Abstract

We present a reformulation of the determination of optical parameters in general film–substrate systems. Developed for interferential films in terms of photometric magnitudes (R, T), the formalism introduced allows us to establish how many parameters can be extracted from a set of measurements and from which type of sample model. These parameters are the refractive index and the absorption of both film and substrate (i.e., ñ1 = n 1 - jk 1 and ñ2 = n 2 - jk 2), the thickness of the film (d), the inhomogeneity of the film (Δn 1), and the surface roughness of the interfaces (σ1, σ2) delimiting the film. The new formalism leads to some new analytical results and confirms others. Among the new results we have the following: (a) The mathematical condition commonly related with extremes (maxima and minima) in an interference pattern defines in fact a condition for envelope extremes. (b) The refractive index of a film can be obtained without prior knowledge of the thickness or the refractive index of the substrate (provided we have an optical interference film). (c) Absorption can be directly extracted from an interference-free magnitude T/(1 - R). (d) Roughness at the inner surface, inhomogeneity in the film, and absorption are correlated in reflection spectral measurements.

© 2000 Optical Society of America

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  1. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), Vol. 1; (1991), Vol. 2; (1998), Vol. 3.
  2. J. C. Martinez-Anton, “Caracterización óptica por espectrogoniometría automática,” Ph. D. dissertation (Universidad Complutense de Madrid, 28040 Madrid, Spain, 1997).
  3. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).
  4. M. Ohring, The Materials Science of Thin Films (Academic, Boston, 1992).
  5. F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, Vol. 2, E. Wolf, ed. (North-Holland, Amsterdam, 1963), pp. 249–287.
  6. F. Abelès, “Optics of thin films,” in Advanced Optical Techniques, A. C. S. Van Heel, ed. (North-Holland, Amsterdam, 1967), Chap. 5.
  7. D. P. Arndt, R. M. A. Azzam, J. M. Bennett, J. P. Borgogno, C. K. Carniglia, W. E. Case, J. A. Drobrowolski, U. J. Gibson, T. Tuttle Hart, F. C. Ho, V. A. Hodgkin, W. P. Klapp, H. A. Macleod, E. Pelletier, M. K. Purvis, D. M. Quinn, D. H. Strome, R. Swenson, P. A. Temple, T. F. Thonn, “Multiple determination of the optical constants of thin-film coating materials,” Appl. Opt. 23, 3571–3596 (1984).
  8. I. Ohlídal, K. Navrátil, Spectroscopic methods for optical analysis of thin films,” Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae Brunensis, Tomus XXV, Physica 37(2), 5–81 (1984).
  9. J. P. Borgogno, E. Pelletier, “Determination of the extinction coefficient of dielectric thin films from spectrophotometric measurements,” Appl. Opt. 28, 2895–2901 (1989).
  10. R. Swanepoel, “Determining refractive index and thickness of thin films from wavelength measurements only,” J. Opt. Soc. Am. 2, 1339–1343 (1985).
    [CrossRef]
  11. F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, Vol. 2, E. Wolf, ed. (North-Holland, Amsterdam, 1963), pp. 249–288.
  12. J. M. Del Pozo, L. Diaz, “Method for the determination of optical constants of thin films: dependence of experimental uncertainties,” Appl. Opt. 31, 4474–4481 (1992).
    [CrossRef] [PubMed]
  13. J. C. Manifacier, J. Gasiot, J. D. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
    [CrossRef]
  14. J. Mouchart, G. Langier, B. Pointu, “Determination des constantes optiques n et k de materiaux faiblement absorbents,” Appl. Opt. 24, 1808–1814 (1985).
    [CrossRef]
  15. I. Ohlídal, “General formulas for the optical characterization of single layers with spectroscopic reflectometry,” J. Mod. Opt. 35, 1373–1381 (1988).
  16. R. Swanepoel, “Determination of thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
    [CrossRef]
  17. D. A. Minkov, “Method for determining the optical constants of a thin film on a transparent substrate,” J. Phys. D 22, 199–205 (1989).
    [CrossRef]
  18. D. A. Minkov, “Computation of the optical constants of a thin dielectric layer on a transmitting substrate from the reflection spectrum at an inclined incidence of light,” J. Opt. Soc. Am. 8, 306–310 (1991).
    [CrossRef]
  19. D. A. Minkov, “Computation of the optical constants of a thin dielectric layer from the envelopes of the transmission spectrum, at an inclined incidence of the radiation,” J. Mod. Opt. 37, 1977–1986 (1990).
    [CrossRef]
  20. D. A. Minkov, “Errors made in the computation of the optical constants of a thin dielectric layer from the envelopes of the reflection spectrum at an inclined incidence of light,” Optik (Stuttgart) 87(4), 137–140 (1991).
  21. D. A. Minkov, “Singularity of the solution when using spectrum envelopes for the computation of the optical constants of a thin dielectric layer,” Optik (Stuttgart) 90(2), 80–84 (1992).
  22. D. A. Minkov, “Method for determining the optical constants of a thin film on a transparent substrate,” J. Phys. D 22, 199–205 (1989).
    [CrossRef]
  23. A. S. Valeev, “Determination of the optical constants of weakly absorbing thin films,” Opt. Spectrosc. (USSR) 15, 269–274 (1963).
  24. A. S. Valeev, “Constants of thin weakly absorbing layers,” Opt. Spectros. (USSR) 18, 498–500 (1965).
  25. V. V. Filippov, “Analytical method for determining optical constants and thickness of absorbing films from absorption spectra,” Opt. Spectrosc. 78, 719–722 (1995).
  26. D. B. Kushev, N. N. Zheleva, M. I. Gyulmezov, M. H. Koparanova, “An envelope method for determination of the optical constants of absorptive substrates,” Infrared Phys. 34, 163–167 (1993).
    [CrossRef]
  27. C. H. Peng, S. B. Desu, “Modified method for obtaining optical properties of weakly absorbing thin films and its application to thin films of Pb(Zr, Ti)O3 solid solutions,” J. Am. Ceram. Soc. 77, 929–938 (1994).
    [CrossRef]
  28. W. T. Beauchamp, J. D. Rancourt, “Refractive-index measurements of moderately reflecting substrates using a wedged film technique,” Appl. Opt. 19, 3239–3244 (1980).
  29. C. K. Carniglia, “Effects of dispersion on the determination of optical constants of thin films,” in Thin Film Technologies II, J. Jacobsson, ed., Proc. SPIE652, 158–165 (1986).
    [CrossRef]
  30. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  31. J. Lekner, Theory of Reflection (Nijhoff, Dordrecht, The Netherlands, 1987).
  32. J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
    [CrossRef] [PubMed]
  33. B. J. Stagg, T. T. Charalampopoulos, “Surface-roughness effects on the determination of materials by the reflection method,” Appl. Opt. 30, 4113–4118 (1991).
    [CrossRef] [PubMed]
  34. R. Schiffer, “Reflectivity of a slightly rough surface,” Appl. Opt. 26, 704–712 (1987).
    [CrossRef] [PubMed]
  35. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).
  36. M. Nieto Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).
  37. W. R. Hunter, “Measurement of optical properties of materials in the vacuum ultraviolet spectral region,” Appl. Opt. 21, 2103–2114 (1982).
  38. J. C. Martínez-Antón, E. Bernabeu, “Spectrogoniometry and the WANTED method for thickness and refractive index determination,” Thin Solid Films 313–314, 85–89 (1998).
  39. J. C. Martínez-Antón, E. Bernabeu, “Interferential spectrogoniometry for simultaneous determination on thickness and refractive index of films,” Opt. Commun. 132, 321–328 (1996).
    [CrossRef]
  40. Y. Hishikawa, N. Nakanura, S. Tsuda, S. Nakano, Y. Kishi, Y. Kuwano, “Inteference-free determination of the optical absorption coefficient and the optical gap of amorphous silicon thin films,” Jpn. J. Appl. Phys. 30, 1008–1014 (1991).
    [CrossRef]

