Abstract

In this work we derive expressions for the orbital angular momentum (OAM) density of light, for both symmetric and nonsymmetric optical fields, that allow a direct comparison between theory and experiment. We present a simple method for measuring the OAM density in optical fields and test the approach on superimposed nondiffracting higher-order Bessel beams. The measurement technique makes use of a single spatial light modulator and a Fourier transforming lens to measure the OAM spectrum of the optical field. Quantitative values for the OAM density as a function of the radial position in the optical field are obtained for both symmetric and nonsymmetric superpositions, illustrating good agreement with the theoretical prediction.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

2011

M. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014(2011).
[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, 48–51 (2011).
[CrossRef]

A. Gatto, M. Tacca, P. Martelli, P. Boffi, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

2010

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

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

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, 153601 (2010).
[CrossRef]

C. Lopez-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. 35, 1215–1217 (2010).
[CrossRef]

2009

2008

2006

2004

2003

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

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, 2285–2287 (2003).
[CrossRef]

2002

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

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, 257901 (2002).
[CrossRef]

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

2001

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

2000

Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 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, 301–308 (2000).
[CrossRef]

1995

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, 826–829 (1995).
[CrossRef]

1993

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, 123–132 (1993).
[CrossRef]

1992

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

1987

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

1964

A. V. Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

Alfano, R. R.

Allen, L.

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, 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, 8185–8189 (1992).
[CrossRef]

Arlt,

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

Arlt, J.

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

Barnett, S. M.

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

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, 5448–5456 (2004).
[CrossRef]

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, 257901 (2002).
[CrossRef]

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, 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, 153601 (2010).
[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, 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, 8185–8189 (1992).
[CrossRef]

Belyi, V.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

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, 153601 (2010).
[CrossRef]

Boffi, P.

A. Gatto, M. Tacca, P. Martelli, P. Boffi, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

Boyd, R. W.

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

Burge, R. E.

Chavez-Cerda, S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

Chen, G.

Chen, Z.

Courtial, J.

M. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014(2011).
[CrossRef]

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, 153601 (2010).
[CrossRef]

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, 5448–5456 (2004).
[CrossRef]

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, 257901 (2002).
[CrossRef]

Curtis, J. E.

Dholakia, K.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

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

Dudley, A.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

M. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014(2011).
[CrossRef]

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

I. Litvin, A. Dudley, and A. Forbes, “Poynting vector and orbital angular momentum density of superpositions of Bessel beams,” Opt. Express 19, 16760–16771 (2011).
[CrossRef]

Dultz, W.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

Durnin, J.

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

Eberly, J. H.

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

Fonseca, E. J. S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Forbes, A.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

M. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014(2011).
[CrossRef]

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

I. Litvin, A. Dudley, and A. Forbes, “Poynting vector and orbital angular momentum density of superpositions of Bessel beams,” Opt. Express 19, 16760–16771 (2011).
[CrossRef]

Franke-Arnold, S.

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

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, 5448–5456 (2004).
[CrossRef]

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, 257901 (2002).
[CrossRef]

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, 826–829 (1995).
[CrossRef]

Gan, X.

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, 48–51 (2011).
[CrossRef]

Garces-Chavez, V.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Gatto, A.

A. Gatto, M. Tacca, P. Martelli, P. Boffi, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

Gibson, G.

Hautakorpi, M.

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, 826–829 (1995).
[CrossRef]

Heckenberg, N. R.

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, 826–829 (1995).
[CrossRef]

Helmerson, K.

Hickmann, J. M.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Ireland, D.

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

Jack, B.

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

Jha, A. K.

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

Kaivola, M.

Khilo, N.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

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

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, 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, 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, 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, 301–308 (2000).
[CrossRef]

Lavery, M.

M. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014(2011).
[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, 153601 (2010).
[CrossRef]

Leach, J.

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

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, 257901 (2002).
[CrossRef]

Lin, J.

Lindberg, J.

Litvin, I.

Liu, S.

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, 48–51 (2011).
[CrossRef]

Lopez-Mariscal, C.

Lugt, A. V.

A. V. Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

Mair, A.

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

Martelli, P.

A. Gatto, M. Tacca, P. Martelli, P. Boffi, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

Martinelli, M.

A. Gatto, M. Tacca, P. Martelli, P. Boffi, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

McGloin, D.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

Miceli, J.

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

Padgett, M.

M. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014(2011).
[CrossRef]

Padgett, M. 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, 153601 (2010).
[CrossRef]

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

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, 5448–5456 (2004).
[CrossRef]

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

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, 257901 (2002).
[CrossRef]

Pas’ko, V.

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, 48–51 (2011).
[CrossRef]

Romero, J.

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

Ropot, P.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

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, 826–829 (1995).
[CrossRef]

Schmitz, C. H. J.

Schmitzer, H.

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

Setälä, T.

Sibbett, W.

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

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, 301–308 (2000).
[CrossRef]

Soares, W. C.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[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, 301–308 (2000).
[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, 8185–8189 (1992).
[CrossRef]

Sztul, H. I.

Tacca, M.

