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Tuning the orbital angular momentum in

optical vortex beams

Christian H.J.Schmitz,Kai Uhrig,Joachim P.Spatz

and Jennifer E.Curtis

Max-Planck-Institute for Metals Research,Dept.of New Materials and Biosystems

Heisenbergstr.3,D-70569Stuttgart,Germany

University of Heidelberg,Dept.of Biophysical Chemistry

Im Neuenheimer Feld253,D-69120Heidelberg,Germany

Jennifer.Curtis@urz.uni-heidelberg.de

Abstract:We introduce a method to tune the local orbital angular mo-

mentum density in an optical vortex beam without changing its topological

charge or geometric intensity distribution.We show that adjusting the

relative amplitudes a and b of two interfering collinear vortex beams of

equal but opposite helicity provides the smooth variation of the orbital

angular momentum density in the resultant vortex beam.Despite the

azimuthal intensity modulations that arise from the interference,the local

orbital angular momentum remains constant on the vortex annulus and

scales with the modulation parameter,c=(a?b)/(a+b).

?2006Optical Society of America

OCIS codes:(050.1970)Diffractive optics;(090.0090)Holography;(120.5060)Phase modu-

lation;(140.7010)Trapping;(230.6120)Spatial light modulators.

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32.The general expression for the phase of a beam resulting from the collinear superposition of two optical vortex

beams of equal but opposite topological charge is

?(θ)=arctan[c tan(?(θ+α))]+πRound[?(θ+α)/π]+2π+(?a+?b)/2,

for c=0orθ+α=nπ/(2?)with n=1,3,5...(4??1)and

?(θ)=(c?/|c|)(θ+α)+2π?+(?a+?b)/2,

forθ+α=nπ/(2?)with n=1,3,5...(4??1).The function Round(x)rounds the argument to its closest integer.

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#70769 - $15.00 USD Received 11 May 2006; revised 30 June 2006; accepted 3 July 2006 (C) 2006 OSA24 July 2006 / Vol. 14, No. 15 / OPTICS EXPRESS 6605

Express14,938–949(2006).

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1.Introduction

Light carries angular momentum comprised of both a spin component associated with polar-ization[1,2],and of an orbital component arising from the spatial pro?le of light intensity and the phase[3].Considerable interest in orbital angular momentum arises from its potential use in multiple applications including quantum information processing[4,5,6,7,8,9],atomic manipulation[10,11,12,13],micromanipulation[14,15,16]and the biosciences[17,18]. Control of the local and total orbital angular momentum in a beam,as well as the quantum mechanical state of the photons is important in many of these applications[4]as is the spatial distribution and photon density of the beam.

In general,any beam with inclined wavefronts carries orbital angular momentum.Allen and coworkers[3]showed that Laguerre-Gaussian laser beams,which have a helical phase structure described byφ=?θ,possess a well-de?ned orbital angular momentum of?ˉh per photon.At the center of the helical phase is a singularity where the phase is unde?ned and the?eld amplitude vanishes,giving rise to a“dark beam”in the core of the light wave.The orbital angular momen-tum in a helical beam(also called an optical vortex beam)can be adjusted by either changing the wavefronts’helicity,?,or by increasing the photon?ux.However,since the radius of an optical vortex scales with its helicity[19,20],applications that require controlled geometry or photon density will be limited to?xed orbital angular momentum(OAM).

This article introduces an OAM-tunable optical vortex which maintains a constant geometry and total intensity during tuning(Fig.1).The OAM-tunable optical vortex is the resultant beam produced by the overlap of two collinear optical vortex beams of equal helicity but opposite chirality.We show that the local orbital angular momentum density of this complex optical vor-tex can be smoothly tuned without adjusting the resultant beam’s topological charge,which is shown to be equivalent to the magnitude of the helicity of the two component beams.Surpris-ingly,despite the rapid variation of the photon?ux with azimuthal position,the local orbital angular momentum density for a given interference vortex remains constant.The?nal result is the decoupling of the orbital angular momentum,the topological charge,the geometry and the power distribution in this class of helical beam.

