2.1 方形導波菅の不連続問題
方形導波菅のモード関数
方形導波管 #1（$a_1 \times b_1$）のTE$_{mn}$モードのスカラ関数
$\Psi^{\#1}_{[mn]}$，
方形導波管 #2（$a_2 \times b_2$）のTE$_{m'n'}$モードのスカラ関数
$\Psi^{\#2}_{[m'n']}$
を，
\begin{eqnarray}
\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{eqnarray}
TEモードの境界条件より，
\begin{gather}
\left. \frac{dh^{\#1}_{x[m]}}{dx} \right|_{x=0,a_1} =0, \ \ \ \ \
\left. \frac{dh^{\#2}_{x[m']}}{dx} \right|_{x=-x_2,a_2-x_2} =0
\\
\left. \frac{dh^{\#1}_{y[n]}}{dy} \right|_{y=0,b_1} =0, \ \ \ \ \
\left. \frac{dh^{\#2}_{y[n']}}{dy} \right|_{y=-y_2,b_2-y_2} =0
\end{gather}
よって，
\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（一様分布は存在しない），
\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，$A^{\#1}_{[mn]}$，$A^{\#2}_{[m'n']}$はTEモードの正規化係数を示し，
\begin{align}
&A^{\#1}_{[mn]}
= \frac{1}{\pi} \sqrt{\frac{a_1 b_1 \epsilon _m \epsilon _n}{(m b_1)^2+(n a_1)^2}}
\\
&A^{\#2}_{[m'n']}
= \frac{1}{\pi}
\sqrt{\frac{a_2 b_2 \epsilon _{m'} \epsilon _{n'}}{(m' b_2)^2+(n' a_2)^2}}
\end{align}
ただし，
\begin{align}
&\epsilon _m = \left\{
\begin {array}{ll}
1 & (m=0) \\
2 & (m=1,2, \cdots )
\end{array} \right., \ \ \ \ \
\epsilon _{m'} = \left\{
\begin {array}{ll}
1 & (m'=0) \\
2 & (m'=1,2, \cdots )
\end{array} \right.
\\
&\epsilon _n = \left\{
\begin {array}{ll}
1 & (n=0) \\
2 & (n=1,2, \cdots )
\end{array} \right., \ \ \ \ \
\epsilon _{n'} = \left\{
\begin {array}{ll}
1 & (n'=0) \\
2 & (n'=1,2, \cdots )
\end{array} \right.
\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{align}
&\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{align}
TMモードの境界条件より，
\begin{align}
&\left. h^{\#1}_{x(m)} \right|_{x=0,a_1} =0, \ \ \ \ \
\left. h^{\#2}_{x(m')} \right|_{x=-x_2,a_2-x_2} =0
\\
&\left. h^{\#1}_{y(n)} \right|_{y=0,b_1} =0, \ \ \ \ \
\left. h^{\#2}_{y(n')} \right|_{y=-y_2,b_2-y_2} =0
\end{align}
よって，
\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，$k_{xm}$，$k_{yn}$の定義は先に示したものと変わらないが，
$m, m', n, n'=1,2,\cdots $
となる，また，
$A^{\#1}_{(mn)}$，$A^{\#2}_{(m'n')}$
はTMモードの正規化係数を示し，
\begin{align}
&A^{\#1}_{(mn)} = \frac{2}{\pi} \sqrt{\frac{a_1b_1}{(mb_1)^2+(na_1)^2}}
\\
&A^{\#2}_{(m'n')} = \frac{2}{\pi} \sqrt{\frac{a_2 b_2}{(m' b_2)^2+(n' a_2)^2}}
\end{align}
TMモードの場合は，$x$，$y$方向ともに分布をもつ．なお，TE$_{mn}$モードは添字$[mn]$，TM$_{mn}$モードは添字$(mn)$を用い，TEモードとTMモードの区別なく$mn$次のモードを表す場合，次のように表すことにする．
\begin{align}
&\Psi^{\#1}_{mn} \equiv A^{\#1}_{mn} h^{\#1}_{xm}(x) h^{\#1}_{yn}(y)
\\
&\Psi^{\#2}_{m'n'} \equiv A^{\#2}_{m'n'} h^{\#2}_{xm'}(x) h^{\#2}_{yn'}(y)
\end{align}
モード関数の内積（#1，#2）
$\Psi^{\#1}_{[mn]}$で求められるTE$_{mn}$モードと
$\Psi^{\#2}_{[m'n']}$で求められるTE$_{m'n'}$モードの内積，
あるいは，
$\Psi^{\#1}_{(mn)}$で求められるTM$_{mn}$モードと
$\Psi^{\#2}_{(m'n')}$で求められるTM$_{m'n'}$モードの内積は次のようになる．
\begin{eqnarray}
&&\iint _{S_A} \nabla_t \Psi^{\#1}_{mn} \cdot \nabla_t \Psi^{\#2}_{m'n'} dS
\nonumber \\
&=& \iint _{S_A} \left( \frac{\partial \Psi^{\#1}_{mn}}{\partial x} \frac{\partial \Psi^{\#2}_{m'n'}}{\partial x}
+ \frac{\partial \Psi^{\#1}_{mn}}{\partial y} \frac{\partial \Psi^{\#2}_{m'n'}}{\partial y} \right) dx dy
\nonumber \\
&=& A^{\#1}_{mn} A^{\#2}_{m'n'} \int _{l_y} \int _{l_x}
\left( \frac{dh^{\#1}_{xm}}{dx} \frac{dh^{\#2}_{xm'}}{dx} h^{\#1}_{yn} h^{\#2}_{yn'} \right.
