Abstract

The nature of changes in the interference pattern caused by the presence of polarization-changing elements in one or both beams of an interferometer, in particular those caused by an effective optical activity due to passage of a polarized beam through a coiled optical fiber, are clarified. It is pointed out that, for an incident state that is not circularly polarized so that the two interfering beams go to different polarization states, there is an observable nonzero Pancharatnam phase shift between them that depends on the incident polarization state and on the solid angle subtended by the track of the k vector at the center of the sphere of k vectors. The behavior of this phase shift is singular when the two interfering states are nearly orthogonal. It is shown that, for zero path difference between the two beams, the amplitude of intensity modulation as a function of optical activity is independent of the incident polarization state.

© 2007 Optical Society of America

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References

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  1. M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. London, Ser. A 392, 45-57 (1984).
    [CrossRef]
  2. S. Pancharatnam, "Generalized theory of interference and its applications," Proc. Indian Acad. Sci., Sect. A 44, 247-262 (1956).
  3. R. Bhandari, "Observation of nonintegrable geometric phase on the Poincaré sphere," Phys. Lett. A 133, 1-3 (1991).
    [CrossRef]
  4. R. Bhandari, "SU(2) phase jumps and geometric phases," Phys. Lett. A 157, 221-225 (1991).
    [CrossRef]
  5. R. Bhandari, "4π spinor symmetry-some new observations," Phys. Lett. A 180, 15-20 (1993).
    [CrossRef]
  6. R. Bhandari, "Interferometry without beam splitters-a sensitive technique for spinor phases," Phys. Lett. A 180, 21-24 (1993).
    [CrossRef]
  7. R. Bhandari, "Observation of Dirac singularities with light polarization--I, II," Phys. Lett. A 171, 262-270 (1992).
    [CrossRef]
  8. R. Bhandari, "Evolution of light beams in polarization and direction," Physica B 175, 111-122 (1991).
    [CrossRef]
  9. R. Bhandari, "Polarization of light and topological phases," Phys. Rep. 281, 1-64 (1997).
    [CrossRef]
  10. R. Y. Chiao and W. S. Wu, "Manifestations of Berry's topological phase for the photon," Phys. Rev. Lett. 57, 933-936 (1986).
    [CrossRef] [PubMed]
  11. E. M. Frins and W. Dultz, "Direct observation of Berry's topological phase by using an optical fiber ring interferometer," Opt. Commun. 136, 354-356 (1997).
    [CrossRef]
  12. 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]
  13. J. Anandan, "Non-adiabatic non-abelian geometric phase," Phys. Lett. A 133, 171-175 (1988).
    [CrossRef]
  14. F. Wilczek and A. Zee, "Appearance of gauge strucures in simple dynamical systems," Phys. Rev. Lett. 52, 2111-2114 (1984).
    [CrossRef]
  15. P. Senthilkumaran, "Berry's phase fiber loop mirror characteristics," J. Opt. Soc. Am. B 22, 505-511 (2005).
    [CrossRef]
  16. P. Senthilkumaran, G. Thursby, and B. Culshaw, "Fiber-optic tunable loop mirror using Berry's geometric phase," Opt. Lett. 25, 533-535 (2000).
    [CrossRef]
  17. P. Senthilkumaran, G. Thursby, and B. Culshaw, "Fiber optic Sagnac interferometer for the observation of Berry's topological phase," J. Opt. Soc. Am. B 17, 1914-1919 (2000).
    [CrossRef]
  18. P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003), p. 57.
  19. After submitting this manuscript we became aware of the work of Tavrov et al. , in which they use the geometric spin redirection phase due to out-of-plane propagation of light to realize an achromatic π-phase shift between the two beams of an astronomical interferometer for "nulling interferometry." In our judgement, the linear phase shift between the beams for circular polarization shown in Fig. 2(a) and the highly nonlinear phase shift for linear polarization shown in Fig. 2(b) of their paper correspond approximately to the curves A and C shown in Fig. of this paper.
  20. When the Jones matrix of a halfwave plate is written without the factor i as done in and in some texbooks on optics, it implies an isotropic phase factor exp(±iπ/2) multiplying the SU(2) part. This must be removed before applying the considerations of Section .
  21. Y. Aharonov and J. Anandan, "Phase change during a cyclic quantum evolution," Phys. Rev. Lett. 58, 1593-1596 (1987).
    [CrossRef] [PubMed]
  22. A. Tavrov, R. Bohr, M. Totzeck, and H. Tiziani, "Achromatic nulling interferometer based on a geometric spin-redirection phase," Opt. Lett. 27, 2070-2072 (2002).
    [CrossRef]

