Abstract

We relate Berry’s topological phase to the polarization rotation of linearly polarized light in helicoidal single-mode ideal fibers where the pitch length and coil radius are allowed to change adiabatically. First we present an alternative derivation for this phase using the Serret–Frenet coordinate system and show that this phase can be derived and interpreted in terms of both solid and planar angles. The results obtained are then applied to various helicoidal fiber structures, and from this we show that the total change in the polarization rotation angle can be tailored through a judicious choice of the fiber geometry. Finally, we propose that certain helicoidal fiber configurations can be used as fiber sensors.

© 1998 Optical Society of America

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    [CrossRef]
  2. A. Tomita, R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fibre,” Phys. Rev. Lett. 57, 937–940 (1986).
    [CrossRef] [PubMed]
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  8. B. Markovski, V. I. Vinitsky, Topological Phases in Quantum Theory (World Scientific, Singapore, 1989).
  9. A. Shapere, F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989).
  10. S. Pancharatnam, “Generalized theory of interference and its application,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).
  11. M. P. Varnham, R. D. Birch, D. N. Payne, “Helical-core circularly-birefringent fibers,” in Proceedings of the International Conference on Integrated Optics and Optical Fibre Communications–European Conference on Optical Communications (Instituto Internazionale delle Communicazione, Genova, Italy, 1985), pp. 135–138.
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  13. M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature (London) 326, 277–278 (1987).
    [CrossRef]
  14. M. V. Berry, “Classical adiabatic angles and quantal adiabatic phase,” J. Phys. A 18, 15–27 (1985).
    [CrossRef]
  15. J. H. Hannay, “Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian,” J. Phys. A 18, 221–230 (1985).
    [CrossRef]
  16. R. Y. Chiao, Y-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
    [CrossRef] [PubMed]
  17. V. S. Liberman, B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium,” Phys. Rev. E 49, 2389–2396 (1994).
    [CrossRef]
  18. A. Yu. Savchenko and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium: three-dimensional case,” Phys. Rev. E 50, 2287–2292 (1994).
    [CrossRef]
  19. V. S. Liberman, B. Ya. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
    [CrossRef] [PubMed]
  20. F. D. M. Haldane, “Path dependence of the geometric rotation of polarization in optical fibers,” Opt. Lett. 11, 730–732 (1986).
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  21. L. H. Ryder, “The optical Berry phase and the Gauss-Bonnet theorem,” Eur. J. Phys. 12, 15–18 (1991).
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    [CrossRef]
  24. A. Altintas, J. D. Love, “Effective cut-offs for modes on helical fibres,” Opt. Quantum Electron. 22, 213–226 (1990).
    [CrossRef]
  25. R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
    [CrossRef] [PubMed]
  26. R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, H. Nathel, “Observation of a topological phase by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
    [CrossRef] [PubMed]
  27. P. Hariharan, M. Roy, “A geometric phase interferometer,” J. Mod. Opt. 39, 1811–1815 (1992).
    [CrossRef]
  28. E. Frins, W. Dultz, “Direct observation of Berry’s topological phase by using an optical fibre ring interferometer,” Opt. Commun. 136, 354–356 (1997).
    [CrossRef]
  29. P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
    [CrossRef]
  30. J. Liñares, M. C. Nistal, D. Baldomin, “Beam modes in graded-index media and topological phases,” Appl. Opt. 33, 4293–4299 (1994).
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  31. D. Subbarao, “Topological phase in Gaussian beam optics,” Opt. Lett. 16, 223–225 (1995).
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    [CrossRef]
  33. G. Davis, A. Mazzolini, “A new biomedical sensor utilizing macrobending losses in optical fibers,” Proc. Aust. Conf. Opt. Fibre Technol. 21, 129–132 (1996).
  34. S. K. Yao, C. K. Asawa, “Microbending fiber optic sensing,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. SPIE412, 9–13 (1983).
    [CrossRef]
  35. N. Lagakos, J. H. Cole, J. A. Bucaro, “Microbend fiber-optic sensor,” Appl. Opt. 26, 2171–2180 (1987).
    [CrossRef] [PubMed]
  36. T. Abe, Y. Mitsunaga, H. Koga, “A strain sensor using twisted optical fibers,” J. Lightwave Technol. 7, 525–529 (1989).
    [CrossRef]
  37. T. Yoshino, K. Inoue, Y. Kobayashi, “Spiral fibre microbend sensors,” IEE Proc. Optoelectron. 144, 145–150 (1997).
    [CrossRef]
  38. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
    [CrossRef]
  39. I. Bassett, “Design principle for a circularly birefringent optical fiber,” Opt. Lett. 13, 844–846 (1988).
    [CrossRef] [PubMed]

