Abstract

To permit unified analysis and simultaneous evaluation of geometrical spin-redirection phase and Pancharatnam phase, the conventional 2×2 Jones matrix calculation is generalized and a new scheme of 3×3 matrix calculation is proposed. With the proposed algorithm one can trace the polarization state changes and the geometric phase shifts caused by beam propagation along an arbitrary optical path that involves both reflection and refraction at surfaces with Fresnel shift and birefringence.

© 2000 Optical Society of America

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References

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  1. M. Berry, “Anticipation of the geometric phase,” Phys. Today 43, 34–40 (1990).
    [CrossRef]
  2. R. Simon, N. Mukunda, “Bargman invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
    [CrossRef] [PubMed]
  3. A. P. Alodjants, S. M. Arakelian, “Quantum phase measurements and nonclassical polarization states of light,” J. Mod. Opt. 46, 475–507 (1999).
    [CrossRef]
  4. H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
    [CrossRef] [PubMed]
  5. P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observation with white light,” J. Mod. Opt. 41, 663–667 (1994).
    [CrossRef]
  6. H. Schmitzer, S. Klein, W. Dultz, “An experimental test of the path dependency of Pancharatnam’s geometric phase,” J. Mod. Opt. 45, 1039–1047 (1998).
    [CrossRef]
  7. T. H. Chyba, L. J. Wang, L. Mandel, R. Simon, “Measurement of the Pancharatnam phase for a light beam,” Opt. Lett. 13, 562–564 (1988).
    [CrossRef] [PubMed]
  8. P. Hariharan, H. Ramachandran, K. Suresh, J. Samuel, “The Pancharatnam phase as a strictly geometric phase: a demonstration using pure projections,” J. Mod. Opt. 44, 707–713 (1997).
    [CrossRef]
  9. R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometrical phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
    [CrossRef] [PubMed]
  10. S. M. Rytov, Dokl. Akad. Nauk SSSR 18, 263 (1938). English translation in Topological Phases in Quantum Theory, B. Markovski, V. I. Vinitsky, eds. (World Scientific, Singapore, 1989), p. 6.
  11. V. V. Vladimirsky, Dokl. Akad. Nauk SSSR 31, 222 (1941). English translation in Topological Phases in Quantum Theory, B. Markovski, V. I. Vinitsky, eds. (World Scientific, Singapore, 1989), p. 11.
  12. A. Tomita, 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. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
    [CrossRef]
  14. A. V. Tavrov, Y. Miyamoto, T. Kawabata, M. Takeda, V. V. Andreev, “A method to evaluate the geometrical spin-redirection phase for a nonplanar ray,” J. Opt. Soc. Am. A 16, 919–921 (1999).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964), pp. 700–703.
  16. B. K. Pierscionek, D. D. Y. C. Chan, “Mathematical description of isogyre formation in refracting structures,” Ophthalmic Physiol. Opt. 13, 212–215 (1993).
    [CrossRef]
  17. W. N. Charman, “Explanation for the observation of isogyres in crystalline lenses viewed between crossed polarizers,” Ophthalmic Physiol. Opt. 13, 209–211 (1993).
    [CrossRef] [PubMed]

1999 (2)

A. P. Alodjants, S. M. Arakelian, “Quantum phase measurements and nonclassical polarization states of light,” J. Mod. Opt. 46, 475–507 (1999).
[CrossRef]

A. V. Tavrov, Y. Miyamoto, T. Kawabata, M. Takeda, V. V. Andreev, “A method to evaluate the geometrical spin-redirection phase for a nonplanar ray,” J. Opt. Soc. Am. A 16, 919–921 (1999).
[CrossRef]

1998 (1)

H. Schmitzer, S. Klein, W. Dultz, “An experimental test of the path dependency of Pancharatnam’s geometric phase,” J. Mod. Opt. 45, 1039–1047 (1998).
[CrossRef]

1997 (1)

P. Hariharan, H. Ramachandran, K. Suresh, J. Samuel, “The Pancharatnam phase as a strictly geometric phase: a demonstration using pure projections,” J. Mod. Opt. 44, 707–713 (1997).
[CrossRef]

