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{array}{rcl} (1) & R_{t}^{E +} & = & {\frac{V_{1}^{-}}{V_{1}^{+}} |}_{V_{2}^{-} = 0} = {\frac{\sqrt{Z_{1}} b_{1}}{\sqrt{Z_{1}} a_{1}} |}_{a_{2} = 0} = {\frac{b_{1}}{a_{1}} |}_{a_{2} = 0} = S_{11} \\ (2) & R_{t}^{E -} & = & {\frac{V_{2}^{+}}{V_{2}^{-}} |}_{V_{1}^{+} = 0} = {\frac{\sqrt{Z_{2}} b_{2}}{\sqrt{Z_{2}} a_{2}} |}_{a_{1} = 0} = {\frac{b_{2}}{a_{2}} |}_{a_{1} = 0} = S_{22} \end{array}

また，接線電界の透過係数

T_{t}^{E \pm}

は，

\begin{array}{rcl} (3) & T_{t}^{E +} & = & {\frac{V_{2}^{+}}{V_{1}^{+}} |}_{V_{2}^{-} = 0} = {\frac{\sqrt{Z_{2}} b_{2}}{\sqrt{Z_{1}} a_{1}} |}_{a_{2} = 0} = \sqrt{Y_{1} Z_{2}} S_{21} \\ (4) & T_{t}^{E -} & = & {\frac{V_{1}^{-}}{V_{2}^{-}} |}_{V_{1}^{+} = 0} = {\frac{\sqrt{Z_{1}} b_{1}}{\sqrt{Z_{2}} a_{2}} |}_{a_{1} = 0} = \sqrt{Y_{2} Z_{1}} S_{12} \end{array}

これより，TE波の接線電界の透過係数

T_{te}^{E \pm}

は，

\begin{array}{rcl} T_{te}^{E +} & = & \sqrt{Y_{1_{TE}} Z_{2_{TE}}} S_{21}^{^{TE}} \\ = & \sqrt{Y_{w 1} \frac{k_{z 1}}{k_{1}} Z_{w 2} \frac{k_{2}}{k_{z 2}}} S_{21}^{^{TE}} \\ (5) & = & \sqrt{\frac{Y_{w 1} \cos θ_{1}}{Y_{w 2} \cos θ_{2}}} S_{21}^{^{TE}} \\ T_{te}^{E -} & = & \sqrt{Y_{2_{TE}} Z_{1_{TE}}} S_{12}^{^{TE}} \\ = & \sqrt{Y_{w 2} \frac{k_{z 2}}{k_{2}} Z_{w 1} \frac{k_{1}}{k_{z 1}}} S_{12}^{^{TE}} \\ (6) & = & \sqrt{\frac{Y_{w 2} \cos θ_{2}}{Y_{w 1} \cos θ_{1}}} S_{12}^{^{TE}} \end{array}

また，TM波の接線電界の透過係数

T_{tm}^{E \pm}

は，

\begin{array}{rcl} T_{tm}^{E +} & = & \sqrt{Y_{1_{TM}} Z_{2_{TM}}} S_{21}^{^{TM}} \\ = & \sqrt{Y_{w 1} \frac{k_{z}}{k_{z 1}} Z_{w 2} \frac{k_{z 2}}{k_{z}}} S_{21}^{^{TM}} \\ (7) & = & \sqrt{\frac{Y_{w 1} \cos θ_{2}}{Y_{w 2} \cos θ_{1}}} S_{21}^{^{TM}} \\ T_{tm}^{E -} & = & \sqrt{Y_{2_{TM}} Z_{1_{TM}}} S_{12}^{^{TM}} \\ = & \sqrt{Y_{w 2} \frac{k_{z}}{k_{z 2}} Z_{w 1} \frac{k_{z 1}}{k_{z}}} S_{12}^{^{TM}} \\ (8) & = & \sqrt{\frac{Y_{w 2} \cos θ_{1}}{Y_{w 1} \cos θ_{2}}} S_{12}^{^{TM}} \end{array}

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{matrix} (9) & n = k_{2} / k_{1} = \sqrt{ϵ_{2} / ϵ_{1}} \end{matrix}

を用いると，スネルの法則

\begin{matrix} (10) & \sin θ_{1} = n \sin θ_{2} \end{matrix}

より，

\begin{array}{rcl} \cos θ_{2} & = & \sqrt{1 - \sin^{2} θ_{2}} \\ = & \sqrt{1 - {(\frac{\sin θ_{1}}{n})}^{2}} \\ (11) & = & \frac{1}{n} \sqrt{n^{2} - \sin^{2} θ_{1}} \end{array}

