118
6 Light Scattering by Dielectric Bodies of Irregular …
Functions D
n
mm (α, β, γ ) can be represented as the product of three factors, each of
which depends on only one Euler angle [17]
D
n
mm (α, β, γ ) = exp(−imα)d
n
mm (β) exp(−im
γ ),
where d
n
mm (β) are Wigner functions and satisfy the conditions of unitarity
D
−1
(α, β, γ )
n
mm =
D
∗
(α, β, γ )
n
mm ,
n
m=−n
D
∗
mm (α, β, γ )D
n
∗
mm (α, β, γ ) =
n
m=−n
D
n
mm (α, β, γ )D
n
−1
mm (α, β, γ ) = δ m m 1
and orthogonality
2n + 1
8π 2
2π
0
dα
π
0
sin βdβ
2π
0
dγ D
n
mm (αβγ )D
n
∗
1
m 1 m
1
(αβγ ) = δ nn δ mm 1 δ m m
1
,
For function d
n
mm (β), we have
π
0
sin βdβd
n
mm (β)d
n
mm (β) =
2
2n + 1
δ nn .
Functions d
n
mm (β) satisfy the following relations
m
sin β
d
n
om (β)|
=1/2δ m 1 [n(n+1)]
1/2
β=0
,
d
dβ
d
n
om (β)|
=1/2mδ m 1 [n(n+1)]
1/2
β=0
,
m
sin β
d
n
om (β) = 1/2[n(n + 1)]
1/2
[d
n
1m (β) + d
n
−1m (β)],
d
dβ
d
n
om (β) = 1/2[n(n + 1)]
1/2
[d
n
1m (β) − d
n
−1m (β)],
d
n
mm (β)d m 1 m
1
(β) =
n+n
n 1 =|n−n |
C
n 1 m+m 1
nmn m 1
C
n 1 m
+m
1
nm n m
1
d
n 1
m+m 1 m +m
1
(β),
where C
n 1 m+m 1
nmn m 1
, C
n 1 m
+m
1
nm n m
1
are the Clebsch–Gordan coefficients
d
n
mm (β) = (−1)
m
−m d
n
−m−m (β) = (−1)
m
−m d
n
m m (β).
The product of two D-functions D
n 1
m 1 m 1
(αβγ ) D
n 2
m 2 m
2
(αβγ ) can be written as the
following sum [17]: