Abstract

We use the concept of turns to provide a geometrical representation of the action of any lossless multilayer, which can be considered to be analogous in the unit disk to sliding vectors in Euclidean geometry. This construction clearly shows the peculiar effects arising in the composition of multilayers. A simple optical experiment revealing the appearance of the Wigner angle is analyzed in this framework.

© 2004 Optical Society of America

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  2. L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics (Addison-Wesley, Reading, Mass., 1981).
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    [CrossRef]
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    [CrossRef]
  5. M. Juárez, M. Santander, “Turns for the Lorentz group,” J. Phys. A 15, 3411–3424 (1982).
    [CrossRef]
  6. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
    [CrossRef] [PubMed]
  7. R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
    [CrossRef]
  8. H. A. Macleod, Thin-Film Optical Filters (Hilger, Bristol, UK, 1986).
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    [CrossRef] [PubMed]
  10. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).
  11. L. M. Brekovskikh, Waves in Layered Media (Academic, New York, 1960).
  12. J. Lekner, Theory of Reflection (Martinus Nijhoff, Dordrecht, The Netherlands, 1987).
  13. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  14. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
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  16. P. C. S. Hayfield, G. W. T. White, “An assessment of the stability of the Drude–Tronstad polarized light method forthe study of film growth on polycrystalline metals,” in Ellipsometry in the Measurements of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ. 256 (U.S. Government Printing Office, Washington, D.C., 1964), pp. 157–200. For a more recent review of the model see Ref. 13, Sec. 4.6.
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    [CrossRef]
  18. I. Ohlı́dal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, Vol. 41, E. Wolf, ed. (Elsevier, North-Holland, Amsterdam, 2000), pp. 181–282.
  19. J. J. Monzón, T. Yonte, L. L. Sánchez-Soto, J. F. Cariñena, “Geometrical setting for the classification of multilayers,” J. Opt. Soc. Am. A 19, 985–991 (2002).
    [CrossRef]
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    [CrossRef]
  21. A. F. Beardon, The Geometry of Discrete Groups (Springer, New York, 1983), Chap. 7.
  22. A. Ben-Menahem, “Wigner’s rotation revisited,” Am. J. Phys. 53, 62–66 (1985).
    [CrossRef]
  23. D. A. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  24. A. A. Ungar, “The relativistic velocity composition paradox and the Thomas rotation,” Found. Phys. 19, 1385–1396 (1989).
    [CrossRef]
  25. J. J. Monzón, L. L. Sánchez-Soto, “Origin of the Thomas rotation that arises in lossless multilayers,” J. Opt. Soc. Am. A 16, 2786–2792 (1999).
    [CrossRef]
  26. J. J. Monzón, L. L. Sánchez-Soto, “A simple optical demonstration of geometric phases from multilayer stacks: the Wigner angle as an anholonomy,” J. Mod. Opt. 48, 21–34 (2001).
    [CrossRef]
  27. A. Shapere, F. Wilczek, eds. Geometric Phases in Physics (World Scientific, Singapore, 1989).
  28. P. K. Aravind, “The Wigner angle as an anholonomy in rapidity space,” Am. J. Phys. 65, 634–636 (1997).
    [CrossRef]

2002

2001

L. L. Sánchez-Soto, J. J. Monzón, T. Yonte, J. F. Cariñena, “Simple trace criterion for classification of multilayers,” Opt. Lett. 26, 1400–1402 (2001).
[CrossRef]

J. J. Monzón, L. L. Sánchez-Soto, “A simple optical demonstration of geometric phases from multilayer stacks: the Wigner angle as an anholonomy,” J. Mod. Opt. 48, 21–34 (2001).
[CrossRef]

1999

1997

P. K. Aravind, “The Wigner angle as an anholonomy in rapidity space,” Am. J. Phys. 65, 634–636 (1997).
[CrossRef]

1989

A. A. Ungar, “The relativistic velocity composition paradox and the Thomas rotation,” Found. Phys. 19, 1385–1396 (1989).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

1985

A. Ben-Menahem, “Wigner’s rotation revisited,” Am. J. Phys. 53, 62–66 (1985).
[CrossRef]

1982

M. Juárez, M. Santander, “Turns for the Lorentz group,” J. Phys. A 15, 3411–3424 (1982).
[CrossRef]

1972

1948

F. Abelès, “Sur la propagation des ondes electromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 504–520 (1948).

Abelès, F.

F. Abelès, “Sur la propagation des ondes electromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 504–520 (1948).

Apfel, J. H.

Aravind, P. K.

P. K. Aravind, “The Wigner angle as an anholonomy in rapidity space,” Am. J. Phys. 65, 634–636 (1997).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Beardon, A. F.

A. F. Beardon, The Geometry of Discrete Groups (Springer, New York, 1983), Chap. 7.

Ben-Menahem, A.

A. Ben-Menahem, “Wigner’s rotation revisited,” Am. J. Phys. 53, 62–66 (1985).
[CrossRef]

Biedenharn, L. C.

