Abstract

We derive certain constraints on the reflection matrix for reflection from a plane, nonmagnetic, optically anisotropic surface using a reciprocity theorem stated long ago by Van de Hulst [Light Scattering by Small Particles (Wiley, 1957)] in the context of scattering of polarized light. The constraints are valid for absorbing and chiral media and can be used as tools to check the consistency of derived expressions for such matrices in terms of the intrinsic parameters of the reflecting medium as illustrated by several examples.

© 2009 Optical Society of America

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References

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  1. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717-754 (2004).
    [Crossref]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957), p. 49.
  3. R. Bhandari, “Transpose symmetry of the Jones matrix and topological phases,” Opt. Lett. 33, 854-856 (2008).
    [Crossref] [PubMed]
  4. R. Bhandari, “Cancellation of simple optical anisotropies without use of a Faraday mirror,” Opt. Lett. 33, 1839-1841 (2008).
    [Crossref] [PubMed]
  5. Z. Sekera, “Scattering matrices and reciprocity relationships for various representations of the state of polarization,” J. Opt. Soc. Am. 56, 1732-1740 (1966).
    [Crossref]
  6. Sekera's result, i.e., Eq. , originates in the work of Van de Hulst.
  7. T. P. Sosnowksi, “Polarization mode filters for integrated optics,” Opt. Commun. 4, 408-412 (1972).
    [Crossref]
  8. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1977).
  9. D. den Engelsen, “Ellipsometry of anisotropic films,” J. Opt. Soc. Am. 61, 1460-1466 (1971).
    [Crossref]
  10. J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121-6133 (1991).
    [Crossref]
  11. M. P. Silverman, “Reflection and refraction at the surface of a chiral medium: comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation,” J. Opt. Soc. Am. A 3, 830-837 (1986).
    [Crossref]
  12. E. Georgieva, “Reflection and refraction at the surface of an isotropic chiral medium: eigenvalue-eigenvector solution using a 4×4 matrix method,” J. Opt. Soc. Am. A 12, 2202-2211 (1995).
    [Crossref]
  13. J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5, 417-443 (1996).
    [Crossref]
  14. M. Born and E. Wolf, Principles of Optics (Pergamon, 1975), p. 38.
  15. F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981), p. 530.
  16. R. W. Ditchburn, Light (Dover, 1991), p. 423.
  17. S. Bassiri, C. H. Papas, and N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450-1459 (1988).
    [Crossref]
  18. N. Vansteenkiste, P. Vignolo, and A. Aspect,“Optical reversibility theorems for polarization: application to remote control of polarization,” J. Opt. Soc. Am. A 10, 2240-2245 (1993).
    [Crossref]

2008 (2)

2004 (1)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717-754 (2004).
[Crossref]

1996 (1)

J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5, 417-443 (1996).
[Crossref]

1995 (1)

E. Georgieva, “Reflection and refraction at the surface of an isotropic chiral medium: eigenvalue-eigenvector solution using a 4×4 matrix method,” J. Opt. Soc. Am. A 12, 2202-2211 (1995).
[Crossref]

1993 (1)

1991 (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121-6133 (1991).
[Crossref]

1988 (1)

1986 (1)

1972 (1)

T. P. Sosnowksi, “Polarization mode filters for integrated optics,” Opt. Commun. 4, 408-412 (1972).
[Crossref]

1971 (1)

1966 (1)

Aspect, A.

Azzam, R. M. A.

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

Bashara, N. M.

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

Bassiri, S.

Bhandari, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975), p. 38.

den Engelsen, D.

Ditchburn, R. W.

R. W. Ditchburn, Light (Dover, 1991), p. 423.

Engheta, N.

Georgieva, E.

E. Georgieva, “Reflection and refraction at the surface of an isotropic chiral medium: eigenvalue-eigenvector solution using a 4×4 matrix method,” J. Opt. Soc. Am. A 12, 2202-2211 (1995).
[Crossref]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981), p. 530.

Lekner, J.

J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5, 417-443 (1996).
[Crossref]

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121-6133 (1991).
[Crossref]

Papas, C. H.

Potton, R. J.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717-754 (2004).
[Crossref]

Sekera, Z.

Silverman, M. P.

Sosnowksi, T. P.

T. P. Sosnowksi, “Polarization mode filters for integrated optics,” Opt. Commun. 4, 408-412 (1972).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957), p. 49.

Vansteenkiste, N.

Vignolo, P.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981), p. 530.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975), p. 38.

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

J. Phys. Condens. Matter (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121-6133 (1991).
[Crossref]

Opt. Commun. (1)

T. P. Sosnowksi, “Polarization mode filters for integrated optics,” Opt. Commun. 4, 408-412 (1972).
[Crossref]

Opt. Lett. (2)

Pure Appl. Opt. (1)

J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5, 417-443 (1996).
[Crossref]

Rep. Prog. Phys. (1)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717-754 (2004).
[Crossref]

Other (6)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957), p. 49.

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

Sekera's result, i.e., Eq. , originates in the work of Van de Hulst.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975), p. 38.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981), p. 530.

R. W. Ditchburn, Light (Dover, 1991), p. 423.

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Figures (2)

Fig. 1
Fig. 1

Geometry of scattering of a plane wave with wave vector k along z ̂ to a wave with wave vector k along z ̂ , where z ̂ and z ̂ lie in the X–Z plane. The coordinate system ( x ̂ , y ̂ , z ̂ ) defining the polarization basis states in the scattered wave is obtained from the ( x ̂ , y ̂ , z ̂ ) system in the incident wave by a rotation about y ̂ through an angle α, which is the scattering angle.

Fig. 2
Fig. 2

Geometry of reflection of a plane wave propagating along z ̂ from a plane surface SS whose normal along n ̂ lies in the X–Z plane. The angle of incidence is θ and the relation between the coordinate systems ( x ̂ , y ̂ , z ̂ ) and ( x ̂ , y ̂ , z ̂ ) is the same as in the scattering problem illustrated in Fig. 1.

Equations (13)

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A ( k , k ) = A ¯ ( k , k )
A ¯ i j = ( 1 ) i + j A j i .
Z = Z ¯ ,
Z i j = Z j i .
R ( ϕ ) Z R ( ϕ ) = Z ,
R ( ϕ ) = ( cos ϕ sin ϕ sin ϕ cos ϕ )
| ψ f = Z | ψ i .
Z R R ( ϕ ) | ψ i = R ( ϕ ) | ψ f .
Z R = R ( ϕ ) Z R ( ϕ ) .
Z 0 = r ( 1 0 0 1 ) ,
U = L 45 ( η ) = R ( 45 ) L 0 ( η ) R ( 45 ) ,
Q = U Z U , Q = U Z U .
Q = Q ¯ .

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