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<Pin
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(10.3.79) (10.3.80)
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-~ . \7<P ou t
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+ ~ . \7<Pin = ~ . -E
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Q~(~)
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To solve <Pin and <Pout, let
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(10.3.81)
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(10.3.82)
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= BQi(~)pl(1])cos
where P~(1]) is the associated Legendre polynomial and endre function of second kind. In (10.3.82)
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is the Leg(10.3.83) (10.3.84)
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2 Applying boundary conditions (10.3.79) and (10.3.82) gives
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Ql(~) =
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v12=1 [_~__ ~ I (~~-11)] + e-1
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where fo = This gives
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(EP-E) fo~o
[~o- ~~-1In(~0+1)]
~0-1
(10.3.85)
Vc2-a 2 . The
internal induced electric field is E ind = - \7<Pin'
-(2) E ind
= C2f2
(10.3.86)
with C 2 and 12 as given in (10.3.72) and (10.3.47), respectively. For the case of 15' let the electric field inside the spheroid be
E = 15
Then
A A) P = (Ep r;;r-E) (ZX + XZ yA 5
(10.3.87)
(10.3.88)
The potentials <Pin and <Pout are proportional to pi(~)pi(1]) cos and Q (OpiC17) cos respectively, where P~(~) is Legendre function of the first kind. Let
<Pin <Pout
= -Al;~'17J(e -1)(1-1]2)cos =
, Bv12=1 [3 e - 2 3~
- -In (~ + -2 ~-1
1)] 31]~cos
(10.3.89) (10.3.90)
10 DENSE MEDIA MODELS AND THREE-DIMENSIONAL SIMULATIONS
Applying boundary conditions of (10.3.79) and (10.3.80) gives
A=_l
{_(~~-1) (EP-E)
(2e-1) [3~~-2 _ 3~Oln(~o+1)]} -1 2 1
(10.3.91)
Using (10.3.89), we get E ind = - V<Pin = C 5 15 where 0 5 and 15 are as given in (10.3.74) and (10.3.50), respectively.
If the particles are closely packed, the near field interactions have large spatial variations over the size of a spheroid that may induce quadrupole fields inside the spheroid. However, the non-near field interactions have small spatial variations over the size of a spheroid and only induce dipole fields inside the spheroid.
-(5) -
Substituting (10.3.43) into (10.3.42), we get
E(r) = Einc(r)
where
L ajoJija(r)
(10.3.92)
j=l a=l
7ija(r) =
k21 dr' g(r, r')!ja(r') (Erj - 1)
dr' V' g(r, r') '!ja(r') (Erj - 1)
(10.3.93)
is the electric field induced by the polarization P j (r) of the spheroid j. Of particular importance is the internal field created by P j (r) on itself. For this self term contribution, the second term in (10.3.93) dominates. Because of the smallness of the spheroid, an electrostatic solution can be sought for the second term in (10.3.93). We have from (10.3.70), for r in Vj, the self term becomes
(10.3.94)
where j is the particle index, a is the basis function index, and Cja's are given by (10.3.71)-(10.3.75) for each particle j. In the expressions of (10.3.70)(10.3.75), we need to replace Ep and ~o by the value for each particle. An approximation sign is used in (10.3.94) to indicate the low frequency approximation.
3.3 Densely Packed Spheroids
(10.3.95) This gives
al{3 = (1 _Ie) { 1{3
r dr fl{3(r) . Eine(r) + t"t aja lVz dr fl{3(r) . (jja(r)} r j=1
(10.3.96) Because of the small spheroid assumption, only the dipole term contributes to the first term in (10.3.96) which is the polarization induced by the incident field. Thus
for (3 = 1,2,3 (10397) " (3 > 3 . . lor After the coefficients al{3, I = 1,2, ... , N, and (3 = 1,2, ... , N b are solved, the far field scattered field in the direction (e s , cPs) is expressed as
Vz 1
d-r -f 1{3 (-r) E me (-r) = {volfl(3 . Eine(rl) 0
ikr Es(r) = k 2 :7fr (VsVs + hs!"s) .
L L aja (Erj - 1) lv
j=la=l
dr' e-iks.r7ja(r')
(10.3.98) where Erj = Ej / E. Under the small spheroid assumption, only the dipole fields will contribute to the far field radiation in (10.3.98). Thus, we have
ikr Es(r) ::::: k 2 :7fr (vsv s + hsh s) .
L L aja (Erj j=la=l
1) VOjfjae-iks rj (10.3.99)
We next illustrate the results of the numerical simulations by using N = 2000 spheroids and up to f = 30% by volume fraction. The relative permittivity used for the spheroids is 3.2 and the size parameter of the
10 DENSE MEDIA MODELS AND THREE-DIMENSIONAL SIMULATIONS
spheroids used is such that ka = 0.2. At this volume fraction, permittivity, and size, we did not include the quadrupole effects in the simulations. For dipole interactions, we replace the integral in the last term of (10.3.93) as follows.
dr fl(Ar) . 7ijaJr) =
(Erj -
1) VOj vOl k2 fl(3' G(rl' rj) fja
(10.3.100)
In the simulations, all the spheroids are prolate and are identical in size with c = ea, where e is the elongation ratio of the prolate spheroid. The size of the box in which the spheroids are placed is Nv
(10.3.101)
where f is the fractional volume, and v = 47fa 2 c/3 is the volume of one spheroid. An incident electric field of Einc(r) = ye ikz (10.3.102) is launched onto the box containing the N spheroids. The matrix equation of (10.3.96) is solved by iteration. After the matrix equation is solved, the scattered field is calculated by (10.3.99). The scattered field is decomposed into vertical and horizontal polarization (10.3.103) We performed N r = 50 realizations in the numerical illustrations. Let (J be the realization index. Decomposition of the field into coherent and incoherent scattered fields is also made as in Section 3.2. The incoherent scattered field is decomposed into vertical and horizontal polarization