Abstract

A method is developed for determining the optical properties of an optically rough coating on an opaque substrate from reflectance measurements. A modified Kubelka–Munk two- flux model is used to calculate the reflectance of the coating as a function of the refractive index, absorption coefficient, scattering coefficient, and thickness. The calculated reflectance is then fitted to measurements using a spectral projected gradient algorithm, allowing the optical properties to be obtained. The technique is applied to titanium dioxide coatings on a titanium substrate. Realistic values of refractive index and absorption coefficients are generally obtained. Quantities that are useful for solar water-splitting applications are calculated, including the depth profile of absorption and the proportion of the incident photon flux absorbed in the coating under solar illumination.

© 2007 Optical Society of America

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  1. D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
    [CrossRef]
  2. G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B.-S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
    [CrossRef]
  3. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, 1955).
  4. E. Shanti, V. Dutta, A. Banerjee, and K. L. Chopra, "Electrical and optical properties of undoped and antimony-doped tin oxide films," J. Appl. Phys. 51, 6243-6251 (1981).
    [CrossRef]
  5. S. P. Lyashenko and V. K. Miloslavskii, "A simple method for the determination of the thickness and optical constants of semiconducting and dielectric layers," Opt. Spectrosc. 16, 80-81 (1964).
  6. R. Swanepoel, "Determination of the thickness and optical constants of amorphous silicon," J. Phys. E 16, 1214-1222 (1983).
    [CrossRef]
  7. K. L. Eskins and K. W. Mitchell, "Simple method to determine optical constants of thin film silicon:hydrogen alloys and its application to device modeling," in Proceedings of the Eighteenth IEEE Photovoltaic Specialists Conference (IEEE, 1985), pp. 720-725.
  8. E. G. Birgin, I. Chambouleyron, and J. M. Martínez, "Estimation of the optical constants and thickness of thin films using unconstrained optimization," J. Comput. Phys. 151, 862-880 (1999).
    [CrossRef]
  9. W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
    [CrossRef]
  10. W. E. Vargas, I. Rojas, D. E. Azofeifa, and N. Clark, "Optical and electrical properties of hydrided palladium thin films studied by an inversion approach from transmittance measurements," Thin Solid Films 496, 189-196 (2006).
    [CrossRef]
  11. A. Ramírez-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]
  12. A. B. Murphy, "Modified Kubelka-Munk model for calculation of the reflectance of coatings with optically rough surfaces," J. Phys. D 39, 3571-3581 (2006).
    [CrossRef]
  13. P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).
  14. P. Kubelka, "New contributions to the optics of intensely light-scattering materials. Part I," J. Opt. Soc. Am. 38, 448-457 (1948).
    [CrossRef] [PubMed]
  15. J. L. Saunderson, "Calculation of the color of pigmented plastics," J. Opt. Soc. Am. 32, 727-736 (1942).
    [CrossRef]
  16. W. E. Vargas, "Inversion methods from Kubelka-Munk analysis," J. Opt. A 4, 452-456 (2002).
    [CrossRef]
  17. 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]
  18. A. B. Murphy, P. R. F. Barnes, L. K. Randeniya, I. C. Plumb, I. E. Grey, M. D. Horne, and J. A. Glasscock, "Efficiency of solar water splitting using semiconductor electrodes," Int. J. Hydrogen Energy 31, 1999-2017 (2006).
    [CrossRef]
  19. P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
    [CrossRef]
  20. J. Akikusa and S. U. M. Khan, "Photoresponse and AC impedance characterization of n-TiO2 films during hydrogen and oxygen evolution reactions in an electrochemical cell," Int. J. Hydrogen Energy 22, 875-882 (1997).
    [CrossRef]
  21. S. U. M. Khan, M. Al-Shahry, and W. B. Ingler, Jr., "Efficient photochemical water splitting by a chemically modified n-TiO2" Science 297, 2243-2245 (2002).
    [CrossRef] [PubMed]
  22. K. Noworyta and J. Augustynski, "Spectral photoresponses of carbon-doped TiO2 film electrodes," Electrochem. Solid-State Lett. 7, E31-33 (2004).
  23. J. Singh, I.-K. Oh, and S. O. Kasap, "Optical absorption, photoexcitation and excitons in solids: fundamental concepts," in Photo-Excited Processes, Diagnostics and Applications, A. Peled, ed. (Kluwer, 2003), pp. 25-55.
  24. B. Maheu, J. N. Letoulouzan, and G. Gouesbet, "Four-flux models to solve the scattering transfer equation in terms of Lorenz-Mie parameters," Appl. Opt. 23, 3353-3362 (1984).
    [CrossRef] [PubMed]
  25. G. Kortüm, Reflectance Spectroscopy (Springer, 1969).
  26. M. Athans, M. L. Dertouzos, R. N. Spann, and S. J. Mason, Systems, Networks and Computation: Multivariable Methods (McGraw-Hill, 1973), pp. 132-143.
  27. M. Raydan, "The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem," SIAM J. Optim. 7, 26-33 (1997).
    [CrossRef]
  28. E. G. Birgin, J. M. Martínez, and M. Raydan, "Nonmonotone spectral projected gradient methods on convex sets," SIAM J. Optim. 10, 1196-1211 (2000).
    [CrossRef]
  29. E. G. Birgin, J. M. Martínez, and M. Raydan, "Algorithm 813: SPG--Software for convex-constrained optimization," ACM Trans. Math. Softw. 27, 340-349 (2001).
    [CrossRef]
  30. L. Grippo, F. Lampariello, and S. Lucidi, "A nonmonotone line search technique for Newton's method," SIAM J. Numer. Anal. 23, 707-716 (1986).
    [CrossRef]
  31. J. Barzalai and J. M. Borwein, "Two-point step size gradient methods," IMA J. Numer. Anal. 8, 141-148 (1988).
    [CrossRef]
  32. M. Dechamps and P. Lehr, "Sur l'oxydation du titane α en atmosphère d'oxygène: Rôle de la couche oxydée et méchanisme d'oxydation," J. Less-Common Met. 56, 193-207 (1977).
    [CrossRef]
  33. D. W. Lynch and W. R. Hunter, "Introduction to the data for several metals," in Handbook of Optical Constants, Vol. III, E. D. Palik, ed. (Academic, 1998), pp. 233-286.
  34. M. Cardona and G. Harbeke, "Optical properties and band structure of wurtzite-type crystals and rutile," Phys. Rev. 137A, 1467-1476 (1965).
    [CrossRef]
  35. J. R. Devore, "Refractive indices of rutile and sphalerite," J. Opt. Soc. Am. 41, 416-419 (1951).
    [CrossRef]
  36. M. W. Ribarsky, "Titanium dioxide (TiO2) (rutile)," in Handbook of Optical Constants, E. D. Palik, ed. (Academic, 1985), pp. 795-800.
  37. B. Karunagaran, R. T. Rajendra Kumar, C. Viswanathan, D. Mangalaraj, S. K. Narayandass, and G. Mohan Rao, "Optical constants of DC magnetron sputtered titanium dioxide thin films measured by spectroscopy ellipsometry," Cryst. Res. Technol. 38, 773-778 (2003).
    [CrossRef]
  38. Y.-Q. Hou, D.-M. Zhuang, G. Zhang, M. Zhao, and M. S. Wu, "Influence of annealing temperature on the properties of titanium oxide thin film," Appl. Surf. Sci. 218, 97-105 (2003).
    [CrossRef]
  39. American Society for Testing and Materials, "Standard tables for reference solar spectral irradiance at air mass 1.5: Direct normal and hemispherical for a 37° tilted surface" (ASTM, 1998), Standard G159-98.
  40. W. E. Vargas, "Two-flux radiative transfer model under nonisotropic propagating diffuse radiation," Appl. Opt. 38, 1077-1085 (1999).
    [CrossRef]
  41. P. R. F. Barnes, L. K. Randeniya, P. F. Vohralik, and I. C. Plumb, "The influence of substrate etching on the photoelectrochemical performance of thermally oxidized TiO2 films," J. Electrochem. Soc. 154, H249-H257 (2007).
    [CrossRef]

