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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 18 — Aug. 29, 2011
  • pp: 16760–16771
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Poynting vector and orbital angular momentum density of superpositions of Bessel beams

Igor A. Litvin, Angela Dudley, and Andrew Forbes  »View Author Affiliations


Optics Express, Vol. 19, Issue 18, pp. 16760-16771 (2011)
http://dx.doi.org/10.1364/OE.19.016760


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Abstract

We study theoretically the orbital angular momentum (OAM) density in arbitrary scalar optical fields, and outline a simple approach using only a spatial light modulator to measure this density. We demonstrate the theory in the laboratory by creating superpositions of non-diffracting Bessel beams with digital holograms, and find that the OAM distribution in the superposition field matches the predicted values. Knowledge of the OAM distribution has relevance in optical trapping and tweezing, and quantum information processing.

© 2011 OSA

1. Introduction

It has been well known for some time now that photons carry spin angular momentum of + ħ (–ħ) per photon for left (right) circularly polarised light, and that the transfer of this momentum can be measured in the laboratory when the light passes through a birefringent plate [1

1. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936). [CrossRef]

]. More recently [2

2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]

,3

3. 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. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

] it has been realised that light may also carry an extrinsic component of angular momentum, orbital angular momentum (OAM), when the electric field or mode has an azimuthal angular dependence of exp(ilφ), where l is the azimuthal mode index. Such fields carry OAM of per photon, and may be found as beams expressed in several basis functions, including Laguerre-Gaussian beams [2

2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]

], Bessel-Gaussian beams [4

4. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000). [CrossRef]

] and Airy beams [5

5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef] [PubMed]

] to name but a few. Since the discovery of light beams carrying OAM many new research areas have emerged, from transferring OAM to matter in optical tweezers [6

6. 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(5), 826–829 (1995). [CrossRef] [PubMed]

] to investigating the conservation and entanglement of OAM in parametric down conversion for quantum information processing [7

7. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

11

11. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]

]. One of the problems has been the measurement of OAM, with various attempts at the efficient sorting of modes carrying OAM using interferometers and computer generated holograms [12

12. S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175(4-6), 301–308 (2000). [CrossRef]

16

16. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003). [CrossRef] [PubMed]

]. Studies into the angular momentum of combined beams during free space propagation have also been made [17

17. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997). [CrossRef]

]. In general, one is interested in the superposition of OAM carrying fields, both at the classical [18

18. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express 15(14), 8619–8625 (2007). [CrossRef] [PubMed]

20

20. D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28(8), 657–659 (2003). [CrossRef] [PubMed]

] and quantum levels [7

7. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

].

In this paper we consider the problem of calculating and then measuring the orbital angular momentum density of a coherent superposition of non-diffracting beams. We derive simple equations for the OAM density from first principles by calculating the Poynting vector of the field, and from this deduce both the linear and angular momentum components. The results indicate that while the intensity pattern of the field is rotating at a fixed angular velocity (dependent on the relative phase velocities of the component fields), the OAM magnitude and direction is dependent on the location within the spatial distribution of the field. That is to say, different parts of the field carry different amounts of OAM, and of various sign. We verify the results by outlining and then executing a new technique for the direct measurement of the OAM density in the laboratory. We show how this approach allows for a quantitative measurement of the OAM spectrum as a function of position within the field.

2. Theory

We wish to consider the OAM density for a general complex scalar field, u(x,y,z), which varies in time (in complex notation) as exp(iωt). This problem has been addressed by others previously [5

5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef] [PubMed]

,21

21. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of high-order Bessel light beams,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002). [CrossRef]

26

26. M. V. Berry, “Paraxial beams of spinning light,” SPIE 3487, 6–11 (1998). [CrossRef]

], but with some inconsistencies in notation and derivation. For this reason, and for the benefit of the reader, we briefly outline the approach for defining OAM density, starting from the Poynting vector, before applying the result to the specific problem of coherent superpositions of non-diffracting Bessel beams. We start by defining a vector potential in the Lorentz gauge as a representation of our linearly polarized (in the x direction) laser mode as [27

27. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (2003). [CrossRef]

