4.5 透過係数
透過波側の自由空間と誘電体の境界面を
$z=0$
にとると，導体素子がない場合の透過波の接線電界
$\VEC{E}_{t,\tan}$
は，
\begin{gather}
\VEC{E}_{t,\tan} \Big| _{z=0}
= \Big\{ T_{te}^{E-} V_{1_{\TE}}^- (\VEC{u}_t \times \VEC{u}_z ) + T_{tm}^{E-} V_{1_{\TM}}^- \VEC{u}_t \Big\}
e^{j\VEC{k}_t \cdot \VECi{\rho}}
\end{gather}
また，導体素子による散乱波の接線電界
$\VEC{E}_{s,\tan}$
は，
\begin{gather}
\VEC{E}_{s,\tan} \Big| _{z=0}
= \frac{1}{d_xd_y} \sum _{m,n} \widetilde{\DYA{G}}_T^{_{(d_o)EJ}} (\VEC{k}_{tmn})
\cdot \SDV{J}_s (\VEC{k}_{tmn}) e^{j\VEC{k}_{tmn} \cdot \VECi{\rho}}
\end{gather}
この境界面での全透過波
$\VEC{E}^{(\FSS)}_{t,\tan}$
を，反射波と同様に次のようにフロケモードで展開する．
\begin{eqnarray}
\VEC{E}^{(\FSS)}_{t,\tan}
&=& \VEC{E}_{t,\tan} \Big| _{z=0} + \VEC{E}_{s,\tan} \Big| _{z=0}
\nonumber \\
&=& \sum _{m,n} \left\{ V_{[mn]}^- \left( \VEC{u}_{tmn} \times \VEC{u}_z \right) + V_{(mn)}^- \VEC{u}_{tmn} \right\}
e^{j\VEC{k}_{tmn} \cdot \VECi{\rho}}
\end{eqnarray}
透過係数を求めるため，両者を等しくおき，両辺に
$\psi _{00}^* (\VECi{\rho})= e^{-j\VEC{k}_{t00} \cdot \VECi{\rho}}$
を乗じ，単位セル$S$にわたり積分して，
\begin{eqnarray}
&&\int _S \Big[
\sum _{m,n} \left\{ V_{[mn]}^- \left( \VEC{u}_{tmn} \times \VEC{u}_z \right) + V_{(mn)}^- \VEC{u}_{tmn} \right\}
e^{j\VEC{k}_{tmn} \cdot \VECi{\rho}} \Big] e^{-j\VEC{k}_{t00} \cdot \VECi{\rho}} dS
\nonumber \\
&=& \int _S \Big[ \Big\{ T_{te}^{E-} V_{1_{\TE}}^- (\VEC{u}_t \times \VEC{u}_z ) + T_{tm}^{E-} V_{1_{\TM}}^- \VEC{u}_t \Big\}
e^{j\VEC{k}_t \cdot \VECi{\rho}} \Big] e^{-j\VEC{k}_{t00} \cdot \VECi{\rho}} dS
\nonumber \\
&&+ \int _S \Big[ \frac{1}{d_xd_y} \sum _{m,n} \widetilde{\DYA{G}}_T^{_{(d_o)EJ}} (\VEC{k}_{tmn})
\cdot \SDV{J}_s (\VEC{k}_{tmn}) e^{j\VEC{k}_{tmn} \cdot \VECi{\rho}} \Big] e^{-j\VEC{k}_{t00} \cdot \VECi{\rho}} dS
\end{eqnarray}
直交性より，
\begin{align}
&\Big\{ V_{[00]}^- \left( \VEC{u}_t \times \VEC{u}_z \right) + V_{(00)}^- \VEC{u}_t \Big\}
\nonumber \\
&= \Big\{ T_{te}^{E-} V_{1_{\TE}}^- \left( \VEC{u}_t \times \VEC{u}_z \right) + T_{tm}^{E-} V_{1_{\TM}}^- \VEC{u}_t \Big\}
+ \frac{1}{d_xd_y} \widetilde{\DYA{G}}_T^{_{(d_o)EJ}} (\VEC{k}_{t}) \cdot \SDV{J}_s (\VEC{k}_{t})
