Abstract

We propose a new and simple procedure to overcome the ambiguity in the determination of optical constants of thin absorbing films from spectroscopic reflectance and transmittance measurements. The basis for the proposed method is an error analysis with the help of an error simulation technique and an error variation technique. We show that in practice (owing to experimental errors) it is not possible to overcome the problem of ambiguity by normal-incidence spectroscopic measurements alone. At least one oblique-incidence measurement is necessary for unambiguously determining the optical constants of the film. We discuss the consequences of experimental errors of the measured transmittance and reflectance values for the determination of the optical constants.

© 1997 Optical Society of America

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References

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  1. G. Leising, “Optics of media,” in Organic Materials for Photonics, G. Zerbi, ed. (North-Holand, Amsterdam, 1993).
    [CrossRef]
  2. 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, 1254–1263, (1981).
    [CrossRef] [PubMed]
  3. E. A. Bondar, Yu. A. Kulyupin, N. N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201–209 (1978).
    [CrossRef]
  4. T. S. Eriksson, A. Hjortsberg, “Determination of optical constants from photometric measurements,” Sol. Energy Mater. 11, 141–148 (1984).
    [CrossRef]
  5. 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 Appl. Phys. 24, 2088–2094 (1991).
    [CrossRef]
  6. Z. Yinping, Z. Chung, G. Xinshi, L. Xingang, “A precise and simple method the relative transmission fringe depth method of determining the optical constants and thickness of semitransparent films,” J. Phys. D Appl. Phys. 25, 1004–1009 (1992).
    [CrossRef]
  7. T. Hashimoto, H. Matsuzaki, H. Tsuchida, K. Yamamoto, “High-precision measurement for refractive index distribution and dispersion using an improved scanning total reflection method,” Jpn. J. Appl. Phys. 1 31, 1602–1605 (1992).
    [CrossRef]
  8. B. J. Stagg, T. T. Charalampopoulos, “Sensitivity of the reflection technique: optimum angles of incidence to determine the optical properties of materials,” Appl. Opt. 31, 4420–4427 (1992).
    [CrossRef] [PubMed]
  9. J. M. del Pozo, L. Diaz, “Method for the determination of optical constants of thin films: dependence on experimental uncertainties,” Appl. Opt. 31, 4474–4481 (1992).
    [CrossRef] [PubMed]
  10. T. Kihara, K. Yokomori, “Simultaneous measurement of the refractive index and thickness of thin films by S-polarized reflectances,” Appl. Opt. 31, 4482–4487 (1992).
    [CrossRef] [PubMed]
  11. H. Sommariva, W. Papousek, “Reflection of non-monochromatic waves from multilayered dispersive media,” in Proceedings of the 1992 (URSI) International Symposium on Electromagnetic Theory [International Union of Radio Science (URSI)Sidney, Australia, 1992], pp. 492–494.
  12. H. Sommariva, “Reflection and transmission of non-monochromatic electromagnetic waves in layered media,” Ph.D. thesis (Institute for Theoretical Physics, University of Technology Graz, Austria, 1992).
  13. R. Uitz, G. Temmel, G. Leising, H. Kahlert, “Infrared optical constants and dichroism of thin polymer films: trans-polyacetylene,” Z. Phys. B Condensed Matter 67, 459–465 (1987).
    [CrossRef]
  14. H. Sommariva, W. Papousek, G. Leising, “Symmetry properties of the transmission of electromagnetic waves through layered media for arbitrary polarization,” Int. J. Electron. Commun. 48, 42–44 (1994).
  15. E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).
  16. R. C. McPhedran, L. C. Botten, D. R. McKenzie, R. P. Netterfield, “Unambiguous determination of optical constants of absorbing films by reflectance and transmittance measurements,” Appl. Opt. 23, 1197–1205 (1984).
    [CrossRef] [PubMed]
  17. K. Lamprecht, “The problem of ambiguity in connection with the determination of optical constants of thin absorbing films from spectroscopic reflectance and transmittance measurements,” Ph.D. thesis (Institute for Theoretical Physics and Institute for Solid State Physics, University of Technology Graz, Austria, 1996).
  18. J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
    [CrossRef]

1994 (1)

H. Sommariva, W. Papousek, G. Leising, “Symmetry properties of the transmission of electromagnetic waves through layered media for arbitrary polarization,” Int. J. Electron. Commun. 48, 42–44 (1994).