2007 (1)

P. R. F. Barnes, L. K. Randeniya, P. F. Vohralik, and I. C. Plumb, "The influence of substrate etching on the photoelectrochemical performance of thermally oxidized TiO2 films," J. Electrochem. Soc. 154, H249-H257 (2007).
[CrossRef]

2006 (4)

W. E. Vargas, I. Rojas, D. E. Azofeifa, and N. Clark, "Optical and electrical properties of hydrided palladium thin films studied by an inversion approach from transmittance measurements," Thin Solid Films 496, 189-196 (2006).
[CrossRef]

A. B. Murphy, "Modified Kubelka-Munk model for calculation of the reflectance of coatings with optically rough surfaces," J. Phys. D 39, 3571-3581 (2006).
[CrossRef]

A. B. Murphy, P. R. F. Barnes, L. K. Randeniya, I. C. Plumb, I. E. Grey, M. D. Horne, and J. A. Glasscock, "Efficiency of solar water splitting using semiconductor electrodes," Int. J. Hydrogen Energy 31, 1999-2017 (2006).
[CrossRef]

P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
[CrossRef]

2004 (2)

K. Noworyta and J. Augustynski, "Spectral photoresponses of carbon-doped TiO2 film electrodes," Electrochem. Solid-State Lett. 7, E31-33 (2004).

A. Ramírez-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]

2003 (4)

B. Karunagaran, R. T. Rajendra Kumar, C. Viswanathan, D. Mangalaraj, S. K. Narayandass, and G. Mohan Rao, "Optical constants of DC magnetron sputtered titanium dioxide thin films measured by spectroscopy ellipsometry," Cryst. Res. Technol. 38, 773-778 (2003).
[CrossRef]

Y.-Q. Hou, D.-M. Zhuang, G. Zhang, M. Zhao, and M. S. Wu, "Influence of annealing temperature on the properties of titanium oxide thin film," Appl. Surf. Sci. 218, 97-105 (2003).
[CrossRef]

W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
[CrossRef]

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B.-S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

2002 (3)

W. E. Vargas, "Inversion methods from Kubelka-Munk analysis," J. Opt. A 4, 452-456 (2002).
[CrossRef]

S. U. M. Khan, M. Al-Shahry, and W. B. Ingler, Jr., "Efficient photochemical water splitting by a chemically modified n-TiO2" Science 297, 2243-2245 (2002).
[CrossRef] [PubMed]

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 (1)

E. G. Birgin, J. M. Martínez, and M. Raydan, "Algorithm 813: SPG--Software for convex-constrained optimization," ACM Trans. Math. Softw. 27, 340-349 (2001).
[CrossRef]

2000 (1)

E. G. Birgin, J. M. Martínez, and M. Raydan, "Nonmonotone spectral projected gradient methods on convex sets," SIAM J. Optim. 10, 1196-1211 (2000).
[CrossRef]

1999 (2)

W. E. Vargas, "Two-flux radiative transfer model under nonisotropic propagating diffuse radiation," Appl. Opt. 38, 1077-1085 (1999).
[CrossRef]

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

1997 (2)

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

J. Akikusa and S. U. M. Khan, "Photoresponse and AC impedance characterization of n-TiO2 films during hydrogen and oxygen evolution reactions in an electrochemical cell," Int. J. Hydrogen Energy 22, 875-882 (1997).
[CrossRef]

