## Angular momentum of multimode and polarization patterns

Optics Express, Vol. 15, Issue 23, pp. 15214-15227 (2007)

http://dx.doi.org/10.1364/OE.15.015214

Acrobat PDF (783 KB)

### Abstract

We study the mechanical properties of a broad class of multimode and polarization light patterns, resulting from the interference and superposition of waves in helical modes. General local and global properties of energy and angular momentum (AM) are identified, in order to define the conditions to optimize the AM with increasing beam complexity. We show the possibility to engineer independently the local densities of optical AM and energy, opening the possibility of an experimental demonstration of their respective effects in light-matter interaction. Multimode Laguerre-Gaussian beams also allows us to tailor the local spin AM through the Gouy phase.

© 2007 Optical Society of America

## 1. Introduction

1. R.A. Beth, “Mechanical Detection and Measurement of the Angular Momentum of Light,” Phys. Rev. **50**, 115–125 (1936) [CrossRef]

2. 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**, 8185–8189 (1992) [CrossRef] [PubMed]

3. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**, 15–28 (2005) [CrossRef]

4. L. Allen, S.M. Barnett, and M.J. Padgett, *Optical angular momentum* (Institute of Physics Publishing, Bristol, 2003) [CrossRef]

5. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri “Overcoming the Rayleigh Criterion Limit with Optical Vortices,” Phys. Rev. Lett. , **97**, 163903 (2006); [CrossRef] [PubMed]

5. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express **14**, 120–127 (2006)

6. C. Maurer, A Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. **9**, 78 (2007) and references therein. [CrossRef]

7. D. G. Grier, “A revolution in optical manipulation,” Nature **424**, 810–816 (2003); [CrossRef] [PubMed]

7. A. Jesacher, S. Fr̈hapter, S. Bernet, and M. Ritsch-Marte, “Size selective trapping with optical cogwheel tweezers,” Opt. Express **12**, 4129–4135 (2004);

7. S. H. Tao, X-C. Yuan, J. Lin, X. Peng, and H. B. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express **13**, 7726–7631 (2005)

8. 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**, 535–541 (2006) [CrossRef] [PubMed]

9. 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**, 6604–6612 (2006) [CrossRef] [PubMed]

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

11. R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco, and M. J. Padgett, “Multiport holographic velocimetry in microfluidic systems,” Phys. Rev. Lett. **96**, 134502 (2006) [CrossRef] [PubMed]

12. G. Molina-Terriza, J.P. Torres, and L. Torner, “Twisted photons,” Nature Phys. **3**, 305–310 (2007) and refernces therein. [CrossRef]

*multimode*beams and

*polarization*patterns are composed by different helical components with azimuthal phase dependence exp(

*iℓϕ*) and can be generated with spatial light modulators, or other linear optical elements [4

4. L. Allen, S.M. Barnett, and M.J. Padgett, *Optical angular momentum* (Institute of Physics Publishing, Bristol, 2003) [CrossRef]

6. C. Maurer, A Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. **9**, 78 (2007) and references therein. [CrossRef]

13. W. Nasalski, “Polarization versus spatial characteristics of optical beams at a planar isotropic interface,” Phys. Rev. E **74**, 056613 (2006) [CrossRef]

14. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express **14**, 3039–3044 (2006) [CrossRef] [PubMed]

15. J. Courtial, R. Zambrini, M. R. Dennis, and M. Vasnetsov, “Angular momentum of optical vortex arrays,” Opt. Express **14**, 938 (2006) [CrossRef] [PubMed]

16. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Applied Optics **46**, 2893–2898 (2007) [CrossRef] [PubMed]

17. J. Masajada, A. Popiolek-Masajada, and M. Leniec, “Creation of vortex lattices by a wavefront division,” Opt. Express **15**, 5196–5207 (2007) [CrossRef] [PubMed]

4. L. Allen, S.M. Barnett, and M.J. Padgett, *Optical angular momentum* (Institute of Physics Publishing, Bristol, 2003) [CrossRef]

19. M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, “Polarization patterns in Kerr media,” Phys. Rev. E **58**, 2992–3007 (1998); [CrossRef]

