Abstract

We describe the rotation of images by means of optical beam steering with use of the concept of geometric phase. The discussion is concentrated on systems composed of discrete reflections but can be generalized to refractive steering systems. Geometric phase reduces the analysis of image rotation to simple geometric constructions and the calculation of areas on a sphere. The analysis also applies to the rotation of linear polarization of the light with ideal mirrors.

© 1999 Optical Society of America

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References

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  1. Optical Design, MIL-HDBK-141, (Washington, D.C., 1962).
  2. D. W. Swift, “Image rotation devices—a comparative study,” Opt. Laser Technol., 175–188 (August1972).
  3. R. E. Hopkins, “Mirror and prism systems,” Appl. Opt. Opt. Eng. 3, 269–308 (1965).
  4. M. V. Berry, “Quantum phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
    [CrossRef]
  5. R. Y. Chiao, Y.-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986);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]
  6. M. Segev, R. Solomon, A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
    [CrossRef] [PubMed]
  7. R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 13, 1211–1213 (1988).
    [CrossRef]
  8. R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
    [CrossRef]
  9. Y. Aharonov, J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
    [CrossRef] [PubMed]
  10. J. Samuel, R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
    [CrossRef] [PubMed]
  11. M. Kitano, T. Yabuzaki, T. Ogawa, “Comment on ‘Observation of Berry’s topological phase by use of an optical fiber’,” Phys. Rev. Lett. 58, 523 (1987).
    [CrossRef]
  12. F. D. M. Haldane, “Path dependence of the geometric rotation of polarization in optical fibers,” Opt. Lett. 11, 730–732 (1986).
    [CrossRef] [PubMed]
  13. J. Segert, “Photon’s Berry’s phase as a classical topological effect,” Phys. Rev. A 36, 10–15 (1987).
    [CrossRef] [PubMed]
  14. L. H. Ryder, “The optical Berry phase and the Gauss–Bonnet theorem,” Eur. J. Phys. 12, 15–18 (1991).
    [CrossRef]
  15. W. J. Smith, “Image formation: geometrical and physical optics,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), pp. 2-41–2-51.
  16. A. Gleichen, The Theory of Modern Optical Instruments (His Majesty’s Stationery Office, London, 1921).
  17. E. J. Galvez, P. M. Koch, “Use of four mirrors to rotate linear polarization but preserve input–output collinearity. II,” J. Opt. Soc. Am. A 14, 3410–3414 (1997).
    [CrossRef]
  18. Empirically we found a commercial dielectric mirror, New Focus model 5102, that conserved linear polarization and had high s and p polarization reflectances for an angle of incidence of 45° at 633 nm.

1997 (2)

1992 (1)

M. Segev, R. Solomon, A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
[CrossRef] [PubMed]

1991 (1)

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

1988 (2)

J. Samuel, R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 13, 1211–1213 (1988).
[CrossRef]

1987 (3)

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

M. Kitano, T. Yabuzaki, T. Ogawa, “Comment on ‘Observation of Berry’s topological phase by use of an optical fiber’,” Phys. Rev. Lett. 58, 523 (1987).
[CrossRef]

J. Segert, “Photon’s Berry’s phase as a classical topological effect,” Phys. Rev. A 36, 10–15 (1987).
[CrossRef] [PubMed]

1986 (2)

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);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]

1984 (1)

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

1972 (1)

D. W. Swift, “Image rotation devices—a comparative study,” Opt. Laser Technol., 175–188 (August1972).

1965 (1)

R. E. Hopkins, “Mirror and prism systems,” Appl. Opt. Opt. Eng. 3, 269–308 (1965).

Aharonov, Y.

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

Anandan, J.

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

Berry, M. V.

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

Bhandari, R.

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

R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 13, 1211–1213 (1988).
[CrossRef]

J. Samuel, R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

Chiao, R. Y.

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

Galvez, E. J.

Gleichen, A.

A. Gleichen, The Theory of Modern Optical Instruments (His Majesty’s Stationery Office, London, 1921).

Haldane, F. D. M.

Hopkins, R. E.

R. E. Hopkins, “Mirror and prism systems,” Appl. Opt. Opt. Eng. 3, 269–308 (1965).

Kitano, M.

M. Kitano, T. Yabuzaki, T. Ogawa, “Comment on ‘Observation of Berry’s topological phase by use of an optical fiber’,” Phys. Rev. Lett. 58, 523 (1987).
[CrossRef]

Koch, P. M.

Ogawa, T.

M. Kitano, T. Yabuzaki, T. Ogawa, “Comment on ‘Observation of Berry’s topological phase by use of an optical fiber’,” Phys. Rev. Lett. 58, 523 (1987).
[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. 13, 1211–1213 (1988).
[CrossRef]

J. Samuel, R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

Segert, J.

J. Segert, “Photon’s Berry’s phase as a classical topological effect,” Phys. Rev. A 36, 10–15 (1987).
[CrossRef] [PubMed]

Segev, M.

M. Segev, R. Solomon, A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
[CrossRef] [PubMed]

Smith, W. J.

