Abstract

The 4 × 4 matrix formalisms developed respectively by Berreman [ D. W. Berreman, J. Opt. Soc. Am. 62, 502 ( 1972)] and by Yeh [P. Yeh, J. Opt. Soc. Am. 69, 742 ( 1979)] for the description of the ellipsometric properties of planar multilayer anisotropic media are compared. Features of both are used to provide a framework that is applicable to the calculation of ellipsometric properties of, e.g., stratified media that incorporate magnetic materials.

© 1984 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 51ff.
  3. D. W. Berreman, “Optics in stratified and anisotropic media,” J. Opt. Soc. Am. 62, 502–510 (1972). References to equations in this paper are given a prefix B for identification.
    [CrossRef]
  4. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977). In this reference (in Sec. 4.7.1) a complete statement of the basic formalism of Ref. 3 is included, but the time dependence is exp(iωt) instead of exp(−iωt).
  5. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979). References to equations in this paper are given a prefix Y for identification.
    [CrossRef]
  6. P. S. Pershan, “Magneto-optical effects,” J. Appl. Phys. 38, 1482–1490 (1967).
    [CrossRef]
  7. See also L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1960), Sec. 60.
  8. The nonparity-preserving dielectric anisotropy of magnetic materials in the visible region arises from a magnetically induced anisotropy in the selection rules for electric-dipole transitions. For nonmagnetic materials, diamagnetic response to a static external magnetic field also gives rise to a nonparity-preserving dielectric anisotropy.
  9. Yeh denoted the equivalent of K(h) by P(h), and Berreman used P(h) for ΨK(h)Ψ−1. We avoid the use of P(h).
  10. The fact that Berreman used M to designate a completely different matrix should not be confusing here since he introduced his M in a completely different context.
  11. J. J. Krebs, W. G. Maisch, G. A. Prinz, D. W. Forester, “Applications of magneto-optics in ring laser gyroscopes,” IEEE Trans. Magn. MAG-16, 1179–1184 (1980).
    [CrossRef]
  12. R. E. McClure, “An electrical equivalent circuit for the transverse magneto-optic effect in thin magnetic films,” IEEE Trans. Magn. MAG-16, 1185 (1980).
    [CrossRef]
  13. J. J. Krebs, W. G. Maisch, “Magneto-optic materials for biasing ring laser gyros,” (Naval Research Laboratory, Washington, D.C., 1980).
  14. D. O. Smith, “Magneto-optical scattering from multi-layer magnetic and dielectric films,” Opt. Acta 12, 13 (1965); Opt. Acta 13, 121 (1966); Opt. Acta 14, 351 (1967).
    [CrossRef]
  15. R. P. Hunt, “Magneto-optic scattering from thin solid films,” J. Appl. Phys. 38, 1652 (1967).
    [CrossRef]
  16. R. S. Seymour, “Light propagation in stratified anisotropic media,” presented at Eighth Australian Workshop on Optical Communications, 1983.

1980 (2)

J. J. Krebs, W. G. Maisch, G. A. Prinz, D. W. Forester, “Applications of magneto-optics in ring laser gyroscopes,” IEEE Trans. Magn. MAG-16, 1179–1184 (1980).
[CrossRef]

R. E. McClure, “An electrical equivalent circuit for the transverse magneto-optic effect in thin magnetic films,” IEEE Trans. Magn. MAG-16, 1185 (1980).
[CrossRef]

1979 (1)

1972 (1)

1967 (2)

R. P. Hunt, “Magneto-optic scattering from thin solid films,” J. Appl. Phys. 38, 1652 (1967).
[CrossRef]

P. S. Pershan, “Magneto-optical effects,” J. Appl. Phys. 38, 1482–1490 (1967).
[CrossRef]

1965 (1)

D. O. Smith, “Magneto-optical scattering from multi-layer magnetic and dielectric films,” Opt. Acta 12, 13 (1965); Opt. Acta 13, 121 (1966); Opt. Acta 14, 351 (1967).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977). In this reference (in Sec. 4.7.1) a complete statement of the basic formalism of Ref. 3 is included, but the time dependence is exp(iωt) instead of exp(−iωt).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977). In this reference (in Sec. 4.7.1) a complete statement of the basic formalism of Ref. 3 is included, but the time dependence is exp(iωt) instead of exp(−iωt).

Berreman, D. W.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 51ff.

Forester, D. W.

J. J. Krebs, W. G. Maisch, G. A. Prinz, D. W. Forester, “Applications of magneto-optics in ring laser gyroscopes,” IEEE Trans. Magn. MAG-16, 1179–1184 (1980).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

Hunt, R. P.

R. P. Hunt, “Magneto-optic scattering from thin solid films,” J. Appl. Phys. 38, 1652 (1967).
[CrossRef]

Krebs, J. J.

J. J. Krebs, W. G. Maisch, G. A. Prinz, D. W. Forester, “Applications of magneto-optics in ring laser gyroscopes,” IEEE Trans. Magn. MAG-16, 1179–1184 (1980).
[CrossRef]

J. J. Krebs, W. G. Maisch, “Magneto-optic materials for biasing ring laser gyros,” (Naval Research Laboratory, Washington, D.C., 1980).

Landau, L. D.

See also L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1960), Sec. 60.

Lifshitz, E. M.

See also L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1960), Sec. 60.

Maisch, W. G.

