Balancing coefficients of e( -kz ), we obtain in Java

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Balancing coefficients of e( -kz ), we obtain
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( -A q (k-L)-k Bp (k-L) 2) 2
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-I -I B -I dk-L -(0) (k-L)F (2) (k-L-k-L)
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(1.2.57)
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=; B~2)(k-L) + Jdk~
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(e(-k z )
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A~)(k~)) F(k-L - k~)
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(1.2.58a)
- ~; Jdk~ Jdk~ Jdk~ J
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(h( -k z ) . B~) (k~)) F(k-L (h( -kz ) . zB~1) (k~)) F(k-L -
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Balancing coefficients of h( -kz ), we have
-kAp (k-L)+B q (k-L) =kAz (k-L)+ik z + ik z
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dk-Lh(-kz) A-L (k-L)F(k-L-k-L) -
-(1)
Jdk~h(-kz) zAF)(k~)F(k-L
1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
~; J dk~h(-kz) A~)(k~)F(2)(k1- - k~)
dk~ e( -kz ) . B~) (k~)F(k1- - k~)
(1.2.58b)
- ik z J
+ ~; J dk~e(-kz) B~)(k~)F(2)(k1- - k~)
Balancing (1.2.191) to the second order and using (1.2.24e) and (1.2.24b), we obtain
Balancing coefficients of
e1(J\~lz)
yields
-A(2)(k )~ + k 1z B(2)(k ) q 1- k k1 P 11
B~2)(k1-) - ik1Z~ Jdk~ (e1(k 1Z ) A~)(k~)) F(k1- - k~) k
- kJz :1 J - ik 1z J - kJz J
dk~ (e1(k1Z) A(O\k~)) F(2)(k1- - k~)
dk~ (h1(k 1z ) . B~\k~) + h1(k 1z ) . zB~l)(k~)) F(k1- - k~)
dk~ (h1(k 1z ) B(O)(k~)) F(2)(k1- - k~)
(1.2.60a)
Balancing coefficient of h 1(k 1z ) gives
2.2 Second-Order Solutions
k 1z k A(2)(k ) + B(2)(k )
kr A z
(2) -
.Jdk~
- kJz
. k (k.l) - zk1z k 1
(h 1(k 1Z ) . A~\k~) (h 1(k 1Z )'
+ h 1(k 1z )' zA~l)(k~)) P(k.l - k~)
(1.2.6Gb)
:1 Jdk~
+ ik 1z
+ kJz
Jdk~ Jdk~
A(O)(k~)) P(2)(k.l - k~)
(e1 (k 1z ) . B(l) (k~)) P(k.l (e1(k1Z)'
B(O)(k~)) P(2)(k.l - k~)
For the scattered wave, the second-order scattered field is, from (1.2.19g) and (1.2.24c) and (1.2.24d),
EsCf) = -
dk.likl-.Tl-eikzz :z . {e(kz)A~2)(k.l)
+ h(k z) [k; A~2)(k.l) - ~ A~2)(k.l)]
- e(kz)ik z
- h(kz)ik z -
Jdk~ Jdk~ e(kz)~; Jdk~ h(kz)~; Jdk~
(e(k z ) . A (1) (k~)) P(k.l (h(k z ) . A (l\k~)) P(k.l (e(k z ) (h(k z )
A(O\k~)) P(2)(k.l- k~)
A(O)(k~)) P(2)(k.l- k~)
h(kz)B~2)(k.l) + e(k z ) (~ B~2)(k.l) - ~z B~2)(k.l))
+ h(kz)ik z
- e(kz)ik z
+ h(k z )kl
- e(k z
Jdk~ Jdk~ Jdk~ /l Jdk~
(e(k z ) . B(l\k~)) P(k.l (h(k z ) . B(l) (k~)) P(k.l (e(k z )'
B(O)(k~)) P(2)(k.l- k~)
(h(k z )' B(O)(k~)) P(2)(k.l -
k~)}
(1.2.61)
1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
Next, we put (1.2.61) in a form that resembles the extinction equation of (1.2.57).
