面電流源に対するグリーン関数

面電流源に対するグリーン関数

スペクトル領域の電界型ダイアディック・グリーン関数

　境界面に接する電界のベクトル

{\tilde{E}}_{\tan}

は，

\begin{matrix} (1) & {\tilde{E}}_{\tan} = {\tilde{E}}_{u} u_{u} + {\tilde{E}}_{v} u_{v} = {\tilde{Z}}_{_{T E}}^{(z)} {\tilde{J}}_{u} u_{u} + {\tilde{Z}}_{_{T M}}^{(z)} {\tilde{J}}_{v} u_{v} \end{matrix}

成分を行列表示すると，

\begin{matrix} (2) & (\begin{matrix} {\tilde{E}}_{\tan} \cdot u_{u} \\ {\tilde{E}}_{\tan} \cdot u_{v} \end{matrix}) = (\begin{matrix} {\tilde{E}}_{u} \\ {\tilde{E}}_{v} \end{matrix}) = (\begin{matrix} {\tilde{Z}}_{_{T E}}^{(z)} & 0 \\ 0 & {\tilde{Z}}_{_{T M}}^{(z)} \end{matrix}) (\begin{matrix} {\tilde{J}}_{u} \\ {\tilde{J}}_{v} \end{matrix}) \end{matrix}

x

，

y

成分を求めると，

\begin{array}{rcl} (\begin{array}{c} {\tilde{E}}_{x} \\ {\tilde{E}}_{y} \end{array}) & = & (\begin{array}{c} ({\tilde{E}}_{v} u_{v} + {\tilde{E}}_{u} u_{u}) \cdot u_{x} \\ ({\tilde{E}}_{v} u_{v} + {\tilde{E}}_{u} u_{u}) \cdot u_{y} \end{array}) \\ = & (\begin{array}{c} u_{u} \cdot u_{x} & u_{v} \cdot u_{x} \\ u_{u} \cdot u_{y} & u_{v} \cdot u_{y} \end{array}) (\begin{array}{c} {\tilde{E}}_{u} \\ {\tilde{E}}_{v} \end{array}) \\ = & [Φ]^{t} (\begin{array}{c} {\tilde{Z}}_{_{T E}}^{(z)} & 0 \\ 0 & {\tilde{Z}}_{_{T M}}^{(z)} \end{array}) (\begin{array}{c} {\tilde{J}}_{u} \\ {\tilde{J}}_{v} \end{array}) \\ = & [Φ]^{t} (\begin{array}{c} {\tilde{Z}}_{_{T E}}^{(z)} & 0 \\ 0 & {\tilde{Z}}_{_{T M}}^{(z)} \end{array}) (\begin{array}{c} ({\tilde{J}}_{x} u_{x} + {\tilde{J}}_{y} u_{y}) \cdot u_{u} \\ ({\tilde{J}}_{x} u_{x} + {\tilde{J}}_{y} u_{y}) \cdot u_{v} \end{array}) \\ = & [Φ]^{t} (\begin{array}{c} {\tilde{Z}}_{_{T E}}^{(z)} & 0 \\ 0 & {\tilde{Z}}_{_{T M}}^{(z)} \end{array}) (\begin{array}{c} u_{x} \cdot u_{u} & u_{y} \cdot u_{u} \\ u_{x} \cdot u_{v} & u_{y} \cdot u_{v} \end{array}) (\begin{array}{c} {\tilde{J}}_{x} \\ {\tilde{J}}_{y} \end{array}) \\ (3) & = & [Φ]^{t} (\begin{array}{c} {\tilde{Z}}_{_{T E}}^{(z)} & 0 \\ 0 & {\tilde{Z}}_{_{T M}}^{(z)} \end{array}) [Φ] (\begin{array}{c} {\tilde{J}}_{x} \\ {\tilde{J}}_{y} \end{array}) \end{array}

