スロット結合パッチアレーFSS

z = d

および

z = - d

の両面にパッチアレー（パッチ領域を面S

_{1}

，面S

_{2}

とする），

z = 0

にスロット（スロット領域を面S

_{3}

とする）を設けた地導体板からなる周波数選択板（FSS）を考える．このとき，

0 < z < d

に比誘電率

ϵ_{1}

，

- d < z < 0

に比誘電率

ϵ_{1}

の誘電体基板で各々パッチを支持するものとする．いま，FSSに平面波（電界

E_{i}

，磁界

H_{i}

）を入射させると，パッチ上に電流，スロットに磁流が誘起される．等価定理より，パッチのかわりに誘電体基板上に等価電流源，スロットのかわりに地導体板上に等価磁流源を考え，生じる散乱電磁界を

E_{s}

，

H_{s}

とする．また，入射波によって生じる誘電体基板および地導体板による反射波を

E_{r}

，

H_{r}

とすると，一般に，全電界

E

および全磁界

H

は，

\begin{matrix} (1) & E = E_{i} + E_{r} + E_{s} \\ (2) & H = H_{i} + H_{r} + H_{s} \end{matrix}

ただし，FSSの透過波

E^{'}

，

H^{'}

については，入射波および反射波の寄与がないことから，散乱波

E_{s}^{'}

，

H_{s}^{'}

のみで次のようになる．

\begin{matrix} (3) & E^{'} = E_{s}^{'} \\ (4) & H^{'} = H_{s}^{'} \end{matrix}

また，境界条件は，境界面の接線成分を添え字

\tan

で表すと，

\begin{aligned} (5) & E_{\tan} = 0 ({on S}_{1}) \\ (6) & H_{\tan} = H_{\tan}^{'} ({on S}_{3}) \\ (7) & E_{\tan}^{'} = 0 ({on S}_{2}) \end{aligned}

これより，

\begin{aligned} (8) & E_{i, \tan} |_{S_{1}} + E_{r, \tan} |_{S_{1}} + E_{s, \tan} |_{S_{1}} = 0 ({on S}_{1}) \\ (9) & H_{i, \tan} |_{S_{3}} + H_{r, \tan} |_{S_{3}} + H_{s, \tan} |_{S_{3}} = H_{s, \tan}^{'} |_{S_{3}} ({on S}_{3}) \\ (10) & E_{s, \tan}^{'} |_{S_{2}} = 0 ({on S}_{2}) \end{aligned}

散乱電界

E_{s, \tan} |_{S_{1}}

は，パッチ領域の面S

_{1}

の等価電流源

J_{s 1}

およびスロット領域の面S

_{3}

上の等価磁流源

M_{s}

より，

\begin{matrix} (11) & E_{s, \tan} |_{S_{1}} = E_{s, \tan} (J_{s 1}) |_{S_{1}} + E_{s, \tan} (M_{s}) |_{S_{1}} \end{matrix}

ここで，

\begin{aligned} (12) & E_{s, \tan} (J_{s 1}) |_{S_{1}} = \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{\bar{\bar{G}}}}_{T}^{_{E J (0)}} (k_{t m n}) \cdot {\tilde{J}}_{s 1} (k_{t m n}) e^{j k_{t m n} \cdot ρ} \\ (13) & {\tilde{\bar{\bar{G}}}}_{T}^{_{E J (0)}} (k_{t m n}) = {\tilde{G}}_{x x}^{^{E J}} u_{x} u_{x} + {\tilde{G}}_{x y}^{^{E J}} u_{x} u_{y} + {\tilde{G}}_{y x}^{^{E J}} u_{y} u_{x} + {\tilde{G}}_{y y}^{^{E J}} u_{y} u_{y} \end{aligned}