1998 (1)

J. C. Martínez-Antón, E. Bernabeu, “Spectrogoniometry and the WANTED method for thickness and refractive index determination,” Thin Solid Films 313–314, 85–89 (1998).

1996 (1)

J. C. Martínez-Antón, E. Bernabeu, “Interferential spectrogoniometry for simultaneous determination on thickness and refractive index of films,” Opt. Commun. 132, 321–328 (1996).
[CrossRef]

1995 (1)

V. V. Filippov, “Analytical method for determining optical constants and thickness of absorbing films from absorption spectra,” Opt. Spectrosc. 78, 719–722 (1995).

1994 (1)

C. H. Peng, S. B. Desu, “Modified method for obtaining optical properties of weakly absorbing thin films and its application to thin films of Pb(Zr, Ti)O3 solid solutions,” J. Am. Ceram. Soc. 77, 929–938 (1994).
[CrossRef]

1993 (1)

D. B. Kushev, N. N. Zheleva, M. I. Gyulmezov, M. H. Koparanova, “An envelope method for determination of the optical constants of absorptive substrates,” Infrared Phys. 34, 163–167 (1993).
[CrossRef]

1992 (2)

D. A. Minkov, “Singularity of the solution when using spectrum envelopes for the computation of the optical constants of a thin dielectric layer,” Optik (Stuttgart) 90(2), 80–84 (1992).

