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(1.2.25d)
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Using (1.2.25b-d) in (1.2.25a) gives
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ei o(k1.. - kil..) = "2 k [e( -kz)e( -k z ) + h( -kz)h( -k z )] z . [A(k1..)
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1 k + "2 k
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(1.2.25e)
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2.1 Zerotll- and First-Order Solutions
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Similarly, (1.2.7b) becomes 0=
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k~)]
+ 2k 1
[-h1(klz)el(kl z ) + el(k1z)hl(k 1z )]
. [B(kl-) - ik 1z - kJz
Jdk~B(k~)F(kl(kl- -
(1.2.25f)
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k~)]
In the following, we will solve (1.2.25e) and (1.2.25f) up to the second order.
2.1 Zeroth- and First-Order Solutions
Zeroth-Order Solution
From (1.2.24), let
= B(O) = 0 z A~\kl-) = (qia~O) + Pia~O )o(kl- - k n ) B~) (kl-) = (qib~O) + Pib~O )o(kl- - k n )
A(O) z
Balance (1.2.25e) and (1.2.25f) to the zeroth order
(1.2.26a) (1.2.26b)
(1.2.26c)
eio(kl- -kn)
2k e(-kz)e(-k z ) + h(-kz)h(-kz ) . Al- (kl-) z
k + 2k [
k [A
-(0)-
-h( -kz)e( -kz ) + e( -kz)h( -kz ) . B l- (kl-)
A A ]
-(0)-
(1.2.26d)
el(k1z)el(k 1z ) + hl(klz)hl(klz) . Al- (kl-) 1z k A A A ] -(0)+ 2k1 [A -hl(klz)el(klz) + el(k1z)hl(k 1z ) . B l- (kl-) (1.2.26e) 1z Substitute (1.2.26b) and (1.2.26c) into (1.2.26d) and (1.2.26e) and make use of (1.2.22a)-(1.2.22f); on taking a dot product of (1.2.26d) with e( -kiz ) and
0= 2k
-(0)-
1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
e(-k o ) . eo = !~a(O) + !b(O) (1.2.27a) tz t 2k q 2 p iz h( -k o ) . eo = !a(O) - !~b(O) (1.2.27b) tZ t 2 p 2k q iz On taking a dot product of (1.2.26e) with el(kl zi ) and h1(klzi) gives
o = !~aq(O) 2 k 1zi
!bp(O) 2 2 k 1zi q
(1.2.27c) (1.2.27d)
2 k1 p Solving (1.2.27a) and (1.2.27c) gives
-!~a(O) - !~b(O)
o;~O) (kiJJ =
b~O) (kiJJ
[e( -kiz) . ei]
k~z (1 -
Rho)
(1.2.28a) (1.2.28b)
= [e( -kiz ) . ei] (1
+ Rho)
where Rho is the Fresnel reflection coefficient for the TE waves: kiz - k 1zi R ho = kiz + k1zi Solving (1.2.27b) and (1.2.27d) gives
(1.2.29)
0;1 ) (kilJ = [h( -kiz ) . ed(1
-(0) -
+ R vo )
kiz (1 - R vo )
(1.2.30a) (1.2.30b)
bq (kilJ = -[h( -kiz ) . ei]
where R vo is the Fresnel reflection coefficient for the TM waves: _ krkiz - k 2 k 1zi R va - 2k k 1 iz + k 2k 1zi The zeroth-order surface fields are o;(O)(r1-) = (qia~O) + Pia1 ))eiki-L:r-L
(1.2.31)
(1.2.32a) (1.2.32b)
b(O\r1-) = (qib~O) E(O) (r) = - _1_
+ Pib1 ))eiki-LOr-L
Substitution into (1.2.13a) gives the zeroth order scattered field
87r 2
dk eik-L or -L +ikz z ~ 1kz
dr' e -ik-L or'.L
. {[e(kz)e(k z ) + h(kz)h(kz )] .
