Abstract

Two classical treatments of the transformation of the polarization of light in twisted birefringent media belong to Neumann and Kuske [see H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979)]. Since these authors have chosen parameters that characterize the state of polarization differently, the equations derived by them have been considered to be independent of one another. We show that duality exists between these equations. By the appropriate exchange of parameters, the first system of equations is transformed into the second one and vice versa. This duality follows from the duality between the two different parametric representations of the unitary unimodular matrix that describes the transformation of polarization in twisted birefringent media.

© 1999 Optical Society of America

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References

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  1. A. R. Stokes, The Theory of the Optical Properties of Inhomogeneous Materials (Spon, London, 1963).
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  3. H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).
  4. H. Aben, “Characteristic directions in optics of twisted birefringent media,” J. Opt. Soc. Am. A 3, 1414–1421 (1986).
    [Crossref]
  5. M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
    [Crossref]
  6. N. Vansteenkiste, P. Vignolo, A. Aspect, “Optical reversibility theorems for polarization: application to remote control of polarization,” J. Opt. Soc. Am. A 10, 2240–2245 (1993).
    [Crossref]
  7. A. Lakhtakia, W. S. Weiglhofer, “On light propagation in helicoidal bianisotropic mediums,” Proc. R. Soc. London Ser. A 448, 419–437 (1995).
    [Crossref]
  8. M. Schubert, B. Rheinländer, C. Cramer, H. Schmiedel, J. A. Woollam, “Generalized transmission ellipsometry for twisted biaxial dielectric media: application to chiral liquid crystals,” J. Opt. Soc. Am. A 13, 1930–1940 (1996).
    [Crossref]
  9. H. Y. Kim, E. H. Lee, B. Y. Kim, “Polarization properties of fiber lasers with twist-induced circular birefringence,” Appl. Opt. 36, 6764–6769 (1997).
    [Crossref]
  10. J. A. Davis, I. Moreno, P. Tzai, “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. 37, 937–945 (1998).
    [Crossref]
  11. J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.
  12. H. Aben, J. Anton, J. Josepson, “Ambiguity of the fringe order in integrated photoelasticity,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany, 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 309–317.
  13. H. Aben, J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent objects,” Appl. Opt. 36, 7172–7179 (1997).
    [Crossref]
  14. H. Aben, L. Ainola, “Interference blots and fringe dislocations in optics of twisted birefringent media,” J. Opt. Soc. Am. A 15, 2404–2411 (1998).
    [Crossref]
  15. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
    [Crossref] [PubMed]
  16. I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
    [Crossref]
  17. F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten oder ungleichförmig erwärmten unkrystallinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3–254 (1843).
  18. A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin, Kl. Math., Phys., und Techn., No. 4, 115–126 (1962).
  19. A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik 19, 261–272 (1962).
  20. A. Kuske, “L’analyse des phénomènes optiques en photoélasticité à trois dimensions par la méthode du cercle de ‘J,’ ” Rev. Franç. Méc. No. 9, 49–58 (1964).
  21. H. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
    [Crossref]
  22. S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, New York, 1969).
  23. H. Poincaré, Théorie mathématique de la lumière (Garré et Naud, Paris, 1892) Vol. 2.

1998 (2)

1997 (2)

1996 (1)

1995 (1)

A. Lakhtakia, W. S. Weiglhofer, “On light propagation in helicoidal bianisotropic mediums,” Proc. R. Soc. London Ser. A 448, 419–437 (1995).
[Crossref]

1994 (1)

1993 (1)

1991 (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[Crossref] [PubMed]

1990 (1)

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[Crossref]

1986 (1)

1966 (1)

H. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
[Crossref]

1964 (1)

A. Kuske, “L’analyse des phénomènes optiques en photoélasticité à trois dimensions par la méthode du cercle de ‘J,’ ” Rev. Franç. Méc. No. 9, 49–58 (1964).

1962 (2)

A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin, Kl. Math., Phys., und Techn., No. 4, 115–126 (1962).

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik 19, 261–272 (1962).

1843 (1)

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten oder ungleichförmig erwärmten unkrystallinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3–254 (1843).

Aben, H.

H. Aben, L. Ainola, “Interference blots and fringe dislocations in optics of twisted birefringent media,” J. Opt. Soc. Am. A 15, 2404–2411 (1998).
[Crossref]

H. Aben, J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent objects,” Appl. Opt. 36, 7172–7179 (1997).
[Crossref]

H. Aben, “Characteristic directions in optics of twisted birefringent media,” J. Opt. Soc. Am. A 3, 1414–1421 (1986).
[Crossref]

H. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
[Crossref]

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

H. Aben, J. Anton, J. Josepson, “Ambiguity of the fringe order in integrated photoelasticity,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany, 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 309–317.

Ainola, L.

Anton, J.

H. Aben, J. Anton, J. Josepson, “Ambiguity of the fringe order in integrated photoelasticity,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany, 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 309–317.

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[Crossref] [PubMed]

Aspect, A.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Bhagavantam, S.

S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, New York, 1969).

Cramer, C.

Davis, J. A.

Freund, I.

