Abstract

Three novel methods for the determination of optical anisotropy are proposed and tested. The first, the special points method, may be applied to any uniaxially anisotropic medium and is based on the measurement of s- and p-polarized light reflectances under near-normal or grazing angles (or both) and of the Brewster angle. The second method is based on the use of the Azzam universal relationship between the Fresnel s- and p-reflection coefficients. For a flat surface and an isotropic medium, the Azzam combination of coefficients becomes zero and thus is independent of the incidence angle, whereas for a uniaxial or biaxial anisotropic sample it acquires a certain angular dependence, which may be used to determine the anisotropy of the sample. Finally, for those cases in which the anisotropy of the material of a film deposited on an isotropic substrate is itself of interest, a third method, the interference method, is suggested. This technique makes use of the different dependences of s- and p-polarized beam optical path-length changes on the variation of the angle of incidence.

© 1998 Optical Society of America

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  1. J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
    [CrossRef]
  2. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  3. B. Djafari Rouhani and J. Sapriel, “Effective dielectric and photoelastic tensors of superlattices in the long-wavelength regime,” Phys. Rev. B 34, 7114–7120 (1986).
    [CrossRef]
  4. R. W. Boyd, J. E. Sipe, “Nonlinear susceptibility of layered composite materials,” J. Opt. Soc. Am. B 11, 297–303 (1994).
    [CrossRef]
  5. F. Ferrieu, A. Halimaoui, D. Bensahel, “Optical characterization of porous silicon layers by spectrometric ellipsometry in the 1.5–5eV range,” Solid State Commun. 84, 293–296 (1992).
    [CrossRef]
  6. P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 223, 131–136 (1993).
    [CrossRef]
  7. P. Basmaji, G. I. Surdutovich, R. Z. Vitlina, J. Kolenda, V. S. Bagnato, H. Mohajeri-Moghaddam, N. Peyghambarian, “Anisotropy investigations and photoluminescence properties of porous silicon,” Solid State Commun. 91, 649–653 (1994).
    [CrossRef]
  8. A. V. Ghiner, G. I. Surdutovich, “Method of integral equations and an extinction theorem for two-dimensional problems in nonlinear optics,” Phys. Rev. A 50, 714–723 (1994);“Linear and nonlinear optical characteristics of porous silicon,” Braz. J. Phys. 24, 344–348 (1994).
    [CrossRef] [PubMed]
  9. R. Z. Vitlina, A. M. Dykhne, “Reflection of electromagnetic waves from a surface with a low relief,” Sov. Phys. JETP 72, 983–990 (1991).
  10. R. Z. Vitlina, “Reflection of light from small stochastic roughness,” Sov. Phys. Opt. Spectrosc. 72, 660–667 (1992).
  11. R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North - Holland, New York, 1977), pp. 356–357.
  12. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  13. J. J. Greffet, “Theoretical model of the shift of the Brewster angle on a rough surface,” Opt. Lett. 17, 238–240 (1992).
    [CrossRef] [PubMed]
  14. M. Saillard, “A characterization tool for dielectric random rough surfaces: Brewster’s phenomenon,” Waves Random Media 2, 67–79 (1992);A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
    [CrossRef]
  15. L. S. Braginskii, I. A. Gilinskii, S. N. Svitasheva, “Light reflection by a rough surface: interpretation of ellipsometric measurements,” Sov. Phys. Dokl. 32, 297–299 (1987).
  16. R. M. Azzam, “Relationship between the p and s Fresnel reflection coefficients of an interface independent of angle of incidence,” J. Opt. Soc. Am. A 3, 928–929 (1986).
    [CrossRef]
  17. A. V. Ghiner, G. I. Surdutovich, Ellipsometry: Theory, Methods and Applications (Nauka, Novosibirsk, 1987), pp. 50–52.
  18. G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
    [CrossRef]
  19. L. D. Landau, E. M. Lifshitz, P. L. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, UK, 1984), Chap. 2.
  20. J. von Behren, L. Tsybeskov, P. M. Fauchet, “Preparation and characterization of ultrathin porous silicon films,” Appl. Phys. Lett. 66, 1662–1664 (1995).
    [CrossRef]
  21. C. Pickering, M. T. Beatle, D. J. Robbins, “Optical properties of porous silicon films,” Thin Solid Films 125, 157–161 (1985).
    [CrossRef]
  22. M. Novak, “Determination of optical constants and average thickness of inhomogeneous-rough thin films using spectral dependence of optical transmittance,” Thin Solid Films 254, 200–210 (1995).
    [CrossRef]
  23. G. Parjadis da Lariviere, T. M. Frigerio, T. Rivory, F. Abeles, “Estimation of the degree of inhomogeneity of the refractive index of dielectric films from spectroscopic ellipsometry,” Appl. Opt. 31, 6056–6062 (1992).
    [CrossRef]

