2.11 面電流源に対するグリーン関数
スペクトル領域の電界型ダイアディック・グリーン関数
境界面に接する電界のベクトル\(\widetilde{\boldsymbol{E}}_{\tan}\)は，
\begin{gather}
\widetilde{\boldsymbol{E}}_{\tan}
= \widetilde{E}_u \boldsymbol{u}_u + \widetilde{E}_v \boldsymbol{u}_v
= \widetilde{Z}_{_{TE}}^{(z)} \widetilde{J}_u \boldsymbol{u}_u + \widetilde{Z}_{_{TM}}^{(z)} \widetilde{J}_v \boldsymbol{u}_v \\
\end{gather}
成分を行列表示すると，
\begin{gather}
\begin{pmatrix}
\widetilde{\boldsymbol{E}}_{\tan} \cdot \boldsymbol{u}_u \\ \widetilde{\boldsymbol{E}}_{\tan} \cdot \boldsymbol{u}_v
\end{pmatrix}
=
\begin{pmatrix}
\widetilde{E}_u \\ \widetilde{E}_v
\end{pmatrix}
=
\begin{pmatrix}
\widetilde{Z}_{_{TE}}^{(z)} & 0 \\
0 & \widetilde{Z}_{_{TM}}^{(z)} \\
\end{pmatrix}
\begin{pmatrix}
\widetilde{J}_u \\ \widetilde{J}_v
\end{pmatrix}
\end{gather}
\(x\)，\(y\)成分を求めると，
\begin{eqnarray}
\begin{pmatrix}
\widetilde{E}_x \\ \widetilde{E}_y
\end{pmatrix}
&=&
\begin{pmatrix}
\big( \widetilde{E}_v \boldsymbol{u}_v + \widetilde{E}_u \boldsymbol{u}_u \big) \cdot \boldsymbol{u}_x \\
\big( \widetilde{E}_v \boldsymbol{u}_v + \widetilde{E}_u \boldsymbol{u}_u \big) \cdot \boldsymbol{u}_y
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\boldsymbol{u}_u \cdot \boldsymbol{u}_x & \boldsymbol{u}_v \cdot \boldsymbol{u}_x \\
\boldsymbol{u}_u \cdot \boldsymbol{u}_y & \boldsymbol{u}_v \cdot \boldsymbol{u}_y
\end{pmatrix}
\begin{pmatrix}
\widetilde{E}_u \\ \widetilde{E}_v
\end{pmatrix}
\nonumber \\
&=& [\Phi]^t
\begin{pmatrix}
\widetilde{Z}_{_{TE}}^{(z)} & 0 \\
0 & \widetilde{Z}_{_{TM}}^{(z)} \\
\end{pmatrix}
\begin{pmatrix}
\widetilde{J}_u \\ \widetilde{J}_v
\end{pmatrix}
\nonumber \\
&=& [\Phi]^t
\begin{pmatrix}
\widetilde{Z}_{_{TE}}^{(z)} & 0 \\
0 & \widetilde{Z}_{_{TM}}^{(z)} \\
\end{pmatrix}
\begin{pmatrix}
\big( \widetilde{J}_x \boldsymbol{u}_x + \widetilde{J}_y \boldsymbol{u}_y \big) \cdot \boldsymbol{u}_u \\
\big( \widetilde{J}_x \boldsymbol{u}_x + \widetilde{J}_y \boldsymbol{u}_y \big) \cdot \boldsymbol{u}_v
\end{pmatrix}
\nonumber \\
&=& [\Phi]^t
\begin{pmatrix}
\widetilde{Z}_{_{TE}}^{(z)} & 0 \\
0 & \widetilde{Z}_{_{TM}}^{(z)} \\
\end{pmatrix}
\begin{pmatrix}
\boldsymbol{u}_x \cdot \boldsymbol{u}_u & \boldsymbol{u}_y \cdot \boldsymbol{u}_u \\
\boldsymbol{u}_x \cdot \boldsymbol{u}_v & \boldsymbol{u}_y \cdot \boldsymbol{u}_v
\end{pmatrix}
\begin{pmatrix}
\widetilde{J}_x \\ \widetilde{J}_y
\end{pmatrix}
\nonumber \\
&=& [\Phi]^t
\begin{pmatrix}
\widetilde{Z}_{_{TE}}^{(z)} & 0 \\
0 & \widetilde{Z}_{_{TM}}^{(z)} \\
\end{pmatrix}
[\Phi]
