Gaussian beams with very high orbital angular momentum

Gaussian beams with very high orbital angular momentum

15 December 1997 OPTICS COMMUNICATIONS ELSEWIER Optics Communications 144 (1997) 210-213 Gaussian beams with very high orbital angular momentum ...

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15 December

1997

OPTICS

COMMUNICATIONS ELSEWIER

Optics Communications

144 (1997) 210-213

Gaussian beams with very high orbital angular momentum J. Courtial

*, K.

Dholakia, L. Allen, M.J. Padgett

School of Physics and Astronomy, Uniuersity of St. Andrew, Fife KY16 9SS. Scotland. UK Received

12 May 1997; revised 2 July 1997: accepted 2 July 1997

Abstract An elliptical Gaussian beam focussed by a cylindrical lens can possess large amounts of orbital angular momentum. We give an expression for the angular momentum, which arises from the azimuthal component of the Poynting vector, and show that it can be as high as 1OOOOfLper photon. We examine the phase distribution of the beams in an interference experiment. 0 1997 Elsevier Science B.V. PACS: 42.60.H Keywords: Astigmatism;

Laser beam propagation;

Angular

momentum

of light

In 1992 Allen et al. [1] predicted that light beams could possess a well defined orbital angular momentum about the beam axis, quite distinct from the spin angular momentum associated with circular polarisation. The proposal that Laguerre-Gaussian beams possessing orbital angular momentum could be created from any of the commonly occurring Hermite-Gaussian modes using a simple two-lens mode-converter was soon demonstrated experimentally by Beijersbergen et al. [2]. The paper also gave a physical explanation for the transfer of angular momentum to a beam by means of a lens, in terms of the force on a dielectric material arising from a field gradient. The effect of a thin lens on an arbitrary light field had been discussed theoretically by van Enk and Nienhuis [3] and equivalent results have been found subsequently by the use of operator algebra [4]. It was also recognised that the amount of angular momentum transferred by a lens can be arbitrarily high [2-41. All beams with an azimuthal phase term, eif4 of which Laguerre-Gaussian beams are an example, have an orbital angular momentum of /h. per photon [S]. The mechanical demonstration of this orbital angular momentum has recently been verified experimentally [6]. Microscopic parti-

* Corresponding

author. E-mail: [email protected].

0030-4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PfI SOO30-4018(97)00376-3

cles held within a Laguerre-Gaussian laser beam have been rotated and the orbital angular momentum quantified by the comparison of this rotation to that induced by the spin angular momentum. These results confirm that the orbital angular momentum is indeed /fi per photon [7]. The mode index of a Laguerre-Gaussian can assume any integer value, but in practical systems it is likely to be low. In this paper we show how a cylindrical lens system can be used to produce light beams with an arbitrary orbital angular momentum as high as 1OOOOh per photon. They are elliptical Gaussian light beams with general astigmatism, the propagation of which has been discussed by Amaud and Kogelnik 181. The intensity and phase structure of astigmatic, elliptical Gaussian beams differs from the Laguerre-Gaussian beams in a number of important ways. Elliptical beams have their peak intensity on the beam axis and consequently have no phase singularity. In this paper we derive an expression for the orbital angular momentum density and perform an interference experiment to investigate the phase structure of the beams. We concentrate on the angular momentum exchange that occurs when an elliptical Gaussian beam is transmitted through a cylindrical lens whose axes are not aligned with the axes of the beam. In this way astigmatism is introduced into the beam. We show that the orbital angular momentum imparted can be orders of magnitude higher than that

J. Courtial et al. /Optics

Communications

associated with that of a realisable Laguerre-Gaussian beam. When a transverse laser mode described by u(x,y,z) is passed through a thin lens it acquires an additional phase factor ,y(x,y) such that the mode function immediately after the lens, u’(x,y,z), is given by u’(x,y,z)

=~(x,~,z)exp[ix(*.y)].

Van Enk and Nienhuis [3] show that the associated change in the z-component of the orbital angular momentum, 6L,, for monochromatic light of angular frequency w is given by

For a cylindrical lens of focal length f, inserted at an angle (Y to the x-axis, the phase factor introduced for light of a wavenumber k is given by

2f

- &,( x’ sin’ (Y+ _v’cos’ a

z

+2gsin

Lycos a),

wq,k = --/

=

3.f

-x

-* 1

wq,k = ----I

_L

x’-_v’)sin2a

r

k

dxd_v((

--*. --z

dx

-$)eq(

w,w,

-$),

-----sin2a. 4

or -----sin2cu per photon.