グリーン関数は先に示したとおりであるので，ここでは省略する．また，

\begin{matrix} (14) & E_{s, \tan} (M_{s}) |_{S_{1}} = \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{\bar{\bar{G}}}}_{T}^{_{E M (d)}} (k_{t m n}) \cdot {\tilde{M}}_{s} (k_{t m n}) e^{j k_{t m n} \cdot ρ} \end{matrix}

グリーン関数は，

\begin{matrix} (15) & {\tilde{\bar{\bar{G}}}}_{T}^{_{E M (d)}} (k_{t m n}) = {\tilde{G}}_{x x}^{^{E M}} u_{x} u_{x} + {\tilde{G}}_{x y}^{^{E M}} u_{x} u_{y} + {\tilde{G}}_{y x}^{^{E M}} u_{y} u_{x} + {\tilde{G}}_{y y}^{^{E M}} u_{y} u_{y} \end{matrix}

ここで，

\begin{array}{rcl} {\tilde{G}}_{x x}^{^{E M}} & = & - {\tilde{G}}_{y y}^{^{E M}} \\ (16) & = & - \frac{j k_{x m n} k_{y m n} (ϵ_{r} - 1) \sin k_{z m n} d}{T_{e}^{(m n)} T_{m}^{(m n)}} \\ (17) & {\tilde{G}}_{x y}^{^{E M}} & = & \frac{k_{z m n}}{T_{e}^{(m n)}} - \frac{j k_{x m n}^{2} (ϵ_{r} - 1) \sin k_{z m n} d}{T_{e}^{(m n)} T_{m}^{(m n)}} \\ (18) & {\tilde{G}}_{y x}^{^{E M}} & = & - \frac{k_{z m n}}{T_{e}^{(m n)}} + \frac{j k_{y m n}^{2} (ϵ_{r} - 1) \sin k_{z m n} d}{T_{e}^{(m n)} T_{m}^{(m n)}} \end{array}

ただし，

\begin{array}{rcl} (19) & 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 \\ (20) & 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}

逆に，散乱電界

E_{s, \tan}^{'} |_{S_{2}}

は，パッチ領域の面S

_{2}

の等価電流源

J_{s 2}

およびスロット領域の面S

_{3}

下の等価磁流源

- M_{s}

より，

\begin{matrix} (21) & E_{s, \tan}^{'} |_{S_{2}} = E_{s, \tan} (J_{s 2}) |_{S_{2}} + E_{s, \tan} (- M_{s}) |_{S_{2}} \end{matrix}

ここで，

\begin{aligned} (22) & E_{s, \tan} (J_{s 2}) |_{S_{2}} = \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{\bar{\bar{G}}}}_{T}^{_{E J (0)}} (k_{t m n}) \cdot {\tilde{J}}_{s 2} (k_{t m n}) e^{j k_{t m n} \cdot ρ} \\ (23) & E_{s, \tan} (- M_{s}) |_{S_{2}} = \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{\bar{\bar{G}}}}_{T}^{_{E M (d)}} (k_{t m n}) \cdot (- {\tilde{M}}_{s} (k_{t m n})) e^{j k_{t m n} \cdot ρ} \end{aligned}

また，散乱磁界

H_{s, \tan} |_{S_{3}}

（

z = 0_{+}

）は，パッチ領域の面S

_{1}

の等価電流源

J_{s 1}

およびスロット領域の面S

_{3}

上の等価磁流源

M_{s}

より，

\begin{matrix} (24) & H_{s, \tan} |_{S_{3}} = H_{s, \tan} (J_{s 1}) |_{S_{3}} + H_{s, \tan} (M_{s}) |_{S_{3}} \end{matrix}

ここで，

\begin{matrix} (25) & H_{s, \tan} (J_{s 1}) |_{S_{3}} = \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{\bar{\bar{G}}}}_{T}^{_{H J (d)}} (k_{t m n}) \cdot {\tilde{J}}_{s 1} (k_{t m n}) e^{j k_{t m n} \cdot ρ} \end{matrix}