1992 (5)

Z. Yinping, Z. Chung, G. Xinshi, L. Xingang, “A precise and simple method the relative transmission fringe depth method of determining the optical constants and thickness of semitransparent films,” J. Phys. D Appl. Phys. 25, 1004–1009 (1992).
[CrossRef]

T. Hashimoto, H. Matsuzaki, H. Tsuchida, K. Yamamoto, “High-precision measurement for refractive index distribution and dispersion using an improved scanning total reflection method,” Jpn. J. Appl. Phys. 1 31, 1602–1605 (1992).
[CrossRef]

B. J. Stagg, T. T. Charalampopoulos, “Sensitivity of the reflection technique: optimum angles of incidence to determine the optical properties of materials,” Appl. Opt. 31, 4420–4427 (1992).
[CrossRef] [PubMed]

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

T. Kihara, K. Yokomori, “Simultaneous measurement of the refractive index and thickness of thin films by S-polarized reflectances,” Appl. Opt. 31, 4482–4487 (1992).
[CrossRef] [PubMed]

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 Appl. Phys. 24, 2088–2094 (1991).
[CrossRef]

1987 (1)

R. Uitz, G. Temmel, G. Leising, H. Kahlert, “Infrared optical constants and dichroism of thin polymer films: trans-polyacetylene,” Z. Phys. B Condensed Matter 67, 459–465 (1987).
[CrossRef]

1984 (2)

1981 (1)

1978 (1)

E. A. Bondar, Yu. A. Kulyupin, N. N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201–209 (1978).
[CrossRef]

1965 (1)

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Bondar, E. A.

E. A. Bondar, Yu. A. Kulyupin, N. N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201–209 (1978).
[CrossRef]

Botten, L. C.

Charalampopoulos, T. T.

Chung, Z.

Z. Yinping, Z. Chung, G. Xinshi, L. Xingang, “A precise and simple method the relative transmission fringe depth method of determining the optical constants and thickness of semitransparent films,” J. Phys. D Appl. Phys. 25, 1004–1009 (1992).
[CrossRef]

del Pozo, J. M.

Diaz, L.

Eriksson, T. S.

T. S. Eriksson, A. Hjortsberg, “Determination of optical constants from photometric measurements,” Sol. Energy Mater. 11, 141–148 (1984).
[CrossRef]

Hashimoto, T.

T. Hashimoto, H. Matsuzaki, H. Tsuchida, K. Yamamoto, “High-precision measurement for refractive index distribution and dispersion using an improved scanning total reflection method,” Jpn. J. Appl. Phys. 1 31, 1602–1605 (1992).
[CrossRef]

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 Appl. Phys. 24, 2088–2094 (1991).
[CrossRef]

Kahlert, H.

R. Uitz, G. Temmel, G. Leising, H. Kahlert, “Infrared optical constants and dichroism of thin polymer films: trans-polyacetylene,” Z. Phys. B Condensed Matter 67, 459–465 (1987).
[CrossRef]

Kihara, T.

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 Appl. Phys. 24, 2088–2094 (1991).
[CrossRef]

Kulyupin, Yu. A.

E. A. Bondar, Yu. A. Kulyupin, N. N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201–209 (1978).
[CrossRef]

Lamprecht, K.

K. Lamprecht, “The problem of ambiguity in connection with the determination of optical constants of thin absorbing films from spectroscopic reflectance and transmittance measurements,” Ph.D. thesis (Institute for Theoretical Physics and Institute for Solid State Physics, University of Technology Graz, Austria, 1996).

Leising, G.

H. Sommariva, W. Papousek, G. Leising, “Symmetry properties of the transmission of electromagnetic waves through layered media for arbitrary polarization,” Int. J. Electron. Commun. 48, 42–44 (1994).

R. Uitz, G. Temmel, G. Leising, H. Kahlert, “Infrared optical constants and dichroism of thin polymer films: trans-polyacetylene,” Z. Phys. B Condensed Matter 67, 459–465 (1987).
[CrossRef]

G. Leising, “Optics of media,” in Organic Materials for Photonics, G. Zerbi, ed. (North-Holand, Amsterdam, 1993).
[CrossRef]

Matsuzaki, H.

