Abstract

We measure the local skew angle of the Poynting vector within a helically-phased, exp(ilφ), beam using a Shack Hartmann wavefront sensor. It is the skew angle of the Poynting vector with respect to the beam axis that gives rise to the orbital angular momentum of a light beam. We confirm that this skew angle is l/kr, corresponding to an orbital angular momentum of per photon. Measurement of orbital angular momentum in this way is an alternative to interferometric techniques giving a non-ambiguous result to both the magnitude and sign of l from a single measurement, without any restriction on the optical bandwidth.

© 2006 Optical Society of America

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  1. L. Allen, M. J. Padgett and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999), M. Padgett, J. Courtial, and L. Allen, "Light's Orbital Angular Momentum," Physics Today,  57, 35-40 (2004).
  2. M. Harris, C. A. Hill and J. M. Vaughan, "Optical helices and spiral interference fringes," Opt. Commun. 106, 161-166 (1994), M. Padgett, J. Arlt, N. Simpson and L. Allen, "An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes," Am. J. Phys. 64,77-82 (1996).
    [CrossRef]
  3. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon," Phys. Rev. Lett. 92, 013601 (2004).
    [CrossRef] [PubMed]
  4. e.g. "Adaptive Optics for Astronomical Telescopes." John W. Hardy, Oxford University Press (1998).
  5. D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
    [CrossRef] [PubMed]
  6. 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, 053601 (2002).
    [CrossRef] [PubMed]
  7. M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995), 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]
  8. J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).
  9. H. I. Sztul and R. R. Alfano, "Double-slit interference with Laguerre-Gaussian beams," Opt. Lett. 31, 999-1001 (2006).
    [CrossRef] [PubMed]
  10. HoloEye Photonics AG, www.holoeye.com/spatial_light_modulators-technology.html
  11. V. Yu. Bazhenov, M. V. Vasnetsov and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52,429-431 (1990).
  12. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian modes," Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
  13. I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Y. Bekshaev, "Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam," Opt. Lett. 28, 1185-1187 (2003).
    [CrossRef] [PubMed]
  14. J. Leach, M. Dennis, J. Courtial and M. Padgett, "Knotted threads of darkness," Nature 432, 165 (2004).
    [CrossRef] [PubMed]
  15. J. Courtial, K. Dholakia, L. Allen and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997).
    [CrossRef]
  16. 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).
    [CrossRef] [PubMed]

2006

2005

2004

J. Leach, M. Dennis, J. Courtial and M. Padgett, "Knotted threads of darkness," Nature 432, 165 (2004).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

L. Allen, M. J. Padgett and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999), M. Padgett, J. Courtial, and L. Allen, "Light's Orbital Angular Momentum," Physics Today,  57, 35-40 (2004).

2003

J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Y. Bekshaev, "Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam," Opt. Lett. 28, 1185-1187 (2003).
[CrossRef] [PubMed]

2002

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, 053601 (2002).
[CrossRef] [PubMed]

2000

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995), 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]

1997

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

1996

M. Harris, C. A. Hill and J. M. Vaughan, "Optical helices and spiral interference fringes," Opt. Commun. 106, 161-166 (1994), M. Padgett, J. Arlt, N. Simpson and L. Allen, "An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes," Am. J. Phys. 64,77-82 (1996).
[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 modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

1990

V. Yu. Bazhenov, M. V. Vasnetsov and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52,429-431 (1990).

Alfano, R. R.

Allen, L.

L. Allen, M. J. Padgett and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999), M. Padgett, J. Courtial, and L. Allen, "Light's Orbital Angular Momentum," Physics Today,  57, 35-40 (2004).

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, 053601 (2002).
[CrossRef] [PubMed]

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995), 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]

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

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

Alonzo, C.

Arlt, J.

J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).

Babiker, M.

L. Allen, M. J. Padgett and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999), M. Padgett, J. Courtial, and L. Allen, "Light's Orbital Angular Momentum," Physics Today,  57, 35-40 (2004).

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Basistiy, I. V.

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. V. Vasnetsov and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52,429-431 (1990).

Beijersbergen, M. W.

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

Bekshaev, A. Y.

Courtial, J.

J. Leach, M. Dennis, J. Courtial and M. Padgett, "Knotted threads of darkness," Nature 432, 165 (2004).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

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

Dennis, M.

J. Leach, M. Dennis, J. Courtial and M. Padgett, "Knotted threads of darkness," Nature 432, 165 (2004).
[CrossRef] [PubMed]

Dholakia, K.