2005 (1)

2002 (1)

2000 (2)

1997 (2)

R. Bhandari, "Polarization of light and topological phases," Phys. Rep. 281, 1-64 (1997).
[CrossRef]

E. M. Frins and W. Dultz, "Direct observation of Berry's topological phase by using an optical fiber ring interferometer," Opt. Commun. 136, 354-356 (1997).
[CrossRef]

1993 (2)

R. Bhandari, "4π spinor symmetry-some new observations," Phys. Lett. A 180, 15-20 (1993).
[CrossRef]

R. Bhandari, "Interferometry without beam splitters-a sensitive technique for spinor phases," Phys. Lett. A 180, 21-24 (1993).
[CrossRef]

1992 (1)

R. Bhandari, "Observation of Dirac singularities with light polarization--I, II," Phys. Lett. A 171, 262-270 (1992).
[CrossRef]

1991 (3)

R. Bhandari, "Evolution of light beams in polarization and direction," Physica B 175, 111-122 (1991).
[CrossRef]

R. Bhandari, "Observation of nonintegrable geometric phase on the Poincaré sphere," Phys. Lett. A 133, 1-3 (1991).
[CrossRef]

R. Bhandari, "SU(2) phase jumps and geometric phases," Phys. Lett. A 157, 221-225 (1991).
[CrossRef]

1988 (1)

J. Anandan, "Non-adiabatic non-abelian geometric phase," Phys. Lett. A 133, 171-175 (1988).
[CrossRef]

1987 (1)

Y. Aharonov and J. Anandan, "Phase change during a cyclic quantum evolution," Phys. Rev. Lett. 58, 1593-1596 (1987).
[CrossRef] [PubMed]

1986 (2)

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]

R. Y. Chiao and W. S. Wu, "Manifestations of Berry's topological phase for the photon," Phys. Rev. Lett. 57, 933-936 (1986).
[CrossRef] [PubMed]

1984 (2)

F. Wilczek and A. Zee, "Appearance of gauge strucures in simple dynamical systems," Phys. Rev. Lett. 52, 2111-2114 (1984).
[CrossRef]

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

1956 (1)

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

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

E. M. Frins and W. Dultz, "Direct observation of Berry's topological phase by using an optical fiber ring interferometer," Opt. Commun. 136, 354-356 (1997).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (6)

J. Anandan, "Non-adiabatic non-abelian geometric phase," Phys. Lett. A 133, 171-175 (1988).
[CrossRef]

R. Bhandari, "Observation of nonintegrable geometric phase on the Poincaré sphere," Phys. Lett. A 133, 1-3 (1991).
[CrossRef]

R. Bhandari, "SU(2) phase jumps and geometric phases," Phys. Lett. A 157, 221-225 (1991).
[CrossRef]

R. Bhandari, "4π spinor symmetry-some new observations," Phys. Lett. A 180, 15-20 (1993).
[CrossRef]

R. Bhandari, "Interferometry without beam splitters-a sensitive technique for spinor phases," Phys. Lett. A 180, 21-24 (1993).
[CrossRef]

R. Bhandari, "Observation of Dirac singularities with light polarization--I, II," Phys. Lett. A 171, 262-270 (1992).
[CrossRef]

Phys. Rep. (1)

R. Bhandari, "Polarization of light and topological phases," Phys. Rep. 281, 1-64 (1997).
[CrossRef]

Phys. Rev. Lett. (4)

