2.12 面磁流源に対するグリーン関数
スペクトル領域の磁界型ダイアディック・グリーン関数
先に示した境界面に接する電界のベクトル\(\widetilde{\boldsymbol{E}}_{\tan}\)は，
\begin{eqnarray}
\widetilde{\boldsymbol{E}}_{\tan}
&=& \widetilde{E}_v \boldsymbol{u}_v + \widetilde{E}_u \boldsymbol{u}_u
\nonumber \\
&=& Z_{_{TM}}^{(z)} \widetilde{J}_v \boldsymbol{u}_v + Z_{_{TE}}^{(z)} \widetilde{J}_u \boldsymbol{u}_u
\end{eqnarray}
成分を行列表示すると，
\begin{gather}
\begin{pmatrix}
\widetilde{E}_u \\ \widetilde{E}_v
\end{pmatrix}
=
\begin{pmatrix}
Z_{_{TE}}^{(z)} & 0 \\
0 & Z_{_{TM}}^{(z)} \\
\end{pmatrix}
\begin{pmatrix}
\widetilde{J}_u \\ \widetilde{J}_v
\end{pmatrix}
\end{gather}
双対性より，
\begin{eqnarray}
\widetilde{\boldsymbol{H}}_{\tan}^f
&=& \widetilde{H}_u^f \boldsymbol{u}_u + \widetilde{H}_v^f \boldsymbol{u}_v
\nonumber \\
&=& \widetilde{Y}_{_{TM}}^{(z)} \widetilde{M}_u \boldsymbol{u}_u + \widetilde{Y}_{_{TE}}^{(z)} \widetilde{M}_v \boldsymbol{u}_v
\end{eqnarray}
成分を行列表示すると，
\begin{gather}
\begin{pmatrix}
\widetilde{\boldsymbol{H}}_{\tan}^f \cdot \boldsymbol{u}_u \\ \widetilde{\boldsymbol{H}}_{\tan}^f \cdot \boldsymbol{u}_v
\end{pmatrix}
=
\begin{pmatrix}
\widetilde{H}_u^f \\ \widetilde{H}_v^f
\end{pmatrix}
=
\begin{pmatrix}
\widetilde{Y}_{_{TM}}^{(z)} & 0 \\
0 & \widetilde{Y}_{_{TE}}^{(z)} \\
\end{pmatrix}
\begin{pmatrix}
\widetilde{M}_u \\ \widetilde{M}_v
\end{pmatrix}
\end{gather}
これより，スペクトル領域のグリーン関数を求めると，
\begin{eqnarray}
&&\begin{pmatrix}
\widetilde{G}_{xx}^{^{HM}} & \widetilde{G}_{xy}^{^{HM}} \\
\widetilde{G}_{yx}^{^{HM}} & \widetilde{G}_{yy}^{^{HM}}
\end{pmatrix}
= [\Phi]^t
\begin{pmatrix}
\widetilde{Y}_{_{TM}}^{(z)} & 0 \\
0 & \widetilde{Y}_{_{TE}}^{(z)} \\
\end{pmatrix}
[\Phi]
\nonumber \\
&=&
\begin{pmatrix}
\sin \Phi & \cos \Phi \\
-\cos \Phi & \sin \Phi
\end{pmatrix}
\begin{pmatrix}
\widetilde{Y}_{_{TM}}^{(z)} & 0 \\
0 & \widetilde{Y}_{_{TE}}^{(z)} \\
\end{pmatrix}
\begin{pmatrix}
\sin \Phi & -\cos \Phi \\
\cos \Phi & \sin \Phi
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\widetilde{Y}_{_{TM}}^{(z)} \sin ^2 \Phi + \widetilde{Y}_{_{TE}}^{(z)} \cos ^2 \Phi
& \big( \widetilde{Y}_{_{TE}}^{(z)} - \widetilde{Y}_{_{TM}}^{(z)} \big) \sin \Phi \cos \Phi \\
\big( \widetilde{Y}_{_{TE}}^{(z)} - \widetilde{Y}_{_{TM}}^{(z)} \big) \sin \Phi \cos \Phi
& \widetilde{Y}_{_{TM}}^{(z)} \cos ^2 \Phi + \widetilde{Y}_{_{TE}}^{(z)} \sin ^2 \Phi
\end{pmatrix}
\end{eqnarray}
よって，
\(\widetilde{\boldsymbol{H}}_{\tan}^f
= \widetilde{\bar{\bar{\boldsymbol{G}}}}_T^{^{HM}} \cdot \widetilde{\boldsymbol{M}}\)
より，
