'IU(P)(k p - kip, -k z + k iz ))1 2 2 desinoj'27r d<jJ(lf(P)(kp-kip,kz+kiz)12) + Dnck in Java

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'IU(P)(k p - kip, -k z + k iz ))1 2 2 desinoj'27r d<jJ(lf(P)(kp-kip,kz+kiz)12) + Dnck
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(3.6.27)
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. IU(P)(k p - kip, k z + k iz ))\2 }
We can see that, in (3.6.27), the second term is from ('I/)s(1)) ('I/)s(1)*) , the third and fourth terms are the outgoing incoherent intensities from the boundary, and the fifth and sixth terms are the outgoing incoherent intensities from the top boundary. The second-order coherent scattered field can be calculated as follows. We use (3.4.19). The double summation of L:l L:#l is decomposed into the sum of scattering from two different scatterers in the same cluster and from two scatterers in two different clusters. Thus taking the average of (3.4.19)
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(3.6.28) By using the plane wave representation of Green's functions in (3.6.28), one can show that the coherent scattered field is only appreciable in region 2. By considering the transmitted intensity associated with the second-order coherent field, one gets
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+ H(k p - kip, -k z + k iz )I(1(P)(kp - kip, -k z + k iz ))1 2}
_ Ifl2 (21f11, o D)2
(3.6.29)
By combining (3.6.27) and (3.6.29), energy conservation is exactly obeyed. For thick-layer problems with layer thickness d, one can divide the problem into many layers of thickness D and perform cascading as done in Section 5. A difference form of radiative transfer equation can be obtained as in (3.5.16), with extinction coefficients and phase matrices. The extinction coefficients can be obtained from the transmitted intensity of the coherent wave:
3 VOLUME SCATTERING: CASCADE OF LAYERS
K,e-(-kDI'l/JinC(T)1 2 )
{_A.l
l}~r m
1[01. (--;)(,.1 8(1)*(-'))+(01 8(1)(--;))01* (~) <Pmc 7 <P T <jJ 7 <jJincT
+ ('l/J 8(1) (1')) ('1//(1)* (1")) + \!'l/Jinc(1')UJ8(2) (1")) + \!('l/fs (2) (r))'l/J7nc(1")] }
(3.6.30) Note that ('l/J 8(1))('l/J 8 (1)*) is used in (3.6.30), which contains the coherent component. Substituting (3.6.27), (3.6.28), and (3.6.29) into (3.6.30) gives the following equations for this ca.."e of clustered point scatterers with two pair distribution functions g8 and 9p
41fn Im(J) 2 K,c = ok - 41f(/0IfI + nc
r io
r dO sin 0 io
d</;
x {(If(P)(k p - kip, k z + kiz )1 2) + (IJ(P)(k p - kip, -k z + k iZ )1 2)}
r/ 2 dB Jo
sin 0
r Jo
d1>(21f)3 { H(k p - kip, k z + k iz )
.I(J(P)(k p -
kip, k z + kiz ))l2
(3.6.31a)
+ H(k p - kip, -k z + "~iz)I(J(I')(kp - kip, -k z + k iZ ))1 2 }
The first two terms in (3.6.31) cancel by invoking the optical theorem. Hence
K,e=n c
7f /2 dB:-:;inB 127f d1> {(It!P)(kp-kip,kz+kiz)12) 1 o 0
+ (If(P)(k p - kip, -k z + k iZ )1 2)}
+ n~
r/ .fo
2 dOsinO
r21r d</;(21fyJ{ H(k p - kip, k z + kiz ) io
(3.6.31b)
. l(f(P)(k p - k zp, k z + "'zz ))1 2 1 .
+ H(k p -
kif" -k z + k iz )I(1(P)(k p - kip, -k z + k iz ))l2}
The phase function is ohtained from the angular distribution of the incoherent intensity term in (3.6.28), (~'.f(1'N),f(1")), in a manner similar to that performed in Section 5. Tlm!'> the phase function for this case of clustering point scatterers is P(O, </;;1f - Oi, </;i) = nc(lf(P)(k p - kip, k z + k iz )1 2)
n~ (21f)3 Hp(k p -
kip, k z + k iz ) 1(1(1') (k p - kip, /,~z
+ kiz )) 1
(3.6.32)
6 Effects of ClusterilJp;
This is quite similar to (3.5.6), of the nonclustering case with 111 2 replaced 2 by (If(l- ) 1) and IU(P))1 2 , respectively, and no replaced by nco When we use the phase function as defined in (3.6.32), the extinction coefficient of (3.6.31) can be written as
r Jo /
dB sin fJ Jo
d<p[P(B,<p;1f-Bi ,<Pi)+P(1f-B,<p; 1f-Bi , <Pi)] (3.6.33)
Note that depending on the two pair functions 98 and 9p, the extinction coefficient K,e can be directionally dependent on (B i ,<Pi). The results of (3.6.32) and (3.6.33) have the following physical interpretation. If we view the random media as consisting of primary scatterers, then the scattering is a result of correlated scattering of primary scatterers similar to dense media theory. The result of scattering can be smaller than the independent scattering of primary scatterers, an effect due to 9p' However, each primary scatterer consists of a cluster of small scatterers. Thus the scattering by a primary scatterer can be larger than the sum of the scattering cross sections of the small scatterers in the cluster, an effect due to