ここで，

\begin{matrix} (4) & (\begin{matrix} {\tilde{E}}_{x} \\ {\tilde{E}}_{y} \end{matrix}) = (\begin{matrix} {\tilde{G}}_{x x}^{^{E J}} & {\tilde{G}}_{x y}^{^{E J}} \\ {\tilde{G}}_{y x}^{^{E J}} & {\tilde{G}}_{y y}^{^{E J}} \end{matrix}) (\begin{matrix} {\tilde{J}}_{x} \\ {\tilde{J}}_{y} \end{matrix}) \end{matrix}

とおくと，

\begin{array}{rcl} (\begin{array}{c} {\tilde{G}}_{x x}^{^{E J}} & {\tilde{G}}_{x y}^{^{E J}} \\ {\tilde{G}}_{y x}^{^{E J}} & {\tilde{G}}_{y y}^{^{E J}} \end{array}) = [Φ]^{t} (\begin{array}{c} {\tilde{Z}}_{_{T E}}^{(z)} & 0 \\ 0 & {\tilde{Z}}_{_{T M}}^{(z)} \end{array}) [Φ] \\ = & (\begin{array}{c} \sin Φ & \cos Φ \\ - \cos Φ & \sin Φ \end{array}) (\begin{array}{c} {\tilde{Z}}_{_{T E}}^{(z)} & 0 \\ 0 & {\tilde{Z}}_{_{T M}}^{(z)} \end{array}) (\begin{array}{c} \sin Φ & - \cos Φ \\ \cos Φ & \sin Φ \end{array}) \\ (5) & = & (\begin{array}{c} {\tilde{Z}}_{_{T E}}^{(z)} \sin^{2} Φ + {\tilde{Z}}_{_{T M}}^{(z)} \cos^{2} Φ & ({\tilde{Z}}_{_{T M}}^{(z)} - {\tilde{Z}}_{_{T E}}^{(z)}) \sin Φ \cos Φ \\ ({\tilde{Z}}_{_{T M}}^{(z)} - {\tilde{Z}}_{_{T E}}^{(z)}) \sin Φ \cos Φ & {\tilde{Z}}_{_{T E}}^{(z)} \cos^{2} Φ + {\tilde{Z}}_{_{T M}}^{(z)} \sin^{2} Φ \end{array}) \end{array}

これより，

\begin{matrix} (6) & {\tilde{\bar{\bar{G}}}}_{T}^{^{E J}} = {\tilde{G}}_{x x}^{^{E J}} u_{x} u_{x} + {\tilde{G}}_{x y}^{^{E J}} u_{x} u_{y} + {\tilde{G}}_{y x}^{^{E J}} u_{y} u_{x} + {\tilde{G}}_{y y}^{^{E J}} u_{y} u_{y} \end{matrix}

とおくと，

{\tilde{E}}_{\tan}

は，

\begin{array}{rcl} {\tilde{E}}_{\tan} & = & ({\tilde{G}}_{x x}^{^{E J}} {\tilde{J}}_{x} + {\tilde{G}}_{x y}^{^{E J}} {\tilde{J}}_{y}) u_{x} + ({\tilde{G}}_{y x}^{^{E J}} {\tilde{J}}_{x} + {\tilde{G}}_{y y}^{^{E J}} {\tilde{J}}_{y}) u_{y} \\ (7) & = & {\tilde{\bar{\bar{G}}}}_{T}^{^{E J}} \cdot \tilde{J} \end{array}

ただし，

\begin{array}{rcl} {\tilde{G}}_{x x}^{^{E J}} & = & {\tilde{Z}}_{_{T E}}^{(z)} \sin^{2} Φ + {\tilde{Z}}_{_{T M}}^{(z)} \cos^{2} Φ \\ (8) & = & \frac{1}{k_{t}^{2}} (k_{y}^{2} {\tilde{Z}}_{_{T E}}^{(z)} + k_{x}^{2} {\tilde{Z}}_{_{T M}}^{(z)}) \end{array}

