Abstract

We select the best combinations of spectrophotometric quantities for the most accurate determination of the optical constants, n (refractive index), k (absorption coefficient), and the thicknesses of thin absorbing films. The basic comparative criteria used are the maximum absolute errors in the determination of n, k, and d that result from experimental errors in photometric measurements and in the optical constants of the substrates. We studied all possible combinations of photometric quantities T, Tsθ, Tpθ, R, Rsθ, Rpθ, R m, Rmsθ, and Rmpθ at 0° < θ ≤ 70°, where T denotes transmission; R, reflection; the subscripts s and p, s- and p-polarized light; m, reflection from a thin film coated upon an opaque substrate; and superscript θ, the angle of incidence of light. The absence of the subscripts s and p implies nonpolarized light; that of the subscript m, a nonabsorbing substrate; and that of superscript θ, normally incident light. The error analysis that is made admits the following conclusions: (1) The best double combinations are (TR), (TR m), (TRp70), and (TRmp70); (2) the best triple combinations are (TRR m), (TRRp70), (TRRmp70), (TR m Rp70), and (TR m Rmp70); (3) the methods indicated above, suitably combined, are quite sufficient to provide the maximum accuracy and reliability of n, k, and d for all practical situations; (4) TRR methods based on measurements with obliquely polarized light are more suitable for thin films with n < 1, such as some metal films; (5) the regions of n, k, and d/λ with the highest and the lowest accuracies do not overlap in either the TR or the TRR methods. Hence more combinations, preferably all, should be applied for the most accurate determination of n, k (and d), and the errors should be evaluated as a criterion for the best combination.

© 2001 Optical Society of America

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References

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  1. J. E. Nestell, R. W. Christy, “Derivation of optical constants of metals from thin-film measurements at oblique incidence,” Appl. Opt. 11, 643–650 (1972).
    [CrossRef] [PubMed]
  2. H. Liddell, Computer-Aided Techniques for Design of Multilayer Filters (Hilger, Bristol, UK, 1981), Chap. 6, pp. 118–133.
  3. 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).
  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. J.-J. Chen, J.-D. Lin, L.-J. Sheu, “Simultaneous measurements of spectral optical properties and thickness of an absorbing thin film on a substrate,” Thin Solid Films 354, 176–186 (1999).
    [CrossRef]
  6. A. B. Djurisic, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
    [CrossRef]
  7. 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]
  8. W. Hansen, “Optical characterization of thin films: theory,” J. Opt. Soc. Am. 63, 793–802 (1973).
    [CrossRef]
  9. 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]
  10. 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]
  11. O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates from transmittance and reflectance measurements,” J. Phys. D 24, 2088–2094 (1991).
    [CrossRef]
  12. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 4, pp. 55–57.
  13. P. O. Nilsson, “Determination of optical constants from intensity measurements at normal incidence,” Appl. Opt. 7, 435–441 (1968).
    [CrossRef] [PubMed]
  14. A. Hjortsberg, “Determination of optical constants of absorbing materials using transmission and reflection of thin films on partially metallized substrates: analysis of the new (T, Rm) technique,” Appl. Opt. 20, 1254–1262 (1981).
    [CrossRef] [PubMed]
  15. 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]
  16. D. Smith, E. Shiles, M. Inokuti, Handbook of Optical constants of Solids, D. Palik, ed. (AcademicOrlando, Fla., 1985), Part II, pp. 377–405.

1999 (2)

J.-J. Chen, J.-D. Lin, L.-J. Sheu, “Simultaneous measurements of spectral optical properties and thickness of an absorbing thin film on a substrate,” Thin Solid Films 354, 176–186 (1999).
[CrossRef]

A. B. Djurisic, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
[CrossRef]

1998 (1)

1997 (1)

1992 (1)

1991 (1)

O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates from transmittance and reflectance measurements,” J. Phys. D 24, 2088–2094 (1991).
[CrossRef]

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]

1981 (1)

1973 (1)

1972 (2)

J. E. Nestell, R. W. Christy, “Derivation of optical constants of metals from thin-film measurements at oblique incidence,” Appl. Opt. 11, 643–650 (1972).
[CrossRef] [PubMed]

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]

1968 (1)

Aspnes, D. E.

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]

Chen, J.-J.

J.-J. Chen, J.-D. Lin, L.-J. Sheu, “Simultaneous measurements of spectral optical properties and thickness of an absorbing thin film on a substrate,” Thin Solid Films 354, 176–186 (1999).
[CrossRef]

Christy, R. W.

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.

Djurisic, A. B.

A. B. Djurisic, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
[CrossRef]

Fritz, T.

A. B. Djurisic, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
[CrossRef]

Hansen, W.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 4, pp. 55–57.

Hjortsberg, A.

Hopfe, V.

O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates from transmittance and reflectance measurements,” J. Phys. D 24, 2088–2094 (1991).
[CrossRef]

Inokuti, M.

D. Smith, E. Shiles, M. Inokuti, Handbook of Optical constants of Solids, D. Palik, ed. (AcademicOrlando, Fla., 1985), Part II, pp. 377–405.

Kitova, S.

Klobes, P.

O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates from transmittance and reflectance measurements,” J. Phys. D 24, 2088–2094 (1991).
[CrossRef]

Konstantinov, I.

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]

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).

Lamprecht, K.

Leising, G.

Leo, K.

A. B. Djurisic, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
[CrossRef]

Liddell, H.

H. Liddell, Computer-Aided Techniques for Design of Multilayer Filters (Hilger, Bristol, UK, 1981), Chap. 6, pp. 118–133.

