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Proc. R. Soc. A (2012) 468, 1825–1838
doi:10.1098/rspa.2011.0573
Published online 28 March 2012
REVIEW
Contributions of John Henry Poynting to the
understanding of radiation pressure
BY R. LOUDON1, *
AND
C. BAXTER2
1 School
of Computer Science and Electronic Engineering, University of Essex,
Colchester, Essex CO4 3SQ, UK
2 Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK
The name of Poynting is universally recognized for his development of the well-known
expression for the flow of electromagnetic energy. Not so well known is Poynting’s series
of papers on radiation pressure, with 2011 marking the centenary of the last of his 15
publications on this topic. This paper reviews and assesses his radiation-pressure work,
with a level of coverage aimed at the reader familiar with the Maxwell electromagnetic
theory and interested in the current understanding of radiation pressure. We begin with
brief details of Poynting’s life, followed by accounts of the relevant publications by others
before and during his period of activity in the field from 1903 to 1911. His contributions
to the understanding of radiation-pressure effects in the solar system, and the linear and
angular momenta of light are discussed, with evaluations from a modern perspective.
Keywords: solar system; linear and angular optical momentum; Abraham–Minkowski
1. Introduction
Poynting’s name is familiar to every student of electromagnetism as the man who,
in 1884, published a theorem for the conservation of energy in an electromagnetic
field, as well as an expression for the flow of energy, now known, respectively, as
Poynting’s theorem and the Poynting vector. At a later stage in his career, in
the early years of the twentieth century, he published a series of groundbreaking
papers on radiation pressure, covering both measurements and theories of the
effect. Our main concerns in this article, 100 years after his final publication in
the field, are with the basis of his work in the Maxwell theory, a review of his
radiation-pressure papers, and an assessment of his current standing in the field.
The historical setting and evolution of Poynting’s experimental and theoretical
activities are well covered in our cited references. It is, however, useful to begin
with a brief account of his life.
John Henry Poynting (1852–1914) was born in Monton, a suburb of
Manchester, UK, about 8 km west of the city centre, where his father was
Unitarian Minister. He was educated initially in his father’s school, and in 1867,
*Author for correspondence ([email protected]).
Received 20 September 2011
Accepted 27 February 2012
1825
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R. Loudon and C. Baxter
he entered Owens College (to become Manchester University in 1880), obtaining
an external London University BSc degree in 1872. He was awarded an entrance
scholarship to Trinity College, Cambridge in the same year, to study for a
bachelor’s degree in mathematics. He achieved the position of third highestscoring student with First Class Honours in the Mathematical Tripos of 1876.
He then returned to Manchester to take up a position as demonstrator in the
physical laboratory at Owens College, where J. J. Thomson, one of the discoverers
of the electron (Pais 1986, pp. 78–86) and winner of the 1906 Nobel Prize in
Physics, was one of his students. Back to Cambridge in 1878, on his election as a
Fellow of Trinity College, he worked under Maxwell in the Cavendish Laboratory
on experiments to determine the mean density of the Earth. In 1880, Poynting
married Maria Adney Cropper, the daughter of another Unitarian Minister at
Stand, part of Whitefield, another suburb about 8 km north of Manchester.
In the same year of 1880, Poynting was appointed to the Chair of Physics at
the new Mason Science College (to become Birmingham University in 1900), and
he stayed there until his early death from diabetes. His career in Birmingham
was marked by a series of distinctions, including the award of a Cambridge ScD
in 1987; election as a Fellow of the Royal Society in 1888; awards of the Adams
and Hopkins prizes at Cambridge in 1893 and 1903; election as President of the
Physical Society in 1905; and awards of Royal Medal and Bakerian Lectureship by
the Royal Society in 1905 and 1910, respectively. He was also one of the founders
of the UK National Physical Laboratory in 1907. In addition to Poynting’s
pioneering work in the new physics of electrodynamics and radiation pressure, he
was active in areas outside the physical sciences, making, for example, a study of
the drunkenness statistics of large towns in Great Britain and a statistical analysis
of changes in commodity prices on the London Stock Exchange. We are not aware
of any full-scale biography of Poynting, but there are extensive accounts of his
life and achievements in obituaries published shortly after his death by Larmor
(1914), Lodge (1914) and Thomson (1916). They all testify not only to Poynting’s
professional qualities, but also to his modesty and to the catholic nature of his
interests.
