2.15 スロット素子のスペクトル領域グリーン関数の導出
単層誘電体基板の地導体面に磁流源がある場合
誘電体基板（厚さ\(d\)，比誘電率\(\epsilon _r\)）の地導体に設けたスロット素子を考える．\(z=0\)の境界条件より，
\begin{gather}
\widetilde{E}_1(0) = -\widetilde{M}
\end{gather}
単層誘電体基板の地導体面に磁流源がある場合
このとき，
\begin{gather}
\widetilde{H}_1(0) = Y_{in}^{(+)} \widetilde{E}_1(0) = -Y_{in}^{(+)} \widetilde{M} \equiv \widetilde{Y}^{(0)} \widetilde{M}
\end{gather}
ここで，
\begin{eqnarray}
\widetilde{Y}^{(0)}
&=& -Y_{in}^{(+)} = -Y_1 \frac{Y_2+jY_1 \tan k_{z1} d }{Y_1 + jY_2 \tan k_{z1} d}
\nonumber \\
&=& -Y_1 \frac{Y_2 \cos k_{z1} d+jY_1 \sin k_{z1} d }{Y_1 \cos k_{z1}d+ jY_2 \sin k_{z1} d}
\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
これより，
\begin{eqnarray}
\widetilde{Y}^{(0)}_{_{TE}}
&=& -Y_{1_{TE}} \frac{Y_{2_{TE}} \cos k_{z1} d+jY_{1_{TE}} \sin k_{z1} d }{Y_{1_{TE}} \cos k_{z1}d+ jY_{2_{TE}} \sin k_{z1} d}
\nonumber \\
&=& -\frac{k_{z1}}{\omega \mu_1} \cdot
\frac{\displaystyle{\frac{k_{z2}}{\omega \mu_2} \cos k_{z1} d+j\frac{k_{z1}}{\omega \mu_1} \sin k_{z1} d }}{\displaystyle{
\frac{k_{z1}}{\omega \mu_1} \cos k_{z1}d+ j\frac{k_{z2}}{\omega \mu_2} \sin k_{z1} d}}
\nonumber \\
&=& -\frac{jk_{z1}}{\omega \mu _0 T_e} \big( -j k_{z2} \cos k_{z1}d + k_{z1} \sin k_{z1}d \big)
\end{eqnarray}
また，
\begin{eqnarray}
\widetilde{Y}^{(0)}_{_{TM}}
&=& -Y_{1_{TM}} \frac{Y_{2_{TM}} \cos k_{z1} d+jY_{1_{TM}} \sin k_{z1} d }{Y_{1_{TM}} \cos k_{z1}d+ jY_{2_{TM}} \sin k_{z1} d}
\nonumber \\
&=& -\frac{\omega \epsilon _1}{k_{z1}} \cdot
\frac{\displaystyle{\frac{\omega \epsilon _2}{k_{z2}} \cos k_{z1} d+j\frac{\omega \epsilon _1}{k_{z1}} \sin k_{z1} d }}{\displaystyle{
\frac{\omega \epsilon _1}{k_{z1}} \cos k_{z1}d+ j\frac{\omega \epsilon _2}{k_{z2}} \sin k_{z1} d}}
\nonumber \\
&=& -\frac{j\omega \epsilon _0 \epsilon _r}{k_{z1} T_m} \big( -jk_{z1} \cos k_{z1}d + k_{z2} \epsilon _r \sin k_{z1}d \big)
\end{eqnarray}
よって，磁界型ダイアディック・グリーン関数の各成分は，
\begin{eqnarray}
\widetilde{G}_{xx}^{^{HM}}
&=& \frac{1}{k_t^2} \Big( k_y^2 \widetilde{Y}_{_{TM}}^{(0)} + k_x^2 \widetilde{Y}_{_{TE}}^{(0)} \Big)
\nonumber \\
&=& -\frac{j\omega \epsilon _0 \epsilon _r k_y^2}{k_t^2 k_{z1} T_m}
\big( k_{z2} \epsilon _r \sin k_{z1}d-jk_{z1} \cos k_{z1}d \big)
\nonumber \\
&&-\frac{jk_{z1} k_x^2}{\omega \mu _0 k_t^2 T_e} \big( k_{z1} \sin k_{z1}d-j k_{z2} \cos k_{z1}d \big)
\\
\widetilde{G}_{xy}^{^{HM}}
&=& \widetilde{G}_{yx}^{^{HM}}
= \frac{k_x k_y}{k_t^2} \Big( \widetilde{Y}_{_{TE}}^{(0)} - \widetilde{Y}_{_{TM}}^{(0)} \Big)
\nonumber \\
&=& \frac{k_x k_y}{k_t^2} \left\{
\frac{j\omega \epsilon _0 \epsilon _r}{k_{z1} T_m} \big( k_{z2} \epsilon _r \sin k_{z1}d-jk_{z1} \cos k_{z1}d \big) \right.
