Abstract

The tuneable reflectivity of a fiber loop mirror consisting of a Sagnac interferometer with a fixed ratio directional coupler is attributed to Berry’s topological phase. Berry’s phase is the phase acquired by a light beam when the direction of propagation is changed, while the optical path length remains unchanged. The nature of the topological phase is studied and exact relations characterizing Berry’s phase fiber-optic mirror are derived.

© 2005 Optical Society of America

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  1. B. Culshaw and J. Dakin, eds., Optical Fiber Sensors: Systems and Applications (Artech House, Norwood, Mass., 1989), Vol. 2.
  2. D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
    [CrossRef]
  3. C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, and P. Urquhart, "Fiber laser with adjustable fiber reflector for wavelength tuning and variable output coupling," Proc. Inst. Electr. Eng. Part J 135, 303-309 (1988).
  4. P. Senthilkumaran, G. Thursby, and B. Culshaw, "Fiber optic tunable loop mirror using Berry's geometric phase," Opt. Lett. 25, 533-535 (2000).
    [CrossRef]
  5. A. Tomita and R. Y. Chiao, "Observation of Berry's topological phase by use of an optical fiber," Phys. Rev. Lett. 57, 937-940 (1986).
    [CrossRef] [PubMed]
  6. M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. London, Ser. A 392, 45-57 (1984).
    [CrossRef]
  7. M. V. Berry, "Interpreting the anholonomy of coiled light," Nature (London) 326, 277-278 (1987).
    [CrossRef]
  8. J. N. Ross, "The rotation of the polarization in low birefringence monomode optical fibers due to geometric effects," Opt. Quantum Electron. 16, 455-461 (1984).
    [CrossRef]
  9. S. G. Lipson, "Berry's phase in optical interferometry: a simple derivation," Opt. Lett. 15, 154-155 (1990).
    [CrossRef] [PubMed]
  10. P. Senthilkumaran, G. Thursby, and B. Culshaw, "New approach to the design of fiber optic devices based on Berry's topological phase," in Photonics 2000: International Conference on Fiber Optics and Photonics, S. K. Lahri, R. Gangopadhyay, A. K. Datta, S. K. Ray, B. K. Mathur, and S. Das, eds., Proc. SPIE 4417, 434-441 (2001).
    [CrossRef]
  11. P. Senthilkumaran, B. Culshaw, and G. Thursby, "Fiber optic Sagnac interferometer for the observation of Berry's topological phase," J. Opt. Soc. Am. B 17, 1914-1919 (2000).
    [CrossRef]
  12. G. Thursby, P. Senthilkumaran, and B. Culshaw, "Possible applications for Berry's topological phase in fiber sensor systems," presented at 14th International Conference on Optical Fiber Sensors, Venice, Italy, October 11-13, 2000.
  13. P. Senthilkumaran, "Interferometric array illuminator with analysis of nonobservable fringes," Appl. Opt. 38, 1311-1316 (1999).
    [CrossRef]
  14. S. Pancharatnam, "Generalized theory of interference, and its applications," Proc. Indian Acad. Sci., Sect. A 44, 247-262 (1956).
  15. M. Martinelli and P. Vavassori, "A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits," Opt. Commun. 80, 166-176 (1990).
    [CrossRef]
  16. H. Schmitzer, S. Klein, and W. Dultz, "An experimental test of the path dependency of Pancharatnam's geometric phase," J. Mod. Opt. 45, 1039-1047 (1998).
    [CrossRef]

2001 (1)

P. Senthilkumaran, G. Thursby, and B. Culshaw, "New approach to the design of fiber optic devices based on Berry's topological phase," in Photonics 2000: International Conference on Fiber Optics and Photonics, S. K. Lahri, R. Gangopadhyay, A. K. Datta, S. K. Ray, B. K. Mathur, and S. Das, eds., Proc. SPIE 4417, 434-441 (2001).
[CrossRef]

2000 (2)

1999 (1)

1998 (1)

H. Schmitzer, S. Klein, and W. Dultz, "An experimental test of the path dependency of Pancharatnam's geometric phase," J. Mod. Opt. 45, 1039-1047 (1998).
[CrossRef]

1990 (2)

M. Martinelli and P. Vavassori, "A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits," Opt. Commun. 80, 166-176 (1990).
[CrossRef]

S. G. Lipson, "Berry's phase in optical interferometry: a simple derivation," Opt. Lett. 15, 154-155 (1990).
[CrossRef] [PubMed]

1988 (2)

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, and P. Urquhart, "Fiber laser with adjustable fiber reflector for wavelength tuning and variable output coupling," Proc. Inst. Electr. Eng. Part J 135, 303-309 (1988).

