Abstract

The application of error analysis within a certain algorithm for the most accurate and unambiguous determination of refractive index n, absorption coefficient k, and thickness d of thin absorbing films in a wide spectral range is illustrated with three examples. Thin films of a dye, Ag, and Au are selected because their optical constants (small n for Ag and Au and considerable variations of n and k for the dye films) along with their thinness make investigating these thin films difficult. The important steps of the algorithm that ensure reliable isolation of the physically correct solutions and maximum accuracy of n and k in the spectral range investigated are also demonstrated.

© 2001 Optical Society of America

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References

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  1. Tz. Babeva, S. Kitova, I. Konstantinov, “Photometric methods for determining the optical constants and the thicknesses of thin absorbing films: selection of a combination of photometric quantities on the basis of error analysis,” Appl. Opt. 40, 2675–2681 (2001).
    [CrossRef]
  2. R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
    [CrossRef]
  3. T. C. Paulick, “Inversion of normal-incidence (R, T) measurements to obtain n + ik for thin films,” Appl. Opt. 25, 562–564 (1986).
    [CrossRef]
  4. K. Lamprecht, W. Papousek, G. Leising, “Problem of ambiguity in the determination of optical constants of thin absorbing films from spectroscopic reflectance and transmittance measurements,” Appl. Opt. 36, 6364–6371 (1997).
    [CrossRef]
  5. V. Panayotov, I. Konstantinov, “Algebraic determination of thin-film optical constants from photometric (T, Rf, Rm) and (T, Rb, Rm) measurements,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 1070–1079 (1994).
    [CrossRef]
  6. T. Yasuda, D. E. Aspnes, “Optical-standard surfaces of single-crystal silicon for calibrating ellipsometers and reflectometers,” Appl. Opt. 33, 7435–7438 (1994).
    [CrossRef] [PubMed]
  7. D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
    [CrossRef]
  8. The standard calibration mirror of protected Al film was made in the Central Laboratory of Photoproceses, Sofia, Bulgaria ( http://www.clf.bas.bg) , by specially developed technology.
  9. I. Konstantinov, Tz. Babeva, S. Kitova, “Analysis of errors in thin-film optical parameters derived from spectrophotometric measurements at normal light incidence,” Appl. Opt. 37, 4260–4267 (1998).
    [CrossRef]
  10. W. Hansen, “Optical characterization of thin films: theory,” J. Opt. Soc. Am. 63, 793–802 (1973).
    [CrossRef]
  11. J. Pozo, L. Diaz, “Method for determination of optical constants of thin films: dependence on experimental uncertainties,” Appl. Opt. 31, 4474–4482 (1992).
    [CrossRef] [PubMed]
  12. W. Press, S. Teukolsky, W. Veterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1992), Chap. 10, pp. 408–410.
  13. F. Abelès, M. Theye, “Methode de calcul des constantes optiques des couches minces absorbantes à partir de mesures de reflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
    [CrossRef]

2001 (1)

1998 (1)

1997 (1)

1994 (1)

1992 (1)

1986 (1)

1983 (1)

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

1973 (1)

1972 (1)

R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
[CrossRef]

1966 (1)

F. Abelès, M. Theye, “Methode de calcul des constantes optiques des couches minces absorbantes à partir de mesures de reflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Abelès, F.

F. Abelès, M. Theye, “Methode de calcul des constantes optiques des couches minces absorbantes à partir de mesures de reflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Aspnes, D. E.

T. Yasuda, D. E. Aspnes, “Optical-standard surfaces of single-crystal silicon for calibrating ellipsometers and reflectometers,” Appl. Opt. 33, 7435–7438 (1994).
[CrossRef] [PubMed]

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Babeva, Tz.

Campbell, R. D.

R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
[CrossRef]

Denton, R. E.

R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
[CrossRef]

Diaz, L.

Hansen, W.

Kitova, S.

Konstantinov, I.

Lamprecht, K.

Leising, G.

Panayotov, V.

