Orbital angular momentum transfer in helical Mathieu beams

Orbital angular momentum transfer inhelical Mathieu beams

Carlos Lopez-Mariscal, Julio C. Gutierrez-VegaPhotonics and Mathematical Optics Group, Tecnologico de Monterrey

Monterrey, Mexico 64849

[email protected]

Graham Milne and Kishan DholakiaSchool of Physics and Astronomy, The University of St Andrews, KY16 9SS, Scotland

Abstract: We observe the transfer of orbital angular momentum totrapped particles in the azimuthally asymmetric transverse intensity dis-tribution of a helical Mathieu beam. The average rotation rate, instantaneousangular displacement and terminal velocity of the trapped particles aremeasured experimentally. The angular dependence of these parametersis found to be in good agreement with the variation of the optical gradi-ent force, the transfer of OAM from the wavefield and the Stokes drag force.

© 2006 Optical Society of America

OCIS codes: (140.7010) Trapping; (170.4520) Optical confinement and manipulation;(050.1970) Diffractive optics; (140.3300) Laser beam shaping; (260.1960) Diffraction theory;(350.4990) Particles.

References and links1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and

the transformation of Laguerre-Gaussian laser modes,” Phys.Rev. A45,, 8185-8189 (1992).2. J. Durnin, J. J. Miceli and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58,1499 (1987).3. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the Angular Momentum of Light: Preparation of

Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett.88, 013601 (2002).4. M. Babiker, W. L. Power, and L. Allen, “Light-induced Torque on Moving Atoms,” Phys. Rev. Lett.73, 1239–

1242 (1994).5. M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Davila Romero, “Orbital Angular Momentum Exchange in

the Interaction of Twisted Light with Molecules,” Phys. Rev. Lett. 89, 143601 (2002).6. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental Observation of Fluidlike Motion of Optical

Vortices,” Phys. Rev. Lett.79, 3399-3402 (1997).7. H. He, M. E. J Friese, N. R. Heckenberg and H. Rubinsztein-Dunlop, “Direct Observation of Transfer of Angular

Momentum to Absorptive Particles from a Laser Beam with a Phase Singularity,” Phys. Rev. Lett.75, 826-829(1995).

8. Berry, M V, “Paraxial beams of spinning light,” inSingular optics, M. S. Soskin, ed., Proc. SPIE3487, 6-11(1998).

9. A.T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital AngularMomentum of a Light Beam,” Phys. Rev. Lett.88, 053601 (2002).

10. V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer and K. Dholakia, “Observation of theTransfer of the Local Angular Momentum Density of a Multiringed Light Beam to an Optically Trapped Particle,”Phys. Rev. Lett.91, 093602 (2003).

11. G. Delannoy, O. Emile, and A. Le Floch, “Direct observation of a photon spin-induced constant acceleration inmacroscopic systems,” App. Phys. Lett.86, 081109 (2005).

12. J. Curtis and D. G. Grier, “Structure of Optical Vortices,” Phys. Rev. Lett.90, 133901 (2003).13. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett.28, 872-874 (2003).14. K. Sasaki, M. Kashioka, H. Misawa, N. Kitamura and H. Masuhara, “Pattern formation and flow control of fine

particles by laser-scanning micromanipulation,” Opt. Lett.16,1463-1465 (1991).

(C) 2006 OSA 1 May 2006 / Vol. 14, No. 9 / OPTICS EXPRESS 4183#68797 - $15.00 USD Received 15 March 2006; revised 17 April 2006; accepted 17 April 2006

mailto:[email protected]

15. N. B. Simpson, K. Dholakia, L. Allen and M. J. Padgett, “Optical tweezers with increased axial trapping effi-ciency,” Opt. Lett.22,52-54 (1997).

16. K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt and K. Dholakia, “Orbital angular momentumof a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt.4, S82-S89 (2002)

17. J. C. Gutierrez-Vega, M. D.Iturbe-Castillo, and S. Chavez-Cerda, “Alternative formulation for invariant opticalfields: Mathieu beams,” Opt. Lett.25,1493-1495 (2000).

18. S. Chavez-Cerda, J. C. Gutierrez-Vega, and G. H. C. New, “Elliptic vortices of electromagnetic wavefields,” Opt.Lett. 26,1803-1805, (2001).

19. S. Chavez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez-Vega, A.T. O’Neil, I. MacVicar, andJ. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B:Quantum Semiclass. Opt.4, S52–S57, (2002).

20. R. C. Hardy and R. L. Cottington, “Viscosity of DeuteriumOxide and Water from 5◦ to 125◦ C,” J. Chem. Phys.,17, 509- 510 (1949).

