light's orbital angular momentum momentum

Like all wave phenomena, light has mechanical proper-ties. Johannes Kepler suggested that comet tails al-

ways point away from the Sun because light carries linearmomentum. In 1905, John Poynting developed the theoryof electromagnetic radiation pressure and momentum den-sity and, in 1921, Albert Einstein showed that Planck’sblackbody law and the motion of molecules in a radiationfield could be explained if the linear momentum of a pho-ton is \k. (The wave number k = 2p/l and \ = h/2p, wherel is the wavelength and h is Planck’s constant.) In moderntimes, light’s linear momentum has been directly exploitedfor trapping and cooling atoms and molecules.

It was also Poynting who, in 1909, realized that po-larized light has angular momentum—spin angular mo-mentum, associated with circular polarization. For a sin-gle photon, it has a value of �\. The idea of light’s orbitalangular moment came only much later: In 1992, a groupat Leiden University in the Netherlands that included oneof us (Allen) recognized that light beams with an az-imuthal phase dependence of exp(⊗i�v) carries an angu-lar momentum independent of the polarization state.1 Theangle v is the azimuthal coordinate in the beam’s crosssection, and � can take any integer value, positive or neg-ative. This orbital angular momentum, they predicted,would have a value of L = �\ per photon. Just as with cir-cularly polarized light, the sign of the orbital angular mo-ment indicates its handedness with respect to the beamdirection.

For any given �, the beam has � intertwined helicalphase fronts, as illustrated in figure 1. A feature of helicallyphased beams is that the phase singularity on the beamaxis dictates zero intensity on the axis. Therefore the cross-sectional intensity pattern of all such beams has an annu-lar character that persists no matter how tightly the beamis focused. The on-axis singularity is a specific instance ofphase dislocation, the general literature for which is recent,extensive, and beyond the scope of this article.2

The concept of optical orbital angular momentum oflight is not altogether new. It is well known that multipo-lar transitions can produce radiation that carries orbitalangular momentum. But such processes are rare and re-late, in the visible, to a few “forbidden” atomic and molec-

ular transitions. What’s new and ex-citing is that it is now possible to pro-duce, rather easily, laboratory lightbeams with quantized orbital angularmomentum. These beams can be usedto investigate all the analogues of po-larized light. For example, one canlook for a photon analogue of thespin–orbit coupling of electrons and,

quite generally, to search for new optical interactions.

Optical angular momentumTo a considerable extent, one can understand light’s mo-mentum properties without reference to photons. A care-ful analytic treatment of the electromagnetic field givesthe total angular momentum of any light field in terms ofa sum of spin and orbital contributions.1 In free space, thePoynting vector, which gives the direction and magnitudeof the momentum flow, is simply the vector product of theelectric and magnetic field intensities. For helical phasefronts, the Poynting vector has an azimuthal component,as shown in figure 1. That component produces an orbitalangular momentum parallel to the beam axis. Because themomentum circulates about the beam axis, such beams aresaid to contain an optical vortex.

The most common form of helically phased beam is theso-called Laguerre–Gaussian (LG) laser mode. In general,lasers emit a beam that gradually expands as it propa-gates. The magnitude and phase of the electric field at dif-ferent positions in the cross section are described by amode function. For most laser beams without helical phas-ing, that function is the product of a Hermite polynomialand a Gaussian. Hermite–Gaussian (HG) modes have sev-eral intensity maxima, depending on the order of the poly-nomials, arrayed in a rectilinear pattern and separated byintensity zeros.

The cylindrical LG modes have an explicit exp(⊗i�v)phase factor. That makes them the natural choice for thedescription of beams carrying orbital angular momentum.Although LG modes have been produced directly in lasersystems,3 they are more easily produced by the conversionof HG beams.