J. J. Krebs, W. G. Maisch, G. A. Prinz, D. W. Forester, “Applications of magneto-optics in ring laser gyroscopes,” IEEE Trans. Magn. MAG-16, 1179–1184 (1980).
[CrossRef]

J. J. Krebs, W. G. Maisch, “Magneto-optic materials for biasing ring laser gyros,” (Naval Research Laboratory, Washington, D.C., 1980).

McClure, R. E.

R. E. McClure, “An electrical equivalent circuit for the transverse magneto-optic effect in thin magnetic films,” IEEE Trans. Magn. MAG-16, 1185 (1980).
[CrossRef]

Pershan, P. S.

P. S. Pershan, “Magneto-optical effects,” J. Appl. Phys. 38, 1482–1490 (1967).
[CrossRef]

Prinz, G. A.

J. J. Krebs, W. G. Maisch, G. A. Prinz, D. W. Forester, “Applications of magneto-optics in ring laser gyroscopes,” IEEE Trans. Magn. MAG-16, 1179–1184 (1980).
[CrossRef]

Seymour, R. S.

R. S. Seymour, “Light propagation in stratified anisotropic media,” presented at Eighth Australian Workshop on Optical Communications, 1983.

Smith, D. O.

D. O. Smith, “Magneto-optical scattering from multi-layer magnetic and dielectric films,” Opt. Acta 12, 13 (1965); Opt. Acta 13, 121 (1966); Opt. Acta 14, 351 (1967).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 51ff.

Yeh, P.

IEEE Trans. Magn. (2)

J. J. Krebs, W. G. Maisch, G. A. Prinz, D. W. Forester, “Applications of magneto-optics in ring laser gyroscopes,” IEEE Trans. Magn. MAG-16, 1179–1184 (1980).
[CrossRef]

R. E. McClure, “An electrical equivalent circuit for the transverse magneto-optic effect in thin magnetic films,” IEEE Trans. Magn. MAG-16, 1185 (1980).
[CrossRef]

J. Appl. Phys. (2)

R. P. Hunt, “Magneto-optic scattering from thin solid films,” J. Appl. Phys. 38, 1652 (1967).
[CrossRef]

P. S. Pershan, “Magneto-optical effects,” J. Appl. Phys. 38, 1482–1490 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Acta (1)

D. O. Smith, “Magneto-optical scattering from multi-layer magnetic and dielectric films,” Opt. Acta 12, 13 (1965); Opt. Acta 13, 121 (1966); Opt. Acta 14, 351 (1967).
[CrossRef]

Other (9)

J. J. Krebs, W. G. Maisch, “Magneto-optic materials for biasing ring laser gyros,” (Naval Research Laboratory, Washington, D.C., 1980).

R. S. Seymour, “Light propagation in stratified anisotropic media,” presented at Eighth Australian Workshop on Optical Communications, 1983.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977). In this reference (in Sec. 4.7.1) a complete statement of the basic formalism of Ref. 3 is included, but the time dependence is exp(iωt) instead of exp(−iωt).

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 51ff.

See also L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1960), Sec. 60.

The nonparity-preserving dielectric anisotropy of magnetic materials in the visible region arises from a magnetically induced anisotropy in the selection rules for electric-dipole transitions. For nonmagnetic materials, diamagnetic response to a static external magnetic field also gives rise to a nonparity-preserving dielectric anisotropy.

Yeh denoted the equivalent of K(h) by P(h), and Berreman used P(h) for ΨK(h)Ψ−1. We avoid the use of P(h).

The fact that Berreman used M to designate a completely different matrix should not be confusing here since he introduced his M in a completely different context.

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Equations (14)

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( / z ) ψ = ( i ω / c ) Δ ψ .
ψ = Ψ ϕ .
ϕ r ( s r - 1 ) = Ψ r - 1 ψ r ( s r - 1 ) .
ϕ r ( s r - 1 + h r ) = K ( h r ) Ψ r - 1 ψ r ( s r - 1 ) .
ψ r ( s r ) = Ψ r K ( h r ) Ψ r - 1 ψ r ( s r - 1 ) .
F r ( s r - 1 , h r ) Ψ r K ( h r ) Ψ r - 1 .
ψ r ( s r - 1 ) = [ F r ( s r - 1 , h r ) ] - 1 ψ r ( s r - 1 + h r ) = Ψ r K ( - h r ) Ψ r - 1 ψ r ( s r - 1 + h r ) ,
[ F r ( s r - 1 , h r ) ] - 1 = F r ( - h r , s r ) .
ψ e ( s ) = ( l = 1 L ] F l ( s l - 1 , h l ) ψ 0 ( 0 ) ,             s = l = 1 L h l .
F ( s , 0 ) ( l = 1 L ] F l ( s l - 1 , h l ) .
[ F ( s , 0 ) ] - 1 = ( l = 1 L ] F l ( - h l , s l ) F ( - s , s ) ,
ψ 0 ( 0 ) = [ 1 - 1 0 0 ( n / cos θ ) ( n / cos θ ) 0 0 0 0 1 1 0 0 n cos θ - n cos θ ] [ E x + ( 0 ) E x - ( 0 ) E y + ( 0 ) E y - ( 0 ) ] .
ϕ 0 ( 0 ) = Ψ 0 - 1 F ( - s , s ) Ψ e ϕ e ( s ) .
M = Ψ 0 - 1 F ( - s , s ) Ψ e .

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