It is shown how the total internal reflection of orbital-angular-momentum-endowed light can lead to the generation of evanescent light possessing rotational properties in which the intensity distribution is firmly localized in the vicinity of the surface. The characteristics of these surface optical vortices depend on the form of the incident light and on the dielectric mismatch of the two media. The interference of surface optical vortices is shown to give rise to interesting phenomena, including pattern rotation akin to a surface optical Ferris wheel. Applications are envisaged to be in atom lithography, optical surface tweezers, and spanners

D. L. Andrews

L. Allen

M. Babiker

V. E. Lembessis

English

Lembessis, V. E.

Babiker, M.

Andrews, D L.

University of East Anglia digital repository

Surface optical vorticesV. E. Lembessis,1,* M. Babiker,1 and D. L. Andrews21Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom2School of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ, United KingdomReceived 18 September 2008; published 14 January 2009It is shown how the total internal reflection of orbital-angular-momentum-endowed light can lead to thegeneration of evanescent light possessing rotational properties in which the intensity distribution is firmlylocalized in the vicinity of the surface. The characteristics of these surface optical vortices depend on the formof the incident light and on the dielectric mismatch of the two media. The interference of surface opticalvortices is shown to give rise to interesting phenomena, including pattern rotation akin to a surface opticalFerris wheel. Applications are envisaged to be in atom lithography, optical surface tweezers, and spanners.DOI: 10.1103/PhysRevA.79.011806 PACS numbers: 42.50.Tx, 37.10.DeRecent years have witnessed the emergence of a branch ofoptics concerned with the production and applications oftwisted light. This is a class of laser light distinguished by anazimuthal phase dependence expil, where l is an integerand  is the azimuthal angle about the beam axis in cylin-drical coordinates. Such light was shown in a pioneeringpaper by Allen et al. 1 to possess an orbital angular mo-mentum l, distinct from the spin angular momentum asso-ciated with wave polarization. The prototypical set of twistedlight beams are the well-defined Laguerre-Gaussian beams,which can now readily be created in the laboratory.Laguerre-Gaussian laser light has been investigated over thelast two decades or so, and its effects on matter, both in thebulk and as single atoms, have been considered. It is nowwell established that twisted light can be used as an opticaldevice capable of rotating small objects 2–5, supplement-ing the capacity it shares with any laser light to influencetranslational motion optical tweezers. The effects of lighton single atoms and molecules have been explored theoreti-cally, identifying cooling and heating effects of translational,as well as the rotational, motion when atoms are subject totwisted light. The rotational motion of the atoms has beenshown to arise from a laser-induced torque 6 associatedwith the azimuthal component of the light pressure force. Ithas also been established that the transfer of orbital angularmomentum to dipole-active atoms and molecules involvesonly the center-of-mass motion, rather than the internal dy-namics 7. Multiple Laguerre-Gaussian beams have beenshown to provide different optical molasses situations in twoand three dimensions 8–10.The interference of different Laguerre-Gaussian lightbeams has recently been studied experimentally, revealing arich variety of intensity distributions. For copropagatingbeams of equal and opposite values of l, but with slightlydiffering frequencies, it has been shown experimentally thatthe interference pattern forms a so-called optical Ferris wheelin that the pattern rotates at an angular frequency that de-pends on l and the frequency separation of the two beams11. The authors of this article point out that the light inten-sity distribution may be used to trap atoms, but the copropa-gation of the two beams is disadvantageous in that there areno optical molasses forces that work to cool the atoms axi-ally, as in the case of counterpropagating beams.A different interference scenario is one involving the in-ternal reflection of twisted light at a planar surface betweenan optically dense medium, within which the light is inci-dent, and a vacuum. As far as we know, this simple andeasily realizable experimental situation has not been consid-ered before in the context of surface-atom manipulation us-ing Laguerre-Gaussian light. Nevertheless, evanescent