Es(r) = -
dk.lik.L0r_dik.z :z . {e(k z ){ A~2)(k.l) + k; B~2)(k.l)
k (2) -TBp (k.l)-zk z
J -I (,
-(1) dk.l e(-kz) A (k.l) F(k.l-k.l)
-I) - -I
- ~; Jdk~ Jdk~ J -I [ ,
- ik z
Z - k2
(e(-k z )' A(O)(k~)) P(2)(k.l -
[-h(-k z )' B~)(k~) + h(-k z )
ZB~l)(k~)] F(k.l - k~)
dk.l - h (- kz) . -(0) (k.l) B.l
+ F (2) (-k.l -
-I] } k.l)
+ h(k z ){ ~ A~2)(k.l) - ~z A12)(k.l)
B~2)(k.l)
Jdk~ - ~; Jdk~ + J -I [ k; +"2 J -I [,
- ik z ik z
Es(r) = -
[-h(-k z )'
A~)(k~) + h(-kz ) zAF)(k~)] F(k.l- k~)
[-h(-k z )' A~)(k~)] P(2)(k.l -
dk.l e( -k z ) . -(1) (k.l) F(k.l - k.l) B
-I] - -I
dk.l e(-k z ) -(0) (k.l) F (2) (k.l - k.l) B.l
-I ]
-I} }
(1.2.62)
Using (1.2.58a) and (1.2.58b) in (1.2.62), we get a factor of 2. Thus
J [A~2)(k.l) + ~ B~2)(k.l) - ~; Jdk~ A~\k~)) k~) + J dk~h(-kz)' B~)(k~)F(k.l k~)]
dk.lik.Lor.L+ik.z :z {e(k z ) (e( -k z ) . F(2)(k.l ik z
+ h(k z ) [~ A~2)(k.l) - B~2)(k.l)
+ ik z
Jdk~ + ~; J dk~
h( -k z ) . A~) (k~)F(k.l -
e( -k z ) . B~) (k~)p(2) (k.l -
k~)] }
(1.2.63)
2.2 Second-Order Solutions
We next eliminate B~2) from (1.2.58a) and (1.2.60a) :
-Aq (k-i) k (kz + k1z) - kk (kz + k1z)B z (k-i) 1 1 -
_ kp
(2) -
k~~~z (k1z + kz ) J dk~ (e( -kz ) . A~\k~)) P(2)(k-i - k~)
ik 1z
+~(kz+k1Z)
J -f ('
-(1) dk-i h(-kz) B-i (k-i) P(k-i-k-i)
-f) - -f
+ ~~: (krz Thus
k;) J
dk~ (h(-k z)' B~)(k~)) P(2)(k-i - k~).
(1.2.64)
A~2)(k-i) = - ~ B~2)(k-i) + k 1;kz J dk~ (e(-k z )' A~\k~)) P(2)(k-i - k~)
- ik 1z J -
dk~ (h( -kz ) . B~) (k~)) P(k-i - k~)
(1.2.65)
k~Z(k1Z-kz) J dk~(h(-kz) B~)(k~))p(2)(k-i-k~)
Similarly, using (1.2.58b) and (1.2.60b)
B(2)(k ) = k kA(2)(k) k 1z + k z q -i p z -i krkz + k 2k 1z
z . -(1) +zk1zk 2 k 2k + k1zk J dk-i h(-kz) A-i (k-i) P(k-i-k-i) k k2 1 z+ 1z 2 2 + k k 1z (kr - k ) J dkf (h(-k). A (0) (kf ) ) p(2) (k - k' ) 2 krkz + k2k1z -i z -i -i -i-i 2 . (kr - k ) J -(1) + zk 1z k z 2 2k dk -i e( -k z ) . B -i (k -i) P(k-i - k-i) k 1k z + k 1z k2 + k zk 1z (kr k 1z+ k z ) Jdk ('(-k ) . B(O)(k P(2)(k - k (1.2.66) 2 krkz + k2k1z -i e z -i -i -i-i