and the orbital angular momentum is M L=-.P

1 w_f-w; - 2cf

xsincu+ycosa)(xcos(Y--sin(Y), f(

where w, and w, are the distances at which the Gaussian term drops to l/e of its on axis value in the x and y directions respectively. It is straightforward to show for such a beam that the change in orbital angular momentum on passing through a cylindrical lens aligned at an angle cy with respect to the x-axis is given by

W

dx

k = --

1-!--exp( T

2.f

dx

x2 -y’)sin2a)

An elliptical Gaussian beam aligned with the x and y axes has an intensity distribution

lu(x.y,:)l” =

--(xsinnfycoscr)‘,

$,=,y=

G=Xd!,-Yz

m /

2f

and is independent of position within the beam. For a collimated input beam, the phase of the field, $, is that introduced immediately after a cylindrical lens and is given by

x lu( s,y,,_)12.

6L. L

-,,““B,u,2. d+

M, =

P

dxdy((

If the mode distribution u( x,y,:) is symmetric about the x and y axes of the lens, as is the case for both a higher-order Hermite-Gaussian beam and an elliptical Gaussian beam, then by symmetry, the term 2 xy cos 2 cx integrates to zero and SL, simplifies to 6L,

For highly elliptical beams a few millimetres in size, at optical wavelengths, this can easily exceed 1OOOOh per photon. Only the f-number and aperture of the cylindrical lens limits the amount of orbital angular momentum that can be transferred. It is worth noting that, unlike the Laguerre-Gaussian beams, the elliptical beams have an on-axis intensity but this does not mean that they have an on-axis orbital angular momentum. It has been shown for a Laguerre-Gaussian beam that not only is the orbital angular momentum well defined, but it is also constant across the beam, unlike the local spin density which varies with intensity gradient [ 1,9]. In general, the z-component of the angular momentum density, MT, is given by [2]

M. --Z Z/Y@

so that 6L,

211

where C$ is the azimuthal position within the beam and 0 is the phase angle of the field. This is, of course, the integrand of our previous expression for 6L,. For a Laguerre-Gaussian beam x is simply /@+ and the z-component of the orbital angular momentum density divided by the energy density, p = o%,lul’, is simply,

k --(xsin(~+ycoscu)~

x(x,y)=

144 (19971210-213

fi

to energy density ratio

k

of

(xsina+.vcosa)(xcoscr-Jzsincu).

For a collimated elliptical beam we can usefully define a new pair of axes aligned with the principal axes of the cylindrical lens, such that x’ = x cos LY- y sin LYand y’ = xsin a+ycos (Y. so that M. L=-

kx’y’

P

o.f’

.

We see that in the case of the astigmatically focussed elliptical Gaussian beam, the ratio of orbital angular momentum to energy density varies across the beam and is zero on axis. Exchange of orbital angular momentum in light/matter interactions may depend crucially on this variation of local angular momentum density. In the transfer of orbital angular momentum to atomic systems, the

position of the atom within the beam is very important

DOI. Any non-circularly symmetric field distribution can exchange orbital angular momentum with a cylindrical lens. For example, a high-order, He~ite-Gaussian beam with its peak intensity off-axis could be used. The off-axis peak intensity would lead to even greater transfer of orbital angular momentum than for a simple Gaussian, The origin of this orbital angular momentum lies in the azimuthal component of the Poynting vector [l, I I] and its associated phase structure. In an earlier experiment, the phase structure of a Laguerre-Gaussian beam was examined by observing the interference pattern between the Laguerre-Gaussian beam and a plane wave of the same frequency [ 121.We have performed a similar experiment to observe the phase structure of ~tjgmatically focussed eliiptical beams. Although interference patterns of general astigmatic waves and their wavefronts were first observed many decades ago [13,14f, such inte~ero~ams have not been interpreted in the knowledge of the orbital angular momentum of the light beam. The experimental arrangement is detailed in Fig. 1. The output from a polarised He-Ne laser is expanded and coupled into a Mach-Zehnder interferometer. One arm of the jnt~~erom~ter contains a beam telescope adjusted to give a collimated, large diameter beam to act as the phase reference. The other arm contains two cylindrical lenses positioned to produce an elliptical beam with beam waists of approximately 2 mm and 0.2 mm, coincident with a third cylindrical lens of focal length 200 mm. This third lens is aligned at an angle of 45” with respect to the major axis of the elliptical beam and in our experiment is calctdated to impart an orbital angular momentum of approximately 2% per photon to

collimated elliptical beam

&4OQmm=2f

&CQ

Fig. 2. The interferogram of a plane wave reference and an elliptical beam which has passed through a cylindrical lens of focal length X00 mm aligned at 45” with respect to the principal axis of the elliptical beam.