\nonumber \\
&&\left. + h^{\#1}_{xm} h^{\#2}_{xm'} \frac{dh^{\#1}_{yn}}{dy} \frac{dh^{\#2}_{yn'}}{dy} \right) dx dy
\nonumber \\
&=& A^{\#1}_{mn} A^{\#2}_{m'n'} \left[ \int _{l_x} (h^{\#1}_{xm})' (h^{\#2}_{xm'})' dx
\int _{l_y} h^{\#1}_{yn} h^{\#2}_{yn'} dy \right.
\nonumber \\
&&\left. + \int _{l_x} h^{\#1}_{xm} h^{\#2}_{xm'} dx
\int _{l_y} (h^{\#1}_{yn})' (h^{\#2}_{yn'})' dy \right]
\end{eqnarray}
ただし，$S_A$は導波管断面内の一部あるいは全部の領域，
$l_x$，$l_y$は$x$，$y$の積分範囲を示す．
両者ともTEモードの場合，$m, n=0,1,2,\cdots (m\neq0$ or $n\neq0$），
$m', n'=0,1,2,\cdots (m'\neq0$ or $n'\neq0$）について，
\begin{eqnarray}
&&\iint _{S_A} \VEC{e}^{\#1}_{[mn]} \cdot \VEC{e}^{\#2}_{[m'n']} dS
\nonumber \\
&=& A^{\#1}_{[mn]} A^{\#2}_{[m'n']}
\left[ \int _{l_x} (h^{\#1}_{x[m]})' (h^{\#2}_{x[m']})' dx
\int _{l_y} h^{\#1}_{y[n]} h^{\#2}_{y[n']} dy \right.
\nonumber \\
&&\left. + \int _{l_x} h^{\#1}_{x[m]} h^{\#2}_{x[m']} dx
\int _{l_y} (h^{\#1}_{y[n]})' (h^{\#2}_{y[n']})' dy \right]
\end{eqnarray}
また，両者ともTMモードの場合，$m, n=1,2,3,\cdots$，$m', n'=1,2,3,\cdots$について，
\begin{eqnarray}
&&\iint _{S_A} \VEC{e}^{\#1}_{(mn)} \cdot \VEC{e}^{\#2}_{(m'n')} dS
\nonumber \\
&=& A^{\#1}_{(mn)} A^{\#2}_{(m'n')}
\left[ \int _{l_x} (h^{\#1}_{x(m)})' (h^{\#2}_{x(m')})' dx
\int _{l_y} h^{\#1}_{y(n)} h^{\#2}_{y(n')} dy \right.
