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

2.4 Inhomogeneous Particles

109

(Appendix D), we approximate the surface fields ei,1 and hi,1 by the finite expansions

$ eN

(r

) %

N

n1(r1)

×

M µ1 (ki,1r1)

i,1

1

N

= c1,µ

hN

(r

)

−

j

εi,1

n

(r )

×

N

1 (k

i,1

r )

i,1

1

µ=1

µi,1

1

1

µ

1

N

n

(r

)

×

N 1

(k

r

)

1

1

µ

i,1

1

+d1,µ

j

εi,1

n

(r

)

×

M 1

(k

i,1

r

)

−

µi,1

1

1

µ

1

n

(r )

×

M 3

(k

r

)

N

1

1

µ

i,1

1

+c1,µ

j

εi,1

n

(r

)

×

N 3 (k

r

)

−

µi,1

1

1

µ

i,1

1

n

(r

)

×

N 3

(k

r

)

N

1

1

µ

i,1

1

+d1,µ

j

(r

)

M 3

(k

r

) . (2.73)

εi,1 n

−

µi,1

1

1

×

µ

i,1

1

The surface fields ei,2 and hi,2 are the tangential components of the electric and magnetic fields in the domain Di,2 and the surface fields approximations can be expressed as linear combinations of regular vector spherical wave functions:

$

eNi,2(r2 ) hNi,2(r2 )

% N

N

n2(r2 ) × M µ1 (ki,2r2 )

= c2,µ

−

j

εi,2

n

(r )

×

N

1 (k

i,2

r )

µ=1

µi,2

2

2

µ

2

n

(r )

×

N 1

(k

r )

N

2

2

µ

i,2

2

+d2,µ

j

(r )

M 1

(k

r ) . (2.74)

εi,2 n

−

µi,2

2

2

×

µ

i,2

2

To express the resulting system of equations in matrix form we introduce the Qpqt (k1, k2) matrix as the Qpq (k1, k2) matrix of the surface St. The elements of the Qpqt matrix are given by (2.11)–(2.14) but with St in place of S. For example, the elements (Qpqt )11νµ read as

(Q

pq

11

jk12

n (r )

q

p

t

)

=

×

M

(k2r )

·

N

(k1r )

νµ

π

St

t

µ

t

ν

t

(k1r )" dS (r ) .

+

ε2

n (r )

×

N q

(k2r )

·

M

p

ε1

t

µ

t

ν

t

t

Inserting (2.73) and (2.74) into (2.70), (2.71) and (2.72), using the identities (cf. (B.23), (B.24) and (B.25)),

Qpp	(k, k) = 0 ,	for p = 1 or p = 3 ,
1
Q13	(k, k) = I ,
1

Q311 (k, k) = −I ,

110 2 Null-Field Method

and taking into account the transformation rules

M

1

(k

i,1

r

1

)

M

1

(k

r )

ν

tr

µ

i,1

2

1

(ki,1r )

=

S12

N

1

(ki,1r )

N

ν

µ

ν

1

µ

2

and

M

3

(k

i,1

r

2

)

M

3

(k

r )

ν

rt

µ

i,1

1

3

(ki,1r

)

=

S21

N

3

(ki,1r )

,

N

ν

µ

ν

2

µ

1

we obtain the system of matrix equations

Q131(ks, ki,1)i1 + Q133(ks, ki,1)i1 = −e ,

−i1 + S12tr Q211(ki,1, ki,2)i2 = 0 ,

S21rt i1 + Q231(ki,1, ki,2)i2 = 0 ,

(2.75)

N

N

T

N

N

T

, i2

N

N

]

T

, and as before,

where i1 = [c1,µ

, d1,µ] , i1

= [c1,µ, d1,µ

]

= [c2,µ, d2,µ

e = [aν , bν ]

T

is the vector containing the expansion coe cients of the incident

field. The S12tr matrix relates the vector spherical wave functions defined with respect to the coordinate system O1x1y1z1 to those defined with respect to the coordinate system O2x2y2z2, and can be expressed as the product of a translation and a rotation matrix:

S12tr

= T

11 (ki,1r12) R (α, β, γ) ,

(2.76)

where T and R are defined in Appendix B. The S21rt

matrix describes the

inverse transformation and is given by

rt =

R

(

−

γ, β,

α)

T

33 ( k

r

12

)

for r

> r

12

.

