Abstract

The reverse engineering problem addressed in the present research consists of estimating the thicknesses and the optical constants of two thin films deposited on a transparent substrate using only transmittance data through the whole stack. No functional dispersion relation assumptions are made on the complex refractive index. Instead, minimal physical constraints are employed, as in previous works of some of the authors where only one film was considered in the retrieval algorithm. To our knowledge this is the first report on the retrieval of the optical constants and the thickness of multiple film structures using only transmittance data that does not make use of dispersion relations. The same methodology may be used if the available data correspond to normal reflectance. The software used in this work is freely available through the PUMA Project web page (http://www.ime.usp.br/~egbirgin/puma/).

© 2008 Optical Society of America

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References

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  1. I. Chambouleyron and J. M. Martínez, “Optical properties of dielectric and semiconductor thin films,” Handbook of Thin Films Materials, H. S. Nalwa ed., (Academic, 2001), Vol. 3, Chap. 12, pp. 593-622
  2. D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” J. Phys. D 36, 1850-1857(2003).
    [CrossRef]
  3. E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862-880 (1999).
    [CrossRef]
  4. M. Mulato, I. Chambouleyron, E. G. Birgin, and J. M. Martínez, “Determination of thickness and optical constants of amorphous silicon films from transmittance data,” Appl. Phys. Lett. 77, 2133-2135 (2000).
    [CrossRef]
  5. I. Chambouleyron, S. D. Ventura, E. G. Birgin, and J. M. Martínez, “Optimal constants and thickness determination of very thin amorphous semiconductor films,” J. Appl. Phys. 92, 3093-3102 (2002).
    [CrossRef]
  6. S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, “Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data,” J. Appl. Phys. 97, 043512 (2005).
    [CrossRef]
  7. TFCalc, Software Spectra Inc., Portland, Oreg., USA, www.sspectra.com, 2008.
  8. Filmwizard, Scientific Computing Int., Encinitas, Calif., USA, www.sci-soft.com, 2008.
  9. F. A. Jenkins and H. E. White, Fundamentals of Optics, (McGraw-Hill, 1981).
  10. E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Optimization problems in the estimation or parameters of thin films and the elimination of the influence of the substrate,” J. Comput. Appl. Math. 152, 35-50 (2003).
    [CrossRef]
  11. M. Born and E. Wolf, Principles of Optics, 1st ed. (Pergamon, 1959).
  12. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, 1950).
  13. R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214-1222 (1983).
    [CrossRef]
  14. T. C. Paulick, “Inversion of normal-incidence (R,T) measurements to obtain n + ik for thin films,” Appl. Opt. 25, 562-564 (1986).
    [CrossRef] [PubMed]
  15. H. M. Liddell, Computer-Aided Techniques for the Design of Multilayer Filters (Adam Hilger, 1980).
  16. S. D. Ventura, “Optimization techniques for film parameter estimation,” Ph.D. thesis (Department of Applied Mathematics, University of Campinas, 2004).
  17. M. Raydan, “The Barzilai and Borwein gradient method for the large unconstrained minimization problem,” SIAM J. Optim. 7, 26-33 (1997).
    [CrossRef]
  18. E. G. Birgin and J. M. Martínez, “A spectral conjugate gradient method for unconstrained optimization,” Appl. Math. Optim. 43, 117-128 (2001).
    [CrossRef]
  19. B. Akaoglu, I. Atilgan, and B. Katircioglu, “Thickness and optical constant distributions of PECVD a-SiCx : H thin films along electrode radial direction,” Thin Solid Films 437, 257-265 (2003).
    [CrossRef]
  20. E. G. Birgin, I. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
    [CrossRef]
  21. F. Curiel, W. E. Vargas, and R. G. Barrera, “Visible spectral dependence of the scattering and absorption coefficients of pigmented coatings from inversion of diffuse reflectance spectra,” Appl. Opt. 41, 5969-5978 (2002).
    [CrossRef] [PubMed]
  22. A. Ramirez-Porras and W. E. Vargas-Castro, “Transmission of visible light through oxidized copper films: feasibility of using a spectral projected gradient method,” Appl. Opt. 43, 1508-1514 (2004).
    [CrossRef] [PubMed]
  23. N. F. Mott and E. A. Davis, Electronic Properties in Non Crystalline Materials (Clarendon, 1979).
  24. G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, I. Chambouleyron, and C. F. O. Graeff, “Local order in hydrogenated amorphous germanium thin films studied by EXAFS,” J. Phys. Condens. Matter 9, 5875-5888 (1997).
    [CrossRef]
  25. A. R. Zanatta, M. Mulato, and I. Chambouleyron, “Exponential absorption edge and disorder in Column IV amorphous semiconductors,” J. Appl. Phys. 84, 5184-5190 (1998)
    [CrossRef]
  26. M. H. Brodsky, D. M. Kaplan, and J. F. Ziegler, Proceedings of the 11th International Conference on the Physics of Semiconductors (PWN-Polish Scientific, 1972), p. 529.
  27. W. Paul, G. A. N. Connell, and R. J. Temkin, “Amorphous germanium I. A model for the structural and optical properties,” Adv. Phys. 22, 531-580 (1973).
    [CrossRef]
  28. T. M. Donovan and K. Heinemann, “High-resolution electron microscope observations of voids in amorphous Ge,” Phys. Rev. Lett. 27, 1794-1796 (1971).
    [CrossRef]