\nonumber \\
&&\left. + \int _{l_x} h^{\#1}_{x(m)} h^{\#2}_{x(m')} dx
\int _{l_y} (h^{\#1}_{y(n)})' (h^{\#2}_{y(n')})' dy \right]
\end{eqnarray}
一方，
$\Psi^{\#1}_{[mn]}$で求められるTE$_{mn}$モードと
$\Psi^{\#2}_{(m'n')}$で求められるTM$_{m'n'}$モードでは，
\begin{eqnarray}
&&\iint _{S_A} \VEC{e}^{\#1}_{[mn]} \cdot \VEC{e}^{\#2}_{(m'n')} dS
\nonumber \\
&=& \iint _{S_A} \big( \nabla_t \Psi^{\#1}_{[mn]} \times \VEC{a}_z \big)
\cdot \nabla_t \Psi^{\#2}_{(m'n')} dS
\nonumber \\
&=& \iint _{S_A} \left( -\VEC{a}_y \frac{\partial \Psi^{\#1}_{[mn]}}{\partial x}
+ \VEC{a}_x \frac{\partial \Psi^{\#1}_{[mn]}}{\partial y} \right)
\cdot \left( \VEC{a}_x \frac{\partial \Psi^{\#2}_{[m'n']}}{\partial x}
+ \VEC{a}_y \frac{\partial \Psi^{\#2}_{[m'n']}}{\partial y} \right) dS
\nonumber \\
&=& \iint_{S_A} \left(
\frac{\partial \Psi^{\#1}_{[mn]}}{\partial y}
\frac{\partial \Psi^{\#2}_{(m'n')}}{\partial x}
- \frac{\partial \Psi^{\#1}_{[mn]}}{\partial x}
\frac{\partial \Psi^{\#2}_{(m'n')}}{\partial y}
\right) dx dy
\nonumber \\
&=& A^{\#1}_{[mn]} A^{\#2}_{(m'n')}
\int _{l_y} \int _{l_x} \left( h^{\#1}_{x[m]} \frac{dh^{\#1}_{y[n]}}{dy} \frac{dh^{\#2}_{x(m')}}{dx} h^{\#2}_{y(n')} \right.
\nonumber \\
&&\left. - \frac{dh^{\#1}_{x[m]}}{dx} h^{\#1}_{y[n]} h^{\#2}_{x(m')} \frac{dh^{\#2}_{y(n')}}{dy} \right) dx dy
\nonumber \\
&=& A^{\#1}_{[mn]} A^{\#2}_{(m'n')}
\left[ \int _{l_x} h^{\#1}_{x[m]} (h^{\#2}_{x(m')})' dx
\int _{l_y} (h^{\#1}_{y[n]})' h^{\#2}_{y(n')} dy \right.
\nonumber \\
&&\left. - \int _{l_x} (h^{\#1}_{x[m]})' h^{\#2}_{x(m')} dx
\int _{l_y} h^{\#1}_{y[n]} (h^{\#2}_{y(n')})' dy \right]
\nonumber
\end{eqnarray}
逆に，
$\Psi^{\#1}_{(mn)}$で求められるTM$_{mn}$モードと
$\Psi^{\#2}_{[m'n']}$で求められるTE$_{m'n'}$モードでは，
\begin{eqnarray}
&&\iint _{S_A} \VEC{e}^{\#1}_{(mn)} \cdot \VEC{e}^{\#2}_{[m'n']} dS
\nonumber \\
&=& A^{\#1}_{(mn)} A^{\#2}_{[m'n']}
\left[ \int _{l_x} (h^{\#1}_{x(m)})' h^{\#2}_{x[m']} dx
\int _{l_y} h^{\#1}_{y(n)} (h^{\#2}_{y[n']})' dy \right.
