OSA's Digital Library

Applied Optics

Applied Optics


  • Editor: James C. Wyant
  • Vol. 46, Iss. 4 — Feb. 1, 2007
  • pp: 502–505

Closed equation for the normal incidence reflectance of thin films on absorbing substrates

William E. Vargas and Diego Castro  »View Author Affiliations

Applied Optics, Vol. 46, Issue 4, pp. 502-505 (2007)

View Full Text Article

Enhanced HTML    Acrobat PDF (61 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The optical constants of thin films can be obtained from inversion of spectrophotometric measurements by using minimization gradient methods. The computational approach of these minimization methods requires closed compact formulas for reflectance and∕or transmittance. For normal incidence closed compact formulations for the direct transmittance, both for thin films on transparent or absorbing substrates, and for the reflectance of thin films on transparent substrates, are available in the literature. We report here a closed compact formula to evaluate reflectance spectra of thin films on absorbing substrates, and it is shown that for vanishing substrate absorption this new, to the best of our knowledge, approach gives the same results obtained from the formulation corresponding to thin films supported by transparent substrates.

© 2007 Optical Society of America

OCIS Codes
(120.5700) Instrumentation, measurement, and metrology : Reflection
(160.4670) Materials : Optical materials
(240.0310) Optics at surfaces : Thin films
(310.6860) Thin films : Thin films, optical properties

ToC Category:
Optical Computing

Original Manuscript: March 16, 2006
Revised Manuscript: July 10, 2006
Manuscript Accepted: September 2, 2006

William E. Vargas and Diego Castro, "Closed equation for the normal incidence reflectance of thin films on absorbing substrates," Appl. Opt. 46, 502-505 (2007)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. 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, 1-12 (2005). [CrossRef]
  2. R. Swanepoel, "Determination of the thickness and optical constants of amorphous silicon," J. Phys. E 16, 1214-1222 (1983). [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. 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]
  5. E. N. Kotlikov and G. V. Tereshchenko "Study of optical constants of films used for the synthesis of broad-band antireflection coatings," Opt. Spectrosc. 82, 603-609 (1997).
  6. B. T. Sullivan, "Optical properties of palladium in the visible and near UV spectral regions," Appl. Opt. 29, 1964-1970 (1990). [CrossRef] [PubMed]
  7. A. Ramírez and W. E. Vargas, "Transmission of visible light through oxidized copper films: feasibility of using a spectral projected gradient method," Appl. Opt. 43, 1508-1514 (2004). [CrossRef]
  8. R. Swanepoel, "Transmission and reflection of an absorbing thin film on an absorbing substrate," S. Afr. Tydskr. Fis. 12, 148-156 (1989).

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