2.Interference of two oppositely charged optical vortex beams

We experimentally and theoretically address the“interference”optical vortex produced by the superposition of two collinear optical vortices with variable relative amplitudes and equal but opposite helicity,?.The scalar?eld of a classic optical vortex is u=A(r)e(i?θ)where?is a positive or negative integer and A(r)is real.An interference vortex is described by the sum of two such scalar?elds,

u iv=A(r)[ae i(?θ+?a)+be?i(?θ??b)]=B(r,θ)e i?(θ),(1) with constant,positive,real amplitudes a,b and phase offsets?a and?b.In agreement with intuitive arguments,the mixing amplitudes a and b govern the relative contribution of each optical vortex to the local and total orbital angular momentum.The quantitative dependence on the mixing amplitudes is derived in Section4.

Collinear optical vortex beams have been presented in various contexts.The optical cog-wheel beam[21,22]is an interference vortex with a=b(Fig.1(f)).It carries no orbital angular momentum,and possesses a modulated intensity pro?le about the azimuthal coordinate and

#70769 - $15.00 USD Received 11 May 2006; revised 30 June 2006; accepted 3 July 2006 (C) 2006 OSA24 July 2006 / Vol. 14, No. 15 / OPTICS EXPRESS 6606

p h a s e m a s k i n t e n s i t y d i s t r i b u t i o n

2

c =

1c =

0.2

c =

0Fig.1.(a-c)Phase masks of interference vortices with ?=20and (d-f)their measured

intensity for mixing amplitudes:c =1,0.2and 0.As c decreases,more intensity is directed

from the minima to the 2?maxima on the ring to the maxima.A mixing amplitude c =0

gives a binary intensity distribution.

with 2?intensity peaks.Other work shows that the interference of two helical beams with different ?-values produces concentric optical vortices of different radii [23,24].In a more general approach,Soskin and coworkers address the orbital angular momentum in multiple collinear vortices of equal amplitude and varying topological charges [25].Maleev and Swart-zlander [26]address the intensity distribution and motion of optical phase singularities of non-collinear overlapping optical vortices.Segev [27]presents a theoretical treatment of the orbital angular momentum content in stable necklace-ring beams in a self-focusing Kerr medium.A subclass of necklace-ring beams are equivalent to the interference vortex introduced here.Very recent work has introduced the above formulation to study thermally activated transport on tilted washboard energy landscape potentials [28].

To investigate the angular momentum content of an interference optical vortex,we cre-ate an optical vortex trap using a holographic optical tweezers (HOT)apparatus [15].Op-tical traps are particularly useful for measuring the angular momentum content of focused beams [14,29,30].The key component of our HOT apparatus,as previously described [31],is a computer-addressable re?ective liquid crystal display called a spatial light modulator (SLM)which imposes the phase pro?le of the desired interference vortex onto an incoming collimated beam.The phase pro?le of an interference vortex extracted from Eq.(1)is ?(θ)=arctan [c tan (?(θ+α))]+(?a +?b )/2,(2)

where α=(?a ??b )/(2?).This expression accurately describes the phase for values between ?π/(2?)<θ+α<π/(2?).For θ+α=π/(2?)the phase is π/2.This wedge of phase plus a multiple of πis repeated around the axis in 2?slices from 0<θ<2πas shown in Fig.1(a-c).The mixing amplitudes a and b determine the modulation amplitude,c =(a ?b )/(a +b ).To explicitly calculate the phase at all values of θ,we introduce an extended expression [32].

When creating an optical vortex with HOT,the beam’s amplitude is determined by the input beam,which is typically Gaussian.Theory shows that the discrepancy between the theoretical amplitude and the true amplitude of the beam does not signi?cantly impact the resultant beam’s intensity distribution or the orbital angular momentum [20].An image of this beam is then transferred to the back focal plane of a high numerical aperture objective lens which tightly focuses the beam into the focal plane of a microscope,where it may

be imaged with and without trapped objects.

#70769 - $15.00 USD Received 11 May 2006; revised 30 June 2006; accepted 3 July 2006

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I /a +b ()2

2a n d ()/M M j /q l 024681012141618

1

234

5q /degree }I/()a +b 22

}()/M M j/q l Fig.2.Azimuthal dependence of the normalized intensity,I /(a 2+b 2)and the normalized

phase derivative and local wavefront helicity,(??/?θ)/?.The lines represent calculated

values using Eqns.3and 5with ?=20and c =0.6(dark lines)and c =0.2(light lines).

The data symbols represent the measured local intensities whose errors are in the range of

the symbol size.

