Linear Rayleigh and Raman scattering to the second order:
analytical results for light scattering by any scatterer of size k
0
d ≲ 1/10
Robert P. Cameron
∗
Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom and
School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
Neel Mackinnon
School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
(Dated: July 9, 2018)
We extend the usual multipolar theory of linear Rayleigh and Raman scattering to include the
second-order correction. The new terms promise a wealth of information about the shape of a
scatterer and yet are insensitive to the scatterer’s chirality. Our extended theory might prove
especially useful for analysing samples in which the scatterers have non-trivial shapes but no chiral
preference overall, as the zeroth-order theory offers little information about shape and the first-order
correction is often quenched for such samples. A basic estimate suggests that our extended theory
can be applied to a scatterer as large as k
0
d ∼ 1/10 with less than ∼ 0.1% error resulting from the
neglect of the third- and higher-order corrections. Our results are entirely analytical.
I. INTRODUCTION
Light scattering is an all-pervasive phenomenon. To-
gether with light absorption, it is largely responsible for
the appearance of the material world [1, 2]. Theoretical
understanding of light scattering is sufficiently advanced
to enable determinations of the nature of interstellar dust
[3], radar [4], studies of the structures of viruses [5] and
the measurement of the salinity of seawater [6], to name
but a few applications. There is much still to be explored,
however, and the study of light scattering remains at the
cutting edge of research [7–15].
One can distinguish between different types of light
scattering [1, 16–21]. This paper is concerned with one
of the simplest and most common of these: a two-photon
process in which a quantum of light collides with an elec-
trically neutral scatterer [22]. Nonlinear light scattering
processes, involving three or more photons, are also pos-
sible and are proving increasingly useful [8, 9, 22]. We
do not consider these here, however: our interest is em-
phatically in linear light scattering.
If the aforementioned collision is elastic, it is usually
referred to as Rayleigh scattering for k
0
d ≪ 1 [16, 23, 24]
or Willis-Tyndall scattering for 110 < k
0
d < 10 [25],
with d a characteristic length of the scatterer and k
0
the
wavenumber of the incident photon. If the collision is
inelastic [19], it is referred to as combination [20] or Ra-
man scattering [21]. For k
0
d ≪ 1, light scattering is of-
ten treated using the zeroth-order Rayleigh theory (or its
extension to Raman scattering), in which multipolar ex-
pansions for the scatterer are truncated at electric-dipole
order. The first-order correction to this theory was in-
troduced relatively recently [26, 27] and has proved ex-
tremely useful in the laboratory for the analysis of chiral
molecules [2, 5, 28–30], as it discriminates between left
∗
robert.p.cameron@strath.ac.uk
and right. For 110 <k
0
d < 10, elastic light scattering by
scatterers with sufficiently simple shapes can be treated
using the mathematical machinery of (analytical) Mie
theory [1, 25]. Numerical approaches are usually used
instead for scatterers with more complicated shapes [3],
although a semi-analytical extension of Mie theory has
recently been put forward [7, 13].
In this paper we introduce the second-order correction
to the zeroth-order Rayleigh / Raman theory. The new
terms promise a wealth of information about the shape
of a scatterer and yet are insensitive to the scatterer’s
chirality (left versus right) if the scatterer happens to be
chiral. Our extended theory might prove especially useful
for analysing samples in which the scatterers have non-
trivial shapes but no chiral preference overall. Consider,
for example, a racemic sample of chiral molecules, as
might be produced in a symmetric chemical reaction us-
ing achiral precursors [31]. The zeroth-order theory offers
little information about the shapes of the molecules and
the first-order correction is quenched by virtue of there
being equal numbers of left- and right-handed molecules.
The second-order correction, however, can still be ex-
ploited as an incisive probe of the shapes of the molecules.
Our extended theory might also help bridge the perceived
divide between small scatterers (k
0
d ≪ 1) and medium-
sized scatterers (110 < k
0
d < 10): a basic estimate sug-
gests that the zeroth-order theory together with its first-
and second-order corrections can be applied to a scat-
terer as large as k
0
d ∼ 110, with less than ∼ 0.1% error
resulting from the neglect of the third- and higher-order
corrections. Let us emphasise here that our results are
entirely analytical.
