Ellipsometric formulas for an inhomogeneous layer with arbitrary refractive-index profile

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Abstract
The reflectances of inhomogeneous layers are usually calculated by numerical solution of Maxwell’s equations. This requires a specific model for the layer structure. We are interested here in the inverse problem: finding the refractive-index profile n(z) from ellipsometric data (ψ and Δ). We have calculated the reflectances explicitly in a first 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 Δ in terms of the Fourier transform of the profile Γ(2q), where q is the normal component of the incident wave vector. The equation Γ(2q) = Γ′ + iΓ″ is complex; one can construct Γ′(2q) and Γ″(2q) in terms of the experimental ψ and Δ for all the accessible span of q vectors. For thick diffuse layers of thickness e ≫ λ/4π, this should allow for a complete reconstruction of the profile. For thin layers, e ≪ λ/4π, what are really measured are the moments Γ0 and Γ1 (of orders 0 and 1) of the index profile. To illustrate these methods, we discuss two specific examples of a slowly decreasing index profile: (1) wall effects in critical binary mixtures and (2) polymer adsorption from a good solvent.