19. G.-L. Oppo, A. J. Scroggie, and W. J. Firth,“Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators,” Phys. Rev. E **63**, 066209 (2001)

20. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature **441**, 946 (2006) [CrossRef] [PubMed]

21. D. Boiko, G. Guerrero, and E. Kapon, “Polarization Bloch waves in photonic crystals based on vertical cavity surface emitting laser arrays,” Opt. Express **12**, 2597–2602 (2004) [CrossRef] [PubMed]

22. A. Ferrando, M. Zacarés, and M.-A. García-March, “Vorticity cutoff in Nonlinear Photonic Crystals,” Phys. Rev. Lett. **95**, 043901 (2005). [CrossRef] [PubMed]

23. R. Zambrini and S.M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. **96**, 113901 (2006) [CrossRef] [PubMed]

*inter ference*of an increasing number of modes. Some recent work reports on the effects of interfering LG modes in the context of singular optics looking, for example, at the topology of the singularities along the propagation beam direction [24

24. 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**, 4064–4074 (1997) [CrossRef]

8. 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**, 535–541 (2006) [CrossRef] [PubMed]

9. 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**, 6604–6612 (2006) [CrossRef] [PubMed]

8. 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**, 535–541 (2006) [CrossRef] [PubMed]

9. 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**, 6604–6612 (2006) [CrossRef] [PubMed]

*superposition*of orthogonal linearly polarized beams, in the context of AM. With the proper choice of relative phases, interference can be used to obtain LG modes carrying orbital AM starting from two orthogonal Hermite-Gaussian beams, with vanishing

*orbital*AM [2

2. 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**, 8185–8189 (1992) [CrossRef] [PubMed]

*spin*AM and also to radially and azimuthally polarized beams [6

6. C. Maurer, A Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. **9**, 78 (2007) and references therein. [CrossRef]

*local*properties, such as the densities of linear and AM [25

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**, 053601 (2002). [CrossRef] [PubMed]

26. R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt. **52**, 1045–1052 (2005). [CrossRef]

27. R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, “Angular momentum paradox in a vortex core,” J. Mod. Opt. , **52**, 1135–1144 (2005). [CrossRef]

*optimize*the amount of AM per photon with increasing beam complexity. In particular, we consider whether it is more efficient to interfere or superpose beams in order to generate orbital and spin AM.

## 2. Linear and angular momenta and energy

**E**. The review in this section aims to point out the principal effects arising when multimode polarization patterns are considered. We consider only monocromatic beams with electric field

**E**= (

*E*,

_{x}*E*,

_{y}*E*) is the complex amplitude, ω the frequency and

_{z}*k*= ω/

*c*. We work throughout within the limit of paraxial propagation. The transversality constraint on the electromagnetic field, ∇ ∙

**e**= 0, fixes the field component in the propagation direction,

*E*, as a function of

_{z}*E*and

_{x}*E*. Therefore the mechanical properties of paraxial waves can be expressed as a function only of the transverse field

_{y}**E**

_{⊥}= (

*E*,

_{x}*E*). The linear momentum density in the direction of propagation is

_{y}*x*,

*y*). The importance of these quantities follows from their immediate relation with fluxes in the paraxial limit [28

28. S. M. Barnett, “Optical angular-momentum flux“, J. Opt. B: Quantum Semiclass. Opt. **4**, S7–S16 (2002) [CrossRef]

*P*= ∫ d

_{z}*x*d

*y*

*p*≫

_{z}*P*

_{x,y}). The flux of linear momentum is naturally associated with the energy by Poynting’s theorem, which leads to the expression of the time averaged energy per unit length

**p**

_{⊥}= (

*p*,

_{x}*p*) are

_{y}*j*=

*x*,

*y*. Even if the average total transverse momentum

**P**

_{⊥}is negligible with respect to

*P*, the local value of the density

_{z}**p**

_{⊥}still has important effects being at the origin of optical AM [29

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

**E**

_{⊥}suggesting that

*no transverse momentum*

**p**

_{⊥}can be exerted

*on the focal plane*of a fundamental Gaussian mode, in apparent contradiction with the well-known property of Gaussian light beams to move small objects towards the beam axis. The consistency of Eq. (4) is ensured, however, by the paraxial evolution causing any field purely real in a plane (for instance

*z*= 0) to become complex on propagation. Therefore the transverse momentum

**p**

_{⊥}would not vanish over an interval ∣

*z*∣ < Δ

*z*, with Δ

*z*being the thickness of the trapped object.

