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

sin Θ

1.5 Transition Matrix

69

cos σ1 = eα · eα = −sin β cos θ − cos β sin θ cos (ϕ − α) , sin Θ

cos σ2 = eϕ · eϕ = cos β sin θ − sin β cos θ cos (ϕ − α) .

For an ensemble of randomly positioned particles, the waves scattered by di erent particles are random in phase, and the Stokes parameters of these incoherent waves add up. Therefore, the scattering matrix for the ensemble is the sum of the scattering matrices of the individual particles:

F = N F ,

where N is number of particles and F denotes the ensemble-average scattering matrix per particle. Similar relations hold for the extinction matrix and optical cross-sections. Because the particles are identical, the ensembleaverage of a scattering quantity X is the orientation-averaged quantity

1			2π 2π π			X (αp, βp, γp) sinβp dβp dαp dγp ,
X =
	8π2		0	0	0
where αp, βp and γp are the particle orientation angles.
In the following			analysis,		the T matrix formulation is used to derive

e cient analytical techniques for computing X . These methods work much faster than the standard approaches based on the numerical averaging of results computed for many discrete orientations of the particle. We begin with the derivation of the rotation transformation rule for the transition matrix and then compute the orientation-averaged transition matrix, optical crosssections and extinction matrix. An analytical procedure for computing the orientation-averaged scattering matrix will conclude our analysis.

Rotation Transformation of the Transition Matrix

To derive the rotation transformation rule for the transition matrix we assume that the orientation of the particle coordinate system Oxyz with respect to the global coordinate system OXY Z is specified by the Euler angles αp, βp and γp.

In the particle coordinate system, the expansions of the incident and scattered field are given by

∞n

Ee (r, θ, ϕ) = amnM 1mn (ksr, θ, ϕ) + bmnN 1mn (ksr, θ, ϕ) ,

n=1 m=−n

∞n

Es (r, θ, ϕ) = fmnM 3mn (ksr, θ, ϕ) + gmnN 3mn (ksr, θ, ϕ) ,

n=1 m=−n

70 1 Basic Theory of Electromagnetic Scattering

while in the global coordinate system, these expansions take the form

∞	n	(ksr, Φ, Ψ ) + bmnN mn1 (ksr, Φ, Ψ ) ,
Ee (r, Φ, Ψ ) = amnM mn1
n=1 m=−n
∞	n
Es (r, Φ, Ψ ) = fmnM mn3		(r, Φ, Ψ ) + gmnN mn3	(ksr, Φ, Ψ ) .
n=1 m=−n

Assuming the T -matrix equations s = T e and s = T e, our task is to express the transition matrix in the global coordinate system T in terms of the transition matrix in the particle coordinate system T . Defining the “augmented” vectors of spherical wave functions in each coordinate system

w1,3	M mn1,3 (ksr, θ, ϕ)
	(ksr, θ, ϕ) =	(k r, θ, ϕ)
	N 1,3
	mn	s
and
	M mn1,3 (ksr, Φ, Ψ )
w1,3 (ksr, Φ, Ψ ) =		,
	N mn1,3	(ksr, Φ, Ψ )

and using the rotation addition theorem for vector spherical wave functions, we obtain

Ee = e	T	w1 = e	T	w1 = e	T	R (αp, βp, γp) w1 ,
T		w3 = s	T	w3 = s	T	R (−γp, −βp, −αp) w3 .
Es = s
Consequently
		e = RT (αp, βp, γp) e ,
and therefore,		s = RT (−γp, −βp, −αp) s ,
T (αp, βp, γp) = RT (−γp, −βp, −αp) T RT (αp, βp, γp) .							(1.115)
The explicit expression of the matrix elements is [169, 228, 233]

nn1

Tmn,mij	1n1 (αp, βp, γp) =		Dmn m (−γp, −βp, −αp) Tmij n,m1n1
		m =−n m1=−n1
		n1	(αp, βp, γp)	(1.116)
		×Dm1m1

for i, j = 1, 2.

Because the elements of the amplitude matrix can be expressed in terms of the elements of the transition matrix, the above relation can be used to express the elements of the amplitude matrix as functions of the particle orientation angles αp, βp and γp. The properties of the Wigner D-functions can then be used to compute the integrals over the particle orientation angles.