Generating the beamsSpin angular momentum depends only on the polarizationof the beam, not on its phase. Therefore both HG and LGbeams can possess spin angular momentum. Beams car-rying spin angular momentum are readily produced byusing a quarter-wave plate to convert linearly into circu-larly polarized light. The Leiden group introduced an anal-ogous trick with cylindrical lenses to transform an HGbeam with no angular momentum into a LG beam that car-ries orbital angular momentum (see figure 2).4

Although this conversion process is highly efficient,each LG mode does require a particular initial HG mode.That requirement limits the range of LG modes one canproduce. Consequently, the most common method for cre-

© 2004 American Institute of Physics, S-0031-9228-0405-010-0 May 2004 Physics Today 35

Miles Padgett and Johannes Courtial are physicists at Glas-gow University in Scotland. Les Allen holds visiting appoint-ments at the Universities of Glasgow, Strathclyde, and Sussex.

The realization that light beams can have quantized orbitalangular momentum in addition to spin angular momentumhas led, in recent years, to novel experiments in quantummechanics and new methods for manipulating microparticles.

Miles Padgett, Johannes Courtial, and Les Allen

Light’s Orbital Angular Momentum

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36 May 2004 Physics Today http://www.physicstoday.org

ating helical beams has been the use ofnumerically computed holograms. Suchholograms can generate beams with any de-sired value of orbital angular momentumfrom the same initial beam (see figure 3).The requisite hologram can be formed byrecording, onto photographic film, the in-terference pattern between a plane waveand the beam one seeks to produce. Illumi-nating the resulting hologram with anotherplane wave produces a first-order diffractedbeam with the intensity and phase patternof the desired beam.

The holographic approach can take ad-vantage of the high-quality spatial lightmodulators (SLMs) that have recently be-come available. These pixelated liquid-crystal devices takethe place of the photographic film. Furthermore, numeri-cally calculated holographic patterns can be displayed onan SLM. These devices produce reconfigurable, computer-controlled holograms that allow a simple laser beam to beconverted into an exotic beam with almost any desiredphase and amplitude structure. And the beam pattern canbe changed many times per second to meet experimentalrequirements. Figure 3 shows how a comparatively simple“forked” holographic pattern can transform the plane-wave output of a conventional laser into a pair of LG beamscarrying orbital angular momentum.5 In recent years,SLMs have been used in applications as diverse as adap-tive optics, real-time holography, and optical tweezing.

Unlike spin angular momentum, which has only twoindependent states corresponding to left- and right-handed circular polarization, orbital angular momentumhas an unlimited number of possible states, correspondingto all integer values of �. Although the link between spinangular momentum and circular polarization is clear, thelink between orbital angular momentum and other waysof describing the beam is less obvious. It’s tempting, for ex-ample, to directly associate the orbital component to the �-value of an optical vortex; but that’s wrong. Because thecenter of the vortex is a position of zero optical intensity,it carries neither linear nor angular momentum. Instead,

the angular momentum is associated with regions of highintensity, which for an LG mode is a bright annular ring.

That association is well illustrated by a recent exper-iment by Lluis Torner and coworkers at the University ofCatalonia in Barcelona, Spain.6 They showed that, afterthe beam passes through the focus of a cylindrical lens, theazimuthal component of the linear momentum near thevortex center is reversed, but the total orbital angular mo-mentum of the beam remains unchanged. The reversal ofthe vortex is simply image inversion in geometrical optics;it has no implications for orbital angular momentum.

Orbital angular momentum arises whenever a beam’sphase fronts are not perpendicular to the propagation di-rection. In the approximation of geometric optics, onewould say that the light rays that make up the beam areskewed with respect to its axis. Simplistic as it is, thisskewed-ray model predicts the correct result in most ex-perimental situations.