(2.77)

S21

− −

− i,1

1

Since R is a block-diagonal matrix and T is a block-symmetric matrix it follows that S12tr and S21rt are also block-symmetric matrices.

Solving the system of matrix equations gives

i1	=	−	Q31	(ks, ki,1) + Q33		(ks, ki,1)	tr T 2	rt −1 e ,
			1		1		S12	S21
i1 = S12tr T 2S21rt i1 ,
i2	=	−	Q31	(ki,1, ki,2) −1		rt i1 ,		(2.78)
			2		S21

where

T 2 = −Q112 (ki,1, ki,2) Q312 (ki,1, ki,2) −1

is the transition matrix of the inhomogeneity imbedded in a medium with relative media constants εi,1 and µi,1.

2.4 Inhomogeneous Particles

111

The Stratton–Chu representation theorem for the scattered field Es in Ds, yields the expansion of the approximate scattered field ENs in the exterior of a sphere enclosing the particle

N

ENs (r1) = fνN M 3ν (ksr1) + gνN N 3ν (ksr1) ,

ν=1

where the expansion coe cients are given by

$ fνN %

jks2

N

$ N

1

(ksr1) %

ν

gN

=

ei,1 (r ) ·

1

(ksr )

π S

M

1

ν

ν

1

$ M

1

(ksr1) %

µs

N

ν

dS (r

+ j

h

(r )

·

) .

εs

i,1

N

1

(ksr )

1

ν

1

Taking into account the expressions of the approximate surface fields given by (2.73), we derive the matrix equation

s = Q111 (ks, ki,1) i1 + Q113 (ks, ki,1) i1 ,		(2.79)
where s = [f N , gN ]T	is the vector containing the expansion coe cients of
ν ν

the scattered field. Combining (2.78) and (2.79) we see that the transition matrix relating the scattered field coe cients to the incident field coe cients, s = T e, is given by

−1

T = − Q111 (ks, ki,1) + Q131 (ks, ki,1) T 2 Q311 (ks, ki,1) + Q331 (ks, ki,1)T 2 ,

where	(2.80)
T 2 = S12tr T 2S21rt .	(2.81)

For a homogeneous particle T 2 = 0, and we obtain the result established in Sect. 2.1

T = −Q111 (ks, ki) Q311 (ks, ki) −1 .

The expression of the transition matrix can also be written as

T =	T 1		−	Q13		(ks, ki,1) T	2	Q31		(ks, ki,1) −1
				1					1
					33	(ks, ki,1)T		Q	31	(ks, ki,1) −	1	−1	(2.82)
	×	I + Q			1		2		1			,

where

T 1 = −Q111 (ks, ki) Q311 (ks, ki) −1

is the transition matrix of the host particle. If the coordinate systems O1x1y1z1 and O2x2y2z2 coincide, (2.82) is identical to the result given by Peterson and

112 2 Null-Field Method

Str¨om [189]. The various terms obtained by a formal expansion of the inverse in (2.82) can be interpreted as various multiple-scattering contributions to the total transition matrix. Indeed, using the representation

T= T 1 − Q131 T 2 Q311 −1 I − Q331 T 2 Q311 −1 + ...

=T 1 − Q131 T 2 Q311 −1 − T 1Q331 T 2 Q311 −1 +Q131 T 2 Q311 −1 Q331 T 2 Q311 −1 + . . .

we see that the term T 1 represents a reflection at S1, Q131 T 2(Q311 )−1 represents a passage of a wave through S1 and a reflection at S2, T 1Q331 T 2(Q311 )−1 represents a refraction of a wave through S1 and two consecutive reflections at S2 and S1, etc.

The expressions of the total transition matrix given by (2.80) or (2.82) are important in practical applications. As it has been shown by Peterson and Str¨om [189], this result can be extended to the case of S1 containing an arbitrary number of separate enclosures by simply replacing T 2 with the system transition matrix of the particles. In the later sections we will derive the transition matrix for a system of particles and the present formalism will enable us to analyze scattering by an arbitrarily shaped, inhomogeneous particle with an arbitrary number of irregular inclusions. In this context it should be mentioned that the separation of variables solution for a single sphere (the Lorenz–Mie theory) can be extended to spheres with one or more eccentrically positioned spherical inclusions by using the translation addition theorem for vector spherical wave functions. Theories of scattering by eccentrically stratified spheres have been derived by Fikioris and Uzunoglu [64], Borghese et al. [22], Fuller [73], Mackowski and Jones [152] and Ngo et al. [180], while treatments for a sphere with multiple spherical inclusions have been rendered by Borghese et al. [20], Fuller [75] and Ioannidou and Chrissoulidis [107]. A detailed review of the separation of variable method for inhomogeneous spheres has been given by Fuller and Mackowski [77]. The separation of variable method has also been employed in spheroidal coordinate systems by Cooray and Ciric [41] and Li et al. [141] to analyze the scattering by inhomogeneous spheroids.