1.5 Transition Matrix

71

Orientation-Averaged Transition Matrix

The elements of the orientation-averaged transition matrix with respect to the global coordinate system are given by

' ij	(	=	1		2π 2π π		ij	(αp, βp, γp) sinβp dβp dαp dγp
Tmn,m1n1				8π2	0	0 0	Tmn,m1n1
			1		n	n1	ij	2π 2π π		n
		=					Tm n,m1n1		Dm m (−γp, −βp, −αp)
				8π2				0 0 0
					m =−n m1=−n1
					n1	(αp, βp, γp) sinβp dβp dαp dγp .				(1.117)
			×Dm1m1

Using the definition of the Wigner D-functions (cf. (B.34)), the symmetry relation of the Wigner d-functions dnm m(−βp) = dnmm (βp), and integrating over αp and γp, yields

' ij	(	1		n	n1
Tmn,m1n1	=		2	m =−n m1=−n1		∆m m∆m1m1 δm m1 δmm1
				ij	π	n	n1
		×Tm n,m1n1 0				dmm (βp) dm1m1		(βp) sinβp dβp ,

where ∆mm is given by (B.36). Taking into account the identities: ∆m m = ∆mm and (∆mm )2 = 1, and using the orthogonality property of the d- functions (cf. (B.43)), we obtain [168, 169]

'(

	Tmn,mij		1n1	= δmm1 δnn1 tnij		(1.118)
	with	1		n
	tij =			T ij	.	(1.119)
		2n + 1
	n			m n,m n
				m =−n

The above relation provides a simple analytical expression for the orientationaveraged transition matrix in terms of the transition matrix in the particle coordinate system. The orientation-averaged T ij matrices are diagonal and their elements do not depend on the azimuthal indices m and m1.

Orientation-Averaged Extinction and Scattering Cross-Sections

In view of the optical theorem, the orientation-averaged extinction crosssection is (cf. (1.88) with |Ee0| = 1)

Cext = 4kπ Im )epol · Es∞ (ez )* . s

72 1 Basic Theory of Electromagnetic Scattering

Considering the expansion of the far-field pattern in the global coordinate system (cf. (1.96)), taking the average and using the expression of the orientationaveraged transition matrix (cf. (1.118) and (1.119)), gives

)e

Es

(ez )* =

1

(

−

j)n+1 t11amn + t12bmn e

mmn (ez )

pol ·

∞

ks n,m

n

n

pol ·

21

a

+ t

22

b

e

+ j t

·

n (e ) ,

n

n mn

pol

mn

z

mn

where the summation over the index m involves the values −1 and 1. In the next chapter we will show that for axisymmetric particles, T−ijmn,−mn =

−Tmn,mnij and T0ijn,0n = 0 for i = j, while for particles with a plane of symmetry, Tmn,mnij = 0 for i = j. Thus, for macroscopically isotropic and mirrorsymmetric media, (1.119) gives t12n = t21n = 0. Further, using the expressions of the incident field coe cients (cf. (1.26))

a	= 4jne		·	m		(e		) ,		(1.120)
mn		pol		mn			z	(e
b	= 4jn+1e			pol ·	n				z	) ,
mn	−					mn

and the special values of the vector spherical harmonics in the forward direction

mmn (ez ) =

√

(jmex − ey ) ,

2n+1

4

(1.121)

nmn (ez ) =

√

(ex + jmey ) ,

2n+1

4

we obtain [163]

2π

# ∞

+

Cext

=

Re

(2n + 1) t11

+ t22

,

− k2

n

n

s

n=1

2π

# ∞

n

+

11

22

= −

ks2

Re

Tmn,mn + Tmn,mn .

(1.122)

n=1 m=−n

The above relation shows that the orientation-averaged extinction crosssection for macroscopically isotropic and mirror-symmetric media is determined by the diagonal elements of the transition matrix in the particle coordinate system. The same result can be established if we consider an ensemble of randomly oriented particles (with t12n = 0 and t21n = 0) illuminated by a linearly polarized plane wave (with real polarization vector epol).

For an arbitrary excitation, the scattering cross-section can be expressed in the global coordinate system as

Cscat =	π	∞ n &	&2	+ gmn	2	=		π
	2	&fmn&						2 s†s ,
				| |
	ks n=1 m=−n &		&				ks

							1.5 Transition Matrix	73
whence, using the T -matrix equation s = T e, we obtain
	Cscat		=	π	e† 'T †	(αp, βp, γp) T (αp, βp, γp)( e .
				2
				ks

Since

T (αp, βp, γp) = RT (−γp, −βp, −αp) T RT (αp, βp, γp) ,

T † (αp, βp, γp) = R (αp, βp, γp) T †R (−γp, −βp, −αp) ,

and in view of (B.54) and (B.55),

R (−γp, −βp, −αp) = RT (−γp, −βp, −αp) −1 ,

R (αp, βp, γp) = RT (−γp, −βp, −αp) ,

we see that

T † (αp, βp, γp) T (αp, βp, γp) = RT (−γp, −βp, −αp) T †T RT (αp, βp, γp) .