]:
A(x,y,z)=u(x,y,z)exp(i(kzωt))x
(1)
from which we can easily calculate the electric and magnetic field components of our complex scalar field:
B(x,y,z)=×A(x,y,z)E(x,y,z)=ic2ω×B(x,y,z),
(2)
where Eand Bare the electric and magnetic fields of the laser mode, respectively, and all other terms have their usual meaning. Equations (2) can readily be solved to find the two fields as functions of the laser mode itself (our complex scalar field):

B(x,y)=ikexp(i(kzωt))(u(x,y,z)y+iku(x,y,z)yz)E(x,y)=iωexp(i(kzωt))(u(x,y,z)x+iku(x,y,z)xz).
(3)

Here we have applied the paraxial limit and assumed that the rate of change of the field in the z direction is small compared to the transverse directions to remove derivatives of the field in the z direction. We are now in a position to calculate the time average of the Poynting vector:
S=Sreal=ε0c2Ereal×Breal,
(4)
where for brevity we will drop the arguments of the functions. The subscript real indicates that we are required to consider only the real part of the fields (cos(kzωt)), easily found to be:

Ereal=12(E+E*)Breal=12(B+B*),
(5)

By substitution of Eqs. (3) and (5) into (4), and after removing all the oscillating terms since they average to zero (over time), one can show that
S=ε0c2Ereal×Breal=ε0c24(E×B*+E*×B)=ε0ω4(i(uu*u*u)+2k|u|2z),
(6)
where it is understood that the gradient applies only in the transverse directions. We now wish to apply Eq. (6) to the problem of a superposition of OAM carrying fields, and we start with the problem of a coherent superposition of two such fields, of opposite handedness (i.e., opposite azimuthal phase), expressed in cylindrical co-ordinates as:
u(r,φ,z)=u0(r,z)exp(iψ(r,z))(exp(ilφ)+α0exp(ilφ)),
(7)
with the parameter α 0 representing the amplitude difference of the two fields (i.e., how the fields are weighted). The amplitude (u 0) and phase (ψ) of the complex scalar field is determined from the characteristics of the mode under study, and will be defined for Bessel and Bessel-Gauss beams a little later in this section. Substitution of Eq. (7) into Eq. (6) and solving for each component of the Poynting vector, we find that:

Sr=ε0ωc22ψr[u02(1+αl2+2αlcos(2lφ))]Sφ=lε0ωc2u022r(αl21)Sz=ε0ωc2ku022(1+αl2+2αlcos(2lφ)).
(8)

In the case of a sum of Bessel and Bessel-Gaussian beams, the parameter αl = α 0(–1)l, whereas for Laguerre-Gaussian beams it takes the form αl = α 0; here α 0 is the constant amplitude weighting of the fields. From these equations we may readily calculate the total angular momentum density of the field, and since by definition our field has a zero spin component, we have the total OAM density (along the direction of propagation, z) as:

Lz=1c2(r×S)z.
(9)

In this expression, the units of S are the conventional W/m2, while the OAM density (Lz) is now expressed as the angular momentum per unit volume, or Ns/m2. With these fundamentals in place, we are ready to calculate the OAM density for superposition fields.

2.1 Superposition of Bessel beams

Consider the general case of the superposition of two Bessel beams (BBs) of differing cone angles and opposite azimuthal phase; then Eq. (7) becomes
u(r,φ,z)=A0[Jl(q1r)exp(iΔkz)exp(ilφ)+α0Jl(q2r)exp(iΔkz)exp(ilφ)],
(10)
where Jl is the Bessel function of order l. Similar expressions can be found in references [28

28. S. Orlov and A. Stabinis, “Propagation of superpositions of coaxial optical Bessel beams carrying vortices,” J. Opt. A, Pure Appl. Opt. 6(5), S259–S262 (2004). [CrossRef]

] and [29

29. S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209(1-3), 155–165 (2002). [CrossRef]

]. The cone angles, γ 1 and γ 2, differ as a result of the generation process (see later), but can be set to be equal if so required. As a result of the slightly different cone angles, both the radial and longitudinal wave numbers also differ for the two beams: the radial wave numbers become qi = ksinγi, while the longitudinal wave numbers, given by ki = kcosγi differ from the central z-dependent wave number (k) by ± Δk (the subscripts i here refers to the first (second) beam in the superposition). We have maintained a normalization constant A 0 in order to later compare the theoretical results to experiment. Thus the Poynting vector becomes:

Sr=A02α0ε0ωc22[(q2Jl1(q2r)Jl(q1r)+q1Jl(q2r)Jl+1(q1r))sin(2[lφ+Δkz])]Sφ=A02lε0ωc22r(Jl2(q1r)α02Jl2(q2r))Sz=A02ε0ωc2k2(Jl2(q1r)+α02Jl2(q2r)+2α0Jl(q1r)Jl(q2r)cos(2[lφ+Δkz])).
(11)

The time average Poynting vector, in both the r and z directions, has an additional term Δkz which is responsible for the slow rotation of the intensity distribution during its propagation. This behavior has been noted previously [30

30. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17(26), 23389–23395 (2009). [CrossRef] [PubMed]

] and we can see that this is a result of the detuning of the radial wave numbers for the superposition of two or more beams. From the equation for Sr, a single non-diffracting beam has no Sr component. However, in the case of a superposition of two non-diffracting beams, the resulting beam has a non-zero Sr component. This results in a small rotation in the z direction, thus changing the intensity distribution of the superposition at a particular radial coordinate. We may easily then compute the OAM density to be:

Lz(r)=lε0ωA022(Jl2(q1r)α02Jl2(q2r)).
(12)

2.2 Superposition of Bessel-Gauss beams

Lz(r,z)=lε0ω2[1+(z/zr)2]×exp(2r2+γ2z2w02(1+(z/zr)2))[|Jl(q1r1+i(z/zr))|2α02|Jl(q2r1+i(z/zr))|2].
(14)

These results are consistent with what might be expected using the more general formalism of ref [24

24. R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express 15(23), 15214–15227 (2007). [CrossRef] [PubMed]

], and by extension of the vectorial results in ref [21

21. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of high-order Bessel light beams,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002). [CrossRef]

]. Indeed, our results here are the special case of these two approaches for scalar fields (linear polarized) and where the superposition is due to interference of the two modes. It remains then to define a method of actually measuring this quantity in the laboratory, a subject that has not been adequately addressed to date.

2.3 Measuring the OAM density

al(r,z)=12π02πu(r,φ,z)exp(ilφ)dφ.
(16)

The relative weighting of the power contained in each azimuthal mode can then be defined as [5

5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef] [PubMed]

]:

Pl(z)=0|al(r,z)|2rdrl0|al(r,z)|2rdr.
(17)

Lz(r)=lε0ω4π(al2(r)al2(r)).
(18)

We will show, for the first time, how this quantity may be measured in the laboratory in an easy to implement procedure with a phase-only spatial light modulator (SLM), thus allowing a direct measurement of the OAM density of the light.

3. Experimental methodology

A HeNe laser (λ ~633 nm) was expanded through a 6 × telescope and directed onto the liquid crystal display (LCD) of a SLM (HoloEye, PLUTO-VIS, with 1920 × 1080 pixels of pitch 8 μm and calibrated for a 2π phase shift at ~633 nm) depicted in Fig. 1
Fig. 1 A schematic of the experimental setup for detecting the OAM density of our superposition modes as a function of r. L: Lens (f1 = 25 mm; f2 = 150 mm; f3 = 200 mm and f4 = 200 mm); M: Mirror; LCD: Liquid Crystal Display; O: Objective; CCD: CCD Camera. The objective, O2, was placed at the focus (or Fourier plane) of lens, L4.
. The first SLM (denoted as LCD 1) was programmed to produce various superposition fields using the concept of Durnin’s ring-slit [31

31. J. Durnin, J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

], but implemented digitally [29

29. S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209(1-3), 155–165 (2002). [CrossRef]

]. Other techniques to experimentally generate superpositions of helically-phased fields using a single LCD exist and have been implemented by others [32

32. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Collinear superposition of multiple helical beams generated by a single azimuthally modulated phase-only element,” Opt. Lett. 30(24), 3266–3268 (2005). [CrossRef] [PubMed]

]. By addressing alternating sets of pixels on the SLM with phase values that are out of phase by π, resulting in the light reflected from the LCD being scattered from its initial propagation axis, we are able to make use of our phase-only SLM to represent the ring-slit (which would usually require an amplitude device or mask).