J. M. Del Pozo, L. Diaz, “Method for the determination of optical constants of thin films: dependence of experimental uncertainties,” Appl. Opt. 31, 4474–4481 (1992).
[CrossRef] [PubMed]

1991 (4)

B. J. Stagg, T. T. Charalampopoulos, “Surface-roughness effects on the determination of materials by the reflection method,” Appl. Opt. 30, 4113–4118 (1991).
[CrossRef] [PubMed]

D. A. Minkov, “Errors made in the computation of the optical constants of a thin dielectric layer from the envelopes of the reflection spectrum at an inclined incidence of light,” Optik (Stuttgart) 87(4), 137–140 (1991).

D. A. Minkov, “Computation of the optical constants of a thin dielectric layer on a transmitting substrate from the reflection spectrum at an inclined incidence of light,” J. Opt. Soc. Am. 8, 306–310 (1991).
[CrossRef]

Y. Hishikawa, N. Nakanura, S. Tsuda, S. Nakano, Y. Kishi, Y. Kuwano, “Inteference-free determination of the optical absorption coefficient and the optical gap of amorphous silicon thin films,” Jpn. J. Appl. Phys. 30, 1008–1014 (1991).
[CrossRef]

1990 (1)

D. A. Minkov, “Computation of the optical constants of a thin dielectric layer from the envelopes of the transmission spectrum, at an inclined incidence of the radiation,” J. Mod. Opt. 37, 1977–1986 (1990).
[CrossRef]

1989 (3)

D. A. Minkov, “Method for determining the optical constants of a thin film on a transparent substrate,” J. Phys. D 22, 199–205 (1989).
[CrossRef]

D. A. Minkov, “Method for determining the optical constants of a thin film on a transparent substrate,” J. Phys. D 22, 199–205 (1989).
[CrossRef]

J. P. Borgogno, E. Pelletier, “Determination of the extinction coefficient of dielectric thin films from spectrophotometric measurements,” Appl. Opt. 28, 2895–2901 (1989).

1988 (1)

I. Ohlídal, “General formulas for the optical characterization of single layers with spectroscopic reflectometry,” J. Mod. Opt. 35, 1373–1381 (1988).

1987 (1)

1985 (2)

J. Mouchart, G. Langier, B. Pointu, “Determination des constantes optiques n et k de materiaux faiblement absorbents,” Appl. Opt. 24, 1808–1814 (1985).
[CrossRef]

R. Swanepoel, “Determining refractive index and thickness of thin films from wavelength measurements only,” J. Opt. Soc. Am. 2, 1339–1343 (1985).
[CrossRef]

1984 (2)

1983 (1)

R. Swanepoel, “Determination of thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
[CrossRef]

1982 (2)

1980 (1)

1976 (1)

J. C. Manifacier, J. Gasiot, J. D. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

1965 (1)

A. S. Valeev, “Constants of thin weakly absorbing layers,” Opt. Spectros. (USSR) 18, 498–500 (1965).

1963 (1)

A. S. Valeev, “Determination of the optical constants of weakly absorbing thin films,” Opt. Spectrosc. (USSR) 15, 269–274 (1963).

Abelès, F.

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, Vol. 2, E. Wolf, ed. (North-Holland, Amsterdam, 1963), pp. 249–287.

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, Vol. 2, E. Wolf, ed. (North-Holland, Amsterdam, 1963), pp. 249–288.

F. Abelès, “Optics of thin films,” in Advanced Optical Techniques, A. C. S. Van Heel, ed. (North-Holland, Amsterdam, 1967), Chap. 5.

Arndt, D. P.

Azzam, R. M. A.

Bashara, N. M.

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

Beauchamp, W. T.

Bennett, J. M.

Bernabeu, E.

J. C. Martínez-Antón, E. Bernabeu, “Spectrogoniometry and the WANTED method for thickness and refractive index determination,” Thin Solid Films 313–314, 85–89 (1998).

J. C. Martínez-Antón, E. Bernabeu, “Interferential spectrogoniometry for simultaneous determination on thickness and refractive index of films,” Opt. Commun. 132, 321–328 (1996).
[CrossRef]

Borgogno, J. P.

Carniglia, C. K.

Case, W. E.

Charalampopoulos, T. T.

Del Pozo, J. M.

Desu, S. B.

C. H. Peng, S. B. Desu, “Modified method for obtaining optical properties of weakly absorbing thin films and its application to thin films of Pb(Zr, Ti)O3 solid solutions,” J. Am. Ceram. Soc. 77, 929–938 (1994).
[CrossRef]

Diaz, L.

Drobrowolski, J. A.

Filippov, V. V.

V. V. Filippov, “Analytical method for determining optical constants and thickness of absorbing films from absorption spectra,” Opt. Spectrosc. 78, 719–722 (1995).