1996 (1)

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

1995 (2)

J. von Behren, L. Tsybeskov, P. M. Fauchet, “Preparation and characterization of ultrathin porous silicon films,” Appl. Phys. Lett. 66, 1662–1664 (1995).
[CrossRef]

M. Novak, “Determination of optical constants and average thickness of inhomogeneous-rough thin films using spectral dependence of optical transmittance,” Thin Solid Films 254, 200–210 (1995).
[CrossRef]

1994 (3)

R. W. Boyd, J. E. Sipe, “Nonlinear susceptibility of layered composite materials,” J. Opt. Soc. Am. B 11, 297–303 (1994).
[CrossRef]

P. Basmaji, G. I. Surdutovich, R. Z. Vitlina, J. Kolenda, V. S. Bagnato, H. Mohajeri-Moghaddam, N. Peyghambarian, “Anisotropy investigations and photoluminescence properties of porous silicon,” Solid State Commun. 91, 649–653 (1994).
[CrossRef]

A. V. Ghiner, G. I. Surdutovich, “Method of integral equations and an extinction theorem for two-dimensional problems in nonlinear optics,” Phys. Rev. A 50, 714–723 (1994);“Linear and nonlinear optical characteristics of porous silicon,” Braz. J. Phys. 24, 344–348 (1994).
[CrossRef] [PubMed]

1993 (1)

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 223, 131–136 (1993).
[CrossRef]

1992 (5)

F. Ferrieu, A. Halimaoui, D. Bensahel, “Optical characterization of porous silicon layers by spectrometric ellipsometry in the 1.5–5eV range,” Solid State Commun. 84, 293–296 (1992).
[CrossRef]

R. Z. Vitlina, “Reflection of light from small stochastic roughness,” Sov. Phys. Opt. Spectrosc. 72, 660–667 (1992).

J. J. Greffet, “Theoretical model of the shift of the Brewster angle on a rough surface,” Opt. Lett. 17, 238–240 (1992).
[CrossRef] [PubMed]

G. Parjadis da Lariviere, T. M. Frigerio, T. Rivory, F. Abeles, “Estimation of the degree of inhomogeneity of the refractive index of dielectric films from spectroscopic ellipsometry,” Appl. Opt. 31, 6056–6062 (1992).
[CrossRef]

M. Saillard, “A characterization tool for dielectric random rough surfaces: Brewster’s phenomenon,” Waves Random Media 2, 67–79 (1992);A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

1991 (1)

R. Z. Vitlina, A. M. Dykhne, “Reflection of electromagnetic waves from a surface with a low relief,” Sov. Phys. JETP 72, 983–990 (1991).

1990 (1)

1987 (1)

L. S. Braginskii, I. A. Gilinskii, S. N. Svitasheva, “Light reflection by a rough surface: interpretation of ellipsometric measurements,” Sov. Phys. Dokl. 32, 297–299 (1987).

1986 (2)

R. M. Azzam, “Relationship between the p and s Fresnel reflection coefficients of an interface independent of angle of incidence,” J. Opt. Soc. Am. A 3, 928–929 (1986).
[CrossRef]

B. Djafari Rouhani and J. Sapriel, “Effective dielectric and photoelastic tensors of superlattices in the long-wavelength regime,” Phys. Rev. B 34, 7114–7120 (1986).
[CrossRef]

1985 (1)

C. Pickering, M. T. Beatle, D. J. Robbins, “Optical properties of porous silicon films,” Thin Solid Films 125, 157–161 (1985).
[CrossRef]

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

1904 (1)

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[CrossRef]

Abeles, F.