In what follows we imagine ourselves to be in an iner-
tial frame of reference with time t; right-handed Carte-
sian coordinates x, y and z with associated unit vectors
ˆ
x,
ˆ
y and
ˆ
z and spherical coordinates r, θ and φ with as-
sociated unit vectors
ˆ
r,
ˆ
θ
θ
θ and
ˆ
φ
φ
φ. The SI system of units
is adopted and the Einstein summation convention [32]
2
is to be understood, with subscripts a, b, c, . . . running
over x, y and z. Complex quantities are indicated using
tildes, except where otherwise stated.
II. GENERAL CALCULATION
Our aim in this paper is to introduce the second-
order correction to the zeroth-order Rayleigh / Raman
theory as simply as possible. We focus our attention,
therefore, upon a semiclassical model in which a single
scatterer is illuminated by weak, quasi-monochromatic
light that has been ‘switched on’ slowly at some distant
time in the past. The scatterer could represent a small
molecule in vacuum [2], for example. We make no spe-
cific assumptions about the scatterer, except that it is
smaller than around one-tenth of the wavelength of the
illuminating light and is localised near the spatial ori-
gin (x = y = z = 0), so that we can perform converging
multipolar expansions about the spatial origin. The re-
lationship of the ‘local multipole approach’ [26, 27] used
in this paper to the ‘distributed dipole approach’ [3] is
examined in [33].
The electric and magnetic fields of the illuminating
light at the spatial origin have the following forms:
E ≈ R
˜
Ee
−iω
0
t
(1)
B ≈ R
˜
Be
−iω
0
t
, (2)
with ω
0
= ck
0
the angular frequency of the illuminat-
ing light. The illuminating light induces oscillations in
the charge and current distributions of the scatterer and
these oscillations are themselves the source of electro-
magnetic radiation: scattered light. The electric and
magnetic fields of the scattered light have the following
forms:
e ≈ R
˜
ee
−iωt
(3)
b ≈ R
˜
be
−iωt
, (4)
with ω = ck the angular frequency of the scattered light.
This is equal to ω
0
for Rayleigh scattering, or ω
0
−ω
fi
for
a Raman-scattering transition f ←i, where ω
fi
is the an-
gular frequency of the transition. In this paper we use a
parameter λ =1 to help us keep track of order in our mul-
tipolar expansions. The powers of λ quoted by us have
their origins in Taylor expansions. Each term in one of
these expansions has an additional spatial derivative (of
the illuminating light or scattered light) and length scale
(the position of some constituent of the scatterer relative
to the chosen origin of multipolar expansion) relative to
the term before it. We thus associate each power of λ
with a factor of ∼ (k
0
d) ∼ (kd) ≲ (110), where d is a