the transmitted beam. Reduction of the focal length of the third lens, and/or increasing the dimensions of the incident beam, causes signi~c~tly larger amounts of orbital anguiar mumen~m to be transferred to the beam. However, in our interference experiment the fringe5 then become too close to resolve and cannot be displayed. A plane

d

astigmatically focussed , elliptical beam

P

z

inte~e~nce pattern between plane wave reference and astigmatically focussed elliptical beam reference incident collimated Gaussian beam Fig. 1. The experimental ~ange~nt beam.

to obtain the in~rfe~~gram of the astig~ti~ally focussed elliptical beam and a collimated reference

J. Courtial et al./Optics

Communications

t intensity

Fig. 3. Wave front and intensity distribution of an astigmatically focussed elliptical Gaussian beam immediately behind the cylindrical lens.

t

144 (1997) 210-213

113

beams, is for the optically induced rotation of micron sized particles. In this case the large particle size compared to the size of the beam ensures that the variation in angular momentum density within the beam is not important. Previously, these so called “optical spanners” have been used to verify the angular momentum content of Laguerre-Gaussian laser beams [6,7]. The significantly higher angular momentum content of these elliptical beams will allow greater torques to be applied and correspondingly greater rotational velocities to be maintained. This may be useful for a number of applications where the mechanical rotation of a trapped particle forms the basis of the overall system requirement. In addition, astigmatically focussed elliptical Gaussian beams have no phase singularity and can be generated directly from the fundamental Gaussian beam with 100% efficiency. There are no requirements for mode matching and the amount of orbital angular momentum transferred is continuously adjustable over a broad range of values by changing the orientation of the cylindrical lens. It is very likely that these beams may also be used in interactions with atoms [15], but specific calculations need to be carried out for the relevant distribution of phase and amplitude.

intensity Acknowledgements

Fig. 4. Wave front and intensity distribution of an astigmatically focussed elliptical Gaussian beam in the focal plane of the cylindrical lens.

This work is supported by EPSRC; MJP is a Royal Society Research Fellow; KD is a Royal Society of Edinburgh Research Fellow.

References a distance d behind this third lens is imaged on a CCD detector array where the interference pattern between the elliptical beam and the collimated reference beam can be recorded. Fig. 2 shows the phase and intensity profile of the elliptical Gaussian beam at various positions behind the plane of the third lens. Immediately behind the plane of the third lens, the phase structure of the beam maps that of the optical thickness of the lens, and hence straight line interference fringes are observed parallel to its principal axis. In the vicinity of the focal plane, the phase structure is more complex; the fringes are hyperbolic. From these interferograms it is possible to deduce the form of the wavefront. Figs. 3 and 4 show schematic representations of the intensity and phase structure of these astigmatically focussed elliptical Gaussian beams, the associated Poynting vectors and resulting z-component of the angular momentum. The z-component of the orbital angular momentum of these beams arises from the azimuthal component of the Poynting vector and its cross-product with the radius vector. A possible application of these astigmatically focussed elliptical Gaussian beams, or high-order Hermite-Gaussian

[ll L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 (1992) 8185. 01 M.W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, J.P. Woerdman, Optics Comm. 96 (1993) 123. [31 S.J. van Enk, G. Nienhuis, Optics Comm. 112 (1994) 225. t41 G. Nienhuis, L. Allen, Phys. Rev. A 48 (1993) 656. [51 SM. Barnett, L. Allen, Optics Comm. 110 (1994) 670. H. RubinszteinEl H. He, M.E.J. Friese, N.R. Heckenberg, Dunlop, Phys. Rev. Lett. 75 (1995) 826. [71 N.B. Simpson, K. Dholakia, L. Allen. M.J. Padgett, Optics Lett. 22 (1997) 12. [81J.A. Arnaud, H. Kogelnik, Appl. Optics 8 (1969) 1687. [91 L. Allen, V.E. Lembessis, M. Babiker, Phys. Rev. A 53 (1996)R2937. [lOI W.L. Power, L. Allen, M. Babiker, V.E. Lembessis, Phys. Rev. A 52 (1995) 479. [Ill M.J. Padgett, L. Allen, Optics Comm. 121 (1995) 36. [121 M. Padgett, 3. Arlt, N. Simpson, Am. J. Phys. 64 (1996) 77. 1131 R. Kingslake, Trans. Opt. Sot. Lond. 28 (1926) 104. [I41 C. Candler, Modem Interferometers (Hilger and Watts Ltd., London, 195 1). 1151 L. Allen, M. Babiker. W.K. Lai, V.E. Lembessis, Phys. Rev. A 54 (1996) 4259.