\begin{pmatrix}
\widetilde{J}_x \\ \widetilde{J}_y
\end{pmatrix}
\end{eqnarray}
ここで，
\begin{gather}
\begin{pmatrix}
\widetilde{E}_x \\ \widetilde{E}_y
\end{pmatrix}
=
\begin{pmatrix}
\widetilde{G}_{xx}^{^{EJ}} & \widetilde{G}_{xy}^{^{EJ}} \\
\widetilde{G}_{yx}^{^{EJ}} & \widetilde{G}_{yy}^{^{EJ}}
\end{pmatrix}
\begin{pmatrix}
\widetilde{J}_x \\ \widetilde{J}_y
\end{pmatrix}
\end{gather}
とおくと，
\begin{eqnarray}
&&\begin{pmatrix}
\widetilde{G}_{xx}^{^{EJ}} & \widetilde{G}_{xy}^{^{EJ}} \\
\widetilde{G}_{yx}^{^{EJ}} & \widetilde{G}_{yy}^{^{EJ}}
\end{pmatrix}
= [\Phi]^t
\begin{pmatrix}
\widetilde{Z}_{_{TE}}^{(z)} & 0 \\
0 & \widetilde{Z}_{_{TM}}^{(z)} \\
\end{pmatrix}
[\Phi]
\nonumber \\
&=&
\begin{pmatrix}
\sin \Phi & \cos \Phi \\
-\cos \Phi & \sin \Phi
\end{pmatrix}
\begin{pmatrix}
\widetilde{Z}_{_{TE}}^{(z)} & 0 \\
0 & \widetilde{Z}_{_{TM}}^{(z)} \\
\end{pmatrix}
\begin{pmatrix}
\sin \Phi & -\cos \Phi \\
\cos \Phi & \sin \Phi
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\widetilde{Z}_{_{TE}}^{(z)} \sin ^2 \Phi + \widetilde{Z}_{_{TM}}^{(z)} \cos ^2 \Phi
& \big( \widetilde{Z}_{_{TM}}^{(z)} - \widetilde{Z}_{_{TE}}^{(z)} \big) \sin \Phi \cos \Phi \\
\big( \widetilde{Z}_{_{TM}}^{(z)} - \widetilde{Z}_{_{TE}}^{(z)} \big) \sin \Phi \cos \Phi
& \widetilde{Z}_{_{TE}}^{(z)} \cos ^2 \Phi + \widetilde{Z}_{_{TM}}^{(z)} \sin ^2 \Phi
\end{pmatrix}
\end{eqnarray}
これより，
\begin{gather}
\widetilde{\bar{\bar{\boldsymbol{G}}}}_T^{^{EJ}}
= \widetilde{G}_{xx}^{^{EJ}} \boldsymbol{u}_x \boldsymbol{u}_x
+ \widetilde{G}_{xy}^{^{EJ}} \boldsymbol{u}_x \boldsymbol{u}_y
+ \widetilde{G}_{yx}^{^{EJ}} \boldsymbol{u}_y \boldsymbol{u}_x
+ \widetilde{G}_{yy}^{^{EJ}} \boldsymbol{u}_y \boldsymbol{u}_y
\end{gather}
とおくと，\(\widetilde{\boldsymbol{E}}_{\tan}\)は，
\begin{eqnarray}
\widetilde{\boldsymbol{E}}_{\tan}
&=& \Big( \widetilde{G}_{xx}^{^{EJ}} \widetilde{J}_x + \widetilde{G}_{xy}^{^{EJ}} \widetilde{J}_y \Big) \boldsymbol{u}_x
+ \Big( \widetilde{G}_{yx}^{^{EJ}} \widetilde{J}_x + \widetilde{G}_{yy}^{^{EJ}} \widetilde{J}_y \Big) \boldsymbol{u}_y
\nonumber \\
&=& \widetilde{\bar{\bar{\boldsymbol{G}}}}_T \hspace{-2.2mm} ^{^{EJ}} \cdot \widetilde{\boldsymbol{J}}
\end{eqnarray}
ただし，
\begin{eqnarray}
\widetilde{G}_{xx}^{^{EJ}}
&=& \widetilde{Z}_{_{TE}}^{(z)} \sin ^2 \Phi + \widetilde{Z}_{_{TM}}^{(z)} \cos ^2 \Phi
\nonumber \\
&=& \frac{1}{k_t^2} \Big( k_y^2 \widetilde{Z}_{_{TE}}^{(z)} + k_x^2 \widetilde{Z}_{_{TM}}^{(z)} \Big)
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{xy}^{^{EJ}}
