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{gather}
\begin{pmatrix}
\widetilde{E}_2(0) \\ \widetilde{H}_2(0)
\end{pmatrix}
= \left[ \boldsymbol{F}^{(+)} \right]
\begin{pmatrix}
\widetilde{E}_2(d_2) \\ \widetilde{H}_2(d_2)
\end{pmatrix}, \ \ \
\left[ \boldsymbol{F}^{(+)} \right] =
\begin{pmatrix}
F_{11}^{(+)} & F_{12}^{(+)} \\ F_{21}^{(+)} & F_{22}^{(+)}
\end{pmatrix}
\end{gather}
多層誘電体基板中に面電流源がある場合
逆は，
\begin{gather}
\begin{pmatrix}
\widetilde{E}_2(d_2) \\ \widetilde{H}_2(d_2)
\end{pmatrix}
= \left[ \boldsymbol{F}^{(+)} \right]^{-1}
\begin{pmatrix}
\widetilde{E}_2(0) \\ \widetilde{H}_2(0)
\end{pmatrix}, \ \ \
\left[ \boldsymbol{F}^{(+)} \right]^{-1} =
\begin{pmatrix}
F_{11}^{(+)\prime} & F_{12}^{(+)\prime} \\
F_{21}^{(+)\prime} & F_{22}^{(+)\prime}
\end{pmatrix}
\end{gather}
このとき，\(z=d_2\)の接線電磁界の連続条件より，
\begin{gather}
\widetilde{E}_2(d_2) = \widetilde{E}_3(d_2)
\\
\widetilde{H}_2(d_2) = \widetilde{H}_3(d_2)
\end{gather}
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{gather}
\widetilde{H}_3(d_2) = Y_3 \widetilde{E}_3(d_2)
\end{gather}
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\(Y_{in}^{(+)}\)は，
\begin{eqnarray}
Y_{in}^{(+)}
&=& \frac{\widetilde{H}_2(0)}{\widetilde{E}_2(0)}
\nonumber \\
&=& \frac{F_{21}^{(+)} \widetilde{E}_2(d_2) + F_{22}^{(+)} \widetilde{H}_2(d_2)}{F_{11}^{(+)} \widetilde{E}_2(d_2) + F_{12}^{(+)} \widetilde{H}_2(d_2)}
\nonumber \\
&=& \frac{F_{21}^{(+)} \widetilde{E}_3(d_2) + F_{22}^{(+)} \widetilde{H}_3(d_2)}{F_{11}^{(+)} \widetilde{E}_3(d_2) + F_{12}^{(+)} \widetilde{H}_3(d_2)}
\nonumber \\
&=& \frac{F_{21}^{(+)} \widetilde{E}_3(d_2) + F_{22}^{(+)} Y_3 \widetilde{E}_3(d_2)}{F_{11}^{(+)} \widetilde{E}_3(d_2) + F_{12}^{(+)} Y_3 \widetilde{E}_3(d_2)}
\nonumber \\
&=& \frac{F_{21}^{(+)} + F_{22}^{(+)} Y_3}{F_{11}^{(+)} + F_{12}^{(+)} Y_3}
\end{eqnarray}
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{gather}
\begin{pmatrix}
\widetilde{E}_1(-d_1) \\ \widetilde{H}_1(-d_1)
\end{pmatrix}
= \left[ F^{(-)} \right]
\begin{pmatrix}
\widetilde{E}_1(0) \\ \widetilde{H}_1(0)
\end{pmatrix}, \ \ \
\left[ F^{(-)} \right] =
\begin{pmatrix}
F_{11}^{(-)} & F_{12}^{(-)} \\ F_{21}^{(-)} & F_{22}^{(-)}
\end{pmatrix}
\end{gather}
逆は，
\begin{gather}
\hspace{-2mm}
\begin{pmatrix}
\widetilde{E}_1(0) \\ \widetilde{H}_1(0)
\end{pmatrix}
= \left[ F^{(-)} \right]^{-1}
\begin{pmatrix}
\widetilde{E}_1(-d_1) \\ \widetilde{H}_1(-d_1)
\end{pmatrix}, \ \ \
\left[ F^{(-)} \right]^{-1}=
\begin{pmatrix}
F_{11}^{(-)\prime} & F_{12}^{(-)\prime} \\ F_{21}^{(-)\prime} & F_{22}^{(-)\prime}
\end{pmatrix}
\end{gather}
このとき，\(z=-d_1\)の接線電磁界の連続条件より，
\begin{gather}
\widetilde{E}_1(-d_1) = \widetilde{E}_0(-d_1)
\\
\widetilde{H}_1(-d_1) = \widetilde{H}_0(-d_1)
\end{gather}
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{gather}
\widetilde{H}_0(-d_1) = -Y_0 \widetilde{E}_0(-d_1)
\end{gather}
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\(Y_{in}^{(-)}\)は，
\begin{eqnarray}
Y_{in}^{(-)}