2005

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, “Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data,” J. Appl. Phys. 97, 043512 (2005).
[CrossRef]

2004

2003

B. Akaoglu, I. Atilgan, and B. Katircioglu, “Thickness and optical constant distributions of PECVD a-SiCx : H thin films along electrode radial direction,” Thin Solid Films 437, 257-265 (2003).
[CrossRef]

E. G. Birgin, I. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Optimization problems in the estimation or parameters of thin films and the elimination of the influence of the substrate,” J. Comput. Appl. Math. 152, 35-50 (2003).
[CrossRef]

D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” J. Phys. D 36, 1850-1857(2003).
[CrossRef]

2002

I. Chambouleyron, S. D. Ventura, E. G. Birgin, and J. M. Martínez, “Optimal constants and thickness determination of very thin amorphous semiconductor films,” J. Appl. Phys. 92, 3093-3102 (2002).
[CrossRef]

F. Curiel, W. E. Vargas, and R. G. Barrera, “Visible spectral dependence of the scattering and absorption coefficients of pigmented coatings from inversion of diffuse reflectance spectra,” Appl. Opt. 41, 5969-5978 (2002).
[CrossRef] [PubMed]

2001

E. G. Birgin and J. M. Martínez, “A spectral conjugate gradient method for unconstrained optimization,” Appl. Math. Optim. 43, 117-128 (2001).
[CrossRef]

2000

M. Mulato, I. Chambouleyron, E. G. Birgin, and J. M. Martínez, “Determination of thickness and optical constants of amorphous silicon films from transmittance data,” Appl. Phys. Lett. 77, 2133-2135 (2000).
[CrossRef]

1999

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862-880 (1999).
[CrossRef]

1998

A. R. Zanatta, M. Mulato, and I. Chambouleyron, “Exponential absorption edge and disorder in Column IV amorphous semiconductors,” J. Appl. Phys. 84, 5184-5190 (1998)
[CrossRef]

1997

G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, I. Chambouleyron, and C. F. O. Graeff, “Local order in hydrogenated amorphous germanium thin films studied by EXAFS,” J. Phys. Condens. Matter 9, 5875-5888 (1997).
[CrossRef]

M. Raydan, “The Barzilai and Borwein gradient method for the large unconstrained minimization problem,” SIAM J. Optim. 7, 26-33 (1997).
[CrossRef]

1986

1983

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214-1222 (1983).
[CrossRef]

1973

W. Paul, G. A. N. Connell, and R. J. Temkin, “Amorphous germanium I. A model for the structural and optical properties,” Adv. Phys. 22, 531-580 (1973).
[CrossRef]

1971

T. M. Donovan and K. Heinemann, “High-resolution electron microscope observations of voids in amorphous Ge,” Phys. Rev. Lett. 27, 1794-1796 (1971).
[CrossRef]

Akaoglu, B.

B. Akaoglu, I. Atilgan, and B. Katircioglu, “Thickness and optical constant distributions of PECVD a-SiCx : H thin films along electrode radial direction,” Thin Solid Films 437, 257-265 (2003).
[CrossRef]

Atilgan, I.

B. Akaoglu, I. Atilgan, and B. Katircioglu, “Thickness and optical constant distributions of PECVD a-SiCx : H thin films along electrode radial direction,” Thin Solid Films 437, 257-265 (2003).
[CrossRef]

Barrera, R. G.

Birgin, E. G.

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, “Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data,” J. Appl. Phys. 97, 043512 (2005).
[CrossRef]

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Optimization problems in the estimation or parameters of thin films and the elimination of the influence of the substrate,” J. Comput. Appl. Math. 152, 35-50 (2003).
[CrossRef]

E. G. Birgin, I. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

I. Chambouleyron, S. D. Ventura, E. G. Birgin, and J. M. Martínez, “Optimal constants and thickness determination of very thin amorphous semiconductor films,” J. Appl. Phys. 92, 3093-3102 (2002).
[CrossRef]

E. G. Birgin and J. M. Martínez, “A spectral conjugate gradient method for unconstrained optimization,” Appl. Math. Optim. 43, 117-128 (2001).
[CrossRef]

M. Mulato, I. Chambouleyron, E. G. Birgin, and J. M. Martínez, “Determination of thickness and optical constants of amorphous silicon films from transmittance data,” Appl. Phys. Lett. 77, 2133-2135 (2000).
[CrossRef]

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862-880 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 1st ed. (Pergamon, 1959).

Brodsky, M. H.

M. H. Brodsky, D. M. Kaplan, and J. F. Ziegler, Proceedings of the 11th International Conference on the Physics of Semiconductors (PWN-Polish Scientific, 1972), p. 529.

Chambouleyron, I.