Figure 1(a-c)shows three phase masks calculated using Eq.(2)with c =1.0,0.2and 0.An interference vortex has the phase of a classical optical vortex when c =±1.When c =0,the phase pro?le is binary with 2?alternating phase segments of 0and π,equivalent to the phase of the optical cogwheel.For 0<|c |<1,the local curvature of the helical wavefront is no longer constant nor linear;however the phase singularity remains,qualifying the beam as a kind of optical vortex [25].

Figure 1(d-f)shows the corresponding intensity distributions imaged with the HOT setup.For all values of the modulation parameter,c ,a ring of light appears with 2?radial intensity fringes that vary with the azimuthal angle,as previously described for both the classical optical vortex trap produced by HOT [33]and the optical cogwheel [21].Adjusting the relative constant of phase of the two overlapping vortices,α,rotates the intensity fringes about the center of the resultant beam.The intensity distribution between the bright and dark fringes varies as a function of the modulation parameter c ,as the measured data show in Fig.2.Measurements show that the total intensity in the beam is conserved with varying c ,and that it is merely redistributed between the dark and bright fringes.

This variation agrees with the theoretical prediction of the intensity distribution of an inter-ference vortex in the image plane of the microscope,

I ∝|?u iv (r ′,θ′)|2=?A

(r ′)2 a 2+b 2+2ab cos 2?(θ′+α) ,(3)plotted as the solid lines overlapping the data points.The ?eld in the SLM plane,u iv (r ,θ)is related to the ?eld in the microscope’s image plane,?u iv (r ′,θ′),by a Fourier transform,which has been shown to preserve the beam’s helical phase and give rise to a radially-dependent amplitude for a single optical vortex [20].Hence:

?u iv (r ′,θ′)=?A (r ′)[ae i (?θ′+?a )+be ?i (?θ′??b )]=?B (r ′,θ′)e i ??(θ′),(4)

where ??

(θ′)≡?(θ)and ???/?θ′≡??/?θ.The radius of the interference vortex trap is independent of c and linearly proportional to ?like that of a classic optical vortex trap [20].However,unlike a conventional optical vortex,variations in the local wavefront helicity ??/?θdo not result in an azimuthally dependent radius R (θ′)∝(R 0+??/?θ)[34].While this ansatz has been used to dramatically vary the geometry of conventional optical vortices [35],the interference vortex whose local helicity is #70769 - $15.00 USD

Received 11 May 2006; revised 30 June 2006; accepted 3 July 2006

(C) 2006 OSA 24 July 2006 / Vol. 14, No. 15 / OPTICS EXPRESS 6608

-1.0-0.50.00.5 1.00.00.2

0.4

0.60.81.0Q c

T (s )W -30-20-100102030

51015

20

Fig.3.(a)The measured period T for the modulation parameter c =1.0,0.3,0.2,0.1de-

pends parabolically on the mean helicity or equivalently the topological charge,Q .(b)The

relative rotational frequency ?=T min /T as a function of c .The line represents s (c )?1

in Eq.(6),which is expressed analytically in Eq.(10).The maximum rotational speed

(c =±1)was 13μm/s.

modulated with 2?peaks (Fig.2),

???θ=c ?cos 2[?(θ+α)]+c 2sin 2[?(θ+α)],(5)

lacks the 2?intensity ?ower-like lobes that would appear on a conventional vortex.The differ-ence is that the amplitude of an interference vortex depends on the azimuthal coordinate while the amplitude of the optical vortex does not.This apparent limitation is in fact what makes the interference vortex a potentially interesting tool.As is shown in the following section,the vortex’s radius remains constant despite changes in ??/?θwhile its local orbital angular mo-mentum density is tuned smoothly.

3.Smooth variation of the orbital angular momentum

Measurements of the local orbital angular momentum density can be made by studying the velocity of a trapped particle as it spins around the annulus of a helical beam [20,30,36,37,38].In practice,only a small percentage of the orbital angular momentum is transmitted through scattering [37].Nevertheless,the particle speed scales as expected with increases in power,P ,and with the predicted orbital angular momentum in the beam which depends on ?.In some regimes,the azimuthal intensity variations interfere with the transport of particles around the vortex [20,28].This occurs when the variations are large enough to locally trap particles in an optical potential well deeper than k B T .To avoid this effect,each c -dependent interference optical vortex was ?lled completely with 2μm diameter polystyrene microspheres to ensure smooth rotation.Their resulting collective frequency of rotation was determined using video microscopy and particle tracking.

