Abstract

The optical properties of diamonds are modeled over a wide spectral range with the modified Adachi’s model. Model parameters were estimated by use of the acceptance-probability-controlled simulated annealing algorithm. The employed model is quite flexible, as it uses an adjustable broadening function at each critical point. The broadening function can vary over a range of functions with similar kernels but different wings, so that extended absorption tails inherent to the conventional Lorentzian broadening can be eliminated. Good agreement with the experimental data is obtained in the entire investigated range. The obtained relative rms error for the real part of the index of refraction equals 4.7%, whereas for the imaginary part of the index-of-refraction relative rms error is 3.6%.

© 1998 Optical Society of America

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References

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  1. D. F. Edwards, H. R. Philipp, “Cubic carbon (diamond),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 665–673.
  2. H. R. Philipp, E. A. Taft, “Kramers–Kronig analysis of reflectance data for diamond,” Phys. Rev. 136, A1445–A1448 (1964).
    [CrossRef]
  3. H. R. Philipp, E. A. Taft, “Optical properties of diamond in the vacuum ultraviolet,” Phys. Rev. 127, 159–161 (1962).
    [CrossRef]
  4. W. C. Walker, J. Osantowski, “Ultraviolet optical properties of diamond,” Phys. Rev. 134, A153–A157 (1964).
    [CrossRef]
  5. D. F. Edwards, E. Ochoa, “Infrared refractive index of diamond,” J. Opt. Soc. Am. 71, 607–608 (1981).
    [CrossRef]
  6. O. Stenzel, R. Petrich, “Flexible construction of error functions and their minimization: application to calculation of optical constants of absorbing and scattering thin-film materials from spectrophotometric data,” J. Phys. D 28, 978–989 (1995).
    [CrossRef]
  7. L. H. Robins, E. N. Farabaugh, A. Feldman, “Determination of optical constants of diamond thin films by transmittance and reflectance spectroscopy: experiment and model calculations,” Diamond Films Technol. 5, 199–224 (1995).
  8. A. D. Papadopoulos, E. Anastassakis, “Optical properties of diamond,” Phys. Rev. B 43, 5090–5097 (1991).
    [CrossRef]
  9. M. Cardona, Modulation Spectroscopy (Academic, New York, 1969), pp. 15–23.
  10. S. Ozaki, S. Adachi, “Spectroscopic ellipsometry and thermoreflectance of GaAs,” J. Appl. Phys. 78, 3380–3386 (1995).
    [CrossRef]
  11. A. D. Rakić, M. L. Majewski, “Modeling the optical dielectric function of GaAs and AlAs: extension of Adachi’s model,” J. Appl. Phys. 80, 5909–5915 (1996).
    [CrossRef]
  12. C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11749–11767 (1992).
    [CrossRef]
  13. O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
    [CrossRef]
  14. A. Franke, A. Stendal, O. Stenzel, C. von Borczyskowski, “Gaussian quadrature approach to the calculation of the optical constants in the vicinity of inhomogenously broadened absorption lines,” Pure Appl. Opt. 5, 845–853 (1996).
    [CrossRef]
  15. C. W. Higginbotham, M. Cardona, F. H. Pollak, “Intrinsic piezoreffingence of Ge, Si and GaAs,” Phys. Rev. 184, 821–829 (1969).
    [CrossRef]
  16. A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
    [CrossRef]

1997

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

1996

A. Franke, A. Stendal, O. Stenzel, C. von Borczyskowski, “Gaussian quadrature approach to the calculation of the optical constants in the vicinity of inhomogenously broadened absorption lines,” Pure Appl. Opt. 5, 845–853 (1996).
[CrossRef]

A. D. Rakić, M. L. Majewski, “Modeling the optical dielectric function of GaAs and AlAs: extension of Adachi’s model,” J. Appl. Phys. 80, 5909–5915 (1996).
[CrossRef]

1995

O. Stenzel, R. Petrich, “Flexible construction of error functions and their minimization: application to calculation of optical constants of absorbing and scattering thin-film materials from spectrophotometric data,” J. Phys. D 28, 978–989 (1995).
[CrossRef]

L. H. Robins, E. N. Farabaugh, A. Feldman, “Determination of optical constants of diamond thin films by transmittance and reflectance spectroscopy: experiment and model calculations,” Diamond Films Technol. 5, 199–224 (1995).

