Enlighten – Research publications by members of the University of Glasgow
http://eprints.gla.ac.uk
Yao, A.M., and Padgett, M.J. (2011) Orbital angular momentum: origins,
behavior and applications. Advances in Optics and Photonics, 3 (2). p.
161. ISSN 1943-8206
http://eprints.gla.ac.uk/67185/
Deposited on: 17
th
July 2012
Orbital angular momentum: origins,
behavior and applications
Alison M. Yao
1
and Miles J. Padgett
2
1
SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG,
Scotland, UK
2
Department of Physics and Astronomy, SUPA, Kelvin Building, University of
Glasgow, Glasgow G12 8QQ, Scotland, UK (m.padgett@physics.gla.ac.uk)
Received October 8, 2010; revised January 5, 2011; accepted January 5, 2011;
published May 15, 2011 (Doc. ID 136333)
As they travel through space, some light beams rotate. Such light beams have
angular momentum. There are two particularly important ways in which a light
beam can rotate: if every polarization vector rotates, the light has spin; if the
phase structure rotates, the light has orbital angular momentum (OAM), which
can be many times greater than the spin. Only in the past 20 years has it been
realized that beams carrying OAM, which have an optical vortex along the
axis, can be easily made in the laboratory. These light beams are able to spin
microscopic objects, give rise to rotational frequency shifts, create new forms
of imaging systems, and behave within nonlinear material to give new insights
into quantum optics.
c
2011 Optical Society of America
OCIS codes: 050.4865, 260.6042
1. The Origins and History of Orbital Angular Momentum ..........163
1.1. Electromagnetic Fields to Carry Angular Momentum .........163
1.2. History of Spin and Orbital Angular Momenta ...............165
1.3. Orbital Angular Momentum and Phase Singularities .........166
2. Generation of Helically Phased Beams ............................168
2.1. Spiral Phase Plates ...........................................168
2.2. Laguerre–Gaussian Modes ...................................169
2.3. Diffractive Optical Elements for Generating Orbital Angular
Momentum ..................................................170
2.4. Mode Converters Formed from Cylindrical Lenses ...........172
2.5. Coherence Requirements for Beams Carrying Orbital Angular
Momentum ..................................................173
2.6. Orbital Angular Momentum beyond Light ...................175
3. Interaction of Helically Phased Beams with Matter ...............176
3.1. Observing the Angular Momentum of Light ..................176
3.2. Mechanisms for Angular Momentum Transfer ...............177
Advances in Optics and Photonics 3, 161–204 (2011) doi:10.1364/AOP.3.000161
1943-8206/11/020161-44/$15.00
c
OSA
161
3.3. Optical Momentum to Drive Micromachines .................178
3.4. Orbital Angular Momentum and the Interaction with Cold
Atoms .......................................................179
4. Analogous Representation and Effects for Helically Phased and
Polarized Beams .................................................180
4.1. Poincar
´
e Sphere for Orbital Angular Momentum Modes ..... 180
4.2. Rotational Doppler Shift .....................................182
4.3. Mechanical Faraday Effect and Image Drag ..................183
5. Orbital Angular Momentum in Nonlinear and Quantum Optics ..183
5.1. Optical Vortices and Orbital Angular Momentum in Kerr
Media .......................................................183
5.2. Second-Order Nonlinear Interactions ........................ 185
5.3. Parametric Down Conversion with OAM Beams .............186
5.4. Quantum Entanglement of Orbital Angular Momentum ......187
5.5. Logic Operations with Orbital Angular Momentum ..........189
6. Measuring the Orbital Angular Momentum of Light ..............189
6.1. Forked Diffraction Gratings to Measure Orbital Angular
Momentum ..................................................189
6.2. Measuring OAM by Interferometry ..........................190
6.3. Orbital Angular Momentum and Diffraction by Apertures and
Angular Uncertainty .........................................191
6.4. Measuring Orbital Angular Momentum by Image
Reformatting ................................................192
6.5. Use of Orbital Angular Momentum in Imaging ...............193
7. Reflections on the Contribution of and Future Opportunities for
Orbital Angular Momentum .....................................195
Acknowledgments ..................................................195
References and Notes ...............................................195
Advances in Optics and Photonics 3, 161–204 (2011) doi:10.1364/AOP.3.000161 162
Orbital angular momentum: origins,
behavior and applications
Alison M. Yao and Miles J. Padgett
1. The Origins and History of Orbital Angular
Momentum
1.1. Electromagnetic Fields to Carry Angular Momentum
Most scientists realize that light carries a linear momentum equivalent
to
¯
hk
0
per photon and, if circularly polarized, a spin angular momentum
(SAM) of ±
¯
h per photon. In 1992, Allen et al. recognized that light beams
with an azimuthal phase dependence of exp(iφ) carry an orbital angular
momentum (OAM) that can be many times greater than the spin [1] and
that such beams were readily realizable. This OAM is completely distinct
from the familiar SAM, most usually associated with the photon spin,
that is manifest as circular polarization [2]; see Fig. 1.
