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2.6 Multiple Particles

129

where i1 = [cN1,µ, dN1,µ]T, i2 = [cN2,µ, dN2,µ]T, and as usually, e = [aν , bν ]T is the vector containing the expansion coe cients of the incident ﬁeld in the global

coordinate system. Further, deﬁning the scattered ﬁeld coe cients

s1 = Q111 (ks, ki,1)i1 ,

s2 = Q112 (ks, ki,2)i2 ,

and introducing the individual transition matrices,

T1 = −Q111 (ks, ki,1) Q311 (ks, ki,1) −1 ,

T2 = −Q112 (ks, ki,2) Q312 (ks, ki,2) −1 ,

we rewrite the matrix system (2.134) as

−

rtr

= T 1

10e ,

12 s2

−

rtr

= T 2

20e ,

21 s1

and ﬁnd the solutions

rtr

T 2

rtr

−1

rtr

T 2

s1 = T 1 I

T 1

− S

rtr

T 2

−1

rtr

T 1

(2.135)

s2 = T 2 I

T 1 12

− S

To compute the T matrix of the two-particles system and to derive a scatteredﬁeld expansion centered at the origin O of the global coordinate system we use the Stratton–Chu representation theorem for the scattered ﬁeld Es in Ds. In the exterior of a sphere enclosing the particles, the expansion of the approximate scattered ﬁeld ENs in terms of radiating vector spherical wave functions reads as

ENs (r) = fνN M 3ν (ksr) + gνN N 3ν (ksr) ,

ν=1

where the expansion coe cients are given by

$ fνN %

jks2

$ N

(ksr ) %

(r )

gνN

π S1

i,1

(ksr )

$ M

(ksr ) %

µs

+ j

hi,1

(r1)

dS (r1)

εs

(ksr )

jks2

(r ) $ N

(ksr ) %

i,2

(ksr )

$ M

(ksr ) %

µs

+ j

hi,2

(r2 )

dS (r2 ) .

(2.136)

εs

(ksr )

130 2 Null-Field Method

Finally, using the addition theorem for the regular vector spherical wave functions

(k r

)

(ksr )

S01

(ksr

)

with

S01tr = T 11 (ksr01) R (α1, β1, γ1) ,

and

)

(k r )

s 2

(ksr )

S02

(ksr )

with

S02tr = T 11 (ksr02) R (α2, β2, γ2) ,

we obtain

s = S01tr Q111(ks, ki,1)i1 + S02tr Q211(ks, ki,2)i2
= S01tr s1 + S02tr s2 ,	(2.137)

where s = [fνN , gνN ]T is the vector containing the expansion coe cients of the scattered ﬁeld in the global coordinate system. Combining (2.135) and (2.137), and using the identities S20rt (S10rt )−1 = S21rtr and S10rt (S20rt )−1 = S12rtr, yields [187]

T =

− S

rtr

−1

rtr

01T 1

I + 12 T

− S

rtr

−1

I +

rtr

02T 2

T 2

T 1

T 1 12

where the explicit expressions of the transformation matrices S21rtr are given by

S12rtr = R (−γ1, −β1, −α1) T 11 (ksr12) R (α2, β2, γ2) ,

(2.138)

and S12rtr

and

S21rtr = R (−γ2, −β2, −α2) T 11 (−ksr12) R (α1, β1, γ1) ,

respectively. Equation (2.138) gives the system transition matrix T in terms of the individual transition matrices T 1 and T 2, and the transformation matrices S and S. S and S involve translations of the regular and radiating vector spherical wave functions, respectively, and geometric constraints are introduced by the S matrices. Obviously, the geometric restrictions r12 > r2

and r12

2.6 Multiple Particles

131

> r1 introduced by the S matrices are fulﬁlled if the smallest circumscribing spheres of the particles do not overlap. The following feature of

equation (2.138) is apparent: if T 2 = 0, then T = S01tr T 1S10rt , and if T 1 = 0, then T = S02tr T 2S20rt , as it should. In general, T is a sum of two terms, each

of which is a modiﬁcation of these limiting values. The various terms in a formal expansion of the inverses occurring in (2.138) can be considered as multiple-scattering contributions. The terms involving only T 1 and T 2 repre-

sent reﬂections at S1	and S2, respectively, the terms involving only T 1		rtr	T 2
		12
rtr		S

and T 2S21 T 1 represent consecutive reﬂections at S2 and S1, and S1 and S2, respectively, etc.

