2.14 マイクロストリップ素子のスペクトル領域グリーン関数の導出
地導体板付き単層誘電体基板の表面に面電流源がある場合
マイクロストリップ素子として，厚み\(d\)の単層誘電体基板（比誘電率\(\epsilon _r\)）の片面に地導体板を付け（\(z=-d\)），
誘電体と空気の境界面（\(z=0\)）に電流源がある場合を考える．このとき，\(-d \leq z \leq 0\)の誘電体を領域(1)として，スペクトル領域の電界および磁界は，
\begin{eqnarray}
\widetilde{E}_1 (z) &=& \widetilde{E}^+_1 e^{-jk_{z1} z} + \widetilde{E}^-_1 e^{jk_{z1} z}
\\
\widetilde{H}_1 (z) &=& Y_1 \big( \widetilde{E}^+_1 e^{-jk_{z1} z} - \widetilde{E}^-_1 e^{jk_{z1} z})
\end{eqnarray}
地導体板付き単層誘電体基板の表面に面電流源がある場合
また，\(z \geq 0\) の自由空間を領域(2)として，電界および磁界は，
\begin{eqnarray}
\widetilde{E}_2 (z) &=& \widetilde{E}^+_2 e^{-jk_{z2} z}
\\
\widetilde{H}_2 (z) &=& Y_2 \widetilde{E}^+_2 e^{-jk_{z2} z}
\end{eqnarray}
地導体板が完全導体のとき，\(z=-d\) の電界の接線成分はゼロゆえ，
\begin{gather}
\widetilde{E}_1 (-d) = \widetilde{E}^+_1 e^{jk_{z1} d} + \widetilde{E}^-_1 e^{-jk_{z1} d} =0
\end{gather}
よって，
\begin{gather}
\widetilde{E}^+_1 = -\widetilde{E}^-_1 e^{-j2k_{z1} d}
\end{gather}
これより，
\begin{eqnarray}
\widetilde{E}_1 (z)
&=& -\widetilde{E}^-_1 e^{-j2k_{z1} d} e^{-jk_{z1} z} + \widetilde{E}^-_1 e^{jk_{z1} z}
\nonumber \\
&=& \widetilde{E}^-_1 e^{-jk_{z1} z} \Big( -e^{-jk_{z1} (z+d)} + e^{jk_{z1} (z+d)} \Big)
\nonumber \\
&=& \widetilde{E}^-_1 e^{-jk_{z1} z} (j2) \sin k_{z1} (z+d)
\nonumber \\
&\equiv& C_1 \sin k_{z1} (z+d)
\end{eqnarray}
また，
\begin{eqnarray}
\widetilde{H}_1 (z)
&=& Y_1 \Big( -\widetilde{E}^-_1 e^{-j2k_{z1} d} e^{-jk_{z} z} - \widetilde{E}^-_1 e^{jk_{z1} z} \Big)
\nonumber \\
&=& Y_1 \widetilde{E}^-_1 e^{-jk_{z1} z} \Big( -e^{-jk_{z1} (z+d)} - e^{jk_{z1} (z+d)} \Big)
\nonumber \\
&=& Y_1 \widetilde{E}^-_1 e^{-jk_{z1} z} (-2) \cos k_{z1} (z+d)
\nonumber \\
&=& jY_1 C_1 \cos k_{z1} (z+d)
\end{eqnarray}
誘電体と空気の境界（\(z=0\)）において，\(z\geq 0\) 自由空間を見た入力アドミタンス\(Y_{in}^{(+)}\)は，
\begin{gather}
Y_{in}^{(+)} = \frac{\widetilde{H}_2(0)}{\widetilde{E}_2(0)}
= \frac{Y_2 \widetilde{E}^-_2}{\widetilde{E}^-_2} = Y_2
\end{gather}
一方，誘電体と空気の境界（\(z=0\)）において，\(z\leq 0\) の誘電体を見た入力アドミタンス\(Y_{in}^{(-)}\)は，\(z=-d\) での電磁界の連続条件を考慮すると，
\begin{eqnarray}
Y_{in}^{(-)}
&=& \frac{-\widetilde{H}_1(0)}{\widetilde{E}_1(0)}
\nonumber \\
&=& -\frac{-\widetilde{E}_1(-d)jY_1 \sin k_{z1} d + \widetilde{H}_1(-d) \cos k_{z1} d}{
\widetilde{E}_1(-d) \cos k_{z1} d - \widetilde{H}_1(-d) j Z_1 \sin k_{z1} d}
\nonumber \\
&=& \frac{\cos k_{z1} d}{j Z_1 \sin k_{z1} d}
\nonumber \\
&=& -j Y_1 \cot k_{z1} d
\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