Our experiments show that the interference optical vortex transfers orbital angular momen-tum to the particles when c =0.Figure 3(a)shows the period of rotation T measured for four different values of c as a function of the topological charge,Q = ??/?θ θ,given by the beam’s mean helicity [39].For c >0,Q =?and for c <0,Q =??.Particle rotation is similar to that of a single particle on a classical optical vortex whose period scales as T ～?2[20]but with an extra dependence on c ,T =s (c ) 1+t ?2 .(6)Here,t is constant for all c and ?,while s depends on the modulation parameter c .The period changes systematically with different c -values even with ?xed ?.This is a rare demonstration #70769 - $15.00 USD Received 11 May 2006; revised 30 June 2006; accepted 3 July 2006

(C) 2006 OSA 24 July 2006 / Vol. 14, No. 15 / OPTICS EXPRESS 6609

of the fact that orbital angular momentum and topological charge are conceptually indepen-dent[25].It is possible to have non-zero orbital angular momentum when Q=0[40]. Particles spin fastest on a classical vortex(c=±1)resulting in the shortest period T min∝?2. The normalized rotational frequency?=T min/T is constant for all?at?xed c as shown in Fig.3(b).Each data point constitutes the average of several experimental?values at different Q.These observations imply that the average orbital angular momentum can be smoothly varied by changing the modulation parameter c independently of?and P.Meanwhile,?and P uniquely determine the maximum rotational frequencyω∝P/?2and the interference vortex’s radius, R∝?.An additional advantage is the possibility to tune the particle velocity from full speed to zero without decreasing the intensity such that thermal forces dominate the radial optical gradient force that con?nes particles to the optical vortex trap.

4.Local orbital angular momentum is constant

Our measurements of colloidal rotation on an interference optical vortex do not de?nitively measure the local orbital angular momentum density in the beam.Given the modulated az-imuthal intensity,the angular momentum may vary with position and yet be effectively aver-aged out by motion on a particle-?lled optical vortex.Thus,we consider theoretically how the local orbital angular momentum density varies.

The angular momentum density in a transverse electromagnetic?eld is M=r×p where the linear momentum density is p=ε0E×B.Following the lead of Allen[3],a collinear inter-ference optical vortex trap has a scalar?eld?u iv,which is related to the vector potential in the Lorentz gauge by A=?u iv exp(?i(kz+ωt))?x with E=??A/?t and B=?×A.In the paraxial approximation,the time average of the real part of p for any scalar?eld u is given by

p=ε0

2

Re[E?×B+E×B?]=iω

ε0

2

(u??u?u?u?)+ωkε0|u|2?z.(7)

Here?z is the unit vector in the z direction and?is the gradient operator in r andθ.Thus,the azimuthal component of linear momentum density for an arbitrary scalar?eld with inclined wavefronts described by any azimuthally-dependent phase?(θ)is

p=ε0ω 1r???θ|u|2 ?θ.(8)

Even when the amplitude has an azimuthal dependence,the momentum depends only on the phase and the total intensity of the beam.

The local orbital angular momentum density about the z-axis of an interference OV in the trapping plane is thus

M z=ε0ω??

?θ

|?u iv|2=ε0ω

4a2c?

(1+c)2

?A(r′)2=ε

0ω

4b2c?

(1?c)2

?A(r′)2,(9)

for u=?u iv.Despite the complicated functional dependence of the intensity|?u iv|2and the local helicity??/?θshown in Fig.2,they exactly cancel out to give a constant local orbital angular momentum density at a given radius,r′.Two examples are shown in Fig.2for c=0.6and c=0.2.The local helicity is an oscillatory function with2?peaks coincident with the intensity minima of the optical vortex atθ′+α=nπ/2?,where n is an odd integer between0and2?. While the magnitude of the minimum photon?ux decreases with decreasing c,the correspond-ing local helicity increases proportionally.The result is that the local orbital angular momentum is constant at all azimuthal positions on the interference vortex at a?xed radius.

