Abstract

An algebraic method to calculate the optical constants for a weakly absorbing thin film from the spectrum of normal reflectance is described. The calculation of the refractive index of the thin film is simplified by constructing a midpoint envelope through the reflection spectrum. If a portion of the spectrum includes a region where the film is nonabsorbing, the results can be used to calculate an algebraic solution for the refractive index and the absorption coefficient of the thin film throughout the entire spectrum. The method is used to determine the constants for a coating of alumina on a glass substrate. The results are compared to the calculation from the extrema of the spectrum.

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References

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  1. C. K. Carniglia, "Effects of dispersion on the determination of optical constants of thin films," in Thin Film Technologies II, J. Roland Jacobsson, ed., Proc. SPIE 652, 158-165 (1986).
  2. J. C. Manifacier, J. Gasiot, and J. P. 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 (1979).
    [CrossRef]
  3. D. E. Morton, "Characterizing optical thin films (I)," Vacuum Technol. Coat. 2(9), 24-31 (2001).
  4. H. G. Tompkins and W. A. McGahan, "The anatomy of a reflectance spectrum," in Spectroscopic Ellipsometry and Reflectometry: A User's Guide (Wiley, 1999), pp. 54-61.
  5. R. Swanepoel, "Determining refractive index and thickness of thin films from wavelength measurements only," J. Opt. Soc. Am. A 2, 1339-1343 (1985).
    [CrossRef]
  6. R. and G. A. J. Amaratunga, "Determination of the optical constants and thickness of thin films on slightly absorbing substrates," Appl. Opt. 34, 7914-7924 (1995).
  7. D. A. Minkov, "Calculation of the optical constants of a thin layer upon a transparent substrate from the reflection spectrum," J. Phys. D 22, 1157-1161 (1989).
    [CrossRef]
  8. D. P. Arndt, R. M. A. Azzam, J. M. Bennett, J. P. Borgogno, C. K. Carniglia, W. E. Case, J. A. Dobrowolski, 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, and T. F. Thonn, "Multiple determination of the optical constants of thin film coating materials," Appl. Opt. 23, 3571-3596 (1984).
    [CrossRef] [PubMed]
  9. W. E. Case, "Algebraic method for extracting thin film optical parameters from spectrophotometer measurements," Appl. Opt. 22, 1832-1836 (1983).
    [CrossRef] [PubMed]
  10. V. Panayotov and I. Konstantinov, "Determination of thin film optical parameters from photometric measurements: an algebraic solution for the (T, Rf, Rb) method," Appl. Opt. 30, 2795-2800 (1991).
    [CrossRef] [PubMed]
  11. O. Stenzel, V. Hopfe, and 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. M. Ylilammi and T. Ranta-aho, "Optical determination of the film thicknesses in multilayer thin film structures," Thin Solid Films 232, 56-62 (1993).
    [CrossRef]
  13. H. Anders, "An introduction to the theory of thin films," Thin Films in Optics (Focal Press, 1967), pp. 13-40.
  14. S. D. Conte and C. de Boor, "Interpolation by Polynomials," Elementary Numerical Analysis: An Algorithmic Approach (McGraw-Hill, 1980), pp. 31-71.
  15. Y. Hishikawa, N. Nakamura, S. Tsuda, S. Nakano, Y. Kishi, and Y. Kuwano, "Interference-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]
  16. A. Starke, H. Schink, J. Kolbe, and J. Ebert, "Laser induced damage thresholds and optical constants of ion plated and ion beam sputtered Al2O3− and HfO2− coatings for the ultraviolet," Proc. SPIE 1270, 299-304 (1990).
    [CrossRef]
  17. S. S. Ballard, J. S. Browder, and J. F. Ebersole, "Refractive index of special crystals and certain glasses: sapphire," in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, 1972), pp. 6-40.

2001

D. E. Morton, "Characterizing optical thin films (I)," Vacuum Technol. Coat. 2(9), 24-31 (2001).

1995

1993

M. Ylilammi and T. Ranta-aho, "Optical determination of the film thicknesses in multilayer thin film structures," Thin Solid Films 232, 56-62 (1993).
[CrossRef]

1991

Y. Hishikawa, N. Nakamura, S. Tsuda, S. Nakano, Y. Kishi, and Y. Kuwano, "Interference-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]

V. Panayotov and I. Konstantinov, "Determination of thin film optical parameters from photometric measurements: an algebraic solution for the (T, Rf, Rb) method," Appl. Opt. 30, 2795-2800 (1991).
[CrossRef] [PubMed]

