2.6 ダブルステップ不連続
積分範囲が導波路 #1の断面と一致している場合，導波管 #1（$a_1 \times b_1$）の管壁$C$の境界条件を適用して積分できる．
そこで，導波管 #1のTE$_{mn}$モードと導波管 #2（$a_2 \times b_2$）のTE$_{m'n'}$モードを，
\begin{gather}
\Psi^{\#1}_{[mn]} \equiv A^{\#1}_{[mn]} h^{\#1}_{x[m]}(x) h^{\#1}_{y[n]}(y), \ \ \ \ \
\Psi^{\#2}_{[m'n']} \equiv A^{\#2}_{[m'n']} h^{\#2}_{x[m']}(x) h^{\#2}_{y[n']}(y)
\end{gather}
ここで，
\begin{align}
&h^{\#1}_{x[m]}(x) = \cos \big( k_{xm} x \big), \ \ \ \ \
h^{\#2}_{x[m']}(x) = \cos \big\{ \hat{k}_{xm'} (x + x_2) \big\}
\\
&h^{\#1}_{y[n]}(y) = \cos \big( k_{yn} y \big), \ \ \ \ \
h^{\#2}_{y[n']}(y) = \cos \big\{ \hat{k}_{yn'} (y + y_2) \big\}
\end{align}
ただし，
\begin{align}
&k_{xm} = \frac{m\pi}{a_1}, \ \ \ \ \
k_{yn} = \frac{n\pi}{b_1} \ \ \ \ \ (m,n=0,1,2,\cdots , \ m=n\neq0)
\\
&\hat{k}_{xm'} = \frac{m'\pi}{a_2}, \ \ \ \ \
\hat{k}_{yn'} = \frac{n'\pi}{b_2} \ \ \ \ \ (m',n'=0,1,2,\cdots , \ m'=n'\neq0)
\end{align}
また，導波管 #1のTM$_{mn}$モードと導波管 #2のTM$_{m'n'}$モードを，
\begin{gather}
\Psi^{\#1}_{(mn)} \equiv A^{\#1}_{(mn)} h^{\#1}_{x(m)}(x) h^{\#1}_{y(n)}(y), \ \ \ \ \
\Psi^{\#2}_{(m'n')} \equiv A^{\#2}_{(mn)} h^{\#2}_{x(m')}(x) h^{\#2}_{y(n')}(y)
\end{gather}
ここで，
\begin{align}
&h^{\#1}_{x(m)}(x) = \sin \big( k_{xm} x \big), \ \ \ \ \
h^{\#2}_{x(m')}(x) = \sin \big\{ \hat{k}_{xm'} (x + x_2) \big\},
\\
&h^{\#1}_{y(n)}(y) = \sin \big( k_{yn} y \big), \ \ \ \ \
h^{\#2}_{y(n')}(y) = \sin \big\{ \hat{k}_{yn'} (y + y_2) \big\}
\end{align}
ただし，
\begin{align}
&k_{xm} = \frac{m\pi}{a_1}, \ \ \ \ \
k_{yn} = \frac{n\pi}{b_1} \ \ \ \ \ (m,n=1,2,\cdots )
\\
&\hat{k}_{xm'} = \frac{m'\pi}{a_2}, \ \ \ \ \
\hat{k}_{yn'} = \frac{n'\pi}{b_2} \ \ \ \ \ (m',n'=1,2,\cdots )
\end{align}
導波管 #2のモードの内積については，積分範囲$S_A$が導波管 #2の断面の一部となり，
\begin{eqnarray}
&&\iint _{S_A} \VEC{e}^{\#2}_{[mn]} \cdot \VEC{e}^{\#2}_{[m'n']} dS
\nonumber \\
&=& A^{\#2}_{[mn]} A^{\#2}_{[m'n']}
\Big( \hat{k}_{xm} \hat{k}_{xm'} \hat{X}_{mm'}^{22,\mathrm{s}} \hat{Y}_{nn'}^{22,\mathrm{c}}
+ \hat{k}_{yn} \hat{k}_{yn'}\hat{X}_{mm'}^{22,\mathrm{c}} \hat{Y}_{nn'}^{22,\mathrm{s}} \Big)
\nonumber \\
&&(m, n=0,1,2,\cdots (m\neq0 \ \mbox{or} \ n\neq0)
\nonumber \\
&&m', n'=0,1,2,\cdots (m'\neq0 \ \mbox{or} \ n'\neq0))
\\
&&\iint _{S_A} \VEC{e}^{\#2}_{(mn)} \cdot \VEC{e}^{\#2}_{(m'n')} dS
\nonumber \\
&=& A^{\#2}_{(mn)} A^{\#2}_{(m'n')}
\Big( \hat{k}_{yn} \hat{k}_{yn'} \hat{X}_{mm'}^{22,\mathrm{s}} \hat{Y}_{nn'}^{22,\mathrm{c}}
+ \hat{k}_{xm} \hat{k}_{xm'} \hat{X}_{mm'}^{22,\mathrm{c}} \hat{Y}_{nn'}^{22,\mathrm{s}} \Big)
\nonumber \\
&&(m, n=1,2,3,\cdots , \ \ m',n'=1,2,3,\cdots)
\\
&&\iint _{S_A} \VEC{e}^{\#2}_{[mn]} \cdot \VEC{e}^{\#2}_{(m'n')} dS