lightdue to conventional beams has been successfully used inatomic mirrors 12 and its use in surface manipulation hasbeen considered for small objects at surfaces 13–17, ratherthan near-resonance atoms and molecules. As we show herefor Laguerre-Gaussian beams, the evanescent light thatemerges in the vacuum region is exponentially decaying withthe distance normal to the surface, but carries the in-planedistribution of the incident beam and its angular momentumproperties. Superimposed on this is a plane wave travelingalong the surface with a wave vector equal to the componentof the axial wave vector of the incident beam. The fielddistribution thus created forms a surface optical vortex, witha well-defined orbital angular momentum. Counterpropagat-ing or copropagating evanescent surface optical vortices can,in principle, also be created. Clearly, there is ample scope forinterference effects, different possibilities for atom trappingand atom manipulation at a surface, and optical surface span-ners and tweezers. In what follows we outline the physicsunderlying surface optical vortices and detail their possibleinvolvement in the applications mentioned above.Consider first the electric field of a Laguerre-GaussianLG beam traveling along z in a medium of a constant re-fractive index n, characterized by the integers l and p, circu-lar frequency , and axial wave vector k=nk0, where k0= /c is the wave vector in vacuum. For LG light planepolarized along yˆ, the field vector can be written in cylindri-cal coordinates asEklpI r,,z = yˆFklpr,zeikz−teil, 1where Fklpr ,z is the standard envelope function of r and z,with r=x2+y2:*On leave of absence from New York College of Athens, 38 Ama-lias Avenue, GR 105 58, Athens, Greece.PHYSICAL REVIEW A 79, 011806R 2009RAPID COMMUNICATIONS1050-2947/2009/791/0118064 ©2009 The American Physical Society011806-1Fklpr,z = Ek00Clp1 + z2/zR21/2 2rwzlLpl 2r2w2ze−r2/w2zeiGRz. 2Here Ek00 is the amplitude for a corresponding plane wave ofwave vector k, Clp is a mode constant, and the beam waist ataxial coordinate z is wz such that w2z=2z2+zR2 /kzR,where zR is the Rayleigh range. The last phase factor in Eq.2 accommodates both the Guoy phase and the change inbeam curvature with axial position. We haveGRz =kr2z2z2 + zR2+ 2p + l + 1tan−1z/zR . 3Note that the plane z=0 corresponds to the minimum beamwaist w0=w0 and on this plane the Guoy and curvatureterms in Eq. 3 both vanish.Such a light field can be arranged, as shown in Fig. 1, tostrike the internal planar surface of a medium, in which it ispropagating—that is, in contact with the vacuum. If the in-terface with the vacuum occupies the plane z=0 and theangle of incidence, , exceeds the total internal reflectionangle, an evanescent mode is created in the vacuum. Themain requirements are the applicability of the standardphase-matching condition of boundary reflection and thecondition that the electric field vector component tangentialto the surface be continuous across the boundary. To be ableto define the evanescent electric field, we must first obtainexpressions appropriate for a beam incident at an angle .This is obtainable by simple rotational transformations of theexpressions in Eqs. 1–3.The fields of a Laguerre-Gaussian beam propagating in ageneral direction can be constructed from the formalism out-lined above by the application of two coordinate transforma-tions, the first transformation rotating the beam as a rigidbody about the y axis by angle  and the second rotates theresultant beam in the z plane about the x axis by an angle .The compound transformation produces the relationsx → x = x cos  + z sin  , 4y → y = − x sin  sin  + y cos  + z cos  sin  , 5z → z = − x sin  cos  − y sin  + z cos  cos  . 6Equations 4–6 are useful in general situations involvingmultiple beams of different orientations incident on the sur-face. For a single beam the expressions simplify consider-ably. To determine the field distributions of the incidentbeam, we choose = and =0, and for the internally re-flected light, we have =	− and =0, as shown in Fig. 1.The continuity of the electric field vector tangential to thesurface, along with the exponential decay with the coordinatez, determines the form of the evanescent field in the vacuumregion. We haveEklpevanx,y,z = 2EklpI x → x cos ;y ;z → − x sin e−zk0n2 sin2 −1e−ik0nx sin . 7The explicit form of the evanescent electric field that dis-plays the angular momentum properties, as well as the modecharacteristics, is as follows:Eklpevanx,y,z= yˆE000Clp1 + x2 sin2 /zR21/2 2x2 cos2  + y2w021 + x2 sin2 /zR21/2lexp− x2 cos2  + y2w021 + x2 sin2 /zR21/2Lpl x2 cos2  + y2w021 + x2 sin2 /zR21/2exp	− zk0n2 sin2  − 1 − ik0nx sin expil arctany/x cos  . 