1988 (1)

J. Barzalai and J. M. Borwein, "Two-point step size gradient methods," IMA J. Numer. Anal. 8, 141-148 (1988).
[CrossRef]

1986 (1)

L. Grippo, F. Lampariello, and S. Lucidi, "A nonmonotone line search technique for Newton's method," SIAM J. Numer. Anal. 23, 707-716 (1986).
[CrossRef]

1984 (1)

1983 (2)

D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

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

1981 (1)

E. Shanti, V. Dutta, A. Banerjee, and K. L. Chopra, "Electrical and optical properties of undoped and antimony-doped tin oxide films," J. Appl. Phys. 51, 6243-6251 (1981).
[CrossRef]

1977 (1)

M. Dechamps and P. Lehr, "Sur l'oxydation du titane α en atmosphère d'oxygène: Rôle de la couche oxydée et méchanisme d'oxydation," J. Less-Common Met. 56, 193-207 (1977).
[CrossRef]

1965 (1)

M. Cardona and G. Harbeke, "Optical properties and band structure of wurtzite-type crystals and rutile," Phys. Rev. 137A, 1467-1476 (1965).
[CrossRef]

1964 (1)

S. P. Lyashenko and V. K. Miloslavskii, "A simple method for the determination of the thickness and optical constants of semiconducting and dielectric layers," Opt. Spectrosc. 16, 80-81 (1964).

1951 (1)

1948 (1)

1942 (1)

1931 (1)

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Akikusa, J.

J. Akikusa and S. U. M. Khan, "Photoresponse and AC impedance characterization of n-TiO2 films during hydrogen and oxygen evolution reactions in an electrochemical cell," Int. J. Hydrogen Energy 22, 875-882 (1997).
[CrossRef]

Al-Shahry, M.

S. U. M. Khan, M. Al-Shahry, and W. B. Ingler, Jr., "Efficient photochemical water splitting by a chemically modified n-TiO2" Science 297, 2243-2245 (2002).
[CrossRef] [PubMed]

Aspnes, D. E.

D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Athans, M.

M. Athans, M. L. Dertouzos, R. N. Spann, and S. J. Mason, Systems, Networks and Computation: Multivariable Methods (McGraw-Hill, 1973), pp. 132-143.

Augustynski, J.

K. Noworyta and J. Augustynski, "Spectral photoresponses of carbon-doped TiO2 film electrodes," Electrochem. Solid-State Lett. 7, E31-33 (2004).

Azofeifa, D. E.

W. E. Vargas, I. Rojas, D. E. Azofeifa, and N. Clark, "Optical and electrical properties of hydrided palladium thin films studied by an inversion approach from transmittance measurements," Thin Solid Films 496, 189-196 (2006).
[CrossRef]

W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
[CrossRef]

Banerjee, A.

E. Shanti, V. Dutta, A. Banerjee, and K. L. Chopra, "Electrical and optical properties of undoped and antimony-doped tin oxide films," J. Appl. Phys. 51, 6243-6251 (1981).
[CrossRef]

Barnes, P. R. F.

P. R. F. Barnes, L. K. Randeniya, P. F. Vohralik, and I. C. Plumb, "The influence of substrate etching on the photoelectrochemical performance of thermally oxidized TiO2 films," J. Electrochem. Soc. 154, H249-H257 (2007).
[CrossRef]

P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
[CrossRef]

A. B. Murphy, P. R. F. Barnes, L. K. Randeniya, I. C. Plumb, I. E. Grey, M. D. Horne, and J. A. Glasscock, "Efficiency of solar water splitting using semiconductor electrodes," Int. J. Hydrogen Energy 31, 1999-2017 (2006).
[CrossRef]

Barrera, R. G.

Barzalai, J.

J. Barzalai and J. M. Borwein, "Two-point step size gradient methods," IMA J. Numer. Anal. 8, 141-148 (1988).
[CrossRef]

Birgin, E. G.

E. G. Birgin, J. M. Martínez, and M. Raydan, "Algorithm 813: SPG--Software for convex-constrained optimization," ACM Trans. Math. Softw. 27, 340-349 (2001).
[CrossRef]

E. G. Birgin, J. M. Martínez, and M. Raydan, "Nonmonotone spectral projected gradient methods on convex sets," SIAM J. Optim. 10, 1196-1211 (2000).
[CrossRef]

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

Boatner, L. A.

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B.-S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

Borwein, J. M.

J. Barzalai and J. M. Borwein, "Two-point step size gradient methods," IMA J. Numer. Anal. 8, 141-148 (1988).
[CrossRef]

Budai, J. D.

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B.-S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

Cardona, M.

M. Cardona and G. Harbeke, "Optical properties and band structure of wurtzite-type crystals and rutile," Phys. Rev. 137A, 1467-1476 (1965).
[CrossRef]

Chambouleyron, I.

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

Chopra, K. L.

E. Shanti, V. Dutta, A. Banerjee, and K. L. Chopra, "Electrical and optical properties of undoped and antimony-doped tin oxide films," J. Appl. Phys. 51, 6243-6251 (1981).
[CrossRef]

Clark, N.

W. E. Vargas, I. Rojas, D. E. Azofeifa, and N. Clark, "Optical and electrical properties of hydrided palladium thin films studied by an inversion approach from transmittance measurements," Thin Solid Films 496, 189-196 (2006).
[CrossRef]

W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
[CrossRef]

Curiel, F.

Dechamps, M.

M. Dechamps and P. Lehr, "Sur l'oxydation du titane α en atmosphère d'oxygène: Rôle de la couche oxydée et méchanisme d'oxydation," J. Less-Common Met. 56, 193-207 (1977).
[CrossRef]

Dertouzos, M. L.

