Abstract

Light emerging from a spiral phase plate with a non-integer phase step has a complicated vortex structure and is unstable on propagation. We generate light carrying fractional orbital angular momentum (OAM) not with a phase step but by a synthesis of Laguerre-Gaussian modes. By limiting the number of different Gouy phases in the superposition we produce a light beam which is well characterised in terms of its propagation. We believe that their structural stability makes these beams ideal for quantum information processes utilising fractional OAM states.

©2008 Optical Society of America

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  1. M. V. Berry, “Much ado about nothing: optical distortion lines (phase singularities, zeros, and vortices),” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 1–5 (1998). http://spie. org/x648.xml?product_id=317693.
    [Crossref]
  2. M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. Royal Society of London, Series A 457(2013),2251–2263 (2001). http://www.journals.royalsoc.ac.uk/content/yv6lqeufbq2phg9l.
    [Crossref]
  3. J. Leach, M. R. Dennis, and M. J. Padgett, “Laser beams: Knotted threads of darkness,” Nature 432(165) (2004). http://www.nature.com/nature/journal/v432/n7014/abs/432165a.html.
    [Crossref] [PubMed]
  4. J. Leach, M. R. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). http://www.iop.org/EJ/abstract/1367–2630/7/1/055/.
    [Crossref]
  5. 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, 8185–8190 (1992). http://link.aps. org/abstract/PRA/v45/p8185.
    [Crossref] [PubMed]
  6. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
    [Crossref]
  7. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefrontnext term laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
    [Crossref]
  8. V. Y. Bazhenov, M. V. Vastnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP Lett. 52, 429–431 (1990).
  9. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–269 (2004). http://www.iop.org/EJ/abstract/1464-4258/6/2/018.
    [Crossref]
  10. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). http://www.iop.org/EJ/abstract/1367-2630/6/1/071/.
    [Crossref]
  11. K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Illustrations of optical vortices in three dimensions,” Journal of European Optical Society - Rapid Publications 1,06008 (2006). https://www.jeos.org/index.php/jeos rp/article/view/06008.
    [Crossref]
  12. J. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007). http://www.informaworld.com/smppcontent~content=a779773614.
    [Crossref]
  13. S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhaus, and J. P. Woerdman, “How to observe High-Dimensional Two-Photon Entanglement with Only Two Detectors,” Physical Review Letters 92, 217901 (2004). http://link.aps.org/abstract/PRL/v92/e217901.
    [Crossref] [PubMed]
  14. G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007). http://link.aps.org/abstract/PRA/v75/e012319.
    [Crossref]
  15. A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, “Nonlocality of high-dimensional two-photon orbital angular momentum states,” Phys. Rev. A 72, 052114 (2005). http://link.aps.org/abstract/PRA/v72/e052114.
    [Crossref]
  16. S. M. Barnett and D. T. Pegg, “Quantum theory of rotation angles,” Phys. Rev. A 41, 3427–3435 (1990). http://link.aps.org/abstract/PRA/v41/p3427.
    [Crossref] [PubMed]
  17. J. Courtial, “Self-imaging beams and the Guoy [sic] effect,” Opt. Commun. 151, 1–4 (1998).
    [Crossref]
  18. R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, “Momentum paradox in a vortex core,” J. Modern Opt. 52(8), 1135–1144 (2005). http://www.informaworld.com/smpp/content~content=a713736931.
    [Crossref]
  19. C. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express. 13, 1749–1760 (2005). http://www.opticsexpress.org/abstract. cfm?URI=oe-13-5-1749.
    [Crossref] [PubMed]

2007 (2)

J. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007). http://www.informaworld.com/smppcontent~content=a779773614.
[Crossref]

G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007). http://link.aps.org/abstract/PRA/v75/e012319.
[Crossref]

2006 (1)

K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Illustrations of optical vortices in three dimensions,” Journal of European Optical Society - Rapid Publications 1,06008 (2006). https://www.jeos.org/index.php/jeos rp/article/view/06008.
[Crossref]