**E**

_{⊥}(

**x**)= (

*αxˆ*+

*βŷ*)

*u*(

**x**), where α and β are complex constants. These beams have a transverse momentum density -(ε

_{0}/2ω)Im(

*u*∇

_{⊥}

*u*

^{*}) that is completely independent of the polarization state of the beam. On the other hand, in the presence of polarization patterns, the linearly polarized components

*E*contribute differently to the linear momentum. As a particular example, a field with vanishing average momentum

_{x,y}**P**

_{⊥}can be obtained by superposing two orthogonally polarized waves with momenta pointing in opposite directions. This is a first example illustrating how the vectorial character of electromagnetic fields can lead to novel capabilities in optical manipulation.

*not*contribute to the integrated total momentum but is, nevertheless, locally non-vanishing. This vector,

_{z}= -

*i*(

*E*

^{*}

_{x}*E*-

_{y}*c*.

*c*.) (that is the local difference between the left and right circularly polarized intensities) [30], taken in the limit of slow variation of the field envelope along the propagating direction

*z*. Because of its derivative form, this term does not contribute to the average linear momentum

**P**

_{⊥}for any physical beam and it has also been shown that it does not produce directly any mechanical effect [26

26. R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt. **52**, 1045–1052 (2005). [CrossRef]

*this derivative term is extremely important because it leads to the spin component of optical AM*.

**j**=

**x**×

**p**and we find the

*total*AM density in the propagation direction in the form

*Optical angular momentum* (Institute of Physics Publishing, Bristol, 2003) [CrossRef]

26. R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt. **52**, 1045–1052 (2005). [CrossRef]

**E**

_{⊥}with a local circular polarization in some region to obtain a non-vanishing spin AM. Novel fields characterized by azimuthal and radial polarizations therefore do not carry any spin AM even if in general they have a non-vanishing orbital AM. It was already noticed in seminal papers about optical AM that linearly polarized beams in helical (LG) modes have a density of orbital AM(8) that is locally proportional to the energy density [2

2. 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**, 8185–8189 (1992) [CrossRef] [PubMed]

*Optical angular momentum* (Institute of Physics Publishing, Bristol, 2003) [CrossRef]

*for multimode beams*,

*neither the spin nor the orbital AM densities are proportional to the local energy density*. It follows also that global properties of these beams are different from the single mode case. In the following, we will consider the average

*L*= ∫dxdy ℓ

_{z}_{z}resulting from the sum of the

*orbital*AM carried separately by the linearly polarized components

*E*and

_{x}*E*and the spin given the spatially averaged Stokes parameter σ

_{y}_{z}(

**x**). We discuss multimode beams comparing beams obtained either by interference or by superposition.

### 2.1. Interference and superposition of optical beams

*add*. This occurs when the superposed fields have orthogonal polarizations. In the following we consider the main differences between the mechanical properties in the

*superposition*of two orthogonal linearly polarized beams

*interference*of the same fields

*U*and

*V*, but now with parallel polarization. Clearly the interference field (11) does not carry any spin AM but does allow us to include in our analysis the properties of

*multimode*fields, from the sum of different helical modes. On the other hand, the superposition field (10) allows us to identify the relations between different mechanical properties (energies and momenta) of polarization patterns.