\nonumber \\
&&\left. - \int _{l_x} h^{\#1}_{x(m)} (h^{\#2}_{x[m']})' dx
\int _{l_y} (h^{\#1}_{y(n)})' h^{\#2}_{y[n']} dy \right]
\end{eqnarray}
これらの式中の$x$，$y$に関する微分は，導波菅 #1のTEモードについては，
\begin{align}
&(h^{\#1}_{x[m]}(x))'
= \frac{d}{dx} \Big\{ \cos \big( k_{xm} x \big) \Big\}
= -k_{xm} \sin \big( k_{xm} x \big)
\\
&(h^{\#1}_{y[n]}(y))'
= \frac{d}{dy} \Big\{ \cos \big( k_{yn} y \big\} \Big\}
= -k_{yn} \sin \big( k_{yn} y \big)
\end{align}
また，TMモードについては，
\begin{align}
&(h^{\#1}_{x(m)}(x))'
= \frac{d}{dx} \Big\{ \sin \big( k_{xm} x \big) \Big\}
= k_{xm} \cos \big( k_{xm} x \big)
\\
&(h^{\#1}_{y(n)}(y))'
= \frac{d}{dy} \Big\{ \sin \big( k_{yn} y \big) \Big\}
= k_{yn} \cos \big( k_{yn} y \big)
\end{align}
導波菅 #2も同様にして，
\begin{align}
&(h^{\#2}_{x[m']}(x))'
= -\hat{k}_{xm'} \sin \big\{ \hat{k}_{xm'} (x + x_2) \big\}
\\
&(h^{\#2}_{y[n']}(y))'
= -\hat{k}_{yn'} \sin \big\{ \hat{k}_{yn'} (y + y_2) \big\}
\\
&(h^{\#2}_{x(m')}(x))'
= \hat{k}_{xm'} \cos \big\{ \hat{k}_{xm'} (x + x_2) \big\}
\\
&(h^{\#2}_{y(n')}(y))'
= \hat{k}_{yn'} \cos \big\{ \hat{k}_{yn'} (y + y_2) \big\}
\end{align}
いま，
\begin{eqnarray}
\hat{X}_{mm'}^{12,\mathrm{s}} &\equiv&
\int _{L_x} \sin \big( k_{xm} x \big)
\sin \big\{ \hat{k}_{xm'} (x + x_2) \big\} dx
\\
\hat{Y}_{nn'}^{12,\mathrm{c}} &\equiv&
\int _{L_y} \cos \big( k_{yn} y \big)
\cos \big\{ \hat{k}_{yn'} (y + y_2) \big\} dy
\\
\hat{X}_{mm'}^{12,\mathrm{c}} &\equiv&
\int _{L_x} \cos \big( k_{xm} x \big)
\cos \big\{ \hat{k}_{xm'} (x + x_2) \big\} dx
\\
\hat{Y}_{nn'}^{12,\mathrm{s}} &\equiv&
\int _{L_y} \sin \big( k_{yn} y \big)
\sin \big\{ \hat{k}_{yn'} (y + y_2) \big\} dy
\end{eqnarray}
とおくと，両者ともTEモードの場合，および両者ともTMモードの場合，
\begin{eqnarray}
\int _{l_x} (h^{\#1}_{x[m]})' (h^{\#2}_{x[m']})' dx
&=& \int _{l_x} k_{xm} \hat{k}_{xm'}
\sin \big( k_{xm} x \big) \sin \big\{ \hat{k}_{xm'} (x + x_2) \big\} dx
\nonumber \\
&=& k_{xm} \hat{k}_{xm'} \hat{X}_{mm'}^{12,\mathrm{s}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_y} h^{\#1}_{y[n]} h^{\#2}_{y[n']} dy
&=& \int _{l_y} \cos \big( k_{yn} y \big) \cos \big\{ \hat{k}_{yn'} (y + y_2) \big\} dy
\nonumber \\
&=& \hat{Y}_{nn'}^{12,\mathrm{c}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_x} h^{\#1}_{x[m]} h^{\#2}_{x[m']} dx
&=& \int _{l_x} \cos \big( k_{xm} x \big) \cos \big\{ \hat{k}_{xm'} (x + x_2) \big\} dx
\nonumber \\
&=& \hat{X}_{mm'}^{12,\mathrm{c}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_y} (h^{\#1}_{y[n]})' (h^{\#2}_{y[n']})' dy
&=& \int _{l_y} k_{yn} \hat{k}_{yn'}
\sin \big( k_{yn} y \big) \sin \big\{ \hat{k}_{yn'} (y + y_2) \big\} dy
\nonumber \\
&=& k_{yn} \hat{k}_{yn'} \hat{Y}_{nn'}^{12,\mathrm{s}}
\end{eqnarray}
また，両者ともTMモードの場合，
\begin{eqnarray}
\int _{l_x} (h^{\#1}_{x(m)})' (h^{\#2}_{x(m')})' dx
&=& \int _{l_x} k_{xm} \hat{k}_{xm'}
\cos \big( k_{xm} x \big) \cos \big\{ \hat{k}_{xm'} (x + x_2) \big\} dx
\nonumber \\
