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 <i>n</i>(<i>z</i>) from ellipsometric data (ψ and Δ). We have calculated the reflectances explicitly in a first Born approximation [i.e., to first order in <i>n</i>(<i>z</i>) – <i>n</i><sub>0</sub>, where <i>n</i><sub>0</sub> is the index of the pure liquid]. The effect of the reflecting wall at <i>z</i> = 0 is incorporated exactly. Finally, we express ψ and Δ in terms of the Fourier transform of the profile Γ(2<b>q</b>), where q is the normal component of the incident wave vector. The equation Γ(2<b>q</b>) = Γ′+<i>i</i>Γ″ is complex; one can construct Γ′(2<b>q</b>) and Γ″(2<b>q</b>) in terms of the experimental ψ and Δ for all the accessible span of <b>q</b> 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 Γ<sub>0</sub> and Γ<sub>1</sub> (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.
© 1983 Optical Society of America
J. C. Charmet and P. G. de Gennes, "Ellipsometric formulas for an inhomogeneous layer with arbitrary refractive-index profile," J. Opt. Soc. Am. 73, 1777-1784 (1983)