\(\widetilde{\bar{\bar{\boldsymbol{G}}}}_T\)を定義して，
\begin{gather}
\widetilde{\bar{\bar{\boldsymbol{G}}}}_T^{^{HM}}
= \widetilde{G}_{xx}^{^{HM}} \boldsymbol{u}_x \boldsymbol{u}_x + \widetilde{G}_{xy}^{^{HM}} \boldsymbol{u}_x \boldsymbol{u}_y +
\widetilde{G}_{yx}^{^{HM}} \boldsymbol{u}_y \boldsymbol{u}_x + \widetilde{G}_{yy}^{^{HM}} \boldsymbol{u}_y \boldsymbol{u}_y
\end{gather}
ここで，
\begin{eqnarray}
\widetilde{G}_{xx}^{^{HM}}
&=& \widetilde{Y}_{_{TM}}^{(z)} \sin ^2 \Phi + \widetilde{Y}_{_{TE}}^{(z)} \cos ^2 \Phi
\nonumber \\
&=& \frac{1}{k_t^2} \Big( k_y^2 \widetilde{Y}_{_{TM}}^{(z)} + k_x^2 \widetilde{Y}_{_{TE}}^{(z)} \Big)
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{xy}^{^{HM}}
&=& \widetilde{G}_{yx}^{^{HM}}
= \big( \widetilde{Y}_{_{TE}}^{(z)} - \widetilde{Y}_{_{TM}}^{(z)} \big) \sin \Phi \cos \Phi
\nonumber \\
&=& \frac{k_x k_y}{k_t^2} \Big( \widetilde{Y}_{_{TE}}^{(z)} - \widetilde{Y}_{_{TM}}^{(z)} \Big)
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{yy}^{^{HM}}
&=& \widetilde{Y}_{_{TM}}^{(z)} \cos ^2 \Phi + \widetilde{Y}_{_{TE}}^{(z)} \sin ^2 \Phi
\nonumber \\
&=& \frac{1}{k_t^2} \Big( k_x^2 \widetilde{Y}_{_{TM}}^{(z)} + k_y^2 \widetilde{Y}_{_{TE}}^{(z)} \Big)
\end{eqnarray}
スペクトル領域の電界型ダイアディック・グリーン関数
同様に，先に示した境界面に接する磁界のベクトル\(\widetilde{\boldsymbol{H}}_{\tan}\)は，
\begin{eqnarray}
\widetilde{\boldsymbol{H}}_{\tan}
&=& \widetilde{H}_v' (-\boldsymbol{u}_v) + \widetilde{H}_u \boldsymbol{u}_u
\nonumber \\
&=& -P_{_{TE}}^{(z)} \widetilde{J}_u \boldsymbol{u}_v + P_{_{TM}}^{(z)} \widetilde{J}_v \boldsymbol{u}_u
\end{eqnarray}
成分を行列表示すると，
\begin{gather}
\begin{pmatrix}
\widetilde{H}_u \\ -\widetilde{H}_v'
\end{pmatrix}
=
\begin{pmatrix}
0 & P_{_{TM}}^{(z)} \\
-P_{_{TE}}^{(z)} & 0 \\
\end{pmatrix}
\begin{pmatrix}
\widetilde{J}_u \\ \widetilde{J}_v
\end{pmatrix}
\end{gather}
双対性より，
\begin{eqnarray}
-\widetilde{\boldsymbol{E}}_{\tan}^f
&=& \widetilde{E}_u^{f \prime \prime} \big( -\boldsymbol{u}_u \big) + \widetilde{E}_v^{f \prime} \boldsymbol{u}_v
\nonumber \\
&=& \widetilde{Q}_{_{TE}}^{(z)} \widetilde{M}_v \big( -\boldsymbol{u}_u \big) + \widetilde{Q}_{_{TM}}^{(z)} \widetilde{M}_u \boldsymbol{u}_v
\end{eqnarray}
成分を行列表示すると，
\begin{gather}
\begin{pmatrix}
\widetilde{\boldsymbol{E}}_{\tan}^f \cdot \boldsymbol{u}_u \\ \widetilde{\boldsymbol{E}}_{\tan}^f \cdot \boldsymbol{u}_v
\end{pmatrix}
= -
\begin{pmatrix}
-\widetilde{E}_u^{f \prime \prime} \\ \widetilde{E}_v^{f\prime}
\end{pmatrix}
= -
\begin{pmatrix}
0 & -\widetilde{Q}_{_{TE}}^{(z)} \\
\widetilde{Q}_{_{TM}}^{(z)} & 0 \\
