On the fundamental differences between 2D and 3D models of Coherence Scanning Interferometry

Abstract

Coherence scanning interferometry (CSI) is a non-contacting, optical method that is widely used to measure the topography of engineering surfaces. Several linear models of CSI have been proposed in the literature including the elementary Fourier optics (EFO) Model that represents CSI as a two-dimensional (2D) filtering operation applied to a phase-modulated object field defined in a reference plane, and three-dimensional (3D) approaches, including the Foil and universal Fourier optics (UFO) Models, that represent the surface explicitly within a volumetric framework and incorporate scattering through a 3D transfer function (TF). Although previous work, has compared the output of virtual instruments based on these models for several test cases, the underlying theory has not been compared directly. By reformulating the 2D EFO Model in an equivalent 3D representation, this paper provides for the first time, a detailed comparison of the TFs that characterize these models. This analysis reveals that all models share a common structure but differ in both their treatment of apodization and more critically the inclusion of a surface scattering term. This term, arising from the imposition of surface boundary conditions, modifies the transfer characteristics and introduces curvature-dependent weighting that is absent in the EFO and UFO formulations. Numerical examples using high-curvature hemispherical features demonstrate the fundamental differences between 2D and 3D models, and a more precise estimate of phase at the boundary with the inclusion of the surface scattering term.