Abstract

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

© 2011 OSA

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  1. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936).
    [CrossRef]
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    [CrossRef]
  3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [CrossRef] [PubMed]
  4. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000).
    [CrossRef]
  5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
    [CrossRef] [PubMed]
  20. D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28(8), 657–659 (2003).
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    [CrossRef]
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    [CrossRef]
  23. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184(1-4), 67–71 (2000).
    [CrossRef]
  24. R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express 15(23), 15214–15227 (2007).
    [CrossRef] [PubMed]
  25. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002).
    [CrossRef] [PubMed]
  26. M. V. Berry, “Paraxial beams of spinning light,” SPIE 3487, 6–11 (1998).
    [CrossRef]
  27. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2011 (2)

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

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

2010 (2)

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

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

2009 (1)

2008 (3)

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

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

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

2007 (2)

2006 (2)

2005 (2)

2004 (2)

2003 (3)

2002 (5)

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

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

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

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

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

2001 (2)

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

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

2000 (3)

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

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

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

1998 (1)

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

1997 (1)

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

1995 (2)

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

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

1993 (1)

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

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 laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

1987 (1)

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

1936 (1)

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

Alfano, R. R.

Alieva, T.

Allen, L.

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

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

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

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

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Arlt, J.

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

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

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

Arnold, A. S.

Barnett, S. M.

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

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

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

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

Barreiro, J. T.

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

Beijersbergen, M. W.

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

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

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Berkhout, G. C. G.

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

Berry, M. V.

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

Beth, R. A.

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

Boyd, R. W.

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

Bryant, P. E.

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

Burge, R. E.

Calvo, M. L.

Caravaca-Aguirre, A. M.

Carrasco, S.

Chávez-Cerda, S.

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

Courtial, J.

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

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

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

Cristóbal, G.

Curtis, J. E.

Davis, L. W.

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

Dholakia, K.

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

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

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

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

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

Dienerowitz, M.

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

Dudley, A.

Durnin, J.

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

Eberly, J. H.

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

Ellinas, D.

Forbes, A.

Franke-Arnold, S.

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

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

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

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

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Gao, C.

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

Garcés-Chávez, V.

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

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

Gibson, G.

Girkin, J. M.

Gorshkov, V. N.

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

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

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

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Ireland, D. G.

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

Jack, B.

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

Jha, A. K.

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

Khilo, N.

Khonina, S. N.

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

Kotlyar, V. V.

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

Kwiat, P. G.

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

Laakkonen, P.

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

Lavery, M. P. J.

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

Leach, J.

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

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

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

Lembessis, V. E.

Lin, J.

Liu, Y.

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

MacDonald, M. P.

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

MacVicar, I.

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

Mair, A.

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

Malos, J. T.

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

Mazilu, M.

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

McGloin, D.

Miceli, J.

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

O’Neil, A. T.

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

Ohberg, P.

Orlov, S.

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

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

Padgett, M. J.

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

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

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

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

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

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

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

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

Pas’ko, V.

Paterson, L.

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

Petrov, D. V.

Qi, X.

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

Regelskis, K.

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

Rodrigo, J. A.

Romero, J.

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

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

Schmitz, C. H. J.

Sibbett, W.

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

Skidanov, R. V.

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

Smilgevicius, V.

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

Soifer, V. A.

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

Soskin, M. S.

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

Spatz, J. P.

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 laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Stabinis, A.

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

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

Sztul, H. I.

Tao, S. H.

Torner, L.

Torres, J. P.

Turunen, J.

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

Uhrig, K.

Van der Veen, H. E. L. O.

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

Vasilyeu, R.

Vasnetsov, M.

Vasnetsov, M. V.

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

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

Vaziri, A.

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

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

Volke-Sepulveda, K.

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

Wang, L.

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

Wei, T.-C.

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

Weihs, G.

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

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

Woerdman, J. P.

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

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Wright, A. J.

Xin, J.

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

Yao, A. M.

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

Yuan, X. C.

Yuan, X.-C.

Zambrini, R.

Zeilinger, A.