characteristic length of the scatterer, as above. Far from
the scatterer (kr ≫1),
˜e
a
≈
µ
0
ω
2
e
ikr
4πr
(δ
ab
−ˆr
a
ˆr
b
)
λ
0
˜µ
b
+λ
1
1
c
bcd
˜
M
c
ˆr
d
−λ
1
ik
3
˜
Θ
bc
ˆr
c
−λ
2
ik
2c
bcd
˜
M
ce
ˆr
e
ˆr
d
−λ
2
k
2
6
˜
Q
bcd
ˆr
c
ˆr
d
(5)
˜
b
a
≈
1
c
abc
ˆr
b
˜e
c
, (6)
where ˜µ
a
,
˜
Θ
ab
and
˜
Q
abc
are the induced electric-dipole,
electric-quadrupole and electric-octupole moments of the
scatterer, and
˜
M
a
and
˜
M
ab
are the induced magnetic-
dipole and magnetic-quadrupole moments. These re-
sults, (5) and (6), constitute an extension of those given
in [2] to include terms of order λ
2
and can be regarded
as a special case of the results given in [34], particular to
harmonic oscillations. The induced multipole moments
of the scatterer are related to the illuminating light by
the scatterer’s property tensors:
λ
0
˜µ
a
≈λ
0
˜α
ab
˜
E
b
+λ
1
1
3
˜
A
a,bc
∂
c
˜
E
b
+λ
1
˜
G
ab
˜
B
b
+λ
2
1
6
˜
B
a,bcd
∂
d
∂
c
˜
E
b
+λ
2
1
2
˜
D
(m)
a,bc
∂
c
˜
B
b
, (7)
λ
1
˜
M
a
≈λ
1
˜
G
ab
˜
E
b
+λ
2
1
3
˜
D
a,bc
∂
c
˜
E
b
+λ
2
˜χ
ab
˜
B
b
, (8)
λ
1
˜
Θ
ab
≈λ
1
˜
A
c,ab
˜
E
c
+λ
2
˜
C
ab,cd
∂
d
˜
E
c
+λ
2
˜
D
c,ab
˜
B
c
, (9)
λ
2
˜
M
ab
≈λ
2
˜
D
(m)
c,ab
˜
E
c
(10)
λ
2
˜
Q
abc
≈λ
2
˜
B
d,abc
˜
E
d
. (11)
These results, (7)-(11), also constitute an extension of
those given in [2] to include terms of order λ
2
. Explicit
quantum-mechanical expressions for multipole moments
and property tensors are given in appendix A, where we
also show how the property tensors reduce under certain
special circumstances.
We consider the Stokes parameters s
ξ
(ξ ∈ {0, 1, 2, 3})
of the scattered light, which can be written succinctly as
follows:
s
ξ
=
˜
f
ξab
˜e
a
˜e
∗
b
, (12)
with
˜
f
0ab
=
ˆ
θ
a
ˆ
θ
b
+
ˆ
φ
a
ˆ
φ
b
, (13)
˜
f
1ab
=
ˆ
θ
a
ˆ
θ
b
−
ˆ
φ
a
ˆ
φ
b
, (14)
˜
f
2ab
=−
ˆ
θ
a
ˆ
φ
b
−
ˆ
φ
a
ˆ
θ
b
(15)
˜
f
3ab
=−i
ˆ
θ
a
ˆ
φ
b
+i
ˆ
φ
a
ˆ
θ
b
. (16)
Note that
˜
f
ξab
=
˜
f
∗
ξba
, which ensures that the s
ξ
are real.
Furthermore
˜
f
0ab
,
˜
f
1ab
, and
˜
f
2ab
are symmetric in a and
b and purely real whereas
˜
f
3ab
is antisymmetric in a and
b and purely imaginary.
3
Working to order λ
2
, we substitute (5) with (7)-(11)
into (12) and find that the Stokes parameters of the scat-
tered light take the following form:
s
ξ
≈λ
0
s
(0)
ξ
+λ
1
s
(1)
ξ
+λ
2
s
(2)
ξ
, (17)
with the s
(0)
ξ
, the s
(1)
ξ
and the s
(2)
ξ
as defined below.
The zeroth-order theory is embodied by the following:
s
(0)
ξ
=s
α−α
ξ
, (18)
with
s
α−α
ξ
=KR
1
2
˜α
ab
˜α
∗
cd
˜
f
ξac
˜
E
b
˜
E
∗
d
(19)
and
K =
µ
2
0
ω
4
8π
2
r
2
, (20)
as is well known [2, 22]. The zeroth-order theory is al-
ready sufficient to account in a basic way for the polar-
isation, depolarisation and colour of the light from the
sky [16, 23, 24], for example.
