Abstract

We investigate the orbital angular momentum (OAM) of paraxial beams containing off-axis phase dislocations and put forward a simple method to calculate the intrinsic orbital angular momentum of an arbitrary paraxial beam. Using this approach we find that the intrinsic OAM of a fundamental Gaussian beam with a vortex imprinted off axis has a Gaussian dependence on the vortex displacement, implying that the expectation value of the intrinsic OAM of a photon can take on a continuous range of values (i.e., integer and noninteger values in units of ℏ). Finally, we investigate, both numerically and experimentally, the far-field profiles of beams carrying half-integer OAM per photon, these beams having been created by the method of imprinting off-axis vortices.

© 2004 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformations of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [Crossref] [PubMed]
  2. S. J. van Enk, G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94, 147–158 (1992).
    [Crossref]
  3. S. J. van Enk, G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 962–978 (1994).
    [Crossref]
  4. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
    [Crossref]
  5. W. J. Firth, D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 97, 2450–2453 (1997).
    [Crossref]
  6. M. Soljac̆ić, M. Segev, “Integer and fractional angular momentum borne on self-trapped necklace-ring beams,” Phys. Rev. Lett. 86, 420–423 (2001).
    [Crossref] [PubMed]
  7. A. T. O’Neill, I. MacVicar, L. Allen, M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053 601 (2002).
    [Crossref]
  8. G. Molina-Terriza, J. P. Torres, L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013 601 (2002).
    [Crossref]
  9. S. Franke-Arnold, S. M. Barnett, M. J. Padgett, L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033 823 (2002).
    [Crossref]
  10. M. J. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. M. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777–785 (2002).
    [Crossref]
  11. A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [Crossref] [PubMed]
  12. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257 901 (2002).
    [Crossref]
  13. H. H. Arnaut, G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett. 85, 286–289 (2000).
    [Crossref] [PubMed]
  14. E. R. Eliel, S. M. Dutra, G. Nienhuis, J. P. Woerdman, “Comment on ‘Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion’,” Phys. Rev. Lett. 86, 5208 (2001).
    [Crossref] [PubMed]
  15. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).
  16. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [Crossref]
  17. F. S. Roux, “Dynamical behavior of optical vortices,” J. Opt. Soc. Am. B 12, 1215–1221 (1995).
    [Crossref]
  18. M. J. Padgett, J. Arlt, N. Simpson, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
    [Crossref]
  19. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
    [Crossref]
  20. N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, 5951–5962 (1992).
    [Crossref]
  21. D. Rozas, C. T. Law, G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
    [Crossref]
  22. Z. S. Sacks, D. Rozas, G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
    [Crossref]
  23. I. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
    [Crossref]
  24. I. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
    [Crossref]
  25. M. V. Berry, “Paraxial beams of spinning light,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 6–11 (1998).
    [Crossref]
  26. M. V. Vasnetsov, I. V. Basistiy, M. S. Soskin, “Free-space evolution of monochromatic mixed screw-edge wavefront dislocations,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 29–33 (1998).
    [Crossref]
  27. S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, G. W. ’t Hooft, “Production and characterization of spiral phase plates,” Appl. Opt. 43, 688–694 (2004).
    [Crossref] [PubMed]
  28. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  29. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  30. J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
    [Crossref]
  31. It is important to realize that separation of the total angular momentum of a beam into a spin part and an orbital part can be done only for the component of the total angular momentum that lies parallel to (or at least makes a paraxial angle with) the propagation axis of the beam.2

2004 (1)

2002 (5)

A. T. O’Neill, I. MacVicar, L. Allen, M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053 601 (2002).
[Crossref]

G. Molina-Terriza, J. P. Torres, L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013 601 (2002).
[Crossref]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033 823 (2002).
[Crossref]

M. J. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. M. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777–785 (2002).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257 901 (2002).
[Crossref]

2001 (3)

M. Soljac̆ić, M. Segev, “Integer and fractional angular momentum borne on self-trapped necklace-ring beams,” Phys. Rev. Lett. 86, 420–423 (2001).
[Crossref] [PubMed]

E. R. Eliel, S. M. Dutra, G. Nienhuis, J. P. Woerdman, “Comment on ‘Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion’,” Phys. Rev. Lett. 86, 5208 (2001).
[Crossref] [PubMed]

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

2000 (1)

H. H. Arnaut, G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett. 85, 286–289 (2000).
[Crossref] [PubMed]

1998 (1)

1997 (4)

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

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

D. Rozas, C. T. Law, G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[Crossref]

W. J. Firth, D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 97, 2450–2453 (1997).
[Crossref]

1996 (1)

M. J. Padgett, J. Arlt, N. Simpson, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[Crossref]

1995 (2)

F. S. Roux, “Dynamical behavior of optical vortices,” J. Opt. Soc. Am. B 12, 1215–1221 (1995).
[Crossref]

I. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

1994 (2)

S. J. van Enk, G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 962–978 (1994).
[Crossref]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

1993 (2)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[Crossref]

I. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

1992 (3)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, 5951–5962 (1992).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformations of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

S. J. van Enk, G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94, 147–158 (1992).
[Crossref]

’t Hooft, G. W.

