On the dragging of light by a rotating
medium
By Jörg B. Götte1, Stephen M. Barnett1 and Miles Padgett2
1Department of Physics, University of Strathclyde, SUPA, John Anderson
Building, Glasgow G4 0NG, UK
2Department of Physics and Astronomy, University of Glasgow, SUPA, Kelvin
Building, Glasgow G12 8QQ, UK
When light is passing through a rotating medium the optical polarisation is rotated.
Recently it has been reasoned that this rotation applies also to the transmitted
image (Padgett et al. 2006). We examine these two phenomena by extending an
analysis of Player (1976) to general electromagnetic fields. We find that in this
more general case the wave equation inside the rotating medium has to be amended
by a term which is connected to the orbital angular momentum of the light. We
show that optical spin and orbital angular momentum account respectively for the
rotation of the polarisation and the rotation of the transmitted image.
Keywords: image rotation, polarisation, rotating dielectric, specific rotary
power
1. Introduction
Jones (1976) studied the propagation of light in a moving dielectric and showed
by experiment that a rotating medium induces a rotation of the polarisation of
the transmitted light. Player (1976) confirmed that this observation could be ac-
counted for through an application of Maxwells equations in a moving medium.
More recently Padgett et al. (2006) reasoned that the rotation of the medium turns
a transmitted image by the same angle as the polarisation. This is in contrast to the
Faraday effect (Faraday 1846), where a static magnetic field in a dielectric medium,
parallel to the propagation of light, causes a rotation of the polarisation but not
a rotation of a transmitted image. Rotation of the plane of polarisation and im-
age rotation in a rotating medium may be attributed respectively to the spin and
orbital angular momentum of light (Allen et al. 1999, 2003).
The first theoretical treatment of this problem was published by Fermi (1923),
who considered plane waves and a non-dispersive medium. The theoretical ana-
lysis of Player (1976) was also restricted to the propagation of plane waves, but
took the dispersion of the medium into account. Player assumed that the dielectric
response does not depend on the motion of the medium. In our treatment we
follow his assumption although a more careful analysis by Nienhuis et al. (1992)
showed that there will be an effect of the motion on the refractive index for a
dispersive medium near to an absorption resonance (see also Baranova & Zel’dovich
(1979) for a discussion on the effect of the Coriolis force on the refractive index).
In contrast to Player we allow for more general electromagnetic fields that can
carry orbital angular momentum (OAM). This leads to an additional term in our
Article submitted to Royal Society TEX Paper
http://arxiv.org/abs/0704.0725v1
2 J. B. Götte, S. M. Barnett and M. Padgett
wave equation, which corresponds to a Fresnel drag term familiar from analysis of
uniform motion. For a rotating medium, however, this drag leads to a rotational
shift of the image. The propagation of light in a rotating medium thus involves
both spin angular momentum (SAM) and OAM. We solve the wave equation for
circularly polarised Bessel beams and consider two different superpositions of such
Bessel beams to quantify the effects of both polarisation and image rotation. For
rotation of the polarisation we examine a superposition of left- and right-circularly
polarised Bessel beams carrying the same amount of OAM. For image rotation we
consider a superposition of Bessel beams with the same circular polarisation but
opposite OAM values. Such a superposition creates an intensity pattern with lobes
or ‘petals’. In both cases the constituent Bessel beams propagate differently in the
medium, which leads to a change in their relative phase. This is the origin of the
rotation of both the polarisation and the transmitted image. For both phenomena
we derive an expression for the angle per unit length of dielectric through which
the image or the polarisation is rotated.
The significance of the total angular momentum can be most easily seen in the
wave equation for the propagation of light in a rotating medium. We derive this
wave equation in section 2. In the remaining sections we calculate the rotation of
polarisation (section 3) and the image rotation (section 4) and reveal their common
form.
2. Wave equations
The wave equation for a general electric displacment D in a rigid dielectric medium
rotating with angular velocity Ω is given by:
−∇2D = −ǫ(ω′)D̈+ 2[ǫ(ω′)− 1]
Ω× Ḋ− (v · ∇)Ḋ
. (2.1)
An analoguous wave equation can be derived for the magnetic induction B. Com-
pared to the form derived by Player (1976), who considered the special case of a
plane wave propagating along the direction of Ω, these wave equations contain an
additional term 2[ǫ(ω′)− 1](v · ∇)Ḋ. This term is responsible for the Fresnel drag
effect which modifies the speed of light in a moving medium (McCrea 1954; Bar-
ton 1999; Rindler 2001). In the following we will derive this wave equation for the
electric displacement.
