マイクロストリップ・リフレクト無限アレー

マイクロストリップ・リフレクト無限アレー

　地導体のある誘電体基板で支持するマイクロストリップパッチの無限アレーでは，グリーン関数は次式で与えられる．

\begin{array}{rcl} (1) & {\tilde{G}}_{x x}^{^{E J}} & = & \frac{- j Z_{0}}{k_{0}} \frac{k_{z m n}^{(a i r)} (ϵ_{r} k_{0}^{2} - k_{x m n}^{2}) \cos k_{z m n} d + j k_{z m n} (k_{0}^{2} - k_{x m n}^{2}) \sin k_{z m n} d}{T_{e}^{(m n)} T_{m}^{(m n)}} \sin k_{z m n} d \\ (2) & {\tilde{G}}_{x y}^{^{E J}} & = & {\tilde{G}}_{y x}^{^{E J}} = \frac{j Z_{0}}{k_{0}} \cdot \frac{k_{x m n} k_{y m n} (k_{z m n}^{(a i r)} \cos k_{z m n} d + j k_{z m n} \sin k_{z m n} d)}{T_{e}^{(m n)} T_{m}^{(m n)}} \sin k_{z m n} d \\ (3) & {\tilde{G}}_{y y}^{^{E J}} & = & \frac{- j Z_{0}}{k_{0}} \frac{k_{z m n}^{(a i r)} (ϵ_{r} k_{0}^{2} - k_{y m n}^{2}) \cos k_{z m n} d + j k_{z m n} (k_{0}^{2} - k_{y m n}^{2}) \sin k_{z m n} d}{T_{e}^{(m n)} T_{m}^{(m n)}} \sin k_{z m n} d \end{array}

ここで，

\begin{array}{rcl} (4) & T_{e}^{(m n)} & = & k_{z m n} \cos k_{z m n} d + j k_{z m n}^{(a i r)} \sin k_{z m n} d \\ (5) & T_{m}^{(m n)} & = & k_{z m n}^{(a i r)} ϵ_{r} \cos k_{z m n} d + j k_{z m n} \sin k_{z m n} d \end{array}

導体素子上の境界条件より，

\begin{matrix} (6) & E_{i, \tan} + E_{r, \tan} + E_{s, \tan} = Z_{s} J_{s} (on S) \end{matrix}

ただし，

E_{i, \tan}

は入射電界，

E_{r, \tan}

は導体素子がない場合の反射電界を示し，

\begin{array}{rcl} E_{i, \tan} |_{S} & = & E_{i, x} (x, y) u_{x} + E_{i, y} (x, y) u_{y} \\ (7) & = & (V_{1 x}^{+} u_{x} + V_{1 y}^{+} u_{y}) e^{j k_{t} \cdot ρ} \\ E_{r, \tan} |_{S} & = & E_{r, x} (x, y) u_{x} + E_{r, y} (x, y) u_{y} \\ (8) & = & (V_{1 x}^{-} u_{x} + V_{1 y}^{-} u_{y}) e^{j k_{t} \cdot ρ} \end{array}

このとき，列ベクトル

V_{x}

，

V_{y}

の各々の要素

v_{x k l}

，

v_{y k l}

は，

\begin{array}{rcl} v_{(\binom{x}{y}) k l} & = & - \int_{S} T_{(\binom{x}{y}) k l}^{*} (E_{i, (\binom{x}{y})} + E_{r, (\binom{x}{y})}) d S \\ = & - (V_{1 (\binom{x}{y})}^{+} + V_{1 (\binom{x}{y})}^{-}) \int_{S} T_{(\binom{x}{y}) k l}^{*} e^{j k_{t} \cdot ρ} d S \\ (9) & = & - (V_{1 (\binom{x}{y})}^{+} + V_{1 (\binom{x}{y})}^{-}) {\tilde{T}}_{(\binom{x}{y}) k l}^{*} \end{array}