Fillard, J. D.

J. C. Manifacier, J. Gasiot, J. D. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Gasiot, J.

J. C. Manifacier, J. Gasiot, J. D. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Gibson, U. J.

Gyulmezov, M. I.

D. B. Kushev, N. N. Zheleva, M. I. Gyulmezov, M. H. Koparanova, “An envelope method for determination of the optical constants of absorptive substrates,” Infrared Phys. 34, 163–167 (1993).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).

Hishikawa, Y.

Y. Hishikawa, N. Nakanura, S. Tsuda, S. Nakano, Y. Kishi, Y. Kuwano, “Inteference-free determination of the optical absorption coefficient and the optical gap of amorphous silicon thin films,” Jpn. J. Appl. Phys. 30, 1008–1014 (1991).
[CrossRef]

Ho, F. C.

Hodgkin, V. A.

Hunter, W. R.

Kishi, Y.

Y. Hishikawa, N. Nakanura, S. Tsuda, S. Nakano, Y. Kishi, Y. Kuwano, “Inteference-free determination of the optical absorption coefficient and the optical gap of amorphous silicon thin films,” Jpn. J. Appl. Phys. 30, 1008–1014 (1991).
[CrossRef]

Klapp, W. P.

Koparanova, M. H.

D. B. Kushev, N. N. Zheleva, M. I. Gyulmezov, M. H. Koparanova, “An envelope method for determination of the optical constants of absorptive substrates,” Infrared Phys. 34, 163–167 (1993).
[CrossRef]

Kushev, D. B.

D. B. Kushev, N. N. Zheleva, M. I. Gyulmezov, M. H. Koparanova, “An envelope method for determination of the optical constants of absorptive substrates,” Infrared Phys. 34, 163–167 (1993).
[CrossRef]

Kuwano, Y.

Y. Hishikawa, N. Nakanura, S. Tsuda, S. Nakano, Y. Kishi, Y. Kuwano, “Inteference-free determination of the optical absorption coefficient and the optical gap of amorphous silicon thin films,” Jpn. J. Appl. Phys. 30, 1008–1014 (1991).
[CrossRef]

Langier, G.

Lazarides, B.

Lekner, J.

J. Lekner, Theory of Reflection (Nijhoff, Dordrecht, The Netherlands, 1987).

Macleod, H. A.

Manifacier, J. C.

J. C. Manifacier, J. Gasiot, J. D. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Martinez-Anton, J. C.

J. C. Martinez-Anton, “Caracterización óptica por espectrogoniometría automática,” Ph. D. dissertation (Universidad Complutense de Madrid, 28040 Madrid, Spain, 1997).

Martínez-Antón, J. C.

J. C. Martínez-Antón, E. Bernabeu, “Spectrogoniometry and the WANTED method for thickness and refractive index determination,” Thin Solid Films 313–314, 85–89 (1998).

J. C. Martínez-Antón, E. Bernabeu, “Interferential spectrogoniometry for simultaneous determination on thickness and refractive index of films,” Opt. Commun. 132, 321–328 (1996).
[CrossRef]

Minkov, D. A.

D. A. Minkov, “Singularity of the solution when using spectrum envelopes for the computation of the optical constants of a thin dielectric layer,” Optik (Stuttgart) 90(2), 80–84 (1992).

D. A. Minkov, “Errors made in the computation of the optical constants of a thin dielectric layer from the envelopes of the reflection spectrum at an inclined incidence of light,” Optik (Stuttgart) 87(4), 137–140 (1991).

D. A. Minkov, “Computation of the optical constants of a thin dielectric layer on a transmitting substrate from the reflection spectrum at an inclined incidence of light,” J. Opt. Soc. Am. 8, 306–310 (1991).
[CrossRef]

D. A. Minkov, “Computation of the optical constants of a thin dielectric layer from the envelopes of the transmission spectrum, at an inclined incidence of the radiation,” J. Mod. Opt. 37, 1977–1986 (1990).
[CrossRef]

D. A. Minkov, “Method for determining the optical constants of a thin film on a transparent substrate,” J. Phys. D 22, 199–205 (1989).
[CrossRef]

D. A. Minkov, “Method for determining the optical constants of a thin film on a transparent substrate,” J. Phys. D 22, 199–205 (1989).
[CrossRef]

Mouchart, J.

Nakano, S.

Y. Hishikawa, N. Nakanura, S. Tsuda, S. Nakano, Y. Kishi, Y. Kuwano, “Inteference-free determination of the optical absorption coefficient and the optical gap of amorphous silicon thin films,” Jpn. J. Appl. Phys. 30, 1008–1014 (1991).
[CrossRef]

Nakanura, N.