Our analysis starts with the same considerations as Player (1976), by intro-
ducing a rest frame and a moving frame. In the rest frame the dielectric medium
rotates with an angular velocity v = Ω× r and in the moving frame the medium is
at rest. We restrict our analysis to small velocities with v ≪ c and use Maxwell’s
equations in both reference frames (Landau & Lifshitz 1975). For the medium at
rest we assume the following constitutive relations:
′ = ǫ(ω′)E′, (2.2a)
′ = H′, (2.2b)
where we have used primes to denote the fields and their frequency ω′ in the moving
frame. The fields in the moving frame can be expressed in the rest frame by a Lorentz
Article submitted to Royal Society
On the dragging of light by a rotating medium 3
transformation (Stratton 1941; Jackson 1998), which gives to first order in v/c:
′ = D+ v ×H, (2.3a)
′ = B− v ×E, (2.3b)
′ = E+ v ×B, (2.3c)
′ = H− v ×D, (2.3d)
where we have set c = 1 and work with units in which ǫ0 = µ0 = 1. The two
constitutive relation in (2.2) in the rest frame are thus given by
D+ v ×H = ǫ(ω′) (E+ v ×B) , (2.4a)
B− v ×E = H− v ×D. (2.4b)
The dielectric constant is still given as a function of the frequency in the moving
frame. We also assume that the dielectric constant depends only on the frequency
and is otherwise independent of the state of motion of the medium. On combining
these two equations we can express D and B with the two other fields E and H to
the first order in v:
D = ǫ(ω′)E+ [ǫ(ω′)− 1]v×H, (2.5a)
B = H− [ǫ(ω′)− 1]v×E. (2.5b)
After taking the curl of (2.5a) we can use the Maxwell equation ∇×E = −Ḃ and
express Ḃ, with the help of (2.5b), in terms of Ḣ and Ė. If we assume v to be
constant (see Appendix A), as in Player’s paper (Player, 1976) this yields
∇×D = −ǫ(ω′)Ḣ+ ǫ(ω′)[ǫ(ω′)− 1]v × Ė+ [ǫ(ω′)− 1]∇× (v ×H). (2.6)
It follows from (2.5a) that ǫ(ω′)v × E = v ×D, to the first order in v, and so we
can rewrite (2.6) as:
∇×D = −ǫ(ω′)Ḣ+ [ǫ(ω′)− 1]v × Ḋ+ [ǫ(ω′)− 1]∇× (v ×H). (2.7)
We can now take the curl of (2.7) to obtain a wave equation for D, as ∇×∇×D =
−∇2D for ∇·D = 0, and the curl of Ḣ is given by ∇× Ḣ = D̈. In order to express
the curl of the vector products we use the identity∇×(a×b) = ∂ibia−∂iaib, where
the doubly occurring index denotes a summation over the Cartesian components.
The operator ∂i represents differentiation with respect to the ith component and
acts on the whole product which gives rise to terms containing the divergences of
v,D and H. These terms are either zero, because ∇ · v = 0 and ∇ ·D = 0 or they
lead to terms which are of second order in v and therefore negligible. The wave
equation for D is thus given by
−∇2D = −ǫ(ω′)D̈+ [ǫ(ω′)− 1]
(Ḋ · ∇)v − (v · ∇)Ḋ
+ [ǫ(ω′)− 1]∇× [(H · ∇)v − (v · ∇)H] .
(2.8)
For a rotation v = Ω×r we can specify terms of the form (a ·∇)v by expressing the
components of the velocity v using the Levi-Civitta symbol εijk as vi = εijkΩjrk.
The components of (a · ∇)v are thus given by
[(a · ∇)v]i = al∂lεijkΩjrk = alεijkΩjδlk = [Ω× a]i . (2.9)
Article submitted to Royal Society
4 J. B. Götte, S. M. Barnett and M. Padgett
If we use the results from (2.9) in (2.8) we find for ∇2D:
−∇2D = −ǫ(ω′)D̈+ [ǫ(ω′)− 1]
Ω× Ḋ− (v · ∇)Ḋ
+ [ǫ(ω′)− 1]∇× [Ω×H− (v · ∇)H] .
(2.10)
The curl of the last bracket requires some some additional calculations. The first
term is given by:
∇× (Ω×H) = (∇ ·H)Ω− (Ω · ∇)H, (2.11)
and the second term can be written as:
∇× (v · ∇)H = Ω (∇ ·H)−∇ (Ω ·H) + (v · ∇) Ḋ, (2.12)
where the last term originates from ∇×H. The terms containing the divergence of
H cancel and the term (v · ∇) Ḋ can be added to the second term in (2.10). The
two remaining terms − (Ω · ∇)H and ∇ (Ω ·H) together give Ω× Ḋ:
Ω× Ḋ = Ω× (∇×H) = ∇ (Ω ·H)− (Ω · ∇)H. (2.13)
This concludes the derivation of the wave equation (2.1). It is possible to derive the
same wave equation for B using similar methods.
For a rotation around the z axis with constant angular velocity Ω = Ωez, the
directional derivative v · ∇ is proportional to an azimuthal derivative, as v · ∇ =
Ω× r · ∇ = Ω∂φ. This allows us to identify the two terms Ω× Ḋ and Ω∂φḊ in the
wave equation
−∇2D = −ǫ(ω′)D̈+ 2[ǫ(ω′)− 1]
Ω× Ḋ− Ω∂φḊ
(2.14)
as the polarisation rotation and rotary Fresnel drag terms, respectively. Player’s
derivation does not contain the term proportional to ∂φḊ because he treated only
the case of a plane wave propagating in the z-direction and for such fields D is
independent of φ.
On substituting a monochromatic ansatz of the form D = D0 exp(−iωt) into
(2.14), where ω is the optical angular frequency in the rest frame, we obtain:
−∇2D0 = ǫ(ω
′)ω2D0 − 2[ǫ(ω
′)− 1]ωΩ [iez ×D0 − i∂φD0] . (2.15)
If we make an ansatz for D0 with a general polarisation given by the complex
numbers α and β (with |α|2 + |β|2 = 1) in the form of D0 = (αex + βey)D+Dzez,
we find that the x and y components of the wave equation (2.15) decouple if β = ±iα
corresponding to left- and right-circularly polarised light respectively. If we restrict
the solutions to these two cases we can write the wave equation as:
∇2D = −ǫ(ω′)ω2D + 2[ǫ(ω′)− 1]ωΩ (±1− i∂φ)D, (2.16)
where the plus sign refers to left-circular polarisation and the minus sign to right-
circular polarisation. We can then identify ±1 as the extreme values of the variable
σ which corresponds to the circular polarisation or SAM of the light beam. Similarly
we can identify −i∂φ = Lz as the OAM operator, so that the wave equation contains
a term which depends on the total angular momentum σ + Lz:
∇2D = −ǫ(ω′)ω2D + 2[ǫ(ω′)− 1]ωΩ (σ + Lz)D. (2.17)
We shall see that it is the dependence on the optical angular momentum that is
responsible for the rotation of both the polarisation and of a transmitted image.