2.6.2 Formulation for a System with N Particles

The generalization of the T -matrix relation to a system with more than two constituents is straightforward. The system of matrix equations consists in the null-ﬁeld equations in the interior of Sl

			N
	−			rtr		S	rt	N
sl	−	T l	S		sp = T l	S	l0 e for l = 1, 2, . . . ,	N	(2.139)
sl		T l	lp		sp = T l		l0 e for l = 1, 2, . . . ,		(2.139)

p=l

and the matrix equation corresponding to the scattered ﬁeld representation

N
s = S0trl sl .	(2.140)

l=1

In practical computer calculations it is convenient to consider the global matrix A with block-matrix components

All = I , l = 1, 2, . . . , N ,

A	lp	−	S	rtr	,		l, p = 1, 2, . . . ,	N	,

		= T l lp				l = p ,

and to express the solution to the system of matrix equations (2.139) as

	$ N	%
sl =	AlpT pSprt0	e , l = 1, 2, . . . , N ,	(2.141)
	p=1

where A stay for A−1, and Alp, l, p = 1, 2, . . . , N, are the block-matrix components of A. In view of (2.140), the system T -matrix becomes

T = S0trl AlpT pSprt0

l=1 p=1

and this transition matrix can be used to compute the scattering characteristics for ﬁxed or random orientations of the ensemble [153, 165].

132 2 Null-Field Method

In order to estimate the computer memory requirement it is important to know the dimensions of the matrices involved in the calculation. If Nrank(l) and Mrank(l) are the maximum expansion order and the number of azimuthal modes for the lth particle, then the dimension of the transition matrix T l is dim(T l) = 2Nmax(l) × 2Nmax(l), where

Nmax(l) = Nrank (l) + Mrank (l) [2Nrank(l) − Mrank(l) + 1] .

The dimension of the global matrix A is

N	N

dim (A) = 2	Nmax(l) × 2 Nmax(l) ,
l=1	l=1
since

dim Alp = dim Alp = 2Nmax(l) × 2Nmax(p) .

If Nmax gives the dimension of the system T -matrix dim (T ) = 2Nmax × 2Nmax ,

we have

dim Sprt0 = 2Nmax(p) × 2Nmax , dim S0trl = 2Nmax × 2Nmax(l) .

In our computer code, the parameters Nmax and Nmax(l), l = 1, 2, . . . , N, are independent. Nmax(l) is given by the size parameter of the lth particle, while Nmax is given by the size parameter of a sphere centered at O and enclosing the particles.

2.6.3 Superposition T -matrix Method

For a system of N particles with αl = βl = γl = 0, l = 1, 2, . . . , N, the transformations of the vector spherical vector wave functions involve only the addition theorem under coordinate translations, i.e.,

S0trl = T 11 (ksr0l) ,

Sprt0 = T 11 (−ksr0p) ,

and

Srtr = T 31 (ksrlp)

for l, p = 1, 2, . . . , N. Consequently, the matrix equation (2.139) takes the form

2.6 Multiple Particles

133

	N
sl = T l T 11 (−ksr0l) e + T 31 (ksrlp) sp ,		(2.142)
	p=l
while (2.140) becomes	N
	N
	s = T 11 (ksr0l) sl .	(2.143)

l=1

The precedent equations can be written in explicit form by indicating the vector and matrix indices

[(sl)ν ] = [(Tl)νν ]

(−ksr0l) [eµ]

(ksrlp) (sp)µ

p=l

(2.144)

[sν ] = T

(ksr0l) (sl)µ

(2.145)

νµ

l=1

and can also be derived by using the so-called superposition T -matrix method [169]. The superposition T -matrix method reproduces the two cooperative effects characterizing aggregate scattering: interaction between particles and far-ﬁeld interference [273]. The interaction e ect take into account that each particle is excited by the initial incident ﬁeld and the ﬁelds scattered by all other particles, while the far-ﬁeld interference is a result of the incident and scattered phase di erences from di erent particles. The superposition T -matrix method involves the following general steps:

1.The expansions of the incident and scattered ﬁelds in each particle coordinate system.

2.The representation of the ﬁeld exciting a particle by a single-ﬁeld expansion (which includes the incident ﬁeld and the scattered ﬁelds from all other particles).

3.The solution of the transmission boundary-value problem for each particle.

4.The derivation of the scattered-ﬁeld expansion in the global coordinate system.

The interaction e ect is taken into account in steps 2 and 3, where the scattered ﬁelds from other particles are transformed and included in the incident ﬁeld on each particle, while the interference e ect is taken into account in steps 1 and 4 that address the incident and scattered path di erences, respectively. For a system of N spheres, the individual component T matrices are diagonal with the standard Lorenz–Mie coe cients along their main diagonal, and in this case, the superposition T -matrix method is also known as the multisphere separation of variables technique or the multisphere superposition method [21, 24, 29, 72, 150]. Solution of (2.144) have been obtained using direct matrix inversion, method of successive orders of scattering, conjugate

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