**E**

^{i}carries an (orbital) AM

*orbital*AM also depends on whether we interfere or superpose the two fields. Indeed, the second main observation drawn from Eqs. (12) and (13) is the distinctive property of the superposition of orthogonally polarized waves, allowing not only to add their input energies but also to add their

*orbital*AM. On the other hand, as for the energy, the AM obtained by the interference of two beams is not the sum of their AM. The difference between the

*orbital*angular momenta of

**E**

^{i}and

**E**

^{s}is

**E**

^{s,i}. High powers, however, are often undesirable and, moreover, there are more interesting possibilities. In order to study the efficiency in producing AM, it is meaningful, therefore, to consider the AM per unit of energy (AMPE) or per photon, these quantities being equivalent apart from a factor

*h*̄ω. This scaling also allows us to distinguish any increase of AM from that arising from the difference in energy for interference and superpositions. For example, in the trivial case

*V*= exp(

*iθ*)

*U*(with

*θ*real constant) the difference (14) is non-vanishing, but this is a simple “side effect” of constructive (

*θ*= 0) or destructive

*(θ*=π) interference. The

*orbital*AMPE is the same in this case (and independent of

*θ*) in either superposition or interference.

*u*[4

_{pℓ}*Optical angular momentum* (Institute of Physics Publishing, Bristol, 2003) [CrossRef]

*C*is a real factor with the dimensions of electric field. We can choose the mode amplitudes such that ∑

_{p,ℓ}∣

*a*∣

_{pℓ}^{2}= ∑

_{p,ℓ}∣

*b*∣

_{pℓ}^{2}= 1 so that

*C*fixes the total energy. We find that the efficiency in producing larger orbital AM is optimized either by interference or superposition

*depending*on the spatial spectral details of the component beams. For example, opposite signs of

*ℓ*can cancel in the superposition case, while in the case of interference everything is complicated by the possibility that the relative phases of the

*a*and

*b*amplitudes play a roll in determining the total contribution to the AM of each LG mode. This can be seen clearly by considering the orbital AMPE in interference and superposition

*a*

_{pℓ}

*b*

^{*}

_{pℓ}). It follows that the largest AMPE is obtained by superposing the fields rather than by interfering them when

*orbital*AM spectra, there is more orbital AMPE in their interference or in their superposition. Naturally the spin AM needs to be included in order to obtain the total AMPE of superposed beams. The averaged total spin AMPE, in terms of the component-mode amplitudes is

*only common mode components in the fields U and V contribute to the average total spin AM*.

## 3. Examples

### 3.1. Interfering and superposing single Laguerre-Gaussian modes

#### 3.1.1. Modes with different helicities ℓ_{1} ≠ ℓ_{2}

_{1}= -ℓ

_{2}and different energies was considered in Ref. [9

**14**, 6604–6612 (2006) [CrossRef] [PubMed]

**14**, 535–541 (2006) [CrossRef] [PubMed]

_{1}and ℓ

_{2}have opposite signs and a large difference in the absolute value. Here we consider the case of composing beams with equal energies,

*different*helical indexes (ℓ

_{1}≠ ℓ

_{2}) (i) focusing on general local and global AM of these beams and (ii) generalizing the analysis by including the vectorial degree of freedom of the electromagnetic field. The superposition of orthogonally polarized LG beams allows us to tailor polarization patterns carrying spin AM.

_{1}≠ ℓ

_{2}, that the

*average*orbital AM of the superimposed beams (as well as the energy) is the same as that obtained by interference. In particular

*z*from the beam waist, the

*average*AM are constant at different planes (independent on

*z*). It follows from the orthogonality of the modes,

*U*and

*V*, that the average total spin AM generated by superposition is

*zero*, (Eq. (18)). Finally,

*modes with different helicities have the same average AM both in superposition and interference, and this is also true of the AM per photon*.

*orbital*AM resulting from constructive

*interference*(a,e) and

*superposition*(b,f) are compared in Fig. 2 and Fig. 3 for different LG beams. The main difference is that the

*azimuthal symmetry in the orbital AM distribution is lost in the interference pattern*. In particular, given (19), it follows that the orbital AM density for the superposition field is

_{1}- ℓ

_{2}∣ positive (or negative) lobes, as illustrated in Fig. 2(a) (3-2= 1 lobe), Fig. 2(e) (3 lobes) and Fig. 3(a) (7+1 = 8 lobes).