V. Panayotov, I. Konstantinov, “Algebraic determination of thin-film optical constants from photometric (T, Rf, Rm) and (T, Rb, Rm) measurements,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 1070–1079 (1994).
[CrossRef]

Papousek, W.

Paulick, T. C.

Pozo, J.

Press, W.

W. Press, S. Teukolsky, W. Veterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1992), Chap. 10, pp. 408–410.

Studna, A. A.

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Teukolsky, S.

W. Press, S. Teukolsky, W. Veterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1992), Chap. 10, pp. 408–410.

Theye, M.

F. Abelès, M. Theye, “Methode de calcul des constantes optiques des couches minces absorbantes à partir de mesures de reflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Tomlin, S. G.

R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
[CrossRef]

Veterling, W.

W. Press, S. Teukolsky, W. Veterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1992), Chap. 10, pp. 408–410.

Yasuda, T.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

J. Phys. D (1)

R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
[CrossRef]

Phys. Rev. B (1)

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Surf. Sci. (1)

F. Abelès, M. Theye, “Methode de calcul des constantes optiques des couches minces absorbantes à partir de mesures de reflexion et de transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Other (3)

The standard calibration mirror of protected Al film was made in the Central Laboratory of Photoproceses, Sofia, Bulgaria ( http://www.clf.bas.bg) , by specially developed technology.

W. Press, S. Teukolsky, W. Veterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1992), Chap. 10, pp. 408–410.

V. Panayotov, I. Konstantinov, “Algebraic determination of thin-film optical constants from photometric (T, Rf, Rm) and (T, Rb, Rm) measurements,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 1070–1079 (1994).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

ΔRp70 as a function of the ratio Rsmp70/Rstand70 obtained for RAl-p70 = 0.8, Rstand70 = 0.5, and ΔR smp = ΔR stand = 0.001.

Fig. 2
Fig. 2

Solutions for thickness d for a dye film as a function of λ, derived from the triple methods shown. The indicated value of d is the best solution of each of the methods. The dashed lines illustrate the most accurate solution for d obtained by the (TRRp70) method.

Fig. 3
Fig. 3

Isolated solutions d(λ) = constant with the corresponding maximum errors obtained for dye, Ag, and Au thin films by the indicated methods, which yield the most accurate values of d. In each case the region marked with an arrow is the most accurate thickness solution for that film.

Fig. 4
Fig. 4

Dispersion curves of n and k and their absolute errors for the dye film with d = 102 ± 1.4 nm obtained by the two triple methods shown.

Fig. 5
Fig. 5

Dispersion curves of n and k and their absolute errors for the Ag film and the Au film obtained at the values of d and by the triple methods shown.

Fig. 6
Fig. 6

Dispersion curves of n and k and their absolute errors for the dye, Ag, and Au films obtained by the designated double methods.

Fig. 7
Fig. 7

Dispersion curve of n and k and the corresponding absolute errors, assembled from the spectral regions with maximum accuracy of the methods discussed for dye, Ag, and Au films.

Tables (2)

Tables Icon

Table 1 T, R, R m , Rp70, and Rmp70 Values (in %) of Ag, Au, and Dye Thin Films Deposited upon BK-7 and Si Wafer Substrates As a Function of Wavelength in Steps of 25 nm

Tables Icon

Table 2 Methods and Corresponding Spectral Ranges with Maximum Accuracy from Which the Dispersion Curves of n and k Are Assembled for the Dye, Ag, and Au Films

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

Rp70=Rsmp70Rstand70 RAl-p70,
ΔRp70=Rp70Rsmp70 ΔRsmp70+Rp70Rstand70ΔRstand70+Rp70RAl-p70ΔRAl-p70,
ΔRp70=RAl-p70Rstand70 ΔRsmp70+Rsmp70Rstand70RAl-p70Rstand70 ΔRstand70+ΔRAl-p70.
F=Tcalc-Texp2+Rcalc-Rexp2+Rcalc-Rexp2

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