(qia~O) + Pia10)) eiki-LOr'.L
(qib~O) + Pib~O)) iki-L or'.L }
(1.2.33)
[-h(kz)e(k z ) + e(kz)h(k z )] .
2.1 Zeroth- and First-Order Solutions
Integrating over df~ gives o(kJ.. - kiJ..). Simplifying we get
-(0) (r)= E
1 k et'-k '-L r~+t'k izZ 2 kiz
. { e(kiz )
(a~O) - k~z b~O))
- h(kiz ) (a~O) k
t + b~O))
} (1.2.34)
Substitutions of (1.2.28a) and (1.2.28b) gives
E~O)
Rho[e( -kiz ) . ei]e(kiz ) + Rvo[h( -kiz ) . ei]h(kiz )} iku r ~ +ikizz
(1.2.35)
Similarly, for the transmitted field we have
E~O)
{(I + Rho) [e(-kiz) . ei]e1(-k 1zi )
+ Rvo)[h(-kiz )' ei]h 1(-k 1zd }eiki~.r~-ik1ziZ
(1.2.36)
The zeroth order solution of (1.2.35) and (1.2.36) are just the reflected and transmitted fields for a flat surface.
First-Order Solution
We first calculate the first-order solution of the z-component of the surface fields A and Busing (1.2.24a) and (1.2.26b). Let
A~l)(kJ..) = iF(kJ.. - kiJ..)Az(kJ..) B~l)(kJ..) = iF(kJ.. - kiJ..)Bz(kJ..)
Then
(1.2.37a) (1.2.37b)
Az(kJ..) = (kJ.. - kiJ..) . qia~O)
= -kpsin( <Pk -
+ (kJ.. - kiJ..) . Pia~O) ( (0) <pi)aq + kpcos( <Pk - <pd -
o k pi ap
(1.2.37c)
Bz(kJ..) = (kJ.. - kiJ..) . qib~O)
+ (kJ.. - kiJ..) . Pib~O)(kiJ..) () ( ) (o) O = -kpsin(<pk - <pi)bq + kpCOS(<pk - <pd - k pi bp
(1.2.37d)
Balancing (1.2.25e) to the first order, we have an equation for A~)(kJ..) and
B~)(kJ..)' We further let
A~)(kJ..) = iF(kJ.. - kiJ..)AJ..(kJ..) B~) (kJ..)
(1.2.38a) (1.2.38b) (1.2.38c) (1.2.38d)
iF(kJ.. - kiJ..)BJ..(kJ..)
AJ..(kJ..) = Aq(kJ..)q(kJ..) + Ap(kJ..)p(kJ..) B J..(kJ..) = Bq(kJ..)q(kJ..)
+ Bp(kJ..YjJ(kJ..)
1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
The first-order equation of (1.2.25e) 'on using (1.2.38) is
lk[ - - 2 k e(-kz)e( -kz ) + h( -kz)h( -kz )]iF(kl- z [Aq(kl-)q(kl-)
+ Ap(kl-)p(kl-)]
- 2.k -h( -kz)e( -k z ) + e( -kz)h( -k z ) iF(kl- - k d
lk[ (1) -
] - -
[Bq(kl-)q(kl-)
+ Bp(kl-)p(kl-)]
2. k p A z
lk[ - + 2 k e( -kz)e( -k z ) + h( -kz)h( -k z)]
z ikzF(kl- - kil-) (qia~O)
1 k p (1) - _ (kl-)h( -kz) + 2 k B z (kl-)e( -kz) z
+ Pia~O))
lk[ + 2k -h( -kz)e( -kz ) + e( -kz)h( -kz )]
ikzF(kl- - kil-) (qib~O)
Simplifying (1.2.39), we have
+ Pib1 ))
(1.2.39)
- :z [e( -kz)Aq(kl-) + h( -kz)Ap(kl-) :z] - :z [-h( -kz)Bq(kl-)