#70769 - $15.00 USD Received 11 May 2006; revised 30 June 2006; accepted 3 July 2006 (C) 2006 OSA24 July 2006 / Vol. 14, No. 15 / OPTICS EXPRESS 6610

(a) 0 s (d) 6 s

(c) 4 s 2 3

Fig.4.An optical vortex array consisting of three optical vortices with ?=8.c 1of vortex

1is varied between 1and -1while vortices 2and 3have ?xed c 2=0.17and c 3=?1.The

time series shows the independent control of the rotation of three colloidal rings:(a)Ring 1

is slowly rotated counterclockwise,(b)stopped,and (c)rotated clockwise with increasing

speed (c,d).The other rings (2,3)are rotated with constant velocity of different magnitude

and sign.The bead in the center is ?xed by a regular optical trap.

The orbital angular momentum carried by a classical optical vortex is M z (c =1)∝a 2?and M z (c =?1)∝?b 2?.Thus,the normalized frequency of rotation for a particle around the vortex is given by

?(c )=M z (c )M z (c =±1)=±4c (1±c )2.(10)

The shape of this curve captures the experimentally measured dependence of the normalized frequency on the modulation parameter c .The agreement between theory and data is shown in Fig.3(b).The solid line represents the theoretical prediction and is equivalent to s ?1as de?ned in Eq.(6).Thus,we have presented a deterministic approach to tune both the local and total orbital angular momentum density in an optical vortex beam of ?xed radius and ?xed total power.

5.Tunable optical vortex arrays

As described previously [15,41,42],one to dozens of classical optical vortices may be cre-ated simultaneously and located independently in the focal volume using HOT.Each of these optical vortices may have a unique topological charge and therefore,size and orbital angular momentum content that is controlled by the input parameter ?i ,where 0≤i ≤N and N is the total number of traps.Here it is shown that by replacing the standard input phase for each de-sired vortex,φi =?i θ,with that of an interference vortex (Eq.(2)),we can create arrays of identically-sized optical vortices with smoothly adjustable rotational frequencies.In the exam-ple shown here,the phase mask for a system consisting of three vortices is calculated.The three interference vortices are given the same topological charge (?=8)but are assigned different amplitude modulation values c i .The total angular momentum of the ?rst vortex was varied by changing c 1from -1to 1at different times as captured in Fig.4,while the angular momenta of the two other vortices were set by ?xing c 2=0.17and c 3=?1.

When ?lling these vortices with microspheres,the measured rotational frequency followed the dependency on c established for a single vortex.The standard deviation of the

measured #70769 - $15.00 USD Received 11 May 2006; revised 30 June 2006; accepted 3 July 2006

(C) 2006 OSA 24 July 2006 / Vol. 14, No. 15 / OPTICS EXPRESS 6611

values relative to predicted values was in the range of2%.Changing c1did not in?uence the rotation of the other two vortices.When c1=0,the rotation stops completely.Thus,indepen-dent control of the speed and direction of angular frequency of particles on multiple interference vortices with?xed diameters in spatially-controlled patterns has been achieved.

6.Conclusion

The collinear superposition of two classic optical vortices with equal helical wavefronts but opposite handedness provides an OAM-tunable class of optical vortices via an independent parameter,c,called the modulating amplitude.This parameter controls the orbital angular mo-mentum content of individual optical vortices without affecting their geometry,which is sep-arately controlled by the topological charge.A closer look at the intensity distribution around a vortex shows that adjusting c changes not only the total angular momentum content but also the depth of the2?intensity modulations on optical vortices.Surprisingly,despite these local intensity variations,classic electromagnetic theory predicts that the local angular momentum remains constant with azimuthal position in the beam due to compensation by increased local orbital angular momentum.Furthermore,c is the?rst knob aside from power capable of tun-ing the optical landscape trapping potential of a?xed-diameter optical vortex trap.This has interesting implications for statistical studies of thermally activated biased transport and the development of novel Brownian ratchets[20,28].The extension of these basic results to col-lections of optical vortices has been demonstrated.Efforts to use optimally positioned groups of optical vortex traps to construct optically-driven micromachines[23]or in-parallel molecular or cell assays[43]may be facilitated by this new degree of control.

Acknowledgments

We gratefully acknowledge David Grier for insightful and stimulating discussions.

#70769 - $15.00 USD Received 11 May 2006; revised 30 June 2006; accepted 3 July 2006 (C) 2006 OSA24 July 2006 / Vol. 14, No. 15 / OPTICS EXPRESS 6612

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