Optical vortices, angular momentum and Heisenberg s uncertainty relationship

Transcription

1 Invited Paper Optical vortices, angular momentum and Heisenberg s uncertainty relationship Miles Padgett* Department of Physics and Astronomy, University of Glasgow, Glasgow, Scotland. G12 8QQ. ABSRACT Light beams with helical phasefronts are characterized by an exp(ilφ) azimuthal phase. The centre of these beams is often termed an optical vortex and the surrounding annulus of high intensity carries an orbital angular momentum with a ratio between angular momentum and energy corresponding to lh per photon. Such beams can induce rotation in microscopic objects and, when they are rotated, undergo a frequency shift. The light s orbital angular momentum is a valid description and measurement on both a macroscopic scale and at the level of single photons. Most recently, orbital angular momentum has been shown to be subject to an uncertainty relationship between angular momentum and angular position. Keywords: Optical vortex, orbital angular momentum, optical spanner, frequency shifts, angular uncertainty 1. INTRODUCTION That lights caries a linear momentum of hk 0 ( k 0 = 2π λ ) and a spin angular momentum of σh (σ =+/ 1 for right and left hand circular polarization, respectively) is well established. Although often described as photon properties, both can be derived classically from the vector products of electric and magnetic fields and, for angular momentum, the radius vector. Usually, light beams have near-planar phasefronts and the Poynting vector is directed along the beam axis. However, other forms of phasefront are possible which also satisfy Maxwell s equations. For example, a phase structure of exp(ilφ) describes l-intertwined helical phasefronts where the Poynting vector is then skewed with respect to the beam axis (see figure 1). For such beams, the associated linear momentum has an azimuthal component giving an orbital angular momentum to energy ratio in the direction of propagation of l/ ω (ω is the angular frequency of the light), i.e. corresponding to lh per photon 1. The azimuthal flow of momentum around the beam axis means that the axis of these helical beams is often referred to as an optical vortex. This vortex marks a phase discontinuity resulting in a zero intensity meaning that all helical beams have an annular intensity structure. Examples of such beams include the Laguerre-Gaussian laser modes and some high-order Bessel beams 2. Fig. 1. The form of the helical phasefronts corresponding to an azimuthal phase dependence of exp(ilφ). The arrow shows the trajectory of the Poynting vector. * m.padgett@physics.gla.ac.uk; phone ; Complex Mediums V: Light and Complexity, edited by Martin W. McCall, Graeme Dewar, Proc. of SPIE Vol (SPIE, Bellingham, WA, 2004) X/04/$15 doi: /

2 It should be emphasized, firstly that this orbital angular momentum is distinct from the spin component and hence completely independent of the polarization state and secondly it is a meaningful quantity at both the classical and quantum limits. 2. GENERATION OF LIGHT BEAMS CONTAINING ORBITAL ANGULAR MOMENTUM Light beams containing spin angular momentum are readily produced using a quarter-wave plate to convert linearly polarized to circularly polarized light. By contrast light beams containing orbital angular momentum are more difficult to produce. The most intuitive method for converting a planar phasefront into a helical one uses a spiral phaseplate, comprising a transparent disc whose optical thickness increases with azimuthal angle 3. Fig. 2. A diffraction grating with an on-axis fork dislocation (right) is the holographic equivalent of the spiral-phaseplate, producing a first-order diffracted beam with helical phasefronts. Although successfully employed at mm-wave frequencies 4, it is only recently that spiral-phaseplates have worked well at optical frequencies, where the shorter wavelength requires extremely high machining tolerance 5. Within a ray optical formulation, it is the refraction of the ray at the inclined surface that introduces the skew angle of the momentum flow; a simplistic but never-the-less accurate picture of the momentum exchange. Within the visible spectrum, the method of choice for producing helical phasefronts has been diffractive optical elements, frequently referred to as computergenerated holograms. The usual design is that of a diffraction grating with the addition of an on-axis fork dislocation, that produces a first-order diffracted beam with helical phasefronts; the l-value being dictated by the order of the dislocation, see figure 2. First employed over 10 years ago 6, the recent advent of high quality spatial light modulators means that such components can be calculated and displayed in real-time allowing dynamic control of various beam parameters such as their l-value. 3. TRANSFER OF LIGHT S ANGULAR MOMENTUM TO MATTER Light s angular momentum is not just a mathematical analogy but has real physical meaning. This has been best exemplified by experiments from a number of groups showing that microscopic particles can be set into motion by the transfer of either or both spin and orbital angular momentum. The torque exerted by spin angular momentum was first observed in the 1930 s using a conventional light beam and a suspended quartz waveplate 7. Recent experiments have centered on the use of optical tweezers that use tightly focused beam of light to trap and manipulate micron-sized objects 8. Since the moment of inertia scales with the fifth power of linear dimension, even the small torques produced by a light beam can have dramatic effects on the motion of these microscopic particles 9. An interesting comparison between spin and orbital angular momentum arises when a small particle interacts with a much larger beam 10. This difference in behavior can be explained with reference to the intrinsic and extrinsic properties of the two kinds of momentum, but is equally understood from a physical argument. A circularly polarized beam has a uniform polarization across its entire cross-section and any torque acting on the particle is independent of the particles position within the beam; the result been that the particle is set to spin always about it s own axis. For helical 2 Proc. of SPIE Vol. 5508