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics (Addison-Wesley, Reading, Mass., 1981).

Brekovskikh, L. M.

L. M. Brekovskikh, Waves in Layered Media (Academic, New York, 1960).

Cariñena, J. F.

Franta, D.

I. Ohlı́dal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, Vol. 41, E. Wolf, ed. (Elsevier, North-Holland, Amsterdam, 2000), pp. 181–282.

Hamilton, W. R.

W. R. Hamilton, Lectures on Quaternions (Hodges & Smith, Dublin, 1853).

Hayfield, P. C. S.

P. C. S. Hayfield, G. W. T. White, “An assessment of the stability of the Drude–Tronstad polarized light method forthe study of film growth on polycrystalline metals,” in Ellipsometry in the Measurements of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ. 256 (U.S. Government Printing Office, Washington, D.C., 1964), pp. 157–200. For a more recent review of the model see Ref. 13, Sec. 4.6.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).

Jackson, D. A.

D. A. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Juárez, M.

M. Juárez, M. Santander, “Turns for the Lorentz group,” J. Phys. A 15, 3411–3424 (1982).
[CrossRef]

Lekner, J.

J. Lekner, Theory of Reflection (Martinus Nijhoff, Dordrecht, The Netherlands, 1987).

López-Lacasta, C.

Louck, J. D.

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics (Addison-Wesley, Reading, Mass., 1981).

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Hilger, Bristol, UK, 1986).

Monzón, J. J.

Mukunda, N.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef] [PubMed]

Ohli´dal, I.

I. Ohlı́dal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, Vol. 41, E. Wolf, ed. (Elsevier, North-Holland, Amsterdam, 2000), pp. 181–282.

Sánchez-Soto, L. L.

Santander, M.

M. Juárez, M. Santander, “Turns for the Lorentz group,” J. Phys. A 15, 3411–3424 (1982).
[CrossRef]

Simon, R.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

Ungar, A. A.

A. A. Ungar, “The relativistic velocity composition paradox and the Thomas rotation,” Found. Phys. 19, 1385–1396 (1989).
[CrossRef]

White, G. W. T.

P. C. S. Hayfield, G. W. T. White, “An assessment of the stability of the Drude–Tronstad polarized light method forthe study of film growth on polycrystalline metals,” in Ellipsometry in the Measurements of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ. 256 (U.S. Government Printing Office, Washington, D.C., 1964), pp. 157–200. For a more recent review of the model see Ref. 13, Sec. 4.6.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Yonte, T.

Am. J. Phys.

A. Ben-Menahem, “Wigner’s rotation revisited,” Am. J. Phys. 53, 62–66 (1985).
[CrossRef]

P. K. Aravind, “The Wigner angle as an anholonomy in rapidity space,” Am. J. Phys. 65, 634–636 (1997).
[CrossRef]

Ann. Phys. (Paris)

F. Abelès, “Sur la propagation des ondes electromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 504–520 (1948).

Appl. Opt.

Found. Phys.

A. A. Ungar, “The relativistic velocity composition paradox and the Thomas rotation,” Found. Phys. 19, 1385–1396 (1989).
[CrossRef]

J. Math. Phys.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

J. Mod. Opt.

J. J. Monzón, L. L. Sánchez-Soto, “A simple optical demonstration of geometric phases from multilayer stacks: the Wigner angle as an anholonomy,” J. Mod. Opt. 48, 21–34 (2001).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

M. Juárez, M. Santander, “Turns for the Lorentz group,” J. Phys. A 15, 3411–3424 (1982).
[CrossRef]

Opt. Commun.

J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef] [PubMed]

Other

W. R. Hamilton, Lectures on Quaternions (Hodges & Smith, Dublin, 1853).

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics (Addison-Wesley, Reading, Mass., 1981).

H. A. Macleod, Thin-Film Optical Filters (Hilger, Bristol, UK, 1986).

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).

L. M. Brekovskikh, Waves in Layered Media (Academic, New York, 1960).

J. Lekner, Theory of Reflection (Martinus Nijhoff, Dordrecht, The Netherlands, 1987).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

A. Shapere, F. Wilczek, eds. Geometric Phases in Physics (World Scientific, Singapore, 1989).

I. Ohlı́dal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, Vol. 41, E. Wolf, ed. (Elsevier, North-Holland, Amsterdam, 2000), pp. 181–282.

P. C. S. Hayfield, G. W. T. White, “An assessment of the stability of the Drude–Tronstad polarized light method forthe study of film growth on polycrystalline metals,” in Ellipsometry in the Measurements of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ. 256 (U.S. Government Printing Office, Washington, D.C., 1964), pp. 157–200. For a more recent review of the model see Ref. 13, Sec. 4.6.

A. F. Beardon, The Geometry of Discrete Groups (Springer, New York, 1983), Chap. 7.

D. A. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

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