S. Ozaki, S. Adachi, “Spectroscopic ellipsometry and thermoreflectance of GaAs,” J. Appl. Phys. 78, 3380–3386 (1995).
[CrossRef]

1992

C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11749–11767 (1992).
[CrossRef]

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

1991

A. D. Papadopoulos, E. Anastassakis, “Optical properties of diamond,” Phys. Rev. B 43, 5090–5097 (1991).
[CrossRef]

1981

1969

C. W. Higginbotham, M. Cardona, F. H. Pollak, “Intrinsic piezoreffingence of Ge, Si and GaAs,” Phys. Rev. 184, 821–829 (1969).
[CrossRef]

1964

W. C. Walker, J. Osantowski, “Ultraviolet optical properties of diamond,” Phys. Rev. 134, A153–A157 (1964).
[CrossRef]

H. R. Philipp, E. A. Taft, “Kramers–Kronig analysis of reflectance data for diamond,” Phys. Rev. 136, A1445–A1448 (1964).
[CrossRef]

1962

H. R. Philipp, E. A. Taft, “Optical properties of diamond in the vacuum ultraviolet,” Phys. Rev. 127, 159–161 (1962).
[CrossRef]

Abad, H.

C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11749–11767 (1992).
[CrossRef]

Adachi, S.

S. Ozaki, S. Adachi, “Spectroscopic ellipsometry and thermoreflectance of GaAs,” J. Appl. Phys. 78, 3380–3386 (1995).
[CrossRef]

Anastassakis, E.

A. D. Papadopoulos, E. Anastassakis, “Optical properties of diamond,” Phys. Rev. B 43, 5090–5097 (1991).
[CrossRef]

Cardona, M.

C. W. Higginbotham, M. Cardona, F. H. Pollak, “Intrinsic piezoreffingence of Ge, Si and GaAs,” Phys. Rev. 184, 821–829 (1969).
[CrossRef]

M. Cardona, Modulation Spectroscopy (Academic, New York, 1969), pp. 15–23.

Djurišic, A. B.

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

Edwards, D. F.

D. F. Edwards, E. Ochoa, “Infrared refractive index of diamond,” J. Opt. Soc. Am. 71, 607–608 (1981).
[CrossRef]

D. F. Edwards, H. R. Philipp, “Cubic carbon (diamond),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 665–673.

Elazar, J. M.

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

Farabaugh, E. N.

L. H. Robins, E. N. Farabaugh, A. Feldman, “Determination of optical constants of diamond thin films by transmittance and reflectance spectroscopy: experiment and model calculations,” Diamond Films Technol. 5, 199–224 (1995).

Feldman, A.

L. H. Robins, E. N. Farabaugh, A. Feldman, “Determination of optical constants of diamond thin films by transmittance and reflectance spectroscopy: experiment and model calculations,” Diamond Films Technol. 5, 199–224 (1995).

Franke, A.

A. Franke, A. Stendal, O. Stenzel, C. von Borczyskowski, “Gaussian quadrature approach to the calculation of the optical constants in the vicinity of inhomogenously broadened absorption lines,” Pure Appl. Opt. 5, 845–853 (1996).
[CrossRef]

Garland, J. W.

C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11749–11767 (1992).
[CrossRef]

Higginbotham, C. W.

C. W. Higginbotham, M. Cardona, F. H. Pollak, “Intrinsic piezoreffingence of Ge, Si and GaAs,” Phys. Rev. 184, 821–829 (1969).
[CrossRef]

Hopfe, V.

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Kim, C. C.

C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11749–11767 (1992).
[CrossRef]

Majewski, M. L.

