Effective ellipsometric thickness of an interfacial layer

Abstract

We calculate the effective ellipsometric thickness d and index of refraction δn of interfacial inhomogeneous layers. Exact formulas are derived in the limit of zero contrast for the two usual geometries in which the light travels toward the surface from the air–liquid medium or from the substrate. d and δn are in general functions of the incident vector q, and, for small-q values and a thin layer, these effective parameters follow a parabolic dependence in q, from which the first four moments of the layer profile can be obtained. We discuss the applicability of this method for different families of profiles. We also test its predictions against the effective thickness and refractive index obtained by the numerical integration of the Maxwell equations, for the parabolic profile describing a grafted layer of polymers.