グリーン関数は，

\begin{matrix} (26) & {\tilde{\bar{\bar{G}}}}_{T}^{_{H J (d)}} (k_{t m n}) = {\tilde{G}}_{x x}^{^{H J}} u_{x} u_{x} + {\tilde{G}}_{x y}^{^{H J}} u_{x} u_{y} + {\tilde{G}}_{y x}^{^{H J}} u_{y} u_{x} + {\tilde{G}}_{y y}^{^{H J}} u_{y} u_{y} \end{matrix}

ここで，

\begin{array}{rcl} {\tilde{G}}_{x x}^{^{H J}} & = & - {\tilde{G}}_{y y}^{^{H J}} \\ = & \frac{j k_{x m n} k_{y m n} (ϵ_{r} - 1) \sin k_{z m n} d}{T_{e}^{(m n)} T_{m}^{(m n)}} \\ = & - {\tilde{G}}_{x x}^{^{E M}} \\ (27) & = & {\tilde{G}}_{y y}^{^{E M}} \\ {\tilde{G}}_{x y}^{^{H J}} & = & \frac{k_{z m n}}{T_{e}^{(m n)}} - \frac{j k_{y m n}^{2} (ϵ_{r} - 1) \sin k_{z m n} d}{T_{e}^{(m n)} T_{m}^{(m n)}} \\ (28) & = & - {\tilde{G}}_{y x}^{^{E M}} \\ {\tilde{G}}_{y x}^{^{H J}} & = & - \frac{k_{z m n}}{T_{e}^{(m n)}} + \frac{j k_{x m n}^{2} (ϵ_{r} - 1) \sin k_{z m n} d}{T_{e}^{(m n)} T_{m}^{(m n)}} \\ (29) & = & - {\tilde{G}}_{x y}^{^{E M}} \end{array}

また，

\begin{matrix} (30) & H_{s, \tan} (M_{s}) |_{S_{3}} = \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{\bar{\bar{G}}}}_{T}^{_{H M (0)}} (k_{t m n}) \cdot {\tilde{M}}_{s} (k_{t m n}) e^{j k_{t m n} \cdot ρ} \end{matrix}

グリーン関数は，

\begin{matrix} (31) & {\tilde{\bar{\bar{G}}}}_{T}^{_{H M (0)}} (k_{t m n}) = {\tilde{G}}_{x x}^{^{H M}} u_{x} u_{x} + {\tilde{G}}_{x y}^{^{H M}} u_{x} u_{y} + {\tilde{G}}_{y x}^{^{H M}} u_{y} u_{x} + {\tilde{G}}_{y y}^{^{H M}} u_{y} u_{y} \end{matrix}

ここで，

\begin{array}{rcl} {\tilde{G}}_{x x}^{^{H M}} & = & - \frac{j ω ϵ_{0} ϵ_{r} k_{y m n}^{2}}{k_{t m n}^{2} k_{z m n} T_{m}^{(m n)}} (k_{z m n}^{(a i r)} ϵ_{r} \sin k_{z m n} d - j k_{z m n} \cos k_{z m n} d) \\ (32) & - \frac{j k_{z m n} k_{x m n}^{2}}{ω μ_{0} k_{t m n}^{2} T_{e}^{(m n)}} (k_{z m n} \sin k_{z m n} d - j k_{z m n}^{(a i r)} \cos k_{z m n} d) \\ {\tilde{G}}_{x y}^{^{H M}} & = & \frac{k_{x m n} k_{y m n}}{k_{t m n}^{2}} {\frac{j ω ϵ_{0} ϵ_{r}}{k_{z m n} T_{m}^{(m n)}} (k_{z m n}^{(a i r)} ϵ_{r} \sin k_{z m n} d - j k_{z m n} \cos k_{z m n} d) \\ - \frac{j k_{z m n}}{ω μ_{0} T_{e}^{(m n)}} (k_{z m n} \sin k_{z m n} d - j k_{z m n}^{(a i r)} \cos k_{z m n} d)} \\ (33) & = & {\tilde{G}}_{y x}^{^{H M}} \\ {\tilde{G}}_{y y}^{^{H M}} & = & - \frac{j ω ϵ_{0} ϵ_{r} k_{x m n}^{2}}{k_{t m n}^{2} k_{z m n} T_{m}^{(m n)}} (k_{z m n}^{(a i r)} ϵ_{r} \sin k_{z m n} d - j k_{z m n} \cos k_{z m n} d) \\ (34) & - \frac{j k_{z m n} k_{y m n}^{2}}{ω μ_{0} k_{t m n}^{2} T_{e}^{(m n)}} (k_{z m n} \sin k_{z m n} d - j k_{z m n}^{(a i r)} \cos k_{z m n} d) \end{array}

