透過係数

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

E_{t, \tan}

は，

\begin{matrix} (1) & E_{t, \tan} |_{z = 0} = {T_{t e}^{E -} V_{1_{TE}}^{-} (u_{t} \times u_{z}) + T_{t m}^{E -} V_{1_{TM}}^{-} u_{t}} e^{j k_{t} \cdot ρ} \end{matrix}

また，導体素子による散乱波の接線電界

E_{s, \tan}

は，

\begin{matrix} (2) & E_{s, \tan} |_{z = 0} = \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{\bar{\bar{G}}}}_{T}^{_{(d_{o}) E J}} (k_{t m n}) \cdot {\tilde{J}}_{s} (k_{t m n}) e^{j k_{t m n} \cdot ρ} \end{matrix}

この境界面での全透過波

E_{t, \tan}^{(FSS)}

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{array}{rcl} E_{t, \tan}^{(FSS)} & = & E_{t, \tan} |_{z = 0} + E_{s, \tan} |_{z = 0} \\ (3) & = & \sum_{m, n} {V_{[m n]}^{-} (u_{t m n} \times u_{z}) + V_{(m n)}^{-} u_{t m n}} e^{j k_{t m n} \cdot ρ} \end{array}

透過係数を求めるため，両者を等しくおき，両辺に

ψ_{00}^{*} (ρ) = e^{- j k_{t 00} \cdot ρ}

を乗じ，単位セル

S

にわたり積分して，

\begin{array}{rcl} \int_{S} [\sum_{m, n} {V_{[m n]}^{-} (u_{t m n} \times u_{z}) + V_{(m n)}^{-} u_{t m n}} e^{j k_{t m n} \cdot ρ}] e^{- j k_{t 00} \cdot ρ} d S \\ = & \int_{S} [{T_{t e}^{E -} V_{1_{TE}}^{-} (u_{t} \times u_{z}) + T_{t m}^{E -} V_{1_{TM}}^{-} u_{t}} e^{j k_{t} \cdot ρ}] e^{- j k_{t 00} \cdot ρ} d S \\ (4) & + \int_{S} [\frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{\bar{\bar{G}}}}_{T}^{_{(d_{o}) E J}} (k_{t m n}) \cdot {\tilde{J}}_{s} (k_{t m n}) e^{j k_{t m n} \cdot ρ}] e^{- j k_{t 00} \cdot ρ} d S \end{array}

直交性より，

\begin{aligned} {V_{[00]}^{-} (u_{t} \times u_{z}) + V_{(00)}^{-} u_{t}} \\ (5) & = {T_{t e}^{E -} V_{1_{TE}}^{-} (u_{t} \times u_{z}) + T_{t m}^{E -} V_{1_{TM}}^{-} u_{t}} + \frac{1}{d_{x} d_{y}} {\tilde{\bar{\bar{G}}}}_{T}^{_{(d_{o}) E J}} (k_{t}) \cdot {\tilde{J}}_{s} (k_{t}) \end{aligned}

これより，

(u_{t} \times u_{z})

成分，

u_{t}

成分は，

\begin{array}{rcl} (6) & V_{[00]}^{-} & = & T_{t e}^{E -} V_{1_{TE}}^{-} + \frac{1}{d_{x} d_{y}} (u_{t} \times u_{z}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{(d_{o}) E J}} (k_{t}) \cdot {\tilde{J}}_{s} (k_{t}) \\ (7) & V_{(00)}^{-} & = & T_{t m}^{E -} V_{1_{TM}}^{-} + \frac{1}{d_{x} d_{y}} u_{t} \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{(d_{o}) E J}} (k_{t}) \cdot {\tilde{J}}_{s} (k_{t}) \end{array}

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{array}{rcl} T_{[00]}^{^{TE \to TE}} & = & \frac{V_{[00]}^{-}}{V_{1_{TE}}^{-}} \\ (8) & = & {T_{t e}^{E -} + \frac{1}{d_{x} d_{y}} (u_{z} \times u_{t}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{(d_{o}) E J}} \cdot {\tilde{J}}_{s} |}_{V_{1_{TE}}^{-} = 1, V_{1_{TM}}^{-} = 0} \\ T_{(00)}^{^{TM \to TM}} & = & \frac{V_{(00)}^{-}}{V_{1_{TM}}^{-}} \\ (9) & = & {T_{t m}^{E -} + \frac{1}{d_{x} d_{y}} u_{t} \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{(d_{o}) E J}} \cdot {\tilde{J}}_{s} |}_{V_{1_{TE}}^{-} = 0, V_{1_{TM}}^{-} = 1} \end{array}