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, “Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data,” J. Appl. Phys. 97, 043512 (2005).
[CrossRef]

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Optimization problems in the estimation or parameters of thin films and the elimination of the influence of the substrate,” J. Comput. Appl. Math. 152, 35-50 (2003).
[CrossRef]

E. G. Birgin, I. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

I. Chambouleyron, S. D. Ventura, E. G. Birgin, and J. M. Martínez, “Optimal constants and thickness determination of very thin amorphous semiconductor films,” J. Appl. Phys. 92, 3093-3102 (2002).
[CrossRef]

M. Mulato, I. Chambouleyron, E. G. Birgin, and J. M. Martínez, “Determination of thickness and optical constants of amorphous silicon films from transmittance data,” Appl. Phys. Lett. 77, 2133-2135 (2000).
[CrossRef]

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862-880 (1999).
[CrossRef]

A. R. Zanatta, M. Mulato, and I. Chambouleyron, “Exponential absorption edge and disorder in Column IV amorphous semiconductors,” J. Appl. Phys. 84, 5184-5190 (1998)
[CrossRef]

G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, I. Chambouleyron, and C. F. O. Graeff, “Local order in hydrogenated amorphous germanium thin films studied by EXAFS,” J. Phys. Condens. Matter 9, 5875-5888 (1997).
[CrossRef]

I. Chambouleyron and J. M. Martínez, “Optical properties of dielectric and semiconductor thin films,” Handbook of Thin Films Materials, H. S. Nalwa ed., (Academic, 2001), Vol. 3, Chap. 12, pp. 593-622

Connell, G. A. N.

W. Paul, G. A. N. Connell, and R. J. Temkin, “Amorphous germanium I. A model for the structural and optical properties,” Adv. Phys. 22, 531-580 (1973).
[CrossRef]

Curiel, F.

Dalba, G.

G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, I. Chambouleyron, and C. F. O. Graeff, “Local order in hydrogenated amorphous germanium thin films studied by EXAFS,” J. Phys. Condens. Matter 9, 5875-5888 (1997).
[CrossRef]

Davis, E. A.

N. F. Mott and E. A. Davis, Electronic Properties in Non Crystalline Materials (Clarendon, 1979).

Donovan, T. M.

T. M. Donovan and K. Heinemann, “High-resolution electron microscope observations of voids in amorphous Ge,” Phys. Rev. Lett. 27, 1794-1796 (1971).
[CrossRef]

Fornasini, P.

G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, I. Chambouleyron, and C. F. O. Graeff, “Local order in hydrogenated amorphous germanium thin films studied by EXAFS,” J. Phys. Condens. Matter 9, 5875-5888 (1997).
[CrossRef]

Graeff, C. F. O.

G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, I. Chambouleyron, and C. F. O. Graeff, “Local order in hydrogenated amorphous germanium thin films studied by EXAFS,” J. Phys. Condens. Matter 9, 5875-5888 (1997).
[CrossRef]

Grisenti, R.

G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, I. Chambouleyron, and C. F. O. Graeff, “Local order in hydrogenated amorphous germanium thin films studied by EXAFS,” J. Phys. Condens. Matter 9, 5875-5888 (1997).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, 1950).

Heinemann, K.

T. M. Donovan and K. Heinemann, “High-resolution electron microscope observations of voids in amorphous Ge,” Phys. Rev. Lett. 27, 1794-1796 (1971).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, (McGraw-Hill, 1981).

Kaplan, D. M.

M. H. Brodsky, D. M. Kaplan, and J. F. Ziegler, Proceedings of the 11th International Conference on the Physics of Semiconductors (PWN-Polish Scientific, 1972), p. 529.

Katircioglu, B.

B. Akaoglu, I. Atilgan, and B. Katircioglu, “Thickness and optical constant distributions of PECVD a-SiCx : H thin films along electrode radial direction,” Thin Solid Films 437, 257-265 (2003).
[CrossRef]

Liddell, H. M.

H. M. Liddell, Computer-Aided Techniques for the Design of Multilayer Filters (Adam Hilger, 1980).

Martínez, J. M.

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, “Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data,” J. Appl. Phys. 97, 043512 (2005).
[CrossRef]

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Optimization problems in the estimation or parameters of thin films and the elimination of the influence of the substrate,” J. Comput. Appl. Math. 152, 35-50 (2003).
[CrossRef]

E. G. Birgin, I. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

I. Chambouleyron, S. D. Ventura, E. G. Birgin, and J. M. Martínez, “Optimal constants and thickness determination of very thin amorphous semiconductor films,” J. Appl. Phys. 92, 3093-3102 (2002).
[CrossRef]

E. G. Birgin and J. M. Martínez, “A spectral conjugate gradient method for unconstrained optimization,” Appl. Math. Optim. 43, 117-128 (2001).
[CrossRef]

M. Mulato, I. Chambouleyron, E. G. Birgin, and J. M. Martínez, “Determination of thickness and optical constants of amorphous silicon films from transmittance data,” Appl. Phys. Lett. 77, 2133-2135 (2000).
[CrossRef]

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862-880 (1999).
[CrossRef]

I. Chambouleyron and J. M. Martínez, “Optical properties of dielectric and semiconductor thin films,” Handbook of Thin Films Materials, H. S. Nalwa ed., (Academic, 2001), Vol. 3, Chap. 12, pp. 593-622

Mott, N. F.