\nonumber \\
&&\left. -\frac{jk_{z1}}{\omega \mu _0 T_e} \big( k_{z1} \sin k_{z1}d-j k_{z2} \cos k_{z1}d \big) \right\}
\\
\widetilde{G}_{yy}^{^{HM}}
&=& \frac{1}{k_t^2} \Big( k_x^2 \widetilde{Y}_{_{TM}}^{(0)} + k_y^2 \widetilde{Y}_{_{TE}}^{(0)} \Big)
\nonumber \\
&=& -\frac{j\omega \epsilon _0 \epsilon _r k_x^2}{k_t^2 k_{z1} T_m}
\big( k_{z2} \epsilon _r \sin k_{z1}d-jk_{z1} \cos k_{z1}d \big)
\nonumber \\
&&-\frac{jk_{z1} k_y^2}{\omega \mu _0 k_t^2 T_e} \big( k_{z1} \sin k_{z1}d-j k_{z2} \cos k_{z1}d \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
また，
\begin{eqnarray}
\widetilde{H}_1(d)
&=& -\widetilde{E}_1 (0) j Y_1 \sin k_{z1} d + \widetilde{H}_1 (0) \cos k_{z1} d
\nonumber \\
&=& -\big( -j Y_1 \sin k_{z1} d + Y_{in}^{(+)} \cos k_{z1} d \big) \widetilde{M}
\nonumber \\
&\equiv& \widetilde{Y}^{(d)} \widetilde{M}
\end{eqnarray}
\begin{eqnarray}
\widetilde{E}_1(d)
&=& Z_2 \widetilde{H}_1(d)
\nonumber \\
&=& Z_2 \widetilde{Y}^{(d)} \widetilde{M}
\nonumber \\
&\equiv& \widetilde{Q}^{(d)} \widetilde{M}
\end{eqnarray}
ここで，
\begin{eqnarray}
\widetilde{Y}^{(d)}
&=& j Y_1 \sin k_{z,1} d - Y_{in}^{(+)} \cos k_{z1} d
\nonumber \\
&=& \left( j \frac{Y_1}{Y_{in}^{(+)}} \sin k_{z1} d - \cos k_{z1} d \right) Y_{in}^{(+)}
\nonumber \\
&=& \frac{-Y_2}{Y_2 \cos k_{z1} d + jY_1 \sin k_{z1} d} Y_{in}^{(+)}
\nonumber \\
&=& \frac{-Y_2}{Y_2 \cos k_{z1} d + jY_1 \sin k_{z1} d} Y_1
\frac{Y_2 \cos k_{z1} d+jY_1 \sin k_{z1} d }{Y_1 \cos k_{z1}d+ jY_2 \sin k_{z1} d}
\nonumber \\
&=& \frac{-Y_1 Y_2}{Y_1 \cos k_{z1}d+ jY_2 \sin k_{z1} d}
\end{eqnarray}
また，
\begin{gather}
\widetilde{Q}^{(d)} = Z_2 \widetilde{Y}^{(d)} = -\frac{Y_1}{Y_1 \cos k_{z1}d+ jY_2 \sin k_{z1} d} = -\widetilde{P}^{(-d)}
\end{gather}
これより，
\begin{eqnarray}
\widetilde{Q}_{_{TE}}^{(d)} &=& -\widetilde{P}_{_{TE}}^{(-d)} = -\frac{k_{z1}}{T_e}\\
\widetilde{Q}_{_{TM}}^{(d)} &=& -\widetilde{P}_{_{TM}}^{(-d)} = -\frac{k_{z2} \epsilon _r}{T_m}
\end{eqnarray}
よって，電界型ダイアディック・グリーン関数の各成分は，
\begin{eqnarray}
\widetilde{G}_{xx}^{^{EM}}
&=& -\widetilde{G}_{yy}^{^{EM}}
= \frac{k_x k_y}{k_t^2} \Big( \widetilde{Q}_{_{TE}}^{(d)} - \widetilde{Q}_{_{TM}}^{(d)} \Big)
\nonumber \\
&=& \frac{k_x k_y}{k_t^2} \left( - \frac{k_{z1}}{T_e} + \frac{k_{z2} \epsilon _r}{T_m} \right)
\nonumber \\
&=& -\frac{jk_x k_y (\epsilon _r-1) \sin k_{z1} d}{T_e T_m}
= -\widetilde{G}_{xx}^{^{HJ}} = \widetilde{G}_{yy}^{^{HJ}}
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{xy}^{^{EM}}
&=& \frac{1}{k_t^2} \Big( k_x^2 \widetilde{Q}_{_{TM}}^{(d)} + k_y^2 \widetilde{Q}_{_{TE}}^{(d)} \Big)
\nonumber \\
&=& -\frac{1}{k_t^2} \left( \frac{\epsilon _r k_x^2 k_{z2}}{T_m} + \frac{k_y^2 k_{z1}}{T_e} \right)
\nonumber \\
&=& \frac{k_{z1}}{T_e} - \frac{jk_x^2 (\epsilon _r-1) \sin k_{z1} d}{T_e T_m}
= -\widetilde{G}_{yx}^{^{HJ}}
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{yx}^{^{EM}}
&=& -\frac{1}{k_t^2} \Big( k_y^2 \widetilde{Q}_{_{TM}}^{(d)} + k_x^2 \widetilde{Q}_{_{TE}}^{(d)} \Big)
\nonumber \\
&=& \frac{1}{k_t^2} \left( \frac{\epsilon _r k_y^2 k_{z2}}{T_m} + \frac{k_x^2 k_{z1}}{T_e} \right)
\nonumber \\
&=& -\frac{k_{z1}}{T_e} + \frac{jk_y^2 (\epsilon _r-1) \sin k_{z1} d}{T_e T_m}
= -\widetilde{G}_{xy}^{^{HJ}}
\end{eqnarray}