Azzam, R. M.

Bagnato, V. S.

P. Basmaji, G. I. Surdutovich, R. Z. Vitlina, J. Kolenda, V. S. Bagnato, H. Mohajeri-Moghaddam, N. Peyghambarian, “Anisotropy investigations and photoluminescence properties of porous silicon,” Solid State Commun. 91, 649–653 (1994).
[CrossRef]

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 223, 131–136 (1993).
[CrossRef]

Baranauskas, V.

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Bashara, N. M.

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North - Holland, New York, 1977), pp. 356–357.

Basmaji, P.

P. Basmaji, G. I. Surdutovich, R. Z. Vitlina, J. Kolenda, V. S. Bagnato, H. Mohajeri-Moghaddam, N. Peyghambarian, “Anisotropy investigations and photoluminescence properties of porous silicon,” Solid State Commun. 91, 649–653 (1994).
[CrossRef]

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 223, 131–136 (1993).
[CrossRef]

Beatle, M. T.

C. Pickering, M. T. Beatle, D. J. Robbins, “Optical properties of porous silicon films,” Thin Solid Films 125, 157–161 (1985).
[CrossRef]

Bensahel, D.

F. Ferrieu, A. Halimaoui, D. Bensahel, “Optical characterization of porous silicon layers by spectrometric ellipsometry in the 1.5–5eV range,” Solid State Commun. 84, 293–296 (1992).
[CrossRef]

Boyd, R. W.

Braginskii, L. S.

L. S. Braginskii, I. A. Gilinskii, S. N. Svitasheva, “Light reflection by a rough surface: interpretation of ellipsometric measurements,” Sov. Phys. Dokl. 32, 297–299 (1987).

Djafari Rouhani and J. Sapriel, B.

B. Djafari Rouhani and J. Sapriel, “Effective dielectric and photoelastic tensors of superlattices in the long-wavelength regime,” Phys. Rev. B 34, 7114–7120 (1986).
[CrossRef]

Dykhne, A. M.

R. Z. Vitlina, A. M. Dykhne, “Reflection of electromagnetic waves from a surface with a low relief,” Sov. Phys. JETP 72, 983–990 (1991).

Fauchet, P. M.

J. von Behren, L. Tsybeskov, P. M. Fauchet, “Preparation and characterization of ultrathin porous silicon films,” Appl. Phys. Lett. 66, 1662–1664 (1995).
[CrossRef]

Ferrieu, F.

F. Ferrieu, A. Halimaoui, D. Bensahel, “Optical characterization of porous silicon layers by spectrometric ellipsometry in the 1.5–5eV range,” Solid State Commun. 84, 293–296 (1992).
[CrossRef]

Fragalli, J. F.

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Frigerio, T. M.

Ghiner, A. V.

A. V. Ghiner, G. I. Surdutovich, “Method of integral equations and an extinction theorem for two-dimensional problems in nonlinear optics,” Phys. Rev. A 50, 714–723 (1994);“Linear and nonlinear optical characteristics of porous silicon,” Braz. J. Phys. 24, 344–348 (1994).
[CrossRef] [PubMed]

A. V. Ghiner, G. I. Surdutovich, Ellipsometry: Theory, Methods and Applications (Nauka, Novosibirsk, 1987), pp. 50–52.

Gilinskii, I. A.

L. S. Braginskii, I. A. Gilinskii, S. N. Svitasheva, “Light reflection by a rough surface: interpretation of ellipsometric measurements,” Sov. Phys. Dokl. 32, 297–299 (1987).

Greffet, J. J.

Griviskas, V.

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 223, 131–136 (1993).
[CrossRef]

Halimaoui, A.

F. Ferrieu, A. Halimaoui, D. Bensahel, “Optical characterization of porous silicon layers by spectrometric ellipsometry in the 1.5–5eV range,” Solid State Commun. 84, 293–296 (1992).
[CrossRef]

Kolenda, J.

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

P. Basmaji, G. I. Surdutovich, R. Z. Vitlina, J. Kolenda, V. S. Bagnato, H. Mohajeri-Moghaddam, N. Peyghambarian, “Anisotropy investigations and photoluminescence properties of porous silicon,” Solid State Commun. 91, 649–653 (1994).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, P. L. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, UK, 1984), Chap. 2.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, P. L. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, UK, 1984), Chap. 2.