\begin{array}{rcl} {\tilde{G}}_{x y}^{^{E J}} & = & {\tilde{G}}_{y x}^{^{E J}} = ({\tilde{Z}}_{_{T M}}^{(z)} - {\tilde{Z}}_{_{T E}}^{(z)}) \sin Φ \cos Φ \\ (9) & = & \frac{k_{x} k_{y}}{k_{t}^{2}} ({\tilde{Z}}_{_{T M}}^{(z)} - {\tilde{Z}}_{_{T E}}^{(z)}) \end{array}

\begin{array}{rcl} {\tilde{G}}_{y y}^{^{E J}} & = & {\tilde{Z}}_{_{T E}}^{(z)} \cos^{2} Φ + {\tilde{Z}}_{_{T M}}^{(z)} \sin^{2} Φ \\ (10) & = & \frac{1}{k_{t}^{2}} (k_{x}^{2} {\tilde{Z}}_{_{T E}}^{(z)} + k_{y}^{2} {\tilde{Z}}_{_{T M}}^{(z)}) \end{array}

スペクトル領域の磁界型ダイアディック・グリーン関数

同様にして，境界面に接する磁界のベクトル

{\tilde{H}}_{\tan}

が，

\begin{array}{rcl} {\tilde{H}}_{\tan} & = & {\tilde{H}}_{u}^{'} (- u_{u}) + {\tilde{H}}_{v} u_{v} \\ (11) & = & {\tilde{P}}_{_{T M}}^{(z)} {\tilde{J}}_{v} (- u_{u}) + {\tilde{P}}_{_{T E}}^{(z)} {\tilde{J}}_{u} u_{v} \end{array}

で与えられている場合を考える．成分を行列表示すると，

\begin{matrix} (12) & (\begin{matrix} {\tilde{H}}_{\tan} \cdot u_{u} \\ {\tilde{H}}_{\tan} \cdot u_{v} \end{matrix}) = (\begin{matrix} - {\tilde{H}}_{u}^{'} \\ {\tilde{H}}_{v} \end{matrix}) = (\begin{matrix} 0 & - {\tilde{P}}_{_{T M}}^{(z)} \\ {\tilde{P}}_{_{T E}}^{(z)} & 0 \end{matrix}) (\begin{matrix} {\tilde{J}}_{u} \\ {\tilde{J}}_{v} \end{matrix}) \end{matrix}

x

，

y

成分を求めると，

\begin{array}{rcl} (\begin{array}{c} {\tilde{H}}_{x} \\ {\tilde{H}}_{y} \end{array}) & = & (\begin{array}{c} ({\tilde{H}}_{v} u_{v} - {\tilde{H}}_{u}^{'} u_{u}) \cdot u_{x} \\ ({\tilde{H}}_{v} u_{v} - {\tilde{H}}_{u}^{'} u_{u}) \cdot u_{y} \end{array}) \\ = & (\begin{array}{c} u_{u} \cdot u_{x} & u_{v} \cdot u_{x} \\ u_{u} \cdot u_{y} & u_{v} \cdot u_{y} \end{array}) (\begin{array}{c} - {\tilde{H}}_{u}^{'} \\ {\tilde{H}}_{v} \end{array}) \\ = & [Φ]^{t} (\begin{array}{c} 0 & - {\tilde{P}}_{_{T M}}^{(z)} \\ {\tilde{P}}_{_{T E}}^{(z)} & 0 \end{array}) (\begin{array}{c} {\tilde{J}}_{v} \\ {\tilde{J}}_{u} \end{array}) \\ = & [Φ]^{t} (\begin{array}{c} 0 & - {\tilde{P}}_{_{T M}}^{(z)} \\ {\tilde{P}}_{_{T E}}^{(z)} & 0 \end{array}) [Φ] (\begin{array}{c} {\tilde{J}}_{x} \\ {\tilde{J}}_{y} \end{array}) \\ (13) & = & (\begin{array}{c} {\tilde{G}}_{x x}^{^{H J}} & {\tilde{G}}_{x y}^{^{H J}} \\ {\tilde{G}}_{y x}^{^{H J}} & {\tilde{G}}_{y y}^{^{H J}} \end{array}) (\begin{array}{c} {\tilde{J}}_{x} \\ {\tilde{J}}_{y} \end{array}) \end{array}