R. Y. Chiao and W. S. Wu, "Manifestations of Berry's topological phase for the photon," Phys. Rev. Lett. 57, 933-936 (1986).
[CrossRef] [PubMed]

F. Wilczek and A. Zee, "Appearance of gauge strucures in simple dynamical systems," Phys. Rev. Lett. 52, 2111-2114 (1984).
[CrossRef]

Y. Aharonov and J. Anandan, "Phase change during a cyclic quantum evolution," Phys. Rev. Lett. 58, 1593-1596 (1987).
[CrossRef] [PubMed]

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]

Physica B (1)

R. Bhandari, "Evolution of light beams in polarization and direction," Physica B 175, 111-122 (1991).
[CrossRef]

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. 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]

Other (3)

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003), p. 57.

After submitting this manuscript we became aware of the work of Tavrov et al. , in which they use the geometric spin redirection phase due to out-of-plane propagation of light to realize an achromatic π-phase shift between the two beams of an astronomical interferometer for "nulling interferometry." In our judgement, the linear phase shift between the beams for circular polarization shown in Fig. 2(a) and the highly nonlinear phase shift for linear polarization shown in Fig. 2(b) of their paper correspond approximately to the curves A and C shown in Fig. of this paper.

When the Jones matrix of a halfwave plate is written without the factor i as done in and in some texbooks on optics, it implies an isotropic phase factor exp(±iπ/2) multiplying the SU(2) part. This must be removed before applying the considerations of Section .

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Figures (3)

Fig. 1
Fig. 1

Interference pattern on a screen formed by interference of two parts of a wavefront W through slits S 1 and S 2 that have undergone polarization transformations U 1 and U 2 , corresponding to an effective optical activity that causes a rotation about the beam axis through angles γ and γ , respectively. Intensity variation on the screen as a function of the distance y along the screen (approximately proportional to the optical path difference α between the two beams) is shown for five different values of γ, i.e., γ = 0 ° , 45 ° , 90 ° , 135 ° , and 180 ° . As γ changes, the visibility of the fringes changes and the fringes shift along the y axis by an amount δ, which is the Pancharatnam phase difference between the beams. Note that when α = n π , the amplitude of the intensity modulation as a function of γ is 1, whereas for an arbitrary value of α, the modulation can be less than 1. The fringes shown in the figure correspond to a polarization state with θ = 60 ° .

Fig. 2
Fig. 2

Pancharatnam phase shift δ in degrees as a function of the optical activity parameter γ, which in the case of propagation through a fiber loop is equal to the solid angle subtended by the track of the k vector at the center of the sphere of k vectors. The curves A, B, C, D, E, and F correspond to incident polarization states with polar angle θ = 30 ° , 75°, 89.9°, 90.1°, 105°, and 150°. The curves for θ = 0 ° and 180 ° are straight lines nearly coincident with the curves A and F, respectively, and are not shown separately. For γ = n π 2 sr, the polarization of the two beams undergoes rotation through π and π on the Poincaré sphere, and the total phase shift has a magnitude π irrespective of θ. Also note the singular behavior for θ = 90 ° when γ has the values ( 2 n + 1 ) π 4 sr.

Fig. 3
Fig. 3

Conventional Sagnac interferometer configuration that would act as a tunable mirror equivalent to the tunable fiber optic mirror. A rotation of the halfwave plate H about the beam axis results in a complementary modulation of intensity at the two output ports. Rotation through an angle γ 2 is equivalent to passage through a fiber loop with solid angle γ in k space.