1997 (2)

E. Frins, W. Dultz, “Direct observation of Berry’s topological phase by using an optical fibre ring interferometer,” Opt. Commun. 136, 354–356 (1997).
[CrossRef]

T. Yoshino, K. Inoue, Y. Kobayashi, “Spiral fibre microbend sensors,” IEE Proc. Optoelectron. 144, 145–150 (1997).
[CrossRef]

1996 (1)

G. Davis, A. Mazzolini, “A new biomedical sensor utilizing macrobending losses in optical fibers,” Proc. Aust. Conf. Opt. Fibre Technol. 21, 129–132 (1996).

1995 (2)

G. Chen, Q. Wang, “Local fields in single-mode helical fibres,” Opt. Quantum Electron. 27, 1069–1074 (1995).
[CrossRef]

D. Subbarao, “Topological phase in Gaussian beam optics,” Opt. Lett. 16, 223–225 (1995).

1994 (4)

J. Liñares, M. C. Nistal, D. Baldomin, “Beam modes in graded-index media and topological phases,” Appl. Opt. 33, 4293–4299 (1994).
[CrossRef] [PubMed]

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

V. S. Liberman, B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium,” Phys. Rev. E 49, 2389–2396 (1994).
[CrossRef]

A. Yu. Savchenko and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium: three-dimensional case,” Phys. Rev. E 50, 2287–2292 (1994).
[CrossRef]

1992 (2)

V. S. Liberman, B. Ya. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[CrossRef] [PubMed]

P. Hariharan, M. Roy, “A geometric phase interferometer,” J. Mod. Opt. 39, 1811–1815 (1992).
[CrossRef]

1991 (1)

L. H. Ryder, “The optical Berry phase and the Gauss-Bonnet theorem,” Eur. J. Phys. 12, 15–18 (1991).
[CrossRef]

1990 (1)

A. Altintas, J. D. Love, “Effective cut-offs for modes on helical fibres,” Opt. Quantum Electron. 22, 213–226 (1990).
[CrossRef]

1989 (1)

T. Abe, Y. Mitsunaga, H. Koga, “A strain sensor using twisted optical fibers,” J. Lightwave Technol. 7, 525–529 (1989).
[CrossRef]

1988 (3)

I. Bassett, “Design principle for a circularly birefringent optical fiber,” Opt. Lett. 13, 844–846 (1988).
[CrossRef] [PubMed]

R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, H. Nathel, “Observation of a topological phase by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

1987 (2)

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature (London) 326, 277–278 (1987).
[CrossRef]

N. Lagakos, J. H. Cole, J. A. Bucaro, “Microbend fiber-optic sensor,” Appl. Opt. 26, 2171–2180 (1987).
[CrossRef] [PubMed]

1986 (3)

F. D. M. Haldane, “Path dependence of the geometric rotation of polarization in optical fibers,” Opt. Lett. 11, 730–732 (1986).
[CrossRef] [PubMed]

R. Y. Chiao, Y-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

A. Tomita, R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fibre,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

1985 (3)

X. S. Fang, Z. Q. Lin, “Field in single-mode helically-wound optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-33, 1150–1154 (1985).
[CrossRef]

M. V. Berry, “Classical adiabatic angles and quantal adiabatic phase,” J. Phys. A 18, 15–27 (1985).
[CrossRef]

J. H. Hannay, “Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian,” J. Phys. A 18, 221–230 (1985).
[CrossRef]

1984 (2)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[CrossRef]

J. N. Ross, “The rotation of the polarization in low birefringence monomode optical fibres due to geometric effects,” Opt. Quantum Electron. 16, 455–461 (1984).
[CrossRef]

1982 (1)

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

1980 (1)

1979 (1)

1976 (1)

1970 (1)

C. H. Tang, “An orthogonal coordinate system for curved pipes,” IEEE Trans. Microwave Theory Tech. MTT-18, 69 (1970).
[CrossRef]

1956 (1)

S. Pancharatnam, “Generalized theory of interference and its application,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Abe, T.

T. Abe, Y. Mitsunaga, H. Koga, “A strain sensor using twisted optical fibers,” J. Lightwave Technol. 7, 525–529 (1989).
[CrossRef]

Altintas, A.

A. Altintas, J. D. Love, “Effective cut-offs for modes on helical fibres,” Opt. Quantum Electron. 22, 213–226 (1990).
[CrossRef]

Antaramian, A.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, H. Nathel, “Observation of a topological phase by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

Asawa, C. K.

S. K. Yao, C. K. Asawa, “Microbending fiber optic sensing,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. SPIE412, 9–13 (1983).
[CrossRef]

Baldomin, D.

Bassett, I.

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987a).
[CrossRef]

M. V. Berry, “Interpreting the anholonomy of coiled light,” Nature (London) 326, 277–278 (1987).
[CrossRef]

M. V. Berry, “Classical adiabatic angles and quantal adiabatic phase,” J. Phys. A 18, 15–27 (1985).
[CrossRef]

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[CrossRef]

Bhandari, R.

R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

Birch, R. D.

M. P. Varnham, R. D. Birch, D. N. Payne, “Helical-core circularly-birefringent fibers,” in Proceedings of the International Conference on Integrated Optics and Optical Fibre Communications–European Conference on Optical Communications (Instituto Internazionale delle Communicazione, Genova, Italy, 1985), pp. 135–138.

Bucaro, J. A.

N. Lagakos, J. H. Cole, J. A. Bucaro, “Microbend fiber-optic sensor,” Appl. Opt. 26, 2171–2180 (1987).
[CrossRef] [PubMed]

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Chang, D. C.

L. Lewin, D. C. Chang, E. F. Kuester, Electromagnetic Waves and Curved Structures (Peregrinus, London, 1977).

Chen, G.

G. Chen, Q. Wang, “Local fields in single-mode helical fibres,” Opt. Quantum Electron. 27, 1069–1074 (1995).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, H. Nathel, “Observation of a topological phase by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

R. Y. Chiao, Y-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

A. Tomita, R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fibre,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

Cole, J. H.

N. Lagakos, J. H. Cole, J. A. Bucaro, “Microbend fiber-optic sensor,” Appl. Opt. 26, 2171–2180 (1987).
[CrossRef] [PubMed]

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Dandridge, A.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Davis, G.

G. Davis, A. Mazzolini, “A new biomedical sensor utilizing macrobending losses in optical fibers,” Proc. Aust. Conf. Opt. Fibre Technol. 21, 129–132 (1996).

Dultz, W.

E. Frins, W. Dultz, “Direct observation of Berry’s topological phase by using an optical fibre ring interferometer,” Opt. Commun. 136, 354–356 (1997).
[CrossRef]

Fang, X. S.

X. S. Fang, Z. Q. Lin, “Field in single-mode helically-wound optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-33, 1150–1154 (1985).
[CrossRef]

Frins, E.

E. Frins, W. Dultz, “Direct observation of Berry’s topological phase by using an optical fibre ring interferometer,” Opt. Commun. 136, 354–356 (1997).
[CrossRef]

Ganga, K. M.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, H. Nathel, “Observation of a topological phase by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

Giallorenzi, T. G.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Haldane, F. D. M.

Hannay, J. H.

J. H. Hannay, “Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian,” J. Phys. A 18, 221–230 (1985).
[CrossRef]

Hariharan, P.

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

P. Hariharan, M. Roy, “A geometric phase interferometer,” J. Mod. Opt. 39, 1811–1815 (1992).
[CrossRef]

Inoue, K.