Y. Hishikawa, N. Nakanura, S. Tsuda, S. Nakano, Y. Kishi, Y. Kuwano, “Inteference-free determination of the optical absorption coefficient and the optical gap of amorphous silicon thin films,” Jpn. J. Appl. Phys. 30, 1008–1014 (1991).
[CrossRef]

Navrátil, K.

I. Ohlídal, K. Navrátil, Spectroscopic methods for optical analysis of thin films,” Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae Brunensis, Tomus XXV, Physica 37(2), 5–81 (1984).

Nieto Vesperinas, M.

M. Nieto Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

Ohlídal, I.

I. Ohlídal, “General formulas for the optical characterization of single layers with spectroscopic reflectometry,” J. Mod. Opt. 35, 1373–1381 (1988).

I. Ohlídal, K. Navrátil, Spectroscopic methods for optical analysis of thin films,” Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae Brunensis, Tomus XXV, Physica 37(2), 5–81 (1984).

Ohring, M.

M. Ohring, The Materials Science of Thin Films (Academic, Boston, 1992).

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), Vol. 1; (1991), Vol. 2; (1998), Vol. 3.

Pelletier, E.

Peng, C. H.

C. H. Peng, S. B. Desu, “Modified method for obtaining optical properties of weakly absorbing thin films and its application to thin films of Pb(Zr, Ti)O3 solid solutions,” J. Am. Ceram. Soc. 77, 929–938 (1994).
[CrossRef]

Pointu, B.

Purvis, M. K.

Quinn, D. M.

Rancourt, J. D.

Schiffer, R.

Stagg, B. J.

Strome, D. H.

Swanepoel, R.

R. Swanepoel, “Determining refractive index and thickness of thin films from wavelength measurements only,” J. Opt. Soc. Am. 2, 1339–1343 (1985).
[CrossRef]

R. Swanepoel, “Determination of thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
[CrossRef]

Swenson, R.

Temple, P. A.

Thonn, T. F.

Tsuda, S.

Y. Hishikawa, N. Nakanura, S. Tsuda, S. Nakano, Y. Kishi, Y. Kuwano, “Inteference-free determination of the optical absorption coefficient and the optical gap of amorphous silicon thin films,” Jpn. J. Appl. Phys. 30, 1008–1014 (1991).
[CrossRef]

Tuttle Hart, T.

Valeev, A. S.

A. S. Valeev, “Constants of thin weakly absorbing layers,” Opt. Spectros. (USSR) 18, 498–500 (1965).

A. S. Valeev, “Determination of the optical constants of weakly absorbing thin films,” Opt. Spectrosc. (USSR) 15, 269–274 (1963).

Zheleva, N. N.

D. B. Kushev, N. N. Zheleva, M. I. Gyulmezov, M. H. Koparanova, “An envelope method for determination of the optical constants of absorptive substrates,” Infrared Phys. 34, 163–167 (1993).
[CrossRef]

Appl. Opt. (9)

W. T. Beauchamp, J. D. Rancourt, “Refractive-index measurements of moderately reflecting substrates using a wedged film technique,” Appl. Opt. 19, 3239–3244 (1980).

W. R. Hunter, “Measurement of optical properties of materials in the vacuum ultraviolet spectral region,” Appl. Opt. 21, 2103–2114 (1982).

J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
[CrossRef] [PubMed]

D. P. Arndt, R. M. A. Azzam, J. M. Bennett, J. P. Borgogno, C. K. Carniglia, W. E. Case, J. A. Drobrowolski, U. J. Gibson, T. Tuttle Hart, F. C. Ho, V. A. Hodgkin, W. P. Klapp, H. A. Macleod, E. Pelletier, M. K. Purvis, D. M. Quinn, D. H. Strome, R. Swenson, P. A. Temple, T. F. Thonn, “Multiple determination of the optical constants of thin-film coating materials,” Appl. Opt. 23, 3571–3596 (1984).

J. Mouchart, G. Langier, B. Pointu, “Determination des constantes optiques n et k de materiaux faiblement absorbents,” Appl. Opt. 24, 1808–1814 (1985).
[CrossRef]

R. Schiffer, “Reflectivity of a slightly rough surface,” Appl. Opt. 26, 704–712 (1987).
[CrossRef] [PubMed]

J. P. Borgogno, E. Pelletier, “Determination of the extinction coefficient of dielectric thin films from spectrophotometric measurements,” Appl. Opt. 28, 2895–2901 (1989).

