## Angular momentum of optical vortex arrays

Optics Express, Vol. 14, Issue 2, pp. 938-949 (2006)

http://dx.doi.org/10.1364/OPEX.14.000938

Acrobat PDF (1140 KB)

### Abstract

Guided by the aim to construct light fields with spin-like orbital angular momentum (OAM), that is light fields with a uniform and intrinsic OAM density, we investigate the OAM of strictly periodic arrays of optical vortices with rectangular symmetry. We find that the OAM per unit cell depends on the choice of unit cell and can even change sign when the unit cell is translated. This is the case even if the OAM in each unit cell is intrinsic, that is independent of the choice of measurement axis. We show that spin-like OAM can be found only if the OAM per unit cell vanishes. Our results are applicable to the *z* component of the angular momentum of any *x*- and *y*-periodic momentum distribution in the *xy* plane, and can also be applied to other periodic light beams and arrays of rotating solids or liquids.

© 2006 Optical Society of America

## 1. Introduction

1. J. H. Poynting, “The Wave Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light,” Proc. R. Soc. London, Ser. A **82**, 560–567 (1909). [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 modes,” Phys. Rev. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

*and*the electric field vectors at every point rotate uniformly corresponds to a total angular momentum state; we do not consider such states here.

3. 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**, 053,601 (2002). [CrossRef]

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

**p**

_{⊥}= (

*p*,

_{x}*p*),

_{y}**r**= (

*x,y*) and ▽

_{⊥}= (

*∂*/

*∂*/

_{x}, ∂*∂*). This corresponds to an OAM density in the

_{y}*z*direction of

*z*. From now on we will simply refer to (orbital) angular momentum, without explicit mention of the

*z*direction.) An area

*A*then has OAM

*h*̄[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 modes,” Phys. Rev. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

*A*- of

*z*component. This property will be important later when we consider the intrinsic OAM of the unit cell of a periodic light beam.

6. I. Dana and I. Freund, “Vortex-lattice wave fields,” Opt. Commun. **136**, 93–113 (1997). [CrossRef]

7. R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A: Pure Appl. Opt. **3**, 527–532 (2001). [CrossRef]

8. A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” J. Opt. Soc. Am. B **19**, 550–556 (2002). [CrossRef]

9. F. S. Roux, “Optical vortex density limitation,” Opt. Commun. **223**, 31–37 (2003). [CrossRef]

9. F. S. Roux, “Optical vortex density limitation,” Opt. Commun. **223**, 31–37 (2003). [CrossRef]

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

11. K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Optics Express **12**, 1144–1149 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144. [CrossRef] [PubMed]

12. M. R. Dennis and J. H. Hannay, “Saddle points in the chaotic analytic function and Ginibre characteristic polynomial,” J. Phys. A: Math. Gen. **36**, 3379–3383 (2003). [CrossRef]

7. R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A: Pure Appl. Opt. **3**, 527–532 (2001). [CrossRef]

13. J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of Vortex Lattices in Bose-Einstein Condensates,” Science **292**, 476–479 (2001). [CrossRef] [PubMed]

16. J. Sommeria, “Experimental study of the two-dimensional inverse energy cascade in a square box,” J. Fluid Mech. **170**, 139–168 (1986). [CrossRef]

## 2. Example

*x*+ isin

*y*), contains a rectangular array of vortices of alternating charges. The second factor, (cos(

*x*/2)cos(

*y*/2))

^{4}, concentrates the intensity around a rectangular array of vortices with a positive charge, which are positioned where cos(

*x*/2)cos(

*y*/2) = ±1 by multiplying the complex vortex function by an intensity factor which vanishes at the positions of negative-charge vortices; the exponent determines how strongly the intensity is concentrated. While the intensity factor does not affect the phase of the field and therefore the topological charge, it does alter the OAM density.

*P*= 0 =

_{x}*P*(see Eq. (7)), and therefore it does not matter with respect to which point a cell’s OAM and OAM per photon, Ω and

_{y}*l*, are calculated. At the same time, all unit cells shown in Fig. 2 are unit cells of the same array. Therefore, as with other quantities of periodic fields, it appears reasonable to think that the OAM per photon of a unit cell is independent of the choice of unit cell.