_{1}- ℓ

_{2}∣. The important difference is in their respective

**radial**distributions. This gives rise to the interesting observation that, in general,

*the regions of maximum AM density, either of spin or orbital origin, are not the regions with maximum intensity (energy density)*. This is evident in Figs. 2e and 3a: The energy is concentrated in an inner dashed region and the orbital AM in an external one. Similarly, for the

*superposition*of waves (19), the ring of maximum energy intensity in Fig. 2(f) as well as in Fig. 3(b) falls in a region of relatively small orbital AM. A comparison of the radial cross sections is given in Fig. 4(a).

**52**, 1045–1052 (2005). [CrossRef]

**52**, 1045–1052 (2005). [CrossRef]

32. K. T. Gahagan and G.A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. **21**, 827–829 (1996) [CrossRef] [PubMed]

*R*. On the other hand, the orbital AM can show oscillations from negative to positive values if the composing beams have helicities with different signs. On focusing the beam, the angular velocity of the trapped object is predicted to change direction following the dashed line curve in Fig. 4(b). The asymptotic values for large

*R*can be compared with Eqs. (20). We note that a naïve picture of the angular velocity in terms of the transfer of some average number of units of orbital AM (

*J*) per photon would translate in a wrong prediction of increasing angular velocity in a fixed direction.

_{z}_{z}(

**x**) is dictated by the relative phases of the superimposed beams (

*E*and

_{x}*E*). The orthogonality of the spatial modes, does not prevent the possibility of finding a polarization pattern with large local spin AM. In particular, regions of left circular polarization are balanced by the ones with right circular polarization, as illustrated in Figs. 2c and g, and Fig. 3(c). The ∣ℓ

_{y}_{1}-ℓ

_{2}∣ lobes [33] with left and right circular polarization are separated by regions of vanishing spin, or C-contours [34], while the energy is azimuthally symmetric. We conclude that the total AM density, and its spin and orbital components, can be concentrated near to regions of darkness in the beam.

#### 3.1.2. Modes with same helicities

*p*

_{1}≠

*p*

_{2}:

*z*, maintaining constant

*average*values at different planes. Here we focus on the local spin generation induced by the respective Gouy phases exp[-

*i*(2

*p*+ ∣ℓ∣ + 1)tan

_{j}^{-1}(

*z*/

*z*)] (

_{R}*j*= 1,2) of superposed beams with different radial indexes. The relative Gouy phase introduces local polarization gradients, and hence non-vanishing spin AM density, in the form of opposite circular polarizations over different circles, as shown in Fig. 5. In the example, there is no spin at the focal plane, as the relative phase of the beams is vanishing everywhere, while for

*z*≠ 0 we see two circles of opposite spin (see the radial plot of the spin oscillating from positive to negative values in the left panel of Fig. 5). The Gouy phase is an odd function of

*z*and so the spin density before and after the focal plane takes opposite values (compare panels for

*z*= -0.5

*z*and +0.5

_{R}*z*in Fig. 5). Trapping

_{R}*bire-fringent*particles near to a liquid surface and moving the focal plane

*away*from the liquid surface should make it possible to show, experimentally, the effects of the Gouy phase and of the variation of the polarization pattern in different planes by observing the variation of the angular velocities of trapped objects. An interesting arrangement, fully exploiting the possibility to change rotation directions by moving the beam focal plane, would be a set-up with two concentric circular gears. Such an arrangement might be useful both for micromachining and microfluidic applications.

*U*and

*V*are in the same spatial mode (ℓ

_{1}= ℓ

_{2},

*p*

_{1}=

*p*

_{2}) we have a single mode beam. Then, the orbital AMPE in superposition and interference is the same in spite of the fact that their energies and orbital AM are generally different. In other words, in this trivial case energy and AM interfere in the same way. No polarization patterns (local polarization gradients) appear in this case and the spin of the wave depends only on the relative (global) phases of the superimposed waves (the local and global properties of the spin are the same).