21. A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett.24,156-159 (1970).

1. Introduction

Optical angular momentum (OAM) and associated studies withoptical vortices are areas of cur-rent widespread interest. Inclined wavefronts that imply an azimuthal component to the Poynt-ing vector thus resulting in OAM of a light wavefield are well known. Typical forms of thesewavefronts have embedded vortices such as those in Laguerre-Gaussian [1] or Bessel-Gaussianbeams [2]. Light fields with OAM are of importance in quantum information processing [3],fundamental light-matter interactions [4], atomic selection rules [5] and vortex propagation [6],amongst other areas of research. Typically, these fields arecharacterised by an azimuthally-dependent term of the form exp(imϕ), wherem is an integer that denotes the number of inter-twined helices within the field. This helical term, in turn, determines the OAM content of thewavefield.

Optical micromanipulation has been the most powerful experimental technique to elucidatethe underlying physics of such light fields: in 1995, transfer of OAM by absorption was di-rectly observed for the first time [7] and subsequently comparisons made between the spinand orbital angular momenta. Such studies have been advanced in recent years with the iden-tification of the exact intrinsic and extrinsic nature of optical angular momentum [8, 9] andhow this manifests itself in the behaviour of an off-axis particle in a circularly symmetric lightfield [10] measuring the local angular momentum density. More recently, several technical im-provements to the transfer of OAM in the context of optical tweezers have been proposed [11].OAM is now encroaching into new emergent areas including microfluidics where such lightfields have potential applications for micropumps and directing particle motion [12, 13] wherethere is a requirement to direct particles along arbitrary trajectories thus suggesting the use ofasymmetric light fields possessing OAM. Recently, spatial light modulators (SLMs) [13] andacousto-optical deflectors (AODs) [14] have been used to tailor specific lightfields with dynam-ically reconfigurable intensity distributions in mesoscopic systems, including shaped vorticesfor micromanipulation.

The use of fundamental beams in micromanipulation experiments results in the basic under-standing of the physical processes by means of which light fields interact with material particlessince the associated electromagnetic field of the beam can beaccurately described in a closed,analytical form since the exact field distribution is knowna priori. To date however, all exper-imental micromanipulation studies of fundamental beams possesing OAM using fundamentalbeams have been restricted to circularly symmetric transverse patterns with azimuthally uni-form OAM densities, such as Laguerre-Gauss beams [7, 15] andBessel beams [10, 16]. A par-ticular class of fundamental beams is that of nondiffracting beams, whose transverse intensityprofile remains unchanged as the beam propagates in free space for a considerable distance.

In this letter, we show the first experimental demonstrationthat Helical Mathieu (HM) beams


possess OAM. We also present the first experimental results of particle dynamics in the wave-field of a nondiffracting HM beam with azimuthally asymmetric OAM density and intensityprofile. In particular, we look at the controlled rotationalmotion of trapped microparticleswithin the elliptical lobes of the wavefield and study, for the first time, the interplay of twodifferent phenomena, namely, thecontrolablerotation induced by transfer of OAM and theoptical confinement that results from the gradient force. Wealso observe the simultaneous bal-ance of these contributions and the Stokes drag force in the motion of particles within the trapand demonstrate a quantitative analysis of particle trajectories under the influence of the forcesinvolved.

2. Helical Mathieu beams

HM beams are fundamental nondiffracting beams which are solutions of Helmholtz equationin elliptical cylindrical coordinates(ξ ,η , z). Hereξ andη are the radial and angular ellipticalcoordinates andz is the propagation axis[17]. The beams are mathematically described by alinear superposition of products of radial and angular Mathieu functions. For a monochromatic,linearly polarized HM beam of orderm propagating in thezdirection, the field is given by

U (ξ ,η , z) = [Cm(q)Jem(ξ ;q)cem(η ;q)+ iSm(q)Jom(ξ ;q)sem(η ;q)]exp(ikzz), (1)

where Jem(·) and Jom(·) are the even and odd radial Mathieu functions and cem(·), sem(·) arethe even and odd angular Mathieu functions.Cm(q) and Sm(q) are weighting constants thatdepend onq, a continuous parameter that determines theellipticity of HM beams. The trans-verse intensity profileI (ξ ,η , z) is characterised by a set of confocal elliptic rings of varyingintensity[17, 18]. HM beams also have a linear array of vortices with unitary topological chargedistributed along the interfocal line of its elliptical rings [18], associated to a transverse phasegradient along the angular coordinate which accounts for its OAM content [19].

