We review the transport and transfer of momentum and angular momentum by electromagnetic waves, and applications of the mechanical effects of scattering

Heckenberg, Norman R.

Nieminen, Timo A.

Rubinsztein-Dunlop, Halina

Stilgoe, Alexander B.

T. A. Nieminen

A. B. Stilgoe

N. R. Heckenberg

H. Rubinsztein-Dunlop

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Not just energy, but momentum and angular momentum too: Mechanical effects in scattering

University of Queensland eSpace

DOI: 10.1478/C1V89S1P010
AAPP | Atti della Accademia Peloritana dei Pericolanti
Classe di Scienze Fisiche, Matematiche e Naturali
ISSN 1825-1242
Vol. 89, Suppl. No. 1, C1V89S1P010 (2011)
NOT JUST ENERGY, BUT MOMENTUM AND ANGULAR
MOMENTUM TOO: MECHANICAL EFFECTS IN SCATTERING
TIMO A. NIEMINEN,a∗ ALEXANDER B. STILGOE,a
NORMAN R. HECKENBERG,a AND HALINA RUBINSZTEIN-DUNLOP a
ABSTRACT. We review the transport and transfer of momentum and angular momentum
by electromagnetic waves, and applications of the mechanical effects of scattering.
1. Introduction
The transport of energy by light is a widely-experienced fundamental aspect of electro-
magnetic waves, familiar to to all who use lasers, cook with microwaves, or perform the
simple act of standing in sunlight.
What is less well-known is the transport of momentum and, especially, angular mo-
mentum. In concept, the transport of momentum and the consequent forces exerted by
radition are old ideas, dating to at least Kepler’s obervations of and speculations on comet
tails [1]. With the development of improved experimental methods, attempts were made
to measure such forces, for example by John Michell in the 1700s, a pioneer of the torsion
balance, whose attempt failed to do other than demonstrate that concentrated sunlight can
destroy one s experimental apparatus [2]. The forces involved are small, and success was
not achieved until Lebedev [3] and Nichols and Hull [4, 5].
Astrophysical implications were realised, such as the Poynting Robertson effect, and
the importance of radiation pressure in the support of stellar atmospheres against gravity.
Terrestrial applications remained elusive, since the forces resulting from optical or electro-
magnetic radiation pressure are small compared to the power required, and are insufficient
to overcome gravitational or frictional forces in most circumstances.
This changed in 1969 when Arthur Ashkin realized that, although the forces resulting
from electromagnetic radiation were small, the forces required to move small particles
were also small [6]. This prompted experiments demonstrating the acceleration and trap-
ping of small particles by radiation pressure [7, 8].
The ability to non-invasively trap and manipulate microscopic particles made optical
tweezers attractive to biologists, and the new possibility of measuring forces in micro-
scopic biological systems was a major advance, enabled a wide variety of measurements
in biophysics and other fields [9, 10, 11].
C1V89S1P010-2 T. A. NIEMINEN ET AL.
Meanwhile, rotational manipulation in optical tweezers was demonstrated by Sato [12],
and led to a wide range of applications of rotation, including practical methods for mi-
crorheology [13, 14]. Some of these applications involved the design and production
of microscopic objects specially designed to produce torque—optically-driven microma-
chines [15, 16].
2. Energy, momentum, and angular momentum
For a mathematical treatment of the transport of energy by electromagnetic fields, we
can consider the work done on the currents within a volume by the electric field E, and
show energy flux density (ie, the power per unit area) is equal to the Poynting vector
S = E×H. (1)
For a monochromatic wave, especially at optical frequencies, we are usually interested in
the time-averaged Poynting vector in terms of the complex amplitudes rather than the real
fields, which is equal to
S =
1
2
E×H⋆ (2)
where ⋆ denotes the complex conjugate. For a plane electromagnetic wave, the Poynting
vector is always in the direction of propagation of the wave, and has a magnitude of
S =
cϵ
2n
|E|2. (3)
If the fields are known, this allows us to calculate the difference between the energy carried
inwards by incoming radiation and the energy carried outwards by outgoing radiation, most
simply done, for a steady-state case, in the far field. The difference between these two
fluxes is the rate at which energy is transferred to the scatterer—usually the rate at which
it is heated by the incident light. This is well-known, and widely taught.
The theoretical treatment of the transport of momentum was initiated by Maxwell s
predictions of radiation pressure [17, 18], and followed by a thorough treatment a decade