\(z=0\) におけるスペクトル領域の電流を\(\widetilde{J}\)とすると，\(z=0\) の電磁界の連続条件より，
\begin{eqnarray}
\widetilde{E}_2(0) - \widetilde{E}_1(0) &=& 0
\\
\widetilde{H}_2(0) - \widetilde{H}_1(0) &=& -\widetilde{J}
\end{eqnarray}
これより，
\begin{gather}
Y_{in}^{(+)} \widetilde{E}_2(0) + Y_{in}^{(-)} \widetilde{E}_2(0) = -\widetilde{J}
\end{gather}
よって，
\begin{gather}
\widetilde{E}_2(0) = \widetilde{E}_1(0) = -\frac{\widetilde{J}}{Y_{in}^{(+)} + Y_{in}^{(-)}} \equiv \widetilde{Z}^{(0)} \widetilde{J}
\end{gather}
ただし，
\begin{eqnarray}
\widetilde{Z}^{(0)}
&=& -\frac{1}{Y_{in}^{(+)} + Y_{in}^{(-)}}
\nonumber \\
&=& -\frac{1}{Y_2 - jY_1 \cot k_{z1} d}
\end{eqnarray}
これより，
\begin{eqnarray}
Z^{(0)}_{_{TE}}
&=& -\frac{1}{Y_{2_{TE}} - jY_{1_{TE}} \cot k_{z1} d}
\nonumber \\
&=& -\frac{1}{\displaystyle{\frac{k_{z2}}{\omega \mu_2} -j \frac{k_{z1}}{\omega \mu _1} \cot k_{z1}d}}
\end{eqnarray}
ここで，\(\mu _2 = \mu _1 = \mu _0\) より，
\begin{eqnarray}
\widetilde{Z}^{(0)}_{_{TE}}
&=& -\frac{\omega \mu _0 \sin k_{z1} d}{k_{z2} \sin k_{z1} d -j k_{z1} \cos k_{z1} d}
\nonumber \\
&=& -\frac{j\omega \mu _0 \sin k_{z1} d}{k_{z1} \cos k_{z1} d +j k_{z2} \sin k_{z1} d}
\nonumber \\
&=& -\frac{j\omega \mu _0 \sin k_{z1} d}{T_e}
\end{eqnarray}
ただし，
\begin{gather}
T_e \equiv k_{z1} \cos k_{z1} d +j k_{z2} \sin k_{z1} d
\end{gather}
また，
\begin{eqnarray}
\widetilde{Z}^{(0)}_{_{TM}}
&=& -\frac{1}{Y_{2_{TM}} - jY_{1_{TM}} \cot k_{z1} d}
\nonumber \\
&=& -\frac{1}{\frac{\omega \epsilon _2}{k_{z2}} -j \frac{\omega \epsilon _1}{k_{z1}} \cot k_{z1}d}
\end{eqnarray}
ここで，\(\epsilon _2 = \epsilon _0\)，\(\epsilon _1 = \epsilon _r \epsilon _0\) より，
\begin{eqnarray}
\widetilde{Z}^{(0)}_{_{TM}}
&=& -\frac{k_{z1} k_{z2}}{\omega \epsilon _0} \cdot \frac{\sin k_{z1} d}{k_{z1} \sin k_{z1} d - j k_{z2} \epsilon _r \cos k_{z1} d}
\nonumber \\
&=& -\frac{k_{z1} k_{z2}}{\omega \epsilon _0} \cdot \frac{j\sin k_{z1} d}{k_{z2} \epsilon _r \cos k_{z1} d + j k_{z1} \sin k_{z1} d}
\nonumber \\
&=& -\frac{jk_{z1} k_{z2} \sin k_{z1} d}{\omega \epsilon _0 T_m}
\end{eqnarray}
ただし，
\begin{gather}
T_m \equiv k_{z2} \epsilon _r \cos k_{z1} d + j k_{z1} \sin k_{z1} d
\end{gather}
よって，ダイアディック・グリーン関数の成分\(\widetilde{G}_{xx}^{^{EJ}}\)は，
\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 \frac{j\omega \mu _0 \sin k_{z1} d}{T_e} - k_x^2 \frac{jk_{z1} k_{z2} \sin k_{z1} d}{\omega \epsilon _0 T_m} \right)
\nonumber \\