A. D. Rakić, M. L. Majewski, “Modeling the optical dielectric function of GaAs and AlAs: extension of Adachi’s model,” J. Appl. Phys. 80, 5909–5915 (1996).
[CrossRef]

Ochoa, E.

Osantowski, J.

W. C. Walker, J. Osantowski, “Ultraviolet optical properties of diamond,” Phys. Rev. 134, A153–A157 (1964).
[CrossRef]

Ozaki, S.

S. Ozaki, S. Adachi, “Spectroscopic ellipsometry and thermoreflectance of GaAs,” J. Appl. Phys. 78, 3380–3386 (1995).
[CrossRef]

Papadopoulos, A. D.

A. D. Papadopoulos, E. Anastassakis, “Optical properties of diamond,” Phys. Rev. B 43, 5090–5097 (1991).
[CrossRef]

Petrich, R.

O. Stenzel, R. Petrich, “Flexible construction of error functions and their minimization: application to calculation of optical constants of absorbing and scattering thin-film materials from spectrophotometric data,” J. Phys. D 28, 978–989 (1995).
[CrossRef]

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Philipp, H. R.

H. R. Philipp, E. A. Taft, “Kramers–Kronig analysis of reflectance data for diamond,” Phys. Rev. 136, A1445–A1448 (1964).
[CrossRef]

H. R. Philipp, E. A. Taft, “Optical properties of diamond in the vacuum ultraviolet,” Phys. Rev. 127, 159–161 (1962).
[CrossRef]

D. F. Edwards, H. R. Philipp, “Cubic carbon (diamond),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 665–673.

Pollak, F. H.

C. W. Higginbotham, M. Cardona, F. H. Pollak, “Intrinsic piezoreffingence of Ge, Si and GaAs,” Phys. Rev. 184, 821–829 (1969).
[CrossRef]

Raccah, P. M.

C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11749–11767 (1992).
[CrossRef]

Rakic, A. D.

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

A. D. Rakić, M. L. Majewski, “Modeling the optical dielectric function of GaAs and AlAs: extension of Adachi’s model,” J. Appl. Phys. 80, 5909–5915 (1996).
[CrossRef]

Robins, L. H.

L. H. Robins, E. N. Farabaugh, A. Feldman, “Determination of optical constants of diamond thin films by transmittance and reflectance spectroscopy: experiment and model calculations,” Diamond Films Technol. 5, 199–224 (1995).

Scharff, W.

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Stendal, A.

A. Franke, A. Stendal, O. Stenzel, C. von Borczyskowski, “Gaussian quadrature approach to the calculation of the optical constants in the vicinity of inhomogenously broadened absorption lines,” Pure Appl. Opt. 5, 845–853 (1996).
[CrossRef]

Stenzel, O.

A. Franke, A. Stendal, O. Stenzel, C. von Borczyskowski, “Gaussian quadrature approach to the calculation of the optical constants in the vicinity of inhomogenously broadened absorption lines,” Pure Appl. Opt. 5, 845–853 (1996).
[CrossRef]

O. Stenzel, R. Petrich, “Flexible construction of error functions and their minimization: application to calculation of optical constants of absorbing and scattering thin-film materials from spectrophotometric data,” J. Phys. D 28, 978–989 (1995).
[CrossRef]

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Taft, E. A.

H. R. Philipp, E. A. Taft, “Kramers–Kronig analysis of reflectance data for diamond,” Phys. Rev. 136, A1445–A1448 (1964).
[CrossRef]

H. R. Philipp, E. A. Taft, “Optical properties of diamond in the vacuum ultraviolet,” Phys. Rev. 127, 159–161 (1962).
[CrossRef]

Tikhonravov, A.

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

von Borczyskowski, C.

A. Franke, A. Stendal, O. Stenzel, C. von Borczyskowski, “Gaussian quadrature approach to the calculation of the optical constants in the vicinity of inhomogenously broadened absorption lines,” Pure Appl. Opt. 5, 845–853 (1996).
[CrossRef]

Walker, W. C.