The main influence on all of the activity in electromagnetic theory during
the later years of the nineteenth century came from Maxwell’s famous treatise
(Maxwell 1873). Poynting was a member of the group of young physicists led by
Heaviside, Fitzgerald, Lodge and Hertz who developed Maxwell’s electromagnetic
theory in the years following his death in 1879. They transformed his 1873
presentation into the formalism recognizable today as Maxwell’s equations.
The detailed historical accounts by Hunt (1991) and Warwick (2003) describe
Poynting’s contributions to electromagnetism, mainly during the 1880s. His
name is more familiar to students of electromagnetic theory than those of
other important members of the group on account of the widespread use of his
eponymous energy-conservation theorem and energy-flow vector. These energy
relations are not directly relevant for the electromagnetic momentum of interest
here, but the expressions for energy density U and Poynting vector S do occur
in the discussions of momentum in §2.
The plan of this review is as follows. In §2, we outline the relevant theories of
Maxwell, which form the main basis for Poynting’s work on radiation pressure.
Despite the apparent lack of any awareness of each other’s publications, we
comment on the very relevant calculations by Minkowski and Abraham published
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during Poynting’s period of activity in the field. In §§3–5, we aim to explain and
clarify the key results and main conclusions in the distinct areas of Poynting’s
radiation-pressure research output. Thus, §3 outlines his work on radiation
pressure in the solar system, §4 reviews his contributions to the theory of the
linear momentum of light, with ingenious laboratory experiments performed in
collaboration with Guy Barlow, and §5 describes his proposal that light can
carry angular as well as linear momentum. His contributions are assessed from a
modern perspective in §6, based on the more complete understanding that has
resulted from the high volume of subsequent research activity. In order to assess
the relevance of Poynting’s radiation-pressure work for present-day research, it
is necessary to discuss its quantum-mechanical interpretation, as is performed in
this separate Epilogue section. Although this may be unsatisfactory to a historian
of science, we feel that the placement of Poynting’s work in a modern setting
best brings to the fore his lasting contributions to current research into the
radiation-pressure effects of light.
Poynting was not the first to think about such effects, but, in regarding light
as simply a wave capable of carrying both energy and momentum, he initiated
an approach in the early 1900s that remains familiar today. The then-current
orthodoxy required the existence of a material medium for transmission of light,
but Poynting assumed only that the ether was a passive bystander in the exchange
of momentum between a light ray and a material body (Poynting 1904a, p. 538),
a role assigned today to the electromagnetic field alone.
2. Background to Poynting’s radiation-pressure work
Poynting’s collection of scientific papers has been published in a single substantial
memorial volume (Shakespear & Barlow 1920). His 15 radiation-pressure
publications during the years 1903–1911 occur together in the reference list. In the
present section, we review the work prior to 1903 on which Poynting’s ideas and
calculations were based, also the relevant papers by others that were published
during his period of activity up to 1911.
The first recognition of the existence of what we would understand today
as radiation pressure appears to date from speculation by Kepler in the early
seventeenth century to the effect that light streaming away from the Sun is
responsible for the orientation of a comet’s tail in a similar direction. The
subsequent history of work on the measurement and theory of radiation pressure
makes a fascinating story, whose twists and turns are associated with the rival
corpuscular and wave theories of light. The story is covered in detail by Jones
(1953), Schagrin (1974) and Worrall (1982), and there are shorter accounts in the
introductions to papers on laboratory measurements of radiation pressure in the
early twentieth century (Nichols & Hull 1903; Lebedew 1910).