M. Athans, M. L. Dertouzos, R. N. Spann, and S. J. Mason, Systems, Networks and Computation: Multivariable Methods (McGraw-Hill, 1973), pp. 132-143.

Devore, J. R.

Dutta, V.

E. Shanti, V. Dutta, A. Banerjee, and K. L. Chopra, "Electrical and optical properties of undoped and antimony-doped tin oxide films," J. Appl. Phys. 51, 6243-6251 (1981).
[CrossRef]

Eskins, K. L.

K. L. Eskins and K. W. Mitchell, "Simple method to determine optical constants of thin film silicon:hydrogen alloys and its application to device modeling," in Proceedings of the Eighteenth IEEE Photovoltaic Specialists Conference (IEEE, 1985), pp. 720-725.

Glasscock, J. A.

A. B. Murphy, P. R. F. Barnes, L. K. Randeniya, I. C. Plumb, I. E. Grey, M. D. Horne, and J. A. Glasscock, "Efficiency of solar water splitting using semiconductor electrodes," Int. J. Hydrogen Energy 31, 1999-2017 (2006).
[CrossRef]

P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
[CrossRef]

Gouesbet, G.

Grey, I. E.

A. B. Murphy, P. R. F. Barnes, L. K. Randeniya, I. C. Plumb, I. E. Grey, M. D. Horne, and J. A. Glasscock, "Efficiency of solar water splitting using semiconductor electrodes," Int. J. Hydrogen Energy 31, 1999-2017 (2006).
[CrossRef]

P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
[CrossRef]

Grippo, L.

L. Grippo, F. Lampariello, and S. Lucidi, "A nonmonotone line search technique for Newton's method," SIAM J. Numer. Anal. 23, 707-716 (1986).
[CrossRef]

Gwan, P. B.

P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
[CrossRef]

Harbeke, G.

M. Cardona and G. Harbeke, "Optical properties and band structure of wurtzite-type crystals and rutile," Phys. Rev. 137A, 1467-1476 (1965).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, 1955).

Horne, M. D.

A. B. Murphy, P. R. F. Barnes, L. K. Randeniya, I. C. Plumb, I. E. Grey, M. D. Horne, and J. A. Glasscock, "Efficiency of solar water splitting using semiconductor electrodes," Int. J. Hydrogen Energy 31, 1999-2017 (2006).
[CrossRef]

Hou, Y.-Q.

Y.-Q. Hou, D.-M. Zhuang, G. Zhang, M. Zhao, and M. S. Wu, "Influence of annealing temperature on the properties of titanium oxide thin film," Appl. Surf. Sci. 218, 97-105 (2003).
[CrossRef]

Hunter, W. R.

D. W. Lynch and W. R. Hunter, "Introduction to the data for several metals," in Handbook of Optical Constants, Vol. III, E. D. Palik, ed. (Academic, 1998), pp. 233-286.

Ingler, W. B.

S. U. M. Khan, M. Al-Shahry, and W. B. Ingler, Jr., "Efficient photochemical water splitting by a chemically modified n-TiO2" Science 297, 2243-2245 (2002).
[CrossRef] [PubMed]

Jellison, G. E.

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B.-S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

Jeong, B.-S.

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B.-S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

Karunagaran, B.

B. Karunagaran, R. T. Rajendra Kumar, C. Viswanathan, D. Mangalaraj, S. K. Narayandass, and G. Mohan Rao, "Optical constants of DC magnetron sputtered titanium dioxide thin films measured by spectroscopy ellipsometry," Cryst. Res. Technol. 38, 773-778 (2003).
[CrossRef]

Kasap, S. O.

J. Singh, I.-K. Oh, and S. O. Kasap, "Optical absorption, photoexcitation and excitons in solids: fundamental concepts," in Photo-Excited Processes, Diagnostics and Applications, A. Peled, ed. (Kluwer, 2003), pp. 25-55.

Khan, S. U. M.

S. U. M. Khan, M. Al-Shahry, and W. B. Ingler, Jr., "Efficient photochemical water splitting by a chemically modified n-TiO2" Science 297, 2243-2245 (2002).
[CrossRef] [PubMed]

J. Akikusa and S. U. M. Khan, "Photoresponse and AC impedance characterization of n-TiO2 films during hydrogen and oxygen evolution reactions in an electrochemical cell," Int. J. Hydrogen Energy 22, 875-882 (1997).
[CrossRef]

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer, 1969).

Kubelka, P.

P. Kubelka, "New contributions to the optics of intensely light-scattering materials. Part I," J. Opt. Soc. Am. 38, 448-457 (1948).
[CrossRef] [PubMed]

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Kumar, R. T. Rajendra

B. Karunagaran, R. T. Rajendra Kumar, C. Viswanathan, D. Mangalaraj, S. K. Narayandass, and G. Mohan Rao, "Optical constants of DC magnetron sputtered titanium dioxide thin films measured by spectroscopy ellipsometry," Cryst. Res. Technol. 38, 773-778 (2003).
[CrossRef]

Lampariello, F.

L. Grippo, F. Lampariello, and S. Lucidi, "A nonmonotone line search technique for Newton's method," SIAM J. Numer. Anal. 23, 707-716 (1986).
[CrossRef]

Lehr, P.

M. Dechamps and P. Lehr, "Sur l'oxydation du titane α en atmosphère d'oxygène: Rôle de la couche oxydée et méchanisme d'oxydation," J. Less-Common Met. 56, 193-207 (1977).
[CrossRef]

Letoulouzan, J. N.

Li, C.

P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
[CrossRef]

Lucidi, S.