### 3.2. Interfering and superposing multimode beams

*U*and

*V*. The minimum degree of complexity that can be added to this is by considering the patterns generated by superposing two beams not in single mode but still containing, over all, only two LG modes (same number of spectral components than in previous section):

*U*and

*V*have the same average energy and (ii) they are clearly not orthogonal, as

*V*results from the (constructive) interference of

*u*

_{0ℓ1}and another orthogonal mode (ℓ

_{1}≠ ℓ

_{2}). This enables us to study the effects of both parallel and orthogonal mode components. The phase θ can be varied to have constructive or destructive interference between

*U*and

*V*. For -π/2<θ < π/2 the interference of the parallel components in

*U*and

*V*is

*constructive*leading to a larger energy than in the superposition:

_{1}is positive then

*L*>

^{i}_{z}*L*, independently on the value of ℓ

^{s}^{z}_{2}. If ℓ

_{1}is negative, on the other hand and in spite of the constructive interference, the orbital AM is smaller in the interference pattern than in the superposition one. The same result is obtained for destructive interference (π/2 < θ < 3/2π) and negative ℓ

_{1}. Assuming for simplicity that ℓ

_{1},ℓ

_{2}> 0, we conclude that interfering constructively the beams (24) gives more average orbital angular momentum than superposing them. Surprisingly, even if there is more orbital AM in the interference case, this does not always mean that there is more orbital AMPE. In particular, it is possible to obtain more AMPE in the superposition than in the constructive interference case depending on the relative values of ℓ

_{1}and ℓ

_{2}. In general we find that (for constructive interference ∣θ∣ < π/2)

_{1}>ℓ

_{2}. Remember that in the case of orthogonal beams (previous section) the orbital AM as well as the orbital AM per photon were the same in either interference or superposition. The study of the interference and superposition of the beams (24), with a low degree of spectral complexity (only two modes), shows that

*even if the average orbital AM is larger for constructively interfering the beams than for superposing them*(cosθ > 0, ℓ

_{1}> 0),

*the superposition can still be more efficient in providing the largest orbital AMPE (if*ℓ

_{1}< ℓ

_{2}

*)*.

*U*and

*V*can contribute to the spin, as discussed above (Eq. 18). From Eqs. (24) it follows that the average spin AM is

*S*/W

^{S}_{Z}^{s}= -sinθ/√2ω and for a relative phase of θ = -π/2, the spin per photon would be

*h*̄/√2. This value is less than expected (

*h*̄ per photon for circularly polarized light) because the same LG modes for

*E*and

_{x}*E*have different amplitudes. The comparison between the interference and superposition beams, including also the spin, is obtained by comparison of the total AM

_{y}## 4. Conclusion

*local*density of orbital AM and energy opens up the possibility to distinguish the respective effects of light intensity and optical AM in light-matter interactions. In particular, the AM density can be maximum in regions of relatively low intensity, where hollow particles would be naturally trapped. This may have important implications in the context of optical trapping including, for example, the possibility of rotating without burning. Similarly, if small absorbers are used to map the transverse profile of a generic beam then the faster rotations will not necessarily be observed in the regions of maximum intensity. We proposed an experiment to clarify the distinctive role of energy and AM by measuring the angular velocity of an absorbing object, trapped by a multimode beam as it is focused. The generic picture of mechanical effects in terms of the absorption of photons carrying an average orbital AM is not appropriate for mutlimode beams and the local AM distribution needs to be evaluated. We also described a simple way to create polarization patterns, with azimuthal symmetry, provided by the Gouy phase. Superposing LG modes with different radial indexes but the same helicity, it is possible to create patterns with rings of opposite spin out of the focal plane, even if the average spin AM is zero.