Allen, L.

A. T. O’Neill, I. MacVicar, L. Allen, M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053 601 (2002).
[Crossref]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033 823 (2002).
[Crossref]

M. J. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. M. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777–785 (2002).
[Crossref]

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

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformations of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Arlt, J.

M. J. Padgett, J. Arlt, N. Simpson, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[Crossref]

Arnaut, H. H.

H. H. Arnaut, G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett. 85, 286–289 (2000).
[Crossref] [PubMed]

Barbosa, G. A.

H. H. Arnaut, G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett. 85, 286–289 (2000).
[Crossref] [PubMed]

Barnett, S. M.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257 901 (2002).
[Crossref]

M. J. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. M. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777–785 (2002).
[Crossref]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033 823 (2002).
[Crossref]

Basistiy, I.

I. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

I. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Basistiy, I. V.

M. V. Vasnetsov, I. V. Basistiy, M. S. Soskin, “Free-space evolution of monochromatic mixed screw-edge wavefront dislocations,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 29–33 (1998).
[Crossref]

Bazhenov, V. Y.

I. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformations of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry, “Paraxial beams of spinning light,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 6–11 (1998).
[Crossref]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Courtial, J.

M. J. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. M. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777–785 (2002).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257 901 (2002).
[Crossref]

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

Dholakia, K.

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

Dutra, S. M.

E. R. Eliel, S. M. Dutra, G. Nienhuis, J. P. Woerdman, “Comment on ‘Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion’,” Phys. Rev. Lett. 86, 5208 (2001).
[Crossref] [PubMed]

Eliel, E. R.

S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, G. W. ’t Hooft, “Production and characterization of spiral phase plates,” Appl. Opt. 43, 688–694 (2004).
[Crossref] [PubMed]

E. R. Eliel, S. M. Dutra, G. Nienhuis, J. P. Woerdman, “Comment on ‘Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion’,” Phys. Rev. Lett. 86, 5208 (2001).
[Crossref] [PubMed]

Firth, W. J.

W. J. Firth, D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 97, 2450–2453 (1997).
[Crossref]

Franke-Arnold, S.

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033 823 (2002).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257 901 (2002).
[Crossref]

M. J. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. M. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777–785 (2002).
[Crossref]

Gorshkov, V. N.

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

Heckenberg, N. R.

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

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, 5951–5962 (1992).
[Crossref]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[Crossref]

Kloosterboer, J. G.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Law, C. T.

Leach, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257 901 (2002).
[Crossref]

MacVicar, I.

A. T. O’Neill, I. MacVicar, L. Allen, M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053 601 (2002).
[Crossref]

Mair, A.

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

Malos, J. T.

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

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, 5951–5962 (1992).
[Crossref]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013 601 (2002).
[Crossref]

Nienhuis, G.

E. R. Eliel, S. M. Dutra, G. Nienhuis, J. P. Woerdman, “Comment on ‘Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion’,” Phys. Rev. Lett. 86, 5208 (2001).
[Crossref] [PubMed]

S. J. van Enk, G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 962–978 (1994).
[Crossref]

S. J. van Enk, G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94, 147–158 (1992).
[Crossref]

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).

O’Neill, A. T.

A. T. O’Neill, I. MacVicar, L. Allen, M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053 601 (2002).
[Crossref]

Oemrawsingh, S. S. R.

Padgett, M. J.

A. T. O’Neill, I. MacVicar, L. Allen, M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053 601 (2002).
[Crossref]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033 823 (2002).
[Crossref]

M. J. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. M. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777–785 (2002).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257 901 (2002).
[Crossref]

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

M. J. Padgett, J. Arlt, N. Simpson, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[Crossref]

Roux, F. S.

Rozas, D.

Rubinsztein-Dunlop, H.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, 5951–5962 (1992).
[Crossref]

Sacks, Z. S.

Segev, M.

M. Soljac̆ić, M. Segev, “Integer and fractional angular momentum borne on self-trapped necklace-ring beams,” Phys. Rev. Lett. 86, 420–423 (2001).
[Crossref] [PubMed]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Simpson, N.

M. J. Padgett, J. Arlt, N. Simpson, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[Crossref]

Skryabin, D. V.