2005 (4)

R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, “Momentum paradox in a vortex core,” J. Modern Opt. 52(8), 1135–1144 (2005). http://www.informaworld.com/smpp/content~content=a713736931.
[Crossref]

C. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express. 13, 1749–1760 (2005). http://www.opticsexpress.org/abstract. cfm?URI=oe-13-5-1749.
[Crossref] [PubMed]

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, “Nonlocality of high-dimensional two-photon orbital angular momentum states,” Phys. Rev. A 72, 052114 (2005). http://link.aps.org/abstract/PRA/v72/e052114.
[Crossref]

J. Leach, M. R. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). http://www.iop.org/EJ/abstract/1367–2630/7/1/055/.
[Crossref]

2004 (4)

J. Leach, M. R. Dennis, and M. J. Padgett, “Laser beams: Knotted threads of darkness,” Nature 432(165) (2004). http://www.nature.com/nature/journal/v432/n7014/abs/432165a.html.
[Crossref] [PubMed]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–269 (2004). http://www.iop.org/EJ/abstract/1464-4258/6/2/018.
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). http://www.iop.org/EJ/abstract/1367-2630/6/1/071/.
[Crossref]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhaus, and J. P. Woerdman, “How to observe High-Dimensional Two-Photon Entanglement with Only Two Detectors,” Physical Review Letters 92, 217901 (2004). http://link.aps.org/abstract/PRL/v92/e217901.
[Crossref] [PubMed]

2001 (1)

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. Royal Society of London, Series A 457(2013),2251–2263 (2001). http://www.journals.royalsoc.ac.uk/content/yv6lqeufbq2phg9l.
[Crossref]

2000 (1)

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

1998 (1)

J. Courtial, “Self-imaging beams and the Guoy [sic] effect,” Opt. Commun. 151, 1–4 (1998).
[Crossref]

1994 (1)

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

1992 (1)

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, 8185–8190 (1992). http://link.aps. org/abstract/PRA/v45/p8185.
[Crossref] [PubMed]

1990 (2)

V. Y. Bazhenov, M. V. Vastnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP Lett. 52, 429–431 (1990).

S. M. Barnett and D. T. Pegg, “Quantum theory of rotation angles,” Phys. Rev. A 41, 3427–3435 (1990). http://link.aps.org/abstract/PRA/v41/p3427.
[Crossref] [PubMed]

Aiello, A.

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, “Nonlocality of high-dimensional two-photon orbital angular momentum states,” Phys. Rev. A 72, 052114 (2005). http://link.aps.org/abstract/PRA/v72/e052114.
[Crossref]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhaus, and J. P. Woerdman, “How to observe High-Dimensional Two-Photon Entanglement with Only Two Detectors,” Physical Review Letters 92, 217901 (2004). http://link.aps.org/abstract/PRL/v92/e217901.
[Crossref] [PubMed]

Allen, L.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[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, 8185–8190 (1992). http://link.aps. org/abstract/PRA/v45/p8185.
[Crossref] [PubMed]

Alonzo, C.

C. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express. 13, 1749–1760 (2005). http://www.opticsexpress.org/abstract. cfm?URI=oe-13-5-1749.
[Crossref] [PubMed]

Barnett, S. M.

J. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007). http://www.informaworld.com/smppcontent~content=a779773614.
[Crossref]

R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, “Momentum paradox in a vortex core,” J. Modern Opt. 52(8), 1135–1144 (2005). http://www.informaworld.com/smpp/content~content=a713736931.
[Crossref]

S. M. Barnett and D. T. Pegg, “Quantum theory of rotation angles,” Phys. Rev. A 41, 3427–3435 (1990). http://link.aps.org/abstract/PRA/v41/p3427.
[Crossref] [PubMed]

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vastnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP Lett. 52, 429–431 (1990).

