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Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006

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270 B Wave Functions

		ρ	η
			η
	L		Σ
			Σ
		O	z, Re z
		O
L			z
Im z

Fig. B.2. Illustration of the complex plane. L is the generatrix of the surface and L is the image of L in the complex plane

(Fig. B.2). The complex plane Σ3 = {z3 = (Re z3, Im z3)/Re z3, Im z3 R} is the dual of the azimuthal plane ϕ = const., Σ = {η = (ρ, z)/ρ ≥ 0, z R}, and is deﬁned by taking the real axis Re z3 along the z-axis. The vector spherical wave functions can be expressed in terms of the coordinates of the source point z3 Σ3 and the ﬁeld point η Σ as

M 1,3

(kr) =

z1,3

(kR) jmπ|m|

θ sin(θ

−

θ)e

2n(n + 1) n

+ cos(θ

θ)e

−

τ |m| θ e

ejmϕ

(B.31)

− 3

and

N 1,3

(kr) =

!n(n + 1)

zn1,3(kR)

|m|(cos θ)

2n(n + 1)

kRz1,3(kR)

× cos(θ − θ3)er − sin(θ − θ3)eθ

τ |m| θ sin(θ

θ)e

+ cos(θ

θ)e

− 3

+ jmπ|m|

θ e

ejmϕ,

(B.32)

where

R2 = ρ2 + (z

z)2

sin θ =

and

cos θ =

z − z3

− 3

B.3 Rotations

We consider two coordinate systems Oxyz and Ox1y1z1 having the same origin. The coordinate system Ox1y1z1 is obtained by rotating the coordinate

B.3 Rotations

271

	z	θ	M
			M

		β z1
		θ1 r
			y1
	O		y
	O
x	α	γ	x1
x		γ

Fig. B.3. Coordinate rotations

system Oxyz thought the Euler angles (α, β, γ) as shown in Fig. B.3. With (θ, ϕ) and (θ1, ϕ1) being the spherical angles of the same position vector r in the coordinate systems Oxyz and Ox1y1z1, the addition theorem for spherical wave functions under coordinate rotations is [58, 213, 239]

	n
umn1,3 (kr, θ, ϕ) =	Dmmn (α, β, γ)um1,3n(kr, θ1, ϕ1),
	m =−n

where the Wigner D-functions are deﬁned as [262]

m+m

jmα n

jm γ

Dmm (α, β, γ) = (−1)

dmm (β)e

The functions d are given by

dmm (β) = ∆mm dmm (β),

where dn

are the Wigner d-functions and

≥

0 ,

≥

(

0 , m

−

≥

< 0

∆mm =

(

−

m < 0 ,

≥

m+m

, m < 0 , m < 0,

(−1)

(B.33)

(B.34)

(B.35)

(B.36)

with the property ∆mm = ∆m m. The expression of Wigner d-functions for positive and negative values of the indices m and m is given by

(β) =

(n + m )!(n

− m )!

(

−

1)n−m −σ Cn−m −σ Cσ

(n + m)!(n

−

m)!

n+m n−m

cos

β m+m +2σ

sin

β 2n−m−m −2σ

272 B Wave Functions

and note that the above equation is valid for β < 0, if

cos		β	=		1 + cos β						sin		β		=			1 − cos β
		β			1 + cos β				,				β				−	1 − cos β
		2				2			,									2
		2				2						2						2
and for β > π, if

cos	β						1 + cos β				sin			β				1 − cos β
	β		=	−			1 + cos β			,				β		=		1 − cos β	.
	2		=					2		,				2		=		2	.
	2							2						2				2

The Wigner d-functions are real and have the following symmetry properties [58]:

dmn −m (β) = (−1)n+m dmmn (β + π),	(B.37)
d−n mm (β) = (−1)n+mdmmn (β + π),	(B.38)
d−n m−m (β) = (−1)m+m dmmn (β),	(B.39)
dmmn (β) = (−1)m+m dmn m(β),	(B.40)
and
dmmn (−β) = (−1)m+m dmmn (β) = dmn m(β).	(B.41)

Taking into account the above symmetry relations, we can express the d-

functions in terms of the d-functions with positive values of the indices m and m

dn (β) =

dmmn

(β) ,

m ≥ 0 ,

m ≥ 0,

n n

(

−

1) dm

(β + π) ,

≥

0 ,

m < 0,

(

−

(β + π) ,

(B.42)

−

1)ndn

m < 0 ,

≥

−mm

d−m−m (β),

m < 0 ,

m < 0.

