6.4 Matrix Formulation of Scattering for the jth Bilayer …
115
In the region r 1 ≤ r ≤ r 2 , the internal field can be written as [14, 15]
E 2 ( j) = −
∞
n=1
n
m=−n
i E mn [α
j
mn N
1
mn + β
j
mn M
1
mn +
+ γ N
3
mn + δ
j
mn M
3
mn ]
(6.37)
Proceeding analogously to the case of scattering from a homogeneous particle
of an irregular shape, we obtain the solution to the scattering problem for a bilayer
particle of an arbitrary geometry:
a
j
b
j
= T
j
2
p
j
q
j
,
(6.38)
T
j
2 = −[Q
11
2 (k, k 2 ) + Q
13
2 (k, k 2 )] · D[[Q
31
2 (k, k 2 ) + Q
33
2 (k, k 2 )] · D]
−1 , where
D = S 12 · T
j
01 · S 21 , T
j
01 = −Q
11
01 (k 2 , k 1 ) · [Q
31
01 (k 2 , k 1 )]
−1
Q
13
2
=
K
13
+ m 2 /m · J
13 I
13
+ m 2 /m · L
13
L
13
+ m 2 /m · I
13 J
13
+ m 2 /m · K
13
,
(6.39)
Q
31
2
=
K
31
+ m 2 /m · J
31 I
31
+ m 2 /m · L
31
L
13
+ m 2 /m · I
31 J
31
+ m 2 /m · K
31
,
(6.40)
Q
11
2
=
K
11
+ m 2 /m · J
11 I
11
+ m 2 /m · L
11
L
11
+ m 2 /m · I
11 J
11
+ m 2 /m · K
11
,
(6.41)
Q
33
2
=
K
33
+ m 2 /m · J
33 I
33
+ m 2 /m · L
33
L
13
+ m 2 /m · I
33 J
33
+ m 2 /m · K
33
,
(6.42)
Q
11
01
=
I
21
2 + m 1 /m 2 · I
12
2
I
22
2 + m 1 /m 2 · I
11
2
I
22
2 + m 1 /m 2 · I
11
2
I
12
2 + m 1 /m 2 · I
21
2
,
(6.43)
Q
31
01
=
I
21
2 + m 1 /m 2 · I
12
2 I
22
2 + m 1 /m 2 · I
11
2
I
22
2 + m 1 /m 2 · I
11
2 I
12
2 + m 1 /m 2 · I
21
2
,
(6.44)
m 1 is the refractive index of the core, m 2 is the refractive index of the plasma membrane, m is the refractive index of the medium and matrix elements Q
31
01 , Q
11
01 , Q
13
2 ,
Q
31
2 , Q
11
2 and Q
33
2 can be expressed in the form of surface integrals:
K
31
mnm n = α(−1)
m
s
[N
3
(−mn) (kr) × M
1
(m n ) (k 2 r )]ndS,
(6.45)