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

Franke-Arnold, S.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Glückstad, J.

Grier, D. G.

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

Harris, M.

M. Harris, C. A. Hill and J. M. Vaughan, "Optical helices and spiral interference fringes," Opt. Commun. 106, 161-166 (1994), M. Padgett, J. Arlt, N. Simpson and L. Allen, "An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes," Am. J. Phys. 64,77-82 (1996).
[CrossRef]

Hill, C. A.

M. Harris, C. A. Hill and J. M. Vaughan, "Optical helices and spiral interference fringes," Opt. Commun. 106, 161-166 (1994), M. Padgett, J. Arlt, N. Simpson and L. Allen, "An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes," Am. J. Phys. 64,77-82 (1996).
[CrossRef]

Leach, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

J. Leach, M. Dennis, J. Courtial and M. Padgett, "Knotted threads of darkness," Nature 432, 165 (2004).
[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, 053601 (2002).
[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, 053601 (2002).
[CrossRef] [PubMed]

Padgett, M.

J. Leach, M. Dennis, J. Courtial and M. Padgett, "Knotted threads of darkness," Nature 432, 165 (2004).
[CrossRef] [PubMed]

Padgett, M. J.

L. Allen, M. J. Padgett and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999), M. Padgett, J. Courtial, and L. Allen, "Light's Orbital Angular Momentum," Physics Today,  57, 35-40 (2004).

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon," Phys. Rev. Lett. 92, 013601 (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, 053601 (2002).
[CrossRef] [PubMed]

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995), 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]

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

Rodrigo, P. J.

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Slyusar, V. V.

Soskin, M. S.

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, 8185-8189 (1992).
[CrossRef] [PubMed]

Sztul, H. I.

Vasnetsov, M. V.

Vaughan, J. M.

M. Harris, C. A. Hill and J. M. Vaughan, "Optical helices and spiral interference fringes," Opt. Commun. 106, 161-166 (1994), M. Padgett, J. Arlt, N. Simpson and L. Allen, "An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes," Am. J. Phys. 64,77-82 (1996).
[CrossRef]

Woerdman, J. P.

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

Am. J. Phys.

M. Harris, C. A. Hill and J. M. Vaughan, "Optical helices and spiral interference fringes," Opt. Commun. 106, 161-166 (1994), M. Padgett, J. Arlt, N. Simpson and L. Allen, "An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes," Am. J. Phys. 64,77-82 (1996).
[CrossRef]

J. Mod. Opt.

J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).

JETP Lett.

V. Yu. Bazhenov, M. V. Vasnetsov and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52,429-431 (1990).

Nature

J. Leach, M. Dennis, J. Courtial and M. Padgett, "Knotted threads of darkness," Nature 432, 165 (2004).
[CrossRef] [PubMed]

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

Opt. Commun.

M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995), 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]

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

Opt. Express

Opt. Lett.

Phys. Rev. A

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

Phys. Rev. Lett.

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, 053601 (2002).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Physics Today

L. Allen, M. J. Padgett and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999), M. Padgett, J. Courtial, and L. Allen, "Light's Orbital Angular Momentum," Physics Today,  57, 35-40 (2004).

Other

HoloEye Photonics AG, www.holoeye.com/spatial_light_modulators-technology.html

e.g. "Adaptive Optics for Astronomical Telescopes." John W. Hardy, Oxford University Press (1998).

Supplementary Material (1)

» Media 1: MOV (201 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

(a) Experimental configuration measuring the local inclination of the Poynting vector of a beam produced with a spatial light modulator using a Shack Hartman wave front sensor. (b) Sample of a forked grating used to create a LG beam. The example shown was used to create an l=5 beam.

Fig. 2.
Fig. 2.

Intensity and vector plots showing the measured inclination of the Poynting vector for different values of l. Both positive and negative values of l are shown. [Media 1]

Fig. 3.
Fig. 3.

The relationship between the measured skew angle of the Poynting vector, the beam radius and azimuthal mode index, l, is in close agreement with the predicted value, γ=l/kr.

Fig. 4.
Fig. 4.

The mean value of γkr calculated from the data in Figure 3. The error bars show the error in the mean value.

Equations (2)

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

u pl LG = C pl LG w ( z ) ( r 2 w ( z ) ) l exp [ r 2 w 2 ( z ) ] L p l
exp ( i k r 2 z 2 ( z 2 + z R 2 ) ) exp ( i l ϕ ) exp ( i ( 2 p + l + 1 ) tan 1 z z R ) ,

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