&=& -\frac{j}{\omega \epsilon _0} \frac{\sin k_{z1} d}{k_t^2} \left( \frac{k_y^2 k_0^2}{T_e} + \frac{k_x^2 k_{z1} k_{z2}}{T_m} \right)
\nonumber \\
&=& \frac{-j}{\omega \epsilon _0} \cdot \frac{\sin k_{z1} d}{k_t^2} \cdot
\frac{ k_y^2 k_0^2 T_m + k_x^2 k_{z1} k_{z2} T_e}{T_e T_m}
\nonumber \\
&=& \frac{-j}{\omega \epsilon _0} \ \frac{\sin k_{z1} d}{k_t^2} \
\frac{k_{z2} (k_x^2 k_{z1}^2+k_y k_0^2 \epsilon _r ) \cos k_{z1}d + k_{z1}(k_x^2 k_{z2}^2+k_y^2 k_0^2 ) j \sin k_{z1} d}{T_e T_m}
\end{eqnarray}
ここで，
\begin{eqnarray}
k_x^2 k_{z1}^2 + k_y^2 k_0^2 \epsilon _r
&=& k_x^2 \big( k_0^2 \epsilon _r - k_t^2 \big) + \big( k_t^2 - k_x^2 \big) k_0^2 \epsilon _r
\nonumber \\
&=& k_t^2 \big( -k_x^2 + k_0^2 \epsilon _r \big)
\end{eqnarray}
\begin{eqnarray}
k_x^2 k_{z2}^2 + k_y^2 k_0^2
&=& k_x^2 \big( k_0^2 - k_t^2 \big) + \big( k_t^2 - k_x^2 \big) k_0^2
\nonumber \\
&=& k_t^2 \big( -k_x^2 + k_0^2 \big)
\end{eqnarray}
\begin{eqnarray}
\frac{1}{\omega \epsilon _0}
&=& \frac{1}{\omega \sqrt{\epsilon _0 \mu _0}} \sqrt{\frac{\mu _0}{\epsilon _0}}
\nonumber \\
&=& \frac{Z_0}{k_0}
\end{eqnarray}
これより，
\begin{eqnarray}
\widetilde{G}_{xx}^{^{EJ}}
&=& \frac{-j Z_0}{k_0} \ \frac{\sin k_{z1} d}{k_t^2}
\frac{k_{z2} k_t^2 (\epsilon _r k_0^2 - k_x^2 ) \cos k_{z1}d + k_{z1} k_t^2 (k_0^2 -k_x^2) j \sin k_{z1} d}{T_e T_m}
\nonumber \\
&=& \frac{-j Z_0}{k_0} \
\frac{k_{z2} (\epsilon _r k_0^2 - k_x^2 ) \cos k_{z1}d + j k_{z1} (k_0^2 -k_x^2) \sin k_{z1} d}{T_e T_m} \sin k_{z1}d
\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}}^{(0)} - \widetilde{Z}_{_{TE}}^{(0)} \Big)
\nonumber \\
&=& \frac{k_x k_y}{k_t^2} \left( -\frac{jk_{z1} k_{z2} \sin k_{z1} d}{\omega \epsilon _0 T_m}
+ \frac{j\omega \mu _0 \sin k_{z1} d}{T_e} \right)
\nonumber \\
&=& -\frac{j}{\omega \epsilon _0} \frac{k_xk_y \sin k_{z1} d}{k_t^2} \left( \frac{k_{z1}k_{z2}}{T_m} - \frac{k_0^2}{T_e} \right)
\nonumber \\
&=& \frac{-j}{\omega \epsilon _0} \cdot \frac{k_x k_y \sin k_{z1} d}{k_t^2}
\cdot \frac{k_{z1} k_{z2} T_e - k_0^2 T_m}{T_e T_m}
\nonumber \\
&=& \frac{-j}{\omega \epsilon _0} \cdot
\frac{k_x k_y \sin k_{z1} d}{k_t^2} \cdot
\frac{k_{z2} (k_{z1}^2 - k_0^2 \epsilon _r) \cos k_{z1}d + k_{z1} (k_{z2}^2-k_0^2) j \sin k_{z1}d }{T_e T_m}
\nonumber \\
&=& \frac{-j}{\omega \epsilon _0} \cdot \frac{k_x k_y \sin k_{z1} d}{k_t^2} \cdot
\frac{k_{z2} (-k_t^2) \cos k_{z1}d + j k_{z1} (-k_t^2) \sin k_{z1}d }{T_e T_m}