To create a superposition of two non-diffracting beams of opposite handedness, the ring-slit was programmed with two opposite azimuthal phases, and the resulting field Fourier transformed at the focal plane after L3, to produce the field described by Eq. (10). The phase pattern in the annular region was separated into two ring-slits, each possessing an azimuthal phase of equal order but opposite handedness (Fig. 2(a)
Fig. 2 (a) Annular ring programmed onto the SLM with two azimuthal phase patterns: in this example, + 3 and –3 l values; (b) observed experimental superposition (intensity) with 6 petals, and (c) the theoretically expected intensity pattern. The intensity shown in (b) is an attenuated experimental image with arbitrary false color units so as not to saturate the camera, while that in (c) shows a false color plot normalized to 1 in peak value for visual comparison.
). When the orders, |l|, of the two azimuthal phases were of equal but of opposite handedness, a ‘petal’-structure was produced, where the number of ‘petals’ is denoted by 2|l| (Fig. 2(b) and (c)), as expected from theory [30

30. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17(26), 23389–23395 (2009). [CrossRef] [PubMed]

]. The slight difference in radius of the two rings resulted in cone angles (defined as R/f with R the ring radius) of γ 1 = 0.0069 rad and γ 2 = 0.0075 rad, so that Δk ~42.8801 m−1. Another consequence of the slightly differing ring areas (see Fig. 2(a)) is that the energy contained in the two Bessel beams is not equal. Since the illuminating Gaussian beam was measured to be of radius w = 13.29 mm on LCD 1, the energy contained in each ring could be computed, and the weighting coefficients found:

α0=ring1exp(2(r/w)2)rdrring2exp(2(r/w)2)rdr.
(19)

The latter requirement is necessary in order that we find the local OAM density rather than the global OAM density. Figure 3
Fig. 3 The superposition field was divided radially, and an annular ring was programmed onto the second SLM in order to execute the inner product at only this radius. The phase within the annular ring was varied in the azimuthal angle for various values of l. The rest of the SLM was programmed with a checkerboard pattern so as to restrict the transmission function to the ring alone. The ten rings applied to LCD 2 have the following radii: r 1 = 600 µm, r 2 = 880 µm, r 3 = 1160 µm, r 4 = 1440 µm, r 5 = 1720 µm, r 6 = 2000 µm, r 7 = 2280 µm, r 8 = 2560 µm, r 9 = 2840 µm and r 10 = 3120 µm. The black and white images in rows 2 and 4 are gray-scale experimental images of the beam, with the attenuation set so as not to saturate the camera. Therefore the intensity units are arbitrary.
illustrates the concept: the experimental field (Fig. 3; second and forth row) was subdivided radially into 10 sections, with each section sequentially apertured by an annular ring of width 20 pixels. The phase within the annular ring was varied according to the conjugate of the azimuthal modes in Eq. (15), i.e., the azimuthal integrand of Eq. (16), from –4 to + 4 in l index. Thus the decomposition of our field to find the weighting coefficients could be executed as a function of radial co-ordinate and azimuthal mode. This is the first time such a technique has been demonstrated experimentally, and allows for the OAM density to be measured directly: the OAM spectrum, al(r,z) can be found at any radial position across the beam (and of course any z plane). The weighting parameter (α 0) of the superposition field was set by a suitable choice of the width of the two rings in the phase pattern on LCD 1: α 0 = 0.96 (nearly equal weighting).

4. Results and discussion

Applying Eq. (12), we calculate the dependence of the OAM density of a superposition of BBs, where the two components differ in their radial wave numbers. In Fig. 4(a)
Fig. 4 Calculated change in the local OAM when (a) the cone angle of the two modes differ, and (b) when the weighting of the two modes differ.
we see the results for the case of a 633 nm laser beam in a superposition of + 3 and –3 BBs, where the cone angle of the + 3 beam is fixed at 0.5° (line A on the graph) and that of the –3 BB is varied from 0.4° to 0.6°. Two results are evident: firstly, if the cone angles of the two BBs differ, so that their radial wave numbers differ, then the OAM density changes sign as one moves radially across the field. Secondly, the oscillation in the OAM density can be made to change value (magnitude and sign) by changing the relative difference in the cone angles (see line B of Fig. 4(a)). This is due to the oscillating nature of the Bessel function, where the period of the oscillation depends on the radial wave number, and in this particular case we have a superposition of two oppositely oscillating functions having opposite signs for their angular momentum. This can be seen by considering line B on the graph (varying cone angles), where the OAM density changes from a large positive value to a large negative value as one considers a single ring in the field (single radial position), passing through a zero value in between. In the case that the two cone angles are equal and the amplitude of the two Bessel beams are equal, the resulting superposition carries no angular momentum (see line A of Fig. 4(a)).