行列表示して，

\begin{array}{rcl} (\begin{array}{c} v_{x k l} \\ v_{y k l} \end{array}) & = & - (\begin{array}{c} {\tilde{T}}_{x k l}^{*} & 0 \\ 0 & {\tilde{T}}_{y k l}^{*} \end{array}) {(\begin{array}{c} V_{1 x}^{+} \\ V_{1 y}^{+} \end{array}) + (\begin{array}{c} V_{1 x}^{-} \\ V_{1 y}^{-} \end{array})} \\ = & - (\begin{array}{c} {\tilde{T}}_{x u, k l}^{*} & {\tilde{T}}_{x t, k l}^{*} \\ {\tilde{T}}_{y u, k l}^{*} & {\tilde{T}}_{y t, k l}^{*} \end{array}) {(\begin{array}{c} V_{1_{TE}}^{+} \\ V_{1_{TM}}^{+} \end{array}) + (\begin{array}{c} V_{1_{TE}}^{-} \\ V_{1_{TM}}^{-} \end{array})} \\ = & - (\begin{array}{c} {\tilde{T}}_{x u, k l}^{*} & {\tilde{T}}_{x t, k l}^{*} \\ {\tilde{T}}_{y u, k l}^{*} & {\tilde{T}}_{y t, k l}^{*} \end{array}) {(\begin{array}{c} V_{1_{TE}}^{+} \\ V_{1_{TM}}^{+} \end{array}) + (\begin{array}{c} R_{t e}^{E +} & 0 \\ 0 & R_{t m}^{E +} \end{array}) (\begin{array}{c} V_{1_{TE}}^{+} \\ V_{1_{TM}}^{+} \end{array})} \\ (10) & = & - (\begin{array}{c} {\tilde{T}}_{x u, k l}^{*} & {\tilde{T}}_{x t, k l}^{*} \\ {\tilde{T}}_{y u, k l}^{*} & {\tilde{T}}_{y t, k l}^{*} \end{array}) {[U] + [R^{E +}]_{d}} (\begin{array}{c} V_{1_{TE}}^{+} \\ V_{1_{TM}}^{+} \end{array}) \end{array}

ここで，

\begin{matrix} (11) & (\begin{matrix} {\tilde{T}}_{x u, k l}^{*} & {\tilde{T}}_{x t, k l}^{*} \\ {\tilde{T}}_{y u, k l}^{*} & {\tilde{T}}_{y t, k l}^{*} \end{matrix}) \equiv (\begin{matrix} {\tilde{T}}_{x k l}^{*} \sin ϕ_{i} & {\tilde{T}}_{x k l}^{*} \cos ϕ_{i} \\ - {\tilde{T}}_{y k l}^{*} \cos ϕ_{i} & {\tilde{T}}_{y k l}^{*} \sin ϕ_{i} \end{matrix}) \end{matrix}

また，

\begin{aligned} (12) & [U] \equiv (\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}) \\ (13) & [R^{E +}]_{d} \equiv (\begin{array}{c} R_{t e}^{E +} & 0 \\ 0 & R_{t m}^{E +} \end{array}) \end{aligned}

ここで，

\begin{array}{rcl} (14) & R_{t e}^{E +} & = & \frac{k_{z}^{(a i r)} \sin k_{z} d + j k_{z} \cos k_{z} d}{k_{z}^{(a i r)} \sin k_{z} d - j k_{z} \cos k_{z} d} \\ (15) & R_{t m}^{E +} & = & \frac{k_{z} \sin k_{z} d + j ϵ_{r} k_{z}^{(a i r)} \cos k_{z} d}{k_{z} \sin k_{z} d - j ϵ_{r} k_{z}^{(a i r)} \cos k_{z} d} \end{array}

これより，列ベクトル

[V_{x}]

，

[V_{y}]

は，

\begin{matrix} (16) & (\begin{matrix} [V_{x}] \\ [V_{y}] \end{matrix}) = - (\begin{matrix} [{\tilde{T}}_{x u}^{*}]^{t} & [{\tilde{T}}_{x t}^{*}]^{t} \\ [{\tilde{T}}_{y u}^{*}]^{t} & [{\tilde{T}}_{y t}^{*}]^{t} \end{matrix}) {[U] + [R^{E +}]_{d}} (\begin{matrix} V_{1_{TE}}^{+} \\ V_{1_{TM}}^{+} \end{matrix}) \end{matrix}

ただし，列ベクトル

[{\tilde{T}}_{x u}^{*}]^{t}

，

[{\tilde{T}}_{x t}^{*}]^{t}

，

[{\tilde{T}}_{y u}^{*}]^{t}

，

[{\tilde{T}}_{y t}^{*}]^{t}

の各々の要素は

{\tilde{T}}_{x u, k l}^{*}

，

{\tilde{T}}_{x t, k l}^{*}

，

{\tilde{T}}_{y u, k l}^{*}

，

{\tilde{T}}_{y t, k l}^{*}

である．