Maxwell Garnett, J. C.

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[CrossRef]

Maystre, D.

Misoguti, L.

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Mohajeri-Moghaddam, H.

P. Basmaji, G. I. Surdutovich, R. Z. Vitlina, J. Kolenda, V. S. Bagnato, H. Mohajeri-Moghaddam, N. Peyghambarian, “Anisotropy investigations and photoluminescence properties of porous silicon,” Solid State Commun. 91, 649–653 (1994).
[CrossRef]

Novak, M.

M. Novak, “Determination of optical constants and average thickness of inhomogeneous-rough thin films using spectral dependence of optical transmittance,” Thin Solid Films 254, 200–210 (1995).
[CrossRef]

Parjadis da Lariviere, G.

Peyghambarian, N.

P. Basmaji, G. I. Surdutovich, R. Z. Vitlina, J. Kolenda, V. S. Bagnato, H. Mohajeri-Moghaddam, N. Peyghambarian, “Anisotropy investigations and photoluminescence properties of porous silicon,” Solid State Commun. 91, 649–653 (1994).
[CrossRef]

Pickering, C.

C. Pickering, M. T. Beatle, D. J. Robbins, “Optical properties of porous silicon films,” Thin Solid Films 125, 157–161 (1985).
[CrossRef]

Pitaevskii, P. L.

L. D. Landau, E. M. Lifshitz, P. L. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, UK, 1984), Chap. 2.

Rivory, T.

Robbins, D. J.

C. Pickering, M. T. Beatle, D. J. Robbins, “Optical properties of porous silicon films,” Thin Solid Films 125, 157–161 (1985).
[CrossRef]

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Saillard, M.

M. Saillard, “A characterization tool for dielectric random rough surfaces: Brewster’s phenomenon,” Waves Random Media 2, 67–79 (1992);A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

Sipe, J. E.

Surdutovich, G. I.

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

P. Basmaji, G. I. Surdutovich, R. Z. Vitlina, J. Kolenda, V. S. Bagnato, H. Mohajeri-Moghaddam, N. Peyghambarian, “Anisotropy investigations and photoluminescence properties of porous silicon,” Solid State Commun. 91, 649–653 (1994).
[CrossRef]

A. V. Ghiner, G. I. Surdutovich, “Method of integral equations and an extinction theorem for two-dimensional problems in nonlinear optics,” Phys. Rev. A 50, 714–723 (1994);“Linear and nonlinear optical characteristics of porous silicon,” Braz. J. Phys. 24, 344–348 (1994).
[CrossRef] [PubMed]

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 223, 131–136 (1993).
[CrossRef]

A. V. Ghiner, G. I. Surdutovich, Ellipsometry: Theory, Methods and Applications (Nauka, Novosibirsk, 1987), pp. 50–52.

Svitasheva, S. N.

L. S. Braginskii, I. A. Gilinskii, S. N. Svitasheva, “Light reflection by a rough surface: interpretation of ellipsometric measurements,” Sov. Phys. Dokl. 32, 297–299 (1987).

Tsybeskov, L.

J. von Behren, L. Tsybeskov, P. M. Fauchet, “Preparation and characterization of ultrathin porous silicon films,” Appl. Phys. Lett. 66, 1662–1664 (1995).
[CrossRef]

Vitlina, R.

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Vitlina, R. Z.

P. Basmaji, G. I. Surdutovich, R. Z. Vitlina, J. Kolenda, V. S. Bagnato, H. Mohajeri-Moghaddam, N. Peyghambarian, “Anisotropy investigations and photoluminescence properties of porous silicon,” Solid State Commun. 91, 649–653 (1994).
[CrossRef]

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 223, 131–136 (1993).
[CrossRef]

R. Z. Vitlina, “Reflection of light from small stochastic roughness,” Sov. Phys. Opt. Spectrosc. 72, 660–667 (1992).

R. Z. Vitlina, A. M. Dykhne, “Reflection of electromagnetic waves from a surface with a low relief,” Sov. Phys. JETP 72, 983–990 (1991).

von Behren, J.

