Tackling the inverse problem in ellipsometry: analytic expressions for supported coatings with nonuniform refractive index profiles in the thin film and weak contrast limits

Ellipsometry is a powerful tool for the evaluation of the refractive index profile of a film or coating supported on a solid substrate. A well-acknowledged problem, however, is the inverse problem: Given a set of data, under what conditions can a refractive index profile be determined unambiguously? To this end, a series expansion of the Abeles matrix method has been applied to an arbitrary refractive index profile to determine analytic expressions of the ellipsometric ratio rho. Two types of expressions are found: The thin film limit in which the film thickness L is much less than the wavelength of the incident light beam (L << lambda) and the weak contrast limit in which the refractive index of the coating is very near the refractive index of the supporting substrate. In the thin film limit, the first two terms in the series expansion are relatively straight-forward, and they depend on two types of integrals involving the difference between the dielectric profile of the coating and the dielectric constant of the substrate. While higher order terms are possible, they are quite convoluted and do not assist in the inverse problem. In the weak contrast limit, however, the series expansion of rho depends on the moments of the difference between the dielectric profile of the coating and the dielectric constant of the substrate, allowing an analytic expression that applies to coatings that are even much larger than the incident light beam. The expressions associated with both limits are verified through comparison to the numerical evaluation of rho with the Abeles matrix method. The results demonstrate that through judicious selection of the substrate refractive index and incident wavelength, conditions can be created that permit critical insights into the inverse problem for either thin coatings or for coatings that are very near the refractive index of the substrate.