反射係数

R_{te}^{E +}

，

R_{tm}^{E +}

は，

\begin{array}{rcl} R_{te}^{E +} & = & \frac{Y_{1_{TE}} - Y_{2_{TE}}}{Y_{1_{TE}} + Y_{2_{TE}}} \\ = & \frac{k_{z 1} - k_{z 2}}{k_{z 1} + k_{z 2}} \\ = & \frac{k_{1} \cos θ_{1} - k_{2} \cos θ_{2}}{k_{1} \cos θ_{1} + k_{2} \cos θ_{2}} \\ = & \frac{\cos θ_{1} - n \cos θ_{2}}{\cos θ_{1} + n \cos θ_{2}} \\ (12) & = & \frac{\cos θ_{1} - \sqrt{n^{2} - \sin^{2} θ_{1}}}{\cos θ_{1} + \sqrt{n^{2} - \sin^{2} θ_{1}}} \\ R_{tm}^{E +} & = & \frac{Z_{2_{TM}} - Z_{1_{TM}}}{Z_{2_{TM}} + Z_{1_{TM}}} \\ = & \frac{k_{z 2} - n^{2} k_{z 1}}{k_{z 2} + n^{2} k_{z 1}} \\ = & \frac{k_{2} \cos θ_{2} - n^{2} k_{1} \cos θ_{1}}{k_{2} \cos θ_{2} + n^{2} k_{1} \cos θ_{1}} \\ = & \frac{\cos θ_{2} - n \cos θ_{1}}{\cos θ_{2} + n \cos θ_{1}} \\ (13) & = & \frac{\sqrt{n^{2} - \sin^{2} θ_{1}} - n^{2} \cos θ_{1}}{\sqrt{n^{2} - \sin^{2} θ_{1}} + n^{2} \cos θ_{1}} \end{array}

また，TE波の接線電界の透過係数

T_{te}^{E +}

，およびTM波の接線電界の透過係数

T_{tm}^{E +}

は，

\begin{array}{rcl} T_{te}^{E +} & = & \frac{2 Y_{1_{TE}}}{Y_{1_{TE}} + Y_{2_{TE}}} \\ = & \frac{2 k_{z 1}}{k_{z 1} + k_{z 2}} \\ = & \frac{2 k_{1} \cos θ_{1}}{k_{1} \cos θ_{1} + k_{2} \cos θ_{2}} \\ = & \frac{2 \cos θ_{1}}{\cos θ_{1} + n \cos θ_{2}} \\ (14) & = & \frac{2 \cos θ_{1}}{\cos θ_{1} + \sqrt{n^{2} - \sin^{2} θ_{1}}} \\ T_{tm}^{E +} & = & \frac{2 Z_{2_{TM}}}{Z_{1_{TM}} + Z_{2_{TM}}} \\ = & \frac{2 k_{z 2}}{n^{2} k_{z 1} + k_{z 2}} \\ = & \frac{2 k_{2} \cos θ_{2}}{n^{2} k_{1} \cos θ_{1} + k_{2} \cos θ_{2}} \\ = & \frac{2 n \cos θ_{2}}{n^{2} \cos θ_{1} + n \cos θ_{2}} \\ (15) & = & \frac{2 \sqrt{n^{2} - \sin^{2} θ_{1}}}{n^{2} \cos θ_{1} + \sqrt{n^{2} - \sin^{2} θ_{1}}} \end{array}

実効比誘電率

　実効比誘電率

ϵ_{_{TE}}

，

ϵ_{_{TM}}

を次式で定義する．

\begin{array}{rcl} ϵ_{_{TE}} & = & \frac{n^{2} - \sin^{2} θ_{1}}{\cos^{2} θ_{1}} \\ (16) & = & \frac{n^{2} - \sin^{2} θ_{1}}{1 - \sin^{2} θ_{1}} \\ ϵ_{_{TM}} & = & \frac{n^{4} \cos^{2} θ_{1}}{n^{2} - \sin^{2} θ_{1}} \\ (17) & = & \frac{n^{4} (1 - \sin^{2} θ_{1})}{n^{2} - \sin^{2} θ_{1}} \end{array}

これより，TE波の接線電界の透過係数

T_{te}^{E +}

，およびTM波の接線電界の透過係数

T_{tm}^{E +}

は，

\begin{array}{rcl} T_{te}^{E +} & = & \frac{2}{1 + \sqrt{\frac{n^{2} - \sin^{2} θ_{1}}{\cos^{2} θ_{1}}}} \\ (18) & = & \frac{2}{1 + \sqrt{ϵ_{_{TE}}}} \\ T_{tm}^{E +} & = & \frac{2}{\sqrt{\frac{n^{4} \cos^{2} θ_{1}}{n^{2} - \sin^{2} θ_{1}}} + 1} \\ (19) & = & \frac{2}{1 + \sqrt{ϵ_{_{TM}}}} \end{array}