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with K given by (10.2.65), Kd = ksin(}icos<pix + ksin(}isin<pi'fl JK2 - k2sin2(}iz = Kxx+Kyy-Kzz, etd = -cos(}tCos<Pix-cos(}tsin<piYsin (}tZ and (}t is the transmitted angle obeying Snell's law sin (}t =
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The Fresnel transmission coefficient for TM waves is TfuM for the magnetic
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10 DENSE MEDIA MODELS AND THREE-DIMENSIONAL SIMULATIONS
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Figure 10.2.3 Plane wave incident on a half-space of dielectric scatterers. Induced dipoles.
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where k iz = k cos Oi and K z = K2 - k 2 sin 2 Oi. Dipoles are induced in the dielectric spheres and the dipole moment Pj of the jth particle, centered at 'Fj is, using (10.2.38),
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The total radiation field of the dipoles is then Es(r) where J j
L WJ
(10.2.70)
= exp [i(K d
ks) . 'Fj] ,
fJe W - W 47fr
x td x
VoEY r,T M _ _ fy 01 K
k k K
k2a3eikr
(k s x
Otd)
x ks
f T 1- Y Ol
(10.2.71)
ks =
sin Os cos /i; + sin Os sin sY + cos Osz = ksxx agation direction of the scattered wave with k s =
+ ksyY + kszz is the propkk s .
2.3 Coherent Reflection and Incoherent Scattering
Coherent Reflected Field
To calculate the coherent scattered field, the configuration average of (10.2.71) is taken
(Es(r)) = W2:(Jj ) = WN(Jj )
(10.2.72)
Since p(rj) = half-space,
we have, on evaluating (Jj) by integrating over the lower
(Jj)
1: 1: 1 00
dXj dYj
dzjei(Rrks)'Tj
giving rise to J(Kx - k sx ) J(K y - k sy ) indicating that the coherent scattered field is in the specular direction.
_ _ 47f 2 i (E s) = noW (K + k ) J(k sin (}s cos <Ps - k sin (}i cos <Pi) z iz . J(k sin (}s sin <Ps - k sin (}i sin <Pi)
(10.2.73)
in specular reflected direction
(k s
x e td ) x
ks = -
COS((}i
+ (}t)ei
(10.2.74a)
We make use of the relation of (10.2.34) so that
noW = :7fr (- COS((}i
+ (}t)et )
k )Tlt K
(10.2. 74b)
Also K 2 - k 2 = (K z + kiz)(Kz - k iz ), so that
47f in o W . K z + k ~z
7fie = -- ( r
COS((}i
+ (}t)et
(K z - k iz )To1
(10.2.74c)
We make use of the relation
(Kz + kiz ) COS((}i
so that
TM T Ol -k
(}t)
k iz K 2 + k 2K z kK
(10.2.74d)
2kiz = -,------,------(K z + kiz ) COS((}i
(}t)
(10.2.75a)
Also we have
K z - kiZ ) ( K z + kiz
COS((}i
+ (}t)
(}t)
= K2kiz - k K z = R
COS((}i -
2 K 2kiz + k 2K z -
(10.2.75b)
10 DENSE MEDIA MODELS AND THREE-DIMENSIONAL SIMULATIONS
Putting (10.2.75) into (10.2.74c) and (10.2.73), we get
Thus the coherent reflected wave is in the specular direction, containing no depolarization and obeying the Fresnel reflection formula for TM waves. The case of horizontally polarized incident waves is treated in a similar manner. For an incident field of E inc = ;Pi Eo exp( ikid . r). The result for the coherent reflected field is
(Es(r))
= -;Pi i21rkiz R 61E ~ c5(k sin (;Is cos<ps r
ksin(;li cos <Pi) (10.2.77)
. c5(k sin (;Is sin <Ps - k sin (;Ii sin <Pi)
where
R TE 01
kiz - K z k iz + K z
(10.2.78)
is the Fresnel reflection coefficient for TE waves.
Incoherent Scattered Field
The incoherent scattered field s is s(r) = Es(r) - (Es(r)). Hence, the incoherent intensity is (10.2.79) Following a similar procedure as in Section 2.2, we have
(E; . ,) =
IWI'{ 2~~.) + n; Jar, Jarj 19(1', - 1'j) -1]
(10.2.80)
where A o is the area of the target area. The integral in (10.2.80) can be carried out readily. Hence,
( :(r) . s(r)) =
IWl 2 2I~~z)
{I + no
dr [g(r) - 1] ei(ReKrksp: }
(10.2.81)
2.3 Coherent Reflection and Incoherent Scattering
where for vertically polarized incidence, W is given by (10.2.71) and for horizontally polarized incidence 2 e ikr 3 W = w 4/-l (k s x ;Pi) x ks 1 VOEfY (1 + R61E ) 7fr - Y 2 3 ikr (k s X 'f'~ X k s --=------:- -::-::--,,----1.) Y 2kiz _ - k a e (10.2.82a) r 1 - fy K z + k iz so that 2kiz e ikr (A A A) 2 2 (1O.2.82b) noW = -4 (ks x cPd x ks (K - k ) K k. 7fr z + ~z In backscattering direction,
Thus
(l sCr)1 2 ) = (K z - kiz)k iz
.{I +
so that
noAo _1 2Im(Kz )r 2 47f2 (10.2.83)
dr(g(r) - l)e i (ReK r k s ).r}
The backscattering coefficient is
(Jhh(Oi) = 47fr (l s(r)1 )
(Jhh = I (K
~~iZ)kiZ 27rI~(Kz) { 1 + no
dr(g(r) - l)e i (ReK r k s).r}
(10.2.84) For the case of vertical polarized incidence, we can perform similar calculations. In the backscattering direction, Os = Oi and cPs = 7f+cPi, ks x (k s x etd ) = - COS(Oi - Ot)e s . Hence, the backscattering coefficients are the same for (Jvv and (Jhh. We have
(Jvv(Oi
= (Jhh(Oi) =
(Kz - kiz)kiz no
{I + no
no 27rIm(K ) z
dr [g(r) -
1] ei(ReKrks).r}
(10.2.85)
The results of coherent reflection in this section agree with the QCA results in the low frequency approximation. The bistatic intensity agrees with QCA combined with distorted Born approximation. The QCA approximation will be treated in Volume III.
10 DENSE MEDIA MODELS AND THREE-DIMENSIONAL SIMULATIONS
2.4 A Simple Dense Media Radiative Transfer Theory
A simple dense media radiative transfer theory can be developed based on the derivations of Sections 2.1 and 2.2. The model is based on the assumption that the particles are small so that ka 1. However because the particle positions can have long-range correlation (e.g., sticky particles), the correlation distance can be comparable to or larger than a wavelength. Specifically, say ka = 0.2 and the correlation distance is up to 5kb, where b = 2a is the diameter. Then 5kb = 10ka = 2 is larger than unity. First we calculate the effective propagation constant which is determined by (10.2.65a)
K 2 = k 2 (1