3 phasefronts, the local orientation of the skewed phasefront depends upon the radial position and is defined with respect to the beam axis. The azimuthal component of the associated linear momentum acts on the particle inducing an orbital motion of the particle around the beam s axis, see figure 3. Fig. 3. Angular momentum can be transferred from light to mater. The orbital angular momentum (left) of associated wit a helical wave induces an orbital motion of the particle around the beam axis. The spin angular momentum (right) causes the particle to spin around its own axis. 4. ROTATIONAL FREQUENCY SHIFTS Within a circularly polarized light beam, the electric field vector rotates at the optical frequency. An additional rotation of the beam around it s own axis results in a frequency shift equal to the rotation frequency of the beam, an effect observed in the late 1970 s 11. Expressed in terms of the spin angular momentum, a beam rotation frequency, Ω, gives frequency shift ω = σω. The rotational symmetry of beams with helical phasefronts means that they are subject to a similar effect 12. The evolution in time of the phasefront is indistinguishable from a rotation, see figure 4. Fig. 4. For a beam with helical phasefront, its temporal evolution is indistinguishable from a rotation. It follows that one full rotation of a beam about its axis advances or retards the phase by l cycles. A circularly polarized beam has one-fold rotational symmetry around its axis. By contrast, a helically phased beam has l-fold rotational symmetry meaning that upon one rotation of the beam, its phase is advanced or retarded by l-cycles. When subject to a continuous rotation about the beam axis, the corresponding frequency shift is ω = lω. Rather than using a waveplate that modifies only the polarization, the rotation of the helical phase structure is akin to an image rotation that experimentally can be achieved using an optical element such as a Dove prism. In recent years this frequency shift has been measured at both mm-wave 13 and optical frequencies 14. The most interesting feature of these frequency shifts occurs when the rotating light beam contains both spin and orbital angular momentum. Rather than producing two separate shifts, it is observed that the spin and orbital components add to give a single frequency shift proportional to the total angular momentum 15, ω = (σ + l)ω. The reason behind this additive shift becomes evident when one considers a cross section through the beam and examines the instantaneous Proc. of SPIE Vol

4 orientation of the electric field vector. The interplay the radial direction of the field and the azimuthal position results in a (σ + l) fold rotational symmetry in the cross section. For both spin and orbital angular momentum, the energy transfer corresponding to the frequency shift arises from the torque on the beam rotator associated azimuthal component of the light s linear momentum MEASURING ORBITAL ANGULAR MOMENTUM Polarization and hence spin angular momentum, is readily measured using a simple polarizer either acting on the beam as a whole, or coupled with an highly sensitive detector, on single photons. Measuring orbital angular momentum is more challenging. The helical nature of the phasefronts mean that for a beam containing many photons in the same state, the interference pattern between the beam and a plane wave comprises l intertwined spiral fringes 17, see figure 5. Fig. 5. The Laguerre-Gaussian laser modes (top) typify those beams with helical phasefronts, having (for l 0) both a optical vortex along the beam axis and an annular intensity structure. When interfered with a plane wave of the same frequency, the number of spiral interference fringes (bottom) indicates their value of l. At the level of single photons, measuring l is more difficult as the complete interferogram can no longer be obtained. One alternative method utilizes the rotational phase shift at the heart of the rotational frequency shift discussed above. When subject to a single rotation through an angle α, the phase of a helically phased beam is shifted by 2πl (2π α). By incorporating a beam rotation within one arm of a Mach-Zehnder interferometer, constructive interference occurs at one port or other depending upon the l-value of the incoming beam/photon. A network of cascaded interferometers has sorted light into four different l- states at the single photon level with near 100% efficiency 18. More conveniently, but with less efficiency, the same type of hologram used to generate a helical phasefront can be used to confirm whether signal photons have a particular l-value 19. If the l-value of the incident photon is opposite to the number of forks within the hologram, the diffracted beam is converted into a planar wave with a non-zero on-axis intensity. Thus a detector positioned behind an on-axis pinhole will only detect a photon when the l-value of the incident beam matches that of the hologram. Particularly, when combined with a computer addressable spatial light modulator to update number of forks on the hologram in real-time, this method allows a wide range of l-values to be tested in quick succession. 4 Proc. of SPIE Vol. 5508