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefrontnext term laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[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, 8185–8190 (1992). http://link.aps. org/abstract/PRA/v45/p8185.
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–269 (2004). http://www.iop.org/EJ/abstract/1464-4258/6/2/018.
[Crossref]

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. Royal Society of London, Series A 457(2013),2251–2263 (2001). http://www.journals.royalsoc.ac.uk/content/yv6lqeufbq2phg9l.
[Crossref]

M. V. Berry, “Much ado about nothing: optical distortion lines (phase singularities, zeros, and vortices),” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 1–5 (1998). http://spie. org/x648.xml?product_id=317693.
[Crossref]

Bramon, A.

G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007). http://link.aps.org/abstract/PRA/v75/e012319.
[Crossref]

Calvo, G. F.

G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007). http://link.aps.org/abstract/PRA/v75/e012319.
[Crossref]

Coerwinkel, R. P. C.

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

Courtial, J.

J. Leach, M. R. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). http://www.iop.org/EJ/abstract/1367–2630/7/1/055/.
[Crossref]

J. Courtial, “Self-imaging beams and the Guoy [sic] effect,” Opt. Commun. 151, 1–4 (1998).
[Crossref]

Dennis, M. R.

K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Illustrations of optical vortices in three dimensions,” Journal of European Optical Society - Rapid Publications 1,06008 (2006). https://www.jeos.org/index.php/jeos rp/article/view/06008.
[Crossref]

J. Leach, M. R. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). http://www.iop.org/EJ/abstract/1367–2630/7/1/055/.
[Crossref]

J. Leach, M. R. Dennis, and M. J. Padgett, “Laser beams: Knotted threads of darkness,” Nature 432(165) (2004). http://www.nature.com/nature/journal/v432/n7014/abs/432165a.html.
[Crossref] [PubMed]

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. Royal Society of London, Series A 457(2013),2251–2263 (2001). http://www.journals.royalsoc.ac.uk/content/yv6lqeufbq2phg9l.
[Crossref]

Eliel, E. R.

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, “Nonlocality of high-dimensional two-photon orbital angular momentum states,” Phys. Rev. A 72, 052114 (2005). http://link.aps.org/abstract/PRA/v72/e052114.
[Crossref]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhaus, and J. P. Woerdman, “How to observe High-Dimensional Two-Photon Entanglement with Only Two Detectors,” Physical Review Letters 92, 217901 (2004). http://link.aps.org/abstract/PRL/v92/e217901.
[Crossref] [PubMed]

Franke-Arnold, S.

J. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007). http://www.informaworld.com/smppcontent~content=a779773614.
[Crossref]

Glückstad, J.

C. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express. 13, 1749–1760 (2005). http://www.opticsexpress.org/abstract. cfm?URI=oe-13-5-1749.
[Crossref] [PubMed]

Götte, J.

J. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007). http://www.informaworld.com/smppcontent~content=a779773614.
[Crossref]

Kristensen, M.

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

Leach, J.

J. Leach, M. R. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). http://www.iop.org/EJ/abstract/1367–2630/7/1/055/.
[Crossref]

J. Leach, M. R. Dennis, and M. J. Padgett, “Laser beams: Knotted threads of darkness,” Nature 432(165) (2004). http://www.nature.com/nature/journal/v432/n7014/abs/432165a.html.
[Crossref] [PubMed]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). http://www.iop.org/EJ/abstract/1367-2630/6/1/071/.
[Crossref]

Nienhaus, G.

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhaus, and J. P. Woerdman, “How to observe High-Dimensional Two-Photon Entanglement with Only Two Detectors,” Physical Review Letters 92, 217901 (2004). http://link.aps.org/abstract/PRL/v92/e217901.
[Crossref] [PubMed]

O’Holleran, K.

K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Illustrations of optical vortices in three dimensions,” Journal of European Optical Society - Rapid Publications 1,06008 (2006). https://www.jeos.org/index.php/jeos rp/article/view/06008.
[Crossref]

Oemrawsingh, S. S. R.