L. Grippo, F. Lampariello, and S. Lucidi, "A nonmonotone line search technique for Newton's method," SIAM J. Numer. Anal. 23, 707-716 (1986).
[CrossRef]

Lyashenko, S. P.

S. P. Lyashenko and V. K. Miloslavskii, "A simple method for the determination of the thickness and optical constants of semiconducting and dielectric layers," Opt. Spectrosc. 16, 80-81 (1964).

Lynch, D. W.

D. W. Lynch and W. R. Hunter, "Introduction to the data for several metals," in Handbook of Optical Constants, Vol. III, E. D. Palik, ed. (Academic, 1998), pp. 233-286.

Maheu, B.

Mangalaraj, D.

B. Karunagaran, R. T. Rajendra Kumar, C. Viswanathan, D. Mangalaraj, S. K. Narayandass, and G. Mohan Rao, "Optical constants of DC magnetron sputtered titanium dioxide thin films measured by spectroscopy ellipsometry," Cryst. Res. Technol. 38, 773-778 (2003).
[CrossRef]

Martínez, J. M.

E. G. Birgin, J. M. Martínez, and M. Raydan, "Algorithm 813: SPG--Software for convex-constrained optimization," ACM Trans. Math. Softw. 27, 340-349 (2001).
[CrossRef]

E. G. Birgin, J. M. Martínez, and M. Raydan, "Nonmonotone spectral projected gradient methods on convex sets," SIAM J. Optim. 10, 1196-1211 (2000).
[CrossRef]

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

Mason, S. J.

M. Athans, M. L. Dertouzos, R. N. Spann, and S. J. Mason, Systems, Networks and Computation: Multivariable Methods (McGraw-Hill, 1973), pp. 132-143.

Miloslavskii, V. K.

S. P. Lyashenko and V. K. Miloslavskii, "A simple method for the determination of the thickness and optical constants of semiconducting and dielectric layers," Opt. Spectrosc. 16, 80-81 (1964).

Mitchell, K. W.

K. L. Eskins and K. W. Mitchell, "Simple method to determine optical constants of thin film silicon:hydrogen alloys and its application to device modeling," in Proceedings of the Eighteenth IEEE Photovoltaic Specialists Conference (IEEE, 1985), pp. 720-725.

Munk, F.

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Murphy, A. B.

A. B. Murphy, "Modified Kubelka-Munk model for calculation of the reflectance of coatings with optically rough surfaces," J. Phys. D 39, 3571-3581 (2006).
[CrossRef]

P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
[CrossRef]

A. B. Murphy, P. R. F. Barnes, L. K. Randeniya, I. C. Plumb, I. E. Grey, M. D. Horne, and J. A. Glasscock, "Efficiency of solar water splitting using semiconductor electrodes," Int. J. Hydrogen Energy 31, 1999-2017 (2006).
[CrossRef]

Narayandass, S. K.

B. Karunagaran, R. T. Rajendra Kumar, C. Viswanathan, D. Mangalaraj, S. K. Narayandass, and G. Mohan Rao, "Optical constants of DC magnetron sputtered titanium dioxide thin films measured by spectroscopy ellipsometry," Cryst. Res. Technol. 38, 773-778 (2003).
[CrossRef]

Norton, D. P.

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B.-S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

Noworyta, K.

K. Noworyta and J. Augustynski, "Spectral photoresponses of carbon-doped TiO2 film electrodes," Electrochem. Solid-State Lett. 7, E31-33 (2004).

Oh, I.-K.

J. Singh, I.-K. Oh, and S. O. Kasap, "Optical absorption, photoexcitation and excitons in solids: fundamental concepts," in Photo-Excited Processes, Diagnostics and Applications, A. Peled, ed. (Kluwer, 2003), pp. 25-55.

Plumb, I. C.

P. R. F. Barnes, L. K. Randeniya, P. F. Vohralik, and I. C. Plumb, "The influence of substrate etching on the photoelectrochemical performance of thermally oxidized TiO2 films," J. Electrochem. Soc. 154, H249-H257 (2007).
[CrossRef]

P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
[CrossRef]

A. B. Murphy, P. R. F. Barnes, L. K. Randeniya, I. C. Plumb, I. E. Grey, M. D. Horne, and J. A. Glasscock, "Efficiency of solar water splitting using semiconductor electrodes," Int. J. Hydrogen Energy 31, 1999-2017 (2006).
[CrossRef]

Ramírez-Porras, A.

Randeniya, L. K.

P. R. F. Barnes, L. K. Randeniya, P. F. Vohralik, and I. C. Plumb, "The influence of substrate etching on the photoelectrochemical performance of thermally oxidized TiO2 films," J. Electrochem. Soc. 154, H249-H257 (2007).
[CrossRef]

P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
[CrossRef]

A. B. Murphy, P. R. F. Barnes, L. K. Randeniya, I. C. Plumb, I. E. Grey, M. D. Horne, and J. A. Glasscock, "Efficiency of solar water splitting using semiconductor electrodes," Int. J. Hydrogen Energy 31, 1999-2017 (2006).
[CrossRef]

Rao, G. Mohan

B. Karunagaran, R. T. Rajendra Kumar, C. Viswanathan, D. Mangalaraj, S. K. Narayandass, and G. Mohan Rao, "Optical constants of DC magnetron sputtered titanium dioxide thin films measured by spectroscopy ellipsometry," Cryst. Res. Technol. 38, 773-778 (2003).
[CrossRef]

Raydan, M.

E. G. Birgin, J. M. Martínez, and M. Raydan, "Algorithm 813: SPG--Software for convex-constrained optimization," ACM Trans. Math. Softw. 27, 340-349 (2001).
[CrossRef]

E. G. Birgin, J. M. Martínez, and M. Raydan, "Nonmonotone spectral projected gradient methods on convex sets," SIAM J. Optim. 10, 1196-1211 (2000).
[CrossRef]

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

Ribarsky, M. W.