8Note that the above expression for the evanescent light bearsthe vorticity nature in that it is characterized by the azi-muthal phase dependence expil arctany /x cos  of the in-cident and internally reflected light. The important point tobear in mind in this context is that the field distribution as-sociated with this vortex is concentrated on the surface,rather than axially, as in normal Laguerre-Gaussian light inan unbounded space. The typical exponential decay of theevanescent light intensity along z i.e., perpendicular to theinterface is shown in Fig. 2. This is governed by the factorexp−2zk0n2 sin2 −1, which is independent of the inte-gers l and p. The length scale of the exponential decay alongz is seen, from Fig. 2, to span a small fraction of the wave-length, indicating that the evanescent light does not play asignificant role in the trapping normal to the surface. Ad-sorbed atoms are generally subject to the a much morestrongly attractive van der Waal potential.zxEiErmaterial region (z<0) vacuum region (z>0)EevanFIG. 1. Color online Laguerre-Gaussian light internally re-flected at an angle  greater than the critical angle schematic. Theincident beam is arranged such that at =0 the beam waist coin-cides with the surface at z=0. The evanescent light possesses angu-lar momentum properties, but is confined near the surface, exponen-tially decaying in the direction normal to the surface.LEMBESSIS, BABIKER, AND ANDREWS PHYSICAL REVIEW A 79, 011806R 2009RAPID COMMUNICATIONS011806-2The intensity distribution corresponding to the evanescentlight created by an incident Laguerre-Gaussian beam forwhich l=1 and p=0 is shown in Fig. 3, plotted in the x ,yplane at z=0. It is seen that the evanescent light possesseswell-defined intensity maxima and minima that can be usedto trap adsorbed atoms with transition frequencies appropri-ately detuned from the frequency  of the light. Note that theprofile of the intensity distribution is, in fact, no longer cir-cular, but elliptical, because the light strikes the surface atthe angle of incidence ; the ellipticity increases with in-creasing .The case l=3 and p=2, with the same parameters as inearlier figures, is shown in Fig. 4. This case presents a morecomplex field distribution for the corresponding surface op-tical vortex. Clearly the field distributions for any values ofthe parameters l and p can be obtained in a similar manner:we can in principle create a surface vortex of any order,confirming that the surface optical vortex phenomenon isquite general.Consider now the case of copropagating incident beamsof opposite helicity creating an interference of two surfacevortices, in a manner similar to that discussed recently forbeams in an unbounded space, where a frequency offsetleads to the phenomenon of the optical Ferris wheel 11. Itis easy to see that the total electric field generated in thevacuum region contains the factorcos	l arctany/x cos  . 9The appearance of the cosine term in Eq. 9 indicates inter-ference of the two evanescent light beams in the azimuthaldirection. Figure 5a shows the field distributions for thez/IFIG. 2. Color online Intensity variations of the evanescentsurface optical vortex arbitrary units with distance normal to thesurface in units of wavelength . This variation is independent ofthe values of l and p.x/w0y/w0IFIG. 3. Color online Intensity distribution arbitrary units onthe surface z=0 for the surface vortex arising from internally re-flecting a Laguerre-Gaussian light beam for which l=1 and p=0.Distances are measured in units of the beam waist w0. The light hasa wavelength =590 nm, and beam waist is taken as w0=35, cor-responding to a long Rayleigh range zR2.2710−3 m. The refrac-tive index is assumed to be n=2, and the angle of internal reflec-tion is taken as =	 /3, which is greater than the critical angle.x/w0y/w0IFIG. 4. Color online As in Fig. 3, but here the Laguerre-Gaussian beam is such that l=3 and p=2.y/w0 x/w0Iy/w0x/w0(a)(b)FIG. 5. a Color online The intensity distribution arbitraryunits on the surface at z=0 arising from two copropagating inter-nally reflected Laguerre-Gaussian light beams for which l1=2, p1=0 and l2=−2, p2=0. Distances along the axes are measured inunits of the beam waist w0=35, with =590 nm. b The corre-sponding contour plot. If the beams are slightly different in frequen-cies such that 1−2=, the intensity distribution pattern rotates ata frequency  /2  l1− l2  .SURFACE OPTICAL VORTICES PHYSICAL REVIEW A 79, 011806R 2009RAPID COMMUNICATIONS011806-3case l1=2, p1=0 and l2=−2, p2=0, and the contour plots areshown in Fig. 