A. Gatto, M. Tacca, P. Martelli, P. Boffi, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

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, 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, 123–132 (1993).
[CrossRef]

Vasilyeu, R.

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

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

Vasnetsov, M.

Vasnetsov, M. V.

Vaziri, A.

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

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

Volke-Sepulveda, K.

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (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, 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, 282–286 (2008).
[CrossRef]

Weihs, G.

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

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

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, 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, 8185–8189 (1992).
[CrossRef]

Wong, D. W. K.

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, 48–51 (2011).
[CrossRef]

Yao, A. M.

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

Yuan, X. C.

Zeilinger, A.

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

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

Zhang, P.

Zhao, J.

Zheng, Y.

Appl. Opt.

IEEE Trans. Inf. Theory

A. V. Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

J. Opt.

A. Gatto, M. Tacca, P. Martelli, P. Boffi, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt.

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Phys.

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

Nature

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

New J. Phys.

M. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014(2011).
[CrossRef]

Opt. Commun.

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, 48–51 (2011).
[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, 301–308 (2000).
[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, 123–132 (1993).
[CrossRef]

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

A. Dudley, R. Vasilyeu, V. Belyi, N. Khilo, P. Ropot, and A. Forbes, “Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator,” Opt. Commun. 285, 5–12 (2012).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[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, 8185–8189 (1992).
[CrossRef]

Phys. Rev. Lett.

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, 826–829 (1995).
[CrossRef]

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

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

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, 153601 (2010).
[CrossRef]

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, 257901 (2002).
[CrossRef]

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

Science

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

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Figures (7)

Fig. 1.
Fig. 1.

(a) Schematic of the concept for generating the optical field and decomposing its OAM spectrum. (b) The Gaussian beam used to illuminate (c) the digital ring slit hologram for the construction of (d) the optical field, mathematically defined by u ( r , θ , z ) . (e) The hologram, having a transmission function of t ( r , θ ) , together with the lens L 2 , performs the decomposition, which produces (f) the inner product at plane P , mathematically defined by u P ( 0 , ϕ , z ) . SLM, spatial light modulator; O, objective—used to magnify the optical field (d).

Fig. 2.
Fig. 2.

Schematic of the experimental setup for measuring the OAM density of symmetric and nonsymmetric superpositions of Bessel beams as a function of the radial position, R . L, lens ( f 1 = 25 mm , f 2 = 150 mm , f 3 = 200 mm , and f 4 = 200 mm ); M, mirror; LCD, liquid-crystal display; O, objective; PM, pop-up mirror; CCD, CCD camera. The objective, O 2 , was placed at the focus (or Fourier plane) of lens L 4 . The corresponding optical fields or holograms are represented at the appropriate planes.

Fig. 3.
Fig. 3.

Digital ring slits [first column—(a), (d), (g)] and the corresponding experimentally produced fields in the Fourier plane [second column—(b), (e), (h)] accompanied by theoretically calculated fields [third column—(c), (f), (i)]. The ten white dots in (c) denote the radial positions of each of the ten ring slits used in the match filters.

Fig. 4.
Fig. 4.

(a) Density plot of the OAM density for the field given in Fig. 3(b). Red denotes negative OAM, and blue denotes positive. Light to dark blue denotes an increase in positive OAM, and light to dark red denotes an increase in negative OAM. (b), (c), (e), The theoretical (blue curve) and experimentally measured (red points) OAM density for the corresponding fields in Figs 3(b), 3(e), and 3(h), respectively. (d), (f), Plots of the OAM density for (c) and (e), respectively, given as insets.

Fig. 5.
Fig. 5.

(a) Digital hologram used to generate (b)  the experimental field; (c) the theoretical field. A magnification of the ring slit is given as an inset in (a). (d) The theoretical (blue curve) and experimentally measured (red points) OAM density. (e) A plot of the OAM density. Red denotes negative OAM, and blue denotes positive. Light to dark blue denotes an increase in positive OAM, and light to dark red denotes an increase in negative OAM.

Fig. 6.
Fig. 6.

(a) Digital hologram used to generate (b) the experimental field; (c) the theoretical field. A magnification of the ring slit is given as an inset in (a). (d) The theoretical (blue curve) and experimentally measured (red points) OAM density. (e) A plot of the OAM density. Red denotes negative OAM, and blue denotes positive. Light to dark blue denotes an increase in positive OAM, and light to dark red denotes an increase in negative OAM.

Fig. 7.
Fig. 7.