\end{pmatrix}
\begin{pmatrix}
\widetilde{M}_u \\ \widetilde{M}_v
\end{pmatrix}
\end{gather}
これより，スペクトル領域のグリーン関数を求めると，
\begin{eqnarray}
&&\begin{pmatrix}
\widetilde{G}_{xx}^{^{EM}} & \widetilde{G}_{xy}^{^{EM}} \\
\widetilde{G}_{yx}^{^{EM}} & \widetilde{G}_{yy}^{^{EM}}
\end{pmatrix}
= -[\Phi]^t
\begin{pmatrix}
0 & -\widetilde{Q}_{_{TE}}^{(z)} \\
\widetilde{Q}_{_{TM}}^{(z)} & 0
\end{pmatrix}
[\Phi]
\nonumber \\
&=& -
\begin{pmatrix}
\sin \Phi & \cos \Phi \\
-\cos \Phi & \sin \Phi
\end{pmatrix}
\begin{pmatrix}
0 & -\widetilde{Q}_{_{TE}}^{(z)} \\
\widetilde{Q}_{_{TM}}^{(z)} & 0
\end{pmatrix}
\begin{pmatrix}
\sin \Phi & -\cos \Phi \\
\cos \Phi & \sin \Phi
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\big( \widetilde{Q}_{_{TE}}^{(z)} - \widetilde{Q}_{_{TM}}^{(z)} \big) \sin \Phi \cos \Phi
& \widetilde{Q}_{_{TM}}^{(z)} \cos ^2 \Phi + \widetilde{Q}_{_{TE}}^{(z)} \sin ^2 \Phi \\
-\widetilde{Q}_{_{TM}}^{(z)} \sin ^2 \Phi - \widetilde{Q}_{_{TE}}^{(z)} \cos ^2 \Phi
& \big( \widetilde{Q}_{_{TM}}^{(z)} - \widetilde{Q}_{_{TE}}^{(z)} \big) \sin \Phi \cos \Phi
\end{pmatrix}
\end{eqnarray}
よって，
\(\widetilde{\boldsymbol{E}}_{\tan}^f
= \widetilde{\bar{\bar{\boldsymbol{G}}}}_T^{^{EM}} \cdot \widetilde{\boldsymbol{M}}\)
より，
\(\widetilde{\bar{\bar{\boldsymbol{G}}}}_T^{^{EM}}\)
を定義して，
\begin{gather}
\widetilde{\widetilde{\bar{\bar{\boldsymbol{G}}}}}_T \hspace{-2.2mm} ^{^{EM}}
= \widetilde{G}_{xx}^{^{EM}} \boldsymbol{u}_x \boldsymbol{u}_x + \widetilde{G}_{xy}^{^{EM}} \boldsymbol{u}_x \boldsymbol{u}_y +
\widetilde{G}_{yx}^{^{EM}} \boldsymbol{u}_y \boldsymbol{u}_x + \widetilde{G}_{yy}^{^{EM}} \boldsymbol{u}_y \boldsymbol{u}_y
\end{gather}
ここで，
\begin{eqnarray}
\widetilde{G}_{xx}^{^{EM}}
&=& -\widetilde{G}_{yy}^{^{EM}}
= \big( \widetilde{Q}_{_{TE}}^{(z)} - \widetilde{Q}_{_{TM}}^{(z)} \big) \sin \Phi \cos \Phi
\nonumber \\
&=& \frac{k_x k_y}{k_t^2} \Big( \widetilde{Q}_{_{TE}}^{(z)} - \widetilde{Q}_{_{TM}}^{(z)} \Big)
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{xy}^{^{EM}}
&=& \widetilde{Q}_{_{TM}}^{(z)} \cos ^2 \Phi + \widetilde{Q}_{_{TE}}^{(z)} \sin ^2 \Phi
\nonumber \\
&=& \frac{1}{k_t^2} \Big( k_x^2 \widetilde{Q}_{_{TM}}^{(z)} + k_y^2 \widetilde{Q}_{_{TE}}^{(z)} \Big)
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{yx}^{^{EM}}
&=& -\widetilde{Q}_{_{TM}}^{(z)} \sin ^2 \Phi - \widetilde{Q}_{_{TE}}^{(z)} \cos ^2 \Phi
\nonumber \\
&=& -\frac{1}{k_t^2} \Big( k_y^2 \widetilde{Q}_{_{TM}}^{(z)} + k_x^2 \widetilde{Q}_{_{TE}}^{(z)} \Big)
\end{eqnarray}