5b. If the frequencies of the beams differslightly, the pattern will rotate at a frequency  /2  l1− l2 with equal to the frequency difference.In the above analysis, we have outlined the essential phys-ics underlying the phenomenon of surface optical vortices,associated particularly with Laguerre-Gaussian light. Wehave shown that the intensity distribution for a typical sur-face optical vortex has a limited spatial extent in the direc-tion normal to the surface in the vacuum region, of the orderof a fraction of a wavelength. However, the in-plane distri-butions mimic those of the incident and internally reflectedlight and so carry the signature of orbital angular momentumin the incident beam. The optical vortices of a Laguerre-Gaussian basis constitute a complete set for the descriptionof vortices of any desired order defined by the integers l andp, so that the phenomenon is quite general. Further work isneeded to uncover the properties of surface optical vortices.Useful information can be gained by determining phase pro-files, wave-front structures, and energy-momentum flow.To conclude, we note that in addition to the intrinsic im-portance of surface optical vortices as physical entities intheir own right, some technical applications can be envisagedat this stage. We have briefly mentioned the manipulation ofadsorbed atoms and molecules held on surfaces by van derWaal force, and in this connection there has recently beenstrong interest in the opportunity for precision manipulationof such atoms. However, it appears that none of the methodssuggested so far includes near-resonance optical manipula-tion along the surface. Optical vortices offer an unprec-edented potential for the manipulation of adsorbed atomscongregating in extremum regions of intensity on the sur-face. The principle can be employed to create patterned sur-faces by employing carefully designed sets of incident beamsto create a lattice of evanescent light wells. The system alsoaffords the opportunity to manipulate larger objects on thesurface, translation and rotation being affected, by movingthe spot at which the incident beam strikes the interface.Finally it is to be noted that surface optical vortices of otherkinds can be generated from other optical beams carryingorbital angular momentum and that highly significant en-hancements of the evanescent light fields can be realized byintroducing a metallic film. In the evanescent surface plas-mon modes the fields can be at least an order of magnitudelarger 18. Work on surface plasmon optical vortices andguided modes is now in progress, and the results will bereported in due course.V.E.L. wishes to thank the ESF for financial support underthe Programme QUDEDIS Exchange Grant No. 1750, whilethis work was carried out.1 L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P.Woerdman, Phys. Rev. A 45, 8185 1992.2 L. Allen, M. J. Padgett, and M. Babiker, in Progress in OpticsXXXIX, edited by E. Wolf Elsevier Science B.V., 1999, pp.291–372.3 L. Allen, S. M. Barnett, and M. J. Padgett, Optical AngularMomentum Institute of Physics Publishing, Bristol, 2003.4 Structured Light and its Applications: An Introduction toPhase-Structured Beams and Nanoscale Optical Forces, editedby D. L. Andrews Academic, Burlington, MA, 2008.5 Special journal issue on optical tweezers J. Mod. Opt. 50,1501ff 2003.6 M. Babiker, W. L. Power, and L. Allen, Phys. Rev. Lett. 73,1239 1994.7 M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. DávilaRomero, Phys. Rev. Lett. 89, 143601 2002.8 D. L. Andrews, A. R. Carter, M. Babiker, and M. Al-Amri,Proc. SPIE 6131, 101 2006.9 A. R. Carter, M. Babiker, M. Al-Amri, and D. L. Andrews,Phys. Rev. A 72, 043407 2005.10 A. R. Carter, M. Babiker, M. Al-Amri, and D. L. Andrews,Phys. Rev. A 73, 021401R 2006.11 S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D.Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Ar-nold, Opt. Express 15, 8619 2007.12 C. R. Bennett, J. B. Kirk, and M. Babiker, Phys. Rev. A 63,033405 2001, and references therein.13 V. Garces-Chavez, K. Dholakia, and G. C. Spalding, Appl.Phys. Lett. 86, 031106 2005.14 P. J. Reece, V. Garces-Chavez, and K. Dholakia, Appl. Phys.Lett. 88, 221116 2006.15 L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, J. Opt.Soc. Am. B 25, 849 2008.16 Djenan Ganic, Xiaosong Gan, and Min Gu, Opt. Express 12,5533 2004.17 Baohua Jia, Xiaosong Gan, and Min Gu, Opt. Express 16,15191 2008.18 W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature London424, 824 2003.LEMBESSIS, BABIKER, AND ANDREWS PHYSICAL REVIEW A 79, 011806R 2009RAPID COMMUNICATIONS011806-4

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