Lin, J.-D.

J.-J. Chen, J.-D. Lin, L.-J. Sheu, “Simultaneous measurements of spectral optical properties and thickness of an absorbing thin film on a substrate,” Thin Solid Films 354, 176–186 (1999).
[CrossRef]

Nestell, J. E.

Nilsson, P. O.

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).

Papousek, W.

Pozo, J.

Sheu, L.-J.

J.-J. Chen, J.-D. Lin, L.-J. Sheu, “Simultaneous measurements of spectral optical properties and thickness of an absorbing thin film on a substrate,” Thin Solid Films 354, 176–186 (1999).
[CrossRef]

Shiles, E.

D. Smith, E. Shiles, M. Inokuti, Handbook of Optical constants of Solids, D. Palik, ed. (AcademicOrlando, Fla., 1985), Part II, pp. 377–405.

Smith, D.

D. Smith, E. Shiles, M. Inokuti, Handbook of Optical constants of Solids, D. Palik, ed. (AcademicOrlando, Fla., 1985), Part II, pp. 377–405.

Stenzel, O.

O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates from transmittance and reflectance measurements,” J. Phys. D 24, 2088–2094 (1991).
[CrossRef]

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]

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]

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

J. Phys. D (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]

O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin film materials on semitransparent substrates from transmittance and reflectance measurements,” J. Phys. D 24, 2088–2094 (1991).
[CrossRef]

Opt. Commun. (1)

A. B. Djurisic, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
[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]

Thin Solid Films (1)

J.-J. Chen, J.-D. Lin, L.-J. Sheu, “Simultaneous measurements of spectral optical properties and thickness of an absorbing thin film on a substrate,” Thin Solid Films 354, 176–186 (1999).
[CrossRef]

Other (4)

H. Liddell, Computer-Aided Techniques for Design of Multilayer Filters (Hilger, Bristol, UK, 1981), Chap. 6, pp. 118–133.

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).

D. Smith, E. Shiles, M. Inokuti, Handbook of Optical constants of Solids, D. Palik, ed. (AcademicOrlando, Fla., 1985), Part II, pp. 377–405.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 4, pp. 55–57.

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Figures (10)

Fig. 1
Fig. 1

Set of contours of the reflections Rmsθ calculated with the n and k values indicated by the intersecting points of dotted lines. The contours are for angles of incidence from 0° to 80° in steps of 10°. The numbers used for the abscissa and the ordinate show the scale in the nk plane. It can be seen that the contours of Rmsθ for all angles overlap entirely in practically every one of the subgraphics.

Fig. 2
Fig. 2

Contours of transmission T with values of 0.01 plotted in the nk plane for four values of d/λ.

Fig. 3
Fig. 3

Absolute errors Δn and Δk as functions of incidence angle θ obtained for the indicated values of n and k at d/λ = 0.1 with (TRpθ) methods.

Fig. 4
Fig. 4

Same as Fig. 3 but obtained with (TRmpθ) methods.

Fig. 5
Fig. 5

Isolines of Δn, Δk, and Δd with indicated values in the θ1–θ2 plane obtained with (TRpθ1Rpθ2) and (TRmpθ1 Rmpθ2) methods at d/λ = 0.1 for selected values of n and k as shown.

Fig. 6
Fig. 6

Same as in Fig. 5 but obtained with the (TRpθ1Rmpθ2) methods.

Fig. 7
Fig. 7

Isolines of maximum absolute errors Δn and Δk in the nk plane obtained with (TR), (TR m ), (TRp70), and (TRmp70) methods at d/λ = 0.1. The isoline values are labeled. Dashed curves outline n and k values for which the matrix determinant M3 of Eq. (4) becomes zero.

Fig. 8
Fig. 8

Selected regions of the nk plane, where contours of T and R as well as of T and Rp70 are tangent to each other.

Fig. 9
Fig. 9

Isolines of maximum absolute errors Δn, Δk, and Δd in the nk plane obtained with (TRR m ), (TRRp70), (TRRmp70), (TR m Rp70), and (TR m TRmp70) methods at d/λ = 0.1. The isoline values are indicated in the figure. The dashed curves outline n and k values for which the matrix determinant M1 of Eq. (3) becomes zero.

Fig. 10
Fig. 10

Contours of T, R, and R m as well as of T, R, and Rp70 plotted in the selected regions of nk, nd/λ, or kd/λ planes where at least two of them are tangent to each other. The R and R m contours overlap in the nd/λ plane, and the R and Rp70 contours overlap in the kd/λ plane.

Tables (1)

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Table 1 Experimental Errors in the Spectrophotometric Quantities and Optical Constants of Substrates Used in this Study

Equations (5)

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Δn=nTΔT+nR1θ1ΔR1θ1+nR2θ2ΔR2θ2+nnsΔns+nnmΔnm+nkmΔkm+nθ1Δθ1+nθ2Δθ2.
ΔnΔkΔd=|M1-1| * ΔTΔR1θ1ΔR2θ2+|M1-1* M2| * ΔnsΔnmΔkmΔθ1Δθ2,
M1=TnTkTdR1θ1nR1θ1kR1θ1dR2θ2nR2θ2kR2θ2d,M2=Tns0000R1θ1nsR1θ1nmR1θ1kmR1θ1θ10R2θ2nsR2θ2nmR2θ2km0R2θ2θ2.
ΔnΔk=|M3-1| * ΔTΔRθ+|M3-1* M4| * ΔnsΔnmΔkmΔθΔd,
M3=TnTkRθnRθk,M4=Tns000TdRθnsRθnmRθkmRθθRθd.

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