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, “Nonlocality of high-dimensional two-photon orbital angular momentum states,” Phys. Rev. A 72, 052114 (2005). http://link.aps.org/abstract/PRA/v72/e052114.
[Crossref]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhaus, and J. P. Woerdman, “How to observe High-Dimensional Two-Photon Entanglement with Only Two Detectors,” Physical Review Letters 92, 217901 (2004). http://link.aps.org/abstract/PRL/v92/e217901.
[Crossref] [PubMed]

Padgett, M.

J. Leach, M. R. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). http://www.iop.org/EJ/abstract/1367–2630/7/1/055/.
[Crossref]

R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, “Momentum paradox in a vortex core,” J. Modern Opt. 52(8), 1135–1144 (2005). http://www.informaworld.com/smpp/content~content=a713736931.
[Crossref]

Padgett, M. J.

K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Illustrations of optical vortices in three dimensions,” Journal of European Optical Society - Rapid Publications 1,06008 (2006). https://www.jeos.org/index.php/jeos rp/article/view/06008.
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). http://www.iop.org/EJ/abstract/1367-2630/6/1/071/.
[Crossref]

J. Leach, M. R. Dennis, and M. J. Padgett, “Laser beams: Knotted threads of darkness,” Nature 432(165) (2004). http://www.nature.com/nature/journal/v432/n7014/abs/432165a.html.
[Crossref] [PubMed]

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

Pegg, D. T.

S. M. Barnett and D. T. Pegg, “Quantum theory of rotation angles,” Phys. Rev. A 41, 3427–3435 (1990). http://link.aps.org/abstract/PRA/v41/p3427.
[Crossref] [PubMed]

Picón, A.

G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007). http://link.aps.org/abstract/PRA/v75/e012319.
[Crossref]

Rodrigo, P. J.

C. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express. 13, 1749–1760 (2005). http://www.opticsexpress.org/abstract. cfm?URI=oe-13-5-1749.
[Crossref] [PubMed]

Soskin, M. S.

V. Y. Bazhenov, M. V. Vastnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP Lett. 52, 429–431 (1990).

Spreeuw, R. J. C.

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, 8185–8190 (1992). http://link.aps. org/abstract/PRA/v45/p8185.
[Crossref] [PubMed]

Thomson, L. C.

R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, “Momentum paradox in a vortex core,” J. Modern Opt. 52(8), 1135–1144 (2005). http://www.informaworld.com/smpp/content~content=a713736931.
[Crossref]

Vastnetsov, M. V.

V. Y. Bazhenov, M. V. Vastnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP Lett. 52, 429–431 (1990).

Woerdman, J. P.

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, “Nonlocality of high-dimensional two-photon orbital angular momentum states,” Phys. Rev. A 72, 052114 (2005). http://link.aps.org/abstract/PRA/v72/e052114.
[Crossref]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhaus, and J. P. Woerdman, “How to observe High-Dimensional Two-Photon Entanglement with Only Two Detectors,” Physical Review Letters 92, 217901 (2004). http://link.aps.org/abstract/PRL/v92/e217901.
[Crossref] [PubMed]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefrontnext term laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[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, 8185–8190 (1992). http://link.aps. org/abstract/PRA/v45/p8185.
[Crossref] [PubMed]

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). http://www.iop.org/EJ/abstract/1367-2630/6/1/071/.
[Crossref]

Zambrini, R.

J. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007). http://www.informaworld.com/smppcontent~content=a779773614.
[Crossref]

R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, “Momentum paradox in a vortex core,” J. Modern Opt. 52(8), 1135–1144 (2005). http://www.informaworld.com/smpp/content~content=a713736931.
[Crossref]

J. Mod. Opt. (1)

J. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007). http://www.informaworld.com/smppcontent~content=a779773614.
[Crossref]

J. Modern Opt. (1)

R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, “Momentum paradox in a vortex core,” J. Modern Opt. 52(8), 1135–1144 (2005). http://www.informaworld.com/smpp/content~content=a713736931.
[Crossref]

J. Opt. A (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–269 (2004). http://www.iop.org/EJ/abstract/1464-4258/6/2/018.
[Crossref]

