Abstract

The polarization state of a light beam is related to its spin angular momentum and can be represented on the Poincaré sphere. We propose a sphere for light beams in analogous orbital angular momentum states. Using the Poincaré-sphere equivalent, we interpret the rotational frequency shift for light beams with orbital angular momentum [Phys.  Rev.  Lett.   80, 3217 (1998)] as a dynamically evolving geometric phase.

© 1999 Optical Society of America

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References

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  1. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
    [CrossRef] [PubMed]
  2. M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
    [CrossRef] [PubMed]
  3. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, Opt. Lett. 22, 52 (1997).
    [CrossRef] [PubMed]
  4. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  5. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  7. S. J. van Enk, Opt. Commun. 102, 59 (1993).
    [CrossRef]
  8. R. Simon, H. J. Kimble, and E. C. G. Sundarshan, Phys. Rev. Lett. 61, 19 (1988).
    [CrossRef] [PubMed]
  9. B. A. Garetz and S. Arnold, Opt. Commun. 31, 1 (1979).
    [CrossRef]
  10. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
    [CrossRef]

1998 (1)

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

1997 (1)

1996 (1)

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

1995 (1)

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

1993 (2)

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

S. J. van Enk, Opt. Commun. 102, 59 (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]

1988 (1)

R. Simon, H. J. Kimble, and E. C. G. Sundarshan, Phys. Rev. Lett. 61, 19 (1988).
[CrossRef] [PubMed]

1979 (1)

B. A. Garetz and S. Arnold, Opt. Commun. 31, 1 (1979).
[CrossRef]

Allen, L.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

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]

Arnold, S.

B. A. Garetz and S. Arnold, Opt. Commun. 31, 1 (1979).
[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]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Courtial, J.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Dholakia, K.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

Enger, J.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

Friese, M. E. J.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

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

Garetz, B. A.

B. A. Garetz and S. Arnold, Opt. Commun. 31, 1 (1979).
[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.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

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

Kimble, H. J.

R. Simon, H. J. Kimble, and E. C. G. Sundarshan, Phys. Rev. Lett. 61, 19 (1988).
[CrossRef] [PubMed]

Padgett, M. J.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

Robertson, D. A.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Rubinsztein-Dunlop, H.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

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

Simon, R.

R. Simon, H. J. Kimble, and E. C. G. Sundarshan, Phys. Rev. Lett. 61, 19 (1988).
[CrossRef] [PubMed]

Simpson, N. B.

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]

Sundarshan, E. C. G.

R. Simon, H. J. Kimble, and E. C. G. Sundarshan, Phys. Rev. Lett. 61, 19 (1988).
[CrossRef] [PubMed]

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]

van Enk, S. J.

S. J. van Enk, Opt. Commun. 102, 59 (1993).
[CrossRef]

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.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Opt. Commun. (3)

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

S. J. van Enk, Opt. Commun. 102, 59 (1993).
[CrossRef]

B. A. Garetz and S. Arnold, Opt. Commun. 31, 1 (1979).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

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

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

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

R. Simon, H. J. Kimble, and E. C. G. Sundarshan, Phys. Rev. Lett. 61, 19 (1988).
[CrossRef] [PubMed]

Other (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980).

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Figures (4)

Fig. 1
Fig. 1

(a) Poincaré sphere and (b) sphere of first-order modes. Some commonly encountered polarization states and modes are indicated.

Fig. 2
Fig. 2

Intensity cross section of (a) LG0±1 modes and (b) a HG1,0 mode. The modes LG0-1 and LG0+1 differ only in the handedness of their azimuthal mode structure.

Fig. 3
Fig. 3

Decompositions of (a) various polarization states and (b) first-order modes in terms of orthogonal linear polarization states and orthogonal Hermite–Gaussian m=1, n=0 modes, respectively. The black and gray shading indicates the relative phase of the components.

Fig. 4
Fig. 4

Generation of evolving geometric phases for light beams containing (a) spin and (b) orbital angular momentum. (c) Path taken on the Poincaré sphere or the sphere of first-order modes.

Equations (4)

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p1=I0°-I90°I0°+I90°,p2=I45°-I135°I45°+I135°,p3=Iright-IleftIright+Ileft,
p12+p22+p32=1.
o1=IHG1,00°-IHG1,090°IHG1,00°+IHG1,090°,o2=IHG1,045°-IHG1,0135°IHG1,045°+IHG1,0135°,o3=ILG01-ILG0-1ILG01+ILG0-1,
o12+o22+o32=1.

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