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.
© 1983 Optical Society of AmericaFull Article | PDF Article
J. Opt. Soc. Am. A 3(1) 9-15 (1986)
W. H. Southwell
J. Opt. Soc. Am. A 5(9) 1558-1564 (1988)
C. M. Marques, J. M. Frigerio, and J. Rivory
J. Opt. Soc. Am. B 8(12) 2523-2528 (1991)