later by Poynting [19] and Heaviside. The general principles of transport of energy in
media by waves were first formulated by Umov in 1874 (and one finds the Poynting vector
called the Umov or Umov–Poynting vector in the Russian literature), who showed that the
transport of momentum was a thermodynamically necessary consequence of the transport
of energy, with the momentum flux density p determined by the energy flux density and
the speed of transport of energy. The momentum transported per unit area in the direction
of propagation of the wave is
p = nS/c. (4)
The magnitude of the momentum flux is numerically equal to the energy density, since
both are related to the energy flux by the same speed. From a modern perspective, this
can be described in terms of the energy–momentum 4-vector, where the components cor-
responding the the momentum will be zero if the energy is stationary, and non-zero in a
frame where the energy is moving.
Noting the dependence of the momentum flux on refractive index n in equation (4),
we can recall the century-old Abraham–Minkowski controversy over the momentum of an
electromagnetic wave in matter—whether the momentum flux density is nS/c, S/(nc),
Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat., Vol. 89, Suppl. No. 1, C1V89S1P010 (2011) [4 pages]
NOT JUST ENERGY, BUT MOMENTUM AND ANGULAR MOMENTUM TOO . . . C1V89S1P010-3
or some other value. The relationships between energy density, flux, and transport speed
resulting in (4) are valid for all waves (and indeed for other forms of transport of energy,
such as thermal conduction etc.), and this momentum flux must be the total momentum
flux associated with the electromagnetic wave [20].
Finally, any transport of momentum implies a transport of angular momentum as well.
While it is tempting to simply write the angular momentum flux density j as the moment
of the linear momentum flux density,
j = r× p = nr× S/c, (5)
this is not correct—electromagnetic waves can carry spin angular momentum, or intrinsic
angular momentum—angular momentum that is independent of the choice of origin about
which the moments are taken, and thus independent of r. In general, the total angular
momentum will be the sum of the spin s and orbital l angular momenta,
j = l+ s. (6)
Curiously, however, while the moment of the linear momentum density cannot be the
angular momentum density, it is known that for fields finite in extent, the total angular mo-
mentum is given by the integral over the entire field of the moment of the linear momentum
density. This integral can also be written in terms of spin and orbital angular momentum
densities [21]. Unsurprisingly, this has led to yet another century-old controversy, with the
fires kept well-stoked by the fuel of optical rotation in laser trapping [22, 23]. For practical
purposes, it should suffice to note that there is a clear physical distinction between spin
and orbital angular momenta for time-harmonic waves the spin can be determined from
the Stokes parameters, and the orbital angular momentum cannot [24].
3. Conclusion
The transport of energy, momentum, and angular momentum by electromagnetic fields
can be directly calculated from the fields, if the fields are known. Therefore, the absorbed
power, the force, and the torque resulting from scattering can be calculated if the fields
can be calculated—that is, if the problem at hand is amenable to computational modelling.
This can enable a new level of theoretical understanding, and allows the quantitative design
and engineering of optically-driven micromachines. In turn, this has driven work on the
modelling of a range of complex structures [25]. In addition, a range of new technologies,
especially involving rotation, are reaching the stage of practical use [11]. The story, of
course, does not end here, as this is an active and developing field, although our review of
this field must.
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a The University of Queensland
Quantum Science Laboratory
School of Mathematics and Physics
Brisbane, QLD 4072, Australia
∗ To whom correspondence should be addressed | Email: timo@physics.uq.edu.au
Paper presented at the ELS XIII Conference (Taormina, Italy, 2011), held under the APP patronage;
published online 15 September 2011.
© 2011 by the Author(s); licensee Accademia Peloritana dei Pericolanti, Messina, Italy. This article is
an open access article, licensed under a Creative Commons Attribution 3.0 Unported License.
Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat., Vol. 89, Suppl. No. 1, C1V89S1P010 (2011) [4 pages]


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Not just energy, but momentum and angular momentum too: Mechanical effects in scattering

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