Abstract

We describe direct observation of a geometrical phase in a noncyclic case as the rotation of the plane of polarization of a linearly polarized beam of light. The beam travels down uniformly wound half-turn single-mode optical fibers with various pitch angles.

© 1991 Optical Society of America

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References

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  1. J. N. Ross, Opt. Quantum Electron. 16, 455 (1984).
    [CrossRef]
  2. M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
    [CrossRef]
  3. J. H. Hannay, J. Phys. A. 18, 221 (1985); M. V. Berry, J. H. Hannay, J. Phys. A 21, L325 (1988).
    [CrossRef]
  4. R. Y. Chiao, Y.-S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita, R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
    [CrossRef] [PubMed]
  5. F. D. M. Haldane, Opt. Lett. 11, 730 (1986); S. G. Lipson, Opt. Lett. 15, 154 (1990).
    [CrossRef] [PubMed]
  6. J. Samuel, R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988); T. F. Jordan, Phys. Rev. A 38, 1590 (1988).
    [CrossRef] [PubMed]
  7. T. Bitter, D. Dubbers, Phys. Rev. Lett. 59, 251 (1987).
    [CrossRef] [PubMed]
  8. H. Weinfurter, G. Badurek, Phys. Rev. Lett. 64, 1381 (1990).
    [CrossRef]
  9. The fiber is represented by l = r(cos φ)x̂ + r(sin φ)ŷ + (cot θ)rφẑ in Cartesian coordinates. If the polarization vector at some point s0 makes an angle θ(s0) with the local normal unit vector n̂ (s0) (s is the distance along the fiber represented by units in which the fiber has unit length), then parallel transport fixes θ(s) elsewhere through ∂θ(s)/∂s = −τ(s), where τ(s) is the local torsion5 and the local normal axis may be used as a reference direction to measure the change of polarization directions. The local normal unit axis û(s) is written as n̂(s) = −[(cos φ)x̂ + (sin φ)ŷ].
  10. A. M. Smith, Appl. Opt. 19, 2606 (1980).
    [CrossRef] [PubMed]

1990

H. Weinfurter, G. Badurek, Phys. Rev. Lett. 64, 1381 (1990).
[CrossRef]

1988

J. Samuel, R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988); T. F. Jordan, Phys. Rev. A 38, 1590 (1988).
[CrossRef] [PubMed]

1987

T. Bitter, D. Dubbers, Phys. Rev. Lett. 59, 251 (1987).
[CrossRef] [PubMed]

1986

F. D. M. Haldane, Opt. Lett. 11, 730 (1986); S. G. Lipson, Opt. Lett. 15, 154 (1990).
[CrossRef] [PubMed]

R. Y. Chiao, Y.-S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita, R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

1985

J. H. Hannay, J. Phys. A. 18, 221 (1985); M. V. Berry, J. H. Hannay, J. Phys. A 21, L325 (1988).
[CrossRef]

1984

J. N. Ross, Opt. Quantum Electron. 16, 455 (1984).
[CrossRef]

M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
[CrossRef]

1980

Badurek, G.

H. Weinfurter, G. Badurek, Phys. Rev. Lett. 64, 1381 (1990).
[CrossRef]

Berry, M. V.

M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
[CrossRef]

Bhandari, R.

J. Samuel, R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988); T. F. Jordan, Phys. Rev. A 38, 1590 (1988).
[CrossRef] [PubMed]

Bitter, T.

T. Bitter, D. Dubbers, Phys. Rev. Lett. 59, 251 (1987).
[CrossRef] [PubMed]

Chiao, R. Y.

R. Y. Chiao, Y.-S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita, R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

Dubbers, D.

T. Bitter, D. Dubbers, Phys. Rev. Lett. 59, 251 (1987).
[CrossRef] [PubMed]

Haldane, F. D. M.

Hannay, J. H.

J. H. Hannay, J. Phys. A. 18, 221 (1985); M. V. Berry, J. H. Hannay, J. Phys. A 21, L325 (1988).
[CrossRef]

Ross, J. N.

J. N. Ross, Opt. Quantum Electron. 16, 455 (1984).
[CrossRef]

Samuel, J.

J. Samuel, R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988); T. F. Jordan, Phys. Rev. A 38, 1590 (1988).
[CrossRef] [PubMed]

Smith, A. M.

Weinfurter, H.

H. Weinfurter, G. Badurek, Phys. Rev. Lett. 64, 1381 (1990).
[CrossRef]

Wu, Y.-S.

R. Y. Chiao, Y.-S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita, R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

Appl. Opt.

J. Phys. A.

J. H. Hannay, J. Phys. A. 18, 221 (1985); M. V. Berry, J. H. Hannay, J. Phys. A 21, L325 (1988).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

J. N. Ross, Opt. Quantum Electron. 16, 455 (1984).
[CrossRef]

Phys. Rev. Lett.

R. Y. Chiao, Y.-S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita, R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

J. Samuel, R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988); T. F. Jordan, Phys. Rev. A 38, 1590 (1988).
[CrossRef] [PubMed]

T. Bitter, D. Dubbers, Phys. Rev. Lett. 59, 251 (1987).
[CrossRef] [PubMed]

H. Weinfurter, G. Badurek, Phys. Rev. Lett. 64, 1381 (1990).
[CrossRef]

Proc. R. Soc. London Ser. A

M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
[CrossRef]

Other

The fiber is represented by l = r(cos φ)x̂ + r(sin φ)ŷ + (cot θ)rφẑ in Cartesian coordinates. If the polarization vector at some point s0 makes an angle θ(s0) with the local normal unit vector n̂ (s0) (s is the distance along the fiber represented by units in which the fiber has unit length), then parallel transport fixes θ(s) elsewhere through ∂θ(s)/∂s = −τ(s), where τ(s) is the local torsion5 and the local normal axis may be used as a reference direction to measure the change of polarization directions. The local normal unit axis û(s) is written as n̂(s) = −[(cos φ)x̂ + (sin φ)ŷ].

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Figures (3)

Fig. 1
Fig. 1

Experimental setup. The optical fiber forms a half-turn uniform helix, in which the pitch angle θ is constant along the optical fiber. The radius r of the cylinder is 11 cm.

Fig. 2
Fig. 2

The k sphere; θ is the pitch angle. Great circles are geodesics in a two-sphere, so arc DH is the geodesic that connects end points A and B of the evolution curve CQ1 + CQ2 The solid angle Ω(CQ1 + CQ2 + DH) gives a geometrical phase. DQ1 and DQ2 are geodesics that join SQ and HQ, respectively.

Fig. 3
Fig. 3

Change of polarization of the beam versus geometrical phase for the photon with helicity σ = +1. The pitch angle θ is described in Fig. 1. The dotted line is the theory.

Equations (2)

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γ ( C 1 + C 2 + D H ) = π σ ( 1 - cos θ ) ,
γ ( C Q 1 + D Q 1 ) = π - π 2 cos θ - 2 arccos ( cos θ 1 + cos 2 θ ) ,

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