Poynting’s early work was mainly based on Maxwell’s treatise, which had
posthumous second and third editions in 1881 and 1891. It is convenient here
to refer to the more accessible third edition (Maxwell 1891), where §§792 and
793 in volume 2 present the theory of radiation pressure exerted by light in
free space, with estimates of its value for sunlight falling on the surface of the
Earth. The theory in §792 shows that, contrary to the claims of proponents of
the corpuscular nature of light, an electromagnetic wave indeed exerts a pressure
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R. Loudon and C. Baxter
in its direction of propagation. The pressure is determined by the components
of the electromagnetic stress Tij , with i, j any two Cartesian coordinates, and
where its magnitude equals the energy per unit volume U in the wave. For the
propagation of a transverse electromagnetic wave parallel to z, with E parallel to
x and H parallel to y, the pressure parallel to the z-axis and the energy density,
re-expressed here in the SI units now current, are
30
2
2
2
2
1
E.
(2.1)
Tzz = U = 2 (30 E + m0 H ) = 30 E = m0 H , as H =
m0
In §793, Maxwell took the value of 83.4 foot pounds per second per square foot for
the energy of sunlight falling on the Earth’s surface. The corresponding pressure
of sunlight on the Earth, obtained on division by the velocity of light, is about
10−7 pounds per square foot, equivalent to about 4.5 g per hectare in metric units.
In his first paper on radiation pressure, Poynting (1903) emphasized the small
value of the radiation pressure of sunlight on the Earth by its size of ‘less than
one two-hundred-thousandth of a grain on the square-inch’, which turns out to
be the same as that quoted above! The practical measurement of solar radiation
pressure on the Earth is clearly a considerable challenge.
Experimental techniques for the measurement of radiation pressure employed
an evacuated chamber, in which a thin vane was suspended from a torsion fibre.
The asymmetric incidence of light from the Sun or a bright laboratory source
on the vane should then cause it to rotate. The earlier observations of such
rotations have mainly been interpreted as caused instead by other forces on the
vane, which is inevitably heated by convection currents in the residual gas (Jones
1953; Schagrin 1974; Worrall 1982). The first reliable laboratory measurements
of radiation pressure (Lebedew 1901; Nichols & Hull 1901, 1903) were made with
carbon-arc light sources and with more sophisticated techniques to overcome the
convection-current problem. The 1901 results showed that the radiation pressure
was observable, but there was only approximate agreement with the Maxwell
theory. Much better agreement, at the 1 per cent level, was claimed in 1903 with
the use of further improvements in experimental techniques, although doubts
have been cast on their true accuracy (Bell & Green 1933; Worrall 1982). Later
laboratory measurements by Lebedew (1910) of the pressure exerted by light on
gases were in rough agreement with the Maxwell theory.
The existence of the radiation-pressure phenomenon shows that electromagnetic waves carry momentum in addition to their more familiar energy content
(Thomson 1893). Poynting (1905a) considered a light beam as a ‘stream of
momentum’, whose flux (momentum per unit cross-sectional area per unit time)
equals its energy density U . Formal theories of the electromagnetic energy
and momentum in dielectric materials were developed by Minkowski (1908)
and by Abraham (1909, 1910). Their papers provide relativistic treatments of
electromagnetic theory in moving dielectric bodies, and the results of interest
here are obtained by setting the material velocity equal to zero. The Minkowski
and Abraham formalisms give identical expressions for the electromagnetic stress
and energy density. Equation (2.1) now becomes
30 3
E,
(2.2)
Tzz = U = 12 (30 3E 2 + m0 mH 2 ) = 30 3E 2 = m0 mH 2 , as H =
m0 m
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1829
where 3 is the relative electric permittivity and m is the relative magnetic
permeability of the material. The energy flow, or Poynting vector, is also the
same in both formalisms, given by
cU
30 3 2
√
S=E×H=
(2.3)
E z̃ =
z̃, where h = 3m
m0 m
h
is the usual refractive index of a dielectric and z̃ is a unit vector. However, the
momentum density vectors (momentum per unit volume) are different, given by
GM = D × B = 30 3m0 mS =
h 2
c
S and
GA =
E×H
S
= 2
2
c
c
(2.4)
for the Minkowski and Abraham theories, respectively. Their magnitudes are
expressed as
GM =
hU
c
and GA =
U
,
hc
(2.5)
with the use of equations (2.2) and (2.3). These forms are used in our analysis
below and they are discussed in detail by Baxter & Loudon (2010). Both
expressions reduce to U /c in free space.
