Abstract

We explore the spatial frequency property of the far-field diffraction intensity pattern of an optical vortex after passing through an annular aperture in detail. The result reveals that the spatial spectrum consists of alternate bright and dark rings and, in particular, the number of the bright rings is nicely identical to the absolute value of the topological charge. Based on this property, we present and demonstrate a simple technique to characterize the topological charge of an optical vortex through its diffraction intensity pattern after an annular aperture.

© 2009 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  2. A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
    [CrossRef] [PubMed]
  3. Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).
    [CrossRef]
  4. A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, Phys. Rev. A 72, 052114 (2005).
    [CrossRef]
  5. Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, Phys. Rev. A 78, 053810 (2008).
    [CrossRef]
  6. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
    [CrossRef] [PubMed]
  7. J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
    [CrossRef]
  8. K. Ladavac and D. G. Grier, Opt. Express 12, 1144 (2004).
    [CrossRef] [PubMed]
  9. J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, Opt. Express 14, 25 (2006).
  10. C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
    [CrossRef]
  11. M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
    [CrossRef] [PubMed]
  12. M. Padgett, J. Arlt, and N. Simpson, Am. J. Phys. 64, 77 (1996).
    [CrossRef]
  13. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
    [CrossRef] [PubMed]
  14. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
    [CrossRef] [PubMed]
  15. H. I. Sztul and R. R. Alfano, Opt. Lett. 31, 7 (2006).
  16. Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
    [CrossRef]
  17. G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
    [CrossRef] [PubMed]
  18. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
    [CrossRef] [PubMed]

2009 (2)

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
[CrossRef]

2008 (2)

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, Phys. Rev. A 78, 053810 (2008).
[CrossRef]

2006 (2)

H. I. Sztul and R. R. Alfano, Opt. Lett. 31, 7 (2006).

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, Opt. Express 14, 25 (2006).

2005 (1)

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, Phys. Rev. A 72, 052114 (2005).
[CrossRef]

2004 (4)

Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).
[CrossRef]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

K. Ladavac and D. G. Grier, Opt. Express 12, 1144 (2004).
[CrossRef] [PubMed]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

2003 (1)

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

2002 (2)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

1996 (1)

M. Padgett, J. Arlt, and N. Simpson, Am. J. Phys. 64, 77 (1996).
[CrossRef]

1994 (1)

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

1992 (1)

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

Aiello, A.

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, Phys. Rev. A 72, 052114 (2005).
[CrossRef]

Alfano, R. R.

H. I. Sztul and R. R. Alfano, Opt. Lett. 31, 7 (2006).

Allen, L.

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

Arlt, J.

M. Padgett, J. Arlt, and N. Simpson, Am. J. Phys. 64, 77 (1996).
[CrossRef]

Barnett, S.

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Beijersbergen, M. W.

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

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

Berkhout, G. C. G.

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

Bouchal, Z.

Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).
[CrossRef]

Celechovsky, R.

Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).
[CrossRef]

Chen, Q. F.

Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, Phys. Rev. A 78, 053810 (2008).
[CrossRef]

Courtial, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Eliel, E. R.

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, Phys. Rev. A 72, 052114 (2005).
[CrossRef]

Franke-Arnold, S.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Gibson, G.

Grier, D. G.

K. Ladavac and D. G. Grier, Opt. Express 12, 1144 (2004).
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Guo, C. -S.

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

Guo, G. C.

Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, Phys. Rev. A 78, 053810 (2008).
[CrossRef]

Harris, M.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Hill, C. A.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Jennewein, T.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

Keen, S.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, Opt. Express 14, 25 (2006).

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Ladavac, K.

Leach, J.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, Opt. Express 14, 25 (2006).

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Lin, Q.

Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
[CrossRef]

Love, G. D.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, Opt. Express 14, 25 (2006).

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Oemrawsingh, S. S. R.

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, Phys. Rev. A 72, 052114 (2005).
[CrossRef]

Padgett, M.

Padgett, M. J.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, Opt. Express 14, 25 (2006).

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Pan, J. W.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

Pas'ko, V.

Saunter, C.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, Opt. Express 14, 25 (2006).

Shi, B. S.

Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, Phys. Rev. A 78, 053810 (2008).
[CrossRef]

Simpson, N.

M. Padgett, J. Arlt, and N. Simpson, Am. J. Phys. 64, 77 (1996).
[CrossRef]

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

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]

Sztul, H. I.

H. I. Sztul and R. R. Alfano, Opt. Lett. 31, 7 (2006).

Tapster, P. R.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Vasnetsov, M.

Vaughan, J. M.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Vaziri, A.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Wang, Z.

Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
[CrossRef]

Wei, G. -X.

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

Weihs, G.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Woerdman, J. P.

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, Phys. Rev. A 72, 052114 (2005).
[CrossRef]

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

Yue, S. -J.

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

Zeilinger, A.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Zhang, Y. S.

Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, Phys. Rev. A 78, 053810 (2008).
[CrossRef]

Zhang, Z.

Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
[CrossRef]

Am. J. Phys. (1)

M. Padgett, J. Arlt, and N. Simpson, Am. J. Phys. 64, 77 (1996).
[CrossRef]

Appl. Phys. Lett. (1)

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

J. Opt. A (1)

Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

New J. Phys. (1)

Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).
[CrossRef]

Opt. Commun. (1)

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

H. I. Sztul and R. R. Alfano, Opt. Lett. 31, 7 (2006).

Phys. Rev. A (4)

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, Phys. Rev. A 72, 052114 (2005).
[CrossRef]

Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, Phys. Rev. A 78, 053810 (2008).
[CrossRef]

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

Phys. Rev. Lett. (4)

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Schematic of the experimental setup for measuring the TC of an optical vortex based on an annulus aperture.

Fig. 2
Fig. 2

Two-dimensional intensity distribution of the spatial spectra expressed by Eq. (4) when the TC values of the input optical vortex are taken as (a) TC = 2 and (b) TC = 5 .

Fig. 3
Fig. 3

Examples of the far-field diffraction intensity patterns recorded in our experiments when the TC values of the input optical vortex are taken as (a) TC = 2 and (b) TC = 5 . (c) and (d) are the corresponding spatial spectra of the recorded intensity patterns shown in (a) and (b).

Equations (5)

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

o ( r , θ ) = [ circ ( r R b ) circ ( r R a ) ] exp ( j l θ ) ,
I ( ρ ) = I 0 | F { [ circ ( r R b ) circ ( r R a ) ] exp ( j l θ ) } | 2 = 4 π 2 I 0 | R a R b r J l ( 2 π r ρ ) d r | 2 ,
g ( r ) = F 1 { I ( ρ ) } = 8 π 3 A 2 I 0 0 + ρ J 0 ( 2 π r ρ ) | R a R b r J l ( 2 π r ρ ) d r | 2 d ρ .
g ( r ) = 4 π 2 I 0 R Δ R F 1 { | J l ( 2 π R ρ ) | 2 } = C [ δ ( r R ) exp ( i l θ ) ] [ δ ( r R ) exp ( i l θ ) ] = C   circ ( r / 2 R ) cos [ 2 l cos 1 ( r / 2 R ) ] ,
r m = 2 R   sin ( m π / 2 l ) ,     m = 1 , 2 , 3 , , | l | .

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