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It is given by the expression
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= exp(iKlr - r'l)
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47rr-r -II
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() 8.2.7
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The mean Green's function CO lm has a source in region 1 and an observation point in region O. It obeys the wave equation
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(\7 2 + k 2 )GO lm (1', 1")
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Since the observation point in region 0 is far away from the scattering layer, the far-field approximation can be made on G Olm :
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_I exp( ikr) ( . - _I) G Olm (1', r) = exp -'lK s ' r
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where (8.2.10) K s = Ksxx + Ks/il + Kszz (8.2.11) K sx = k sin Os cos >s (8.2.12) K sy = k sin Os sin cPs 2 - k 2 sin 2 0 )1/2 = K + iK" K sz = (K (8.2.13) s sz sz ' and (Os, cPs) is the scattered direction. In the backscattering direction, K s = -K i so that Os = Oi and >s = 7r + cPi. The first-order scattering term 1 is, from (8.2.5),
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L l = no
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Mj M lM2 M 3M 4C Olm(1', rl)Tj(rl, r2) (8.2.14)
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. 'l/Jm(1'2)C Olm (1', 1'3)T;(1'3, 1'4)'l/J~(r4)
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where T j is the transition operator of the jth particle and 1'j is its position. The integrations of the space variables are carried out over region 1. For point scatterers,
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T j (1'l,1'2)
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= 47rfJ(1'l-1'j)J(1'2 -1'j).
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Substituting (8.2.2), (8.2.9), and (8.2.15) into (8.2.14), the four Dirac delta functions in TjTj* cancel the 4-fold MlM2M3M4 integrations. We obtain
1 = no
Mj(47r)2IfI 2IG Olm(1',1'j)'l/'m(1'j)1 2
= n olfl r2
J /0
M J..j
dz j e(2K:'z+2K;:)zJ
Integrating (8.2.16) gives
Ifl 2 n oA
2(K~'z + K;~)'
2 Second-Order Volume Scattering Theory
where A is the illuminated area of the layer of scatterers. Note from (8.2.4b)
K,2 - k 2 sin 2 ()i = K:; - K:~2 2K'K" = 2K:zK:~ For this case K' ~ k and K:~ KIz' Then K iz "" K' cos, ' ().
(8.2.18a) (8.2.18b) (8.2.18c)
Since the attenuation rate is much smaller than the wavenumber, we have the following approximate relation from (8.2.18b) and (8.2.18c):
K~~ = K" / /-li
where /-Li = cos Bi . Similarly,
" K sz
K 1/ //-ls
where /-Ls = cos()s. Hence (8.2.21 ) where
47fn o lfl 2 2K"
is the albedo of isotropic point scatterers. The second-order ladder term is, from (8.2.5)
L2 =
drjdrldrldr2dr3dr4drsdr6dr7drSGOlm("r, rl)Tj(rl, r2)G llm (r2, r3)
. Tz(r3' r4)'ljJm(r4) . GOlm(r, rs)Tj*(rs, r6)Gilm(r6, r7 )~*(r7, rs)'ljJ~(rs)
Substituting in the expressions for Tj and 'TI, we get
2 =
drl(47f)4IfI4IGOlm(r, rj)1 2 IG llm(rj, rl)1 21'ljJm(rlW
From (8.2.24) and using the expressions for
GOl m
(8.2.24) we have
-2K"IT-rtl 2K ) 2 e .:'zzi e2K:~ZI
Irj -
To perform the integral in (8.2.25), let
rd = rj - rl _ rj + rl
(8.2.26a) (8.2.26b)
r A = ---"---
ZA =
+ zl
2 Zd = Zj Zd
(8.2.26c) (8.2.26d)
Since Zj and Zl only extend over the lower half-space, we have
[00 dZj [00 dZI
-2 0 dZd [00 dZA + [00 dZd [ : dZA
= rdsinOdcos<Pd,
Using spherical coordinates for rd,J:d Zd = rdcosOd, we have
n21fl4joo 1 L2 =~ MAl..
= rd sin 0d sin <Pd,
21" 1 00 7 d<Pd drdrd 2 (171"/2 dOd sin Od j-Td COS
00 27fn~1f14A . 1 1 d (171"/2 dO r2 2(K" + K" ) rd
12 71"/
dOd sin 0d
jTdCOS 8d/2 )e- 2K "Td dZA 2 e(2K:'z+2K:~)ZA e(K :'z-K:~hJ cos8d
+ 171"
dOd sin Ode -2K"Td+ 2K:'zTdCOS8d )
Carrying out the integrals in (8.2.28), gives
4K" " " -/-l-d + 2Kiz + 2Ksz 2K.~z + 2K;~ lid -(:-2-K--:'-:-''----=-2-K-:---,')-(:-::2-=K"""::-'-2-K-'-'.,-) + 2Z + sz /-Ld /-ld (8.2.29) By using the relations of (8.2.19) and (8.2.20) and integrating the expression in (8.2.29), we have n51fl 4A 2= L r2
r1 d/-ld
L2 = (
w) 2 27f2 A
1 /-ls
1 Iii
) [Iii In (1 +
~) + /-ls In (1 + ~)] Iii /-Ls
The sum of the first two ladder terms L1 and L 2 is identical to the result of a second iteration of the radiative-transfer equation for a half-space of point particles. The second-order cyclical term C2 is, on making use of the expressions of Tj and Tl from (8.2.15),
C 2 = n5
M[(47f)4IfI 4G 01m (r, rj)G l1m (rj, rl)'ljJm(rz) (8.2.31)
. G;hm(r, rl)Gi1m(rl, rj)'l/J:n(rj)
2 Second-Order Volume Scattering TlJeor.v
Comparing (8.2.24) and (8.2.31), we note the difference between the secondorder ladder term and the second-order cyclical term. In the second-order cyclical term C 2, the argument of 'ljJ m is rl and the argument of 'lfJ:n is rj. Also C Olm and C Olm have different scatterer arguments rj and rl, respectively. These different arguments are due to the cyclical arrangement of the scatterers. Although the integrand for L2 contains no phase terms, the integrand for C2 contains a phase dependence that results from this cyclical arrangement of the scatterers. If we use the expressions of Calm and CUm and make the substitutions, we have 2K ~ iK.r,-iK*.r ,} (8.2.32) C = n~1f14 A dZA ur d e- "T,/ e -iK ,.r+iK* .1', e '
Let K s and K i denote the real parts of K s and K i , respectively. Then "" C2 = n51fl4AJ dZA J drd exp(-2K"rd) xexp(-iKd'Td)exp[2(Kiz+Ksz)ZAJ 2 2 r r
where K d = K:
+ K:
= (Ki~;