散乱磁界

H_{s, \tan}^{'} |_{S_{3}}

（

z = 0_{-}

）は，パッチ領域の面S

_{2}

の等価電流源

J_{s 2}

およびスロット領域の面S

_{3}

下の等価磁流源

- M_{s}

より，

\begin{matrix} (35) & H_{s, \tan}^{'} |_{S_{3}} = H_{s, \tan} (J_{s 2}) |_{S_{3}} + H_{s, \tan} (- M_{s}) |_{S_{3}} \end{matrix}

ここで，

\begin{aligned} (36) & H_{s, \tan} (J_{s 2}) |_{S_{3}} = \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{\bar{\bar{G}}}}_{T}^{_{H J (d)}} (k_{t m n}) \cdot {\tilde{J}}_{s 2} (k_{t m n}) e^{j k_{t m n} \cdot ρ} \\ (37) & H_{s, \tan} (- M_{s}) |_{S_{3}} = \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{\bar{\bar{G}}}}_{T}^{_{H M (0)}} (k_{t m n}) \cdot (- {\tilde{M}}_{s} (k_{t m n})) e^{j k_{t m n} \cdot ρ} \end{aligned}

ガラーキン法

　電流

J_{s 1} (x, y)

，

J_{s 2} (x, y)

，磁流

M_{s} (x, y)

を，基底関数

f_{i}^{J 1}

，

f_{i}^{J 2}

，

f_{i}^{M}

を用いて次のように展開する．

\begin{array}{rcl} (38) & J_{s 1} (x, y) & = & \sum_{i} I_{1, i} f_{i}^{J 1} (x, y) \\ (39) & J_{s 2} (x, y) & = & \sum_{i} I_{2, i} f_{i}^{J 2} (x, y) \\ (40) & M_{s} (x, y) & = & \sum_{i} I_{3, i} f_{i}^{M} (x, y) \end{array}

パッチS

_{1}

上の境界条件の式の両辺に，そのパッチの電流に関わる基底関数

f_{j}^{J 1 *}

を乗じて面S

_{1}

にわたって積分すると，

\begin{matrix} (41) & \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{i, \tan} d S_{1} + \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{r, \tan} d S_{1} + \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{s, \tan} d S_{1} = 0 \end{matrix}

上式の第1項は，

\begin{array}{rcl} \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{i, \tan} d S_{1} & = & \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{i, 0} |_{S_{1}} e^{j k (k_{x} x + k_{y} y)} d S_{1} \\ = & \int_{S_{1}} f_{j}^{J 1 *} (x, y) e^{j (k_{x} x + k_{y} y)} d S_{1} \cdot E_{i, 0} |_{S_{1}} \\ (42) & = & {\tilde{f}}_{j}^{J 1 *} (k_{x}, k_{y}) \cdot E_{i, 0} |_{S_{1}} \end{array}

また，第2項は，

\begin{array}{rcl} \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{r, \tan} d S_{1} & = & \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{r, 0} |_{S_{1}} e^{j k (k_{x} x + k_{y} y)} d S_{1} \\ = & \int_{S_{1}} f_{j}^{J 1 *} (x, y) e^{j (k_{x} x + k_{y} y)} d S_{1} \cdot E_{r, 0} |_{S_{1}} \\ (43) & = & {\tilde{f}}_{j}^{J 1 *} (k_{x}, k_{y}) \cdot E_{r, 0} |_{S_{1}} \end{array}

