1.8 地導体のある誘電体板による反射・透過
地導体のある単層誘電体板
厚み\(d\)の単層誘電体の場合，
\begin{gather} \left.
\begin{pmatrix}
b_1 \\ a_1
\end{pmatrix} =
[\boldsymbol{R}_{s1}] [\boldsymbol{R}_{u2}]
\begin{pmatrix}
a_2 \\ b_2
\end{pmatrix} \right| _{a_2 = -b_2}
\end{gather}
ここで，
\begin{align}
&[\boldsymbol{R}_{s1}] = \frac{1}{\sqrt{1-\Gamma ^2}}
\begin{pmatrix}
1 & \Gamma \\ \Gamma & 1
\end{pmatrix}, \ \ \ \ \
\Gamma = \frac{Y_1 - Y_2}{Y_1 + Y_2}
\\
&[\boldsymbol{R}_{u2}] =
\begin{pmatrix}
e^{jk_{z2} d} & 0 \\ 0 & e^{-jk_{z2} d}
\end{pmatrix}
\end{align}
ただし，\([\boldsymbol{R}_{s1}]\)は自由空間と誘電体の境界面でのRマトリクス，\([\boldsymbol{R}_{u2}]\)は誘電体の均一領域でのRマトリクスを示す．また，\(Y_1\)は自由空間のアドミタンス，\(Y_2\)は誘電体基板のアドミタンス，\(k_{z2}\)は誘電体中の波数ベクトルの$z$成分を示す．よって，
\begin{eqnarray}
\begin{pmatrix}
b_1 \\ a_1
\end{pmatrix} &=& \frac{1}{\sqrt{1-\Gamma ^2}}
\begin{pmatrix}
1 & \Gamma \\ \Gamma & 1
\end{pmatrix}
\begin{pmatrix}
e^{jk_{z2} d} & 0 \\ 0 & e^{-jk_{z2} d}
\end{pmatrix}
\begin{pmatrix}
-b_2 \\ b_2
\end{pmatrix} \nonumber \\
&=& \frac{1}{\sqrt{1-\Gamma ^2}}
\begin{pmatrix}
e^{jk_{z2} d} & \Gamma e^{-jk_{z2} d} \\ \Gamma e^{jk_{z2} d} & e^{-jk_{z2} d}
\end{pmatrix}
\begin{pmatrix}
-b_2 \\ b_2
\end{pmatrix}
\end{eqnarray}
したがって，反射係数\(R_t^{E+}\)は，
\begin{eqnarray}
R_t^{E+}
&=& \left. \frac{b_1}{a_1} \right| _{a_2 = -b_2}
= \frac{-e^{jk_{z2} d} + \Gamma e^{-jk_{z2} d}}{-\Gamma e^{jk_{z2} d} + e^{-jk_{z2} d}}
\nonumber \\
&=& \frac{-(Y_1+Y_2) e^{jk_{z2} d} + (Y_1-Y_2) e^{-jk_{z2} d}}{-(Y_1-Y_2)e^{jk_{z2} d} + (Y_1+Y_2) e^{-jk_{z2} d}}
\nonumber \\
&=& \frac{-Y_1 (e^{jk_{z2} d} - e^{-jk_{z2} d}) - Y_2 (e^{jk_{z2} d} + e^{-jk_{z2} d})}{
-Y_1 (e^{jk_{z2} d} - e^{-jk_{z2} d}) + Y_2 (e^{jk_{z2} d} + e^{-jk_{z2} d})}
\nonumber \\
&=& \frac{-Y_1 j \sin k_{z2}d - Y_2 \cos k_{z2}d}{-Y_1 j \sin k_{z2}d + Y_2 \cos k_{z2}d}
\nonumber \\
&=& \frac{Y_1 \sin k_{z2}d -j Y_2 \cos k_{z2}d}{Y_1 \sin k_{z2}d +j Y_2 \cos k_{z2}d}
\end{eqnarray}
これより，TE波の電界の反射係数\(R^{E+}_ \mathrm{te}\)は，
\begin{eqnarray}
R_ \mathrm{te}^{E+}
&=& \frac{Y_{1_{\mathrm{TE}}} \sin k_{z2}d -j Y_{2_{\mathrm{TE}}} \cos k_{z2}d}{Y_{1_{\mathrm{TE}}} \sin k_{z2}d +j Y_{2_{\mathrm{TE}}} \cos k_{z2}d}
\nonumber \\
&=& \frac{k_{z1} \sin k_{z2} d - j k_{z2} \cos k_{z2} d}{k_{z1} \sin k_{z2} d + j k_{z2} \cos k_{z2} d}
\end{eqnarray}
ただし，
\begin{gather}
Y_{1_{\mathrm{TE}}} = \frac{k_{z1}}{\omega \mu}, \ \ \ \ \
Y_{2_{\mathrm{TE}}} = \frac{k_{z2}}{\omega \mu}
\end{gather}
また，TM波の電界の反射係数\(R^{E+}_ \mathrm{tm}\)は，
\begin{eqnarray}
R_ \mathrm{tm}^{E+}
&=& \frac{Y_{1_{\mathrm{TM}}} \sin k_{z2}d -j Y_{2_{\mathrm{TM}}} \cos k_{z2}d}{Y_{1_{\mathrm{TM}}} \sin k_{z2}d +j Y_{2_{\mathrm{TM}}} \cos k_{z2}d}
\nonumber \\
&=& \frac{k_{z2} \sin k_{z2} d - j \epsilon _r k_{z1} \cos k_{z2} d}{k_{z2} \sin k_{z2} d + j \epsilon _r k_{z1} \cos k_{z2} d}
\end{eqnarray}
ただし，
\begin{gather}
Z_{1_{\mathrm{TM}}} = \frac{k_{z1}}{\omega \epsilon}, \ \ \ \ \
Z_{2_{\mathrm{TM}}} = \frac{k_{z2}}{\omega \epsilon \epsilon _r}
\end{gather}
地導体のある多層誘電体板
\(N\)層誘電体の場合，
地導体のある誘電体（厚み\(d\)，比誘電率\(\epsilon _r\)）を除いた\((N-1)\)層誘電体のRマトリクスを\([\boldsymbol{R}_{N-1}]\)とおくと，
\begin{gather} \left.