\nonumber \\
&=& \frac{j Z_0}{k_0} \cdot
\frac{k_x k_y (k_{z2} \cos k_{z1}d + j k_{z1} \sin k_{z1}d)}{T_e T_m} \sin k_{z1} d
\end{eqnarray}
\(\widetilde{G}_{xx}^{^{EJ}}\)の計算と同様にして，
\begin{eqnarray}
\widetilde{G}_{yy}^{^{EJ}}
&=& \frac{1}{k_t^2} \Big( k_x^2 \widetilde{Z}_{_{TE}}^{(0)} + k_y^2 \widetilde{Z}_{_{TM}}^{(0)} \Big)
\nonumber \\
&=& \frac{1}{k_t^2}
\left( -k_x^2 \frac{j\omega \mu _0 \sin k_{z1} d}{T_e} - k_y^2 \frac{jk_{z1} k_{z2} \sin k_{z1} d}{\omega \epsilon _0 T_m} \right)
\nonumber \\
&=& -\frac{j}{\omega \epsilon _0} \frac{\sin k_{z1} d}{k_t^2} \left( \frac{k_x^2 k_0^2}{T_e} + \frac{k_y^2 k_{z1} k_{z2}}{T_m} \right)
\nonumber \\
&=& \frac{-j Z_0}{k_0} \
\frac{k_{z2} (\epsilon _r k_0^2 - k_y^2 ) \cos k_{z1}d + j k_{z1} (k_0^2 -k_y^2) \sin k_{z1} d}{T_e T_m} \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
一方，\(z=-d\) の地導体面上では，
\begin{gather}
\widetilde{E}_1(-d) =0
\end{gather}
また，
\begin{eqnarray}
\widetilde{H}_1(-d)
&=& \widetilde{E}_1 (0) jY_1 \sin k_{z1} d + \widetilde{H}_1 (0) \cos k_{z1} d
\nonumber \\
&=& \widetilde{E}_1 (0) jY_1 \sin k_{z1} d - Y_{in}^{(-)} \widetilde{E}_1 (0) \cos k_{z1} d
\nonumber \\
&=& -\frac{\widetilde{J}}{Y_{in}^{(+)} + Y_{in}^{(-)}} \Big( jY_1 \sin k_{z1} d - Y_{in}^{(+)} \cos k_{z1} d \Big)
\nonumber \\
&=& -\frac{\widetilde{J}}{Y_2 -jY_1 \cot k_{z1} d} \Big( jY_1 \sin k_{z1} d +j Y_1 \cot k_{z1} d \cos k_{z1} d \Big)
\nonumber \\
&=& -\frac{jY_1 \widetilde{J}}{Y_2 \sin k_{z1} d -jY_1 \cos k_{z1} d}
\nonumber \\
&=& \frac{Y_1 \widetilde{J}}{Y_1 \cos k_{z1} d +jY_2 \sin k_{z1} d} \equiv \widetilde{P}^{(-d)} \widetilde{J}
\end{eqnarray}
ここで，
\begin{gather}
\widetilde{P}^{(-d)} = \frac{Y_1}{Y_1 \cos k_{z1} d +jY_2 \sin k_{z1} d}
\end{gather}
これより（\(\mu _2 = \mu _1 = \mu _0\)），
\begin{eqnarray}
\widetilde{P}^{(-d)}_{_{TE}}
&=& \frac{Y_{1_{TE}}}{Y_{1_{TE}} \cos k_{z,1} d + j Y_{2_{TE}} \sin k_{z,1} d}
\nonumber \\
&=& \frac{\displaystyle{\frac{k_{z1}}{\omega \mu _1}}}{
\displaystyle{\frac{k_{z1}}{\omega \mu _1} \cos k_{z1} d + j \frac{k_{z2}}{\omega \mu _2} \sin k_{z1} d}}
\nonumber \\
&=& \frac{k_{z1}}{k_{z1} \cos k_{z1} d + jk_{z2} \sin k_{z1} d}
\nonumber \\
&=& \frac{k_{z1}}{T_e}
\end{eqnarray}
また（\(\epsilon _2 = \epsilon _0\)，\(\epsilon _1 = \epsilon _r \epsilon _0\)），
\begin{eqnarray}
\widetilde{P}^{(-d)}_{_{TM}}
&=& \frac{Y_{1_{TM}}}{Y_{1_{TM}} \cos k_{z1} d + j Y_{2_{TM}} \sin k_{z1} d}