1987 (1)

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

1986 (1)

A. Tomita and R. Y. Chiao, "Observation of Berry's topological phase by use of an optical fiber," Phys. Rev. Lett. 57, 937-940 (1986).
[CrossRef] [PubMed]

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 fibers due to geometric effects," Opt. Quantum Electron. 16, 455-461 (1984).
[CrossRef]

1956 (1)

S. Pancharatnam, "Generalized theory of interference, and its applications," Proc. Indian Acad. Sci., Sect. A 44, 247-262 (1956).

Ainslie, B. J.

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, and P. Urquhart, "Fiber laser with adjustable fiber reflector for wavelength tuning and variable output coupling," Proc. Inst. Electr. Eng. Part J 135, 303-309 (1988).

Berry, M. V.

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

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

Chiao, R. Y.

A. Tomita and R. Y. Chiao, "Observation of Berry's topological phase by use of an optical fiber," Phys. Rev. Lett. 57, 937-940 (1986).
[CrossRef] [PubMed]

Culshaw, B.

P. Senthilkumaran, G. Thursby, and B. Culshaw, "New approach to the design of fiber optic devices based on Berry's topological phase," in Photonics 2000: International Conference on Fiber Optics and Photonics, S. K. Lahri, R. Gangopadhyay, A. K. Datta, S. K. Ray, B. K. Mathur, and S. Das, eds., Proc. SPIE 4417, 434-441 (2001).
[CrossRef]

P. Senthilkumaran, B. Culshaw, and G. Thursby, "Fiber optic Sagnac interferometer for the observation of Berry's topological phase," J. Opt. Soc. Am. B 17, 1914-1919 (2000).
[CrossRef]

P. Senthilkumaran, G. Thursby, and B. Culshaw, "Fiber optic tunable loop mirror using Berry's geometric phase," Opt. Lett. 25, 533-535 (2000).
[CrossRef]

Dultz, W.

H. Schmitzer, S. Klein, and W. Dultz, "An experimental test of the path dependency of Pancharatnam's geometric phase," J. Mod. Opt. 45, 1039-1047 (1998).
[CrossRef]

Klein, S.

H. Schmitzer, S. Klein, and W. Dultz, "An experimental test of the path dependency of Pancharatnam's geometric phase," J. Mod. Opt. 45, 1039-1047 (1998).
[CrossRef]

Lipson, S. G.

Martinelli , M.

M. Martinelli and P. Vavassori, "A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits," Opt. Commun. 80, 166-176 (1990).
[CrossRef]

Millar, C. A.

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, and P. Urquhart, "Fiber laser with adjustable fiber reflector for wavelength tuning and variable output coupling," Proc. Inst. Electr. Eng. Part J 135, 303-309 (1988).

Miller, I. D.

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, and P. Urquhart, "Fiber laser with adjustable fiber reflector for wavelength tuning and variable output coupling," Proc. Inst. Electr. Eng. Part J 135, 303-309 (1988).

Mortimore, D. B.

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, and P. Urquhart, "Fiber laser with adjustable fiber reflector for wavelength tuning and variable output coupling," Proc. Inst. Electr. Eng. Part J 135, 303-309 (1988).

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, "Generalized theory of interference, and its applications," Proc. Indian Acad. Sci., Sect. A 44, 247-262 (1956).

Ross, J. N.

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

Schmitzer, H.

H. Schmitzer, S. Klein, and W. Dultz, "An experimental test of the path dependency of Pancharatnam's geometric phase," J. Mod. Opt. 45, 1039-1047 (1998).
[CrossRef]

Senthilkumaran, P.