J. von Behren, L. Tsybeskov, P. M. Fauchet, “Preparation and characterization of ultrathin porous silicon films,” Appl. Phys. Lett. 66, 1662–1664 (1995).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. von Behren, L. Tsybeskov, P. M. Fauchet, “Preparation and characterization of ultrathin porous silicon films,” Appl. Phys. Lett. 66, 1662–1664 (1995).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Philos. Trans. R. Soc. London (1)

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[CrossRef]

Phys. Rev. A (1)

A. V. Ghiner, G. I. Surdutovich, “Method of integral equations and an extinction theorem for two-dimensional problems in nonlinear optics,” Phys. Rev. A 50, 714–723 (1994);“Linear and nonlinear optical characteristics of porous silicon,” Braz. J. Phys. 24, 344–348 (1994).
[CrossRef] [PubMed]

Phys. Rev. B (1)

B. Djafari Rouhani and J. Sapriel, “Effective dielectric and photoelastic tensors of superlattices in the long-wavelength regime,” Phys. Rev. B 34, 7114–7120 (1986).
[CrossRef]

Solid State Commun. (2)

F. Ferrieu, A. Halimaoui, D. Bensahel, “Optical characterization of porous silicon layers by spectrometric ellipsometry in the 1.5–5eV range,” Solid State Commun. 84, 293–296 (1992).
[CrossRef]

P. Basmaji, G. I. Surdutovich, R. Z. Vitlina, J. Kolenda, V. S. Bagnato, H. Mohajeri-Moghaddam, N. Peyghambarian, “Anisotropy investigations and photoluminescence properties of porous silicon,” Solid State Commun. 91, 649–653 (1994).
[CrossRef]

Sov. Phys. Dokl. (1)

L. S. Braginskii, I. A. Gilinskii, S. N. Svitasheva, “Light reflection by a rough surface: interpretation of ellipsometric measurements,” Sov. Phys. Dokl. 32, 297–299 (1987).

Sov. Phys. JETP (2)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

R. Z. Vitlina, A. M. Dykhne, “Reflection of electromagnetic waves from a surface with a low relief,” Sov. Phys. JETP 72, 983–990 (1991).

Sov. Phys. Opt. Spectrosc. (1)

R. Z. Vitlina, “Reflection of light from small stochastic roughness,” Sov. Phys. Opt. Spectrosc. 72, 660–667 (1992).

Thin Solid Films (4)

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 223, 131–136 (1993).
[CrossRef]

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

C. Pickering, M. T. Beatle, D. J. Robbins, “Optical properties of porous silicon films,” Thin Solid Films 125, 157–161 (1985).
[CrossRef]

M. Novak, “Determination of optical constants and average thickness of inhomogeneous-rough thin films using spectral dependence of optical transmittance,” Thin Solid Films 254, 200–210 (1995).
[CrossRef]

Waves Random Media (1)

M. Saillard, “A characterization tool for dielectric random rough surfaces: Brewster’s phenomenon,” Waves Random Media 2, 67–79 (1992);A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

Other (3)

L. D. Landau, E. M. Lifshitz, P. L. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, UK, 1984), Chap. 2.

A. V. Ghiner, G. I. Surdutovich, Ellipsometry: Theory, Methods and Applications (Nauka, Novosibirsk, 1987), pp. 50–52.

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North - Holland, New York, 1977), pp. 356–357.

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Figures (12)

Fig. 1
Fig. 1

Equivalence of a low-relief rough surface or an inhomogeneous film to an anisotropic film.

Fig. 2
Fig. 2

Dependence of the Brewster angle θ B on the refractive index n x for different positive and negative values of the anisotropy parameter β. The regions below and above the dashed curve correspond to greater and lesser solutions n x of Eq. (1) for a given θ B and β < 0. Point Ca 1 indicates the experimental data (θ B = 60.5°, β = -9.6%) obtained by the AUR method [see Fig. 10(a) in Section 3 below] for the natural calcite sample, and point Ca 2 B = 60.78°, n x = 1.658, β = -10.3%) corresponds to the literature data. Points Si(|β| < 0.5%) and Sirough B = 74.38°, β = -47.5%) correspond to the AUR measurements [Fig. 10(b)] for highly polished and rough surfaces of the c-Si sample, respectively.