The first-order correction is due to interference be-
tween light waves scattered via the familiar property ten-
sor ‘α’ and light waves scattered via the optical activity
property tensors ‘A’ and ‘G’, as embodied by the follow-
ing:
s
(1)
ξ
=s
α−A
ξ
+s
α−G
ξ
, (21)
with
s
α−A
ξ
=KR
1
3
˜α
ab
˜
A
∗
c,de
˜
f
ξac
˜
E
b
∂
d
˜
E
∗
e
+
ik
3
˜α
ab
˜
A
∗
c,de
˜
f
ξad
˜
E
b
˜
E
∗
c
ˆr
e
(22)
s
α−G
ξ
=KR
˜α
ab
˜
G
∗
cd
˜
f
ξac
˜
E
b
˜
B
∗
d
+
1
c
˜α
ab
˜
G
∗
cd
ecf
˜
f
ξae
˜
E
b
˜
E
∗
d
ˆr
f
, (23)
as is also well known [2, 22]. The first-order correc-
tion accounts for the leading-order contributions to op-
tical activity in an isotropic sample of chiral molecules
[26, 27]. Natural Raman optical activity has been devel-
oped into an incisive spectroscopic tool for chiral scatter-
ers both large and small [2, 5, 28–30]. In contrast, natural
Rayleigh optical activity has been reported for a hand-
ful of large chiral biological structures, including octopus
sperm [35], but has thus far proved elusive for small chi-
ral molecules [2], in spite of potential applications such
as the robust assignment of absolute configuration [36].
The difficulties here might be partially overcome using
structured light [37, 38]. Interestingly, orientated achi-
ral molecules can also exhibit natural optical activity via
the first-order correction, embodied by the s
(1)
ξ
[2], and
partially orientated chiral molecules can exhibit natural
optical activity via the zeroth-order theory, embodied by
the s
(0)
ξ
[39].
We find that the second-order correction is due to mu-
tual interference between light waves scattered via the
optical activity property tensors ‘A’ and ‘G’, together
with equally important contributions due to interference
between light waves scattered via the familiar property
tensor ‘α’ and light waves scattered via the more exotic
property tensors ‘B’, ‘C’, ‘D’, ‘D
m
’ and ‘χ’, as embodied
by the following:
s
(2)
ξ
=s
A−A
ξ
+s
G−A
ξ
+s
G−G
ξ
+s
α−B
ξ
+s
α−C
ξ
+s
α−D
ξ
+s
α−D
m
ξ
+s
α−χ
ξ
, (24)
with
s
A−A
ξ
=KR
1
18
˜
A
a,bc
˜
A
∗
d,ef
˜
f
ad
∂
b
˜
E
c
∂
e
˜
E
∗
f
+
ik
9
˜
A
a,bc
˜
A
∗
d,ef
˜
f
ae
∂
b
˜
E
c
˜
E
∗
d
ˆr
f
+
k
2
18
˜
A
a,bc
˜
A
∗
d,ef
˜
E
a
˜
f
ξbe
ˆr
c
˜
E
∗
d
ˆr
f
, (25)
s
G−A
ξ
=KR
1
3
˜
G
ab
˜
A
∗
c,de
˜
f
ξac
˜
B
b
∂
d
˜
E
∗
e
+
1
3c
˜
G
ab
˜
A
∗
c,de
fag
ˆr
g
˜
E
b
∂
d
˜
E
∗
e
˜
f
ξf c
+
ik
3
˜
G
ab
˜
A
∗
c,de
˜
f
ξad
˜
B
b
˜
E
∗
c
ˆr
e
+
ik
3c
˜
G
ab
˜
A
∗
c,de
fag
˜
f
ξf d
ˆr
g
˜
E
b
˜
E
∗
c
ˆr
e
(26)
s
G−G
ξ
=KR
1
2
˜
G
ab
˜
G
∗
cd
˜
f
ξac
˜
B
b
˜
B
∗
d
+
1
c
˜
G
ab
˜
G
∗
cd
˜
f
ξae
ecf
˜
B
b
˜
E
∗
d
ˆr
f
+
1
2c
2
˜
G
ab
˜
G
∗
cd
aef
cgh
˜
f
ξeg
˜
E
b
˜
E
∗
d
ˆr
f
ˆr
h
(27)
4
the optical activity cross terms and
s
α−B
ξ
=KR
1
6
˜α
ab
˜
B
∗
c,def
˜
f
ξac
˜
E
b
∂
d
∂
e
˜
E
∗
f
−
k
2
6
˜α
ab
˜
B
∗
c,def
˜
f
ξad
˜
E
b
˜
E
∗
c
ˆr
e
ˆr
f
, (28)
s
α−C
ξ
=KR
ik
3
˜α
ab
˜
C
∗
cd,ef
˜
f
ξac
˜
E
b
ˆr
d
∂
e
˜
E
∗
f
, (29)
s
α−D
ξ
=KR
ik
3
˜α
ab
˜
D
∗
c,de
˜
f
ξad
˜
E
b
˜
B
∗
c
ˆr
e
+
1
3c
˜α