Equations (30)

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η = ( cos ( θ 2 ) sin ( θ 2 ) exp ( i ϕ ) ) ,
η 1 = I 1 η , η 2 = I 2 η .
J 1 = exp ( i β 1 ) U 1 , J 2 = exp ( i β 2 ) U 2 .
ψ 1 = J 1 exp ( i α 1 ) η 1 = I 1 exp i ( β 1 + α 1 ) ψ 1 ̃ ,
ψ 2 = J 2 exp ( i α 2 ) η 2 = I 2 exp i ( β 2 + α 2 ) ψ 2 ̃ ,
ψ 1 ̃ = U 1 η , ψ 2 ̃ = U 2 η , ψ 2 ̃ ψ 2 ̃ = ψ 1 ̃ ψ 1 ̃ = 1 .
I = ψ 1 ψ 1 + ψ 2 ψ 2 + 2 Re [ ψ 2 ψ 1 ] = I 1 + I 2 + 2 ( I 1 I 2 ) 1 2 Re [ ψ 2 ̃ ψ 1 ̃ exp i ( β + α ) ] = I 1 + I 2 + 2 ( I 1 I 2 ) 1 2 Re [ ξ exp i ( δ + β + α ) ] ,
β = β 1 β 2 , α = α 1 α 2 , ψ 2 ̃ ψ 1 ̃ = ξ exp ( i δ ) .
α = ( 2 π s λ ) ( y d ) ,
I m a x = I 1 + I 2 + 2 ( I 1 I 2 ) 1 2 ξ ,
I m i n = I 1 + I 2 2 ( I 1 I 2 ) 1 2 ξ ,
V = ( I m a x I m i n ) ( I m a x + I m i n ) = [ 2 ( I 1 I 2 ) 1 2 ( I 1 + I 2 ) ] ξ
J 1 = U 1 = ( exp ( i γ ) 0 0 exp ( i γ ) ) ,
J 2 = U 2 = ( exp ( i γ ) 0 0 exp ( i γ ) ) ,
ψ 2 ̃ ψ 1 ̃ = η U 2 U 1 η = ξ exp ( i δ ) = cos 2 ( θ 2 ) exp ( 2 i γ ) + sin 2 ( θ 2 ) exp ( 2 i γ ) .
ξ cos δ = cos ( 2 γ ) , ξ sin δ = cos θ sin ( 2 γ )
ξ = [ cos 2 ( 2 γ ) + cos 2 ( θ ) sin 2 ( 2 γ ) ] 1 2 , tan δ = cos θ tan ( 2 γ ) .
U 1 = H ( 0 ) R ( γ ) = i ( 0 1 1 0 ) ( exp ( i γ ) 0 0 exp ( i γ ) ) = i ( 0 exp ( i γ ) exp ( i γ ) 0 ) .
U 2 = R ( γ ) H ( 0 ) = i ( 0 exp ( i γ ) exp ( i γ ) 0 ) .
I = ( 1 2 ) [ 1 + Re ( ψ 2 ̃ ψ 1 ̃ exp i α ) ] ,
ψ 2 ̃ ψ 1 ̃ = η U 2 U 1 η = ξ exp ( i δ ) = cos 2 ( θ 2 ) exp ( 2 i γ ) + sin 2 ( θ 2 ) exp ( 2 i γ ) .
I = ( 1 2 ) [ 1 + ξ cos ( δ + α ) ] = ( 1 2 ) [ 1 + cos 2 ( θ 2 ) cos ( α 2 γ ) + sin 2 ( θ 2 ) cos ( α + 2 γ ) ] = ( 1 2 ) [ 1 + cos α cos ( 2 γ ) + sin α sin ( 2 γ ) cos θ ] .
I = ( 1 2 ) [ 1 + M cos ( 2 γ χ ) ] ,
M cos χ = cos α ,
M sin χ = cos θ sin α ,
M = [ cos 2 α + cos 2 θ sin 2 α ] 1 2 .
I = ( 1 2 ) [ 1 + cos ( 2 γ ± α ) ] ,
H ( τ 2 ) = i ( 0 exp ( i τ ) exp ( i τ ) 0 ) .
H ( τ 2 ) = i ( cos τ sin τ sin τ cos τ ) .
H ( τ 2 ) H ( τ 2 ) = ( 1 0 0 1 ) = exp ( ± i π ) ( 1 0 0 1 ) .

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