Fig. 3
Fig. 3

Regions of negative (vertical lines) and positive (horizontal lines) anisotropy are shown in the planes (a) [R(0), tan θ B ], (b) (α, tan θ B ). (a) The boundaries of the positive anisotropy are outlined by curves B, R 0 = tan   θ B - 1 tan   θ B + 1 2 ,   β = 0 , and C, R 0 = 1 + tan 2   θ B 1 / 2 - 1 1 + tan 2   θ B 1 / 2 + 1 2 ,     β . Points Ca 1[R(0) = 0.064, θ B = 60.5°, β = -8%], Sirough[R(0) = 0.0443, θ B = 74.38°, β = -31%], and Si(|β| < 0.5%) represent the results of R(0), tanθ B measurements of a natural calcite sample and the c-Si sample with rough and highly polished surfaces, respectively. Point Ca 2 represents literature data for the calcite sample. (b) Line A (α = tan θ B ) and curve C [α = tan2 θ B (1 + cot2θ B )1/2] correspond to the boundaries of the physical domain. Curve B (α = tan2 θ B ) corresponds to an isotropic medium. Points Ca 1 (α = 2.95, θ B = 60.5°, β = -8.4%) and Ca 2 correspond to the α, tan θ B measurements and the literature data for a natural calcite crystal. Point Sirough represents the experimental result for a c-Si sample with a rough surface (α = 10.5, θ B = 74.38°, β = -46.5%).

Fig. 4
Fig. 4

Critical curve ε1 = ε1 crit obtained from Eq. (16).

Fig. 5
Fig. 5

Dependence of the dielectric permittivity ε of a film on the parameter Γ for three standard substrates: ε = 2.72 (glass), 5.7 (diamond), and 14.9 (Si). The best accuracy in the determination of the film permittivity ε on the basis of the experimentally measured parameter Γ may be achieved outside the vicinity of the curve given by Eq. (16).

Fig. 6
Fig. 6

Contours κ = const for the substrates with ε = 2.72 (dashed–dotted curves), ε = 5.7 (dashed curves), and ε = 14.9 (solid curves).

Fig. 7
Fig. 7

Angular dependences of the R s and R p coefficients for two inhomogeneous thin film on the substrate samples: 1, ε = 15, ε1 = 9, κ = 0.2, θ B = 75.522°, θ m = 74.96°, n x ef = 3.740, βef = +0.0974, θ B ef = 74.95°; 2, ε = 4, ε1 = 6, κ = 0.2, θ B = 63.435°, θ m = 63.98°, n x ef = 2.067, βef = +0.0974, θ B ef = 63.99°. The solid curves represent the numerical simulations according to the exact formula of Eq. (9) and the dashed curves correspond to the anisotropic uniaxial fictitious crystals with the parameters given by Eqs. (18) and (20). In the presented examples the solid and the dashed curves converge for all angles, except for those in the immediate vicinity of θ B .

Fig. 8
Fig. 8

Contours of the effective anisotropy β̃ = βef2 in the plane ε, ε1. In the vicinity of the bisectrix ε = ε1, the sign of the effective anisotropy coincides with the sign of the difference ε1 - ε. Below the bisectrix, for ε > 5.8, there is another region of positive anisotropy with asymptotes ε1 = 1 and ε1 = ε - 2 for ε → ∞. A very narrow region of negative anisotropy exists where ε < 1.5, but in practice this region has no physical importance.

Fig. 9
Fig. 9

(a) Angular dependence of the coefficients R s and R p (arbitrary units, logarithmic scale) for the highly polished (solid curve) and unpolished (dashed curve) c-Si surfaces, (b) the region of grazing angles is shown on a linear scale.

Fig. 10
Fig. 10

Angular dependences of the function K(θ) drawn on the basis of the reflectometry data and the jump-phase model and the Brewster angle measurements [● and ○ represent the values of K(θ) calculated with Eq. (21) and experimentally measured reflection coefficients R p and R s . The function of Eq. (22) (solid curve) is fitted with the value K(π/2) and the measured value of tan θ B ] for (a) a natural calcite crystal sample: K(π/2) = -0.196, θ B = 60.5°. The use of Eqs. (27) and (28) gives β = -9.6% and n x = 1.648 (literature data β = -10.3%, n x = 1.658). (b) An unpolished rough surface c-Si sample (see Fig. 9) with K(π/2) = -2.1, θ B = 74.38°. The use of Eqs. (27) and (28) leads to β = -47.5% and n x = 2.84.