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On the dragging of light by a rotating medium 5
3. Specific rotary power
The rotation of the polarisation arises from the difference in the refractive indices
for left- and right-circularly polarised light. The angle per unit length by which the
polarisation is rotated is called the specific rotary power. For an optically active
medium at rest the specific rotary power is characteristic for a given material, but
from (2.17) it can be seen that light propagates differently in a rotating medium,
depending on whether the circular polarisation turns in the same rotation sense
as the dielectric or in the opposite sense. This phenomenon is described by the
effective specific rotary power (Jones 1976; Player 1976).
The specific rotary power, defined as (Fowles 1975):
δpol(ω) = (nr(ω)− nl(ω))
= (nr(ω)− nl(ω))
, (3.1)
is the angle of rotation of the plane of polarisation in an optical active medium. Here,
the indices r and l refer to right- and left-circularly polarised light. It was convenient
to set c = 1 for our derivation in section 2 but we reintroduce it here to facilitate
the calculation of measurable quantities. In order to illustrate the effect of the OAM
of light we choose a Bessel beam as an ansatz for the electrical displacement in the
x− y plane:
D = Jm(κρ) exp(imφ) exp(ikzz), (3.2)
where κ and kz are the transverse and longitudinal components of the wavevector.
Bessel beams of this form carry OAM of m~ per photon (Allen et al. 1992, 1999,
2003). Substituting the Bessel beam ansatz in the wave equation (2.17) yields the
following result for the overall wavenumber k =
κ2 + k2z :
k2l/r(ω) = ǫ(ω
− 2[ǫ(ω′)− 1]
(σ +m). (3.3)
The indices l and r denoting the circular polarisation correspond respectively to
σ = 1 and σ = −1. With the help of the relations ǫ(ω′) = n2(ω′) and k(ω) =
n(ω)ω/c we can turn the equation for the wavenumbers into an equation for the
effective refractive indices for left- and right-circularly polarised light:
n2l/r(ω) = n
2(ω′)− 2[n2(ω′)− 1]
(σ +m). (3.4)
Following Player (1976) we assume that Ω ≪ ω and we can therefore approximate
the square root for the refractive indices nl/r by a small parameter expansion to
the first order in Ω/ω:
nl/r(ω) ≃ n(ω
n(ω′)−
n(ω′)
(σ +m) . (3.5)
The frequency in the moving frame ω′ is different for left- and right-circularly po-
larised light (Garetz 1981) and, more generally, the azimuthal or rotational Doppler
shift is proportional to the total angular momentum (σ + m) (Allen et al. 1994;
Bialynicki-Birula & Bialynicka-Birula 1997; Courtial et al. 1998; Allen et al. 2003).
For left-circularly polarised light with σ = 1 the frequency is thus ω′ = ω−Ω(1+m),
Article submitted to Royal Society
6 J. B. Götte, S. M. Barnett and M. Padgett
and for right-circularly polarised light with σ = −1 the frequency changes to
ω′ = ω − Ω(−1 + m). Following Player (1976) we expand the refractive index
of the dielectric in a Taylor series to calculate the difference nr − nl:
nl(ω) ≃ n(ω)−
Ω(1 +m)−
n(ω)−
(1 +m) , (3.6a)
nr(ω) ≃ n(ω)−
Ω(−1 +m)−
n(ω)−
(−1 +m) . (3.6b)
Higher order derivatives of n become comparable in magnitude if n′(ω)Ω ≃ n(ω).
This will only be case for a strongly dispersive medium, such as atomic or molecular
gases, near a resonance. For such gaseous media the dielectric response in a rotating
medium has to examined more closely (Nienhuis et al. 1992). For solid materials,
such as a rotating glass rod, and for optical frequencies this condition is not fulfilled
and we can neglect higher order derivatives in the expansion (3.6). Within Player’s
assumption that the refractive index is independent of the motion of the medium
we find for the effective specific rotary power:
δpol(ω) =
ωn′(ω) + n(ω)−
. (3.7)
On introducing the group refractive index ng(ω) = n(ω) + ωn
′(ω) and the phase
refractive index nϕ(ω) = n(ω), we can rewrite the rotary power as
δpol(ω) =
ng(ω)− n
ϕ (ω)
(Ω/c), (3.8)
which is identical to Player’s (1976) expression. In this form the specific rotary
power (3.8) can be used directly with experimental data in the SI unit system. In
the next section we look at image rotation caused by a difference in the effective
refractive indices for different values of m.
4. Image rotation
The specific rotary power describes the rotation of the propagation, but we can
define, analogously, a rotary power of image rotation. The image can simply be
created by the superposition of two light beams carrying different values of OAM
which leads to an azimuthal variation of the intensity pattern. In particular we
consider an incident superposition of two similarly circularly polarised Bessel beams
with opposite OAM values of the form
D = D+ +D−
= Jm(κρ) exp(imφ) exp(ikzz) + J−m(κρ) exp(−imφ) exp(ikzz).