ab
˜
D
∗
c,de
˜
f
ξaf
˜
E
b
∂
d
˜
E
∗
e
fcg
ˆr
g
, (30)
s
α−D
m
ξ
=KR
1
2
˜α
ab
˜
D
(m)∗
c,de
˜
f
ξac
˜
E
b
∂
e
˜
B
∗
d
+
ik
2c
˜α
ab
˜
D
(m)∗
c,de
˜
f
ξaf
˜
E
b
˜
E
∗
c
fdg
ˆr
e
ˆr
g
(31)
s
α−χ
ξ
=KR
1
c
˜α
ab
˜χ
∗
cd
˜
f
ξae
˜
E
b
˜
B
∗
d
ecf
ˆr
f
(32)
the exotic interference terms. The second-order correc-
tion does not appear to have been described explicitly
before and is the central result of this paper. Accounted
for by the s
(2)
ξ
are the “terms in G
2
and A
2
” alluded
to in [2]. Light scattering to second order has also been
touched upon in [33], where the possibility of new rota-
tional Raman lines with zero background is highlighted.
It is important to note that each of the optical activ-
ity property tensors ‘A’ and ‘G’ and each of the exotic
property tensors ‘B’, ‘C’, ‘D’, ‘D
m
’ and ‘χ’ is implic-
itly dependent upon our choice of origin for multipolar
expansions: they differ when calculated about different
origins. In appendix B we show that our physical predic-
tions (based upon the complete Stokes parameters s
ξ
of
the scattered light, with all terms of order λ
0
, λ
1
and λ
2
considered simultaneously) are nevertheless independent
of our choice of origin for multipolar expansions, as they
should be.
As mentioned earlier, a basic estimate based on the
orders of the Taylor expansions reveals that s
(0)
ξ
∼
(k
0
d)
−1
s
(1)
ξ
∼(k
0
d)
−2
s
(2)
ξ
. . . . This suggests that even for
a scatterer with k
0
d ∼ 110, the second-order correction
will yield a modification of only ∼1% to the zeroth-order
theory. That is to say, exploitation of the second-order
correction in the laboratory will demand precision mea-
surements. This estimate also implies that neglect of the
third- and higher-order corrections gives rise to ≲ 0.1%
error for k
0
d ≲ 110, which suggests that the zeroth-order
theory together with its first- and second-order correc-
tions might serve as a precise alternative to numerical
approaches for k
0
d ≲ 110.
To better appreciate the validity of such estimates,
note first the well-established fact [2, 5, 28–30, 36] that
the first-order correction, which has contributions of the
form ‘α × A’ and ‘α × G’, is typically smaller than the
zeroth-order theory, which has contributions of the form
‘α ×α’, by the predicted factor of ∼(k
0
d). It follows im-
mediately (from ‘A ∼(k
0
d)α’ and ‘G ∼(k
0
d)α’) that the
optical activity cross terms, which have contributions of
the form ‘A ×A’, ‘A ×G’ and ‘G ×G’, will typically be
smaller than the first-order correction by the same factor
of ∼(k
0
d). One expects the order of magnitude of the ex-
otic interference terms to be similar because a change in
the choice of origin for mutlipolar expansion intermixes
these with the optical activity cross terms as shown in ap-
pendix B. Preliminary calculations performed by us us-
ing a dynamic coupling model [40] support these claims.
The results we have given thus far are rather general.