T. Yoshino, K. Inoue, Y. Kobayashi, “Spiral fibre microbend sensors,” IEE Proc. Optoelectron. 144, 145–150 (1997).
[CrossRef]

Jiao, H.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, H. Nathel, “Observation of a topological phase by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

Kobayashi, Y.

T. Yoshino, K. Inoue, Y. Kobayashi, “Spiral fibre microbend sensors,” IEE Proc. Optoelectron. 144, 145–150 (1997).
[CrossRef]

Koga, H.

T. Abe, Y. Mitsunaga, H. Koga, “A strain sensor using twisted optical fibers,” J. Lightwave Technol. 7, 525–529 (1989).
[CrossRef]

Kuester, E. F.

L. Lewin, D. C. Chang, E. F. Kuester, Electromagnetic Waves and Curved Structures (Peregrinus, London, 1977).

Lagakos, N.

Larkin, K. G.

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

Lewin, L.

L. Lewin, D. C. Chang, E. F. Kuester, Electromagnetic Waves and Curved Structures (Peregrinus, London, 1977).

Liberman, V. S.

V. S. Liberman, B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium,” Phys. Rev. E 49, 2389–2396 (1994).
[CrossRef]

V. S. Liberman, B. Ya. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[CrossRef] [PubMed]

Lin, Z. Q.

X. S. Fang, Z. Q. Lin, “Field in single-mode helically-wound optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-33, 1150–1154 (1985).
[CrossRef]

Liñares, J.

Love, J. D.

A. Altintas, J. D. Love, “Effective cut-offs for modes on helical fibres,” Opt. Quantum Electron. 22, 213–226 (1990).
[CrossRef]

Marcuse, D.

Markovski, B.

B. Markovski, V. I. Vinitsky, Topological Phases in Quantum Theory (World Scientific, Singapore, 1989).

Mazzolini, A.

G. Davis, A. Mazzolini, “A new biomedical sensor utilizing macrobending losses in optical fibers,” Proc. Aust. Conf. Opt. Fibre Technol. 21, 129–132 (1996).

Mitsunaga, Y.

T. Abe, Y. Mitsunaga, H. Koga, “A strain sensor using twisted optical fibers,” J. Lightwave Technol. 7, 525–529 (1989).
[CrossRef]

Nathel, H.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, H. Nathel, “Observation of a topological phase by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

Nistal, M. C.

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference and its application,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Payne, D. N.

M. P. Varnham, R. D. Birch, D. N. Payne, “Helical-core circularly-birefringent fibers,” in Proceedings of the International Conference on Integrated Optics and Optical Fibre Communications–European Conference on Optical Communications (Instituto Internazionale delle Communicazione, Genova, Italy, 1985), pp. 135–138.

Priest, R. G.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Rashleigh, S. C.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Ross, J. N.

J. N. Ross, “The rotation of the polarization in low birefringence monomode optical fibres due to geometric effects,” Opt. Quantum Electron. 16, 455–461 (1984).
[CrossRef]

Roy, M.

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

P. Hariharan, M. Roy, “A geometric phase interferometer,” J. Mod. Opt. 39, 1811–1815 (1992).
[CrossRef]

Ryder, L. H.

L. H. Ryder, “The optical Berry phase and the Gauss-Bonnet theorem,” Eur. J. Phys. 12, 15–18 (1991).
[CrossRef]

Samuel, J.

R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

Savchenko and B. Ya. Zel’dovich, A. Yu.

A. Yu. Savchenko and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium: three-dimensional case,” Phys. Rev. E 50, 2287–2292 (1994).
[CrossRef]

Shapere, A.

A. Shapere, F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989).

Sigel, G. H.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical fiber sensor technology,” IEEE J. Quantum Electron. QE-18, 626–665 (1982).
[CrossRef]

Simon, A.

Smith, A. M.

Subbarao, D.

Tang, C. H.

C. H. Tang, “An orthogonal coordinate system for curved pipes,” IEEE Trans. Microwave Theory Tech. MTT-18, 69 (1970).
[CrossRef]

Tomita, A.

A. Tomita, R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fibre,” Phys. Rev. Lett. 57, 937–940 (1986).
[CrossRef] [PubMed]

Ulrich, R.

