4.4 反射係数
入射波側の自由空間と誘電体の境界面（$z=0$）での入射平面波の横断面内電界
$\VEC{E}_{i,\tan}^{(1)}$
を，
\begin{eqnarray}
\VEC{E}_{i,\tan}^{(1)} \Big| _{z=0}
&=& \Big\{ V_{1_{\TE}}^- \big( \VEC{u}_t \times \VEC{u}_z \big) + V_{1_{\TM}}^- \VEC{u}_t \Big\}
e^{j\VEC{k}_t \cdot \VECi{\rho}}
\nonumber \\
&=& E_{i,x} \VEC{u}_x + E_{i,y} \VEC{u}_y
\end{eqnarray}
とする．入射角を$(\theta _i, \phi _i)$とすると，入射波の波数ベクトル$\VEC{k}^{inc}$は，
\begin{eqnarray}
\VEC{k}^{inc}
&=& k_x^{inc} \VEC{u}_x + k_y^{inc} \VEC{u}_y + k_z \VEC{u}_z
\nonumber \\
&=& \VEC{k}_t + k_z \VEC{u}_z = k_t \VEC{u}_t + k_z \VEC{u}_z
\nonumber \\
&=& k ( \sin \theta _i \cos \phi _i \VEC{u}_x + \sin \theta _i \sin \phi _i \VEC{u}_y + \cos \theta _i \VEC{u}_z )
\end{eqnarray}
このとき，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_i)EJ}} (\VEC{k}_{tmn})
\cdot \SDV{J}_s (\VEC{k}_{tmn}) e^{j\VEC{k}_{tmn} \cdot \VECi{\rho}}
\end{gather}
ここで，
\begin{align}
&\VEC{k}_{tmn} = k_{tmn} \VEC{u}_{tmn} = k_{xm} \VEC{u}_x + k_{yn} \VEC{u}_y
\\
&k_{xm} = \frac{2\pi m}{d_x} + k_x^{inc}
\\
&k_{yn} = \frac{2\pi n}{d_y} + k_y^{inc}
\\
&k_{zmn} = \sqrt{k^2 - k_{xm}^2 - k_{yn}^2}
\end{align}
また，導体素子がない場合の反射波の接線電界$\VEC{E}_{r,\tan}$は，
\begin{gather}
\VEC{E}_{r,\tan} \Big| _{z=0}
= \Big\{ R_{te}^{E-} V_{1_{\TE}}^- (\VEC{u}_t \times \VEC{u}_z ) + R_{tm}^{E-} V_{1_{\TM}}^- \VEC{u}_t \Big\}
e^{j\VEC{k}_t \cdot \VECi{\rho}}
\end{gather}
境界面での全反射波$\VEC{E}^{(\FSS)}_{r,\tan}$は上の両電界からなり，これをフロケモードで展開すると，
\begin{eqnarray}
\VEC{E}^{(\FSS)}_{r,\tan}
&=& \VEC{E}_{r,\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}
反射係数を求めるため，両者を等しくおき，両辺に$\psi _{00}^* (\VECi{\rho})= e^{-j\VEC{k}_{t00} \cdot \VECi{\rho}}$を乗じて，
単位セル（面$S$）にわたって次のように積分する．
\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\{ R_{te}^{E-} V_{1_{\TE}}^- (\VEC{u}_t \times \VEC{u}_z ) + R_{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_i)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}
ここで，$k_{t00} = k_t$ゆえ，
\begin{eqnarray}
\int _S e^{j\VEC{k}_{tmn} \cdot \VECi{\rho}} e^{-j\VEC{k}_{t00} \cdot \VECi{\rho}} dS
&=& \int _S e^{j \left( \frac{2\pi m}{d_x}x + \frac{2\pi n}{d_y}y \right) } dS
\nonumber \\
&=& d_x d_y \delta _{m0} \delta _{n0}
\end{eqnarray}
より，$\VEC{u}_t^{(00)} = \VEC{u}_t$，$k_{z00} = k_z$を考慮して，
\begin{eqnarray}
&&\Big\{ V_{[00]}^+ \left( \VEC{u}_t \times \VEC{u}_z \right) + V_{(00)}^+ \VEC{u}_t \Big\}
\nonumber \\
&=& \Big\{ R_{te}^{E-} V_{1_{\TE}}^- \left( \VEC{u}_t \times \VEC{u}_z \right) + R_{tm}^{E-} V_{1_{\TM}}^- \VEC{u}_t \Big\}
+ \frac{1}{d_xd_y} \widetilde{\DYA{G}}_T^{_{(d_i)EJ}} (\VEC{k}_{t}) \cdot \SDV{J}_s (\VEC{k}_{t})
\end{eqnarray}
これより，$(\VEC{u}_t \times \VEC{u}_z)$成分，$\VEC{u}_t$成分は，
\begin{eqnarray}
V_{[00]}^+ &=& R_{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_i)EJ}} (\VEC{k}_{t}) \cdot \SDV{J}_s (\VEC{k}_{t}) \\
V_{(00)}^+ &=& R_{tm}^{E-} V_{1_{\TM}}^- + \frac{1}{d_xd_y}
\VEC{u}_t \cdot \widetilde{\DYA{G}}_T^{_{(d_i)EJ}} (\VEC{k}_{t}) \cdot \SDV{J}_s (\VEC{k}_{t})
\end{eqnarray}
したがって，主偏波成分の反射係数
$R_{[00]}^{^{\TE \to \TE}}$（TE波），
$R_{(00)}^{^{\TM \to \TM}}$（TM波）は，
\begin{eqnarray}
R_{[00]}^{^{\TE \to \TE}}
&=& \frac{V_{[00]}^+}{V_{1_{\TE}}^-}
\nonumber \\
&=& \left. R_{te}^{E-} + \frac{1}{d_x d_y}
\left( \VEC{u}_t \times \VEC{u}_z \right) \cdot \widetilde{\DYA{G}}_T^{_{(d_i)EJ}} \cdot \SDV{J}_s
\right| _{V_{1_{\TE}}^- = 1, V_{1_{\TM}}^- = 0}
\\
R_{(00)}^{^{\TM \to \TM}}
&=& \frac{V_{(00)}^+}{V_{1_{\TM}}^-}
\nonumber \\
&=& \left. R_{tm}^{E-} + \frac{1}{d_x d_y}
\VEC{u}_t \cdot \widetilde{\DYA{G}}_T^{_{(d_i)EJ}} \cdot \SDV{J}_s
\right| _{V_{1_{\TE}}^- = 0, V_{1_{\TM}}^- = 1}
\end{eqnarray}
また，交差偏波成分の反射係数
$R_{(00)}^{^{\TE \to \TM}}$，
$R_{[00]}^{^{\TM \to \TE}}$は，
\begin{eqnarray}
R_{(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_i)EJ}} \cdot \SDV{J}_s
\right| _{V_{1_{\TE}}^- = 1, V_{1_{\TM}}^- = 0}
\\
R_{[00]}^{^{\TM \to \TE}}
&=& \frac{V_{[00]}^+}{V_{1_{\TM}}^-}
\nonumber \\
&=& \frac{1}{d_x d_y}
\left. \left( \VEC{u}_t \times \VEC{u}_z \right) \cdot \widetilde{\DYA{G}}_T^{_{(d_i)EJ}} \cdot \SDV{J}_s
\right| _{V_{1_{\TE}}^- = 0, V_{1_{\TM}}^- = 1}
\end{eqnarray}