The orthogonality property of the Wigner d-functions is similar to that of the associated Legendre functions and is given by

π	n	n	2
	dmm (β)dmm (β) sin βdβ =			δnn .	(B.43)
0	dmm (β)dmm (β) sin βdβ =		2n + 1	δnn .	(B.43)

The Wigner d-functions are related to the generalized spherical functions Pmmn by the relation [78]

dn	(β) = jm −mP n	(cos β) .	(B.44)
mm	mm

The generalized spherical functions are complex and have the following symmetry properties:

B.3 Rotations

273

Pmn −m (x) = (−1)nPmmn (−x),

P−nmm (x) = (−1)nPmmn (−x),

P−nm−m (x) = Pmmn (x),

Pmmn (x) = Pmn m(x),

where x = cos β. The orthogonality relation for the generalized spherical functions follows from the orthogonality relation for the Wigner d-function:

π	n	n	( 1)m+m
Pmm (cos β) Pmm (cos β) sin βdβ =			−	δnn .
Pmm (cos β) Pmm (cos β) sin βdβ =			2n + 1	δnn .
0			2n + 1

In practice, the generalized spherical functions can be found from the recurrence relation [162]

(n + 1)2

−

(n + 1)2

−

P n+1

(x)

= (2n + 1) [n(n + 1)x

−

mm ] P n

(x)

−(n + 1) n2 − m2

n2 − m 2Pmmn−1 (x)

with the initial values

Pmmn0−1(x) = 0,

(−j)|m−m |

P n0

(x) =

(2n0)!

2n0

(|m − m |)! (|m + m |)!

m−m

m+m

× (1 − x)

(1 + x)

and n0 = max(|m|, |m |). From (B.42) and (B.44) we see that it is su cient to

compute the generalized spherical functions for positive values of the indices m and m .

If the Euler angles (α1, β1, γ1) and (α2, β2, γ2) describe two consecutive rotations of a coordinate system and the Euler angles (α, β, γ) describes the resulting rotation, the addition theorem for the D-functions is [169, 239]

	n
Dmmn (α, β, γ) =	Dmmn (α1, β1, γ1)Dmn m (α2, β2, γ2)	(B.45)
	m =−n

and the unitarity condition read as

Dmmn (α1, β1, γ1)Dmn m (−γ1, −β1, −α1) = Dmmn (0, 0, 0) = δmm .

m =−n

(B.46)

274 B Wave Functions

If in (B.45) we set α1 = α2 = 0 and γ1 = γ2 = 0, then β = β1 + β2, and we obtain the addition theorem for the d-functions

			n
	dmmn (β) =	dmmn (β1)dmn m (β2).
		m =−n
In particular, when β2 = −β1 we derive the unitarity condition
n	(β1)dmn m (−β1) =		n
dmmn	(β1)dmn m (−β1) =		(−1)m +m dmmn (β1)dmn m (β1)
m =−n			m =−n

=dnmm (β1)dnm m (β1) = dnmm (0) = δmm .

m =−n

The product of two d-functions can be expanded in terms of the Clebsch–
Gordan coe cients Cm+m1u
		mn,m1n1
		n+n1		m +m1u
n	n1		m+m1u u	m +m1u	, (B.47)
dmm (β)dm1m1		(β) =	Cmn,m1n1 dm+m1m +m1	(β) Cm n,m1n1	, (B.47)

u=|n−n1|

and note that the Clebsch–Gordan coe cients are nonzero only when |n − n1| ≤ u ≤ n + n1. The following symmetry properties of the Clebsch–Gordan coe cients are used in our analysis [169, 239]:

m+m1u

= (−1)

n+n1+u

−m−m1u

Cmn,m1n1

C−mn,−m1n1 ,

m+m1u

= (−1)

n+n1+u

m+m1u

Cmn,m1n1

Cm1n1,mn,

m+m1u

= (−1)

2u + 1

−m1n1

m+n

Cmn,m1n1

Cmn,−m−m1u,

2n1 + 1

m1+n+u

m+m1u

= (−1)

2u + 1

Cmn,m1n1

Cm+m1u,−m1n1 .

2n + 1

(B.48)

(B.49)

(B.50)

(B.51)

To compute the Clebsch–Gordan coe cients we ﬁrst deﬁne the coe cients Smn,mu 1n1 by the relation [77]

	Cm+m1u		= gu		Su	,
	mn,m1n1			mn,m1n1	mn,m1n1
where

				(u + m + m1)!(u − m − m1)!
gu	= √
		2u + 1
		2u + 1		(n − m)!(n + m)!
mn,m1n1

×(n + n1 − u)! (u + n − n1)! (u − n + n1)! (n1 − m1)!(n1 + m1)! (n + n1 + u + 1)!