Measuring the angular momentum of a light beam isnot easy. The first demonstration of the transfer of spin an-gular momentum from a light beam was carried out in1936 by Richard Beth at Princeton University.7 The ex-periment was extremely demanding. A suspended quarter-wave plate took angular momentum from a circularly po-larized beam. The plate’s macroscopic size andcorresponding high moment of inertia, however, meant

Figure 1. Orbital angular momentum of alight beam, unlike spin angular momentum,is independent of the beam’s polarization. It

arises from helical phase fronts (left col-umn), at which the Poynting vector (green

arrows) is no longer parallel to the beamaxis. At any fixed radius within the beam,the Poynting vector follows a spiral trajec-

tory around the axis. Rows are labeled by �,the orbital angular-momentum quantum

number. L = �\ is the beam’s orbital angularmomentum per photon. For each �, the left

column is a schematic snapshot of thebeam’s instantaneous phase. An instant later,the phase advance is indistinguishable froma small rotation of the beam. By themselves,beams with helical wavefronts have simple

annular intensity profiles (center column).But when such a beam is made to interfere

with a plane wave, it produces a telltale spiral intensity pattern (right column). The number of spiral arms equals the

number � of intertwined helical phase fronts of the helical beam.

� = –1

� = 0

� = 1

� = 2

� = 3


that its resultant rotation was tiny.The analogous experiment for transfer-

ring orbital angular momentum to a sus-pended cylindrical lens has proved too diffi-cult. Instead, a number of groups haveexamined angular-momentum transfer tomicroscopic particles held by optical tweez-ers.8 Optical tweezers rely on the strong in-tensity gradient at the tight focus of a laserbeam. At such a focus, any small, lightweightdielectric particle experiences a gradientforce sufficient to attract it to the axis. So theparticle is held in place without mechanicalsuspension.

In 1995, Halina Rubinsztein-Dunlop andcoworkers at the University of Queensland inBrisbane, Australia, used an unpolarized, he-lically phased laser beam to impart orbitalangular momentum to a small ceramic parti-cle held by optical tweezers.9 In 1998, theBrisbane group repeated the experiment,10 this time im-parting spin angular momentum from a polarized beam toa birefringent particle—a microscopic encore to Beth’s1936 experiment.

The previous year, two of us (Padgett and Allen) per-formed an experiment with Neil Simpson that demon-strated the use of a circularly polarized and helicallyphased beam as an optical wrench (in Britain we call it anoptical spanner).11 We showed, for � = 1, that when the spinand orbital angular-momentum components of the beamhad the same sense, they combined to induce a rapid ro-tation of a small transparent particle. But when they wereof opposite sense, summing to zero angular momentum,the particle did not rotate. This later experiment demon-strated that the orbital angular momentum associatedwith �⊂1 is mechanically equivalent to the angular mo-mentum \ associated with photon spin.

Although spin and orbital angular momentum areequivalent in many ways, they have, in general, differentinteraction properties. Because the orbital angular mo-mentum of a light beam arises from the inclination of itsphase fronts, its interactions with particles away from theaxis are most easily understood in terms of the azimuthalcomponent of the beam’s linear momentum.

At the microscopic level, such interactions have beenobserved with polarized helical beams acting also as opti-cal tweezers (see figure 4). In several experiments, a smalltransparent particle was confined away from the axis inthe beam’s annular ring of light.12 The particle’s tangen-tial recoil due to the helical phase fronts caused it to orbitaround the beam axis. At the same time, the beam’s spin

angular momentum caused the particle to rotate on itsown axis.

At the level of individual atoms, things are somewhatdifferent. Spin angular momentum per photon is a con-stant at every position within the beam. For example, aZeeman transition can be excited by circularly polarizedlight of the appropriate frequency anywhere an atom findsitself within the beam’s cross section. By contrast, orbitalangular momentum is not transferred in the same way toatoms positioned at different radial distance from thebeam axis. It is the Poynting vector’s tangential compo-nent that makes the atom orbit the beam axis.