## Acknowledgments

## References and links

1. | R.A. Beth, “Mechanical Detection and Measurement of the Angular Momentum of Light,” Phys. Rev. |

2. | 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 |

3. | D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. |

4. | L. Allen, S.M. Barnett, and M.J. Padgett, |

5. | F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri “Overcoming the Rayleigh Criterion Limit with Optical Vortices,” Phys. Rev. Lett. , W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express |

6. | C. Maurer, A Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. |

7. | D. G. Grier, “A revolution in optical manipulation,” Nature A. Jesacher, S. Fr̈hapter, S. Bernet, and M. Ritsch-Marte, “Size selective trapping with optical cogwheel tweezers,” Opt. Express S. H. Tao, X-C. Yuan, J. Lin, X. Peng, and H. B. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express |

8. | 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 |

9. | C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express |

10. | S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express |

11. | R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco, and M. J. Padgett, “Multiport holographic velocimetry in microfluidic systems,” Phys. Rev. Lett. |

12. | G. Molina-Terriza, J.P. Torres, and L. Torner, “Twisted photons,” Nature Phys. |

13. | W. Nasalski, “Polarization versus spatial characteristics of optical beams at a planar isotropic interface,” Phys. Rev. E |

14. | K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express |

15. | J. Courtial, R. Zambrini, M. R. Dennis, and M. Vasnetsov, “Angular momentum of optical vortex arrays,” Opt. Express |

16. | S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Applied Optics |

17. | J. Masajada, A. Popiolek-Masajada, and M. Leniec, “Creation of vortex lattices by a wavefront division,” Opt. Express |

18. | S.M. Barnett and R. Zambrini, “Orbital angular momentum of light“ in |

19. | M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, “Polarization patterns in Kerr media,” Phys. Rev. E G.-L. Oppo, A. J. Scroggie, and W. J. Firth,“Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators,” Phys. Rev. E |

20. | E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature |

21. | D. Boiko, G. Guerrero, and E. Kapon, “Polarization Bloch waves in photonic crystals based on vertical cavity surface emitting laser arrays,” Opt. Express |

22. | A. Ferrando, M. Zacarés, and M.-A. García-March, “Vorticity cutoff in Nonlinear Photonic Crystals,” Phys. Rev. Lett. |

23. | R. Zambrini and S.M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. |

24. | 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 |

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. |

26. | R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt. |

27. | R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, “Angular momentum paradox in a vortex core,” J. Mod. Opt. , |

28. | S. M. Barnett, “Optical angular-momentum flux“, J. Opt. B: Quantum Semiclass. Opt. |

29. | L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density,” Opt. Commun. |

30. | M. Born and E. Wolf, |

31. | A. Siegman, |

32. | K. T. Gahagan and G.A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. |

33. |
The same periodicity was found for the energy and orbital AM of an interference field, because the imaginary part of the phase sensitive term |

34. | J. F. Nye, |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 12, 2007

Revised Manuscript: October 9, 2007

Manuscript Accepted: October 9, 2007

Published: November 2, 2007

**Citation**

Roberta Zambrini and Stephen M. Barnett, "Angular momentum of multimode and polarization patterns," Opt. Express **15**, 15214-15227 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-23-15214