They might be applied to Rayleigh or Raman scatter-
ing, on or off resonance, for any scatterer in any orien-
tation. Moreover, they can be extended to account for
the presence of static fields, by considering distortions of
the property tensors [2, 22]. Let us also emphasise that
our results can be applied for different forms of (quasi-
monochromatic) illuminating light: plane-wave illumina-
tion, considered below, is but one possibility. Illumina-
tion by more exotic forms of light could open the door
to new possibilities, one example of which is highlighted
in section IV. Evanescent fields play important roles in
scattering-type near-field optical microscopy techniques
[10] and it has recently been shown that illumination by
standing waves yields new possibilities for optical activ-
ity [37, 38], to give two more examples of non-plane-wave
illumination in light scattering.
III. PLANE-WAVE ILLUMINATION AND
ROTATIONAL AVERAGING
A scattering experiment often involves a Gaussian
beam of light illuminating a fluid sample. The Stokes
parameters s
ξ
of the scattered light are measured as a
function of the Stokes parameters S
ξ
(ξ ∈ {0, 1, 2, 3}) of
the illuminating light and the scattering angle θ. With
such a setup in mind, we now consider the following spe-
cific example.
With regards to the illuminating light, we consider a
plane wave propagating in the +z direction:
˜
E =
˜
E
0x
ˆ
x +
˜
E
0y
ˆ
y
(33)
˜
B =
1
c
−
˜
E
0y
ˆ
x +
˜
E
0x
ˆ
y
. (34)
We define the Stokes parameters S
ξ
of the illuminating
light as follows:
S
0
=
˜
E
0x
˜
E
∗
0x
+
˜
E
0y
˜
E
∗
0y
, (35)
S
1
=
˜
E
0x
˜
E
∗
0x
−
˜
E
0y
˜
E
∗
0y
, (36)
S
2
=−
˜
E
0x
˜
E
∗
0y
+
˜
E
0y
˜
E
∗
0x
(37)
S
3
=−i
˜
E
0x
˜
E
∗
0y
−
˜
E
0y
˜
E
∗
0x
. (38)
5
We also average over all possible orientations of the scat-
terer, with this average denoted using angular brackets.
With regards to the observation geometry, we choose
φ = π2 which restricts us to the y > 0 region of the
y −z plane, with
ˆ
r = sin θ
ˆ
y +cos θ
ˆ
z, (39)
ˆ
θ
θ
θ = cos θ
ˆ
y −sin θ
ˆ
z (40)
ˆ
φ
φ
φ = −
ˆ
x. (41)
Note that this choice does not limit the generality of the
results below.
Writing down the rotationally averaged Stokes param-
eters s
ξ
of the scattered light in terms of the Stokes
parameters S
ξ
of the incident light involves calculating
the rotational averages of (25)-(32) and using (33)-(41).
Some of the terms in the second-order correction, em-
bodied here by the s
(2)
ξ
, have the same dependences on
the S
ξ
and the scattering angle θ as terms in the zeroth-
order theory, embodied here by the s
(0)
ξ
. We therefore
consider the s
(0)
ξ
and the s
(2)
ξ
simultaneously. The
general results are listed in appendix C. For the special
case of Rayleigh scattering of far-off-resonance light by a
time-reversible scatterer (see appendix A 3), they reduce
to the following forms:
s
(0)
0
+s
(2)
0
= KS
0
A +cos θB
′
+cos
2
θC +cos
3
θD
′′
+S
1
sin
2
θ (E +cos θF
′′
), (42)
s
(0)
1
+s
(2)
1
= KS
1
G +cos θH
′
+cos
2
θI +cos
3
θJ
′′
+S
0
sin
2
θ (K +cos θL
′′
), (43)
s
(0)
2
+s
(2)
2
= KS
2
M
′
+cos θN +cos
2
θO
′
(44)
s
(0)
3
+s
(2)
3
= KS
3
P
′
+cos θQ +cos
2
θR
′
. (45)
Explicit expressions for the coefficients A, . . . , R
′
are listed
in appendix D. Note that s
(0)
0
+s
(2)
0
and s
(0)
1
+s
(2)
1
are independent of S
2
and S
3
and that s
(0)
2
+s
(2)
2
and
s
(0)
3
+s
(2)
3
are independent of S
0
and S
1
. Furthemore
for θ =0 and θ =π we obtain s
(0)
0
+s
(2)
0
∝ S
0
, s
(0)
1
+
s
(2)
1
∝ S
1
, s
(0)
2
+s
(2)
2
∝ S
2
and s
(0)
3
+s
(2)
3
∝ S
3
.