N. F. Mott and E. A. Davis, Electronic Properties in Non Crystalline Materials (Clarendon, 1979).

Mulato, M.

M. Mulato, I. Chambouleyron, E. G. Birgin, and J. M. Martínez, “Determination of thickness and optical constants of amorphous silicon films from transmittance data,” Appl. Phys. Lett. 77, 2133-2135 (2000).
[CrossRef]

A. R. Zanatta, M. Mulato, and I. Chambouleyron, “Exponential absorption edge and disorder in Column IV amorphous semiconductors,” J. Appl. Phys. 84, 5184-5190 (1998)
[CrossRef]

Paul, W.

W. Paul, G. A. N. Connell, and R. J. Temkin, “Amorphous germanium I. A model for the structural and optical properties,” Adv. Phys. 22, 531-580 (1973).
[CrossRef]

Paulick, T. C.

Poelman, D.

D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” J. Phys. D 36, 1850-1857(2003).
[CrossRef]

Ramirez-Porras, A.

Raydan, M.

M. Raydan, “The Barzilai and Borwein gradient method for the large unconstrained minimization problem,” SIAM J. Optim. 7, 26-33 (1997).
[CrossRef]

Rocca, F.

G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, I. Chambouleyron, and C. F. O. Graeff, “Local order in hydrogenated amorphous germanium thin films studied by EXAFS,” J. Phys. Condens. Matter 9, 5875-5888 (1997).
[CrossRef]

Smet, P. F.

D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” J. Phys. D 36, 1850-1857(2003).
[CrossRef]

Swanepoel, R.

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214-1222 (1983).
[CrossRef]

Temkin, R. J.

W. Paul, G. A. N. Connell, and R. J. Temkin, “Amorphous germanium I. A model for the structural and optical properties,” Adv. Phys. 22, 531-580 (1973).
[CrossRef]

Vargas, W. E.

Vargas-Castro, W. E.

Ventura, S. D.

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, “Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data,” J. Appl. Phys. 97, 043512 (2005).
[CrossRef]

E. G. Birgin, I. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

I. Chambouleyron, S. D. Ventura, E. G. Birgin, and J. M. Martínez, “Optimal constants and thickness determination of very thin amorphous semiconductor films,” J. Appl. Phys. 92, 3093-3102 (2002).
[CrossRef]

S. D. Ventura, “Optimization techniques for film parameter estimation,” Ph.D. thesis (Department of Applied Mathematics, University of Campinas, 2004).

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, (McGraw-Hill, 1981).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 1st ed. (Pergamon, 1959).

Zanatta, A. R.

A. R. Zanatta, M. Mulato, and I. Chambouleyron, “Exponential absorption edge and disorder in Column IV amorphous semiconductors,” J. Appl. Phys. 84, 5184-5190 (1998)
[CrossRef]

Ziegler, J. F.

M. H. Brodsky, D. M. Kaplan, and J. F. Ziegler, Proceedings of the 11th International Conference on the Physics of Semiconductors (PWN-Polish Scientific, 1972), p. 529.

Adv. Phys.

W. Paul, G. A. N. Connell, and R. J. Temkin, “Amorphous germanium I. A model for the structural and optical properties,” Adv. Phys. 22, 531-580 (1973).
[CrossRef]

Appl. Math. Optim.

E. G. Birgin and J. M. Martínez, “A spectral conjugate gradient method for unconstrained optimization,” Appl. Math. Optim. 43, 117-128 (2001).
[CrossRef]

Appl. Numer. Math.

E. G. Birgin, I. Chambouleyron, J. M. Martínez, and S. D. Ventura, “Estimation of optical parameters of very thin films,” Appl. Numer. Math. 47, 109-119 (2003).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

M. Mulato, I. Chambouleyron, E. G. Birgin, and J. M. Martínez, “Determination of thickness and optical constants of amorphous silicon films from transmittance data,” Appl. Phys. Lett. 77, 2133-2135 (2000).
[CrossRef]

J. Appl. Phys.

I. Chambouleyron, S. D. Ventura, E. G. Birgin, and J. M. Martínez, “Optimal constants and thickness determination of very thin amorphous semiconductor films,” J. Appl. Phys. 92, 3093-3102 (2002).
[CrossRef]

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, “Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data,” J. Appl. Phys. 97, 043512 (2005).
[CrossRef]

A. R. Zanatta, M. Mulato, and I. Chambouleyron, “Exponential absorption edge and disorder in Column IV amorphous semiconductors,” J. Appl. Phys. 84, 5184-5190 (1998)
[CrossRef]

J. Comput. Appl. Math.

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Optimization problems in the estimation or parameters of thin films and the elimination of the influence of the substrate,” J. Comput. Appl. Math. 152, 35-50 (2003).
[CrossRef]

J. Comput. Phys.

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862-880 (1999).
[CrossRef]

J. Phys. Condens. Matter

G. Dalba, P. Fornasini, R. Grisenti, F. Rocca, I. Chambouleyron, and C. F. O. Graeff, “Local order in hydrogenated amorphous germanium thin films studied by EXAFS,” J. Phys. Condens. Matter 9, 5875-5888 (1997).
[CrossRef]

J. Phys. D

D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” J. Phys. D 36, 1850-1857(2003).
[CrossRef]

J. Phys. E

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214-1222 (1983).
[CrossRef]

Phys. Rev. Lett.