2.4.2 Formulation without Addition Theorem

In our previous analysis we assumed the geometric constraint r1 > r12 (cf. (2.77)), which originates in the use of the translation addition theorem for radiating vector spherical wave functions. In this section we present a formalism that avoids the use of any local origin translation. To simplify our analysis, we assume that the coordinate systems O1x1y1z1 and O2x2y2z2 have the

same spatial orientation and set α = β = γ = 0. We begin by defining the Qpqt (k1, i, k2, j) matrix

2.4 Inhomogeneous Particles

113

	(Qpq )11			(Qpq )12
Qtpq	(k1, i, k2, j) =	t	νµ		t		νµ	,	(2.83)
		pq	21	(Q	pq	)	22
		(Q )	νµ		t		νµ
		t

as

(Qpqt )11νµ =

(Qpqt )12νµ =

(Qpqt )21νµ =

and

(Qpqt )22νµ =

jk12

n

r

M q k2r

·

N

p

(k1r )

π

St

j

×

µ

j

ν

i

"

ε2

q

p

+

n r

×

N

k2r

M

(k1r )

dS (r ) , (2.84)

ε1

j

µ

j

·

ν

i

jk12

n

r

N q k2r

N

p

(k1r )

π

St

j

×

µ j

·

ν

i

(k1r )" dS (r ) , (2.85)

+

ε2

n r

×

M q k2r

M

p

ε1

j

µ

j

·

ν

i

jk12

n r

M q k2r

·

M

p

(k1r )

π

St

j

×

µ

j

ν

i

(k1r )" dS (r ) , (2.86)

+

ε2

n r

×

N q k2r

N

p

ε1

j

µ

j

·

ν

i

jk12

n

r

N q k2r

·

M

p

(k1r )

π

St

j

×

µ

j

ν

i

"

ε2

q

p

+

n r

×

M

k2r

N

(k1r )

dS (r ) . (2.87)

ε1

j

µ

j

·

ν

i

The vectors r , ri and rj are the position vectors of a point M on the surface St, and are defined with respect to the origins O, Oi and Oj , respectively (Fig. 2.3). In terms of the new Qpqt matrices, the system of matrix equations (2.75) can be written as

Q133(ks, 1, ki,1, 1)i1 + Q131(ks, 1, ki,1, 1)i1 = −e ,
−i1 + Q211(ki,1, 1, ki,2, 2)i2 = 0 ,
−Q131(ki,1, 2, ki,1, 1)i1 + Q231(ki,1, 2, ki,2, 2)i2 = 0 ,	(2.88)
while the matrix equation (2.79) reads as
s = Q113 (ks, 1, ki,1, 1) i1 + Q111 (ks, 1, ki,1, 1) i1 .	(2.89)

Using the substitution method we eliminate the unknown vector i2 from the last two matrix equations in (2.88) and obtain

114 2 Null-Field Method

	M
ri	r	St
	j
	r
Oi		Oj
	O

Fig. 2.3. Position vectors of a point M on the surface St O, Oi and Oj

i1 = T 2i1

with

with respect to the origins

(2.90)

T	2	= Q11	(ki,1, 1, ki,2, 2)	Q31	(ki,1, 2, ki,2, 2) −1	Q31	(ki,1, 2, ki,1, 1) . (2.91)
		2		2		1

Equation (2.89) and the first matrix equation in (2.88), written in compact matrix notation as

	s			1 (ks, ki,1)	i				(2.92)
	e	=	Q		1	,
					i1
where
Q1 (ks, ki,1) =	Q113 (ks, 1, ki,1, 1) Q111 (ks, 1, ki,1, 1)							,	(2.93)
	−Q133(ks, 1, ki,1, 1) −Q131(ks, 1, ki,1, 1)
then yield
T = − Q111 (ks, 1, ki,1, 1) + Q113 (ks, 1, ki,1, 1) T 2
× Q131(ks, 1, ki,1, 1) + Q133(ks, 1, ki,1, 1)T 2							−1 .		(2.94)

The expression of the transition matrix is identical to that given by (2.80), but the T 2 matrix is now given by (2.91) instead of (2.81). If the origins O1 and O2 coincide

Q311 (ki,1, 1, ki,1, 1) = −I

and we see that the T 2 matrix is the transition matrix of the inhomogeneity:

T 2 = T 2 = −Q112 (ki,1, 1, ki,2, 1) Q312 (ki,1, 1, ki,2, 1) −1 .