1994 (1)

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

1993 (3)

B. K. Pierscionek, D. D. Y. C. Chan, “Mathematical description of isogyre formation in refracting structures,” Ophthalmic Physiol. Opt. 13, 212–215 (1993).
[CrossRef]

W. N. Charman, “Explanation for the observation of isogyres in crystalline lenses viewed between crossed polarizers,” Ophthalmic Physiol. Opt. 13, 209–211 (1993).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, “Bargman invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

1990 (1)

M. Berry, “Anticipation of the geometric phase,” Phys. Today 43, 34–40 (1990).
[CrossRef]

1989 (1)

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

1988 (2)

R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometrical phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[CrossRef] [PubMed]

T. H. Chyba, L. J. Wang, L. Mandel, R. Simon, “Measurement of the Pancharatnam phase for a light beam,” Opt. Lett. 13, 562–564 (1988).
[CrossRef] [PubMed]

1986 (1)

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

1985 (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

1941 (1)

V. V. Vladimirsky, Dokl. Akad. Nauk SSSR 31, 222 (1941). English translation in Topological Phases in Quantum Theory, B. Markovski, V. I. Vinitsky, eds. (World Scientific, Singapore, 1989), p. 11.

1938 (1)

S. M. Rytov, Dokl. Akad. Nauk SSSR 18, 263 (1938). English translation in Topological Phases in Quantum Theory, B. Markovski, V. I. Vinitsky, eds. (World Scientific, Singapore, 1989), p. 6.

Alodjants, A. P.

A. P. Alodjants, S. M. Arakelian, “Quantum phase measurements and nonclassical polarization states of light,” J. Mod. Opt. 46, 475–507 (1999).
[CrossRef]

Andreev, V. V.

Arakelian, S. M.

A. P. Alodjants, S. M. Arakelian, “Quantum phase measurements and nonclassical polarization states of light,” J. Mod. Opt. 46, 475–507 (1999).
[CrossRef]

Berry, M.

M. Berry, “Anticipation of the geometric phase,” Phys. Today 43, 34–40 (1990).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964), pp. 700–703.

Chan, D. D. Y. C.

B. K. Pierscionek, D. D. Y. C. Chan, “Mathematical description of isogyre formation in refracting structures,” Ophthalmic Physiol. Opt. 13, 212–215 (1993).
[CrossRef]

Charman, W. N.

W. N. Charman, “Explanation for the observation of isogyres in crystalline lenses viewed between crossed polarizers,” Ophthalmic Physiol. Opt. 13, 209–211 (1993).
[CrossRef] [PubMed]

Chiao, R. Y.

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

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

Chow, W. W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Chyba, T. H.

Dultz, W.

H. Schmitzer, S. Klein, W. Dultz, “An experimental test of the path dependency of Pancharatnam’s geometric phase,” J. Mod. Opt. 45, 1039–1047 (1998).
[CrossRef]

Gea-Banacloche, J.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Hariharan, P.

P. Hariharan, H. Ramachandran, K. Suresh, J. Samuel, “The Pancharatnam phase as a strictly geometric phase: a demonstration using pure projections,” J. Mod. Opt. 44, 707–713 (1997).
[CrossRef]

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

Jiao, H.

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

Kawabata, T.

Kimble, H. J.

R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometrical phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[CrossRef] [PubMed]

Klein, S.

H. Schmitzer, S. Klein, W. Dultz, “An experimental test of the path dependency of Pancharatnam’s geometric phase,” J. Mod. Opt. 45, 1039–1047 (1998).
[CrossRef]

Larkin, K. G.

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

Mandel, L.

Miyamoto, Y.

Mukunda, N.

R. Simon, N. Mukunda, “Bargman invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

Nathel, H.

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

Pedrotti, L. M.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Pierscionek, B. K.

B. K. Pierscionek, D. D. Y. C. Chan, “Mathematical description of isogyre formation in refracting structures,” Ophthalmic Physiol. Opt. 13, 212–215 (1993).
[CrossRef]

Ramachandran, H.

P. Hariharan, H. Ramachandran, K. Suresh, J. Samuel, “The Pancharatnam phase as a strictly geometric phase: a demonstration using pure projections,” J. Mod. Opt. 44, 707–713 (1997).
[CrossRef]

Roy, M.

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

Rytov, S. M.

S. M. Rytov, Dokl. Akad. Nauk SSSR 18, 263 (1938). English translation in Topological Phases in Quantum Theory, B. Markovski, V. I. Vinitsky, eds. (World Scientific, Singapore, 1989), p. 6.

Samuel, J.

P. Hariharan, H. Ramachandran, K. Suresh, J. Samuel, “The Pancharatnam phase as a strictly geometric phase: a demonstration using pure projections,” J. Mod. Opt. 44, 707–713 (1997).
[CrossRef]

Sanders, V. E.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Schleich, W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Schmitzer, H.

H. Schmitzer, S. Klein, W. Dultz, “An experimental test of the path dependency of Pancharatnam’s geometric phase,” J. Mod. Opt. 45, 1039–1047 (1998).
[CrossRef]

Scully, M. O.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Simon, R.

R. Simon, N. Mukunda, “Bargman invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

T. H. Chyba, L. J. Wang, L. Mandel, R. Simon, “Measurement of the Pancharatnam phase for a light beam,” Opt. Lett. 13, 562–564 (1988).
[CrossRef] [PubMed]

R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometrical phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[CrossRef] [PubMed]

Sudarshan, E. C. G.

R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometrical phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[CrossRef] [PubMed]

Suresh, K.

P. Hariharan, H. Ramachandran, K. Suresh, J. Samuel, “The Pancharatnam phase as a strictly geometric phase: a demonstration using pure projections,” J. Mod. Opt. 44, 707–713 (1997).
[CrossRef]

Takeda, M.

Tavrov, A. V.

Tomita, A.

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

Vladimirsky, V. V.

V. V. Vladimirsky, Dokl. Akad. Nauk SSSR 31, 222 (1941). English translation in Topological Phases in Quantum Theory, B. Markovski, V. I. Vinitsky, eds. (World Scientific, Singapore, 1989), p. 11.

Wang, L. J.

Wilkinson, S. R.

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964), pp. 700–703.

Dokl. Akad. Nauk SSSR (2)

S. M. Rytov, Dokl. Akad. Nauk SSSR 18, 263 (1938). English translation in Topological Phases in Quantum Theory, B. Markovski, V. I. Vinitsky, eds. (World Scientific, Singapore, 1989), p. 6.

V. V. Vladimirsky, Dokl. Akad. Nauk SSSR 31, 222 (1941). English translation in Topological Phases in Quantum Theory, B. Markovski, V. I. Vinitsky, eds. (World Scientific, Singapore, 1989), p. 11.

J. Mod. Opt. (4)

A. P. Alodjants, S. M. Arakelian, “Quantum phase measurements and nonclassical polarization states of light,” J. Mod. Opt. 46, 475–507 (1999).
[CrossRef]

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

H. Schmitzer, S. Klein, W. Dultz, “An experimental test of the path dependency of Pancharatnam’s geometric phase,” J. Mod. Opt. 45, 1039–1047 (1998).
[CrossRef]

P. Hariharan, H. Ramachandran, K. Suresh, J. Samuel, “The Pancharatnam phase as a strictly geometric phase: a demonstration using pure projections,” J. Mod. Opt. 44, 707–713 (1997).
[CrossRef]

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

Ophthalmic Physiol. Opt. (2)

B. K. Pierscionek, D. D. Y. C. Chan, “Mathematical description of isogyre formation in refracting structures,” Ophthalmic Physiol. Opt. 13, 212–215 (1993).
[CrossRef]

W. N. Charman, “Explanation for the observation of isogyres in crystalline lenses viewed between crossed polarizers,” Ophthalmic Physiol. Opt. 13, 209–211 (1993).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (1)

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

R. Simon, N. Mukunda, “Bargman invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometrical phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[CrossRef] [PubMed]

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

Phys. Today (1)

M. Berry, “Anticipation of the geometric phase,” Phys. Today 43, 34–40 (1990).
[CrossRef]

Rev. Mod. Phys. (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964), pp. 700–703.

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Figures (5)

Fig. 1
Fig. 1

Ray path through an optical system with reflection and refraction.

Fig. 2
Fig. 2

Definition of the directions of p- and s-field components with the pair of normalized wave vectors (ki, kr) in the case of reflection and (ki, kt) in the case of refraction.

Fig. 3
Fig. 3

(a) Three-dimensional interferometer for observation of spin-redirection phase. (b) Representation of spin-redirection phase on the unit sphere swept by the tip of a wave vector.

Fig. 4
Fig. 4

(a) Sagnac interferometer for observation of the Pancharatnam phase. (b) Representation of Pancharatnam phase on the Poincaré sphere.

Fig. 5
Fig. 5

(a) Ray path through a spherical homogeneous and isotropic lens. (b) Result of the computer simulation of isogyre image.

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

kn=MK0,Pn=MP0,
Pn=MP0=λP0,
λ=exp(±iΩS),
(Pn+P0)t(Pn+P0)*=2(1+cos ΩS),
Pn=MP0=[cos ΩS, sin ΩS, 0]t.
A=100000001,
(APn)t(APn)*=(1+cos 2ΩS)/2.
M=Fn+1EntFn  E1tF1E0t,
El=[kli, nl×kli,nl]t
=[kli, pli, sli]t,
Fl+1=[kl+1r, nl×kl+1r, nl]t
=[kl+1r, pl+1r, sl+1r]t,
nl=kli×kl+1rkli×kl+1r
Fl+1=[kl+1t, nl×kl+1t, nl]t
=[kl+1t, pl+1t, sl+1t]t,
L=LppLpsLspLss,
Tl=1000(Ll)0,
M˜=FnTnEn-1tF1T1E0t,
El=[kli, nl-1×kli, nl-1]t
=[kli, pli, sli]t.
Rl=[al, bl, kl]t.
M˜=M˜lnRlt(Ll)00001RlM˜0l,
M˜P0=P0 exp(±iΩp).
L=100-1.
M˜α=Fα4TEα3tFα3TEα2tFα2TEα1tFα1TEα0t
=00-1-sin θ-cos θ0-cos θ sin θ0,
M˜β=Fβ3TEβ2tFβ2TEβ1tFβ1TEβ0t
=00-1010-100.
R=[Z,Y,kn]t
=Fα5Eα4t=Fβ4Eβ3t
=001010-100.
RM˜α=cos(π+θ)-sin(π+θ)0sin(π+θ)cos(π+θ)0001,
RM˜β=-100010001.
RM˜αP0=exp[±i(π+θ)]P0.
RM˜βP0=P0*.
[ARM˜αP0]t[ARM˜βP0]*=exp[±i(π+θ)]P0tAtAP0*=12exp[i(±θ)].
Lα2=R-1AR
=cos θ-sin θsin θcos θ1000cos θsin θ-sin θcos θ
=cos2 θsin θ cos θsin θ cos θsin2 θ,
T2=1000(L2)0=1000cos2 θsin θ cos θ0sin θ cos θsin2 θ.
M˜α=Fα4TEα3tFα3Tα2Eα2tFα2TEα1tFα1TEα0t
=00-1sin θ cos θsin2 θ0-cos2 θ-sin θ cos θ0,
M˜β=Fβ6TEβ5tFβ5TEβ4tFβ4TEβ3tFβ3Tβ2×Eβ2tFβ2TEβ1tFβ1TEβ0t
=00-1sin θ cos θ-sin2 θ0+cos2 θ-sin θ cos θ0.
Pα4=M˜α[±i/2, 1/2,0]t
=±(i/2)exp(iθ)[0, sin θ, cos θ]t,
Pβ6=M˜β[±i/2, 12, 0]t
=±(i/2)exp(±iθ)[0, sin θ, -cos θ]t.
Pβ6tPα4*=(-1/2)cos 2θ exp(±i2θ).
P=000010001F3E2tF21000LpB000LsB×E1tF11000LpA000LsAE0t100,

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