OSA's Digital Library

Applied Optics

Applied Optics


  • Editor: James C. Wyant
  • Vol. 46, Iss. 16 — Jun. 1, 2007
  • pp: 3133–3143

Optical properties of an optically rough coating from inversion of diffuse reflectance measurements

Anthony B. Murphy  »View Author Affiliations

Applied Optics, Vol. 46, Issue 16, pp. 3133-3143 (2007)

View Full Text Article

Enhanced HTML    Acrobat PDF (812 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(000.3860) General : Mathematical methods in physics
(120.4530) Instrumentation, measurement, and metrology : Optical constants
(120.5700) Instrumentation, measurement, and metrology : Reflection
(310.6860) Thin films : Thin films, optical properties

ToC Category:
Instrumentation, Measurement, and Metrology

Original Manuscript: October 9, 2006
Revised Manuscript: January 12, 2007
Manuscript Accepted: January 15, 2007
Published: May 15, 2007

Anthony B. Murphy, "Optical properties of an optically rough coating from inversion of diffuse reflectance measurements," Appl. Opt. 46, 3133-3143 (2007)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited