Effective absorption coefficient and effective thickness in attenuated total reflection spectroscopy

Since the introduction of attenuated total reflection (ATR) spectroscopy for the characterization of materials, attempts have been made to relate the measured reflectivity (R) to the absorption coefficient (α) of the absorbing material of interest. The common approach is limited to the low absorption case under the assumption R∼exp(-αd_e), where d_e is an effective thickness, which is evaluated for the lossless case. In this Letter, a more detailed derivation leads to R=exp(-βd_p/2), enabling the definition of an ATR-effective absorption coefficient β and the penetration depth d_p of the electric field in the absorbing material. It is found that β∼4πε₂/λ, where ε₂ is the imaginary part of the complex dielectric function of the absorbing material, and λ is the wavelength. An alternative formulation is R=exp(-αd_ef), where d_ef is a generalized effective thickness for arbitrary strength of absorption which reduces to d_e in the low absorption limit. The experimental data for water, the biopolymer chitosan, and soda-lime glass prove the reliability of the ATR-effective absorption coefficient in the infrared range.