P. Senthilkumaran, G. Thursby, and B. Culshaw, "New approach to the design of fiber optic devices based on Berry's topological phase," in Photonics 2000: International Conference on Fiber Optics and Photonics, S. K. Lahri, R. Gangopadhyay, A. K. Datta, S. K. Ray, B. K. Mathur, and S. Das, eds., Proc. SPIE 4417, 434-441 (2001).
[CrossRef]

P. Senthilkumaran, B. Culshaw, and G. Thursby, "Fiber optic Sagnac interferometer for the observation of Berry's topological phase," J. Opt. Soc. Am. B 17, 1914-1919 (2000).
[CrossRef]

P. Senthilkumaran, G. Thursby, and B. Culshaw, "Fiber optic tunable loop mirror using Berry's geometric phase," Opt. Lett. 25, 533-535 (2000).
[CrossRef]

P. Senthilkumaran, "Interferometric array illuminator with analysis of nonobservable fringes," Appl. Opt. 38, 1311-1316 (1999).
[CrossRef]

Thursby, G.

P. Senthilkumaran, G. Thursby, and B. Culshaw, "New approach to the design of fiber optic devices based on Berry's topological phase," in Photonics 2000: International Conference on Fiber Optics and Photonics, S. K. Lahri, R. Gangopadhyay, A. K. Datta, S. K. Ray, B. K. Mathur, and S. Das, eds., Proc. SPIE 4417, 434-441 (2001).
[CrossRef]

P. Senthilkumaran, B. Culshaw, and G. Thursby, "Fiber optic Sagnac interferometer for the observation of Berry's topological phase," J. Opt. Soc. Am. B 17, 1914-1919 (2000).
[CrossRef]

P. Senthilkumaran, G. Thursby, and B. Culshaw, "Fiber optic tunable loop mirror using Berry's geometric phase," Opt. Lett. 25, 533-535 (2000).
[CrossRef]

Tomita , A.

A. Tomita and R. Y. Chiao, "Observation of Berry's topological phase by use of an optical fiber," Phys. Rev. Lett. 57, 937-940 (1986).
[CrossRef] [PubMed]

Urquhart, P.

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, and P. Urquhart, "Fiber laser with adjustable fiber reflector for wavelength tuning and variable output coupling," Proc. Inst. Electr. Eng. Part J 135, 303-309 (1988).

Vavassori, P.

M. Martinelli and P. Vavassori, "A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits," Opt. Commun. 80, 166-176 (1990).
[CrossRef]

Appl. Opt. (1)

J. Lightwave Technol. (1)

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

J. Mod. Opt. (1)

H. Schmitzer, S. Klein, and W. Dultz, "An experimental test of the path dependency of Pancharatnam's geometric phase," J. Mod. Opt. 45, 1039-1047 (1998).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (London) (1)

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

Opt. Commun. (1)

M. Martinelli and P. Vavassori, "A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits," Opt. Commun. 80, 166-176 (1990).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

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

Phys. Rev. Lett. (1)

A. Tomita and R. Y. Chiao, "Observation of Berry's topological phase by use of an optical fiber," Phys. Rev. Lett. 57, 937-940 (1986).
[CrossRef] [PubMed]

Proc. Indian Acad. Sci., Sect. A (1)

S. Pancharatnam, "Generalized theory of interference, and its applications," Proc. Indian Acad. Sci., Sect. A 44, 247-262 (1956).

Proc. Inst. Electr. Eng. Part J (1)

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, and P. Urquhart, "Fiber laser with adjustable fiber reflector for wavelength tuning and variable output coupling," Proc. Inst. Electr. Eng. Part J 135, 303-309 (1988).

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

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

Proc. SPIE (1)

P. Senthilkumaran, G. Thursby, and B. Culshaw, "New approach to the design of fiber optic devices based on Berry's topological phase," in Photonics 2000: International Conference on Fiber Optics and Photonics, S. K. Lahri, R. Gangopadhyay, A. K. Datta, S. K. Ray, B. K. Mathur, and S. Das, eds., Proc. SPIE 4417, 434-441 (2001).
[CrossRef]

Other (2)

G. Thursby, P. Senthilkumaran, and B. Culshaw, "Possible applications for Berry's topological phase in fiber sensor systems," presented at 14th International Conference on Optical Fiber Sensors, Venice, Italy, October 11-13, 2000.

B. Culshaw and J. Dakin, eds., Optical Fiber Sensors: Systems and Applications (Artech House, Norwood, Mass., 1989), Vol. 2.

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Figures (5)

Fig. 1
Fig. 1

Fiber optics Sagnac interferometer as a tuneable loop mirror. H is a helically wound fiber, C is a directional coupler, and λ/2 is the half-wave plate.

Fig. 2
Fig. 2

Plot of IR, IO versus γ for a polarized light.