Similarly for BGBs, Eq. (14), we calculate the dependence of the OAM density of a superposition of BGBs, where the two components differ in their weighting. In Fig. 4(b) we see the results for the case of a 633 nm laser beam in a superposition of + 5 and –5 BGBs, both having a cone angle of 0.5 °, but with various weightings of the two beams. Clearly even with equal phase velocities (Δk = 0) the OAM density can be engineered to be non-zero as long as the weighting of the two fields differ: the larger the difference, the greater the OAM density. By considering line D on the graph (varying weighting by varying the input Gaussian beam width), the OAM density changes from a large positive value to a large negative value as one considers a single ring in the field (single radial position), passing through a zero value in between. However for a given weighting of the two fields, as one moves radially across the field, so the OAM density remains either positive or negative (passing through nulls between the rings), but does not oscillate in sign as in the case of Fig. 4(a). The variations in the angular momentum, both in terms of sign and magnitude, results from the variations in the amplitude and cone angles of the superimposed Bessel beams (refer to Fig. 6(a) and (b)
Fig. 6 (a) – (c): Density plots of the three Bessel beams plotted as (r,ϕ) for the + 3 case, the –3 case, and the superposition, respectively; (d) plot of the orbital angular momentum density as a function of radial position on the field – the theory is the solid curve and the experimental data is overlaid as red bars; (e) theoretical prediction of the coefficient |al|2 (or the on axis intensity of the inner product) as a function of the radius of the match filter. Rings 1 to 10 denote those given in the first and third rows of Fig. 3; (f) corresponding experimental data.
). The manipulation of the angular momentum can be implemented efficiently by the method described above and is a useful tool for the controlling of optical forces [33

33. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 12, 1–32 (2008).

].

The predicted OAM density was measured in the laboratory by measuring directly the coefficients given by Eqs. (16) and (17), with the OAM density calculated from Eq. (20). The measured OAM spectrum for the equally weighted superposition beam is shown in Fig. 5
Fig. 5 The measured OAM spectrum (for l values of –4 to + 4) as a function of the radial ring on the beam (equivalent to the radial position on the beam), from 1 to 10, given by: r 1 = 600 µm, r 2 = 880 µm, r 3 = 1160 µm, r 4 = 1440 µm, r 5 = 1720 µm, r 6 = 2000 µm, r 7 = 2280 µm, r 8 = 2560 µm, r 9 = 2840 µm and r 10 = 3120 µm. The height of each bar represents the measured coefficients |al|2.

From the spectrum in Fig. 5, the OAM density could be inferred directly and compared to theory, with the results shown in Fig. 6. In Fig. 6 (a) and (b) we see a density plot of the two fields (for + 3 and –3), and the slight shift in rings is evident due to the slightly different radial wave numbers of the two fields. The superposition field is shown in Fig. 6 (c). In all three plots the measured radial positions (r 1 through r 10) are overlaid as vertical lines. The core result is Fig. 6 (d), where the analytical OAM density prediction (solid curve) as a function of the radial position on the superposition field is shown with the measured OAM density (red bars). It is very clear that there is excellent quantitative agreement between the two (with the exception of the first measurement radius which appears to have a large error). This agreement can also be noted by considering Fig. 6 (e) and (f), where the theoretical and experimental data are shown for the |a 3|2 and |a –3|2 coefficients as a function of the radial position on the beam.