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[Crossref] [PubMed]

Josepson, J.

H. Aben, J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent objects,” Appl. Opt. 36, 7172–7179 (1997).
[Crossref]

H. Aben, J. Anton, J. Josepson, “Ambiguity of the fringe order in integrated photoelasticity,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany, 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 309–317.

J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.

Kim, B. Y.

Kim, H. Y.

Kuske, A.

A. Kuske, “L’analyse des phénomènes optiques en photoélasticité à trois dimensions par la méthode du cercle de ‘J,’ ” Rev. Franç. Méc. No. 9, 49–58 (1964).

A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin, Kl. Math., Phys., und Techn., No. 4, 115–126 (1962).

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik 19, 261–272 (1962).

Lakhtakia, A.

A. Lakhtakia, W. S. Weiglhofer, “On light propagation in helicoidal bianisotropic mediums,” Proc. R. Soc. London Ser. A 448, 419–437 (1995).
[Crossref]

Lee, E. H.

Mansuripur, M.

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[Crossref]

Moreno, I.

Neumann, F. E.

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten oder ungleichförmig erwärmten unkrystallinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3–254 (1843).

Poincaré, H.

H. Poincaré, Théorie mathématique de la lumière (Garré et Naud, Paris, 1892) Vol. 2.

Ramazza, P. L.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[Crossref] [PubMed]

Residori, S.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[Crossref] [PubMed]

Rheinländer, B.

Schmiedel, H.

Schubert, M.

Stokes, A. R.

A. R. Stokes, The Theory of the Optical Properties of Inhomogeneous Materials (Spon, London, 1963).

Tzai, P.

Vansteenkiste, N.

Venkatarayudu, T.

S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, New York, 1969).

Vignolo, P.

Weiglhofer, W. S.

A. Lakhtakia, W. S. Weiglhofer, “On light propagation in helicoidal bianisotropic mediums,” Proc. R. Soc. London Ser. A 448, 419–437 (1995).
[Crossref]

Woollam, J. A.

Abh. Dtsch. Akad. Wiss. Berlin, Kl. Math., Phys., und Techn. (1)

A. Kuske, “Beiträge zur spannungsoptischen Untersuchung von Flächentragwerken,” Abh. Dtsch. Akad. Wiss. Berlin, Kl. Math., Phys., und Techn., No. 4, 115–126 (1962).

Abh. Kön. Akad. Wiss. Berlin (1)

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten oder ungleichförmig erwärmten unkrystallinischen Körpern,” Abh. Kön. Akad. Wiss. Berlin 2, 3–254 (1843).

Appl. Opt. (3)

Exp. Mech. (1)

H. Aben, “Optical phenomena in photoelastic models by the rotation of principal axes,” Exp. Mech. 6, 13–22 (1966).
[Crossref]

J. Appl. Phys. (1)

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[Crossref]

J. Opt. Soc. Am. A (5)

Optik (1)

A. Kuske, “Die Gesetzmässigkeiten der Doppelbrechung,” Optik 19, 261–272 (1962).

Phys. Rev. Lett. (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[Crossref] [PubMed]

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

A. Lakhtakia, W. S. Weiglhofer, “On light propagation in helicoidal bianisotropic mediums,” Proc. R. Soc. London Ser. A 448, 419–437 (1995).
[Crossref]

Rev. Franç. Méc. No. 9 (1)

A. Kuske, “L’analyse des phénomènes optiques en photoélasticité à trois dimensions par la méthode du cercle de ‘J,’ ” Rev. Franç. Méc. No. 9, 49–58 (1964).

Other (7)

A. R. Stokes, The Theory of the Optical Properties of Inhomogeneous Materials (Spon, London, 1963).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.

H. Aben, J. Anton, J. Josepson, “Ambiguity of the fringe order in integrated photoelasticity,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany, 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 309–317.

S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Application to Physical Problems (Academic, New York, 1969).

H. Poincaré, Théorie mathématique de la lumière (Garré et Naud, Paris, 1892) Vol. 2.

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Equations (82)