M. W. Ribarsky, "Titanium dioxide (TiO2) (rutile)," in Handbook of Optical Constants, E. D. Palik, ed. (Academic, 1985), pp. 795-800.

Rojas, I.

W. E. Vargas, I. Rojas, D. E. Azofeifa, and N. Clark, "Optical and electrical properties of hydrided palladium thin films studied by an inversion approach from transmittance measurements," Thin Solid Films 496, 189-196 (2006).
[CrossRef]

Saunderson, J. L.

Shanti, E.

E. Shanti, V. Dutta, A. Banerjee, and K. L. Chopra, "Electrical and optical properties of undoped and antimony-doped tin oxide films," J. Appl. Phys. 51, 6243-6251 (1981).
[CrossRef]

Singh, J.

J. Singh, I.-K. Oh, and S. O. Kasap, "Optical absorption, photoexcitation and excitons in solids: fundamental concepts," in Photo-Excited Processes, Diagnostics and Applications, A. Peled, ed. (Kluwer, 2003), pp. 25-55.

Spann, R. N.

M. Athans, M. L. Dertouzos, R. N. Spann, and S. J. Mason, Systems, Networks and Computation: Multivariable Methods (McGraw-Hill, 1973), pp. 132-143.

Studna, A. A.

D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Swanepoel, R.

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

Vargas, W. E.

W. E. Vargas, I. Rojas, D. E. Azofeifa, and N. Clark, "Optical and electrical properties of hydrided palladium thin films studied by an inversion approach from transmittance measurements," Thin Solid Films 496, 189-196 (2006).
[CrossRef]

W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
[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]

W. E. Vargas, "Inversion methods from Kubelka-Munk analysis," J. Opt. A 4, 452-456 (2002).
[CrossRef]

W. E. Vargas, "Two-flux radiative transfer model under nonisotropic propagating diffuse radiation," Appl. Opt. 38, 1077-1085 (1999).
[CrossRef]

Vargas-Castro, W. E.

Viswanathan, C.

B. Karunagaran, R. T. Rajendra Kumar, C. Viswanathan, D. Mangalaraj, S. K. Narayandass, and G. Mohan Rao, "Optical constants of DC magnetron sputtered titanium dioxide thin films measured by spectroscopy ellipsometry," Cryst. Res. Technol. 38, 773-778 (2003).
[CrossRef]

Vohralik, P. F.

P. R. F. Barnes, L. K. Randeniya, P. F. Vohralik, and I. C. Plumb, "The influence of substrate etching on the photoelectrochemical performance of thermally oxidized TiO2 films," J. Electrochem. Soc. 154, H249-H257 (2007).
[CrossRef]

Wu, M. S.

Y.-Q. Hou, D.-M. Zhuang, G. Zhang, M. Zhao, and M. S. Wu, "Influence of annealing temperature on the properties of titanium oxide thin film," Appl. Surf. Sci. 218, 97-105 (2003).
[CrossRef]

Zhang, G.

Y.-Q. Hou, D.-M. Zhuang, G. Zhang, M. Zhao, and M. S. Wu, "Influence of annealing temperature on the properties of titanium oxide thin film," Appl. Surf. Sci. 218, 97-105 (2003).
[CrossRef]

Zhao, M.

Y.-Q. Hou, D.-M. Zhuang, G. Zhang, M. Zhao, and M. S. Wu, "Influence of annealing temperature on the properties of titanium oxide thin film," Appl. Surf. Sci. 218, 97-105 (2003).
[CrossRef]

Zhuang, D.-M.

Y.-Q. Hou, D.-M. Zhuang, G. Zhang, M. Zhao, and M. S. Wu, "Influence of annealing temperature on the properties of titanium oxide thin film," Appl. Surf. Sci. 218, 97-105 (2003).
[CrossRef]

ACM Trans. Math. Softw. (1)

E. G. Birgin, J. M. Martínez, and M. Raydan, "Algorithm 813: SPG--Software for convex-constrained optimization," ACM Trans. Math. Softw. 27, 340-349 (2001).
[CrossRef]

Appl. Opt. (4)

Appl. Surf. Sci. (1)

Y.-Q. Hou, D.-M. Zhuang, G. Zhang, M. Zhao, and M. S. Wu, "Influence of annealing temperature on the properties of titanium oxide thin film," Appl. Surf. Sci. 218, 97-105 (2003).
[CrossRef]

Cryst. Res. Technol. (1)

B. Karunagaran, R. T. Rajendra Kumar, C. Viswanathan, D. Mangalaraj, S. K. Narayandass, and G. Mohan Rao, "Optical constants of DC magnetron sputtered titanium dioxide thin films measured by spectroscopy ellipsometry," Cryst. Res. Technol. 38, 773-778 (2003).
[CrossRef]

Dev. Chem. Eng. Miner. Process. (1)

P. R. F. Barnes, L. K. Randeniya, A. B. Murphy, P. B. Gwan, I. C. Plumb, J. A. Glasscock, I. E. Grey, and C. Li, "TiO2 photoelectrodes for water splitting: Carbon doping by flame pyrolysis?" Dev. Chem. Eng. Miner. Process. 14, 51-70 (2006).
[CrossRef]

IMA J. Numer. Anal. (1)

J. Barzalai and J. M. Borwein, "Two-point step size gradient methods," IMA J. Numer. Anal. 8, 141-148 (1988).
[CrossRef]

Int. J. Hydrogen Energy (2)

