Abstract

We study, experimentally as well as theoretically, the spatial coherence function and the Wigner distribution function for one-dimensional projections of optical vortices of different orders. The information entropy derived from the spatial coherence functions has been used to quantify the information content of the vortices and compared with those obtained for the Gaussian beam. The experimental results verify the theoretical findings of Agarwal and Banerji [Opt. Lett. 27, 800 (2002)].

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
    [CrossRef]
  2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  3. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
    [CrossRef]
  4. D. G. Grier, Nature 424, 810 (2003).
    [CrossRef] [PubMed]
  5. G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
    [CrossRef]
  6. T. Brunet, J.-L. Thomas, and R. Marchiano, Phys. Rev. Lett. 105, 034301 (2010).
    [CrossRef] [PubMed]
  7. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
    [CrossRef] [PubMed]
  8. C. Iaconis and I. A. Walmsley, Opt. Lett. 21, 1783 (1996).
    [CrossRef] [PubMed]
  9. C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).
  10. R. P. Singh, S. Roychowdhury, and V. K. Jaiswal, J. Mod. Opt. 53, 1803 (2006).
    [CrossRef]
  11. G. S. Agarwal and J. Banerji, Opt. Lett. 27, 800 (2002).
    [CrossRef]
  12. J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
    [CrossRef] [PubMed]
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  14. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
    [CrossRef]
  15. G. Indebetouw, J. Opt. 24, 3 (1993).
    [CrossRef]
  16. A. Kumar, J. Banerji, and R. P. Singh, Opt. Lett. 35, 3841 (2010).
    [CrossRef] [PubMed]

2010 (2)

T. Brunet, J.-L. Thomas, and R. Marchiano, Phys. Rev. Lett. 105, 034301 (2010).
[CrossRef] [PubMed]

A. Kumar, J. Banerji, and R. P. Singh, Opt. Lett. 35, 3841 (2010).
[CrossRef] [PubMed]

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

2006 (1)

R. P. Singh, S. Roychowdhury, and V. K. Jaiswal, J. Mod. Opt. 53, 1803 (2006).
[CrossRef]

2003 (2)

D. G. Grier, Nature 424, 810 (2003).
[CrossRef] [PubMed]

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (1)

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

2000 (1)

C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

1996 (1)

1993 (2)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

G. Indebetouw, J. Opt. 24, 3 (1993).
[CrossRef]

1992 (1)

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

Agarwal, G. S.

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

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

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

Banerji, J.

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

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

Brunet, T.

T. Brunet, J.-L. Thomas, and R. Marchiano, Phys. Rev. Lett. 105, 034301 (2010).
[CrossRef] [PubMed]

Cheng, C. C.

C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).

Curtis, J. E.

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

Grier, D. G.

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

D. G. Grier, Nature 424, 810 (2003).
[CrossRef] [PubMed]

Heier, H.

C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Iaconis, C.

Indebetouw, G.

G. Indebetouw, J. Opt. 24, 3 (1993).
[CrossRef]

Jaiswal, V. K.

R. P. Singh, S. Roychowdhury, and V. K. Jaiswal, J. Mod. Opt. 53, 1803 (2006).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Kumar, A.

Mair, A.

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

Mandel, L.

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

Marchiano, R.

T. Brunet, J.-L. Thomas, and R. Marchiano, Phys. Rev. Lett. 105, 034301 (2010).
[CrossRef] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Padgett, M. J.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

Raymer, M. G.

C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).

Roychowdhury, S.

R. P. Singh, S. Roychowdhury, and V. K. Jaiswal, J. Mod. Opt. 53, 1803 (2006).
[CrossRef]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Singh, R. P.

A. Kumar, J. Banerji, and R. P. Singh, Opt. Lett. 35, 3841 (2010).
[CrossRef] [PubMed]

R. P. Singh, S. Roychowdhury, and V. K. Jaiswal, J. Mod. Opt. 53, 1803 (2006).
[CrossRef]

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]

Thomas, J.-L.

T. Brunet, J.-L. Thomas, and R. Marchiano, Phys. Rev. Lett. 105, 034301 (2010).
[CrossRef] [PubMed]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

Vaziri, A.

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

Walmsley, I. A.

Weihs, G.

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

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

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

Wolf, E.

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

Zeilinger, A.

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

J. Mod. Opt. (2)

C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000).

R. P. Singh, S. Roychowdhury, and V. K. Jaiswal, J. Mod. Opt. 53, 1803 (2006).
[CrossRef]

J. Opt. (1)

G. Indebetouw, J. Opt. 24, 3 (1993).
[CrossRef]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Nature (2)

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

D. G. Grier, Nature 424, 810 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (3)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

T. Brunet, J.-L. Thomas, and R. Marchiano, Phys. Rev. Lett. 105, 034301 (2010).
[CrossRef] [PubMed]

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef] [PubMed]

Other (2)

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

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Experimental setup of a shearing Sagnac interferometer for obtaining the TPCFs and the WDFs for the optical vortices. (Details are given in the text.)

Fig. 2
Fig. 2

TPCF | Γ n m ( 2 ) ( x , y , x , y ) | 2 for (a) a Gaussian beam and (b) an optical vortex of order 1, for fixed x and y.

Fig. 3
Fig. 3

Plots of the TPCF | Γ n m ( 1 ) ( x , x ) | 2 for the 1D projections of beams with topological charges 0, 1, 2, and 3: (a)–(d) theoretical; (e)–(h) experimental.

Fig. 4
Fig. 4

Plots of (a) degree of coherence and (b) information entropy for 1D projections of a Gaussian beam ( l = 0 ) and the optical vortices of different orders.

Fig. 5
Fig. 5

Plots of the WDF of 1D projections of a Gaussian beam and optical vortices of orders 1, 2, and 3 with r = [ ( x / w ) 2 + ( w p / 2 λ ) 2 ] 1 / 2 : (a) theoretical; (b) experimental.

Equations (9)

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

u n , m L G ( x , y ) = 2 π w 2 min ( n , m ) ! n ! m ! ( 1 ) min ( n , m ) exp [ i ( n m ) φ ] × exp [ r 2 w 2 ] ( r 2 w ) | n m | L min ( n , m ) | n m | ( 2 r 2 w 2 ) ,
Γ n m ( 1 ) ( x , x ) = u n , m * L G ( x , y ) u n , m L G ( x , y ) d y .
u n , m L G ( x , y ) = k = 0 m + n i k b ( n , m , k ) u m + n k , k H G ( x , y ) ,
b ( n , m , k ) = [ ( n + m k ) ! k ! 2 n + m n ! m ! ] 1 / 2 1 k ! d k d t k [ ( 1 t ) n ( 1 + t ) m ] | t = 0 .
Γ n m ( 1 ) ( x , x ) = j = 0 m + n b 2 ( n , m , m + n j ) Φ j ( x ) Φ j ( x ) ,
Φ n ( x ) = ( 2 π 2 n w n ! ) 1 / 2 H n ( 2 x / w ) exp ( x 2 / w 2 ) , d x Φ n ( x ) Φ m ( x ) = δ n m .
I n m = j = 0 m + n b 2 ( n , m , m + n j ) log [ b 2 ( n , m , m + n j ) ] .
μ n m = d x d x | Γ n m ( x , x ) | 2 [ d x Γ n m ( x , x ) ] 2 = j = 0 m + n b 4 ( n , m , m + n j ) [ j = 0 m + n b 2 ( n , m , m + n j ) ] 2 .
W n m ( x , p ) = 1 2 π λ d ξ exp ( i p ξ / λ ) Γ n m ( x ξ / 2 , x + ξ / 2 ) .

Metrics