6.2 Matrix Formulation of Scattering for the jth Particle …
107
Following [12], we substitute expansions (6.10), (6.11) and (6.12) with allowance
for expressions (6.3), (6.4), and boundary conditions (6.1) into integral equation (6.2);
this gives
ik
2
π
s
∞
n=1
n
m=−n
(−1)
m
[c
j
mn n × M
1
m n + d
j
mn n × N
1
m n ]
N
3
−mn
M
3
−mn
ds+
+
ik
2
π
ε 1
μ 1
s
∞
n=1
n
m=−n
(−1)
m
[c
j
mn n × N
1
m n + d
j
mn n × M
1
m n ]×
×
M
3
−mn
N
3
−mn
ds = −
p
j
mn
q
j
mn
or, in matrix form,
I
21
1 +
m · I
12
1 I
22
1 +
m · I
11
1
I
22
1 +
m · I
11
1 I
12
1 +
m · I
21
1
d
j
c
j
= −i
p
j
q
j
,
(6.13)
where
m is the relative refraction index of the particle and
ik
2
π
s
∞
n=1
n
m=−n
(−1)
m
[c
j
mn n × M
1
m n + d
j
mn n × N
1
m n ]
N
1
−mn
M
1
−mn
ds+
+
ik
2
π
ε 1
μ 1
s
∞
n=1
n
m=−n
(−1)
m
[c
j
mn n × N
1
m n + d
j
mn n × M
1
m n ]×
×
M
1
−mn
N
1
−mn
ds = −
a
j
mn
b
j
mn
or, in matrix form,
a
j
b
j
= −i
I
21
1 +
m · I
12
1
I
22
1 +
m · I
11
1
I
22
1 +
m · I
11
1
I
12
1 +
m · I
21
1
d
j
c
j
.
(6.14)
Combining expressions (6.13) and (6.14), we obtain
a
j
b
j
= −
I
21
1 +
m · I
12
1
I
22
1 +
m · I
11
1
I
22
1 +
m · I
11
1
I
12
1 +
m · I
21
1
I
21
1 +
m · I
12
1 I
22
1 +
m · I
11
1
I
22
1 +
m · I
11
1 I
12
1 +
m · I
21
1
−1
×
×
p
j
q
j
.
(6.15)