境界面での反射・透過
TE/TM波の反射係数および透過係数
電界の接線成分の反射係数 \(R_t^{E\pm}\) は，
\begin{eqnarray}
R_t^{E+} &=& \left. \frac{V_1^-}{V_1^+} \right| _{V_2^- =0}
= \left. \frac{\sqrt{Z_1} b_1}{\sqrt{Z_1} a_1} \right| _{a_2 =0}
= \left. \frac{b_1}{a_1} \right| _{a_2 =0}
= S_{11}
\\
R_t^{E-} &=& \left. \frac{V_2^+}{V_2^-} \right| _{V_1^+ =0}
= \left. \frac{\sqrt{Z_2} b_2}{\sqrt{Z_2} a_2} \right| _{a_1 =0}
= \left. \frac{b_2}{a_2} \right| _{a_1 =0}
= S_{22}
\end{eqnarray}
また，接線電界の透過係数 \(T_t^{E\pm}\) は，
\begin{eqnarray}
T_t ^{E+}
&=& \left. \frac{V_2^+}{V_1^+} \right| _{V_2^- =0}
= \left. \frac{\sqrt{Z_2} b_2}{\sqrt{Z_1} a_1} \right| _{a_2 =0}
= \sqrt{Y_1 Z_2} S_{21}
\\
T_t ^{E-}
&=& \left. \frac{V_1^-}{V_2^-} \right| _{V_1^+ =0}
= \left. \frac{\sqrt{Z_1} b_1}{\sqrt{Z_2} a_2} \right| _{a_1 =0}
= \sqrt{Y_2 Z_1} S_{12}
\end{eqnarray}
これより，TE波の接線電界の透過係数 \(T^{E\pm}_ \mathrm{te}\) は，
\begin{eqnarray}
T_ \mathrm{te}^{E+}
&=& \sqrt{Y_{1_{\mathrm{TE}}} Z_{2_{\mathrm{TE}}}} S_{21}^{^{\mathrm{TE}}}
\nonumber \\
&=& \sqrt{Y_{w1} \frac{k_{z1}}{k_1} Z_{w2} \frac{k_2}{k_{z2}}} S_{21}^{^{\mathrm{TE}}}
\nonumber \\
&=& \sqrt{\frac{Y_{w1} \cos \theta _1}{Y_{w2} \cos \theta _2}} S_{21}^{^{\mathrm{TE}}}
\\
T_ \mathrm{te}^{E-}
&=& \sqrt{Y_{2_{\mathrm{TE}}} Z_{1_{\mathrm{TE}}}} S_{12}^{^{\mathrm{TE}}}
\nonumber \\
&=& \sqrt{Y_{w2} \frac{k_{z2}}{k_2} Z_{w1} \frac{k_1}{k_{z1}}} S_{12}^{^{\mathrm{TE}}}
\nonumber \\
&=& \sqrt{\frac{Y_{w2} \cos \theta _2}{Y_{w1} \cos \theta _1}} S_{12}^{^{\mathrm{TE}}}
\end{eqnarray}
また，TM波の接線電界の透過係数 \(T^{E\pm}_ \mathrm{tm}\) は，
\begin{eqnarray}
T_ \mathrm{tm}^{E+}
&=& \sqrt{Y_{1_{\mathrm{TM}}} Z_{2_{\mathrm{TM}}}} S_{21}^{^{\mathrm{TM}}}
\nonumber \\
&=& \sqrt{Y_{w1} \frac{k_{z}}{k_{z1}} Z_{w2} \frac{k_{z2}}{k_z}} S_{21}^{^{\mathrm{TM}}}
\nonumber \\
&=& \sqrt{\frac{Y_{w1} \cos \theta _2}{Y_{w2} \cos \theta _1}} S_{21}^{^{\mathrm{TM}}}
\\
T_ \mathrm{tm}^{E-}
&=& \sqrt{Y_{2_{\mathrm{TM}}} Z_{1_{\mathrm{TM}}}} S_{12}^{^{\mathrm{TM}}}
\nonumber \\
&=& \sqrt{Y_{w2} \frac{k_{z}}{k_{z2}} Z_{w1} \frac{k_{z1}}{k_z}} S_{12}^{^{\mathrm{TM}}}
\nonumber \\
&=& \sqrt{\frac{Y_{w2} \cos \theta _1}{Y_{w1} \cos \theta _2}} S_{12}^{^{\mathrm{TM}}}
\end{eqnarray}
異なる誘電体の境界面での平面波の反射・透過
境界面での反射係数・透過係数を求める（\(d=0\)）．相対屈折率
\begin{gather}
n=k_2/k_1 = \sqrt{\epsilon _2 / \epsilon _1}
\end{gather}
を用いると，スネルの法則
\begin{gather}
\sin \theta _1 = n \sin \theta _2
\end{gather}
より，
\begin{eqnarray}
\cos \theta _2
&=& \sqrt{1-\sin ^2 \theta _2}
\nonumber \\
&=& \sqrt{1-\left( \frac{\sin \theta _1}{n} \right) ^2}
\nonumber \\
&=& \frac{1}{n} \sqrt{n^2 - \sin ^2 \theta _1}
\end{eqnarray}
反射係数 \(R^{E+}_ \mathrm{te}\)，\(R^{E+}_ \mathrm{tm}\) は，
\begin{eqnarray}
R^{E+}_ \mathrm{te}
&=& \frac{Y_{1_{\mathrm{TE}}} - Y_{2_{\mathrm{TE}}}}{Y_{1_{\mathrm{TE}}} + Y_{2_{\mathrm{TE}}}}