The spatial separation of the vortices is also determined bythe value ofq. For low valuesof q, the OAM per photon for an HM beam of orderm≥ 1 originates in itsm interfocal vor-tices. Since the phase gradient between any two adjacent vortices is null, the transverse posi-tion of the vortices remains constant as the beam propagates. Interestingly, asq decreases, thefoci gradually approach the origin and forq = 0, all the vortices concur into a single vortexof topological chargem, resulting in the well-known case of Bessel beams. Clearly,this is aconsequence of the symmetry of the elliptical-cylindricalcoordinate system, which collapsesto circular-cylindrical coordinates when the interfocal distance vanishes. In contrast to Besselbeams however, the OAM density of HM beams isnot independent of the azimuthal coordinatebut it varies with the elliptic angular coordinateη . For HM beams of the same order, an increasein the ellipticity factor results in the horizontal stretching of the ringed structure of the beam,deviating gradually from circular symmetry. After a critical valueqc, the beam elliptical ringsare broken, the transverse profile symmetry becomes rather hyperbolic and the vortex structurebecomes more complex compared to its original in-line configuration [19].

3. Experimental Setup

In order to study and characterise the transfer of OAM from anHM beam to dielectric mi-croscopic particles, we have used an optical tweezer setup with a HM beam (m = 7, q = 5)generated with an off-axis blazed phase computer-generated hologram (CGH)[19]. The holo-gram was backlit by a collimated beam from a linearly polarized 1064 nm Ytterbium fiber laserand the resulting field distribution immediately after the hologram was then focused by meansof a converging Fourier lens. An image of the angular spectrum of the HM beam was formed inthe+1 diffraction order at the Fourier plane. The remaining orders of diffraction were blocked


here by means of an iris diaphragm and a second converging lens was then used to reconstructthe HM beam from the image of its angular spectrum. The beam was then directed downwardsand focused tightly into a sample chamber with the particlesin solution and the trapping planewas imaged with an inverted microscope (Nikon TE2000E).

The sample chamber was filled with a diluted solution of monodisperse spherical polystyreneparticles 3.0 microns in size, suspended in a mixture of anionic, non-anionic and amphotericsurfactants (1% in volume) in D2O. The solution is intended to circumvent the absorption ofIR radiation by using D2O instead of water and, at the same time, to reduce the viscosity ofthe medium [20] as seen by the particles moving within the chamber. While the density ofthe solvent is essentially unchanged by adding a small amount of surfactants, its viscosity isreduced to 61% relative to the value for pure D2O as assessed by means of a Beral pipet.Interestingly, OAM transfer is only observed effectively in the presence of surfactants, thus anupper bound for the local value of the scattering force in ourexperiment is the Stokes drag force

FD = −6πµRdrdt

, (2)

for a spherical particle of radiusR moving with velocitydr/dt, wherer (t) = [x(t) ,y(t)] im-mersed within a fluid of viscosity coefficientµ . Because the transverse intensity profile of theHM beam varies in the azimuthal coordinate, there exists a variable OAM transfer as from thebeam to a trapped particle as it moves about the trapping plane. This allows for the observationof variations in the terminal velocity of the particles as their motion is influenced differentlyby both, the gradient force[21]

Fgrad = −2πn3R3

c

(

n2−1n2 +2

)

∇I , (3)

and also by the Stokes drag force (Eq. 2). Heren is the relative refractive index of the particlesin the solution andc the speed of light.

As the beam traversed the sample chamber, the particles in the sample were quickly alignedwith the beam elliptical rings as shown in Fig 1(a), and experienced a strong tendency to drifttowards the transverse intensity regions of maxima in the transverse plane due to the gradientforce. In the longitudinal direction, the upwards force acting on the particles due to buoyancywas counteracted by the radiation pressure of the beam acting downwards within the samplevolume. Particles find their longitudinal position of equilibrium immediately below the planethat defines half the maximum propagation distance of the beam zMAX as defined in Ref. [2].This was verified by displacing thez = zMAX/2 plane downwards in the chamber until theparticles were pressed against the bottom of the sample cell.

We observed the immediate onset of rotational motion in accordance with the sense of thewavefront inclination of the beam. We verified that this rotational behaviour was indeed dueto the OAM of the beam by rotating the CGH by 180 degrees about an axis perpendicular tothe direction of propagation [7] thus reversing the rotation direction of the particles. Trappedparticles described a curve ofξ = ξ0, over a full rotation overη . The observed paths of thetrapped particles were consistent with the elliptical orbit predicted by the intensity distributionof the HM beam. Figure 1(b) shows the measured position of onetrapped particle sampled at arate of 4 Hz over a sampling time ofTs = 100 s.

The instantaneous positionr (t) = (x(t) ,y(t)) of the particle was then tracked using 10-minute video samples captured at 15 frames per second by means of a CCD camera. The videosamples were processed using dedicated software implemented using LabVIEW in order toextract the position of the particles within the trapping plane. Our tracking routines use dynamicpattern recognition algorithms to compute the position of the centers of individual particles.