3. Radiation pressure in the solar system
His first publication on radiation pressure (Poynting 1903) is remarkable partly
for the choice of journal, The Inquirer, a Unitarian periodical still in existence
today. The choice was possibly influenced by his religious background and by the
wish to write an article accessible to the layman. Poynting outlined the thenrecent laboratory experiments (Lebedew 1901) and described them as a triumph
for the theoretical predictions of Maxwell (1873). He also considered the solar
radiation pressure, which is much smaller than the solar gravitational force for
the Earth. However, he pointed out that their relative sizes are reversed for smaller
particles sufficiently close to the Sun, with astronomical consequences.
Poynting (1904a) later suggested that the solar radiation pressure would cause
small orbiting particles gradually to lose momentum and plunge into the Sun. The
effect arises because light from the Sun impinges on the particle not directly sideon, but slightly on its leading surface. The absorbed light energy is subsequently
re-radiated isotropically, thus leading to a net force with component against the
direction of motion. The magnitude of the retarding force is proportional to the
rate of energy re-radiation in Poynting’s calculation. He also published shorter
papers on the same topic in the following two years (Poynting 1904b,c, 1905b,
1906a). The gravitational force and the energy radiation rate are proportional to
the cube and square, respectively, of the particle’s linear dimension. The retarding
effect is thus relatively greater for smaller particles. Poynting’s original paper
contained a number of important typographical mistakes, which were corrected
in his collected papers (Shakespear & Barlow 1920). However, the calculations are
not easy to follow and Poynting’s own in situ notes and corrections can add to the
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confusion. The several criticisms and the contributions by later authors, including
Larmor (1912, 1918), were carefully evaluated and summarized by Robertson
(1937).
Robertson also provided a fuller treatment of the problem consistent with
relativity theory, and he established the existence of a drag of the same nature
as that predicted by Poynting on classical grounds. This radiation-pressureinduced retardation is now known as the Poynting–Robertson effect or Poynting–
Robertson drag. The drag component of the radiation-pressure force has the form
Fdrag = −
S sv
r2
∼
,
c2
r2.5
(3.1)
where S is the Poynting vector of the solar radiation, s is the particle cross
section, v is its velocity, r is the particle radius and r is its orbital radius. The
improved expressions (Robertson 1937) for the retarding force on a spherical
particle reflecting or absorbing/re-emitting the radiation received from the Sun
are modified by numerical factors from those obtained both by Poynting and
Larmor, even in the approximation where only terms of first order in the particle
velocity are retained. Nevertheless, despite these corrections, it was Poynting,
who first identified the force on a particle in the solar system as the sum of
gravitational, radiation-pressure and momentum-drag components.
More recent work (Burns et al. 1979) on the Poynting–Robertson effect has
clarified and simplified the Robertson derivation. The earlier expression was also
generalized to allow for the physical properties of interplanetary dust available
from subsequent measurements.
4. Theory and experiments on the linear momentum of light
This section reviews Poynting’s contributions to the measurement and theory of
the radiation pressure or linear momentum of light in free space and in dielectric
media. Much of his work provided the first detailed accounts of the radiationpressure phenomena he studied.
With the electromagnetic momentum flux in free space equal to the energy
density, Poynting (1905a) quoted theoretical values of
normal pressure : (1 + R)U cos2 q
and
tangential stress :
1
(1
2
− R)U sin 2q
(4.1)
for the components of the force exerted by light incident at angle q to the normal
on a surface that reflects a fraction R of the incident intensity. These results are
straightforwardly calculated by resolution of the incident momentum flux U cos q
on unit area of surface into its normal and tangential components. Insertion of
the reflected contributions with appropriate signs then leads to equation (4.1).
