Abstract

We demonstrate the Fourier relationship between angular position and angular momentum for a light mode. In particular we measure the distribution of orbital angular momentum states of light that has passed through an aperture and verify that the orbital angular momentum distribution is given by the complex Fourier-transform of the aperture function. We use spatial light modulators, configured as diffractive optical components, to define the initial orbital angular momentum state of the beam, set the defining aperture, and measure the angular momentum spread of the resulting beam. These measurements clearly confirm the Fourier relationship between angular momentum and angular position, even at light intensities corresponding to the single photon level.

© 2006 Optical Society of America

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  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 45,8185-8189 (1992).
    [CrossRef] [PubMed]
  2. 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]
  3. A. E. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
    [CrossRef] [PubMed]
  4. E. Merzbacher, Quantum Mechanics (John Wiley & Sons, Brisbane, 1998).
  5. L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," in Progress in Optics XXXIX, E. Wolf, ed. (Elsevier Science, New York,1999), pp. 291-372.
  6. S. Franke-Arnold, S. M. Barnett, M. J. Padgett and L. Allen, "Two-photon entanglement of orbital angular momentum states," Phys. Rev. A 65, 033823 (2002).
    [CrossRef]
  7. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, Bristol, 2003)
  8. V.Y. Bazhenov, M.V. Vasnetsov, and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52, 429-431 (1990).
  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. Exp. 12, 5448-5456 (2004).
    [CrossRef]
  10. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial and M. J. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
    [CrossRef]
  11. S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman "Experimental demonstration of fractional orbital angular momentum entanglement of two photons," Phys. Rev. Lett. 95, 240501 (2005)
    [CrossRef] [PubMed]
  12. S. M. Barnett and D.T. Pegg, "Quantum theory of rotation angles," Phys. Rev. A 41, 3427-3435 (1990).
    [CrossRef] [PubMed]
  13. 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]

2005

S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman "Experimental demonstration of fractional orbital angular momentum entanglement of two photons," Phys. Rev. Lett. 95, 240501 (2005)
[CrossRef] [PubMed]

2004

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. Exp. 12, 5448-5456 (2004).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial and M. J. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
[CrossRef]

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]

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]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett and L. Allen, "Two-photon entanglement of orbital angular momentum states," Phys. Rev. A 65, 033823 (2002).
[CrossRef]

2001

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

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] [PubMed]

1990

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

S. M. Barnett and D.T. Pegg, "Quantum theory of rotation angles," Phys. Rev. A 41, 3427-3435 (1990).
[CrossRef] [PubMed]

’t Hooft, G.W.

S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman "Experimental demonstration of fractional orbital angular momentum entanglement of two photons," Phys. Rev. Lett. 95, 240501 (2005)
[CrossRef] [PubMed]

Aiello, A.

S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman "Experimental demonstration of fractional orbital angular momentum entanglement of two photons," Phys. Rev. Lett. 95, 240501 (2005)
[CrossRef] [PubMed]

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, 053601 (2002)
[CrossRef] [PubMed]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett and L. Allen, "Two-photon entanglement of orbital angular momentum states," Phys. Rev. A 65, 033823 (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] [PubMed]

Barnett, S. M.

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. Exp. 12, 5448-5456 (2004).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial and M. J. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
[CrossRef]

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]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett and L. Allen, "Two-photon entanglement of orbital angular momentum states," Phys. Rev. A 65, 033823 (2002).
[CrossRef]

S. M. Barnett and D.T. Pegg, "Quantum theory of rotation angles," Phys. Rev. A 41, 3427-3435 (1990).
[CrossRef] [PubMed]

Bazhenov, V.Y.

V.Y. 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 laser modes." Phys. Rev. A 45,8185-8189 (1992).
[CrossRef] [PubMed]

Courtial, J.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial and M. J. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
[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. Exp. 12, 5448-5456 (2004).
[CrossRef]

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]

Eliel, E.R.

S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman "Experimental demonstration of fractional orbital angular momentum entanglement of two photons," Phys. Rev. Lett. 95, 240501 (2005)
[CrossRef] [PubMed]

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]

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. Exp. 12, 5448-5456 (2004).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial and M. J. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett and L. Allen, "Two-photon entanglement of orbital angular momentum states," Phys. Rev. A 65, 033823 (2002).
[CrossRef]

Gibson, G.

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. Exp. 12, 5448-5456 (2004).
[CrossRef]

Leach, J.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial and M. J. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
[CrossRef]

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]

Ma, X.

S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman "Experimental demonstration of fractional orbital angular momentum entanglement of two photons," Phys. Rev. Lett. 95, 240501 (2005)
[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]

Mair, A. E.

A. E. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[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]

Oemrawsingh, S.S.R.

S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman "Experimental demonstration of fractional orbital angular momentum entanglement of two photons," Phys. Rev. Lett. 95, 240501 (2005)
[CrossRef] [PubMed]

Padgett, M. J.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial and M. J. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
[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. Exp. 12, 5448-5456 (2004).
[CrossRef]

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]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett and L. Allen, "Two-photon entanglement of orbital angular momentum states," Phys. Rev. A 65, 033823 (2002).
[CrossRef]

Pas’ko, V.

