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Thus, substituting (1.2.65) and (1.2.66) into (1.2.63), we have
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= - J dk-i eik1-.r1- +ikzz :z
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. { e(k z) [~z (k 1z - k z ) J - i(k 1z - k z ) J -
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1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
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2 + h(k ) [k p A (2)(k ) kz(kr - k ) z k z .1 krkz + k 2k 1z + 2k i k 2k k;(k 2 - kr) J dk~ (h( -kz ) . A~) (k~)) F(k.l - k~) k1 z + 1z 2 + k zkr(k - kr) J dk' ('(-k). BCO)(k' )) F (2)(k - k ' ) 2(kr k z + k 2k 1z ) .1 e z .1.1 .1.1 2 2 _ k k 1z (kr - k ) Jdk' (h(-k). ACO)(k' )) F (2)(k _ k ' ) 2 krkz + k2k1z .1 Z.l.l .1.1
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-I) - -I] } (1.2.67)
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AF)(k.l) = J
dk~F(k.l - k~)i(k.l - k~) . A~)(k~)
dk~ F(k.l - k~)F(k~ -
ku )
. [(k.l - k~) . (Aq(k~)q(k~)
+ Ap(k~)])(k~))] (1.2.68)
Substituting the zeroth- and the first-order solutions in (1.2.67) and using (1.2.68), we have
E~2)(r) =
J dk.lik.L.1-.L+ikzz :z
. {e(k,){ k; (kj, - k,) [(e(-k,) . qi HOI + (c(-k,) . Pi) a~OI] F(21(k~
+ (k1z - k z ) J
-I -I
- kn )
dk~ [(h( -k z ) . q(k~)) Bq(k~) + (h( -kz ) . p(k~)) Bp(k~)]
. F(k.l - ku)F(k.l - k.l)
k~z (k 1z -
k z ) [(h(-k z )' qi)b~O) + (h(-k z ). Pi)
2 , { k p kz(kr - k ) + h(k z ) -k (krkz + k2k1z)
00 1 -I -00
b~O)]FC2)(k.l
- k u )}
dk.l(k.l - k.l) F(k.l - k.l)
- -I
. F(k~ - ki.l) (Aq(k~)q(k~) + Ap(k~)p(k~))
k~(k2 - kr) J - I -I - krkz + k 2k dk.lF(k.l-k.l) 1z
. [(h( -k z ) . q(k~)) Aq(k~)
+ (h( -k z ) . p(k~)) Ap(k~)] F(k~ -
ku )
2.2 Second-Order Solutions
E~2)(r) =
dk.lik~.r~+ikzz
. { e(k z ) [f~;) (k.l, ki.l) (e( -kiz ) . ei)
+ f~i) (k.l, ki.l) (h( -kiz ) . ei)]
( h( -k A
A A) (2) + h(kz ) [(2) (k.l' -: (A -kiz ) . ei + f hh (k.l' -ki.l) fhe ki.l) e(
A iz ) . ei )]}
(1.2.70) Then comparing (1.2.69) and (1.2.70), we obtain,
(1.2.71a)
1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
(2) fhe (k.i,kilJ = -
~ kz
(q(k~)A~(k~) + p(k~)A;(k~) )
i: i:
k (kr - k 2 ) k z (krkz + k2k 1z )
dk~F(k.i - k~)F(k~ -
ku)(k.i -
dk~F(k.i - k~)F(k~ -
h( -kz) . q(k~)) A~(k~) + (h( -kz ) . p(k~)) A;(k~)]
- k;k r F(2) (k.i - ki.i)
(e( -kz ) . Pi) (1 + Rho)
2 k kl z ' , k iz - -2- F (2)(- - -ki.i) (h( -k z ) qi ) T(1- Rho) k.i + kzkl z
Jdk~F(k.i
k~)F(k~ 2
ki.i)
. [(e( -k z ) . q(k~)) B~(k~)
+ (e( -kz ) P(k~)) B;(k~)]}
(1.2071c)
(2) k --:::. hh ( .i,kt.t)
__ ~
{ - ; kz
(q(k~)A~(k~) + p(k~)A;(k~))
i: i:
(kr - k ) k z (krkz + k2k 1z )
dk~F(k.i - k~)F(k~ -
ki.i)(k.i -
dk~ F(k.i - k~)F(k~ -
ki.i)
. [(h( -kz ) . q(k~) )A~(k~) + (h( -kz ) . p(k~) )A;(k~)]
kzkr , - -2- F (2) (k.i - -ki.i) (, -kz ) qi ) e(
k iZ -T )
(1 - R vo )
k k 1z ' - -2- F (2) (k.i - -ku) (h( -k z ) . Pi ) (1 + R vo )
+ kzk 1z
Jdk~F(k.i
k~)F(k~ -
ku )
.[(e( -kz ) . q(k~)) B~(k~)+ (e( -kz) p(k~)) B;(k~)] } (1.2071d)
Note that in the above there are wavevectors, k.i, k.i' and ki.io The wavevec-/
2.2 Second-Order Solutions
tor kl.. denotes scattered direction. The wavevector kil.. denotes incident direction. Since this is the second-order scattering, there is an intermediate direction k~, of which an integration is required in (1.2.71). Substituting (1.2.48) into (1.2.71a) and also using k1z - kz = (kr - k2)j(k 1z + kz), we obtain