Varnham, M. P.

M. P. Varnham, R. D. Birch, D. N. Payne, “Helical-core circularly-birefringent fibers,” in Proceedings of the International Conference on Integrated Optics and Optical Fibre Communications–European Conference on Optical Communications (Instituto Internazionale delle Communicazione, Genova, Italy, 1985), pp. 135–138.

Vinitsky, V. I.

B. Markovski, V. I. Vinitsky, Topological Phases in Quantum Theory (World Scientific, Singapore, 1989).

Wang, Q.

G. Chen, Q. Wang, “Local fields in single-mode helical fibres,” Opt. Quantum Electron. 27, 1069–1074 (1995).
[CrossRef]

Wilczek, F.

A. Shapere, F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989).

Wilkinson, S. R.

R. Y. Chiao, A. Antaramian, K. M. Ganga, H. Jiao, S. R. Wilkinson, H. Nathel, “Observation of a topological phase by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. Lett. 60, 1214–1217 (1988).
[CrossRef] [PubMed]

Wu, Y-S.

R. Y. Chiao, Y-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[CrossRef] [PubMed]

Yao, S. K.

S. K. Yao, C. K. Asawa, “Microbending fiber optic sensing,” in Fiber Optic and Laser Sensors I, E. L. Moore, O. G. Ramer, eds., Proc. SPIE412, 9–13 (1983).
[CrossRef]

Yoshino, T.

T. Yoshino, K. Inoue, Y. Kobayashi, “Spiral fibre microbend sensors,” IEE Proc. Optoelectron. 144, 145–150 (1997).
[CrossRef]

Zel’dovich, B. Ya.

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Figures (7)

Fig. 1
Fig. 1

Schematic of a uniform (cylindrical) helix.

Fig. 2
Fig. 2

Conical helix with (a) increasing radius and (b) sinusoidally varying radius.

Fig. 3
Fig. 3

Cylindrical helix with (a) increasing pitch and (b) sinusoidally varying pitch.

Fig. 4
Fig. 4

Variation in φ with z for a conical helix with R 2 varying sinusoidally.

Fig. 5
Fig. 5

Projection of l̂ t onto the surface of a sphere for a conical helix with R 2 varying sinusoidally for (a) large α2 and (b) small α2.

Fig. 6
Fig. 6

Variation in φ with z for a cylindrical helix with P 2 varying (a) linearly and (b) quadratically for values of Γ = P 0/S 0 corresponding to 1, 1/4; 2, 1/2; 3, 3/4.

Fig. 7
Fig. 7

Variation in φ with z for a cylindrical helix with P 2 varying (a) linearly and (b) quadratically for values of α corresponding to 1, 1/10; 2, 1/4; 3, 1/2; 4, 1.

Equations (44)

