E y = Ely H y = H ly n x E s = n x El s n x H s = n x HIs in .NET

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E y = Ely H y = H ly n x E s = n x El s n x H s = n x HIs
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(3.2.33a) (3.2.33b) (3.2.33c) (3.2.33d)
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where E s , H s , El s , and HIs are related to E y , H y , Ely, and H ly by (3.2.2). Equations (3.2.33a) and (3.2.33b) relate the four unknowns. Next we need to put (3.2.33c) and (3.2.33d) as conditions on the eight unknowns. From n x E s = n x Els, (3.2.2a) and (3.2.2b),
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2n x VsEy + Jiin x V s x H y = y;;'2n x
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sEly + Vn x
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(3.2.34)
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3 SCATTERING AND EMISSION BY A PERIODIC ROUGH SURFACE
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Note that where Ay =
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n x (V's x A y ) = -yn V'sA y yAy(Ps)' Thus,
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(3.2.35)
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kiynxV'sEy-wJLy(n'V'sHy) =
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[kiynxV'sEly-wJLly(n'V'sHlY)] (3.2.36)
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From (3.2.33a) and (3.2.33b) we have two relations
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Ey[x, z
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f(x)]
Ely[x, z
f(x)]
(3.2.37a)
Hy[x, z = f(x)] = Hly[x, z = f(x)] (3.2.37b) Note that the surface fields (with z = f(x)) E y , H y , Ely, and H ly are functions of x only. If we take total derivative of (3.2.37a) with respect to x, we have OEl y oEly df(x) dEy oEy oEy df(x) dEly + & ~ = -----;{;: (3.2.38) dx = ox + oz ~ = with z = f(x). Since
A [df A A] n = - dx x + z
[1 + (df)2]-~ dx
[OE y ox
we get
nX V'sE y =
[1+ (df)2]-~
~ [1+ (:
dH dHl Similarly, dxY = dx y, so that
dfOE y ] dx OZ z=f(x)
(3239)
From (3.2.38) and (3.2.39), it follows that
n x V'sEy = n x V'sEl Y
(3.2.40a)
n x V'sHy = n x V'sHl Y Putting (3.2.40a) and (3.2.40b) in (3.2.36), we have y(n V'sH y ) = -don x V'sEl Y + d2y(n V'sH ly ) where
(3.2.40b) (3.2.41)
(3.2.42a) (3.2.42b)
2.2 Surface Field Integral Equations and Coupled Matrix Equations
Equation (3.2.41) is a result of applying n x E s = n x E1s' If we apply the same procedure to n x H s = n x HIs, we get (3.2.43) where
(3.2.44a) (3.2.44b)
Using (3.2.39) and the like in (3.2.41) and (3.2.43), we get respectively
n (\JsHy) = -
do dE 1y dx
+ d2(n \J sH 1y )
(3.2.45a)
[1+ (1x)']
Co--
d H 1y dx
n (\JsEy) =
+ C2(n \J sE 1y )
(3.2.45b)
[1+ (1x)']
Equations (3.2.37a), (3.2.37b), (3.2.45a), and (3.2.45b) provide the four relations for the eight surface fields E y , H y , Ely, H 1y , n . \JsEy, n . \JsHy, n . \JsE1y, and n . \JsH1y . The integral equations (3.2.32a-d) provide the additional four integral equations.
Surface Field Expansion
As indicated in Section 2.1, the surface fields obey Floquet's theorem; that is, 1jJ(x) = exp(ikixX)W(X) , where w(x) is a periodic function with period P. Thus one can expand w(x) in a Fourier series. We use such expansion for the surface field components as follows. With Ps(x) = xx + zf(x), let
Ey[Ps(x)] = E 1y [Ps(x)] =
L 2a~ exp [ikiXX + in ~ x]
(3.2.46a) (3.2.46b) (3.2.46c)
dO"n \J sE 1y [Ps(x)] = ik 1s dx Hy[Ps(x )]
L 2(J~ exp [ikiXX + in ~ x]
= H 1y [Ps(x )] = L
21'~ exp [ikiXX + in ~ x]
3 SCATTERING AND EMISSION BY A PERIODIC ROUGH SURFACE
dO"n VsHly[Ps(X)] = ik1sdx
L 2b~ exp [ikiXX + in ~ X]
(3.2.46d)
From (3.2.45a), (3.2.45b), and (3.2.46a-d),
dO"n . V sHy = dx
L [-do200~i (k iX + n~7f) + d2iklS2b~]
exp [ikiXX
+ in ~ X]
(3.2.46e)
dO"n' V sEy = dx
L [co2r~i (kiX + n~7f) + C2ikls2(3~]
exp [ikiXX
+ in ~ X]
(3.2.46f)
Thus all eight surface field components are now expressed in terms of four sets of unknown coefficients oo~, (3~, r~, and b~. Substitute the surface field expressions of (3.2.46a-d) in the two surface integral equations of (3.2.32c) and (3.2.32d). Define QD 2 , QN2 as the Dirichlet and Neumann matrices with elements
= =
[Q~2 ]mn
P Jp dx
e-ik~m'Ps(x)
[ 27f ] exp ikixx + infix n) 27f x P
~ Jp dx exp [-i(m r (3in
iklS( (3~)f(X)]
(3.2.47a)
[Q~2] mn
== -'k-z lsP
dO" n . V s
Cik~m'Ps(X)
V (3in
I7.iI
[ exp ikixx
+ in-x
27f ]
dxexp [ 2 - n)-p x - ikls( (3~)f(x) ] (3.2.47b) -i(m 7 f (3in (3in P p where the integrations are performed over one period. We obtain the two matrix equations
-QD 2 (3 - QN2 as
a' a' = -1 +~ n
-(h)
(3.2.48a)
-QD 2 b - Q N2 r = B
(3.2.48b)
In deriving the second equality in (3.2.47b), we have performed an inte-
2.2 Surface Field Integral Equations and Coupled Matrix Equations
gration by parts. Here the vectors B, B(h), as, 7J , ;:ys, and 3 contain the (h) S . e1enlents B n, B n, an' (3s In' Un' respectlve1 S'lml'1 ar 1 sub ' n' S 5:S y. y, stltutlng t h e surface field expressions of (3.2.46a-d) in the surface integral equations of (3.2.32a) and (3.2.32b), we get the equations in the following matrix forms
a; = CoQhyl a;(h) =