B. J. Stagg, T. T. Charalampopoulos, “Surface-roughness effects on the determination of materials by the reflection method,” Appl. Opt. 30, 4113–4118 (1991).
[CrossRef] [PubMed]

J. M. Del Pozo, L. Diaz, “Method for the determination of optical constants of thin films: dependence of experimental uncertainties,” Appl. Opt. 31, 4474–4481 (1992).
[CrossRef] [PubMed]

Infrared Phys. (1)

D. B. Kushev, N. N. Zheleva, M. I. Gyulmezov, M. H. Koparanova, “An envelope method for determination of the optical constants of absorptive substrates,” Infrared Phys. 34, 163–167 (1993).
[CrossRef]

J. Am. Ceram. Soc. (1)

C. H. Peng, S. B. Desu, “Modified method for obtaining optical properties of weakly absorbing thin films and its application to thin films of Pb(Zr, Ti)O3 solid solutions,” J. Am. Ceram. Soc. 77, 929–938 (1994).
[CrossRef]

J. Mod. Opt. (2)

I. Ohlídal, “General formulas for the optical characterization of single layers with spectroscopic reflectometry,” J. Mod. Opt. 35, 1373–1381 (1988).

D. A. Minkov, “Computation of the optical constants of a thin dielectric layer from the envelopes of the transmission spectrum, at an inclined incidence of the radiation,” J. Mod. Opt. 37, 1977–1986 (1990).
[CrossRef]

J. Opt. Soc. Am. (2)

R. Swanepoel, “Determining refractive index and thickness of thin films from wavelength measurements only,” J. Opt. Soc. Am. 2, 1339–1343 (1985).
[CrossRef]

D. A. Minkov, “Computation of the optical constants of a thin dielectric layer on a transmitting substrate from the reflection spectrum at an inclined incidence of light,” J. Opt. Soc. Am. 8, 306–310 (1991).
[CrossRef]

J. Phys. D (2)

D. A. Minkov, “Method for determining the optical constants of a thin film on a transparent substrate,” J. Phys. D 22, 199–205 (1989).
[CrossRef]

D. A. Minkov, “Method for determining the optical constants of a thin film on a transparent substrate,” J. Phys. D 22, 199–205 (1989).
[CrossRef]

J. Phys. E (2)

R. Swanepoel, “Determination of thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
[CrossRef]

J. C. Manifacier, J. Gasiot, J. D. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Jpn. J. Appl. Phys. (1)

Y. Hishikawa, N. Nakanura, S. Tsuda, S. Nakano, Y. Kishi, Y. Kuwano, “Inteference-free determination of the optical absorption coefficient and the optical gap of amorphous silicon thin films,” Jpn. J. Appl. Phys. 30, 1008–1014 (1991).
[CrossRef]

Opt. Commun. (1)

J. C. Martínez-Antón, E. Bernabeu, “Interferential spectrogoniometry for simultaneous determination on thickness and refractive index of films,” Opt. Commun. 132, 321–328 (1996).
[CrossRef]

Opt. Spectros. (USSR) (1)

A. S. Valeev, “Constants of thin weakly absorbing layers,” Opt. Spectros. (USSR) 18, 498–500 (1965).

Opt. Spectrosc. (1)

V. V. Filippov, “Analytical method for determining optical constants and thickness of absorbing films from absorption spectra,” Opt. Spectrosc. 78, 719–722 (1995).

Opt. Spectrosc. (USSR) (1)

A. S. Valeev, “Determination of the optical constants of weakly absorbing thin films,” Opt. Spectrosc. (USSR) 15, 269–274 (1963).

Optik (Stuttgart) (2)

D. A. Minkov, “Errors made in the computation of the optical constants of a thin dielectric layer from the envelopes of the reflection spectrum at an inclined incidence of light,” Optik (Stuttgart) 87(4), 137–140 (1991).

D. A. Minkov, “Singularity of the solution when using spectrum envelopes for the computation of the optical constants of a thin dielectric layer,” Optik (Stuttgart) 90(2), 80–84 (1992).

Thin Solid Films (1)

J. C. Martínez-Antón, E. Bernabeu, “Spectrogoniometry and the WANTED method for thickness and refractive index determination,” Thin Solid Films 313–314, 85–89 (1998).

Tomus XXV, Physica (1)

I. Ohlídal, K. Navrátil, Spectroscopic methods for optical analysis of thin films,” Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae Brunensis, Tomus XXV, Physica 37(2), 5–81 (1984).

Other (12)

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, Vol. 2, E. Wolf, ed. (North-Holland, Amsterdam, 1963), pp. 249–288.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), Vol. 1; (1991), Vol. 2; (1998), Vol. 3.

J. C. Martinez-Anton, “Caracterización óptica por espectrogoniometría automática,” Ph. D. dissertation (Universidad Complutense de Madrid, 28040 Madrid, Spain, 1997).

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).

M. Ohring, The Materials Science of Thin Films (Academic, Boston, 1992).

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, Vol. 2, E. Wolf, ed. (North-Holland, Amsterdam, 1963), pp. 249–287.