いま，

\begin{matrix} (44) & V_{1, j} \equiv - {\tilde{f}}_{j}^{J 1 *} (k_{x}, k_{y}) \cdot (E_{i, 0} |_{S_{1}} + E_{r, 0} |_{S_{1}}) \end{matrix}

とおくと，

\begin{array}{rcl} V_{1, j} & = & \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{s, \tan} d S_{1} \\ = & \int_{S_{1}} f_{j}^{J 1 *} \cdot (E_{s, \tan} (J_{s 1}) |_{S_{1}} + E_{s, \tan} (M_{s}) |_{S_{1}}) d S_{1} \\ = & \int_{S_{1}} f_{j}^{J 1 *} \cdot \sum_{i} I_{1, i} E_{s, \tan} (f_{i}^{J 1}) d S_{1} + \int_{S_{1}} f_{j}^{J 1 *} \cdot \sum_{i} I_{3, i} E_{s, \tan} (f_{i}^{M}) d S_{1} \\ = & \sum_{i} I_{1, i} \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{s, \tan} (f_{i}^{J 1}) d S_{1} + \sum_{i} I_{3, i} \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{s, \tan} (f_{i}^{M}) d S_{1} \\ (45) & \equiv & \sum_{i} I_{1, i} z_{j i}^{11} + \sum_{i} I_{3, i} z_{j i}^{13} \end{array}

ここで，

\begin{array}{rcl} (46) & z_{j i}^{11} & = & \int_{S_{1}} f_{j}^{J 1 *} \cdot E_{s, \tan} (f_{i}^{J 1}) d S_{1} \\ (47) & z_{j i}^{13} & = & \int_{S_{1}} f_{j}^{J 1} \cdot E_{s, \tan} (f_{i}^{M}) d S_{1} \end{array}

　次に，スロットS

_{3}

上の境界条件の式の両辺に，そのスロットの磁流に関わる基底関数

f_{j}^{M}

を乗じて面S

_{3}

にわたって積分すると，

\begin{aligned} \int_{S_{3}} f_{j}^{M *} \cdot H_{i, \tan} d S_{3} + \int_{S_{3}} f_{j}^{M *} \cdot H_{r, \tan} d S_{3} + \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} d S_{3} \\ (48) & = \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan}^{'} d S_{3} \end{aligned}

上式の第1項は，

\begin{array}{rcl} \int_{S_{3}} f_{j}^{M *} \cdot H_{i, \tan} d S_{3} & = & \int_{S_{3}} f_{j}^{M *} \cdot H_{i, 0} |_{S_{3}} e^{j k (k_{x} x + k_{y} y)} d S_{3} \\ = & \int_{S_{3}} f_{j}^{M *} (x, y) e^{j (k_{x} x + k_{y} y)} d S_{3} \cdot H_{i, 0} |_{S_{3}} \\ (49) & = & {\tilde{f}}_{j}^{M *} (k_{x}, k_{y}) \cdot H_{i, 0} |_{S_{3}} \end{array}

また，第2項は，同様にして，

\begin{matrix} (50) & \int_{S_{3}} f_{j}^{M *} \cdot H_{r, \tan} d S_{3} = {\tilde{f}}_{j}^{M *} (k_{x}, k_{y}) \cdot H_{r, 0} |_{S_{3}} \end{matrix}

いま，

\begin{matrix} (51) & V_{3, j} \equiv - {\tilde{f}}_{j}^{M *} (k_{x}, k_{y}) \cdot (H_{i, 0} |_{S_{3}} + H_{r, 0} |_{S_{3}}) \end{matrix}