(a) Digital hologram used to generate (b) the experimental field; (c) the theoretical field. A magnification of the ring slit is given as an inset in (a). (d) The theoretical (blue curve) and experimentally measured (red points) OAM density. (e) A plot of the OAM density. Red denotes negative OAM, and blue denotes positive. Light to dark blue denotes an increase in positive OAM, and light to dark red denotes an increase in negative OAM.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

u ( r , θ , z ) = A 0 ( J l ( q 1 r ) exp ( i Δ k z ) exp ( i l θ ) + α 0 J l ( q 2 r ) exp ( i Δ k z ) exp ( i l θ ) ) ,
A 0 = ring 1 exp ( 2 ( r / ω ) 2 ) r d r ,
α 0 = ring 2 exp ( 2 ( r / ω ) 2 ) r d r / ring 1 exp ( 2 ( r / ω ) 2 ) r d r ,
S = ε 0 ω c 2 4 ( i ( u u * u * u ) + 2 k | u | 2 z ^ ) ,
L z = 1 c 2 ( r × S ) z ,
L z THEORY ( r ) = l ε 0 ω A 0 2 2 ( J l 2 ( q 1 r ) + α 0 2 J l 2 ( q 2 r ) ) ,
a n ( r , z ) = 1 2 π 0 2 π u ( r , θ , z ) t ( r , θ ) d θ .
u P ( ρ , ϕ , z ) = exp ( i 2 k f ) i λ f 0 0 2 π t ( r , θ ) u ( r , θ , z ) exp ( i k f r ρ cos ( θ ϕ ) ) r d r d θ ,
u l p ( ρ = 0 , z ) = exp ( i 2 k f ) i λ f 2 π A 0 R 1 R 2 J l ( q 1 r ) exp ( i Δ k z ) r d r ,
u l P ( ρ = 0 , z ) = exp ( i 2 k f ) i λ f 2 π A 0 α 0 R 1 R 2 J l ( q 2 r ) exp ( i Δ k z ) r d r .
u l p ( ρ = 0 , z , r = R ) = exp ( i 2 k f ) i λ f 2 π A 0 2 Δ R · R J l ( q 1 R ) exp ( i Δ k z ) ,
u l P ( ρ = 0 , z , r = R ) = exp ( i 2 k f ) i λ f 2 π A 0 α 0 2 Δ R · R J l ( q 2 R ) exp ( i Δ k z ) .
I l ( 0 ) ε 0 c ( f λ 4 π Δ R · R A 0 ) 2 = J l 2 ( q 1 R ) ,
I l ( 0 ) ε 0 c ( f λ 4 π Δ R · R A 0 ) 2 = α 0 2 J l 2 ( q 2 R ) .
L z EXP ( R ) = l ω 2 c η ( λ f S Ring ) 2 ( I l ( 0 ) I l ( 0 ) ) ,
u ( r , θ , z ) = A 0 l = N N α l J l ( q l r ) exp ( i Δ k l z ) exp ( i l θ ) + α l J l ( q l r ) exp ( i Δ k l z ) exp ( i l θ ) ,
L z THEORY ( r ) = ε 0 ω A 0 2 2 l = N N l α l 2 J l 2 ( q l r ) l α l 2 J l 2 ( q l r ) ,
L z EXP ( R ) = ω 2 c η ( λ f S Ring ) 2 l = N N l ( I l ( 0 ) I l ( 0 ) ) .
u ( r , θ , z ) = A 0 ( J l ( q 1 r ) exp ( i Δ k z ) exp ( i l θ ) + α 0 J m ( q 2 r ) exp ( i Δ k z ) exp ( i m θ ) ) ,
L z THEORY ( r , ϕ , z ) = ε 0 ω A 0 2 2 ( l J l 2 ( q 1 r ) + m α 0 2 J m 2 ( q 2 r ) + ( l + m ) α 0 cos ( ( l m ) ϕ + 2 Δ k z ) J l ( q 1 r ) J m ( q 2 r ) ) .
I l ( 0 ) ε 0 c ( f λ 4 π Δ R · R A 0 ) 2 = J l 2 ( q 1 R ) ,
I m ( 0 ) ε 0 c ( f λ 4 π Δ R · R A 0 ) 2 = α 0 2 J m 2 ( q 2 R ) .
L z EXP ( R , ϕ , z ) = ω 2 c η ( λ f S Ring ) 2 ( l I l ( 0 ) + m I m ( 0 ) + ( l + m ) cos ( ( l m ) ϕ + 2 Δ k z ) I l ( 0 ) I m ( 0 ) ) .
L z EXP ( R ) ¯ = ω 2 c η ( λ f S Ring ) 2 ( l I l ( 0 ) + m I m ( 0 ) ) ,
u ( r , θ , z ) = A 0 l = N N m = N N α l J l ( q l r ) exp ( i Δ k l z ) exp ( i l θ ) + α m J m ( q m r ) exp ( i Δ k m z ) exp ( i m θ ) .
S ϕ = ε 0 ω c 2 i 4 ( u u * u * u ) ,
u + u * = A 0 ( J l exp ( i Δ k z ) exp ( i l θ ) + α 0 J l exp ( i Δ k z ) exp ( i l θ ) ) + A 0 ( J l exp ( i Δ k z ) exp ( i l θ ) + α 0 J l exp ( i Δ k z ) exp ( i l θ ) ) u + u * = ( A 0 J l + A 0 α 0 J l ) 2 cos ( Δ k z + l θ ) u * = 2 A 0 ( J l + α 0 J l ) cos ( Δ k z + l θ ) X u ,
S ϕ u u * u * u u ( X u ) ( X u ) u u X u u X u + u u u X X u ,

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