The tangential stress, more easily measured than the normal pressure, vanishes
for perfect reflection and it has a maximum value of U /2 for total absorption of
light incident at 45◦ . These predictions were confirmed to reasonable accuracy in
experiments with light incident on blackened and silvered discs hung on quartz
fibres, all performed with the assistance of Guy Barlow. They used light from a
Nernst lamp, which consists of a ceramic rod electrically heated to incandescence
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Review. Poynting on radiation pressure
(Mendelssohn 1973). More details of the experiments are given in Poynting
(1910c, pp. 53–58), where an improved 5 per cent accuracy is reported. This
work confirmed the free-space value of radiation pressure.
A second paper (Poynting 1905b) is important for its treatment of the radiation
pressure within a dielectric material. It presented the first ever such discussion,
predating the more comprehensive calculations (Minkowski 1908; Abraham 1909,
1910). Poynting also discussed the pressures associated with different kinds of
wave motion, particularly elastic waves, and the effects of radiation pressure in
the solar system. He argued that because ‘the pressure is the momentum given
out or received per second, and the pressure is equal to the energy density
in the train, the momentum density is equal to the energy-density ÷ wavevelocity’. The momentum density for electromagnetic waves is then the same
as the Minkowski value from equation (2.5), and this assumption is inherent in
Poynting’s subsequent calculations.
Poynting’s main concern, however, was with the reflection and transmission of
light for general incidence angles at the interface between two transparent media,
both assumed to be non-magnetic with m = 1. We consider here only normal
incidence, where equations (2.2) and (2.3) give
Tzz = U = 30 h2 E 2
and
S = 30 chE 2 =
cU
,
h
where
h=
√
3.
(4.2)
Thus, with the radiation pressure equal to the electromagnetic stress or energy
density as shown above, the pressure on the interface, determined by the
stresses (4.2) in the two media with signs appropriate to the propagation
directions of the incident, reflected and transmitted light beams, is
Tinter = (1 + R)U − U =
2U (h − h )
.
(h + h )
(4.3)
The variables in transmission and incidence media are here denoted by primed
and unprimed quantities, respectively, and the reflection coefficient is given by the
usual expression (Jackson 1999). The force on the interface is thus predicted to
act outwards from the material with the higher refractive index, as the push from
the reflected light is overwhelmed by the pull from the transmitted light. The
magnitude of Tinter /U for h /h = 1.5 agrees with the tabulated value (Poynting
1905b) of 0.4 for q = 0, but equation (4.3) did not appear in his paper. The
values tabulated for oblique incidence agree with a generalized version of the
theory outlined above (L. Allen 1994, personal communication). The expression in
equation (4.3) is again based on a choice equivalent to the Minkowski momentum,
and this result is criticized in §6.
Poynting (1905b) also reported two further measurements of the free-space
momentum associated with a light beam, performed in collaboration with Guy
Barlow. Thus, one end of a suspended rectangular glass block was illuminated
by a beam that is twice internally reflected before its emergence from the other
end of the block. A more accurate experiment (Poynting 1905b, 1910c) used a
pair of glass prisms attached to a brass arm suspended from a fibre. Further
measurements by Barlow (1912) had the glass block or prism pair replaced by a
glass cube. In all three versions, the double internal reflection within the glass
block or prism pair ensured that the momentum stream of the transmitted beam
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R. Loudon and C. Baxter
had a parallel shift from that of the incident beam, equivalent to a counterclockwise couple on the light beam. The observed clockwise reaction couple on the
glass in the series of experiments was in increasing agreement with calculations.