The detailed behavior of the atom is modified by fre-quency shifts, relaxation times, and the atom’s degree ofexcitation. In 1994, such effects were analytically exam-ined for atoms in LG beams by Allen, Mohamed Babiker,and coworkers.11 Their calculation showed that, in addi-tion to the torque around the axis, the atom experiences ashift in transition frequency. Both effects are proportionalto the beam’s orbital angular momentum.

Spin–orbit coupling of electrons determines much ofthe atom’s energy-level structure. In an attempt to see ifsuch a coupling can also exist for photons, Allen and com-pany have shown that a light beam exerts a dissipative az-imuthal force on atoms proportional to the product of thebeam’s spin and orbital angular momentum.13 That forceis very small—of the same order as terms invariably neg-lected in atom-trapping theory. But the prediction that thistiny force should be reversed when the beam’s spinchanges handedness was a novel one.

http://www.physicstoday.org May 2004 Physics Today 37

Figure 2. A pair of cylindrical lenses canserve as a converter that transforms a

Hermite–Gaussian (HG) mode into a La-guerre–Gaussian (LG) mode carrying orbital

angular momentum, much as a quarter-waveplate converts linearly polarized light to circu-

lar polarization. Because the same lens pairworks for any HG mode, it can produce a

wide range of LG modes. Increasing the sepa-ration between the cylindrical lenses can re-

verse the handedness of the LG mode, just asa half-wave plate reverses circular polariza-tion. In practice, LG reversal is more conve-

niently achieved with an image inverter suchas a Dove prism.

Linearpolarization

Circularpolarization

Reversedcircular

polarization

ReversedLG modes

Quarter-waveplate

Half-waveplate

HG modes

LG modes

/ modeconverters

p modeconverters

p2

or

Dove prism� = 0

� = –1

� =–2

� = –1

� = 0

� = +1

� = +1

� =+2

...

......

...

...

...

PT_Padgett0504.qxd 4/7/2004 9:16 AM 7

Rotational frequency shiftsIn 1979, Bruce Garetz and Stephen Arnold at thePolytechnic Institute in New York observed that acircularly polarized light beam is frequency shiftedif it is steadily rotated about its own axis.14 Classi-cally, this shift is simply what one would observe ifa clock were laid face up at the center of a rotatingturntable. Looking down on the clock’s face, oneseems to see the second hand sweeping at the wrongangular speed—the sum of the clockwork motionand the turntable’s rotation.

Circularly polarized light behaves in just thesame way. The rotation of the beam on its axisslightly speeds up or slows down the much faster opticalrotation of the electric field vector. The mechanical rota-tion can be achieved with a rotating half-wave plate. Thisrotational frequency shift is, in some sense, an angularDoppler shift. But it is not the same as the Doppler shiftassociated with rotating bodies viewed from the side,which is simply a manifestation of the usual translational

Doppler effect in which the rotation of an object producesa linear-velocity component along the line of sight. In fact,the angular Doppler shift is maximal when the line of sightis the rotation axis—precisely the direction in which thelinear Doppler shift vanishes.

In 1998, our Glasgow group managed to observe therotational frequency shift with a linearly polarized millimeter-wave beam that carried orbital angular mo-mentum; millimeter-wave beams are more forgiving thanshorter wavelengths of inevitable misalignments in suchexperiments.11 The key to understanding the frequencyshift in a helical beam is to recognize that the time evolu-tion of a helical phase front is indistinguishable from ro-tation about the beam axis. So, a single rotation of thebeam advances or retards its phase by � cycles.

