Tolerance on sphere radius for the calibration of the transfer function of coherence scanning interferometry

Although coherence scanning interferometry (CSI) commonly achieves a sub-nanometre noise level in surface topography measurement, the absolute accuracy is difficult to determine when measuring a surface that contains varying local slope angles and curvatures. Recent research has shown that it is possible to use a single sphere with a radius much greater than the source wavelength to calibrate the three-dimensional transfer function of a CSI system. A major requirement is the accurate knowledge of the sphere radius, but the three-dimensional measurement of a sphere with nanometre level uncertainty is a highly challenging metrology problem, and is not currently feasible. Perfect spheres do not exist and every measurement has uncertainty. Without having a quantitative understanding of the tolerance of the sphere radius, the calibration method cannot be used confidently for calibration of the transfer function of a CSI system that may be used in research laboratories or industry. In this paper, the effects of the tolerance of the radius of the calibration sphere on surface topography measurements are quantitatively analysed through a computational approach. CSI measurements of spherical, sinusoidal and rough surfaces are investigated in the presence of various degrees of radius error. A lookup table that relates the surface height error as a function of the radius error and surface slope angle is provided. The users may estimate the required tolerances of the sphere radius for their specific surface measurements if this calibration approach is used. The output of this paper provides a feasibility analysis for this calibration method for further development and applications.