T. Hashimoto, H. Matsuzaki, H. Tsuchida, K. Yamamoto, “High-precision measurement for refractive index distribution and dispersion using an improved scanning total reflection method,” Jpn. J. Appl. Phys. 1 31, 1602–1605 (1992).
[CrossRef]

McKenzie, D. R.

McPhedran, R. C.

Mead, R.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Nelder, J. A.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Netterfield, R. P.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

Papousek, W.

H. Sommariva, W. Papousek, G. Leising, “Symmetry properties of the transmission of electromagnetic waves through layered media for arbitrary polarization,” Int. J. Electron. Commun. 48, 42–44 (1994).

H. Sommariva, W. Papousek, “Reflection of non-monochromatic waves from multilayered dispersive media,” in Proceedings of the 1992 (URSI) International Symposium on Electromagnetic Theory [International Union of Radio Science (URSI)Sidney, Australia, 1992], pp. 492–494.

Popovich, N. N.

E. A. Bondar, Yu. A. Kulyupin, N. N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201–209 (1978).
[CrossRef]

Sommariva, H.

H. Sommariva, W. Papousek, G. Leising, “Symmetry properties of the transmission of electromagnetic waves through layered media for arbitrary polarization,” Int. J. Electron. Commun. 48, 42–44 (1994).

H. Sommariva, W. Papousek, “Reflection of non-monochromatic waves from multilayered dispersive media,” in Proceedings of the 1992 (URSI) International Symposium on Electromagnetic Theory [International Union of Radio Science (URSI)Sidney, Australia, 1992], pp. 492–494.

H. Sommariva, “Reflection and transmission of non-monochromatic electromagnetic waves in layered media,” Ph.D. thesis (Institute for Theoretical Physics, University of Technology Graz, Austria, 1992).

Stagg, B. J.

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 Appl. Phys. 24, 2088–2094 (1991).
[CrossRef]

Temmel, G.

R. Uitz, G. Temmel, G. Leising, H. Kahlert, “Infrared optical constants and dichroism of thin polymer films: trans-polyacetylene,” Z. Phys. B Condensed Matter 67, 459–465 (1987).
[CrossRef]

Tsuchida, H.

T. Hashimoto, H. Matsuzaki, H. Tsuchida, K. Yamamoto, “High-precision measurement for refractive index distribution and dispersion using an improved scanning total reflection method,” Jpn. J. Appl. Phys. 1 31, 1602–1605 (1992).
[CrossRef]

Uitz, R.

R. Uitz, G. Temmel, G. Leising, H. Kahlert, “Infrared optical constants and dichroism of thin polymer films: trans-polyacetylene,” Z. Phys. B Condensed Matter 67, 459–465 (1987).
[CrossRef]

Xingang, L.

Z. Yinping, Z. Chung, G. Xinshi, L. Xingang, “A precise and simple method the relative transmission fringe depth method of determining the optical constants and thickness of semitransparent films,” J. Phys. D Appl. Phys. 25, 1004–1009 (1992).
[CrossRef]

Xinshi, G.

Z. Yinping, Z. Chung, G. Xinshi, L. Xingang, “A precise and simple method the relative transmission fringe depth method of determining the optical constants and thickness of semitransparent films,” J. Phys. D Appl. Phys. 25, 1004–1009 (1992).
[CrossRef]

Yamamoto, K.

T. Hashimoto, H. Matsuzaki, H. Tsuchida, K. Yamamoto, “High-precision measurement for refractive index distribution and dispersion using an improved scanning total reflection method,” Jpn. J. Appl. Phys. 1 31, 1602–1605 (1992).
[CrossRef]

Yinping, Z.

Z. Yinping, Z. Chung, G. Xinshi, L. Xingang, “A precise and simple method the relative transmission fringe depth method of determining the optical constants and thickness of semitransparent films,” J. Phys. D Appl. Phys. 25, 1004–1009 (1992).
[CrossRef]

Yokomori, K.

Appl. Opt. (5)

Comput. J. (1)

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Int. J. Electron. Commun. (1)

H. Sommariva, W. Papousek, G. Leising, “Symmetry properties of the transmission of electromagnetic waves through layered media for arbitrary polarization,” Int. J. Electron. Commun. 48, 42–44 (1994).

