ELLIPSOMETRIC FORMULAS FOR AN INDEX PROFILE OF SMALL AMPLITUDE BUT ARBITRARY SHAPE

The reflectance of non homogeneous layers is usually calculated by numerical solution of the Maxwell equations. This requires a specific model for the layer structure. We are interested here in the inverse problem : to find the refraction index profile n(z) from the ellipsometric data (ψ and Ɗ). We have calculated the reflectances explicitly in a 1st Born approximation (i.e. to first order in n(z) - n0 where n0 is the index of the pure liquid). The effect of the reflecting wall at z = 0 is incorporated exactly. Finally we express ψ and Ɗ in terms of the complex Fourier transform Ɖ(2q) = Ɖ' + iƉ" of the profile (where q is the normal component of the incident wave vector). For thick diffuse layers (e ≫ λ/4π) this should allow for a complete reconstruction of the profile. For thin layers (e ≪ λ/4π) what is really measured is the moments Ɖ0 and Ɖ1 (of order 0 and 1) of the index profile. To illustrate these methods, we discuss two specific examples, which are associated with a slowly decreasing index profile : (i) wall effects in critical binary mixtures ; (ii) polymer adsorption from a good solvent