自由空間中の電流素子のスペクトル領域グリーン関数の導出
誘電体基板がなく，自由空間中の\(xy\)面上に電流源がある場合を考える．
\begin{gather}
\widetilde{Z}^{(0)} = -\frac{1}{Y_0 + Y_0} = -\frac{1}{2Y_0}
\end{gather}
これより，
\begin{gather}
Z^{(0)}_{_{TE}} = -\frac{1}{2Y_{0_{TE}}}
= -\frac{1}{2\frac{k_{z0}}{\omega \mu_0}} = -\frac{\omega \mu_0}{2k_{z0}} \\
Z^{(0)}_{_{TM}} = -\frac{1}{2Y_{0_{TM}}}
= -\frac{1}{2\frac{\omega \epsilon _0}{k_{z0}}} = -\frac{k_{z0}}{2\omega \epsilon _0}
\end{gather}
よって，
\begin{eqnarray}
\widetilde{G}_{xx}^{^{EJ}}
&=& \frac{1}{k_t^2} \Big( k_y^2 \widetilde{Z}_{_{TE}}^{(0)} + k_x^2 \widetilde{Z}_{_{TM}}^{(0)} \Big)
\nonumber \\
&=& \frac{1}{k_t^2} \left\{ k_y^2 \left( -\frac{\omega \mu_0}{2k_{z0}} \right)
+ k_x^2 \left( -\frac{k_{z0}}{2\omega \epsilon _0} \right) \right\}
\nonumber \\
&=& - \frac{k_0^2 k_y^2 + k_{z0}^2 k_x^2}{2 \omega \epsilon _0 k_t^2 k_{z0} }
\end{eqnarray}
ここで，
\begin{eqnarray}
k_0^2 k_y^2 + k_{z0} k_x^2
&=& k_0^2 \big( k_t^2 - k_x^2 \big) + \big( k_0^2 - k_t^2 \big) k_x^2
\nonumber \\
&=& k_t^2 \big( k_0^2 - k_x^2 \big)
\end{eqnarray}
より，
\begin{eqnarray}
\widetilde{G}_{xx}^{^{EJ}}
&=& - \frac{k_0^2 - k_x^2}{2 \omega \epsilon _0 k_{z0} }
= \frac{k_0^2 - k_x^2}{j \omega \epsilon _0} \ \frac{1}{j2k_{z0}}
\nonumber \\
&=& -\frac{j\omega \mu_0}{k_0^2} \ \frac{1}{j2k_{z0}} \ \big( k_0^2 - k_x^2 \big)
\end{eqnarray}
また，
\begin{eqnarray}
\widetilde{G}_{xy}^{^{EJ}}
&=& \widetilde{G}_{yx}^{^{EJ}}
= \frac{k_x k_y}{k_t^2} \Big( \widetilde{Z}_{_{TM}}^{(z)} - \widetilde{Z}_{_{TE}}^{(z)} \Big)
\nonumber \\
&=& \frac{k_x k_y}{k_t^2} \left( -\frac{k_{z0}}{2\omega \epsilon _0} + \frac{\omega \mu_0}{2k_{z0}} \right)
\nonumber \\
&=& \frac{k_x k_y \big( -k_{z0}^2 + k_0^2 \big)}{2\omega \epsilon _0 k_t^2 k_{z0}}
= \frac{k_x k_y}{2\omega \epsilon _0 k_{z0}}
\nonumber \\
&=& -\frac{k_x k_y}{j\omega \epsilon _0} \ \frac{1}{j2k_{z0}}
= \frac{j\omega \mu _0}{k_0^2} \ \frac{1}{j2k_{z0}} \ k_x k_y
\end{eqnarray}
\begin{eqnarray}
\widetilde{G}_{yy}^{^{EJ}}
&=& \frac{1}{k_t^2} \Big( k_x^2 \widetilde{Z}_{_{TE}}^{(z)} + k_y^2 \widetilde{Z}_{_{TM}}^{(z)} \Big)
\nonumber \\
&=& - \frac{k_0^2 - k_y^2}{2 \omega \epsilon _0 k_{z0} }
= \frac{k_0^2 - k_y^2}{j \omega \epsilon _0} \ \frac{1}{j2k_{z0}}
\nonumber \\
&=& -\frac{j\omega \mu_0}{k_0^2} \ \frac{1}{j2k_{z0}} \ \big( k_0^2 - k_y^2 \big)
\end{eqnarray}