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. Exp. 12, 5448-5456 (2004).
[CrossRef]

Pegg, D.T.

S. M. Barnett and D.T. Pegg, "Quantum theory of rotation angles," Phys. Rev. A 41, 3427-3435 (1990).
[CrossRef] [PubMed]

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]

Soskin, M. S.

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

Spreeuw, R. J.C.

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

Vasnetsov, M.

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. Exp. 12, 5448-5456 (2004).
[CrossRef]

Vasnetsov, M.V.

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

Vaziri, A.

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

Voigt, D.

S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman "Experimental demonstration of fractional orbital angular momentum entanglement of two photons," Phys. Rev. Lett. 95, 240501 (2005)
[CrossRef] [PubMed]

Weihs, G.

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

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 laser modes." Phys. Rev. A 45,8185-8189 (1992).
[CrossRef] [PubMed]

Woerdman, J.P.

S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman "Experimental demonstration of fractional orbital angular momentum entanglement of two photons," Phys. Rev. Lett. 95, 240501 (2005)
[CrossRef] [PubMed]

Yao, E.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial and M. J. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
[CrossRef]

Zeilinger, A.

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

JETP Lett.

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

Nature

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

New J. Phys.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial and M. J. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103 (2004).
[CrossRef]

Opt. Exp.

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. Exp. 12, 5448-5456 (2004).
[CrossRef]

Phys. Rev. A

S. Franke-Arnold, S. M. Barnett, M. J. Padgett and L. Allen, "Two-photon entanglement of orbital angular momentum states," Phys. Rev. A 65, 033823 (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] [PubMed]

S. M. Barnett and D.T. Pegg, "Quantum theory of rotation angles," Phys. Rev. A 41, 3427-3435 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett.

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]

S.S.R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E.R. Eliel, G.W. ’t Hooft, and J.P. Woerdman "Experimental demonstration of fractional orbital angular momentum entanglement of two photons," Phys. Rev. Lett. 95, 240501 (2005)
[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]

Other

E. Merzbacher, Quantum Mechanics (John Wiley & Sons, Brisbane, 1998).

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," in Progress in Optics XXXIX, E. Wolf, ed. (Elsevier Science, New York,1999), pp. 291-372.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, Bristol, 2003)

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Figures (4)

Fig. 1.
Fig. 1.

Spatial light modulators are used to create specific OAM states (SLM1), aperture them and measure the resulting OAM distribution (SLM2).

Fig. 2.
Fig. 2.

The measured (bar) and predicted envelope (line) distribution of OAM sidebands generated from an incident =0 (left) and =2 (right) beam after transmission through a hard-edge aperture of angular width π/4.

Fig. 3.
Fig. 3.

Experimental result for two equi-spaced segment apertures of angular width 2π/9. Panels A, B and C show the observed angular momentum spectra for δΦ=0, π/2 and π, respectively, with the predicted envelope function (solid line) and experimentally measured data (bars). Even sidebands are plotted in red and odd sidebands in cyan. The main panel shows the sums of coefficients of even (oe-14-20-9071-i001) and odd (oe-14-20-9071-i002) OAM sidebands as a function of phase difference, δΦ, between the segments. Solid curves show the predicted variation.

Fig. 4.
Fig. 4.

Experimental result for four equi-spaced apertures of angular width 2π/9. Sums of coefficients of all 4N (oe-14-20-9071-i003), 4N+1 (oe-14-20-9071-i004), 4N+2 (oe-14-20-9071-i005) and 4N+3 (oe-14-20-9071-i006), where N is an integer, angular momentum sidebands as a function of phase difference between the two angular apertures. Solid curves show the predicted variation.

Equations (11)

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

ψ = 1 2 π π + π d ϕ Ψ ( ϕ ) exp ( i ϕ ) ,
Ψ ( ϕ ) = 1 2 π = + ψ exp ( i ϕ ) .
Ψ Transmitted ( ϕ ) = Ψ Incident ( ϕ ) × Ψ Mask ( ϕ ) ,
Ψ Transmitted ( ϕ ) = exp ( i ϕ ) × Ψ Mask ( ϕ ) .
Ψ Transmitted ( ϕ ) = n = + A n exp ( i ( + n ) ϕ ) ,
ψ δ = A n = δ .
Ψ Mask ( ϕ ) = Θ 2 π n = + sinc ( n Θ 2 ) exp ( i n ϕ ) ,
ψ δ = A n = δ = Θ 2 π sinc ( δ Θ 2 ) .
Ψ Mask ( m ) ( ϕ ) = q = 1 m Ψ Mask ( ϕ + q 2 π m ) ,
ψ δ ( m ) = A δ ( m ) = A δ q = 1 m exp ( i q m 2 π δ ) = { m A δ for δ = N m 0 otherwise ,
ψ δ ( m , α ) = A δ q = 1 m exp ( i q m 2 π ( δ + α ) ) = { m A δ for δ = m N α A δ exp ( i 2 π ( δ + α ) ) 1 1 exp ( i 2 π ( δ + α ) m ) otherwise .

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