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

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.