The above equation is similar to (1.115), and taking the average, we obtain

' T

†T

ij

( = δmm

δnn

tij ,

mn,m1n1

1

1

n

where

1

n

ij

tij =

T †T

n

2n + 1

m =−n

m n,m n

or explicitly,

t11

=

1

n

∞

n1

&2 +

&T 21

&2

&T 11

,

n

2n + 1

m =−n n1=1 m1

&

m1n1,m n&

& m1n1,m n&

=−n1

t12

=

1

n

∞

n1

11

T 12

T

n

2n + 1

m =−n n1=1 m1

m1n1,m n

m1n1,m n

=−n1

+T 21

T 22

,

m1n1,m n

m1n1,m n

t21

= t12

,

n

n

t22

=

1

n

∞

n1

&2 +

&T 22

&2 .

&T 12

n

2n + 1

m =−n n1=1 m1

&

m1n1,m n&

& m1n1,m n&

=−n1

74 1 Basic Theory of Electromagnetic Scattering

The orientation-averaged scattering cross-section then becomes

C

scat

=

π

t11

a

2 + t12a

b

k2

n

| mn|

n mn mn

s

n,m

21

22

&

&2

,

(1.123)

+tn amnbmn

+ tn

&bmn&

&

&

where as before, the summation over the index m involves the values −1 and

1. For macroscopically isotropic and mirror-symmetric media, t12 = t21 = 0,

n n

and using (1.120) and (1.121), we obtain [120, 162]

2π

∞

Cscat

=

(2n + 1)

t11

+ t22

ks2 n=1

n

n

=

2π

∞

n

∞

n1

&2

+

&T 12

&2

&T 11

ks2

n=1 m=−n n1=1 m1=−n1

&

m1n1,mn&

& m1n1,mn&

+ &T 21

&2

+

&T

22

&2 .

(1.124)

&

m1n1,mn&

& m1n1,mn&

Thus, the orientation-averaged scattering cross-section for macroscopically isotropic and mirror-symmetric media is proportional to the sum of the squares of the absolute values of the transition matrix in the particle coordinate system. The same result holds true for an ensemble of randomly oriented particles illuminated by a linearly polarized plane wave.

Despite the derivation of simple analytical formulas, the above analysis shows that the orientation-averaged extinction and scattering cross-sections for macroscopically isotropic and mirror-symmetric media do not depend on the polarization state of the incident wave. The orientation-averaged extinction and scattering cross-sections are invariant with respect to rotations and translations of the coordinate system and using these properties, Mishchenko et al. [169] have derived several invariants of the transition matrix.

Orientation-Averaged Extinction Matrix

To compute the orientation-averaged extinction matrix it is necessary to evaluate the orientation-averaged quantities Spq (ez , ez ) . Taking into account the expressions of the elements of the amplitude matrix (cf. (1.97)), the equation of the orientation-averaged transition matrix (cf. (1.118) and (1.119)) and the expressions of the vector spherical harmonics in the forward direction (cf. (1.121)), we obtain

∞

Sθβ (ez , ez ) = Sϕα (ez , ez ) = −2kj s (2n + 1) t11n + t22n ,

n=1

∞

Sθα (ez , ez ) = − Sϕβ (ez , ez ) = −21ks (2n + 1) t12n + t21n .

n=1

1.5 Transition Matrix

75

Inserting these expansions into the equations specifying the elements of the extinction matrix (cf. (1.79)), we see that the nonzero matrix elements are

2π

# ∞

+

11

22

i = 1, 2, 3, 4

(1.125)

Kii = −

Re

(2n + 1)

tn

+ tn

,

k2

s

n=1

and

π

! ∞

"

K14

= K41

=

2

Re

, (2n + 1) t12

+ t21

,

2

ks

n=1

n

n

"

(1.126)

2π

! ∞

K23

=

−

K32

=

Im

, (2n + 1)

t12

+ t21 .

k2

s

n=1

n

n

In terms of the elements of the extinction matrix, the orientation-averaged extinction cross-section is (cf. (1.89))

1

Cext = Ie [ K11 Ie + K14 Ve] ,

while for macroscopically isotropic and mirror-symmetric media, the identities t12n = t21n = 0, imply

K14 = K41 = K23 = K32 = 0 .