It should be noted that this is the first direct measurement of the OAM density, or OAM spectrum, to date. The importance of measuring the spectrum has been pointed out by others, where knowledge of the spectrum or local OAM is critical for imaging [34

34. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005). [CrossRef] [PubMed]

] and optical funnels [35

35. J. A. Rodrigo, A. M. Caravaca-Aguirre, T. Alieva, G. Cristóbal, and M. L. Calvo, “Microparticle movements in optical funnels and pods,” Opt. Express 19(6), 5232–5243 (2011). [CrossRef] [PubMed]

], but the few measurements of the OAM either only consider the global OAM using an optical trapping set-up [36

36. S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges,” Opt. Express 14(2), 535–541 (2006). [CrossRef] [PubMed]

,37

37. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express 14(15), 6604–6612 (2006). [CrossRef] [PubMed]

], or are unduly complicated by making use of Doppler shifts in due to a rotating detector [16

16. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003). [CrossRef] [PubMed]

]. In the method proposed here, the same principles (and optical elements) applied in creating the superposition fields can be used to decompose them.

It is interesting to consider the role of the Gouy phase shift in such measurements, since in general this is a function of the azimuthal order of the field. This is not the case for a superposition of Bessel beams, as reported here, and so may be neglected. However, in the case of Laguerre-Gaussian modes (which are used extensively in quantum information studies) the Gouy phase shift would be identical for fields of opposite handedness, proportional to 2p + |l| + 1, where p is the radial order. This means that modes of + l and a –l will have the same phase shift, so again it can be considered as a constant phase offset (but dependent on z). However, while the OAM arises from the azimuthal phase variation, the rotation of the Poynting vector is equal to arctan(z/zr) when considering the radius of maximum field of p = 0, l ≠ 0 modes (since such modes are a single ring of light), i.e., the rotation looks like the Gouy phase shift of a Gaussian mode. The salient point is that the Poynting vector rotation is proportional to arctan(z/zr), and that the maximum rotation is fixed, in the case of a p = 0 mode, to π/2 either side of the beam waist. As the radial order is increased, so the maximum rotation of the field also increases [22

22. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995). [CrossRef]

,23

23. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184(1-4), 67–71 (2000). [CrossRef]

]. It is however interesting to consider what happens to the OAM density if non-zero radial modes are excited, as will surely be the case for the typical laboratory practices of generating such modes by multiplication of a Gaussian with a purely azimuthal phase function. This case has not yet been treated fully in the literature. The result would, most likely, be a superposition not only in l but also in p, resulting in an OAM spectrum with weighting coefficients different to that of a purely azimuthal superposition, with a larger contribution from the a 0 term in Eq. (16) due to the azimuthal symmetry of the radial orders.

5. Conclusion

References and links

1.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936). [CrossRef]

2.

M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]

3.

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. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

4.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000). [CrossRef]

5.

H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef] [PubMed]

6.

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(5), 826–829 (1995). [CrossRef] [PubMed]

7.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

8.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002). [CrossRef] [PubMed]

9.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008). [CrossRef]

10.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010). [CrossRef] [PubMed]

11.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]

12.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175(4-6), 301–308 (2000). [CrossRef]

13.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105(15), 153601 (2010). [CrossRef] [PubMed]

14.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002). [CrossRef] [PubMed]

15.

C. Gao, X. Qi, Y. Liu, J. Xin, and L. Wang, “Sorting and detecting orbital angular momentum states by using a Dove prism embedded Mach–Zehnder interferometer and amplitude gratings,” Opt. Commun. 284(1), 48–51 (2011). [CrossRef]

16.

M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003). [CrossRef] [PubMed]

17.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997). [CrossRef]

18.

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express 15(14), 8619–8625 (2007). [CrossRef] [PubMed]

19.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001). [CrossRef] [PubMed]

20.

D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28(8), 657–659 (2003). [CrossRef] [PubMed]

21.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of high-order Bessel light beams,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002). [CrossRef]

22.

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995). [CrossRef]

23.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184(1-4), 67–71 (2000). [CrossRef]

24.

R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express 15(23), 15214–15227 (2007). [CrossRef] [PubMed]

25.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002). [CrossRef] [PubMed]

26.

M. V. Berry, “Paraxial beams of spinning light,” SPIE 3487, 6–11 (1998). [CrossRef]

27.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (2003). [CrossRef]

28.

S. Orlov and A. Stabinis, “Propagation of superpositions of coaxial optical Bessel beams carrying vortices,” J. Opt. A, Pure Appl. Opt. 6(5), S259–S262 (2004). [CrossRef]

29.