The other Poynting papers that include material on radiation pressure in the
laboratory environment consist of a review article (Poynting 1906a), an abstract
of a review lecture (Poynting 1906b), a conference review talk (Poynting 1909a)
substantially repeated in French (Poynting 1910b) and a preliminary account
(Poynting 1909b) of material later expanded (Poynting 1910a). The latter paper
reproduces the 1910 Royal Society Bakerian lecture by Poynting and Barlow,
where they considered the recoil momentum imparted to a source in the emission
of light. They presented the theory of a set of experiments in which a beam of
light fell on one of an array of four thin discs mounted in a mica plate suspended
from a quartz fibre. The discs had front and back surfaces that were either totally
absorbing (blackened: B) or totally reflecting (silvered: S). Schematically,
⎫
(1) (2) (3) (4) ⎪
⎪
⎬
P → B|B B|S S|S S|B
(4.4)
⎪
5P
⎪
2P 2P ⎭
P
3
showing the direction of a normally incident beam of light that exerts a steady
pressure P on the front surface of the disc. Each disc had an asphaltum layer
heated by the incident light until a steady state was reached. With the front and
back surfaces at the same temperature, the layer then acted as a source of reradiated energy and produced a recoil pressure on the disc. The calculated values
in the last line of equation (4.4) are obtained as follows: disc (1) has equal amounts
of re-radiation from its two sides and only the incident pressure contributes, disc
(2) re-radiates only from its front side with a cosine-law spatial distribution that
adds 2P/3 to the incident pressure, discs (3) and (4) do not absorb and the
pressure is that of specular reflection from their front surfaces. Although there
was reasonable agreement only between calculated and measured values, this is an
important experiment as it clearly demonstrated the presence of recoil momentum
associated with re-emission of radiation. A particularly clear and succinct account
of the work occurs in Poynting’s (1910b) review lecture. This is yet another
experiment that confirmed the free-space value of the radiation pressure.
His last publication (Poynting 1911) on radiation pressure calculated the
strength of the longitudinal elastic waves generated by the pressure from an
electromagnetic plane wave in a dielectric. The pressure is proportional to
the square of the field, as in equation (2.2), and the elastic wave had twice
the frequency of the light wave. He showed that the energy content of the
material waves was an infinitesimal fraction of the energy in the electromagnetic
wave itself.
Remarkably, none of the later Poynting papers reviewed here makes any
reference to the work of Minkowski (1908) and Abraham (1909, 1910) on the
electromagnetic energy and momentum densities in dielectric materials. It may
be that papers published in Göttingen or Palermo in German or Italian were not
easily available or understandable in Birmingham. However, in compensation,
although Minkowski and Abraham both mention the Poynting vector, neither of
them refers to the relevant papers of Poynting on radiation pressure.
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5. Optical angular momentum
The notion that a beam of circularly polarized light carries angular momentum
was an original contribution by Poynting, apparently not previously mentioned
in the literature. Much of his single publication on this topic (Poynting 1909c)
presented a mechanical analogue of the optical effect, in which he treated the
propagation of torsional elastic waves in a cylindrical shaft. The calculation
followed his work (Poynting 1905b) on the linear pressure associated with
transverse waves in elastic solids, mentioned above. It led to the expression
cU
lU
=
(5.1)
2p
u
for the torque exerted on unit area of an absorbing surface by a circularly
polarized free-space light beam of wavelength l. The paper ends with an ingenious
proposal for the practical measurement of optical angular momentum by the
passage of a plane-polarized beam upwards through a stack of quarter-wave
plates. Each of the odd-numbered set of alternate plates, suspended from a
torsion fibre, generates circular polarization and therefore experiences a torque.
Each of the interleaving even-numbered fixed set of plates restores the linear
polarization. The addition of all the torques on the suspended set of plates thus
amplifies the angular momentum transfer. However, Poynting concluded that
‘. . . even with such multiplications, my present experience of light forces does
not give me much hope that the effect could be detected . . .’.
Despite Poynting’s pessimism concerning the practical detection of optical
angular momentum, it was successfully measured by Beth (1936); this paper and
that of Poynting (1909c) are the first two in a modern compilation (Allen et al.
2003). The experiment followed Poynting’s original idea, except that Beth used a
single 25 mm half-wave plate with direct and reflected light from a filament lamp.
The measured angular momentum of circularly polarized light agreed with the
prediction in equation (5.1).
G=
6. Epilogue
Studies of the radiation pressure and torque, or equivalently the optical linear
and angular momenta associated with light beams have burgeoned in the 100
years following Poynting’s pioneering contributions, particularly since about 1970.
