多層誘電体基板中に面電流源がある場合

多層誘電体基板中に面電流源がある場合

　図のように誘電体（厚み

d_{1}

の多層誘電体基板）と誘電体（厚み

d_{2}

の多層誘電体基板）の境界面（

z = 0

）に面電流源がある場合を考える．そして，領域(2)の基本行列が次のように与えられているものとする．

\begin{matrix} (1) & (\begin{matrix} {\tilde{E}}_{2} (0) \\ {\tilde{H}}_{2} (0) \end{matrix}) = [F^{(+)}] (\begin{matrix} {\tilde{E}}_{2} (d_{2}) \\ {\tilde{H}}_{2} (d_{2}) \end{matrix}), [F^{(+)}] = (\begin{matrix} F_{11}^{(+)} & F_{12}^{(+)} \\ F_{21}^{(+)} & F_{22}^{(+)} \end{matrix}) \end{matrix}

多層誘電体基板中に面電流源がある場合

逆は，

\begin{matrix} (2) & (\begin{matrix} {\tilde{E}}_{2} (d_{2}) \\ {\tilde{H}}_{2} (d_{2}) \end{matrix}) = {[F^{(+)}]}^{- 1} (\begin{matrix} {\tilde{E}}_{2} (0) \\ {\tilde{H}}_{2} (0) \end{matrix}), {[F^{(+)}]}^{- 1} = (\begin{matrix} F_{11}^{(+)'} & F_{12}^{(+)'} \\ F_{21}^{(+)'} & F_{22}^{(+)'} \end{matrix}) \end{matrix}

このとき，

z = d_{2}

の接線電磁界の連続条件より，

\begin{matrix} (3) & {\tilde{E}}_{2} (d_{2}) = {\tilde{E}}_{3} (d_{2}) \\ (4) & {\tilde{H}}_{2} (d_{2}) = {\tilde{H}}_{3} (d_{2}) \end{matrix}

また，領域(3)（

z \geq d_{2}

）では，後進波が存在しないことから，

\begin{matrix} (5) & {\tilde{H}}_{3} (d_{2}) = Y_{3} {\tilde{E}}_{3} (d_{2}) \end{matrix}

これより，

z = 0

において

z \geq 0

の誘電体を見た入力アドミタンス

Y_{i n}^{(+)}

は，

\begin{array}{rcl} Y_{i n}^{(+)} & = & \frac{{\tilde{H}}_{2} (0)}{{\tilde{E}}_{2} (0)} \\ = & \frac{F_{21}^{(+)} {\tilde{E}}_{2} (d_{2}) + F_{22}^{(+)} {\tilde{H}}_{2} (d_{2})}{F_{11}^{(+)} {\tilde{E}}_{2} (d_{2}) + F_{12}^{(+)} {\tilde{H}}_{2} (d_{2})} \\ = & \frac{F_{21}^{(+)} {\tilde{E}}_{3} (d_{2}) + F_{22}^{(+)} {\tilde{H}}_{3} (d_{2})}{F_{11}^{(+)} {\tilde{E}}_{3} (d_{2}) + F_{12}^{(+)} {\tilde{H}}_{3} (d_{2})} \\ = & \frac{F_{21}^{(+)} {\tilde{E}}_{3} (d_{2}) + F_{22}^{(+)} Y_{3} {\tilde{E}}_{3} (d_{2})}{F_{11}^{(+)} {\tilde{E}}_{3} (d_{2}) + F_{12}^{(+)} Y_{3} {\tilde{E}}_{3} (d_{2})} \\ (6) & = & \frac{F_{21}^{(+)} + F_{22}^{(+)} Y_{3}}{F_{11}^{(+)} + F_{12}^{(+)} Y_{3}} \end{array}

同様に，領域(1)の基本行列が次のように与えられているものとする．

\begin{matrix} (7) & (\begin{matrix} {\tilde{E}}_{1} (- d_{1}) \\ {\tilde{H}}_{1} (- d_{1}) \end{matrix}) = [F^{(-)}] (\begin{matrix} {\tilde{E}}_{1} (0) \\ {\tilde{H}}_{1} (0) \end{matrix}), [F^{(-)}] = (\begin{matrix} F_{11}^{(-)} & F_{12}^{(-)} \\ F_{21}^{(-)} & F_{22}^{(-)} \end{matrix}) \end{matrix}

逆は，

\begin{matrix} (8) & (\begin{matrix} {\tilde{E}}_{1} (0) \\ {\tilde{H}}_{1} (0) \end{matrix}) = {[F^{(-)}]}^{- 1} (\begin{matrix} {\tilde{E}}_{1} (- d_{1}) \\ {\tilde{H}}_{1} (- d_{1}) \end{matrix}), {[F^{(-)}]}^{- 1} = (\begin{matrix} F_{11}^{(-)'} & F_{12}^{(-)'} \\ F_{21}^{(-)'} & F_{22}^{(-)'} \end{matrix}) \end{matrix}

このとき，

z = - d_{1}

の接線電磁界の連続条件より，

\begin{matrix} (9) & {\tilde{E}}_{1} (- d_{1}) = {\tilde{E}}_{0} (- d_{1}) \\ (10) & {\tilde{H}}_{1} (- d_{1}) = {\tilde{H}}_{0} (- d_{1}) \end{matrix}

また，領域(0)（

z \leq - d_{1}

）では，前進波が存在しないことから，

\begin{matrix} (11) & {\tilde{H}}_{0} (- d_{1}) = - Y_{0} {\tilde{E}}_{0} (- d_{1}) \end{matrix}

これより，

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_{i n}^{(-)}

は，

\begin{array}{rcl} Y_{i n}^{(-)} & = & \frac{- {\tilde{H}}_{1} (0)}{{\tilde{E}}_{1} (0)} \\ = & - \frac{F_{21}^{(-)'} {\tilde{E}}_{0} (- d_{1}) + F_{22}^{(-)'} {\tilde{H}}_{0} (- d_{1})}{F_{11}^{(-)'} {\tilde{E}}_{0} (- d_{1}) + F_{12}^{(-)'} {\tilde{H}}_{0} (- d_{1})} \\ = & - \frac{F_{21}^{(-)'} {\tilde{E}}_{1} (- d_{1}) + F_{22}^{(-)'} {\tilde{H}}_{1} (- d_{1})}{F_{11}^{(-)'} {\tilde{E}}_{1} (- d_{1}) + F_{12}^{(-)'} {\tilde{H}}_{1} (- d_{1})} \\ = & - \frac{F_{21}^{(-)'} {\tilde{E}}_{1} (- d_{1}) - F_{22}^{(-)'} Y_{0} {\tilde{E}}_{1} (- d_{1})}{F_{11}^{(-)'} {\tilde{E}}_{1} (- d_{1}) - F_{12}^{(-)'} Y_{0} {\tilde{E}}_{1} (- d_{1})} \\ (12) & = & - \frac{F_{21}^{(-)'} - F_{22}^{(-)'} Y_{0}}{F_{11}^{(-)'} - F_{12}^{(-)'} Y_{0}} \end{array}

そして，

z = 0

におけるスペクトル領域の電流を

\tilde{J}

とすると，

z = 0

の電磁界の連続条件より，

\begin{array}{rcl} (13) & {\tilde{E}}_{2} (0) - {\tilde{E}}_{1} (0) & = & 0 \\ (14) & {\tilde{H}}_{2} (0) - {\tilde{H}}_{1} (0) & = & - \tilde{J} \end{array}

これより，

\begin{matrix} (15) & Y_{i n}^{(+)} {\tilde{E}}_{2} (0) + Y_{i n}^{(-)} {\tilde{E}}_{2} (0) = - \tilde{J} \end{matrix}

よって，

\begin{matrix} (16) & {\tilde{E}}_{2} (0) = {\tilde{E}}_{1} (0) = \frac{- \tilde{J}}{Y_{i n}^{(+)} + Y_{i n}^{(-)}} \equiv {\tilde{Z}}^{(0)} \tilde{J} \end{matrix}

ただし，

\begin{array}{rcl} (17) & {\tilde{Z}}^{(0)} & = & - \frac{1}{Y_{i n}^{(+)} + Y_{i n}^{(-)}} \\ (18) & Y_{i n}^{(+)} & = & \frac{F_{21}^{(+)} + F_{22}^{(+)} Y_{3}}{F_{11}^{(+)} + F_{12}^{(+)} Y_{3}} \\ (19) & Y_{i n}^{(-)} & = & - \frac{F_{21}^{(-)'} - F_{22}^{(-)'} Y_{0}}{F_{11}^{(-)'} - F_{12}^{(-)'} Y_{0}} \end{array}

また，

\begin{array}{rcl} {\tilde{H}}_{2} (0) & = & Y_{i n}^{(+)} {\tilde{E}}_{2} (0) \\ (20) & = & - \frac{Y_{i n}^{(+)} \tilde{J}}{Y_{i n}^{(+)} + Y_{i n}^{(-)}} \end{array}

\begin{array}{rcl} {\tilde{H}}_{1} (0) & = & {\tilde{H}}_{2} (0) + \tilde{J} \\ = & - \frac{Y_{i n}^{(+)} \tilde{J}}{Y_{i n}^{(+)} + Y_{i n}^{(-)}} + \tilde{J} \\ (21) & = & - \frac{Y_{i n}^{(-)} \tilde{J}}{Y_{i n}^{(+)} + Y_{i n}^{(-)}} \end{array}

z = d_{2}

の誘電体と空気の境界面では，

\begin{array}{rcl} {\tilde{E}}_{2} (d_{2}) & = & F_{11}^{(+)'} {\tilde{E}}_{2} (0) + F_{12}^{(+)'} {\tilde{H}}_{2} (0) \\ = & F_{11}^{(+)'} {\tilde{E}}_{2} (0) + F_{12}^{(+)'} Y_{i n}^{(+)} {\tilde{E}}_{2} (0) \\ = & (F_{11}^{(+)'} + F_{12}^{(+)'} Y_{i n}^{(+)}) {\tilde{Z}}^{(0)} \tilde{J} \\ (22) & \equiv & Y_{t r a n s} {\tilde{Z}}^{(0)} \tilde{J} \equiv {\tilde{Z}}^{(d_{2})} \tilde{J} \end{array}

\begin{array}{rcl} {\tilde{H}}_{2} (d_{2}) & = & Y_{3} {\tilde{E}}_{2} (d_{2}) = Y_{3} {\tilde{Z}}^{(d_{2})} \tilde{J} \\ (23) & \equiv & {\tilde{P}}^{(d_{2})} \tilde{J} \end{array}

ここで，

\begin{matrix} (24) & Y_{t r a n s} \equiv F_{11}^{(+)'} + F_{12}^{(+)'} Y_{i n}^{(+)} \end{matrix}

したがって，

\begin{array}{rcl} (25) & {\tilde{Z}}^{(d_{2})} & = & Y_{t r a n s} {\tilde{Z}}^{(0)} = - \frac{Y_{t r a n s}}{Y_{i n}^{(+)} + Y_{i n}^{(-)}} \\ (26) & {\tilde{P}}^{(d_{2})} & = & Y_{3} {\tilde{Z}}^{(d_{2})} \end{array}