(4.1)
Outside the medium the superposition can be written as one Bessel beam with a
trigonometric modulation
D = Jm(κρ) (exp(imφ) + (−1)
m exp(−imφ)) exp(ikzz), (4.2)
but inside the medium the effective refractive index is different for the two com-
ponents of the superposition (Allen & Padgett 2007). On propagation this leads to
Article submitted to Royal Society
On the dragging of light by a rotating medium 7
HaL HbL ∆
Figure 1. Image rotation
(Intensity pattern created by the superposition of Bessel beams (a) in (4.1) for m = 2.
On propagation the relative phase between the constituent Bessel beams changes which
leads to a rotation of the pattern (b). The angle of rotation at a propagation distance L
is given by δimgL.)
phase difference which causes a rotation of the image (see figure 1). We define
δimg(ω) = (n−(ω)− n+(ω))
, (4.3)
which is the angle per unit length by which the image is rotated. The factorm in the
expression for δimg appears because of the exp(imφ) and exp(−imφ) phase structure
of the interfering beams and the resulting 2m-fold symmetry of the created image
(Pagdett et al. 2006).
The different effective refractive indices for the components of the superposition
(4.1) are given by:
n2+/−(ω) = n
2(ω′)− 2[n2(ω′)− 1]
(σ ±m). (4.4)
Here, σ is fixed in contrast to (3.4). The roles of σ and m are reversed for the image
rotation and the refractive indices for positive and negative OAM are given by:
n+(ω) ≃ n(ω)−
Ω(σ +m)−
n(ω)−
(σ +m) , (4.5a)
n−(ω) ≃ n(ω)−
Ω(σ −m)−
n(ω)−
(σ −m) . (4.5b)
On substituting (4.5) into (4.3) we find:
δimg(ω) =
+ n(ω)−
, (4.6)
which can be written in terms of the group and phase refractive indices as:
δimg(ω) =
ng(ω)− n
ϕ (ω)
(Ω/c). (4.7)
Article submitted to Royal Society
8 J. B. Götte, S. M. Barnett and M. Padgett
This verifies the reasoning of Padgett et al. (2006) that the polarisation and the
image are turned by the same amount when passing through a rotating medium.
It is the total angular momentum that determines the phase shifts and a linearly
polarised image will undergo rotations of both the plane of polarisation and the
intensity pattern or image.
5. Conclusion
We have extended a theoretical study by Player (1976) on the propagation of light
through a rotating medium to include general electromagnetic fields. In the original
analysis Player (1976) showed that the rotation of the polarisation inside a rotating
medium can be understood in terms of a difference in the propagation for left- and
right-circularly polarised light. Player’s (1976) analysis was thus concerned solely
with the spin angular momentum (SAM) of light.
Our treatment has shown that the general wave equation has an additional term,
which is of the same form as the Fresnel drag term for a uniform motion. In the
context of rotating motion, however, this term is connected to the orbital angular
momentum (OAM) of the light. By extending the theoretical analysis to include
OAM we have been able to attribute polarisation rotation and image rotation to
SAM and OAM respectively. We have shown that a superposition of Bessel beams
with the same OAM but opposite SAM states leads to the rotation of the polari-
sation, whereas a superposition of Bessel beams with the same SAM and opposite
OAM values gives rise to a rotation of the transmitted image. We have obtained
quantitative expressions for the rotation of the polarisation and of the transmitted
image and have verified that both are turned through the same angle, as recently
suggested by Padgett et al. (2006).
Player (1976) remarked that the derivation by Fermi (1923) appears to be in
error. The mistake in Fermi’s treatment seems to be in missing the transformation
of the magnetic fields. Whereas the change in the electric fields induced by the
motion of the medium is explicitly given in terms of the electric polarisation P†, a
similar transformation for the magnetic field is missing. In terms of our derivation
this would mean that (2.5b) changes to B = H in the rest frame. This in turn
causes that the term v × Ḋ would be missing in (2.6). This term and the term
∇ × (v × H) contribute equally to the wave equation (2.1), which explains why
Fermi’s result for the specific rotary power is smaller than Player’s and ours by a
factor of two. As pointed out by Player (1976) this missing factor is cancelled by
an additional factor of two in Fermi’s definition of the specific rotary power.
We would like to thank Amanda Wright and Jonathan Leach whose experiments on this
problem motivated our work. This work was supported by the UK Engineering and Phys-
ical Sciences Research Council.
Appendix A. Accelerated motion
The assumption that v = Ω × r is steady is problematic for a rotating motion;
if we assume Ω to be constant over time, then v̇ = (Ω · r)Ω − Ω2r. In princi-
ple this would invalidate our initial considerations for the transformation of the
† Fermi (1923) denotes the electric polarisation by S
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On the dragging of light by a rotating medium 9
electromagnetic fields (2.3) which strictly hold only for uniform motion. Includ-
ing the time-derivative of v would lead to additional terms in (2.6) of the form
ǫ(ω′)[ǫ(ω′) − 1]v̇ × E. If we proceed in taking the curl of this vector product we
produce four terms which either can be neglected because they are second order in
v/c, or they do not contain the time derivative of an optical field. The latter are
smaller than terms that do contain a time derivative by ∼ Ω/ω. For our assumption
Ω ≪ ω all such terms are negligible.
References
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Allen, L., Babiker, M. & Power, W. L. 1994 Azimuthal Doppler-shift in light-beams with
orbital angular-momentum. Opt. Commun. 112, 141–144.