## 3. Angular momentum of a periodic momentum distribution

*n,m*= 0,±1,±2,… and

*X*and

*Y*are the respective lattice periods in the

*x*and

*y*directions. From Eq. (1) it follows that the momentum density

**p**

_{⊥}(

*x,y*) is also a periodic function. In fact, the calculation in this section is applicable to any rectangularly periodic momentum distribution, for example those due to circular polarization in light beams and to arrays of rotating liquids or solids.

*X*and

*Y*. We restrict ourselves to the case of light fields in which the transverse linear momentum in any unit cell

*D*vanishes, that is

**p**

_{⊥}= (

*p*,

_{x}*p*). Then Υ

_{y}_{D}, the OAM in

*D*according to Eq. (3), is intrinsic [4, 3

3. 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**, 053,601 (2002). [CrossRef]

*x,y*) position of the OAM-calculation axis from (0,0), which was implicitly used in Eq. (2), to arbitrary coordinates (

*R*,

_{x}*R*) results in the same angular momentum:

_{y}*A*and

*B*with side lengths

*X*and

*Y*whose lower left corners are positioned respectively at the origin and at (

*x*

_{0},

*y*

_{0}), as shown in Fig. 4. Note that it is natural in this context to characterize a unit cell in terms of the coordinates of one of its corners, whereas in the context of section 4 it is conventional to characterize a cell in terms of its center coordinates. We already observed in section 2 that the angular momenta

**r**. In spite of the fact that the angular momentum carried by any cell is independent on the axis position (and indeed on any change

**r**→

**r**+

**R**) the dependence of the density

*ω*on

**r**leads to differences between Ω

_{A}and Ω

_{B}. In the following we analytically evaluate the difference in angular momentum carried by two such unit cells at arbitrary positions.

*X*and

*Y*in the

*x*and

*y*direction, have the same angular momentum. This is simply due to the periodicity of the linear momentum density and the intrinsic character of the angular momentum carried by a cell. Without loss of generality we can therefore restrict ourselves to overlapping unit cells

*A*and

*B*, as shown in Fig. 4.

*A*and

_{i}*B*(

_{i}*i*= 1,…,4) as the rectangular parts that respectively add up to

*A*and

*B*as shown in Fig. 4. Periodicity guarantees that the momentum distribution in the cell

*B*is the same as inside the corresponding cell

_{i}*A*, for any

_{i}*i*. It follows that

**p**

_{⊥}give the same angular momentum Ω in

*any*cell. It is clear that Eq. (11) is independent of the choice of

*x*

_{0}and

*y*

_{0}if and only if

*x*and

*y*. As

_{A}= 0. Therefore the angular momentum carried by a unit cell is independent of the cell’s position if and only if the angular momentum for all cell positions is zero. As this is clearly not the case for the vortex-array example in section 2, its orbital angular momentum depends on the choice of unit cell.

*x*

_{0},

*y*

_{0}) is the position of the lower left corner of the cell. It is easily found that

*P*,

_{x}*P*) is the total transverse momentum introduced in Eq. (7). As we are considering arrays with intrinsic angular momentum, Eq. (7) implies that

_{y}*positive*angular momentum, there is a shifted cell carrying a

*negative*angular momentum. Clearly, when we consider continuous spatial arrays, there is also a cell position giving vanishing angular momentum. An illustrative example of this general result is given in Fig. 5.

17. D. L. Boiko, G. Guerrero, and E. Kapon, “Polarization Bloch waves in photonic crystals based on vertical cavity surface emitting laser arrays,” Optics Express **12**, 2597–2602 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597. [CrossRef]

## 4. OAM of periodic light beams on propagation

*x*and

*y*direction can also be periodic in the

*z*direction - the Talbot effect [18]. It is therefore natural to investigate the OAM contained in the (three-dimensional) unit cells of such a beam.

*x*and

*y*. Not only does this enable an alternative proof for the disappearance of the average OAM over choice of cell positions, but it allows the investigation of the OAM on propagation of the beams in the

*z*direction.

*m,n*is over all integers (positive and negative), the

*a*are complex coefficients, and all sums are assumed to converge.