\nonumber \\
&=& \frac{k_{z1} - k_{z2}}{k_{z1} + k_{z2}}
\nonumber \\
&=& \frac{k_1 \cos \theta _1 - k_2 \cos \theta _2}{k_1 \cos \theta _1 + k_2 \cos \theta _2}
\nonumber \\
&=& \frac{\cos \theta _1 - n \cos \theta _2}{\cos \theta _1 + n \cos \theta _2}
\nonumber \\
&=& \frac{\cos \theta _1 - \sqrt{n^2 - \sin ^2 \theta _1}}{\cos \theta _1 + \sqrt{n^2 - \sin ^2 \theta _1}}
\\
R^{E+}_ \mathrm{tm}
&=& \frac{Z_{2_{\mathrm{TM}}} - Z_{1_{\mathrm{TM}}}}{Z_{2_{\mathrm{TM}}} + Z_{1_{\mathrm{TM}}}}
\nonumber \\
&=& \frac{k_{z2} - n^2 k_{z1}}{k_{z2} + n^2 k_{z1}}
\nonumber \\
&=& \frac{k_2 \cos \theta _2 - n^2 k_1 \cos \theta _1}{k_2 \cos \theta _2 + n^2 k_1 \cos \theta _1}
\nonumber \\
&=& \frac{\cos \theta _2 - n \cos \theta _1}{\cos \theta _2 + n \cos \theta _1}
\nonumber \\
&=& \frac{\sqrt{n^2 - \sin ^2 \theta _1} - n^2 \cos \theta _1}{\sqrt{n^2 - \sin ^2 \theta _1} + n^2 \cos \theta _1}
\end{eqnarray}
また，TE波の接線電界の透過係数 \(T^{E+}_ \mathrm{te}\)，およびTM波の接線電界の透過係数 \(T^{E+}_ \mathrm{tm}\) は，
\begin{eqnarray}
T^{E+}_ \mathrm{te}
&=& \frac{2 Y_{1_{\mathrm{TE}}}}{Y_{1_{\mathrm{TE}}} + Y_{2_{\mathrm{TE}}}}
\nonumber \\
&=& \frac{2k_{z1}}{k_{z1} + k_{z2}}
\nonumber \\
&=& \frac{2 k_1 \cos \theta _1}{k_1 \cos \theta _1 + k_2 \cos \theta _2}
\nonumber \\
&=& \frac{2 \cos \theta _1}{\cos \theta _1 + n \cos \theta _2}
\nonumber \\
&=& \frac{2 \cos \theta _1}{\cos \theta _1 + \sqrt{n^2 - \sin ^2 \theta _1}}
\\
T^{E+}_ \mathrm{tm}
&=& \frac{2 Z_{2_{\mathrm{TM}}}}{Z_{1_{\mathrm{TM}}} + Z_{2_{\mathrm{TM}}}}
\nonumber \\
&=& \frac{2 k_{z2}}{n^2 k_{z1} + k_{z2}}
\nonumber \\
&=& \frac{2 k_2 \cos \theta _2}{n^2 k_1 \cos \theta _1 + k_2 \cos \theta _2}
\nonumber \\
&=& \frac{2 n \cos \theta _2}{n^2 \cos \theta _1 + n \cos \theta _2}
\nonumber \\
&=& \frac{2\sqrt{n^2 - \sin ^2 \theta _1}}{n^2 \cos \theta _1 + \sqrt{n^2 - \sin ^2 \theta _1}}
\end{eqnarray}
実効比誘電率
実効比誘電率 \(\epsilon _{_{\mathrm{TE}}}\)，\(\epsilon _{_{\mathrm{TM}}}\) を次式で定義する．
\begin{eqnarray}
\epsilon _{_{\mathrm{TE}}}
&=& \frac{n^2-\sin ^2 \theta _1}{\cos ^2 \theta _1}
\nonumber \\
&=& \frac{n^2-\sin ^2 \theta _1}{1-\sin ^2 \theta _1}
\\
\epsilon _{_{\mathrm{TM}}}
&=& \frac{n^4 \cos ^2 \theta _1}{n^2 - \sin ^2 \theta _1}
\nonumber \\
&=& \frac{n^4 (1-\sin ^2 \theta _1)}{n^2 - \sin ^2 \theta _1}
\end{eqnarray}
これより，TE波の接線電界の透過係数 \(T^{E+}_ \mathrm{te}\)，およびTM波の接線電界の透過係数 \(T^{E+}_ \mathrm{tm}\) は，
\begin{eqnarray}
T^{E+}_ \mathrm{te}
&=& \frac{2}{1+\sqrt{\frac{n^2-\sin ^2 \theta _1}{\cos ^2 \theta _1}}}
\nonumber \\
&=& \frac{2}{1+\sqrt{\epsilon _{_{\mathrm{TE}}}}}
\\
T^{E+}_ \mathrm{tm}
&=& \frac{2}{\sqrt{\frac{n^4 \cos ^2 \theta _1}{n^2 - \sin ^2 \theta _1}}+1}
\nonumber \\
&=& \frac{2}{1+\sqrt{\epsilon _{_{\mathrm{TM}}}}}
\end{eqnarray}