Allen, L., Padgett, M. J. & Babiker M. 1999 The Orbital Angular Momentum of Light.
Prog. Opt. 39, 291–372.
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Allen, L. & Padgett, M. J. 2007 Equivalent geometric transformation for spin and orbital
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1. Chicago: University of Chicago Press.
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Article submitted to Royal Society
	Introduction
	Wave equations
	Specific rotary power
	Image rotation
	Conclusion
	Accelerated motion

  When light is passing through a rotating medium the optical polarisation is
rotated. Recently it has been reasoned that this rotation applies also to the
transmitted image (Padgett et al. 2006). We examine these two phenomena by
extending an analysis of Player (1976) to general electromagnetic fields. We
find that in this more general case the wave equation inside the rotating
medium has to be amended by a term which is connected to the orbital angular
momentum of the light. We show that optical spin and orbital angular momentum
account respectively for the rotation of the polarisation and the rotation of
the transmitted image.

Introduction
Jones (1976) studied the propagation of light in a moving dielectric and showed
by experiment that a rotating medium induces a rotation of the polarisation of
the transmitted light. Player (1976) confirmed that this observation could be ac-
counted for through an application of Maxwells equations in a moving medium.
More recently Padgett et al. (2006) reasoned that the rotation of the medium turns
a transmitted image by the same angle as the polarisation. This is in contrast to the
Faraday effect (Faraday 1846), where a static magnetic field in a dielectric medium,
parallel to the propagation of light, causes a rotation of the polarisation but not
a rotation of a transmitted image. Rotation of the plane of polarisation and im-
age rotation in a rotating medium may be attributed respectively to the spin and
orbital angular momentum of light (Allen et al. 1999, 2003).
The first theoretical treatment of this problem was published by Fermi (1923),
who considered plane waves and a non-dispersive medium. The theoretical ana-
lysis of Player (1976) was also restricted to the propagation of plane waves, but
took the dispersion of the medium into account. Player assumed that the dielectric
response does not depend on the motion of the medium. In our treatment we
follow his assumption although a more careful analysis by Nienhuis et al. (1992)
showed that there will be an effect of the motion on the refractive index for a
dispersive medium near to an absorption resonance (see also Baranova & Zel’dovich
(1979) for a discussion on the effect of the Coriolis force on the refractive index).
In contrast to Player we allow for more general electromagnetic fields that can
carry orbital angular momentum (OAM). This leads to an additional term in our
Article submitted to Royal Society TEX Paper
http://arxiv.org/abs/0704.0725v1
2 J. B. Götte, S. M. Barnett and M. Padgett
wave equation, which corresponds to a Fresnel drag term familiar from analysis of
uniform motion. For a rotating medium, however, this drag leads to a rotational
shift of the image. The propagation of light in a rotating medium thus involves
both spin angular momentum (SAM) and OAM. We solve the wave equation for
circularly polarised Bessel beams and consider two different superpositions of such
Bessel beams to quantify the effects of both polarisation and image rotation. For
rotation of the polarisation we examine a superposition of left- and right-circularly
polarised Bessel beams carrying the same amount of OAM. For image rotation we
consider a superposition of Bessel beams with the same circular polarisation but
opposite OAM values. Such a superposition creates an intensity pattern with lobes
or ‘petals’. In both cases the constituent Bessel beams propagate differently in the
medium, which leads to a change in their relative phase. This is the origin of the
rotation of both the polarisation and the transmitted image. For both phenomena
we derive an expression for the angle per unit length of dielectric through which
the image or the polarisation is rotated.
The significance of the total angular momentum can be most easily seen in the
wave equation for the propagation of light in a rotating medium. We derive this
wave equation in section 2. In the remaining sections we calculate the rotation of
polarisation (section 3) and the image rotation (section 4) and reveal their common
form.
2. Wave equations
The wave equation for a general electric displacment D in a rigid dielectric medium
rotating with angular velocity Ω is given by:
−∇2D = −ǫ(ω′)D̈+ 2[ǫ(ω′)− 1]
Ω× Ḋ− (v · ∇)Ḋ
. (2.1)
An analoguous wave equation can be derived for the magnetic induction B. Com-
pared to the form derived by Player (1976), who considered the special case of a
plane wave propagating along the direction of Ω, these wave equations contain an
additional term 2[ǫ(ω′)− 1](v · ∇)Ḋ. This term is responsible for the Fresnel drag
effect which modifies the speed of light in a moving medium (McCrea 1954; Bar-
ton 1999; Rindler 2001). In the following we will derive this wave equation for the
electric displacement.