O. Stenzel, V. Hopfe, and 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]

1990

A. Starke, H. Schink, J. Kolbe, and J. Ebert, "Laser induced damage thresholds and optical constants of ion plated and ion beam sputtered Al2O3− and HfO2− coatings for the ultraviolet," Proc. SPIE 1270, 299-304 (1990).
[CrossRef]

1989

D. A. Minkov, "Calculation of the optical constants of a thin layer upon a transparent substrate from the reflection spectrum," J. Phys. D 22, 1157-1161 (1989).
[CrossRef]

1986

C. K. Carniglia, "Effects of dispersion on the determination of optical constants of thin films," in Thin Film Technologies II, J. Roland Jacobsson, ed., Proc. SPIE 652, 158-165 (1986).

1985

1984

1983

1979

J. C. Manifacier, J. Gasiot, and J. P. 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 (1979).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

J. Phys. D

O. Stenzel, V. Hopfe, and 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]

D. A. Minkov, "Calculation of the optical constants of a thin layer upon a transparent substrate from the reflection spectrum," J. Phys. D 22, 1157-1161 (1989).
[CrossRef]

J. Phys. E

J. C. Manifacier, J. Gasiot, and J. P. 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 (1979).
[CrossRef]

Jpn. J. Appl. Phys.

Y. Hishikawa, N. Nakamura, S. Tsuda, S. Nakano, Y. Kishi, and Y. Kuwano, "Interference-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]

Proc. SPIE

A. Starke, H. Schink, J. Kolbe, and J. Ebert, "Laser induced damage thresholds and optical constants of ion plated and ion beam sputtered Al2O3− and HfO2− coatings for the ultraviolet," Proc. SPIE 1270, 299-304 (1990).
[CrossRef]

Thin Solid Films

M. Ylilammi and T. Ranta-aho, "Optical determination of the film thicknesses in multilayer thin film structures," Thin Solid Films 232, 56-62 (1993).
[CrossRef]

Vacuum Technol. Coat.

D. E. Morton, "Characterizing optical thin films (I)," Vacuum Technol. Coat. 2(9), 24-31 (2001).

Other

H. G. Tompkins and W. A. McGahan, "The anatomy of a reflectance spectrum," in Spectroscopic Ellipsometry and Reflectometry: A User's Guide (Wiley, 1999), pp. 54-61.

C. K. Carniglia, "Effects of dispersion on the determination of optical constants of thin films," in Thin Film Technologies II, J. Roland Jacobsson, ed., Proc. SPIE 652, 158-165 (1986).

H. Anders, "An introduction to the theory of thin films," Thin Films in Optics (Focal Press, 1967), pp. 13-40.

S. D. Conte and C. de Boor, "Interpolation by Polynomials," Elementary Numerical Analysis: An Algorithmic Approach (McGraw-Hill, 1980), pp. 31-71.

S. S. Ballard, J. S. Browder, and J. F. Ebersole, "Refractive index of special crystals and certain glasses: sapphire," in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, 1972), pp. 6-40.

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Figures (2)

Fig. 1
Fig. 1

Reflectance spectrum of a single thin film of ion-beam sputtered aluminum oxide on superpolished glass.

Fig. 2
Fig. 2

Optical constants for an ion-beam sputtered aluminum oxide thin film: (a) refractive index n 1 , (b) absorption coefficient k 1 . (1) Calculation without δ 2 correction, using R add and R sub ; (2) calculation using δ 2 correction, convergent solution for x and R 1 ; (3) calculation using δ 2 correction, convergent solution for x and R 2 ; (◊) IBS film, Ref. [16]; (□) crystalline sapphire, Ref. [17].

Tables (2)

Tables Icon

Table 1 Tangent Envelope Reflectances of Ion-Beam Sputtered Alumina on Superpolished Glass

Tables Icon

Table 2 Comparison of Physical Thickness Calculations Using Traditional Extrema Method and Midpoint Envelope Method

Equations (197)