&=& k_{xm} \hat{k}_{xm'} \hat{X}_{mm'}^{12,\mathrm{c}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_y} h^{\#1}_{y(n)} h^{\#2}_{y(n')} dy
&=& \int _{l_y} \sin \big( k_{yn} y \big) \sin \big\{ \hat{k}_{yn'} (y + y_2) \big\} dy
\nonumber \\
&=& \hat{Y}_{nn'}^{12,\mathrm{s}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_x} h^{\#1}_{x(m)} h^{\#2}_{x(m')} dx
&=& \int _{l_x} \sin \big( k_{xm} x \big) \sin \big\{ \hat{k}_{xm'} (x + x_2) \big\} dx
\nonumber \\
&=& \hat{X}_{mm'}^{12,\mathrm{s}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_y} (h^{\#1}_{y(n)})' (h^{\#2}_{y(n')})' dy
&=& \int _{l_y} k_{yn} \hat{k}_{yn'}
\cos \big( k_{yn} y \big) \cos \big\{ \hat{k}_{yn'} (y + y_2) \big\} dy
\nonumber \\
&=& k_{yn} \hat{k}_{yn'} \hat{Y}_{nn'}^{12,\mathrm{c}}
\end{eqnarray}
TEモードとTMモードの場合，
\begin{eqnarray}
\int _{l_x} h^{\#1}_{x[m]} (h^{\#2}_{x(m')})' dx
&=& \int _{l_x} \hat{k}_{xm'}
\cos \big( k_{xm} x \big) \cos \big\{ \hat{k}_{xm'} (x + x_2) \big\} dx
\nonumber \\
&=& k_{xm} \hat{X}_{mm'}^{12,\mathrm{c}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_y} (h^{\#1}_{y[n]})' h^{\#2}_{y(n')} dy
&=& \int _{l_y} -k_{yn}
\sin \big( k_{yn} y \big) \sin \big\{ \hat{k}_{yn'} (y + y_2) \big\} dy
\nonumber \\
&=& -k_{yn} \hat{Y}_{nn'}^{12,\mathrm{s}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_x} (h^{\#1}_{x[m]})' h^{\#2}_{x(m')} dx
&=& \int _{l_x} -k_{xm}
\sin \big( k_{xm} x \big) \sin \big\{ \hat{k}_{xm'} (x + x_2) \big\} dx
\nonumber \\
&=& -k_{xm} \hat{X}_{mm'}^{12,\mathrm{s}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_y} h^{\#1}_{y[n]} (h^{\#2}_{y(n')})' dy
&=& \int _{l_y} \hat{k}_{yn'}
\cos \big( k_{yn} y \big) \cos \big\{ \hat{k}_{yn'} (y + y_2) \big\} dy
\nonumber \\
&=& \hat{k}_{yn'} \hat{Y}_{nn'}^{12,\mathrm{c}}
\end{eqnarray}
TMモードとTEモードの場合，
\begin{eqnarray}
\int _{l_x} (h^{\#1}_{x(m)})' h^{\#2}_{x[m']} dx
&=& \int _{l_x} k_{xm}
\cos \big( k_{xm} x \big) \cos \big\{ \hat{k}_{xm'} (x + x_2) \big\} dx
\nonumber \\
&=& k_{xm} \hat{X}_{mm'}^{12,\mathrm{c}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_y} h^{\#1}_{y(n)} (h^{\#2}_{y[n']})' dy
&=& \int _{l_y} -\hat{k}_{yn'}
\sin \big( k_{yn} y \big) \sin \big\{ \hat{k}_{yn'} (y + y_2) \big\} dy
\nonumber \\
&=& -\hat{k}_{yn'} \hat{Y}_{nn'}^{12,\mathrm{s}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_x} h^{\#1}_{x(m)} (h^{\#2}_{x[m']})' dx
&=& \int _{l_x} -\hat{k}_{xm'}
\sin \big( k_{xm} x \big) \sin \big\{ \hat{k}_{xm'} (x + x_2) \big\} dx
\nonumber \\
&=& -\hat{k}_{xm'} \hat{X}_{mm'}^{12,\mathrm{s}}
\end{eqnarray}
\begin{eqnarray}
\int _{l_y} (h^{\#1}_{y(n)})' h^{\#2}_{y[n']} dy
&=& \int _{l_y} k_{yn}
\cos \big( k_{yn} y \big) \cos \big\{ \hat{k}_{yn'} (y + y_2) \big\} dy
\nonumber \\
&=& k_{yn} \hat{Y}_{nn'}^{12,\mathrm{c}}
\end{eqnarray}
モード関数の内積は次のようになる．
\begin{align}
%------------------- 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{align}