したがって，

\begin{array}{rcl} (\begin{array}{c} {\tilde{G}}_{x x}^{^{H J}} & {\tilde{G}}_{x y}^{^{H J}} \\ {\tilde{G}}_{y x}^{^{H J}} & {\tilde{G}}_{y y}^{^{H J}} \end{array}) = [Φ]^{t} (\begin{array}{c} 0 & - {\tilde{P}}_{_{T M}}^{(z)} \\ {\tilde{P}}_{_{T E}}^{(z)} & 0 \end{array}) [Φ] \\ = & (\begin{array}{c} \sin Φ & \cos Φ \\ - \cos Φ & \sin Φ \end{array}) (\begin{array}{c} 0 & - {\tilde{P}}_{_{T M}}^{(z)} \\ {\tilde{P}}_{_{T E}}^{(z)} & 0 \end{array}) (\begin{array}{c} \sin Φ & - \cos Φ \\ \cos Φ & \sin Φ \end{array}) \\ (14) & = & (\begin{array}{c} ({\tilde{P}}_{_{T E}}^{(z)} - {\tilde{P}}_{_{T M}}^{(z)}) \sin Φ \cos Φ & - {\tilde{P}}_{_{T E}}^{(z)} \cos^{2} Φ - {\tilde{P}}_{_{T M}}^{(z)} \sin^{2} Φ \\ {\tilde{P}}_{_{T E}}^{(z)} \sin^{2} Φ + {\tilde{P}}_{_{T M}}^{(z)} \cos^{2} Φ & ({\tilde{P}}_{_{T M}}^{(z)} - {\tilde{P}}_{_{T E}}^{(z)}) \sin Φ \cos Φ \end{array}) \end{array}

よって，

{\tilde{H}}_{\tan} = {\tilde{\bar{\bar{G}}}}_{T}^{^{H J}} \cdot \tilde{J}

とおいて，

{\tilde{\bar{\bar{G}}}}_{T}

を定義すると，

\begin{matrix} (15) & {\tilde{\tilde{\bar{\bar{G}}}}}_{T}^{^{H J}} = {\tilde{G}}_{x x}^{^{H J}} u_{x} u_{x} + {\tilde{G}}_{x y}^{^{H J}} u_{x} u_{y} + {\tilde{G}}_{y x}^{^{H J}} u_{y} u_{x} + {\tilde{G}}_{y y}^{^{H J}} u_{y} u_{y} \end{matrix}

各成分は次のようになる．

\begin{array}{rcl} {\tilde{G}}_{x x}^{^{H J}} & = & - {\tilde{G}}_{y y}^{^{H J}} = ({\tilde{P}}_{_{T E}}^{(z)} - {\tilde{P}}_{_{T M}}^{(z)}) \sin Φ \cos Φ \\ (16) & = & \frac{k_{x} k_{y}}{k_{t}^{2}} ({\tilde{P}}_{_{T E}}^{(z)} - {\tilde{P}}_{_{T M}}^{(z)}) \end{array}

\begin{array}{rcl} {\tilde{G}}_{x y}^{^{H J}} & = & - {\tilde{P}}_{_{T E}}^{(z)} \cos^{2} Φ - {\tilde{P}}_{_{T M}}^{(z)} \sin^{2} Φ \\ (17) & = & - \frac{1}{k_{t}^{2}} (k_{x}^{2} {\tilde{P}}_{_{T E}}^{(z)} + k_{y}^{2} {\tilde{P}}_{_{T M}}^{(z)}) \end{array}

\begin{array}{rcl} {\tilde{G}}_{y x}^{^{H J}} & = & {\tilde{P}}_{_{T E}}^{(z)} \sin^{2} Φ + {\tilde{P}}_{_{T M}}^{(z)} \cos^{2} Φ \\ (18) & = & \frac{1}{k_{t}^{2}} (k_{y}^{2} {\tilde{P}}_{_{T E}}^{(z)} + k_{x}^{2} {\tilde{P}}_{_{T M}}^{(z)}) \end{array}