Surface scattering and the 3D transfer characteristics of optical profilometers

Electromagnetic radiation scattered from an engineering surface carries the information that is related to surface topography by surface measuring instruments such as coherence scanning interferometers, confocal and focus variation microscopes. Although the operating principles of these instruments appear quite disparate, their performance is fundamentally limited by the properties of the illumination and the optics used to measure the scattered field and can be remarkably similar in practice. In recent work we have attempted to characterize the performance of optical instruments using 3D linear systems theory. In this way the measured field is related to the surface form by the 3D point-spread function or equivalently the transfer characteristics expressed in the frequency domain. This paper illustrates and extends this concept by examining traditional contacting metrology and non-contacting optical metrology using the same linear systems framework. Linear systems theory is discussed with reference to the measurement of objects with varying surface gradient and discontinuities and in both cases, similar methods to measure and compensate the transfer characteristics using spherical calibration artefacts can be employed. Finally, we consider the non-linear step of estimating the surface form from raw measurements. We discuss inverse morphological filtering in the case of contacting measurements and inversion using a rigorous vector scattering model with the potential to improve measurements using optical profilometers.