Fig. 3
Fig. 3

(a) Plane-polarized light decomposed into left and right circularly polarized light after passing through the helix and the half-wave plate experiences Berry’s phase shifts. (b) Plane-polarized light decomposed into left and right circularly polarized light after passing through the half-wave plate and then the helix experiences Berry’s phase shifts, which is opposite to that of the case presented in (a).

Fig. 4
Fig. 4

(a) Trace of SOP changes on the Poincaré sphere for the two interfering beams, when the input light is plane polarized. OL, equatorial axis about which SOPs are rotated owing to linear birefringence. (b) Trace of SOP changes on the Poincaré sphere for the two beams, when the input light is elliptically polarized.

Fig. 5
Fig. 5

(a) Geodesic trajectory on the Poincare sphere and (b) a nongeodesic trajectory on the momentum sphere for realizing topological phases.

Equations (44)

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γ=-2πNσ1-ps,
IR=M Iin2(1+cos 2γ),
Io=Iin-M Iin2(1+cos 2γ),
Io=Iin2(1-cos 2γ)
Iin=IR+Io.
γ=-σΩ=-2πσ(1-cos θ),
|C=a1|R+a2|L.
a12=Iin cos2(C, R)2,
a22=Iin sin2(C, R)2,
|A=|B=12(a1|R+a2|L).
|A2=12[a1|Lexp(-iγ)+a2|Rexp(iγ)],
|B2=12[a1|Lexp(iγ)+a2|Rexp(-iγ)].
|A2+|B2=12{a1|L[exp(iγ)+exp(-iγ)]+a2|R[exp(iγ)+exp(-iγ)]},
|A2+|B2={a1|L+a2|R}cos γ=|Dcos γ,
IR=MIin cos2 γ=M Iin2(1+cos 2γ),
IR=Iin cos2 γ=Iin2(1+cos 2γ),
I=I1+I2+2I1I2 cos(1/2)c cos δ,
I=2I11+cos12c=2I1(1+cos 2γ).
10000cos 4βsin 4β00sin 4β-cos 4β0000-1
×10000cos 2γsin 2γ00-sin 2γcos 2γ000011100
=1cos 2γ cos 4β-sin 2γ sin 4βcos 2γ sin 4β+sin 2γ cos 4β0.
1cos 2γ cos 4β+sin 2γ sin 4βcos 2γ sin 4β-sin 2γ cos 4β0.
cos(c)=(cos 2γ cos 4β-sin 2γ sin 4β)(cos 2γ cos 4β+sin 2γ sin 4β)+(cos 2γ sin 4β+sin 2γ cos 4β)(cos 2γ sin 4β-sin 2γ cos 4β)=cos(4γ).
I=I1+I2+2I1I2 cos(2γ),
I=2I1(1+cos 2γ).
1lmn,
l2+m2+n2=1
1cos 2γ(l cos 4β+m sin 4β)+sin 2γ(l sin 4β-m cos 4β)-sin 2γ(l cos 4β+m sin 4β)+cos 2γ(l sin 4β-m cos 4β)-n.
 1cos 4β(l cos 2γ+m sin 2γ)+sin 4β(m cos 2γ-l sin 2γ)sin 4β(l cos 2γ+m sin 2γ)-cos 4β(m cos 2γ-l sin 2γ)-n,
cos c=(l2+m2)cos 4γ+n2.
I=2I11+cos12 cos-1[(l2+m2)cos 4γ+n2].
(Imin)R=2I11+cos12 cos-1(n2-l2-m2),
(Imax)R=4I1.
(Imin)o=0,
(Imax)o=2I11-cos12 cos-1(n2-l2-m2)=(Imax)R-(Imin)R.
Io+IR=Iin=(Imin)o+(Imax)R=(Imin)R+(Imax)o.
M=1-cos12 cos-1(n2-l2-m2).
1000000-10010010010000cos 4βsin 4β00sin 4β-cos 4β0000-1×1000000-1001001001100=1100.
12 1ii1cos 2βsin 2βsin 2β-cos 2β 12 1ii110
=cos 2β+i sin 2β0.
10+cos 2β+i sin 2β0,
(1+cos 2β+i sin 2β)(1+cos 2β+i sin 2β)*
=2[1+cos(2β)].
cos 2βsin 2βsin 2β-cos 2βcos 2βsin 2βsin 2β-cos 2β10=10.

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