F. Abelès, “Optics of thin films,” in Advanced Optical Techniques, A. C. S. Van Heel, ed. (North-Holland, Amsterdam, 1967), Chap. 5.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

M. Nieto Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

C. K. Carniglia, “Effects of dispersion on the determination of optical constants of thin films,” in Thin Film Technologies II, J. Jacobsson, ed., Proc. SPIE652, 158–165 (1986).
[CrossRef]

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

J. Lekner, Theory of Reflection (Nijhoff, Dordrecht, The Netherlands, 1987).

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Figures (10)

Fig. 1
Fig. 1

Model of the sample and the measurement scheme. We indicate the optical parameters, the variables, and the magnitudes involved in the formalism. These include absorption, both in the film and the substrate, surface roughness at both interfaces, and a certain inhomogeneity in the film (represented by ñ10 and ñ12). R i and T i are the local (i.e., at each interface) photometric magnitudes.

Fig. 2
Fig. 2

Spectral (top) and angular (bottom) reflectance of a film of SiO2 (d = 1.5 µm) on silicon (flat surfaces). The spectral scan is at ϕ = 0°, and the angular scan is at λ = 0.45 µm. The dashed curves are the envelopes [calculated from Eq. (32)].

Fig. 3
Fig. 3

Wavelength reflectance scan of a film of amorphous silicon (αSi:H) on glass (d = 1 µm). The optical constants for computing the reflectance curve are taken from Swanepoel.16 The envelopes are shown by dashed curves calculated from Eq. (32).

Fig. 4
Fig. 4

Angular reflectance for different films of SiO2 on silicon (λ = 400 nm, d = 0.825 µm). The curve with the stronger modulation represents the tabulated data.1 The lower curve shows results when we added absorption to the SiO2 film (k 1 = 0.06). This strongly reduces the interference modulation and average reflectance. The respective analytic envelopes are shown by the dashed curves and calculated by Eq. (32). Note that the envelope extremes located at 38.8° and 56.7° are practically independent of the introduction of the absorption in the film.

Fig. 5
Fig. 5

Angular reflectance and envelopes in the P polarization of a film of SiO2 on silicon (flat surfaces) at 0.450 µm with two different thicknesses: d 1 = 0.64 µm and d 2 = 0.731 µm. Brewster’s angle for SiO2 is ϕ B = 55.7°. The thickness difference represents λ/4 at 55.7°. The envelopes (dashed curves) are the same for both films, and they have a discontinuous derivative at ϕ B due to the abrupt phase change (φ1 = π around ϕ B ).

Fig. 6
Fig. 6

Same as Fig. 5, but now the sample has slightly absorbing SiO2 films (n 1 = 1.4656 - j0.05, d 1 = 0.74 µm, n 2 = 1.4656 - j0.0567, d 2 = 0.653 µm). The film thicknesses and extinction coefficient values are chosen to have the same envelopes [according to Eq. (32)]. Although Eq. (32) is derived for φ1 ≈ 0, π, we see that the predicted envelopes (dashed curves) are still valid in the current situation, where φ1(55.7°) ≈ 1.7 for both films.

Fig. 7
Fig. 7

Scheme of the procedure for the new formulation. From a reflectance measurement we may extract envelopes E +, E - and derive from them the local magnitudes R 1, R 2a and the envelope extreme λ i or ϕ j . These sources of information permit us to obtain the encircled parameters following the presented order. The correlated parameters (third frame) are tied by a line. The correlation is broken by using χ2 = R 2Pa /R 2Sa and Γ = T/(1 - R) (fourth frame). The estimation of d when the film is inhomogeneous is especially critical. Then we must assume an inhomogeneity profile model (e.g., linear) to estimate 〈n 1〉. See the text for more details.

Fig. 8
Fig. 8

Reflectance measurement (circles) and fit (continuous curve) of a film of SiO2 on BK7 glass. The fit parameters are the following: flat surfaces, homogeneous film, d 1 = 878 nm, and n 1, n 2 taken from tabulated data.1 Here we assume that the film of SiO2 is slightly absorbing. The obtained extinction coefficient is represented in the lower figure. The dashed curve (top) represents the reflectance of the bare substrate, which should coincide with envelope E + for a nonabsorbing homogeneous film.

Fig. 9
Fig. 9

Same measurement as in Fig. 8. Now we assume for the fit that the film is nonabsorbing, and we introduce a hypothetical inhomogeneity. We obtain a thickness of d 1 = 892 nm and an inhomogeneity represented as circles (bottom). (The continuous curve is tabulated data.1 In the bottom figure, the upper circles correspond to the inner refractive index (n 12) and the lower circles to n 10. Note that the fit is as good as in Fig. 8, making indistinguishable inhomogeneity from absorption.