A. B. Murphy, P. R. F. Barnes, L. K. Randeniya, I. C. Plumb, I. E. Grey, M. D. Horne, and J. A. Glasscock, "Efficiency of solar water splitting using semiconductor electrodes," Int. J. Hydrogen Energy 31, 1999-2017 (2006).
[CrossRef]

J. Akikusa and S. U. M. Khan, "Photoresponse and AC impedance characterization of n-TiO2 films during hydrogen and oxygen evolution reactions in an electrochemical cell," Int. J. Hydrogen Energy 22, 875-882 (1997).
[CrossRef]

J. Appl. Phys. (2)

G. E. Jellison, Jr., L. A. Boatner, J. D. Budai, B.-S. Jeong, and D. P. Norton, "Spectroscopic ellipsometry of thin film and bulk anatase (TiO2)," J. Appl. Phys. 93, 9537-9541 (2003).
[CrossRef]

E. Shanti, V. Dutta, A. Banerjee, and K. L. Chopra, "Electrical and optical properties of undoped and antimony-doped tin oxide films," J. Appl. Phys. 51, 6243-6251 (1981).
[CrossRef]

J. Comput. Phys. (1)

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

J. Electrochem. Soc. (1)

P. R. F. Barnes, L. K. Randeniya, P. F. Vohralik, and I. C. Plumb, "The influence of substrate etching on the photoelectrochemical performance of thermally oxidized TiO2 films," J. Electrochem. Soc. 154, H249-H257 (2007).
[CrossRef]

J. Less-Common Met. (1)

M. Dechamps and P. Lehr, "Sur l'oxydation du titane α en atmosphère d'oxygène: Rôle de la couche oxydée et méchanisme d'oxydation," J. Less-Common Met. 56, 193-207 (1977).
[CrossRef]

J. Opt. A (1)

W. E. Vargas, "Inversion methods from Kubelka-Munk analysis," J. Opt. A 4, 452-456 (2002).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. D (1)

A. B. Murphy, "Modified Kubelka-Munk model for calculation of the reflectance of coatings with optically rough surfaces," J. Phys. D 39, 3571-3581 (2006).
[CrossRef]

J. Phys. E (1)

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

Opt. Spectrosc. (1)

S. P. Lyashenko and V. K. Miloslavskii, "A simple method for the determination of the thickness and optical constants of semiconducting and dielectric layers," Opt. Spectrosc. 16, 80-81 (1964).

Phys. Rev. (1)

M. Cardona and G. Harbeke, "Optical properties and band structure of wurtzite-type crystals and rutile," Phys. Rev. 137A, 1467-1476 (1965).
[CrossRef]

Phys. Rev. B (1)

D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Science (1)

S. U. M. Khan, M. Al-Shahry, and W. B. Ingler, Jr., "Efficient photochemical water splitting by a chemically modified n-TiO2" Science 297, 2243-2245 (2002).
[CrossRef] [PubMed]

SIAM J. Numer. Anal. (1)

L. Grippo, F. Lampariello, and S. Lucidi, "A nonmonotone line search technique for Newton's method," SIAM J. Numer. Anal. 23, 707-716 (1986).
[CrossRef]

SIAM J. Optim. (2)

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

E. G. Birgin, J. M. Martínez, and M. Raydan, "Nonmonotone spectral projected gradient methods on convex sets," SIAM J. Optim. 10, 1196-1211 (2000).
[CrossRef]

Thin Solid Films (2)

W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
[CrossRef]

W. E. Vargas, I. Rojas, D. E. Azofeifa, and N. Clark, "Optical and electrical properties of hydrided palladium thin films studied by an inversion approach from transmittance measurements," Thin Solid Films 496, 189-196 (2006).
[CrossRef]

Z. Tech. Phys. (1)

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Other (9)

K. Noworyta and J. Augustynski, "Spectral photoresponses of carbon-doped TiO2 film electrodes," Electrochem. Solid-State Lett. 7, E31-33 (2004).

J. Singh, I.-K. Oh, and S. O. Kasap, "Optical absorption, photoexcitation and excitons in solids: fundamental concepts," in Photo-Excited Processes, Diagnostics and Applications, A. Peled, ed. (Kluwer, 2003), pp. 25-55.

G. Kortüm, Reflectance Spectroscopy (Springer, 1969).

M. Athans, M. L. Dertouzos, R. N. Spann, and S. J. Mason, Systems, Networks and Computation: Multivariable Methods (McGraw-Hill, 1973), pp. 132-143.

K. L. Eskins and K. W. Mitchell, "Simple method to determine optical constants of thin film silicon:hydrogen alloys and its application to device modeling," in Proceedings of the Eighteenth IEEE Photovoltaic Specialists Conference (IEEE, 1985), pp. 720-725.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, 1955).

M. W. Ribarsky, "Titanium dioxide (TiO2) (rutile)," in Handbook of Optical Constants, E. D. Palik, ed. (Academic, 1985), pp. 795-800.

D. W. Lynch and W. R. Hunter, "Introduction to the data for several metals," in Handbook of Optical Constants, Vol. III, E. D. Palik, ed. (Academic, 1998), pp. 233-286.

American Society for Testing and Materials, "Standard tables for reference solar spectral irradiance at air mass 1.5: Direct normal and hemispherical for a 37° tilted surface" (ASTM, 1998), Standard G159-98.

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Figures (9)

Fig. 1
Fig. 1

(Color online) Geometry, showing boundary conditions at z = 0 and z = h . Collimated light is denoted by solid arrows and diffuse light by dotted arrows.

Fig. 2
Fig. 2

(Color online) (a) Measured, initial calculated, and best-fit calculated values of diffuse reflectance R c d for sample 3. (b) Best-fit values obtained by three methods: with the refractive index fixed at its initial value, with the absorption coefficient fixed at its initial value, and without any parameters fixed.