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n 0 + i k 0
n 1 + i k 1
n 2 + i k 2
R = R 1 + R 2 + 2 ( R 1 R 2 ) 1 / 2  cos ( x δ 1 ) 1 + R 1 R 2 + 2 ( R 1 R 2 ) 1 / 2  cos ( x + δ 1 ) ,
R 1 = ( n 1 n 0 ) 2 + ( k 1 k 0 ) 2 ( n 1 + n 0 ) 2 + ( k 1 + k 0 ) 2 ,
R 2 = exp ( 2 a d ) ( n 2 n 1 ) 2 + ( k 2 k 1 ) 2 ( n 2 + n 1 ) 2 + ( k 2 + k 1 ) 2 ,
x = 4 π n 1 d λ + δ 2 ,
a = 4 π k 1 λ ,
tan   δ j = 2 n j 1 k j 2 n j k j 1 n j 1 2 n j 2 + k j 1 2 k j 2 .
n 0 = 1
k 0 = 0
R 1 = ( n 1 1 ) 2 + k 1 2 ( n 1 + 1 ) 2 + k 1 2 .
δ 1
x = 1
x = 1
R a d d = ( r 1 + r 2 1 + r 1 r 2 ) 2 ,
| r 1 | = R 1 1 / 2 , | r 2 | = R 2 1 / 2 .
R s u b = ( r 1 r 2 1 r 1 r 2 ) 2 .
n 1 < n 2
R a d d
n 1 > n 2
R s u b
x = 0
R m i d = R 2 + R 1 1 + R 1 R 2 .
R add
R sub
R m i d = R ¯ R ^ 2 1 R ¯ ,
R ¯ = ( R a d d + R s u b ) / 2 , R ^ = ( R a d d R s u b ) 1 / 2 .
R mid
R R mid
cos   x = ½ ( C + 1 / C ) R / R mid 1 1 R ,
C = r 1 / r 2 .
R 1
R 2
r max = ( 1 R a d d ) 1 / 2 ( 1 R s u b ) 1 / 2 R a d d 1 / 2 ( 1 R s u b ) 1 / 2 R s u b 1 / 2 ( 1 R a d d ) 1 / 2 .
r min = ( 1 R a d d ) 1 / 2 ( 1 R s u b ) 1 / 2 R a d d 1 / 2 ( 1 R s u b ) 1 / 2 + R s u b 1 / 2 ( 1 R a d d ) 1 / 2 .
R 1 < R 2
r 1 = r min
r 2 = r max
r 2 = r min
r 1 = r max
R 2 < R 1
R add
R sub
+ 1
1
cos   x
cos   x = N avg Y ( R ) ,
Y ( R ) = R / R mid 1 1 R ,
N avg = 2 / [ Y ( R add ) Y ( R sub ) ] .
n 1
k 1 = 0
k 2 = 0
R add = ( n 2 1 n 2 + 1 ) 2 .
n 2
n 2 = 1 + R add 1 / 2 1 R add 1 / 2 .
R sub = n 1 2 n 2 n 1 2 + n 2 .
n 2
n 1
n 1 = n 2 ( 1 + R s u b 1 / 2 ) 1 R s u b 1 / 2 .
n 1
d = x λ 4 π n 1 ,
n 1
n 1
n 1 = ( x δ 2 ) λ 4 π d x λ 4 π d .
k 1
k 1 n 1
k 2 n 2
k 1
k 1 = 2 [ ln | n 2 n 1 | ln ( n 2 + n 1 ) ] ln ( R 2 ) 8 π d λ .
k 1
n 1
n 2
R 2
k 1
R 1
k 1
k 1 2 = 2 n 1 1 + R 1 1 R 1 ( 1 + n 1 2 ) .
1 R
k 2
n 2
k 2
δ 2
n 1
δ 2
k 1
n 1
n 1
δ 2
R add
R sub
R 1
R 2
n 1
k 1
R 1
R 2
δ 2
n 1
n 1
k 1
R 1
R 2
R mid
n 1
k 1
R 2
R 1
δ 2 < 0.01 n 1
n 1
δ 2
δ 2 > 0.01 n 1
δ 2
δ 2
n 1
k 1
n 1
k 1
R 1
R 2
R add
R sub
R mid
R / ( 1 T )
335   nm
R add
632   nm
632   nm
350   nm
569.2 ± 0.4   nm
569.08 ± 0.03   nm
0.328   nm
0.314   nm
n 2
R add
n 1
R sub
n 2
δ 2
n 1
k 1
R 1
δ 2
n 1
k 1
R 2
5.7   mm
217.5   nm
k 2
3 × 10 7
k 2
n 1
R 2
k 1
k 1
R 1
217.5   nm
k 1 = 0.142
R 2
25   mm
5.7   mm
R 2
n D = 1.490
800   nm
200   nm
0.5   nm
2.0   nm
360   nm
| R add R sub |
R add
R sub
k 1
R 2
n 1
δ 2
k 1
R 1
δ 2
k 1
n 1
δ 2
R 1
R mid
x = 0
n 1
k 1
δ 1
δ 2
n 1
k 1
δ 2
R add
R sub
δ 2
R 1
δ 2
R 2

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