Our analysis starts with the same considerations as Player (1976), by intro-
ducing a rest frame and a moving frame. In the rest frame the dielectric medium
rotates with an angular velocity v = Ω× r and in the moving frame the medium is
at rest. We restrict our analysis to small velocities with v ≪ c and use Maxwell’s
equations in both reference frames (Landau & Lifshitz 1975). For the medium at
rest we assume the following constitutive relations:
′ = ǫ(ω′)E′, (2.2a)
′ = H′, (2.2b)
where we have used primes to denote the fields and their frequency ω′ in the moving
frame. The fields in the moving frame can be expressed in the rest frame by a Lorentz
Article submitted to Royal Society
On the dragging of light by a rotating medium 3
transformation (Stratton 1941; Jackson 1998), which gives to first order in v/c:
′ = D+ v ×H, (2.3a)
′ = B− v ×E, (2.3b)
′ = E+ v ×B, (2.3c)
′ = H− v ×D, (2.3d)
where we have set c = 1 and work with units in which ǫ0 = µ0 = 1. The two
constitutive relation in (2.2) in the rest frame are thus given by
D+ v ×H = ǫ(ω′) (E+ v ×B) , (2.4a)
B− v ×E = H− v ×D. (2.4b)
The dielectric constant is still given as a function of the frequency in the moving
frame. We also assume that the dielectric constant depends only on the frequency
and is otherwise independent of the state of motion of the medium. On combining
these two equations we can express D and B with the two other fields E and H to
the first order in v:
D = ǫ(ω′)E+ [ǫ(ω′)− 1]v×H, (2.5a)
B = H− [ǫ(ω′)− 1]v×E. (2.5b)
After taking the curl of (2.5a) we can use the Maxwell equation ∇×E = −Ḃ and
express Ḃ, with the help of (2.5b), in terms of Ḣ and Ė. If we assume v to be
constant (see Appendix A), as in Player’s paper (Player, 1976) this yields
∇×D = −ǫ(ω′)Ḣ+ ǫ(ω′)[ǫ(ω′)− 1]v × Ė+ [ǫ(ω′)− 1]∇× (v ×H). (2.6)
It follows from (2.5a) that ǫ(ω′)v × E = v ×D, to the first order in v, and so we
can rewrite (2.6) as:
∇×D = −ǫ(ω′)Ḣ+ [ǫ(ω′)− 1]v × Ḋ+ [ǫ(ω′)− 1]∇× (v ×H). (2.7)
We can now take the curl of (2.7) to obtain a wave equation for D, as ∇×∇×D =
−∇2D for ∇·D = 0, and the curl of Ḣ is given by ∇× Ḣ = D̈. In order to express
the curl of the vector products we use the identity∇×(a×b) = ∂ibia−∂iaib, where
the doubly occurring index denotes a summation over the Cartesian components.
The operator ∂i represents differentiation with respect to the ith component and
acts on the whole product which gives rise to terms containing the divergences of
v,D and H. These terms are either zero, because ∇ · v = 0 and ∇ ·D = 0 or they
lead to terms which are of second order in v and therefore negligible. The wave
equation for D is thus given by
−∇2D = −ǫ(ω′)D̈+ [ǫ(ω′)− 1]
(Ḋ · ∇)v − (v · ∇)Ḋ
+ [ǫ(ω′)− 1]∇× [(H · ∇)v − (v · ∇)H] .
(2.8)
For a rotation v = Ω×r we can specify terms of the form (a ·∇)v by expressing the
components of the velocity v using the Levi-Civitta symbol εijk as vi = εijkΩjrk.
The components of (a · ∇)v are thus given by
[(a · ∇)v]i = al∂lεijkΩjrk = alεijkΩjδlk = [Ω× a]i . (2.9)
Article submitted to Royal Society
4 J. B. Götte, S. M. Barnett and M. Padgett
If we use the results from (2.9) in (2.8) we find for ∇2D:
−∇2D = −ǫ(ω′)D̈+ [ǫ(ω′)− 1]
Ω× Ḋ− (v · ∇)Ḋ
+ [ǫ(ω′)− 1]∇× [Ω×H− (v · ∇)H] .
(2.10)
The curl of the last bracket requires some some additional calculations. The first
term is given by:
∇× (Ω×H) = (∇ ·H)Ω− (Ω · ∇)H, (2.11)
and the second term can be written as:
∇× (v · ∇)H = Ω (∇ ·H)−∇ (Ω ·H) + (v · ∇) Ḋ, (2.12)
where the last term originates from ∇×H. The terms containing the divergence of
H cancel and the term (v · ∇) Ḋ can be added to the second term in (2.10). The
two remaining terms − (Ω · ∇)H and ∇ (Ω ·H) together give Ω× Ḋ:
Ω× Ḋ = Ω× (∇×H) = ∇ (Ω ·H)− (Ω · ∇)H. (2.13)
This concludes the derivation of the wave equation (2.1). It is possible to derive the
same wave equation for B using similar methods.
For a rotation around the z axis with constant angular velocity Ω = Ωez, the
directional derivative v · ∇ is proportional to an azimuthal derivative, as v · ∇ =
Ω× r · ∇ = Ω∂φ. This allows us to identify the two terms Ω× Ḋ and Ω∂φḊ in the
wave equation
−∇2D = −ǫ(ω′)D̈+ 2[ǫ(ω′)− 1]
Ω× Ḋ− Ω∂φḊ
(2.14)
as the polarisation rotation and rotary Fresnel drag terms, respectively. Player’s
derivation does not contain the term proportional to ∂φḊ because he treated only
the case of a plane wave propagating in the z-direction and for such fields D is
independent of φ.
On substituting a monochromatic ansatz of the form D = D0 exp(−iωt) into
(2.14), where ω is the optical angular frequency in the rest frame, we obtain:
−∇2D0 = ǫ(ω
′)ω2D0 − 2[ǫ(ω
′)− 1]ωΩ [iez ×D0 − i∂φD0] . (2.15)
If we make an ansatz for D0 with a general polarisation given by the complex
numbers α and β (with |α|2 + |β|2 = 1) in the form of D0 = (αex + βey)D+Dzez,
we find that the x and y components of the wave equation (2.15) decouple if β = ±iα
corresponding to left- and right-circularly polarised light respectively. If we restrict
the solutions to these two cases we can write the wave equation as:
∇2D = −ǫ(ω′)ω2D + 2[ǫ(ω′)− 1]ωΩ (±1− i∂φ)D, (2.16)
where the plus sign refers to left-circular polarisation and the minus sign to right-
circular polarisation. We can then identify ±1 as the extreme values of the variable
σ which corresponds to the circular polarisation or SAM of the light beam. Similarly
we can identify −i∂φ = Lz as the OAM operator, so that the wave equation contains
a term which depends on the total angular momentum σ + Lz:
∇2D = −ǫ(ω′)ω2D + 2[ǫ(ω′)− 1]ωΩ (σ + Lz)D. (2.17)
We shall see that it is the dependence on the optical angular momentum that is
responsible for the rotation of both the polarisation and of a transmitted image.