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l ˆ t = r ,     l ˆ n = l ˆ t / κ ,     l ˆ b = l ˆ n + κ l ˆ t / τ .
κ 2 = r · r ,     τ = r × r · r r · r .
r s = x l ˆ x + y l ˆ y + z l ˆ z ,
x = R z cos   ϕ ,     y = R z sin   ϕ .
τ z = 2 π P z S 2 z = 2 π   cos 2   θ p z P z ,
φ t = - 0 s   τ s d s .
φ n = 2 π   0 s 1 S s d s .
φ = 0 s 2 π S s - τ s d s = 0 ϕ 1 - cos   θ p ϕ d ϕ
φ = 2 π   0 z 1 P z - 1 S z d z .
φ n = 2 π   0 z d z P z
φ t = - 2 π   0 z d z S z
Δ φ z = φ z - φ 0 z ,
φ 0 = 2 π P 0 S 0 - P 0 S 0
L z = S 0 P 0 S 0 - P 0 P 0 P 0     R 0 P 0 3 / 2 π 2 R 0 2 P 0     R 0 .
φ = 2 π z P 0 - S 0 π α 1 + 4 π 2 α z S 0 2 - 1   linear variation , 2 π z P 0   -   1 α arcsinh 2 π α z S 0   α >   quadratic variation , 2 π z P 0 - 1 | α | arcsin 2 π | α | z S 0   α < 0
φ S 0 - P 0 2 2 π | α | P 0 as   z R 0 2 / | α |   linear   variation , 1 | α | 2 π R 0 P 0   -   arcsin 2 π R 0 S 0 as   z R 0 / | α | quadratic   variation .
φ 2 π z P 0 if   z     S 0 2 / 4 π 2 α ,   P 0 > 2 π R 0 / 3 linear   variation , 2 π z P 0 if   z     S 0 / 4 π α ,   P 0 > 2 π R 0 / 3 quadratic   variation .
φ = 2 π z P 0 + 4 π α 1 P 0 arctanh P 0 2 P 0 2 + 2 π R 0 2 exp α 1 z - arctanh P 0 S 0   case   a 2 π z P 0 + 4 π α 1 G 0 arctanh G 0 2 - 4 π 2 α 2 exp α 1 z G 0 2 - arctanh S 0 G 0   case   b
φ 2 π z P 0 α 1 > 0 - 4 π | α 1 | P 0 ln P 0 π R 0 S 0 - P 0 S 0 + P 0 α 1 < 0
φ - 4 π | α 1 | G 0 ln G 0 π α 2 G 0 - S 0 G 0 + S 0     α 1 < 0 2 π α 1 P 0 ln 1 + R 0 2 α 2 + 2 P 0 G 0 arctanh P 0 G 0 - arctanh S 0 G 0     α 1 > 0 .
φ = 2 π z P 0 - 1 S 0 α 2   F α 2 z | m ,
φ = 4 π α P 0 2 + α z - S 0 2 + α z - P 0 - S 0   linear   variation 2 π α arcsinh α z P 0 - arcsinh α z S 0     α > 0 . 2 π | α | arcsin | α | z P 0 - arcsin | α | z S 0     α < 0     quadratic variation
φ 4 π | α | P 0 + 2 π R 0 - S 0 as   z P 0 2 / | α | linear   variation 2 π | α | arccos P 0 S 0 as   z P 0 / | α | quadratic variation .
φ 4 π α S 0 - P 0 if   z     S 0 2 / α linear   variation 2 π α ln S 0 P 0 if   z     S 0 / α quadratic   variation .
Δ φ = 4 π α   S 0 x 2 + ξ - 1 + ξ + 1 - x 1 - ξ 2 x ,
φ = 2 π α ln S 0 + P 0 P 0 - ln P 0 + α z + S 0 2 + 2 α P 0 z + α 2 z 2 1 / 2 P 0 + α z .
φ 2 π α ln S 0 + P 0 2 P 0 .
φ = 8 π 3 α 2 α z - 2 P 0 2 P 0 2 + α z - α z - 2 S 0 2 × S 0 2 + α z + 2 P 0 3 - S 0 3 .
φ 4 π 2 π R 0 2 α 3 / 2   z 1 / 4 ,
φ = 2 π   z 0 1 P z - 1 S z d z
Δ φ = - 2 π   z 0 Δ P z P 0 - Δ S z S 0 d z ,
z 0 R 2 z P 3 z d z .
Δ φ = 2 π   z 0 1 S 0 - 1 S z d z
θ F = α NV     H · d l = α NVI ,
Δ φ t = φ f - φ i = 2 π R 0 Z i / S i - Z f / S f ,
Δ φ t = - 4 π N 3 Z i - 2 - Z f - 2 .
d Ω C = ± sin   θ ρ 2 d A = ± | l ˆ t × l ˆ u | ρ 2 d A .
γ C = N Ω C = ± N   0 2 π 1 - cos   θ p φ d φ
φ z = ± 2 π   0 z 1 P z - 1 S z d z .
d Ω C = ± sin   θ 1 ρ 2 d A = ± sin   θ 1 cos   θ 2 ρ 2 d S .
C z = R C z l ˆ t z · z ˆ = R C z cos   θ p z ,
R C z = 1 - cos   θ p z cos   θ p z = S z - P z P z .
C z = 1 - cos   θ p z .
ψ C = 0 2 π   C ϕ d ϕ = 2 π   0 P C C z P z d z ,

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