Fig. 11
Fig. 11

Calculated angular dependences of the R s and R p coefficients for the films (n x = 2.664) with (a) positive (β = +10%), (b) zero (β = 0), (c) negative (β = -10%) anisotropy. Refractive index of the substrate n 2 = 1.35, d/λ = 10. Hardly noticeable in the isotropic case, β = 0, the negative and the positive shifts θ p - θ s (equal to -0.4° and +0.6° for the first minima and maxima, respectively) are connected with the derivatives R s,p ′(θ) as described in the text.

Fig. 12
Fig. 12

Effective anisotropy βef of a two-component (with refractive indices n 1 and n 2) medium as a function of the concentration of the second constituent for the layered (curves 1 and 2) and columnar (curves 3 and 4) mesostructures. The different signs of βef for curves 1–4 demonstrate the importance of the geometry of the mesostructure. The distinction between curves 3 and 4 demonstrates the importance of the form of the inclusions (columnar structure of c-Si in vacuum or vacuum holes in a bulk Si material). Point A corresponds to the calculated anisotropy β = -4.5% of a-SiC:H (n 1 = 2.4) and a-Si:H (n 2 = 3.) layered structure with c = 0.2 used in the IM measurements.18

Equations (51)

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tan   θ B = n z n x 2 - 1 n z 2 - 1 1 / 2 = n x β + 1 n x 2 - 1 n x 2 β + 1 2 - 1 1 / 2 .
n x 2 + 1 n z 2 > 2 ,
R 0 = n x - 1 n x + 1 2 ,
α = d R p d θ dR s d θ | θ     π / 2 = n x n z n x 2 - 1 n z 2 - 1 1 / 2 = n x tan   θ B .
n x = 1 + R 0 1 / 2 1 - R 0 1 / 2 , n z = tan   θ B 1 - R 0 1 / 2 { tan 2   θ B 1 - R 0 1 / 2 2 - 4 R 0 1 / 2 } 1 / 2 ,
n x = α   cot   θ B ,     n z = tan 2   θ B tan 4   θ B + tan 2   θ B - α 2 1 / 2 .
β = - n x 4 - 1 n x   δ θ B + n x 2 - 1 2 4 n x δ R 0 R 0 ,
β = - 2 n x 4 - 1 n x   δ θ B + n x 2 - 1 δ α α ;
R s , p = r 01 s , p + r 12 s , p exp - i δ 1 + r 01 s , p r 12 s , p exp - i δ 2 ,
Δ θ = - κ 2 ε ε 8 ε 1 2 ε - ε 1 ε 1 - 1 ε 2 - 1 2 ε + 1 ε 2 + 1 ε 1 - 1 ε - ε 1 + 2 ε 1 2 ε 2 + ε 1 2 - 2 ε 2 ,
R 0 = q 1 - κ 2 ε ε 1 - 1 ε - ε 1 ε - 1 2 , q = ε - 1 ε + 1 2 ,
ε 1 2 ε ε 2 + 1 - 8 q Γ ε + 1 3 + ε 1 ε ε + 1 ε 2 + 1 - ε 2 ε 2 + 4 ε + 1 = 0 .
Γ = Δ θ R 0 - q ,
Γ 1 = ε ε 2 + 1 8 q ε + 1 3 .
Γ 2 = ε 2 + 1 ε 2 + 1 ε + 1 2 + 4 ε ε 2 + 4 ε + 1 32 q ε + 1 3 ε 2 + 4 ε + 1 ,
ε 1 crit = 2 ε ε 2 + 4 ε + 1 ε 2 + 1 ε + 1 ,
