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

Abstract
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