&=& \widetilde{G}_{yx}^{^{EJ}}
= \big( \widetilde{Z}_{_{TM}}^{(z)} - \widetilde{Z}_{_{TE}}^{(z)} \big) \sin \Phi \cos \Phi
\nonumber \\
&=& \frac{k_x k_y}{k_t^2} \Big( \widetilde{Z}_{_{TM}}^{(z)} - \widetilde{Z}_{_{TE}}^{(z)} \Big)
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{yy}^{^{EJ}}
&=& \widetilde{Z}_{_{TE}}^{(z)} \cos ^2 \Phi + \widetilde{Z}_{_{TM}}^{(z)} \sin ^2 \Phi
\nonumber \\
&=& \frac{1}{k_t^2} \Big( k_x^2 \widetilde{Z}_{_{TE}}^{(z)} + k_y^2 \widetilde{Z}_{_{TM}}^{(z)} \Big)
\end{eqnarray}
スペクトル領域の磁界型ダイアディック・グリーン関数
同様にして，境界面に接する磁界のベクトル\(\widetilde{\boldsymbol{H}}_{\tan}\)が，
\begin{eqnarray}
\widetilde{\boldsymbol{H}}_{\tan}
&=& \widetilde{H}_u' \big( -\boldsymbol{u}_u \big) + \widetilde{H}_v \boldsymbol{u}_v
\nonumber \\
&=& \widetilde{P}_{_{TM}}^{(z)} \widetilde{J}_v \big( -\boldsymbol{u}_u \big) + \widetilde{P}_{_{TE}}^{(z)} \widetilde{J}_u \boldsymbol{u}_v
\end{eqnarray}
で与えられている場合を考える．成分を行列表示すると，
\begin{gather}
\begin{pmatrix}
\widetilde{\boldsymbol{H}}_{\tan} \cdot \boldsymbol{u}_u \\ \widetilde{\boldsymbol{H}}_{\tan} \cdot \boldsymbol{u}_v
\end{pmatrix}
=
\begin{pmatrix}
-\widetilde{H}_u' \\ \widetilde{H}_v
\end{pmatrix}
=
\begin{pmatrix}
0 & -\widetilde{P}_{_{TM}}^{(z)} \\
\widetilde{P}_{_{TE}}^{(z)} & 0 \\
\end{pmatrix}
\begin{pmatrix}
\widetilde{J}_u \\ \widetilde{J}_v
\end{pmatrix}
\end{gather}
\(x\)，\(y\)成分を求めると，
\begin{eqnarray}
\begin{pmatrix}
\widetilde{H}_x \\ \widetilde{H}_y
\end{pmatrix}
&=&
\begin{pmatrix}
\big( \widetilde{H}_v \boldsymbol{u}_v - \widetilde{H}_u' \boldsymbol{u}_u \big) \cdot \boldsymbol{u}_x \\
\big( \widetilde{H}_v \boldsymbol{u}_v - \widetilde{H}_u' \boldsymbol{u}_u \big) \cdot \boldsymbol{u}_y
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\boldsymbol{u}_u \cdot \boldsymbol{u}_x & \boldsymbol{u}_v \cdot \boldsymbol{u}_x \\
\boldsymbol{u}_u \cdot \boldsymbol{u}_y & \boldsymbol{u}_v \cdot \boldsymbol{u}_y
\end{pmatrix}
\begin{pmatrix}
-\widetilde{H}_u' \\ \widetilde{H}_v
\end{pmatrix}
\nonumber \\
&=& [\Phi]^t
\begin{pmatrix}
0 & -\widetilde{P}_{_{TM}}^{(z)} \\
\widetilde{P}_{_{TE}}^{(z)} & 0
\end{pmatrix}
\begin{pmatrix}
\widetilde{J}_v \\ \widetilde{J}_u
\end{pmatrix}
\nonumber \\
&=& [\Phi]^t
\begin{pmatrix}
0 & -\widetilde{P}_{_{TM}}^{(z)} \\
\widetilde{P}_{_{TE}}^{(z)} & 0
\end{pmatrix}
[\Phi]
\begin{pmatrix}
\widetilde{J}_x \\ \widetilde{J}_y
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\widetilde{G}_{xx}^{^{HJ}} & \widetilde{G}_{xy}^{^{HJ}} \\