S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209(1-3), 155–165 (2002). [CrossRef]

30.

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17(26), 23389–23395 (2009). [CrossRef] [PubMed]

31.

J. Durnin, J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

32.

J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Collinear superposition of multiple helical beams generated by a single azimuthally modulated phase-only element,” Opt. Lett. 30(24), 3266–3268 (2005). [CrossRef] [PubMed]

33.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 12, 1–32 (2008).

34.

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005). [CrossRef] [PubMed]

35.

J. A. Rodrigo, A. M. Caravaca-Aguirre, T. Alieva, G. Cristóbal, and M. L. Calvo, “Microparticle movements in optical funnels and pods,” Opt. Express 19(6), 5232–5243 (2011). [CrossRef] [PubMed]

36.

S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges,” Opt. Express 14(2), 535–541 (2006). [CrossRef] [PubMed]

37.

C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express 14(15), 6604–6612 (2006). [CrossRef] [PubMed]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(270.0270) Quantum optics : Quantum optics
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Physical Optics

History
Original Manuscript: April 21, 2011
Revised Manuscript: June 2, 2011
Manuscript Accepted: July 26, 2011
Published: August 15, 2011

Citation
Igor A. Litvin, Angela Dudley, and Andrew Forbes, "Poynting vector and orbital angular momentum density of superpositions of Bessel beams," Opt. Express 19, 16760-16771 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16760


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References

  1. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936). [CrossRef]
  2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]
  3. 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. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
  4. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000). [CrossRef]
  5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef] [PubMed]
  6. 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(5), 826–829 (1995). [CrossRef] [PubMed]
  7. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]
  8. A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002). [CrossRef] [PubMed]
  9. J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008). [CrossRef]
  10. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010). [CrossRef] [PubMed]
  11. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]
  12. S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175(4-6), 301–308 (2000). [CrossRef]
  13. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105(15), 153601 (2010). [CrossRef] [PubMed]
  14. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002). [CrossRef] [PubMed]
  15. C. Gao, X. Qi, Y. Liu, J. Xin, and L. Wang, “Sorting and detecting orbital angular momentum states by using a Dove prism embedded Mach–Zehnder interferometer and amplitude gratings,” Opt. Commun. 284(1), 48–51 (2011). [CrossRef]
  16. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28(23), 2285–2287 (2003). [CrossRef] [PubMed]
  17. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997). [CrossRef]
  18. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express 15(14), 8619–8625 (2007). [CrossRef] [PubMed]
  19. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001). [CrossRef] [PubMed]
  20. D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28(8), 657–659 (2003). [CrossRef] [PubMed]
  21. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of high-order Bessel light beams,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002). [CrossRef]
  22. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995). [CrossRef]
  23. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184(1-4), 67–71 (2000). [CrossRef]
  24. R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express 15(23), 15214–15227 (2007). [CrossRef] [PubMed]
  25. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002). [CrossRef] [PubMed]
  26. M. V. Berry, “Paraxial beams of spinning light,” SPIE 3487, 6–11 (1998). [CrossRef]
  27. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (2003). [CrossRef]
  28. S. Orlov and A. Stabinis, “Propagation of superpositions of coaxial optical Bessel beams carrying vortices,” J. Opt. A, Pure Appl. Opt. 6(5), S259–S262 (2004). [CrossRef]
  29. S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209(1-3), 155–165 (2002). [CrossRef]
  30. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17(26), 23389–23395 (2009). [CrossRef] [PubMed]
  31. J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]
  32. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Collinear superposition of multiple helical beams generated by a single azimuthally modulated phase-only element,” Opt. Lett. 30(24), 3266–3268 (2005). [CrossRef] [PubMed]
  33. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 12, 1–32 (2008).
  34. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005). [CrossRef] [PubMed]
  35. J. A. Rodrigo, A. M. Caravaca-Aguirre, T. Alieva, G. Cristóbal, and M. L. Calvo, “Microparticle movements in optical funnels and pods,” Opt. Express 19(6), 5232–5243 (2011). [CrossRef] [PubMed]
  36. S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges,” Opt. Express 14(2), 535–541 (2006). [CrossRef] [PubMed]
  37. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express 14(15), 6604–6612 (2006). [CrossRef] [PubMed]

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