W. J. Firth, D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 97, 2450–2453 (1997).
[Crossref]

Smith, C. P.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, 5951–5962 (1992).
[Crossref]

Soljac?ic, M.

M. Soljac̆ić, M. Segev, “Integer and fractional angular momentum borne on self-trapped necklace-ring beams,” Phys. Rev. Lett. 86, 420–423 (2001).
[Crossref] [PubMed]

Soskin, M. S.

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

I. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

I. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

M. V. Vasnetsov, I. V. Basistiy, M. S. Soskin, “Free-space evolution of monochromatic mixed screw-edge wavefront dislocations,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 29–33 (1998).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformations of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Swartzlander, G. A.

Torner, L.

G. Molina-Terriza, J. P. Torres, L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013 601 (2002).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013 601 (2002).
[Crossref]

van Enk, S. J.

S. J. van Enk, G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 962–978 (1994).
[Crossref]

S. J. van Enk, G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94, 147–158 (1992).
[Crossref]

van Houwelingen, J. A. W.

Vasnetsov, M. V.

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

I. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

I. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

M. V. Vasnetsov, I. V. Basistiy, M. S. Soskin, “Free-space evolution of monochromatic mixed screw-edge wavefront dislocations,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 29–33 (1998).
[Crossref]

Vaziri, A.

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

Verstegen, E. J. K.

Wegener, M. J.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, 5951–5962 (1992).
[Crossref]

Weihs, G.

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

Woerdman, J. P.

S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, G. W. ’t Hooft, “Production and characterization of spiral phase plates,” Appl. Opt. 43, 688–694 (2004).
[Crossref] [PubMed]

E. R. Eliel, S. M. Dutra, G. Nienhuis, J. P. Woerdman, “Comment on ‘Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion’,” Phys. Rev. Lett. 86, 5208 (2001).
[Crossref] [PubMed]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformations of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Zeilinger, A.

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

Am. J. Phys. (1)

M. J. Padgett, J. Arlt, N. Simpson, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[Crossref]

Appl. Opt. (1)

J. Mod. Opt. (3)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[Crossref]

S. J. van Enk, G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 962–978 (1994).
[Crossref]

M. J. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. M. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777–785 (2002).
[Crossref]

J. Opt. Soc. Am. B (3)

Nature (1)

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

Opt. Commun. (5)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

S. J. van Enk, G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94, 147–158 (1992).
[Crossref]

I. Basistiy, V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

I. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

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

Opt. Quantum Electron. (1)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, 5951–5962 (1992).
[Crossref]

Phys. Rev. A (3)

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033 823 (2002).
[Crossref]

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

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformations of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (7)

W. J. Firth, D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 97, 2450–2453 (1997).
[Crossref]

M. Soljac̆ić, M. Segev, “Integer and fractional angular momentum borne on self-trapped necklace-ring beams,” Phys. Rev. Lett. 86, 420–423 (2001).
[Crossref] [PubMed]

A. T. O’Neill, I. MacVicar, L. Allen, M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053 601 (2002).
[Crossref]

G. Molina-Terriza, J. P. Torres, L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013 601 (2002).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257 901 (2002).
[Crossref]

H. H. Arnaut, G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett. 85, 286–289 (2000).
[Crossref] [PubMed]

E. R. Eliel, S. M. Dutra, G. Nienhuis, J. P. Woerdman, “Comment on ‘Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion’,” Phys. Rev. Lett. 86, 5208 (2001).
[Crossref] [PubMed]

Other (6)

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).

It is important to realize that separation of the total angular momentum of a beam into a spin part and an orbital part can be done only for the component of the total angular momentum that lies parallel to (or at least makes a paraxial angle with) the propagation axis of the beam.2

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

M. V. Berry, “Paraxial beams of spinning light,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 6–11 (1998).
[Crossref]

M. V. Vasnetsov, I. V. Basistiy, M. S. Soskin, “Free-space evolution of monochromatic mixed screw-edge wavefront dislocations,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 29–33 (1998).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Binary phase hologram with a dislocation in its center. Black and white indicate the two values of the shift in phase imprinted on the incident field.

Fig. 2
Fig. 2

Schematic illustration of an experiment in which a fork hologram is positioned off axis with respect to an incident Gaussian beam. The incident beam is off center in the near field, as indicated by the black cross that lies above the fork in the hologram. The diffraction axis (dashed line) does not coincide with the propagation axis, with respect to which there is zero net transverse momentum (solid black line). In the far-field intensity profile, black and white indicate regions of high and low intensity, respectively.