JETP Lett. (1)

V. Y. Bazhenov, M. V. Vastnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP Lett. 52, 429–431 (1990).

Journal of European Optical Society - Rapid Publications (1)

K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Illustrations of optical vortices in three dimensions,” Journal of European Optical Society - Rapid Publications 1,06008 (2006). https://www.jeos.org/index.php/jeos rp/article/view/06008.
[Crossref]

Nature (1)

J. Leach, M. R. Dennis, and M. J. Padgett, “Laser beams: Knotted threads of darkness,” Nature 432(165) (2004). http://www.nature.com/nature/journal/v432/n7014/abs/432165a.html.
[Crossref] [PubMed]

New J. Phys. (2)

J. Leach, M. R. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). http://www.iop.org/EJ/abstract/1367–2630/7/1/055/.
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). http://www.iop.org/EJ/abstract/1367-2630/6/1/071/.
[Crossref]

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L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
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M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefrontnext term laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
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J. Courtial, “Self-imaging beams and the Guoy [sic] effect,” Opt. Commun. 151, 1–4 (1998).
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C. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express. 13, 1749–1760 (2005). http://www.opticsexpress.org/abstract. cfm?URI=oe-13-5-1749.
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Phys. Rev. A (4)

G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007). http://link.aps.org/abstract/PRA/v75/e012319.
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A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, “Nonlocality of high-dimensional two-photon orbital angular momentum states,” Phys. Rev. A 72, 052114 (2005). http://link.aps.org/abstract/PRA/v72/e052114.
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Physical Review Letters (1)

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhaus, and J. P. Woerdman, “How to observe High-Dimensional Two-Photon Entanglement with Only Two Detectors,” Physical Review Letters 92, 217901 (2004). http://link.aps.org/abstract/PRL/v92/e217901.
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M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. Royal Society of London, Series A 457(2013),2251–2263 (2001). http://www.journals.royalsoc.ac.uk/content/yv6lqeufbq2phg9l.
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M. V. Berry, “Much ado about nothing: optical distortion lines (phase singularities, zeros, and vortices),” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE3487, 1–5 (1998). http://spie. org/x648.xml?product_id=317693.
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. Fractional OAM beam with M=6.5 realised as the superposition of 20 LG modes. a) The distribution of the modulus square of coefficients cm . b) The number of total modes in the superposition, determines the distribution of the indices p for the contributing LG modes. c) The annular index p is chosen such that the sum 2p+|m|+1 takes on one of two values, giving only two different Gouy phases in the superposition at each propagation distance.

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Fig. 2.
Fig. 2. Illustration of the steps necessary, implicit in Eq. (7), to generate the hologram which produces the non-integer OAM beams according to Eq. (5). A carrier phase representing a blazed grating is added to the phase of the superposition modulo 2π. This combined phase is multiplied by an intensity mask which takes account of the correct mapping between phase depth and diffraction intensity. The result is a hologram containing the required phase and intensity profiles. The various cross-sections are plotted over a range ±3w 0.

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Fig. 3.
Fig. 3. Schematic representation of the experimental setup. To record the intensity and phase profiles an electronic shutter either blocks the reference beam or lets it trough. The mirrors M 3 and M 4 are mounted on a rack which can be moved to select the desired propagation distance. The SLM, the shutter, the movement of the track and CCD array are all automated.

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Fig. 4.
Fig. 4. Intensity and phase profiles on propagation for a superposition of 10 modes and M=6.5. a) The sequence of numerical plots for three different propagation distances at z=0, 2zR and z=4zR shows the changes in the phase and intensity on propagation from the waist plane into the far field. The various cross-sections are plotted over a range ±3w(z) for each value of z. b) shows the corresponding experimental profiles.