とおくと，

\begin{array}{rcl} V_{3, j} & = & \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} d S_{3} - \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan}^{'} d S_{3} \\ = & \int_{S_{3}} f_{j}^{M *} \cdot (H_{s, \tan} (J_{s 1}) |_{S_{3}} + H_{s, \tan} (M_{s}) |_{S_{3}}) d S_{3} \\ - \int_{S_{3}} f_{j}^{M *} \cdot (H_{s, \tan} (J_{s 2}) |_{S_{3}} + H_{s, \tan} (- M_{s}) |_{S_{3}}) d S_{3} \\ = & \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} (J_{s 1}) d S_{3} - \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} (J_{s 2}) d S_{3} \\ + 2 \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} (M_{s}) d S_{3} \\ = & \int_{S_{3}} f_{j}^{M *} \cdot \sum_{i} I_{1, i} H_{s, \tan} (f_{i}^{J 1}) d S_{3} \\ - \int_{S_{3}} f_{j}^{M *} \cdot \sum_{i} I_{2, i} H_{s, \tan} (f_{i}^{J 2}) d S_{3} \\ + 2 \int_{S_{3}} f_{j}^{M *} \cdot \sum_{i} I_{3, i} H_{s, \tan} (f_{i}^{M}) d S_{3} \\ = & \sum_{i} I_{1, i} (\int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} (f_{i}^{J 1}) d S_{3}) \\ + \sum_{i} I_{2, i} (- \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} (f_{i}^{J 2}) d S_{3}) \\ + \sum_{i} I_{3, i} (2 \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} (f_{i}^{M}) d S_{3}) \\ (52) & \equiv & \sum_{i} I_{1, i} z_{j i}^{31} + \sum_{i} I_{2, i} z_{j i}^{32} + \sum_{i} I_{3, i} z_{j i}^{33} \end{array}

ここで，

\begin{array}{rcl} (53) & z_{j i}^{31} & = & \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} (f_{i}^{J 1}) d S_{3} \\ (54) & z_{j i}^{32} & = & - \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} (f_{i}^{J 2}) d S_{3} \\ (55) & z_{j i}^{33} & = & 2 \int_{S_{3}} f_{j}^{M *} \cdot H_{s, \tan} (f_{i}^{M}) d S_{3} \end{array}

パッチS

_{2}

上の境界条件の式の両辺に，そのパッチの電流に関わる基底関数

f_{2, j}

を乗じて面S

_{2}

にわたって積分すると，

\begin{matrix} (56) & \int_{S_{2}} E_{s, \tan}^{'} \cdot f_{2, j} d S_{2} = 0 \end{matrix}

同様に

V_{2, j} \equiv 0

を定義し，

\begin{array}{rcl} V_{2, j} & = & - \int_{S_{2}} f_{j}^{J 2 *} \cdot (E_{s, \tan} (J_{s 2}) |_{S_{2}} + E_{s, \tan} (- M_{s}) |_{S_{2}}) d S_{2} \\ = & - \int_{S_{2}} f_{j}^{J 2 *} \cdot E_{s, \tan} (J_{s 2}) d S_{2} + \int_{S_{2}} f_{j}^{J 2 *} \cdot E_{s, \tan} (M_{s}) d S_{2} \\ = & - \int_{S_{2}} f_{j}^{J 2 *} \cdot \sum_{i} I_{2, i} E_{s, \tan} (f_{i}^{J 2}) d S_{2} \\ + \int_{S_{2}} f_{j}^{J 2 *} \cdot \sum_{i} I_{3, i} E_{s, \tan} (f_{i}^{M}) d S_{2} \\ = & \sum_{i} I_{2, i} (- \int_{S_{2}} f_{j}^{J 2 *} \cdot E_{s, \tan} (f_{i}^{J 2}) d S_{2}) \\ + \sum_{i} I_{3, i} (\int_{S_{2}} f_{j}^{J 2 *} \cdot E_{s, \tan} (f_{i}^{M}) d S_{2}) \\ (57) & \equiv & \sum_{i} I_{2, i} z_{j i}^{22} + \sum_{i} I_{3, i} z_{j i}^{23} \end{array}