T. M. Donovan and K. Heinemann, “High-resolution electron microscope observations of voids in amorphous Ge,” Phys. Rev. Lett. 27, 1794-1796 (1971).
[CrossRef]

SIAM J. Optim.

M. Raydan, “The Barzilai and Borwein gradient method for the large unconstrained minimization problem,” SIAM J. Optim. 7, 26-33 (1997).
[CrossRef]

Thin Solid Films

B. Akaoglu, I. Atilgan, and B. Katircioglu, “Thickness and optical constant distributions of PECVD a-SiCx : H thin films along electrode radial direction,” Thin Solid Films 437, 257-265 (2003).
[CrossRef]

Other

M. H. Brodsky, D. M. Kaplan, and J. F. Ziegler, Proceedings of the 11th International Conference on the Physics of Semiconductors (PWN-Polish Scientific, 1972), p. 529.

H. M. Liddell, Computer-Aided Techniques for the Design of Multilayer Filters (Adam Hilger, 1980).

S. D. Ventura, “Optimization techniques for film parameter estimation,” Ph.D. thesis (Department of Applied Mathematics, University of Campinas, 2004).

N. F. Mott and E. A. Davis, Electronic Properties in Non Crystalline Materials (Clarendon, 1979).

I. Chambouleyron and J. M. Martínez, “Optical properties of dielectric and semiconductor thin films,” Handbook of Thin Films Materials, H. S. Nalwa ed., (Academic, 2001), Vol. 3, Chap. 12, pp. 593-622

M. Born and E. Wolf, Principles of Optics, 1st ed. (Pergamon, 1959).

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, 1950).

TFCalc, Software Spectra Inc., Portland, Oreg., USA, www.sspectra.com, 2008.

Filmwizard, Scientific Computing Int., Encinitas, Calif., USA, www.sci-soft.com, 2008.

F. A. Jenkins and H. E. White, Fundamentals of Optics, (McGraw-Hill, 1981).

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Figures (7)

Fig. 1
Fig. 1

True and retrieved values of (a) the transmittance, (b) the refractive indices, and (c) the absorption coefficients of the numerically generated system AgB.

Fig. 2
Fig. 2

True and retrieved values of (a) the transmittance, (b) the refractive indices, and (c) the absorption coefficients of the numerically generated system BgD.

Fig. 3
Fig. 3

True and retrieved values of (a) the transmittance, (b) the refractive indices, and (c) the absorption coefficients of the numerically generated system CgB.

Fig. 4
Fig. 4

True and retrieved values of (a) the transmittance, (b) the refractive indices, and (c) the absorption coefficients of the numerically generated system CgD.

Fig. 5
Fig. 5

Transmittances of (a) the c-Si substrate and the double-film structure, and of (b) the single-film structures.

Fig. 6
Fig. 6

Retrieved optical constants of the a-Si and a-Ge semiconductor films using PUMA from the single-film and the double-film structure.

Fig. 7
Fig. 7

All figures correspond to systems with two films deposited onto different sides of a glass substrate and with d 1 + d 2 = 1000 nm . The first column [(a)–(c)] corresponds to films of the same material ( a - Si : H ) and the same thickness ( d ¯ 1 = d ¯ 2 = 500 nm ). The second column corresponds to films of the same material ( a - Si : H ) and different thicknesses ( d ¯ 1 = 200 nm and d ¯ 2 = 800 nm ). The third column corresponds to films of different materials ( a - Si : H and a - Ge : H ) and the same thickness ( d ¯ 1 = d ¯ 2 = 500 nm ). (a)–(c) correspond to transmittances generated varying d 1 { d ¯ 1 10 nm , d ¯ 1 , d ¯ 1 + 10 nm } (remember that d 1 + d 2 = 1000 nm ). The two following columns correspond to variations of d 1 within an increasing range. It can be observed that in the case of same-material films with similar thicknesses the variation in the transmittance is smaller than in the other cases. A possible explanation for this observation is the symmetry of the system that does not occur in the other situations.