\end{align}
これより，$(\VEC{u}_t \times \VEC{u}_z)$成分，$\VEC{u}_t$成分は，
\begin{eqnarray}
V_{[00]}^- &=& T_{te}^{E-} V_{1_{\TE}}^- + \frac{1}{d_xd_y}
\left( \VEC{u}_t \times \VEC{u}_z \right) \cdot \widetilde{\DYA{G}}_T^{_{(d_o)EJ}} (\VEC{k}_{t}) \cdot \SDV{J}_s (\VEC{k}_{t})
\\
V_{(00)}^- &=& T_{tm}^{E-} V_{1_{\TM}}^- + \frac{1}{d_xd_y}
\VEC{u}_t \cdot \widetilde{\DYA{G}}_T^{_{(d_o)EJ}} (\VEC{k}_{t}) \cdot \SDV{J}_s (\VEC{k}_{t})
\end{eqnarray}
したがって，主偏波成分の透過係数
$T_{[00]}^{^{\TE \to \TE}}$（TE波），
$T_{(00)}^{^{\TM \to \TM}}$（TM波）は，
\begin{eqnarray}
T_{[00]}^{^{\TE \to \TE}}
&=& \frac{V_{[00]}^-}{V_{1_{\TE}}^-}
\nonumber \\
&=& \left. T_{te}^{E-} + \frac{1}{d_x d_y}
\left( \VEC{u}_z \times \VEC{u}_t \right) \cdot \widetilde{\DYA{G}}_T^{_{(d_o)EJ}} \cdot \SDV{J}_s
\right| _{V_{1_{\TE}}^- = 1, V_{1_{\TM}}^- = 0}
\\
T_{(00)}^{^{\TM \to \TM}}
&=& \frac{V_{(00)}^-}{V_{1_{\TM}}^-}
\nonumber \\
&=& \left. T_{tm}^{E-} + \frac{1}{d_x d_y}
\VEC{u}_t \cdot \widetilde{\DYA{G}}_T^{_{(d_o)EJ}} \cdot \SDV{J}_s
\right| _{V_{1_{\TE}}^- = 0, V_{1_{\TM}}^- = 1}
\end{eqnarray}
また，交差偏波成分の透過係数
$T_{(00)}^{^{\TE \to \TM}}$，
$T_{[00]}^{^{\TM \to \TE}}$は，
\begin{eqnarray}
T_{(00)}^{^{\TE \to \TM}}
&=& \frac{V_{(00)}^-}{V_{1_{\TE}}^-}
\nonumber \\
&=& \left. \frac{1}{d_x d_y}
\VEC{u}_t \cdot \widetilde{\DYA{G}}_T^{_{(d_o)EJ}} \cdot \SDV{J}_s
\right| _{V_{1_{\TE}}^- = 1, V_{1_{\TM}}^- = 0}
\\
T_{[00]}^{^{\TM \to \TE}}
&=& \frac{V_{[00]}^-}{V_{1_{\TM}}^-}
\nonumber \\
&=& \left. \frac{1}{d_x d_y}
\left( \VEC{u}_z \times \VEC{u}_t \right) \cdot \widetilde{\DYA{G}}_T^{_{(d_o)EJ}} \cdot \SDV{J}_s
\right| _{V_{1_{\TE}}^- = 0, V_{1_{\TM}}^- = 1}
\end{eqnarray}
ここで，
\begin{eqnarray}
&&\widetilde{\DYA{G}}_T^{_{EJ}} (\VEC{k}_{t}) \cdot \SDV{J}_s (\VEC{k}_{t})
\nonumber \\
&=&
\begin{pmatrix}
\VEC{u}_x & \VEC{u}_y
\end{pmatrix}
\begin{pmatrix}
\SDS{G}_{xx} & \SDS{G}_{xy} \\ \SDS{G}_{yx} & \SDS{G}_{yy}
\end{pmatrix}
\begin{pmatrix}
\SDS{J}_{x} \\ \SDS{J}_{y}
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\VEC{u}_t \times \VEC{u}_z & \VEC{u}_t