J. Phys. D Appl. Phys. (2)

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 Appl. Phys. 24, 2088–2094 (1991).
[CrossRef]

Z. Yinping, Z. Chung, G. Xinshi, L. Xingang, “A precise and simple method the relative transmission fringe depth method of determining the optical constants and thickness of semitransparent films,” J. Phys. D Appl. Phys. 25, 1004–1009 (1992).
[CrossRef]

Jpn. J. Appl. Phys. 1 (1)

T. Hashimoto, H. Matsuzaki, H. Tsuchida, K. Yamamoto, “High-precision measurement for refractive index distribution and dispersion using an improved scanning total reflection method,” Jpn. J. Appl. Phys. 1 31, 1602–1605 (1992).
[CrossRef]

Sol. Energy Mater. (1)

T. S. Eriksson, A. Hjortsberg, “Determination of optical constants from photometric measurements,” Sol. Energy Mater. 11, 141–148 (1984).
[CrossRef]

Thin Solid Films (1)

E. A. Bondar, Yu. A. Kulyupin, N. N. Popovich, “The inverse problem of the phenomenological theory of the optical properties of thin films,” Thin Solid Films 55, 201–209 (1978).
[CrossRef]

Z. Phys. B Condensed Matter (1)

R. Uitz, G. Temmel, G. Leising, H. Kahlert, “Infrared optical constants and dichroism of thin polymer films: trans-polyacetylene,” Z. Phys. B Condensed Matter 67, 459–465 (1987).
[CrossRef]

Other (5)

H. Sommariva, W. Papousek, “Reflection of non-monochromatic waves from multilayered dispersive media,” in Proceedings of the 1992 (URSI) International Symposium on Electromagnetic Theory [International Union of Radio Science (URSI)Sidney, Australia, 1992], pp. 492–494.

H. Sommariva, “Reflection and transmission of non-monochromatic electromagnetic waves in layered media,” Ph.D. thesis (Institute for Theoretical Physics, University of Technology Graz, Austria, 1992).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

G. Leising, “Optics of media,” in Organic Materials for Photonics, G. Zerbi, ed. (North-Holand, Amsterdam, 1993).
[CrossRef]

K. Lamprecht, “The problem of ambiguity in connection with the determination of optical constants of thin absorbing films from spectroscopic reflectance and transmittance measurements,” Ph.D. thesis (Institute for Theoretical Physics and Institute for Solid State Physics, University of Technology Graz, Austria, 1996).

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Figures (13)

Fig. 1
Fig. 1

Thin film on a thick substrate.

Fig. 2
Fig. 2

Simulated experimental reflection coefficient Rexp (film-side incidence) of the sample calculated from Eqs. (4) and (7) in dependence on the angle of incidence. The film parameters are nF = n1 = 5, κF = κ1 = 1, dF = 4.25 × 10-7. The substrate parameters are nS = 3.39, κS = 0.0003, dS = 5.3 × 10-4 m. Wave number ν̅0 = 750 cm-1; pol, polarization.

Fig. 3
Fig. 3

Simulated experimental transmission coefficient Texp of the sample calculated from Eqs. (5) and (8) in dependence on the angle of incidence. Film and substrate parameters as in Fig. 2.

Fig. 4
Fig. 4

Simulated experimental reflection coefficient exp (substrate-side incidence) of the sample calculated from Eqs. (6) and (9) in dependence on the angle of incidence. Film and substrate parameters as in Fig. 2.

Fig. 5
Fig. 5

Contour diagram of the contour lines Rn = Rexpn = 0.55 and Tn = Texpn = 0.19 at ν̅0 = 750 cm-1.

Fig. 6
Fig. 6

Contour diagram of the contour lines Rn = Rexpn = 0.55, Tn = Texpn = 0.19, and n = expn = 0.38 at ν̅0 = 750 cm-1.

Fig. 7
Fig. 7

Contour diagram of the contour lines Rn = Rexpn = 0.55 - 5% = 0.52, Texpn = 0.19 - 5% = 0.18, and n = expn = 0.38 - 5% = 0.36 at ν̅0 = 750 cm-1.