In this specific case, the orientation-averaged extinction matrix becomes diagonal with diagonal elements being equal to the orientation-averaged extinction cross-section per particle, K = Cext I.

Orientation-Averaged Scattering Matrix

By definition, the orientation-averaged scattering matrix is the orientationaveraged phase matrix with β = 0 and α = ϕ = 0. In the present analysis we consider the calculation of the general orientation-averaged phase matrixZ(er , ek ; αp, βp, γp) without taking into account the specific choice of the incident and scattering directions. We give guidelines for computing the quantities of interest, but we do not derive a final formula for the average phase matrix.

According to the definition of the phase matrix we see that the orientation-

averaged quantities Spq (er , ek )Sp1q1 (er , ek ) , with p, p1 = θ, ϕ and q, q1 = β, α, need to be computed. In view of (1.99), we have

)Spq (er , ek ) Sp1q1 (er , ek )*

16

-

T

)

.

=

v†

(e ) T (α , β , γ ) v

(e

vT (e ) T (α , β , γ ) v

(e

)

ks2

q1

k

p p p

p1

r

p

r

p p p

q

k

16

T

'

†

T

(

=

v

(e

) T

(α

, β

, γ

) v (e

) v

(e

) T (α

, β

, γ

) v (e

) ,

ks2

q1

k

p

p

p

p1

r

p

r

p

p

p

q

k

76 1 Basic Theory of Electromagnetic Scattering

where, as before, T stands for the transition matrix in the global coordinate system. Defining the matrices

V pp1 (er ) = vp1 (er ) vTp (er )

and
	'	†					(
App1 (er ) = T (αp, βp, γp) V pp1 (er )T (αp, βp, γp) ,
we see that
)Spq (er , ek ) Sp q	(er , ek )*		=	16	vqT (ek ) App1 (er ) vq (ek ) .			(1.127)
1	1			k2		1
				s
Using the block-matrix decomposition
		X =	X11 X12				,
			X21			X22
where X stands for V pp1		and App1 , we express the submatrices of App1 as

A11pp1 = 'T 11†V 11pp1 T 11 + T 11†V 12pp1 T 21 + T 21†V 21pp1 T 11 + T 21†V 22pp1 T 21( , A12pp1 = 'T 11†V 11pp1 T 12 + T 11†V 12pp1 T 22 + T 21†V 21pp1 T 12 + T 21†V 22pp1 T 22( , A21pp1 = 'T 12†V 11pp1 T 11 + T 12†V 12pp1 T 21 + T 22†V 21pp1 T 11 + T 22†V 22pp1 T 21( , A22pp1 = 'T 12†V 11pp1 T 12 + T 12†V 12pp1 T 22 + T 22†V 21pp1 T 12 + T 22†V 22pp1 T 22( .

(1.128) It is apparent that each matrix product in the above equations is of the form

'	kl†	uv	ij	(	(1.129)
W pp1 (er ) =	T	(αp, βp, γp) V pp1	(er ) T	(αp, βp, γp) ,

where the permissive values of the index pairs (i, j), (k, l) and (u, v) follow

from (1.128). The elements of the W pp1

matrix are given by

(W

pp1

)

m1n1

,m1n1

(e

) =

'T ij

(α

, β

, γ

) T kl

(α

, β , γ )(

r

mn,m1n1

p

p

p

mn,m1n1

p

p p

n,m n,m

(er ) ,

(1.130)

V uv

mn,mn

×

pp1

and the rest of our analysis concerns with the computation of the term

'

T

ij

(α

, β , γ ) T

kl

(α

, β

, γ

(

(1.131)

T

=

p

p

p

p

) .

mn,m1n1

p

p mn,m1n1

1.5 Transition Matrix

77

It should be mentioned that for notation simplification we omit to indicate the dependency of T on the matrix indices.

Using the rotation transformation rule for the transition matrix (cf. (1.116)), we obtain

1

n

n1

n

n1

2π 2π π

n

T =

Dm m (−γp, −βp, −αp)

8π2

m =−n m1=−n1 m =

n m1=

n1

0

0

0

−

−

×

Dn1

(α , β , γ ) Dn

(

−

γ ,

−

β , α )

1

p p p

m

p

p

−

p

m1m

m

T ij

×

Dn1

1

(α , β , γ ) sinβ

dβ dα dγ

T kl

1

.