The coefficients A, . . . , R
′
can be grouped into three
types by their dependence on different subsets of the
property tensors. The unprimed coefficients (A, C, E, G, I,
K, N and Q) each have contributions of the following types:
‘α −α’, ‘A −A’, ‘G −A’, ‘G −G’, ‘α −B’ and ‘α −D
m
’.
The singly-primed coefficients (B
′
, H
′
, M
′
, O
′
, P
′
and R
′
)
each have contributions of the following types: ‘A −A’,
‘G −A’, ‘G −G’, ‘α −C’, ‘α −D’ and ‘α −χ’. Note that
there are no contributions of the ‘α −α’ type here: the
singly primed coefficients are of pure second-order char-
acter. Finally, the doubly-primed coefficients (D
′′
, F
′′
, J
′′
and L
′′
) each have contributions of the types ‘A −A’ and
‘α −C’. Note that the doubly primed coefficients are also
of pure second-order character.
The first-order correction, embodied here by the s
(1)
ξ
,
is of a rather different character to the zeroth-order the-
ory and its second-order correction, embodied here by
the s
(0)
ξ
and the s
(2)
ξ
. In particular, it has different,
optically active dependencies upon the Stokes parame-
ters S
ξ
of the incident light. In contrast, each of the
coefficients A, . . . , R
′
is unchanged when the scatterer is
inverted through the spatial origin. That is to say, the
rotationally averaged zeroth-order theory and its second-
order correction are independent of the scatterer’s chiral-
ity: they don’t discriminate between left and right. For
a particularly clear discussion of the s
(1)
ξ
, see [41].
IV. OUTLOOK
In this paper we have focussed upon a single scatterer.
Our results are most relevant to elastic light scattering
in samples for which the scatterers can be regarded as
independent and in which multiple scattering is not im-
portant (a rarefied medium such as an ideal gas being the
prototypical example) and to inelastic light scattering at
essentially all sample densities, again provided that mul-
tiple scattering is not important [1, 2]. It remains to in-
corporate our results into more realistic, sample-specific
theories, where the motions of the scatterers, local field
corrections and other subtleties are taken into account.
This is especially important for small scatterers and / or
long wavelengths, as the second-order correction will be
especially small in such cases.
An obvious next step is to explore potential applica-
tions. A group-theoretical analysis of the coefficients
A, . . . , R
′
could prove useful here, as it might facilitate a
better understanding of their dependence upon the shape
and other properties of a scatterer (we already know that
A, . . . , R
′
are independent of a scatterer’s chirality, for ex-
ample). One might hope to find a measurable combina-
tion of A, . . . , R
′
that distinguishes between chirality and
achirality to directly probe the chirality of scatterers in
racemic mixtures, or a combination that is uniquely sen-
sitive to icosahedral scatterers for the purposes of virus
detection, for example. It is also necessary to identify ex-
perimental arrangements optimised towards the second-
order correction, as the signatures of interest will invari-
ably be small. Spatially structured light could prove use-
ful here. Consider the rotational average of our results
for a scatterer located in the node of a linearly polarised
standing wave, for example: there is no scattering to
zeroth-order (as E =0) or first-order (as the illuminating
light is achiral [37, 38]) and the second-order correction
describes the scattered light to leading order.
It is natural, perhaps, to enquire about the third-order
correction to the zeroth-order Rayleigh / Raman theory,
although a basic estimate reveals that this will be smaller
still than the second-order correction by a factor of ∼k
0
d.
It seems that some progress in this direction has already
been made, however: we believe that the novel diamag-