5 Fig. 6. A computer-generated hologram can be used to test a beam of light for a particular l-value. It is only when the l-value of the beam matches that of the hologram (centre) that the resulting beam has a non-zero on-axis intensity, i.e. in this case only for l=2. 6. ANGULAR UNCERTAINTY Heisenberg s uncertainty principle is one of the best known of physical relationships. In its most common form it relates the precision to which momentum and position of a system can be simultaneously defined. Another example include the uncertainty relationship between time and energy. Common to all is that the exact relationship between the conjugate variables is that of a Fourier-transform. Angular momentum is more complicated than linear momentum since it s conjugate variable is angular position, normally expressed within a 0 to 2 π range and the corresponding assignment of an angular uncertainty is contentious. Using a computer generated hologram to measure the distribution of l-values arising from the introduction of an angular aperture (a cake slice) into a light beam it is possible to study the relationship between aperture function and angular momentum states 20, see figure 7. Fig. 7. A computer-generated hologram used to measure the distribution in l-values resulting from the inclusion of an angular aperture into a beam. For ease of alignment, the aperture is combined into the hologram design. Treating the angular distribution as a 2 π cyclic function, the corresponding Fourier-transform accurately describes the measured distribution in l-values, see figure 8. If, as in the Pegg-Barnett phase 21, the calculation of angular uncertainty is restricted to the 0 to 2 π range, then for narrow Gaussian-profiled apertures, one obtains a relationship between aperture width and angular momentum spread of l φ = 0.5. Remembering that the angular momentum is L = lh, this spread is L φ = h 2 ; an angular uncertainty relationship. Proc. of SPIE Vol

6 Fig. 8. For narrow Gaussian-profile apertures (left), the measured distribution of l-values (right-bars) agrees with that predicted by a Fourier-relationship (right-line) having a distribution supporting the uncertainty relationship l φ = 0.5, i.e. L φ = h 2 For wider Gaussian-profile apertures, where the aperture has a finite transmission at all angles, the situation is more complicated. The distributions of aperture and angular momentum are still related as Fourier-pairs, but the product in their uncertainties falls below 0.5. Following again the Pegg-Barnett approach, one obtains an uncertainty relationship l φ = 0.5(1 β), where β is related to the lowest transmission point of the Gaussian-profile aperture. Indeed, the experimental results as a whole are in extremely close agreement with a Pegg-Barnett interpretation. Fig. 9. For wider apertures (left), the measured distribution of l-values (right-bars) again agrees with that predicted by a Fourierrelationship (right-line) having a distribution supporting the uncertainty relationship l φ = 0.5(1 β) 7. CONCLUSIONS Lights beams carry two distinct forms of angular momentum; spin which is associated with circular polarization and orbital that is associated with helical phasefronts. In certain situations, such as the rotational frequency shift, they behave in an additive and equivalent fashion, more generally they do not. A key distinction is that unlike spin angular momentum for which there are only two orthogonal states, the number of states of orbital angular momentum is unbounded. Coupled with the fact that the orbital angular momentum can be readily measured, this multitude of states create interesting and exciting opportunities for new experiments in quantum physics. REFERENCES 1 L Allen M W Beijersbergen, R J C Spreeuw and J P Woerdman, Phys. Rev. A 45, 8185, See collected papers in Optical Angular Momentum L Allen, S M Barnett and M J Padgett, IOP Publishing Ltd, Bristol, M W Beijersbergen, R P C Coerwinkel, M Kristensen and J P Woerdman, Opt. Commun. 112, 321, G A Turnbull, D A Robertson, G M Smith, L Allen and M J Padgett, Opt. Commun. 127, 183, S S R Oemrawsingh, E R Eliel, J P Woerdman, E J K Verstegen, L G Kloosterboer and G W 't Hooft, J. Opt. A 6, S288, V Yu Bazhenov, M V Vasnetsov and M S Soskin, JETP Letts. 52, 429, R A Beth, Phys. Rev. 50, 115, Proc. of SPIE Vol. 5508

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