Tables (2)

Tables Icon

Table 1 Retrievals for Systems AgB, BgD, CgB and CgD

Tables Icon

Table 2 Retrieved Thicknesses of the Single-Film a - Si / c - Si and a - Ge / c - Si and the Double-Film a - Si / a - Ge / c - Si Structures a

Equations (33)

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d i t = thickness   of   top   film   i , n i t ( λ ) = refractive   index   of   top   film   i , κ i t ( λ ) = extinction   coefficient   of   top   film   i .
d i b = thickness   of   bottom   film   i , n i b ( λ ) = refractive   index   of   bottom   film   i , κ i b ( λ ) = extinction   coefficient   of   bottom   film   i ,
T theoretical ( λ ) = T [ s ( λ ) , { d i t } i = 1 m t , { n i t ( λ ) } i = 1 m t , { κ i t ( λ ) } i = 1 m t , { d i b } i = 1 m b , { n i b ( λ ) } i = 1 m b , { κ i b ( λ ) } i = 1 m b ] .
d top = { d i t } i = 1 m t , n top ( λ ) = { n i t ( λ ) } i = 1 m t , κ top ( λ ) = { κ i t ( λ ) } i = 1 m t , d bottom = { d i b } i = 1 m b , n bottom ( λ ) = { n i b ( λ ) } i = 1 m b , κ bottom ( λ ) = { κ i b ( λ ) } i = 1 m b ,
d all = { d top , d bottom } , n all = { n top , n bottom } , κ all = { κ top , κ bottom } .
T theoretical ( λ ) = T [ s ( λ ) , d all , n all ( λ ) , κ all ( λ ) ] .
T FFS = 64 s ( n 1 2 + κ 1 2 ) ( n 2 2 + κ 2 2 ) x 1 x 2 x 1 x 2 ( A ^ x 1 x 2 + B ^ x 1 + C ^ x 2 + D ^ + E ^ ) + x 1 ( F ^ x 1 + G ^ ) + x 2 ( H ^ x 2 + I ^ ) + J ^ ,
A ^ = [ ( n 1 1 ) 2 + κ 1 2 ] [ ( n 2 + n 1 ) 2 + ( κ 2 + κ 1 ) 2 ] [ ( s 2 n 2 ) ( n 2 1 ) κ 2 2 ] , B ^ = 2 [ ( n 1 1 ) 2 + κ 1 2 ] [ B ^ s sin ( φ 2 ) B ^ c cos ( φ 2 ) ] , B ^ s = κ 2 ( s 2 + 1 ) ( n 2 2 n 1 2 + κ 2 2 κ 1 2 ) 2 ( n 2 2 + κ 2 2 s 2 ) ( κ 2 n 1 κ 1 n 2 ) , B ^ c = ( κ 2 2 + n 2 2 s 2 ) ( n 2 2 n 1 2 + κ 2 2 κ 1 2 ) + 2 κ 2 ( s 2 + 1 ) ( κ 2 n 1 κ 1 n 2 ) , C ^ = 2 [ ( s 2 n 2 ) ( n 2 1 ) κ 2 2 ] [ 2 C ^ s sin ( φ 1 ) C ^ c cos ( φ 1 ) ] , C ^ s = κ 1 [ ( n 1 2 + κ 1 2 + n 2 ) ( n 2 1 ) + κ 2 2 ] κ 2 n 1 ( n 1 2 + κ 1 2 1 ) , C ^ c = ( n 1 2 + κ 1 2 1 ) ( n 2 2 n 1 2 + κ 2 2 κ 1 2 ) + 4 κ 1 ( κ 2 n 1 κ 1 n 2 ) , D ^ = 2 [ ( n 2 + n 1 ) 2 + ( κ 2 + κ 1 ) 2 ] [ D ^ s sin ( φ 1 + φ 2 ) + D ^ c cos ( φ 1 + φ 2 ) ] , D ^ s = 2 κ 1 ( n 2 2 + κ 2 2 s 2 ) + κ 2 ( s 2 + 1 ) ( n 1 2 + κ 1 2 1 ) , D ^ c = 2 κ 1 κ 2 ( s 2 + 1 ) ( n 2 2 + κ 2 2 s 2 ) ( n 1 2 + κ 1 2 1 ) , E ^ = 2 [ ( n 2 n 1 ) 2 + ( κ 2 κ 1 ) 2 ] [ E ^ s sin ( φ 2 φ 1 ) E ^ c cos ( φ 2 φ 1 ) ] , E ^ s = κ 2 ( s 2 + 1 ) ( n 1 2 + κ 1 2 1 ) 2 κ 1 ( n 2 2 + κ 2 2 s 2 ) , E ^ c = 2 κ 1 κ 2 ( s 2 + 1 ) + ( n 2 2 + κ 2 2 s 2 ) ( n 1 2 + κ 1 2 1 ) , F ^ = [ ( n 1 1 ) 2 + κ 1 2 ] [ ( n 2 n 1 ) 2 + ( κ 2 κ 1 ) 2 ] [ ( s 2 + n 2 ) ( n 2 + 1 ) + κ 2 2 ] , G ^ = 2 [ ( s 2 + n 2 ) ( n 2 + 1 ) + κ 2 2 ] [ 2 G ^ s sin ( φ 1 ) G ^ c cos ( φ 1 ) ] , G ^ s = κ 1 ( n 2 2 n 1 2 + κ 2 2 κ 1 2 ) + ( κ 2 n 1 κ 1 n 2 ) ( n 1 2 + κ 1 2 1 ) , G ^ c = 4 κ 1 ( κ 2 n 1 κ 1 n 2 ) + ( n 2 2 n 1 2 + κ 2 2 κ 1 2 ) ( n 1 2 + κ 1 2 1 ) , H ^ = [ ( n 1 + 1 ) 2 + κ 1 2 ] [ ( n 2 n 1 ) 2 + ( κ 2 κ 1 ) 2 ] [ ( s 2 n 2 ) ( n 2 1 ) κ 2 2 ] , I ^ = 2 [ ( n 1 + 1 ) 2 + κ 1 2 ] [ I ^ s sin ( φ 2 ) + I ^ c cos ( φ 2 ) ] , I ^ s = 2 ( n 2 2 + κ 2 2 s 2 ) ( κ 2 n 1 κ 1 n 2 ) + ( s 2 + 1 ) κ 2 ( n 2 2 n 1 2 + κ 2 2 κ 1 2 ) , I ^ c = 2 κ 2 ( s 2 + 1 ) ( κ 2 n 1 κ 1 n 2 ) ( n 2 2 + κ 2 2 s 2 ) ( n 2 2 n 1 2 + κ 2 2 κ 1 2 ) , J ^ = [ ( n 1 + 1 ) 2 + κ 1 2 ] [ ( n 2 + n 1 ) 2 + ( κ 2 + κ 1 ) 2 ] [ ( s 2 + n 2 ) ( n 2 + 1 ) + κ 2 2 ] ,
β 1 = 4 π d 1 λ , β 2 = 4 π d 2 λ , φ 1 = β 1 n 1 , φ 2 = β 2 n 2 , x 1 = exp ( β 1 κ 1 ) , x 2 = exp ( β 2 κ 2 ) .
T FSF = 64 s ( n 1 2 + κ 1 2 ) ( n 2 2 + κ 2 2 ) x 1 x 2 x 1 x 2 ( A ˜ x 1 x 2 + B ˜ x 1 + C ˜ x 2 + D ˜ + E ˜ ) + x 1 ( F ˜ x 1 + G ˜ ) + x 2 ( H ˜ x 2 + I ˜ ) + J ˜ ,
A ^ = [ ( n 1 1 ) 2 + κ 1 2 ] [ ( n 2 1 ) 2 + κ 2 2 ] [ n 2 ( n 1 2 + κ 1 2 + s 2 ) + n 1 ( n 2 2 + κ 2 2 + s 2 ) ] , B ^ = 2 [ ( n 1 1 ) 2 + κ 1 2 ] [ B ˜ s sin ( φ 2 ) + B ˜ c cos ( φ 2 ) ] , B ^ s = κ 2 [ 2 n 1 ( s 2 n 2 2 κ 2 2 ) + ( n 1 2 + κ 1 2 + s 2 ) ( n 2 2 + κ 2 2 1 ) ] , B ^ c = n 1 ( n 2 2 + κ 2 2 1 ) ( n 2 2 + κ 2 2 s 2 ) + 2 κ 2 2 ( n 1 2 + κ 1 2 + s 2 ) , C ^ = 2 [ ( n 2 1 ) 2 + κ 2 2 ] [ C ˜ s sin ( φ 1 ) + C ˜ c cos ( φ 1 ) ] , C ^ s = κ 1 [ ( n 2 2 + κ 2 2 + s 2 ) ( n 1 2 + κ 1 2 1 ) 2 n 2 ( n 1 2 + κ 1 2 s 2 ) ] , C ^ c = 2 κ 1 2 ( n 2 2 + κ 2 2 + s 2 ) + n 2 ( n 1 2 + κ 1 2 s 2 ) ( n 1 2 + κ 1 2 1 ) , D ^ = 2 [ κ 2 ( n 1 2 + κ 1 2 s 2 ) + κ 1 ( n 2 2 + κ 2 2 s 2 ) ] [ D ˜ s sin ( φ 1 + φ 2 ) 2 D ˜ c cos ( φ 2 + φ 1 ) ] , D ^ s = 4 κ 1 κ 2 ( n 1 2 + κ 1 2 1 ) ( n 2 2 + κ 2 2 1 ) , D ^ c = κ 2 ( n 1 2 1 ) + κ 1 ( n 2 2 1 ) + κ 1 κ 2 ( κ 1 + κ 2 ) , E ^ = 2 [ κ 1 ( n 2 2 + κ 2 2 s 2 ) κ 2 ( n 1 2 + κ 1 2 s 2 ) ] [ E ˜ s sin ( φ 2 φ 1 ) 2 E ˜ c cos ( φ 2 φ 1 ) ] , E ^ s = 4 κ 1 κ 2 + ( n 2 2 + κ 2 2 1 ) ( n 1 2 + κ 1 2 1 ) , E ^ c = κ 1 ( n 2 2 1 ) κ 2 ( n 1 2 1 ) + κ 1 κ 2 ( κ 2 κ 1 ) , F ^ = [ ( n 1 1 ) 2 + κ 1 2 ] [ ( n 2 + 1 ) 2 + κ 2 2 ] [ n 2 ( n 1 2 + κ 1 2 + s 2 ) n 1 ( n 2 2 + κ 2 2 + s 2 ) ] , G ^ = 2 [ ( n 2 + 1 ) 2 + κ 2 2 ] [ G ˜ s sin ( φ 1 ) + G ˜ c cos ( φ 1 ) ] , G ^ s = κ 1 [ 2 n 2 ( n 1 2 + κ 1 2 s 2 ) + ( n 2 2 + κ 2 2 + s 2 ) ( n 1 2 + κ 1 2 1 ) ] , G ^ c = n 2 ( n 1 2 + κ 1 2 s 2 ) ( n 1 2 + κ 1 2 1 ) 2 κ 1 2 ( n 2 2 + κ 2 2 + s 2 ) , H ^ = [ ( n 1 + 1 ) 2 + κ 1 2 ] [ ( n 2 1 ) 2 + κ 2 2 ] [ n 2 ( n 1 2 + κ 1 2 + s 2 ) n 1 ( n 2 2 + κ 2 2 + s 2 ) ] , I ^ = 2 [ ( n 1 + 1 ) 2 + κ 1 2 ] [ I ˜ s sin ( φ 2 ) + I ˜ c cos ( φ 2 ) ] , I ^ s = κ 2 [ 2 n 1 ( n 2 2 + κ 2 2 s 2 ) + ( n 1 2 + κ 1 2 + s 2 ) ( n 2 2 + κ 2 2 1 ) ] , I ^ c = n 1 ( n 2 2 + κ 2 2 s 2 ) ( n 2 2 + κ 2 2 1 ) 2 κ 2 2 ( n 1 2 + κ 1 2 + s 2 ) , J ^ = [ ( n 1 + 1 ) 2 + κ 1 2 ] [ ( n 2 + 1 ) 2 + κ 2 2 ] [ n 2 ( n 1 2 + κ 1 2 + s 2 ) + n 1 ( n 2 2 + κ 2 2 + s 2 ) ] ,
β 1 = 4 π d 1 λ , β 2 = 4 π d 2 λ , φ 1 = β 1 n 1 , φ 2 = β 2 n 2 , x 1 = exp ( β 1 κ 1 ) , x 2 = exp ( β 2 κ 2 ) .
λ min λ 1 < < λ N λ max .
T meas ( λ i ) = T theoretical [ s ( λ i ) , d all , n all ( λ i ) , κ all ( λ i ) ] .
Minimize i = 1 N [ T theoretical ( s ( λ i ) , d all , n all ( λ i ) , κ all ( λ i ) ) T meas ( λ i ) ] 2
n ( λ max ) 1 , κ ( λ max ) 0 , n ( λ max ) 0 , κ ( λ max ) 0 , n ( λ ) 0 for     all   λ [ λ min , λ max ] , κ ( λ ) 0 for     all   λ [ λ infl , λ max ] , κ ( λ ) 0 for     all   λ [ λ min , λ infl ] , and κ ( λ min ) 0.
n ( λ max ) = 1 + u 2 , κ ( λ max ) = v 2 ,
n ( λ max ) = u 1 2 , κ ( λ max ) = v 1 2 ,
n ( λ ) = w ( λ ) 2 for     all   λ [ λ min , λ max ] ,
κ ( λ ) = z ( λ ) 2 for     all   λ [ λ infl , λ max ] ,
κ ( λ ) = z ( λ ) 2 for     all   λ [ λ min , λ infl ] .
h = ( λ max λ min ) / ( N 1 ) and λ i = λ min + ( i 1 ) h for     i = 1 , , N .
n N = 1 + u 2 , v N = v 2 ,
n N 1 = n N + u 1 2 h , κ N 1 = κ N + v 1 2 h ,
n i = w i 2 h 2 + 2 n i + 1 n i + 2 for     i = 1 , , N 2 ,
κ i = z i 2 h 2 + 2 κ i + 1 κ i + 2 , if     λ i + 1 λ infl ,
κ i = z i 2 h 2 + 2 κ i + 1 κ i + 2 , if     λ i + 1 < λ infl .
s glass ( λ ) = 1 + ( 0.7568 7930 / λ 2 ) 1 ,
s Si ( λ ) = 3.71382 8.6912310 5 λ 2.4712510 8 λ 2 + 1.0467710 11 λ 3 .
n true ( λ ) = 1 + ( 0.09195 12600 / λ 2 ) 1 .
ln ( α true ( E ) ) = { 6.594410 6 exp ( 9.0846 E ) 16.102 , 0.60 < E < 1.40 ; 20 E 41.9 , 1.40 < E < 1.75 ; 59.56 E 102.1 8.391 , 1.75 < E < 2.20.
n true ( λ ) = 1 + ( 0.065 ( 15000 / λ 2 ) 1 .
ln ( α true ( E ) ) = { 6.594410 6 exp ( 13.629 E ) 16.102 , 0.50 < E < 0.93 ; 30 E 41.9 , 0.93 < E < 1.17 ; 89.34 E 102.1 8.391 , 1.17 < E < 1.50.

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