\nonumber \\
&=& \frac{\displaystyle{\frac{\omega \epsilon _1}{k_{z1}}}}{
\displaystyle{\frac{\omega \epsilon _1}{k_{z1}} \cos k_{z1} d + j \frac{\omega \epsilon _2}{k_{z2}} \sin k_{z1} d}}
\nonumber \\
&=& \frac{k_{z2} \epsilon _r}{k_{z2} \epsilon _r \cos k_{z1} d + jk_{z1} \sin k_{z1} d}
\nonumber \\
&=& \frac{k_{z2} \epsilon _r}{T_m}
\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}^{^{HJ}}
&=& -\widetilde{G}_{yy}^{^{HJ}}
= \frac{k_x k_y}{k_t^2} \Big( \widetilde{P}_{_{TE}}^{(-d)} - \widetilde{P}_{_{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{k_x k_y}{k_t^2} \cdot \frac{k_{z1} T_m - \epsilon _r k_{z2} T_e}{T_e T_m}
\nonumber \\
&=& \frac{k_x k_y}{k_t^2} \ \frac{k_{z1} (k_{z2} \epsilon _r \cos k_{z1} d + j k_{z1} \sin k_{z1} d)
- \epsilon _r k_2 (k_1 \cos k_1 d + j k_2 \sin k_1 d)}{T_e T_m}
\nonumber \\
&=& \frac{k_x k_y}{k_t^2} \ \frac{(jk_{z1}^2 - j\epsilon _r k_{z2}^2) \sin k_{z1} d}{T_e T_m}
\nonumber \\
&=& \frac{jk_x k_y}{k_t^2} \
\frac{\{ (k_0^2 \epsilon _r - k_t^2) - \epsilon _r (k_0^2-k_t^2) \} \sin k_{z1} d}{T_e T_m}
\nonumber \\
&=& \frac{jk_x k_y (\epsilon _r-1) \sin k_{z1} d}{T_e T_m}
\end{eqnarray}
また，
\begin{eqnarray}
\widetilde{G}_{xy}^{^{HJ}}
&=& -\frac{1}{k_t^2} \Big( k_x^2 \widetilde{P}_{_{TE}}^{(-d)} + k_y^2 \widetilde{P}_{_{TM}}^{(-d)} \Big)
= -\frac{1}{k_t^2} \left( \frac{k_x^2 k_{z1}}{T_e} + \frac{\epsilon _r k_y^2 k_{z2}}{T_m} \right)
\nonumber \\
&=& -\frac{1}{k_t^2} \left\{ \frac{(k_t^2 - k_y^2)k_{z1}}{T_e} + \frac{\epsilon _r k_y^2 k_{z2}}{T_m} \right\}
\nonumber \\
&=& \frac{k_{z1}}{T_e} - \frac{k_y^2}{k_t^2}
\left( \frac{k_{z1}}{T_e}-\frac{\epsilon _r k_{z2}}{T_m} \right)
\nonumber \\
&=& \frac{k_{z1}}{T_e} - \frac{jk_y^2 (\epsilon _r-1) \sin k_{z1} d}{T_e T_m}
\end{eqnarray}
同様にして，
\begin{eqnarray}
\widetilde{G}_{yx}^{^{HJ}}
&=& \frac{1}{k_t^2} \Big( k_y^2 \widetilde{P}_{_{TE}}^{(-d)} + k_x^2 \widetilde{P}_{_{TM}}^{(-d)} \Big)
= \frac{1}{k_t^2} \left( \frac{k_y^2 k_{z1}}{T_e} + \frac{\epsilon _r k_x^2 k_{z2}}{T_m} \right)
\nonumber \\
&=& \frac{1}{k_t^2} \left\{ \frac{(k_t^2 - k_x^2)k_{z1}}{T_e} + \frac{\epsilon _r k_x^2 k_{z2}}{T_m} \right\}
\nonumber \\
&=& -\frac{k_{z1}}{T_e} + \frac{k_x^2}{k_t^2}
\left( \frac{k_{z1}}{T_e}-\frac{\epsilon _r k_{z2}}{T_m} \right)
\nonumber \\
&=& -\frac{k_{z1}}{T_e} + \frac{jk_x^2 (\epsilon _r-1) \sin k_{z1} d}{T_e T_m}
\end{eqnarray}