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Fig. 5.
Fig. 5. Theoretical intensity and phase profiles at the beam waist for different numbers of modes in the superposition. The various cross-sections are plotted over a range ±3w 0. Adding more modes leads to a higher p index for the dominant LG modes. This explains why the intensity profiles show a larger number of rings for more modes in the superposition. Between each ring of an LG modes is a π phase step which explains the higher number of rings in the phase profile. Increasing the number of modes also leads to a separation of the on-axis high charge vortex in several vortices with charge ±1.

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Fig. 6.
Fig. 6. Three dimensional view of the vortex structure for a superposition of 10 modes and M=6.5. a) shows the numerical results and b) the experimental measurements. Both vortex structures exhibit a number of topological features such as formation of a line of vortices and the existence of ‘hairpins’, connected nodal lines which cumulate in a turning point.

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Fig. 7.
Fig. 7. Intensity and phase profiles for a superposition of light beams in which all modes have the same Gouy phase. The superposition consists of 5 modes and M=6.5. a) shows the numerical results for three different propagation distances of z=0, 2zR and 4zR . The various cross-sections are plotted over a range ±3w(z)Apart from dilation the intensity structure remains invariant. In the phase profiles one can see that in the scaled variables the vortices remain at the same locations. b) shows the experimental results.

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Fig. 8.
Fig. 8. Graphs showing the modulus square of the overlap 〈M(α)|M(α′)〉 for different values of β=α-α′. The overlap is calculated on a finite set of OAM eigenstates and for different values of the total number of OAM eigenstates to show the effect a finite number of states has on the properties of the non-integer OAM states.

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Equations (13)

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∣ M ( α ) 〉 = ∑ m ′ c m ′ [ M ( α ) ] ∣ m 〉 ,
c m ′ [ M ( α ) ] = exp ( − i μ α ) i exp [ i ( M − m ′ ) θ 0 ] 2 π ( M − m ′ ) [ exp [ i ( M − m ′ ) α ] ( 1 − exp ( i μ 2 π ) ) ] .
u p m ( ρ , ϕ , z ) ∝ C mp w ( z ) ( ρ 2 w ( z ) ) ∣ m ∣ exp ( − ρ 2 w 2 ( z ) ) L p ∣ m ∣ ( 2 ρ 2 w 2 ( z ) )
× exp ( i ρ 2 w 2 ( z ) z z R ) exp ( i m φ ) exp [ − i ( 2 p + ∣ m ∣ + 1 ) tan − 1 ( z z R ) ] ,
w ( z ) = 2 z R 2 + z 2 kz R = w 0 1 + z 2 z R 2 ,
Ψ M ( α ) ( ρ , ϕ , z ) = ∑ m ′ = m min m max c m ′ [ M ( α ) ] u p m ′ ( ρ , ϕ , z ) .
p m ′ = Floor [ ( ∣ M ∣ + ( n modes 2 ) − ∣ m ′ ∣ ) 2 ] .
Φ ( x , y ) holo = [ ( Φ ( x , y ) beam + Φ ( x , Λ ) grating ) mod 2 π − π ] sinc 2 [ ( 1 − I ( x , y ) beam ) π ] + π .
∣ M + ( α ) 〉 = 1 2 ( 1 + cos ( μ π ) ) ( ∣ M ( α ) 〉 + ∣ M ( α + π ) 〉 ) .
M ¯ = 2 M π [ 1 + cos ( μ π ) ] − sin ( 2 π μ ) − 2 sin ( μ π ) 2 π [ 1 + cos ( π μ ) ] .
∣ M ( α ) 〉 = ∑ m ′ = m min m max c m ′ [ M ( α ) ] ∣ m ′ 〉 .
〈 M ( α ) ∣ M ( α ′ ) 〉 = exp ( i μ β ) 2 π ( 1 − cos ( 2 π μ ) ) ∑ m ′ = m min m max exp [ i ( M − m ′ ) β ] ( M − m ′ ) 2 ,
〈 M ( α ) ∣ M ( α ′ ) 〉 = exp ( i μ β ) 2 π ( 1 − cos ( 2 π μ ) ) ∑ n = n min n max exp [ i ( n + μ ) β ] ( n + μ ) 2 ,

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