Much of the theoretical and experimental work on radiation pressure has been
motivated by an interest in the fundamental nature of the effect. There are
also important practical applications, as in the development of the photon-drag
detector (Gibson et al. 1970), which relies on the transfer of momentum from
light to the electrons and holes in semiconductors and provides a useful highspeed room-temperature device for optical intensity measurements. Radiation
pressure and torque facilitate the manipulation of particles in suspension, both
linear, as in the optical tweezers of Ashkin et al. (1986), and rotational, as in
the optical spanner of He et al. (1995), Simpson et al. (1997) and Friese et al.
(1998). Radiation-pressure effects are also important in the practical use of global
positioning systems (Pratap & Enge 2001; Leick 2004), where solar and albedo
perturbing forces provide the largest error in determining their accuracy. The
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optical stretcher (Guck et al. 2002) uses two counter-propagating laser beams
to trap and stretch individual biological cells. Finally, details of light-driven
micromachines are provided by Rubinzstein-Dunlop & Friese (2002), and there
has been newspaper coverage (e.g. Guardian and Financial Times) of proposals
for spacecraft driven by the radiation pressure of sunlight.
Nevertheless, one may reasonably wonder at the sheer volume of work on a
quite narrow research topic, as surveyed in a series of detailed review articles
(Robinson 1975; Brevik 1979; Pfeifer et al. 2007; Baxter & Loudon 2010;
Mansuripur 2010; Milonni & Boyd 2010; Kemp 2011). A main reason for
the popularity of radiation-pressure studies lies in a conflict that developed
between the physical interpretations of the two forms of momentum density
in equation (2.5), known as the Abraham–Minkowski controversy. The work of
Poynting represents the first blow in a debate that has been continued to the
present time by many of the papers reviewed in the above articles. The currently
available experiments predominantly measure the Minkowski momentum, the
result assumed by Poynting. However, several later calculations show that an
Abraham momentum transfer should occur in some interactions between light
and dielectrics, often for transfers that are difficult to measure in practice.
The interpretation of radiation pressure at the time of Poynting was unaffected
by the first developments of the quantum and relativity theories. However,
modern discussions of radiation pressure and electromagnetic field momentum
involve quantum and relativistic ideas. The quantum theory initiated by Planck
in 1900 showed that a quantization of the energy of harmonic motion in units
of h̄u accounted for the observed spectral distribution of radiation by a thermal
source. Many recent publications on radiation pressure describe light in terms of
the photon, its quantum of energy as named by Lewis in 1928. The equivalence
of mass and energy identified by Einstein in 1905 has clear relevance to the
propagation of energy and momentum by light, important in radiation-pressure
theory. Einstein (1917) also introduced the momentum of the light quantum, with
magnitude
h̄u
(6.1)
p=
c
in free space, equivalent to the de Broglie relation between particle wavelength
and momentum. This is analogous to the classical electromagnetic momentum
density U /c familiar to Poynting, but the photon momentum p has the nature
of a particle momentum rather than a field momentum density. Equation (5.1)
converts similarly to the angular momentum carried by a single photon, analogous
to the single-photon linear momentum in equation (6.1), as
s = h̄,
(6.2)
where the conventional s notation reflects the spin-like nature of circular
polarization. Beth (1936) himself expressed the results of his measurements as
assigning an angular momentum ±h̄ to each photon of left or right circularly
polarized light, respectively.
The dielectric momentum densities in equation (2.5) convert to the singlephoton momenta
hh̄u
h̄u
and pA =
.
