Abstract

An evaluation method is suggested for determining the Berry geometrical spin-redirection phase in systems with a set of separated optical elements. The evaluation considers the coordinate transformation, taking into account the directions of the beam. The spin-redirection phase contains information about the nonplanarity of beam propagation. This geometrical phase can be determined even if the ray trajectory is traced as an unclosed topological contour on the unit sphere associated with wave-vector directions.

© 1999 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. 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]
  3. P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observation with white light,” J. Mod. Opt. 41, 663–667 (1994).
    [CrossRef]
  4. S. Klein, W. Dultz, H. Schmitzer, “Pancharatnam’s topological phase in relation to the dynamical phase in polarization optics,” in Frontiers of Fundamental Physics, M. Barone, F. Selli, eds. (Plenum, New York, 1994), pp. 437–442.
  5. 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]
  6. S. M. Rytov, in Russian in 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.
  7. V. V. Vladimirsky, in Russian in 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.
  8. 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]
  9. A. Tavrov, V. Andreev, D. Ublinsky, D. Orlov, K. Kogan, “The role of topological phases in the microimage formation,” in Optics in Science and New Technology, J.-S. Chang, J.-H. Lee, S.-Y. Lee, C.-H. Nam, eds., Proc. SPIE2778, 1072–1073 (1996).
  10. V. Andreev, A. Tavrov, D. Ublinsky, D. Orlov, K. Kogan, “Topological phases by Rytov–Vladimirsky and the wavefront dislocations,” Bulletin of the Lebedev Physical Institute (Russian Academy of Science, 1996), Vol. 5–6, pp. 44–52.

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]

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 (1)

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]

1941 (1)

V. V. Vladimirsky, in Russian in 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, in Russian in 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.

Andreev, V.

V. Andreev, A. Tavrov, D. Ublinsky, D. Orlov, K. Kogan, “Topological phases by Rytov–Vladimirsky and the wavefront dislocations,” Bulletin of the Lebedev Physical Institute (Russian Academy of Science, 1996), Vol. 5–6, pp. 44–52.

A. Tavrov, V. Andreev, D. Ublinsky, D. Orlov, K. Kogan, “The role of topological phases in the microimage formation,” in Optics in Science and New Technology, J.-S. Chang, J.-H. Lee, S.-Y. Lee, C.-H. Nam, eds., Proc. SPIE2778, 1072–1073 (1996).

Berry, M.

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

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]

Chyba, T. H.

Dultz, W.

S. Klein, W. Dultz, H. Schmitzer, “Pancharatnam’s topological phase in relation to the dynamical phase in polarization optics,” in Frontiers of Fundamental Physics, M. Barone, F. Selli, eds. (Plenum, New York, 1994), pp. 437–442.

Hariharan, P.

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]

Klein, S.

S. Klein, W. Dultz, H. Schmitzer, “Pancharatnam’s topological phase in relation to the dynamical phase in polarization optics,” in Frontiers of Fundamental Physics, M. Barone, F. Selli, eds. (Plenum, New York, 1994), pp. 437–442.

Kogan, K.

A. Tavrov, V. Andreev, D. Ublinsky, D. Orlov, K. Kogan, “The role of topological phases in the microimage formation,” in Optics in Science and New Technology, J.-S. Chang, J.-H. Lee, S.-Y. Lee, C.-H. Nam, eds., Proc. SPIE2778, 1072–1073 (1996).

V. Andreev, A. Tavrov, D. Ublinsky, D. Orlov, K. Kogan, “Topological phases by Rytov–Vladimirsky and the wavefront dislocations,” Bulletin of the Lebedev Physical Institute (Russian Academy of Science, 1996), Vol. 5–6, pp. 44–52.

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.

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]

Orlov, D.

A. Tavrov, V. Andreev, D. Ublinsky, D. Orlov, K. Kogan, “The role of topological phases in the microimage formation,” in Optics in Science and New Technology, J.-S. Chang, J.-H. Lee, S.-Y. Lee, C.-H. Nam, eds., Proc. SPIE2778, 1072–1073 (1996).

V. Andreev, A. Tavrov, D. Ublinsky, D. Orlov, K. Kogan, “Topological phases by Rytov–Vladimirsky and the wavefront dislocations,” Bulletin of the Lebedev Physical Institute (Russian Academy of Science, 1996), Vol. 5–6, pp. 44–52.

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, in Russian in 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.

Schmitzer, H.

S. Klein, W. Dultz, H. Schmitzer, “Pancharatnam’s topological phase in relation to the dynamical phase in polarization optics,” in Frontiers of Fundamental Physics, M. Barone, F. Selli, eds. (Plenum, New York, 1994), pp. 437–442.

Simon, R.

Tavrov, A.

A. Tavrov, V. Andreev, D. Ublinsky, D. Orlov, K. Kogan, “The role of topological phases in the microimage formation,” in Optics in Science and New Technology, J.-S. Chang, J.-H. Lee, S.-Y. Lee, C.-H. Nam, eds., Proc. SPIE2778, 1072–1073 (1996).

V. Andreev, A. Tavrov, D. Ublinsky, D. Orlov, K. Kogan, “Topological phases by Rytov–Vladimirsky and the wavefront dislocations,” Bulletin of the Lebedev Physical Institute (Russian Academy of Science, 1996), Vol. 5–6, pp. 44–52.

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]

Ublinsky, D.

A. Tavrov, V. Andreev, D. Ublinsky, D. Orlov, K. Kogan, “The role of topological phases in the microimage formation,” in Optics in Science and New Technology, J.-S. Chang, J.-H. Lee, S.-Y. Lee, C.-H. Nam, eds., Proc. SPIE2778, 1072–1073 (1996).

V. Andreev, A. Tavrov, D. Ublinsky, D. Orlov, K. Kogan, “Topological phases by Rytov–Vladimirsky and the wavefront dislocations,” Bulletin of the Lebedev Physical Institute (Russian Academy of Science, 1996), Vol. 5–6, pp. 44–52.

Vladimirsky, V. V.

V. V. Vladimirsky, in Russian in 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]

Dokl. Akad. Nauk SSSR (2)

S. M. Rytov, in Russian in 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, in Russian in 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. (1)

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

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. (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]

Phys. Today (1)

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

Other (3)

S. Klein, W. Dultz, H. Schmitzer, “Pancharatnam’s topological phase in relation to the dynamical phase in polarization optics,” in Frontiers of Fundamental Physics, M. Barone, F. Selli, eds. (Plenum, New York, 1994), pp. 437–442.

A. Tavrov, V. Andreev, D. Ublinsky, D. Orlov, K. Kogan, “The role of topological phases in the microimage formation,” in Optics in Science and New Technology, J.-S. Chang, J.-H. Lee, S.-Y. Lee, C.-H. Nam, eds., Proc. SPIE2778, 1072–1073 (1996).

V. Andreev, A. Tavrov, D. Ublinsky, D. Orlov, K. Kogan, “Topological phases by Rytov–Vladimirsky and the wavefront dislocations,” Bulletin of the Lebedev Physical Institute (Russian Academy of Science, 1996), Vol. 5–6, pp. 44–52.

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Figures (2)

Fig. 1
Fig. 1

Geometry of the skew reflection.

Fig. 2
Fig. 2

Delay of the circular polarized beam resulting from a nonplanar propagation.

Equations (16)

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

AP=P exp(iφPanch).
Φcomb=φ+ϕ.
k=kxkykzP=PxPyPz.
k1=M1k0,
P1=M1P0.
M1=cos ξ-sin ξ0sin ξcos ξ0001 cos 2η0-sin 2η010sin 2η0cos 2η×cos ξsin ξ0-sin ξcos ξ0001,
M1=[ξ][η][-ξ].
M=M2M1=[ξ21][η21][-ξ21].
M=Fi+1EiTFiE1TF1E0T.
n=ki×ki+1ki×ki+1:
Ei=[ki, (n×ki), n],
Fi+1=[ki+1, (n×ki+1), n],
k0T=(001).
M=cos 2ξ-sin 2ξ0sin 2ξcos 2ξ0001.
PM=P0 exp(±2jξ).
MPlinear  M10=cos 2ξ-sin 2ξ.

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