Article submitted to Royal Society
On the dragging of light by a rotating medium 5
3. Specific rotary power
The rotation of the polarisation arises from the difference in the refractive indices
for left- and right-circularly polarised light. The angle per unit length by which the
polarisation is rotated is called the specific rotary power. For an optically active
medium at rest the specific rotary power is characteristic for a given material, but
from (2.17) it can be seen that light propagates differently in a rotating medium,
depending on whether the circular polarisation turns in the same rotation sense
as the dielectric or in the opposite sense. This phenomenon is described by the
effective specific rotary power (Jones 1976; Player 1976).
The specific rotary power, defined as (Fowles 1975):
δpol(ω) = (nr(ω)− nl(ω))
= (nr(ω)− nl(ω))
, (3.1)
is the angle of rotation of the plane of polarisation in an optical active medium. Here,
the indices r and l refer to right- and left-circularly polarised light. It was convenient
to set c = 1 for our derivation in section 2 but we reintroduce it here to facilitate
the calculation of measurable quantities. In order to illustrate the effect of the OAM
of light we choose a Bessel beam as an ansatz for the electrical displacement in the
x− y plane:
D = Jm(κρ) exp(imφ) exp(ikzz), (3.2)
where κ and kz are the transverse and longitudinal components of the wavevector.
Bessel beams of this form carry OAM of m~ per photon (Allen et al. 1992, 1999,
2003). Substituting the Bessel beam ansatz in the wave equation (2.17) yields the
following result for the overall wavenumber k =
κ2 + k2z :
k2l/r(ω) = ǫ(ω
− 2[ǫ(ω′)− 1]
(σ +m). (3.3)
The indices l and r denoting the circular polarisation correspond respectively to
σ = 1 and σ = −1. With the help of the relations ǫ(ω′) = n2(ω′) and k(ω) =
n(ω)ω/c we can turn the equation for the wavenumbers into an equation for the
effective refractive indices for left- and right-circularly polarised light:
n2l/r(ω) = n
2(ω′)− 2[n2(ω′)− 1]
(σ +m). (3.4)
Following Player (1976) we assume that Ω ≪ ω and we can therefore approximate
the square root for the refractive indices nl/r by a small parameter expansion to
the first order in Ω/ω:
nl/r(ω) ≃ n(ω
n(ω′)−
n(ω′)
(σ +m) . (3.5)
The frequency in the moving frame ω′ is different for left- and right-circularly po-
larised light (Garetz 1981) and, more generally, the azimuthal or rotational Doppler
shift is proportional to the total angular momentum (σ + m) (Allen et al. 1994;
Bialynicki-Birula & Bialynicka-Birula 1997; Courtial et al. 1998; Allen et al. 2003).
For left-circularly polarised light with σ = 1 the frequency is thus ω′ = ω−Ω(1+m),
Article submitted to Royal Society
6 J. B. Götte, S. M. Barnett and M. Padgett
and for right-circularly polarised light with σ = −1 the frequency changes to
ω′ = ω − Ω(−1 + m). Following Player (1976) we expand the refractive index
of the dielectric in a Taylor series to calculate the difference nr − nl:
nl(ω) ≃ n(ω)−
Ω(1 +m)−
n(ω)−
(1 +m) , (3.6a)
nr(ω) ≃ n(ω)−
Ω(−1 +m)−
n(ω)−
(−1 +m) . (3.6b)
Higher order derivatives of n become comparable in magnitude if n′(ω)Ω ≃ n(ω).
This will only be case for a strongly dispersive medium, such as atomic or molecular
gases, near a resonance. For such gaseous media the dielectric response in a rotating
medium has to examined more closely (Nienhuis et al. 1992). For solid materials,
such as a rotating glass rod, and for optical frequencies this condition is not fulfilled
and we can neglect higher order derivatives in the expansion (3.6). Within Player’s
assumption that the refractive index is independent of the motion of the medium
we find for the effective specific rotary power:
δpol(ω) =
ωn′(ω) + n(ω)−
. (3.7)
On introducing the group refractive index ng(ω) = n(ω) + ωn
′(ω) and the phase
refractive index nϕ(ω) = n(ω), we can rewrite the rotary power as
δpol(ω) =
ng(ω)− n
ϕ (ω)
(Ω/c), (3.8)
which is identical to Player’s (1976) expression. In this form the specific rotary
power (3.8) can be used directly with experimental data in the SI unit system. In
the next section we look at image rotation caused by a difference in the effective
refractive indices for different values of m.
4. Image rotation
The specific rotary power describes the rotation of the propagation, but we can
define, analogously, a rotary power of image rotation. The image can simply be
created by the superposition of two light beams carrying different values of OAM
which leads to an azimuthal variation of the intensity pattern. In particular we
consider an incident superposition of two similarly circularly polarised Bessel beams
with opposite OAM values of the form
D = D+ +D−
= Jm(κρ) exp(imφ) exp(ikzz) + J−m(κρ) exp(−imφ) exp(ikzz).
