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= T(M) = _ J~(kipa)
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(9.3.54) (9.3.55)
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= T(N) = _ In(kipa)
H n (kipa) For plane wave incidence in the x-y plane, the incident field is E i (-) = r
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(9.3.56)
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Then the incident wave is, from Eq. (1.6.54) in 1 of Volume I and setting kip = k and k iz = 0, 00 'n -in > Ed'F) = Z ek '[iEhiRgMn(k,O,r) -Ev,RgNn(k,O,'F)] (9.3.58)
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9 SIMULATIONS OF TWO-DIMENSIONAL DENSE MEDIA
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Thus for this case, the incident field coefficients are in+1e-incjJi a(M) = Eh n k ' ine-incjJi a(N) = E Vi n k
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(9.3.59) (9.3.60)
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3.2 Foldy-Lax Multiple Scattering Equations for Cylinders
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The cylinders are centered at 7'1,7'2, ... ,7'N. Let the incident direction be in the x-y plane so that Bi = 90 0 , k iz = 0, kip = k. We use the Foldy-Lax multiple scattering equations which states that
Eex(q) = E i
p=l,polq
ES(P)
(9.3.61)
The equation states that the final exciting field of particle q is equal to the incident wave plus the scattered waves to particle q from all other particles p except q itself. The incident wave can also be expressed in terms of cylindrical waves centered at the cylinder q.
E i (7') = (ViEvi
._ _
+ hiEhi )
ik,.p = (ViEvi
+ hiEhi )
iki'Pqeiki,(p-Pq)
= etki'Pq
ine-incjJi k [iEhiRgMn (k, P -pq) - EViRgNn (k, P- Pq)]
n=-CXl
(9.3.62)
The exciting field of cylinder q is equal to
Eex(q) =
n=-CXl
[w~M)(q)Mn (k,p-p q) +w~N)(q)Nn (k,p-p q)]
(9.3.63)
where w~M)(q) and w~N)(q) are the final exciting field coefficients. The exciting field of cylinder p is
Eex(P) =
n'=-CXJ
[w~~)(P)RgMnl (k,p-pp) +w~!'j)(p)RgNnl (k,p-pp)]
(9.3.64)
The scattered field from particle pis,
ES(P)
n'=:.-C>Q
[T~~1)w~~)(P)Mnl (k,p-pp) +T~~)w~!'j)(P)Nnl (k,p-pp)]
(9.3.65)
3.2 Foldy-Lax Multiple Scattering Equations for Cylinders
Figure 9.3.2 Translational addition theorem in the cylindrical coordinate system.
where T~;"f) and T~;.f) are the T -matrix coefficients. To obtain equations for the exciting field coefficents from the Foldy-Lax multiple scattering equations, we need the translational addition theorem of
(l) H n'
(k 1- - -II) ein'h p p pp
(9.3.66)
where pp' is the aziumthal angle that P Fig. 9.3.2). Let
makes with the x-axis (see (9.3.67) (9.3.68)
P- pi = P- Po - (pi - Po)
Ip' - Pol 2:: Ip - Pol
Then
(l) H n'
(k \- - -II) ein'h p p pp
n=-oo
\p - Pol) ein~H~~n'
Ip' - Pol) e-i(n-n')</>-;;T;;;;"
(9.3.69)
For the vector cylindrical wave functions with (9.3.70) we have
n=-oo
RgMn
(k, P- Pq) H~~n' (k Ipp - Pql) e-i(n-n')cPp ppq
(9.3.71)
9 SIMULATIONS OF TWO-DIMENSIONAL DENSE MEDIA
N n, (k,p-p p)
n=-oo
RgNn(k,P-Pq)H~~n,(kIPp-pql)e-i(n-n'),ppppq
(9.3.72)