This formalism will be used in Sect. 2.5 to derive a recurrence relation for the transition matrix of a multilayered particle.

2.5 Layered Particles

115

2.5 Layered Particles

A layered particle is an inhomogeneous particle consisting of several consecutively enclosing surfaces Sl, l = 1, 2, . . . , N. Each surface Sl is defined with respect to a coordinate system Olxlylzl and we assume that the coordinate systems Olxlylzl have the same spatial orientation. The layered particle is immersed in a medium with optical constants εs and µs, while the relative media constants and the wave number in the domain between Sl and Sl+1 are εi,l, µi,l and ki,l, respectively. The geometry of a (multi)layered particle is shown in Fig. 2.4. The case N = 2 has been treated in the previous section and the objective of the present analysis is to extend the results established for two-layered particles to multilayered particles.

2.5.1 General Formulation

For a particle with N layers, the system of matrix equations consists in the null-field equations in the interior of S1,

Q133(ks, 1, ki,1, 1)i1 + Q131(ks, 1, ki,1, 1)i1 = −e ,						(2.95)
the null-field equations in the exterior of Sl−1 and the interior of Sl
−il−1 + Ql13(ki,l−1, l − 1, ki,l, l)il
+Ql11(ki,l−1, l − 1, ki,l, l)il = 0 ,						(2.96)
−Ql31−1(ki,l−1, l, ki,l−1, l − 1)il−1 + Ql33(ki,l−1, l, ki,l, l)il
+Q31	(k	, l, k	, l)i	l	= 0	(2.97)
l	i,l−1	i,l

l = 2, 3, . . . , N − 1

the null-field equations in the exterior of SN−1 and the interior of SN

Ds

		S1
	SN	Sl
		Ol
	Ol
	O1	Ol-1
	ON	Di,l Sl

Di,l-1 Sl-1

Di,l-2

Fig. 2.4. Geometry of a multilayered particle

116 2 Null-Field Method

−iN−1 + QN11(ki,N−1, N − 1, ki,N , N)iN = 0 ,

(2.98)

−Q31N−1(ki,N−1, N, ki,N−1, N − 1)iN−1 + Q31N (ki,N−1, N, ki,N , N)iN = 0

(2.99) and the matrix equation corresponding to the scattered field representation

s = Q113 (ks, 1, ki,1, 1) i1 + Q111 (ks, 1, ki,1, 1) i1 .

(2.100)

For two consecutive layers, the surface fields il−1 and il−1 are related to the surface fields il and il by the matrix equation

i

=

(ki,l

1, ki,l)

i

(2.101)

l−1

Q

l

−

l

,

il−1

il

where

Ql (ki,l−1, ki,l) =

I

0

(2.102)

0

Q31

(ki,l

−

1, l, ki,l

−

1, l

−

1) −1

l−1

Ql13(ki,l−1, l − 1, ki,l, l) Ql11(ki,l−1, l − 1, ki,l, l)

×

Ql33(ki,l−1, l, ki,l, l)

Ql31(ki,l−1, l, ki,l, l)

and l = 2, 3, . . . , N − 1. For the surface fields iN−1and iN−1, that is, for l = N, we have

with

iN−1 = T N iN−1

(2.103)

T

= Q11(ki,

N−

1

,

N −

1, ki,

N

,

N

) Q31(ki,

1,

N

, ki,

N

,

N

) −1

N

N

N

N−

×QN31−1(ki,N−1, N, ki,N−1, N − 1) .

(2.104)

Then, using the matrix equation

s

=

1 (ks, ki,1)

i

e

Q

1

i1

with Q1 being given by (2.93), we see that

s

=

i

N−1

,

e

Q iN−1

where

Q = Q1 (ks, ki,1) Q2 (ki,1, ki,2) . . . QN−1 (ki,N−2, ki,N−1) .

Finally, using (2.103) and denoting by (Q)ij , i, j = 1, 2, the block-matrix components of Q, we obtain the expression of the transition matrix in terms of Q and T N :

2.5 Layered Particles

117

−1

T = (Q)12 + (Q)11 T N (Q)22 + (Q)21 T N .

