flWLTIPLE SCATTERING THEORY FOR DISCRETE SCATTERERS in Java

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5 flWLTIPLE SCATTERING THEORY FOR DISCRETE SCATTERERS
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Derivation of Radiative Transfer Equation from Ladder Approximation
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In this section, we illustrate the derivation of the radiative transfer equation from the ladder approximation. To simplify the derivation, we use isotropic point scatterers. We assume that the extinction coefficient of the coherent wave is much smaller than the wavenumber. Consider a wave 'lj)inc("r) incident on a single point scatterer located at l'a' Then the total field 'lj) is given by
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'1/)(1') = 'lj)inc(r)
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1'lj)inc(ra )
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(5.5.1)
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where f is the scattering amplitude and k is the wavenumber of the background medium. Since C(O)(r) = exp(ih)j41IT is the Green's function for the unperturbed problem, (5.5.1) can be written as
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'ljJ(r)
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= 'ljJinc(r) + ./df'
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C(O) (1', r')To,{r', 1''')'ljJinc ("r")
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(5.5.2)
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Tn (1", 1''') = 47f f 0(1" - 1'0') 0(1''' - 1'0')
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is the transition operator for the point scatterer at C (I)) (1' 1") = -(-,'----;-:, 47flr oiklr-r'l
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(5.5.3)
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We also use (5.5.4)
In diagrammatic notation, let (5.5.5) Then (5.5.2) becomes '1/) =
1)irw
+ --0
'I/)inc
(5.5.6)
where - - stands for C(O). For the case of isotropic point scatterers, we a,ssume that the particle positions are independent. For the first and second moment equations, retaining the first terms in the mass operator and the intensity operator leads, respectively, to
= --
+ ~'C'-----
(5.5.7)
for the Foldy-approximated Dyson's equation, and
(CC*) =
(CC*)
(5.5.8)
5 Derivation of Radiative Transfer Equation
for the ladder-approximated Bethe-Salpeter equation, where the diagrammatic nota.tions are used. In (5.5.7) and (5.5.8), stands for transition operator and a solid line joining two denotes that the two belong to the same scatterer. Equation (5.5.7) in analytic form is
(G(1',1' o)) = G(O) (1', 1'o)+no
JJ J
dfi df 1
df2 G(O) (1', 1'I)Ti(1'1' 1'2)(G(1'2' 1'0)) (5.5.9)
and (5.5.8) in analytic form is
(G(1',1'o)G*(r', 1'~)) = (G(1', 1'0)) (G*(r', ~))
+ no
JJJJJ
dfi df 1 df2 df3
(G(1', 1'1)) (G*(1", 1'2) )Ti (1'1' 1'3)Tt(1'2, 1'4)(G(1'3' 1'o)G*(1'4, 1'~))
Using (5.5.3) in (5.5.9) and (5.5.10), we have
(5.5.10)
(G(1',1' o)) = G(O) (1', 1'0)
+ 47rn of
dfi G(O) (1', 1'i)(G(1'i, 1'0))
(5.5.11)
(G(1', 1'o)G*(r', 1'~)) = (G(1', 1'0)) (G*(r', 1'~)) + n o(47r)2IfI 2 dfi(G(1', 1'i))(G*(r', 1'i)) (G(1'i, 1'o)G*(1'i, ~)) (5.5.12)
The Foldy-approximated Dyson's equation in (5.5.11) can be solved readily. Operating on (5.5.11) by (\72 + k 2 ), we have
(\72
+ k 2 + 47rn of) (G(1', 1'0))
= -<5(1' - 1'0)
(5.5.13)
Hence, the effective propagation constant K obeys the equation
K 2 = k 2 + 47rn of
(5.5.14)
K = k
+ 27r~of
(5.5.15)
and the at tenuation rate
"'e for the coherent wave is "'e = 2K" = 47rkn o1m f (5.5.16) Generally the extinction rate "'e is much smaller than the wavenumber K'
(5.5.17)
Next consider a plane wave incident upon a slab of point scatterers of thickness d in the direction ((}i, <Pi) (Fig. 5.5.1). The slab of scatterers lies between z = 0 and z = -d. Let k be the wavenumber of the upper halfspace (region 0) and the lower half-space (region 2) and also the wavenumber
5 fl'fULTIPLE SCATTERING THEORY FOR DISCRETE SCATTERERS
Regioll 0
z =-d
Regioll 1
Rt'gioll 2
Figure 5.5.1 Scattering of waves by a slab of isotropic point scatterers.
of the background medium of the slab (region 1). It is assumed that the concentration of particles is low so that K' is approximately equal to k. The mean field in region 1 is (5.5.18) where
K i = k sin Oi cos 4>/i~
with
+ k sin Oi sin 4>i'l1 -
K i;):
(5.5.19)
K iz = (K 2 - k 2 sin 2 0;) 1/2 ::::
kCOSOi
+ iK" ILi
(5.5.20)
and ILi = cos 0i. The approximate relation in (5.5.20) is a result of (5.5.17) that attenuation rate is much less than the wavenumber. Let h(r) denote intensity in region 0, which is a summation of all the ladder terms. Diagrammatically,
(5.5.21) where
VVVV\
= mean Green's function (C ll (r, r')) with
both observation and source points in region 1
= mean = mean
field 'ljJ m
Green's function (COl (r, r')) with observation point in region 0 and source in region 1
.5 Derivation of Radiative Transfer Equation
Note that (5.5.21) is consistent with (5.5.8). The mean Green's function (C II ) is
(C ll (r'1,'h))
= 4ei;lrl-r~11 = 8 i
7r1'1-72 7r
dkl.. ikl-.(ru-r;)+iIClzl-Z21
(5.5.22) with kl.. = kxx + kyY and K z = (K2 - k;, - k;)1/2. Because the observation point is in far field in region 0, the mean Green's function (COl (1', 1")) is e ik l' _ (C 011',1' = - e -iK ., .1" (- -')) (5.5.23)
47rT
where K s denotes scattered direction and is
K s = k sin Os cos <P.,x + k sin Os sin <PsY + Kzsz
where
(5.5.24)
K zs =
J K2 -
k 2 sin 2 Os ::::::' k cos Os
+ i K"
/-Ls
(5.5.25)
with lis = cos Os' Next we define [] as
h with ends stripped. Hence diagrammatically,
h(1')
=::J:Ir=
(5.5.26)
By summing the ladder series, the operator equation for [] is
(5.5.27a)
We also let
(1'1, 1'2;r:l, 1'4) = [ ]
:1 I
(5.5.27b)
To solve the ladder intensity, let
(1'1,1'2;1'3,1'4) = F(1'I,1'2)8(1'I-1':J)8(1'2 -1'tl) + Tl. o (47r)2IfI 2 8(1'1 - 1'2) 8(1'1 - 1':l) 8(1'2 - 1'4)
(5.5.28)
On substituting (5.5.28) into (5.5.27) and using the transition operator for point scatterers as given by (5.5.3a), the F function in (5.5.28) obeys the equation
F(1'I,1'2) =