B.3 Rotations

275

The S-coe cients obey the three-term downward recurrence relation
				Su−1	= puSu	+ quSu+1
				mn,m1n1	mn,m1n1		mn,m1n1
for u = n + n1, n + n1 − 1, . . . , max(\|m + m1\|, \|n − n1\|) with
pu =			(2u + 1) {(m − m1)u(u + 1) − (m + m1) [n (n + 1) − n1 (n1 + 1)]}					,
				(u + 1) (n + n1 − u + 1) (n + n1 + u + 1)
q	u	=	−	u(u + n − n1 + 1) (u − n + n1 + 1)
	u		−	(u + 1) (n + n1 − u + 1)
			×	(u + m + m1 + 1) (u − m − m1 + 1)
			×	n + n1 + u + 1
and the starting values
					Sn+n1+1	= 0,
					mn,m1n1
					Sn+n1	= 1.
					mn,m1n1

The rotation addition theorem for vector spherical wave functions is [213]

	n
M mn1,3 (kr, θ, ϕ) =	Dmmn	(α, β, γ)M m1,3n(kr, θ1, ϕ1),						(B.52)
m =−n
	n
N mn1,3 (kr, θ, ϕ) =	Dmmn	(α, β, γ)N m1,3n(kr, θ1, ϕ1) ,						(B.53)
m =−n
and in matrix form we have
M mn1,3 (kr, θ, ϕ)	= R (α, β, γ)		M m1,3n (kr, θ1, ϕ1)				,
N 1,3 (kr, θ, ϕ)	= R (α, β, γ)		N 1,3	(kr, θ	, ϕ	)	,
mn			m n	1	1

where R is the rotation matrix. The rotation matrix has a block-diagonal structure and is given by

R (α, β, γ) =	Rmn,m n (α, β, γ)	0
R (α, β, γ) =	0	Rmn,m n (α, β, γ)	,
where
Rmn,m n (α, β, γ) = Dmmn		(α, β, γ)δnn .
The unitarity condition for the D-functions yields
R (−γ, −β, −α) = R−1 (α, β, γ)			(B.54)

276 B Wave Functions

and since

Dmn m (−γ, −β, −α) = Dmmn (α, β, γ),

we also have

R (−γ, −β, −α) = R† (α, β, γ) ,

(B.55)

where X† stands for the complex conjugate transpose of the matrix X. Properties of Wigner functions and generalized spherical functions (which

have been introduced in the quantum theory of angular momentum) are also discussed in [27, 160].

B.4 Translations

We consider two coordinate systems Oxyz and Ox1y1z1 having identical spatial orientations but di erent origins (Fig. B.4). The vectors r and r1 are the position vectors of the same ﬁeld point in the coordinate systems Oxyz and Ox1y1z1, respectively, while the vector r0 connects the origins of both coordinate systems and is given by r0 = r − r1.

In general, the addition theorem for spherical wave functions can be written as [27, 70]

∞	n

umn (kr) =	Cmn,m n (kr0) um n (kr1) .

n =0 m =−n

Integral and series representations for the translation coe cients Cmn,m n can be obtained by using the integral representations for the spherical wave

z
z1		M
r	r1
r0	O1	y1	y
O

x x1

Fig. B.4. Coordinate translation

B.4 Translations

277

functions. For regular spherical wave functions, we use the integral representation (B.1), the relation r = r0 + r1, and the spherical wave expansion of the plane wave exp(jk · r1) to obtain

∞

(kr

) u1

(kr

)

(B.56)

(kr) =

mn,m n

m n

n =0 m =−n

with

jn −n 2π π

jk(β,α) r

Cmn,m n (kr0) =

Ymn (β, α) Y−m n (β, α) e

sin β dβ dα .

2π

(B.57) Taking into account the spherical wave expansion of the plane wave exp(jk·r0) and integrating over α, we ﬁnd the series representation

Cmn,m1 n (kr0) = 2jn −n jn a (m, m | n , n, n ) u1m−m n (kr0) (B.58)

with the expansion coe cients

a (m, m | n , n, n ) = π Pn|m| (cos β) Pn|m | (cos β) Pn|m−m | (cos β) sin βdβ.