W. J. Smith, “Image formation: geometrical and physical optics,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), pp. 2-41–2-51.

Solomon, R.

M. Segev, R. Solomon, A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
[CrossRef] [PubMed]

Swift, D. W.

D. W. Swift, “Image rotation devices—a comparative study,” Opt. Laser Technol., 175–188 (August1972).

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);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]

Yabuzaki, T.

M. Kitano, T. Yabuzaki, T. Ogawa, “Comment on ‘Observation of Berry’s topological phase by use of an optical fiber’,” Phys. Rev. Lett. 58, 523 (1987).
[CrossRef]

Yariv, A.

M. Segev, R. Solomon, A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
[CrossRef] [PubMed]

Appl. Opt. Opt. Eng. (1)

R. E. Hopkins, “Mirror and prism systems,” Appl. Opt. Opt. Eng. 3, 269–308 (1965).

Eur. J. Phys. (1)

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

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

Opt. Laser Technol. (1)

D. W. Swift, “Image rotation devices—a comparative study,” Opt. Laser Technol., 175–188 (August1972).

Opt. Lett. (1)

Phys. Rep. (1)

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

Phys. Rev. A (1)

J. Segert, “Photon’s Berry’s phase as a classical topological effect,” Phys. Rev. A 36, 10–15 (1987).
[CrossRef] [PubMed]

Phys. Rev. Lett. (6)

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

J. Samuel, R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

M. Kitano, T. Yabuzaki, T. Ogawa, “Comment on ‘Observation of Berry’s topological phase by use of an optical fiber’,” Phys. Rev. Lett. 58, 523 (1987).
[CrossRef]

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

M. Segev, R. Solomon, A. Yariv, “Manifestation of Berry’s phase in image-bearing optical beams,” Phys. Rev. Lett. 69, 590–592 (1992).
[CrossRef] [PubMed]

R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 13, 1211–1213 (1988).
[CrossRef]

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

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

Other (4)

Optical Design, MIL-HDBK-141, (Washington, D.C., 1962).

W. J. Smith, “Image formation: geometrical and physical optics,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), pp. 2-41–2-51.

A. Gleichen, The Theory of Modern Optical Instruments (His Majesty’s Stationery Office, London, 1921).

Empirically we found a commercial dielectric mirror, New Focus model 5102, that conserved linear polarization and had high s and p polarization reflectances for an angle of incidence of 45° at 633 nm.

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Figures (9)

Fig. 1
Fig. 1

System to test the case of cyclic geometric phase with an even number of reflections: (a) experimental arrangement of mirrors with k0=(0, 1, 0), k1=(1, 0, 0), k2=(0, 0, 1), k3=(-sin θ, -cos θ, 0), and k4=k0 and (b) the k˜-sphere construction that predicts ϕ=90°+θ.

Fig. 2
Fig. 2

Measurements of image rotation as a function of the solid angle Ω (the calculated area in the k˜ sphere) for case 1(a) (see text).

Fig. 3
Fig. 3

Optical system to verify the predictions of a geometric phase for a figure-eight path in the helicity sphere: (a) experimental arrangement of mirrors with k0=(0, 1, 0), k1=(5, 0, 3)/(34)1/2, k2=(0, 0, 1), k3=(3, 4, 0)/5, and k4=k0 and (b) the corresponding k˜-sphere construction.

Fig. 4
Fig. 4

Analysis of the geometric phase of the variable-angle Porro system: (a) prism arrangement and (b) the corresponding k˜-sphere construction that predicts ϕ=2θ.

Fig. 5
Fig. 5

Analysis of the geometric phase of the consecutive Dove system: (a) prism arrangement and (b) the corresponding k˜-sphere construction that predicts ϕ=360°-2θ.

Fig. 6
Fig. 6

System to test the case of noncyclic geometric phase with an even number of reflections: (a) experimental arrangement of mirrors with k0=(0, 1, 0), k1=(-1, -1, 0)/21/2, k2=(-1, 0, 0), k3=(0, 0, 1), and k4=(sin θ, cos θ, 0), (b) the k˜-sphere construction, with the (dotted) closing geodesic that predicts ϕ=-(90°+θ), and (c) experimental results.

Fig. 7
Fig. 7

Analysis of the geometric phase of the Dove prism: (a) prism schematic and (b) the corresponding k˜-sphere construction, with the (dotted) closing geodesic that predicts ϕ=180°-2θ.

Fig. 8
Fig. 8

Analysis of the geometric phase of a popular polarization rotator made of orthogonal reflections, a noncyclic case with an odd number of mirrors and antiparallel input and output k˜ vectors: (a) mirror schematic with k0=(0, 1, 0), k1=(sin θ, 0, -cos θ), k2=(cos θ, 0, sin θ), and k3=k0 and (b) the corresponding k˜-sphere construction, with the (dotted) closing geodesic that predicts ϕ=90°+2θ.

Fig. 9
Fig. 9

Measurement of the polarization (squares) and image (circle) rotation for the mirror arrangement of Fig. 8(a). The results are consistent with the results of the k˜-sphere construction of Fig. 8(b).

Equations (1)

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k˜n=(-1)nkn,

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