Abstract

In this work we implement a Mach–Zehnder interferometer with an image rotator in one of its arms to measure the orbital angular momentum (OAM) spectrum of a partially coherent beam. By measuring the visibility of the interference as a function of the angle of rotation, the OAM distribution can be recovered via a Fourier transform. Theoretical calculations based on the coherent mode decomposition of the cross-spectral density are in excellent agreement with the experimental data.

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
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  4. J. Serna and J. M. Movilla, Opt. Lett. 26, 405 (2001).
    [CrossRef]
  5. G. V. Bogatyryova, C. V. Felde, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, Opt. Lett. 28, 878 (2003).
    [CrossRef] [PubMed]
  6. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Phys. Rev. Lett. 92, 143905 (2004).
    [CrossRef] [PubMed]
  7. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  8. Y. D. Liu, C. Gao, M. Gao, and F. Li, Opt. Commun. 281, 1968 (2007).
  9. W. H. Carter and E. Wolf, J. Opt. Soc. Am. A 67, 785 (1977).
    [CrossRef]
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  11. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, 2007).
  13. R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
    [CrossRef] [PubMed]
  14. G. Toraldo Di Francia, J. Opt. Soc. Am. 45, 497 (1955).
    [CrossRef]

2007

Y. D. Liu, C. Gao, M. Gao, and F. Li, Opt. Commun. 281, 1968 (2007).

2006

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

2004

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

2003

2001

1999

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

1977

W. H. Carter and E. Wolf, J. Opt. Soc. Am. A 67, 785 (1977).
[CrossRef]

1955

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Agarwal, G. S.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Barnett, S. M.

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Bogatyryova, G. V.

Carter, W. H.

W. H. Carter and E. Wolf, J. Opt. Soc. Am. A 67, 785 (1977).
[CrossRef]

Felde, C. V.

Gao, C.

Y. D. Liu, C. Gao, M. Gao, and F. Li, Opt. Commun. 281, 1968 (2007).

Gao, M.

Y. D. Liu, C. Gao, M. Gao, and F. Li, Opt. Commun. 281, 1968 (2007).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, 2007).

Li, F.

Y. D. Liu, C. Gao, M. Gao, and F. Li, Opt. Commun. 281, 1968 (2007).

Liu, Y. D.

Y. D. Liu, C. Gao, M. Gao, and F. Li, Opt. Commun. 281, 1968 (2007).

Maleev, I. D.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Marathay, A. S.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

Movilla, J. M.

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

Polyanskii, P. V.

Ponomarenko, S. A.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, 2007).

Serna, J.

Soskin, M. S.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Swartzlander, G. A.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

Toraldo Di Francia, G.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Wolf, E.

G. V. Bogatyryova, C. V. Felde, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, Opt. Lett. 28, 878 (2003).
[CrossRef] [PubMed]

W. H. Carter and E. Wolf, J. Opt. Soc. Am. A 67, 785 (1977).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Zambrini, R.

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

Y. D. Liu, C. Gao, M. Gao, and F. Li, Opt. Commun. 281, 1968 (2007).

Opt. Lett.

Phys. Rev. A

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett.

R. Zambrini and S. M. Barnett, Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

Other

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, 2007).

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Figures (4)

Fig. 1
Fig. 1

Experimental setup used to create a partially coherent beam and to measure its OAM spectrum. See details in the text.

Fig. 2
Fig. 2

Measured visibility V ( θ ) for two different circular apertures. The circles correspond to a diameter d 1 = 1.2 mm ( N = 3.1 ) , and the triangles correspond to d 1 = 2.2 mm ( N = 5.7 ) . The dashed curves are the theoretical curves, calculated by substituting P l from Eq. (3) in Eq. (5).

Fig. 3
Fig. 3

Measured interference pattern for (a) θ = 0 ° and (b) θ = 180 ° . Beam splitter BS2 was on purpose misaligned in order to allow the visualization of vertical fringes. The dashed line delimits the theoretical coherence area.

Fig. 4
Fig. 4

OAM spectrum obtained for two different circular apertures (a) d 1 = 1.2 mm ( N = 3.1 ) and (b) d 1 = 2.2 mm ( N = 5.7 ) . The bars represent the experimental results, and the circles represent the theoretical predictions, according to Eqs. (3, 4).

Equations (6)

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W ( ρ 1 , ρ 2 ) = l = + p = 0 λ l , p f l , p ( ρ 1 ) f l , p * ( ρ 2 ) e i l ( ϕ 1 ϕ 2 ) 2 π ,
W ( ρ 1 , ρ 2 ) = T ( ρ 1 ) T ( ρ 2 ) J 1 ( α | ρ 1 ρ 2 | ) α | ρ 1 ρ 2 | e i π λ L ( ρ 1 2 ρ 2 2 ) ,
P l = i = 0 k j = 0 k ( 1 + | l | + 2 i ) ( 1 + | l | + 2 j ) × [ H ( α d 2 2 , 1 + | l | + 2 i , 1 + | l | + 2 j ) ] 2 ,
H ( γ , m , n ) = 2 m n γ m + n m ! n ! ( m + n ) F 3 2 [ { m + n 2 , m + n + 1 2 } ] ,
[ { 1 + m , 1 + n , 1 + m + n } , γ 2 ] ,
V ( θ ) = l = l = + P l cos ( l θ )

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