\nonumber \\
&=& A^{\#2}_{[mn]} A^{\#2}_{(m'n')}
\Big( \hat{k}_{xm} \hat{k}_{yn'} \hat{X}_{mm'}^{22,\mathrm{s}} \hat{Y}_{nn'}^{22,\mathrm{c}}
- \hat{k}_{xm'} \hat{k}_{yn} \hat{X}_{mm'}^{22,\mathrm{c}} \hat{Y}_{nn'}^{22,\mathrm{s}} \Big)
\nonumber \\
&&(m, n=0,1,2,\cdots (m\neq0 \ \mbox{or} \ n\neq0), \ \
m', n'=1,2,3,\cdots)
\\
&&\iint _{S_A} \VEC{e}^{\#2}_{(mn)} \cdot \VEC{e}^{\#2}_{[m'n']} dS
\nonumber \\
&=& A^{\#2}_{(mn)} A^{\#2}_{[m'n']}
\Big( \hat{k}_{xm'} \hat{k}_{yn} \hat{X}_{mm'}^{22,\mathrm{s}} \hat{Y}_{nn'}^{22,\mathrm{c}}
- \hat{k}_{xm} \hat{k}_{yn'} \hat{X}_{mm'}^{22,\mathrm{c}} \hat{Y}_{nn'}^{22,\mathrm{s}} \Big)
\nonumber \\
&&(m, n=1,2,3,\cdots, \ \
m', n'=0,1,2,\cdots (m'\neq0 \ \mbox{or} \ n'\neq0))
\end{eqnarray}
積分範囲$S_A$は，導波管 #1の断面
$0 \leq x \leq a_1$，$0 \leq y \leq b_1$と一致しているので，
導波管 #1のモードの正規直交性より，
\begin{eqnarray}
\iint _{S_A} \VEC{e}^{\#1}_{[mn]} \cdot \VEC{e}^{\#1}_{[m'n']} dS
&=& \delta _{mm'} \delta _{nn'}
\\
\iint _{S_A} \VEC{e}^{\#1}_{(mn)} \cdot \VEC{e}^{\#1}_{(m'n')} dS
&=& \delta _{mm'} \delta _{nn'}
\\
\iint _{S_A} \VEC{e}^{\#1}_{[mn]} \cdot \VEC{e}^{\#1}_{(m'n')} dS
&=& 0
\\
\iint _{S_A} \VEC{e}^{\#1}_{(mn)} \cdot \VEC{e}^{\#1}_{[m'n']} dS
&=& 0
\end{eqnarray}
また，導波管 #1と #2のモードの内積については，
\begin{eqnarray}
%------------------- TE-TE
&&\iint _{S_A} \VEC{e}^{\#1}_{[mn]} \cdot \VEC{e}^{\#2}_{[m'n']} dS
\nonumber \\
&=& A^{\#1}_{[mn]} A^{\#2}_{[m'n']}
\Big( k_{xm} \hat{k}_{xm'} \hat{X}_{mm'}^{12,\mathrm{s}} \hat{Y}_{nn'}^{12,\mathrm{c}}
+ k_{yn} \hat{k}_{yn'}\hat{X}_{mm'}^{12,\mathrm{c}} \hat{Y}_{nn'}^{12,\mathrm{s}} \Big)
\\
%------------------- TM-TM
&&\iint _{S_A} \VEC{e}^{\#1}_{(mn)} \cdot \VEC{e}^{\#2}_{(m'n')} dS
\nonumber \\
&=& A^{\#1}_{(mn)} A^{\#2}_{(m'n')}
\Big( k_{yn} \hat{k}_{yn'} \hat{X}_{mm'}^{12,\mathrm{s}} \hat{Y}_{nn'}^{12,\mathrm{c}}
+ k_{xm} \hat{k}_{xm'} \hat{X}_{mm'}^{12,\mathrm{c}} \hat{Y}_{nn'}^{12,\mathrm{s}} \Big)
\\
%------------------- TE-TM
&&\iint _{S_A} \VEC{e}^{\#1}_{[mn]} \cdot \VEC{e}^{\#2}_{(m'n')} dS
\nonumber \\
&=& A^{\#1}_{[mn]} A^{\#2}_{(m'n')}
\Big( k_{xm} \hat{k}_{yn'} \hat{X}_{mm'}^{12,\mathrm{s}} \hat{Y}_{nn'}^{12,\mathrm{c}}
- \hat{k}_{xm'} k_{yn} \hat{X}_{mm'}^{12,\mathrm{c}} \hat{Y}_{nn'}^{12,\mathrm{s}} \Big)
\\
%------------------- TM-TE
&&\iint _{S_A} \VEC{e}^{\#1}_{(mn)} \cdot \VEC{e}^{\#2}_{[m'n']} dS
\nonumber \\
&=& A^{\#1}_{(mn)} A^{\#2}_{[m'n']}
\Big( \hat{k}_{xm'} k_{yn} \hat{X}_{mm'}^{12,\mathrm{s}} \hat{Y}_{nn'}^{12,\mathrm{c}}
- k_{xm} \hat{k}_{yn'} \hat{X}_{mm'}^{12,\mathrm{c}} \hat{Y}_{nn'}^{12,\mathrm{s}} \Big)
\end{eqnarray}