106
6 Light Scattering by Dielectric Bodies of Irregular …
The required properties of invariance are satisfied by the following spherical
harmonics [12]:
M
J
mn (kr) = (−1)
m d n z
J
n (kr)C mn (θ ) exp(imϕ),
(6.5)
N
J
mn (kr) = (−1)
m d n
n(n + 1)
kr
z
J
n (kr)P mn (θ ) +
1
kr
z
J
n (kr)B mn (θ )
×
× exp(imϕ),
(6.6)
B mn (θ ) = i θ
d
dθ
d
n
om (θ ) + i ϕ
im
sin(θ )
d
n
om (θ ),
(6.7)
C mn (θ ) = i θ
im
sin(θ )
d
n
om (θ ) − i ϕ
d
dθ
d
n
om (θ ),
(6.8)
P mn (θ ) = i r d
n
om (θ ), d n =
(2n + 1)
4n(n + 1)
,
(6.9)
where z
J
n is any of four spherical functions form (3.4),
d
n
om (θ ) =
(−1)
n−m
2 n n!
(n + m)!
(n − m)!
1/2
(1 − cos
2
(θ ))
−m/2
×
×
d
n−m
(dcos(θ )) n−m [(1 − cos
2
(θ )
n
].
Let us write the expansion of the incident wave on the surface of the jth particle
in the vector spherical harmonics:
E I ( j) = −
∞
n=1
n
m=−n
i E mn [ p
j
mn N
1
mn + q
j
mn M
1
mn ].
(6.10)
The expression for the internal field at the jth particle in vector spherical harmonics is:
E i ( j) = −
∞
n=1
n
m=−n
i E mn [d
j
mn N
1
mn + c
j
mn M
1
mn ],
(6.11)
The expansion for the field scattered by the jth particle in vector spherical harmonics has the form
E s ( j) =
∞
n=1
n
m=−n
i E mn [a
j
mn N
3
mn + b
j
mn M
3
mn ],
(6.12)