Putting the cylindrical wave translational formula into Foldy-Lax multiple scattering equations, we have
n=-oo
[w~M)(q) RgMn (k, P - Pq) + w~N)(q) RgNn (k, P - pq)]
._ _
= eZki'Pq
ine-in,pi _ _ k [iEhiRgM n (k,p - Pq) - EviRgN n (k,p -Pq)]
n=-oo
+L L
vi'q
' " [T(M) w (M)(P)R 9 M n (k , - - - q) L..J n' P P n'
n'=-oo n=-oo
+T~;V)w~~)(p)RgNn (k, P - pq)] x
H~~n' (k Ip p - pql) e-i(n-n'),ppppq
(9.3.73) Balancing the coefficients for the TE waves gives
(M)(q) Wn
= ezk"P q
.- _ i n+ 1 e- in ,pi E k hi
L L H~~n' (k Ipp n'=-oo
pqJ) e-i(n-nl),ppppqT~~1)w~~1)(p)
(9.3.74)
Pi'q
and for TM waves
w(N)(q)
.- _ ine-in,pi _eZki'Pq E . k ~
L L H~~n' (k Ipp - Pql) e-i(n-nl) pppqT~;V)w~~)(p)
n'=-oo
(9.3.75)
Pi'q
The above equations can be written in a matrix form. For example, for TE case with N max = 1 and N = 2, we have
3.2 Foldy-Lax Multiple Scattering Equations for Cylinders
W_ I
(M)(I)
(M)(I)
(M)(I) WI
(9.3.76)
(M)(2) W_ I
(M)(2)
(M)(2) W_I
After the exciting field coefficients are solved, the "final" scattered field from cylinder q and the final internal field of the cylinder q can be obtained.
Es(q)
n=-oo
[a~M)S(q)RgMn (k,p-P q ) +a~N)s(q)RgNn (k,p-pq )]
(9.3.77)
9 SIMULATIONS OF TWO-DIMENSIONAL DENSE MEDIA
a(M)s(q)
a(N)s(q)
= T(M)W(M)(q) n n = T(N)W(M)(q) n n
(9.3. 78a)
(9.3.78b)
The total scattered field is
ES=D E
",,-s(q)
(9.3.79)
For the final internal field of cylinder q
E~~~ =
where
n=-oo
[c~M)(q) RgMn (k p , P- Pq ) + c~N)(q) RgNn (k p , P- pq )]
(9.3.80)
C(M)(q)
B(M)w(M)(q)
c~N)(q)
B~N)w~M)(q)
(9.3.81)
After the "final" exciting field coeffcients are obtained, we can calculate the "final" scattered field in the direction ks = k(cos /i: + sin sY). Using the asymptotic form of Hankel functions and derivatives, we have, in the limit of p --+ 00,
(9.3.82)
(9.3.83)
~ V f,ei(kP-~)~ ntoo [_;Pi7;\M)w~M)(q) + Z1;\N)w~N)(q)]
x ein(cPs-'!j)e-iks,pq
(9.3.84)
Coherent Field, Incoherent Field and Scattering Coefficient
Let A be the area in which the cylinders are placed. The fractional area occupied by the cylinders is
f = --y
N7ra 2
(9.3.85)
The Monte Carlo simulations of scattering by the cylinders are performed for N r realizations. Let E sr be the scattered field E s for the rth realization. The
3.4 Scattered Field and Internal Field Formulations
coherent scattered field (E s ) is obtained by averaging over N r realizations 1
(E s ) = -LEsr
(9.3.86)
The averaged intensity is given by
(9.3.87)
and the incoherent intensity is equal to
(9.3.88)
The scattering rate is the scattering cross section per unit volume. To obtain unit volume, we can consider a unit length in the z-direction. Then the scattering coefficient is
"'s = "'-"-------;----------;--2
12Jrd sp ((IEsI2)
-1( EslI 2)
(IEvi12 + IEhi1)
(9.3.89)
Only the incoherent intensity is used in calculating the scattering coefficient for reasons as discussed in 7. If the cylinders are not absorptive, then "'e = "'s' 3.4 Scattered Field and Internal Field Formulations Instead of using equations for the exciting field coefficients, the scattered field coefficients can also be used in Foldy-Lax equations. For TE waves