In a follow-up experiment later that year, with a beamthat had both circular polarization and a helical phasefront, we showed that spin and orbital angular momentumcombine to give a rotational frequency shift proportionalto the total angular momentum. That result is a general-ization of the frequency shift predicted by Iwo and ZofiaBialynicki-Birula at Warsaw University in 1997 for thetransition frequency of a rotating atom.15

Interaction with nonlinear crystalsThe high power density achievable with focused laserbeams has made nonlinear optics a common phenomenonin the optics laboratory. In much the same way that largeinput voltages cause audio amplifiers to distort, yielding


Figure 3. A hologram can produce a light beam withhelical phase fronts and associated orbital angularmomentum �\ per photon. The appropriate holo-

gram can be calculated, or it can be generated fromthe interference pattern between the desired beamform and a plane wave. The resulting holographic

pattern resembles a diffraction grating, but it has an�-pronged dislocation at the beam axis. (The one

shown here has three prongs.) When the hologram isilluminated with a plane wave, the first-order dif-

fracted beams have the desired helical phase fronts.

= –3�

Incidentplanewave

= +3�

Threefold dislocationhologram

First-order diffracted beams

� = 0

Figure 4. Transferring angular momentum from light to mattercan make the matter rotate. When a circularly polarized laserbeam with helical phase fronts traps a micron-sized dielectricparticle (yellow ball) in an annular ring of light around the beamaxis, one can observe the transfer of both orbital and spin angu-lar momentum.12 The trapping is a form of optical tweezing ac-complished, without mechanical constraints, by the ring’s inten-sity gradient. The orbital angular momentum transferred to theparticle makes it orbit around the beam axis (top). The spin an-gular momentum, on the other hand, sets the particle spinningon its own axis (bottom).


output that contains extraneous frequencies, intense lightbeams distort the dielectric response of many optical crys-tals. This distortion, or nonlinearity, causes the light emit-ted by the crystal to include frequencies that were not inthe optical input.

The most efficient nonlinear processes involve inter-action among three waves. One either combines two inci-dent waves to form a third, in a process known as fre-quency up-conversion, or one performs frequencydown-conversion by splitting a single incoming wave intotwo. In both cases, conservation of energy at the photonlevel dictates that the largest of the three frequencies isthe sum of the two lesser frequencies.

Conservation of momentum also plays a role. Whenthe interacting beams are plane waves, all the Poyntingvectors are collinear, and momentum conservation dictatesa relationship among the three refractive indices. The req-uisite balancing of refractive indices is called phase match-ing. Remember that the Poynting vector in helicallyphased beams describes a spiral trajectory around thebeam axis. One might think that this spiraling modifiesthe phase-matching conditions; but that’s not the case.

Consider, for example, second-harmonic generation inwhich a single beam, split in two, constitutes both inputs,and the output beam is doubled in frequency. The orbitalangular momentum is also doubled (see figure 5). In thephoton picture, this means that two photons combine toform one photon of twice the energy, twice the linear mo-mentum, and twice the orbital angular momentum. Con-trast this conservation of orbital angular momentumwithin the light fields against what happens to the pho-tons’ spin. There has to be a transfer of spin angular mo-mentum to the frequency-doubling crystal, because a sin-gle photon cannot have a spin angular momentum of 2\.

For down-conversion, energy conservation allows theinitial beam to be split into two input beams with any com-bination of frequencies, so long as their sum equals the fre-quency of the initial beam. The phase-matching condition iswhat controls that frequency-splitting ratio. But there isnothing like the phase-matching restriction to constrain thedivision of the initial orbital angular momentum betweenthe two input beams. In principle, any combination of or-bital angular momenta that conserves the initial beam’s or-bital angular momentum is allowed. In fact, each of the splitbeams is in a mixture of orbital angular momentum stateswith a well-defined mean value. But as we shall see, the pe-culiarly quantum mechanical relationship between thebeams can only be observed at the single-photon level.

Any number of fields can interact nonlinearly within

the constraints of energy and momentum conservation.But efficiency imposes practical limits. Take, for example,four-wave mixing in which three light waves combine toproduce a fourth. One such process is phase conjugation,which occurs when a nonlinear material is optically ex-cited by two pump beams so that an additional signal beamcreates a beam traveling in the opposite direction with op-posite phase. It’s a kind of mirror.