Fig. 10
Fig. 10

Spectral scans of a film of amorphous silicon (αSi:H) on glass (d = 1000 nm, S pol. ϕ0 = 0°). The represented magnitudes are R (inside the dashed envelopes), T (the upper oscillating curve), and Γ = T/(1 - R) (the thicker curve). The optical constants are taken from Ref. 16. Note that the magnitude Γ is interference free.

Equations (58)

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r=r1+r2 exp-j2β1+r1r2 exp-j2β,
t=t1t2 exp-jβ1+r1r2 exp-j2β,
rSi=q˜i-q˜i+1/q˜i+q˜i+1,
rPi=Q˜i-Q˜i+1/Q˜i+Q˜i+1,
q˜i=ñi cosϕi,
Q˜i=cosϕi/ñi,
q˜i=qi-jpi,
Q˜i=Qi-jPi.
ñi=ni-jki,
q˜i=ñi2-n0 sinϕ21/2
q˜i2-q˜j2=ñi2-ñj2.
β=2πdλñ12-n0 sinϕ21/2=2πdλ q˜1,
R=R1+R2a+2R1R2a1/2 cos2βr+φ1-φ21+R1R2a+2R1R2a1/2 cos2βr-φ1-φ2,
T=T1T2a1+R1R2a+2R1R2a1/2 cos2βr-φ1-φ2,
R2a=R2 exp-4βi,  T2a=T2 exp-2βi,  βi=Imβ,  βr=Reβ,  ri=Ri expjφi,
q1=n12-k12-n0 sinϕ21/2,
p1=n1k1q1,
β=2πλ0dñ12z-n0 sinϕ21/2dz=2πλ0d q1zdz.
β2π dλñ12-n0 sinϕ21/2
βr=2π dλn12-k12-n0 sinϕ21/2,
βi=2π dλn1k1n12-k12-n0 sinϕ21/2.
ri=riFGσi/λ,
ri=riF1-22πqi-1σiλ2+oσi/λ2.
ti=tiF1-122πqi-1-qiσiλ2+oσi/λ2,
Ri=RiF1-42πqi-1σiλ2+oσi/λ2,
Ti=TtF1-2πqi-1-qiσiλ2+oσi/λ2.
Fx1, , xj, , xt-Ex1, , xj, , xt=0,  xj Fx1, , xj, , xt-xj Ex1, , xj, , xt=0.
φ10, π.
cos2βr+φ1-φ2=cos2βr-φ1-φ2=±cos2βr-φ2,
0=E-11+R1R2a±2R1R2a1/2 cos2βr-φ2+1+R1R2a-R1-R2a,0=E1+R1R2a±2R1R2a1/2cos2βr-φ2+E-1R1R2a+R1R2a1±cos2βr-φ2R1R2a1/2-2R1R2a1/2 sen2βr-φ22βr-φ2,
cos2βr-φ2=±1,
ER±=R11/2±R2a1/21±R1R2a1/22.
ET±=T1T2a1±R1R2a1/22.
R=R1+R2a+2R1R2a1/2 cos2βr+φ1-φ2,
T=T1T2a.
cos2βr+φ1-φ2=±1,
R11/2=1+ER+ER-1/2-1-ER+1-ER-1/2ER+1/2+ER-1/2,
R2a1/2=1-ER+ER-1/2-1-ER+1-ER-1/2ER+1/2-ER-1/2.
q1=q01±R11/2/1R11/2.
n1=q12-q02+n021/2
n12=n021±R11/21R11/22 cos2ϕ0+sin2ϕ0.
R1=q0-q12+p12/q0+q12+p12.
R1=q0-q12q0+q12+4q0q1q0+q14 p12.
q1=q01±R11/21R11/21-R14R11/2p1q02,
p1=k11-n02 sin2ϕ/n121/2.
n1=q12+n02 sin2ϕ1-k12n12-n02 sin2ϕ1/2.
χi=|rPiFGqi-1σ/λ|2|rSiFGqi-1σ/λ|2=RPiFRSiF.
q1=q0 tan2ϕ01±χ11/21χ11/2,
4πdλn12-k12-n0 sinϕ21/2-φ2=Nπ,  N=0, 1, 2, ,
d=m+Δφ2/π4qk+mλk+m-qkλk,
Δdd=-λq12dn1dλλk,
R2a=R2 exp-8πn1q1k1dλ.
χ2=R2aPR2aS=R2PFR2SF1-42πq1σ2/λ2exp-4βi1-42πq1σ2/λ2exp(-4βi=R2PFR2SF.
Γ=T1-R.
ΓT1-R=T11-R1T2a1-R2a.
Γ=1-R21-R2a exp-2βi.
Γexp-2βi.
exp-2βi=Γ+Γ2+4R2a1-Γ1/22.

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