Fig. 1. (a) Trapped particles in the elliptical rings of the HM beam. (b) Horizontal and ver-tical position of a single particle in the beam, note the periodic regular motion. Samplingtime is 400 s. (c) Variation of the terminal velocity of a single particle in the sample asa function of the polar angle during one full orbital displacement. Velocity minima corre-spond to intensity maxima.

In order to extract information about the transfer of OAM from the beam to the particles,we have used a multiple-point moving average method for analyzing the data time series thusisolating the local mean value of the statistical position,in turn decoupling the uncorrelatedrandom displacements associated to Brownian motion from the elliptical motion. Dynamicvariables of motion were subsequently calculated by takingsuccessive numerical derivativesof the measured instantaneous position. Here, we present quantitative assessments of the dy-namic variables of the particles in motion and explain thesein terms of the forces involved inthe experiment.

4. Results

Due to the structure of the transverse field of HM beams, trapped particles orbit about the inter-focal line of the elliptical rings of the transverse field of the HM beam, in contrast to previousexperiments [9, 10], in which off-axis rotation was observed around a single point in circularorbits. The elliptical orbital motion results from a combination of the confinement of particleswithin the ellipse due to the gradient force, and to the induction of rotation due to the transferof OAM. Overall, OAM transfer dominated the orbital dynamics of the particles. However, lo-cal variations of the instantaneous angular velocity were observed due to the interplay betweentwo forces. On one side, the optical gradient force increases in magnitude in the proximity ofhigh intensity regions and thus tends to displace the trapped particles to local maxima withinthe transverse extent of the beam. On the other side, the OAM transfer rate increases as theparticles approach regions of varying intensity and are transfered OAM from the angular mo-mentum component in the propagation direction. The dynamics of the particles thus arise fromthe delicate interplay between the transverse gradient force, the transfer of OAM and the Stokesdrag force. Clearly, Brownian motion also contributes to the instantaneous displacements of thetrapped particles. This is evidenced in the points of intensity minima, where the confinement isless tight and thus particles deviate more from their elliptic orbit due to random motion.

Since the transverse field of HM beams is not azimuthally symmetric, particles trappedwithin the beam are expected to increase their angular speedas they approach intensity maximaand subsequently slow down due to the local gradient force. Since the scattering mechanismin turn becomes more significant in high-intensity regions,it competes with the local gradi-ent force and OAM transfer thus propels the trapped particles away from intensity maxima and


http://www.opticsexpress.org/viewmedia.cfm?id=89625&seq=1

Fig. 2. Rotation rates as a function of beam power. The straight line showsthe linear fit.

into an elliptical orbit with a terminal angular velocity determined by the Stokes drag force. Thetuning of each of these contributions can be used to configuremicrofuidic devices that couldvary the terminal speed of trapped particles. In the particular case of transparent polystyreneparticles, the mechanism of OAM transfer is that of scattering, as absorption at this wavelengthis essentially negligible. In this case, the transfer of OAMis proportional to the intensity ofthe wavefield, this was experimentally observed as an increase in the beam power resulted ina linear variation of the average angular velocity of the particles in the sample (see Fig. 2 ).Particles typically completed one cycle in 89 s at a beam power of 700 mW, while doubling thepower resulted in a velocity increase of nearly 270% yielding a new rotation period of 39 s.

5. Conclusions

We have demonstrated the transfer of OAM from a HM beam to trapped transparent particlesand measured the evolution of the particles under the influence of the azimuthally asymmet-ric beam profile. We have observed variations in the terminalvelocity of individual particlesand associated the motion of the particles to the interplay of the optical gradient force, the az-imuthally varying OAM transfer and the Stokes drag force. Wehave also observed the linearincrease of the OAM transfer rate with beam power by measuring the average rotation rateof the particles in the beam. The random displacements associated to Brownian dynamics aredecoupled from the motion of interest by using multiple-point moving average data processing.

The modulation of the OAM transfer can be achieved by adjusting the ellipticity parameterqof the transverse field. Increasing the orderm of the beam, will also increase the net OAM perphoton, thus increasing the orbital rotation speed for a given effective scattering rate and beampower, which would be useful for microfluidics applications.

Acknowledgments

This work was partially funded by Tecnologico de Monterrey research grant CAT007 and Cona-cyt Mexico grant 42808. This work, as part of the European ScienceFoundation EUROCORESProgramme (grant: 02-PE-SONS-063-NOMSAN), was supportedby funds from the UK Engi-neering and Physical Sciences Research Council and the EC Sixth Framework Programme.


Documents

Orbital angular momentum transfer in helical Mathieu beams