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dEdz=VE,
E=E1E2,
V=-i 12 C(1-2)dφ/dz-dφ/dzi 12 C(1-2).
E=UE0,
U=α1+iα2β1+iβ2-β1+iβ2α1-iα2,
α12+α22+β12+β22=1.
α1=cos ξ cos θ,α2=sin ξ cos θ,
β1=cos ζ sin θ,β2=sin ζ sin θ.
U=cosξcosθ+isinξcosθcosζsinθ+isinζsinθ-cos ζsin θ+isinζsinθcosξcosθ-isinξcosθ.
tan ξ=α2α1,tan ζ=β2β1,
cos2 θ=α12+α22,sin2 θ=β12+β22.
α1=cos ρ cos ν,α2=cos κ sin ν,
β1=sin ρ cos ν,β2=sin κ sin ν.
U=cosρcosν+icosκsinνsinρcosν+isinκsinν-sinρcosν+isinκsinνcosρcosν-icosκsinν.
tan ρ=β1/α1,tan κ=β2/α2,
cos2 ν=α12+β12,sin2 ν=α22+β22.
α2β1,β1α2,
ξρ,ζκ,θν,
cos ξ cos θ=cos ρ cos ν,
sin ξ cos θ=cos κ sin ν,
cos ζ sin θ=sin ρ cos ν,
sin ζ sin θ=sin κ sin ν.
tan ξ=cos κcos ρ tan ν,
tan ζ=sin κsin ρ tan ν,
cos 2θ=cos 2ρ cos2 ν+cos 2κ sin2 ν,
tan2 θ=sin2 ρ+sin2 κ tan2 νcos2 ρ+cos2 κ tan2 ν.
tan ρ=cos ζcos ξ tan θ,
tan κ=sin ζsin ξ tan θ,
cos 2ν=cos 2ξ cos2 θ+cos 2ζ sin2 θ,
tan2 ν=sin2 ξ+sin2 ζ tan2 θcos2 ξ+cos2 ζ tan2 θ.
U=G(λ)=eiλ00e-iλ,
U=S(ϑ)=cos ϑ-sin ϑsin ϑcos ϑ.
G-1(λ)=G(-λ),
S-1(ϑ)=S(-ϑ).
U=G(λ)S(ϑ)G(-μ),
λ=ξ+ζ2,μ=ζ-ξ2,ϑ=-θ,
ξ=λ-μ,ζ=λ+μ,θ=-ϑ.
E=S(ϑ)E0,
E=G(-λ)E,E0=G(-μ)E0.
dG(λ)dz S(ϑ)G(-μ)+G(λ) dS(ϑ)dz G(-μ)
+G(λ)S(ϑ) dG(-μ)dzG(μ)S(-ϑ)G(-λ)=V.
dG(λ)dz=ie-iλ00-e-iλ dλdz,
dS(ϑ)dz=-sin ϑcos ϑ-cos ϑsin ϑ dϑdz,
dG(-μ)dz=i-e-iμ00eiμ dμdz.
i100-1 dλdz-0e2iλ-e-2iλ0 dϑdz
-icos 2ϑe2iλ sin 2ϑ-e-2iλ sin 2ϑ-cos 2ϑ dμdz=V.
dλdz-cos 2ϑ dμdz=-12 C(1-2),
cos 2λ dϑdz-sin 2λ sin 2ϑ dμdz=-dφdz,
sin 2λ dϑdz+cos 2λ sin 2ϑ dμdz=0.
dλdz+tan 2λ cot 2ϑ dϑdz=-12 C(1-2),
1cos 2λ dϑdz=-dφdz.
U=S(α)G(γ)S(-α0),
α0=ρ+κ2,α=κ-ρ2,γ=ν,
ρ=α0-α,κ=α0+α,ν=γ.
E=G(γ)E0,
E=S(-α)E,E0=S(-α0)E0.
dS(α)dz G(γ)S(-α0)+S(α) dG(γ)dz S(-α0)+S(α)G(γ) dS(-α0)dz
×S(α0)G(-γ)S(-α)=V.
0-110 dαdz+icos 2αsin 2αsin 2α-cos 2α dγdz
+-i sin 2γ sin 2αcos 2γ+i sin 2γ cos 2α-cos 2γ+i sin 2γ cos 2αi sin 2γ sin 2α
×dα0dz=V.
cos 2α dγdz-sin 2γ sin 2α dα0dz=-12 C(1-2),
dαdz-cos 2γ dα0dz=-dφdz,
sin 2α dγdz+sin 2γ cos 2α dα0dz=0.
1cos 2α dγdz=-12 C(1-2),
dαdz+tan 2α cot 2γ dγdz=dφdz.
cos(λ-μ)cos ϑ=cos(α0-α)cos γ,
sin(λ-μ)cos ϑ=cos(α0+α)sin γ,
-cos(λ+μ)sin ϑ=sin(α0-α)cos γ,
sin(λ+μ)sin ϑ=sin(α0+α)sin γ.
tan 2λ=2 sin 2α0 tan γsin 2(α0-α)-sin 2(α0+α)tan2 γ,
tan 2μ=2 sin 2α tan γsin 2(α0-α)+sin 2(α0+α)tan2 γ.
cos 2ϑ=cos 2(α0-α)cos2 γ+cos 2(α0+α)sin2 γ,
tan2 ϑ=sin2(α0-α)+sin2(α0+α)tan2 γcos2(α0-α)+cos2(α0+α)tan2 γ.
tan 2α0=-2 sin 2λ tan ϑsin 2(λ-μ)-sin 2(λ+μ)tan2 ϑ,
tan 2α=-2 sin 2μ tan ϑsin 2(λ-μ)+sin 2(λ+μ)tan2 ϑ,
cos 2γ=cos 2(λ-μ)cos2 ϑ+cos 2(λ+μ)sin2 ϑ,
tan2 γ=sin2(λ-μ)+sin2(λ+μ)tan2 ϑcos2(λ-μ)+cos2(λ+μ)tan2 ϑ.
λα0,μα,ϑ-γ,
12 C(1-2)dφdz,dφdz12 C(1-2),
λα,μα0,ϑγ,
V=dUdz U-1,

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