κ 2 = q - R 0 q ε ε - 1 2 ε 1 - 1 ε - ε 1 .
R 0 = n x ef - 1 n x ef + 1 2 , i . e . ,   n x ef = ε 1 - κ 2 4 ε - ε 1 ε 1 - 1 ε - 1 .
Δ θ = θ n x β = 0 δ n x + θ β n x = ε δ β = ε ε 2 - 1 ε - 1 ε   δ n x - δ β .
β ef = - 1 8   κ 2 ε 1 - 1 ε - ε 1 ε + 1 2 ε - 1 ε 1 2 ε ε 2 + 1 ε 1 - 1 ε 1 - ε + 2 2 ε ε 2 - ε 1 2 + ε 1 2 ε 2 - 1 .
Δ θ = κ 2 4 ε 3 / 2 ε + 1 2 ε 1 - ε ,
β ef = κ 2 4 ε - 1 ε + 1 ε 1 - ε ,
β film ef = - ε - 1 2 ε + 1
K θ = U s 2 cos 2   θ - U s U p + sin 2   θ ,
K θ = n x 2 1 - n z n x n x 2 - sin 2   θ n z 2 - sin 2   θ 1 / 2 .
K θ β   n x 2 sin 2   θ n x 2 - sin 2   θ ,
K θ β   sin 2   θ
K θ = β + β 2 / 2 β + 1 2 sin 2   θ ,
n x = sin   θ K θ K θ - β   sin 2   θ 1 / 2
K π / 2 = n x n x - tan   θ B ,
β = tan   θ B tan   θ B 2 ± tan   θ B 2 2 + K π 2 1 / 2 tan 2   θ B 2 + 1 - K π 2 ± tan   θ B tan   θ B 2 2 + K π 2 1 / 2 1 / 2 ,
K π / 2 > K crit π / 2 = - 1 + tan 2   θ B 2 1 / 2 × tan   θ B - 1 + tan 2   θ B 2 1 / 2 ,
β = β crit = - 1 - tan   θ B 2 1 + tan 2   θ B ,
n x 2 ± = 1 + tan 2   θ B 2 ± 1 + tan 2   θ B 2 2 - tan 2   θ B 1 + β 2 1 / 2 .
β = K π 2 tan 2   θ B + tan   θ B - 2 2   tan   θ B ,
K x θ = n y 2 1 - n x n z n y 2 n y 2 - sin 2   θ n z 2 - sin 2   θ 1 / 2 ,
n y n x = β yx + 1 = - K x 0 K y 0 .
α p θ = θ p = 2 π   d λ n x n z n z 2 - sin 2   θ p 1 / 2 ,
α s θ = θ s = 2 π   d λ n x 2 - sin 2   θ s ,
Δ p = α p 0 - α p θ p = 2 π d λ n x - n x n z n z 2 - sin 2   θ p 1 / 2
Δ s = α s 0 - α s θ s = 2 π d λ n x - n x 2 - sin 2   θ s 1 / 2
Δ θ p ,   θ s = Δ p - Δ s = 2 π d λ n x 2 - sin 2   θ s 1 / 2 - n x n z n z 2 - sin 2   θ p 1 / 2
β = sin   θ p sin   θ s - 1 .
β = θ p - θ s cot   θ s .
n z 2 = ε zz ef = 1 - c ε 1 + c ε 2 - 1 ,
β ef = ε ε + c 1 - c ε - 1 2 1 / 2 - 1 .
ε xx ef = ε yy ef = ε + 1 + c ε - 1 ε + 1 - c ε - 1 ,     ε zz ef = 1 + c ε - 1 , β ef = ε + 1 + c ε ε - 1 - c 2 ε - 1 2 ε + 1 + c ε - 1 1 / 2 - 1 .
ε xx ef = ε yy ef = ε   2 + c ε - 1 2 ε - c ε - 1 ,     ε zz ef = 1 + c ε - 1 , β ef = 2 ε + c 2 ε - 1 ε - 1 - c 2 ε - 1 2 2 ε + c ε ε - 1 1 / 2 - 1 .
ε xx ef = ε ε - 1 ln   ε ,     ε zz ef = ε - 1 ln   ε ,     β ef = ε - 1 ε ln   ε - 1 .
R 0 = tan   θ B - 1 tan   θ B + 1 2 ,   β = 0 ,
R 0 = 1 + tan 2   θ B 1 / 2 - 1 1 + tan 2   θ B 1 / 2 + 1 2 ,     β .

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