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

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

J. Nanophoton. (1)

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

J. Opt. A, Pure Appl. Opt. (1)

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

J. Opt. B Quantum Semiclassical Opt. (1)

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

Nat. Phys. (1)

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

Nature (1)

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

Opt. Commun. (7)

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

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

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

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

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

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

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

Opt. Express (9)

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

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (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(2), 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(15), 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(14), 8619–8625 (2007).
[CrossRef] [PubMed]

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

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

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

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

Opt. Lett. (3)

Phys. Rev. (1)

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

Phys. Rev. A (3)

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

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

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (6)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[CrossRef] [PubMed]

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

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

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

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

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

Science (2)

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

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

SPIE (1)

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

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Figures (6)

Fig. 1
Fig. 1

A schematic of the experimental setup for detecting the OAM density of our superposition modes as a function of r. L: Lens (f1 = 25 mm; f2 = 150 mm; f3 = 200 mm and f4 = 200 mm); M: Mirror; LCD: Liquid Crystal Display; O: Objective; CCD: CCD Camera. The objective, O2, was placed at the focus (or Fourier plane) of lens, L4.

Fig. 2
Fig. 2

(a) Annular ring programmed onto the SLM with two azimuthal phase patterns: in this example, + 3 and –3 l values; (b) observed experimental superposition (intensity) with 6 petals, and (c) the theoretically expected intensity pattern. The intensity shown in (b) is an attenuated experimental image with arbitrary false color units so as not to saturate the camera, while that in (c) shows a false color plot normalized to 1 in peak value for visual comparison.

Fig. 3
Fig. 3

The superposition field was divided radially, and an annular ring was programmed onto the second SLM in order to execute the inner product at only this radius. The phase within the annular ring was varied in the azimuthal angle for various values of l. The rest of the SLM was programmed with a checkerboard pattern so as to restrict the transmission function to the ring alone. The ten rings applied to LCD 2 have the following radii: r 1 = 600 µm, r 2 = 880 µm, r 3 = 1160 µm, r 4 = 1440 µm, r 5 = 1720 µm, r 6 = 2000 µm, r 7 = 2280 µm, r 8 = 2560 µm, r 9 = 2840 µm and r 10 = 3120 µm. The black and white images in rows 2 and 4 are gray-scale experimental images of the beam, with the attenuation set so as not to saturate the camera. Therefore the intensity units are arbitrary.

Fig. 4
Fig. 4

Calculated change in the local OAM when (a) the cone angle of the two modes differ, and (b) when the weighting of the two modes differ.

Fig. 6
Fig. 6

(a) – (c): Density plots of the three Bessel beams plotted as (r,ϕ) for the + 3 case, the –3 case, and the superposition, respectively; (d) plot of the orbital angular momentum density as a function of radial position on the field – the theory is the solid curve and the experimental data is overlaid as red bars; (e) theoretical prediction of the coefficient |al |2 (or the on axis intensity of the inner product) as a function of the radius of the match filter. Rings 1 to 10 denote those given in the first and third rows of Fig. 3; (f) corresponding experimental data.

Fig. 5
Fig. 5

The measured OAM spectrum (for l values of –4 to + 4) as a function of the radial ring on the beam (equivalent to the radial position on the beam), from 1 to 10, given by: r 1 = 600 µm, r 2 = 880 µm, r 3 = 1160 µm, r 4 = 1440 µm, r 5 = 1720 µm, r 6 = 2000 µm, r 7 = 2280 µm, r 8 = 2560 µm, r 9 = 2840 µm and r 10 = 3120 µm. The height of each bar represents the measured coefficients |al |2.