W. C. Walker, J. Osantowski, “Ultraviolet optical properties of diamond,” Phys. Rev. 134, A153–A157 (1964).
[CrossRef]

Diamond Films Technol.

L. H. Robins, E. N. Farabaugh, A. Feldman, “Determination of optical constants of diamond thin films by transmittance and reflectance spectroscopy: experiment and model calculations,” Diamond Films Technol. 5, 199–224 (1995).

J. Appl. Phys.

S. Ozaki, S. Adachi, “Spectroscopic ellipsometry and thermoreflectance of GaAs,” J. Appl. Phys. 78, 3380–3386 (1995).
[CrossRef]

A. D. Rakić, M. L. Majewski, “Modeling the optical dielectric function of GaAs and AlAs: extension of Adachi’s model,” J. Appl. Phys. 80, 5909–5915 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. D

O. Stenzel, R. Petrich, “Flexible construction of error functions and their minimization: application to calculation of optical constants of absorbing and scattering thin-film materials from spectrophotometric data,” J. Phys. D 28, 978–989 (1995).
[CrossRef]

Phys. Rev.

H. R. Philipp, E. A. Taft, “Kramers–Kronig analysis of reflectance data for diamond,” Phys. Rev. 136, A1445–A1448 (1964).
[CrossRef]

H. R. Philipp, E. A. Taft, “Optical properties of diamond in the vacuum ultraviolet,” Phys. Rev. 127, 159–161 (1962).
[CrossRef]

W. C. Walker, J. Osantowski, “Ultraviolet optical properties of diamond,” Phys. Rev. 134, A153–A157 (1964).
[CrossRef]

C. W. Higginbotham, M. Cardona, F. H. Pollak, “Intrinsic piezoreffingence of Ge, Si and GaAs,” Phys. Rev. 184, 821–829 (1969).
[CrossRef]

Phys. Rev. B

A. D. Papadopoulos, E. Anastassakis, “Optical properties of diamond,” Phys. Rev. B 43, 5090–5097 (1991).
[CrossRef]

C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11749–11767 (1992).
[CrossRef]

Phys. Rev. E

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

Pure Appl. Opt.

A. Franke, A. Stendal, O. Stenzel, C. von Borczyskowski, “Gaussian quadrature approach to the calculation of the optical constants in the vicinity of inhomogenously broadened absorption lines,” Pure Appl. Opt. 5, 845–853 (1996).
[CrossRef]

Thin Solid Films

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Other

D. F. Edwards, H. R. Philipp, “Cubic carbon (diamond),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 665–673.

M. Cardona, Modulation Spectroscopy (Academic, New York, 1969), pp. 15–23.

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Figures (2)

Fig. 1
Fig. 1

Real and imaginary parts of the dielectric function of diamonds versus energy. Triangles, experimental data; solid curve, model.

Fig. 2
Fig. 2

Real and imaginary parts of the index of refraction of diamonds versus energy. Triangles, experimental data; solid curve, model.

Tables (1)

Tables Icon

Table 1 Values of the Model Parameters

Equations (11)

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

ω = + I ω + II ω + III ω ,
I ω = AE 0 - 3 / 2 f χ 0 + 1 2 E 0 E 0 + Δ 0 3 / 2 f χ 0 s ,
f y = y - 2 [ 2 - 1 + y 1 / 2 - 1 - y - 1 / 2 ,
χ 0 = ω + i Γ 0 E 0 ,
χ 0 s = ω + i Γ 0 E 0 + Δ 0 ,
II ω = - B 1 χ 1 - 2   ln 1 - χ 1 2 - B 1 s χ 1 s - 2   ln 1 - χ 1 s 2 ,
χ 1 = ω + i Γ 1 E 1 ,
χ 1 s = ω + i Γ 1 E 1 + Δ 1 ,
III ω = j = 2 4 f j E j 2 - ω 2 - i ω Γ j .
Γ i ω = Γ i   exp - α i ω - E i Γ i 2 .
F = i = 1 N n ω i n expt ω i - 1 + k ω i k expt ω i - 1 2 ,

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