(6.3)
pM =
c
hc
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1835
The controversy between the two formulations is conveniently discussed in terms
of these single-photon values, with one of them sometimes claimed to be correct
and the other as incorrect. The arguments in favour of one or the other are
summarized by Barnett & Loudon (2010). At the simplest level, the Minkowski,
but not the Abraham, momentum satisfies the de Broglie relation between particle
momentum and wavelength. However, the Abraham momentum is supported
by a general condition for conservation of the centre of mass energy when a
single-photon pulse is transmitted through a block of transparent medium with
antireflection coatings (Balazs 1953). The condition for an unchanged centre of
mass energy relative to an identical pulse that travels the same distance outside
the block requires a displacement of the block, equivalent to its acquisition of a
linear momentum
1 h̄u
h̄u
, so that pblock + pA =
(6.4)
pblock = 1 −
h c
c
during transmission of the photon. The total momentum with the pulse inside
thus equals the photon momentum in free space before and after its transmission.
A resolution of the conflict between the two momenta by Barnett (2010)
identifies the Abraham form as the kinetic momentum, which is associated
with the motion of a dielectric specimen as a whole. This association is clearly
illustrated by the example in equation (6.4), where the block displacement is
a consequence of a velocity imparted to it and hence a kinetic momentum.
The Minkowski form is identified as the canonical momentum, which is
associated with the motion of objects embedded in the dielectric (Barnett
2010). The accompanying de Broglie relation led Schrödinger to represent the
quantum-mechanical momentum by the operator −i h̄V, and the usual position–
momentum uncertainty relation then follows. This canonical momentum operator
is a generator of translations and, for a body immersed in a transparent
medium, Minkowski provides the relevant momentum, as observed in a range
of experiments. The original Barnett paper should be consulted for a full
understanding of the two momenta, and there is also useful subsequent discussion
of their subtleties (Barnett & Loudon 2010; Baxter & Loudon 2010; Milonni &
Boyd 2010; Kemp 2011). In summary, the expressions attributed to Minkowski
and Abraham in equation (6.3) represent different physical forms of momentum,
and both are correct.
The very early contribution by Poynting (1905b) should now be evaluated
against the background of these two varieties of momentum measurement. More
recent calculations of radiation pressure are based on evaluations of the Lorentz
force (Loudon 2002), a method initiated long ago by Lorentz himself (Lorentz
1909, §§23 and 24). These calculations find that the force on a dielectric interface
acts into the material with the higher refractive index, consistent with the
Abraham form of momentum in the dielectric. A simple example is provided
by the transmission of a single-photon pulse through an antireflection-coated
surface, where a calculation based on the Lorentz force leads to the same results
as in equation (6.4). Thus, pblock is the momentum transferred to a dielectric
as the pulse passes through its surface and pA is the momentum of the photon
within the block. This inward force on the material disagrees with the outward
force predicted by Poynting in equation (4.3) and with his assumption of a
momentum equivalent to the Minkowski form inside a material. However, the
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1836
R. Loudon and C. Baxter
measured forces on mirrors suspended in dielectric liquids (Jones & Leslie 1978)
and on charge carriers in semiconductors (Gibson et al. 1970) are consistent
with the Minkowski form of photon momentum, as taken effectively by Poynting,
and these observations agree with the Lorentz-force calculations (Loudon 2002;
Loudon et al. 2005).
Poynting’s results for the three areas of radiation-pressure theory and
experiment covered in §§3–5, frequently the first contributions in their fields,
have largely withstood the tests of time. For the solar system, he was the
first to propose the effect now known as the Poynting–Robertson drag, a force
important in the motions of small particles closer to the Sun. His estimate of its
size was in error by a numerical factor, but its nature was correctly identified.
For the radiation pressure or linear momentum of light, Poynting presented the
theories for a range of laboratory experiments. These were mainly carried out in
collaboration with Guy Barlow and they verified the size of the optical momentum
in free space, whose flux equals the energy density U and whose density equals
U /c. Poynting was also the first to recognize the property that circularly polarized
light carries an angular momentum. His calculated value of the associated torque,
given in equation (5.1), agrees with that currently accepted. Poynting’s papers
are regularly cited in relevant current publications and his work from at least 100
years ago remains a most important contribution to radiation-pressure studies.
We thank Professor Les Allen for communicating his careful analysis of the calculations outlined
in Poynting (1905b), and also the anonymous referees whose suggestions significantly improved our
presentation of Poynting’s work.
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