\begin{pmatrix}
b_1 \\ a_1
\end{pmatrix} =
[\boldsymbol{R}_{N-1}] [\boldsymbol{R}_{u,N}]
\begin{pmatrix}
a_2 \\ b_2
\end{pmatrix} \right| _{a_2 = -b_2}
\end{gather}
ここで，
\begin{eqnarray}
[\boldsymbol{R}_{N-1}] &=&
\begin{pmatrix}
R_{11} & R_{12} \\ R_{21} & R_{22}
\end{pmatrix}
\\
[\boldsymbol{R}_{u,N}] &=&
\begin{pmatrix}
e^{jk_{z} d} & 0 \\ 0 & e^{-jk_{z} d}
\end{pmatrix}
\end{eqnarray}
ただし，\(k_{z}\)は誘電体中の波数ベクトルの$z$成分を示す．よって，
\begin{eqnarray}
\begin{pmatrix}
b_1 \\ a_1
\end{pmatrix} &=&
\begin{pmatrix}
R_{11} & R_{12} \\ R_{21} & R_{22}
\end{pmatrix}
\begin{pmatrix}
e^{jk_{z} d} & 0 \\ 0 & e^{-jk_{z} d}
\end{pmatrix}
\begin{pmatrix}
-b_2 \\ b_2
\end{pmatrix} \nonumber \\
&=&
\begin{pmatrix}
R_{11} e^{jk_{z} d} & R_{12} e^{-jk_{z} d} \\ R_{21} e^{jk_{z} d} & R_{22} e^{-jk_{z} d}
\end{pmatrix}
\begin{pmatrix}
-b_2 \\ b_2
\end{pmatrix}
\end{eqnarray}
したがって，反射係数\(R_t^{E+}\)は，
\begin{eqnarray}
R_t^{E+}
&=& \left. \frac{b_1}{a_1} \right| _{a_2 = -b_2}
= \frac{-R_{11} e^{jk_{z} d} + R_{12} e^{-jk_{z} d}}{-R_{21} e^{jk_{z} d} + R_{22} e^{-jk_{z} d}}
\nonumber \\
&=& \frac{R_{11} - R_{12} e^{-j2k_{z} d}}{R_{21} - R_{22} e^{-j2k_{z} d}}
\end{eqnarray}
いま，\((N-1)\)層誘電体の散乱行列\([\boldsymbol{S}_{N-1}]\)を，
\begin{gather}
[\boldsymbol{S}_{N-1}] =
\begin{pmatrix}
S_{11} & S_{12} \\ S_{21} & S_{22}
\end{pmatrix}
\end{gather}
とすると，\([\boldsymbol{R}_{N-1}]\)マトリクスは，この散乱行列要素より
\begin{eqnarray}
[\boldsymbol{R}_{N-1}] &=&
\begin{pmatrix}
R_{11} & R_{12} \\ R_{21} & R_{22}
\end{pmatrix}
\nonumber \\
&=& \frac{1}{S_{21}}
\begin{pmatrix}
-S_{11}S_{22} + S_{21}^2 & S_{11} \\ -S_{22} & 1
\end{pmatrix}
\end{eqnarray}
で表すことができ，これより，
\begin{eqnarray}
R_t^{E+}
&=& \frac{-S_{11}S_{22} + S_{21}^2 - S_{11} e^{-j2k_{z} d}}{-S_{22} - e^{-j2k_{z} d}}
\nonumber \\
&=& S_{11} - \frac{S_{21}^2}{S_{22} + e^{-j2k_{z} d}}
\end{eqnarray}