Fig. 3
Fig. 3

Experimental setup for examining the far-field intensity profile of a l=1/2 beam. A fundamental Gaussian impinges on a displaced hologram, carrying a unity-strength phase dislocation. The displacement between the center of the hologram and that of the input beam equals w0(ln(2)/2)1/2 [see Eq. (22)], where w0 equals the radius of the input beam. The lens L1 with focal length f images the far field of the first diffraction order, while the other orders are blocked.

Fig. 4
Fig. 4

Intensity profiles of an l=1/2 optical beam obtained in the setup of Fig. 3. (a) Calculated near-field profile, where ⊕ indicates the position of the vortex unity charge in the near field. The intersection of the dashed lines indicates the center of gravity of the beam. (b) Experimentally obtained far-field intensity profile at a wavelength of 633 nm. (c) Calculated far-field profile, where ⊕ indicates the position of a unity-charge vortex and the intersection of the dashed lines indicates where the diffraction axis crosses the far-field plane.

Fig. 5
Fig. 5

Fundamental Gaussian beam impinges on a hologram. Lenses L1 and L2 image the near field of the first diffraction order onto a second hologram. Lens L3 then images the far-field intensity profiles of the three lowest diffraction orders onto a CCD camera. Both holograms are displaced with respect to the incident beam, so that each hologram imprints a single vortex in the same location or in opposite locations.

Fig. 6
Fig. 6

Two vortices are imprinted opposite to each other in a fundamental Gaussian beam. In the top row the vortices have opposite charge, whereas in the bottom row their charges are both positive. In each row the first column shows a calculated near-field profile, where the intersection of the dashed lines indicates the center of the beam while ⊕ (Q=+1) and ⊖ (Q=-1) symbols indicate the approximate positions of the positively and negatively charged vortices, respectively. The second column shows the experimentally obtained far-field intensity profile corresponding to the near fields shown in the first column. The last column shows far-field profiles that have been calculated by Fourier transforming the near fields. Although (c) does not contain any vortices, the two lobes differ in phase by π.

Fig. 7
Fig. 7

Two vortices are imprinted on top of each other in a fundamental Gaussian beam. In the top row the vortices have opposite charge, thus effectively annihilating each other, whereas in the bottom row their charges are positive, thus being summed. In each row the first column shows a theoretical near-field profile where the intersection of the dashed lines indicates the center of the beam and the ⊙ (Q=0, overlapping oppositely charged vortices) and ⊗ (Q=2, overlapping equally charged vortices) symbols indicate the approximate positions of the annihilated and summed vortices, respectively. The second column shows the experimentally obtained far-field intensity profiles corresponding to the near fields shown in the first column. The last column shows far-field profiles that have been calculated by Fourier transforming the near fields.

Equations (23)

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Q=12πdϕ,
E(r, t)=σu(r)exp(ikz-iωt)+c.c.
p(ρ)=0iω [u*(ρ)u(ρ)-c.c.],
L=[ρ×p(ρ)]zdρ=20iωu*(ρ)(ρ×)zu(ρ)dρ,
l=ω20|u(ρ)|2dρ L
=u*(ρ) iθ u(ρ)dρ/|u(ρ)|2dρ,
ulpLG(r, θ)=ClpLG(r2/w0)|l|Lp|l|(2r2/w02)×exp(-r2/w02)exp(ilθ),
u(ρ)=l=-p=0ClpulpLG(ρ),
Clp=002πu(ρ)[ulpLG(ρ)]*dρ.
l=l=-p=0l|Clp|2,
uin(r)umin(r)exp(-img  r),
umout(ρ)=Cmumin(ρ)exp(imQθ),
ρ|u(ρ)|2dρ=0,
p(ρ)dρ=0.
L0=[ρ×p(ρ)]zdρ.
u(ρ)=u(ρ)exp(ik0ρ),
p(ρ)=0iω [u*(ρ)u(ρ)-u(ρ)u*(ρ)+2ik0|u(ρ)|2]=p(ρ)+20ω   k0|u(ρ)|2.
L=[(ρ-ρ0)×p(ρ)]zdρ=L0-20ω (ρ0×k0)z|u(ρ)|2dρ,
uin(ρ)=B exp(-|ρ|2/w02),
umout(ρ)=Cmumin(ρ+ρv)exp(imQθ),
Lm0=20ω   umout*(ρ)[(ρ+ρv)×]zumout(ρ)dρ.
[(ρ+ρv)×]z   exp(imQθ)=imQ ρ(ρ+ρv)r2×exp(imQθ),
Lm0=20ω Cm2mQB2×   exp-2 r2+2rrvcos θ+rv2w02×(r+rv   cos θ)drdθ=Cm2mQ 0ω w02πB2   exp-2 rv2w02.

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