ここで，

\begin{array}{rcl} (58) & z_{j i}^{22} & = & - \int_{S_{2}} f_{j}^{J 2 *} \cdot E_{s, \tan} (f_{i}^{J 2}) d S_{2} \\ (59) & z_{j i}^{23} & = & \int_{S_{2}} f_{j}^{J 2 *} \cdot E_{s, \tan} (f_{i}^{M}) d S_{2} \end{array}

積分を実行して，グリーン関数を用いて表すと，

\begin{array}{rcl} (60) & z_{j i}^{11} & = & \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{f}}_{j}^{J 1 *} (k_{t m n}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{E J (0)}} (k_{t m n}) \cdot {\tilde{f}}_{i}^{J 1} (k_{t m n}) \\ (61) & z_{j i}^{12} & = & 0 \\ (62) & z_{j i}^{21} & = & 0 \\ (63) & z_{j i}^{13} & = & \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{f}}_{j}^{J 1 *} (k_{t m n}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{E M (d)}} (k_{t m n}) \cdot {\tilde{f}}_{i}^{M} (k_{t m n}) \\ (64) & z_{j i}^{22} & = & - \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{f}}_{j}^{J 2 *} (k_{t m n}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{E J (0)}} (k_{t m n}) \cdot {\tilde{f}}_{i}^{J 2} (k_{t m n}) \\ (65) & z_{j i}^{23} & = & \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{f}}_{j}^{J 2 *} (k_{t m n}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{E M (d)}} (k_{t m n}) \cdot {\tilde{f}}_{i}^{M} (k_{t m n}) \\ (66) & z_{j i}^{31} & = & \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{f}}_{j}^{M *} (k_{t m n}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{H J (d)}} (k_{t m n}) \cdot {\tilde{f}}_{i}^{J 1} (k_{t m n}) \\ (67) & z_{j i}^{32} & = & - \frac{1}{d_{x} d_{y}} \sum_{m, n} {\tilde{f}}_{j}^{M *} (k_{t m n}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{H J (d)}} (k_{t m n}) \cdot {\tilde{f}}_{i}^{J 2} (k_{t m n}) \\ (68) & z_{j i}^{33} & = & \frac{2}{d_{x} d_{y}} \sum_{m, n} {\tilde{f}}_{j}^{M *} (k_{t m n}) \cdot {\tilde{\bar{\bar{G}}}}_{T}^{_{H M (0)}} (k_{t m n}) \cdot {\tilde{f}}_{i}^{M} (k_{t m n}) \end{array}

行列表示式は，

\begin{matrix} (69) & (\begin{matrix} V_{1} \\ V_{2} \\ V_{3} \end{matrix}) = (\begin{matrix} [Z_{11}] & [Z_{12}] & [Z_{13}] \\ [Z_{21}] & [Z_{22}] & [Z_{23}] \\ [Z_{31}] & [Z_{32}] & [Z_{33}] \end{matrix}) (\begin{matrix} I_{1} \\ I_{2} \\ I_{3} \end{matrix}) \end{matrix}

ただし，

\begin{matrix} (70) & z_{j i}^{12} = z_{j i}^{21} = 0 \end{matrix}

したがって，次式を解けば電磁流分布を求めることができる．

\begin{matrix} (71) & (\begin{matrix} I_{1} \\ I_{2} \\ I_{3} \end{matrix}) = {(\begin{matrix} [Z_{11}] & [Z_{12}] & [Z_{13}] \\ [Z_{21}] & [Z_{22}] & [Z_{23}] \\ [Z_{31}] & [Z_{32}] & [Z_{33}] \end{matrix})}^{- 1} (\begin{matrix} V_{1} \\ V_{2} \\ V_{3} \end{matrix}) \end{matrix}