(4.1)
Outside the medium the superposition can be written as one Bessel beam with a
trigonometric modulation
D = Jm(κρ) (exp(imφ) + (−1)
m exp(−imφ)) exp(ikzz), (4.2)
but inside the medium the effective refractive index is different for the two com-
ponents of the superposition (Allen & Padgett 2007). On propagation this leads to
Article submitted to Royal Society
On the dragging of light by a rotating medium 7
HaL HbL ∆
Figure 1. Image rotation
(Intensity pattern created by the superposition of Bessel beams (a) in (4.1) for m = 2.
On propagation the relative phase between the constituent Bessel beams changes which
leads to a rotation of the pattern (b). The angle of rotation at a propagation distance L
is given by δimgL.)
phase difference which causes a rotation of the image (see figure 1). We define
δimg(ω) = (n−(ω)− n+(ω))
, (4.3)
which is the angle per unit length by which the image is rotated. The factorm in the
expression for δimg appears because of the exp(imφ) and exp(−imφ) phase structure
of the interfering beams and the resulting 2m-fold symmetry of the created image
(Pagdett et al. 2006).
The different effective refractive indices for the components of the superposition
(4.1) are given by:
n2+/−(ω) = n
2(ω′)− 2[n2(ω′)− 1]
(σ ±m). (4.4)
Here, σ is fixed in contrast to (3.4). The roles of σ and m are reversed for the image
rotation and the refractive indices for positive and negative OAM are given by:
n+(ω) ≃ n(ω)−
Ω(σ +m)−
n(ω)−
(σ +m) , (4.5a)
n−(ω) ≃ n(ω)−
Ω(σ −m)−
n(ω)−
(σ −m) . (4.5b)
On substituting (4.5) into (4.3) we find:
δimg(ω) =
+ n(ω)−
, (4.6)
which can be written in terms of the group and phase refractive indices as:
δimg(ω) =
ng(ω)− n
ϕ (ω)
(Ω/c). (4.7)
Article submitted to Royal Society
8 J. B. Götte, S. M. Barnett and M. Padgett
This verifies the reasoning of Padgett et al. (2006) that the polarisation and the
image are turned by the same amount when passing through a rotating medium.
It is the total angular momentum that determines the phase shifts and a linearly
polarised image will undergo rotations of both the plane of polarisation and the
intensity pattern or image.
5. Conclusion
We have extended a theoretical study by Player (1976) on the propagation of light
through a rotating medium to include general electromagnetic fields. In the original
analysis Player (1976) showed that the rotation of the polarisation inside a rotating
medium can be understood in terms of a difference in the propagation for left- and
right-circularly polarised light. Player’s (1976) analysis was thus concerned solely
with the spin angular momentum (SAM) of light.
Our treatment has shown that the general wave equation has an additional term,
which is of the same form as the Fresnel drag term for a uniform motion. In the
context of rotating motion, however, this term is connected to the orbital angular
momentum (OAM) of the light. By extending the theoretical analysis to include
OAM we have been able to attribute polarisation rotation and image rotation to
SAM and OAM respectively. We have shown that a superposition of Bessel beams
with the same OAM but opposite SAM states leads to the rotation of the polari-
sation, whereas a superposition of Bessel beams with the same SAM and opposite
OAM values gives rise to a rotation of the transmitted image. We have obtained
quantitative expressions for the rotation of the polarisation and of the transmitted
image and have verified that both are turned through the same angle, as recently
suggested by Padgett et al. (2006).
Player (1976) remarked that the derivation by Fermi (1923) appears to be in
error. The mistake in Fermi’s treatment seems to be in missing the transformation
of the magnetic fields. Whereas the change in the electric fields induced by the
motion of the medium is explicitly given in terms of the electric polarisation P†, a
similar transformation for the magnetic field is missing. In terms of our derivation
this would mean that (2.5b) changes to B = H in the rest frame. This in turn
causes that the term v × Ḋ would be missing in (2.6). This term and the term
∇ × (v × H) contribute equally to the wave equation (2.1), which explains why
Fermi’s result for the specific rotary power is smaller than Player’s and ours by a
factor of two. As pointed out by Player (1976) this missing factor is cancelled by
an additional factor of two in Fermi’s definition of the specific rotary power.
We would like to thank Amanda Wright and Jonathan Leach whose experiments on this
problem motivated our work. This work was supported by the UK Engineering and Phys-
ical Sciences Research Council.
Appendix A. Accelerated motion
The assumption that v = Ω × r is steady is problematic for a rotating motion;
if we assume Ω to be constant over time, then v̇ = (Ω · r)Ω − Ω2r. In princi-
ple this would invalidate our initial considerations for the transformation of the
† Fermi (1923) denotes the electric polarisation by S
Article submitted to Royal Society
On the dragging of light by a rotating medium 9
electromagnetic fields (2.3) which strictly hold only for uniform motion. Includ-
ing the time-derivative of v would lead to additional terms in (2.6) of the form
ǫ(ω′)[ǫ(ω′) − 1]v̇ × E. If we proceed in taking the curl of this vector product we
produce four terms which either can be neglected because they are second order in
v/c, or they do not contain the time derivative of an optical field. The latter are
smaller than terms that do contain a time derivative by ∼ Ω/ω. For our assumption
Ω ≪ ω all such terms are negligible.
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Article submitted to Royal Society
	Introduction
	Wave equations
	Specific rotary power
	Image rotation
	Conclusion
	Accelerated motion