In a recent phase-conjugation experiment at Pernam-buco Federal University in Recife, Brazil, Sergio Barreiroand Jose Wellington Tabosa used cold cesium atoms as thenonlinear material.16 They reported that if the signal beamcarries orbital angular momentum, its phase informationand angular momentum is transferred first to the atomsand then, after a deexcitation time, to the “reflected” beam.Such a mechanism might one day be exploited for memorystorage in multidimensional information processing.

Quantum effectsThe first truly quantum-mechanical experiment usinglight beams with orbital angular momentum was reportedin 2001 by Anton Zeilinger’s group at the University of Vi-enna.17 In a down-conversion experiment, the groupdemonstrated that the conservation of orbital angular mo-mentum applied individually to each pair of emitted pho-tons. This demonstration extended to orbital angular mo-mentum the analogous tests of quantum mechanicscarried out by Alain Aspect’s group in the early 1980s. Theyhad shown that the spin angular momentum of down-converted photon pairs was an entangled quantum state.The spin of a photon is defined by two states. But, as we’veseen, the number of possible orbital angular-momentumstates is unlimited. That difference presents the prospectof a deeper examination of quantum entanglement be-tween photon states than has previously been possible.

The degree of a beam’s polarization is readily meas-ured with a polarizer. But orbital angular momentum, be-cause of the arbitrarily large number of quantum states,is more difficult to quantify. One can check for a specificvalue of � by means of the same holograms one uses to cre-ate helical phase fronts. But that method doesn’t permit the� of individual photons to be measured unambiguously.

Last year, however, our group devised a method forusing interferometers to sort single photons into their dif-ferent orbital angular-momentum states.11 Our interfer-ometer had a beam rotator in one arm. That introduced an�-dependent phase shift that allowed the photons to besorted into states of odd and even �. Subsequent interfer-ometer stages then permitted further sorting into specific

http://www.physicstoday.org May 2004 Physics Today 39

Figure 5. Second-harmonic generation with helicalphase-front light beams in a nonlinear crystal con-serves orbital angular momentum as well as linearmomentum and energy within the light field. The

two input beams are from the same source, and theoutput frequency is double that of the input. Theprocess also doubles �\, the orbital angular mo-

mentum per photon. Nonlinear effects in a dielec-tric crystal can also produce a reverse three-photon

interaction that results in frequency down-conver-sion. A single high-energy photon becomes two

photons of lower energy. That phenomenon alsoconserves energy, linear momentum, and orbital

angular momentum within the light field.


orbital angular-momentum states.An essential aspect of the quantum world is the fun-

damental limit that Heisenberg’s uncertainty principlesets on the accuracy with which one can simultaneouslyknow the values of a pair of conjugate variables. Angularposition and orbital angular momentum are such a pair.In work not yet published, Steve Barnett and our grouphave recently demonstrated that this aspect of the Heisen-berg principle can actually introduce orbital angular mo-mentum—albeit with an expectation value of zero—into alight beam that starts out with none (see figure 6).

The experiment defines the angular position of a pho-ton in the beam’s cross section by making it pass througha V-shaped sector aperture whose apex is coincident withthe beam axis. We have shown that, when a beam in aneigenstate of orbital angular momentum L = �\ per pho-ton passes through such an aperture, it acquires a distri-bution of orbital angular-momentum states. For narrowapertures, we showed that the product of aperture width,Dv, and angular momentum spread, DL, is bounded by \/2,as quantum mechanics tells us it must be.

The exploration and exploitation of light with orbitalangular momentum are still in early days. There’s muchto be learned about the interaction of such light withatoms. Exciting applications with rotating optical micro-machines are in prospect. For quantum communicationand information processing, the expansion of the Hilbertspace of angular-momentum states offers opportunities fornew approaches to encryption and data storage.