_{m,n}*ψ*,

**p**, etc.) for

*ψ*(

*x,y*) in the form (18). Substitution into equations (1) and (2) gives the

*z*-component of the OAM density:

**P**

_{⊥}, which vanishes when the angular momentum is intrinsic (see Eq. (8)), is

*x*,

_{c}*y*). The (

_{c}*x*,

_{c}*y*)-dependent total OAM in the unit cell is easily calculated to be

_{c}**P**

_{⊥}is zero and the first term in (21) vanishes. In that case, there is no constant term in the right-hand side of Eq. (21), which is a Fourier series, so the average of Ω

_{(xc,yc)}over all cell positions is zero (as expected - see Eq. (17)).

*ψ*in other planes, we substitute for the expression for each Fourier component in the

*z*= 0 plane the expression that explicitly takes its

*z*dependence into account:

*k*= 2

_{x}*πm*/

*X*and

*k*= 2

_{y}*πn*/

*Y*are the transverse wave numbers. For

*k*we use the approximation

_{z}*m*and

*n*. The wave

*ψ*can then be calculated at all points in space:

*ikz*) as it affects only the phase of the whole beam. Specifically it does not affect the transverse momentum and the angular momentum in the

*z*direction.

_{(xc,yc)}is now a (periodic) function of

*z*whose average over

*z*does not necessarily vanish.

*X*=

*Y*(= 2

*π*)), which simplifies the calculation. Figure 6 shows the beam’s intensity cross section for different values of the propagation distance

*z*and the OAM per photon for different unit cells as a function of

*z*. It can be seen that propagation through half a Talbot period (

*z*= τ/2) shifts this beam, indeed any beam for which both

*M*and

*N*are odd, through half a transverse period in the

*x*and

*y*direction, i.e.,

_{(xc,yc)}(

*z*) over a Talbot distance is not necessarily 0, but given for any arbitrary beam by

_{(xc,yc)}over its propagation period is doubly periodic in

*x*and

_{c}*y*, as Eq. (27) would suggest. As before, the average over the transverse position (

_{c}*x*,

_{c}*y*) vanishes as this Fourier series has no constant term.

_{c}## 5. Finite arrays and edge effects

*finite*size and therefore can only consist of a finite number of unit cells. Figure 7 shows an example of a finite vortex-array: a circular section of the example shown in Eq. (5). It is clear that for any particular choice of unit cell only part of the beam that can be covered by complete unit cells; an area near the edge of the beam remains uncovered. The angular momentum in the whole beam is the sum of the angular momenta contained in the part of the beam that is covered by the unit cells, which is simply the angular momentum per unit cell times the number of unit cells, and that in the edge of the beam. The shape of the edge part changes depending on the choice of unit cell; the angular momentum contained in it is always the difference between the angular momentum in the whole beam and that contained in the part covered by unit cells.

20. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. **144**, 210–213 (1997). [CrossRef]

*r*. We assume that the edge area is on average of a width that is independent of

*r*and that the average momentum density is also constant. The edge area is then approximately the average width times the circumference of the beam, so it is approximately proportional to

*r*. The angular momentum density near the edge, calculated with respect to the centre of the beam, is proportional to the momentum density and

*r*. This implies that the angular momentum in the edge is proportional to

*r*

^{2}. The number of unit cells covering the centre of the beam also goes approximately with

*r*

^{2}, so we conclude that the angular momentum in the edge remains significant compared to that in the main part of the beam, even for large beams.

## 6. Conclusions and ideas for generalizations

*h*̄), and a unit cell with a non-square shape (in fact a non-connected shape) with a modulus of the OAM per photon that is significantly higher (-6.84

*h*̄). The latter cell is constructed by splitting up a square unit cell into four parts and moving those by integer multiples of the lattice periods. In fact, by increasing the separation of the parts the OAM per photon in the unit cell can be made arbitrarily high.

## Acknowledgments

*Visiting Fellow*at the University of Glasgow (grant no. EP/B000028/1). Roberta Zambrini is supported by EPSRC (grant no. SO3898/01). Johannes Courtial and Mark Dennis are Royal Society University Research Fellows.

