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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References

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  1. W. R. Hamilton, Lectures on Quaternions (Hodges & Smith, Dublin, 1853).
  2. L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics (Addison-Wesley, Reading, Mass., 1981).
  3. J. J. Monzón, L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
    [Crossref]
  4. J. J. Monzón, L. L. Sánchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
    [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).
  9. J. H. Apfel, “Graphics in optical coating design,” Appl. Opt. 11, 1303–1312 (1972).
    [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).
  15. F. Abelès, “Sur la propagation des ondes electromagnétiques dans les milieux stratifiés,” Ann. Phys. (Paris) 3, 504–520 (1948).
  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.
  17. T. Yonte, J. J. Monzón, L. L. Sánchez-Soto, J. F. Cariñena, C. López-Lacasta, “Understanding multilayers from a geometrical viewpoint,” J. Opt. Soc. Am. A 19, 603–609 (2002).
    [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]
  20. 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]
  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 (2)

2001 (2)

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 (3)

1997 (1)

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

1989 (3)

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 (1)

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

1982 (1)

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

1972 (1)

1948 (1)

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, “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]

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. (2)

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) (1)

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

Appl. Opt. (1)

Found. Phys. (1)

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

J. Math. Phys. (1)

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. (1)

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 (4)

J. Phys. A (1)

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

Opt. Commun. (1)

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. (1)

Phys. Rev. Lett. (1)

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 (13)

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).

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

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.

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

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

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

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Figures (6)

Fig. 1
Fig. 1

Wave vectors of the input [Ea(+) and Es(-)] and output [Ea(-) and Es(+)] fields in a multilayer sandwiched between two identical, semi-infinite ambient and substrate media.

Fig. 2
Fig. 2

Representation of the sliding turn Tγ,ζ/2 in terms of two reflections in two lines Γ1 and Γ2 orthogonal to the axis of the translation γ, which has two fixed points zf1 and zf2. The transformation of a typical off-axis point zs is also shown.

Fig. 3
Fig. 3

Composition of two hyperbolic turns Tγ1,ζ1/2 and Tγ2,ζ2/2 by using a parallelogramlike law when the axes γ1 and γ2 of the translations intersect.

Fig. 4
Fig. 4

Composition of two multilayers represented by Hermitian matrices H1 and H2. H2 maps the point zs=-R2 into the origin, while H1 maps the origin into za=R1. We show also the associated turns T1 and T2 as well as the resulting turn T(12) obtained through the parallelogram law. The composite multilayer H1H2 transforms the point zs into za. The data of the corresponding multilayers are given in the text.

Fig. 5
Fig. 5

Same as Fig. 4, but now the resulting turn T(12) has been slid to three different positions along the axis. The corresponding points are also transformed by H1H2. All the geodesic triangles plotted have the same hyperbolic area Ψ.

Fig. 6
Fig. 6

Scheme of two Hermitian multilayers H1 and H2. The compound multilayer H1H2 obtained by putting together these two components induces a Wigner rotation of angle 2τ.

Equations (28)

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Ea(+)Ea(-)=MasEs(+)Es(-),
Mas=1/TasRas*/Tas*Ras/Tas1/Tas*αββ*α*.
Ras= |Ras|exp(iρ),Tas= |Tas|exp(iτ),
zs=Es(-)/Es(+),za=Ea(-)/Ea(+).
za=Φ[Mas, zs]=β*+α*zsα+βzs,
zf=Φ[Mas, zf],
zf=12β(-2i Im(α)±{[Tr(Mas)]2-4}1/2).
ζ=2 ln[12(Tr(Mas)+{[Tr(Mas)]2-4}1/2)]
(Mas)=1{2[Re(α)+1]}1/2α+1ββ*α*+1.
ζ(Mas)=2ζ(Mas).
Tγ,ζ/2Mas.
cosh ζ=cosh ζ1cosh ζ2+sinh ζ1sinh ζ2cos θ.
tan(Ω/2)=tanh(ζ1/2)tanh(ζ2/2)sin θ1-tanh(ζ1/2)tanh(ζ2/2)cos θ.
L1(v1)L2(v2)=L(12)(v)R(Ψ),
M=HU,
M=HU=1/|T|R*/|T|R/|T|1/|T|×exp(-iτ)00exp(iτ).
R=tanh(ζ/2)exp(iρ),T=sech(ζ/2)exp(iτ),
Φ[H, 0]=R,Φ[H-1, R]=0.
H1H2=H(12)U=1/|T(12)|R(12)*/|T(12)|R(12)/|T(12)|1/|T(12)|×exp(-iΨ/2)00exp(iΨ/2),
R(12)=R1+R21+R1*R2,T(12)=|T1T2|1+R1*R2,
Ψ2=arg[T(12)]=arg(1+R1R2*),
Φ[H2, -R2]=0,Φ[H1, 0]=R1.
Ω=Ψ,
T(12)H1H2.
R0j0=r0j[1-exp(-i2δj)]1-r0j2exp(-i2δj),
T0j0=t0jtj0exp(-iδj)1-r0j2exp(-i2δj),
δj=2πλ njdjcos θj,
M=j=1mM0j0.

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