3.4 Scattering by a Group of Spherical Objects
49
T
j
12 = T
j
1 + T
j
2 , T
j
1 =
a
j
n 1 p 0
0 b
j
n 1q
, T
j
2 =
0 a
j
n 1q
b
j
n 1 p 0
,
System (3.70) must be solved with the aid of the method of reduction in which a
finite number of equations and a finite number of unknown quantities are considered
and a stable algorithm of biconjugate gradients [10, 11] is employed.
When coefficients a
j
mn and b
j
mn are found using system (3.70), the expressions for
the scattered field in the principal coordinate system can be written as
E s =
∞
n=1
n
m=−n
i E mn [a
j
mn N
3
mn + b
j
mn M
3
mn ],
(3.71)
H s =
k
ωμ
∞
n=1
n
m=−n
i E mn [b
j
mn N
3
mn + a
j
mn M
3
mn ],
(3.72)
where
a
j
mn =
L
l=1
∞
ν=1
ν
μ=−ν
[a
l
μν A
μν
mn (l, j 0 ) + b
l
μν B
μν
mn (l, j 0 )],
b
j
mn =
L
l=1
∞
ν=1
ν
μ=−ν
[a
l
μν B
μν
mn (l, j 0 ) + b
l
μν A
μν
mn (l, j 0 )].
The system for the coefficients a
j
mn , b
j
mn can be simplified if we consider a part of the
field that is forward-or backward-scattered by the particles at small angles relative
to the Z axis.
The expressions for the scattered field in the far-field zone are represented as
E sθ ∼ E 0
e
ikr
−ikr
∞
n=1
n
m=−n
(2n + 1)
(n − m)!
(n + m)!
[a
j
mn τ mn + b
j
mn π mn ]e
imφ
(3.73)
E sφ ∼ E 0
e
ikr
−ikr
∞
n=1
n
m=−n
(2n + 1)
(n − m)!
(n + m)!
[a
j
mn π mn + b
j
mn τ mn ]e
imφ
,
(3.74)
where
τ mn =
∂
∂θ
P
m
n (cos θ), π mn =
m
sin θ
P
m
n (cos θ)
Symbol (∼) indicates asymptotic interpretation of expressions (3.73) and (3.74),
which follow from expression (3.71) at (kr 1). We consider the scattering at
relatively large distances from the jth particle. Therefore, the electric vectors of the
scattered field are parallel to the electric field of the incident field, so that only the
49
T
j
12 = T
j
1 + T
j
2 , T
j
1 =
a
j
n 1 p 0
0 b
j
n 1q
, T
j
2 =
0 a
j
n 1q
b
j
n 1 p 0
,
System (3.70) must be solved with the aid of the method of reduction in which a
finite number of equations and a finite number of unknown quantities are considered
and a stable algorithm of biconjugate gradients [10, 11] is employed.
When coefficients a
j
mn and b
j
mn are found using system (3.70), the expressions for
the scattered field in the principal coordinate system can be written as
E s =
∞
n=1
n
m=−n
i E mn [a
j
mn N
3
mn + b
j
mn M
3
mn ],
(3.71)
H s =
k
ωμ
∞
n=1
n
m=−n
i E mn [b
j
mn N
3
mn + a
j
mn M
3
mn ],
(3.72)
where
a
j
mn =
L
l=1
∞
ν=1
ν
μ=−ν
[a
l
μν A
μν
mn (l, j 0 ) + b
l
μν B
μν
mn (l, j 0 )],
b
j
mn =
L
l=1
∞
ν=1
ν
μ=−ν
[a
l
μν B
μν
mn (l, j 0 ) + b
l
μν A
μν
mn (l, j 0 )].
The system for the coefficients a
j
mn , b
j
mn can be simplified if we consider a part of the
field that is forward-or backward-scattered by the particles at small angles relative
to the Z axis.
The expressions for the scattered field in the far-field zone are represented as
E sθ ∼ E 0
e
ikr
−ikr
∞
n=1
n
m=−n
(2n + 1)
(n − m)!
(n + m)!
[a
j
mn τ mn + b
j
mn π mn ]e
imφ
(3.73)
E sφ ∼ E 0
e
ikr
−ikr
∞
n=1
n
m=−n
(2n + 1)
(n − m)!
(n + m)!
[a
j
mn π mn + b
j
mn τ mn ]e
imφ
,
(3.74)
where
τ mn =
∂
∂θ
P
m
n (cos θ), π mn =
m
sin θ
P
m
n (cos θ)
Symbol (∼) indicates asymptotic interpretation of expressions (3.73) and (3.74),
which follow from expression (3.71) at (kr 1). We consider the scattering at
relatively large distances from the jth particle. Therefore, the electric vectors of the
scattered field are parallel to the electric field of the incident field, so that only the