Equations (20)

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A ( x , y , z ) = u ( x , y , z ) exp ( i ( k z ω t ) ) x
B ( x , y , z ) = × A ( x , y , z ) E ( x , y , z ) = i c 2 ω × B ( x , y , z ) ,
B ( x , y ) = i k exp ( i ( k z ω t ) ) ( u ( x , y , z ) y + i k u ( x , y , z ) y z ) E ( x , y ) = i ω exp ( i ( k z ω t ) ) ( u ( x , y , z ) x + i k u ( x , y , z ) x z ) .
S = S r e a l = ε 0 c 2 E r e a l × B r e a l ,
E r e a l = 1 2 ( E + E * ) B r e a l = 1 2 ( B + B * ) ,
S = ε 0 c 2 E r e a l × B r e a l = ε 0 c 2 4 ( E × B * + E * × B ) = ε 0 ω 4 ( i ( u u * u * u ) + 2 k | u | 2 z ) ,
u ( r , φ , z ) = u 0 ( r , z ) exp ( i ψ ( r , z ) ) ( exp ( i l φ ) + α 0 exp ( i l φ ) ) ,
S r = ε 0 ω c 2 2 ψ r [ u 0 2 ( 1 + α l 2 + 2 α l cos ( 2 l φ ) ) ] S φ = l ε 0 ω c 2 u 0 2 2 r ( α l 2 1 ) S z = ε 0 ω c 2 k u 0 2 2 ( 1 + α l 2 + 2 α l cos ( 2 l φ ) ) .
L z = 1 c 2 ( r × S ) z .
u ( r , φ , z ) = A 0 [ J l ( q 1 r ) exp ( i Δ k z ) exp ( i l φ ) + α 0 J l ( q 2 r ) exp ( i Δ k z ) exp ( i l φ ) ] ,
S r = A 0 2 α 0 ε 0 ω c 2 2 [ ( q 2 J l 1 ( q 2 r ) J l ( q 1 r ) + q 1 J l ( q 2 r ) J l + 1 ( q 1 r ) ) sin ( 2 [ l φ + Δ k z ] ) ] S φ = A 0 2 l ε 0 ω c 2 2 r ( J l 2 ( q 1 r ) α 0 2 J l 2 ( q 2 r ) ) S z = A 0 2 ε 0 ω c 2 k 2 ( J l 2 ( q 1 r ) + α 0 2 J l 2 ( q 2 r ) + 2 α 0 J l ( q 1 r ) J l ( q 2 r ) cos ( 2 [ l φ + Δ k z ] ) ) .
L z ( r ) = l ε 0 ω A 0 2 2 ( J l 2 ( q 1 r ) α 0 2 J l 2 ( q 2 r ) ) .
u ( r , z , φ ) = ( 1 1 + ( z / z r ) 2 ) exp [ ( i k 2 z ( 1 + ( z / z r ) 2 ) 1 w 0 2 ( 1 + ( z / z r ) 2 ) ) ( r 2 + γ 2 z 2 ) ] × ( J l ( q 1 r 1 + i ( z / z r ) ) exp ( i k z 1 z ) exp ( i l φ ) + α 0 J l ( q 2 r 1 + i ( z / z r ) ) exp ( i k z 2 z ) exp ( i l φ ) ) χ ( z ) ,
L z ( r , z ) = l ε 0 ω 2 [ 1 + ( z / z r ) 2 ] × exp ( 2 r 2 + γ 2 z 2 w 0 2 ( 1 + ( z / z r ) 2 ) ) [ | J l ( q 1 r 1 + i ( z / z r ) ) | 2 α 0 2 | J l ( q 2 r 1 + i ( z / z r ) ) | 2 ] .
u ( r , φ , z ) = 1 2 π l a l ( r , z ) exp ( i l φ ) ,
a l ( r , z ) = 1 2 π 0 2 π u ( r , φ , z ) exp ( i l φ ) d φ .
P l ( z ) = 0 | a l ( r , z ) | 2 r d r l 0 | a l ( r , z ) | 2 r d r .
L z ( r ) = l ε 0 ω 4 π ( a l 2 ( r ) a l 2 ( r ) ) .
α 0 = r i n g 1 exp ( 2 ( r / w ) 2 ) r d r r i n g 2 exp ( 2 ( r / w ) 2 ) r d r .
L z = l ω η c ( λ f S r i n g ) 2 ( I 3 ( 0 ) I 3 ( 0 ) ) ,

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