray}
さらに，第1式を第2式に代入すると，
\begin{align}
&\big[ \bar{P}_{11} \big] \VECi{b}_1
= \big[ \bar{P}_{12} \big] \big( \VECi{a}_2 -
\big\{ \big[ \bar{P}_{12} \big]_T \big( \VECi{a}_1 +\VECi{b}_1 \big) - \VECi{a}_2 \big\}
\big) + \big[ \bar{P}_{11} \big] \VECi{a}_1
\nonumber \\
&\Big( \big[ \bar{P}_{11} \big]
+ \big[ \bar{P}_{12} \big] \big[ \bar{P}_{12} \big]_T \Big) \VECi{b}_1
= 2 \big[ \bar{P}_{12} \big] \VECi{a}_2
+ \Big( \big[ \bar{P}_{11} \big] - \big[ \bar{P}_{12} \big] \big[ \bar{P}_{12} \big]_T \Big)
\VECi{a}_1
\end{align}
よって，
\begin{eqnarray}
\VECi{b}_1
&=& \Big( \big[ \bar{P}_{11} \big]
+ \big[ \bar{P}_{12} \big] \big[ \bar{P}_{12} \big]_T \Big)^{-1}
\nonumber \\
&&\cdot \Big\{ 2 \big[ \bar{P}_{12} \big] \VECi{a}_2
+ \Big( \big[ \bar{P}_{11} \big] - \big[ \bar{P}_{12} \big] \big[ \bar{P}_{12} \big]_T \Big)
\VECi{a}_1 \Big\}
\nonumber \\
&\equiv& \big[ S_{12} \big] \VECi{a}_2 + \big[ S_{11} \big] \VECi{a}_1
\end{eqnarray}
したがって，
\begin{gather}
\big[ S_{12} \big]
= 2 \Big( \big[ \bar{P}_{11} \big] + \big[ \bar{P}_{12} \big] \big[ \bar{P}_{12} \big]_T \Big)^{-1} \big[ \bar{P}_{12} \big]
\end{gather}
これより，
\begin{gather}
\big[ S_{22} \big]
= \big[ \bar{P}_{12} \big]_T \big[ S_{12} \big] - \big[ U \big]
\end{gather}
逆に，第2式を第1式に代入して$\VECi{b}_1$を消去して整理すると次式が得られる，
\begin{gather}
\big[ S_{21} \big] = 2\Big( \big[ U \big] + \big[ \bar{P}_{12} \big]_T
\big[ \bar{P}_{11} \big]^{-1} \big[ \bar{P}_{12} \big] \Big)^{-1} \big[ \bar{P}_{12} \big]_T
\end{gather}
ただし，散乱行列の対称性より$\big[ S_{21} \big] $は次のように転置で求めることができる．
\begin{gather}
\big[ S_{21} \big] = \big[ S_{12} \big]_T
\end{gather}
さらに，
\begin{gather}
\big[ S_{11} \big]
= \big[ U \big] - \big[ \bar{P}_{11} \big]^{-1}
\big[ \bar{P}_{12} \big] \big[ S_{21} \big]
\end{gather}
積分範囲$S_0$が導波管 #1の断面と同じ場合，$[\bar{P}_{11} \big] = \big[ U \big]$となり，
不連続部でSelf-Reactionが連続となる．