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### References

- R. A. Beth, "Mechanical Detection and Measurement of the Angular Momentum of Light," Phys. Rev. 50, 115 - 125 (1936) [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 - 8189 (1992) [CrossRef] [PubMed]
- D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15 - 28 (2005) [CrossRef]
- L. Allen, S. M. Barnett, M. J. Padgett, Optical angular momentum (Institute of Physics Publishing, Bristol, 2003) [CrossRef]
- F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini and C. Barbieri "Overcoming the Rayleigh Criterion Limit with Optical Vortices," Phys. Rev. Lett. 97, 163903 (2006); W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120 - 127 (2006). [CrossRef] [PubMed]
- C. Maurer, A Jesacher, S. Furhapter, S. Bernet and M. Ritsch-Marte, "Tailoring of arbitrary optical vector beams," New J. Phys. 9, 78 (2007) and references therein. [CrossRef]
- D. G. Grier, "A revolution in optical manipulation," Nature 424, 810 - 816 (2003);A. Jesacher, S. Frhapter, S. Bernet, and M. Ritsch-Marte, "Size selective trapping with optical cogwheel tweezers," Opt. Express 12, 4129 - 4135 (2004); S. H. Tao, X-C. Yuan, J. Lin, X. Peng, H. B. Niu, "Fractional optical vortex beam induced rotation of particles," Opt. Express 13, 7726 - 7631 (2005) [CrossRef] [PubMed]
- 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, 535 - 541 (2006) [CrossRef] [PubMed]
- 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, 6604 - 6612 (2006) [CrossRef] [PubMed]
- 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, 8619 - 8625 (2007) [CrossRef] [PubMed]
- R. Di Leonardo, J. Leach, H. Mushfique, J. M. Cooper, G. Ruocco and M. J. Padgett, "Multiport holographic velocimetry in microfluidic systems," Phys. Rev. Lett. 96, 134502 (2006) [CrossRef] [PubMed]
- G. Molina-Terriza, J. P. Torres and L. Torner, "Twisted photons," Nature Phys. 3, 305 - 310 (2007) and refernces therein. [CrossRef]
- W. Nasalski, "Polarization versus spatial characteristics of optical beams at a planar isotropic interface," Phys. Rev. E 74, 056613 (2006) [CrossRef]
- K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four, and five plane waves," Opt. Express 14, 3039-3044 (2006) [CrossRef] [PubMed]
- J. Courtial, R. Zambrini, M. R. Dennis and M. Vasnetsov, "Angular momentum of optical vortex arrays," Opt. Express 14, 938 (2006) [CrossRef] [PubMed]
- S. Vyas and P. Senthilkumaran, "Interferometric optical vortex array generator," App. Opt. 46, 2893 - 2898 (2007) [CrossRef] [PubMed]
- J. Masajada, A. Popiolek-Masajada, and M. Leniec, "Creation of vortex lattices by a wavefront division," Opt. Express 15, 5196 - 5207 (2007) [CrossRef] [PubMed]
- S. M. Barnett and R. Zambrini, "Orbital angular momentum of light" in Quantum Imaging, Mikhail I. Kolobov Ed., (Springer-Verlag New York, 2006) and references in Section 12.6
- M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, "Polarization patterns in Kerr media," Phys. Rev. E 58, 2992 - 3007 (1998);G.-L. Oppo, A. J. Scroggie, and W. J. Firth,"Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators," Phys. Rev. E 63, 066209 (2001) [CrossRef]
- E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006) [CrossRef] [PubMed]
- D. Boiko, G. Guerrero, and E. Kapon, "Polarization Bloch waves in photonic crystals based on vertical cavity surface emitting laser arrays," Opt. Express 12, 2597 - 2602 (2004) [CrossRef] [PubMed]
- A. Ferrando, M. Zacares, and M.-A. Garcia-March, "Vorticity cutoff in Nonlinear Photonic Crystals," Phys. Rev. Lett. 95, 043901 (2005). [CrossRef] [PubMed]
- R. Zambrini and S. M. Barnett, " Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006) [CrossRef] [PubMed]
- 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, 4064 - 4074 (1997) [CrossRef]
- A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002). [CrossRef] [PubMed]
- R. Zambrini and S. M. Barnett, "Local transfer of angular momentum to matter," J. Mod. Opt. 52, 1045 - 1052 (2005). [CrossRef]
- R. Zambrini, L. C. Thomson, S. M. Barnett, M. Padgett, "Angular momentum paradox in a vortex core," J. Mod. Opt., 52, 1135 - 1144 (2005). [CrossRef]
- S. M. Barnett, "Optical angular-momentum flux", J. Opt. B: Quantum Semiclass. Opt. 4, S7 - S16 (2002) [CrossRef]
- L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density," Opt. Commun. 184, 67 - 71 (2000). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
- A. Siegman, Lasers (University Science Books, Sausalito, 1986)
- K. T. Gahagan, G. A. Swartzlander, "Optical vortex trapping of particles," Opt. Lett. 21, 827 - 829 (1996) [CrossRef] [PubMed]
- The same periodicity was found for the energy and orbital AM of an interference field, because the imaginary part of the phase sensitive term UV appears in the spin AM of a superposition field while the real part appears in the orbital AM of the interference one.
- J. F. Nye, Natural focusing and the fine structure of light (Institute of Physics Publishing, Bristol, 1999)

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