## References and links

1. | J. H. Poynting, “The Wave Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light,” Proc. R. Soc. London, Ser. A |

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 modes,” Phys. Rev. A |

3. | 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. |

4. | M. Berry, “Paraxial beams of spinning light,” in |

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

6. | I. Dana and I. Freund, “Vortex-lattice wave fields,” Opt. Commun. |

7. | R. M. Jenkins, J. Banerji, and A. R. Davies, “The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides,” J. Opt. A: Pure Appl. Opt. |

8. | A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” J. Opt. Soc. Am. B |

9. | F. S. Roux, “Optical vortex density limitation,” Opt. Commun. |

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

11. | K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Optics Express |

12. | M. R. Dennis and J. H. Hannay, “Saddle points in the chaotic analytic function and Ginibre characteristic polynomial,” J. Phys. A: Math. Gen. |

13. | J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of Vortex Lattices in Bose-Einstein Condensates,” Science |

14. | R. Donnelly, |

15. | P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, England, 1992). |

16. | J. Sommeria, “Experimental study of the two-dimensional inverse energy cascade in a square box,” J. Fluid Mech. |

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

18. | K. Patorski, “The self-imaging phenomenon and its applications,” Progr. Opt. |

19. | L. Mandel and E. Wolf, |

20. | J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. |

**OCIS Codes**

(260.0260) Physical optics : Physical optics

(350.0350) Other areas of optics : Other areas of optics

**ToC Category:**

Physical Optics

**Citation**

Johannes Courtial, Roberta Zambrini, Mark R. Dennis, and Mikhail Vasnetsov, "Angular momentum of optical vortex arrays," Opt. Express **14**, 938-949 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-2-938

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### References

- J. H. Poynting, "The Wave Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light," Proc. R. Soc. London, Ser. A 82, 560-567 (1909). [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian modes," Phys. Rev. A 45, 8185-8189 (1992). [CrossRef] [PubMed]
- 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, 053,601 (2002). [CrossRef]
- M. Berry, "Paraxial beams of spinning light," in Singular Optics, M. S. Soskin, ed., vol. 3487 of Proc. SPIE, pp. 1-5 (SPIE - the International Society for Optical Engineering, Bellingham, Wash., USA, 1998).
- S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclass. Opt. 4, S7-S16 (2002). [CrossRef]
- I. Dana and I. Freund, "Vortex-lattice wave fields," Opt. Commun. 136, 93-113 (1997). [CrossRef]
- R. M. Jenkins, J. Banerji, and A. R. Davies, "The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides," J. Opt. A: Pure Appl. Opt. 3, 527-532 (2001). [CrossRef]
- A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, "Generation of lattice structures of optical vortices," J. Opt. Soc. Am. B 19, 550-556 (2002). [CrossRef]
- F. S. Roux, "Optical vortex density limitation," Opt. Commun. 223, 31-37 (2003). [CrossRef]
- R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, "Momentum paradox in a vortex core," J. Mod. Opt. 52, 1135-1144 (2005). [CrossRef]
- K. Ladavac and D. G. Grier, "Microoptomechanical pumps assembled and driven by holographic optical vortex arrays," Optics Express 12, 1144-1149 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144</a>. [CrossRef] [PubMed]
- M. R. Dennis and J. H. Hannay, "Saddle points in the chaotic analytic function and Ginibre characteristic polynomial," J. Phys. A: Math. Gen. 36, 3379-3383 (2003). [CrossRef]
- J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, "Observation of Vortex Lattices in Bose-Einstein Condensates," Science 292, 476-479 (2001). [CrossRef] [PubMed]
- R. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, 1991).
- P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, England, 1992).
- J. Sommeria, "Experimental study of the two-dimensional inverse energy cascade in a square box," J. Fluid Mech. 170, 139-168 (1986). [CrossRef]
- D. L. Boiko, G. Guerrero, and E. Kapon, "Polarization Bloch waves in photonic crystals based on vertical cavity surface emitting laser arrays," Optics Express 12, 2597-2602 (2005). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597</a>. [CrossRef]
- K. Patorski, "The self-imaging phenomenon and its applications," Progr. Opt. XXVII, 3-108 (1989).
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
- J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997). [CrossRef]

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