The relationship between linear and angular momentum is a simple one;
L = r ×p, where r is the particle’s position from the origin, p = mv is its
linear momentum, and ×denotes the cross product. For example, a laser
pointer shone at a door can exert a torque about the hinge, albeit not
usually enough to open it! However, when we discuss orbital angular
momentum we refer instead to an angular momentum component
parallel to the propagation direction, z, and hence for which r × p is
notionally zero [3]. In relation to the door example given above, we seek
to identify an angular momentum capable of twisting the door knob.
Any angular momentum component in the z direction, by definition,
requires a component of linear momentum in the x, y plane, i.e., a light
beam with transverse momentum components. The angular momentum
density, j, is related to the linear momentum density p = ε
0
E×B through
j = r × p, where ε
0
is the dielectric permittivity, and E and B are the
electric and magnetic fields, respectively. The linear momentum of a
transverse plane wave is then in the propagation direction, z, and there
cannot be any component of angular momentum in the same direction.
Hence it follows that, at the most fundamental level, an angular
momentum in the z direction requires a component of the electric and/or
magnetic field also in the z direction. One sees immediately that, even if
circularly polarized, a plane wave cannot carry an angular momentum of
any type. This last statement has led to some debate, but the resolution
of this seeming paradox is simply that the perfect plane wave is only
ever found in textbooks. Real beams are limited in extent either by
the beams themselves or by the measurement system built to observe
Advances in Optics and Photonics 3, 161–204 (2011) doi:10.1364/AOP.3.000161 163
Figure 1
The spin angular momentum (SAM) of light is connected to the polarization of
the electric field. Light with linear polarization (left) carries no SAM, whereas
right or left circularly polarized light (right) carries a SAM of ±
¯
h per photon.
them, and this finite aperture always gives rise to an axial component
of the electromagnetic field [4]. For the case of circular polarization,
the axial component of the electromagnetic field is an unavoidable
consequence of the radial gradient in intensity that occurs at the edge of
the beam or the measurement system. A detailed treatment of these edge
effects, for any arbitrary geometry, always returns a value of the angular
momentum, when integrated over the whole beam, of ±
¯
h per photon [4]
for right-handed and left-handed circular polarization, respectively.
Strangely, the origin of OAM is easier to understand. The simplest
example of a light beam carrying OAM is one with a phase in the
transverse plane of φ(r,φ) = exp(iφ), where φ is the angular coordinate
and can be any integer value, positive or negative. As shown in Fig. 2,
such beams have helical phase fronts with the number of intertwined
helices and the handedness depending on the magnitude and the sign
of , respectively ( = 3 corresponds to a pasta fusilli). One can see
immediately that an electromagnetic field transverse to these phase
fronts has axial components. Equivalently, the Poynting vector, which
is at all times parallel to the surface normal of these phase fronts,
has an azimuthal component around the beam and hence an angular
momentum along the beam axis.
It was a breakthrough of the 1992 paper [1] that they recognized
that all helically phased beams carried an OAM equivalent to a value
of
¯
h per photon. However, perhaps what is most surprising is not
that helically phased beams carry an angular momentum—a simple
ray-optical picture suggests just that from the azimuthal component of
the momentum flow—but that this OAM, just like spin, should be in
units of
¯
h.
That the OAM should be quantized in units of
¯
h follows from a simple
geometrical argument. At a radius r, the inclination of the phase front,
and hence of the Poynting vector, with respect to the beam axis is simply
λ/2π r. This, in turn, sets the azimuthal component of the light’s linear
momentum as
¯
hk
0
λ/2π r per photon [5], which, when multiplied by
the radius vector, gives an angular momentum of
¯
h per photon [6].
For comparison, we note that a circular path of circumference λ has a
radius of λ/2π. A linear momentum of
¯
hk
0
directed around this circle
gives an angular momentum of
¯
h, i.e., the SAM of the photon. Within the
Advances in Optics and Photonics 3, 161–204 (2011) doi:10.1364/AOP.3.000161 164