We note that the expansion coe cients a(·) are deﬁned by the spherical harmonic expansion theorem [70]

Pn|m| (cos β) Pn|m | (cos β) = a (m, −m | n , n, n ) Pn|m+m | (cos β) ,

where the summation over n is ﬁnite covering the range |n − n |, |n − n | + 2, ..., n + n . These coe cients can be identiﬁed with a product of two Wigner 3j symbols, which are associated with the coupling of two angular momentum eigenvectors. The azimuthal integration in (B.57) can be analytically performed by using the identity

k · r0 = kρ0 sin β cos (α − ϕ0) + kz0 cos β

and the standard integral

2π

ejx cos(α−ϕ0)ej(m−m )α dα = 2πjm −mJm −m(x)e−j(m −m)ϕ0 ,

where (ρ0, ϕ0, z0) are the cylindrical coordinates of r0, and the result is

C1	(kr	) = jn +m −n−me−j(m −m)ϕ0				π J	m −m	(kρ sin β)
mn,m n	0					0	m −m	0
						0
		×	P \|m\| (cos β) P	\|m \|	(cos β) ejkz0 cos β sin β dβ . (B.59)
		×	n	n

Cmn,m3 n

278 B Wave Functions

If the translation is along the z-axis, the addition theorem involves a single summation

			∞
		umn1 (kr) =	Cmn,mn1	(kz0) umn1 (kr1)
			n =0
and the translation coe cients simplify to
C1	(kz	) = jn −n π P \|m\| (cos β) P \|m\|			(cos β) ejkz0 cos β sin β dβ. (B.60)
mn,mn	0	0	n	n
		0

For radiating spherical wave functions we consider the integral representation (B.2) and the relation r = r0 + r1. For r1 > r0, we express u3mn as (cf. (B.5))

umn3 (kr) =	1	2π π Ymn (β, α) Q(k, β, α, r1)ejk(β,α)·r0 sin βdβdα
	2πjn
		0 0

and use the spherical wave expansion of the quasi-plane wave Q(k, β, α, r1) to derive

				∞ n
		(kr) =			(kr		) u3	(kr		) .
u3		(kr) =		C1	(kr	0	) u3	(kr	1	) .
mn				mn,m n		0	m n		1
				n =0 m =−n
For r1 < r0, we represent the radiating spherical wave functions as
umn3 (kr) =		1	2π π Ymn (β, α) Q(k, β, α, r0)ejk(β,α)·r1 sin βdβdα
umn3 (kr) =	2πjn		2π π Ymn (β, α) Q(k, β, α, r0)ejk(β,α)·r1 sin βdβdα
	2πjn		0	0

and use the spherical wave expansion of the plane wave exp(jk · r1) to obtain

	∞	n
			(kr	) u1	(kr	)
u3	(kr) =	C3	(kr	) u1	(kr	)
mn		mn,m n	0	m n	1

n =0 m =−n

with
C	3		(kr	0	)
mn,m n
=	jn −n 2π π					Ymn (β, α) Y−m n (β, α) Q(k, β, α, r0) sin β dβ dα . (B.61)
		π	0 0

Inserting the spherical wave expansion of the quasi-plane wave Q(k, β, α, r0) into (B.61) yields

Cmn,m3 n (kr0) = 2jn −n jn a (m, m | n , n, n ) u3m−m n (kr0) . (B.62)

Finally, we note the integral representation for the translation coe cients in the speciﬁc case of axial translation:

B.4 Translations

279

C3	(kz	) = 2jn −n π2 −j∞ P \|m\| (cos β) P	\|m\|	(cos β) ejkz0 cos β sin βdβ .
mn,mn	0	n	n
		0		(B.63)
				(B.63)

Recurrence relations have been derived for the Cmn,mn coe cients [150]. The derivation is simple for the case of axial translation and positive m. Using the integral representation (B.60) (or (B.63)) and the recurrence relations for

the normalized associated Legendre functions (A.15) and (A.16), give

(n − m − 1)(n − m + 1) Cmn

1,mn (kz0)

(2n − 1)(2n + 1)

−

(n + m)(n + m + 1)

Cmn+1,mn

(kz0)

(2n + 1)(2n + 3)

(n + m − 1)(n + m)

m−1n,m−1n −1

(kz )

(2n − 1)(2n + 1)

(n − m + 1)(n − m + 2)

m−1n,m−1n

(kz

) ,

(B.64)

(2n + 1)(2n + 3)

while the recurrence relation (A.14), yields

(n − m + 1)(n + m + 1)

Cmn+1,mn (kz0)

(2n + 1)(2n + 3)

−

(n − m)(n + m)

Cmn

−

1,mn (kz0)

(2n

−

1)(2n + 1)

(n − m)(n + m)

(kz )

(2n − 1)(2n + 1)

mn,mn −1

−

(n − m + 1)(n + m + 1)

Cmn,mn +1 (kz0) .

(B.65)

(2n + 1)(2n + 3)

We note that the convention Cmn,mn = 0 for m > n or m > n , is assumed in the above equations. For the recurrence relationships to be of practical use, initial values are needed. This is accomplished by using the integral representations

jn (kz0) =

Pn (cos β) e

jkz0 cos β

sin β dβ,

2(2n + 1)

π2 −j∞ P

h(1)

(kz ) =

(cos β) ejkz0 cos β sin β dβ ,

2n + 1

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