\widetilde{G}_{yx}^{^{HJ}} & \widetilde{G}_{yy}^{^{HJ}}
\end{pmatrix}
\begin{pmatrix}
\widetilde{J}_x \\ \widetilde{J}_y
\end{pmatrix}
\end{eqnarray}
したがって，
\begin{eqnarray}
&&\begin{pmatrix}
\widetilde{G}_{xx}^{^{HJ}} & \widetilde{G}_{xy}^{^{HJ}} \\
\widetilde{G}_{yx}^{^{HJ}} & \widetilde{G}_{yy}^{^{HJ}}
\end{pmatrix}
= [\Phi]^t
\begin{pmatrix}
0 & -\widetilde{P}_{_{TM}}^{(z)} \\
\widetilde{P}_{_{TE}}^{(z)} & 0
\end{pmatrix}
[\Phi]
\nonumber \\
&=&
\begin{pmatrix}
\sin \Phi & \cos \Phi \\
-\cos \Phi & \sin \Phi
\end{pmatrix}
\begin{pmatrix}
0 & -\widetilde{P}_{_{TM}}^{(z)} \\
\widetilde{P}_{_{TE}}^{(z)} & 0
\end{pmatrix}
\begin{pmatrix}
\sin \Phi & -\cos \Phi \\
\cos \Phi & \sin \Phi
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\big( \widetilde{P}_{_{TE}}^{(z)} - \widetilde{P}_{_{TM}}^{(z)} \big) \sin \Phi \cos \Phi
& -\widetilde{P}_{_{TE}}^{(z)} \cos ^2 \Phi - \widetilde{P}_{_{TM}}^{(z)} \sin ^2 \Phi \\
\widetilde{P}_{_{TE}}^{(z)} \sin ^2 \Phi + \widetilde{P}_{_{TM}}^{(z)} \cos ^2 \Phi
& \big( \widetilde{P}_{_{TM}}^{(z)} - \widetilde{P}_{_{TE}}^{(z)} \big) \sin \Phi \cos \Phi
\end{pmatrix}
\end{eqnarray}
よって，
\(\widetilde{\boldsymbol{H}}_{\tan}
= \widetilde{\bar{\bar{\boldsymbol{G}}}}_T \hspace{-2.2mm} ^{^{HJ}} \cdot \widetilde{\boldsymbol{J}}\)
とおいて，
\(\widetilde{\bar{\bar{\boldsymbol{G}}}}_T\)を定義すると，
\begin{gather}
\widetilde{\widetilde{\bar{\bar{\boldsymbol{G}}}}}_T^{^{HJ}}
= \widetilde{G}_{xx}^{^{HJ}} \boldsymbol{u}_x \boldsymbol{u}_x
+ \widetilde{G}_{xy}^{^{HJ}} \boldsymbol{u}_x \boldsymbol{u}_y
+ \widetilde{G}_{yx}^{^{HJ}} \boldsymbol{u}_y \boldsymbol{u}_x
+ \widetilde{G}_{yy}^{^{HJ}} \boldsymbol{u}_y \boldsymbol{u}_y
\end{gather}
各成分は次のようになる．
\begin{eqnarray}
\widetilde{G}_{xx}^{^{HJ}}
&=& -\widetilde{G}_{yy}^{^{HJ}}
= \big( \widetilde{P}_{_{TE}}^{(z)} - \widetilde{P}_{_{TM}}^{(z)} \big) \sin \Phi \cos \Phi
\nonumber \\
&=& \frac{k_x k_y}{k_t^2} \Big( \widetilde{P}_{_{TE}}^{(z)} - \widetilde{P}_{_{TM}}^{(z)} \Big)
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{xy}^{^{HJ}}
&=& -\widetilde{P}_{_{TE}}^{(z)} \cos ^2 \Phi - \widetilde{P}_{_{TM}}^{(z)} \sin ^2 \Phi
\nonumber \\
&=& -\frac{1}{k_t^2} \Big( k_x^2 \widetilde{P}_{_{TE}}^{(z)} + k_y^2 \widetilde{P}_{_{TM}}^{(z)} \Big)
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{yx}^{^{HJ}}
&=& \widetilde{P}_{_{TE}}^{(z)} \sin ^2 \Phi + \widetilde{P}_{_{TM}}^{(z)} \cos ^2 \Phi
\nonumber \\
&=& \frac{1}{k_t^2} \Big( k_y^2 \widetilde{P}_{_{TE}}^{(z)} + k_x^2 \widetilde{P}_{_{TM}}^{(z)} \Big)
\end{eqnarray}