1

p p p

p

p

p

p

m n,m

n1

m m

m n,m

n1

1

Taking into account the definition of the Wigner D-functions (cf. (B.34)) and integrating over αp and γp, yields

1

n

n1

n

n1

T =

δm1

mδm1

m ,m1

m ∆

2

−

m,m1

−

m =−n m1=−n1 m

=

n m1=

n1

−

−

π dn

−

−

×

(

−

β ) dn

(

β ) dn1

(β ) dn1

1

(β ) sinβ dβ

m m

p

−

p

m1m

p

1

p

p

p

0

m m

1

m m

×

T ij

T kl

,

m n,m1n1

m n,m1n1

where dn

are the Wigner d-functions defined in Appendix B,

mm

∆ = ∆m m∆m m∆m1m1 ∆m1m ,

1

and ∆mm is given by (B.36). To compute the integral

I =

1

δm1

mδm1

m,m1

m ,m1

m

2

−

−

−

−

π dn

(

−

β ) dn

(

−

β ) dn1

(β ) dn1

1

(β ) sinβ dβ

,

×

m m

p

p

m1m

p

1

p

p

p

0

m m

1

m m

we use the symmetry relations (cf. (B.39) and (B.41))

dmn m(−βp) = dmmn

(βp) = (−1)m+m d−n m−m (βp) ,

dn

(

−

βp) = dn

(βp) = ( 1)m+m dn

(βp) ,

m

−

m m

m

mm

−

−

n1

d

n

m m

and d

n1

d

n

the expansions of the d-functions products dm1m

−

−

1

1

1

− −

m m

m m

given by (B.47), and the orthogonality property of the d-functions (cf. (B.43)).

78 1 Basic Theory of Electromagnetic Scattering

We obtain

I = (−1)m+m +m+m (−1)n+n1+n+n1 δm1−m,m1−mδm1−m ,m1−m

umax	1			m −m1u		m1−mu		m −m1u
		m1	mu						,
× u=umin		Cm1n−1, mnC			m1n1,m nCm n , mnC m n ,m n
			−
	2u + 1			−		1 1 −		− 1 1

where Cm+m1u are the Clebsch–Gordan coe cients defined in Appendix B,

mn,m1n1

and

umin = max (|n − n1| , |n − n1| , |m1 − m| , |m1 − m| , |m − m1| , |m − m1|) ,

umax = min (n + n1, n + n1) .

Further, using the symmetry properties of the Clebsch–Gordan coe cients (cf. (B.48) and (B.51)) we arrive at

I = (−1)n+n1+n+n1 δm1

−

m,m1

mδm1

−

m ,m1

−

m

umax

−

2u + 1

−m1n1

×

m1n1

u=umin

(2n

+ 1) (2n

+ 1)

Cm1−mu,mnCm −m1u,−m n

1

1

m

n1

m

n

1

1

×

Cm 1

mu,mnC−

m n

1

m

−

m1u,

and

−

−

=

n

n1

n

n1

∆

T

ij

T

kl

. (1.132)

T

I

m =−n m1=−n1 m =

n m1

=

n1

m n,m1n1

m n,m1

n1

−

−

The orientation-averaged quantities Spq (er , ek )Sp1q1 (er , ek ) can be computed from the set of equations (1.127)–(1.132).

For an incident wave propagating along the Z-axis, the augmented vector of spherical harmonics vq (ez ) can be computed by using (1.121). Choosing the XZ-plane as the scattering plane, i.e., setting ϕ = 0, we see that the matrices V pp1 (er ) involve only the normalized angular functions πn|m|(θ) and τn|m|(θ). The resulting orientation-averaged scattering matrix can be computed at a set of polar angles θ and polynomial interpolation can be used to evaluate the orientation-averaged scattering matrix at any polar angle θ.

For macroscopically isotropic media, the orientation-averaged scattering matrix has sixteen nonzero elements (cf. (1.113)) but only ten of them are independent. For macroscopically isotropic and mirror-symmetric media, the orientation-averaged scattering matrix has a block-diagonal structure (cf. (1.114)), so that only eight elements are nonzero and only six of them are independent. In this case we determine the six quantities |Sθβ (θ)|2 ,