The structure of the above equations is such that a recurrence relation for computing the transition matrix can be established. For this purpose, we define the matrix T l+1,l+2,...,N as

	il = T l+1,l+2,...,N il .
and use (2.101) to obtain
il−1	= (Ql)12 + (Ql)11 T l+1,l+2,...,N il ,
il−1	= (Ql)22 + (Ql)21 T l+1,l+2,...,N il ,

where (Ql)ij , i, j = 1, 2, are the block-matrix components of Ql(ki,l−1, ki,l). Hence, the matrix T l,l+1,...,N , satisfying il−1 = T l,l+1,...,N il−1, can be computed by using the downward recurrence relation

T l,l+1,...,N = (Ql)12 + (Ql)11 T l+1,l+2,...,N
× (Ql)22 + (Ql)21 T l+1,l+2,...,N −1	(2.105)

for l = N − 1, N − 2, . . . , 1. For l = N − 1, T N is given by (2.104), while for

l = 1, Ql is the matrix Q1(ks, ki,1) and T l,l+1,...,N is the transition matrix of the layered particle

T = T 1,2,...,N = − Q111 (ks, 1, ki,1, 1) + Q131 (ks, 1, ki,1, 1) T 2,3,...,N

−1

× Q311 (ks, 1, ki,1, 1) + Q331 (ks, 1, ki,1, 1)T 2,3,...,N .

If the origins coincide, the above relations simplify considerably, since

Ql13(ki,l−1, 1, ki,l, 1) Ql11(ki,l−1, 1, ki,l, 1)

Ql (ki,l−1, ki,l) =

−Ql33(ki,l−1,

1, ki,l, 1) −Ql31(ki,l−1, 1, ki,l, 1)

(2.106)

and

T

= T

N

=

−

Q11

(ki,

N−

1, 1, ki,

N

, 1)

Q31(ki,

N−

1, 1, ki,

N

, 1) −1 .

N

N

N

We obtain

Tl,l+1,...,N = − Q11l (ki,l−1, 1, ki,l, 1) + Q13l (ki,l−1, 1, ki,l, 1)T l+1,l+2,...,N

×Q31l (ki,l−1, 1, ki,l, 1) + Q33l (ki,l−1, 1, ki,l, 1)T l+1,l+2,...,N −1

(2.107)

l,l+1,...,N

118 2 Null-Field Method and further

T l,l+1,...,N = T l − Q13l (ki,l−1, 1, ki,l, 1)T

× Q31l (ki,l−1, 1, ki,l, 1) −1

l+1,l+2,...,N

I + Q33l (ki,l−1, 1, ki,l, 1)

×	T l+1,l+2,...,	Q31		(ki,l	−	1, 1, ki,l, 1) −1 −1	(2.108)
		N	l

of which (2.82) is the simplest special case. Note that in (2.107) and (2.108), T is the total transition matrix of the layered particle with outer surface Sl.

2.5.2 Practical Formulation

In practical computer calculations it is simpler to solve the system of matrix equations (2.95)–(2.99) for all unknown vectors il and il, l = 1, 2, . . . , N − 1, and iN . For this purpose, we consider the global matrix

A1

0

0 ...

0

0

A12 A21

0 ...

0

0

0

A23

A32 ...

0

0

A =

,

0

0 ... AN−

1,

N−

2

0

0

0

0

0 ...

AN−1,N

AN

where

A1 = Q133(ks, 1, ki,1, 1) Q131(ks, 1, ki,1, 1) ,

Q13(ki,l

−

1, l

−

1, ki,l, l) Q11(ki,l

−

1, l

−

1, ki,l, l)

Al,l−1 =

l

, l)

l

, l)

Q33

(k

i,l−1

, l, k

i,l

Q31

(k

, l, k

i,l

l

l

i,l−1

0

Al−1,l =

−I

,

0 −Ql31−1(ki,l−1, l, ki,l−1, l − 1)

and

Q11(ki,N−1, N − 1, ki,N , N)

AN = N .

Q31N (ki,N−1, N, ki,N , N)

Then, denoting by A the inverse of A,

(2.109)

,(2.110)

(2.111)

(2.112)

				A11		A12 ...		A1,2N−1
A	= A−1	=		A	21	A	22 ...	A	2,2N−1		,

A2N−1,1 A2N−1,2 ... A2N−1,2N−1