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{array}{rcl} T_{(00)}^{^{TE \to TM}} & = & \frac{V_{(00)}^{-}}{V_{1_{TE}}^{-}} \\ (10) & = & {\frac{1}{d_{x} d_{y}} u_{t} \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{(d_{o}) E J}} \cdot {\tilde{J}}_{s} |}_{V_{1_{TE}}^{-} = 1, V_{1_{TM}}^{-} = 0} \\ T_{[00]}^{^{TM \to TE}} & = & \frac{V_{[00]}^{-}}{V_{1_{TM}}^{-}} \\ (11) & = & {\frac{1}{d_{x} d_{y}} (u_{z} \times u_{t}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{(d_{o}) E J}} \cdot {\tilde{J}}_{s} |}_{V_{1_{TE}}^{-} = 0, V_{1_{TM}}^{-} = 1} \end{array}

ここで，

\begin{array}{rcl} {\tilde{\bar{\bar{G}}}}_{T}^{_{E J}} (k_{t}) \cdot {\tilde{J}}_{s} (k_{t}) \\ = & (\begin{array}{c} u_{x} & u_{y} \end{array}) (\begin{array}{c} {\tilde{G}}_{x x} & {\tilde{G}}_{x y} \\ {\tilde{G}}_{y x} & {\tilde{G}}_{y y} \end{array}) (\begin{array}{c} {\tilde{J}}_{x} \\ {\tilde{J}}_{y} \end{array}) \\ = & (\begin{array}{c} u_{t} \times u_{z} & u_{t} \end{array}) (\begin{array}{c} \sin ϕ_{i} & - \cos ϕ_{i} \\ \cos ϕ_{i} & \sin ϕ_{i} \end{array}) (\begin{array}{c} {\tilde{G}}_{x x} & {\tilde{G}}_{x y} \\ {\tilde{G}}_{y x} & {\tilde{G}}_{y y} \end{array}) (\begin{array}{c} {\tilde{J}}_{x} \\ {\tilde{J}}_{y} \end{array}) \\ = & (\begin{array}{c} u_{t} \times u_{z} & u_{t} \end{array}) (\begin{array}{c} {\tilde{Z}}_{_{TE}} & 0 \\ 0 & {\tilde{Z}}_{_{TM}} \end{array}) (\begin{array}{c} \sin ϕ_{i} & - \cos ϕ_{i} \\ \cos ϕ_{i} & \sin ϕ_{i} \end{array}) (\begin{array}{c} {\tilde{J}}_{x} \\ {\tilde{J}}_{y} \end{array}) \\ = & (\begin{array}{c} u_{t} \times u_{z} & u_{t} \end{array}) (\begin{array}{c} {\tilde{Z}}_{_{TE}} \sin ϕ_{i} & - {\tilde{Z}}_{_{TE}} \sin \cos_{i} \\ {\tilde{Z}}_{_{TM}} \cos ϕ_{i} & {\tilde{Z}}_{_{TM}} \sin ϕ_{i} \end{array}) (\begin{array}{c} \sum_{p, q} {\tilde{B}}_{x p q} I_{x p q} \\ \sum_{p, q} {\tilde{B}}_{y p q} I_{y p q} \end{array}) \\ (12) & = & (\begin{array}{c} u_{t} \times u_{z} & u_{t} \end{array}) (\begin{array}{c} {\tilde{Z}}_{_{TE}} {\sin ϕ_{i} \sum_{p, q} {\tilde{B}}_{x p q} I_{x p q} - \cos ϕ_{i} \sum_{p, q} {\tilde{B}}_{y p q} I_{y p q}} \\ {\tilde{Z}}_{_{TM}} {\cos ϕ_{i} \sum_{p, q} {\tilde{B}}_{x p q} I_{x p q} + \sin ϕ_{i} \sum_{p, q} {\tilde{B}}_{y p q} I_{y p q}} \end{array}) \end{array}

これより，

\begin{aligned} (u_{t} \times u_{z}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{E J}} (k_{t}) \cdot {\tilde{J}}_{s} (k_{t}) \\ (13) & = {\tilde{Z}}_{_{TE}} {\sin ϕ_{i} \sum_{p, q} {\tilde{B}}_{x p q} I_{x p q} - \cos ϕ_{i} \sum_{p, q} {\tilde{B}}_{y p q} I_{y p q}} \\ u_{t} \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{E J}} (k_{t}) \cdot {\tilde{J}}_{s} (k_{t}) \\ (14) & = {\tilde{Z}}_{_{TM}} {\cos ϕ_{i} \sum_{p, q} {\tilde{B}}_{x p q} I_{x p q} + \sin ϕ_{i} \sum_{p, q} {\tilde{B}}_{y p q} I_{y p q}} \end{aligned}