Abstract

We show how a vortex structure manifests itself in the one-dimensional projection of a vortex field. We calculate the extent of spatial coherence and entropy of such projections. We quantify the spatial coherence and discuss the properties of the Wigner functions for the projected field.

© 2002 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 165, 336 (1974).
  2. J. M. Vaughan and D. V. Willetts, Opt. Commun. 73, 403 (1979).
  3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [Crossref] [PubMed]
  4. K. T. Gahagan and G. A. Swartzlander, Opt. Lett. 21, 827 (1996).
    [Crossref] [PubMed]
  5. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
    [Crossref] [PubMed]
  6. For a comprehensive review, see L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. 39, p. 291.
    [Crossref]
  7. J. Scheuer and M. Orenstein, Science 285, 230 (1999).
    [Crossref] [PubMed]
  8. R. M. Jenkins, J. Banerji, and A. R. Davies, J. Opt. A 3, 527 (2001).
    [Crossref]
  9. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
    [Crossref]
  10. It is also possible for vortex-carying fields to be partially coherent, as described recently by S. A. Ponomarenko, J. Opt. Soc. Am. A 18, 150 (2001).
    [Crossref]
  11. See, for example, www.photonics.com/Spectra/Tech/Nov99/techVortices.html .
  12. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).
    [Crossref]
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
    [Crossref]
  14. M. Harris, C. A. Hill, and J. M. Vaughan, Opt. Commun. 106, 161 (1994).
    [Crossref]
  15. R. Simon and N. Mukunda, J. Opt. Soc. Am. A 17, 2440 (2000).
    [Crossref]
  16. The expression for the Wigner function of the full two-dimensional field [Eq. (1)] is given in R. Simon and G. S. Agarwal, Opt. Lett. 25, 1313 (2000).
    [Crossref]
  17. G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
    [Crossref]

2002 (1)

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[Crossref]

2001 (3)

It is also possible for vortex-carying fields to be partially coherent, as described recently by S. A. Ponomarenko, J. Opt. Soc. Am. A 18, 150 (2001).
[Crossref]

R. M. Jenkins, J. Banerji, and A. R. Davies, J. Opt. A 3, 527 (2001).
[Crossref]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).
[Crossref]

2000 (2)

1999 (1)

J. Scheuer and M. Orenstein, Science 285, 230 (1999).
[Crossref] [PubMed]

1996 (1)

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

1994 (1)

M. Harris, C. A. Hill, and J. M. Vaughan, Opt. Commun. 106, 161 (1994).
[Crossref]

1993 (1)

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

1979 (1)

J. M. Vaughan and D. V. Willetts, Opt. Commun. 73, 403 (1979).

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 165, 336 (1974).

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]

For a comprehensive review, see L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. 39, p. 291.
[Crossref]

Babiker, M.

For a comprehensive review, see L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. 39, p. 291.
[Crossref]

Banerji, J.

R. M. Jenkins, J. Banerji, and A. R. Davies, J. Opt. A 3, 527 (2001).
[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]

Berry, M. V.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 165, 336 (1974).

Davies, A. R.

R. M. Jenkins, J. Banerji, and A. R. Davies, J. Opt. A 3, 527 (2001).
[Crossref]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).
[Crossref]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).
[Crossref]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

Gahagan, K. T.

Gbur, G.

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[Crossref]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).
[Crossref]

Harris, M.

M. Harris, C. A. Hill, and J. M. Vaughan, Opt. Commun. 106, 161 (1994).
[Crossref]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

Hill, C. A.

M. Harris, C. A. Hill, and J. M. Vaughan, Opt. Commun. 106, 161 (1994).
[Crossref]

Jenkins, R. M.

R. M. Jenkins, J. Banerji, and A. R. Davies, J. Opt. A 3, 527 (2001).
[Crossref]

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).
[Crossref]

Mandel, L.

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

Mukunda, N.

Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 165, 336 (1974).

Orenstein, M.

J. Scheuer and M. Orenstein, Science 285, 230 (1999).
[Crossref] [PubMed]

Padgett, M. J.

For a comprehensive review, see L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. 39, p. 291.
[Crossref]

Ponomarenko, S. A.

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).
[Crossref]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

Scheuer, J.

J. Scheuer and M. Orenstein, Science 285, 230 (1999).
[Crossref] [PubMed]

Simon, R.

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]

Swartzlander, G. A.

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]

Vaughan, J. M.

M. Harris, C. A. Hill, and J. M. Vaughan, Opt. Commun. 106, 161 (1994).
[Crossref]

J. M. Vaughan and D. V. Willetts, Opt. Commun. 73, 403 (1979).

Visser, T. D.

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[Crossref]

Willetts, D. V.

J. M. Vaughan and D. V. Willetts, Opt. Commun. 73, 403 (1979).

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.

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[Crossref]

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

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B 72, 109 (2001).
[Crossref]

J. Opt. A (1)

R. M. Jenkins, J. Banerji, and A. R. Davies, J. Opt. A 3, 527 (2001).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

M. Harris, C. A. Hill, and J. M. Vaughan, Opt. Commun. 106, 161 (1994).
[Crossref]

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

J. M. Vaughan and D. V. Willetts, Opt. Commun. 73, 403 (1979).

Opt. Lett. (2)

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. (2)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[Crossref] [PubMed]

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[Crossref]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 165, 336 (1974).

Science (1)

J. Scheuer and M. Orenstein, Science 285, 230 (1999).
[Crossref] [PubMed]

Other (3)

For a comprehensive review, see L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. 39, p. 291.
[Crossref]

See, for example, www.photonics.com/Spectra/Tech/Nov99/techVortices.html .

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

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Figures (4)

Fig. 1
Fig. 1

Plot of In,N-n as a function of n for N=20 (open circles), N=21 (filled circles), and N=22 (open squares).

Fig. 2
Fig. 2

Left, three-dimensional bar chart of μnm for various values of n and m. Also shown, right, is a bar chart of μn0 (shaded black) and μnn (shaded gray) as functions of n.

Fig. 3
Fig. 3

Contour plots of correlation function Γnmx,x for (a) n=1,m=0; (b) n=5,m=0; (c) n=m=1; (d) n=m=5.

Fig. 4
Fig. 4

Wigner function Wnmx,p plotted as a function of r=x/w2+wp/2λ21/2 for various values of n and m.

Equations (13)

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

un,mLGx,y=2πw2minn,m!n!m!-1minn,m×exp-in-mϕ×exp-r2w2r2wn-mLminn,mn-m2r2w2,
Γnmx,x=-un,m*LGx,yun,mLGx,ydy.
un,mHGx,y=ΦnxΦmy,
Φnx=2π2nwn!1/2Hn2x/w×exp-x2/w2,
-dxΦnxΦmx=δnm.
un,mLGx,y=k=0m+nikbn,m,kum+n-k,kHGx,y,
bn,m,k=n+m!k!2n+mn!m!1/21k!dkdtk1-tn1+tmt=0.
Γnmx,x=j=0m+nb2n,m,m+n-jΦjxΦjx.
Inm=-j=0m+nb2n,m,m+n-jlogb2n,m,m+n-j.
μnm=dxdxΓnmx,x2dxΓnmx,x2=j=0m+nb4n,m,m+n-jj=0m+nb2n,m,m+n-j2.
Wnmx,p=12πλ-dξ exp-ipξ/λΓnmx-ξ/2,x+ξ/2.
Wnmx,p=j=0m+nb2n,m,m+n-jWjhox,p,
Wjhox,p=-1nπλLn4x2w2+w2p24λ2×exp-2x2w2+w2p24λ2

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