Fig. 3
Fig. 3

(Color online) Initial and best-fit values of absorption coefficient and refractive index for sample 3. Best-fit values obtained by three methods are shown: with the refractive index fixed at its initial value, with the absorption coefficient fixed at its initial value, and without any parameters fixed.

Fig. 4
Fig. 4

(Color online) Initial and best-fit values of reflection coefficients r c c f , r c d f , r d d b , and r d d s for sample 3.

Fig. 5
Fig. 5

(Color online) Measured, initial calculated, and best-fit calculated values of diffuse reflectance R c d for samples 1 and 2.

Fig. 6
Fig. 6

(Color online) Best-fit values of absorption coefficient and refractive index for all three samples.

Fig. 7
Fig. 7

(Color online) Sensitivity of best-fit value of diffuse reflectance R c d to changes in the fitted values of absorption coefficient, scattering coefficient, refractive index, and coating thickness for sample 3. The values are normalized to give a dimensionless sensitivity.

Fig. 8
Fig. 8

(Color online) Wavelength dependence of reflectance, transmittance, and absorbance for sample 3. The initial values and best-fit values are shown.

Fig. 9
Fig. 9

(Color online) Depth distribution of absorbed radiation flux for all three samples. The inset shows detail near the surface of the coating.

Tables (2)

Tables Icon

Table 1 Oxidation Conditions and Properties of Rutile TiO2 Coatings on Ti for Coatings of Thickness h , rms Surface Roughness σ 0 and Autocorrelation Length τ

Tables Icon

Table 2 Calculated Thickness and Efficiencies for Each Sample

Equations (36)

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

S = 2 ( 1 ζ ) s ,
K = ε k ,
d I d d z = ( S + K ) I d + S J d ,
d J d d z = ( S + K ) J d S I d ,
R c d = r c d f I c 0 + ( 1 r d d b ) J d ( 0 ) I c 0 .
R c d = r c d f + ( 1 r c d f r c c f ) ( 1 r d d b ) R K M 1 r d d b R K M ,
R K M = 1 r d d s [ a b coth ( b S h ) ] a + b coth ( b S h ) r d d s ,
a = ( S + K ) / S ,
b = a 2 1 .
R c c = r c c f .
T = ( 1 r d d s ) I d ( h ) I c 0 ,
T = b ( 1 r c c f r c d f ) ( 1 r d d s ) b ( 1 r d d b r d d s ) cosh ( S b h ) + ( a r d d b r d d s + a r d d b r d d s ) sinh ( S b h ) .
A = 1 R c c R c d T .
A 0 h α ( z ) d z ,
α ( z ) = K [ I d ( z ) + J d ( z ) ] / I c 0 ,
α ( z ) = K ( 1 r c c f r c d f ) [ b ( 1 + b r d d s ) cosh ( S b h S b z ) + ( 1 + a ) ( 1 r d d s ) sinh ( S b h S b z ) ] b ( 1 r d d b r d d s ) cosh ( S b h ) + ( a r d d b r d d s + a r d d b r d d s ) sinh ( S b h ) .
J z ( z ) = 0 I λ α λ ( z ) d λ ,
K = 2 k ,
k = 4 π κ c / λ .
S = s .
F = i = 1 p { R c d   exp ( λ i ) R c d [ s ( λ i ) , κ c ( λ i ) , n c ( λ i ) , h ] } 2 ,
Z i = { s ( λ i ) , i = 1 , 2 ,   …   ,   p κ c ( λ i ) , i = p + 1 , p + 2 ,   …   ,   2 p n c ( λ i ) , i = 2 p + 1 , 2 p + 2 ,   …   ,   3 p h , i = 3 p + 1.
g i ( ) = F ( ) Z i | Z j , j i ,
( l + 1 ) = ( l ) β l ( ( l ) ) ,
β 1 = ( ( l ) ( l 1 ) ) T ( ( l ) ( l 1 ) ) ( ( l ) ( l 1 ) ) T [ ( ( l ) ) ( ( l 1 ) ) ] ,
F X = 2 { R c d   exp ( λ i ) R c d [ s ( λ i ) , κ c ( λ i ) , n c ( λ i ) , h ] } × R c d [ s ( λ i ) , κ c ( λ i ) , n c ( λ i ) , h ] X ,
R c d = R c d [ S , K ( κ c ) , r c c f ( κ c , n c ) , r c d f ( κ c , n c ) , r d d b ( κ c , n c ) , r d d s ( κ c , n c ) , h ] ,
R c d κ c ( λ i ) = R c d K K κ c ( λ i ) + R c d r c c f r c c f κ c ( λ i ) + R c d r c d f r c d f κ c ( λ i ) + R c d r d d b r d d b κ c ( λ i ) + R c d r d d s r d d s κ c ( λ i ) ,
R c d n c ( λ i ) = R c d r c c f r c c f n c ( λ i ) + R c d r c d f r c d f n c ( λ i ) + R c d r d d b r d d b n c ( λ i ) + R c d r d d s r d d s n c ( λ i ) .
1 n c 5 , 0 κ c 2 , κ c ( λ i ) < κ c * ( λ i 50   nm ) , d κ c ( λ ) d λ 0 , 0 s 10 6 m 1 , 0 .9 h * h 1.1 h * ,
R = R c c + R c d = r c c f + R c d .
η = 1 J s 0 h J z ( z ) d z ,
J s = 0 I λ d λ
η 1 = 1 J s 0 I λ { 1 exp [ k ( λ ) h ] } d λ .
d I d d z = ε [ k + ( 1 ζ ) s ] I d + ε ( 1 ζ ) s J d ,
d J d d z = ε [ k + ( 1 ζ ) s ] J d ε ( 1 ζ ) s I d .

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