Fig. 8
Fig. 8

Minima of f(nF, κF, e) according to Eq. (25) in their dependence on the assumed error e of the spectroscopic measurements. f1 is the minimum in the area of solution 1, and f2 is the minimum in area of solution 2 (see Fig. 7).

Fig. 9
Fig. 9

Contour diagram of the contour lines Rn = Rexpn = 0.55, Tn = Texpn = 0.19 and RP(α = 70°) = RexpP = 0.14 at ν̅0 = 750 cm-1.

Fig. 10
Fig. 10

Contour diagram of the contour lines Rn = Rexpn = 0.55 - 5% = 0.52, Tn = Texpn = 0.19 - 5% = 0.18 and RP(α = 70°) = RexpP = 0.14 - 5% = 0.13 at ν̅0 = 750 cm-1.

Fig. 11
Fig. 11

Minima of f(nF, κF, e) according to Eq. (26) in their dependence on the assumed error e of the spectroscopic measurements. f1 is the minimum in the area of solution 1, and f2 is the minimum in area of solution 2 (see Fig. 10).

Fig. 12
Fig. 12

Minima of f(nF, κF, e) according to Eq. (27) in their dependence on the assumed error e of the spectroscopic measurements. f1 is the minimum in the area of solution 1, and f2 is the minimum in area of solution 2. For e < -12% f(nF, κF, e) has only a single minimum in area of solution 1 (no minimum in area of solution 2).

Fig. 13
Fig. 13

Optical constants nF and κF of the film in their dependence on the assumed error e of the spectroscopic measurements.

Equations (28)

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Jν¯=Jξ2πexp-ν¯-ν¯022ξ2
lC=1Δν¯0,
2nFdFlC2nSdS;
RS=r¯ASF2+t¯SAt¯AS2t¯ASF2t¯ASF2r¯SA2 exp-4dS Im kzS1-r¯SA|2r¯SAF2 exp-4dS Im kzS,
TS=t¯SA2t¯ASF2 exp-2dS Im kzS1-r¯SA2r¯SAF2 exp-4dS Im kzS,
R¯S=r¯ASF2+t¯AS|2t¯SA2r¯SAF2 exp-4dS Im kzS1-r¯SA2r¯SAF2 exp-4dS Im kzS.
RP=rASF2+tSAtAS2tASF2tASF2rSA2 exp-4dS Im kzS1-rSA2rSAF2 exp-4dS Im kzS,
TP=tSA2tASF2 exp-4dS Im kzS1-rSA2rSAF2 exp-4dS Im kzS,
R¯P=rASF2+tAS2tSA2rASF2 exp-4dS Im kzS1-rSA2rSAF2 exp-4dS Im kzS.
rASF=rAF+rFStAFtFA expikzFdFexpikzFdF-rFSrFA expikzFdF,
tASF=tAFtFSexp-ikzFdF-rFSrFA expikzFdF,
rSAF=rSF+rFAtSFtFS expikzFdFexp-ikzFdF-rFSrFA expikzFdF.
rln=nkzl-lkznnkzl+lkzn,
tln=2nkzlnkzl+lkzn,
rln¯=kzl-kznkzl+kzn,
tln¯=2kzlkzl+kzn,
l=l-iσlω00=l-iσl2π0cν¯0,
kzl=-k0l-sin2 α1/2, Re(·)1/2>0.
k0=ω/c=2πν¯0,
fnF, κF=RnnF, κF-Rexpn2+TnnF, κF-Texpn2+R¯nnF, κF-R¯expn2,
fnF, κFmin=fnF1, κF1=f1;
fnF, κFmin=fnF2, κF2=f2;
f1<f2,
f2<f1.
fnF, κF=RnnF, κF-Rexpn1+e1002+TnnF, κF-Texpn1+e1002+R¯nnF, κF-R¯expn1+e1002.
fnF, κF=RnnF, κF-Rexpn1+e1002+TnnF, κF-Texpn1+e1002+RPnF, κF-RexpP1+e1002,
fnF, κF=RnnF, κF-Rexpn1+e1002+TnnF, κF-Texpn1+e1002+TPnF, κF-TexpP1+e1002.
fnF, κF=RnnF, κF-Rexpn2+TnnF, κF-Texpn2+TPnF, κF-TexpP2.

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