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~ k 2 [e iKz 'iJ2 yp -
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yp'iJ2 eiKz]
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(6.1.31)
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Use (6.1.31) and Green's theorem to transform the volume integral into three surface integrals over Bd, Be, and 5 00 (Fig. 6.1.2). The integral over Boo can be shown to vanish. iKZ It = eiKz1 (-l)P {_ dB (eiKZ oYp _ y oe ] K2 - k 2 j s" ar p ar 7"=b iKZ _ { dB [eiKzOYP _ YP oe ]} (6.1.32) } Sd OZ OZ Z=-ZI
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1.1
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t----t-Ir-'7",..-.-jr-.--------t-........ z
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Figure 6.1.2 Volume integration bounded by surface Sel, Se, m1(1 Soo'
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The integral over Se is
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iKZ r dS[eiKzOYP _ YP oeor ] 1.=b = -47fi (K is" or
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k 2 )L p(k, Klb)
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(6.1.33)
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Lp(k, Klb) = - (K2 _ k 2) [~~h~(kb)jp(Kb) - Khp(kb)j;(Kb)]
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(6.1.34)
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For the surface integral over Sd, the following Kasterin's representation [Waterman and Truell, 1961 J can be used.
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Yp = hp(kr)Pp(cosB) = (-i)PPp C~~) ho(kr)
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(6.1.35)
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i Sri
r dS [eiK Zoyp _ Yp oeiK Z]
{)z {)z
z=-Zt
e-iK Zt ( -i)P+ 1 (X) P dP{
[!..- _iK] P (-!:-!..-) eik~} {)z ~k {)z Vp2 + z2
Z=-Zt
27f~'p+l (k + K) e -iKzt+ikzt k2
(6136) ..
Substituting (6.1.33) and (6.1.36) into (6.1.32) gives
2 '( ')P ikz t 7f~ -~ _e_ (K - k) k2
+ 47f(-i)P L
(k Klb) eiKzt
(6.1.37)
6 QUASI-CRYSTALLINE APPROXIMATION IN DENSE MEDIA SCATTERING
To evaluate [2 of (6.1.30), we note that the volume of integration in (6.1.30) consists of the half-space Z2 > 0 minus a sphere of radius b centered at the point 1"1. We have seen that for 0 :::; f :::; 0.4, g(r2 - 1"1) - 1 is practically zero for 11"2 - 1"11 larger than a few b's. If the point 1"1 is at least several diameters deep in the scattering medium (thus ignoring boundary layer effects), the volume of integration in (6.1.30) can be extended to infinite space. Thus, letting 1" = 1"2 - 1"1 in (6.1.30) and making the above assumption give (6.1.38) where (6.1.39) Substituting [ = h
+ [2
in (6.1.26), we have
[( _1)m
a!f,<!"f)e iKz1 =
no 1'mv
1'mn
a(m,
vl- m, nlp)a(v, n,p)TSM)a!f,~M)
1)b(v,
+( _1)m+1
L a(m, vl- m, nlp,p p
n,p)TSN)a~~N)]
+ Mp(k, I<lb)
. { (I< _ k)k2
211'i( _i)Peikz
+ 411'( _i)Pe,Kz
Lp(k, I<lb)
+ eikzla~~)
(6.1.40)
There are two kinds of wave dependences in (6.1.40) as characterized by their respective phase terms. One type of term has a exp( ikz 1 ) dependence corresponding to waves traveling with the propagation constant of the incident wave. The other type of term has a exp( iI< zI) dependence corresponding to waves travelling with the propagation constant of the effective medium. The terms with propagation constant k should balance each other, giving the generalized EwaldOseen extinction theorem. The medium generates a wave that extinguishes the original incident wave. Balancing the terms with propagation constant I< gives the generalized Lorentz-Lorenz law. These laws were derived for the case of point dipoles in Born and Wolf [1975]. In the treatment here, the Ewald-Oseen extinction theorem and Lorentz-Lorenz law are generalized to the use of finite-size spheres that give scattering attenuation of the coherent wave. The generalized Lorentz-Lorenz law is, on balancing eiKz1 terms in
1.1 Collerent Wave Propagation
(6.1.40) and replacing (/1, v)
(m, n),
o,:~M) = L no 'YJln
n 'YJlV
[( -l)Jl
L o,(it, nl- /1, vlp)o,(n, v,p)T2tJ)o,:~M)
+ (-1)"+ l ~ a(l', nl . 47r( -i)p [Lp(k, Klb)
1', vip, P - 1)b(n, v, P
)T~N)a~~N)]
(6.1.410,)
+ Mp(k, Klb)]
for /1 = l. Equation (6.1.410,) is the result of starting with (6.1.180,). If we start with (6.1.18b) and use similar derivation, we get
o,:~N) = L no 'Yw
t [(
_l)Jl+1
'YJlv
L 0,(/1, nl-IL, vlp,p p
l)b(n,
v,p)T~M)o,:~M)
+ (-1)" ~ a(l', nl . 47r( -i)P [Lp(k, Klb)
1', vip)a(n, v, P
)T~N) a~~N)]
(6.1.41b)
+ Mp(k, Klb)J
The generalized Ewald-Oseen extinction theorem is obtained by balancing the exp(ikzt} terms in (6.1.20).
1 v 47r(2v + 1)-2 z <5Jll
+ no L.,.. (-1)
"P+/l 'YIn (27ri) i P 'Ylv (I{ _ k)k2
(M) x { Tn o, E(M) 0,(/1, n I - IL, vlp)o, (v, p n, ) ln
(N) E(N) _ - Tn o,l n o,(IL, nl- /1, v!p,p - l)b(n, v,p) } - 0
(6.1.420,)
If we start with (6.1.18b) and use similar derivation, we get
P vi47r (2v + 1)2z v UJll + no ,,( -1 )P+Jl -'YIn (K _ k)k 2 1 ~ (27ri) i L.,..
n ,p
'Ylv
X { -
T~M) o,~~M) o,(IL, nl
/1, vip, p - 1)b(n, v, p)
(N) ( )( + Tn o, E(N) aIL, .n I - /1, v Ipan, v, p)} - 0 ln
(6.1.42b)
a (n, v,p)
_ (2v
+ l)n(n + 1) 'v-n+PA( ) 2v(v + 1) z n, v,p
(6.1.430,) (6.1.43b)
__ (2v + l)n(n + 1) .v-n+PB( ) b( n,v,p) 2v(v+ 1) z n,v,p
6 QUASI-CRYSTALLINE APPROXIMATION IN DENSE MEDIA SCATTERING
where
A(n, v,p)
B(n, v,p) =
+ 1~( 2v+ 1) [2v(v + 1)(2v + 1) + (v + 1)(n + v . (n + p - v + 1) - v(n + v + P + 2)(v + p - n + 1)] v(v + 1) + n(n + 1) - p(p + 1)