Martin Harwit has proposed that the orbital angularmomentum of light from celestial sources might provide anew window on the cosmos.18 For example, the observationof orbital angular momentum in light scattered by blackholes could be very instructive. And we could search forsignals from extraterrestrials who might be availing them-selves of the high information density made possible by or-bital angular momentum. At the birth of modern astron-omy, Kepler studied the role of light’s linear momentum.Perhaps astronomers will soon find something equallyvaluable in its orbital angular momentum.

References1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerd-

man, Phys. Rev. A 45, 8185 (1992); L. Allen, M. J. Padgett, M. Babiker, Prog. Opt. 39, 291 (1999).

2. J. F. Nye, Natural Focusing and Fine Structure of Light:Caustics and Wave Dislocation, Institute of Physics,Philadelphia (1999).

3. C. Tamm, C. O. Weiss, J. Opt. Soc. Am. B 7, 1034 (1990); M. Harris, C. A. Hill, J. M. Vaughan, Opt. Commun. 106, 161(1994).

4. M. W. Beijersbergen, L. Allen, H. van der Veen, J. P. Woerd-man, Opt. Commun. 96, 123 (1993).

5. V. Yu. Bazhenov, M. V. Vasnetsov, M. S. Soskin, J. Expt.

Theor. Phys. Lett. 52, 429 (1990).6. G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner,

E. Wright, Phys. Rev. Lett. 87, 023902 (2001).7. R. A. Beth, Phys. Rev. 50, 115 (1936).8. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, Opt. Lett.

11, 288 (1986); D. G. Grier, Nature 424, 810 (2003).9. M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop,

Phys. Rev. Lett. 75, 826 (1995).10. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, H.

Rubinsztein-Dunlop, Nature 394, 348 (1998).11. See collected historical and recent papers in Optical Angular

Momentum, L. Allen, S. M. Barnett, M. J. Padgett, eds., In-stitute of Physics, Philadelphia (2003).

12. A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, Phys. Rev.Lett. 88, 053601 (2002); V. Garces-Chavez, D. McGloin, M. J.Padgett, W. Dultz, H. Schmitzer, K. Dholakia, Phys. Rev.Lett. 91, 093602 (2003).

13. L. Allen, V. E. Lembessis, M. Babiker, Phys. Rev. A 53, R2937(1996).

14. B. A. Garetz, S. Arnold, Opt. Commun. 31, 1 (1979); B. A.Garetz, J. Opt. Soc. Am. 71, 609 (1981).

15. I. Bialynicki-Birula, Z. Bialynicki-Birula, Phys. Rev. Lett. 78,2539 (1997).

16. S. Barreiro, J. W. R. Tabosa, Phys. Rev. Lett. 90, 133001 (2003).17. A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Nature 412, 313

(2001); A. Aspect, J. Dalibard, G. Roger, Phys. Rev. Lett. 49,1804 (1982).

18. A. Swartzlander, Opt. Lett. 26, 497 (2001); M. Harwit, Astro-phys. J. 597, 1266 (2003). �


Figure 6. Heisenberg’s uncertainty principle is manifested ina recent experiment at Glasgow University with a light beamthat initially carries no orbital angular momentum. The range

Dv of azimuthal angular positions for a photon in a crosssection of the beam is defined by a sector aperture. Up-

stream of the aperture, the beam is in an � = 0 eigenstate oforbital angular momentum (upper panel). But the uncertaintyprinciple dictates that the restriction in v introduce a spread

(lower panel) in the orbital angular momentum L = �\ perphoton. For narrow apertures, the relationship is DvDL�\/2.

PR

OB

AB

ILIT

YP

RO

BA

BIL

ITY

0

0

–2

–2

–1

–1

0

0

1

1

2

2

2p

2p

ANGULARPOSITION v

ANGULARPOSITION v

ANGULARMOMENTUM �

ANGULARMOMENTUM �

Dv

D�

Dv

Sectionaperture

Beamin

Beamout


light's orbital angular momentum momentum

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