\end{pmatrix}
\begin{pmatrix}
\sin \phi _i & -\cos \phi _i \\ \cos \phi _i & \sin \phi _i
\end{pmatrix}
\begin{pmatrix}
\SDS{G}_{xx} & \SDS{G}_{xy} \\ \SDS{G}_{yx} & \SDS{G}_{yy}
\end{pmatrix}
\begin{pmatrix}
\SDS{J}_{x} \\ \SDS{J}_{y}
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\VEC{u}_t \times \VEC{u}_z & \VEC{u}_t
\end{pmatrix}
\begin{pmatrix}
\SDS{Z}_{_{\TE}} & 0 \\ 0 & \SDS{Z}_{_{\TM}}
\end{pmatrix}
\begin{pmatrix}
\sin \phi _i & -\cos \phi _i \\ \cos \phi _i & \sin \phi _i
\end{pmatrix}
\begin{pmatrix}
\SDS{J}_{x} \\ \SDS{J}_{y}
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\VEC{u}_t \times \VEC{u}_z & \VEC{u}_t
\end{pmatrix}
\begin{pmatrix}
\SDS{Z}_{_{\TE}} \sin \phi _i & -\SDS{Z}_{_{\TE}} \sin \cos _i \\ \SDS{Z}_{_{\TM}} \cos \phi _i & \SDS{Z}_{_{\TM}} \sin \phi _i
\end{pmatrix}
\begin{pmatrix}
\displaystyle{\sum _{p,q} \SDS{B}_{xpq} I_{xpq}} \\ \displaystyle{\sum _{p,q} \SDS{B}_{ypq} I_{ypq}}
\end{pmatrix}
\nonumber \\
&=&
\begin{pmatrix}
\VEC{u}_t \times \VEC{u}_z & \VEC{u}_t
\end{pmatrix}
\begin{pmatrix}
\SDS{Z}_{_{\TE}} \Big\{ \sin \phi _i \displaystyle{\sum _{p,q} \SDS{B}_{xpq} I_{xpq}}
- \cos \phi _i \displaystyle{\sum _{p,q} \SDS{B}_{ypq} I_{ypq}} \Big\} \\
\SDS{Z}_{_{\TM}} \Big\{ \cos \phi _i \displaystyle{\sum _{p,q} \SDS{B}_{xpq} I_{xpq}}
+ \sin \phi _i \displaystyle{\sum _{p,q} \SDS{B}_{ypq} I_{ypq}} \Big\}
\end{pmatrix}
\end{eqnarray}
これより，
\begin{align}
&\left( \VEC{u}_t \times \VEC{u}_z \right) \cdot \widetilde{\DYA{G}}_T^{_{EJ}} (\VEC{k}_{t}) \cdot \SDV{J}_s (\VEC{k}_{t})
\nonumber \\
&= \SDS{Z}_{_{\TE}} \Big\{ \sin \phi _i \displaystyle{\sum _{p,q} \SDS{B}_{xpq} I_{xpq}}
- \cos \phi _i \displaystyle{\sum _{p,q} \SDS{B}_{ypq} I_{ypq}} \Big\}
\\
&\VEC{u}_t \cdot \widetilde{\DYA{G}}_T^{_{EJ}} (\VEC{k}_{t}) \cdot \SDV{J}_s (\VEC{k}_{t})
\nonumber \\
&= \SDS{Z}_{_{\TM}} \Big\{ \cos \phi _i \displaystyle{\sum _{p,q} \SDS{B}_{xpq} I_{xpq}}
+ \sin \phi _i \displaystyle{\sum _{p,q} \SDS{B}_{ypq} I_{ypq}} \Big\}
\end{align}