Abstract

The geometric rotation of the polarization angle in a ideal cylindrical optical fiber without birefringence is obtained for arbitrary fiber paths in terms of the image of the path in the tangent vector space. The result extends a recent result of Chiao and Wu [Phys. Rev. Lett. 57, 933 (1986)] restricted to paths with parallel ends, which was derived from Berry’s phase in the adiabatic limit of quantum mechanics [Proc. R. Soc. London Ser. A 392, 45 (1984)]. The treatment given here is purely classical and uses differential geometry to extend earlier work by Ross [Opt. Quantum Electron. 16, 455 (1984)] on the uniform helix.

© 1986 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. A. Tomita, R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
    [CrossRef] [PubMed]
  3. R. Y. Chiao, Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986).
    [CrossRef] [PubMed]
  4. M. V. Berry, Proc. R. Soc. London Ser A 392, 45 (1984).
    [CrossRef]
  5. The result obtained here requires only that t̂(s) be continuous, i.e., that κ(s) be everywhere finite, as also required by the elastic and optical properties of physical fibers. For mathematical convenience, the derivation also tacitly assumes that κ(s) never vanishes and that τ(s) is continuous. A more-general fiber configuration can be arbitrarily closely approximated by one with these properties, with an arbitrarily small correction to the result for the polarization-rotation angle.

1986 (2)

A. Tomita, R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

R. Y. Chiao, Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986).
[CrossRef] [PubMed]

1984 (2)

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

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

Berry, M. V.

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

Chiao, R. Y.

A. Tomita, R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

R. Y. Chiao, Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986).
[CrossRef] [PubMed]

Ross, J. N.

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

Tomita, A.

A. Tomita, R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

Wu, Y. S.

R. Y. Chiao, Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986).
[CrossRef] [PubMed]

Opt. Quantum Electron. (1)

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

Phys. Rev. Lett. (2)

A. Tomita, R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).
[CrossRef] [PubMed]

R. Y. Chiao, Y. S. Wu, Phys. Rev. Lett. 57, 933 (1986).
[CrossRef] [PubMed]

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

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

Other (1)

The result obtained here requires only that t̂(s) be continuous, i.e., that κ(s) be everywhere finite, as also required by the elastic and optical properties of physical fibers. For mathematical convenience, the derivation also tacitly assumes that κ(s) never vanishes and that τ(s) is continuous. A more-general fiber configuration can be arbitrarily closely approximated by one with these properties, with an arbitrarily small correction to the result for the polarization-rotation angle.

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Figures (1)

Fig. 1
Fig. 1

Geometrical construction for the net rotation of polarization [relative to (0) × (1)] in terms of the motion of the tangent vector (s) as a function of distance 0 < s < 1 along the fiber. The rotation is given (modulo 2π) by the solid angle subtended by the closed curve obtained by closing (s) along an arbitrary great-circle path in the space of the tangent vector.

Equations (3)

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Δ ϕ = [ 0 1 d u b ^ · n ^ u ] s = 0 s = 1 - [ 0 1 d s b ^ · n ^ s ] u = 0 u = 1 .
Δ ϕ = 0 1 d s 0 1 d u { b ^ s · n ^ u - b ^ u · n ^ s } .
Δ ϕ = 0 1 d s 0 1 d u [ { t ^ · b ^ s } { t ^ · n ^ u } - { t ^ · b ^ u } { t ^ · n ^ s } ] = 0 1 d s 0 1 d u [ { b ^ · t ^ s } { n ^ · t ^ u } - { n ^ · t ^ s } { b ^ · t ^ u } ] = - 0 1 d s 0 1 d u { t ^ · t ^ s × t ^ u } .

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