Determination of oxidized metals' oxide layer thickness from local extrema of reflectance spectra: Theoretical basis and application to anodized titanium

The present paper concerns the oxide layer thickness determination of oxidized metals in the case where an optical interference phenomenon occurs due to multiple reflections inside the oxide layer. The paper focuses on anodized titanium but can be extended to the layer thickness determination of any material composed of a non-absorbing layer over an absorbing substrate. It establishes theoretical formulae to compute the oxide layer thickness from the positions of the local extrema of the material reflectance spectra. In contrast with many publications these formulae take into account the air/oxide and oxide/metal interfaces' electromagnetic phase-shift. They make also the distinction between TE-, TM- and non-polarized light and are valid for all incidence angles. By applying these formulae to simulated reflectance spectra with known oxide thicknesses, we show that neglecting the interface phase-shift is not appropriate for determining the oxide thickness of samples with thin oxide thicknesses, with for example a relative error on the thickness higher than 85% for a 20 nm thick layer for normally incident light. When the interface phase-shift is taken into account this relative error goes down to 3%. Nevertheless the light polarization has to be carefully selected for incidence angles close to the Brewster angle of the air/oxide interface. This paper shows also that the relative error on the oxide thickness determination is relatively well correlated with the relative standard deviation characterizing the discrepancy of the oxide thickness values obtained from different positions of extrema. The typical behavior with polarization and incidence angle of this standard deviation predicted by the theoretical study is in good agreement with the experimental behavior observed for anodized titanium samples.