&=& \frac{-\widetilde{H}_1(0)}{\widetilde{E}_1(0)}
\nonumber \\
&=& -\frac{F_{21}^{(-)\prime} \widetilde{E}_0(-d_1) + F_{22}^{(-)\prime} \widetilde{H}_0(-d_1)}{
F_{11}^{(-)\prime} \widetilde{E}_0(-d_1) + F_{12}^{(-)\prime} \widetilde{H}_0(-d_1)}
\nonumber \\
&=& -\frac{F_{21}^{(-)\prime} \widetilde{E}_1(-d_1) + F_{22}^{(-)\prime} \widetilde{H}_1(-d_1)}{
F_{11}^{(-)\prime} \widetilde{E}_1(-d_1) + F_{12}^{(-)\prime} \widetilde{H}_1(-d_1)}
\nonumber \\
&=& -\frac{F_{21}^{(-)\prime} \widetilde{E}_1(-d_1) - F_{22}^{(-)\prime} Y_0 \widetilde{E}_1(-d_1)}{
F_{11}^{(-)\prime} \widetilde{E}_1(-d_1) - F_{12}^{(-)\prime} Y_0 \widetilde{E}_1(-d_1)}
\nonumber \\
&=& -\frac{F_{21}^{(-)\prime} - F_{22}^{(-)\prime} Y_0}{F_{11}^{(-)\prime} - F_{12}^{(-)\prime} Y_0}
\end{eqnarray}
そして，\(z=0\) におけるスペクトル領域の電流を\(\widetilde{J}\)とすると，\(z=0\) の電磁界の連続条件より，
\begin{eqnarray}
\widetilde{E}_2(0) - \widetilde{E}_1(0) &=& 0
\\
\widetilde{H}_2(0) - \widetilde{H}_1(0) &=& -\widetilde{J}
\end{eqnarray}
これより，
\begin{gather}
Y_{in}^{(+)} \widetilde{E}_2(0) + Y_{in}^{(-)} \widetilde{E}_2(0) = -\widetilde{J}
\end{gather}
よって，
\begin{gather}
\widetilde{E}_2(0) = \widetilde{E}_1(0) = \frac{-\widetilde{J}}{Y_{in}^{(+)} + Y_{in}^{(-)}} \equiv \widetilde{Z}^{(0)} \widetilde{J}
\end{gather}
ただし，
\begin{eqnarray}
\widetilde{Z}^{(0)} &=& -\frac{1}{Y_{in}^{(+)} + Y_{in}^{(-)}}
\\
Y_{in}^{(+)} &=& \frac{F_{21}^{(+)} + F_{22}^{(+)} Y_3}{F_{11}^{(+)} + F_{12}^{(+)} Y_3}
\\
Y_{in}^{(-)} &=& -\frac{F_{21}^{(-)\prime} - F_{22}^{(-)\prime} Y_0}{F_{11}^{(-)\prime} - F_{12}^{(-)\prime} Y_0}
\end{eqnarray}
また，
\begin{eqnarray}
\widetilde{H}_2(0)
&=& Y_{in}^{(+)} \widetilde{E}_2(0)
\nonumber \\
&=& -\frac{Y_{in}^{(+)} \widetilde{J}}{Y_{in}^{(+)} + Y_{in}^{(-)}}
\end{eqnarray}
\begin{eqnarray}
\widetilde{H}_1(0)
&=& \widetilde{H}_2(0) + \widetilde{J}
\nonumber \\
&=& -\frac{Y_{in}^{(+)} \widetilde{J}}{Y_{in}^{(+)} + Y_{in}^{(-)}} + \widetilde{J}
\nonumber \\
&=& -\frac{Y_{in}^{(-)} \widetilde{J}}{Y_{in}^{(+)} + Y_{in}^{(-)}}
\end{eqnarray}
\(z=d_2\)の誘電体と空気の境界面では，
\begin{eqnarray}
\widetilde{E}_2(d_2)
&=& F_{11}^{(+)\prime } \widetilde{E}_2 (0) + F_{12}^{(+)\prime} \widetilde{H}_2 (0)
\nonumber \\
&=& F_{11}^{(+)\prime } \widetilde{E}_2 (0) + F_{12}^{(+)\prime} Y_{in}^{(+)} \widetilde{E}_2 (0)
\nonumber \\
&=& \Big( F_{11}^{(+)\prime } + F_{12}^{(+)\prime} Y_{in}^{(+)} \Big) \widetilde{Z}^{(0)} \widetilde{J}
\nonumber \\
&\equiv& Y_{trans} \widetilde{Z}^{(0)} \widetilde{J} \equiv \widetilde{Z}^{(d_2)} \widetilde{J}
\end{eqnarray}
\begin{eqnarray}
\widetilde{H}_2(d_2)
&=& Y_3 \widetilde{E}_2 (d_2) = Y_3 \widetilde{Z}^{(d_2)} \widetilde{J}
\nonumber \\
&\equiv& \widetilde{P}^{(d_2)} \widetilde{J}
\end{eqnarray}
ここで，
\begin{gather}
Y_{trans} \equiv F_{11}^{(+)\prime } + F_{12}^{(+)\prime} Y_{in}^{(+)}
\end{gather}
したがって，
\begin{eqnarray}
\widetilde{Z}^{(d_2)} &=& Y_{trans} \widetilde{Z}^{(0)} = -\frac{Y_{trans}}{Y_{in}^{(+)} + Y_{in}^{(-)}}
\\
\widetilde{P}^{(d_2)} &=& Y_3 \widetilde{Z}^{(d_2)}
\end{eqnarray}